Measuring Technical Efficiency

Lecture XIV

Basic Concepts of Production Efficiency

Lovell, C. A. Knox. “Production Frontiers and Productive Efficiency.” In Harold O. Fried, C. A. Knox Lovell and Shelton S. Schmitz (eds.) The Measurement of Productive Efficiency (New York: Oxford University Press, 1993): 3-67.

The most basic concept of the production function is that they represent some kind of frontier.

For example, in our discussion of Diewert, we defined the production function as:

Thus, y=f(x) was the largest output possible for a given set of inputs.

( ) max : belongs to f x x L

Univariate Case

x

f x y

Level Set

1x

2x

L y

These formulations appear to acknowledge that some firms may be performing sub-optimally. That they could obtain a higher amount of output for the same bundle of inputs. This concept underlies the notion of technical inefficiency.

However, even if the firm is operating on the frontier, we also must recognize that they may be using inputs non-optimally.

A basic notion from the production function is that

If inputs do not correspond to this allocation, then the firms could trade one input for another and reduce cost.

1 1 2 1

2 1 2

2

f xMPP x dx w

f xMPP dx wx

Again revisiting our discussion of Diewert:

Thus, we have a graphic depiction of the allocative inefficiency:

, min : belongs to x

c w y w x x L y

Allocative Inefficiency1x

2x

0 , min :x

AE w x c w y w x x L y

Total Inefficiency1x

2x

0 , min :x

AE w x c w y w x x L y

0TE w x w x

A Mathematical Formulation

Restatement of the level set. Defining the production technology as:

This is the basic production possibility set in Diewert:

: , is feasibleL y x y x

: , nonnegativeL y x f x y x

This also leads to the definition of the isoquant:

This definition rules out the interior points to the level set. The efficient subset can then be defined as:

: , , 0,1Iso L y x x L y x L y

: , ,Eff L y x x L y x L y x x

Based on this definition, the input distance function is written as:

For the isoquant:

, max :IxD y x L y

: , 1IIso L y x D y x

The Debreu-Farrell input oriented measure of technical efficiency can then be expressed as:

, min :

, 1

1,,

I

I

II

DF y x x L y

DF y x

DF y xD y x

A slightly different development is given by Färe and Primont:

In the univarate case:

where T is the technology set. Or

:

max : ,

N

y

F R R

F x y x y T

, : ,T x y F x y y R

The distance function is then given by

Alternatively, the distance function can be written in terms of the technology set:

, min :oyD x y F x

, min : ,oyD x y x T

The last representation is then expandable into multivariate space:

: :

, inf 0 : ,

N M N Mo

o

F R R D R R R

yD x y x T

Färe and Primont

0x

0y

0

00 0,o

yF x D x y

The Färe-Primont formulation depicts the output augmentation point of view, while the first formulation depicts the input distance formulation.

Properties of Debreu-Farrell Measures

DFI(y,x) is homogeneous of degree –1 in inputs and DFO(y,x) is homogeneous of degree –1 in outputs. DFI(y,x) is weakly monotonically decreasing in inputs and DFO(y,x) is weakly monotonically decreasing in outputs. DFI(y,x) and DFO(y,x) are invariant with respect to changes in units of measurement.

Measurement with cost and profit functions:

, ; min : , ; 1Ixc y w w x D y x

Empirical Estimation

General formulation:

; exp

exp; exp

i i i i

ii i

i i

y f x

yTE

f x

Econometric Methods One-sided error terms–Gamma distributions and corrected OLS.Composed error term–Stochastic frontier Models.

Data Envelope Analysis

*

1

*1 1

1

*

1

*1 1

1

*

1

minN

i iz i

N

i ii

N

i mi mi

N

i ii

N

i ki ki

z c c

st z x x

z x x

z y y

z y y