Ecology, Economy and Society–the INSEE Journal 1 (1): 17–31, April 2018 THEMATIC ESSAY On the Normalization of Dimensioned Variables in Ecological Economics Deepak Malghan  1. INTRODUCTION The fundamental concern of ecological economics is to accurately model all aspects of the economy–ecosystem interaction problem — the myriad ways in which the economic and ecological systems are connected to each other. Almost all the monetary and physical variables used to describe economy– ecosystem interactions are dimensional in nature. The exact cardinal value taken by dimensioned variables is contingent on the particular measurement unit used. While several papers on the subject have pointed to the care required in using dimensioned variables in ecological economics, there is little consensus on how dimensional variables must be incorporated in economy–ecosystem interaction models (Mayumi and Giampietro 2010; Malghan 2011; Chilarescu and Viasu 2012; Baiocchi 2012; Mayumi and Giampietro 2012). Mayumi and Giampietro (2010) inaugurated the debate by making the provocative claim that many models in economics and ecological economics that make use of transcendental functions like the logarithm are fundamentally flawed when these functions use what are apparently dimensioned variables. Malghan (2011) claimed that several popular biophysical sustainability indicators are dimensionally inconsistent because they neglect the ‗qualitative residual‘ that is the defining characteristic of any social–ecological system (Georgescu-Roegen 1971). In a brief comment, Chilarescu and Viasu (2012) showed that the critique that a neoclassical production function (Arrow et al. 1961) is dimensionally inconsistent does not consider that the parameters of a production function  Indian Institute of Management Bangalore, Bannerghatta Road, Bengaluru 560076, India and Ashoka Trust for Research in Ecology and the Environment, Srirampura, Bengaluru 560064, India; dmalghan@iimb.ac.in, dmalghan@atree.org Copyright © Malghan 2018. Released under Creative Commons Attribution- NonCommercial 4.0 International licence (CC BY-NC 4.0) by the author. Published by Indian Society for Ecological Economics (INSEE), c/o Institute of Economic Growth, University Enclave, North Campus, Delhi 110007. ISSN: 2581-6152 (print); 2581-6101 (web). DOI: https://doi.org/10.37773/ees.v1i1.7 https://doi.org/10.37773/ees.v1i1.7 Ecology, Economy and Society–the INSEE Journal [18] are dimensioned variables, too. Thus, in the familiar Cobb-Douglas production function, of the form  LAKLKFY  ),( , the parameter A has appropriate dimensions (contingent on  and  ) such that the function itself has the exact same dimension as Y (Chilarescu and Viasu 2012). In an earlier debate on a similar subject, Folsom and Gonzalez (2005) had shown, in response to the dimensional inconsistency claim made by Barnett-II (2003), how the parameters of the Cobb-Douglas production function are assumed to have implicit dimensions required by dimensional consistency. To critique the claim in Mayumi and Giampietro (2010), Baiocchi (2012) used examples from a variety of disciplines, including the IPAT identity and Environmental Kuznets Curve, and also offered a critical historical literature review of dimensional analysis. Unfortunately, this debate on dimensional consistency in ecological economics has only helped to muddy the waters rather than provide a consistent framework for achieving dimensional consistency while studying the economy-ecosystem interaction problem. It is trivial to demonstrate that a logarithmic function cannot have dimensioned variables as its argument. The more pertinent question is whether it might be possible to non-dimensionalize basic models of economy-ecosystem interaction that are of interest to ecological economists. We illustrate the problem with the transcendental logarithm function that has been at the centre of the recent debate. In their rejoinder to Chilarescu and Viasu (2012), Mayumi and Giampietro use the familiar Maclaurian expansion of )1ln( z and )1ln( z to obtain a polynomial expansion for the natural logarithm of any positive real number,  z (Mayumi and Giampietro 2012, equation 5):                          .... 