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Response to “dimensions and logarithmic function in economics: A comment”
Ecological Economics 75 (2012) 12–14
Contents lists available at SciVerse ScienceDirect
Ecological Economics journal homepage: www.elsevier.com/locate/ecolecon
Commentary
Response to “dimensions and logarithmic function in economics: A comment” Kozo Mayumi a,⁎, Mario Giampietro b a b
Faculty of Integrated Arts and Sciences, The University of Tokushima, Minami-Josanjima 1–1, Tokushima City 770–8502, Japan ICREA Research Professor, Institute of Environmental Science and Technology (ICTA), Autonomous University of Barcelona (UAB), 08193 Bellaterra, Barcelona, Spain
a r t i c l e
i n f o
Available online 9 February 2012 JEL classification: B41 C01 E01 Keywords: Dimensions Logarithmic function Nicholas Georgescu-Roegen Macroeconomics Cobb–Douglas function
a b s t r a c t This piece is a response to Chilarescu and Viasu’s comment on our previously published article in Ecological Economics. We give an analytical expression of the logarithmic function and show that putting a number with dimensions in logarithmic function does violate the principle of dimensional homogeneity: dimensionally different numbers cannot be summed up. We present other examples of this analytical error, including several by Nobel Prize winners in Economics. We show that Proposition 1 by Chilarescu and Viasu is too obvious to be necessary to prove. We briefly touch on the hidden analytical fallacy associated with empirical works in economics that extensively use Cobb-Douglas or CES production function. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Constantin Chilarescu and Ioana Viasu wrote (2012) a comment on our previously published paper in Ecological Economics (Mayumi and Giampietro, 2010) and they claimed: (1) putting a number with dimensions in logarithmic function causes no analytical problems; and (2) there are no dimensional errors in the Cobb–Douglas or the CES production function. At this moment it must be emphasized that we have not mentioned anything about their second point on our paper. However, it is very instructive to investigate the nature of the production process in the economic process and its analytical representation, so we will touch on these two issues. Section 1 gives an analytical expression of the logarithmic function and shows that putting a number with dimensions in a logarithmic function does violate the principle of dimensional homogeneity: dimensionally different numbers cannot be summed up. This section presents other examples of this type of analytical error committed by several Nobel Prize winners in Economics. Section 2 shows that Proposition 1 by Chilarescu and Viasu (2012) is too obvious to be necessary to prove. Section 2 also touches on the hidden analytical fallacy associated with empirical works in economics that extensively use Cobb–Douglas or CES production function. A conclusion follows.
⁎ Corresponding author. E-mail address: [email protected] (K. Mayumi). 0921-8009/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2012.01.018
2. The Logarithmic Function and Dimensions: A Fatal Analytical Fallacy Revisited First, start with the following well known expression (Mayumi and Giampietro, 2010; Mayumi et al., 2011),
loge ð1 þ xÞ ¼ lnð1 þ xÞ ¼ x−
x2 x3 ð−1Þn−1 xn þ þ; ; ; þ þ; ; ; ; 2 3 n
where−1b x ≤ 1:
ð1Þ ð2Þ
Replacing x by −x in relation (1) produces the following,
lnð1−xÞ ¼ −x−
x2 x3 xn − −; ; ; − þ; ; ; ; : 2 3 n
ð3Þ
Combining these two expressions (1) and (3) we have the following, ! 3 5 2m−1 1þx x x x ¼ 2 x þ þ þ; ; ; þ þ; ; ; : ln 1−x 3 5 2m−1
ð4Þ
Therefore, a unique value of x (− 1 b x b 1) exists corresponding to z which is positive.
