Bio-based and Applied Economics 8(1): 3-19, 2019 ISSN 2280-6180 (print) © Firenze University Press ISSN 2280-6172 (online) www.fupress.com/bae Full Research Article DOI: 10.13128/bae-8144 How did farmers act? Ex-post validation of linear and positive mathematical programming approaches for farm- level models implemented in an agent-based agricultural sector model Gabriele Mack*, ali Ferjani, anke MöhrinG, albert von ow, SteFan Mann Agroscope, Socioeconomics Research Group, 8356 Ettenhausen, Switzerland Abstract. This study evaluates linear programming (LP) and positive mathemati- cal programming (PMP) approaches for 3,400 farm-level models implemented in the SWISSland agent-based agricultural sector model. To overcome limitations of PMP regarding the modelling of investment decisions, we further investigated whether the forecasting performance of farm-level models could be improved by applying LP to animal production activities only, where investment in new sectors plays a major role, while applying PMP to crop production activities. The database used is the Swiss Farm Accountancy Data Network. Ex-post evaluation was performed for the period from 2005 to 2012, with the 2003-2005 three-year average as a base year. We found that PMP applied to crop production activities improves the forecasting performance of farm-level models compared to LP. Combining PMP for crop production activi- ties with LP for modelling investment decisions in new livestock sectors improves the forecasting performance compared to PMP for both crop and animal production activities, especially in the medium and long term. For short-term forecasts, PMP for all production activities and PMP combined with LP for animal production activities produce similar results. Keywords. Agent-based sector model, farm-level model, linear programming, posi- tive mathematical programming, ex-post validation. JEL codes. C61, Q18, Q19. 1. Introduction Agricultural policy models apply either linear programming (LP) or positive math- ematical programming (PMP) approaches to analyse the impact of policy changes. The main advantages of PMP models over conventional LP models are that they guarantee exact calibration to the base year and avoid predicting overspecialisation without adding weakly justified constraints to the model formulation (Kanellopoulos et al., 2010). Further *Corresponding author: Gabriele.mack@agroscope.admin.ch 4 G. Mack et alii advantages of PMP models are that they do not require large datasets and can be viewed as a bridge between econometric models, with substantial data requirements, and more limited LP models (Heckelei and Britz, 2005; Howitt et al., 2012). Studies evaluating the practice of PMP more than 15 years after Howitt published the first paper on this subject in 1995 show that PMP has become very popular in aggregated policy-decision support models (Garnache et al., 2015; Heckelei et al., 2012). The popular- ity of PMP is underscored by the fact that the majority of both European and non-Euro- pean aggregated sector models1 have used it for the calibration of crop and animal pro- duction since 2000. However, PMP is much less popular in farm-level models. One reason for the limited use of PMP in this context is that farm-level models generally only take into account the activities observed during the reference period, even though new policies and market con- ditions allow farmers to undertake new production activities. To date, only a few farm- level models have used PMP to calibrate the crop activities of arable farms (Iglesias et al., 2008; Kanellopoulos et al., 2010) or both animal and crop production activities of dairy- farm models (Buysse et al., 2007, Louhichi et al., 2010). Iglesias et al. (2008) extended the PMP approach by incorporating new irrigation technologies for crop production activities in farm-level models using PMP. Farm-level models implemented in agent-based models, which use mathematical pro- gramming methods to determine the production decisions of the farm agents (Happe, 2004; Röder and Kantelhardt, 2009; Lobianco and Esposti, 2010; Schreinemachers et al., 2011), also prefer an LP approach over PMP. To our knowledge, there have been, to this point, no farm-level models implemented in agent-based models which use PMP. The aim of this study is to assess the best mathematical programming approach for farm-level models implemented in the SWISSland2 agent-based agricultural sector model on an empirical basis, i.e. going beyond theoretical considerations. We analysed the fore- casting performance of a linear optimisation approach compared to a PMP approach. Because there is no single PMP approach in practice, but several different mathematical versions of PMP which all influence the forecasting performance of farm-level models, this study reviewed the most frequently used approaches for application in single farm models. To overcome limitations of the PMP approach regarding the modelling of invest- ment decisions, we further investigated whether the forecasting performance