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Comment on “σ: The long and short of it”
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Journal of Macroeconomics 30 (2008) 687–690 www.elsevier.com/locate/jmacro
Comment on ‘‘r: The long and short of it’’
q
Theodore Palivos Department of Economics, University of Macedonia, Salonica 540 06, Greece Received 7 September 2007; accepted 18 September 2007 Available online 25 September 2007
Abstract I comment on some of the issues discussed by Robert Chirinko (Chirinko, R.S., 2008. r: The long and short of it. Journal of Macroeconomics 30 (2), 671–686). I also offer some additional evidence and discuss the theoretical implications of the empirical findings regarding r. Ó 2007 Elsevier Inc. All rights reserved. JEL classification: D24; E23; O41 Keywords: Elasticity of substitution; Production function; Economic growth
1. Introduction The underlying relation between the inputs that participate in the production process plays a pivotal role in several areas of economics. Some of the most important aspects of this relation are succinctly summarized by the concept of the elasticity of (factor) substitution (r). It is thus important to understand the key implications of this concept as well as to obtain a robust estimate of its magnitude. The article by Chirinko (2008) does an excellent job in bringing the reader up-to-date in the literature. Specifically, the article provides first some background on the concept of r and the constant elasticity of substitution (CES) production function. Next, it summarizes some of the theoretical issues that were raised during the conference and presents a comprehensive survey of the existing estimation methods of the elasticity of substitution used q
I would like to thank Anastasia Litina for helpful comments. E-mail address: [email protected]
0164-0704/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2007.09.004
688
T. Palivos / Journal of Macroeconomics 30 (2008) 687–690
in some of the papers of this volume and elsewhere. Finally, it offers a list of issues that should be given further consideration. This short note not only comments on some of the issues raised by Chirinko but also supplements the paper by presenting some additional empirical and theoretical aspects regarding r. The discussion centers mainly around two questions: (a) is r a parameter or an endogenous variable? and (b) is r lower or higher than unity? I offer some additional evidence drawn from cross-country studies and discuss the theoretical implications of the empirical findings. 2. Empirical Issues First of all, the empirical estimation of a CES production function for the recovery of r should pay more attention to the issue of normalization. The reason for this is that the comparison of different CES functions is meaningful only if they belong to the same family; that is, if they differ only with respect to r. Hence, a common reference point is needed to ensure that all other baseline parameters are the same (for a more detailed presentation of this issue see the discussion in Klump et al. (2008), and the references cited therein). Second, as mentioned in the concluding section of the Arrow et al. (1961) paper, quoted also in the concluding section of the Chirinko paper, the value of the elasticity of substitution might shift during the development process. Indeed, Miyagiwa and Papageorgiou (2007) investigate this conjecture within a multi-sector growth model. They find that the aggregate elasticity of substitution (AES) varies over time, despite the fact that the value of r within each sector is constant. Moreover, AES is positively related to the level of economic development. These theoretical results find also empirical support in the literature. Specifically, Duffy and Papageorgiou (2000), using a panel data sample of countries, estimate the elasticity of substitution to be 0.83 in the sub-sample of 23 low-capital countries, whereas it jumps up to 1.09 in the sub-sample of 21 high-capital countries. These findings are important for two reasons. First, they provide evidence for the above-mentioned conjecture that the value of r may not be independent of the stage of economic development and, in particular, of the capital accumulation per worker. Second, as explained in the next section, a value of r that is greater than one can have profound implications for the behavior of the economy. Finally, some authors have tackled the issue of a varying r following a different but related approach. They have attempted to estimate more general production functions, in which r is an endogenous variable right from the start. Karagiannis et al. (2005), henceforth KPP, survey several empirical studies that have estimated a variable elasticity of substitution (VES) production function. Most of these studies have rejected either the Cobb–Douglas or/and the CES in favor of a VES production function, using either cross-section data sets consisting of different sectors or time-series for an entire country. KPP estimate the following VES production function Y ¼ AK av ½L þ baKð1aÞv ;
ð1Þ
proposed by Revankar (1971), using the same cross-country panel data set as in Duffy and Papageorgiou (2000). In doing so, they, as do Duffy and Papageorgiou, use raw and adjusted for human capital labor data and allow for constant and non-constant returns to scale. In all four cases, they reject the Cobb–Douglas specification in favor of (1). More-
T. Palivos / Journal of Macroeconomics 30 (2008) 687–690
689
over, in all four cases, they find a value of r in the entire sample that is in general greater than unity. If these results as well as the ones in Duffy and Papageorgiou survive further scrutiny then they may change the way economists think about economic growth. Hence, I think that this is an important topic that should be given further attention.