K. Mayumi, M. Giampietro / Ecological Economics 75 (2012) 12–14
Thus, for every positive real number z, we can safely define the logarithmic function as follows using the relation x = (z − 1) / (z + 1): (
z−1 1 z−1 3 1 z−1 5 þ þ þ; ; ; zþ1 3 zþ1 5 zþ1 ) 1 z−1 2m−1 þ; ; ; : þ 2m−1 z þ 1
ð5Þ
lnz ¼ 2
Fig. 1 shows the mapping z = (1 + x) / (1 − x) for − 1 b x b 1. The right hand side is the analytical expression of the logarithmic function. It is obvious that if the value of z is expressed in US$, this operation will create both “a square dollar” and “a cubic dollar”, which are nonsensical, let alone “higher order dollars”. Clearly this is a violation of the principle of dimensional homogeneity. Therefore, Chilarescu and Viasu's claim (2012) that “Of course this is not true” is completely wrong. So contrary to Chilarescu and Viasu's claim, log(V/ L) or log W in Arrow et al.'s paper (1961) can never be used as a scientific representation. If the readers still cannot understand why the dimensional arguments cannot be put into the logarithmic function, please visit the following sites: 1. http://van.physics.illinois.edu/qa/listing.php?id=7238 2. http://en.wikipedia.org/wiki/Dimensional_analysis#Polynomials_ and_transcendental_functions To further illustrate our point we introduce additional examples of this analytical error committed by several Nobel Prize winners in Economics. The first example is from Leontief (1982, p. 104). Leontief states “A typical example of a theoretical “production function” intended to describe the relationship between, say, the amount of steel produced, y1, and the quantities of the four different inputs, y2, y3, y4, and y5 needed to produce it is, for instance, described as follows” (italics added) and he expressed the following equation, 2 3 lny1 ¼ a1 lnG þ ð1−a1 Þ lnG :
ð6Þ
Since y1 must be a physical quantity (weight) expressed in kg or ton, for example, the expression ln y1 violates the principle of dimensional homogenieity. Similarly macroeconomics often uses the logarithmic specification. Consider three papers of Lobert Lucas, Jr. that we happened to encounter. In the paper, “Making A Miracle” (Lucas 1993), perhaps without any doubt Allan D. Searle's result (1945) is cited in Fig. 1. According to Lucas, “Searle plotted man-hours per vessel against number of vessels completed to date in that yard on log–log paper (Lucas 1993,
z
z=
x= 1
-1
0
1
1+ x 2 = −1 − 1− x x −1
z −1 z+1
x
Fig. 1. The monotonically increasing function z(x) for − 1 b x b 1.
13
pp. 259–260, italics added). Man-hours per vessel has the dimension of time, i.e., hours. So, it is not allowed to put this argument in a logarithmic function. In another paper, “Macroeconomic Priorities”, Lucas (2003) states that “[u]sing annual U.S. data for the period 1947–2001, the standard deviation of the log of real per capita consumption about a linear trend is 0.0032” (Lucas 2003, p. 4, italics added). In yet another paper, “Trade and the Diffusion of the Industrial Revolution” (Lucas 2009), he mentions that we “consider a world of one sector ‘AK’ economies in which an economy's GDP per capita is proportional to its stock of human capital, knowledge capital, or whatever term you like” (Lucas 2009, p. 5). Lucas created four figures (Figs. 11–15 in that paper) all of which have the same horizontal axis, Log per capita GDP. All these figures are nonsensical according to what has been said thus far. 3. Analytical Representation of the Economic Process Physics dictates that both sides of any expression must have the same dimension. In principle, all dimensions of both a set of parameters and a set of variables can be selected freely as long as both sides of the expression must have the same dimension. For example, given a relation w = kxy, suppose that k is a parameter and that x, y and w are variables. Naturally w has the same dimension of kxy. Suppose that both w and kx have the dimension of time, then it is obvious that y must be a pure number. Proposition 1 given by Chilarescu and Viasu (2012) is then too obvious to be necessary to prove. However, Proposition 1 does not tell us whether or not the selected dimensional choice for a given expression has an operational meaning or relevance for the purpose of the analysis. There are two important issues associated with the aggregated production functions used in economics such as the Cobb–Douglas function that have not been examined critically. First of all, let's consider the standard Cobb–Douglas function as follows (Mayumi et al., 2011), α 1−α
Y ¼ AK L
ð7Þ