of farm- level models could be improved by applying LP only to those production activities where investment in new sectors plays a major role. This is why we also validated an approach which combines PMP for crop production activities and LP for animal production activi- ties. The ex-post evaluation was carried out for the period from 2005 to 2012, with the 2003-2005 three-year average as a base year. Over this period, Swiss agricultural policy changed decisively, particularly for milk and meat production. To cite an example, Swit- 1 Examples of PMP-based, aggregated models representing either farm-type groups or whole regions are the Ger- man FARMIS model (Offermann et al., 2005), the Italian FIPIM model (Arfini et al., 2011), the Spanish PRO- MAPA model (Júdez et al., 2008), the European CAPRI-FARM model (Gocht and Britz, 2011), the Swiss SILAS model (Mann et al., 2003), the German-Austrian Glowa-Danubia Decision-Support System model (Winter, 2005), the European CAPRI-REG model (Britz and Witzke, 2014), the Dutch DRAM model (Helming, 2005), the USDA REAP model (Johansson et al., 2007), the California SWAP model (Howitt et al., 2012) and the New Zealand model NZFARM (Daigneault et al., 2014). 2 SWISSland’ is the German acronym for ‘Structural Change Information System Switzerland. 5How did farmers act? zerland concluded a free-trade agreement for cheese with the EU in 2007. The same year saw the country’s gradual withdrawal from the milk quota system (Flury et al., 2005), as well as the introduction of direct payments for dairy cows. Section 2 of this paper gives a brief overview of an LP approach for single farm optimisation models and describes the most relevant PMP versions considered for the evaluation. Section 3 gives an overview of the SWISSland agent-based sector model and describes the different PMP and LP modelling options tested for the 3,400 single farm models implemented in the SWISSland model for the ex-post period from 2005 to 2012. By drawing a comparison with the historical pathway, Section  4 illustrates the forecasting performance of the single farm models at the farm and sectoral scales, and Section 5 pro- vides conclusions as to how PMP and LP could be used in farm-based modelling. 2. Overview of LP and PMP approaches Mathematical programming has been used in agricultural economics for more than fifty years. Mathematical programming starts from a decision rule of the decision maker, which determines the levels of the different variables when aiming to optimise the objec- tive set by the decision maker (Hazell and Norton, 1986). Mathematical programming applied to farm models maximises the farm profit. max Z =∑ipixi – cixi (1a) subject to: ∑iAwixi ≤ Bw and xi ≥ 0 (1b) In Equation 1a, parameter Z denotes the farm profit to be maximised, p is the vector of product prices, c is the vector of variable costs, x is the vector of production levels and i is the index for the production activities. The optimal solution must fulfil the constraints in Equation 1b, where Bw is the available quantity of resource endowments w, and A is the demand of resource endowments of one unit of x. Mathematical programming models assuming constant marginal costs in the objective function became generally known as LP models. A main disadvantage of LP models is a tendency to overspecialise in crop produc- tion (Howitt, 1995). This was the main reason why Howitt (1995) developed models based on the PMP technique. Howitt et al. (2012; 245) describe PMP as a ‘deductive approach to simulating the effects of policy changes on cropping patterns at the extensive and inten- sive margins. The term positive implies the use of observed data as part of the model cali- bration process’. PMP models use information contained in shadow values of an LP model which is bound to observed activity levels by calibration constraints (Step 1). Based on these shad- ow values, a non-linear objective function is specified such that observed activity levels are reproduced by the optimal solution of the new programming problem without bounds (Step 2). Many PMP-models use a quadratic, decreasing marginal gross margin function (Equation 2) that assumes increasing marginal costs in the objective function, whilst returns to scale remain constant. This functional form was proposed by Howitt (1995) because of increasing variable costs per unit of production due to inadequate machinery 6 G. Mack et alii and management capacity, and due to decreasing yields related to land heterogeneity. max Z = ∑ipixi – dixi – 1/2 xiQiixi (2) Q revenue x ii ii i = ∗ 1 ρ * * (3) di = ci – λi – Qiixi* (4) In Equation 2, parameter di denotes the vector of the linear term for crop and animal production activity i of the quadratic objective function, whilst Qii denotes the symmetric, positive (semi-) definite matrix of the quadratic cost term. Most PMP models estimate the matrix coefficients Qii and di of the quadratic cost terms based on exogenous supply elas- ticities ρii from the literature, according to Equation 3. In