1 1 3 1 1 1 2)ln( 3 z z z z z (1) It is straightforward to see that equation 1 cannot take a dimensioned z . Thus, Mayumi and Giampietro (2012) argue that a regression model that includes a term like )ln( LV used by Arrow et al. (1961) in their labour–capital substitution model is problematic because the logarithm takes on a dimensioned quantity (measured in US dollars per person–year of labour unit, for example). Even in the 1960s, the classic paper by Arrow et al. (1961) had been critiqued for not considering the dimensional consistency of production function specifications (De Jong 1967; De Jong and Kumar 1972; Cantore and Levine 2012). However, Mayumi and Giampietro (2010, 2012) ignore the fact that it is possible in theory to obtain non-dimensional versions of V and L through the well-established process of normalization. In principle, there should be no objection to using an expression like )ln( L V [19] Deepak Malghan if value added ( V ) and quantity of labour ( L ) are expressed as non- dimensional variables. While Mayumi and Giampietro (2012) cite several examples from prominent economists committing the apparent error of using dimensioned quantities in the logarithmic functions, we demonstrate in section 2 that normalization or non-dimensionalization can in principle address this problem. We argue that this is a relatively minor technical point, and that the more fundamental problem is that of representing the economy– ecosystem interaction problem in a dimensionally consistent fashion. The remainder of this paper is organized as follows: the next section will review normalization and non-dimensionalization using canonical examples from economics and ecology. It is not merely sufficient for an ecological economics model to be dimensionally consistent. The key question is ‘whether or not the selected dimensional choice for a given expression has an operational meaning or relevance for the purpose [of] analysis’ (Mayumi and Giampietro 2012; emphasis in original). While this was the true import of Mayumi and Giampietro (2010), the subsequent papers in the debate have missed the forest for the trees by focussing exclusively on narrow technical dimensional consistency. In section 3, we discuss the limitations of normalization and non-dimensionalization procedures. In particular, we show that it is nontrivial to normalize dimensioned variables in analytically accurate models of economy-ecosystem interaction. 2. NORMALIZATION AND NON-DIMENSIONALIZATION Using several canonical (and elementary) examples from ecology and economics, we demonstrate in this section that normalization and non- dimensionalization can address dimensional consistency issues in ecological economics. We examine the production function and the consumer‘s utility maximization problem from elementary microeconomics; the logistic population growth model from ecology; and the normalisation of the Gaussian distribution in statistics. 2.1. Normalization of the Canonical Cobb-Douglas Production Function The standard Cobb-Douglas production function for two inputs K and L can be represented as:  LAKY  (2) Now consider a simple constant-returns version of equation 2 such that (  1 ) and ( 0,;10  LK ):  LAKY   1 (3) Ecology, Economy and Society–the INSEE Journal [20] The central dimensional concern with the Cobb-Douglas function in equation 3 is that capital ( K ), and labor ( L ) are measured in units that are different from each other, and from the output ( Y ). The constant A has a dimension that is contingent on the factor-share parameter  , such that equation 3 is dimensionally consistent. To make this point of A being dimensional even more explicit, equation-3 can be rewritten as:  LAKAY LK   1 (4a) LK AAA  (4b) where the dimensional constants K A and LA are the so-called efficiency parameters. The presence of these two dimensioned quantities makes analytical work and interpretation difficult. However, as shown by De Jong (1967) and Cantore and Levine (2012), equation 4 is most easily normalised and rendered into a non-dimensional form. Consider a normalization-point 0 Y such that: ))(( 0 1 00  LAKAY LK   (5) Now dividing equation 4 by equation 5 we readily obtain the non- dimensional version of the constant-returns Cobb-Douglas function:  lky   1 (6a) 000 ;; L L l K K k Y Y y  (6b) Any econometric model involving logarithms of the non-dimensional variables ( lky ,, ) will pose no dimensional issues — for example, a log-log model to estimate factor share,  . While equation 6 eliminates the dimensional constants, it offers no clarity on how to pick the normalization point ( 0 Y ). In the context of a neoclassical economic growth model, it would be most intuitive to use the steady state value as the normalization point. While the choice of normalization point is easily determined for the present problem, we show below how this can be non-trivial