Suppose that K, L, and Y are represented in terms of the US dollar. Since α + (1 − α) = 1, the dimension of the left-hand side, the US dollar, is compatible with that of the right-hand side as a whole if A is a dimensionless pure number. However, each term on the right-hand side, i.e., K α and L 1 − α, does not make any sense unless pffiffiffiffiffiffiffiffiffiα = 0 or 1. Suppose α = 1/2, the reader can 1 1 wonder if K 2 ¼ L2 ¼ 100 has any operational meaning. We must reflect ourselves whether or not “cubic dollars” or “square dollars” or “root dollars” has any meaning for economic life. Secondly, there is an analytical misrepresentation of the aggregate economic process over time: what is the actual product of the economic process and its relevant analytical representation? Those neoclassical economists adopting the substitution assumption and using Cobb–Douglass or CES production functions have not paid due attention to the essential distinction between flows and funds in the material production process (Georgescu-Roegen, 1992). This distinction leads to the heart of the issue which is the length of a time horizon. It is the pre-analytical selection of a time horizon for the analysis, a descriptive domain associated with the choice of a given time scale, that defines what is produced by an economy. On a short time horizon one can decide to focus the analysis on the production of goods and services (performing an analysis of the flows). On a longer time horizon, when accounting for economic sustainability, one can decide to focus the analysis on the very processes required to produce and consume goods and services by performing an analysis of the reproduction and expansion of the funds. These two different types of analysis will provide different conclusions to the modeler and would require a different selection of models, variables and parameters. Neglecting the distinction between funds and flows (and neglecting the need of
14
K. Mayumi, M. Giampietro / Ecological Economics 75 (2012) 12–14
representing their production and reproduction using different attributes and models referring to different time scales) results in a systematic indifference to the biophysical foundation of economic activities. It is not surprising then that the curve fitting practice typical of aggregated production functions prevails. On this point Georgescu-Roegen states: “several standard economists have used the partial derivatives of F [the production function] with respect to t [technological progress], completely ignoring when they were trapped the fact that neither the function F, nor the arguments K [capital], H [labor], are the same in 1980 as in 1960: F1980(K'80,H'80) − F1960(K'60,H'60) is not a difference on which the derivative is based. This fumble proves that even in mathematics we cannot do without epistemology” (Georgescu-Roegen, 1992). Since the real outcome of the economic process is the production of the production and consumption process, any analytical representation cannot capture this evolutionary nature of the economic process in analytical terms (Mayumi 2009; Giampietro et al., 2011).
4. Conclusion Concerning the issue of dimensions we have shown that it is an analytical fallacy to put dimensional arguments in logarithmic functions or the meaningless variables in Cobb–Douglas functions. Surprisingly there is one example in which these two types of analytical fallacy simultaneously have been committed. Paul A. Samuelson (1974, p. 1268) wrote the following:
1
U ¼ logtea þ ðsalt Þ2 :
ð8Þ
We do hope that our response is useful to reorient our field toward a more analytically sound basis and to reach more realistic representations of the economic process.
Acknowledgements We would like to express our sincere thanks to Richard B. Howarth, Editor in Chief, and to Anne Carter Aitken, Managing Editor, for their invitation to writing this response. We would like to express our gratitude to John Polimeni for his valuable and tremendous help in improving the language in this response. We would like to emphasize that all responsibility for the way in which we have taken advice and criticism into consideration remains with us. References Arrow, K.J., Chenery, H.B., Minhas, B.S., Solow, R.M., 1961. Capital–labor substitution and economic efficiency. The Review of Economics and Statistics 43 (3), 225–250. Chilarescu, C. and Viasu, I. 2012. Dimensions and logarithmic function in economics: a comment. Ecological Economics. Georgescu-Roegen, N., 1992. Nicholas Georgescu-Roegen about himself. In: Szenberg, M. (Ed.), Eminent Economists: Their Life Philosophies. Cambridge University Press, Cambridge, pp. 128–159. Giampietro, M., Mayumi, K., Sorman, A., 2011. The Metabolic Pattern of Societies: Where Economists Fall Short. Roultedge, London. Leontief, W., 1982. Academic economists. Science 217, 104–107. Lucas Jr., R., 1993. Making miracle. Econometrica 61 (2), 251–272. Lucas Jr., R., 2003. Macroeconomic priorities. American Economic Review 93 (1), 1–14. Lucas Jr., R., 2009. Trade and the diffusion of the industrial revolution. American Economic Journal: Macroeconomics 1 (1), 1–25. Mayumi, K., 2009. Nicholas Georgescu-Roegen: his bioeconomics approach to development and change. Development and Change 40 (6), 1235–1254. Mayumi, K., Giampietro, M., 2010. Dimensions and logarithmic function in economics: a short critical analysis. Ecological Economics 69, 1604–1609. Mayumi, K., Giampietro, M., Ramos-Martin, J., 2011. Reconsideration of dimensions and curve fitting practice in economics elaborating on Georgescu-Roegen's economic methodology” UHE Working Paper 2011_05. Department of Economics and Economic History, Autonomous University of Barcelona, Barcelona, Spain http:// www.h-economica.uab.es/wps/2011_05.pdf. Samuelson, P.A., 1974. Complementarity: an essay on the 40th anniversary of the Hicks– Allen revolution in demand theory. Journal of Economic Literature 12 (4), 1255–1289. Searle, A.D., 1945. Productivity changes in selected wartime shipbuilding programs. Monthly Labor Reviews 61, 1132–1147.
