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Dimensions and logarithmic function in economics: A comment
Ecological Economics 75 (2012) 10–11
Contents lists available at SciVerse ScienceDirect
Ecological Economics journal homepage: www.elsevier.com/locate/ecolecon
Commentary
Dimensions and logarithmic function in economics: A comment Constantin Chilarescu a,⁎, Ioana Viasu b a b
Laboratory CLERSE, University of Lille1, France Faculty of Economics and Business Administration, West University of Timisoara, Romania
a r t i c l e
i n f o
Article history: Received 21 October 2011 Accepted 16 January 2012 Available online 3 February 2012
a b s t r a c t In this paper we give some comments onto the paper of Kozo Mayumi and Mario Giampietro, recently published in this journal and, finally we present some conclusions. © 2012 Elsevier B.V. All rights reserved.
JEL classifications: C02 C51 C62 Keywords: Production functions Dimensions
1. Introduction Mayumi and Giampietro (henceforth MG) (2010, Ecological Economics, 69, 1604–1609) analyzing the different aspects of economic interactions, consider that this approach requires a special attention to be paid to dimensional issues, especially when we work with exponential functions or with logarithmic functions. What they mean by dimensions are the elementary units (such as mass, length, time, or currency) which are required for data of the quantitative values assigned to the variables. They claim that it is an analytical error to put a dimensional number x into exponential functions and logarithmic functions. They give several examples of this analytical error both in ecological economics and conventional economics. Among the several examples given in this paper, MG consider the famous paper of Arrow et al. (1961) where they tried to investigate the substitution between capital and labor within the neoclassical production theory. In the first section of that paper the fourth authors used a regression analysis of the following variables: • V value added in thousands of U.S. dollars; • L labor input in man-years; • W wages (total labor cost divided by L) in dollars per man-year.
⁎ Corresponding author. E-mail address: [email protected] (C. Chilarescu). 0921-8009/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2012.01.017
Arrow et al. statistically tested the following simple relation using these three variables: log
V ¼ loga þ blogW þ ε L
ð1Þ
MG claim that this relation cannot be used judging by the dimensions V/L and W which they used. Of course this is not true. In our opinion, MG's discussion on the CES production function merits further examination. That economics often ignores dimensions and units does not necessarily mean that unstated but implied dimensions and units are wrong or invalid. Variables must have dimensions. Parameters may or may not have dimensions. In the case of the Cobb–Douglas production function (see Cobb and Douglas, 1928) β α
F ðK; LÞ ¼ AK L ; or in the case of the CES production function −ρ −ρ −1ρ F ðK; LÞ ¼ γ δK þ ð1−δÞL parameters A and respectively γ have dimensions that force the function's left and right side dimensions to match. A long and complete discussion on this subject can be found in the paper of Folsom and Gonzalez (2005). In the next section of our paper we give a complete characterization of the two famous production functions: the Cobb–Douglas production function and the CES production function and we show
C. Chilarescu, I. Viasu / Ecological Economics 75 (2012) 10–11
11
From here it immediately follows that the unit dimension of A will
that these two production functions are consistent with respect to the dimension units.
be
2. The Neoclassical Production Functions
v ¼ uu1 u2 :
We start this section with the study of the Cobb–Douglas production function.
