Productivity Measurement under Uncertainty

Stochastic productivity indicators are de ned, and superlative measures of these indicators are derived for risk-neutral producers. It is shown that in a rational-expectations equilibrium, di¤erences in the market values of rms are superlative indicators of cross-sectional productivity di¤erences. An illustrative regression-based method for estimating the intertemporal superlative productivity indicators is discussed and applied to data for postwar US agriculture. Comparison with existing nonstochastic productivity indicators suggests that properly accounting for the stochastic environment in which rms operate could have important empirical implications for measuring productivity growth. Key words: productivity measurement, uncertainty, risk Productivity Measurement under Uncertainty Most currently computed economic productivity indexes are based on the speci cation of a primal productivity index, the assumption of price-taking behavior on the part of producers, and superlative approximation results originally due to Diewert (1976). Caves, Christensen, and Diewerts (1982, 1982) classic derivations of the Törnqvist productivity index from a transcendental logarithmic radial distance function de ne and epitomize this approach. In deriving these indexes, the basic tactic is to reduce a productivity index speci ed in primal terms to a di¤erential approximation and then to use the assumption of price-taking behavior to obtain easily calculable measures.1 A salient assumption in these approximation arguments is that both the primal technology and the decision environment are nonstochastic. This paper develops productivity measures for producers facing stochastic technologies and a stochastic decision environment. Superlative approximations to these stochastic productivity measures are then derived for risk-neutral producers. Among other results, it is shown that, in a rational expectations equilibrium, di¤erences in rm values provide superlative measures of productivity di¤erences. If one is to consider stochastic productivity measures seriously, some prima facie evidence should exist that current productivity measures, which ignore the stochastic nature of technology and the decision environment, are potentially misleading. Even though it is a truism that the world is a very uncertain place, uncertainty may not a¤ect practical productivity measurement. To this end, Figure 1 portrays the logarithmic di¤erences of the o¢ cial total factor productivity measure for US agriculture reported by the United States Department of Agriculture (USDA) for the years 1949 to 1999.2 These di¤erences approximate year-to-year percentage changes in measured US agricultural total factor productivity. Through the 1950s and 1960s, measured productivity seemed to grow at a reasonably stable 1Diewert (1987) and Balk (1998) contain summaries of the state of the art in the economic computation of index numbers. 2USDA was the rst federal agency to introduce multifactor productivity measures into its statistical program starting in 1960. The total factor productivity measure from which this logarithmic di¤erence is derived as well as the data underlying the calculation of the total factor productivity measure are publicly available at www.ers. usda.gov/data/AgProductivity. This measure is also reported annually in the data appendix to the Economic Report of the President.