Input substitution and technological development on Finnish dairy farms for 1965-1991 Empirical application on bookkeeping dairy farms Matti Ryhänen Department of Economics and Management, University of Helsinki ACADEMIC DISSERTATION To he presented, with the permission of the Faculty ofAgriculture and Forestry of the University ofHelsinki, for public criticism in Auditorium XII, University of Helsinki, Aleksanterinkatu 5, Helsinki, on February 3, 1995, at 12 noon. https://www.c-info.fi/en/info/?token=exNpvvcVtwfKrc5-.AE4RzGvwUlxjWKD8BWxs5g.R3O8Sg0C3umGsYr9gzRy41L-zHEObwUo-xUe4VzOn2Dr6HE0V5eZUr2e1UefNakE-3KQ5qIMyunCrxtzxHzMdqq0kxj3xUFCv4vLhmtCabGDRBSXIERGbRueM_UhF0Ns_0HL-81SxhEzxwcc-wBW9Puk8V3HTrdGrHqHvrdloXQGKvQfK36xQ_MdDPFW9AuqZEMx67YlpRiU5y-3nEHoeG79T2F9139tPSxvMXSw7TbDMWcbezhEVPIvD1DMTBKmc7HbVSPC4mzV ACKNOWLEDGEMENTS Now that this academic dissertation is completed, the author has the pleasant task of ex- pressing his gratitude to all persons and institutions that contributed to the work. The study was made while working as an assistant and acting senior assistant at the Department of Agricultural Economics, Faculty of Agriculture and Forestry of the University of Helsinki, at present called the Department of Economics and Management. I wish to express my deepest gratitude to my teacher and supervisor of my research work, Professor Emeritus Viljo Ryynänen, the former Head of the Department of Agricultur- al Economics, for his expert guidance and friendly support and encouragement at all stages of my studies. His influence was significant in arousing my interest in agricultural produc- tion economics and, in particular, in the topic of this study. I wish to thank Professor Matti Ylätalo, teacher of my main subject and second supervi- sor of this study. His contribution was invaluable, especially in the early stages of the empirical study. Also, his support, expertise, and the numerous discussions we had were highly significant in completing this task. To the Head of the Department of Economics and Management, Professor Karl Johan Weckman, I wish to express my sincere gratitude for his support. I am also very grateful for the good working atmosphere that, thanks to him, prevails in our Department. Further, I wish to express my gratitude to Professor Yrjö Vartia, Head of the Department of Economics of the University of Helsinki, for his support and personal guidance in utiliz- ing the panel data. I am very grateful to Professor Jouko Sirén, Head of the Agricultural Economics Research Institute, for his advice and the critical and constructive comments, which helped to improve the study. I also wish to thank Ph.D. Mikael Ingberg for inspecting my licentiate’s dissertation and for his suggestions for amplifying it into a doctoral thesis. I wish to express my gratitude to the whole staff of our Department, and especially to my colleagues Lie. Sc. Timo Sipiläinen, M. Sc. Arto Latukka, and M.Sc. Perttu Pyykkönen for their constructive criticism, help, and support. Their fair and encouraging attitude both at work and in free-time will never be forgotten. Mr. Jari Siirilä helped me in recording and preliminary processing of the research data collected from the bookkeeping farms. Together with the Head of the Bureau for Profitabili- ty Studies, M.Sc. Juhani Ikonen and the staff of the Institute, he deserves my deepest gratitude for collecting the data, which took almost a year. I thank M.Sc. Asko Simojoki for finishing the figures included in the study. The original Finnish text was translated by M.A. Jaana Kola, and the final checking of the language was done by M.Sc. Terese Forster. I wish to thank both of them for a job well done. 1 have received grants for this study from the August Johannes and Aino Tiura Founda- tion of Agricultural Research, the MTK funds of the Finnish Cultural Foundation, the Academy ofFinland, and the Kyösti Haataja Foundation of Okobank Group, and 1 wish to express my sincere gratitude for the economic support. I also thank the Editorial Board of Agricultural Science in Finland for including my study in theirpublication series. Also, I wish to thank my parents, Katri and Vilho Ryhänen, for the practical knowledge and skills I gained when growing and working on a dairy farm in Northern Savo. Practical knowledge has been an invaluable asset in my research work. Helsinki, October 1994 Matti Ryhänen CONTENTS 1 Introduction 525 1.1 Background for the study 525 1.2 Objectives and significance of the study 526 1.3 Approach to the research problem 527 2 Application of the dual approach of the neoclassical production and cost theory on Finnish dairy farms 528 2.1 Theoretical description of the production technology in milk production 529 2.2 Traditional approach in empirical study of production technology 530 2.3 Application of the dual approach in the study of the economy of dairy farms 531 2.3.1 Economic behaviour 532 2.3.2 Study of economic activity 532 2.3.3 Cost function study 534 2.3.4 Properties of the cost function 534 2.4 Comparative statics 536 2.4.1 Effect of changes in input prices 536 2.4.2 Effect of changes in output 537 2.5 Duality between the production and cost function 538 2.6 Definition of elasticities of substitution from the cost function 538 2.6.1 Marginal rate of technical substitution 538 2.6.2 Relationship between the isoquant and isocost curves 539 2.6.3 Elasticities of substitution 540 2.7 Cost flexibility 542 3 Short run in a long-run study 543 3.1 Static examination 543 3.2 Dynamic examination 544 4 Earlier studies 546 4.1 Finnish studies 546 4.2 Foreign studies 548 5 Theoretical model for the study 551 5.1 Choice of the cost function 551 5.2 Theoretical model 553 5.3 Research method 555 6 Agricultural production during the research period 556 7 Data and variables 558 8 Results of the study 562 8.1 Descriptive statistics of the farms included in the study 562 8.2 Representative dairy farm approach 563 8.2.1 Short run in a long-run study 563 8.2.1.1 Autocorrelation functions 564 8.2.1.2 DF-and ADF-tests 564 8.2.2 Empirical models and hypothesis tests 566 8.2.2.1 Regression statistics and tests 568 8.2.2.2 Empirical models and the theoretical assumptions 570 8.2.2.3 Empirical models for the post energy crisis period 571 8.2.3 Elasticities of substitution 574 8.2.3.1 Allen partial elasticities of substitution and derived demand elasticities.... 574 8.2.3.2 Morishima partial elasticities ofsubstitution 576 8.2.3.3 Shadow elasticities of substitution 577 8.2.3.4 Summary of the elasticities of substitution 578 8.2.4 Technical change and cost flexibility 579 8.3 Dairy farm level approach 582 8.3.1 Estimation procedure for panel data set 582 8.3.2 Empirical models and hypothesis tests 583 8.3.3 Elasticities of substitution 584 8.3.4 Technical change and cost flexibility 586 9 Examination of the results and conclusions 588 10 Summary 593 References 595 Selostus 599 Appendices Agricultural Science in Finland 3 (1994) 524 Input substitution and technological development on Finnish dairy farms for 1965-1991 Empirical application on bookkeeping dairy farms Matti Ryhänen Ryhänen, M. 1994. Input substitution and technological development on Finn- ish dairy farms for 1965-1991. Empirical application on bookkeeping dairy farms. Agricultural Science in Finland 3: 519-601. (Department of Economics and Management, P.O. Box 27, FIN-00014 University of Helsinki, Finland.) The study presents an attempt to gain a better understanding of the input substitu- tion and technological development on Finnish dairy farms. The dual approach of the neoclassical production and cost theory is applied. A system of derived demand and cost functions is estimated using a representative dairy farm data and panel data of bookkeeping Finnish dairy farms. The flexible translog cost function is utilized to solve the empirical research problem. The cost function study is chosen, because it makes it possible to study production of farms operating in the area of decreasing average costs. According to the results, inputs are for the most part substitutes with each other. With the existing production technology, the substitution of inputs for other inputs is inelastic. The own price elasticities are also inelastic. Technical change is pur- chased feed-saving and other inputs-using. The average annual rate of technical change was 1.3 percent. The new production chains resulting from technical change have made it possible to increase the size of dairy farms. Increasing the size of dairy farms should be allowed so in order to make it possible to utilize the advan- tages related to the economies of size. Key words: dual approach, elasticity of substitution, flexible functional forms, input demand, panel data, representative farm data, technical change 1 Introduction 1.1 Background for the study Agricultural production in Finland has undergone a rapid change since the early 19605. Changes in agriculture are closely linked to the development of the national economy. The rise in the standard of living created new jobs, which attracted a la- bour force, especially from small farms. The pop- ulation of the countryside has decreased by over half a million people during this period of time, although the population of Finland has increased by about 300,000 persons. As a result of the change, the labour force tied to agriculture has dropped from 450,000 to the present estimation of 150,000persons (Etla 1993, p. 106). At the same time, the net capital stock of agricultural production buildings, machinery, and implements has increased over 1.5 times. The rise 525 Agricultural Science in Finland 3 (1994) in ihe total productivity in agriculture has varied between 2% and 4%. However, the changes have not led to improved profitability in agriculture (Ylätalo 1987,p. 2, 70-71). The incomes of agricultural entrepreneurs have remained smaller than those of people working in other sectors (Puurunen 1990, p. 77-88). The low incomes and the fact that farmers are very much tied to their work has made it difficult to find young people who are willing to continue farming, especially on dairy farms (Ryhänen 1989, p. 50). This has resulted in a rapid de- crease in the number of dairy farms. During the research period close to 200,000 dairy farms stopped production. At present, the number of dairy farms is under 35,000. Finnish agricultural policy has been character- ized by conflicting interests and widely different views on agricultural policy between interest groups like political parties and labour market organizations. In this kind of atmosphere, the re- alization of any long-term agricultural policy has been problematic. In addition, as a result of the vast overproduction, the objectives in the regula- tion of milk production have mainly concerned restricting production. Consequently, dairy farms were largely excluded from the active develop- ment measures, for example, size restrictions were placed on dairy farms. How did the problems come about? After the Second World War, it was not possible to import food. Instead, the Finnish government encour- aged more efficient and increased agricultural pro- duction to make up for the food shortages. At the same time, the “social problem” of homeless fam- ilies was addressed. This resulted in growth in the number of farms and a decrease in the aver- age farm size. As a result of the increased arable land area and the more efficient production in the 19605, the shortage of food gradually turned to overproduction. In addition to the need to solve the present problem of overproduction, the need to develop the dairy farms increased at the same time. De- spite measures to restrict production, in the long run economic factors will determine the trends in agricultural production. Small, labour intensive dairy farms will become unprofitable and give up production. The present small average farm size and the relatively large share of small farms make it impossible to use inputs in an efficient way at the level of the national economy. When developing dairy farms, it is essential to know how the resources should be allocated to milk production. There are very few reliable re- search results available on the use of production inputs on dairy farms in Finland. Consequently, the study of the use of these inputs was chosen as the central objective of this study. The initial assumption of the study is that a milk producer acts so as to optimize the econom- ic result of production. In addition, it is assumed that labour and capital are the factors by means of which milk producers organize the production of their farms. As a result, the use of labour and capital also describes indirectly the abilities and entrepreneurship of the milk producer. In addition to optimizing the economic result, the milk producer may have other objectives, for example, enough leisure time, avoiding risk, and independence (Brandes et al. 1980). These fac- tors are impossible to measure in an exact and reliable way, so that research on the effects of these factors is excluded from the study. 1.2 Objectives and significance of the study The objective of the study is to examine the ef- fect of the change in the relative prices of inputs and the technological development of production on the derived demand for inputs of dairy farms. In other words, the aim is to study how the change in the relative prices of inputs and technological development affect the production technology of a dairy farm. The study provides information on to what extent inputs can substitute each other, as well as on how technological development has influenced the demand of individual inputs, re- sulting in an increase in the demand for certain inputs and a decrease in the demand for the others. 526 Agricultural Science in Finland 3 (1994) There has been very little research on the pos- sibilities for input substitution on Finnish dairy farms. However, awareness of these possibilities is important from the viewpoint of the needs of both dairy farms and agricultural policy. Knowl- edge on the relations between the inputs is essen- tial in developing dairy farms and in directing agricultural production. According to the production and cost theory, a change in the relative prices of inputs affects the actions of dairy farms. When relative prices change, the milk producer should know which inputs are substitutes and which ones are com- plements of each other, and the extent to which input substitution is possible so that the producer could strive to adjust production in an optimal way. In the study it is tested if milk producers follow the assumption of rational behaviour, which is assumed in the production and cost theory. It is also examined to what extent the production tech- nology of the farm has changed, along with the technological development. It is necessary to know the earlier technological development in estimat- ing the effect of future changes on the demand for inputs. In addition, in making decisions on agricultural policy, e.g., incidence of taxation, it is useful to know how the change in the price of an individual input influences the demand. In most Finnish studies and publications relat- ed to the use of labour and capital, the use of capital in agriculture has mainly been examined at the macro level (Ihamuotila 1972, 1983, Ylätalo 1987). In the macro level studies the main emphasis has been on finding out the capi- tal stock of agriculture. In addition, the studies have concentrated on research into the produc- tivity of agriculture, as well as of the changes in the amounts of capital and labour inputs used. In the study of the reasons for change in the agri- cultural sector, evidence for the theories has usu- ally been searched for in the observation series included in the total statistics. The studies made so far have provided impor- tant results on the significance of capital in agri- cultural production, but these results give only a rough picture of the use of inputs on dairy farms. There is very little empirical research results avail- able on the use of labour and capital, and the relations between them on Finnish dairy farms. 1.3 Approach to the research problem In this study, the neoclassical production and cost theory is applied in examining the use of inputs on dairy farms. The dual approach is used to solve the research problem. Based on the neo- classical production and cost theory, it is assumed that the production function describes the techni- cal relationship that transforms inputs into out- put, dairy farms aim at minimizing costs, and the prices of inputs and products are exogenous. The neoclassical framework is well suited for both theoretical and empirical factor substitution studies, and thus it was chosen as the theoretical point of departure for the study. According to Varian (1992, p. 23), profit maximization has been the basic assumption in most economic anal- yses of firm behaviour. This assumption provides an exact framework for the analysis and testing of the results. The neoclassical production and cost theory describes the production process of an enterprise, which means that conclusions can be made logically on the basis of the theory. The study is based on a long-run examination, which means that all costs are assumed to be variable costs. In the short run all costs of milk production cannot be considered to be variable, but some of the costs, e.g., the cowshed, is in the short run quasi-fixed. By means of the cointe- grated time series and the error correction model linking these, it is possible to incorporate a long- run and short-run study through the theory of statistics (Hendry 1991, Engle and Granger 1991). In connection with the attempt to solve the research problem, it will be examined if the input demand system of milk production can be modelled into an error correction model. in the beginning of the 19705, after the energy crisis, research in general economics and in agri- cultural economics was to an increasing extent directed to the investigation of the substitution possibilities among inputs. The study of the sub- 527 Agricultural Science in Finland 3 (1994) stitution possibilities also gave a powerful impe- tus to the development of new functional forms. In this century economists have strived to find a connection between the mathematical methods of representation and the neoclassical production function. However, some economists took a dif- ferent view of the problem (e.g., Diewert 1971, de Janvry 1972, Christensen et al. 1971, 1973). They started developing flexible functional forms, which greatly increased the possibilities for an empirical application of the neoclassical produc- tion and cost theory. These functional forms are used to solve the research problem. The theory of the study and the grounds for the approach chosen are presented in Chapter 2. The first part of the chapter is based on litera- ture, and it deals with the properties of the pro- duction possibility set and the production func- tion by examining these from the viewpoint of the milk producer. Then the dual approach, which is the main approach in the study, is presented. In the dual approach the main emphasis is on exam- ining the derived demand of inputs and input sub- stitution, as well as in determining the elasticities of substitution. Chapter 3 provides a theoretical examination of the possibilities to incorporate the short-run and the long-run study. The Finnish and international publications related to the research topic are reviewed in Chapter 4, and the theoreti- cal model of the study is presented in Chapter 5. The development of Finnish agriculture during the past three decades is described in Chapter 6, and the research data is presented in Chapter 7. The results of the study are presented in Chap- ter 8. Chapter 9 presents an examination of the results and conclusions, based on the research results and the theory, and on the basis of the information provided by the results, the possibil- ities to develop the production of dairy farms are discussed. 2 Application of the dual approach of the neoclassical production and cost theory on Finnish dairy farms The neoclassical production and cost theory, and the basic findings it has provided, have estab- lished their position in economics during the past decades. No serious alternatives to the neoclassi- cal production and cost theory have been devel- oped. The most remarkable progress in the re- search on production economics has been achieved in improving the generality of the results of the neoclassical production and cost theory. In the past few years the main objective in the research of the production and cost theory has been to make it possible to describe more complicated production processes than earlier, and to improve the testability of the models used in the empirical study. The dual approach and the flexible functional forms developed in the past two decades have expanded the application possibilities of the pro- duction and cost theory considerably. There are concrete advantages in the dual approach in ap- plied economics, in particular. The demand and supply functions determineddirectly from the dual function simplify the analyses considerably, com- pared to the primal studies, because in determin- ing them no nonlinear equation systems that are difficult or impossible to solve are needed. In the dual approach it is possible to describe the pro- duction technology in certain circumstances equiv- alently by means of both the primal function and the dual function (Diewert 1982, p. 535-547). In this case, it is assumed that the prices of inputs indicate the same things of production technolo- gy as the amounts of inputs, and that enterprises aim at maximizing profits and/or minimizing costs. It is essential for a milk producer to be famil- iar with the production technology, because this determines the limits within which the actions of the dairy farm are possible. In the beginning of the study, a brief description of the production 528 Agricultural Science in Finland 3 (1994) technology in milk production is presented. In this study the economic behaviour of the milk producer receives special emphasis, so that an exhaustive account of the dual approach that is central in the study is presented, to the extent that it is needed to solve the empirical research problem,. 2.1 Theoretical description of the production technology in milk production When planning milk production, the first task of a milk producer is to examine the combinations of inputs by which production is feasible, within the framework of the existing production tech- nology. The technical production possibilities fac- ing dairy farms can be described by a production possibility set, which gives all feasible input and output combinations. Thus the production possi- bility set is a subset of R". Consequently, the production possibility set provides a complete pic- ture of die production possibilities of a dairy farm. The production plan of a milk producer can be presented as a list that includes both inputs and products. Some goods, like barley, on a dairy farm may be, simultaneously, input (feed for ani- mals) and product (cultivation of grains). Thus, it is expedient to present the inputs and products as netputs. If the netput of a good i is positive (neg- ative), the dairy farm produces more (less) of the good i than it uses. This can be presented exactly as a production plan on a netput vector yeR", where y, is negative if the good i is a net input, and positive if the good i is a positive netput. At the loss of generality, the inputs and out- puts are dealt with separately, because this kind of representation is considered intuitively useful (e.g., McFadden 1978a, p. 6, Chambers 1988, p. 252, Varian 1992, p. 2-3). Thus, x = (x„...,xn) e R" describes a positive input vector, and y = (yi,...,ym)GR+ describes a positive output vector. With the existing production technology the pro- duction possibility set T, which gives all techni- cally feasible combinations of (x,y) can be de- fined as follows: (1)T = {(x,y): x can produce y} The properties required of T are presented in Appendix I. The technological knowledge and the laws of nature determine largely the production possibil- ity set of a dairy farm. However, the production possibility set may be more limited in practical decision-making on a dairy farm due to, e.g., re- strictions on production, natural conditions, and environmental factors. Also, on a dairy farm the production possibility set in the short run may differ from that of a long run, so that in an inves- tigation of dairy farms it is essential to separate short-run and long-run production plans from each other. In short-run production plans quasi-fixed factors are considered to be factors that restrict production. In the long run all inputs can be con- sidered variable inputs. In practice, distinguish- ing the short run and the long run, in milk pro- duction, from each other is not a dichotomous phenomenon. The short-run study can be presented exactly as the short-run production possibility set T(z) C T, in which the vector ze R+ describes the production restrictions in the short run. Vector z is, for example, a list of maximum amounts of inputs and products that are possible in the short run. A more detailed account of issues related to short-run and long-run studies is presented in Chapter 3. When the production technology in milk pro- duction is examined, the production possibility set presented above can be simplified. Other prod- ucts related to milk production can be considered by-products of milk production, in which case we shift from the technically efficient production plans, transformation function, to the production function, which describes the maximum scalar output as a function of inputs, instead of maxi- mal vectors of netput. Thus, the model for the production technology on farms that specialize in milk production can be formulated so that the output is presented as one aggregate output. In general, heavily aggregate data have been used in empirical studies based on the production and cost theory that have been applied to agricul- 529 Agricultural Science in Finland 3 (1994) ture (e.g., Ray 1982, Glass and McKillop 1990, Ryhänen 1992). According to the production and cost theory, as little aggregation as possi- ble is an objective, so that in more recent stud- ies the degree of aggregation has been reduced, and the studies mainly concentrate on examining the production technology of one production line (e.g., Tiffin 1991, Thijssen 1992a, b). In these studies the netput bundle (x,y) has been formed from the scalar y and vector x, so that x can produce y. From the viewpoint of this study, it is expedi- ent to present the production technology as an input requirement set, which can be presented as follows: (2)V(y) = (xeRJ: (y,x)eT} = {x: f(x) >y}. According to the definition given above, V(y) is the set of positive input vectors x = (x,,...,x„), which produce at least output y. The input re- quirement set corresponds to the traditional iso- quant, except that it also includes non-efficient input bundles. The boundary of V(y) at a certain output level is the same as the isoquant of this output level (see Uzawa 1964, p. 216). Thus, the isoquant is the set of efficient input vectors, which produce exactly output y. In the traditional approach of the neoclassical production and cost theory, production technolo- gy is presented by means of the production func- tion indicating the physical and technical rela- tions (e.g., Heady 1952, Bradford and Johnson 1953). Production function describes the techni- cal relation between the inputs and output, which means that it does not include any economic con- tent. The production function is presented as fol- lows: (3)y = f(x). where the scalar y is the maximum output that can be produced by means of the existing pro- duction technology in a certain period of time. Positive output can be achieved by utilizing the non-negative input vector x = (x,,...,xn) in pro- duction. Independent of how the milk producer defines the supply of his product, it is profitable for him to produce this output at as low a cost as possi- ble. From an economic viewpoint, the production possibility set and the production function deter- mine the limit to the optimization problem. The production function defines unambiguously the technical restriction of the optimization problem. This study is based on the assumption that there is a production function for dairy farms, which describes the connection between the inputs used during a certain period of time and the maximum output produced by means of these inputs. When analysed from the economic viewpoint, the pro- duction function is expected to have certain prop- erties, which means that the production function must be theoretically well defined (see Appendix I and 2). 2.2 Traditional approach in empirical study of production technology The properties of the production function pre- sented in Appendix 2 have usually been adequate for the purposes of theoretical analyses of the traditional or primal approach, but in most cases they have not been fully adequate for empirical analyses (Chambers 1988, p. 36). In empirical studies it has often been necessary to set addi- tional restrictions on the production functions in order to make the production and cost theory and the research techniques consistent with each oth- er. In connection with this, assumptions on the homogeneity, homotheticity, and separability have been made. In empirical studies homogeneous production functions of degree k have frequently been used, in which f(ax) = akf(x). The best known homo- geneous production functions are the Leontief, Cobb-Douglas and CES production functions (Nadiri 1982, p. 457-459). Homogeneous pro- duction functions have proven useful in empiri- cal applications. For example, the proportional changes in all individual inputs that change the scale of production are reflected exactly by the 530 Agricultural Science in Finland 3 (1994) same proportional change in an aggregate in- put. Agricultural economists have frequently used the Cobb-Douglas production function in their analyses. The production function is defined as homoth- etic, if it can be presented in the form f(x) = g[h(x)], where h(*) is homogeneous ofdegree one, g(») is a monotonic function, and g and h are twice differentiable (Varian 1992, p. 18). Ho- mothetic production functions are functions which generate linear expansion paths emanating from the origin, when the prices of inputs are constant. Every homogeneous production function is also homothetic, but homothetic production functions also include non-homogeneous production func- tions, in which the returns to scale may vary as a function of output (Sandler and Swimmer 1978, p. 357). When the number of inputs increases, the esti- mation of an empirical model has usually been considered difficult, even impossible. This restric- tion has often forced economists to employ a smaller number of inputs in empirical analyses than in theoretical analyses. Homogeneous tech- niques and a relatively small number of inputs has often proven to be a useful approach in solv- ing empirical problems. As a definition, inputs X; and Xj are separable from the input xk (Blackorby et al. 1977, p. 197), if a a//aV dxt [df/dxjj (4) i.e., the slope of the isoquant in the dimension i,j is not affected by what occurs in dimension k. The location of the isoquant in the dimension i,j may change as a result of a change in the dimen- sion k. The assumptions of the homogeneity, homo- theticity, and separability of the production func- tion restrict the possibilities to use the different functional forms in empirical analyses considera- bly. In general, this has not been considered a major problem. Econometric techniques have been applied, according to the restrictions mentioned above. The assumption of strict separability and homogeneity (Cobb-Douglas, CES) provided a good point of departure for the estimation of eco- nomic parameters for many decades. However, compromises had to be made between the gener- ality, and the analytical or empirical flexibility. 2.3 Application of the dual approach in the study of the economy of dairy farms In the research on agricultural economics based on the production and cost theory practiced in Finland, the physical and technological produc- tion possibilities of agriculture have traditionally been examined by means of the production func- tion (e.g., Ryynänen 1970, Hemilä 1983, Yläta- lo 1987). The dual approach was chosen for this study because it makes it possible to formulate the model applying the production and cost theo- ry directly into a form that describes the causal economic relations. Also, in an empirical study the research methods of the dual approach can be more easily and exactly dealt with mathematical- ly and in the data processing than the methods of the traditional approach, which means that there are significant advantages in the use of the dual functions (profit and cost function), compared to the traditional approach. The development of the dual approach of the production and cost theory was started by Hotel- ling and Samuelson in the 1930 s and 1940 s (Mc- Fadden 1978a, p. 5). According to McFadden, the publication of Shephard in 1953, where the duality between the cost and production function is presented, can be considered the breakthrough of the dual approach. In the 19605, the dual ap- proach was further developed (e.g., McFadden 1963, Uzawa 1964). From the viewpoint of em- pirical studies, the development of flexible func- tional forms was decisive (Diewert 1971, Chris- tensen et al. 1971, 1973). In the 19705, McFad- den developed, with his colleagues at the Univer- sity of Berkeley, the foundations for the applica- tion of the dual approach in empirical study. Ac- cording to Chambers (1988, p. 121), the research 531 Agricultural Science in Finland 3 (1994) results of McFadden and his colleagues caused the research in empirical production economics to turn in completely new directions. 2.3.1 Economic behaviour According to the neoclassical production and cost theory, the basic assumption in this study is that entrepreneurs aim at maximizing profit. Accord- ing to the theory, the actions of an entrepreneur are optimal, when the marginal revenue is equal to the marginal cost of production. This deter- mines the output level chosen for production and the quantity of inputs used by an optimally act- ing entrepreneur. When an entrepreneur deter- mines his optimal activity, he also has to take into account the factors restricting production. The technical production possibilities and the mar- kets (the consumption of the products) are the most common factors restricting production in the enterprises. The behaviour of the milk producer in the mar- kets can be approximated by means of the as- sumption of pure competition (see e.g., Debertin 1986, p. 9-11). In the conditions of pure compe- tition the entrepreneur is assumed to possess com- plete knowledge of the prices of both products and inputs. From the viewpoint of a milk producer, the input markets in milk production may be consid- ered to fulfill the preconditions for pure competi- tion, at least with respect to the fact that it is not possible for them to influence very much the prices of inputs they have acquired through their own actions. The product markets of a dairy farm are not based on pure competition, rather the prices of products are agreed on in advance during the farm income negotiations. At the farm level the quantity of milk production is regulated by the state, which means that the equilibrium of milk production can be described by means of profit maximization at a certain level of output, instead of unrestricted profit maximization. If the conditions mentioned above prevail, milk producers cannot influence the prices of inputs or products in a decisive way through their own actions. Thus, from the viewpoint of a milk pro- ducer, the prices of both inputs and products are exogenous. 2.3.2 Study of economic activity On the basis of the previous chapter it can be assumed that a milk producer takes the prices of both products and inputs as given. The aggrega- tion assumption made in connection with the the- oretical description of the production technology in milk production makes it possible to write the problem of profit maximization on a dairy farm in the following form: H(p,w) = Max (pf(x)-wx) = Max (py-C(w,y)), (5) x>o y > 0 in which the scalar p is the price of output, vec- tor w = (W|,...,w n ) presents the prices of inputs, and the quantity of inputs used is described by the vector x = (x,,...xn ). The profit function of a dairy farm Tl(p,w) gives the maximum profit as the function of prices. Applied to the research problem, the restricted profit function can be pre- sented equivalently with the cost function (6)C(w,y) = Min {wx: xe V(y)), x >o where the cost function indicates the minimum cost at a certain output level, when the prices of inputs are equal to w. When the level of output has been determined in advance, the return is also fixed, in which case the profit of a dairy farm can be maximized by minimizing costs. The connection between the ways of presenting the equation (5) is presented in Appendix 3. The cost function is more useful than the prof- it function in the study of the economic actions ofFinnish dairy farms, because in the case of the cost function both decreasing and increasing av- erage costs are possible. The profit function can be used in the study of the economic actions of a dairy farm only in the area of the average cost curve where the average costs increase. The use of the profit function in the study of the economic actions of dairy farms can be illus- trated as follows. Let us assume that conditions of pure competition prevail, and the behaviour of 532 Agricultural Science in Finland 3 (1994) the milk producer is in accordance with the profit maximization assumption. In this case, the mar- ginal cost is equal to the marginal revenue. In these conditions the marginal revenue is constant, and is equal to the price of product p, so that the elasticity of size can be calculated as follows: (7)AC/p = C(w,y)/(py), where AC is the average cost. In formula (7) it can be seen directly that a rationally behaving milk producer produces milk only if the ratio according to formula (7) in the long run (all in- puts are variable) is not larger than one, because profit cannot be negative in the long run (cf. Chambers 1988, p. 124-125). If the ratio is more than one, a rational milk producer should expand the enterprise or give up production gradually. Support for this reasoning can be found in the current situation in milk production in Finland. Within the framework of the technology it would be possible to increase milk production on indi- vidual dairy farms, but the milk quotas prevent the increase of the farm size. The development of dairy farms seems con- sistent with the theory. Since an increase in farm size has been prevented, many farms have given up milk production. During 1985-1993, over 30,000 dairy farms quited production. At the end of 1993, the number of dairy farms was a little over 34,000. The rapid decrease in the number of dairy farms provides evidence for the fact that the preconditions for profitable milk production have been weak. On the basis of this, it seems that dairy farms operate, on the average, in the area of decreasing average costs, for which the profit function has not been defined. In this case the restricted profit function, which can be pre- sented equivalently with the cost function, can be used to solve the research problem. In the literature on agricultural economics the economies of size is a problem area that has been dealt with extensively. According to the literature the research results deviate from each other, but most studies have arrived at the conclusion that in agriculture the average costs decrease as a func- tion of output (see e.g.. Heady 1952, p. 349- 350, Hoch 1976, p. 746-748, Smith et al. 1986, p. 719-720, Castle 1989, p. 574-577). It should be noted, however, that the empirical research results related to the economies of size are local, which means that they do not tell the absolute truth. According to Quiggin (1991, p. 36), in the world of risk and uncertainty the optimum output level will be either in the constant or increasing returns to scale area. This is characteristic in ag- ricultural production, and most studies on the economies of size of agricultural enterprises have provided evidence for this (cf. Heady 1952, p. 350). In addition to the traditional U-shaped average cost curve, evidence for an L-shaped av- erage cost curve has also been found, in which case the production technology is first described by the increasing returns to size and finally by the constant returns to size. Even if, according to the theory, the cost func- tion study is a more useful approach in the exam- ination of the economic behaviour ofFinnish dairy farms than the profit function study, there are also limitations in its use. In the cost function study it is assumed that changes in the prices of inputs do not affect the level of output, so that the indirect effect of the changes in the prices of inputs on the output level is not taken into ac- count. The profit function study makes it possible to define the change in endogenous output. Milk production restrictions began in 1970, and the measures to restrict production finally led to the production quota system in 1985. In the produc- tion quota system it is not profitable for a dairy farm to exceed its milk quota, which means that the theoretical basis for the definition of the change in endogenous output does not exist. The measures to restrict milk production have been presented in detail, e.g., in Kola’s (1991, p. 122- 126) study on the regulation of production. The cost function study is appropriate for the examination of the behaviour of Finnish dairy farms, because it is well suited for the study of the behaviour of enterprises that do not operate in the conditions of pure competition in their prod- uct markets (Jorgenson 1986, p. 1884). In regu- 533 Agricultural Science in Finland 3 (1994) lated production the price of a product has been determined, e.g., by agreements, as is the case in milk production. Thus, the demand for milk is determined according to a regulated price, which means that, under these preconditions, the level of output is exogenous. In this case, the neces- sary conditions for the optimum in milk pro- duction can be derived from the cost minimiza- tion. It should be noted that the cost function study is not incompatible with profit maximization be- haviour. The profit function can be presented, so that at a certain output level, the difference be- tween the total revenue and total costs is maxi- mized. In other words, when the profit is maxi- mized at a certain output level on a dairy farm, the costs are, at the same time, minimized at this output level. If this was not the case, it would be possible to produce milk cheaper, which would no longer be a question ofprofit maximization. This can also be presented by means of elas- ticities. Let us assume that the maximum profit is achieved at the output level y*. In this case the elasticity of size solved from the profit function is equal to the elasticity of size solved from the cost function at the same output level, i.e. AC(y*)/ MC(y*). This connection is realized since the profit maximizing first order condition states that the output price is equal to the marginal cost at this point (formula 7). 