3. Theoretical considerations As mentioned in the introduction of the Chirinko paper, there is clearly a strong case for the importance of r in several areas of economics. In this section I will attempt to survey the role played by r just in a few of them.1 First of all, there is a link, unnoticed until recently, between the properties of the production function and the asymptotic value of r. Indeed, each time we assume that the production function satisfies the Inada conditions, we also assume that the elasticity of substitution is asymptotically equal to unity, which is a rather restrictive assumption (see Barelli and Pessoˆa, 2003; Litina and Palivos, 2007a). However, the converse relationship is not true; that is, a production function may exhibit an elasticity of substitution with an asymptotic value equal to unity and still not satisfy all the Inada conditions (an example of such a production function is given in Palivos and Karagiannis, 2007). Second, as Chirinko mentions a consensus is lacking regarding the value of r, and in particular if it is less or greater than unity. As noted by Solow (1956) and Pitchford (1960), a CES production function with a value of r that exceeds unity can result in perpetual endogenous growth, even in the absence of technical progress (see also the references in footnote 2 of the Chirinko paper). In fact, Palivos and Karagiannis (2007) have recently extended this result by showing that any production function whose elasticity of substitution is only asymptotically greater than unity can counteract the diminishing returns to capital and result in unbounded endogenous growth, despite the absence of technical progress and the presence of non-reproducible factors (labor). This finding accords well with the interpretation of r not as a curvature but as an efficiency parameter, given by de La Grandville (1997) and Klump and de La Grandville (2000). Third, one of the criticisms of the neoclassical growth model is that it predicts differences in rates of return to capital that are excessive. For example, if we assume a Cobb–Douglas production function, then poor countries with about one-tenth the income of rich countries should exhibit rates of return that are one hundred times as large. However, as Mankiw (1995) argues this is an artifact of the underlying production technology. If, instead of a Cobb–Douglas, we assume a CES production function with a value of r equal to four, then the return to capital in a poor country falls from 100 to only 3.2 times as large as that in a rich country, a return differential that is empirically much more plausible. Finally, the value of r can affect other important variables of the economy. Recently, Smetters (2003) has shown that contrary to the Cobb–Douglas case, where the saving rate increases monotonically during the transition to the steady state, a CES production function with a value of r different from unity may give rise to a non-monotonic behavior of the saving rate. Go´mez (2008) confirms these results by characterizing qualitatively the 1
See Turnovsky (2008) for an analysis on how r affects (a) the speed of convergence and the distribution of income and wealth and (b) the relative merits of tied and un-tied foreign aid.
