PRODUCTION FUNCTION AND ESTIMATION OF EFFICIENCY) Konstantinos - PowerPoint PPT Presentation

LECTURE 3- PRODUCTION, TECHNOLOGY AND COST FUNCTIONS (TRANSLOG PRODUCTION FUNCTION AND ESTIMATION OF EFFICIENCY) Konstantinos Kounetas School of Business Administration Department of Economics Master of Science in Applied Economic Analysis

Introduction • Please recall in your mind the basic definitions of technical, allocative and total efficiency. • In the relevant literature, there are two distinct approaches to estimate technical and scale efficiency,stochastic frontier analysis (SFA) and data envelopment analysis (DEA). The main difference between these two methodological approaches is that the stochastic frontier (SF), given that it is parametric, allows the coexistence of inefficiencies and random errors, while the DEA, being nonparametric, attributes the total deviation from the frontier to inefficiency. • The approach explicitly recognizes that production function represents technically maximum feasible output level for a given level of output. • The Stochastic Frontier Analysis (SFA) technique may be used in modelling functional relationships where you have theoretical bounds: • Estimation of cost functions and the study of cost efficiency • Estimation of revenue functions and revenue efficiency • This technique is also used in the estimation of multi-output and multi-input distance functions • Potential for applications in other disciplines • Farrell (1957) poses that the production function is never well known suggesting the CD function while Aigner and Chu (1968) considered an estimation of it.

Going back…. • Much of the work on stochastic frontiers began in 70’s. Major contributions from Aigner, Schmidt, Lovell, Battese and Coelli and Kumbhakar. • Ordinary least squares (OLS) regression production functions: • fit a function through the centre of the data • assumes all firms are efficient • Deterministic production frontiers: • fit a frontier function over the data • assumes there is no data noise • SFA production frontiers are a “mix” of these two

Methodology • Following Aigner and Chu (1968)        ln y x u      Min ln u y x i i i i i i     1 row vector, x K i i i    subject to 0 . u    1 column vector, K i non negative error term u i   x u l y e • Thus, estimation using LP i i     u TE i e   i x x e e i i • The specific measure is an output oriented Farrell measure of technical efficiency taking values between zero and one. • It is important, however, to know the distribution of errors. Afriat (1972) assuming gamma estimates using a ML method, Richmond added Corrected Ordinary Least Squares while Schmidt (1976) assumes exponential or half normal random variables (If we apply OLS, intercept estimate is biased downwards, all other parameters are unbiased)

Deterministic Frontier models • So COLS suggests that the OLS estimator from OLS be corrected. • If we do not wish to make use of any probability distribution for y i then   ˆ ˆ      ˆ ( ) ( ) max : 1 , 2 ,..., COLS OLS imum u i N o o i i ˆ where is the OLS residual for i-th firm. u i • If we assume that u i is distributed as Gamma then ˆ ˆ 2      ˆ ( ) ( ) [ ] COLS OLS OLS o o • It is a bit more complicated if u i follows half-normal distribution.

The Stochastic Production Function I • It is a relationship between output and a set of input quantities. • We use this when we have a single output • In case of multiple outputs: • people often use revenue (adjusted for price differences) as an output measure • It is possible to use multi-output distance functions to study production technology. • Aigner et al (1977) and Meeusen and van den Broeck (1977) proposed independently the functional relationship is usually written in the form:       ln y x v u i i i i     1 row vector, x K i      1 column vector, K     non negative error term. ln u Therefor e y x i i i measurement e rro rs v i

The Stochastic Production Function II • Crucial assumption by Aigner et al (1977)       ln y x v u i i i i     1 row vector, x K i      1 column vector, K non negative error term u i measurement errors v i that   1. v i = “noise” error term – symmetric i.i.d (eg. normal distribution )  2 0, v i v 2. u i = “inefficiency error term” - non-negative (eg. half-normal distribution) Do you think that these assumptions are correct? Criticism!! The ,model is called stochastic because the output values are bounded by the stochastic variable.

Production functions/frontiers Deterministic q SFA × × × OLS × × × × × × × x

Stochastic frontiers-graphical representation deterministic frontier y i q i = exp( β 0 + β 1 ln x i ) * ? exp( β 0 + β 1 ln x A + v A ) q A noise effect noise effect * ? exp( β 0 + β 1 ln x B + v B ) q B inefficiency effect q B ? exp( β 0 + β 1 ln x B + v B – u B ) inefficiency effect q A ? exp( β 0 + β 1 ln x A + v A – u A ) x A x B

Maximum Likelihood Estimation Ι Let X1, X2,..., Xn be a random sample from a distribution that depends on one or more unknown parameters θ1, θ2,..., θm with probability density (or mass) function f(xi; θ1, θ2,..., θm). Suppose that (θ1, θ2,..., θm ) is restricted to a given parameter space Ω. When regarded as a function of θ 1 , θ 2 ,..., θ m , the joint probability density (or mass) function of X 1 , X 2 ,..., X n : n       θ , θ ,..., θ ;θ , θ ,..., θ L f x 1 2 m i 1 2 m  1 i (( θ 1 , θ 2 ,..., θ m ) in Ω) is called the likelihood function . If now         , ,..., , , ,..., ,..., , ,..., u x x x u x x x u x x x   1 1 2 n 2 1 2 n 1 2 n m is the m -tuple that maximizes the likelihood function, then ^       , Χ ,..., Χ u   is the maximum likelihood estimator of θ i , for i =   1 2 n 1, 2, ..., m . The corresponding observed values of the statistics in         , ,..., , , ,..., ,..., , ,..., u x x x u x x x u x x x   1 1 2 n 2 1 2 n 1 2 n m are called the maximum likelihood estimates of θi , for i = 1, 2, ..., m.

Maximum Likelihood Estimation Ι I Let us now have a look at some well-known distributions: n    2   x   2 i   x  1  n 1  i 1             ; θ  2 θ ;θ  2 2 2 f x e L f x e   i   i   n /2 2 2 n 2 2  1 i    θ log 1 n L n n                2   θ log 2 log 2 log L x    i 2 2 2 2  1 i n  x i   x n         i 1          ; θ θ ;θ n f x e L f x e i i ! !,..., ! x x x  i 1 1 n    θ log n L              θ log !,..., ! log L n x x x   i 1 n  1 i   1 1      θ  ,θ=max , ,.., n ,0 1 x x x x             ; θ θ ;θ   1 2  i  n n n f x L f x  i i        ώ ώ i 1  0,  0, 1  x        ; θ 1 f x x e      i

Maximum Likelihood Estimation ΙΙ I • The parameters of the stochastic frontier production function can be estimated using the ML method (or COLS by Richmond (1974)). • Parameters to be estimated in a standard SF model are:   β 2 2 , and v u • Likelihood methods are used in estimating the unknown parameters. Coelli (1995)’s Monte Carlo study shows that in large samples MLE is better than COLS. • According to Aigner et al., (1977) expressed the likelihood  function in terms of     σ 2 σ 2 σ , 2 u  s u v v • While Battese and Corra (1977) shows that while testing for the σ 2 σ 2 presence of technical inefficiency depends upon the    u u  σ 2 σ 2 σ 2 parametrization used s u v

Mean Technical Efficiency • Battese and Corra (1977) shows that in terms of parameterization the log-likelihood function is equal to:      N N 1 N N                     2 2 ln ln log ln 1 ln L z y x   2  s i i i 2 2 2 2   1 1 i i s      ln y x  i i , z    i 1 s     distribution function of the standard normal random variable Mean Technical Efficiency         2        s   u β 2 2 2 1 2 , and E e  e  i   s v u  