Efficiency Measurement
William Greene
Stern School of Business
New York University
Session 4
Functional Form and
Efficiency Measurement
Single Output Stochastic Frontier
yi
= f (xi )TE i e vi
ln y i =  + xi + vi  ui
=  + xi +  i .
ui > 0, but vi may take any value. A symmetric
distribution, such as the normal distribution, is usually
assumed for vi. Thus, the stochastic frontier is
+’xi+vi
and, as before, ui represents the inefficiency.
The Normal-Half Normal Model
Estimating ui
No direct estimate of ui
 Data permit estimation of yi – β’xi.
Can this be used?




εi = yi – β’xi = vi – ui
Indirect estimate of ui, using
E[ui|vi – ui] = E[ui|yi,xi]
vi – ui is estimable with ei = yi – b’xi.
Fundamental Tool - JLMS
(i ) 
i
   
E[ui | it ]  
 
 , i 
2 i
(i ) 

1    
We can insert our maximum likelihood estimates
of all parameters.
Note: This estimates E[u|vi – ui], not ui.
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

m 1  y
M
q
m
q
i,t,m

1/q
  T xit  v it  u it
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
C ( y,w) = min{w x : f (x)  y}
T
Implied Cost Frontier Function
Stochastic Cost Frontier
Cobb-Douglas Cost Frontier
Translog Cost Frontier
Cost frontier with K variable inputs, one fixed input (F) and
output, y.
ln C F    Kk=1 k lnw k  F ln F  y ln y
 12 Kk =1 Kl =1  kl lnw k lnw l  12  FF ln2 F  12  yy ln2 y
 Kk =1  kF lnw k ln F  Kk =1  ky lnw k ln y  Fy ln F ln y  v i  u i
Cost functions fit subject to theoretical homogeneity in prices
restriction:
lnC
 k=1 lnw  1. Imposed by dividing C and all but
k
K
one of the input prices by the "last" (numeraire) price.
Restricted Translog Cost Function
 C 
 PK 
 PL 
2
1
ln 




ln


ln


ln
y


ln
y
K
yy 2


 L 
 y
 PF 
 PF 
 PF 
 PK 
 PK   PL 
2  PL 
1
  KK 12 ln 2 


ln


ln
 LL 2

 KL 
 ln 

 PF 
 PF 
 PF   PF 
 PK 
 PL 
  yK ln y ln 
   yL ln y ln 
vu
 PF 
 PF 
Cost Application to C&G Data
Estimates of Economic Efficiency
Duality – Production vs. Cost
Multiple Output Cost Frontier
 C
ln 
 wK

M
1 M
M




ln
y


 m1 my m 2  l 1  m1 lm ln yl ln ym

 wk 
  k 1 k ln 

 wK 
K 1
  m 1
M
 wk 
 k 1 mk ln ym ln  w 
 K
K 1
 wk
1 K 1 K 1
  k 1  l 1  kl ln 
2
 wK
  wl 
 ln 
 + vu
  wK 
Banking Application
Economic Efficiency
Allocative Inefficiency and
Economic Inefficiency
Technical inefficiency:
Off the isoquant.
Allocative
inefficiency: Wrong
input mix.
Cost Structure – Demand System
Cost Function
Cost = f(output, input prices) = C(y,w)
Shephard's Lemma Produces Input Demands
C
x*(y,w ) = Cost minimizing demands =
w
Cost Frontier Model
Stochastic cost frontier
lnC(y,w) = g(lny,lnw) + v + u
u = cost inefficiency
Factor demands in the form of cost shares
 ln C(y,w)
sk 
 h(lny,lnw) + ek
lnw k
ek  allocative inefficiency
The Greene Problem





Factor shares are derived from the cost function by
differentiation.
Where does ek come from?
Any nonzero value of ek, which can be positive or
negative, must translate into higher costs. Thus, u
must be a function of e1,…,eK such that
∂u/∂ek > 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, xw/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