[Part 1] 1/18 Stochastic FrontierModels Efficiency Measurement Stochastic Frontier Models William Greene Stern School of Business New York University 0Introduction 1Efficiency Measurement 2Frontier Functions 3Stochastic Frontiers 4Production and Cost 5Heterogeneity 6Model Extensions 7Panel Data 8Applications

[Part 1] 2/18 Stochastic FrontierModels Efficiency Measurement The Production Function “A single output technology is commonly described by means of a production function f(z) that gives the maximum amount q of output that can be produced using input amounts (z 1,…,z L-1 ) > 0. “Microeconomic Theory,” Mas-Colell, Whinston, Green: Oxford, 1995, p See also Samuelson (1938) and Shephard (1953).

[Part 1] 3/18 Stochastic FrontierModels Efficiency Measurement Thoughts on Inefficiency Failure to achieve the theoretical maximum  Hicks (ca. 1935) on the benefits of monopoly  Leibenstein (ca. 1966): X inefficiency  Debreu, Farrell (1950s) on management inefficiency All related to firm behavior in the absence of market restraint – the exercise of market power.

[Part 1] 4/18 Stochastic FrontierModels Efficiency Measurement A History of Empirical Investigation  Cobb-Douglas (1927)  Arrow, Chenery, Minhas, Solow (1963)  Joel Dean (1940s, 1950s)  Johnston (1950s)  Nerlove (1960)  Berndt, Christensen, Jorgenson, Lau (1972)  Aigner, Lovell, Schmidt (1977)

[Part 1] 5/18 Stochastic FrontierModels Efficiency Measurement Inefficiency in the “Real” World Measurement of inefficiency in “markets” – heterogeneous production outcomes:  Aigner and Chu (1968)  Timmer (1971)  Aigner, Lovell, Schmidt (1977)  Meeusen, van den Broeck (1977)

[Part 1] 6/18 Stochastic FrontierModels Efficiency Measurement Production Functions

[Part 1] 7/18 Stochastic FrontierModels Efficiency Measurement Defining the Production Set Level set: The Production function is defined by the isoquant The efficient subset is defined in terms of the level sets:

[Part 1] 8/18 Stochastic FrontierModels Efficiency Measurement Isoquants and Level Sets

[Part 1] 9/18 Stochastic FrontierModels Efficiency Measurement The Distance Function

[Part 1] 10/18 Stochastic FrontierModels Efficiency Measurement Inefficiency in Production

[Part 1] 11/18 Stochastic FrontierModels Efficiency Measurement Production Function Model with Inefficiency

[Part 1] 12/18 Stochastic FrontierModels Efficiency Measurement Cost Inefficiency y* = f(x)  C* = g(y*,w) (Samuelson – Shephard duality results) Cost inefficiency: If y < f(x), then C must be greater than g(y,w). Implies the idea of a cost frontier. lnC = lng(y,w) + u, u > 0.

[Part 1] 13/18 Stochastic FrontierModels Efficiency Measurement Specifications

[Part 1] 14/18 Stochastic FrontierModels Efficiency Measurement Corrected Ordinary Least Squares

[Part 1] 15/18 Stochastic FrontierModels Efficiency Measurement COLS Cost Frontier

[Part 1] 16/18 Stochastic FrontierModels Efficiency Measurement Modified OLS An alternative approach that requires a parametric model of the distribution of u i is modified OLS (MOLS). The OLS residuals, save for the constant displacement, are pointwise consistent estimates of their population counterparts, - u i. Suppose that u i has an exponential distribution with mean λ. Then, the variance of u i is λ 2, so the standard deviation of the OLS residuals is a consistent estimator of E[u i ] = λ. Since this is a one parameter distribution, the entire model for u i can be characterized by this parameter and functions of it. The estimated frontier function can now be displaced upward by this estimate of E[u i ].

[Part 1] 17/18 Stochastic FrontierModels Efficiency Measurement COLS and MOLS

[Part 1] 18/18 Stochastic FrontierModels Efficiency Measurement Principles  The production function model resembles a regression model (with a structural interpretation).  We are modeling the disturbance process in more detail.