by Sebastian Gechert, Tomas Havranek, Zuzana Irsova, and Dominika Kolcunova
A key parameter in economics is the elasticity of substitution between capital and labor, but empirical estimates vary. This column takes stock of the literature. Among 3,186 estimates produced by 121 studies, the mean elasticity is 0.9, not far from the Cobb-Douglas assumption of 1. But the mean is biased by 3 factors: publication selection, use of aggregated data, and omission of the first-order condition for capital. The mean corrected for these biases is 0.3. 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 depends on the value of the elasticity (Solow, 1956). Klump and de La Grandeville (2000) suggest that a larger elasticity in a country results in higher per capita income at any stage of development. Turnovsky (2002) argues that a smaller elasticity leads to faster convergence. The explanation for the decline of the labor share in income during the recent decades that was put forward by Piketty (2014) and Karabarbounis and Neiman (2013) holds only when the elasticity surpasses one. Cantore et al. (2014) show how the effect of technology shocks on hours worked is sensitive to the elasticity. Nekarda and Ramey (2013) argue that the countercyclicality of the price markup over marginal cost also depends on the elasticity of substitution. In addition, the elasticity represents an important parameter in analyzing the effects of fiscal policies, including the effect of corporate taxation on capital formation, and in determining optimal taxation of capital (Chirinko, 2002).
Figure 1: The elasticity of substitution matters for monetary policy
Note: The figure shows simulated impulse responses of inflation to a monetary policy shock. We use the SIGMA model of Erceg et al. (2008) developed for the Federal Reserve Board and vary the value of the capital-labor substitution elasticity while leaving other parameters at their original values. The model does not have a stable solution for the elasticity larger than one.
The size of the elasticity has practical consequences for monetary policy, as Figure 1 illustrates. For example, in the SIGMA model used by the Federal Reserve Board (Erceg et al., 2008), the effectiveness of interest rate changes in steering inflation doubles when one assumes the elasticity to equal 0.9 instead of 0.5, yielding wildly different policy implications. We choose the SIGMA model for the illustration because, as one of very few models employed by central banks, it actually allows for different values of the elasticity of substitution. 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 should imply a more aggressive campaign of interest rate cuts in response to a recession.
Empirical estimates of the elasticity vary widely both within and between studies (Figure 2). Solid references can be found for calibrating the elasticity anywhere between 0 and 1.5, which effectively means that the empirical literature does not discipline calibrations at all – despite decades of research on the value of the elasticity and the work of dozens of prominent economists.
Figure 2: No consensus on the value of the elasticity
To take stock of the voluminous literature and provide concrete guidelines for the calibration of the elasticity, we conduct a meta-analysis (Gechert et al., 2019). We collect 3,186 coefficients from 121 studies, which produce a mean estimate of 0.9. But we show that the picture is seriously distorted by publication bias. After correcting for the bias, the mean reported elasticity shrinks to 0.5. This correction alone implies halving the effectiveness of monetary policy in a structural model, as shown by Figure 1. Moreover, some data and method choices bias the estimated elasticity systematically. If one agrees that sector-level data dominate more aggregated country- or state-level data and that including information from the first-order condition for capital dominates ignoring it, the implied mean estimate further decreases to 0.3. Thus we recommend 0.3 for the calibration of the elasticity, consistent with burying the Cobb-Douglas production function.
The finding of strong publication bias predominates in our results and is responsible for most of the reduction from 0.9 (the simple mean) to 0.3 (our recommended calibration). The bias arises when different estimates have a different probability of being reported depending on sign and statistical significance. The identification we use builds on the fact that almost all econometric techniques used to estimate the elasticity assume that the ratio of the estimate to its standard error has a symmetrical distribution, typically a t-distribution. So the estimates and standard errors 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.
Because empirical economists command plenty of degrees of freedom, a large estimate of the elasticity can always emerge if the researcher looks for it long enough, and an upward bias in the literature arises. 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. Apart from linear techniques based on the Lombard effect, we employ recently developed methods by Ioannidis et al. (2017), Andrews and Kasy (2019), Bom and Rachinger (2019), and Furukawa (2019), which account for the potential nonlinearity between the standard error and selection effort.
Figure 3: Publication bias in the literature
Figure 3 provides a graphical illustration of the mechanism outlined in the previous paragraph. In the scatter plot 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. The figure shows the predicted funnel shape, still with plenty of heterogeneity at the top (see the paper for a detailed analysis of heterogeneity using both Bayesian and frequentist model averaging) – but also shows asymmetry. 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. The finding of strong publication selection and the corrected mean effect around 0.5 is confirmed by all techniques we apply.
We are not the first to highlight the disconnect between the Cobb-Douglas specification commonly used in macroeconomic models and the empirical literature estimating the elasticity of substitution. Chirinko (2008) and Knoblach et al. (2019) provide useful surveys of portions of the literature, and both studies suggest that the Cobb-Douglas production function is not backed by the available evidence. We argue that after controlling for publication bias the case against Cobb-Douglas strengthens to the point where one must warn against the continued use of this convenient simplification. As we show in Figure 1, a structural model built to aid monetary policy is biased from the beginning if it uses an elasticity of one for capital-labor substitution. Computational convenience should yield to the stylized fact established by half a century of meticulous research: capital and labor are gross complements.
Andrews, I. & M. Kasy (2019): “Identification of and Correction for Publication Bias.” Americal Economic Review 109(8): pp. 2766-2794.
Bom, P. R. D. & H. Rachinger (2019): “A Kinked Meta-Regression Model for Publication Bias Correction.” Research Synthesis Methods, forthcoming.
Cantore, C., M. Leon-Ledesma, P. McAdam, & A. Willman (2014): “Shocking Stuff: Technology, Hours, and Factor Substitution.” Journal of the European Economic Association 12(1): pp. 108-128.
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.
Furukawa, C. (2019): “Publication Bias under Aggregation Frictions: Theory, Evidence, and a New Correction Method.” Unpublished paper, MIT.
Gechert, S., T. Havranek, Z. Irsova, & D. Kolcunova (2019): “Death to the Cobb-Douglas Production Function.” FMM working paper 51, Hans-Böckler-Stiftung.
Ioannidis, J., T. Stanley, & H. Doucouliagos (2017): “The Power of Bias in Economics Research.” Economic Journal 127(605): F236-F265.
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 (2019): “The Elasticity of Substitution Between Capital and Labour in the US Economy: A Meta-Regression Analysis.” Oxford Bulletin of Economic and Statistics, forthcoming.
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, forthcoming.
Nekarda, C. J. & V. A. Ramey (2013): “The Cyclical Behavior of the Price-Cost Markup.” NBER Working Paper 19099.
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.
 The full paper, together with data and codes, is available at http://meta-analysis.cz/sigma.