2.3.3 Cost function study In the latter part of this study the economic be- haviour of a dairy farm is examined on the basis of the cost function. In the study it is assumed that a milk producer chooses the quantity of in- puts to be used, so that the costs are minimized at the output level determined in advance. In the form of a cost function the research problem is presented as follows: (8)c = C(w,y) = Min {wx: xeV(y)), x £ 0 where w = (w,,...,wn) is the vector of the positive input prices. A milk producer chooses for pro- duction the input bundle x, which minimizes the N cost c = wx = ZwjX| at a given level of output. i = I Thus, the cost function defines the minimum cost of production, when output y, which is determined in advance, is produced during a certain period of time at input prices w. Consequently, in theory and in practice minimizing costs is equal to max- imizing profit at a certain level of output. 2.3.4 Properties of the cost function In order to achieve economic content, restrictions must be made on the cost function, as was the case with the production function. C(w,y) depends on the production technology, which means that the production technology determines the limits within which the minimization of costs is possi- ble (see Appendix 2). The properties of the cost function according to the technology restrictions are (Chambers 1988, p. 52-59): 1. C(w,y) >O, w,y >O, non-negativity; 2. if w’ > w, then C(w’,y) > C(w,y); 3. concave and continuous in w; 4. C(aw,y) = aC(w,y), a > 0; 5. if y > y’, then C(w,y) > C(w,y’); 6. C(w,o) = 0, no fixed costs; and 7. 3C(w,y)/9w|= X|(w,y), X| is the demand vector of inputs. C(w,y) is twice continuously differ- entiable. According to property 1 of the cost function, it is impossible to produce a positive output with- out costs. According to property 2, the rise in the price of any input x, cannot reduce costs. Property 3 is difficult to perceive intuitively. Let us assume that the price of one input chang- es, while the prices of other inputs remain un- changed. If the price of an input rises, the costs cannot decrease (property 2), but they may in- crease at a decreasing rate. This is possible be- cause, as the input becomes more expensive and the prices of other inputs remain the same, the cost minimizing dairy farm substitutes other in- puts for the relatively more expensive input. Ac- cording to property 3, inputs are substitutes with each other, if the dairy farm can shift from the relatively expensive inputs to the use of other 534 Agricultural Science in Finland 3 (1994) inputs. In this case, the increase in the price of input Xj reduces its use, and increases the use of other inputs. If C(w,y) is linear, the input bundle is completely fixed, which means that substitu- tion between inputs does not occur. Property 4 describes positive linear homoge- neity, where only relative changes in the prices have an effect on the optimum. If the prices of inputs change in the same proportion, the choice of inputs that minimizes costs does not change. According to property 5, increasing output can- not result in lower costs. According to property 6, a zero output does not cause any costs, so that the examination is directed to the long-run cost function. According to 7, the twice differentiable cost function has a property that is called the Shep- hard’s Lemma. By means of the Shephard’s Lem- ma it is possible to determine the cost minimiz- ing derived demand functions from the cost func- tion. The demand for inputs x*, which minimizes the costs, is equal to the partial derivatives of the cost function with respect to input prices W;. Due to the Shephard’s Lemma it is not necessary to define a production function corresponding to the cost function, which means that it is not neces- sary to solve the complex algebra involved in deriving the input demand functions using the production function and Lagrangian techniques (Diewert 1987, p. 692). According to Binswanger (1974, p. 377), in applied production analysis there are certain ad- vantages in the cost function study, compared with the production function study: 1. In a cost function study it is not necessary to impose the homogenous production function of degree one. The cost function is homoge- nous of degree one in input prices regardless of the homogeneity properties of the produc- tion function, because doubling the prices of all inputs also doubles the costs, but does not affect the input ratios. 2. The prices of inputs can be considered more independent variables than the quantities of in- puts. An entrepreneur makes decisions on the use of inputs on the basis of the exogenous prices, which means that the quantities of in- puts used become endogenous variables. 3. Inversion of the matrix is not needed in solv- ing the elasticities of substitution, and thus estimation errors can be avoided. 4. The cost function study reduces estimation problems and it is well suited for a translog- function, because it is linear in logarithms. Pope (1982) discussed in his article the signif- icance and applicability of the dual approach in the applied economic study. According to him, the approach used in the study should be chosen based on the objectives of the study. When defin- ing the elasticities of substitution the cost func- tion is more useful than the production function (Pope 1982, p. 347). The estimation of marginal products from the cost function involves the same problems as the estimation of the elasticities of substitution from the production function. The cost function study also makes it possible to solve the cost minimizing demand vector as a multi-stage cost minimization problem, if the number of inputs analysed is great. In this case, it is assumed that production technology is ho- mothetically separable (Fuss 1977, p. 89). For example, in a two-stage cost minimization problem, the aggregate price index of different types of feed, for example, is first optimized by means of the translog-function. Hulten (1973) has shown that the Divisia index is the best choice among the index numbers. Fuss (1977, p. 96-97) has shown that the Divisia index, which is an ideal index, can be described exactly as a linear- ly homogenous translog-function, so that the ag- gregate prices of the first stage can be used in optimization in the second stage. In an empirical study the modelling of the mul- ti-stage cost minimization problem for milk pro- duction in Finland is problematic, because accu- rate data on, e.g., the cost shares of the different types of feed, are not available. The multi-stage method has been used in the empirical studies of the demand behaviour of consumers. It is suita- ble for determining the elasticities of demand and substitution of the submodels of beverages, food, clothes, etc., after which the elasticities of de- 535 Agricultural Science in Finland 3 (1994) prices, and twice continuously differentiable with respect to input prices at the point (w*,y*), the Hessian matrix must be a negative semidefinite matrix. Consequently, the derived demand has the following properties (McFadden 1978a, p. 47-48): mand and substitution of aggregate consumer goods can be examined (see Laurila 1994). 2.4 Comparative statics 2.4.1 Effect of changes in input prices 3x|(w,y)/3wi < O (10) According to the properties of the cost function, the relative increase in the price of an input re- duces its use. Let us assume that the prices of all inputs increase simultaneously, so that the price vector of inputs is multiplied by the positive sca- lar. Thus, according to the linear homogeneity of the cost function, the costs rise in the same pro- portion as prices. The linear homogeneity of the cost function (k = 1) means that the derived de- mand functions are homogeneous of degree zero in input prices, because the partial derivatives of a function homogeneous of degree k are homo- geneous of degree k-1 (Chiang 1984, p. 411- 413), i.e., the cost minimizing demand for inputs does not change, because an equal proportionate change in the prices of inputs does not lead to a change in the demand for inputs. 3xi(w,y)/3w j = öxjCw^yVäWj (H) Property (10) is a restatement of property 2 of the cost function, where a rise in the price of an input cannot increase its demand. Property (11) is a symmetry property, which can be used in testing the properties of the cost function, as well as for reducing the number of parameters to be estimated (Fuss 1987, p. 996-997). Property (11) is a technical result from the differentiability as- sumption of the cost function and the derived demand functions, and its use in empirical study may reduce problems related to statistical mathe- matics (Fuss et al. 1978, p. 229). The elasticities of substitution are usually con- sidered a suitable way of measuring the substi- tutability of inputs with each other, because they are unit free measures. The derived demand elas- ticity is regarded as a natural measurement for the substitution between inputs. Using the terms of the cost function, the derived demand elastici- ty is presented as follows: Derived demand can be derived directly from the cost function. Let us assume that C(w,y) is twice continuously differentiable at (w*,y*). Ap- plying Shephard’s Lemma, the assumption of dif- ferentiability ensures that the cost minimizing in- put demand function Xj(w,y) exists, and it is once continuously differentiable at (w*,y*) (Diewert 1987, p. 692). Define 5xi(w*,y*)/5wj to be the NxN matrix of the partial derivatives of the N derived demand functions with respect to N prices Wj (i,j = L---.N). According to the Shephard’s Lemma, eg = = 9ln Xi(w,y)/3in Wj. (12) According to the Euler’s Theorem and equa- tion (12), with a linearly homogenous cost func- tion (Fuss et al. 1978, p. 232) 3xi(w*,y')/3wj = 02 C(w*,y*)/3w i 3wj (9) N 2e# = O (13) where 92C(w*,y*)/3w i3 is the Hessian matrix of second order partial derivatives of the cost func- tion at (w*,y*). According to the Young’s Theo- rem, the twice continuously differentiable cost function has a property according to which 32C(w*,y*)/3w|3wj is a symmetric NxN matrix. Since the cost function is concave in input and according to equations (10) and (12) e„ < 0. (14) These results are important from the viewpoint of the empirical study, because they make it pos- 536 Agricultural Science in Finland 3 (1994) sible to examine the derived demand behaviour in a systematic way on the basis of the properties of the cost function. The elasticities of substitu- tion derived from the cost function are presented in Chapter 2.6.3. 2.4.2 Effect of changes in output According to Hanoch (1975, p. 492), returns to scale can be presented in two different ways: 1) Returns to scale is the relative increase in out- put, as all input quantities are increased pro- portionally along the scale line from the ori- gin in input space (measured by means of the elasticity of scale). 2) Returns to scale (size) is the increase in out- put relative to costs for variations along the expansion path in input space, which means that returns to size are determined from the expansion path of the enterprise. In this case, the prices of inputs are fixed, and costs are minimized at all output levels (measured by means of the elasticity of size). The elasticity of scale and size are equivalent everywhere when the cost minimizing points are located along the scale line (Stefanou and Mad- den 1988, p. 126). This is realized only for ho- mothetic production functions. Usually the elas- ticity of scale and size deviate from each other. The difference is presented in Figure 1. Let us assume that a dairy farm operates at point A, where the elasticity of scale and size are equivalent. In addition, let us assume that the output increases, whereas the prices of inputs re- main unchanged, in which case the dairy farm shifts to point C on a new isoquant. The shift results from the change in the output on the scale line (A->B), and the reallocation of the cost min- imizing input bundle (B->C). The elasticity of size measures the change of costs from point A to point C. The elasticity of scale measures the change on the scale line when we shift from one isoquant to another (A->B). According to this, the measurements are equivalent only if the cost minimizing points are on the scale line. In that case, the elasticity of size is dependent only on output (see proof in Sandler and Swimmer 1978, p. 354-355). In milk production the input ratios, presented by the minimum cost combination of inputs, usu- ally varies as a function of output. An example of this is the different combinations of the labour and capital inputs on small and large dairy farms. Thus, the elasticity of size is more appropriate in examining the economy of dairy farms than is the elasticity of scale, because the latter does not usually correspond to the economically best choice (cf. Stanton 1978, p. 729-730). However, homothetic techniques have been popular in applied production analysis. First, be- cause homotheticity implies that all inputs be- have so that they do not reduce output; and, secondly, in homothetic production functions the optimum input ratios are independent of the level ofoutput. In addition, the assumption of the homo- thecity of the production function has made it possible to treat the inputs as one aggregate in- put. When the cost function is consistent with the homothetic production function, the elasticity of size is independent of the prices of inputs (e.g., Ohta 1974, p. 63-65). Usually the returns to scale definitions are glo- bal in nature. However, it is possible that in- creasing returns to scale (size) prevail in the case of certain quantities of inputs used, and decreas- Fig. I. Elasticity of scale and size (Chambers 1988,p. 73). 537 Agricultural Science in Finland 3 (1994) ing returns to scale (size) in the case of other quantities of inputs used (Varian 1992, p. 16- 17, 88-89). In such a case, a local definition is necessary. The elasticity of scale (size) measure- ment can be utilized in the definition. When dif- ferential calculus is applied, all measurements are local, i.e., they measure what happens in the small neighbourhood of the measurement point. In this case, the elasticity of scale (size) measures the change in output in the small neighbourhood of the input space. 2.5 Duality between the production and cost function According to the dual approach, all economically relevant information on production technology can be solved from a well defined cost function, if its definition is consistent with the well defined pro- duction function (McFadden 1978a, p. 3-4, 19- 20). The cost function has been expressed as the function of output and input prices, which makes the cost function an efficient tool for research in production and cost theory, and, in particular, for the economic applications. According to equation (8), C(w,y) is the mini- mum cost for producing output y, which means that a cheaper way of producing output y does not exist. When the prices of inputs vary over all possible price vectors, the optimal derived de- mand varies over all possible Xj(w,y), in which case the optimum points form a curve that re- sembles the isoquant, which forms the lower boundary for the set marked V*(y). V*(y) can be presented as follows: (15)V'(y) = |x:wx> C(w,y), Vw > 0} where the original input requirement set V(y) is included in the approximated input requirement set V*(y). V'(y) is always convex, independent of the shape of V(y). Due to this property, V*(y) corresponds to the quasi-concave production func- tion. In the dual approach, the following connec- tion is central: (16)V'(y) = V(y). If V(y) is convex and monotonic, V'(y) is iden- tical with V(y). If V(y) is nonconvex or nonmo- notonic, V*(y) can be represented as a convexi- fied and monotonized version of V(y), so that V*(y) has the same cost function as V(y). This connection makes it possible to solve the optimal production technology, from the viewpoint ofeco- nomic decision-making, directly from the cost function. The connection has been proved by Diewert (1971) and McFadden (1978a). The minimization of costs guarantees the gen- erality of the theory, because V’(y) is always con- vex, and in the case of cost minimization it is equivalent to V(y). Consequently, the cost func- tion of a dairy farm summarizes all economically relevant aspects of production technology (see Diewert 1982, p. 537-545, Varian 1992, p. 84). 2.6 Definition of elasticities of substitution from the cost function 2.6.1 Marginal rate of technical substitution Agricultural economists have long been aware of the concept of the marginal rate of technical sub- stitution (MRTS). In the 1820s, von Thunen started collecting data, which provided the first empiri- cal evidence of the fact that an input could sub- stitute for another maintaining a constant output (Chambers 1988, p. 2). Since the 19605, the in- terest in the study of the MRTS gained new depth, since the development of automatic data process- ing (ADP) made it possible to process more com- plex models by means of computers. Pioneering work in this sector was performed by Heady (1963) and his colleagues. The MRTS indicates to what an extent the use of an input can be replaced by increasing the use of another input, when we move from one point to another on the isoquant. In the case of at least three inputs, the isoquants are (hyper)surfaces. Isoquants are essential in the study of the MRTS. Isoquants describe the physical and tech- 538 Agricultural Science in Finland 3 (1994) nical relationship between the inputs, when the output remains constant. Consequently, an iso- quant is a curve or surface where every point represents a possible combination of inputs, at a given level of output. According to the produc- tion and cost theory, the use of inputs is assumed to be efficient, which means that the isoquant indicates the minimum quantities of inputs need- ed to produce a certain level of output. The shape of the isoquant may vary a great deal, depending on the substitutionpossibilities among inputs. Sub- stitution between inputs is easier, the closer the shape of the isoquant is to linear. 2.6.2 Relationship between the isoquant and isocost curves As was noted earlier, any economically relevant production technology can be described complete- ly by starting from different concepts or func- tions, such as the production function (produc- tion technology of a dairy farm) and the cost function (economic behaviour of a milk produc- er). From the viewpoint of this study, it is expe- dient to present the dual relationship between these functions using the isoquant and isocost curves. In economics a transformation function is of- ten used to characterize the production possibili- ty set. McFadden (1978a, p. 34-37) has proved the duality between the input distance function F(x,y) = Max |X,: (xA)g V(y)), (17) which is a transformation function, and the cost function. The value of the input distance function is given by the ratio of the lenght of the input vector observed to the lenght of an input vector defined by the intersection of the given isoquant and the ray through the input vector observed. In other words, it measures the largest possible ra- dial contraction of the input vector when the in- put vector is held at V(y). The input bundle is efficient in producing y when F(x,y) = 1. Ac- cording to the duality, V(y) has an image set in the space of input prices, and this can be present- ed as follows: R(y) = (w:wx> F(x,y), Vx > 0} = |w: C(w,y) > I}. (18) The set R(y) has been called the factor price requirement set. Its boundary forms the isocost. The duality between V(y) and R(y) can be pre- sented as the following relationship (McFadden 1978a, p. 37). (19)wx> 1, V weß(y) and xe V(y). Since the cost function is concave, and thus quasiconcave, R(y) must be convex, and the iso- cost must be convex to the origin. Thus, the iso- cost curve describes the boundary (w: C(w,y) = 1} of the factor price requirement set. Figure 2 presents the boundaries of the sets V(y) and R(y), i.e. the isoquant and isocost curves for the Fig. 2. Isoquant and isocost curves (Varian 1992,p. 89). 539 Agricultural Science in Finland 3 (1994) same level of output y, and the slopes of the curves. The slope of the isocost curve at w* can be solved as follows: dC (w*,y) 3w2 OWj Xj(w*,y) 3W| 3C(w*,y) x2 (w*,y) dw2 (20) The second stage of equation (20) results from Shephard’s Lemma. According to the equation, the ratio of the respective marginal prices of in- puts is minus the ratio of the optimal inputs. If (Xi*,x2*) is the cost minimizing point on the iso- quant at prices (W|*,w 2*), then according to the first order conditions 3f(x*) 3xj d\2 w2 * dx2 3f(x*) w ( * 3xj (21) There is a duality between equations (20) and (21). The ratio of the input prices can be ob- tained from the slope of the isoquant, and the ratio of the input quantities from the slope of the isocost curve. The slope of the isocost curve, ac- cording to equation (20) in Figure 2., is equal to -x,/x 2. This relationship makes it possible to de- rive the slope of the isoquant and the relative use of inputs directly from the cost function, which makes it possible to derive the elasticities of sub- stitution directly from the cost function. An inverse relationship can be presented be- tween the isoquant and isocost curves. The straight isocost line is the dual of the rectangular iso- quant, and vice versa. The dual isocost curve of isoquant y’, which is more curved than y”, is flatter than the isocost curve of y”, and vice ver- sa. This relationship can be illustrated by choos- ing point (W|,w 2) on the isocost curve, and then moving along the curve to point (w,’,w 2 ’), which differs clearly from point (w,,w 2 ). If the slope of the isocost curve changes only little, the isocost curve is rather flat. Since the demand for inputs is obtained from the slope of the isocost curve. when costs are minimized, the demand for inputs remains almost unchanged. As a result, the iso- quant must be strongly curved. 2.6.3 Elasticities of substitution The concept of the elasticity of substitution was developed by Joan Robinson and John Hicks, without being aware of each other’s work, in the 1930 s (Helm 1987, p. 127). Developing the con- cept of elasticity of substitution was an important step forward in the marginal theory, because it combined the concepts of elasticity and substitu- tion, which had earlier been presented separately in literature. The pure number which is independent of the units of measurement was considered the ideal measurement for the elasticity of substitution. Since the 19305,the various elasticities of substi- tution have been generated. Originally, the defi- nitions of the various elasticities of substitution were mainly developed on the basis of the primal approach (see Appendix 4). The duality between the production and cost functions makes it possible to determine the elas- ticities of substitution directly from the cost func- tion. Implicitly, the elasticity of substitution was originally introduced to measure the relative change in the ratio of two inputs brought about by a relative change in their input price ratio, or marginal product ratio for a given level of output (cf. Appendix 4, equation 64). Hicks defined the elasticity of substitution for a 2-dimensional in- put vector as follows: 3 ln (x 2/X|) ° 3ln (Wj/w,)' (22) Mundlak (1968, p. 225-231) generated the foundation for the definitions of the various elas- ticities of substitution into a general theory of derived demand for inputs, and examined the re- lationships between the various definitions for n- dimensional input vectors. He gave three defini- tions: one input one price; two inputs one price; and two inputs two prices measures. The following presentation of the elasticities 540 Agricultural Science in Finland 3 (1994) of substitution is mainly based on the publica- tions of Allen (1950), Mundlak (1968), Koizu- mi (1976), McFadden (1978b, p. 78-80), Ball and Chambers (1982, p. 700), Nadiri (1982, p. 467-468), and Chambers (1988, p. 95-97). On the basis of these publications, the properties of the elasticities of substitution obtained by means of the different measurements are evaluated and compared with each other. The discussion is in terms of the cost function under the assumptions of cost minimization, twice continuous differen- tiability of cost function, and that the cost mini- mizing derived demand functions for inputs x( and \j are available via Shephard’s Lemma. I. One input one price elasticities of substitution (OOES) OOES can be referred to as the Allen framework. We start assuming a constant output, and that the vector of input prices, exept wj? is constant. Writ- ing equation (9) as an elasticity we get: äin X. (w,y) 3Xj(w,y)w. = g. . 1 äin Wj 'J 3WjX|(w,y) (23) Thus, all derived demand responses to input prices can be solved directly from the Hessian matrix of the cost function. Thus, a unit free meas- ure was obtained for the derived demandelastici- ty. The measurement is not symmetric, because C|j * Cjj. The symmetric version of OOES is the Allen partial elasticity of substitution, in which O,l. It can be shown that o = = -..--il--.'J CjCj Sj (w,y) (24) in which q = oC(w,y)/3w|, c y = d2 C(w,y)/dWjdWj, and Sj= [WjXj/C(w,y)] (Uzawa 1962, Binswanger 1974, p. 378-379). Note, that as elasticities of cost function with respect to input prices we can write dlnQw.yVdlnw, = 3C(w,y)wj/dw JC(w,y) - W|Xj(w,y)/C(w,y) = S,(w,y). Sj(w,y) measures the cost share of the input Xj of the total cost. Input X| is called the Allen substitute for input Xj, if Ojj > 0 and > 0. Correspondingly, input x, is called the Allen complement for input Xj, if öl, < 0 and By < 0. With respect to the sign, e y and Cjj are equivalent, even if e,| * Cji, which means that the classification of inputs into Allen substi- tutes and complements is not dependent on which one of the elasticities ey or is used in the clas- sification. If the increase in the price of input x; increases (reduces) the use of input x y and the output level remains constant, input x, is a substitute (com- plement) for input Xj. Input’s own price elasticity (e H) measures the percentage change in the use of input Xj, resulting from a percentage change in input’s own price Wj. According to the concavity of the cost function, e H < 0, and according to the homogeneity of the derived demand for inputs N -eH. Consequently, no input can be Allen’s complement to all other inputs. 2. 7vvo inputs one price elasticity of substitution (TOES) It is possible that the price ratio changes when the price of only one of the prices change, keep- ing the other prices and output constant. This kind of substitution between inputs is measured by the TOES measurement. This measurement corresponds to the concept of Morishima partial elasticity of substitution. We let S|(w,y) = S, and Sj(w,y) = Sr M _ ,n (w,y) /xj (w,y) ] a din Wjn (25) = S (o .-o ) = e - eU jr u JJ It is easy to see by using equation (69) from Appendix 4 and equation (24) that = TOES measurement can be considered a more complete measurement for describing economic phenomena than OOES measurement, because it takes into account the change in the i,j input ratio when the price of input Xj changes. Inputs x{ and Xj are Morishima substitutes (complements), if o“> 0 (a“ < 0). Inputs x( and Xj are Morishima substitutes, if a rise in the price of input \} causes a rise in the input ratio X|(w,y)/Xj(w,y). If inputs are Allen substitutes, they are also 541 Agricultural Science in Finland 3 (1994) Morishima substitutes, but the opposite is not true. Morishima elasticity of substitution is not neces- sarily symmetric with respect to the sign, which means that the classification of inputs x; and Xj into substitutes and complements depends on which input price changes. No input can be Morishima complement for all other inputs, as was also the case withAllen. 3. Two inputs two prices elasticity of substitu- tion (TTES) Finally, we consider the case where two input prices (W| and Wj) change. We assume a constant output and the vector of input prices, exept W| and Wj, is constant. The effect of the change in the price of two inputs on the substitution be- tween inputs can be measuredby the TTES meas- urement. S _ din [Xj (w,y) /Xj {w,y) ] (26)o 3 In (Wj/Wj)u It is possible to impose a symmetry property for the Morishima elasticity of substitution. This can be done by taking the weighted average of Morishima elasticities of substitution o“ and a“. The cost shares S, and Sj are related to the defini- tion of these Morishima elasticities of substitu- tion in equation (25), so that 8/(81+ Sj) and 8/(81 + Sj) can be considered the logical weights when the weighted average for Morishima elasticity of substitution is determined (Koizumi 1976, p, 154). In this case TTES is s S i M S i M 0i J " Sj +Sj GiJ +Sj+ Sj °j' (27) = s?s: (2G ii- GH-V which McFadden (1963, p. 76) called the shad- ow elasticity of substitution, because in this case costs are held constant, i.e. the elasticity is deter- mined from the isocost curve. The shadow elas- ticity of substitution is a more complete meas- urement of the elasticity of substitution than GOES and TOES. Intuitively, it can be regarded as closest to the elasticity of substitution defined by Hicks, since it gives the percentage adjust- ment in input ratio to change in input price ratio. As has been noted, GOES and TTES are sym- metric, i.e. Oy - Oj| and a-j - o-,. In this chapter, various measurements for the elasticity of substitution have been discussed. It can be asked which is the most appropriate. It has not been possible to give an unambiguous answer to this question. Even if the different meas- urements of substitution do not always give equiv- alent elasticities in cases where there are more than two inputs, this does not mean that the meas- urements were wrong. It shows that there have been difficulties in defining a measurement with a single concept in cases where there are more than two inputs, which means that the substitu- tion between inputs should be examined by means of different measurements for the elasticity of substitution. 2.7 Cost flexibility The effect of output on the level of costs is meas- ured by the cost flexibility measurement, which measures the ratio of marginal cost (MC) to aver- age cost (AC). The cost flexibility corresponds to the measurement of the elasticity of scale (size) in the primal approach. The cost flexibility is presented as follows; r|(w,y) = [öC(w,y)/öy]/[C(w,y)/y] = öin C(w,y)/3ln y. (28) The classical AC curve is assumed to be U- shaped as a function of output (Truett and Ro- berts 1973, p. 976). The U-shaped AC curve reaches its minimum at the point where the MC curve intersects it, i.e., the point where the cost flexibility is one. If the AC curve is U-shaped, the elasticity of size decreases as a function of output. In agriculture, in practice, the AC curve may deviate from the shape of the classical AC curve (Doll and Orazem 1984, p. 198-199). In long-run studies it can be assumed that all costs of a dairy farm are variable. This means 542 Agricultural Science in Finland 3 (1994) that increasing average costs seem unrealistic, be- cause a milk producer can be assumed to be able to increase production by, for example, multiply- ing the existing production process. On the other hand, it can be assumed that some production inputs, (e.g., management of a dairy farm), in the long run, can be like a fixed input in character. In this case, it is expedient to assume that the AC curve is U-shaped. According to this, it is likely that both increasing and decreasing average costs may occur, and this is when the properties of the cost function are needed. 3 Short run in a long-run study In most studies it has been assumed that the equi- librium states of the long-run optimization prob- lem can be approximated directly from the data, and that all inputs are variable in the long run, and, therefore, they can be optimized at all ob- servation points (e.g., Ray 1982, p. 492-493, Glass and McKillop 1990, p. 284-285). This thinking is based on comparative statics. Long-run assumptions are not directly appli- cable to short run, so that the short run is exam- ined separately. In the short run costs can be di- vided into different groups according to their fix- ity (Chambers 1988, p. 100, Fernandez-Cor- nejo et al. 1992, p. 331). If changes occur in the prices of inputs or the demand for the product, the quantities of quasi-fixed inputs cannot be im- mediately adjusted to the optimal level, which means that long-run and short-run optimization may differ from each other. In the short run all costs of milk production cannot be considered variable, but part of the costs are quasi-fixed in character, e.g., the cost of the cowshed. In the very short run it is not possi- ble for a milk producer to increase the capacity of the cowshed, so that the use of variable inputs is optimized on the condition that the use of the cowshed remains unchanged. In the long run it is possible for a milk producer to increase the ca- pacity, in which case the restrictions of the short run are removed. Defining an unambiguous link in economic terms, between the short and long run has not been suc- cessful. Adjustment costs and asset fixity have aroused a lot of discussion and argument among agricultural economists (e.g., Johnson, G.L. 1982, Johnson, M.A. and Pasour Jr. 1981, 1982, Cham- bers and Vasavada 1983, 1985, Vasavada and Chambers 1986, Hsu and Chang 1990). 3.1 Static examination In the production and cost theory the problem of the short and long run has been approached by dividing the costs into groups on the basis of their fixity (e.g.. Chambers 1988, Varian 1992). In the traditional short-run production function study it is assumed that the input vector can be divided into the vectors of variable and fixed in- puts (e.g.. Heady 1952, Debertin 1986). In the short run V(y) is written as follows: (29)L(y,x") = {x': f(x',x") > y). where x" is the maximum available quantity of fixed inputs used, and x’ is the set of variable input vectors. The cost minimization problem of the short run can be presented as follows: (30)C(w',y,x") = Min{w'x': x'eL(y,x")|, x' >o in which w' is the vector of the variable input prices, L(y,x") is a non-empty set, C(w',y,x”) sat- isfies the properties of the cost function, and is non-increasing in x". The cost function studies of the long run (the starting point in milk production, unless giving up production is planned) and the short run differ from each other, because in the short run fixed inputs do not always make it possible to mini- 543 Agricultural Science in Finland 3 (1994) mize costs. However, by theory, variable costs are minimized for any choice of x", and the cost minimization problem can be written as follows: (31)C(w,y) = Min C(w',y,x") + w"x". In equation (31) the problem of the long run is divided into two parts so that variable costs are minimized at the current quantity of fixed inputs, and after this a new quantity of fixed inputs is chosen. Thus the equation presents the effect of the fixity of inputs on a dairy farm. According to the first order condition, 3C(w',y,x") , „ n—\ :- +w- = o, (32) j e N" (N" is the set of inputs x" ), which means that in the long run fixed inputs are purchased to the extent that the reduction in the variable costs (the shadow price) is at equilibri- um with the marginal cost, i.e., Wj". According to the second order condition, it is required that the matrix with the elements „V ( w',V ,X") . . a»; a»; ■'■ jeN (33) is positive semidefinite. The function of the short- run variable costs is convex with respect to fixed inputs, so that the cost minimization of the long run is realized. The convexity of C(w',y,x") means that the shadow prices approach zero, when the use of fixed inputs increases towards the infinite, i.e., the greater the use of fixed inputs is, the smaller is the reduction in the short-run variable costs, when the use of fixed inputs is increased. According to the envelope theorem, the long- run cost function is tangent to the short-run cost function from below (see Birchenhall and Grout 1984, p. 225-228, Chambers 1988, p. 104). The short-run and long-run cost functions share the point in which the fixed input bundle minimizes costs. According to Shephard’s Lemma, (34)Xj(w,y) = Xi(w,y,x"(w,y)), in which x"(w,y) is the solution to equation (31). According to Le Chatelier’s Law, the long-run cost function is more concave than the short-run cost function, which means that long-run derived demand is more own price elastic than short-run derived demand, i.e., 3x. (w,y) 3x.(w,y,x"(w.y)) < 3wj 3wj (35) During different time periods, inputs can be divided into several groups on the basis of their fixity, so that this reasoning can be applied gen- erally (Chambers 1988, p. 106-107). In empirical studies the short-run elasticities have been defined by deriving the demand equa- tions of variable inputs with respect to prices of variable inputs by holding the fixed inputs con- stant (e.g., Thussen 1992b, Helming et al. 1993). Dynamic models make it possible to define the elasticities of the short run, intermediate run, and long run (Ito 1991, p. 11). Elasticities of the short run have been defined by holding the fixed inputs constant, those of the intermediate run by including an adjustment coefficient, and those of the long run by assuming that fixed inputs have been fully adjusted to the desired level of the use of inputs. 