The Substitution Elasticity, Factor Shares, Long-Run Growth, and the Low-Frequency Panel Model

The value of the elasticity of substitution between labor and capital (σ) is a crucial assumption in the study of factor incomes (e.g., Piketty (2014a), Piketty and Zucman (forthcoming), Karabarbounis and Neiman (2014)) and long-run growth (Solow, 1956). This paper begins by examining the role of σ in the analyses of these two issues. It then develops and implements a new strategy for estimating this crucial parameter by combining a low-pass filter with panel data techniques to identify the low-frequency/long-run relations appropriate to production function estimation. Our low-frequency/long-run approach is in the spirit of Friedman's permanent income theory of consumption and Eisner's related permanent income theory of investment. While their estimation strategies and ours are similar in relying on permanent (long-run) components, we identify these unobservable components with a low-pass filter. Using spectral analysis, we assess the extent to which our choices of the critical periodicity and window defining the low-pass filter are successful in emphasizing long-run variation. The empirical results are based on the comprehensive panel industry dataset constructed by Dale Jorgenson and his research associates. Our preferred estimate of σ is 0.40. This result is robust to variations in the two spectral parameters that define the low-pass filter, instrumental and split-sample estimates, and disaggregation by industry. Moreover, we document that standard estimation methods, which do not filter-out transitory variation, generate downwardly biased estimates. As medium and high frequency variation is introduced into the model variables, σ declines by over 40% relative to the benchmark value. Despite correcting for this bias, our preferred estimate of σ is less than one, and we find no support for the Cobb-Douglas assumption.