A Test of HTML Production in SWP4

Philip A. Viton

January 13, 2002

1 Econometric Approaches
2 The DEA Approach

1 Econometric Approaches

Econometric approaches to cost efficiency estimate a parametric cost function and then identify cost efficiency via differences between observed cost and the minimum predicted cost. Consider carrier k in year t. It has a V -vector z_kt = (z_kt1,...,z_ktV) of system characteristics and produces the M-vector of outputs y_kt = (y_kt1,...,y_ktM) from the N-vector of inputs x_kt = (x_kt1,...,x_ktN) according to the transformation function F(y_kt,x_kt;z_kt) = 0. It is assumed to face the N factor prices w_kt = (w_kt1,...,w_ktN). Its (minimum) cost function C^*(y_kt,w_kt;z_kt) is the indirect objective function of the problem: choose inputs x_kt to

sum minimize nwktnxktn subject to: F(ykt,xkt;zkt) = 0.

If C_kt is the observed cost of carrier k in year t, we allow for a failure to attain the minimum cost by writing

* ekt Ckt = C (ykt,wkt;zkt)e

(1)

where

_kt is a random variable, so that

ln C = lnC*(y ,w ;z )+ e kt kt kt kt kt

(2)

Because the observed cost C_kt cannot exceed the theoretical minimum cost C_kt^*,

_kt is necessarily non-negative. We obtain a parametric cost function by specifying a functional form for C^*(y_kt,w_kt;z_kt). For the remainder of this section, we shall assume that the cost function is translog in inputs and outputs; but that system characteristics enter linearly.¹ Then we may write:

* sum V sum N M sum ln C (ykt,xkt;zkt) = a0 + aizkti + bn ln yktn + gm ln wktm (3) i=1 n=1 m=1 1 N sum N sum 1 sum M M sum + 2 dnpln yktnlnyktp + 2 hmjlnwktm lnwktj n=1 p=1 m=1 j=1 1 N sum sum M + 2 qnm ln wktnlnwktm n=1 m=1

The theory of cost minimization implies that C^*(y_kt,w_kt;z_kt) is homogeneous of degree zero in input prices, which places restrictions on the parameters of the cost function (see, eg, Spady and Friedlaender (1978)); in addition the parameters

and

are symmetric, so that, for example

_np =

_pn.

2 The DEA Approach

Data Envelopment Analysis, or DEA, see, eg Banker, Charnes and Cooper (1984), or Färe, Grosskopf and Lovell (1994), begins by constructing the set of feasible input-output combinations. It then finds, with reference to that set, the input bundle that minimizes the cost of producing the observed output. DEA is non-parametric in that it makes no assumption about the structure of the cost function — indeed it does not explicitly obtain the cost function at all — and, being non-statistical, it has no need for parametric distributional assumptions.² To construct the feasible set it begins by assuming that the observed input-output combinations (the data) are feasible and then adds unobserved combinations by considerations of additivity (scale) and disposal.³

The upshot is that cost-minimizing carrier selects an input bundle _kt to minimize total cost, sum _n_ktnw_ktn = _kt^'w_kt subject to feasibility; and this is the solution to the non-linear program

sum minimize ~xktnwktn subject to : n sum yktm = m l,srlsylsm A m = 1,2,...,M sum c~xktn = l,strlsxlsn A n = 1,2,...,N sum l,srls = 1 0 < c < 1 0 < m < 1

and we define the cost-efficiency of carrier k at period t by

w'kt~xkt CEkt = w'ktxkt

the ratio of observed to minimal cost.

References

Banker, R., A. Charnes, and W. W. Cooper, “Some models for estimating technical and scale inefficiencies in data envelopment analysis,” Mangement Science, 1984, 30 (9), 1078–1092.

Banker, Rajiv D., “Maximum Likelihood, Consistency and Data Envelopment Analysis: A Statistical Foundation,” Management Science, 1993, 39 (10), 1265–1273.

Färe, R., S. Grosskopf, and C. A. K. Lovell, Production Frontiers, Cambridge: Cambridge University Press, 1994.

Spady, R. H. and A. F. Friedlaender, “Hedonic cost functions for the regulated trucking industry,” Bell Journal of Economics, 1978, 9 (1), 159–179.