3.2 Dynamic examination In the dynamic adjustment cost model no assump- tions are made on the fixity of inputs, but the change in the quantity of use of quasi-fixed in- puts is modelled as more expensive than the change in the quantity of variable inputs (Vasa- vada and Chambers 1986, Ito 1991, Fernan- dez-Cornejo et al. 1992). Thus, the adjustment cost is part of the theoretical model, in which it is assumed to be strictly convex, i.e., its marginal cost is assumed to be increasing (Jorgenson 1986, p. 1905, Ito 1991, p. 3). If the adjustment cost were assumed to be linear or strictly concave, delayed adjustment would not occur (Rothschildt 1971, p. 621, Maccini 1987, p. 25). The dynamic problem of a dairy farm is the 544 Agricultural Science in Finland 3 (1994) minimization of the present value of future total costs. Cost minimization is subject to the produc- tion function and the increasing marginal cost of the adjustment cost. Adjustment costs are assumed to ensue from institutional factors, imperfect mar- kets, and internal factors of firms. Internal adjust- ment costs are assumed to ensue from directing resources from production to investments, reorga- nization of production, as well as changing the level of knowledge and skills. Adjustment costs mean that a rapid increase of capital is more expensive than a slow increase. Without the adjustment cost the dynamic model becomes a static model. According to Hsu and Chang (1990), the ex- amination of the fixity of inputs should be com- bined with the adjustment cost model, in order to obtain a realistic approach for the examination of the fixity of inputs. The convexity of the adjust- ment cost in the neighbourhood of the origin does not guarantee the fixity of inputs, so that the popu- lar quadratic adjustment cost function captures only part of the slowness of the adjustment of inputs. The fixity of inputs is guaranteed by the dis- continuity of the first derivative of the adjust- ment cost function. According to Hsu and Chang, this explains why the adjustment cost theory has had only little significance in the study of the fixity of inputs. According to them, testing the fixity of inputs by means of the traditional econo- metric techniques is bound to fail (see also Cham- bers and Vasavada 1983, p. 768, 1985, p. 139). The smoothness of the adjustment cost at the ori- gin and the symmetry of the adjustment cost should be tested empirically. Only through this is it possible to obtain information on the fixity of inputs and the degree of fixity, as well as the investment decisions of the enterprise (Roth- schildt 1971, p. 609, 621). In the literature the adjustment cost is usually presented as differentiable everywhere and sym- metric with respect to, for example, the change in capital input, which is probably not true (Hsu and Chang 1990, p. 305-307). According to em- pirical studies, no support has been found for the assumption that the adjustment cost is symmetric (e.g., Chang and Stefanou 1988,Pfann and Ver- spagen 1989, p. 368-370). According to Macci- Ni (1987, p. 25), the foundations of the adjust- ment cost theory are weak. The problem is not the adjustment cost itself, but the assumption of the increasing marginal cost of the adjustment cost. According to Rothschildt (1971, p. 608), decreasing marginal costs are as likely and feasi- ble as increasing adjustment costs. However, the quadratic adjustment cost assump- tion has been popular in empirical time series models. Models in which this assumption has not been made have tended to have poor statistical properties (Hetemäki 1990, p. 15). According to Hetemäki, rejecting the non-quadratic adjustment cost models for this reason may have been based on erroneous statistical inference. The studies presented above question the reli- ability and applicability of the traditional adjust- ment cost models. In milk production the labour input may in the short run be more fixed than the capital input, which means that the preconditions for constructing a dynamic demand model of in- puts in this study are quite problematic. In ad- justment cost models the labour input is usually assumed to be variable, and capital is considered quasi-fixed (Maccini 1987, p. 23). Debertin (1986, p. 285) notes that in a short-run examina- tion tractors, machinery, full-time labour force, and land should be considered fixed inputs. In the literature it remains unclear how long- run and short-run examinations should be com- bined. The error correction model (ECM) and information provided by cointegration have been offered as solutions to this problem (e.g., He- temäki 1990, Engle and Granger 1991). Ac- cording to advocates of this approach, in empiri- cal studies the properties of time series do not receive enough attention (Hendry 1991). ECM is based on the idea that, in the long run, equilibrium between the factors exists. Short run is seen as stochastic noise, which cannot be ex- plained by economic theory. The ECM cannot answer the question of the mechanism in the eco- nomic theory that brings about the dynamics of the model. However, its assumptions are not un- realistic. In this study, the applicability of the ECM in the short-run dynamic modelling of milk production will be examined. 545 Agricultural Science in Finland 3 (1994) Granger’s concept of integrated series refers to the stationarity properties of the time series. The time series x„ t = 1,2,...,T is integrated of order d, denoted x, ~ 1(d), if the time series (1- L)dx, has a stationary representation after differ- encing d times (Virén 1992, p. 121). Cointegra- tion can be presented as follows: the components of vector x, are integrated of degree d,b (x, ~ I(d,b)), if all components of x, are of degree d and there is a vector a (* 0), so that z, = a'x, ~ I(d-b), b > 0. Vector a is called the cointegration vector (Granger 1991, p. 72). Intuitively this means that in the long run two or more series move in a parallel direction. Even if the series are not stationary, their linear combi- nation is stationary. Thus, the series have a com- mon stochastic trend. When the common trend is subtracted out, the difference between variables is stationary. The series can be considered a long- run “equilibrium” relationship. In this connec- tion, equilibrium differs from the concept of the economic theory. In the literature on cointegra- tion equilibrium is understood as an observed re- lationship that a set of variables has formed in the long run. Cointegration is a necessary condition for the existence of the ECM. The formation of a long- run relationship between the series 1(0) and 1(1) is impossible, because series 1(0) has a fixed av- erage, whereas in the case of series 1(1) the aver- age moves towards the infinite. If the series are 1(1) and they are cointegrated, according to Grang- er’s representation theorem (Engle and Grang- er 1991, p. 86-87), an ECM exists. In the case of two series the ECM can be presented as follows: (36)Ay, = Po-Y (y,-i-Rx,_,) + P,Ax, + e„ in which e, is an error term with mean zero, constant variance, and zero covariance, y mea- sures the extent of correction of errors by adjust- ments in y, p measures the long-run equilibrium relationship between x and y, P, measures the short-run effect on y of changes in x, and Ay„ Ax, and are 1(0). If (yt_,-pxM) is 1(1), the error correction mechanism does not exist. Implicitly, the ECM can be thought to operate so that the short-run shocks Ax, feed Ay„ and the long-run error (y,_|-px M) attaches the long run into the model. If Ax, =O, (y,_,-pxM) determines the steady state in the long run. Thus, the ECM can be regarded as a description of a stochastic process, according to which the economy cor- rects the error in the equilibrium. Granger’s representation theorem provides the theoretical basis for the ECM. It makes it possi- ble to model the short run dynamics for the de- rived demand model, provided that the variables in the cost function and the cost share equations are cointegrated (Hetemäki 1990, p. 25). 4 Earlier studies In international studies of agricultural economics the dual approach was used to solve empirical problems quite soon after the flexible functional forms were developed (e.g., Binswanger 1974). The dual approach has rarely been applied in Finn- ish agricultural economic studies. 4.1 Finnish studies Finnish studies in which the neoclassical produc- tion and cost theory has been applied, and in which the substitution between inputs and elas- ticities of substitution have been examined are dealt with in this study. The aim is to link the present study to earlier studies made in Finland. At the same time, it is reflected on what new information this study possibly brings to what is known at present. The first study in the field of agricultural eco- nomics in Finland in which the elasticities of sub- stitution were examined empirically was HemilA’s (1983) doctoral thesis. The main emphasis was in the examination of the technological change in 546 Agricultural Science in Finland 3 (1994) Finnish agriculture, but it also included an analy- sis of the substitution possibilities between in- puts. In his study Hemilä used two inputs, labour and capital, so that he aggregated all inputs ex- cept the labour input into the capital input. The data consisted of cross section data collected from the bookkeeping farms from the years 1947/48, 1956/57, 1968, and 1979, which were normal years in terms of weather conditions. The CES production function was applied in the study. He observed that the elasticity of substitution was between 1.6 and 4.2. He concluded that on the basis of the elasticity of substitution estimates calculated in his study, no far-reaching conclu- sions can be drawn (Hemilä 1983, p. 211-212). The substitution of capital input for labour input was easier in the latter part of the research period than in the beginning. Ylätalo (1987, p. 69-71) touched on the sub- stitution of the labour input through the capital input in agriculture. In his study the main em- phasis was on examining the productivity and investments in agriculture. Ylätalo used the Cobb- Douglas production function, and the data used were aggregate time series data from the years 1961-1984. He examined the substitution between labour and capital by calculating the change per- centages of each factor, and by putting the values into the following equation: (37)AK = AA + IAB, in which AK = change in the productivity of cap- ital, % AA = change in the total productivity, % 1 = income share of labour input AB = change in labour input/capital stock Ylätalo used 0.7, 0.6, and 0.5, alternatively as the income shares of labour input. In equation (37) term IAB indicates the substitution effect. According to Ylätalo’s research results, the sub- stitution effect between labour and capital has remained almost the same. According to him, in terms of its absolute value, the substitution be- tween labourand capital has been bigger than the change in the productivity of capital. Since the total productivity has grown at the same time, substituting capital for labour has been economi- cally profitable. Lumiaho (1990, p. 58, 63-65) examined the elasticities of substitution by the same methods as Hemilä. He chose cross section data from the bookkeeping farms from the years 1979, 1982, 1984, and 1986. According to Lumiaho, analys- ing the elasticities of substitution is problematic, because some of the substitutions receive nega- tive values. The elasticities of substitution deter- mined by Lumiaho deviated from those deter- mined by Hemilä with respect to their level, but they gave a similar result in the development of the substitution of the labour and capital input. Ryhänen (1992) examined the effect of the change in relative prices of inputs and the tech- nological development on the derived demand for inputs by means of the translog cost function. He used aggregate time series data from the years 1962-1989. Variables chosen for the cost func- tion were: output, prices of capital, labour, pur- chased feed and purchased fertilizers, and a line- ar time trend to describe technological develop- ment. The results show that the inputs used in agri- cultural production are for the most part substi- tutes for each other. The values of the elasticities of substitution are small, which means that by means of the existing production technology the substitutionof inputs for other inputs is inelastic. The own price elasticities are also inelastic. According to the research results, it seems that in Finnish agriculture there are advantages relat- ed to the farm size, which means that expanding agricultural enterprises should be allowed. Tech- nical change in agriculture has been capital-us- ing, and labour- and purchased feed-saving. Sub- stituting the cheaper capital input for the expen- sive labour input has been profitable. In addition, through investments new technology has been ob- tained for production. Ylätalo (1987) and Ryhänen (1992) only had at their disposal data that was aggregate from the whole agricultural sector. The production and cost theory is based on decision-making of individual agricultural entrepreneurs, which means that the 547 Agricultural Science in Finland 3 (1994) use of data consisting of information collected from individual farms would be theoretically on firmer ground than the use of aggregate data. Hemilä (1983) and Lumiaho (1990) used cross section data consisting of information collected from individual farms. In their studies solving the elasticities of substitution was restricted by the primal approach, as well as the restrictions on the substitution possibilities set by the pro- duction functions used (see e.g., Lau 1986, p. 1519). In this study the dual approach and flexible functional forms are used, and the data are col- lected from bookkeeping dairy farms. This should make it possible to define the elasticities of sub- stitution more accurately than earlier, and pro- vide a better theoretical foundation for the analy- sis. Time series data are formed from the infor- mation collected for this study, which is based on the idea of a representative dairy farm. The data set collected makes it possible to utilize a panel data set, too, for some dairy farms. The results from the panel data set will be compared with the results from the representative dairy farm data set. 4.2 Foreign studies In the case of foreign studies, the examination is restricted to recent studies in which the dual ap- proach has been applied and flexible functional forms have been used. Studies in which the de- rived demands and the elasticities of substitution in agriculture have been examined are dealt with in this study. Cost function studies Binswanger (1974) estimated the elasticities of input demand and elasticities of substitution in agriculture in the United States, utilizing the trans- log cost function. As data he used cross section data collected from different states from the years 1949, 1954, 1959, and 1964. He assumed the scale economies to be neutral, so that he excluded the output from his analysis. Technical change was assumed to be non-neutral. As variables for the cost function he chose the prices of five aggre- gate inputs (land, labour, machinery, fertilizers, and ‘other inputs’), and he chose a linear time trend to describe the technical change. According to the research results, the best sub- stitutes were land and fertilizers. ‘Other inputs’ was a substitute for all inputs, except for land. Machinery was a substitute for both land and labour. A complementary relationship existed be- tween labour and fertilizers, machinery and ferti- lizers, and land and ‘other inputs’. Technical change was labour-saving and machinery-using. The derived demand for arable land was inelas- tic. The own price elasticities for other inputs was close to -1. The Cobb-Douglas form was not an appropriate production and cost function spec- ification in this data. Ray (1982) chose the translog cost function for the analysis of agricultural production in the United states. For data he chose aggregate time series data from the years 1939-1977. In the cost function he chose as variables the crop output and livestock output, as well as the prices of hired labour, capital, fertilizers, ‘purchased feed, seeds, animals’, and miscellaneous inputs (pesticides, detergents, electricity, telephone). According to Ray, this kind of multi-output and multi-input model is problematic, because the number of pa- rameters to be estimated is quite large. Ray’s primary objective was to measure the elasticities of substitution between inputs. He as- sumed technical change to be Hicks neutral. The annual technical change was estimated at 1.8% (increase in productivity). The possible non-neu- tral technical change was excluded from the anal- ysis, because he wanted to have only one meas- urement for the increase in productivity. He may also have excluded non-neutral technical change because of the problems related to model specifi- cation, as well as the rapid growth in the number of parameters to be estimated. According to Ray, capital was a substitute for all other inputs. In addition to capital, fertilizers and ‘purchased feed, seeds and animals’ were substitutes for hired labour. The elasticity of sub- stitution between hired labour and capital de- 548 Agricultural Science in Finland 3 (1994) creased, whereas, at the same time, it increased between hired labour and fertilizers, as well as hired labour and ‘purchased feed, seeds and ani- mals’. The own price elasticities for all the in- puts were negative and less than one in their ab- solute value. The absolute value of the own price elasticity for hired labour was the highest. As a function of time, the absolute values of the own price elasticities for all inputs increased. This was likely to be caused by the increase in the use of purchased inputs compared with inputs produced on the farm. The non-homothetic technology in- dicated that the cost function is not separable. The labour input of the farm family was ex- cluded from Ray’s data. Usually the labour of the farm family is priced according to the market prices (e.g., Binswanger 1974). According to Ray (1982, p. 497), this kind of pricing should not be used, because it does not take into account the demand for agricultural workers in the market and the management work included in the labour input of the farm family. According to him, the labour input of the farm family should be a sepa- rate input, and it should not be added to the hired labour. Lopez (1980) examined the production struc- ture and the derived demand for inputs in Cana- dian agriculture by means of a generalized Leon- tief cost function. As data he chose time series data from the years 1946-1977. In the cost func- tion he chose as variables the output and prices of four aggregate inputs [labour (own and hired), capital (machinery, implements, farm vehicles, animals), ‘land and structures’, and aggregate in- termediate inputs (fertilizers, seeds, feed, fuel, electricity)]. According to Lopez, the Leontief, Cobb-Doug- las, and CES functional forms were not appropri- ate for the data. The constant returns to scale hypothesis was also rejected. It was surprising that the null hypothesis, neutral technical change, could not be rejected, even if technological progress has been important in Canadian agricul- ture, being labour-saving and capital-using. Ac- cording to Lopez (1980, p. 43) this phenomenon reflects the fact that the increase in the demand for capital and the decrease in the demand for the labour input has been caused by changes in the relative prices, in which case the expansion path of a non-homothetic production function bends into a higher capital/labour ratio as the output increases, rather than into non-neutral technical change. This observation is supported by the fact that when the model was specified into a con- stant returns to scale model, the null hypothesis, neutral technical change, was rejected. The own price elasticities for inputs were inelastic -(0.280- 0.897). The elasticities of substitution between all pairs of inputs were positive, but quite small. The highest degree of substitution occurred be- tween labourand capital, and between capital and intermediate inputs. Glass and McKillop (1990) examined the pro- duction structure of Irish agriculture by means of a generalized translog cost function. As data they chose aggregate time series data from the years 1953-1986. In the cost function they chose as variables the output and prices of four inputs (cap- ital, ‘feed and seeds’, fertilizers, and labour). The objective of the study was to obtain econometric measures of substitution between inputs, elastici- ties of input demand, neutral and non-neutral tech- nical change, and scale economies. According to Glass and McKillop, labour is a substitute for capital, fertilizers and ‘feed and seeds’. A complementary relationship exists be- tween fertilizers and ‘feed and seeds’. The own price elasticities for inputs were inelastic -(0.08- 0.85). Technical change was labour-saving and fertilizer-using. It also seemed to be capital- using and ‘feed and seeds’-using. Glass and McKillop estimated the average an- nual technical change to be 1.33%. Technical change slowed down during the research period, and the average non-neutral technical change was negative. According to them, the result was prob- ably caused by the fact that agricultural entrepre- neurs assumed the demand for agricultural prod- ucts to be higher than the realized demand and increased the capacity too much, so that the sub- stitution of capital for labour input seems to have been inefficient. Agriculture in Ireland is described by the dis- economies of scale. Technological development 549 Agricultural Science in Finland 3 (1994) seems to have influenced the change in the farm size. According to Glass and McKillop (1990, p. 278), along with the progress of the technolo- gy, savings of costs are achieved in larger enter- prises. The average annual increase in the total productivity was 2.0% (technical change 1.33%, scale economies -0.22%, and residual 0.89%). Finally, Glass and McKillop note that the increase in productivity has been hindered by the relative- ly small average farm size and by inefficient use of capital. Profit function studies Higgins (1986) derived a set of demand and out- put supply equations from the short-run translog profit function. From the estimated parameters he solved the own and cross price elasticities of the demand and supply. The cross section data included information from 1454 bookkeeping farms from the year 1982. In the profit function he chose as variables the prices of three outputs and four variable inputs, as well as quantities of five fixed inputs. Pig and poultry farms, as well as farms in which there were zero or negative profits were, excluded from the study. Higgins’ empirical result did not support the profit maximization assumption. According to him, this may have been caused by e.g., errors in the approximation and assumptions, problems in the measurement, as well as by the fact that the data could include farms that did not maximize profit (see also McKay et al. 1983, Shumway 1983, Weawer 1983, Lopez 1984). However, it should be noted that in Higgins’ results only own price elasticity for crop output had the wrong sign, which automatically meant rejection of the assumption of convexity and, hence, the profit maximization hypothesis. According to Higgins, milk output was the most price elastic. Decreases in the price of milk rela- tive to other prices reduces milk production, whereas the effect of relative decreases in the prices of beef and mutton on meat production was a lot smaller. According to Higgins, a de- crease in the price of industrial fodder had a great- er impact on milk production than on meat pro- duction. He concludes that price policy directed to inputs and products is an efficient means of controlling production in milk production, but not in meat production. Tiffin (1991) constructed a theoretical model to describe how production decisions that can be realized in the short run are made on a farm. The model that was based on the theoretical profit function was applied empirically. Research data consisted of information from the years 1964/5- 1983/4, which had been collected from farms in England and Wales, where 50-75% of the labour input was used for milk production. Information after 1984 was not included in the data, because this is when the quota system was introduced, and the output of milk became a fixed factor (exogenous). The annual number of farms included in the study varied between 237 and 377 farms. Tiffin’s model was based on the idea of a representative farm, which was arrived at by calculating the averages from the data separately for each year. As variables Tiffin chose the prices of crops, ani- mals, and milk, as well as the prices of variable inputs (fertilizers, industrial fodder, labour), and as fixed inputs arable land, physical capital, ani- mal capital, as well as the time trend to describe the change in technological factors. Tiffin’s empirical result did not support the profit maximization assumption. He considered the reliability of his results to be weak, so he made his conclusions at a general level. Accord- ing to the results, on a dairy farm the variable factors change only little, when the prices change, holding the production technology constant. Tiff- in points out that it is important to understand the long-run adjustment mechanisms of the produc- tion technology, by means of which milk produc- ers operate during each period of time. Accord- ing to him, the control of the output and input quantities, in order to change production that has been the objective of agricultural policy, is not a reasonable measure, unless the production tech- nology of a dairy farm can be changed at the same time. Finally, he notes that in empirical stud- ies there generally occurs a “trade-off” between theoretical exactness and statistical reliability. 1550 Agricultural Science in Finland 3 (1994) Thijssen (1992b) examined supply response and input demand in milk production in the Neth- erlands using incomplete panel data. He used the quadratic profit function as the flexible function- al form in the short-run examination. He applied Hausman’s test for testing the estimates obtained by means of the fixed effects estimator and ran- dom effects estimator. According to the test, the random effect model is based on the wrong spec- ification, so that he ended up using the fixed effects model. Thijssen had at his disposal infor- mation from the bookkeeping dairy farms from the years 1970-1982. The farms were followed usually for five years, so that the data became incomplete panel data. There were altogether 2,196 observations from 568 dairy farms. As variables Thijssen chose the normalized price, for which he used the ratio of Törnqvist price index of the variable input and the Törn- qvist price index of the output, as well as three fixed inputs (labour, capital, arable land). The aggregate variable input includes feed, fertiliz- ers, seeds, pesticides, fuel, and the output includes milk and meat. He chose a linear trend to de- scribe the technical change. Finally, he added a variable describing the weather index into the demand function for the variable input and to the output supply function. The model estimated by Thijssen fulfilled sta- tistically the assumptions set by the theory. The incomplete panel data made it possible that the intercept in the demand function for the variable input and the intercept in the output supply func- tion vary over the dairy farms. The value of the intercept varied between farms, which shows that there are differences caused by the managerial abilities and the quality of land between farms. As the own price elasticity of the output Thijssen estimated 0.10, and as the price elasticity of the demand for the variable input -0.25. These elas- ticities are smaller than those arrived at in earlier studies based on the profit function. According to the results, the possibilities of the traditional agricultural and environmental policies, which are based on taxation and support, to influence the production quantities are small in the short run. Helming et al. (1993) examined the produc- tion relationship between the short-run outputs and inputs from Thijssen’s (1992a, b) data set before and after the introduction of the milk quo- ta system was presented. They used the restricted quadratic profit function in their examination. The estimated model fulfilled the theoretical assump- tions well. Price elasticities were smaller during the milk quotas than during the time prior to the quotas, but it should be noted that all short-run elasticities were inelastic. According to Helming et al. (1993, p. 358), the overcapacity of quasi- fixed inputs resulting from the milk quotas would influence milk production a great deal, if the quota system were abolished. In the comparison between the research results presented above it must be taken into account that the studies differ from each other with re- spect to their assumptions, the model used, as well as the aggregation of the variables. Empiri- cal analyses have for the most part been problem specific, which means that each research prob- lem had the circumstances of the research period as its starting point. Consequently, the formula- tion of the problem, with its restrictions, in, for example, different countries and in different pro- duction lines has differed a great deal, as the preconditions for production have been different. 5 Theoretical model for the study 5.1 Choice of the cost function for this study. The functional form presented al- gebraically should fulfill all theoretical proper- The theory and the research problem impose re- ties required of economic relationships, in order quirements on the functional form to be chosen to make it possible to include the appropriate 551 Agricultural Science in Finland 3 (1994) parameters into the model (Lau 1986, p. 1520). The theory of statistics does not provide any un- ambiguous guidelines for the choice of the func- tional form. Consequently, the researcher has to choose the functional form and the restrictions on the parameters set for the function to be esti- mated. The theoretical discussion of the first part of the study provides the foundations for the em- pirical analysis. The theory receives particular emphasis in this study, because illogicality is excluded by means of theoretical analysis. By means of the empirical analysis, the values of the parameters are estimated, and the structure of the model is tested. Thus, the values of the pa- rameters arrived at in the empirical analysis de- scribe the economic behaviour of the decision- maker measured from milk production in prac- tice. The Cobb-Douglas production function, which included two inputs, can be considered the start- ing point for the development of advanced func- tional forms (Cobb and Douglas 1928).The prop- erties of the Cobb-Douglas production function include constant returns to scale, neutral techni- cal change, and a constant value, one, for the elasticity of substitution. The CES function, which allows the elasticity of substitution to have other values than one, can be considered the next step forward (Arrow et al. 1961). It is perfect in the case of two inputs, but its generalization to cases with three or more inputs involves too many re- strictions on the substitution possibilities. The development of flexible functional forms was a major improvement compared to the earli- er functions (Jorgenson 1986,p. 1845). The trans- log functions (Christensen et al. 1971, 1973) and the generalized Leontief functions (Diewert 1971) became the most widely used flexible func- tional forms. Flexible functional forms refer to functional forms with the second order approxi- mation property (Lau 1986, p. 1540, Chambers 1988, p. 164). Thus, the function includes an ad- equate number of parameters, so that the derived demands and the matrix of derived demand elas- ticities can be solved. According to Diewert (1987, p. 693), the fol- lowing properties must be taken into account in the choice of the functional form: I. Flexibility: The functional form should be so flexible that the amount of free parameters required by the theory can be estimated. 2. Parsimony; Only the minimum amount of free parameters required for the flexibility proper- ty is used. 3. Linearity: The unknown parameters of the cost function should be solved linearly-in-parame- ters from the system of equations to be esti- mated. 4. Consistency: The functional form of the cost function should be consistent with properties 1-6 (see Chapter 2.3.4). According to Diewert (1987, p. 693) the trans- log function fulfills the first three properties, but the fourth gives cause for critique, as in empiri- cal analyses flexible functional forms have not fulfilled this condition at every point of the data. Locally flexible functional forms are well-behaved with respect to the fourth condition, but not nec- essarily at all points of the data. Since the econo- metric statistical conclusions are based on the behaviour of the model at all points, the lack of global properties restricts the usefulness of flexi- ble functional forms. The translog function is globally valid only when the strict conditions set on the parameters prevail, but locally it may be valid under rela- tively more relaxed restrictions. Even if the glo- bal theoretical consistency is not fulfilled, there still exists a set of input prices in which the theo- retical consistency is satisfied, and this may be large enough for all practical purposes (Lau 1986, p. 1527-1539). The lack of global properties has led to the development of new functional forms, in order to obtain globally better behaved functions. Despite these attempts, it has not been possible to devel- ope a new generation of functional forms that behave as desired (e.g., Hetemäki 1990, p. 13). On the basis of the previous discussion, the translog cost function is chosen for the estima- tion of the research model. The translog cost func- tion is 552 Agricultural Science in Finland 3 (1994) N In C(w,y) = p () + ZPilnWi + p Iny + P XT i = I N N + V 2 £ 2 8.-lnWilnW: (38) + V2(3 yy(lny)2 + V^P-n-CT)2 N N + Z(3 jlnWjlny + ZpjTInWjT + p xlnyT, I=l i=i in which p tj = pjh and T describes the technical change. By the properties of a cost function, the cost function (38) must be consistent with linear homogeneity in the prices of inputs. Therefore, the following restrictions have to be imposed on the parameters to be estimated: N N N N N Zpi = I and Zp yi = Zp,, = Zp |i = ZplT =O. (39) i-l i-l i-l i-l i-l By Shephard’s Lemma the cost shares of in- puts can be solved from equation (38). The cost shares of inputs can be presented as elasticities of the cost function with respect to prices of in- puts (see Jorgenson 1986, p. 1885). ainC(w.y) _ dC(w,y) Wi ölnwj oWjC(w,y) (40) wi x i(w.y) = -ékTT = s - (w ’ y) ’ so that the cost shares of inputs are N S|(w,y) =P; + ZPijlnWj + P yi lny + P Ti T. j“l (41) The cost shares of inputs are determinedas the function of the input prices, output, and technical change. According to formula (41), the unknown parameters of the cost shares can be solved line- arly-in-parameters from the system of equations to be estimated. The approximation of the production technol- ogy is homothetic, if the cost function is separa- ble with respect to output and the prices of in- puts, i.e., if Pyi = 0 in all i. Homotheticity means that the cost minimizing input bundle is deter- mined as a function of the prices of inputs and technical change so that the cost minimizing in- put bundle is independent of the level of output. A homothetic cost function is homogeneous, if cost flexibility is independent of output, i.e., pyy = 0. When these restrictions apply, the degree of the homogeneity can be obtained from the coeffi- cient Py . If py = 1, the cost function is linearly homogenous, and production is described by con- stant returns to scale (Jorgenson 1986,p. 1892- 1893). The restrictions of the production and cost the- ory set into the model reduce the space of the parameters so that it is possible to find out the economic relations. In addition, it should be not- ed that the properties of flexible functional forms are local, which means that the interpretation of the results is also local. 5.2 Theoretical model The theoretical framework of the study is illus- trated in Figure 3. In the study it is assumed that milk production can be described by a produc- tion function, which has the properties presented in Appendix 1, and that a typical dairy farm op- erates in conditions restricted by the technical relationship between the inputs and output deter- mined by the production function. The production function can be presented as follows: (42)Y = f(K,L,M,H,T). The total output Y, and the quantities of the capital items K, labour L, purchased feed M, and purchased fertilizers H used on a dairy farm, as well as the technical change T are included in the model. In the study, it is assumed that the prices of inputs and products are determined exogenously, and that a dairy farm minimizes costs. Based on the duality, there exist a cost function which re- flects the production technology. The cost func- tion can be presented as follows: (43)c = g(WK,WL,W M,W H,Y.T), in which c is the total cost of production, WK, WL , W M, W H are the prices of inputs, Y is the 553 Agricultural Science in Finland 3 (1994) total output, and T is a linear time trend, which describes the technical change. The cost function is assumed to fulfill the prop- erties presented in Chapter 2.3.4. After the choice of the functional form, the theoretical model of the study is as follows: In C(w,y) = p„ + pylnY + p K lnWK + P,.InW L + p JnWM + p H lnWH + p TI + V4MT) 2 + 1/2P YY(InY) 2 + >/2P KK (InWK ) 2 + P KLInWK InWL + p KM InWK lnW M + P KHInWK InW H + V 2p LL (lnWL ) 2 + pLM lnWL lnWM + PLHInWLInWH (44) + y 2p MM(lnW M ) 2 + p MHInWM lnWH + y2p HH(lnW H ) 2 + pYK lnYlnWK + p YLInY!nWL + p YM lnYlnWM + P YHInYInW H + PTYTInY + PTKTInWK + P TLTInWL + PTMTInWM + P THTInW H , in which lnWK , lnWL , lnW M, lnWH are the loga- rithms of the prices of capital items, labour, pur- chased feed, and purchased fertilizers, InY is the logarithm of the quantity of total output, and T is the time trend describing the technical change. Fig. 3. Theoretical framework of the study. 554 Agricultural Science in Finland 3 (1994) Technical change is described by coefficients (3Ti, Technical change is Hicks neutral, if (3 Ti = 0 for all inputs X|. Technical change is said to be input xr using (xrsaving), if (iTj > 0(< 0) (Nadiri 1982, p. 444-445, 467). Linear homogeneity in prices of input imposes the following restrictions on formula (44). Pkk + Pkl + Pkm + Pkh - O Pk +Pl+Pm+ Ph - 1 Pll + Plk + Plm + Plh = 0 Ptk + Ptl + Ptm + Pth = 9 (45) PmM + pMK + PmL + PmH = 0 Pyk + PyL + PyM + PyH = 0 Phh + Phk + Phl + Phm = 0 Under these restrictions, the cost function al- lows for non-homotheticity and non-neutral tech- nical change. It should be noted that the translog cost function cannot be made globally concave by imposing restrictions on the parameters. According to Christensen and Greene (1976, p. 662), the simultaneous estimation of the equa- tion system formed by the cost function and the cost share equations is an optimal procedure in order to solve for efficient parameters. Thus, in this study the cost shares of inputs from the trans- log cost function, are also solved. By Shephard’s Lemma, we arrive at the following equations for the cost shares. Sk = Pk + PkklhWk + P KLInWL + P KM InW M + PKH InW„ + PyklhY + P tkT SL =PI + Pu.lnW L + p LK InWK + P LMInW M + PLH InW H + pYL lnY + P TI.T SM =Pm + PmmlhWm + P MKInWK + P„L InWL + PmhliWh + PymloY + PtmT S H =Ph + P HHInW H + p HKlnWK + pHL lnWL + P„M InWM + p YH lnY + pTHT, in which SK , SL, SM, S H are the cost shares of capital items, labour, purchased feed, and pur- chased fertilizers, respectively. The equations for the cost shares sum up into one, because their total must be equal to the total costs. The cost share equations have to fulfill the assumptions set by the production theory (45), as well as the symmetry assumption 3,,= 3,i- It is implicitly assumed in the study that, as a group, capital items, labour, purchased feed, and purchased fertilizers are weakly separable from the inputs excluded from the study. 