The Shape of Aggregate Production Functions: Evidence from Estimates of the World Technology Frontier

The objective of the current paper is to estimate the aggregate, country-level production function as a relationship between countries' aggregate inputs and their maximum attainable output, computed on the basis of the World Technology Frontier -- the best-practice frontier at each moment in time. By doing so, we are able to single out technological aspects of the production processes from their institutional background, at least up to a multiplicative constant. Such estimates of the aggregate production function will be then used as a convenient starting point for further analyses, aimed at deriving this function's crucial characteristics, and discussing which parametric form agrees most with the available empirical evidence. As crucial features of the estimated aggregate production function, we shall investigate its implications for the cross-country distribution of technical inefficiency, the pattern of dependence of its (variable) partial elasticities on factor endowments, (variable) returns-to-scale properties, and its implied (Morishima and Allen--Uzawa) elasticities of substitution.We estimate the aggregate production function with two alternative methods. First, we apply the nonparametric Data Envelopment Analysis (DEA) approach, augmented with the Simar and Wilson (1998, 2000) bootstrap procedure which enables us to adjust for the bias of DEA efficiency estimates as well as to compute standard errors and confidence intervals for these estimates. The advantage of this first approach is that it does not require one to make a priori assumptions on the functional form of the aggregate production function -- and yields testable predictions instead. Unfortunately, the DEA approach is not suited to providing predictions on the function's curvature features such as elasticities of substitution. Second, we also apply the Stochastic Frontier Analysis (SFA) methodology which allows us to estimate the production function directly, under certain predefined (parametric) functional specifications. Such parametric models are estimated with Bayesian techniques, particularly well-suited to production function estimation due to their relative robustness under collinearity and measurement error. The advantage of the SFA approach is that it allows to test several parametric specifications (such as the Cobb--Douglas and translog) directly. It is also useful for drawing precise conclusions on the aggregate production function's elasticities of substitution.1. The Cobb--Douglas production function fails to reproduce the important properties of our data (regarding the distribution of inefficiency levels, partial elasticities and elasticities of substitution). 2. The (non-parametric) bootstrap-augmented DEA frontier is not only markedly different from the Cobb--Douglas production function specification, but also from the translog, even though the latter offers much more flexibility and can be fitted to the data relatively well. 3. Partial elasticities of the aggregate production function are correlated with inputs both in the DEA and in the translog case, and they vary substiantially across countries and time, providing strong evidence against the Cobb--Douglas specification and lending support to the skill-biased technical change hypothesis. 4. Tests of returns to scale based on the DEA, Cobb--Douglas and translog representations of the frontier provide mixed evidence on this property, although DRS seems more prevalent in larger economies, and IRS -- in smaller economies. 5. Unskilled and skilled labor are not perfectly substitutable. 6. (Morishima and Allen--Uzawa) elasticities of substitution vary largely across countries and time, staying in broad agreement with the hypothesis of capital--skill complementarity. 7. According to DEA, differences in GDP per worker between the USA and most Western European countries in 1980 have been mostly due to differences in efficiency and skilled labor endowments, whereas in 2004 they have been mostly due to differences in efficiency and physical capital endowments. Average efficiency differences have grown visibly between 1980 and 2004. 8. According to the Cobb--Douglas production function specification, the differences in GDP per worker between the USA and other countries in the sample have been predominantly Total Factor Productivity (TFP)-driven, with a few exceptions where physical capital differences played an equally important role. 9. According to DEA, factor accumulation and technological progress have provided significant positive contributions to GDP growth in 1980--2004, with technological progress being particularly powerful in 1990--2004. Average efficiency levels have been declining, providing negative contributions to GDP growth. 10. According to the Cobb--Douglas production function specification, TFP growth, physical capital accumulation, and human capital accumulation have all provided positive contributions to GDP growth throughout 1980--2004. The variance of their relative strength across countries and time was large.