Applications in Production
Economics
Lecture XXVII
Akridge, Jay T. “Measuring Productive
Efficiency in Multiple Product
Agribusiness Firms: A Dual
Approach.” American Journal of
Agricultural Economics 71(1)(Feb.
1989): 116–25.
The purpose of this article is to determine
how effectively a sample of retail
multiproduct agribusinesses achieve an
objective cost minimization.”
“In particular, the frontier multiproduct cost
function, which reflects the minimum cost of
producing any given output vector as defined
by the sample’s least-cost firms, provides the
benchmark to make valid cost comparisons in
multiple product firms.”
“The frontier cost function can be used
to compare the observed cost of any
sample firm against the cost which the
sample’s least-cost producers would
incur if producing an identical output
vector and provides the basis for
computing Farrell-type indexes of
productive, technical, and allocative
efficiency.
Measuring Productive Efficiency
Let CR(Y,W,K) be the total variable costs
of producing output vector Y using input
vector XR, where W is a vector of
variable input prices and K is a vector of
quasi-fixed factors of production
XR is the vector of inputs actually used.
CT(Y,W,K) is the cost of producing the
output vector Y using the input vector XR
XR is the efficient input vector.
CP(Y,W,K) is the cost associated with
input vector XP to produce output vector
XP is allocatively and technically efficient.
X
P
X
T
X
R
Farrell measures of efficiency
Y

TE  R

C Y 
C
T
XT
X
R
Y

AE  T

C Y 
XP
Y

OPE  R

C Y 
XP
C
P
C
P
X
T
XR
Single-factor technical efficiency
(STEi) relates the technically efficient
use of xi holding all other inputs
constant to the actual application of xi.
R
Z
I
C
CM
R
ZI
STE1 
ZR
C Y 
STCE1  R
C Y 
M
Estimation
Akridge estimates the frontier assuming
a non-stochastic Translog cost function
with associated share equations:
ln  C    0   w  1 wAw    y  1 yBy  wy  vi
2
2
s j   j  wA., j    j ,. y  uij
The residual from the cost function is
assumed to be distributed Gamma
f  vi  
 v
exp   vi 
  P
P P 1
i
The residual vector in the share
equations are assumed to be normal
f  ui  
1
 2 
N

2
1
2

exp  1 ui 1ui
2
ˆ
Z 
MLE

1 n
  ui ui
n i 1
The likelihood function is then

NT
L    
ln  2   1  T ln    P    P ln   

2
n
n
i 1
i 1

  P  1  ln  Ci    X i      Ci   xi  
T
ln Z
2
Table 3. Descriptive Statistics for Retail Fertilizer Plant Multifactor
Efficiency Indexes
Statistic
Technical
Efficiency
Allocative
Efficiency
Overall
Efficiency
Mean
0.900
0.996
0.897
Std. Deviation
0.068
0.005
0.068
Maximum
1.005
1.000
0.993
Minimum
0.705
0.977
0.705
Featherstone, Allen M. and Charles B.
Moss “Measuring Economies of Scale
and Scope in Agricultural Banking.”
American Journal of Agricultural
Economics 76(3)(Aug. 1994): 655–61.
“Study of the production technology of
financial institutions can determine
whether and to what degree
economies of size exist and how
agricultural lending will fit into the
overall business plans of consolidated
banks.”
Multiproduct Cost Concepts
Product-specific economies are
measured by incremental cost.
The incremental cost of the ith output (ICi) is
defined as the cost of producing the entire
multiproduct output bundle [C(Y) ] minus the
cost of producing all the output except the ith
output
ICi  C Y   C YN i 
s.t.YN i  Y1 ,
Yi 1 , 0, Yi 1 ,
YN 
Product-specific economies of scale (Si )
are the average incremental cost of
producing the ith output ICi/Yi divided by
the marginal incremental cost of
producing the ithoutput
ICi
Si 
C
Yi
Yi
If Siis greater than 1, then product-specific
economies of scale exist.
Product-specific economies of scale are
analogous to the single-output case of scale
economies.
Economies of scope (diversification)
arise from savings obtained from the
simultaneous production of several
outputs.
Economies of scope [SCi(Y)] exist if the cost
of producing the optimal level of outputs in
“individual firms” is greater than the cost of
producing the same optimal output levels in
a multiproduct firm.
C Y1   C Y2   C Y 
SCN Y  
C Y1   C Y2   C Y 
C Y 
Both the economies of scope [SCi(Y)]
and product-specific economies (Si)
can be combined to give an overall
measure of the returns to scale for an
individual firm:
S N Y  
1S1 Y   1  1  S2 Y 
1  SCN Y 
Table 3. Marginal Costs and Product-Specific Economies of
Scale for Bank Outputs
Output
Marginal
Cost
Product
Specific
Economies of
Scale
Agricultural Loans
-0.702
1.0035
Nonagricultural Real Estate Loans
1.717
0.9955
Other Nonagricultural Loans
1.221
0.9942
Transaction Deposits
14.773
0.9995
Nontransaction Deposits
6.570
0.9978
Other Bank Output
-1.467
1.0008