5.3 Research method In econometrics it is crucial to estimate the rela- tionships between the observed economic varia- bles so that they can be quantified. In general, the relationships to be estimated are assumed to be stochastic. In practice, this means that the re- lationships between variables are specified so that they can be divided into a deterministic and sto- chastic part (Lau 1986, p. 1516). The cost share equations and the cost function presented in the previous chapter form the deter- ministic part of the econometric model of this study. This is where the cost share equations and the cost function are presented as a function of the observed variables and the unknown parame- ters. In addition to the deterministic part, a sto- chastic part is added to the right-hand side of the equations. The cost function (44) and the cost share equations (46) form the equation system to be estimated. Parametric restrictions are set between the equa- tions of the equation system so that the assump- tions set by the neoclassical production and cost theory are fulfilled. The symmetry assumption will be tested. The cost shares are the same amount as the total costs, which means that the equation system to be estimated is singular. The singulari- ty of the equation system is avoided by taking away one cost share equation from the equation system, and by choosing an estimation procedure that estimates the parameters so that the values of the parameters estimated are not dependent on which equation is left out of the equation system. Taking away one cost share equation from the system does not reduce the information, because the parameters of the excluded cost share equa- tion can be solved by means of the parametric restrictions. According to Pindyck and Rubinfeld (1991, p. 314), the choice of an appropriate estimation method depends on the use of the method, as well as the ability of the method to provide con- 555 Agricultural Science in Finland 3 (1994) sistent and asymptotically efficient estimates. Methodologically the easiest way would be to estimate the translog cost function directly by means of the OLS procedure, but in this case the additional information included in the cost share equations would not be utilized. In addition to this problem, in OLS estimation the multicolline- arity may cause problems in the estimation of an individual cost function (Christensen and Greene 1976, p. 662). It has been noted that the direct application of the OLS and GLS procedures into equation sys- tems usually leads to biased and nonconsistent parametric estimates (Mardia et al. 1979, p. 208). According to Monte Carlo simulations, the esti- mation methods for equation systems have given smaller variance to estimates than the estimation methods for individual equations. Therefore, in- stead of estimating equation by equation, an esti- mation method in which the whole equation sys- tem can be estimated simultaneously is chosen. Modern statistical programs (see Hall 1993) make it possible to use GLS estimation, so that the advanced estimation methods can be taken advantage of in solving the equation system. Zell- ner’s iterative estimation procedure is chosen for solving the research problem of this study (see Pindyck and Rubinfeld 1991, p. 326-327). The method is appropriate for estimating the equation system of the study, because the cost share equa- tions form a system of seemingly unrelated equa- tions (SUR). In these equations the cross-equa- tion error terms of the cost share equations are correlated, and none of the cost share variables, acting as dependent variables, occurs on the right- hand side of equations to be estimated. In iterative SUR estimation each equation is first estimated separately by means of the OLS procedure. The estimations provide the estimates for the error terms, from which the covariance matrix of the error terms is formed. After this, GLS estimation is used. In GLS estimation the error terms are estimated again, and the covari- ance matrix of error terms is updated. GLS itera- tion is continued until convergence is achieved (p = 0.000001). In system estimation the correlation between residuals can be taken into account, which im- proves the efficiency of the estimation (e.g., Won- nacott and Wonnacott 1979, p. 503). System estimation also makes it possible to set the pa- rameters of one equation to be estimated to be the same as the parameters of another equation or, other equations of the system. According to Monte Carlo simulations made by Kmenta and Gilbert (1968, p. 1191-1192), by iterating Zellner’s procedure, ML estimates (maximum likelihood) can be solved for the pa- rameters. Determining the ML estimates makes it possible, for example, to test the homotheticity hypothesis by the likelihood ratio test. The deter- minants of the error covariance matrix estimated from an unrestricted and a restricted equation sys- tem are marked and QR , respectively. The likelihood ratio test is as follows (e.g., Spanos 1986, p. 299-303): x= ( I ql, I / laR I )-T/2 , (47) in which T is the number of observations. The testing of the hypotheses is based on the test sta- tistic,' -21 n X, which has asymptotically a %2 dis- tribution, with degrees of freedom equal to the number of restrictions imposed (Christensen et al. 1975, p. 378-389). 6 Agricultural production during the research period The importance of agriculture for employment and in the gross domestic product of the national economy has decreased considerably since the beginning of the 19605. In 1960, the share of agriculture in the GDP was 10.7% (Tilastollinen päätoimisto 1970, p. 287), and in 1991, it was 3.3% (Tilastokeskus 1993, p. 37). The decrease of the share of agriculture in the GDP means that 556 Agricultural Science in Finland 3 (1994) the share of people earning their livelihood from agriculture has also decreased. During the same period of time, the share of the population em- ployed in agriculture, out of the total active la- bour force, has fallen from 29.2% (Tilastollinen päätoimisto 1970, p. 36) to 7.1% (Tilastokeskus 1993, p. 106). In the beginning of the 19605, the number of agricultural enterprises started to decrease, and the average arable land area started to increase. Thus, the early part of the 1960 s can be consid- ered a turning point in the structural develop- ment of agriculture (Mäkinen 1990, p. 101). The decrease in the number of farms can be seen, in particular, in the rapid decrease of the number of agricultural enterprises with less than 10 hectares arable land. At the same time, the number of agricultural enterprises with over 10 hectares ar- able land increased slightly. However, the increase in the average arable land area has been quite slow. During the research period, Finnish agricul- ture became increasingly specialized. Farms no longer raise different species of animals, and they have specialized in either crop production or live- stock production. At present, most agricultural enterprises engaged in livestock production raise only one species of animal. During the research period, it was also characteristic that the share of dairy farms decreased continuously, and the share of crop producing farms increased. The development has been rapid in milk pro- duction. The number of dairy cattle has fallen from 240,000, at the beginning of the 19605, to under 35,000 at present. Despite the rapid change, there are less than 10 cows on more than half of the dairy farms, which means that this develop- ment is likely to continue into the future (Ry- hänen 1989, p. 51). The trend in the yields of field crops has been increasing, but the total yields have varied a great deal because of the weather conditions (Siltanen 1977, p. 5-8, Kettunen 1993, p. 12-15). The years 1976 and 1990 were exceptionally good, and in 1981 and 1987 the yields were exception- ally poor. One of the most obvious features of the struc- tural change in agriculture has been the rapid decrease in the labour force in agriculture. The decrease in the number of agricultural enterpris- es has not been nearly as drastic, so that the dif- ference in the trends must at least partly be caused by the fact that the shift of the labour force from agriculture to other sectors has mainly concerned the assisting family members and hired labour (Ylätalo 1987, p. 30). As the labour force in agriculture has decreased, the use of other inputs has increased clearly, except for agricultural land. The area of agricultural land increased slowly in the 19605, but in the 1970 s it started to decrease. In addition to the changes in the structure and number of agricultural enterprises, significant changes have also occurred in the use of inputs and in investments. From 1960 to 1984, the use of production factors (e.g., purchased feed, pur- chased fertilizers) tripled, and the investments in- creased 1.5 times (Ylätalo 1987, p. 34). The net capital stock increased steadily, resulting from the fact that depreciations were smaller than in- vestments. Total agricultural production has continued to grow, despite various measures to restrict pro- duction. At the beginning of the 19605, the foun- dation of new agricultural enterprises was still supported with state funds. During the rapid changes in society in the 1960 s and 19705, agri- cultural enterprises decreased considerably. The overall development of the economy, which at- tracted a labour force, especially from small farms, is considered the most important individual fac- tor behind the change. As a result, the structural development of agriculture was favourable in the early part of the 19705. Since the mid-19705, the central objective of agricultural policy has been to reduce the over- production of agriculture. Reducing overproduc- tion has been difficult, despite the rapid decrease in the number of farmers, because production has at the same time become more productive, and the output and yield levels have increased. The measures to restrict production have slowed down the structural development of agriculture (Yläta- lo 1987, p. 41, Ryhänen 1989, p. 47, Kola 1991, p. 112). 557 Agricultural Science in Finland 3 (1994) 7 Data and variables The theory of the first part of the study forms the basis for the empirical analysis. The empirical observation data consists of the total return of dairy farms, the prices of inputs, and the cost shares of inputs. The total return of dairy farms and the cost shares of inputs were collected from the bookkeeping farms. Four aggregate inputs were chosen for the study: labour (the farm fami- ly and hired labour), capital items (machinery, implements, buildings), purchased fertilizers, and purchased feed. Forestry, non-agricultural produc- tion, secondary incomes, and the private house- hold of dairy farms were not included in the study. The data were collected from the entries of the accounting year, which were marked down in terms of flows, so that the output and the inputs used in producing the output are located during the same time period. The year 1965 was chosen as the beginning of theresearch period, because this is when the book- keeping farms shifted from the accounting period beginning in July, to an accounting period start- ing from the beginning of the year. In the ac- counting year 1964-1965, a complete balancing of the accounts had not been made on all farms, so that the data of this period remained small. In addition, a so-called incomplete balancing of the accounts had been made for some of the farms. These changes made it impossible to obtain reli- able data from the beginning of the 19605. In the course of time, certain changes have been made in the bookkeeping form, and the changes affecting the data of this study have to be dealt with. The accounting period 1965 forms the basis for the whole research period, so that the variables of this accounting period are pre- sented in detail in Appendix 5. The changes made in the forms and a detailed account of the data until 1991, which is the last year included in the study, are also presented in Appendix 5. The number of bookkeeping farms has varied during the research period, farms have been left out, and new farms have joined the bookkeeping. Two different sets of data were collected from the bookkeeping dairy farms: A dairy farms (44), which were included in the study during the whole research period (in 1962 and 1989 the milk production of these farms ac- counted for a minimum of 50% of the total return of agriculture, which does not include direct sub- sidies), and B dairy farms (168-364), on which the milk production of the year in question accounted for a minimum of 70% of the total return of agricul- ture, which does not include direct subsidies. The idea of a “representative” dairy farm was applied to both data sets (cf. Tiffin 1991, p. 397). The observations of representative A and B dairy farms were obtained by calculating separately the averages from the A and B dairy farm data sets for each year. The data set collected from the A dairy farms forms a panel data set. The production and cost theory is based on the decision-making of an in- dividual entrepreneur. This means that the use of panel data consisting of information collected from individual dairy farms are theoretically on firmer ground and makes the application of theory more reliable than the use of aggregate data. During theresearch period the development of ADP made it possible to solve the research problem with panel data. Thus, the A dairy farms data set will be used as the panel data set, too. The following information was collected from the bookkeeping farms: Returns: return of milk, livestock, crop and garden production, and other agricultural return. Costs: purchased feed, purchased fertilizers, maintenance of buildings and implements, rents of machinery and implements, insurance (tractor and combine harvester), and the labour costs of agricultural enterprises performed by a hired la- bour force (wages and social security payments). Consumption oflabour: hours of labour ofhired workers and the men, women, and children of the farm family currently working and investing in agriculture. Investments: investments in machinery, imple- ments and production buildings, as well as in- come from sales of machinery and implements. 558 Agricultural Science in Finland 3 (1994) The quantity ofoutput The quantity of output was determined from the total return of agriculture, which includes the return of livestock production, crop production, and garden production. The percentage share of livestock production of the total return of agri- culture on a representative A dairy farm varied annually from 85.6% to 97.1%, the average be- ing 93.3%, and on a representative B dairy farm between 92.0% and 97.9%, with an average of 96.1%. The quantities of output had not been written down on the farms included in the study, so that the aggregate output quantity was determined as follows. The producer price and quantity series of milk and other livestock products were used for determining the aggregate quantity of output, because it was not possible to determine the price and output series for the products forming the latter part of the total return of agriculture on the basis of the data. Also, the share of the products forming the latter part of the total return of agri- culture was small. The producer price series of beef was used as the price series of other live- stock products, because the meat from animals removed from production forms the largest share of other livestock products. This also includes the animals supplied to other farms, but these could not be separated into a group of their own in the data. On the basis of this, the aggregate quantity of output was determined so that the total return of agriculture was regarded as consisting of the re- turn of milk and other livestock products. The total return of agriculture was divided into the returns of milk and other livestock products in 2 proportion to their shares of the return (E = 1). i= 1 By means of the producer price series of milk and beef, the produced quantity series of milk and other livestock products were calculated. The aggregate output quantity is calculated from the price and quantity series as a Divisia index (1985 = 1 .0000), M In Q, -InQ,, = E Vi (v it + v it_,) (In qjt -In qit_,), (48) in which q,,...,q„ are output quantities and Q is the Divisia quantity index. It is obtained by cu- mulating over time a weighted sum of rates of change of the component quantities. The weights are the arithmetic average of the output shares, M vit - (Pitdi/ p jtq jt), in each of the two periods j=i for which the rate of change is calculated. p,,...,p„ are producer prices. The Divisia index is used in aggregation, because as an ideal index it was con- sidered the best choice among the index numbers (see Hulten 1973, Diewert 1976, 1978, Yläta- lo 1987, p. 18). The use of the Divisia index for aggregation has been a general starting point in empirical de- mand studies (e.g., Ray 1982, Higgins 1986). When the Divisia index is used, linear homoge- neity of this aggregate is assumed. The Divisia quantity indices for the representative A and B dairy farms are presented in Appendix 6. Costs Labour cost: Labour cost is arrived at by adding up the labour costs of agricultural enterprises per- formed by a hired labour force (wages and social security payments) and the costs of the labour of the farm family. The labour cost of the farm fam- ily is determined by multiplying the hours of la- bour by the unit wages of agricultural workers. The share of hired labour in the total costs of the farms included in the study was small. On the bookkeeping farms the pension payments (MYEL) were separated from other payments only after 1989, so they could not be included in the labour cost. Until 1990, the pension payments were de- termined according to the area (“MYEL” hec- tares), specified separately for each farm, and in- cluded the share of forests as well. Cost of machinery, implements, and buildings: For determining the cost of machinery, imple- ments, and buildings, information on the capital stock, investments, and depreciations of dairy farms is needed. In this study the depreciations for agricultural machinery and implements is de- termined by assuming that after 15 years of use, machinery and implements have 10% of their in- 559 Agricultural Science in Finland 3 (1994) itial value left. The research results on the age and time of use of machinery were taken into account when determining the depreciations (e.g., Bolin 1971, p. 66, Juvonen 1982, p. 44, Kämäräinen 1988, p. 55-57). When economic plans are made in business economics, shorter periods of time than the tech- nical time of use are often used in order to make sure that the property shares are depreciated in full before they are no longer used. In this study, the times of use of machinery and implements are based on research results, which means that they can be considered more accurate than the times of use based on rough approximations. How- ever, certain reservations have to be made, be- cause it is very difficult to determine the exact time of use of long-run property shares. The depreciation of production buildings is de- termined in the same way as that of agricultural machinery and implements, and the time of use is 40 years (see Vihavainen et al. 1980, Ylätalo and Pyykkönen 1991). The property series used in the study are formulated so that the amount of capital at the end of the period t is equal to Kt . The series is formulated as follows (Boyle 1981, p. 158): (49)Kt = It + (l-d)K,_„ in which I, = investments in period t d = constant rate of capital depreciation In order to define Kt , the depreciation rate of capital d must be known. According to the as- sumptions presented above, the depreciation rate is determined as follows: (50)(l-d) v = 0.1, in which V is the years of use of capital, when there is 10% left of the initial value of the capital item. As a result of these assumptions and calcula- tions, the depreciation rates for agricultural ma- chinery, implements, and buildings are as follows: machinery and implements d = 0.1423 buildings d = 0.0559 For practical reasons, in 1968, the bookkeep- ing farms started to use the same property values used in the tax system (MTTL 1970, p. 3). In this connection, an initial inventory of capital assets was made according to section 18 in the Act on the Taxation of Farm Income (MVL). In the ini- tial inventory of the capital assets, the replace- ment cost of agricultural buildings, machinery, and implements corresponding to their economic use value deducted by the following value reduc- tions for each year, from the year when the goods were purchased or built, until 1968, is used as the purchase cost of agricultural buildings, ma- chinery, and implements (Verohallitus 1988, p. 107-108): production buildings: made of wood 5.0 % 4.0 %made of stone machinery and implements 20.0 % In the study, the values of the capital items at the end of 1967, determined in the initial inven- tory, are considered the starting point in formu- lating the capital series. The capital values at the end of 1964 for machinery, implements, and build- ings were determined as follows: machinery and implements K, = (K t+,-I t+ |)/(l-0.2) (51)buildings K, = (Kt+l -I t+I )/(l-0.045) The capital series for machinery and imple- ments was determined by means of equations (49) and (50) with the values at the end of 1964 as the starting point. The value of the investment work of the farm family was added to the building investment series of the bookkeeping farms. The value of the investment work on a dairy farm was calculated in the same way as the cost of the agricultural labour of the farm family. The cost of machinery, implements, and build- ings is obtained by multiplying the value of the capital series each year by the sum of the depre- ciation rate and the real interest rate, and by then adding the repair costs and rent of machinery. 560 Agricultural Science in Finland 3 (1994) The real interest rate was defined as the effective yields on tax free government bonds, deducted by the rate of inflation calculated from the cost- of-living index. The bonds were considered an alternative to investments. Cost of purchased fertilizers and purchased feed: Costs collected directly from bookkeeping farms are used as costs of purchased fertilizers and purchased industrial feed. Cost shares Si =ci / Zc|, S| = the cost share of input i C| =cost of the use of input i Prices In the base year 1985, all indices are normalized so that they receive the value 1.0000. Labour: The price of the labour input of men and women was collected from the publications “Studies on the profitability of Finnish agricul- ture” by the Agricultural Economics Research In- stitute (AERI) for the years 1965-1991. The total average income of agricultural workers (FIM/h) in quarters of a year was used as the price of the labour input. Because of the change in the statis- tical basis in 1982, the price of the labour input for 1965-1981 was chained to correspond to the series of 1982-1991. The use of labour in agriculture varies accord- ing to the season. According to the research re- sults, grazing reduces the use of labour during the summer, but, at the same time, the cultivation of crops increases the use of labour on dairy farms. On the basis of earlier research results, it was concluded that in the first and fourth quarter of a year the use of labour on dairy farms is about 20%, and in the second and third quarters it is about 30% of the use of labour during the whole year (Ryynänen and Pölkki 1982, p. 83, Alas- talo 1991, p. These percentages were used as weights in formulating the annual labour input price series. The quantities of labour input were collected from the hours of labour done by men, women, and children on the farms included in the re- search. In the profitability study of agriculture the work done by under fourteen-year-olds is es- timated at 50% of the work done by men, so that the hours of labour of men and children were calculated by dividing the children’s hours of la- bour by two, and adding this to the men’s hours of labour. This was followed by the calculation of the Divisia price indices of the labour input, presented in Appendix 6. Purchasedfertilizers and purchasedfeed: price indices calculated in the AERI, presented in Ap- pendix 6. Price of capital: The price of capital Wk is equal to the user cost of capital. It is considered an appropriate definition for the price of the cap- ital input, because it describes the cost shares related to the ownership of the capital input. Ac- cording to Boyle (1981, p. 159), the price series of capital in question can be formulated as fol- lows: (52)W K = q(r +d) in which q = cost index of capital in question r = real interest rate d = fixed depreciation rate The user costs of capital were calculated as Divisia indices, which are presented in Appendix 6 (Note! when inflation is high, (r+d) can be negative). In 1973-1975, it was negative in the case of production buildings. Thus, Divisia indi- ces cannot be calculated for the years 1973-1975, and for these years WK was calculated as a weight- ed average of capital items with weights of 0.3 for production buildings and 0.7 for machinery and implements (average cost shares during the research period). It would be logical for the study to include arable land as well. However, determining the price and quality of arable land on the farms in- cluded in the study was impossible. Also, there were considerable differences between different statistical sources in the development of the pric- es of arable land (Ylätalo and Pyykkönen 1991, p. 3-7). For example, changes in the land use, mechanization of agriculture, the change on the 561 Agricultural Science in Finland 3 (1994) trade of arable land into sales of supplementary land, as well as periods of inflation during the research period have affected the price of arable land. No reliable research data was available, and, consequently, arable land had to be excluded from the study. Thus, the assumption of weak separability is not completely valid for arable land. However, in empirical studies the assumption of weak separa- bility has to be made, and it has assumed a so- called “State of Art” status in applied economic study (Chambers 1988, p. 157). The problems related to the determination of the price of arable land and the capital stock, as well as the reliability of the data available from different sources, are extensively covered in the publications of Ihamuotila and Stanton (1970, p. 10-28), Ryynänen (1978, p. 77-85), Ihamuoti- la (1983, p. 9-37), and Ylätalo and Pyykkönen (1991, p. 3-7). 8 Results of the study 8.1 Descriptive statistics of the farms included in the study The data consists of farms participating in the profitability research ofFinnish agriculture. Book- keeping is realized on a voluntary basis, which means that these farms cannot be regarded as representing all Finnish dairy farms. The average farm size in terms of both the number of dairy cows and the area is considerably larger than in the whole country on the average. In 1990, the average number of cows on dairy farms of group A was 14.3 and in group B 15.6, whereas on all dairy farms the average number of cows was 10.8. The level of output is also above the average on the bookkeeping farms. Since the bookkeeping is voluntary, the agricultural entrepreneurs partici- pating in it are more active than usual. Information from the bookkeeping data was used, because the assumptions related to the pro- duction theory are more likely to be fulfilled on the farms included in the study than in sample data. Bookkeeping farms include fewer farms that give up production, which distorts the picture of the production technology on farms that continue production, than on all Finnish farms. Changes in the prices, measures to regulate production, changes in the production conditions, and other factors affect the bookkeeping farms in the same way as other farms are affected. This means that from the viewpoint of solving the research prob- lem, the data is as good as sample data. Also, using sample data was not possible, because sam- ple data was not recorded. The data consisted of two sets: by means of dairy farm group A, the distortions caused by the change of farms could be eliminated, so that the data describes the development of the production technology in the same farm group. In dairy farm group B, part of the farms and the number of farms varied annually, so that the data were col- lected to describe the development of the pro- duction technology in the whole sector. The de- crease in the number of the weakest and smallest dairy farms in the data should describe the gener- al development in the whole sector reasonably well. It can be assumed that the change in the production technology has been more rapid on dairy farms of group B than in group A. The development of the production technolo- gy on farms included in the study should be seen as development on farms that continue milk pro- duction, which should describe the future devel- opment quite well. The development of the ara- ble land area and the number of dairy cows on a representative dairy farm of group A and B for 1965-1991 is presented in Figures 4 and 5. Figures 4 and 5 show that on dairy farms of group B the farm size has grown more rapidly than in group A, measured as both the number of dairy cows and the arable land area. The devel- opment is similar when measured as the quantity 562 Agricultural Science in Finland 3 (1994) of output. The number of cows, arable land area, and their standard deviations every five years are presented in Tables la and lb. The development of the output quantities, costs, and cost shares is presented in Appendix 6. 8.2 Representative dairy farm approach 8.2.1 Short run in a long-run study Economic theory does not provide any unambig- uous solution to the combination of short-run and long-run studies. As was noted in Chapter 3, this problem has aroused a lot of discussion both in economics in general and in agricultural econom- ics. The adjustment cost model and the disconti- nuity of the first derivative function of the ad- justment cost function linked to the fixity of in- puts are difficult to combine into the same mod- el, so that the study of the short-run adjustment by means of the traditional econometric techniques has been seen to be impossible. It is known that economic phenomena include an adjustment cost, but its theoretical foundations are not yet known accurately enough. The error correction model has been offered as the solution to the combination of the short-run and long-run examination. The error correction model does not provide a solution to the problem of the economic theory, but it provides for the possibility to examine the linkage between the long-run equilibrium and the short-run dynamic development by means of empirical data. In the Table la. Number of cows, arable land area, and their standard deviations on dairy farms of group A every five years. Cows S.dev. A.l.area S.dev. 1965 7.04 2.43 15.77 6.83 1970 8.03 2.29 16.03 5.85 1975 10.62 3.87 18.69 7.31 1980 13.32 5.30 20.40 7,10 1985 15.02 5.10 22.88 8.02 1990 14.31 4.88 25.99 11.59 Table lb. Number of cows, arable land area, and their standard deviations on dairy farms of group B every five years. Cows S.dev. A.l.area S.dev, 1965 7.01 2.73 12.47 5.77 1970 8.04 3.38 14.19 6.29 1975 11.88 5.55 17.98 7.95 1980 14.41 7.04 20.02 10.40 1985 15.66 6.66 22.59 11.23 1990 15.57 5.89 24.53 10.96 563 Fig. 4. Development of the arable land area on farms included in the study for 1965-1991. Fig. 5, Development of the number of dairy cows on farms included in the study for 1965-1991. Agricultural Science in Finland 3 (1994) following, it will be examined if the time series of the study are cointegrated, i.e., if it is possible to construct an error correction model to study the derived demand of inputs in milk production. 8.2.1.1 Autocorrelation functions The autocorrelation function indicates the amount of correlation between the data points in the time series. In a stationary process the average, vari- ance, and covariances are invariant with respect to time. The autocorrelation function can also be used for testing the stationarity of the time series (Pindyck and Rubinfeld 1991, p. 449). If the valueof the autocorrelation function does not de- crease rapidly along with the increase in the lag, the result indicates nonstationarity. On the basis of the autocorrelation functions, all the level terms of the time series of the study seemed nonstationary. This is, because the first autocorrelation coefficient is fairly high, and the coefficient decreases slowly when the lag is in- creased, but S H and SK seem to be questionable series. The autocorrelation functions of differenced series seem to be close to zero in the case of all other series, except for the In WL and In C time series. The autocorrelation functions of the level terms and differenced time series are presented in Appendix 7. According to the autocorrelation functions, the differencing of the series removes the nonsta- tionarity from all series, except for the In WL and In C series. From these series, second differences were taken, and the autocorrelation functions were calculated (see Appendix 7). According to the result, these series would seem to follow the 1(2) process. It is possible that the time series are cointe- grated, if they are integrated of the same order, or if the higher order series are cointegrated with the order of the low order series. According to the autocorrelation functions, all series are not integrated of the same order. 8.2.1.2 DF-and ADF-tests In a more accurate study of the order of integra- tion. the Dickey-Fuller (DF) and Augmented Dick- ey-Fuller (ADF) tests are used. The following regressions are made in the tests (Granger 1991, p. 71): (53)DF: Ax, = (p-l)x,_, + e, p ADF: Ax t = (p-l)x t_, + ZyjAx( _j +e„ j = i (54) in which A is x,-x,_,. In the DF- and ADF-tests the H 0 is that p = 1, in which case x, ~ 1(1). The DF-test is used in the first order autoregressive case, and the ADF-test in the higher order au- toregressive processes. Values defined by Mackin- non (1991, p. 267-275) are used as the critical values for cointegration. To start with, the time series are examined by regressing each variable in the level form by their five lags. The estimated results are presented in Appendix 8. According to the results, the series seem to follow the AR(1) process, except for the In WH series. ADF-series is thus used only for In W H series, because otherwise the test becomes overparametrized in the AR(1) series (Hetemäki 1990, p. 35). Theoretically the ADF regression should include an adequate number of lagged first differences to make sure that the error term of the ADF regression would be white noise. In an empirical study this objective may be difficult to achieve, if the process has an ARMA representa- tion, because the corresponding AR process usu- ally has an infinite number of terms. The DF- and ADF-tests were made with the intercept and the trend terms, because economic time series involve a trend component. On the basis of the results, all series seem to be 1(1), except for In WL , In W M, and In W H series, which seem to be 1(2) series, but In WH may be an 1(3) series. On dairy farms of group A In C and SL series, and on dairy farms of group B SM series are 1(1) series at the 10% significance level, but they are 1(2) series at the 1% significance level. The results of the DF- and ADF-tests are present- ed in Tables 2a and 2b. The results from the autocorrelation functions are parallel to the results of the DF- and ADF- tests. In both cases In WL looks like an 1(2) se- ries, and In C possibly like an 1(2) series. The 1(2) series according to the DF- and ADF-tests, 564 Agricultural Science in Finland 3 (1994) Table 2a. DF- and ADF-tests on dairy farms of group A. Serie Levels l.diff. 2. diff. 3. diff. In WK -2.199 -4.265 In WL -0.120 -2.381 -5.685 In W M -0.895 -2.525 -4.591 In W H -3.220 -2.777 -3.385 -5.422 In Y -0.867 -4.921 In C 0.121 -3.437 -7.030 S K -1,837 -4.199 SL -1.604 -3.322 -5.618 S M -0.865 -4.164 S„ -1.875 -5.422 Table 2b. DF- and ADF-tests on dairy farms of group B. Serie Levels l.diff. 2. diff. 3. diff. In WK -2.197 -4.275 In WL -0.050 -2.140 -5.179 In WM -0.895 -2.525 -4.591 In WH -3.220 -2.777 -3.385 -5.422 In Y -0.598 -4.701 In C 0.332 -3.821 S K -1.803 -4.422 SL -1.590 -5.067 S M -0.909 -3.531 -5.470 S H -1.011 -5.334 Table 3. Mackinnon’s critical values of cointegration for Tables 2a and 2b. C.value Levels l.diff. 2. diff. 3. diff. I % -4.374 -4.394 -4.417 -4.442 5 % -3.603 -3.612 -3.622 -3.633 10% -3.237 -3.242 -3.247 -3.254 In W M and In W H , received the highest values of the autocorrelation function when differenced once, after the In WL series. According to the theory, cointegration is pos- sible if the variables are integrated of the same order. It is possible that a valid cointegration re- lationship exists between series of different or- ders, if the higher order series are cointegrated with the lower order series (Hetemäki 1990, p. 39). According to the results, In WL , In W M, and In W tl are 1(2) series, but if a cointegration vector, which is 1(1), exists between them, the combination of these series and all 1(1) series may form a valid cointegration relationship. Therefore, it will be examined with yet anoth- er cointegration test if the first differences of the prices of labour, purchased feed, and purchased fertilizers (In WL , In W M, and In WH ) are cointe- grated. The test is made as an Engle-Granger cointegration test, in which one series at a time is regressed by the other series (Engle and Grang- er 1991). The test is based on testing the station- arity of the error term. If the error term is station- ary, the variables are cointegrated. In the first stage of the test the cointegration regression be- tween once differenced variables is estimated. In the second stage the stationarity of the obtained error term is tested by the DF-test. The results are presented in the following arrangement (A:=dairy farms of group A and B:=dairy farms of group B). Depen- Indepen- Indepen- DF t value dent dent dent A: InW, In W M In W H -2.841 A: In W M In WL In W H -3.571 A: In W H In WL In W M -2.526 B: In WL In WM In W H -2.755 B: In W M In WL In W H -3.572 B: In W H In WL In W M -2.553 Mackinnon’s critical values: 1% -5.486 5%-4.621 10%-4.210 In addition, in the second stage of the cointe- gration test the ADF-test was performed on those error terms of the cointegration regressions in which the dependent variable of the cointegra- tion regression on dairy farms of group A was In WL and In W M, and in group B In W M , because the autoregressive process of these error terms seemed a higher process than the AR(1) process (see Appendix 8). According to the ADF-test, the DF t values were -2.364 (A: In WL ), -2.800 (A: In W M ) and -2.783 (B; In W M), when Mackin- 565 Agricultural Science in Finland 3 (1994) non’s critical values at the 10% significance lev- el were between -4.2 and -4.3. According to the cointegration tests, 1(2) se- ries are not cointegrated, which means that they cannot form a cointegration relationship with the 1(1) series and, consequently, an error correction model cannot be constructed to study the derived demand of inputs in milk production. 