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Comment on “C.E.S. production functions in the light of the Cambridge critique”
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Journal of Macroeconomics 30 (2008) 798–800 www.elsevier.com/locate/jmacro
Comment on ‘‘C.E.S. production functions in the light of the Cambridge critique” Giancarlo Gandolfo Sapienza University of Rome, Department of Economics, Via Castro Laurenziano 9, 00161 Roma, Italy Received 17 December 2007; accepted 18 December 2007 Available online 15 January 2008
Abstract The controversy revived in Bertram Schefold’s paper is based on three common assumptions: (1) the underlying techniques are linear (2) perfect competition obtains (3) the economy operates in a putty-putty context. The consequences of dropping these assumptions are discussed, and the relation between nonlinearity and the error due to the use of an ‘‘imprecise” production function is examined. Ó 2008 Elsevier Inc. All rights reserved. JEL classification: E13; D57; B51 Keyword: CES production function
This is a difficult and interesting paper. It deals with the controversy between the two Cambridges on capital theory and neoclassical production functions, and exemplifies it with CES production functions. (1) The theoretical debate revived in the paper focussed on the phenomena of reswitching (at the profit rate 0 < r < r1 technique 1 is chosen; at the profit rate r1 < r < r2 technique 2 is chosen; and at the profit rate r > r2 the technique chosen is again technique (1) and reverse capital deepening (as the rate of profit increases, more capital intensive, instead of more labour intensive, techniques are chosen). Reswitching is a sufficient condition for reverse capital deepening (RCD), but the latter can occur also in the absence of reswitching.
E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2007.12.006
G. Gandolfo / Journal of Macroeconomics 30 (2008) 798–800
799
These phenomena, that have been shown to be theoretically possible (reswitching requires the wage curve to be non-linear, as is indeed the case with heterogeneous products), undermine the possibility of working with an aggregate production function of the surrogate type (in fact, a surrogate production function can only be constructed if the wage curves are all linear). A companion paper (Hahn and Schefold 2006) shows that these phenomena are empirically possible, although in a very limited fraction of the cases, namely 3.65% (1% RCD), while the remaining 96.35% are consistent with the neoclassical paradigm. Let me now come to a brief discussion. The first concerns the assumption that the underlying techniques are linear, an assumption accepted by both opposing parties. This is certainly not true for many production processes. The second concerns the assumption that perfect competition obtains, again an assumption accepted by both parties. This assumption entails, for example, the uniformity of the wage rate and the uniformity of the rate of profit across industries. Again, this is certainly not the case. The third concerns the (implicit) assumption that we are in a putty–putty context, namely that there is both ex ante and ex post substitutability. Actually we live in a world of the putty-clay type, namely ex ante substitutability, but no ex post substitutability, which means the possibility of choice among the various techniques before the installation of the capital goods, but once the choice has been made and the capital goods relative to the technique chosen have been installed, substitutability disappears. These factual observations bring me to the following comments, one theoretical and the other empirical. From the theoretical point of view, as soon as we drop the assumption of perfect competition, the assumption that the rate of interest is identical with the rate of profit becomes untenable. The rate of interest is, like the wage rate, an element that has to be counted on the side of the costs of a technique. This has two consequences. First, prices are no longer determined according to the formula of normal prices p ¼ ð1 þ rÞAp þ wl used in the paper. Second, if the rate of profit is kept distinct from the rate of interest, there is nothing paradoxical that an increase in the rate of profit-given the rate of interest and the wage rate, namely costs- causes an increase in the amount of capital being used, because this increase is a way to increase profits. From the empirical point of view, the data used, being based on real-life observations, presumably include nonlinearities and non-uniformity, as well as putty-clay phenomena. If so, they are not suitable to either confirm or disprove the theory behind the controversy. In particular, I feel that the debate on reswitching and reverse capital deepening makes sense only ex ante. Ex post, changes in the wage rate (for example determined by bargains between firms and trade unions) do not lead to changes in techniques, but to changes in the rate of profit, at least in the short run. (2) An important idea set forth in the paper is to calculate an ‘‘imprecise” surrogate production function (or spurious production function) by computing a linear approximation to the underlying non-linear wage curves, and to study whether this approximation is good enough. Assuming that this is the case, the author sees ‘‘no obvious reason why the rate of substitution should remain constant in the face of large changes of the intensity of capital in a given state of technical knowledge”. This is, of course, a blow to CES, and, in my opinion, points to VES (Variable Elasticity of Substitution) production functions as
800
G. Gandolfo / Journal of Macroeconomics 30 (2008) 798–800
more plausible functional forms. Although VES production functions have been around for quite a while, they are not considered in the paper (but, after all, the Conference was devoted to CES production functions). Apart from this, the author points out that relying on a spurious production function may give rise to ‘‘constellations which cause the empirical results obtained by actual states of the economy to be misleading”. (3) I conclude with some observations on one of the points raised by the author, namely the problem of non-linearity and linear approximations. Usually, economists in all fields start from general non-linear models, and then linearize them around some reference point or path and examine the properties of the model locally valid. But how small is the neighbourhood of the linearization point for the linear approximation to be valid? This question is (almost) never asked, and economists go on as if such a neighbourhood had a practical relevance. Unfortunately this is not always the case, and this problem lies behind the measure of the errors involved in the construction of an ‘‘imprecise” surrogate production function, a measure that the author calls ‘‘declination”. If declination is sufficiently small, then the linear approximation to the non-linear wage curves can be reasonably used. The study of the conditions that might make declination small appears to be essential.