To prove that γ is a constant whose dimension unit depends on the dimension units of K, L and F we consider the following differential equation:
−β −α
Definition 1. A production function is a a mapping σ¼
2
F : Rþ →Rþ such that Y = F(K, L) is the maximum output that can be produced for a given set of inputs K and L. We suppose that F∈C2 , that is F has continuous partial derivatives until the second orders. Under these hypotheses, for all ðK; LÞ∈R2þ we can write: ∂F ∂F ∂F dF ðK; LÞ ∂K dK ∂F dL ¼ F þ ∂L : dK þ dL⇒ dF ðK; LÞ ¼ F L F ðK; LÞ K ∂K ∂L K L
ð2Þ
∂F K
∂F ∂L F L
k¼
and σ is the elasticity of substitution between capital and labor. It is just a simple exercise to prove that σ is a pure number with no unit dimension. The above equation can be written as:
ð3Þ
This is a classical differential equation with separable variables whose solution is given by F ðK; LÞ ¼ AK L :
ð4Þ
Proposition 1. Let F(K, L) be the Cobb–Douglas production function or the CES production function. The unit dimension of the constants A and γ depends on the unit dimension of the variables K and L, and of the unit dimension of F. Proof of the Proposition 1. Now we suppose that the dimension unit of F is denoted by u and the dimension units of K and L are denoted by u1 and respectively by u2. First observe that β and α are pure numbers, (invariant constants) and they do not have dimensions. Instead, the parameter A is also constant but not invariant constants because it needs to have a dimension. Let v be the unit dimension of A. Now we can find the dimension of A. We denote Zu the value of F expressed in the dimension of u units, by Xu1 the value of K expressed in the dimension of u1 units and by Yu2 the value of L expressed in the dimension of u2 units, where Z, X, Y are pure numbers. Consequently we have: β β
α α
Zu ¼ AX u1 Y u2 :
ð6Þ
−ρ −ρ −1ρ F ðK; LÞ ¼ γ δK þ ð1−δÞL ; where ρ¼
β α
K F ðK; LÞ ; f ðkÞ ¼ L L
whose solution, after some algebraic manipulations is given by
we can write dF ðK; LÞ dK dL ¼β þα : F ðK; LÞ K L
ð5Þ
where
f″ 1 1 f′ ¼ þ f′ σ k f
Denoting by β ¼ ∂K α¼ F
f ′ðf −kf ′Þ kf ′f ″
1−σ : σ
Proceeding in the same way as in the case of the Cobb–Douglas production function we conclude that unit dimension of the constant γ depends on the unit dimension of the variables K and L, and of the unit dimension of textitF and thus the proof is completed. 3. Some Conclusions In our opinion MG's short critical analysis has not shown us any dimensional errors in the Cobb–Douglas production function or in the CES production function. We cannot comment the other examples presented the paper. 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. Cobb, C.W., Douglas, P.H., 1928. A theory of production. The American Economic Review 18 (1), 139–165. Folsom, R.N., Gonzalez, R.A., 2005. Dimensions and economics: some answers. The Quarterly Journal of Austrian Economics 4, 45–65. Mayumi, K., Giampietro, M., 2010. Dimensions and logarithmic function in economics: a short critical analysis. Ecological Economics 69, 1604–1609.
Contents lists available at SciVerse ScienceDirect
Ecological Economics journal homepage: www.elsevier.com/locate/ecolecon
Commentary
Dimensions and logarithmic function in economics: A comment Constantin Chilarescu a,⁎, Ioana Viasu b a b
Laboratory CLERSE, University of Lille1, France Faculty of Economics and Business Administration, West University of Timisoara, Romania
a r t i c l e
i n f o
Article history: Received 21 October 2011 Accepted 16 January 2012 Available online 3 February 2012
a b s t r a c t In this paper we give some comments onto the paper of Kozo Mayumi and Mario Giampietro, recently published in this journal and, finally we present some conclusions. © 2012 Elsevier B.V. All rights reserved.
JEL classifications: C02 C51 C62 Keywords: Production functions Dimensions
1. Introduction Mayumi and Giampietro (henceforth MG) (2010, Ecological Economics, 69, 1604–1609) analyzing the different aspects of economic interactions, consider that this approach requires a special attention to be paid to dimensional issues, especially when we work with exponential functions or with logarithmic functions. What they mean by dimensions are the elementary units (such as mass, length, time, or currency) which are required for data of the quantitative values assigned to the variables. They claim that it is an analytical error to put a dimensional number x into exponential functions and logarithmic functions. They give several examples of this analytical error both in ecological economics and conventional economics. Among the several examples given in this paper, MG consider the famous paper of Arrow et al. (1961) where they tried to investigate the substitution between capital and labor within the neoclassical production theory. In the first section of that paper the fourth authors used a regression analysis of the following variables: • V value added in thousands of U.S. dollars; • L labor input in man-years; • W wages (total labor cost divided by L) in dollars per man-year.