8.2.2 Empirical models and hypothesis tests An empirical investigation is carried out on the basis of the theoretical model. In order to solve the research problem, alternative model specifi- cations made possible by the production and cost theory were estimated from the empirical data, and the parameter restrictions set in the theoreti- cal examination were tested. The parameters of the models were estimated from the equation sys- tem, consisting of the cost function (44) and the cost share equations (46), on which restrictions according to the production and cost theory were set between the equations. The restrictions were released according to each test situation. The sym- metry assumption could not be tested from this equation system because of the lack of the de- grees of freedom, so that they were tested from the equation system of the cost share equations (46). The dependent cost shares sum up into one, i.e., the cost share equations form a singular equa- tion system. As a result, the covariance matrix of the error terms of the cost share equations is sin- gular. The singularity problem was solved by leav- ing out one cost share equation in the estimation (fertilizer input). The parameters of the equation left out were calculated from the parameter re- strictions, and their t values by replacing the cost share equation of purchased feed by the cost share equation of fertilizers. The following models made possible by the production technology were esti- mated from the equation system of the cost func- tion and the cost share equations: 1. non-homothetic function - non-neutral tech- nical change (NH-NN) 2. homothetic function - non-neutral technical change (H-NN) 3. homogenous function - non-neutral technical change (HG-NN) 4. non-homothetic function - neutral technical change (NH-N) 5. constant return to scale - non-neutral techni- cal change (C-NN) The models derived in the theoretical part of the study and the parameter restrictions set for them are tested by means of the empirical data. If statistical tests reject the models derived from the theory and the theoretical parameter restric- tions, the theory and the models derived from the theory become questionable. In empirical study “theory” is also used to refer to the functional form and the estimation technique chosen, as well as to the economic theory on which the model to be estimated is based (Tiffin 1991, p. 398). The model specifications and parameter re- strictions according to the production and cost theory were tested by the likelihood ratio test. The first row in Table 4 gives the % 2 test value for H 0 hypothesis, for example, supporting the symmetry assumption (SYM) |3j| in all i * j, against the alternative H, hypothesis of no sym- metry. The calculated %2 values of the symmetry assumption on farms of groupA was 9.09, and in group B 2.12, which are smaller than the critical values at the 1% and 5% significance level. There- fore, the H 0 hypothesis on the symmetry of the Hessian matrix is not rejected. The alternative model specifications made possible by the pro- duction and cost theory were tested with the sym- metry assumption in force. The H 0 hypotheses of the homotheticity and homogeneity of the production technology, neu- tral technical change, and constant returns to scale are rejected at the 1% significance level. Accord- ing to the likelihood ratio test, the structure of the cost function cannot be simplified from the general model (NH-NN), so that in the follow- ing, a cost function that is consistent with a non- homothetic production process and describes non- neutral technical change is used. The specification of the cost function to de- 566 Agricultural Science in Finland 3 (1994) Table 4. Critical values of y 1 test in testing H 0 hypotheses. H - Solved value Degrees Critical value hypo- of thesis A farms B farms freedom 1% 5% SYM 9,09 2.12 6 16.81 12.59 H-NN 24.96 36.81 5 15.09 11.07 HG-NN 25.00 36.92 6 16.81 12.59 NH-N 16.58 26.17 4 13.28 9.47 C-NN 53.83 71.76 7 18.48 14.07 Table sa. Estimated parameters solved from the equation system of the cost function and the cost share equations on dairy farms of group A (p values in parenthesis). Pkk 0.1321 pKL -0.0455 PKM -0.0772 Pkh -0 0094 (0.000) (0.000) (0.000) (0.230) PLL 0.0204 PLM 0.0326 PLH -0.0075 PMM 0.0187 (0.399) (0.157) (0.631) (0.545) P MH 0.0259 PHH -0,0090 P„ 1,5757 pK 0.1119 (0.140) (0.612) (0.169) (0.005) P L 0.3827 PM 0.3398 PH 0.1656 PY 4.5892 (0.000) (0.000) (0.002) (0.073) PyY 5.0288 PYK -0.1019 PYL -0.1445 PYM 0.1742 (0.062) (0.008) (0.007) (0.004) P YH 0.0722 PT -0.1380 P„ 0.0061 Pxv -0.2103 (0.001) (0.182) (0.191) (0.079) PXK 0.0056 PXL 0.0034 PXM -0.0056 PXH -0.0034 (0.004) (0.103) (0.020) (0.028) Yk+ 0.0100 Yk- -0.0159 Yl+ 0.0271 Yl- -0.0210 (0.032) (0.002) (0.032) (0.002) Ym+ -0.0275 Ym- 0.0319 Yh + -0.0096 Yh- 0.0050 (0.000) (0.000) (0.017) (0.174) Yk, 0.0743 Yu 0.0251 Ymi -0.0805 Ym -0.0189 (0.000) (0.129) (0.000) (0.111) Yc, 0.5711 (0.069) scribe non-homothetic production technology and non-neutral technical change means that changes in the prices of inputs and the level of output affect significantly economies of scale (size) and technical development. The result of the likeli- hood ratio test is supported by the fact that the parameters pYi and p Xi, presented in Tables 5a and sb, are all statistically significant, except for PXL on A dairy farms, and pXL and PXH in group B. The values of the parameters estimated from the NH-NN model and their asymptotic p values (in parentheses) are presented in Tables 5a and sb. In addition to the relative prices, change in output, and technical change, the weather condi- tions also have an effect on the derived demand for inputs in milk production. The weather con- ditions affect directly the yields of silage, hay, pasture and feed grain, as well as the quality of the crops. Thus, the weather conditions must be estimated in the model as a dummy variable. Dum- 567 Agricultural Science in Finland 3 (1994) Table sb. Estimated parameters solved from the equation system of the cost function and the cost share equations on dairy farms of group B (p values in parenthesis). Pkk 0.1350 P KL -0.0451 pKM -0.0729 PKH -0.0170 (0.000) (0.000) (0.000) (0.011) Pu. 0.0943 P, M -0.0404 p LH -0.0088 PMM 0.1105 (0.001) (0.120) (0.434) (0.036) P MH 0.0028 P HH 0.0231 Po -0.8462 PK 0.1007 (0.833) (0.051) (0.189) (0.005) P L 0.4453 P M 0.3377 P„ 0.1163 PY -0.4611 (0.000) (0.000) (0.001) (0.624) P YY -0.5333 P YK -0.0810 p YL -0.1265 PYM 0.1802 (0.490) (0.000) (0.000) (0.000) p YH 0.0273 P x 0.0804 -0.0037 PXY 0.0307 (0.046) (0.188) (0.197) (0.489) P XK 0.0059 P XI 0.0002 P XM -0.0051 PXH -0.0010 (0,000) (0.876) (0.001) (0.283) yK+ 0,0098 yK_ -0.0170 yL+ 0.0230 yL _ -0.0118 (0.048) (0.002) (0.000) (0.017) yM+ -0.0170 yM_ 0.0278 y H+ -0.0158 y„_ 0.0010 (0.012) (0.000) (0.000) (0.695) yKI 0.1046 yLI -0.0218 yMI -0.0576 yHI -0.0252 (0.000) (0.125) (0.008) (0.007) yCI 0.0451 (0.088) my variables refer to either-or situations. Thus, something either happened or did not happen. According to Intriligator (1978, p. 58-61), dum- my variables can encompass qualitative variables and qualitative shifts over time or space (e.g., war or peace time). A dummy variable receives only two values (0,1), of which one means one possibility, and the other the other possibility. In this study the weather conditions were in- cluded in the cost share equations as dummy var- iables by formulating two distinct dummy varia- bles: 1) an exceptionally good year and the year following it were marked by one and other years by zeros, 2) an exceptionally poor year and the year following it were marked by one, and other years by zeros. It was important to include the weather conditions because almost all of the weather dummy variables of the cost share func- tions of the capital item, purchased feed, and pur- chased fertilizers were statistically significant at the 5% significance level. A dummy variable was also used for the period of high inflation from 1973-1975 (> 15%). In this study, it is assumed that dummy variables affect the intercepts of the estimated functions. The values of the dummy variables yi+ (good years), y,_ (poor years), and y, (period of high inflation) are presented in the lower part of Tables 5a and sb. 8.2.2.1 Regression statistics and tests The validity of the model depends on the overall fit of the estimated system, the significance of the coefficients, and whether the estimated cost function satisfies the theoretical assumptions (Weaver 1983, p. 51). The overall fit of the esti- mated NH-NN models is good, when the meas- urement is made using the R 2 and F values. When measured by the corrected R 2 (R£), the good- ness-of-fit of the cost functions on A farms are 0.994 and on B farms 0.997 (F values: 220.1 and 358.2), and the goodness-of-fit of the cost share equations vary between 0.871 and 0.959 (F val- ues vary between 22.9-77.1). The use of the cor- 568 Agricultural Science in Finland 3 (1994) Table 6a. Goodness-of-fit, F values, and significance of the cost function (C) and the cost share equations (S*), and the standard error of the regression (SER) on farms of group A. Function R 2 F value Signific- SER ance C 0.999 0.994 220.1 0.000 0.077 S K 0.972 0.959 77.1 0.000 0,010 SL 0.938 0.911 34.2 0.000 0.013 S M 0.910 0.871 22.9 0.000 0.013 Table 6b. Goodness-of-fit, F values, and significance of the cost function (C) and the cost share equations (S,), and the standard error of the regression (SER) on farms of group B. Function R 2 R£ F value Signific- SER ance C 0.999 0.997 358.2 0.000 0.064 S K 0.969 0.956 71.4 0.000 0.013 SL 0.961 0.944 55.6 0.000 0.009 S M 0.941 0.914 35.7 0.000 0.012 reeled R 2 in examining the goodness-of-fit of the model is desirable, because when a corrected R 2 is used, the improving effect of the number of dependent variables on R 2 is eliminated (Pindyck and Rubinfeld 1991, pp. 76-79). According to Pindyck and Rubinfeld, the standard error of the regression (SER) in relation to the average of the dependent variable should not exceed 15%. In the cost function and the cost share equations this ratio varies between 1.8% and 7.6%. The goodness-of-fit measures, F values, and SER of the cost function and the cost share equations are presented in Tables 6a and 6b. In evaluating the estimated model, the statisti- cal significance of the estimated parameters, i.e., the significant deviation from zero, is generally used. The number of statistically significant pa- rameters on dairy farms of group A is 23 (18), and on dairy farms of group B 26 (20) of the possible total of 41 parameters at the 5 (1) % significance level. Of the parameters related to the elasticities of substitution and own price elas- ticities (marked in equation (38) by on A farms three (P KK (1%), PKL (1%) and P KM (1%)) and on B farms six (p KK (1%), pKL (1%), pKM (1%), P KH (5%), Pll (1%) and p MM (5%)) out of ten deviated statistically from zero. The statistical significance criterion of param- eters is appropriate for evaluating the parameters of ‘ad hoc’ models, in which the statistical sig- nificance of parameters is used as justification for including them in the model. In theoretically derived models, like the model of this study, the inclusion of the parameters into the model is jus- tified by the theory, which is considered a more powerful justification than the statistical signifi- cance of ‘ad hoc’ parameters (Tiffin 1991, p. 399). The number of statistically significant param- eters may usually give some indication of the degree of reliability of the results provided by the measurements derived from the model. It should be noted, however, that in this model, which is based on the idea of estimating the pa- rameters that are first and second order logarith- mic derivatives of the cost function evaluated at the approximation point, no prior expectations about their sign are set by the production and cost theory (see Weaver 1983, p. 51). According to the results, the weather condi- tions influenced the demand for inputs on dairy farms in a significant way. This result can be regarded as correct, because the shortage of feed resulting from poor crops is satisfied by means of purchased feed and, correspondingly, in good years the need for purchased feed decreases. This means that at the same input prices the cost shares of inputs are not the same in good and poor years. The H 0 hypothesis, the error term of the ex- plained equations includes a first order positive autocorrelation, was testedby the Durbin-Watson test. The critical values of the DW-test in the case of 27 observations and five independent var- iables (Pindyck and Rubinfeld 1991, p. 568) and the DW-values of the estimated functions are presented in Table 7. According to Pindyck and Rubinfeld (1991, p. 144-145), in time series the independent vari- 569 Agricultural Science in Finland 3 (1994) Table 7. Critical values and estimated values of the DW- test. Critical values Est. DW-value Function d, du A farms B farms C 1.01 1.86 1,30 1.16 S K 1.01 1.86 1.55 1.29 SL 1.01 1.86 1,20 1.67 S M 1.01 1.86 1.34 1.35 ables are likely to be autocorrelated, in which case the use of the d, value may be the more significant one of the two values. In connection with the testing of the time series properties of the independent variables, it was observed that they include an AR(1) process, and in one case an AR(2) process. The DW-test values of the es- timated functions remain between the critical val- ues, which means that the H 0 hypothesis can be neither accepted nor rejected. If d, is considered the limit of the test, the H 0 hypothesis can be rejected. When the covariance matrix of the error terms is singular, the significant autocorrelation of the error terms causes the estimated parameters to become dependent on which cost share equation is left out of the system to be estimated (Berndt and Savin 1975). When the cost share equation to be left out was varied, no changes were ob- served to occur in the values of the parameters, so that in this respect the possible autocorrela- tion of the error terms does not hinder the esti- mation and interpretation of the parameters. As a summary of the general examination, it can be noted that the models are at least satisfactory. 8.2.2.2 Empirical models and the theoretical assumptions The cost function estimated empirically is well behaved, if it is consistent with a theoretically well defined cost function. In a well behaved cost function, the cost shares, cost flexibility, and tech- nical change are homogenous of degree zero with respect to input prices. In addition, a well be- haved cost function satisfies the conditions of monotonicity and concavity. Also, the cost shares must sum up into one, and the Hessian matrix of the second partial derivatives of the cost function must be a symmetric matrix, according to Young’s Theorem. According to the production and cost theory, the cost shares and cost flexibility should be non- negative. Parametric restrictions that make sure the non-negativity of the cost shares and cost flexibility cannot be set in advance to the trans- log cost function without losing the flexibility of the cost function. Instead, restrictions that imply monotonicity of the cost shares wherever they are non-negative are considered (Jorgenson 1986, p. 1858). The matrix of the cost shares derived from the translog cost function ((Jjj+SS’-diagfS]), in which diag[S] is the diagonal matrix with S/s on the diagonal, must be negative semidefinite (Lau 1986, p. 1543). The negative semidefinite- ness of the matrix ensures that the cost function is concave as a function of input prices when the cost shares are non-negative. The symmetry assumption was tested from the cost share equations, and it was accepted in both farm groups. On dairy farms of groups A and B the estimated cost shares were positive at all ob- servation points. On B farms the cost flexibility was positive at all observation points, but on A farms 14 out of 26 observations were negative. According to Ball and Chambers (1982, p. 706), it is possible that the cost flexibility and techni- cal change are confounded when equation (38) is estimated, in which case it is doubtful whether the cost flexibility estimated from equation (38) can be interpreted appropriately. According to Diamond et al. (1978, p. 147), reliable cost flexibility cannot be defined from time series data, rather cross section data is need- ed. In spite of this, equations like (38) are fre- quently used in empirical studies to solve the cost flexibility from time series data (e.g., Ray 1982, Glass and McKillop 1990). In empirical studies in economics, restrictive assumptions are also generally made on, for ex- ample, the neutrality of cost flexibility, in which case the output has not been included in equation 570 Agricultural Science in Finland 3 (1994) (38) (e.g., Binswanger 1974), or constant returns to scale (e.g., Berndt and Wood 1975). If the cost function of the dairy farms of groups A and B is restricted by the homogeneity assumption (HG-NN), the cost flexibility on A farms is 0.33 and on B farms 0.39. When the cost function to be estimated is defined as homogenous, explain- ing the variation of costs by input prices and technical change only may lead to an incorrect interpretation, if cost variation is partly caused by variation in the factors of cost flexibility, too. The technical change and cost flexibility are dealt with in more detail in a chapter of their own. A necessary condition for the concavity of the cost function as a function of input prices is that all own price elasticities are negative. A neces- sary and sufficient condition on concavity is that the Hessian matrix formed from the parameters estimated from the cost function is negative sem- idefinite. The sign of the own price elasticity of labour, purchased feed, and purchased fertilizers is at all observation points in accordance with the theory. The sign of the own price elasticity of capital items is in accordance with the theory, except during the years of exceptionally high in- flation 1973-1975, and in 1980, when the infla- tion was also high, so that the necessary condi- tion on concavity is not satisfied in these years. In other years the Hessian matrix formed from the parameters estimated from the cost function on dairy farms of group A is negative semidefi- nite. In group B the Hessian matrix is negative semidefinite in 1978 and 1983-1991, and in other years it is indefinite. A possible reason for the deviation of the model from the assumption for 1973-1975 and 1980 is the rapid and drastic change in the relative prices caused by inflation. The indefiniteness of the Hes- sian matrix of B dairy farms for 1965-1977 and 1979-1982 may be caused partly by the great annual variation in the number of farms included in the study during this period of time (Appendix 8). According to the research results, during rap- id and drastic changes it is not possible to adjust the production of dairy farms immediately in a way the changes would require. However, it should be noted that a situation in which the concavity assumption is not satisfied is not uncommon in empirical study (see e.g., Glass and McKillop 1990, p. 276, Chambers and Pope 1994, p. 110). According to Wales (1977, p. 191), slight deviations from the assump- tions of concavity and monotonicity do not nec- essarily weaken the assumption of cost minimi- zation or the reliability of the estimated parame- ters. 8.2.2.3 Empirical modelsfor the post energy crisis period At the beginning of the research period there was a powerful economic shock, the energy crisis, which greatly affected the preconditions for ac- tion of the national economy and private enter- prises, such as dairy farms. In addition, there was extensive annual variation in the data for the first part of the study on dairy farms of group B. Thus, it is also necessary to examine the economic ac- tivity of dairy farms during a less dramatic peri- od in order to provide more uniform data. Models are estimated for the period after the energy crisis, 1976-1991. The time span is so short that the cost function cannot be estimated due to the lack of the degrees of freedom, so that the models are estimatedfrom the cost share equa- tions. Flexible functional forms and the dual ap- proach make it possible to estimate the elastici- ties of substitution without the cost function. It is not possible to solve the cost flexibility and the technical change from the cost share equations. The calculated % 2 value of the symmetry as- sumption on farms of group A was 12.85, and in group B 4.97, which are smaller than 16.81, the critical value at the 1% significance level. There- fore, the H 0 hypothesis on the symmetry of the Hessian matrix is not rejected. The cost share equation system (46) is estimated holding the parameter restrictions (45) in force. The values of the parameters estimated from the cost share equations, and their asymptotic p values in pa- renthesis, are presented in Tables 8a and Bb. The estimated models of cost share equation systems fit the data well, when the measurement is made using the R 2 and F values. The corrected 571 Agricultural Science in Finland 3 (1994) Table Ba. The estimated parameters of the cost share equation system on dairy farms of group A for 1976-1991 (p values in parenthesis). P KK 0.1060 P KL -0.0256 PKM -0.1029 pKH 0.0225 (0.000) (0.014) (0.000) (0.041) P LL 0.0584 P LM 0.0158 PLH -0.0486 P MM 0.0609 (0.015) (0.424) (0.011) (0.073) P MH 0.0262 P HH -0.0001 PK 0.1650 PL 0.3559 (0.225) (0.998) (0.000) (0.000) P M 0.3128 Ph 0.1663 PYK 0.0387 PYL -0.2394 (0.000) (0.014) (0.314) (0.002) P YM 0.1387 P YH 0.0620 PTK 0.0031 PXL 0,0044 (0.096) (0.371) (0.046) (0.072) pTM -0.0044 P TH -0.0031 (0.134) (0.245) yK+ 0.0144 yK_ -0.0095 yL+ 0.0148 yL_ -0.0174 (0.001) (0,004) (0.002) (0.000) yM+ -0.0251 yM _ 0.0237 yH+ -0.0041 yH 0.0032 (0.002) (0.001) (0.403) (0.442) Table Bb. The estimated parameters of the cost share equation system on dairy farms of group B for 1976-1991 (p values in parenthesis). PKK 0.0991 (3 kl -0.0366 P KM -0.0658 P KH 0.0033 (0.000) (0.004) (0.001) (0.653) Pll 0,0892 PIM -0.0243 P,, H -0.0283 P MM 0.1021 (0.001) (0.226) (0.037) (0,005) PMH -0.0120 PH„ 0.0370 P K 0.0991 PL 0.4278 (0.364) (0.019) (0.004) (0.000) P M 0.3854 P„ 0.0877 PYK -0.0168 P YL -0.1697 (0.000) (0.014) (0.621) (0.003) P YM 0.2062 PYH -0.0197 PXK 0.0058 P TL 0.0010 (0.005) (0.540) (0.001) (0.547) p TM -0.0074 PTH 0.0006 (0.006) (0.671) yKt 0.0116 Yk- -0.0118 yL+ 0.0196 yL. -0.0132 (0.005) (0.003) (0.001) (0.004) yM+ -0.0176 yM 0.0279 y„+ -0.0136 yH -0.0029 (0.007) (0.000) (0.001) (0.272) R 2 of the cost share equations varies between 0.862 and 0.976 (F value varies between 14.4 and 89.7). In the cost share equations the stand- ard error of the regression in relation to the aver- age of the dependent variable varied between 1.3% and 4.4%. The goodness-of-fit measures, F val- ues, and SER of the cost share equations are pre- sented in Tables 9a and 9b. The number of statistically significant param- eters on dairy farms of group A is 18 (12), and in group B 22 (19) out of a possible total of 30 parameters at the 5 (1) % significance level. Of the parameters related to the elasticities of sub- stitution and own price elasticities, on A farms six ((3 kk (1%), P KL (5%), PKM (1%), pKH (5%), Pll (5%), and PLH (5%)); and on B farms seven (P KK 572 Agricultural Science in Finland 3 (1994) Table 9a. Goodness-of-fit, F values, and significance, as well as the standard error of the regression (SER) of the cost share equations (S|) on dairy farms of group A. Table 9b. Goodness-of-fit, F values, and significance, as well as the standard error of the regression (SER) of the cost share equations (Sj) on dairy farms of group B. Function R 2 F value Signific- SER Function R 2 F value Signific- SER ance ance S K 0.987 0.976 89.7 0,000 0,023 SK 0,986 0.973 78.8 0.000 0.029 SL 0.954 0.914 23.7 0.000 0.013 SL 0.954 0.913 23.6 0,000 0.014 S M 0.927 0.862 14.4 0.000 0.044 SM 0.930 0.869 15.2 0.000 0.040 (1%), (3 KL (1%), (3 KM (1%), PLL (1%), PLH (5%), P HH (1%), and PMM (5%)) out of ten deviated sta- tistically from zero. According to the results, the weather conditions affect the demand for inputs in a significant way, so that at the same input prices the cost shares vary due to good and poor years. The results obtained from the post energy crisis period is consistent with the results of the whole research period. Table 10presents the critical values of the DW- test in the case of 16 observations and five inde- pendent variables (Pindyck and Rubinfeld 1991, p. 568), as well as the estimated values when the H 0 hypothesis, the error term of the explained equations includes a first order positive autocor- relation, is tested. The area between d, and du values is wide, due to the small number of obser- vations. The number of observations cannot be increased, because the data are available only as annual data. The DW values of the estimated func- tions remain between the critical values in two out of three functions, which means that the H 0 hypothesis cannot be accepted or rejected. When the cost share equation to be left out was varied, no changes were observed in the values of the parameters. As a summary of the general exami- nation, it can be noted that the models are at least satisfactory. The cost share equation system estimated em- pirically is well-behaved, if it is consistent with the theoretical assumptions. Homogeneity of de- gree zero with respect to input prices was en- sured by means of parameter restrictions. In ad- dition, it was ensured that the cost shares sum up to one, and the Hessian matrix is a symmetric matrix according to Young’s Theorem. Table 10. Critical values of the DW-test and values esti- mated from the cost share functions. Critical values Est. DW value Function d, d„ A farms B farms S K 0.62 2.15 2.62 2.06 S L 0.62 2.15 1.78 2.95 S M 0.62 2.15 1.90 2.07 According to the production and cost theory, the cost shares should be non-negative. Parame- ter restrictions that guarantee the non-negativity of the cost shares cannot be set in advance in the cost share equations solved from the translog cost function without losing the flexibility of the cost function. The matrix of the cost shares derived from the translog cost function + SS’-diag[S]) should be negative semidefinite. The negative semidefiniteness of the matrix makes sure that the cost function is concave, as a function of input prices when the cost shares are non-nega- tive. The cost shares estimated from dairy farms of groups A and B are positive at all observation points. The sign of the own price elasticity of the inputs included in the model is in accordance with the theory, so that the necessary condition for the concavity of the cost function with re- spect to input prices is satisfied at all observation points. The Hessian matrix of the parameters es- timated on dairy farms of group A is negative semidefinite, except in 1976 and 1980. On dairy farms of group B the Hessian matrix is negative semidefinite in all years included in the research, 573 Agricultural Science in Finland 3 (1994) so that the concavity of the cost function with respect to input prices is realized at all observa- tion points. On the basis of the result, milk pro- ducers seem to behave according to the cost min- imization assumption. The deviation of the mod- el of the dairy farms of group A from the as- sumption in 1976 and 1980 may partly be caused by the rapid and powerful economic shocks in the 19705, as well as the inflation shock of 1980. 8.2.3 Elasticities of substitution In the theoretical part of the study it was present- ed analytically that there may be differences be- tween the values (measurements) of substitution defined in different ways. Because of these pos- sible differences, the elasticities of substitutions should be solved from the data by means of sev- eral measurements for the elasticity of substitu- tion. In this study the derived demand elastici- ties, Allen and Morishima partial elasticities of substitution, and shadow elasticities of substitu- tion are solved from the data. In the study the elasticities of substitution are solved from a sys- tem of four inputs: capital items K, labour L, purchased feed M, and purchased fertilizers H. 8.2.3.1 Allen partial elasticities ofsubstitution and derived demand elasticities The determination ofAllen partial elasticities of substitution and derived demand elasticities is based on both the estimated parameters present- ed in Tables sa, sb, Ba, and 8b and the estimated values of the cost shares solvedby means of Shep- hard’s Lemma. The economic information includ- ed in parameters and (3,j in the tables cannot be seen directly from their values, so that elasticity formulas for the translog cost function have been derived for the calculation of the elasticities of substitution. Allen partial elasticities of substitu- tion can be calculated from the translog cost func- tion by the following formulas: 8.. c- = p±+l IJ (55) " S 2 S . (56) These elasticity formulas have been derived by Binswanger (1974). According to formula (24), the Allen partial elasticity of substitution is equal to the derived demand elasticity divided by the cost share. The derived demand elasticities can be determined from formula (24), and these are presented in Tables 1la-1 Id. The first col- umn in the tables gives the value of eKj , the sec- ond the value of eLj , and so forth. The elasticities have been solved from the values of the parame- ters presented in the elasticity formulas and the averages of the cost shares. The results of the tables can be interpreted, ceteris paribus, as the average demand behaviour of the research period when the price in question changes. Tables Ha- lid present the derived demand elasticities, be- cause these are more informative than Allen par- tial elasticities of substitution. The elasticities of substitution are positive (neg- ative), if inputs are substitutes (complements) of each other. These relationships are easier to ex- amine from Allen partial elasticities of substitu- tion than from derived demand elasticities, be- cause Allen partial elasticities of substitution are symmetric. All inputs seem to be Allen substi- tutes with each other, except the capital items and purchased feed. On dairy farms of group A, in the data for the post energy crisis period, the Allen partial elasticity of substitution of labour and purchased fertilizers was also negative. In all examinations the relationship of complementari- ty was found between only one pair of inputs, so that when relative prices change in milk produc- tion. it is possible to substitute cheaper inputs for an input that becomes more expensive, at least to some extent. According to the cross price elasticity ey, the inputs are for the most part substitutes with each other, as should be the case according to Allen partial elasticities of substitution. It can be seen in Tables 1 la-1Id that the cross price elasticities are not symmetric. The asymmetry can be inter- preted so that, for example, the demand for la- 574 Agricultural Science in Finland 3 (1994) Table 11a. Derived demand elasticities on dairy farms of group A. K L HM K -0.1500.114 -0.1890.107 (0.040) (0.020) (0.054) (0.081) L 0.281 -0.4560.670 0.424 (0.040) (0.048) (0.105) (0.165) M -0.1810.261 -0.7090.467 (0.056) (0.045) (0.157) (0.179) H 0.0500.081 0.228 -0.998 (0.037) (0.031) (0.034) (0.185) Standard errors in parentheses: s.e.(eij) = s.e.((3 i j)/S j Table I lb. Derived demand elasticities on dairy farms of group B. HMK L K -0.1100.103 -0.1440.013 (0.053) (0.018) (0.065) (0.064) L 0.264 -0.3130.309 0.403 (0.051) (0.045) (0.115) (0.118) M -0.1600.134 -0.2720.245 (0.028) (0.050) (0.153) (0.140) H 0.0060.076 0.107 -0.661 (0.012) (0.022) (0.061) (0.117) Table lie. Derived demand elasticities for post energy crisis period on dairy farms of group A. K HL M K -0.2810.155 -0.2830.444 (0.035) (0,017) (0.048) (0.099) L 0.364 -0.3930.563 -0.024 (0.039) (0.040) (0.082) (0.157) M -0.2860.242 -0.5000.486 (0.053) (0.040) (0.130) (0.212) H 0.203 -0.0040.220 -0.906 (0.044) (0.031) (0.087) (0.317) Table lid. Derived demand elasticities for post energy crisis period on dairy farms of group B. HMLK K -0.3030.122 -0.0810.234 (0.044) (0.019) (0.057) (0.075) L 0.288 -0.3390.368 0.171 (0.046) (0.037) (0.079) (0.121) M -0.0950.183 -0.3300.106 (0.067) (0.039) (0.114) (0.133) H 0.1100.034 0.043 -0.512 (0.035) (0.024) (0.053) (0.134) hour increases when the price of capital items rises less than the demand for capital items when the price of labourrises. According to the results, the use of capital items is more sensitive to a change in the price of labour than the use of labour to a change in the price of capital items, which means that the substitution of labour for capital items is very inelastic. The cross price elasticity between labour and capital items is inelastic in the other direction, too, and the elasticity varies between 0.264 and 0.364. Similarly, the cross price elasticities be- tween capital items and purchased fertilizers and, labour and purchased fertilizers are, on the aver- age, very inelastic, so that the substitution of these inputs with each other is very difficult. On the average, the cross price elasticities are more elastic between labourand purchased feed, and between purchased fertilizers and purchased feed. The elas- ticities vary between 0.134 and 0.670, and be- tween 0.043 and 0.486, respectively, which shows that there is inelasticity between these inputs, too, and thus their substitutability with each other is weak. The signs of the own price elasticities (e H ) of all inputs in Tables 1 la-1 Id are in accordance with the theory. The demand for capital items was the most inelastic; the elasticity varied be- tween -0.110 and -0.303. The demand for labour was almost as inelastic, so that a change of 1% in the own price of these inputs caused a change of only less than 0.5% in the use of these inputs. The demand for purchased fertilizers is the most elastic; the elasticity varies between -0.512 and -0.998. According to the results, a price change of 1% leads to, on the average, a change of clear- ly under 1% in the demand for the input in ques- tion, when all other factors are held constant. 575 Agricultural Science in Finland 3 (1994) Table 12a. Morishima partial elasticities of substitution on dairy farms of group A. K L M H K 0.265 -0.039 0.257 L 0.737 1.126 0.881 M 0.528 0.970 1.176 H 1.048 1.079 1.226 Table 12b. Morishima partial elasticities of substitution on dairy farms of group B. K L M H K 0.213 -0.034 0.123 L 0.577 0.623 0.716 M 0.112 0.406 0.518 H 0.667 0.737 0.768 Table 12c. Morishima partial elasticities of substitution for the post energy crisis period on dairy farms of group A. K L M H K 0.436 -0.001 0.725 L 0.757 0.955 0.369 M 0.214 0.742 0.986 H 1.110 0.901 1.126 Table 12d. Morishima partial elasticities of substitution for the post energy crisis period on dairy farms of group B. K L M H K 0.426 0.223 0.538 L 0.628 0.707 0.510 M 0.235 0.513 0.436 H 0.622 0.546 0.555 8.2.3.2 Morishima partial elasticities of substitution The Allen partial elasticity of substitution and the derived demand elasticity measure solely the adjustment of a single input to a change in the price of a single input. Instead, the Morishima partial elasticity of substitution measures how the i,j input ratio responds to a change in Wj (see formula 25). Morishima partial elasticities of sub- stitution are presented in Tables 12a-12d. The first column gives the value of a{g, the second the value of and so forth. It can be seen in the tables that, like the derived demand elasticity, the Morishima partial elasticity of substitution is also asymmetric. According to the Morishimapar- tial elasticity of substitution, the inputs seem to be substitutes with each other, except capital items and purchased feed on dairy farms of groups A and B when the price of capital items changes. Due to the asymmetry, the interpretation of the Morishima partial elasticity of substitution dif- fers from the interpretation of the Allen partial elasticity of substitution. According to the Allen partial elasticity of substitution, capital items and purchased feed are complements with each other. so that an increase in the price of purchased feed reduces the use of capital items. According to the Morishimapartial elasticity of substitution when the price of purchased feed rises, capital items and purchased feed are substitutes with each other, which can be explained through the in- crease in the ratio of capital items and purchased feed. The increase in the ratio of capital items and purchased feed can be explained by the fact that, in percentages, the use of purchased feed decreases more than the use of capital items. On the whole, the opposite signs ofAllen and Morishima partial elasticities of substitution be- tween capital items and purchased feed, when the price of purchased feed rises, is interpreted so that, when the price of purchased feed rises, the use of capital items is reduced, but, at the same time, the use of purchased feed is also re- duced according to the production and cost theo- ry (concavity). As can be seen in Tables 1 la-1 Id, the use of capital items and purchased feed is reduced so that the K/M ratio increases. Corre- spondingly, according to gKJk, when the price of capital items increases, the K/M ratio increases so that the rise in the price of capital items caus- es a smaller reduction in the use of capital items 576 Agricultural Science in Finland 3 (1994) Table 13a. Shadow elasticities of substitution on dairy farms of group A in the lower triangle, and on dairy farms of group B in the upper triangle. K L M H K - 0.316 -0,035 0.490 L 0.401 - 0.4720.734 M 0.2511.013 - 0.692 H 0.7961.047 1.210 - Table 1 3b. Shadow elasticities of substitution for the post energy crisis period on dairy farms of group A in the lower triangle, and on dairy farms of group B in the upper triangle. K L M H K 0.486 0.228 0.595 L 0.532 0.577 0.540 M 0.105 0.807 0.521 H 0.989 0.815 1.082 than in the use purchased feed. This means that, when the price of capital items rises, capital items and purchased feed are complements according to the Morishima partial elasticity of substitution. According to Tables 12a, 12b, and 12c, on dairy farms of groups A and B Ok M and have opposite signs. Thus, on these farms capital items and purchased feed are substitutes (complements) according to the Morishima partial elasticity of substitution, when the price of purchased feed (capital) increases. According to the result, the demand for purchased feed is more sensitive to changes in the price than the demand for capital items, in the case of changes with respect to the prices of both inputs wK or wL . According to the Morishima partial elasticity of substitution, the elasticities of substitution be- tween inputs are more elastic than the derived demand elasticity. Similarly, there is less com- plementarity between inputs with the Morishima partial elasticity of substitution than with the de- rived demand elasticity. Morishima partial elasticities of substitution related to capital items show complementarity or inelastic substitution between capital items and other inputs everywhere in the data, except on dairy farms of group A between capital items and purchased fertilizers, when the price of purchased fertilizers changes. The elastic substitution be- tween capital items and purhased fertilizers in group A, when the price of purchased fertilizers changes, is quite surprising. In all other cases, the results show complementarity or inelasticity between capital items and other inputs. Note, that in group B they are in all cases inelastic. On dairy farms of group B Morishima partial elasticities of substitution show inelastic substi- tution between other inputs (L, M and H); the elasticity varies between 0.406 and 0.768. In group A Morishima partial elasticities of substitution between these inputs are, on the average, more elastic than in group B; the elasticity varies be- tween 0.369 and 1.226. 8.2.3.3 Shadow elasticities of substitution The shadow elasticity of substitution measures the change in the ratio of the use of two inputs in percentages, when the relative prices of these in- puts change in percentages. The shadow elastici- ties of substitution are presented in Tables 13a and 13b. According to the results presented in the tables, all inputs seem to be substitutes with each other according to the shadow elasticity of substitution, except that a complementary rela- tionship seems to exist between capital items and purchased feed on dairy farms of group B for the model of the whole research period. Determining the shadow elasticity of substitu- tion differs from determining the other elastici- ties of substitution in that the shadow elasticity of substitution is determined from the isocost curve, in which case costs are held constant. In addition, it should be noted that in all other meas- urements of the elasticity of substitution used in this study the change in the price of only one input is taken into account. Theoretically, the shad- ow elasticity of substitution can be considered an elasticity of substitution of a longer run than the derived demand elasticity. This conclusion is based 577 Agricultural Science in Finland 3 (1994) on the fact that the inputs excluded from the ex- amination of shadow elasticities of substitution can be purchased freely at fixed prices. According to the shadow elasticity of substitu- tion, on dairy farms of group B the elasticities of substitution between inputs are inelastic; the elas- ticity varies between, -0.035 and 0.734. On dairy farms of group A the shadow elasticities of sub- stitution are more elastic than in group B. The shadow elasticities of substitution between capi- tal items and other inputs are, on the average inelastic on dairy farms of groupA, too; the elas- ticity varies between 0.105 and 0.989. The shad- ow elasticities of substitution between other in- puts (L, M and H) on dairy farms of group A are relatively elastic; the elasticity varies between 0.807 and 1.210. When shadow elasticities of substitution are compared to the derived demand elasticities it is observed that in the so-called long run the use of inputs is more elastic than in the short run, which is in accordance with the law ofLe Chatelier. 