Equation 3, the parameter rev- enue* denotes the observed revenues from product sales in the base year and parameter xi* denotes the production levels of the base year. To determine the coefficients di and Qii (Equations 3 and 4), the shadow values λi of the calibration constraints for both marginal and preferential activities need to be recovered from the primal LP model described in Equation 1. In ‘standard’ PMP the cost functions are estimated for each production activ- ity xi separately, whilst Röhm et al. (2003) consider the elasticity of substitution among interrelated crops. Because, in ‘standard’ PMP, increasing marginal costs are only assumed for preferen- tial activities whilst constant costs are applied for marginal activities, PMP has often been criticised for its arbitrary assumptions (Howitt et al., 2012; Kanellopoulos et al., 2010). Thus, two PMP versions (Howitt et al., 2012) have been developed to overcome these lim- itations. The first PMP version, the ‘extended PMP variant’, was published by Kanellopou- los et al. (2010). It solves this problem by estimating a Q matrix for either marginal or preferential activities by using exogenous land rents β in the linear objective function for the available area y according to Equation 5: max Z =∑ipixi – cixi – β * y (5) Another variant of PMP was proposed by Paris and Howitt (1998). This variant esti- mates the resource and calibration constraint shadow values based on maximum entropy (ME). 3. Methods and database 3.1 Overview of the agent-based sector model SWISSland The agent-based SWISSland model depicts 3,400 farms from the Swiss Farm Account- ancy Data Network [FADN] data pool as realistically as possible in terms of their opera- tional and cost structures, as well as their social behaviour, as a representative sample of the estimated 50,000 family farms in Switzerland. The key objects of the model are agents representing FADN farms. For each farm, we model production and investment decisions, farm takeover and farm exit decisions, as well as lease decisions for land plots and inter- 7How did farmers act? action among agents on the land market (Table 1, categorised according to An [2012]). Table 1 also lists the various data sources and the methods we use for modelling the deci- sion-making of the agents. For the modelling of lease decisions, a spatial structure of representative reference municipalities was implemented in the model (Mack et al., 2013). This allows the farms to interact on the land market. These interactions are only possible within the lease regions and with (constructed) neighbouring agents, however. A lease algorithm enables the plot- by-plot allocation of exiting farms’ land to the remaining farms operating in the imme- diate vicinity. A plot-by-plot bidding process models which neighbouring agent receives the freed-up land and at what lease price. The neighbouring agent achieving the highest expected increase in income with the lease of the plot receives the lease plot. Exiting farms are those where the farm manager is not passing on the farm to a suc- cessor, or where the potential successor decides against farm takeover on economic grounds. Two income parameters, (1) household income per farm and (2) agricultural income per total labour input, were selected to model farm takeover decisions. Income criteria to model farm exits and farm entries were derived from the regional income levels in the previous period from 2005 to 2012. A detailed description of the different modules of the SWISSland model can be found in Möhring et al. (2016). Because this paper focuses on the modelling of production and investment decisions, we present this issue in detail in Section 3.2. The model simulates a forecast period of up to thirty calendar years, corresponding more or less to a generational cycle of the farming family. The adaptive reactions of the individual agents and their behaviour when interacting with other agents are depicted in annual steps. Table 1. Behavioural and decision submodels included in the SWISSland agent-based sector model and data collection sources. Submodels Behaviour Data Collection Decision Model Sa m pl e su rv ey ( FA D N ) Sa m pl e su rv ey ( re pr es en ta tiv e) C en su s da ta G IS d at a B ay es ia n ne tw or k M ic ro ec on om ic H eu ri st ic r ul e- ba se d Sp ac e th eo ry -b as ed In st itu tio n- ba se d Pr ef er en ce -b as ed H yp ot he tic al r ul es Agent rational decision module Production decisions x x Farm manager’s life cycle Farm takeover, Farm exit x x x Land market Lease decisions for land plots x x x x x x Growth and investment Investment decisions x x Strategy for shifts in labour input x x x x x 8 G. Mack et alii SWISSland calculates sectoral output indicators via an extrapolation algorithm. Zim- mermann et al. (2015) have compared various extrapolation alternatives for the model. Product quantities and prices, land-use and labour trends, income trends according to the Economic Accounts for Agriculture, sectoral input and output factors for calculating envi- ronmental impacts, and key structural figures, such as number of farms, size and type of farm or number of farms changing their farming system, are all sectoral output indicators. 