[Part 4] 1/25 Stochastic FrontierModels Production and Cost 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 4] 2/25 Stochastic FrontierModels Production and Cost Single Output Stochastic Frontier u i > 0, but v i may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for v i. Thus, the stochastic frontier is  +  ’x i +v i and, as before, u i represents the inefficiency.

[Part 4] 3/25 Stochastic FrontierModels Production and Cost The Normal-Half Normal Model

[Part 4] 4/25 Stochastic FrontierModels Production and Cost Estimating u i  No direct estimate of u i  Data permit estimation of y i – β’x i. Can this be used? ε i = y i – β’x i = v i – u i Indirect estimate of u i, using E[u i |v i – u i ] = E[u i |y i,x i ]  v i – u i is estimable with e i = y i – b’x i.

[Part 4] 5/25 Stochastic FrontierModels Production and Cost Fundamental Tool - JLMS We can insert our maximum likelihood estimates of all parameters. Note: This estimates E[u|v i – u i ], not u i.

[Part 4] 6/25 Stochastic FrontierModels Production and Cost 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).

[Part 4] 7/25 Stochastic FrontierModels Production and Cost 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.

[Part 4] 8/25 Stochastic FrontierModels Production and Cost Duality Between Production and Cost

[Part 4] 9/25 Stochastic FrontierModels Production and Cost Implied Cost Frontier Function

[Part 4] 10/25 Stochastic FrontierModels Production and Cost Stochastic Cost Frontier

[Part 4] 11/25 Stochastic FrontierModels Production and Cost Cobb-Douglas Cost Frontier

[Part 4] 12/25 Stochastic FrontierModels Production and Cost Translog Cost Frontier

[Part 4] 13/25 Stochastic FrontierModels Production and Cost Restricted Translog Cost Function

[Part 4] 14/25 Stochastic FrontierModels Production and Cost Cost Application to C&G Data

[Part 4] 15/25 Stochastic FrontierModels Production and Cost Cost Application to C&G Data

[Part 4] 16/25 Stochastic FrontierModels Production and Cost Estimates of Economic Efficiency

[Part 4] 17/25 Stochastic FrontierModels Production and Cost Duality – Production vs. Cost

[Part 4] 18/25 Stochastic FrontierModels Production and Cost Multiple Output Cost Frontier

[Part 4] 19/25 Stochastic FrontierModels Production and Cost Banking Application

[Part 4] 20/25 Stochastic FrontierModels Production and Cost Economic Efficiency

[Part 4] 21/25 Stochastic FrontierModels Production and Cost Allocative Inefficiency and Economic Inefficiency

[Part 4] 22/25 Stochastic FrontierModels Production and Cost Cost Structure – Demand System

[Part 4] 23/25 Stochastic FrontierModels Production and Cost Cost Frontier Model

[Part 4] 24/25 Stochastic FrontierModels Production and Cost The Greene Problem  Factor shares are derived from the cost function by differentiation.  Where does e k come from?  Any nonzero value of e k, which can be positive or negative, must translate into higher costs. Thus, u must be a function of e 1,…,e K such that ∂u/∂e k > 0  Noone had derived a complete, internally consistent equation system  the Greene problem.  Solution: Kumbhakar in several papers. (E.g., JE 1997) Very complicated – near to impractical Apparently of relatively limited interest to practitioners Requires data on input shares typically not available

[Part 4] 25/25 Stochastic FrontierModels Production and Cost A Less Direct Solution (Sauer,Frohberg JPA, 27,1, 2/07)  Symmetric generalized McFadden cost function – quadratic in levels  System of demands, x w /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 system Errors are decoupled from cost by the ‘averaging.’  Application to rural water suppliers in Germany