8.2.3.4 Summary of the elasticities of substitution The elasticities of substitution presented above have been solved from the values of the parame- ters presented in the elasticity formulas, and the averages of the cost shares. The values of the elasticities of substitution at five year intervals are presented in Appendix 9. The year 1976 was chosen, instead of 1975, because it is the first year of the empirical model for the post energy crisis period. Thus it is possible to compare the elasticities of substitution estimated from the whole research period and the post energy crisis period with each other. The different elasticities of substitution calcu- lated above give similar information on the sub- stitution between inputs. The elasticities of sub- stitution calculated empirically supported the the- oretical conception that in milk production in- puts are at least to some extent substitutes with each other. Complementarity was observed mainly between capital items and purchased feed. In the years of high inflation 1973-1975, and in 1980, there was complementarity in the elasticities be- tween capital items and other inputs. This is like- ly to be caused by the fact that on dairy farms production cannot be adjusted according to rapid changes in the relative prices. An average development trend during the re- search period was that the inelasticity of produc- tion decreased slightly up to the mid-19705, and after that the elasticity increased so that at the beginning and end of the research period the elas- ticities of substitution and own price elasticities of inputs were almost equal (see Appendix 9). The elasticities of substitution estimated from the post energy crisis data give similar information, i.e., on the average, the elasticity of production increased slightly in the post energy crisis peri- od. It should be noted, however, that the changes in the elasticity have been very small, and for the most part the production has been relatively ine- lastic. When the elasticities of substitution and own price elasticities on dairy farms of groups A and B are compared with each other from Tables 1 la- nd, 12a-12d, and 13a-13b, it can be seen that on dairy farms of group B the elasticities are more inelastic in 56 cases, and on farms of group A in 12 cases. The comparison included the de- rived demand elasticity, the Morishima partial elasticity of substitution, and the shadow elastic- ity of substitution for both the whole research period and the post energy crisis period (the total of 68). Dairy farms of group B are more strongly specialized in milk production than of group A, which probably explains the greater inelasticity of the production in group B. In general, the elas- ticity of production seems to decrease as a result of specialization. The relatively small values of the elasticity of substitution solved in the study mean that with the existing production technology the substitu- tion of inputs for other inputs has been quite ine- lastic. This is illustrated in Figure 6. Figure 6 describes an efficiently used combi- nation of inputs by which an output determined in advance is produced. The isoquant presented in the figure is strongly curved. With this kind of production technology, for example, an increase 578 Agricultural Science in Finland 3 (1994) in the price of labour in relation to the price of capital items causes only a small change in the relative use of labour and capital items. This is illustrated in Figure 6 by the shift from point A to point B. According to the elasticities solved in the study, substitution between inputs with the existing pro- duction technology seems inelastic. Inelastic elas- ticities of substitution and own price elasticities mean that, with the production technology in ques- tion, changes in the prices of inputs have little effect on their demand. Thus, the effect of input taxes (e.g., environmental taxes) on the demand for inputs is small, so that, by their possible in- troduction, costs are mainly increased, and the farm income of the milk producer is reduced. The results of this study do not support the traditionally used assumption on the applicabili- ty of Cobb-Douglas and CES functional forms in the modelling of the production technology of milk production. The elasticities of substitution seem to deviate from one, so that the Cobb-Doug- las functional form assumption cannot be accept- ed. As the elasticities of substitution seem to vary between different inputs, CES functional form assumption cannot be accepted, either. On the basis of this, Cobb-Douglas and CES functional forms are not appropriate for modelling milk pro- duction in Finland. 8.2.4 Technical change and cost flexibility Cost flexibility and technical change seem to be interrelated, so that they are dealt with together. A change in the scale of production is usually understood as global. Traditionally the economies of scale have been interpreted from the advan- tage or disadvantage related to the size of the enterprise. In this study the economies of scale (size) are local, and they are presented according to the cost flexibility concept. The technical change measures to what extent the changes in the total productivity have been caused by the increased efficiency of the produc- tion. The total productivity is measured as the ratio between the output quantity and the amount of inputs used. Thus, technical change measures the part of growth in production that does not result from an increase in the use of inputs. Tech- nical change usually occurs on dairy farms when new production technology is introduced, and also through an increase in crop yields and the milk output level, as a result of progress in livestock and plant breeding. Technical change can be de- scribed by the shift of the isoquant in the input space. Temporally, technical change is usually seen in two different ways. It can be short-term devel- opment, in which case the effect of measures on the current production is taken into account. Al- ternatively, structural technical change also takes into account the long-term effects, which are re- alized only through reorganization of production. An example of a structural technical change in milk production is the introduction of milking machines that replace milking by hand, which is labour intensive. This investment increases the production capacity of a dairy farm, which is re- alized through future outputs. The estimated parameters indicate that techni- cal change in milk production is not Hicks neu- tral. The H 0 hypothesis on neutral technical change was rejected by the likelihood ratio test. The esti- mated parameters describing technical change PTi have been presented in Tables 5a and sb, as well as 8a and Bb. Figure 7 illustrates the technical change in milk production in Finland. Fig. 6. Isoquant of the labour and capital inputs. 579 Agricultural Science in Finland 3 (1994) Table 14. Estimated PTi and (IY| parameters from NH-NN, NH-N, and H-NN dairy farm models. Model Pyk Pyl Pym Pyh A:NH-NN -0,1019 -0.1445 0.1742 0.0721 B:NH-NN -0.0810 -0.1265 0.1802 0.0273 A: NH-N 0.0019 -0.1035 0.0722 0.0294 B: NH-N 0.0113 -0.1246 0.1038 0.0095 Ptk Ptl Ptm Pth A: NH-NN 0.0056 0.0034 -0.0056 -0.0034 B: NH-NN 0.0059 0.0002 -0.0051 -0.0010 A: H-NN 0.0010 -0,0023 0.0007 0.0006 B: H-NN 0.0021 -0,0030 0.0003 0.0006 In Figure 7, along with the technical change, the isoquant Y 0 shifts in the input space to posi- tion Y|. When the relative prices of inputs and output remain constant, the ratio of the use of inputs moves as a result of the technical change from OA to 08, which causes the share of the use of one input to increase, and the share of the other input to decrease. Rejecting the Hicks neutral technical change means that, as a result of technical change the isoquant moves along the expansion path of the enterprise towards the origin so that the cost shares of inputs change simultaneously. In all estimated models technical change in milk production has been capital items-using. The parameters describ- ing technical change estimated from different models give a different picture of labour-, pur- chased feed-, and purchased fertilizer-saving or -using technical change. It is obvious that the problem in estimating the cost function (38) is with the small number of degrees of freedom, and thus the possible multi- collinearity. This means that the identification and indication of the effects of parameters (3XY, PYY, Pxx , (3Xi , and (3Yi separately is difficult, even im- possible. PYi parameters describe the effects of the farm size on the relative cost shares of in- puts. If the biases related to the farm size are positive (negative), the cost share of the corre- sponding input increases (decreases) as the out- put level changes. The capital items-using technical change turned out to be as assumed. The technical change de- scribing the use of labour, purchased feed, and purchased fertilizers varies according to the spec- ification of the model, so that they are examined separately. It is likely that technical change and increasing the farm size are interrelated, so that it is impossible to separate their effects from each other. The purchased feed- and purchased fertilizer- saving technical change estimated from the NH- NN models seems somewhat surprising. Self-suf- ficiency in feed has been an objective in Finnish milk production. The production technology of milk production has been developed so that it is suitable for the production of own feed on dairy farms, and thus the purchased feed-saving tech- nical change appears logical. The purchased fer- tilizer-saving technical change may be caused by the increased accuracy in fertilization resulting from technical change, and the more efficient use of manure. To study the linkages between technical change and the farm size, parameters pxi and P Yi estimat- ed from models NH-N and H-NN are also exam- ined. The parameters are presented in Table 14. When the model is specified as homothetic, tech- nical change can be examined separately from the output quantity. However, in this case a mod- el that was rejected by the likelihood ratio test is examined. Technical change according to the H- Fig. 7. Non-neutral Hicks technical change. 580 Agricultural Science in Finland 3 (1994) NN model is capital items-, purchased feed-, and purchased fertilizer-using and labour-saving. Ac- cording to the NH-N model, increasing the out- put quantities greatly increases the cost share of purchased feed, whereas the cost share of capital items and fertilizers increases only slightly, and a strong reduction occurs in the cost share of labour. When parameters describing technical change estimated from NH-NN and H-NN models are compared, it is observed that the signs of the parameters related to labour, purchased feed, and purchased fertilizers change places with each other. The same phenomenon can be observed in the case of capital, when parameter P YK estimat- ed from NH-NN and NH-N models is examined. This phenomenon has also been observed by Lopez (1980, p. 43) in his study on the produc- tion structure in Canadian agriculture and the de- rived demand for inputs. According to PYj param- eters in NH-NN models, the cost shares of capi- tal items and labour decrease, and those of pur- chased feed and purchased fertilizers increase when the farm size grows. On the basis of this examination, technical change and increasing the farm size are interre- lated, so that parameters PTi and PYi should be interpreted together. However, the joint interpre- tation of parameters (3Ti and PYj does not mean that the original model could be simplified, be- cause the simplified models were rejected by the likelihood ratio test. According to the estimated results, parameter PTK is positive in all models, which means that technical change in milk production has been cap- ital items-using. The signs ofparameters (3TL, pTM, and PTH change in the restricted models (NH-N and H-NN). It seems that in milk production tech- nical change is linked to the farm size, which is a logical result. According to Lund and Hill (1979, p. 146-148), changes in the efficiency may have a similar effect on costs as does the increase in the farm size. Technical change, such as the introduction of milking machines, mechanical manure disposal, new machine chains for feed crops, etc., has made it possible to increase the size of dairy farms. According to the estimated parameters, it seems that, as a result of the increase in the farm size and technical change, other inputs have been sub- stituted for the labour input. The interpretation of technical change in an empirical study is also difficult, because it has not been possible to develop an unequivocal meas- urement for it. In examining technical change, strongly simplified variables must be used, for example, the linear time trend. In addition, in empirical studies there are problems in separat- ing the effects of technical change and the farm size from each other, because part of the techni- cal change is realized through investments and inputs, and part through working methods and improved know-how. In this study it was logical to link the exami- nation of technical change to the cost function study in order to be able to utilize calculation in terms of derivatives. It is also possible to exam- ine the technical change (productivity) by utiliz- ing the index theory directly. According to Cham- bers (1988, p. 203-249), neither of the possible approaches is perfect, and it is not clear which is preferable. According to the research results, in the re- search period the average cost flexibility on dairy farms of group A was 0.06, and on farms of group B 0.20. On dairy farms of group A half of the observations of cost flexibility were negative, which is not acceptable theoretically. When the cost flexibility was solved from the HG-NN mod- el, it was 0.33 on farms of group A, and 0.39 on farms of group B. On the basis of the results, an increase of 1% in output (ceteris paribus) increases costs by un- der 0.4%, so that increasing the size of dairy farms should be allowed. The estimated, relative- ly high economies of scale (size) are likely to be partly caused by structural technical change, which causes calculatory economies of scale (size) in econometric time series analyses (cf. Lehto 1991, p. 15). The economies of scale (size) may also indi- cate inefficient use of the production capacity. Since milk production is restricted very strongly, there is unused production capacity on dairy farms, which cannot be utilized efficiently in the pro- 581 Agricultural Science in Finland 3 (1994) duction. In addition, the estimated economies of scale (size) also indicate that it is necessary to maintain production capacity for short peak sea- sons in the production to be able to meet the quality requirements of the products during these periods (timeliness cost). To summarize the economies of scale (size), it can be noted that the size of dairy farms should increase so that the existing advantages could be utilized. In particular, the underutilization of the production capacity resulting from the increase in the yield level of field crops and the average milk output should be replaced rapidly by, at least, a full degree of use of the existing capacity. The possible multicollinearity hidden in the cost function (38), as well as the fact that the farm size and technical change are interrelated have an effect on the estimated values of the parameters related to the economies of scale (size) and technical change. Thus, the parameters relat- ed to the technical change and economies of scale (size) and their interpretations should be taken with caution. In addition, parameters related to economies of scale (size) estimated from time series data are less reliable than parameters esti- mated from cross section or panel data sets. 8.3 Dairy farm level approach The data set collected from the A dairy farms makes it possible to utilize panel data and its properties. The panel data set possesses many advantages over time series or cross sectional data sets when enlarging the number of data points, and making possible more detailed analysis of economic problems (Hsiao 1986, p. 1-2). In this study the panel data set is used since the neoclassical production and cost theory is based on the farm level argument. So having panel data makes the application of the theory more reliable. On the other hand, the panel data set is used to confirm and specify the results obtained from the analysis of the representative dairy farm data set because the time series used was non- stationary but not cointegrated and, therefore, ECM cannot be applied. Thus, in time series anal- yses there was the possibility of spurious regres- sion. In this study the panel data set gives new in- formation about the variation of output, costs, and cost shares across farms. However, farm lev- el prices have not been recorded. Thus, the sam- ple over the research period contains only 27 ob- servations of the prices. In other words, prices differ over the years but not across the farms. This restricts the potential advantages provided by the analysis of panel data. On the other hand, the panel data set is more effective in studying the size economies and technical change than is the representative dairy farm data set. 8.3.1 Estimation procedure for panel data set In the equations, the intercept may vary over the dairy farms. The model was assumed to be a fixed effect model and the intercepts were treated as fixed parameters. When the indicator (dummy) variables are used, the inference is conditional on the individuals in the sample. The indicator model is appropriate because the sample of dairy farms cannot be regarded as a random sample from some larger population (see Judge et al. 1988, p. 489). According to Thijssen (1992, p. 220), the intercepts reflect quality differences in the inputs between farms and consist mainly of managerial differences and differences in the quality of land. The fixed effect model can be written in a general form as follows (Judge et al. 1988, p. 469): R K y it j=l k =2 (57) i = 1.2,...,R, t = 1,2,...,T where y it is the dependent variable, xkit are the explanatory variables, e it are assumed to inde- pendent and identically distributed random varia- bles with mean zero and variance Gk, R is a number of farms, T is a number of time-series observations, p is the K vector of parameters, is the fixed effect of farm i, and Djt are indica- tor variables and assume values oor 1; Djt = lif j= i, and D |t = oif j j. 2582 Agricultural Science in Finland 3 (1994) Equation (57) cannot be estimated, because the model becomes too large for the computer pack- ages available. According to Judge et al. (1988, p. 471-472), the model can be solved by trans- forming the observations on each dairy farm so that they are in terms of the deviation from the mean for that dairy farm. This can be expressed as follows: K yi-y,=£Pk(xki,-xk,) + e l,-§ 1k -2 (58) TTT where y( = l/TSyit, x( = 1/Tlxki„ Ej = l/TZeit . t=lt=I t = 1 Both equations (57) and (58) give the same re- sidual vector, but the estimated standard errors of the coefficients need to be corrected by multiply- ing them by [(RxT-K)/(RxT-R-K)] M‘. In exam- ining the goodness-of-fit of the model, the R 2 from equation (58) also needs to be corrected. The F-test is applied to test the fixed effect model against the model of all dairy farms having the same intercept (see Judge et al., p. 415-416) (SSEr -SSEu)/(R- 1) F = SSE/(RxT-R-K) ’ (59) where SSE r is the sum of squared residuals ob- tained from the estimation of the equation in which all dairy farms have the same intercept. SSEU is the sum of squared residuals obtained from equa- tion (58). According to the F-test, the H 0 hypoth- esis that all the fixed effects are equal is rejected. For the model of the whole research period F-test values for SK, SL , S M, and C were 16.66, 61.27, 66.31, and 24.66, respectively. For the post ener- gy crisis model F-test values for S K , SL , and S M were 16.23, 67.15, and 58.85, respectively. The critical F-test value at the 1% significance level for (43,1135), (43,1122), and (43,653) degrees of freedom is about 1.7. The two models, for the whole research period and for the post energy crisis period, are estimat- ed using the A dairy farm data set, as in the case of representative dairy farm models above. Be- cause the cost share equations form a system of seemingly unrelated equations, SUR is an appro- priate estimation technique. 8.3.2 Empirical models and hypothesis tests The calculated %2 value of the symmetry assump- tion for the whole research period model is 12.90, and for the post energy crisis period model 14.53. These values are smaller than 16.81, the critical value at the 1% significance level, therefore, the H 0 hypothesis on the symmetry of the Hessian matrix is not rejected. The system of equations are estimated according to equation (58), holding the parametric restrictions (45) in force. The val- ues of parameters estimated from the equation systems are presented in Appendix 10. The pa- rameter values related to the elasticities of sub- Fig. Ba. The parameters related to the elasticities of sub- stitution and own price elasticities for the whole research period. Fig. Bb. The parameters related to the elasticities of sub- stitution and own price elasticities for the post energy crisis period. 583 Agricultural Science in Finland 3 (1994) stitution and own price elasticities are compared with the parameter values solved from the repre- sentative dairy farm data set in Figures 8a and Bb. The parameter values estimated from the pan- el data set differ only slightly from the parameter values obtained from the representative dairy farm, except for (3HH , (3LL , and pLH in the whole research period model (arrows in Figures show the largest differences). The economic information included in parameters PH and cannot be seen directly from their values, so the possible differences in the elasticities of substitution will be examined later in Chapter 8.3.3. The estimated models fit the data satisfactori- ly, when the evaluation is made using the R 2 val- ues. The calculated R 2 values for the whole re- search period model for SK, SL , S M, and C are 55.60, 72.58, 73.24, and 98.62, respectively; and for the post energy crisis period model for S K, SL , and S M are 59.55, 82.25, and 80.49, respectively. The number of statistically significant param- eters in the whole research period model is 28 (24) out of a possible total of 41 parameters at the 5 (1) % significance level, and in the post energy crisis period model 16(13) out of a possi- ble total of 26 parameters. Of the parameters re- lated to the elasticities of substitution and own price elasticities, in the whole research period model seven (p KK (1%), (3KL (1%), (3 KM (1%), (3KH (1%), pLL (1%), pLH (1%), and pHH (1%)); and in the post energy crisis period model five (pKK (1%), Pkm (1%), Pll (I%X Plh d%), and pMH (1%)) out of ten deviated statistically from zero. The cost shares estimated from the whole re- search period model and from the post energy crisis period model are not positive at all sample points, 12 out of a total of 4752 cost shares in the whole research period and 17 out of a total of 2816 cost shares on the post energy crisis period are negative. Thus, monotonicity is not satisfied at these observations. The sign of the own price elasticity of the inputs included in the model is in accordance with the theory, evaluated at the sam- ple mean. But they do not have the correct sign at all sample points, 434 out of a total of 4752 own price elasticities in the whole research peri- od model and 164 out of 2816 own price elastici- ties in the post energy crisis period model are positive. On the basis of this result, some of the milk producers do not seem to behave according to the cost minimization assumption, but in the ma- jority of cases the own price elasticities are nega- tive as assumed. The deviationof the models from the assumptions may partly be caused by the rap- id and powerful economic shocks during the re- search period, as well as the risk and uncertainty in milk production. The weather conditions seem to affect the demand for inputs in a significant way, so that at the same input prices the cost shares vary depending on good and poor years. The results obtained from the panel data set is parallel with the results obtained from the repre- sentative dairy farm data set, except for the pa- rameters PHH and PLL that differ from the whole research period representative dairy farm param- eters. 8.3.3 Elasticities of substitution In this study the panel data set gives a more reli- able picture of the effects of the price variables than does the representative dairy farm data set, although the price variations across farms have not been recorded. Furthermore, using the panel data set increases the reliability of the parame- ters related to output changes. Thus, if there is multicollinearity between price series and output series in the representative dairy farm data set, analysis of panel data gives more reliable infor- mation about the effects of the price variables (Vartia 1994). The cross price and own price elasticities solved from the panel data set are quite similar to the elasticities solved from the representative dairy farm data set, except the own price elasticity for purchased fertilizers and the cross price elasticity for labour and purchased fertilizers when the price of labour changes, which are more inelastic. It seems that there is multicollinearity between pur- chased fertilizers, output and technological devel- opment in the representative dairy farm data set. Otherwise, the results obtained from the panel data set are interpreted the same as are the results 584 Agricultural Science in Finland 3 (1994) obtained from the representative dairy farm data set. The cross price and own price elasticities solved from the panel data set are compared with the elasticities solved from the representative dairy farm data set in Figures 9a-9c. The cross price and own price elasticities are presented in Ap- pendix 11. The Morishima partial elasticities of substitu- tion and shadow elasticities of substitution solved from the panel data set differ only slightly from the elasticities solved from the representative dairy farm data set, except the elasticities related to purchased fertilizers which are, on average, much more inelastic than those solved from the repre- sentative dairy farm data set, particularly in the whole research period model. One reason for this surprisingly large difference in the substitution possibilities related to purchased fertilizers in the whole research period model may be caused by the possible multicollinearity hidden in the time series data, as mentioned before. Otherwise, the results are interpreted the same as the results of the representative dairy farm, but the elasticities are, on average, more inelas- tic. The Morishima partial elasticities of substi- tution and shadow elasticities of substitution are compared with the elasticities solved from the representative dairy farm data set in Figures 10a- 10c. The Morishima partial elasticities of substi- tution and shadow elasticities of substitution are presented in Appendix 11. The different elasticities of substitution calcu- lated above from the panel data set give similar information on the substitution possibilities be- tween inputs. The elasticities of substitution solved empirically support the theoreticalconception that, in milk production, inputs are at least, to some extent, substitutes with each other. When the elas- ticities of substitution and own price elasticities solved from the panel data set and representative dairy farm data set are compared with each other, it can be seen that the elasticities solved from the panel data set are more inelastic in 46 cases, and from the representative dairy farm data set in 22 cases. The comparison included the derived de- mand elasticity, Morishima partial elasticity of substitution, and shadow elasticity of substitution. Fig. 9a. Own price elasticities in the whole research peri- od and in the post energy crisis period. Fig. 9b. Cross price elasticities in the whole research peri- od. Fig. 9c. Cross price elasticities in the post energy crisis period. 585 Agricultural Science in Finland 3 (1994) According to the elasticities solved from the panel data set, substitution between inputs with the existing production technology seems more inelastic than in the case of the representative dairy farm; in particular elasticities related to pur- chased fertilizers. On the other hand, the differ- ences in results between the panel data set and the representative dairy farm data set are mainly small, and there is no systematic twist. 8.3.4 Technical change and cost flexibility Cost flexibility and technical change are interre- lated in the models solved from the representa- tive dairy farm data set. The panel data set pos- sesses large amounts of new information con- cerning output variation across farms. New in- formation of cost and cost shares are obtained, too. Thus, the panel data set provides more relia- ble results concerning the cost flexibility, and more effectively separates the effects of the farm size economies and technical change from each other, as compared to the representative dairy farm data set (Vartia 1994). In this study, the panel data set does not give a more reliable picture of the effects of technical change than does the representative dairy farm data set, because there are no farm specific meas- ures for technical change. In other words, the linear time trend is the same for every farm in a certain year. If multicollinearity existed between the time trend and output series in the represent- ative dairy farm data set, it would be possible to get more reliable information about the effects of technical change from the panel data set, too. The estimated parameters indicate that techni- cal change in milk production is not Hicks neu- tral. The H 0 hypothesis on neutral technical change is rejected by the likelihood ratio test. The calcu- lated x 2 value of the neutral technical change as- sumption is 19.34, which is larger than 13.28, the critical value at the 1% significance level. The estimated parameters describing technical change pTI are presented in Appendix 10. Technical change in milk production seems to be purchased feed-saving and other inputs-using. Self-sufficiency in feed has been an objective in Fig. 10a. Morishima partial elasticities of substitution in the whole research period. Fig. 10b. Morishima partial elasticities of substitution in the post energy crisis period. Fig. 10c. Shadow elasticities of substitution in the whole research period and post energy crisis period. 586 Agricultural Science in Finland 3 (1994) Finnish milk production, and thus, the purchased feed-saving technical change appears logical. The average annual rate of technical change for the whole research period is 1.30, and it is increas- ing of the level of technology. On the basis of the result, the technological development has had a positive contribution to the dairy farms. The use of the panel data set makes it possible to obtain much more reliable information con- cerning the size economies than does the use of the representative dairy farm data set. The H 0 hypotheses of the homotheticity and homogenei- ty of the production technology, and constant re- turns to scale are rejected by the likelihood ratio test at the 1% significance level. The calculated X 2 value of the homotheticity is 301.30, which is very much larger than 15.09, the critical value at the 1% significance level. According to the like- lihood ratio test, the structure of the cost func- tion cannot be simplified from the general model (NH-NN). According to the results, in the research peri- od the average cost flexibility solved from the panel data set is 0.58, and it is increasing of the level of output. All of the observations are pos- itive, and thus in accordance with the theory. For a comparison, the cost flexibility solved from the representative dairy farm data set was 0.06, and half of the observations were negative, which is not acceptable theoretically. On the ba- sis of the results obtained from the panel data set, an increase of 1% in output increases costs about 0.6%, so that increasing the size of dairy farms should be allowed. The empirical result on milk producer behaviour under increasing returns to size is consistent with the theoretical assump- tion made earlier in the theoreticalpart of the study. Biases of scale, p yi parameters in Appendix 10, describe the effects of economies of size for the relative distribution of costs among inputs. When the biases related to the farm size is posi- tive (negative), the cost share of the correspond- ing input increases (decreases) with a change in the level of output. The cost shares of labour decrease, and those of other inputs increase when the farm size grows. Thus, the cost flexibility increases with the prices of purchased feed, capi- tal items, and purchased fertilizers, and decreas- es with the price of labour. For a comparison, the PYK solved from therepresentative dairy farm data differed from the result solved from the panel data set. It can be assumed that small dairy farms differ from other farms regarding the variable Iny. It is examined using rough indicator variables, such as 1 x Iny for the small farms and zero for the other farms. The estimated parameters which are connected to these variables do not deviate sta- tistically from zero by the t-test. On the basis of the result, the coefficient of Iny for the small farms does not differ from the coefficient for the other farms. Fig. 11. The parameters of technical change solved from the panel data set and representative dairy farm data set. Fig. 12. The parameters of size economies solved from the panel data set and representative dairy farm data set. 587 Agricultural Science in Finland 3 (1994) To complete the discussion of technical change and cost flexibility, (iYj and (3xi parameters solved from the panel data set and representative dairy farm data set are compared in Figures 11 and 12. From Figures 11 and 12 we can see that the parameters differ greatly from each other. The multicollinearity hidden in the cost function of the representative dairy farm (38) has an effect on the estimated values of the parameters (3 Yi , (3Xi , (3yy , (J tt, (J ty, (3 X , and pY . Thus, these parameters and their interpretations should be viewed with caution when the parameters are solved from the representative dairy farm data set. 9 Examination of the results and conclusions The study was initiated because of the need for new information on the input substitution and technological development in Finnish agri- culture. There has been very little research on the effects of the changes in relative prices and technical change on the production technology of dairy farms, so that the study was considered necessary. The dual approach of the neoclassical production and cost theory was chosen for the examination of the elasticities of substitution between inputs and the technical change on dairy farms. The flexible translog cost function and cost share equations derived from it were utilized to solve the empirical research problem. The cost function study was chosen, because it made it possible to examine the production of farms operating in the area of decreasing average costs. Data collected from dairy farms participating in the profitability research of Finnish agricul- ture, for the years 1965-1991, were used in the study. For this study, two sets of data were col- lected: dairy farms of group A (44), which were included during the whole research period, and dairy farms of group B (168-364), the number of which varied annually. Models were estimated separately for the post energy crisis period 1976- 1991. The data were well suited for the study, because the data consists of dairy farms practic- ing active entrepreneurship. The idea of a representative dairy farm was applied in both data sets. The dairy farms of group A form a panel data set. The applications of the theory are more reliable with the panel data set than with the representative dairy farm data set because the panel data set consists of informa- tion from individual dairy farms. Thus, the re- sults obtained from the panel data set are theoret- ically more reliable than the results obtained from the aggregate data set. During the research peri- od the development of ADP made it possible to solve the research problem with panel data. Thus, the data set from dairy farms of group A was also used as a panel data set. The potential of dairy farms to influence the price of milk and the inputs needed to produce it is very limited, which means that the adjustment to the changes in the prices of inputs is realized mainly by changing the production technology, if possible. The own price elasticities and the elas- ticities of substitution, as well as the technical change indicate the possibilities to adjust the pro- duction to the new price relations. Therefore, it is important for milk producers to be aware of the elasticities, but, in particular, it is important for the agricultural policy makers, because they need to know the effects of the decisions on the economy of the enterprises. In this study, the elasticities of substitution be- tween the capital items, labour, purchased feed, and purchased fertilizers were solved. The inputs were classified as substitutes or complements on the basis of the estimated elasticities of substitu- tion. According to the Allen partial elasticities of substitution, the inputs are, for the most part, sub- stitutes with each other. In all cases, the capital items and purchased feed were complements. In some parts of the data, complementarity between 588 Agricultural Science in Finland 3 (1994) the labour and purchased fertilizers, and capital items and purchased fertilizers was observed. According to the Morishima partial elasticity of substitution and shadow elasticities of substi- tution, the inputs are substitutes with each other, except that complementarity was observed be- tween the capital items and purchased feed. Ac- cording to the elasticities of substitution, the sub- stitution between inputs is mostly inelastic, i.e., it is quite difficult to substitute an input for an- other. The demand for the capital items was the most inelastic when measured by the own price elas- ticity. The demand for the labour was almost as inelastic. According to Marttila (1991, p. 72), the capital tied to agriculture is very fixed in Finland, which means that its substitution by other production factors is inelastic. When the elasticities of substitution and own price elasticities on representative dairy farms of groups A and B were compared with each other, the elasticities are on average more inelastic on dairy farms of group B than on dairy farms of group A. Dairy farms of group B are more strongly specialized in milk production than dairy farms of group A, which may explain the greater in- elasticity of the production in group B. In gener- al, substitution possibilities on dairy farms seem to decrease as a result of specialization. The elasticities solved from the panel data set are quite similar to the elasticities solved from the representative dairy farm data set. Thus, re- sults support each other. However, the elasticities related to purchased fertilizers are on average much more inelastic than those obtained from the representative dairy farm, particularly with the whole research period model. One reason for this may be the multicollinearity hidden in the time series data. Differences in the other elastici- ties solved from the panel data set and the repre- sentative dairy farm data set are mainly minor and there is no systematic twist. The results obtained from the panel data set are more reliable than the results obtained from the representative dairy farm data set, because the use of panel data increased the information about variation of output, costs, and cost shares across dairy farms. Thus, if there is multicolline- arity between the series in the representative dairy farm data set, the analysis of panel data gives the means to correct information obtained from the time series data. The inelastic elasticities of substitution and own price elasticities mean that, with the existing pro- duction technology, the changes in the prices and/ or relative prices of inputs have little effect on their demand. Consequently, it is not possible to regulate the demand for inputs in a flexible way by, for example, taxation. Thus, taxation meas- ures are mainly visible in the farm income of the milk producer. The possibilities of traditional ag- ricultural policies and environmental policies to have an influence on the level of demand of the inputs of production, in Finnish agriculture, with levies and/or subsidies are small. The elasticities of substitution estimated in the study were nearly similar to those estimated in other countries where substitution between in- puts also dominated. Lopez (1980) found a rela- tionship of substitution between all inputs. Bin- swanger (1974) found a relationship of comple- mentarity between the labour input and fertiliz- ers, machinery and fertilizers, and arable land and other inputs. According to Ray (1982), mis- cellaneous inputs (e.g., cleaners, electricity) are complements for the labour input, fertilizers, and feed. According to Glass and McKillop (1990), the aggregate input consisting of feed and seeds was a complement for fertilizers and the capital input. Differences between the research results are probably caused, in part, by the different con- ditions and production technology in different countries. Comparison of the research results was difficult, because they differed from each other in terms of the estimated model and the aggrega- tion of inputs. The elasticities estimated in other countries were also relatively inelastic. Binswanger’s re- sults differed from the other research results with respect to the elasticity of the elasticities of sub- stitution, which is probably caused by the fact that Binswanger’s data dates from an earlier pe- riod of time (1949-1964) than the data of the other studies. At that time production was not as 589 Agricultural Science in Finland 3 (1994) specialized as later, so that the possibilities for flexibility were greater than at present. Technical change on dairy farms has in- creased the use of capital. The technical change describing the use of the labour, purchased feed, and purchased fertilizers solved from the repre- sentative dairy farms data set varied according to the model specification. Technical change solved from the panel data has been