3.2 Options for modelling production and investment decisions Rational agent behaviour is taken as an important basic assumption of the model. Hence, each agent maximises its annual household income for each time period t (Equa- tion 6). In keeping with the theory of adaptive expectations, the agents (a) make their pro- duction decisions based on price (p) and yield (ε) expectations from the previous year (t-1) for the various animal (l) and crop production (g) activities. Prices and yields were estimated for each agent on an individual-farm basis using the FADN data for the base year, with the observed price trends and average annual yield changes (∆) resulting from 2000 to 2012 being stipulated exogenously for each time period. Household income results from the sale of agricultural products originating from land use (LAND g) and livestock farming (ANIMAL l), from off-farm work (OFFFARM o), and from the proceeds of direct payments (PAYMENT d) less the means-of-production costs (COSTFUNCTION). The level of direct payments corresponds to the year-specific, pro- duction-dependent and production-independent approaches in each case, in accordance with current agricultural-policy provisions. Because this study tests various linear and PMP-based quadratic cost functions for crop and animal production activities, the cost functions are described in detail in the Equations 7-12 below. Max INCOMEa,t = ∑gpa,g * ∆pt-1,g * εa,g * ∆εt-1,g * LANDa,t,g + ∑lpa,l * ∆pt-1,l * εa,l * ∆εt-1,l * ANIMALa,t,l + ∑opa,o * ∆pt-1,o * OFFFARMa,t,o + ∑dpd,a * ∆pt,d * PAYMENTa,t,d – OSTFUNCTIONa,t subject to ∑gωa,g,w * LANDa,t,g ≤ Areaa,t ∑lωa,l,w * ANIMALa,t,l ≤ Placesa,t ∑fωa,f,w * LABOURa,t,f * LANDa,t,g + LABOURa,t,f * ANIMALa,t,l ≤ LABOURCAPa,t (6) The resource endowment (ω) of a farm consists of the available area (Area), animal places on the farm (Places), other capacities limiting animal and crop production (e.g. sugar beet quota, milk quota up to 2007, provisions on the receipt of direct payments), and labour force (LABOURCAP). The use of individual-farm FADN data ensures that various factors influencing the objective-function and production-coefficient matrix are automatically taken into account, 9How did farmers act? allowing the depiction of numerous management options that are typical for Switzerland. The cost and output parameters of the production activities are therefore heterogeneous and influence the agents’ decision-making scope. Five different options for modelling animal and crop production decisions were analysed in this study (Table 2). Option 1 determines both crop and animal production decisions based on linear cost functions for 17 crops and 8 animal production activities according to Equation 7: Max INCOMEa,t = REVENUEa,t – ∑lcl,a * ∆ct-1,l * ANIMALa,t,l – ∑gcg,a * ∆ct-1,g * LANDa,t,g (7) Option 1 does not calibrate the production activities to base-year levels. It takes into account the uptake of crop production activities which were not observed in the base year, but which occur in the farm’s historic crop mix. For animal production activities, it con- siders the adoption of new production sectors. For modelling new production activities, which were not observed in the base-year, missing information must be added with the help of average values for other farms, or extrapolated using standard data. For all agent activities occurring in the production programme of the forecast years rather than in the base year, the yield and price coefficients are estimated with the aid of a random distri- bution based on the means and standard deviations of the values for all agents from the same region and farm type (see Möhring et al. [2016]). Options 2a and 2b apply linear cost functions for animal production activities only, while PMP-based quadratic cost functions are used to determine crop production deci- sions (Equation 8). These options consider only crop production activities which were observed in the base year, whereas, for animal production activities, investment activities in new production sectors are taken into account. Max INCOMEa,t = REVENUEa,t – ∑gcg,a * ∆ct-1,g * LANDa,t,g – ∑gda,g * LANDa,t,g – 0.5 ∑gQa,g * LAND2a,t,g – ∑lcl,a * ∆ct-1,l * ANIMALa,t,l (8) Option 2a estimates the matrix coefficients Q of the non-linear cost term based on base-year revenues (revenue*) and base-year crop production levels (LAND*), and uses supply elasticities equal to one owing to the lack of empirical data (Equation 9). Q revenue LAND g a g a g a , * * , , = (9) For those production activities where the output is used on the farm itself, is calcu- lated based on linear costs and shadow values according to the German farm type model FARMIS (Schader, 2009): Qg,a = (cg,a + λg,a) / LAND*g,a (10) The linear term d of the quadratic cost function is calculated according to Equation 11. dg,a = λg,a – Qg,a LAND*g,a (11) 10 G. Mack et alii Option 2b estimates the matrix coefficients of the quadratic cost functions for crop production activities on the basis of maximum entropy. The maximum entropy technique in combination with the PMP calibration allows us to recover a quadratic activity vari- able cost function accommodating complementarity and substitution relations between activities. To estimate the parameter vector dg,a and the matrix Qg,a of the variable cost support points for the parameters were defined. As a starting point, the linear param- eters dg,a could be centred around the observed accounting cost per unit of the activity. For example, the two unknown parameters are specified as an additive function of a num- ber of support points. We could choose five support points Zd (d1,..d5) and ZQ (zq1,.. zq5) for parameter dg,a and the matrix Qg,a. The entropy problem is maximised using sup- port-points consisting of a Zd vector and a ZQ matrix. Because no cross cost effects are expected between crop and animal activities, the linear vector d of the quadratic activ- ity cost function is partitioned into one vector which includes the crop activities and a second vector which includes the animal activities. Similarly, the quadratic matrix Q is partitioned into one matrix which includes the crop activities and a second matrix which includes the animal activities. Both PMP approaches guarantee exact calibration of supply decisions at farm and aggregated levels, taking into account the trade of factors among farms. Nevertheless, different approaches can produce different results when used to pre- dict the future behaviour of the farmer. Options 2a and 2b combine the advantages of both PMP and LP modelling, with PMP calibrating crop production activities to observed base-year levels taking into account the pedoclimatic conditions of the individual farms, and LP enabling modelling of the adoption of new animal production sectors. In all models with a linear cost func- tion in animal husbandry, agents can invest in new barns, allowing them to expand their herd size considerably even within a specific time period, provided that all other neces- sary resources are available in sufficient quantity. Moreover, switching to new production activities is easily possible in the animal husbandry sector. In order to avoid an objective function with an integer formulation, however, individual barn construction variants (pre- viously selected and evaluated according to plausibility) are tested iteratively with the aid of the loop process for each agent entitled to investment. Here, the annual external costs of the entire building (depreciation, repair, insurance and interest) are taken into account, irrespective of whether the barn can be fully utilised. If the agent is entitled to receive investment credits or investment aid, these lower the interest charges. Ultimately, the var- iant with the highest positive objective-function value is implemented. In the following year, all animal places resulting from the investment in the barn are available to the farm- er. In this case, further use of the old barn is ruled out. Investment activities in new ani- mal sectors are taken into account when a farm successor takes over from his predecessor. Only for older agents it was assumed that investment was primarily in the animal sectors pursued to date. Options 3a and 3b test PMP-based quadratic production-cost functions for both ani- mal and crop production activities (Equation 12): Max INCOMEa,t = REVENUEa,t – ∑gcg,a * ∆ct-1,g * LANDa,t,g – ∑gda,g * LANDa,t,g – 0.5 ∑gQa,g * LAND2a,t,g – ∑lcl,a * ∆ct-1,l * ANIMALa,t,l – ∑lda,l * ANIMALa,t,l – 0.5 ∑lQa,l * ANIMAL2a,t,l (12) 11How did farmers act? Because investments in new barns radically alter the cost structure, the PMP-based cost function completely changes the function values derived in the base year. Since no methods were previously available to estimate the change in the PMP-based cost functions derived from the base year, a continuous model approach in which the agents continu- ously expand their barns by individual animal places was chosen for Options 3a and 3b. Table 2. Modelling options for determining production and investment decisions in the farm-level models of the SWISSland agent-based sector model. Option No Name Cost function for crop production activities Cost function for animal production activities PMP calibration method Estimate of matrix coefficients of quadratic cost function Investments 1 Linear Linear Linear - - Investment activities for new buildings 2a Linear-Quad- Revenues PMP-based quadratic Linear Extended Revenues Investment activities for new buildings 2b Linear-Quad- Entropy PMP-based quadratic Linear Extended Maximum entropy Investment activities for new buildings 3a Quad- Revenues PMP-based quadratic PMP-based quadratic Extended Revenues Continuous investment costs for buildings 3b Quad- Entropy PMP-based quadratic PMP-based quadratic Extended Maximum entropy Continuous investment costs for buildings