Contents lists available at SciVerse ScienceDirect
Ecological Economics journal homepage: www.elsevier.com/locate/ecolecon
Commentary
Response to “dimensions and logarithmic function in economics: A comment” Kozo Mayumi a,⁎, Mario Giampietro b a b
Faculty of Integrated Arts and Sciences, The University of Tokushima, Minami-Josanjima 1–1, Tokushima City 770–8502, Japan ICREA Research Professor, Institute of Environmental Science and Technology (ICTA), Autonomous University of Barcelona (UAB), 08193 Bellaterra, Barcelona, Spain
a r t i c l e
i n f o
Available online 9 February 2012 JEL classification: B41 C01 E01 Keywords: Dimensions Logarithmic function Nicholas Georgescu-Roegen Macroeconomics Cobb–Douglas function
a b s t r a c t This piece is a response to Chilarescu and Viasu’s comment on our previously published article in Ecological Economics. We give an analytical expression of the logarithmic function and show that putting a number with dimensions in logarithmic function does violate the principle of dimensional homogeneity: dimensionally different numbers cannot be summed up. We present other examples of this analytical error, including several by Nobel Prize winners in Economics. We show that Proposition 1 by Chilarescu and Viasu is too obvious to be necessary to prove. We briefly touch on the hidden analytical fallacy associated with empirical works in economics that extensively use Cobb-Douglas or CES production function. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Constantin Chilarescu and Ioana Viasu wrote (2012) a comment on our previously published paper in Ecological Economics (Mayumi and Giampietro, 2010) and they claimed: (1) putting a number with dimensions in logarithmic function causes no analytical problems; and (2) there are no dimensional errors in the Cobb–Douglas or the CES production function. At this moment it must be emphasized that we have not mentioned anything about their second point on our paper. However, it is very instructive to investigate the nature of the production process in the economic process and its analytical representation, so we will touch on these two issues. Section 1 gives an analytical expression of the logarithmic function and shows that putting a number with dimensions in a logarithmic function does violate the principle of dimensional homogeneity: dimensionally different numbers cannot be summed up. This section presents other examples of this type of analytical error committed by several Nobel Prize winners in Economics. Section 2 shows that Proposition 1 by Chilarescu and Viasu (2012) is too obvious to be necessary to prove. Section 2 also touches on the hidden analytical fallacy associated with empirical works in economics that extensively use Cobb–Douglas or CES production function. A conclusion follows.
⁎ Corresponding author. E-mail address: [email protected] (K. Mayumi). 0921-8009/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2012.01.018
2. The Logarithmic Function and Dimensions: A Fatal Analytical Fallacy Revisited First, start with the following well known expression (Mayumi and Giampietro, 2010; Mayumi et al., 2011),
loge ð1 þ xÞ ¼ lnð1 þ xÞ ¼ x−
x2 x3 ð−1Þn−1 xn þ þ; ; ; þ þ; ; ; ; 2 3 n
where−1b x ≤ 1:
ð1Þ ð2Þ
Replacing x by −x in relation (1) produces the following,
lnð1−xÞ ¼ −x−
x2 x3 xn − −; ; ; − þ; ; ; ; : 2 3 n
ð3Þ
Combining these two expressions (1) and (3) we have the following, ! 3 5 2m−1 1þx x x x ¼ 2 x þ þ þ; ; ; þ þ; ; ; : ln 1−x 3 5 2m−1
ð4Þ
Therefore, a unique value of x (− 1 b x b 1) exists corresponding to z which is positive.