purchased feed- saving and other inputs-using. Self-sufficiency in feed has been an objective, and thus the pur- chased feed-saving technical change appears log- ical. The average annual rate of technical change for the whole research period has been 1.3%, and it has been increasing of the level of technology. On the basis of this result, the technological de- velopment has had a positive contribution to the dairy farms. In the long run, profitability is the driving force of production. In the short run, liquidity may re- strict production. Maintaining the profitability and continuity of production requires that the dairy farm keeps up with the technical development. Without utilizing the technical development, a milk producer cannot compete with other entre- preneurs in the same field or in other fields in the long run, but the production becomes unprofita- ble and must be discontinued. According to the results, new technology should be utilized in in- creasing the farm size. In the EU system, how- ever, increasing the farm size may be difficult because of the weakening profitability (Latukka etal. 1994). Increasing the output quantities and substitut- ing the relatively cheaper capital items for the more expensive labour has been profitable. In- vestments have brought new technology to dairy farms. Ylätalo (1987, p. 84) came to the same conclusion. He noted that, at the level of agricul- ture as a whole, the arguments of too high invest- ments in agriculture were not true. As a result of technical development, the in- troduction of new machine and production chains has made it possible to increase the size of dairy farms. The size of the farms included in this study increased considerably, but compared to, for ex- ample, Sweden and Denmark, Finnish dairy farms are still very small. In the research period the output of dairy farms of group A increased over 2.5 times, whereas the use of labour input de- creased by about 20%. On dairy farms of group B output grew over 3.5 times, but because of the strong growth in output the decrease in the use of the labour input remained below 5%. According to the results, there are advantages related to the size (scale) of the enterprise in milk production. On the basis of the results ob- tained from the panel data set, an increase of 1.0% in output increases costs about 0.6%. So the unit costs decrease as production is expanded and thus, increasing the size of dairy farms should be allowed. When the farm size grows, the cost shares of labour decrease, and those of other in- puts increase. In milk production it has not been possible to utilize the economies of size (scale) in full dur- ing the last two decades, because the main focus in agricultural policy has been in abolishing the overproduction of milk, and developing produc- tion has remained in the background. In the representative dairy farm data set technical change and farm size are confounded, and there is multicollinearity between them and purchased fertilizers, which made it impossible to separate their effects clearly from each other. The panel data set made it possible to obtain more reliable information concerning technical change and size economies than did the repre- sentative dairy farm data set. The increasing re- turns to size assumption made in the theoretical part of the study is consistent with empirical re- sults. In developing the production of dairy farms it is essential that they are allowed to grow. This means that the number of dairy farms will de- crease. Ryynänen (1972, p. 11) brought this for- ward very clearly: the growth of an enterprise and an entrepreneur is not possible, unless the society provides conditions that favour change. Finnish society faces a time in which it has to promote the growth of the farm size in a more determinate way than earlier. The gap between the farm size required for development and the 590 Agricultural Science in Finland 3 (1994) prevailing farm size may otherwise become too wide. According to the results, this prognosis seems to have come true. Measures to restrict production at the farm level (mandatory fallowing, production quotas) have increased the unit costs, because the production capacity of the farm has not been fully utilized. Also, these measures restrict the production to the area where the unit costs decrease most strong- ly when the farm size increases (cf. Heikkilä 1987). According to Ryynänen and Pyykkönen (1988), costs can be lowered and the profitability of the production can be secured only by increas- ing the farm size and the efficiency of the use of production factors. In Finland the farm size is determined by more than justeconomic factors. Stanton (1978, p. 727) notes that the size of agricultural enterprises must be seen as a political issue. According to him, agricultural production has a special role com- pared with other sectors. This was/is the prevail- ing view in Finland, too. When examining the current situation, it should be kept in mind that the objectives of agricultural policy also include objectives that do not belong to agriculture. A special feature in Finnish policy was the settlement after the wars, which increased the number of farms and decreased the average farm size considerably. Through the settlement, the “social problem” was transferred to the agri- cultural sector and agricultural policy, which led to the splitting up of dairy farms, and thus to the small farm size. According to the research results, milk pro- ducers seem to have operated rationally. By means of the growth of the farm size and the new pro- duction technology they have been able to in- crease output, so it has not been necessary to increase the use of the labour input, the price of which has increased the most. Milk producers seem to have operated principally in an economi- cal way. Thus, milk producers have operated ac- cording to the theoretical assumptions within the framework provided by the legislation. In times of powerful economic shocks it has not been pos- sible to adjust the production of dairy farms in an optimal way, which can be considered a logical result because of the inelastic substitution of in- puts. In 104 cases out of 108 the signs of the own price elasticities of inputs in the representative dairy farm models for the whole research period were in accordance with the theory. In times of powerful economic shocks, the years 1973-1976 and 1980, the sign of the own price elasticity of the derived demand for the capital input deviated from the assumption. In the post energy crisis peri- od representative dairy farm models all own price elasticities were in accordance with the theory. According to the negative semidefiniteness of the Hessian matrix and the estimated positive cost shares, the theoretical concavity and monotonici- ty assumptions were realized in all post energy crisis period representative dairy farm models at all observation points, except on dairy farms of group A in 1976 and 1980. In the representative dairy farm models for the whole research period the concavity assumption was not realized in times of powerful economic shocks. On representative dairy farms of group B the deviation of the nega- tive semidefiniteness of the Hessian matrix at the beginning of the research period may also have been caused by the great variation in the number of farms included in the research. The cost shares estimated from the the panel data set for the whole research period and for the post energy crisis period are not positive at all sample points, 12 out of a total of 4752 cost shares in the whole research period and 17 out of a total 2816 cost shares in the post energy crisis period are negative. Thus, monotonicity is not satisfied at these observations. The sign of the own price elasticity of the inputs included in the model is in accordance with the theory, evaluated at the sample mean. But, they do not have the correct sign at all sample points, 434 out of a total of 4752 own price elasticities in the whole research period model and 164 out of a total of 2816 own price elasticities in the post energy crisis period model are positive. Even ifmilk producers were aware of the mar- ginal rate of technical substitution and price rela- tions between inputs and would strive to operate in an optimal way, this cannot always be ob- 591 Agricultural Science in Finland 3 (1994) served from ex-post data, because the situations examined ex-ante and ex-post differ from each other. Milk producers make ex-ante decisions, but research is based on ex-post data, which means that observations deviating from optimum behav- iour are possible because of risk and uncertainty. This can be seen in the effects of the powerful economic shocks during the research period, to which milk producers could not adjust themselves immediately. Thus, on the basis of ex-post obser- vations it cannot be concluded that milk produc- ers would behave irrationally, although the data includes some observations that deviate from the optimum behaviour (cf. Singh 1987, p. 444-445). The empirical research results should be viewed cautiously. The research data and research meth- ods do not allow experimental research, and thus repetition is not possible. In spite of the possible defects in the existing data, the bookkeeping data was the best possible data for solving the empiri- cal problem of the study. In interpreting the results obtained from the representative dairy farm data set it should be kept in mind that time series data is deficient for determining the advantages and disadvantages re- lated to the size (scale) of the enterprise. On the other hand, the panel data set used in the study possessed a large amount of new information con- cerning output, costs, and cost shares variation across farms. Thus, the panel data set provided more reliable results concerning the size (scale) economies, and made it possible to separate the effects of the farm size economies and technical change from each other. In recent years the production and cost theory, as well as the research methods, have developed rapidly, but in spite of this, simplification of the research problem is still necessary in empirical studies. Alternatively, one would have to aban- don the empirical approach. In this study, it was essential to obtain an empirical answer to the research problem. Theoretical study alone does not give quantitative information needed in poli- cy making. In empirical analyses the estimated function is necessarily an approximation of the real cost func- tion. Also, the number of variables included in the analysis must be restricted, because the data and the research methods do not allow for exam- ination of large groups of variables simultane- ously. In this study, too, the estimated cost func- tion is an approximation of the real cost function. Aggregate price indices of inputs and the aggre- gate output were used in the estimation. The prob- lems related to the number of variables in milk production can be illustrated by, for example, sev- eral kinds of feed (e.g., protein feed, energy feed, minerals, vitamins). Various kinds of machines are also used on dairy farms, so that obtaining and using detailed data is impossible with the current knowledge and data. In this study the demand for the capital items, labour, purchased fertilizers, and purchased feed was assumed to be weakly separable. In practice, it cannot be assumed, a priori, that this restric- tion would be completely true. The choice of the combination of purchased feed, for example, may depend on the choice of the combination of capi- tal items. The assumption on the separability of inputs had to be made, even if it could not be tested, because it makes the theory and the em- pirical research methods compatible with each other. The assumption of separability is needed, because in empirical studies it must be solved, a priori, which individual inputs are connected to which aggregate input, and thus it has assumed a “State ofArt" position in applied economic study (Chambers 1988, p. 157). The fact that the adjustment to new price rela- tions has been assumed to occur without adjust- ment costs may also have influenced the research results. On the other hand, in economics it has not been succeeded to solve the dependency be- tween the adjustment costs and the fixity of in- puts, so that the use of a dynamic approach in- volves a theoretical problem. The small number of observations (a year) is problematic, because the estimation methods are based on asymptotic properties, and thus the results may include a small sample bias. To improve the reliability of the empirical re- sults it would be essential to obtain better re- search data. The price indices used in this study are the same for all dairy farms for one year. 592 Agricultural Science in Finland 3 (1994) Thus, regional differences in prices are treated in the same way as quality differences. If the dairy farm level prices would be available, it would be possible to distinguish more inputs than in this study. On the other hand, by using different price indices, it is possible that these indices become endogenous variables. Improving the quality of the data would be important, but the requirements set for an ade- quate amount of data shouldalso be kept in mind. Developing the statistical methods would also im- prove the reliability of the results. 10 Summary In this study, the effect of the change in the rela- tive prices of inputs and the technical change of production on the derived demand for inputs of dairy farms was examined. The dual approach of the neoclassical production and cost theory is used in the study. Based on the theory, it is assumed that the production function describes the techni- cal relationship that transforms inputs into out- put, dairy farms aim at minimizing costs, and the prices of inputs and products are determined ex- ogenously. To solve the empirical research problem, an equation system in which parameter restrictions according to the neoclassical production and cost theory were set between the equations. The sys- tem of equations was constructed from the cost function and the cost share equations. The trans- log cost function was chosen as the cost function to be estimated. The equation system consisting of the translog cost function and the cost share equations was estimated by Zellner’s iterative es- timation method. On the basis of the likelihood ratio test, the cost function had to be specified as non-homothetic, and it also had to be possible to estimate non-neutral technical change from it. On the basis of the research results, the inputs used in milk production are for the most part substitutes with each other. The values of the elas- ticities of substitution are small, so that, with the existing production technology, the substitution of inputs for other inputs is inelastic. The own price elasticities of the derived demand for in- puts were also inelastic. This means that, the pos- sibilities of traditional agricultural policies and environmental policies to have an influence on the level of demand of inputs of production, in Finnish agriculture, with levies and/or subsidies are small. Thus, changes in the prices of inputs will affect mostly the farm income of the milk producer. Technical change on dairy farms has increased the use of capital. According to the results, tech- nical change and the farm size were interrelated in time series data, in representative dairy farm data, which made it impossible to separate their effects clearly from each other. Technical change solved from the panel data has been purchased feed-saving and other inputs using. Self-sufficien- cy in feed has been an objective, and thus the purchased feed-saving technical change appears logical. The panel data set made it possible to obtain more reliable information concerning tech- nical change and size economies than did the representative dairy farm data set. The average annual rate of technical change for the whole research period has been 1.3%, and it has been increasing of the level of technology. On the basis of this result, technological development has been advantageous to dairy farms. The new production chains resulting from tech- nical change have made it possible to increase the size of dairy farms. Maintaining the profita- bility and continuity of dairy farms requires that they keep up with the technical development. Sub- stituting the cheaper capital input for the more expensive labour input has been a profitable meas- ure, in particular, as new technology has been introduced to production through investments. According to the results, there are advantages related to the size economies in milk production. 593 Agricultural Science in Finland 3 (1994) In milk production it has not been possible to utilize the economies of size in full during the last two decades, because the main focus in agri- cultural policy has been in abolishing the over- production, and developing production has re- mained in the background. On the basis of the results obtained from the panel data set, a 1.0% increase in output increases costs about 0.6%. So, unit costs decrease as production is expand- ed. 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Pel- lervon taloudellisen tutkimuslaitoksen julkaisuja 10: 1-74. 598 Agricultural Science in Finland 3 (1994) SELOSTUS Tuotantopanosten substituutio sekä tekninen kehitys Suonien maitotiloilla vuosina 1965-1991 Empiirinen sovellutus kirjanpitoa pitäville maitotiloille Matti Ryhänen Helsingin yliopisto Tutkimuksen tausta ja tavoitteet Suomen maataloustuotanto on muuttunut nopeasti 1960- luvun alusta lähtien. Maatalouden muutokset ovat kytkey- tyneet läheisesti kansantalouden kehitykseen. Taloudelli- sen kasvun ja elintason kohoamisen myötä syntyneet työ- paikat ovat houkutelleet työvoimaa etenkin pientiloilta. Kehityksen mukana maaseudun väestö on vähentynyt yli puolella miljoonalla henkilöllä. Samanaikaisesti maitoti- lojen lukumäärä on laskenut 240 000 tilasta nykyiseen alle 35 000 tilaan. Maitotilojen nopea väheneminen ei ole poistanut mai- don ylituotantoa. Vaikean ylituotannon vuoksi maidon- tuotantoa on rajoitettu voimakkaasti, jonka seurauksena maitotilojen aktiivinen kehittäminen on jäänyt vähälle huo- miolle, Tulevaisuudessa tarve maitotilojen kehittämiseen näyttää kasvavan. Tällöin tarvitaan uutta tietoa tuotanto- panosten substituutiomahdollisuuksista (tuotantopanosten välisestä korvattavuudesta) sekä maitotilojen teknisestä ke- hityksestä. Tuotantopanosten substituutiomahdollisuuksien tunteminen ei ole oleellista pelkästään maidontuottajille, joiden mahdollisuudet vaikuttaa tuotantopanosten ja tuot- teiden hintoihin ovat vähäiset, vaan se on erityisen tär- keää poliittisille päättäjille, joiden tulisi tuntea kehittä- mistoimenpiteiden yritystaloudelliset vaikutukset. Tutkimuksen tavoitteena on selvittää, kuinka maidon- tuottaja reagoi ja muuttaa tuotantopanosten käyttöä niiden suhteellisten hintojen muuttuessa. Maidontuottajat eivät voi siirtää suhteellisten panoshintojen muutosta tuotteil- lensa hintoihin, koska ne on sovittu etukäteen maatalous- tuloneuvotteluissa, joten sopeutuminen tuotantopanosten hintojen suhteellisiin muutoksiin tapahtuu pääosin maito- tilan tuotantoteknologiaa muuttamalla, siinä mitassa kuin se vain on mahdollista. Tutkimuksen keskeisenä tavoitteena on selvittää tuo- tantopanosten kysynnän substituutio-ja oman hinnan jous- tojen arvot. Tuotantopanosten välisten substituutiomah- dollisuuksien tutkiminen suomalaisilla maitotiloilla on ol- lut vähäistä. Maitotiloja kehitettäessä ja maidontuotantoa ohjattaessa tuotantopanosten keskinäisten suhteiden ja subs- tituutiomahdollisuuksien tunteminen on oleellista. Ilman yritystaloudellista tietoa kehittämis- ja ohjaustoimenpiteet voivat olla ristiriidassa asetettujen tavoitteiden kanssa. Lisäksi tavoitteena on tutkia, kuinka tekninen kehitys on vaikuttanut yksittäisten tuotantopanosten kysyntään sekä minkä tuotantopanosten kysyntä on lisääntynyt ja minkä vähentynyt teknisen kehityksen myötä. Tutkimuksen teoreettinen kehys Tässä tutkimuksessa hyödynnetään neoklassista tuotanto- ja kustannusteoriaa. Tutkimus pohjautuu pitkän aikajän- teen tarkasteluun, jolloin kaikki kustannukset oletetaan muuttuviksi kustannuksiksi. Tuotanto-ja kustannusteorian mukaisesti oletetaan, että maitotilalle on olemassa tuotan- tofunktio, joka täyttää hyvin käyttäytyvän tuotantofunkti- on ominaisuudet. Koska maidontuotantoa on rajoitettu maassamme vuodesta 1970 lähtien ja koska maitotilat ei- vät toimi täydellisen kilpailun oloissa tuotemarkkinoil- laan, niiden taloudellista käyttäytymistä kuvataan tuotan- tokustannusten minimointiongelmana. Tutkimusongelman ratkaisemiseen käytetään duaalilä- hestymistapaa. Teorian mukaan oletetaan, että tuotanto- funktio kuvaa tuotantopanosten ja tuotoksen välisen yhtey- den, maitotilat toimivat kustannusten minimoijina ja että tuotantopanosten ja tuotteiden hinnat määräytyvät ekso- geenisesti. Duaalilähestymistapa jakustannusfunktion käyt- tö mahdollistavat tutkimusongelman ratkaisemisen. Aineistoja tutkimusmenetelmät Tutkimusaineistona käytetään Suomen maatalouden kan- nattavuustutkimukseen osallistuvilta maitotiloilta kerätty- jä tietoja vuosilta 1965-1991. Tutkimusta varten kerättiin kaksi aineistoa: A-maitotilat (44 kpl), jotka ovat mukana koko tutkimusaikajänteen ja B-maitotilat (168-364 kpl), joiden määrä vaihtelee vuosittain. Energiakriisin jälkei- selle ajalle 1976-1991 estimoidaan mallit erikseen. Empiirinen havaintoaineisto koostuu maitotilojen ko- konaistuotosta, kustannuksista, tuotantopanosten hinnois- ta sekä tuotantopanosten kustannusosuuksista. Tutkimuk- seen valittiin neljä aggregoitua tuotantopanosta; työ (mai- dontuottajaperheen ja palkkaväen työ), pääoma (koneet, kalusto, laitteet ja rakennukset), ostolannoite ja ostoväki- rehu. Maitotilojen metsätalous, muu tuotantotoiminta, si- vuansiotalous ja yksityistalous eivät kuuluneet tutkimuk- sen piiriin. Aineisto kerättiin kirjanpitotilojen tilivuoden tapahtumista, jotka on merkitty muistiin virtasuureina si- 599 Agricultural Science in Finland 3 (1994) ten, että tuotos ja sen tuottamiseen käytetyt tuotantopa- nokset sattuvat samalle ajanjaksolle. Tutkimusjakson aluksi valittiin vuosi 1965, koska täl- löin kirjanpitotiloilla siirryttiin kesä-heinäkuun vaihteen tilikaudesta vuoden alusta alkavaan tilikauteen. Tilivuon- na 1964-1965 kaikille tiloille ei ole laadittu täydellistä tilinpäätöstä, joten tilikauden aineisto on jäänyt pieneksi. Luotettavan aineiston saanti vuotta 1965 edeltävältä aika- jänteeltä ei ollut mahdollista. ”Edustava” maitotila periaatetta sovelletaan molempiin tutkimusaineistoihin. A- ja B-maitotiloja kuvaavan edus- tavan tilan havainnot saadaan laskemalla A- ja B-maitoti- la -aineistoista keskiarvot kullekin vuodelle erikseen. A-maitotila -aineisto kerätään paneeliaineistoksi, koska tietoja halutaan erityisesti yritystasolta. Tuotanto- ja kus- tannusteoria perustuu yksittäisten yrittäjien päätöksente- koon, joten paneeliaineiston käyttö on teoreettisesti vah- vemmalla pohjalla kuin ”edustava” maitotila -aineiston käyttö. Tutkimusongelman ratkaisemiseksi kustannusfunktios- ta ja kustannusosuusyhtälöistä muodostetaan yhtälösys- teemi, jossa yhtälöiden välille asetetaan tuotanto- ja kus- tannusteorian mukaiset parametrirajoitukset. Symmetri- syysoletus testataan. Malli saadaan estimoitavaan muo- toon approksimoimalla joustavamuotoisella translogfunk- tiolla kustannusfunktiota ja kustannusosuusyhtälöitä. Yhtälöryhmän singulaarisuus vältetään poistamalla yksi kustannusosuusyhtälö yhtälösysteemistä sekä valitsemal- la estimointiproseduuri,joka estimoi parametrit siten, että estimoitujen parametrien arvot eivät riipu siitä, mikä kus- tannusosuusyhtälöistä pudotetaan yhtälösysteemistä. Pois- tetun kustannusosuusyhtälön parametrit ratkaistaan para- metrirajoitusten avulla. Yhtälösysteemi estimoidaan Zellnerin iteratiivisella es- timointimenetelmällä. Se sopii hyvin tutkimuksen yhtälö- systeemin estimointiin, koska kustannusosuusyhtälöt muo- dostavat SUR-yhtälösysteemin (system of seemingly un- related equations), jossa kustannusosuusyhtälöiden virhe- termit ovat keskenään korreloituneita sekä siksi, että yh- tälöiden oikealla puolella olevat muuttujat ovat eksegee- ttisiä. SUR-estimointitekniikkaa käytetään sekä "edusta- va” maitotila -aineistoon että paneeliaineistoon. Estimoiduista parametri- ja kustannusosuusarvoista rat- kaistaan pääoma-, työ-, ostorehu- ja ostolannoitepanosten väliset substituutiojoustot Alienin osittaissubstituutiojous- ton, Morishiman osittaissubstituutiojouston ja varjosubs- tituutiojouston mittareilla sekä näiden tuotantopanosten ristijoustot ja oman hinnan joustot. Tutkimustulokset Tutkimustulosten mukaan maidontuotannossa tuotantopa- nokset ovat pääosin substituutteja keskenään. Substituu- tiojoustojen arvot ovat suhteellisen pieniä ja tuotantopa- nosten johdetun kysynnän oman hinnan joustot ovat myös itseisarvoltaan pieniä, joten vallitsevalla tuotantoteknolo- gialla tuotantopanosten korvaaminen toisilla tuotantopa- noksilla on joustamatonta. Tämä merkitsee sitä, että pe- rinteisen maatalouspolitiikan ja ympäristöpolitiikan mah- dollisuudet vaikuttaa maitotilojen tuotantopanosten kysyn- tään tuotantopanosten hintoihin liittyvien verojen ja/tai tukien avulla ovat vähäiset. Alienin osittaissubstituutiojouston mittarilla mitattuna tuotantopanokset ovat pääosin substituutteja keskenään. Kaikissa tutkimusaineistoissa pääoma- ja ostorehupanos ovatkomplementtejä (tuotantopanosten välillä täydennys- suhde). Osassa tutkimusaineistoja havaitaan komplemen- taarisuutta myös työ- ja ostolannoitepanoksen sekä pää- oma- ja ostolannoitepanoksen välillä, Morishiman osit- taissubstituutiojouston ja varjosubstituutiojouston mitta- reilla mitattuna tuotantopanokset ovat pääosin substituut- teja keskenään. Pääoma- ja ostorehupanoksen välillä ha- vaitaan myös komplementaarisuutta. Verrattaessa ”edustavalle” A- ja B-maitotilalle estimoi- tuja substituutio- ja oman hinnan joustoja havaitaan, että B-maitotilojen joustot ovat keskimäärin jäykempiä kuin A-maitotilojen. B-maitotilat ovat erikoistuneet A-maitoti- loja voimakkaammin maidontuotantoon, mikä selittänee joustojen eroja. Paneeliaineistosta ratkaistut joustot antavat samanlaista informaatiota kuin ”edustava” maitotila -aineistosta rat- kaistut joustot muutoin paitsi ostolannoitepanokseen liit- tyvien joustojen osalta, jotka ovat keskimäärin huomatta- vasti jäykempiä kuin ”edustava” maitotila -aineistosta rat- kaistut joustot. Syynä tähän näyttäisi olevan aikasarjoihin kätkeytyvä multikollineaarisuusongelma, jota voidaan kor- jata hankkimalla lisää aineistoa. Paneeliaineistosta saatavat tulokset ovat luotettavampia kuin edustava maitotila -aineistosta saatavat tulokset, koska paneeliaineiston käyttö lisää informaatiota tiloittaisten tuo- tosten, kustannusten ja kustannusosuuksien vaihtelusta. Jos ”edustava” maitotila-aineistoon liittyy multikollineaarisuus- ongelma, paneeliaineisto mahdollistaa aikasarja-aineistosta saatavan informaation korjaamisen. Tutkimuksessa estimoituja substituutiojoustoja verrat- taessa muissa maissa estimoituihin substituutiojoustoihin havaitaan niissä huomattavaa samankaltaisuutta. Kaikissa läpikäydyissä tutkimuksissa tuotantopanosten välillä subs- tituutiosuhde on vallitsevana. Muissa maissa estimoidut joustot ovat myös suhteellisen jäykkiä. Tekninen kehitys maitotiloilla on ollut pääoman käyt- töä lisäävää. Työ-, ostorehu- ja ostolannoitepanoksen käyt- töä kuvaava tekninen kehitys vaihtelee ”edustava” maito- tila -aineistossa mallin spesifioinnin mukaan. Tulosten mu- kaan ”edustava” maitotila -aineistossa tekninen kehitys ja yrityskoko kietoutuvat toisiinsa, joten niiden vaikutusten erottaminen toisistaan on mahdotonta. Paneeliaineistosta ratkaistu tekninen kehitys on ostorehujen käyttöä säästä- vää ja muita tuotantopanoksia käyttävää. Suomalaisessa maidontuotannossa on pyritty rehuomavaraisuuteen, joten ostorehujen käyttöä säästävä tekninen kehitys on loogi- nen tulos. Maidontuotannon tuotantoteknologiaa ei voida kuvata neutraalilla teknisellä kehityksellä. Vuosittainen tek- ninen kehitys maidontuotannossa tutkimusaikajänteellä on ollut keskimäärin 1.3 prosenttia. Tutkimustulosten mukaan maidontuotannossa on yri- tyskokoon liittyviä etuja. Tulosten mukaan tuotoksen nos- taminen prosentilla lisää kustannuksia 0.6 prosenttia, jo- 600 Agricultural Science in Finland 3 (1994) ten yksikkökustannukset alenevat tuotantoa laajennettaes- sa. Tulosten mukaan maidontuotannossa olevia yritysko- koon liittyviä etuja ei ole täysin voitu hyödyntää, koska maatalouspolitiikan pääpaino on ollut ylituotannon vähen- tämiseen tähtäävässä ohjaamisessa, jolloin tuotannon ke- hittäminen on jäänyt taka-alalle. Yrityskoon kasvaessa työ- panoksen kustannusosuus pienenee ja muiden tuotantopa- nosten kustannusosuus vastaavasti kasvaa. Maitotilojen tuotannon kehittämiselle on välttämätöntä, että niille annetaan kasvun mahdollisuus. Tämä merkitsee samalla maitotilojen lukumäärän vähenemistä. Tilakoh- taiset tuotannonrajoitukset (kesannointivelvoite, tuotanto- kiintiö) ovat lisänneet yksikkökustannuksia, koska maito- tiloilla oleva tuotantokapasiteetti on jäänyt vajaakäyttöön. Lisäksi tuotannonrajoitukset rajaavat useimmilla tiloilla yrityskoon alueelle, jossa yksikkökustannukset alenevat voimakkaasti yrityskoon kasvaessa. Mahdollisessa EU-jä- senyydessä yrityskoon kasvattaminen voi olla vaikeaa heik- kenevän kannattavuuden vuoksi. Tutkimus poikkeaa keskeisesti aiemmista panoskysyn- tätutkimuksista, koska tutkimuksessa verrataan ”edusta- va” maitotila -tarkastelun tulosten japaneeliaineiston ana- lyysistä saatavien tulosten eroja sekä selvitetään, missä määrin paneeliaineisto varmentaa tai mahdollisesti kumoaa ”edustava” maitotila -tarkastelun tulokset. Suoritettujen estimointien perusteella paneeliaineisto tukee ja tarkentaa ”edustava” maitotila -aineistoista esti- moituja tuloksia. Estimoitujen substituutioparametrien erot ovat yleensä pieniä eikä niissä ole systemaattista vääntöä. Vain ostolannoitteisiin liittyvät substituutiojoustot ovat sys- temaattisesti jäykempiä paneeliaineistosta ratkaistuna kuin ”edustava” maitotila -aineistosta ratkaistuna. 601 Agricultural Science in Finland 3 (1994) Properties of production possibility set T The production possibility set T contains all technically feasible combinations of (x,y). In empirical study the observa- tions are a set of observations collected from R dairy farms. Thus, it is possible to have only some points of (x,y), but not a whole convex set in applied study in production analysis. Therefore, to determine the production possibility set on the basis of these points requires the followingrestrictions on T (Chambers 1988, p. 252, Jacobsen and Paulsen 1991, p. 5-8): 1. T is nonempty; 2. T is a closed set; 3. T is a convex set; 4. if (x,y)eT, x’> x, then (x’,y)eT; 5. if (x,y) eT, y’ < y, then (x,y’) eT; 6. for every finite x, Tis bounded from above; and 7. (0,0) eT, but (o,y) éT, if y > 0. In the literature on agricultural economics these restrictions are generally accepted. There also exists a weak disposability concept of properties 4 and 5, when there is a possibility that some inputs may have an adverse effect on output, if they are used in too high proportion (e.g., ByRNES et al. 1987, p. 369-373). It is assumed in this study that milk producers do not accept negative marginal products, so that the concept of weak disposability will not be needed. According to property 1 of the production possibility set, there has to be at least one point where drawing conclusions from the production technology would be possible. Property 2 is purely a technical restriction that the optimum solution should exist. According to property 2, all of the boundary points of T belong to T. Property 3 implies that for any two technically feasible input-output combinations in set T, the line segment connecting these two points lies entirely in T and is also technically feasible. Properties 4 and 5 rule out negative marginal product, i.e., overutilization of some input(s) to the point where output falls, is ruled out. Thus, marginal product is equal or greater than zero. According to property 4, if an input bundle can produce a given output bundle, a larger input bundle can also produce that output bundle. According to property 5, if an input bundle can produce a given output bundle, it can also produce all smaller output bundles, e.g., zero output is possible. According to property 6, T is bounded from above. This implies that there is some limit that can be produced. In other words, for a given input bundle it is impossible to produce an infinite amount of output. So a limit must exist. Property 7 is a “no free lunch” property. The origin belongs to T, because producing zero output is always possible, but it is not possible to produce anything if there is no input. The production technology described above is quite abstract. Thus, it is illustrated by Figure I. Let us assume that we have a data set collected randomly from two dairy farms; A = (x‘,y') e T and B =(x2 2,2 ) eT, which describe input-output combinations from these dairy farms. In Figure I, illustration of production technology is started using property 4 to the point A, According to property 4, y 1 can be produced using xl . It can also be produced using more input, for example x 3 (point C). This means that if a farmer can milk 20 cows during a day, two farmers can milk these 20 cows during a day, too. According to this property, we can draw the horizontal line from point A via point C to the right. According to property 5, x 1 can produce yl , but the same amount of x' can also produce less output, for example, zero output is possible. This means that if a farmer can milk 20 cows during a day, it is possible that he/she can milk 10 cows during a day, too. Then we can draw the vertical line between points x 1 and A. We can continue with point B the same way as we did with pointA. We can see in Figure I that there is a hole between points A and B. Thus, we use property 3. If the production possibility frontier is concave, the frontier and the set under it form a convex set. Convex combinations are weighted averages which sum up to one. So we can draw the conclusion that with two known points, A = (x',y')eT and B = (x2 ,y2 )eT, so that 0 (0 < 0 < 1), we will be able to find points (for example D) where D = (x,y) = (1-0) (x’,y') + 0(x 2,y 2)eT. So we can draw a line between points A and B because all convex combinations belong to the production possibility set T. By using the free disposa- bility of inputs and outputs on this line we can fill the whole convex hole. The production possibility set in Figure I illustrates the production technology of these two dairy farms. In the literature it is said that T forms a free disposal convex hull with respect to the collected data set. In Figure I the bound- ary of the production possibility set describes a traditional production function because it is defined as the locus of the technically efficient input-output combinations. Fig. I. Production possibility setT. Appendix 1. Properties of the production function The assumption that the production function of a dairy farm yields the maximum obtainable output from a given input vector does not provide a sufficient basis for economic analysis. Thus, it is necessary to use further restrictions on the production function (e.g., Chambers 1988, p. 8-14); la. if x’ > x, then f(x’) > f(x), monotonicity; lb. if x’ > x, then f(x’) > f(x), strict monotonicity; 2a. V(y) = (x:f(x) > y) is a convex set, quasi-concavity; 2b. f(Xx,+ (l-X,)x 2 ) i> Xf(x,)+ (1-X)f(x2 ), O>X > 1, concavity; 3a. f(0 n) =O, null vector; 3b. f(x|t.O, xit| ,...,xn ) = O, Vx( ; 4, V(y) is closed and nonempty, y > 0; 5. f(x) is finite, nonnegative, real valued, and single valued Vx which are nonnegative and finite; and 6. f(x) is everywhere twice continuously differentiable. These properties do not represent the set of hypotheses to be maintained universally, but these properties are generally agreed to characterize the real world and economic behaviour of a farm. They represent a catalog of assumptions that are useful and necessary in the presentation of the production and cost theory. According to property 1, all marginal products are nonnegative or additional units of any input cannot decrease the level of output. Property 2a implies that V(y) is convex. Convexity of V(y) implies that, if x, and x 2 can produce y, then any weighted average of these input bundles can do this, too; or x’ = 3.x, + (l-3,)x 2 is an interior point of V(y). V(y) is a (strictly) convex set if f(x) is (strictly) quasi-concave. f(x) is quasi-concave, if upper contour sets are convex for all y. Property 2b is a mathematical representation of the law of diminishing marginal productivity. It means that when f(x) is twice continuously differentiable the Hessian matrix of f(x) is negative semidefinite. Property 3a means that production of a strictly positive output without the use of scarce resources is ruled out. It is physically possible to produce output without economically scarce resources. Such instances are omitted here because they do not represent an economic problem. Property 3b is more restrictive than 3a, It states that the production of output without all essential inputs is impossible. In the real world the production is partly contradictory with properties 3a and 3b. For example, the use of fertilizers increases the yield, but is not essential to achieve some level of output. Property 3 means that V(y) does not intersect the axis. Property 4 is a technical assumption. It means that it is always possible to produce positive output, and it rules out the possibility of holes in the boundary of V(y). So it is used to guarantee the existence of a well-defined constrained optimum. Property 5 is self-explanatory. Property 6 is necessary since it makes possible the use of differential calculus in the analysis. Appendix2. Profit function The profit function in conditions ofpure competition can be presented as: n(p,w) = Max (py-wx;(y,x) eT), x>o, y>o (60) = Max {pf(x)-wx}, x>o (61) This profit function is derived from the production function so that the constraint f(x) = y is binding, since the farmer aims at maximizing profit for any given level of output. Thus, f(x) is substituted for y, and a constrained problem is turned to an unconstrained one. As an alternative, it is possible to leave the revenue part as a function of y, and present the cost part as a function of y. The latter part of the function is the cost function, since it is related to the optimal level of cost to output. Thus, the profit function can be written as: n(p,w) = Max (py-C(w,y)J, y > O (62) According to the discussion above, the profit function can be derived directly from the cost function. Appendix 3. Definitions of various elasticities of substitution According to Heady (1952, p. 144-145), an elasticity of substitution between two inputs can be defined as an elasticity of production. Thus, the elasticity of substitution is equal to the percentage change in the use of x, divided by the percentage change in the use of x 2. E,h = (Ax,/x,)/(Ax2/x 2 ) . (63) This measure failed to determine the slope of the isoquant map. Thus, the elasticity of substitution defined by Hicks (1963) became a more generally accepted definition (Dedertin 1986, p. 197). Its mathematical definition is more complicated, but the interpretation of the solved values relative to the shape of the isoquant map is clear. The elasticity of substitution defined by Hicks is given by: _ 9(x 2/x,) f|/f2 _ dln (x 2/X|) * 9(f,/f2 ) x 2 /x, 3ln (f,/f2 ) (64) in which y =f(xr x2 ) is fixed, and f, = 3y/3x ] and f 2 = 3y/3x2 are marginal products of inputs x [ and x 2. The elasticity of substitution Es between two inputs is defined as the elasticity of the ratio of the inputs with respect to the marginal rate of technical substitution between them. Intuitively, E s tells how the ratio of inputs changes as the slope of isoquant changes. For example, if a small change in slope causes a large change in the input ratio, the Es gets a large value. The first order conditions for cost minimization say that the dln(f | /f2) equals the 3ln(w/w 2 ) which gives another familiar definition. Elasticity of substitution can be solved directly on the basis of first and second derivatives of the production function in two inputs case (Henderson and Quandt 1971, p. 62): g _ (f| x i + f; xz) X|X 2 (2f 12f If2- f 1 fz2 -1) (65) in which fM = 3 2y/3xJ; f22 = 3 2y/3x 2 ; and fl 2 = f2l = 32 y/3x,3x2 (Young’s theorem) Equation (65) makes it possible to calculate the elasticity of substitution at a certain point on an isoquant for a twice continuously differentiable production function. Equation (65) can be written in matrix notation as: x,f, + x 2fx F l20,2 = (66)x,x 2 F ' ' in which Fis the bordered Hessian determinant and F l 2 is the cofactor. The measures above are directly applicable only for two input cases. They are equivalent measurements, except for Heady’s measurement. In the case that the production function has three