⁎ Corresponding author. E-mail address: [email protected] (C. Chilarescu). 0921-8009/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2012.01.017
Arrow et al. statistically tested the following simple relation using these three variables: log
V ¼ loga þ blogW þ ε L
ð1Þ
MG claim that this relation cannot be used judging by the dimensions V/L and W which they used. Of course this is not true. In our opinion, MG's discussion on the CES production function merits further examination. That economics often ignores dimensions and units does not necessarily mean that unstated but implied dimensions and units are wrong or invalid. Variables must have dimensions. Parameters may or may not have dimensions. In the case of the Cobb–Douglas production function (see Cobb and Douglas, 1928) β α
F ðK; LÞ ¼ AK L ; or in the case of the CES production function −ρ −ρ −1ρ F ðK; LÞ ¼ γ δK þ ð1−δÞL parameters A and respectively γ have dimensions that force the function's left and right side dimensions to match. A long and complete discussion on this subject can be found in the paper of Folsom and Gonzalez (2005). In the next section of our paper we give a complete characterization of the two famous production functions: the Cobb–Douglas production function and the CES production function and we show
C. Chilarescu, I. Viasu / Ecological Economics 75 (2012) 10–11
11
From here it immediately follows that the unit dimension of A will
that these two production functions are consistent with respect to the dimension units.
be
2. The Neoclassical Production Functions
v ¼ uu1 u2 :
We start this section with the study of the Cobb–Douglas production function.
To prove that γ is a constant whose dimension unit depends on the dimension units of K, L and F we consider the following differential equation:
−β −α
Definition 1. A production function is a a mapping σ¼
2
F : Rþ →Rþ such that Y = F(K, L) is the maximum output that can be produced for a given set of inputs K and L. We suppose that F∈C2 , that is F has continuous partial derivatives until the second orders. Under these hypotheses, for all ðK; LÞ∈R2þ we can write: ∂F ∂F ∂F dF ðK; LÞ ∂K dK ∂F dL ¼ F þ ∂L : dK þ dL⇒ dF ðK; LÞ ¼ F L F ðK; LÞ K ∂K ∂L K L
ð2Þ
∂F K
∂F ∂L F L
k¼
and σ is the elasticity of substitution between capital and labor. It is just a simple exercise to prove that σ is a pure number with no unit dimension. The above equation can be written as:
ð3Þ
This is a classical differential equation with separable variables whose solution is given by F ðK; LÞ ¼ AK L :
ð4Þ
Proposition 1. Let F(K, L) be the Cobb–Douglas production function or the CES production function. The unit dimension of the constants A and γ depends on the unit dimension of the variables K and L, and of the unit dimension of F. Proof of the Proposition 1. Now we suppose that the dimension unit of F is denoted by u and the dimension units of K and L are denoted by u1 and respectively by u2. First observe that β and α are pure numbers, (invariant constants) and they do not have dimensions. Instead, the parameter A is also constant but not invariant constants because it needs to have a dimension. Let v be the unit dimension of A. Now we can find the dimension of A. We denote Zu the value of F expressed in the dimension of u units, by Xu1 the value of K expressed in the dimension of u1 units and by Yu2 the value of L expressed in the dimension of u2 units, where Z, X, Y are pure numbers. Consequently we have: β β
α α
Zu ¼ AX u1 Y u2 :
ð6Þ
−ρ −ρ −1ρ F ðK; LÞ ¼ γ δK þ ð1−δÞL ; where ρ¼
β α
K F ðK; LÞ ; f ðkÞ ¼ L L
whose solution, after some algebraic manipulations is given by
we can write dF ðK; LÞ dK dL ¼β þα : F ðK; LÞ K L
ð5Þ
where
f″ 1 1 f′ ¼ þ f′ σ k f
Denoting by β ¼ ∂K α¼ F
f ′ðf −kf ′Þ kf ′f ″
1−σ : σ
Proceeding in the same way as in the case of the Cobb–Douglas production function we conclude that unit dimension of the constant γ depends on the unit dimension of the variables K and L, and of the unit dimension of textitF and thus the proof is completed. 3. Some Conclusions In our opinion MG's short critical analysis has not shown us any dimensional errors in the Cobb–Douglas production function or in the CES production function. We cannot comment the other examples presented the paper. 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. Cobb, C.W., Douglas, P.H., 1928. A theory of production. The American Economic Review 18 (1), 139–165. Folsom, R.N., Gonzalez, R.A., 2005. Dimensions and economics: some answers. The Quarterly Journal of Austrian Economics 4, 45–65. Mayumi, K., Giampietro, M., 2010. Dimensions and logarithmic function in economics: a short critical analysis. Ecological Economics 69, 1604–1609.