Journal of Macroeconomics 30 (2008) 798–800 www.elsevier.com/locate/jmacro
Comment on ‘‘C.E.S. production functions in the light of the Cambridge critique” Giancarlo Gandolfo Sapienza University of Rome, Department of Economics, Via Castro Laurenziano 9, 00161 Roma, Italy Received 17 December 2007; accepted 18 December 2007 Available online 15 January 2008
Abstract The controversy revived in Bertram Schefold’s paper is based on three common assumptions: (1) the underlying techniques are linear (2) perfect competition obtains (3) the economy operates in a putty-putty context. The consequences of dropping these assumptions are discussed, and the relation between nonlinearity and the error due to the use of an ‘‘imprecise” production function is examined. Ó 2008 Elsevier Inc. All rights reserved. JEL classification: E13; D57; B51 Keyword: CES production function
This is a difficult and interesting paper. It deals with the controversy between the two Cambridges on capital theory and neoclassical production functions, and exemplifies it with CES production functions. (1) The theoretical debate revived in the paper focussed on the phenomena of reswitching (at the profit rate 0 < r < r1 technique 1 is chosen; at the profit rate r1 < r < r2 technique 2 is chosen; and at the profit rate r > r2 the technique chosen is again technique (1) and reverse capital deepening (as the rate of profit increases, more capital intensive, instead of more labour intensive, techniques are chosen). Reswitching is a sufficient condition for reverse capital deepening (RCD), but the latter can occur also in the absence of reswitching.
E-mail address: [email protected] 0164-0704/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2007.12.006
G. Gandolfo / Journal of Macroeconomics 30 (2008) 798–800
799
These phenomena, that have been shown to be theoretically possible (reswitching requires the wage curve to be non-linear, as is indeed the case with heterogeneous products), undermine the possibility of working with an aggregate production function of the surrogate type (in fact, a surrogate production function can only be constructed if the wage curves are all linear). A companion paper (Hahn and Schefold 2006) shows that these phenomena are empirically possible, although in a very limited fraction of the cases, namely 3.65% (1% RCD), while the remaining 96.35% are consistent with the neoclassical paradigm. Let me now come to a brief discussion. The first concerns the assumption that the underlying techniques are linear, an assumption accepted by both opposing parties. This is certainly not true for many production processes. The second concerns the assumption that perfect competition obtains, again an assumption accepted by both parties. This assumption entails, for example, the uniformity of the wage rate and the uniformity of the rate of profit across industries. Again, this is certainly not the case. The third concerns the (implicit) assumption that we are in a putty–putty context, namely that there is both ex ante and ex post substitutability. Actually we live in a world of the putty-clay type, namely ex ante substitutability, but no ex post substitutability, which means the possibility of choice among the various techniques before the installation of the capital goods, but once the choice has been made and the capital goods relative to the technique chosen have been installed, substitutability disappears. These factual observations bring me to the following comments, one theoretical and the other empirical. From the theoretical point of view, as soon as we drop the assumption of perfect competition, the assumption that the rate of interest is identical with the rate of profit becomes untenable. The rate of interest is, like the wage rate, an element that has to be counted on the side of the costs of a technique. This has two consequences. First, prices are no longer determined according to the formula of normal prices p ¼ ð1 þ rÞAp þ wl used in the paper. Second, if the rate of profit is kept distinct from the rate of interest, there is nothing paradoxical that an increase in the rate of profit-given the rate of interest and the wage rate, namely costs- causes an increase in the amount of capital being used, because this increase is a way to increase profits. From the empirical point of view, the data used, being based on real-life observations, presumably include nonlinearities and non-uniformity, as well as putty-clay phenomena. If so, they are not suitable to either confirm or disprove the theory behind the controversy. In particular, I feel that the debate on reswitching and reverse capital deepening makes sense only ex ante. Ex post, changes in the wage rate (for example determined by bargains between firms and trade unions) do not lead to changes in techniques, but to changes in the rate of profit, at least in the short run. (2) An important idea set forth in the paper is to calculate an ‘‘imprecise” surrogate production function (or spurious production function) by computing a linear approximation to the underlying non-linear wage curves, and to study whether this approximation is good enough. Assuming that this is the case, the author sees ‘‘no obvious reason why the rate of substitution should remain constant in the face of large changes of the intensity of capital in a given state of technical knowledge”. This is, of course, a blow to CES, and, in my opinion, points to VES (Variable Elasticity of Substitution) production functions as
800
G. Gandolfo / Journal of Macroeconomics 30 (2008) 798–800
more plausible functional forms. Although VES production functions have been around for quite a while, they are not considered in the paper (but, after all, the Conference was devoted to CES production functions). Apart from this, the author points out that relying on a spurious production function may give rise to ‘‘constellations which cause the empirical results obtained by actual states of the economy to be misleading”. (3) I conclude with some observations on one of the points raised by the author, namely the problem of non-linearity and linear approximations. Usually, economists in all fields start from general non-linear models, and then linearize them around some reference point or path and examine the properties of the model locally valid. But how small is the neighbourhood of the linearization point for the linear approximation to be valid? This question is (almost) never asked, and economists go on as if such a neighbourhood had a practical relevance. Unfortunately this is not always the case, and this problem lies behind the measure of the errors involved in the construction of an ‘‘imprecise” surrogate production function, a measure that the author calls ‘‘declination”. If declination is sufficiently small, then the linear approximation to the non-linear wage curves can be reasonably used. The study of the conditions that might make declination small appears to be essential.