when studying the economy-ecosystem interaction problem. Indeed, as we discuss below in section 3 below, in ecological economics mass balance problems (say when studying stock of timber in a forest), the choice of normalization point is critical for model specification. Further, we show why the selection of the appropriate normalization point is fundamentally a non-technical choice that is value laden. Before we take up another canonical example — the logistic growth equation from ecology to illustrate the process of non-dimensionalization (a homologue of the normalization process discussed here) — it is important to note that more general production functions (CES, for example) can be [21] Deepak Malghan normalized in the same manner as the pedagogically simple case of Cobb- Douglas discussed here (Klump and La Grandville 2000; Klump and Saam 2008; Cantore and Levine 2012; Temple 2012). 2.2. The Logistic Equation Consider the logistic equation that has been the pedagogical model of choice for students of ecology from the time Alfred Lotka formalised the original Verhulst formulation in the context of population growth of parasite colonies (Lotka 1925). The simple population growth logistic equation with a fixed carrying capacity, K and population growth rate, r can be written as: 0 )0(;1 PP K P rP dt dP        (7) where P is the population at any time t ; and the initial population is known such that 0 )0( PP  . In the above equation, all the four variables are dimensional — P and K have the dimension of [N] (number of individual plasmodium parasites in a colony for example); t has the [T] dimension (time, measured in hours or minutes); and r has the dimension of [T−1] (inverse time dimension, measured in per-hour or per-minute, persevering with the plasmodium colony growth example). The units in which population and time are measured are arbitrary and the parameter values in equation 7 will change if we went from measuring time in hours to say, in minutes or years. It is straightforward to non-dimensionalize equation 7 so that it is invariant to particular choices of units for population and time. This is achieved by scaling or normalizing the time and population variables as:        r t 1  (8a) K P x  (8b)  d dx rK rd Kxd dt dP  )/( )( (8c) K P x 0 0  (8d) The new variables X and  defined in equation 8 are non-dimensional. Substituting equation 8 in equation 7 we obtain the non-dimensionalized form of the logistic equation: Ecology, Economy and Society–the INSEE Journal [22] 0 )0();1( xxxx d dx   (9) Unlike the original dimensioned variables, P , K , t and r , the scaled non- dimensional variables X and  can be used in any transcendental functions like the natural logarithm or the exponential function. Like any non- dimensionalization process, the scaled variables X and  are related to the intrinsic property of the physical phenomenon being studied. The scaled population, X represents the population relative to the carrying capacity, K and is the intrinsic unit for measuring population in a simple logistic model. 1 By measuring population using non-dimensional X , we have scaled the problem so that equation 9 applies to a wide variety of phenomena following the logistic growth pattern. Similarly r 1 that we used to scale time, t to obtain the non-dimensional  is the intrinsic unit for measuring time in the context of population growth models. In an exponential growth model (the initial part of the logistic growth curve when KP  , the population grows by a factor of e in the time interval r 1 – an intrinsic unit for measuring time in any exponential growth problems. Besides being an intrinsic representation, the scaled non-dimensional form of the logistic equation is also the most parsimonious representation of the problem of carrying capacity constrained growth. 2.3. The Standard Normal, and the Box-Cox Transform Consider a random variable X that is distributed with mean  and variance ),(~( 22  NX . The Gaussian distribution for )(, XfX is given by: 2 2 1 2 1 )(            X exf (10) Any normally distributed variable can be expresses in terms of the standard normal, Z (where )1,0(~ NZ ). 