[Part 1] 1/18 Stochastic FrontierModels Efficiency Measurement Stochastic Frontier Models William...
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![Page 1: [Part 1] 1/18 Stochastic FrontierModels Efficiency Measurement Stochastic Frontier Models William Greene Stern School of Business New York University 0Introduction.](https://reader036.fdocuments.us/reader036/viewer/2022062401/5a4d1b0e7f8b9ab05998d38b/html5/thumbnails/1.jpg)
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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.