William Greene Stern School of Business New York University 0 Introduction 1 Efficiency Measurement 2 Frontier Functions 3 Stochastic Frontiers 4 Production and Cost 5 Heterogeneity ID: 316644
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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 ApplicationsSlide2
Single Output Stochastic Frontier
u
i > 0, but vi may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for vi. Thus, the stochastic frontier is +’xi+vi and, as before,
ui represents the inefficiency.Slide3
The Normal-Half Normal ModelSlide4
Estimating ui
No direct estimate of uiData permit estimation of yi – β’xi. Can this be used?
εi = yi – β’xi = vi – ui Indirect estimate of ui, using E[ui|vi – ui] = E[ui|yi,xi]vi – ui is estimable with ei = yi – b’xi.Slide5
Fundamental Tool - JLMS
We can insert our maximum likelihood estimates of all parameters.
Note: This estimates E[u|vi – ui], not ui.Slide6
Multiple Output Frontier
The formal theory of production departs from the transformation function that links the vector of outputs, y to the vector of inputs, x; T
(y,x) = 0.As it stands, some further assumptions are obviously needed to produce the framework for an empirical model. By assuming homothetic separability, the function may be written in the form A(y) = f(x).Slide7
Multiple Output Production Function
Inefficiency in this setting reflects the failure of the firm to achieve the maximum aggregate output attainable. Note that the model does not address the economic question of whether the chosen output mix is optimal with respect to the output prices and input costs. That would require a profit function approach. Berger (1993) and Adams et al. (1999) apply the method to a panel of U.S. banks – 798 banks, ten years
.Slide8
Duality Between Production and CostSlide9
Implied Cost Frontier FunctionSlide10
Stochastic Cost FrontierSlide11
Cobb-Douglas Cost FrontierSlide12
Translog Cost FrontierSlide13
Restricted Translog Cost FunctionSlide14
Cost Application to C&G DataSlide15
Estimates of Economic EfficiencySlide16
Duality – Production vs. CostSlide17
Multiple Output Cost FrontierSlide18
Banking ApplicationSlide19
Economic EfficiencySlide20
Allocative Inefficiency and Economic InefficiencySlide21
Cost Structure – Demand SystemSlide22
Cost Frontier ModelSlide23
The Greene Problem
Factor shares are derived from the cost function by differentiation.Where does ek come from?Any nonzero value of ek, which can be positive or negative, must translate into higher costs. Thus, u must be a function of e1
,…,eK such that ∂u/∂ek > 0Noone had derived a complete, internally consistent equation system the Greene problem.Solution: Kumbhakar in several papers. (E.g., JE 1997)Very complicated – near to impracticalApparently of relatively limited interest to practitionersRequires data on input shares typically not availableSlide24
A Less Direct Solution(Sauer,Frohberg JPA, 27,1, 2/07)
Symmetric generalized McFadden cost function – quadratic in levelsSystem of demands, xw/y = * + v, E[v]=0.Average input demand functions are estimated to avoid the ‘Greene problem.’ Corrected wrt a group of firms in the sample.
Not directly a demand systemErrors are decoupled from cost by the ‘averaging.’Application to rural water suppliers in Germany