2 2 1 2 1 )( Z eZf    (11) As every beginning student of statistics is taught, for any random variable      X ZNX ),,(~ 2 is a standard normal, or )1,0(~ N X   . Besides helping with statistical inference, this normalization process is of 1 One could have also carried out the non-dimensionalization of equation-7 by setting x = P/P0. A non-dimensional x that is a scaled by the initial population is however not intrinsic to the system as the carrying capacity (for a system with time invariant K). [23] Deepak Malghan significance for our discussion about dimensioned variables. X is a dimensioned variable (has the dimensions of [T] for example if X was measuring some temporal phenomenon). However, the normalized variable Z is dimensionless as  and  have the same dimensions as X – [T] in the present example. Thus, while X cannot be used as an argument in transcendental functions, an expression of the form )ln(ZY  can be evaluated using equation 1. This normalization process is even more significant if one considers the fact that the sum of a sufficiently large set of independent random variables (with finite variance) will converge to a normal distribution (the central limit theorem). Statisticians have long recognized the centrality of transformations in studying dimensioned variables. While the pedagogical example presented here is familiar even to a beginning student, there is a well-developed literature on generalizing transformations starting with the seminal contribution of John Tukey (1957), and the celebrated paper of Box and Cox (1964). The Box-Cox transformation ( X ) of variable X is simply:    1  X X (12) It is straightforward to show that the CES production function (of which the Cobb Douglas corresponds to a specific parametric value) is a special case of the Box-Cox transformation. 2 Further, as 0 , the Box-Cox transform is the log transform ( )log( XX  ). It is for this reason that a Box-Cox transform with 0 finds numerous applications in applied economics. 2.4. The Numéraire Good and Consumer’s Utility Maximization Problem The most widely used example of normalization in economics – by a wide margin – is the numéraire good. All prices in the pure theory of exchange are relative prices — prices that have been normalised by an appropriate numéraire. Money (dollars for example) is simply the most common choice for the numéraire. In principle, any other commodity can be used as a numéraire. Consider an individual‘s utility function defined by a Cobb-Douglas function (in a simple two-good case) as follows:  YXU   1 (13) 2 I thank an anonymous referee for this pedagogical suggestion. Ecology, Economy and Society–the INSEE Journal [24] Following our discussion in equation 5, we can write out a corresponding utility normalization point as: 0,; 000 1 00   YXYXU  (14) Dividing equation 13 by equation 14 we obtain a non-dimensional analogue of equation-6:  yxu   1 (15a) 000 ;; Y Y y X X x U U u  (15b) All three variables (utility, and the quantity of two goods that are consumed) in equation 15 are non-dimensional. While )ln(U is not defined, equation 1 can be used to evaluate )ln(u . Before we consider the consumer‘s utility maximization problem, we write out the budget constraint faced by the consumer: MYPXP Yx  (16) In equation 16, M is the disposable income available to the consumer; and X P and Y P are respectively prices (say in dollars per unit) of goods X and Y respectively. The budget constraint when expressed using dimensionless x and y (instead of dimensioned quantities X and Y ) can be written out as: MyPxP YX  ~~ (17a) XX PXP 0 ~  (17b) YY PYP 0 ~  (17c) In equation 17 XP ~ and 0 ~ Y are simply prices corresponding to normalised (and dimensionless) quantities of X and Y . Money, measured in dollars ($) is the numéraire in both equations (16) and (17). While X P and Y P have the dimensions of XP quantity dollar ~ , and ~ Y P have dimensions of dollars. One of the fundamental insights from consumer‘s problem is that the neither the budget set nor the budget constraint is affected by our choice of numéraire. Now, if we normalize equation 17 using x as the numéraire good, we can rewrite the budget constraint as: __ MyPx Y  (18a) ~ ~ _ X Y Y P P P  (18b) [25] Deepak Malghan ~ _ X P M M  (18c) Every term in the budget constraint represented by equation 18 is dimensionless. As the relative price _ y P and _ M are dimensionless they can be used as arguments in a transcendental function. Thus a regression equation that uses )ln( _ iM poses no dimensional problems (where _ i M the disposable income of household i). We can now write out the consumer‘s utility maximization problem using equations (15) and (18). The dimensionless Lagrangian is simply: __ 1 )()( MyPxyxL Y     (19) By setting 0,0       y L x L and eliminating  we obtain the dimensionless first order condition for the consumer‘s utility maximization problem:                  _ 11 Y P x y   (20) Every single variable in equation 20 is dimensionless. 3. OBJECT LESSONS FOR BIOPHYSICAL AND ECOLOGICAL ECONOMICS We have demonstrated using canonical examples from economics, ecology, and statistics that normalization and non-dimensionalization can transform dimensional forms into their dimensionless counterparts. The examples presented in the previous section show that in theory, normalization can circumvent the objections raised by Mayumi and Giampietro (2010) in the recent debate over dimensions. However, as pointed out by Mayumi and Giampietro (2012) in their rejoinder, the more relevant question is one of delineating the physical basis for normalization. In the examples that we have considered, the non-dimensionalization procedure for the logistic equation or the construction of the standard normal statistic is well- grounded. From the two economics‘ examples we have considered, normalization using an arbitrary choice of the numéraire good in the consumer problem is well-established. A production function on the other hand must not only be dimensionally consistent but also reflect the physical basis for production. Normalization only solves the technical problem of Ecology, Economy and Society–the INSEE Journal [26] dimensional consistency but the normalized representation of the production process is only as good as the original dimensioned representation. An accurate physical representation of the production process has been one of the founding tenets of ecological economics (Georgescu-Roegen 1971; Kraev 2002; Røpke 2004). As an illustration of the difficulties involved in selecting a normalization point in realistic models of economy-ecosystem interaction, consider any model that includes a throughput variable ( . X ), say measured in kilograms per year so that . X has the dimensions of [MT−1]. The throughput . X cannot be an argument in any transcendental function. It is trivial to normalize the throughput with some reference throughput, . X to obtain a non- dimensional version  X Xx . . such that the normalized throughput, . x has no physical dimensions and can be used as arguments in transcendental functions. Indeed, such a measure is homologous to the rapidity measure used in physics to characterize speed relative to the speed of light.3 Unlike relativity-physics however, the choice of reference throughput,  X is not universal but highly context dependent. A possible candidate for reference throughput is the maximum sustainable throughput — the throughput above which the integrity of the underlying biophysical system in jeopardy. Consider an illustrative example — throughput of timber from a forest. The maximum sustainable throughput is a function of the health of the underlying forest ecosystem and will vary across both space and time. A tropical forest will necessarily have a different maximum sustainable throughput from a temperate forest. Even in a single location, maximum sustainable throughput will vary with time. The determination of maximum sustainable throughput is a function of ecosystems as funds rather than stocks (Malghan 2011). The fundamental economy-ecosystem interaction problem, or the biophysical connections between the economy and the ecosystem is not reducible to cardinal arithmetic that governs stocks and flows.4 Instead, any reasonably complete account of the economy–ecosystem interaction problem must also account for funds and fluxes that cannot be described in a cardinal space (Georgescu-Roegen 1971; Malghan 2011). A fund is a 'special configuration of a given stock of materials(s)' (Malghan 2011). Consider, for example, an automobile. It is a stock of various material 3 In physics, rapidity, φ is defined as )(tanh 1 c v  where c is the speed of light. 4 This part of the essay is abstracted from the presentation in Malghan (2011). [27] Deepak Malghan stocks – steel, aluminium, plastic, etc. However, these stocks have to come together in a specific configuration to constitute a fund of useful transportation services. An automobile that has been ‗totalled‘ in an accident still retains all the original stock but is no longer a fund of transportation services. Thus, while simple mass conservation laws can account for stocks, funds are best described by laws that 'follow the spirit of entropy law in thermodynamics' (Malghan 2011). More formally, equation- 21 below fully describes the evolution of stock over time, funds are not reducible to the cardinal space. dtxxxtx out t in )()0()( . 0 .   (21) where )(tx is the quantity of stock at any time t ; )0(x is the initial stock at 0t ; and inx . and o u tx . are inflows and outflows respectively. The stock x is in steady state if ttxtx outin  )()( .. . No such simple steady state condition can written out for a fund, as illustrated in the totalled automobile example above where all the stocks in the car in an approximate steady state. In addition to providing the original exposition of the concept of ‗fund‘, Georgescu-Roegen (1971) also speculated on an entropy law modelled on the Second Law of thermodynamics for matter. This so-called ‗fourth law‘ has been hotly contested (Cleveland and Ruth 1997; Ayres 1998, 1999; Hammond and Winnett 2009). For our purposes here, it is sufficient to note that conservations laws (like the first law of thermodynamics) alone cannot completely describe a fund, and we need to invoke some mechanism like the Second Law that allows for qualitative degradation of energy and matter (Malghan 2011). The distinction between stocks and funds introduced here, using the automobile as a pedagogical example, has direct implications for dimensional consistency in even the most elementary models of the economy–ecosystem interaction in biophysical economics or ecological economics. Continuing with the sustainable throughput example discussed above, consider a very simple measure of the scale of a forestry industry:   Y x S . (22) where ^ Y is the rate at which timber regenerates in the forest, and . x is the throughput. Now suppose we have S measured for two different places: 1 S = 0.8 in a tropical forest; and 2S = 0.7 in a temperate forest. Can we automatically conclude that the forestry industry in the temperate forest is Ecology, Economy and Society–the INSEE Journal [28] more sustainable than the one in the tropics because 2 S < 1 S ? We cannot because equation 22 does not contain any information about the underlying fund. S is dimensionless only in the stock-flow space in the sense that both . x and ^ Y have the same dimensions – say, tons/year. In order to be able to make comparisons across space and time, we will need metrics that are appropriately normalized to render them dimensionless in both the stock- flow space and the fund–flux space. In particular, it is important to note that S ≤ 1 does not automatically imply sustainable throughput. During regeneration of a degraded forest, a sustainable throughput might as well be S = 0 (Malghan 2011). Several extant aggregate biophysical metrics are not dimensionless in the fund–flux space (Malghan 2011). Examples of such metrics include the ecological footprint (Rees 1992; Wackernagel and Rees 1996; Wackernagel et al. 2004); human appropriation of the products of photosynthesis (Vitousek et al. 1986; Rojstaczer et al. 2001); and aggregate material throughput metrics (National-Research-Council, 2004; Adriaanse et al. 1997; Matthews et al. 2000; Klee and Graedel 2004; Gordon et al. 2006; Wernick and Ausubel 1995). These aggregate metrics are all appropriately normalized in the stock-flow space but not in the fund-flux space. For example aggregating multiple flows (as is done in the material throughput) is simple enough in the stock-flow space but fraught with problems in the fund-flux space. Consider a simple material flow metric:     ni i i xX 1 .. (23) The sum in equation 23 is dimensionally valid only in the stock flow space, but not in the fund–flux space. If the n elementary flows are summed into the aggregate throughput, . X , the sum is defined as long as all the n flows are measured in a common unit – say tons/year. However, in the fund flux space, it makes little sense to sum up tons/year of some metal and tons/year of water. For the sum to be defined in the fund-flux space, the individual flows have to be appropriately normalized so that they are 'strictly dimensionless' in the fund-flux space (Malghan 2011). The difficulty with determining an appropriate normalization point in the throughput example above is related to a more general problem of mapping ordinal and cardinal variables in a dimensionally consistent fashion. An accurate representation of the economy-ecosystem interaction problem requires accounting for ecosystem as a fund in addition to ecosystem as simply a collection of stocks. Unlike stocks and flows, funds and fluxes are ordinal and subject to additional dimensional consistency constraints. The [29] Deepak Malghan mapping between an ordinal fund-flux space and the cardinal stock-flow space is at the heart of economics of ecosystem services (Farley 2012; Malghan 2011). In the current debate of dimensioned variables in