PMP: Positive Mathematical Programming 3.3 Assessing forecasting performance In this study, we assess the forecasting performance of the options based on the aver- age forecasting error (AFE) measuring the difference between forecasted and historical parameters at the farm and sectoral scales. The farm-scale parameters assess the forecast- ing performance only of those agents who remained in the sample for the entire simula- tion period (2005 to 2012). In contrast, sectoral parameters represent changes in the total Swiss farm population over the period from 2005 to 2012 and take into account the farm sample changes due to farm exits and entries. Therefore, the simulation results from all agents were extrapolated to the sectoral scale based on Zimmermann et al. (2015). At the farm scale, the AFE measures the percentage difference between historical and forecasted average production levels for each activity. The weighted average forecasting error (WAFE) of crops aggregates the AFE of all crops based on average production share in the FADN farm sample. The WAFE is calculated analogously for animals. Finally, the total weighted average annual forecasting error (TWAFE) aggregates the WAFE for crops 12 G. Mack et alii and animals equally. At the farm scale, average crop and animal production levels from all FADN farms over a period of three years represent historical parameters. At the sectoral scale, we calculate the production changes from 2003-2005 and 2010- 2012 in percent. The forecasting error measures the deviation from historical values. At sectoral scale, historical values are based on production changes in the total Swiss farm population over this period. 4. Results The SWISSland results were obtained for each specification rule of the cost function. Table 3 presents the historical average production levels of the corresponding FADN- farms and the AFE for crop and animal production activities in the short and long term. Linear cost functions for both crop and animal production activities (Option 1) lead at farm scale to the WAFE of almost 50% for crops and to the TWAFE for both animal and crop production in both time periods (Table 3). The results in Table 3 also show that crop activities supported by direct payments, such as extensive grassland, fallow land, oilseed rape, soya and sunflower, are highly overestimated in the linear version (Option 1), whilst PMP for crop production activities significantly reduces the AFE in both time periods. In the short term, the approaches with quadratic production costs for crop activities and linear production costs for animal activities (Options 2a and 2b) show, on average, the same WAFE as Options 3a and 3b with quadratic production costs for both animal and crop production activities. However, in the long term, Options 2a and 2b show better forecasting performance than Options 3a and 3b. The forecasting performance of Options 2a and 2b improves, in particular, for the livestock categories of cattle, dairy cows, suckler cows, horses and hens, which showed above-average production increases from 2005 to 2012 due to investment activities. Furthermore, the AFE of fodder and grassland activities decreases in Options 2a and 2b because these activities are highly influenced by the cattle production level. Only for marginal animal activities, such as sheep and goats, which are underrepresented in the Swiss FADN farm sample, is the AFE higher in the linear version than in the PMP variants. For crop activities as a whole, the entropy versions and the rev- enue versions lead to similar results in the short and long term. The results also show that both PMP variants (based on revenues or entropy) do not influence forecasting perfor- mance where PMP is combined with LP. Where PMP is used for both production catego- ries, the entropy method leads to slightly better forecasting performance in the long term. Table 4 shows that all model options using PMP (Options 2a to 3b) reproduce the observed farm exits in the long term much better than the linear version (Option 1), which significantly underestimates farm exits. Because high farm income reduces the probability of a farm exit, these results indicate that the linear version (Option 1) signifi- cantly overestimates farm specialisation and farm income. Comparing the extrapolated production changes of all agents with the historical production changes in the agricultural sector shows that the options with linear cost functions for animals (Options 2a and 2b) lead to better results in the long term, particularly in the sectors where the highest pro- duction increases were previously observed, such as suckler cows, hens, horses, goats and poultry. In these animal sectors, above-average investments in new housing, which over- compensate for the reduced production owing to farm exits, were observed in the past. 13How did farmers act? Table 3. Short- and long-term results at farm scale: Historical crop and animal production levels of all Swiss FADN farms and forecasting errors of the modelling