690
T. Palivos / Journal of Macroeconomics 30 (2008) 687–690
global dynamics of the saving rate using a phase-diagram analysis. Finally, Litina and Palivos (2007b) extend these results for the case of any production function, allowing thus for the case where the elasticity of substitution becomes an endogenous variable. It should be noted that a non-monotonic behavior finds strong empirical support in, among others, the empirical work of Maddison (1992). 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, 225–250. Barelli, P., Pessoˆa, S.de A., 2003. Inada conditions imply that production function must be asymptotically Cobb– Douglas. Economics Letters 81, 361–363. Chirinko, R.S., 2008. r: The long and short of it. Journal of Macroeconomics, 30 (2), 671–686. de La Grandville, O., 1997. Curvature and the elasticity of substitution: Straightening it out. Journal of Economics 66, 23–34. Duffy, J., Papageorgiou, C., 2000. A cross-country empirical investigation of the aggregate production function specification. Journal of Economic Growth 5, 87–120. Go´mez, M.A., 2008. Dynamics of the saving rate in the neoclassical growth model with CES production. Macroeconomic Dynamics, forthcoming. Karagiannis, G., Palivos, T., Papageorgiou, C., 2005. Variable elasticity of substitution and economic growth. In: Diebolt, C., Kyrtsou, C. (Eds.), New Trends in Macroeconomics. Springer, pp. 21–37. Klump, R., de La Grandville, O., 2000. Economic growth and the elasticity of substitution: Two theorems and some suggestions. American Economic Review 90, 282–291. Klump, R., McAdam, P., Willman, A., 2008. Unwrapping some Euro area growth puzzles: Factor substitution, productivity and unemployment. Journal of Macroeconomics 30 (2), 645–666. Litina, A., Palivos, T., 2007a. Do Inada conditions imply that production function must be asymptotically Cobb– Douglas? A comment. Economics Letters, in press. doi:10.1016/j.econlet.2007.09.035. Litina, A., Palivos, T., 2007b. The behavior of the saving rate in the neoclassical optimal growth model. Unpublished manuscript. Maddison, A., 1992. A long-run perspective on saving. The Scandinavian Journal of Economics 94, 181–196. Mankiw, N.G., 1995. The growth of nations. Brookings Papers on Economic Activity, 275–326. Miyagiwa, K., Papageorgiou, C., 2007. Endogenous aggregate elasticity of substitution. Journal of Economic Dynamics and Control 31, 2899–2919. Palivos, T., Karagiannis, G., 2007. The elasticity of substitution as an engine of growth. Unpublished manuscript. Pitchford, J.D., 1960. Growth and the elasticity of substitution. Economic Record 36, 491–504. Revankar, N.S., 1971. A class of variable elasticity of substitution production functions. Econometrica 39, 61–71. Smetters, K., 2003. The (interesting) dynamic properties of the neoclassical growth model with CES production. Review of Economic Dynamics 6, 697–707. Solow, R.M., 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94. Turnovsky, S.J., 2008. The role of factor substitution in the theory of economic growth and income distribution: Two examples. Journal of Macroeconomics 30 (2), 604–629.
Journal of Macroeconomics 30 (2008) 687–690 www.elsevier.com/locate/jmacro
Comment on ‘‘r: The long and short of it’’
q
Theodore Palivos Department of Economics, University of Macedonia, Salonica 540 06, Greece Received 7 September 2007; accepted 18 September 2007 Available online 25 September 2007
Abstract I comment on some of the issues discussed by Robert Chirinko (Chirinko, R.S., 2008. r: The long and short of it. Journal of Macroeconomics 30 (2), 671–686). I also offer some additional evidence and discuss the theoretical implications of the empirical findings regarding r. Ó 2007 Elsevier Inc. All rights reserved. JEL classification: D24; E23; O41 Keywords: Elasticity of substitution; Production function; Economic growth
1. Introduction The underlying relation between the inputs that participate in the production process plays a pivotal role in several areas of economics. Some of the most important aspects of this relation are succinctly summarized by the concept of the elasticity of (factor) substitution (r). It is thus important to understand the key implications of this concept as well as to obtain a robust estimate of its magnitude. The article by Chirinko (2008) does an excellent job in bringing the reader up-to-date in the literature. Specifically, the article provides first some background on the concept of r and the constant elasticity of substitution (CES) production function. Next, it summarizes some of the theoretical issues that were raised during the conference and presents a comprehensive survey of the existing estimation methods of the elasticity of substitution used q
I would like to thank Anastasia Litina for helpful comments. E-mail address: [email protected]
0164-0704/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmacro.2007.09.004
688
T. Palivos / Journal of Macroeconomics 30 (2008) 687–690
in some of the papers of this volume and elsewhere. Finally, it offers a list of issues that should be given further consideration. This short note not only comments on some of the issues raised by Chirinko but also supplements the paper by presenting some additional empirical and theoretical aspects regarding r. The discussion centers mainly around two questions: (a) is r a parameter or an endogenous variable? and (b) is r lower or higher than unity? I offer some additional evidence drawn from cross-country studies and discuss the theoretical implications of the empirical findings. 