Death to the Cobb-Douglas Production Function?

The elasticity of substitution between capital and labor is a key parameter in economics, measuring how easy it is to substitute one factor of production for the other. The value of the elasticity plays an essential role in the theory of economic growth as well as in analyses that constitute the starting point for monetary and fiscal policy decisions. But empirical estimates vary widely in the literature. Among 3,186 estimates produced by 121 studies, the mean elasticity is not far from the value of 1 assumed in the Cobb-Douglas production function,¹  which is often used in macro-models. However, as we show in our research paper, Death to the Cobb-Douglas Production Function? A Quantitative Survey of the Capital-Labor Substitution Elasticity , the mean can be seriously distorted. The estimates are distorted by publication selection (i.e., estimates that are close to one are published preferentially) and specific method and data choices. The mean elasticity corrected for these biases is 0.3, one-third of the simple average of all estimates. The weight of evidence accumulated in the empirical literature thus emphatically rejects the Cobb-Douglas specification.

The elasticity of substitution between capital and labor is central to a host of economic problems. Our understanding of long-run growth and its sustainability depends on the value of the elasticity (Solow, 1956). Some studies suggest that the higher is the elasticity, i.e., the easier it is to substitute one production factor for the other, the higher is per capita income at any stage of development (Klump and De La Grandville, 2000). Other studies argue that a smaller elasticity leads to faster convergence (Turnovsky, 2002). The explanation for the decline of the labor share in income during recent decades that was put forward by Piketty (2014) holds only when the elasticity surpasses one. Aside from the theory of economic growth, the elasticity also represents an important parameter in analyzing the effects of fiscal policies, including the effect of corporate taxation on capital formation (Chirinko, 2002).

The size of the elasticity may have practical consequences for monetary policy as well. For example, in the model used by the Fed² (Erceg et al., 2008), the effectiveness of interest rate changes in steering inflation is higher when larger elasticity values are assumed. However, almost all models use the convenient simplification of the Cobb-Douglas production function, which implicitly assumes that the elasticity equals one. If the true elasticity is smaller, these models overstate the strength of monetary policy and in fact should imply more aggressive interest rate cuts in response to a recession and more aggressive interest rate hikes in response to an expansion.

The elasticity of substitution is a key parameter, but empirical estimates of the elasticity vary widely both within and between studies. In order to take stock of the voluminous literature and to find a true, undistorted mean value of the elasticity, we use the method of meta-analysis. Meta-analysis offers a quantitative survey of the available literature and is often used in economics to explain differences in the results of particular studies and to explore the patterns of publication selection. We collect 3,186 coefficients from 121 studies, which produce a mean estimate of 0.9. However, the values are very heterogeneous, lying mostly in the range of 0 to 1.5 (Figure 1). Thus, the empirical literature does not clearly guide the calibrations of this parameter in macroeconomic models.

Figure 1: No consensus on the value of the elasticity

(Open the whole chart in a new window.)

The mean estimate in the empirical literature is seriously distorted by publication selection (publication bias) and by method and data choices. After correcting for publication bias, the mean elasticity shrinks to 0.5. When we adjust the mean elasticity for the systematic effect of some estimation methods used by researchers, the value further decreases to 0.3. Publication bias arises when different estimates have a different probability of being reported depending on sign and statistical significance. Almost all econometric techniques used to estimate the elasticity assume that the ratio of estimates to standard errors has a symmetric distribution, so the estimate and its standard error should represent independent quantities. But if statistically significant positive estimates are preferentially selected for publication, large standard errors (given by noise in data or imprecision in estimation) become associated with large estimates. An upward bias thus arises in the literature. A useful analogy appears in McCloskey and Ziliak (2019), who liken publication bias to the Lombard effect in biology: speakers increase their effort in the presence of noise. 

The funnel plot (Figure 2), which provides a graphical representation of the mechanism described in the previous paragraph, shows that negative and zero estimates are underreported in the literature. The horizontal axis measures the magnitude of the estimated elasticities, and the vertical axis measures their precision. In the absence of publication bias, the scatter plot will form an inverted funnel: the most precise estimates will lie close to the true mean elasticity, imprecise estimates will be more dispersed, and both small and large imprecise estimates will appear with the same frequency. Our figure shows that the funnel is asymmetrical: for the funnel to be symmetrical, and hence consistent with the absence of publication bias, we should observe many more reported negative and zero estimates. 

Figure 2: Publication bias in the literature

The Cobb-Douglas production function, which assumes the elasticity of substitution between capital and labor to be equal to one, does not correspond to the empirical estimates corrected for the publication bias and thus is not a proper representation of economic relations. We are not the first to highlight the disconnect between the Cobb-Douglas specification and the empirical literature. There are other studies pointing to the fact that the use of the aggregate Cobb-Douglas production function is not backed by the available evidence (Chirinko, 2008, and Knoblach et al., 2020). Nevertheless, we argue that after controlling for publication bias, the case against Cobb-Douglas strengthens even more. Half a century of research establishes a clear stylized fact: capital and labor are gross complements. This means that they are used together, and an increase in the price of one factor of production (in our case, the price is the wage rate or the cost of capital) leads to a fall in demand for both factors of production.

References

Cobb, C. W. & P. H. Douglas (1928): "A Theory of Production." American Economic Review 18(1): pp. 139-165.

Chirinko, R. S. (2002): "Corporate Taxation, Capital Formation, and the Substitution Elasticity between Labor and Capital." National Tax Journal 55(2): pp. 339-355.

Chirinko, R. S. (2008): "σ: The Long and Short of it." Journal of Macroeconomics 30(2): pp. 671-686.

Erceg, C. J., L. Guerrieri, & C. Gust (2008): "Trade Adjustment and the Composition of Trade." Journal of Economic Dynamics and Control 32(8): pp. 2622-2650.

Karabarbounis, L. & B. Neiman (2013): "The Global Decline of the Labor Share." Quarterly Journal of Economics 129(1): pp. 61-103.

Klump, R. & O. De La Grandville (2000): "Economic Growth and the Elasticity of Substitution: Two Theorems and Some Suggestions." American Economic Review 90(1): pp. 282-291.

Knoblach, M., M. Rossler, & P. Zwerschke (2020): "The Elasticity of Substitution Between Capital and Labour in the US Economy: A Meta-Regression Analysis." Oxford Bulletin of Economic and Statistics 82(1): pp. 62-82.

McCloskey, D. N. & S. T. Ziliak (2019): "What Quantitative Methods Should We Teach to Graduate Students? A Comment on Swann's 'Is Precise Econometrics an Illusion?'" Journal of Economic Education 50(4): pp. 356-361.

Piketty, T. (2014): "Capital in the 21st Century." Cambridge, MA: Harvard University Press.

Solow, R. M. (1956): "A Contribution to the Theory of Economic Growth." Quarterly Journal of Economics 70(1): pp. 65-94.

Turnovsky, S. J. (2002): "Intertemporal and Intratemporal Substitution, and the Speed of Convergence in the Neoclassical Growth Model." Journal of Economic Dynamics and Control 26(9-10): pp. 1765-1785.

¹ A production function relates the quantity of output (GDP) to the quantity of inputs (for example, labor and capital). Due to its simplicity, the Cobb-Douglas production function is the most frequently used production function specification.

² For illustrative purposes we chose the SIGMA model used by the Fed, as it is one of the very few models employed by central banks that actually allows for different values of the elasticity of substitution