K. Mayumi, M. Giampietro / Ecological Economics 75 (2012) 12–14
Thus, for every positive real number z, we can safely define the logarithmic function as follows using the relation x = (z − 1) / (z + 1): (
z−1 1 z−1 3 1 z−1 5 þ þ þ; ; ; zþ1 3 zþ1 5 zþ1 ) 1 z−1 2m−1 þ; ; ; : þ 2m−1 z þ 1
ð5Þ
lnz ¼ 2
Fig. 1 shows the mapping z = (1 + x) / (1 − x) for − 1 b x b 1. The right hand side is the analytical expression of the logarithmic function. It is obvious that if the value of z is expressed in US$, this operation will create both “a square dollar” and “a cubic dollar”, which are nonsensical, let alone “higher order dollars”. Clearly this is a violation of the principle of dimensional homogeneity. Therefore, Chilarescu and Viasu's claim (2012) that “Of course this is not true” is completely wrong. So contrary to Chilarescu and Viasu's claim, log(V/ L) or log W in Arrow et al.'s paper (1961) can never be used as a scientific representation. If the readers still cannot understand why the dimensional arguments cannot be put into the logarithmic function, please visit the following sites: 1. http://van.physics.illinois.edu/qa/listing.php?id=7238 2. http://en.wikipedia.org/wiki/Dimensional_analysis#Polynomials_ and_transcendental_functions To further illustrate our point we introduce additional examples of this analytical error committed by several Nobel Prize winners in Economics. The first example is from Leontief (1982, p. 104). Leontief states “A typical example of a theoretical “production function” intended to describe the relationship between, say, the amount of steel produced, y1, and the quantities of the four different inputs, y2, y3, y4, and y5 needed to produce it is, for instance, described as follows” (italics added) and he expressed the following equation, 2 3 lny1 ¼ a1 lnG þ ð1−a1 Þ lnG :
ð6Þ
Since y1 must be a physical quantity (weight) expressed in kg or ton, for example, the expression ln y1 violates the principle of dimensional homogenieity. Similarly macroeconomics often uses the logarithmic specification. Consider three papers of Lobert Lucas, Jr. that we happened to encounter. In the paper, “Making A Miracle” (Lucas 1993), perhaps without any doubt Allan D. Searle's result (1945) is cited in Fig. 1. According to Lucas, “Searle plotted man-hours per vessel against number of vessels completed to date in that yard on log–log paper (Lucas 1993,
z
z=
x= 1
-1
0
1
1+ x 2 = −1 − 1− x x −1
z −1 z+1
x
Fig. 1. The monotonically increasing function z(x) for − 1 b x b 1.
13
pp. 259–260, italics added). Man-hours per vessel has the dimension of time, i.e., hours. So, it is not allowed to put this argument in a logarithmic function. In another paper, “Macroeconomic Priorities”, Lucas (2003) states that “[u]sing annual U.S. data for the period 1947–2001, the standard deviation of the log of real per capita consumption about a linear trend is 0.0032” (Lucas 2003, p. 4, italics added). In yet another paper, “Trade and the Diffusion of the Industrial Revolution” (Lucas 2009), he mentions that we “consider a world of one sector ‘AK’ economies in which an economy's GDP per capita is proportional to its stock of human capital, knowledge capital, or whatever term you like” (Lucas 2009, p. 5). Lucas created four figures (Figs. 11–15 in that paper) all of which have the same horizontal axis, Log per capita GDP. All these figures are nonsensical according to what has been said thus far. 3. Analytical Representation of the Economic Process Physics dictates that both sides of any expression must have the same dimension. In principle, all dimensions of both a set of parameters and a set of variables can be selected freely as long as both sides of the expression must have the same dimension. For example, given a relation w = kxy, suppose that k is a parameter and that x, y and w are variables. Naturally w has the same dimension of kxy. Suppose that both w and kx have the dimension of time, then it is obvious that y must be a pure number. Proposition 1 given by Chilarescu and Viasu (2012) is then too obvious to be necessary to prove. However, Proposition 1 does not tell us whether or not the selected dimensional choice for a given expression has an operational meaning or relevance for the purpose of the analysis. There are two important issues associated with the aggregated production functions used in economics such as the Cobb–Douglas function that have not been examined critically. First of all, let's consider the standard Cobb–Douglas function as follows (Mayumi et al., 2011), α 1−α
Y ¼ AK L
ð7Þ
