The aggregate production function of the finnish economy in the twentieth century.

• Author: Luoma, Arto

1. Introduction

   The elasticity of substitution between labor and capital is one of the key parameters in modern economic theory because the data generating processes describing output growth and capital formation seem to vary relative to it. For instance, in the standard neoclassical growth model, which typically generates exogenous growth, there is a possibility of long-run endogenous growth when the elasticity of substitution is above unity. This property exists in the neoclassical growth model because the marginal product of capital remains bounded away from zero in the limit as capital stock grows large. On the other hand, if the elasticity of substitution is below unity, there may exist either one positive and distinct steady-state value for capital or no such values, while a value above (or equal to) unity always generates one unique steady state path for capital. This property prevails in Diamond's (1965) overlapping generations model (though there may also occur two steady states in his model), as well as in Solow's neoclassical growth model (see Solow 1956; Duffy and Papageorgiou 2000).

   Motivated mainly by these issues, a range of studies, beginning with Arrow et al. (1961), have tried to estimate the elasticity of substitution between capital and labor for the United States and other industrialized countries. (1) The differences in the results presented in this literature are striking because there are a host of theoretical, statistical, and data issues involved in empirical analysis. For instance, there are restricting assumptions regarding the nature of technological progress, the most common of which is the assumption that technological change is neutral. (2) Most of the studies also rely on derived first-order conditions using the hypothesis of profit maximization by firms in a competitive framework, which may not be valid (see, for example, Chetty and Sankar 1969). The use of first-order conditions also requires calculation of labor- and capital-income series (3) and, under imperfect competition, the markup component (or pure profit component), which is very problematic.

   Recent empirical research has focused mainly on the nature of technological progress. For example, David and van de Klundert (1965), Ripatti and Vilmunen (2001), Antras (2004), Jalava et al. (2006), and Klump, McAdam, and Willman (2007a, b) relax the typical assumption of Hicks-neutral technological change (4) and estimate the elasticity of substitution to be somewhere between 0.3 and 0.8 in the post-World War II period using U.S., Euro-area, and Finnish data. These authors, however, base their analysis on the derived first-order conditions of profit maximization by firms in a competitive framework and restrict factor-augmenting technological progress to be linear, leaving a need for further analysis of the form of the production technology. A positive exception is the intuitive study of Klump, McAdam, and Willman (2007a) (see also Klump, McAdam, and Willman 2007b), which allows imperfect competition and technological progress to be log-linear or hyperbolic (not just linear). In this study, the authors also pay close attention to the importance of proper normalization of the constant elasticity of substitution (CES) functions. The issue, originally considered by Klump and de La Grandville (2000) and mainly neglected in the earlier empirical CES studies, gives one explanation for the mixed empirical results concerning the elasticity of substitution parameter.

   We contribute to the literature by directly estimating the parameters of the normalized CES function with factor-augmenting technical progress, rather than using the derived first-order conditions of profit maximizing behavior (see, for example, Zellner [1971] and Duff), and Papageorgiou [2000], for direct Bayesian and classical estimation of the CES production function with neutral technical progress, respectively). Operating in this manner allows us to relax the hypothesis of profit maximization in a competitive framework and to avoid the problems of calculating the labor- and capital-income series and under imperfect competition, the profit component. The approach is thus particularly applicable when the income data are of weak quality or even not available at all, which is common with very long-run time series or with gross-country data. We base our estimation on the Bayesian approach because the maximum likelihood (ML) estimation of the CES production function with factor-augmenting technical progress turned out to be a very challenging task. In particular, the ML estimation is very sensitive to the starting values of maximization, and the parameters typically fail to converge to the global optimum because of a multimodal likelihood. Although Bayesian inference is based on the likelihood function, it is weighted by a prior density. Priors provide a straightforward approach to guide the parameters' posteriors away from economically nonmeaningful peaks. Furthermore, even weakly informative priors have been observed to facilitate numerical maximization markedly. Bayesian methods also take into account the additional uncertainty present in long-tailed and possibly multimodal likelihoods (see, for example, Koop and Potter 1999).

   Posterior results indicate that, in the long run (over 100 years), the estimates for the elasticity of substitution and capital income share are intimately linked to the assumed shape of capital-augmenting technological progress. The presence of linear or log-linear restrictions in this progress leads to upwardly biased estimates for the elasticity of substitution and income share parameters. However, when the econometric specification is modified to allow hyperbolic technological progress, we generally obtain significantly lower estimates of these parameters. We argue that the reason for this bias is very likely linked to omitted curvature parameters because linear restriction excludes the possibility that the speed of capital-augmenting technological progress converges to zero. This omitted variable bias is probably more critical when we deal with economies that are in transit to their balanced growth path during the estimation period because the fact that capital can be accumulated while labor cannot should imply a dominant role for labor-augmenting technical change along the long-term balanced growth path and only a transitory role for capital. (5)

   Finally, our analysis is based on the Finnish output, capital, and working hours series from 1902 to 2004. We use Finnish data because during the twentieth century, Finland developed from a relatively backward agricultural society to a modern postindustrial state, giving us an interesting test case of an economy where the transition process has been strong during the estimation period (see further discussion on the special features of the Finnish economy in Jalava et al. 2006).

   The rest of the article is organized as follows: Section 2 introduces the estimated model and the estimation method, section 3 presents the data and the estimation results, and section 4 concludes.

2. Estimation Method

   We base our Bayesian analysis on the normalized production function specification of Klump and de La Grandville (2000), thoroughly studied by Klump, McAdam, and Willman (2007a), which allows for nonneutral technological change. It is given by

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

   where [rho] is the substitution parameter, [[alpha].sub.0] the distribution parameter, Y the aggregate output, K the aggregate capital, and L the aggregate labor, while [Y.sub.0], [K.sub.0], and [L.sub.0] are their baseline values (see de La Grandville 1989; Klump and de La Grandville 2000; Klump and Preissler 2000). Under factor remuneration at the marginal product, the distribution parameter [[alpha].sub.0] is identified as the capital income share of total factor income at a fixed point.

   The derivation of Equation 1 is based on a few fundamental assumptions (see Klump, McAdam, and Willman 2007a). First, the levels of efficiency of input factors are measured as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the terms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent...