stochastic frontier models

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Stochastic FrontierModels

Efficiency Measurement

Stochastic Frontier Models

William GreeneStern School of BusinessNew York University

0 Introduction1 Efficiency Measurement2 Frontier Functions3 Stochastic Frontiers4 Production and Cost5 Heterogeneity6 Model Extensions7 Panel Data8 Applications

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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 (z1,…,zL-1) > 0.

“Microeconomic Theory,” Mas-Colell, Whinston, Green: Oxford, 1995, p. 129. See also Samuelson (1938) and Shephard (1953).

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Stochastic FrontierModels

Efficiency Measurement

Thoughts on InefficiencyFailure 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 marketpower.

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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)

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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)

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Stochastic FrontierModels

Efficiency Measurement

Production Functions

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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:

L .y x y x( ) = { : ( , ) is producible}

I( ) = { : L( ) and ( ) if 0 <1}.y x x y x yL

k k k j

ES( )={ : L( ) and ' L( ) for ' when k and < for some j}.

y x x y x y xx x x x

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Stochastic FrontierModels

Efficiency Measurement

Isoquants and Level Sets

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Stochastic FrontierModels

Efficiency Measurement

The Distance Function

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Stochastic FrontierModels

Efficiency Measurement

Inefficiency in Production

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Stochastic FrontierModels

Efficiency Measurement

Production Function Model with Inefficiency

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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.

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Stochastic FrontierModels

Efficiency Measurement

Specifications

1

121 1 1

Cobb Douglas ln lnTranslog ln ln ln lnBox-Cox transformations to cope with zerosRegularity Conditions: Monotonicity and ConcavityTranslog Cost Modelln

Kk kk

K K Kk k km k mk k m

k

y x

y x x x

C

121 1 1

L L1st21 s 1 t 1

1 1

ln ln ln ln ln ln ln ln ,

K K Kk km k mk k m

Ls s s ts

K Lks k sk s

w w w

y y y

w y

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Stochastic FrontierModels

Efficiency Measurement

Corrected Ordinary Least Squares

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Stochastic FrontierModels

Efficiency Measurement

COLS Cost Frontier

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Stochastic FrontierModels

Efficiency Measurement

Modified OLSAn 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].

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Stochastic FrontierModels

Efficiency Measurement

COLS and MOLS

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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.