2. Empirical Issues First of all, the empirical estimation of a CES production function for the recovery of r should pay more attention to the issue of normalization. The reason for this is that the comparison of different CES functions is meaningful only if they belong to the same family; that is, if they differ only with respect to r. Hence, a common reference point is needed to ensure that all other baseline parameters are the same (for a more detailed presentation of this issue see the discussion in Klump et al. (2008), and the references cited therein). Second, as mentioned in the concluding section of the Arrow et al. (1961) paper, quoted also in the concluding section of the Chirinko paper, the value of the elasticity of substitution might shift during the development process. Indeed, Miyagiwa and Papageorgiou (2007) investigate this conjecture within a multi-sector growth model. They find that the aggregate elasticity of substitution (AES) varies over time, despite the fact that the value of r within each sector is constant. Moreover, AES is positively related to the level of economic development. These theoretical results find also empirical support in the literature. Specifically, Duffy and Papageorgiou (2000), using a panel data sample of countries, estimate the elasticity of substitution to be 0.83 in the sub-sample of 23 low-capital countries, whereas it jumps up to 1.09 in the sub-sample of 21 high-capital countries. These findings are important for two reasons. First, they provide evidence for the above-mentioned conjecture that the value of r may not be independent of the stage of economic development and, in particular, of the capital accumulation per worker. Second, as explained in the next section, a value of r that is greater than one can have profound implications for the behavior of the economy. Finally, some authors have tackled the issue of a varying r following a different but related approach. They have attempted to estimate more general production functions, in which r is an endogenous variable right from the start. Karagiannis et al. (2005), henceforth KPP, survey several empirical studies that have estimated a variable elasticity of substitution (VES) production function. Most of these studies have rejected either the Cobb–Douglas or/and the CES in favor of a VES production function, using either cross-section data sets consisting of different sectors or time-series for an entire country. KPP estimate the following VES production function Y ¼ AK av ½L þ baKð1aÞv ;
ð1Þ
proposed by Revankar (1971), using the same cross-country panel data set as in Duffy and Papageorgiou (2000). In doing so, they, as do Duffy and Papageorgiou, use raw and adjusted for human capital labor data and allow for constant and non-constant returns to scale. In all four cases, they reject the Cobb–Douglas specification in favor of (1). More-
T. Palivos / Journal of Macroeconomics 30 (2008) 687–690
689
over, in all four cases, they find a value of r in the entire sample that is in general greater than unity. If these results as well as the ones in Duffy and Papageorgiou survive further scrutiny then they may change the way economists think about economic growth. Hence, I think that this is an important topic that should be given further attention.
3. Theoretical considerations As mentioned in the introduction of the Chirinko paper, there is clearly a strong case for the importance of r in several areas of economics. In this section I will attempt to survey the role played by r just in a few of them.1 First of all, there is a link, unnoticed until recently, between the properties of the production function and the asymptotic value of r. Indeed, each time we assume that the production function satisfies the Inada conditions, we also assume that the elasticity of substitution is asymptotically equal to unity, which is a rather restrictive assumption (see Barelli and Pessoˆa, 2003; Litina and Palivos, 2007a). However, the converse relationship is not true; that is, a production function may exhibit an elasticity of substitution with an asymptotic value equal to unity and still not satisfy all the Inada conditions (an example of such a production function is given in Palivos and Karagiannis, 2007). Second, as Chirinko mentions a consensus is lacking regarding the value of r, and in particular if it is less or greater than unity. As noted by Solow (1956) and Pitchford (1960), a CES production function with a value of r that exceeds unity can result in perpetual endogenous growth, even in the absence of technical progress (see also the references in footnote 2 of the Chirinko paper). In fact, Palivos and Karagiannis (2007) have recently extended this result by showing that any production function whose elasticity of substitution is only asymptotically greater than unity can counteract the diminishing returns to capital and result in unbounded endogenous growth, despite the absence of technical progress and the presence of non-reproducible factors (labor). This finding accords well with the interpretation of r not as a curvature but as an efficiency parameter, given by de La Grandville (1997) and Klump and de La Grandville (2000). Third, one of the criticisms of the neoclassical growth model is that it predicts differences in rates of return to capital that are excessive. For example, if we assume a Cobb–Douglas production function, then poor countries with about one-tenth the income of rich countries should exhibit rates of return that are one hundred times as large. However, as Mankiw (1995) argues this is an artifact of the underlying production technology. If, instead of a Cobb–Douglas, we assume a CES production function with a value of r equal to four, then the return to capital in a poor country falls from 100 to only 3.2 times as large as that in a rich country, a return differential that is empirically much more plausible. Finally, the value of r can affect other important variables of the economy. Recently, Smetters (2003) has shown that contrary to the Cobb–Douglas case, where the saving rate increases monotonically during the transition to the steady state, a CES production function with a value of r different from unity may give rise to a non-monotonic behavior of the saving rate. Go´mez (2008) confirms these results by characterizing qualitatively the 1
See Turnovsky (2008) for an analysis on how r affects (a) the speed of convergence and the distribution of income and wealth and (b) the relative merits of tied and un-tied foreign aid.
