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Stochastic Frontier Models

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

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Stochastic Frontier Models

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

  2. The Production Function “A single output technology is commonly described by means of a production functionf(z) that gives the maximum amount q of output that can be produced using input amounts (z1,…,zL-1) > 0. “Microeconomic Theory,” Mas-Colell, Whinston, Green: Oxford, 1995, p. 129. See also Samuelson (1938) and Shephard (1953).

  3. 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.

  4. 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)

  5. 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)

  6. Production Functions

  7. 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:

  8. Isoquants and Level Sets

  9. The Distance Function

  10. Inefficiency in Production

  11. Production Function Model with Inefficiency

  12. 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.

  13. Specifications

  14. Corrected Ordinary Least Squares

  15. COLS Cost Frontier

  16. Modified OLS An alternative approach that requires a parametric model of the distribution of ui is modified OLS (MOLS). The OLS residuals, save for the constant displacement, are pointwise consistent estimates of their population counterparts, - ui. Suppose that ui has an exponential distribution with mean λ. Then, the variance of ui is λ2, so the standard deviation of the OLS residuals is a consistent estimator of E[ui] = λ. Since this is a one parameter distribution, the entire model for ui can be characterized by this parameter and functions of it. The estimated frontier function can now be displaced upward by this estimate of E[ui].

  17. COLS and MOLS

  18. Principles • The production function model resembles a regression model (with a structural interpretation). • We are modeling the disturbance process in more detail.

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