or more inputs, a number of alternative definitions for elasticity of substitution have been developed. For example, in a three-input case it is possible to derive three separate isoquants when holding one of them in turn constant. This makes it possible to generalize the equation (64). According to Chambers (1988, p. 33), this short-run elasticity is called direct elasticity of substitution. It measures the degree of substitutability between inputs x; and x; while all x k (k * i,k j) are fixed. The Allen partial elasticity of substitution (a) is a generalization for n inputs and it can be calculated using a generalization of equation (66)(Allen 1950, p. 503-505): S v»/< F S *»/» a = UJ LM = iii ( r'\ x.x, F x,x, U (67) in which (F 1 is the ijth element of f_l (f is bordered Hessian from the equation (3)). Inputs are complements if the elasticity of substitution is negative, and substitutes if the elasticity is positive. a )2 and a., in the case of a two-input production function, are equivalent. The Morishima partial elasticity of substitution (o“) is another generalization given by (Koizumi 1976, p. 152): o" =Li Lm-Lj Lm11 Xi F Xj F (68) 4 All other marginal rate of technical substitutions are intact. According to equations (67) and (68), there is a connection between Allen and Morishima partial elastisities of substitution (Koizumi 1976, p. 153): (69) k = I According to equation (69), the Morishima elasticity of substitution is asymmetric since a“. It is possible that Allen complements can be Morishima substitutes. On the other hand, if inputs are Allen substitutes, they are also Morishima substitutes. Appendix 4. Sources of data The data was collected from the bookkeeping dairy farms for 1965-1991. The data from 1965-1975 was collected manually from the original bookkeeping forms, and from 1976-1991 mainly directly from the databases recorded at the Agricultural Economics Research Institute (AERI), and the rest by hand from the original bookkeeping forms. The manually collected observations were recorded by the DBASE program. The SURVO program was used for checking and preliminary processing of the data. Accounting year 1965 Returns were collected from the Gross Return Form of Agriculture, which means that they indicate the return of the financial year in question in FIM (the change in the storage, consumption of the private household, and other transfers affecting the return have been taken into account). The costs were collected from the Production Expenses in Agricul- ture Form, so that they are the costs of the production of the financial year (change in the storage, account receivables, etc., have been taken into account). Insurance payments (tractor and the combine harvester) were collected from the Cash Receipts and Cash Expenses Form. The use of labour of the farm family in current work and investment work in agriculture, the costs of the current work done by hired labour, and its use of labour in current work and investment work were collected from the Labour Costs Form. Maintenance costs do not include the costs of maintenance work done by hired labour or the farm family, because they are included in the costs of current work in agriculture. Since residential buildings were included in the property of agriculture, the repair costs had been included in the costs of agriculture. Investments were collected from the Agricultural Assets and Liabilities Form, and own timber as well as the costs of hired labour are also included in them. The sales of machinery and implements were collected from the Implements Form. The arable land areas of farms were collected from the Gross Return Form, and the average number of cows from the Livestock Form. If the average number of cows was not reported, the arithmetic average of the number of cows at the beginning and the end of the year was marked as the average number of cows in the research data. Accounting year 1966 Residential buildings were separated from agriculture as a property group of their own. The repairs were no longer included in the expenditure of agriculture or the use of labour in current work. The repair costs of tractors and combine harvesters were not specified in the Production Expenses in Agriculture Form, but in the Cash Receipts and Cash Expenses Form. Only the insurances of tractors and combine harvesters could be specified. Other insurances are reported as unspecified in the group ‘Other Expenditure ofAgriculture’. Accountingyear 1967 No changes compared to the previous year. Accountingyear 1968 The values of machinery, implements, and buildings in taxation started to be used as the bookkeeping values of machinery, implements, and production buildings. In this connection, an initial inventory of the capital assets according to section 18 in the Act on the Taxation of Farm Income (MVL) was made, so that from this year the capital values of buildings, machinery, and implements were included in the data. In addition to the insurance of tractors, combine harvesters and cars, it was again possible to separate other insurances, but their contents were not specified. The rents of machinery and implements were unspecified, as was the case in the earlier years, too. The rent incomes from residential buildings were included in the total returns, and correspondingly, the costs ensuing from the residential buildings in the production expenses in agriculture, and the repair work of residential buildings in the current work of agriculture. In building investments residential buildings are still classified as a group of their own. Due to the lack of the Gross Return Form from the Southern Ostrobothnia and Northern Finland, the returns had to be collected from the Cash Receipts and Cash Expenses Form, which means that it was not possible to determine the cash value of deliveries in kind and the changes in the storage. Accounting years 1969-1975 No changes compared to the previous year. Accounting years 1976-1988 Since 1976 the information were mainly obtained from the databases recorded at the Agricultural Economics Research Institute. The insurance payments for hired labour and the number of cows remained to be collected by hand. Pension payments were not specified. Appendix 5. Accounting years 1989-1991 All necessary data was obtained from the databases recorded at the Agricultural Economics Research Institute. Appendix of other data sources SourceVariable Effective yield on bonds, % p.a. Inflation percentage Price of labour input Wage and salary index, Agriculture Prices of purchased fertilizers and industrial feed Machinery and implement cost index and the cost index for agricultural buildings Producer price of milk Producer price of beef Statistics of the Bank of Finland Central Statistical Office (Rise of cost-of-living index from December till December) Central Statistical Office as wages of agricultural workers FIM/h in 1982-1991 Central Statistical Office Indices calculated at the AERI Indices calculated at the AERI Statistics of the AERI Statistics of the AERI Appendix 5. Data appendix Dairy farms of group A (a representative dairy farm). obs InW, InW InW, InW, InY InCM _H_k^ _L 1965 -1.635320 -1.857451 -1,990625 -1.848647 -1.620246 -1.660784 -2.037839 -1.889218 -3.036138 -2.938785 -2.590400 -1.226525 -1.043755 -0.672365 -0.833605 -1.121778 -0.601827 -0.472679 -0.308674 -0.089433 0.000000 0,033686 0.022720 -0.143132 -0,063132 0.188453 -2.595923 -2.491835 -2.408590 -2.318186 -2.251559 -2.135716 -2.004661 -1.838027 -1.569146 -1.353297 -1.095878 -0.936305 -0.818255 -0.720518 -0.607443 -0.504032 -0.389620 -0.275477 -0.224244 -0.108476 0.000000 0.064832 0.132194 0.195550 0.305785 0.386228 0.491484 -1.832582 -1.801810 -1.714798 -1.639897 -1.624552 -1.619488 -1.594549 -1.565421 -1.427116 -1.224175 -1.038458 -0.879477 -0.681219 -0.701179 -0.705220 -0.621757 -0.421594 -0.298406 -0.164875 -0.048140 0.000000 -0.020203 0.020783 0.048790 0.089841 0.145830 0.186480 -1.771957 -1.737271 -1.703749 -1.604450 -1.604450 -1.604450 -1.604450 -1.570217 -1.482805 -1.301953 -1.044124 -0.791863 -0.606969 -0.556870 -0.531028 -0.441611 -0.236989 -0.175545 -0.154317 -0.076881 0.000000 0.005982 -0.133531 -0.215672 -0.199671 -0.121038 0.099845 -0.839075 -2.566017 -0.876974 -2.561434 -0.834273 -2.519051 -0.908248 -2.361094 -0,806898 -2.243486 1966 1967 1968 1969 1970 -0.774401 -2.179138 -0.693727 -2.100142 -0.656777 -1.978605 -0.655659 -1.858169 -0.569303 -1.545567 -0.466857 -1.337809 -0.349955 -1.081269 -0.310623 -0.942027 -0.273542 -0.779730 -0.207959 -0.696274 -0.168454 -0.605048 -0.150102 -0.419304 -0.098782 -0.271756 -0.047396 -0.208894 -0.020693 -0.109827 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 I9BS 0.0000000.000000 -0.007387 0.061505 0.0085530.103969 -0.017502 0.116172 1986 1987 1988 1989 0.077368 0.195841 0.103576 0.266342 0.096355 0.338459 1990 1991 0.311110 Dairy farms of group B (a representative dairy farm). obs InW InW, InW, InW, InY InC K L M H 1965 -1.597954 -1.822631 -1.957720 -1.815663 -1.580608 -1.623993 -2.021626 -1.869375 -3,036138 -2,938785 -2.590401 -1.207446 -1.029403 -0.668513 -0.823438 -1.102711 -0.594551 -0.467000 -0.304516 -0.088700 0.000000 0.033947 0.023003 -0.145014 -0.064496 0.187574 0.310290 -2.589348 -2.496328 -2.398336 -2.314789 -2.251065 -2.138546 -1.989513 -1.829835 -1.568166 -1.349851 -1.100797 -0.928576 -0.817808 -0.711319 -0.601383 -0.497694 -0.390161 -0.276017 -0.212055 -0.112173 0.000000 0.064223 0.133201 0.204099 0.309908 0.382845 0.488684 -1.832582 -1.801810 -1.714798 -1,639897 -1.624552 -1.619488 -1.594549 -1.565421 -1.427116 -1.224175 -1.038458 -0.879477 -0.681219 -0.701179 -0.705220 -0.621757 -0.421594 -0.298406 -0.164875 -0.048140 0.000000 -0.020203 0.020783 0.048790 0.089841 0.145830 0.186480 -1.771957 -1.137623 -2.756070 1966 -1.737271 -1.116756 -2.748029 1967 -1.703749 -1.115565 -2.713158 1968 -1.604450 -1.187312 -2.576233 1969 -1.604450 -1.037583 -2.444437 1970 -1.604450 -0.908124 -2.336763 1971 -1,604450 -0.797619 -2.239817 -1.570217 -0.681792 -2.0341471972 -0.681792 -2.034147 1973 -1.482805 -0.642606 -1.904534 1974 -1.301953 -0.533958 -1.611721 1975 -1.044124 -0.485970 -1.382024 1976 -0.791863 -0.353665 -1.110938 1977 -0.606969 -0.305683 -0.973674 1978 -0.556870 -0.239184 -0.807640 -0.531028 -0.194190 -0.7108451979 -0.194190 -0.710845 1980 -0.441611 -0.140458 -0.638220 1981 -0.236989 -0.114457 -0.424095 1982 -0.175545 -0.072022 -0.271466 1983 -0.154317 -0.030428 -0.205892 1984 -0.076881 -0.015743 -0,098354 1985 0.000000 0.005982 0.000000 -0.006602 -0.011142 -0.026899 0.118441 0.125275 0.126271 0.000000 0.044843 0.083095 0,103065 0.210227 0.269486 0.342104 1986 1987 -0.133531 -0.215672 -0.199671 -0.121038 1988 1989 1990 1991 0.099845 Appendix 6. Dummy variables obs D + D D. obs D + D D, 1965 000 1966 0 0 0 1967 0 0 0 1968 0 0 0 1969 0 0 0 1970 0 0 0 1971 000 1972 000 1973 0 0 1 1974 0 0 1 1975 0 0 1 1976 I 0 0 1977 1 0 0 1978 0 0 0 1979 000 1980 0 0 0 1981 0 1 0 1982 0 1 0 1983 000 1984 000 1985 0 0 0 1986 000 1987 0 1 0 1988 0 1 0 1989 000 1990 1 0 0 1991 1 0 0 D + = a dummy variable for an exceptionally good year D = a dummy variable for an exceptionally poor year D! = a dummy variable for the period of high inflation (> 15 %) Appendix 6. Cost shares and cost (actual, fitted, residual, and residual plot) Dairy farms of group A (a representative dairy farm). S, Residual Plot obs RESIDUAL ACTUAL FITTED : * : 1965 0.00668 0.26997 0.26329 : * : 1966 0.00457 0.24059 0.23602 : * : 1967 -0.00434 0.20456 0.20890 *: : 1968 -0.01080 0.21920 0.23000 : * : 1969 0.00411 0.25538 0.25127 ; * : 1970 -0.00480 0.23778 0.24258 : : * 1971 0.01962 0.20194 0.18232 : * : 1972 -0.00372 0.18995 0.19367 : * : 1973 -0.00280 0.09545 0.09826 : * : 1974 0.00543 0.08621 0.08077 : * : 1975 -0.00263 0.09089 0.09352 *: : 1976 -0.01101 0.17020 0.18122 * : : 1977 -0.01443 0.17015 0.18458 * : : 1978 -0.01184 0.21027 0.22211 : * : 1979 -0.00717 0.18753 0.19470 : : * 1980 0.01712 0.16339 0.14627 : * : 1981 0.00759 0.18779 0.18020 : * 1982 0.00997 0.19237 0.18239 : * : 1983 -0.00097 0.20658 0.20755 : * : 1984 0.00128 0.22569 0.22442 : * : 1985 0.00101 0.23140 0.23039 : * : 1986 -0.00339 0.23639 0.23979 : * : 1987 0.00145 0.22295 0.22150 : * : 1988 -0.00528 0.19834 0.20362 : * : 1989 -0.00485 0.21290 0.21776 : * : 1990 0.00163 0.25687 0.25524 : *: 1991 0.00893 0.27674 0.26782 ===================== =========== ======== ==================== SL Residual Plot obs RESIDUAL ACTUAL FITTED . : * 1965 0.01757 0.49989 0.48232 j : * 1966 0.01355 0.51771 0.50416 ' : :* 1967 0.01360 0.52534 0.51173 | * : 1968 -0.00948 0.51341 0.52290 * j : 1969 -0.02033 0.48280 0.50313 * : ; 1970 -0.01966 0.48654 0.50620 | * : 1971 0.00222 0.52081 0.51859 j *: 1972 0.01195 0.52593 0.51398 j* : 1973 -0.01120 0.59260 0.60380 J :* 1974 0.01354 0.61349 0.59996 j * : 1975 -0.00234 0.57974 0.58207 J * ; 1976 -0.00276 0.51237 0.51513 j* : 1977 -0.01133 0.50069 0.51201 | * : 1978 0.00231 0.46932 0.46701 " : * : 1979 0.00062 0.47087 0.47025 \ * ; 1980 -0.00413 0.48107 0.48520 " : * : 1981 0.00303 0.45166 0.44863 : * : 1982 -0.01131 0.43332 0.44463 * : ; 1983 -0.01600 0.44337 0.45937 j * ; 1984 -0.00215 0.45237 0.45452 '■ * : 1985 -0.00375 0.45032 0.45407 \ * : 1986 0.00335 0.46097 0.45762 J : * 1987 0.01685 0.45884 0.44200 j * 1988 0.01222 0.47174 0.45953 '. * : 1989 0.01019 0.48022 0.47004 ■ * : 1990 -0.00137 0.48682 0.48820 * : 1991 -0.00705 0.48182 0.48887 Appendix 6. S K Residual Plot obs RESIDUAL ACTUAL FITTED * : 1965 -0.01287 0.13654 0.14940 * : : 1966 -0.01545 0.14372 0.15917 : * : 1967 -0.00162 0.17483 0.17645 : : * 1968 0.01774 0.17161 0.15387 : :* 1969 0.01334 0.16405 0.15070 : : * 1970 0.01807 0.17578 0.15772 * : : 1971 -0.02497 0.17500 0.19998 : * : 1972 -0.00869 0.18747 0.19615 : * : 1973 -0.00069 0.21166 0.21235 : * : 1974 -0.00590 0.22384 0.22974 : * : 1975 0.00659 0.24018 0.23359 : * : 1976 0.00486 0.21554 0.21068 : : * 1977 0.01761 0.22773 0.21012 : *: 1978 0.01111 0.22504 0.21392 : * : 1979 0.00459 0.24103 0.23643 : * : 1980 -0.00812 0.25905 0.26717 : * : 1981 0.00171 0.27094 0.26923 : * : 1982 -0.00329 0.26688 0.27017 : * : 1983 0.00925 0.24288 0.23363 ; * : 1984 0.00388 0.22755 0.22366 : * : 1985 -0.00462 0.21653 0.22114 : * : 1986 -0.00963 0.20387 0.21350 * : : 1987 -0.01726 0.22544 0.24271 : * : 1988 -0.00240 0.24338 0.24578 : * : 1989 0.00571 0.22907 0.22337 : * : 1990 0.00044 0.18145 0.18101 ; * : 1991 9.9E-05 0.17465 0.17455 InC Residual Plot Obs RESIDUAL ACTUAL FITTED : * : 1965 0.06312 -2.56602 -2.62913 : * : 1966 0.02470 -2.56143 -2.58614 : * : 1967 -0.00557 -2.51905 -2.51349 : * : 1968 -0.03014 -2.36109 -2.33095 : * : 1969 -0.01034 -2.24349 -2.23315 : * : 1970 -0.03466 -2.17914 -2.14447 : * : 1971 0.03197 -2.10014 -2.13211 : * : 1972 -0.00166 -1.97860 -1.97695 : * : 1973 -0.04271 -1.85817 -1.81545 : * : 1974 0.04858 -1.54557 -1.59415 : * : 1975 -0.00587 -1.33781 -1.33194 : * : 1976 -0.02341 -1.08127 -1.05785 : * : 1977 -0.06012 -0.94203 -0.88191 : * : 1978 -0.03658 -0.77973 -0.74315 : * : 1979 -0.00346 -0.69627 -0.69282 : * : 1980 0.04305 -0.60505 -0.64809 : * : 1981 0.01241 -0.41930 -0.43172 : * : 1982 0.02927 -0.27176 -0.30102 : * : 1983 -0.01392 -0.20889 -0.19497 : * : 1984 -0.05048 -0.10983 -0.05935 : * : 1985 -0.02627 0.00000 0.02627 : * : 1986 0.00882 0.06151 0.05268 : * : 1987 0.02573 0.10397 0.07824 : * : 1988 0.03500 0.11617 0.08117 : * : 1989 0.04895 0.19584 0.14689 : * : 1990 0.00723 0.26634 0.25911 : * : 1991 -0.03464 0.33846 0.37309 Appendix 6. Cost shares (actual, fitted, residual, and residual plot) Dairy farms of group B (a representative dairy farm). S„ Residual Plot obs RESIDUAL ACTUAL FITTED : : * 1965 0.01847 0.28201 0.26354 : * : 1966 0.00622 0.23657 0.23035 : * : 1967 -0.00340 0.20315 0.20655 * : : 1968 -0.01570 0.21081 0.22651 : * : 1969 -0.00568 0.24233 0.24801 : * : 1970 -0.00293 0.22917 0.23210 : : * 1971 0.01302 0.17980 0.16678 : * : 1972 0.00428 0.17821 0.17393 : * : 1973 -0.00602 0.09425 0.10027 : * : 1974 0.00816 0.09093 0.08277 : * : 1975 -0.00215 0.10050 0.10265 : * ; 1976 -0.00857 0.15764 0.16621 : * : 1977 -0.00838 0.16127 0.16965 * ; : 1978 -0.01612 0.18870 0.20482 *: : 1979 -0.01169 0.16934 0.18103 : : * 1980 0.01314 0.14569 0.13255 : *: 1981 0.00882 0.17384 0.16502 : * : 1982 0.00185 0.17137 0.16951 : * : 1983 -0.00558 0.19241 0.19799 : * : 1984 0.00228 0.21977 0.21749 ; * : 1985 0.00645 0.23065 0.22420 : * : 1986 -0.00658 0.22709 0.23367 : * : 1987 0.00342 0.22112 0.21770 ; * : 1988 0.00098 0.19929 0.19832 : * : 1989 0.00228 0.21454 0.21226 ; * : 1990 0.00143 0.25422 0.25278 : * : 1991 0.00268 0.26634 0.26367 SL Residual Plot Obs RESIDUAL ACTUAL FITTED : * : 1965 0.00358 0.51042 0.50685 : : * 1966 0.01942 0.54115 0.52174 : *: 1967 0.00703 0.54031 0.53328 * : 1968 -0.00842 0.53167 0.54009 * : : 1969 -0.01418 0.50191 0.51609 * : : 1970 -0.01305 0.49919 0.51224 : * : 1971 0.00538 0.53481 0.52942 : * : 1972 -0.00091 0.52073 0.52164 : * : 1973 8.6E-05 0.56612 0.56603 : * : 1974 -0.00520 0.55366 0.55886 : * : 1975 0.00512 0.55606 0.55095 : * : 1976 0.00160 0.52591 0.52431 : * : 1977 -0.00088 0.51030 0.51118 : * : 1978 0.00063 0.47472 0.47409 : * : 1979 0.00067 0.48653 0.48586 * : 1980 -0.00853 0.48893 0.49746 : * : 1981 0.00504 0.46493 0.45989 * : : 1982 -0.01304 0.44113 0.45418 : * : 1983 -0.00158 0.45238 0.45396 : * : 1984 0.00208 0.44863 0.44655 : * : 1985 0.00557 0.45424 0.44867 : * : 1986 -0.00117 0.45379 0.45496 : *: 1987 0.00755 0.45806 0.45051 : * : 1988 0.00120 0.46773 0.46653 : * : 1989 0.00214 0.46676 0.46461 :* : 1990 -0.00708 0.47233 0.47941 : * : 1991 0.00573 0.48603 0.48031* Appendix 6. S H Residual Plot Obs RESIDUAL ACTUAL FITTED * : 1965 -0.01235 0.12884 0.14119 * : : 1966 -0.01844 0.13752 0.15596 : * : 1967 -0.00049 0.16616 0.16665 : : * 1968 0.01919 0.16261 0.14342 : : * 1969 0.01549 0.16274 0.14725 : * : 1970 0.00921 0.17384 0.16464 * : : 1971 -0.02032 0.18485 0.20517 : * : 1972 -0.00442 0.20226 0.20668 : * : 1973 0.00109 0.24219 0.24110 : * : 1974 0.00279 0.26537 0.26257 : * : 1975 -0.00388 0.24799 0.25187 : * : 1976 0.00258 0.22422 0.22164 : * : 1977 0.00266 0.23279 0.23013 : *: 1978 0.01205 0.23334 0.22128 : * : 1979 0.01062 0.24138 0.23076 : * : 1980 0.00248 0.26345 0.26097 : * : 1981 -0.00527 0.26433 0.26960 : *: 1982 0.01134 0.28334 0.27200 * : 1983 0.00487 0.25184 0.24697 ; * : 1984 -0.00311 0.23472 0.23783 * : : 1985 -0.01485 0.21523 0.23008 ■ * : 1986 -0.00271 0.21377 0.21648 *: : 1987 -0.01423 0.22625 0.24047 : * : 1988 0.00615 0.25092 0.24477 : * : 1989 0.00364 0.23612 0.23249 : * : 1990 0.00317 0.19985 0.19668 : * : 1991 -0.00670 0.17693 0.18363 InC Residual Plot Obs RESIDUAL ACTUAL FITTED : * : 1965 0.05515 -2.75607 -2.81122 : * : 1966 0.01685 -2.74803 -2.76487 : * : 1967 -0.02388 -2.71316 -2.68928 : * : 1968 0.01216 -2.57623 -2.58839 : * : 1969 -0.03051 -2.44444 -2.41393 : * : 1970 -0.03985 -2.33676 -2.29691 : * : 1971 -0.00439 -2.23982 -2.23542 : * : 1972 0.02705 -2.03415 -2.06120 : * : 1973 -0.01211 -1.90453 -1.89243 : * : 1974 0.02054 -1.61172 -1.63226 : * : 1975 -0.00843 -1.38202 -1.37359 : * : 1976 -0.00925 -1.11094 -1.10169 : * : 1977 -0.05117 -0.97367 -0.92251 : * : 1978 -0.03206 -0.80764 -0.77558 : * : 1979 0.01571 -0.71084 -0.72655 : * : 1980 0.02859 -0.63822 -0.66681 : * : 1981 0.02145 -0.42409 -0.44554 : * : 1982 0.04291 -0.27147 -0.31437 : * : 1983 -0.00303 -0.20589 -0.20286 : * : 1984 -0.03096 -0.09835 -0.06740 : * : 1985 -0.02841 0.00000 0.02841 : * : 1986 -0.01628 0.04484 0.06113 : * : 1987 0.00064 0.08309 0.08245 : * : 1988 0.03769 0.10306 0.06537 : * : 1989 0.03997 0.21023 0.17026 : * : 1990 -0.00280 0.26949 0.27229 : * : 1991 -0.02425 0.34210 0.36636 Appendix 6. Cost shares (actual, fitted, residual, and residual plot) Dairy farms of group A for 1976-1991 (a representative dairy farm). S K Residual Plot obs RESIDUAL ACTUAL FITTED : * : 1976 3.2E-06 0.17020 0.17020 * : 1977 -0.00482 0.17015 0.17497 : * 1978 0.00505 0.21027 0.20522 :* : 1979 -0.00438 0.18753 0.19191 : : * 1980 0.00656 0.16339 0.15683 : * : 1981 0.00044 0.18779 0.18735 : * : 1982 0.00042 0.19237 0.19195 : * : 1983 -0.00281 0.20658 0.20939 : * : 1984 0.00213 0.22569 0.22357 : * : 1985 0.00041 0.23140 0.23099 : * : 1986 -0.00157 0.23639 0.23797 : * : 1987 0.00098 0.22295 0.22197 : * : 1988 -0.00184 0.19834 0.20018 *: : 1989 -0.00538 0.21290 0.21829 : * : 1990 -0.00055 0.25687 0.25742 : :* 1991 0.00537 0.27674 0.27138 SL Residual Plot obs RESIDUAL ACTUAL FITTED : * : 1976 0.00355 0.51237 0.50882 : * : 1977 0.00049 0.50069 0.50019 : * : 1978 -0.00506 0.46932 0.47438 : * : 1979 -0.00165 0.47087 0.47252 : * : 1980 0.00320 0.48107 0.47788 : * : 1981 0.00466 0.45166 0.44700 * : : 1982 -0.00815 0.43332 0.44146 * : : 1983 -0.00759 0.44337 0.45096 : * : 1984 0.00415 0.45237 0.44822 : * : 1985 0.00156 0.45032 0.44876 : * : 1986 0.00371 0.46097 0.45726 : :* 1987 0.00681 0.45884 0.45203 : * : 1988 -0.00333 0.47174 0.47507 : * : 1989 0.00168 0.48022 0.47854 : * : 1990 9.7E-06 0.48682 0.48681 : * : 1991 -0.00406 0.48182 0.48587 S„ Residual Plot obs RESIDUAL ACTUAL FITTED : * : 1976 -0.00780 0.21554 0.22334 : * : 1977 0.00336 0.22773 0.22436 : :* 1978 0.01140 0.22504 0.21364 : * : 1979 0.00389 0.24103 0.23714 * : : 1980 -0.01787 0.25905 0.27692 : * : 1981 0.00637 0.27094 0.26457 : * : 1982 0.00197 0.26688 0.26491 : * : 1983 0.00628 0.24288 0.23659 : * : 1984 0.00325 0.22755 0.22430 : * : 1985 -0.00369 0.21653 0.22022 : * : 1986 -0.00741 0.20387 0.21128 : * : 1987 -0.00834 0.22544 0.23378 : * : 1988 -7.5E-06 0.24338 0.24338 : * : 1989 0.00417 0.22907 0.22491 : * : 1990 0.00157 0.18145 0.17988 : * : 1991 0.00287 0.17465 0.17178 Appendix 6. Cost shares (actual, fitted, residual, and residual plot) Dairy farms of group B for 1976-1991 (a representative dairy farm). Sj Residual Plot obs RESIDUAL ACTUAL FITTED : * : 1976 0.00157 0.15764 0.15607 : * | : 1977 -0.00097 0.16127 0.16224 : * : 1978 -3.3E-05 0.18870 0.18873 :* : 1979 -0.00543 0.16934 0.17477 : * : 1980 0.00266 0.14569 0.14303 : * : 1981 0.00327 0.17384 0.17057 : * : 1982 -0.00488 0.17137 0.17624 * : 1983 -0.00577 0.19241 0.19817 : * 1984 0.00571 0.21977 0.21405 : : * 1985 0.00927 0.23065 0.22138 : * ; 1986 -0.00259 0.22709 0.22967 : * : 1987 0.00408 0.22112 0.21704 ; * : 1988 -0.00247 0.19929 0.20177 : * : 1989 -0.00383 0.21454 0.21837 : * : 1990 -0.00029 0.25422 0.25451 : * : 1991 -0.00031 0.26634 0.26666 SL Residual Plot Obs RESIDUAL ACTUAL FITTED : * : 1976 0.00099 0.52591 0.52492 : * : 1977 -0.00082 0.51030 0.51112 : * : 1978 -0.00184 0.47472 0.47657 : * : 1979 0.00173 0.48653 0.48480 : * : 1980 -0.00269 0.48893 0.49162 : : * 1981 0.00958 0.46493 0.45535 * : : 1982 -0.00882 0.44113 0.44996 * : 1983 -0.00066 0.45238 0.45305 : * : 1984 0.00107 0.44863 0.44756 : * : 1985 0.00492 0.45424 0.44932 : * : 1986 -0.00249 0.45379 0.45627 : * : 1987 0.00370 0.45806 0.45436 : * : 1988 -0.00445 0.46773 0.47218 : * : 1989 -3.6E-05 0.46676 0.46679 * : : 1990 -0.00766 0.47233 0.47999 : : * 1991 0.00748 0.48603 0.47855 S„ Residual Plot obs RESIDUAL ACTUAL FITTED : * : 1976 -0.00302 0.22422 0.22724 : * : 1977 -0.00052 0.23279 0.23331 : * : 1978 0.00515 0.23334 0.22819 : * : 1979 0.00455 0.24138 0.23683 : * : 1980 -0.00031 0.26345 0.26376 : * : 1981 -0.00714 0.26433 0.27147 : * 1982 0.00988 0.28334 0.27346 : * : 1983 0.00397 0.25184 0.24787 : * : 1984 -0.00310 0.23472 0.23782 * : : 1985 -0.01382 0.21523 0.22905 : * : 1986 -0.00055 0.21377 0.21432 * : : 1987 -0.01246 0.22625 0.23871 : * 1988 0.00972 0.25092 0.24119 : * : 1989 0.00412 0.23612 0.23200 : * : 1990 0.00504 0.19985 0.19481 : * : 1991 -0.00150 0.17693 0.17843 Appendix 6. Autocorrelation functions of the level series Dairy farms of group A Series Lags lnWK 0.871 0.721 0.566 0.476 0.407 InW, 0.903 0.802 0.696 0.589 0.476 lnW M 0.914 0.817 0.719 0.620 0.515 lnWH 0.913 0.824 0.730 0.638 0.533 lnY 0.869 0.717 0.559 0.469 0.400 In C 0.914 0.818 0.714 0.611 0.503 S K 0.699 0.376 0.141 0.046 -0.019 S, 0.841 0.581 0.339 0.161 0.066 S M 0.787 0.534 0.413 0.353 0.289 S„ 0.509 0.181 0.072 -0.089 -0.167 Dairy farms of group B Series Lags lnWK 0.869 0.717 0.559 0.469 0.400 InW, 0.903 0.802 0.697 0.589 0.476 lnW M 0.914 0.817 0.719 0.620 0.515 lnWH 0.913 0.824 0.730 0.638 0.533 In Y 0.908 0.804 0.687 0.563 0.446 In C 0.914 0.818 0.713 0.607 0.497 S K 0.709 0.433 0.243 0.153 0.063 S, 0.862 0.726 0.551 0.443 0.338 S M 0.787 0.528 0.371 0.280 0.206 S„ 0.657 0.351 0.191 -0.028 -0.179 Autocorrelation functions of the first differences Dairy farms of group A Series Lags lnWK 0.095 0.065 -0.383 -0.190 -0.120 lnWL 0.696 0.521 0.225 -0.032 -0.134 lnWM 0.521 0.095 -0.148 -0.276 -0.374 lnWH 0.603 0.147 -0.115 -0.146 -0.090 lnY 0.093 0.069 -0.386 -0.184 -0.124 In C 0.482 0.298 0.131 0.133 -0.008 S K 0.173 0.008 -0.316 -0.221 -0.163 SL 0.319 -0.068 -0.208 -0.267 -0.244 S M 0.254 -0.093 -0.236 -0.136 0.047 S H -0.131 -0.279 0.192 -0.033 -0.168 Dairy farms of group B Series Lags lnWK 0.093 0.069 -0.386 -0.184 -0.124 lnWL 0.738 0.545 0.272 -0.005 -0.153 lnWM 0.521 0.095 -0.148 -0.276 -0.374 lnWH 0.603 0.147 -0.115 -0.146 -0.090 lnY 0.151 0.142 -0.009 0.035 0.096 In C 0.476 0.383 0.237 0.191 0.001 S K 0.167 0.018 -0.250 -0.185 -0.094 SL -0.075 0.136 -0.242 0.013 -0.089 S M 0.364 -0.081 -0.287 -0.188 -0.011 S H 0.084 -0.057 0.231 0.130 -0.111 Autocorrelation functions of the second differences Dairy farms of group A Scries Lags InW, -0.230 0.214 -0.087 -0.240 0.005 In C -0.325 0.090 -0.215 0.072 -0.131 Dairy farms of group B Series Lags InW, -0.143 0.158 -0.016 -0.221 -0.126 In C -0.440 0.159 -0.139 0.141 -0.184 7Appendix Autoregressive processes of dairy farm series (p-values in parenthesis) Dairy farms of group A Series Intercept AR(1) AR(2) AR(3) AR(4) AR(S) lnW K 3.199 1.036 0.069 -0.497 0.234 0.133 (0.88) (0.00) (0.85) (0.15) (0.52) (0.62) lnWL 2.175 1.350 -0.052 -0.441 -0.140 0.255 (0.07) (0.00) (0.90) (0.26) (0.73) (0.24) lnWM 1.993 1.585 -0.778 0.095 -0.140 0.104 (0.51) (0.00) (0.90) (0.26) (0.73) (0.24) lnWH 1.063 2.019 -1.628 0.653 -0.122 0.052 (0.59) (0.00) (0.01) (0.32) (0.83) (0.84) InY 0.370 0.992 -0.054 -0.092 0.239 -0.143 (0.37) (0.00) (0.86) (0.73) (0.42) (0.48) In C 1.235 1.371 -0.265 -0.311 0.338 -0.161 (0.47) (0.00) (0.52) (0.43) (0.41) (0.49) SK 0.192 0.952 -0.009 -0.427 0.093 0.039 (0.00) (0.00) (0.98) (0.20) (0.79) (0.81) SL 0.493 1.199 -0.421 0.057 -0.118 0.090 (0.00) (0.00) (0.29) (0.89) (0.76) (0.72) SM 0.227 1.064 -0.343 -0.264 0.161 0.097 (0.00) (0.00) (0.40) (0.51) (0.68) (0.71) SH 0.092 0.789 -0.227 0.306 -0.250 -0.047 (0.00) (0.01) (0.50) (0.39) (0.47) (0.89) Dairy farms of group B Series Intercept AR(1) AR(2) AR(3) AR(4) AR(S) lnWK 2.729 1.036 0.071 -0.504 0.242 0.126 (0.87) (0.00) (0.84) (0.15) (0.50) (0.63) lnWL 2.105 1.377 -0.089 -0.406 -0.183 0.272 (0.05) (0.00) (0.82) (0.32) (0.65) (0.19) lnWM 1.993 1.585 -0.778 0.095 -0.140 0.104 (0.51) (0.00) (0.90) (0.26) (0.73) (0.24) lnWH 1.063 2.019 -1.628 0.653 -0.122 0.052 (0.59) (0.00) (0.01) (0.32) (0.83) (0.84) InY 0.289 0.573 0.276 -0.083 0.052 0.047 (0.04) (0.03) (0.27) (0.73) (0.85) (0.78) In C 0.946 1.153 0.009 -0.243 0.182 -0.138 (0.42) (0.00) (0.98) (0.52) (0.64) (0.55) SK 0.184 0.964 0.012 -0.347 0.019 0.072 (0.00) (0.00) (0.97) (0.31) (0.96) (0.76) SL 0.490 0.851 0.227 -0.292 0.098 -0.029 (0.00) (0.00) (0.49) (0.36) (0.76) (0.90) SM 0.234 1.152 -0.489 -0.206 0.153 0.077 (0.00) (0.00) (0.21) (0.60) (0.68) (0.74) S H 0.094 0.940 -0.156 0.298 -0.242 -0.257 (0.00) (0.00) (0.66) (0.40) (0.55) (0.37) Appendix 8. Autoregressive processes of the dairy farms residuals (p-values in parenthesis) Dairy farms of group A and B Series Intercept AR(I) AR(2) AR(3) AR(4) AR(S) A:eL 0.004 0.316 0.210 -0.003 -0.311 -0.056 (0.69) (0.24) (0.42) (0.99) (0.24) (0.82) A:e M 0.003 0.220 -0.333 -0.074 -0.268 -0.179 (0.68) (0.43) (0.24) (0.80) (0.34) (0.52) A:e H -0.002 0.508 -0.227 -0.227 -0.175 0.051 (0.90) (0.11) (0.50) (0.48) (0.58) (0.89) B:e, 0.005 0.333 0.160 0.026 -0.227 -0.162 (0.63) (0.21) (0.54) (0.92) (0.39) (0.51) B:e M 0.003 0.219 -0.324 -0.073 -0.264 -0.181 (0.71) (0.45) (0.26) (0.80) (0.35) (0.52) B:e H -0.002 0.527 -0.254 -0.237 -0.139 0.018 (0.89) (0.10) (0.45) (0.46) (0.66) (0.96) The number of dairy farms of group B in research period 1965 228 1966 243 1967 187 1968 335 1969 262 1970 181 1971 168 1972 192 1973 190 1974 188 1975 211 1976 270 1977 256 1978 300 1979 314 1980 304 1981 332 1982 322 1983 364 1984 349 1985 346 1986 321 1987 362 1988 324 1989 305 1990 303 1991 338 8 ALLEN PARTIAL ELASTICITIES OF SUBSTITUTION Dairy farms of group A. KK LL MM HH 1965 -0.89291 -0.98550 -4.85716 -9.34119 1970 -0.87798 -0.89578 -4.58997 -10.7246 1976 -0.49661 -0.86427 -3.32599 -10.7974 1981 -0.48219 -1.12751 -2.45678 -9.67576 1986 -0.87341 -1.08767 -3.27433 -11.3585 1991 -0.89257 -0.96006 —4.11636 -15.4469 Dairy farms of group A for 1976-1991. KK LL MM HH 1976 -1.21762 -0.73976 -2.25706 -9.24222 1981 -1.31880 -0.94486 -1.91004 -8.89364 1986 -1.33111 -0.90763 -2.36934 -9.69690 1991 -1.24611 -0.81078 -2.75842 -13.0914 Dairy farms of group B. KK LL MM HH 1965 -0.84994 -0.60582 -0.53923 -7.37109 1970 -0.80145 -0.59275 -0.99709 -7.21351 1976 -0.12775 -0.56417 -1.26230 -7.40683 1981 -0.10037 -0.72848 -1.18884 -6.41507 1986 -0.80608 -0.74232 -1.26132 -6.98701 1991 -0.84999 -0.67315 -1.16856 -8.42997 Dairy farms of group B for 1976-1991. KK LL MM HH 1976 -1.33945 -0.58133 -1.42373 -5.49331 1981 -1.45699 -0.76591 -1.29845 -5.22339 1986 -1.47560 -0.76322 -1.44349 -5.29820 1991 -1.35662 -0.70015 -1.39803 -5.73572 Dairy farms of group A. KL KM KH LM LH MH 1965 0.641845 -0.96301 0.661033 1.452616 0.850707 2.653373 1970 0.629606 -1.01820 0.586883 1.408514 0.840269 2.758616 1976 0.512788 -1.02245 0.443851 1.300521 0.842143 2.324046 1981 0.437403 -0.59158 0.489917 1.270025 0.834693 1.944933 1986 0.585519 -0.50826 0.561388 1.333820 0.814566 2.363460 1991 0.652620 -0.65175 0.491184 1.382210 0.775098 3.160796 Dairy farms of group A for 1976-1991. KL KM KH LM LH MH 1976 0.704391 -1.70674 2.355852 1.139440 0.020957 2.199986 1981 0.694311 -1.07576 2.189818 1.133989 -0.07651 1.978508 1986 0.764736 -1.04640 2.012773 1.164020 -0.13779 2.324789 1991 0.805847 -1.20710 2.169895 1.189857 -0.41058 3.146461 Dairy farms of group B. KL KM KH LM LH MH 1965 0.662031 -0.96032 0.272171 0.435922 0.803640 1.224446 1970 0.620288 -0.90883 0.197187 0.521355 0.811256 1.186980 1976 0.481967 -0.98003 -0.16165 0.652635 0.808925 1.143922 1981 0.405144 -0.63954 0.025733 0.674427 0.818608 1.098522 1986 0.575355 -0.44197 0.235103 0.590144 0.796160 1.136404 1991 0.643532 -0.50651 0.111440 0.542325 0.746905 1.210788 Appendix 9. Dairy farms of group B for 1976-1991. KL KM KH LM LH MH 1976 0.553123 -0.85448 1.227613 0.796533 0.412520 0.423127 1981 0.528641 -0.42037 1.186262 0.803662 0.394308 0.568128 1986 0.650639 -0.33616 1.142312 0.751809 0.378136 0.437226 1991 0.713110 -0.38229 1.160100 0.715766 0.225550 0.117058 OWN PRICE ELASTICITIES Dairy farms of group A. KK LL MM HH 1965 -0.23509 -0.47532 -0.72566 -0.98073 1970 -0.21298 -0.45344 -0.72393 -1.00275 1976 -0.08999 -0.44521 -0.70072 -1.00383 1981 -0.08689 -0.50583 -0.66143 -0.98634 1986 -0.20943 -0.49774 -0.69907 -1.01193 1991 -0.23905 -0.46934 -0.71851 -1.06213 Dairy farms of group A for 1976-1991. KK LL MM HH 1976 -0.20723 -0.37640 -0.50409 -0.90241 1981 -0.24707 -0.42235 -0.50533 -0.89896 1986 -0.31676 -0.41502 -0.50059 -0.90656 1991 -0.33817 -0.39393 -0.47384 -0.92910 Dairy farms of group B. KK LL MM HH 1965 -0.22399 -0.30706 -0.07613 -0.65175 1970 -0.18601 -0.30363 -0.16416 -0.65657 1976 -0.02123 -0.29580 -0.27977 -0.65061 1981 -0.01656 -0.33502 -0.32051 -0.67672 1986 -0.18835 -0.33772 -0.27305 -0.66299 1991 -0.22411 -0.32332 -0.21458 -0.61024 Dairy farms of group B for 1976-1991. KK LL MM HH 1976 -0.20904 -0.30515 -0.32353 -0.50412 1981 -0.24851 -0.34876 -0.35249 -0.53597 1986 -0.33890 -0.34823 -0.30936 -0.52844 1991 -0.36175 -0.33505 -0.24945 -0.43797 CROSS PRICE ELASTICITIES Dairy farms of group A. KL LK KM MK KH HK 1965 0.309574 0.168991 -0.14387 -0.25355 0.069401 0.174043 1970 0.318706 0.152729 -0.16059 -0.24699 0.054873 0.142366 1976 0.264152 0.092927 -0.21541 -0.18528 0.041264 0.080434 1981 0.196232 0.078820 -0.15927 -0.10660 0.049942 0.088283 1986 0.267945 0.140401 -0.10851 -0.12187 0.050014 0.134615 1991 0.319046 0.174784 -0.11376 -0.17455 0.033773 0.131548 LM ML LH HL MH HM 1965 0.217020 0.700625 0.089315 0.410313 0.278577 0.396414 1970 0.222150 0.712989 0.078565 0.425344 0.257930 0.435088 1976 0.273993 0.669937 0.078294 0.433813 0.216066 0.489630 1981 0.341928 0.569771 0.085088 0.374468 0.198266 0.523634 1986 0.284770 0.610382 0.072569 0.372761 0.210560 0.504598 1991 0.241264 0.675721 0.053295 0.378922 0.217336 0.551716 Appendix 9. Dairy farms of group A for 1976-1991. KL LK KM MK KH HK 1976 0.358408 0.119887 -0.38118 -0.29048 0.230025 0.400966 1981 0.310357 0.130079 -0.28461 -0.20154 0.221346 0.410262 1986 0.349683 0.181984 -0.22108 -0.24901 0.188174 0.478979 1991 0.391537 0.218691 -0.20735 -0.32758 0.153997 0.588866 LM ML LH HL MH HM 1976 0.254482 0.579770 0.002046 0.010663 0.214806 0.491344 1981 0.300019 0.506893 -0.00773 -0.03420 0.199987 0.523454 1986 0.245934 0.532260 -0.01288 -0.06300 0.217344 0.491181 1991 0.204393 0.578115 -0.02913 -0.19949 0.223304 0.540499 Dairy farms of group B. KL LK KM MK KH HK 1965 0.335550 0.174471 -0.13558 -0.25308 0.024065 0.071728 1970 0.317736 0.143969 -0.14963 -0.21094 0.017948 0.045767 1976 0.252700 0.080107 -0.21721 -0.16289 -0.01419 -0.02686 1981 0.186321 0.066856 -0.17242 -0.10553 0.002714 0.004246 1986 0.261763 0.134443 -0.09567 -0.10327 0.022309 0.054936 1991 0.309095 0.169680 -0.09301 -0.13355 0.008067 0.029383 LM ML LH HI. MH HM 1965 0.061547 0.220947 0.071057 0.407325 0.108265 0.172879 1970 0.085835 0.267058 0.073840 0.415557 0.108038 0.195424 1976 0.144650 0.342183 0.071056 0.424127 0.100482 0.253538 1981 0.181825 0.310162 0.086354 0.376469 0.115883 0.296161 1986 0.127754 0.268491 0.075547 0.362221 0.107833 0.246008 1991 0.099587 0.260484 0.054068 0.358746 0.087648 0.222337 Dairy farms of group B for 1976-1991. KL LK KM MK KH HK 1976 0.290345 0.086326 -0.19417 -0.13335 0.112658 0.191593 1981 0.240716 0.090170 -0.11411 -0.07170 0.121722 0.202340 1986 0.296867 0.149432 -0.07204 -0.07720 0.113934 0.262354 1991 0.341259 0.190158 -0.06821 -0.10194 0.088585 0.309352 LM ML LH HL MH HM 1976 0.181004 0.418116 0.037857 0.216540 0.038830 0.096151 1981 0.218170 0.365947 0.040460 0.179548 0.058295 0.154229 1986 0.161127 0.343028 0.037715 0.172532 0.043608 0.093706 1991 0.127714 0.342530 0.017223 0.107937 0.008938 0.020886 MORISHIMA PARTIAL ELASTICITIES OF SUBSTITUTION Dairy farms of group A. KL LK KM MK KH HK 1965 0.7849030.404087 0.581786 -0,01845 1.0501340.409139 1970 0.7721530.365710 0.563339 -0.03401 1.057630 0.355347 1976 0.7093680.182924 0.485310 -0.09529 1.045100 0.170432 1981 0.7020700.165712 0.502166 -0.01971 1.036289 0.175175 1986 0.7656870.349838 0.5905550.087558 1.0619450.344051 1991 0.7883920.413835 0.6047480.064497 1.0959030.370599 LM ML LH HL MH HM 1965 0.9426811.175954 1.0700480.885641 1.2593101.122074 1970 0.9460811.166436 1.0813210.878790 1.2606871.159020 1976 0.9747151.115153 1.0821290.879029 1.2199021.190351 1981 1.0033671.075609 1.0714350.880306 1.1846131.185073 1986 0.9838421.108125 1.0845010.870504 1.2224921.203670 1991 0.9597771.145067 1.1154250.848268 1.2794661.270229 Appendix 9. Dairy farms of group A for 1976-1991. KL LK KM MK KH HK 1976 0.7348170.327126 0.122909 0.7327130.377157 0.220723 0.7647100.498748 0.279509 0.7854740.556862 0.266486 -0.08324 0.045533 0,067750 0.010588 HL 1.1324360.608205 1.1203160.657341 1.0947370.795744 1.0830970.927038 1981 1986 1991 LM ML LH 0.7585760.956178 0.904457 0.8053590.929249 0.891235 0.7465290.947287 0.893680 0.6782350.972053 0.899961 MH HM 1976 0.387072 0.388152 0.352017 0.194447 1.1172170.995438 1.0989571.028793 1.1239080.991776 1.1524041.014341 1981 1986 1991 Dairy farms of group B. KL LK KM 0.6426140.398466 -0.05945 0.6213680.329986 0,014531 0.5485000.101341 0.062561 0.5213430.083421 0.148090 0.5994930.322800 0.177373 0.6324160.393798 0.121571 MK KH HK 1965 -0.02908 -0.02492 -0.14165 -0.08897 0.085080 0.090563 HL 0.6758170.295723 0.6745210.231784 0.636416 -0.00563 0.6794400.020810 0.6853070.243293 0.6183120.253501 1970 1976 1981 1986 1991 LM ML LH 0.1376820.528010 0.722809 0.2499980.570690 0.730414 0.4244260.637983 0.721672 0.5023360.645184 0.763081 0.4008060.606221 0.738545 0.3141700.583805 0.664314 MH HM 1965 0.714388 0.719189 0.719928 0.711491 0.699950 0.682067 0.7600170.249013 0.7646120.359586 0.7510980.533315 0.7926090.616673 0.7708310.519061 0.6978940.436920 1970 1976 1981 1986 1991 Dairy farms of group B for 1976-1991. KL LK KM MK KH HK 1976 0.5954980.295374 0.129356 0.5894770.338689 0.238372 0.6451030.488334 0.237321 0.6763160.551916 0,181238 0.075688 0.176815 0.261694 0.259814 HL 0.6167790.400641 0.6576950.450859 0.6423770.601257 0.5265650.671110 1981 1986 1991 LM ML LH 0.5045340.723269 0.541978 0.5706620.714708 0.576432 0.4704960.691264 0.566158 0.3771660.677588 0.455202 MH HM 1976 0.521693 0.528309 0.520768 0.442994 0.5429520.419681 0.5942680.506722 0.5720520.403075 0.4469180.270338 1981 1986 1991 SHADOW ELASTICITIES OF SUBSTITUTION Dairy farms of group A. KL KM KH 0.5385610.364489 0.867398 0,497384 0.3279780.862249 0.3199270.173186 0.748525 0.3194130.189536 0.725160 0.4928190.353643 0.867476 0.5464040.391576 0.947731 LM LH MH 1965 0.997849 0.998428 1.015480 1.030461 1.023379 1.008528 1.0370821.202671 1.0497441.222848 1.0510781.210854 1.0360471.184740 1.0496291.216950 1.0824831.276855 1970 1976 1981 1986 1991 Dairy farms of group A for 1976-1991. KL KM KH 0.429316 0.005911 0.941330 0.482168 0.118161 0.958067 0.589784 0.179920 1.010405 0.638791 0.167293 1.050746 LM LH MH 1976 0.818853 0.851422 0.809975 0.754981 0.8211581.080173 0.7984531.079561 0.8017331.083376 0.8100421.112040 1981 1986 1991 Appendix 9. Dairy farms of group B. KL KM KH 0.481986 -0.04886 0.580329 0.420844 -0.00184 0.549806 0.208973 -0.05414 0.414421 0.1990630.001037 0.422596 0.4166890.132989 0.557651 0.4783650.108842 0.539729 Dairy farms of group B for 1976-1991. KL KM KH 0.364157 0.097540 0.536748 0.407031 0.200568 0,580005 0.540824 0.249086 0.629927 0.596430 0.212738 0.558742 LM 0.222723 0.328001 0,487879 0.555129 0.467034 0.388745 LM 0.570618 0.624464 0.541053 0.458758 LH MH 0.7215580.563236 0.7287200.620415 0.7214220.689284 0.7534550.743129 0.7318850.694104 0,666639 0.624103 LH MH 0.5389600.507490 0.5675820.570254 0.5580160.518388 0.4535230.393998 Appendix 9. Estimated parameters solved from the equation system of the cost function and the cost share equa- tions on dairy farms of group A from the panel data set (t values in parenthesis). Pkk 0.1423 P KL -0.0614 Pkm -0.0562 Pkh -0.0247 (16.48) (6.657) (5.247) (4.837) Pu. 0.0689 Plm 0.0320 PLH -0.0395 Pmm 0.0059 (3.862) (1.792) (3.906) (0.235) Pmh P HH 0.0460 Po PK 0.18180.0182 (1.682) (4.771) (5.739) Pl 0.5527 Pm 0.3953 P„ -0.1299 pY 0.5519 (10.12) (6.169) (2.327) (10.47) P YK 0.0166Pyy 0.0125 PYL -0.1283 PYM 0.0973 (0.268) (2.418) (16.41) (13.68) Pyh 0.0143 Px 0.0063 P„ -0.0013 PTY 0.0011 (3.877) (1.323) (4.571) (0.366) Ptk 0.0004 Ptl 0.0018 PTM -0.0030 PXH 0.0008 (0,841) (2.117) (3.765) (1.712) yK4 0.0068 yK_ -0.0132 Yl+ 0.0188 Yl_ -0.0158 (1.345) (2.588) (3.342) (2.814) Y Mt -0.0187 Y„_ 0.0268 y Ht -0.0069 Yh_ 0.0022 (3.752) (5.356) (2.676) (0.869) Yk, 0.1134 Yu -0.0231 Ymi —0.0570 Yh. -0.0333 (7.946) (1.557) (3.696) (4.417) yCI 0.0102 (0.264) Estimated parameters of the cost share equation system on dairy farms of group A from the panel data set for the post energy crisis period (t values in parenthesis). Pkk 0.1029 P KL -0.0216 P Km -0.0834 PKH 0.0021 (7.884) (1.691) (6.032) (0.283) Pll 0.0901 P UM -0.0080 PI H -0.0605 Pmm 0.0530 (3.901) (0.344) (5.220) (1.664) Pmh 0.0383 Phh 0.0201 Pk p. (2.721) (1.655) Pm Ph Pyk 0.0656 PYL -0.1648 (5.258) (12.18) Pym 0.0990 PYH 0.0001 P 1K PXL 0.00050.0016 (7.911) (0.020) (2.298) (0.571) PXM -0.0021 p XH -0.0001 (2.166) (0.136) Yk+ 0.0106 Yk_ -0.0086 Yl+ 0.0164 Yl- -0.0175 (2.329) (1.895) (3.306) (3.588) Ym* -0.0196 Ym- 0.0246 Yh+ -0.0073 Yh- 0.0015 (4,269) (5.418) (2.958) (0.632) Appendix 10. Derived demand elasticities on dairy farms of group A from the panel data set for the whole research period. M HK L K -0.1000.083 -0.082 -0.053 L 0.203 -0.3600.667 0.091 M -0.0780.260 -0.7740.386 H -0.0250.017 0.189 -0.424 Derived demand elasticities on dairy farms of group A from the panel data set for the post energy crisis period. K L M H K -0.2960.163 -0.1900.229 L 0.383 -0.3280.450 -0.149 M -0.1920.194 -0.5380.613 H 0.105 -0.0290.278 -0.693 Morishima partial elasticities of substitution on dairy farms of group A from the panel data set for the whole research period. K L M H K - 0,183 0.0190.047 L 0.563 - 1.0260.451 M 0.6961.033 - 1.160 H 0.3990.441 0.613 Morishima partial elasticities of substitution on dairy farms of group A from the panel data set for the post energy crisis period. K L M H K - 0.4590.107 0.525 L 0.711 - 0.7770.179 M 0.3450.731 - 1.151 H 0.7990.664 0.971 Shadow elasticities ofsubstitution on dairy farms of group A from the panel data set for the whole research period in the lower triangle, and for the post energy crisis period in the upper triangle. K L M H K - 0.5340.225 0.713 L 0.293 - 0.7450.585 M 0.3641.031 - 1.027 H 0.2870.443 0.792 4 Appendix 11.