Suppose that K, L, and Y are represented in terms of the US dollar. Since α + (1 − α) = 1, the dimension of the left-hand side, the US dollar, is compatible with that of the right-hand side as a whole if A is a dimensionless pure number. However, each term on the right-hand side, i.e., K α and L 1 − α, does not make any sense unless pffiffiffiffiffiffiffiffiffiα = 0 or 1. Suppose α = 1/2, the reader can 1 1 wonder if K 2 ¼ L2 ¼ 100 has any operational meaning. We must reflect ourselves whether or not “cubic dollars” or “square dollars” or “root dollars” has any meaning for economic life. Secondly, there is an analytical misrepresentation of the aggregate economic process over time: what is the actual product of the economic process and its relevant analytical representation? Those neoclassical economists adopting the substitution assumption and using Cobb–Douglass or CES production functions have not paid due attention to the essential distinction between flows and funds in the material production process (Georgescu-Roegen, 1992). This distinction leads to the heart of the issue which is the length of a time horizon. It is the pre-analytical selection of a time horizon for the analysis, a descriptive domain associated with the choice of a given time scale, that defines what is produced by an economy. On a short time horizon one can decide to focus the analysis on the production of goods and services (performing an analysis of the flows). On a longer time horizon, when accounting for economic sustainability, one can decide to focus the analysis on the very processes required to produce and consume goods and services by performing an analysis of the reproduction and expansion of the funds. These two different types of analysis will provide different conclusions to the modeler and would require a different selection of models, variables and parameters. Neglecting the distinction between funds and flows (and neglecting the need of
14
K. Mayumi, M. Giampietro / Ecological Economics 75 (2012) 12–14
representing their production and reproduction using different attributes and models referring to different time scales) results in a systematic indifference to the biophysical foundation of economic activities. It is not surprising then that the curve fitting practice typical of aggregated production functions prevails. On this point Georgescu-Roegen states: “several standard economists have used the partial derivatives of F [the production function] with respect to t [technological progress], completely ignoring when they were trapped the fact that neither the function F, nor the arguments K [capital], H [labor], are the same in 1980 as in 1960: F1980(K'80,H'80) − F1960(K'60,H'60) is not a difference on which the derivative is based. This fumble proves that even in mathematics we cannot do without epistemology” (Georgescu-Roegen, 1992). Since the real outcome of the economic process is the production of the production and consumption process, any analytical representation cannot capture this evolutionary nature of the economic process in analytical terms (Mayumi 2009; Giampietro et al., 2011).
4. Conclusion Concerning the issue of dimensions we have shown that it is an analytical fallacy to put dimensional arguments in logarithmic functions or the meaningless variables in Cobb–Douglas functions. Surprisingly there is one example in which these two types of analytical fallacy simultaneously have been committed. Paul A. Samuelson (1974, p. 1268) wrote the following:
1
U ¼ logtea þ ðsalt Þ2 :
ð8Þ
We do hope that our response is useful to reorient our field toward a more analytically sound basis and to reach more realistic representations of the economic process.
Acknowledgements We would like to express our sincere thanks to Richard B. Howarth, Editor in Chief, and to Anne Carter Aitken, Managing Editor, for their invitation to writing this response. We would like to express our gratitude to John Polimeni for his valuable and tremendous help in improving the language in this response. We would like to emphasize that all responsibility for the way in which we have taken advice and criticism into consideration remains with us. References Arrow, K.J., Chenery, H.B., Minhas, B.S., Solow, R.M., 1961. Capital–labor substitution and economic efficiency. The Review of Economics and Statistics 43 (3), 225–250. Chilarescu, C. and Viasu, I. 2012. Dimensions and logarithmic function in economics: a comment. Ecological Economics. Georgescu-Roegen, N., 1992. Nicholas Georgescu-Roegen about himself. In: Szenberg, M. (Ed.), Eminent Economists: Their Life Philosophies. Cambridge University Press, Cambridge, pp. 128–159. Giampietro, M., Mayumi, K., Sorman, A., 2011. The Metabolic Pattern of Societies: Where Economists Fall Short. Roultedge, London. Leontief, W., 1982. Academic economists. Science 217, 104–107. Lucas Jr., R., 1993. Making miracle. Econometrica 61 (2), 251–272. Lucas Jr., R., 2003. Macroeconomic priorities. American Economic Review 93 (1), 1–14. Lucas Jr., R., 2009. Trade and the diffusion of the industrial revolution. American Economic Journal: Macroeconomics 1 (1), 1–25. Mayumi, K., 2009. Nicholas Georgescu-Roegen: his bioeconomics approach to development and change. Development and Change 40 (6), 1235–1254. Mayumi, K., Giampietro, M., 2010. Dimensions and logarithmic function in economics: a short critical analysis. Ecological Economics 69, 1604–1609. Mayumi, K., Giampietro, M., Ramos-Martin, J., 2011. Reconsideration of dimensions and curve fitting practice in economics elaborating on Georgescu-Roegen's economic methodology” UHE Working Paper 2011_05. Department of Economics and Economic History, Autonomous University of Barcelona, Barcelona, Spain http:// www.h-economica.uab.es/wps/2011_05.pdf. Samuelson, P.A., 1974. Complementarity: an essay on the 40th anniversary of the Hicks– Allen revolution in demand theory. Journal of Economic Literature 12 (4), 1255–1289. Searle, A.D., 1945. Productivity changes in selected wartime shipbuilding programs. Monthly Labor Reviews 61, 1132–1147.