Efficiency Measurement William Greene Stern School of Business New York University

Session 4 Functional Form and Efficiency Measurement

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.

The Normal-Half Normal Model

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.

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.

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).

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.

Duality Between Production and Cost

Implied Cost Frontier Function

Stochastic Cost Frontier

Cobb-Douglas Cost Frontier

Translog Cost Frontier

Restricted Translog Cost Function

Cost Application to C&G Data

Estimates of Economic Efficiency

Duality – Production vs. Cost

Multiple Output Cost Frontier

Banking Application

Economic Efficiency

Allocative Inefficiency and Economic Inefficiency Technical inefficiency: Off the isoquant. Allocative inefficiency: Wrong input mix.

Cost Structure – Demand System

Cost Frontier Model

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

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