ecological economics, the true import of the first salvo fired by Mayumi and Giampietro (2010) was the fact that several empirical models are not careful about making the distinction between fund and stock functions of ecosystem. We would be missing the forest for the trees if we focused the dimensions debate exclusively on technical aspects of normalization. Normalization procedure, as demonstrated using elementary examples is well-established for cardinal variables but the cardinal stock-flow space alone is inadequate for accurately modelling the economy-ecosystem interaction problem. There is a need for ecological economics to develop models of economy-ecosystem interaction that are at once realistic representation of the problem and are dimensionally consistent. ACKNOWLEDGEMENTS I thank Kanchan Chopra, Herman Daly, Federico Demaria, Joshua Farley, Philip Lawn, Joan Martinez Alier, Matthias Ruth, Anish Sugathan, Hema Swaminathan, Sneha Thapliyal, and Robert Ulanowicz for critical comments on earlier versions. Suggestions and comments from the two anonymous referees were pivotal in further improving the exposition. I am most grateful to Nandan Nawn and the editorial team at EES for help with typesetting the paper that was originally written using a typesetting program that is not compatible with the normal journal production process at EES. Usual disclaimers apply. REFERENCES Adriaanse, A.S. Bringezu, A. Hammond, Y. Moriguchi, E. Rodenburg, D. Rogich and H. Schtz. 1997. Resource Flows: The Material Basis of Industrial Economies. World Resources Institute. Arrow, K. J., H. B. Chenery, B.S. Minhas and R.M. Solow. 1961. ―Capital-labor substitution and economic efficiency.‖ The Review of Economics and Statistics 43 (3): 225–250. https://doi.org/10.2307/1927286 Ayres, R. U. 1998. ―Eco-thermodynamics: economics and the second law.‖ Ecological Economics 26: 189–209. https://doi.org/10.1016/S0921-8009(97)00101-8 Ayres, R. U. 1999. ―The second law, the fourth law, recycling and limits to growth.‖ Ecological Economics 29: 473–483. https://doi.org/10.1016/S0921- 8009(98)00098-6 Baiocchi, G. 2012. ―On dimensions of ecological economics.‖ Ecological Economics 75: 1–9. https://doi.org/10.1016/j.ecolecon.2012.01.016 https://doi.org/10.2307/1927286 https://doi.org/10.1016/S0921-8009(97)00101-8 https://doi.org/10.1016/S0921-8009(98)00098-6 https://doi.org/10.1016/S0921-8009(98)00098-6 https://doi.org/10.1016/j.ecolecon.2012.01.016 Ecology, Economy and Society–the INSEE Journal [30] Barnett-II, W. 2003. ―Dimensions and economics: Some problems.‖ The Quarterly Journal of Austrian Economics 6 (3): 27–46. https://doi.org/10.1007/s12113-003- 1022-2 Box, G. E., and D.R. Cox. 1964. ―An analysis of transformations.‖ Journal of the Royal Statistical Society. Series B (Methodological) 26(2): 211–252. https://doi.org/10.1111/j.2517-6161.1964.tb00553.x Cantore, C., Levine, P., 2012. ―Getting normalization right: Dealing with ‗dimensional constants‘ in macroeconomics.‖ Journal of Economic Dynamics & Control 36 (12): 1931-1949 https://doi.org/10.1016/j.jedc.2012.05.009 Chilarescu, C. and I. Viasu. 2012. ―Dimensions and logarithmic function in economics: A comment.‖ Ecological Economics 75: 10–11. https://doi.org/10.1016/j.ecolecon.2012.01.017 Cleveland, C. and M. Ruth. 1997. ―When, where, and by how much do biophysical limits constrain the economic process? a survey of Nicholas Georgescu-Roegen‘s contribution to ecological economics.‖ Ecological Economics 22: 203–223. https://doi.org/10.1016/S0921-8009(97)00079-7 De Jong, F., 1967. Dimensional Analysis for Economists. North Holland. De Jong, F. and T. Kumar. 1972. ―Some considerations on a class of macro- economic production functions.‖ De Economist 120: 134–152. https://doi.org/10.1007/BF01718126 Farley, J. 2012. ―Ecosystem services: The economics debate.‖ Ecosystem Services 1 (1): 40 – 49. https://doi.org/10.1016/j.ecoser.2012.07.002 Folsom, R. and R. Gonzalez. 2005. ―Dimensions and economics: Some answers.‖ The Quarterly Journal of Austrian Economics 8 (4): 45–65. https://doi.org/10.1007/s12113-005-1003-8 Georgescu-Roegen, N. 1971. The Entropy Law and the Economic Process. 2nd Edition. Harvard University Press. https://doi.org/10.4159/harvard.9780674281653 Gordon, R., Bertram, M. and T. Graedel. 2006. ―Metal stocks and sustainability.‖ Proceedings of the National Academy of Sciences 103: 1209–1214. https://doi.org/10.1073/pnas.0509498103 Hammond, G.P. and A.B. Winnett. 2009. ―The influence of thermodynamic ideas on ecological economics: An interdisciplinary critique.