options. Historical parameters Average production levels of all FADN farms Average forecasting error of modelling options [AFE in %] 2003- 2005 2006- 2008 2010- 2012 No 1 Linear§ No 2a Linear- Quad- Revenues‡ No 2b Linear- Quad- Entropy† No 3a Quad- Revenues¶ No 3b Quad- Entropy¤ Base year S L S L S L S L S L S L UNIT (ha) (ha) (ha) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Bread grain 1.39 1.40 1.46 50 52 10 13 8 12 8 10 6 10 Feed grain 1.07 1.13 0.92 84 80 14 5 14 6 12 12 9 12 Grain maize 0.22 0.20 0.21 28 21 8 2 8 2 4 6 11 4 Silage maize 0.88 0.91 1.00 9 17 7 3 7 3 2 6 3 6 Sugar beet 0.30 0.32 0.32 11 12 3 4 3 4 2 3 3 3 Potatoes 0.30 0.27 0.25 299 326 2 4 2 5 13 8 3 9 Oilseed rape 0.21 0.23 0.30 237 162 14 34 14 33 12 33 12 32 Sunflower 0.04 0.05 0.04 321 444 11 15 11 15 15 15 11 15 Legumes 0.07 0.08 0.05 130 236 19 18 18 19 12 24 15 24 Vegetables 0.09 0.10 0.11 237 224 13 16 13 16 11 15 12 15 Fallow land 0.04 0.05 0.03 73 148 13 25 15 23 2 29 14 24 Temporary grassland 2.86 2.92 3.34 10 22 5 8 3 10 1 11 0 13 Extensive grassland 1.30 1.30 1.31 68 64 1 1 3 5 20 1 3 5 Less-intens. grassland 0.69 0.68 0.65 8 13 16 21 17 23 2 25 18 24 Intensive grassland 8.66 8.79 8.92 9 11 1 2 0 2 14 3 1 2 Extensive pastures 0.21 0.25 0.25 20 21 11 13 7 9 3 16 8 9 Intensive pastures 1.77 1.78 1.69 2 3 2 3 5 0 3 2 6 1 UNIT (LU) (LU) (LU) (%) (%) (%) (%) (%) (%) (%) (%) (%) (%) Livestock (total) 26.98 27.65 29.83 10 13 5 5 4 6 5 11 5 13 Cattle (total) 21.60 22.07 24.00 13 16 7 8 6 8 7 14 8 14 Dairy cows 14.80 14.97 16.21 11 15 6 6 3 6 4 11 4 11 Suckler cows 1.28 1.57 1.86 15 5 1 2 7 5 22 34 22 35 Horses 0.19 0.22 0.20 16 2 1 11 10 2 6 27 4 34 Sheep 0.21 0.22 0.22 14 33 29 33 14 30 4 7 3 8 Goats 0.05 0.05 0.06 8 6 11 10 7 11 5 23 2 23 Sows 3.98 4.14 4.08 5 7 9 10 7 10 6 9 5 7 Fattening pigs 2.52 2.61 2.67 7 1 2 4 8 3 14 13 14 2 Hens 0.34 0.35 0.59 1 25 19 18 0 21 1 40 3 39 Poultry 0.60 0.60 0.67 1 11 13 23 6 12 0 10 0 12 14 G. Mack et alii The results show that modelling investment decisions in new animal capacities based on linear cost functions (Options 2a and 2b) leads to better results than using continuous investment activities combined with quadratic cost functions (Options 3a and 3b). The results also show that PMP used for crop production activities underestimates produc- tion increases which are above-average (such as rapeseed, sugar beet, field vegetables etc.). These results are caused by two characteristics of PMP. On the one hand, the farm-level models only take into account the activities observed during the 2005 reference period, so the adoption of new crop production activities in subsequent years could not be taken into account. On the other hand, the quadratic cost functions prevent overspecialisation and above-average production increases for single activities. We can only assess the per- formance of the model based on its forecasting capacity. 5. Conclusions This ex-post validation at farm scale clearly shows that, in the short term, supply curve specifications based on PMP only or on PMP combined with LP for selected Historical parameters Average production levels of all FADN farms Average forecasting error of modelling options [AFE in %] 2003- 2005 2006- 2008 2010- 2012 No 1 Linear§ No 2a Linear- Quad- Revenues‡ No 2b Linear- Quad- Entropy† No 3a Quad- Revenues¶ No 3b Quad- Entropy¤ Base year S L S L S L S L S L S L Weighted average forecasting error [WAFE in %] Crop production       50 55 4 5 4 5 4 7 4 7 Animal production     10 14 6 10 6 10 7 13 7 11 Total weighted average forecasting error [TWAFE in %] Average       30 34 5 8 5 8 5 10 5 9 S = Short term; L = long term; LU: Livestock Unit; FADN: Swiss Farm Accountancy Data Network Data Pool; § Linear = Linear cost functions for crop and animal production activities; ‡ Linear-Quad-Revenues = Linear cost functions for animal production activities and PMP-based quad- ratic cost functions for crop production activities. Estimate of PMP coefficients based on revenues; † Linear-Quad-Entropy = Linear cost functions for animal production activities and PMP-based quadratic cost functions for crop production activities. Estimate of PMP coefficients based on maximum entropy; ¶ Quad-Revenues = PMP-based quadratic cost functions for animal and crop production activities. Esti- mate of PMP coefficients based on revenues; ¤ Quad-Entropy = PMP-based quadratic cost functions for animal and crop production activities. Esti- mate of PMP coefficients based on maximum entropy. 