690
T. Palivos / Journal of Macroeconomics 30 (2008) 687–690
global dynamics of the saving rate using a phase-diagram analysis. Finally, Litina and Palivos (2007b) extend these results for the case of any production function, allowing thus for the case where the elasticity of substitution becomes an endogenous variable. It should be noted that a non-monotonic behavior finds strong empirical support in, among others, the empirical work of Maddison (1992). 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, 225–250. Barelli, P., Pessoˆa, S.de A., 2003. Inada conditions imply that production function must be asymptotically Cobb– Douglas. Economics Letters 81, 361–363. Chirinko, R.S., 2008. r: The long and short of it. Journal of Macroeconomics, 30 (2), 671–686. de La Grandville, O., 1997. Curvature and the elasticity of substitution: Straightening it out. Journal of Economics 66, 23–34. Duffy, J., Papageorgiou, C., 2000. A cross-country empirical investigation of the aggregate production function specification. Journal of Economic Growth 5, 87–120. Go´mez, M.A., 2008. Dynamics of the saving rate in the neoclassical growth model with CES production. Macroeconomic Dynamics, forthcoming. Karagiannis, G., Palivos, T., Papageorgiou, C., 2005. Variable elasticity of substitution and economic growth. In: Diebolt, C., Kyrtsou, C. (Eds.), New Trends in Macroeconomics. Springer, pp. 21–37. Klump, R., de La Grandville, O., 2000. Economic growth and the elasticity of substitution: Two theorems and some suggestions. American Economic Review 90, 282–291. Klump, R., McAdam, P., Willman, A., 2008. Unwrapping some Euro area growth puzzles: Factor substitution, productivity and unemployment. Journal of Macroeconomics 30 (2), 645–666. Litina, A., Palivos, T., 2007a. Do Inada conditions imply that production function must be asymptotically Cobb– Douglas? A comment. Economics Letters, in press. doi:10.1016/j.econlet.2007.09.035. Litina, A., Palivos, T., 2007b. The behavior of the saving rate in the neoclassical optimal growth model. Unpublished manuscript. Maddison, A., 1992. A long-run perspective on saving. The Scandinavian Journal of Economics 94, 181–196. Mankiw, N.G., 1995. The growth of nations. Brookings Papers on Economic Activity, 275–326. Miyagiwa, K., Papageorgiou, C., 2007. Endogenous aggregate elasticity of substitution. Journal of Economic Dynamics and Control 31, 2899–2919. Palivos, T., Karagiannis, G., 2007. The elasticity of substitution as an engine of growth. Unpublished manuscript. Pitchford, J.D., 1960. Growth and the elasticity of substitution. Economic Record 36, 491–504. Revankar, N.S., 1971. A class of variable elasticity of substitution production functions. Econometrica 39, 61–71. Smetters, K., 2003. The (interesting) dynamic properties of the neoclassical growth model with CES production. Review of Economic Dynamics 6, 697–707. Solow, R.M., 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94. Turnovsky, S.J., 2008. The role of factor substitution in the theory of economic growth and income distribution: Two examples. Journal of Macroeconomics 30 (2), 604–629.