‖ Sustainability 1: 1195– 1225. https://doi.org/10.3390/su1041195 Klee, R. and T. Graedel. 2004. ―Elemental cycles: A status report on human or natural dominance.‖ Annual Review of Environment and Resources 29: 69–107. https://doi.org/10.1146/annurev.energy.29.042203.104034 Klump, R. and O.D. La Grandville. 2000. ―Economic growth and the elasticity of substitution: two theorems and some suggestions.‖ American Economic Review 90: 282–291. https://doi.org/10.1257/aer.90.1.282 Klump, R and M. Saam. 2008. ―Calibration of normalised ces production functions in dynamic models.‖ Economics Letters 99: 256–259. https://doi.org/10.1016/j.econlet.2007.07.001 https://doi.org/10.1007/s12113-003-1022-2 https://doi.org/10.1007/s12113-003-1022-2 https://doi.org/10.1111/j.2517-6161.1964.tb00553.x https://doi.org/10.1016/j.jedc.2012.05.009 https://doi.org/10.1016/j.ecolecon.2012.01.017 https://doi.org/10.1016/S0921-8009(97)00079-7 https://doi.org/10.1007/BF01718126 https://doi.org/10.1016/j.ecoser.2012.07.002 https://doi.org/10.1007/s12113-005-1003-8 https://doi.org/10.4159/harvard.9780674281653 https://doi.org/10.1073/pnas.0509498103 https://doi.org/10.3390/su1041195 https://doi.org/10.1146/annurev.energy.29.042203.104034 https://doi.org/10.1257/aer.90.1.282 https://doi.org/10.1016/j.econlet.2007.07.001 [31] Deepak Malghan Kraev, E. 2002. ―Stocks, flows and complemantarity: formalizing a basic insight of ecological economics.‖ Ecological Economics 43: 277–286. https://doi.org/10.1016/S0921-8009(02)00220-3 Lotka, A. J. 1925. Elements of Physical Biology. Baltimore: Williams & Wilkins Company. Malghan, D. 2011. ―A dimensionally consistent aggregation framework for biophysical metrics.‖ Ecological Economics 70: 900–909. https://doi.org/10.1016/j.ecolecon.2010.12.015 Matthews, E., C. Amann, S. Bringezu, M. Fischer-Kowalski, W. Httler, R. Kleijn, Y. Moriguchi, C. Ottke, E. Rodenburg, D. Rogich, H. Schandl, H. Schtz, E. Van Der Voet, and H. Weisz. 2000. The Weight of Nations: Material Outflows from Industrial Economies. World Resources Institute. Mayumi, K. and M. Giampietro. 2010. ―Dimensions and logarithmic function in economics: A short critical analysis.‖. Ecological Economics 69: 1604–1609. https://doi.org/10.1016/j.ecolecon.2010.03.017 Mayumi, K. and M. Giampietro. 2012. ―Response to ―dimensions and logarithmic function in economics : A comment .‖ Ecological Economics 75: 12–14. https://doi.org/10.1016/j.ecolecon.2012.01.018 National Research Council, 2004. Materials Count: The Case for Material Flow Analysis. The National Academies Press. Rees, W. E. 1992. ―Ecological footprints and appropriated carrying capacity: what urban economics leaves out.‖ Environment and Urbanization 4 (2): 121–30. https://doi.org/10.1177/095624789200400212 Rojstaczer, S., Sterling, S. and N. Moore. 2001. ―Human appropriation of photosynthesis products.‖ Science 294: 2549–2552. https://doi.org/10.1126/science.1064375 Røpke, I. 2004. ―The early history of modern ecological economics.‖ Ecological Economics 50: 293–314. https://doi.org/10.1016/j.ecolecon.2004.02.012 Temple, J. 2012. ―The calibration of CES production functions.‖ Journal of Macroeconomics 34: 294–303. https://doi.org/10.1016/j.jmacro.2011.12.006 Tukey, J. W. 1957. ―On the comparative anatomy of transformations.‖ The Annals of Mathematical Statistics 28: 602–632. https://doi.org/10.1214/aoms/1177706875 Vitousek, P.M., P.R. Ehrilch, A.H. Ehrlich, and P.A. Matson. 1986. ―Human appropriation of the products of photosyntheis.‖ Bioscience 36: 368–373. https://doi.org/10.2307/1310258 Wackernagel, M., C. Monfreda, D. Moran, S. Goldfinger, D. Deumling and M. Murray. 2004. National footprint and biocapacity accounts 2004: the underlying calculation method. Tech. rep., Global Footprint Network. Wackernagel, M. and W. Rees. 1996. Our ecological footprint: reducing human impact on the earth. New society. Wernick, I., and J. Ausubel. 1995. ―National material metrics for industrial ecology.‖ Resources Policy 21: 189–98. https://doi.org/10.1016/0301- 4207(96)89789-3 https://doi.org/10.1016/S0921-8009(02)00220-3 https://doi.org/10.1016/j.ecolecon.2010.12.015 https://doi.org/10.1016/j.ecolecon.2010.03.017 https://doi.org/10.1016/j.ecolecon.2012.01.018 https://doi.org/10.1177/095624789200400212 https://doi.org/10.1126/science.1064375 https://doi.org/10.1016/j.ecolecon.2004.02.012 https://doi.org/10.1016/j.jmacro.2011.12.006 https://doi.org/10.1214/aoms/1177706875 https://doi.org/10.2307/1310258 https://doi.org/10.1016/0301-4207(96)89789-3 https://doi.org/10.1016/0301-4207(96)89789-3