15How did farmers act? Table 4. Long-term results at sectoral scale: Historical sectoral production changes from base year 2003/05 to 2010/12 and deviation of model results from historical sectoral changes (+/- %) of the modelling options. Unit Observed sectoral change from 2003/05 - 2010/12 No 1 Linear§ No 2a Linear- Quad- Revenues‡ No 2b Linear- Quad- Entropy† No 3a Quad- Revenues¶ No 3b Quad- Entropy¤ Historical change (+/-%) Deviation from historical sectoral change (+/-%) of the modelling options Farm exits Total farms  Qty. -11% 5% 1% 0% 2% 3% Valley region  Qty. -12% 5% 4% 2% 4% 6% Hill region  Qty. -9% 4% -3% -4% 0% -1% Mountain region  Qty. -10% 3% 1% 1% 1% 1% Farm size < 20 ha  Qty. -18% 0% -3% 1% -1% 6% Farm size 20-30 ha  Qty. +4 8% 8% -4% 6% -2% Farm size > 30 ha  Qty. +15% 9% -4% -13% -4% -19% Crop production Bread grain ha  -4% -17% -11% -14% -1% -3% Fodder crop ha -17% -36% -2% -5% 12% 9% Potatoes  ha -17% 28% -7% -7% -1% 1% Rapeseed  ha 35% -52% -48% -50% -41% -43% Sunflower  ha -32% 23% 12% 12% 18% 15% Field vegetables  ha 11% 173% -14% -17% -9% -10% Silage maize  ha 12% 2% -6% -5% -14% -6% Sugar beet  ha 6% -23% -12% -12% -11% -7% Open arable land  ha -6% 8% -6% -8% 0% 1% Temporary ley  ha 9% 2% -3% -7% -15% -13% Total arable area  ha -2% 5% -4% -7% -4% -3% Permanent grassland  ha -2% 5% 2% -2% 3% -2% Total utilised agricultural area  ha -2% 5% 0% -3% 1% -2% Total livestock  LU 3% 1% -3% -5% -9% -11% Dairy cows  LU -6% 4% 2% 1% -3% -2% Suckler cows  LU 55% -6% -17% -20% -60% -60% Pigs  LU -3% -1% -6% -6% -3% -34% Fattening calves  LU -13% 2% 1% 2% 12% 24% Fattening bulls  LU -6% 14% 5% 2% 2% 2% Cattle total  LU 2% 0% -4% -4% -11% -9% Sheep  LU -1% -19% -21% -21% -5% -3% Goats  LU 25% 78% 78% 78% -39% -42% Horses  LU 13% 93% 81% -11% 88% 122% Broilers  LU 31% 18% 13% 8% -38% -40% Hens  LU 19% 10% 5% 2% -20% -20% 16 G. Mack et alii production activities significantly improve the forecasting performance of an agent- based model compared with specifications based on LP only. For short-term forecasts, where investment decisions do not play a major role, PMP for all production activi- ties and PMP combined with LP produce similar results. For long-term forecasts, the results at farm scale and at sectoral scale show that combining LP for animal produc- tion activities with PMP for crop production activities leads to the best forecasting performance. The combined approach could mitigate some limitations of PMP which are relevant mainly in the medium and long term, such as the adoption of new produc- tion activities, while still exploiting the advantages of PMP in order to avoid overspe- cialised model results. This study confirms also the finding of Buysse et al. (2007) that, in sectors where new production activities are expected to be adopted owing to market and policy changes (i.e. switching from direct payments towards market support or opening borders of an iso- lated country), the LP approach could represent an appropriate solution, in particular in long-term forecasts, whereas, in the case of minor policy changes or in the short term (i.e. slight modifications of direct payments or tariffs), PMP could improve the forecast- ing results. The underlying reason for this might lie in the fact that farmers have to take both gradual and binary decisions. In animal production, either a new house will be built or it will not. After a radical reform of agricultural policy, the farming business will be continued or not. Our results have shown that, for such binary decisions, LP is effective. For situations where price fluctuations suggest an increase in potatoes at the expense of a farmer’s wheat acreage, PMP is a more suitable instrument. The results show that supply curve specifications based on the extended variant of PMP and that revenues and specifications based on PMP and maximum entropy lead Unit Observed sectoral change from 2003/05 - 2010/12 No 1 Linear§ No 2a Linear- Quad- Revenues‡ No 2b Linear- Quad- Entropy† No 3a Quad- Revenues¶ No 3b Quad- Entropy¤ Historical change (+/-%) Deviation from historical sectoral change (+/-%) of the modelling options Average of absolute deviation from historical sectoral change (%) All attributes 20% 12% 10% 13% 16% LU: Livestock Unit; § Linear = Linear cost functions for crop and animal production activities; ‡ Linear-Quad-Revenues = Linear cost functions for animal production activities and PMP-based quad- ratic cost functions for crop production activities. Estimate of PMP coefficients based on revenues; † Linear-Quad-Entropy = Linear cost functions for animal production activities and PMP-based quad- ratic cost functions for crop production activities. Estimate of PMP coefficients based on maximum entropy; ¶ Quad-Revenues = PMP-based quadratic cost functions for animal and crop production activities. Esti- mate of PMP coefficients based on revenues; ¤ Quad-Entropy = PMP-based quadratic cost functions for animal and crop production activities. Esti- mate of PMP coefficients based on maximum entropy. 17How did farmers act? to similar results. The results support other studies by Gocht (2005) and Winter (2005), both of whom discovered that the different PMP versions led to similar model results. Although all tested approaches lead to deviations in the actual observable trends, we may conclude that PMP for crop production activities combined with LP for animal produc- tion activities is preferable to full PMP when assessing the forecasting performance of sec- toral production changes in the medium or long term. 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