Applications in Production Economics
Lecture XXVII
I.
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.
A. “The purpose of this article is to determine how effectively a sample of retail
multiproduct agribusinesses achieve an objective cost minimization.”
1. “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.”
2. “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.
B. Measuring Productive Efficiency
1. Let C R (Y , W , K ) be the total variable costs of producing output vector
Y using input vector X R , where W is a vector of variable input prices
and K is a vector of quasi-fixed factors of production – X R is the
vector of inputs actually used.
2. C T (Y ,W , K ) is the cost of producing the output vector Y using the
input vector X T – X T is the efficient input vector.
3. C P (Y , W , K ) is the cost associated with input vector X P to produce
output vector Y – X P is allocatively and technically efficient.
XP
X
XR
T
1
AEB 6184 – Production Economics
Professor Charles Moss
Lecture XXVII
Fall 2005
4. Farrell measures of efficiency
XT
C T (Y )
TE = R
=
C (Y )
XR
AE =
XP
C P (Y )
=
C T (Y )
XT
XP
C P (Y )
OPE = R
=
C (Y )
XR
5. 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
CR
CM
ZI
ZR
C M (Y )
STE1 =
STCE1 =
C R (Y )
C. Estimation
1. Akridge estimates the frontier assuming a non-stochastic Translog cost
function with associated share equations:
ln ( C ) = α 0 + α ′w + 1 w′Aw + β ′ y + 1 y′By + w′Ψy + vi
2
2
s j = α j + w′A[., j ] + Ψ [ j ,.] y + uij
2. The residual from the cost function is assumed to be distributed
Gamma
2
AEB 6184 – Production Economics
Professor Charles Moss
Lecture XXVII
Fall 2005
f ( vi ) =
λ P viP −1 exp ( −λ vi )
Γ ( P)
3. The residual vector in the share equations are assumed to be normal
1
1 n
−1
−1
2
ˆ
1
′
f ( ui ) =
Ω exp − ui Ω ui ⇒ Z = Ω MLE = ∑ ui ui′
N
2
2
n i =1
( 2π )
)
(
4. The likelihood function is then
NT
L (θ ) = −
( ln ( 2π ) + 1) − T ln ( Γ ( P ) ) − P ln ( λ )
2
(
n
n
i =1
i =1
)
+ ( P − 1) ∑ ln ( Ci − β ′ X i ) − λ ∑ ( Ci − β ′xi ) −
T
ln Z
2
D. Results
Table 3. Descriptive Statistics for Retail Fertilizer Plant Multifactor Efficiency Indexes
Statistic
Technical
Allocative
Overall
Efficiency
Efficiency
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
II.
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.
A. “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.”
B. Multiproduct Cost Concepts
1. Product-specific economies are measured by incremental cost. The
incremental cost of the i th 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 i th output
ICi = C (Y ) − C (YN −i )
s.t.YN −i = (Y1 , KYi −1 , 0, Yi +1 , KYN )
2. Product-specific economies of scale ( Si ) are the average incremental
ICi
cost of producing the i th output
divided by the marginal
Yi
incremental cost of producing the i th output.
ICi
Yi
Si =
∂C
∂Yi
3
AEB 6184 – Production Economics
Professor Charles Moss
Lecture XXVII
Fall 2005
a. If Si is greater than 1, then product-specific economies of scale
exist.
b. Product-specific economies of scale are analogous to the
single-output case of scale economies.
3. Economies of scope (diversification) arise from savings obtained from
the simultaneous production of several outputs.
a. 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.
b. Mathematically
C (Y1 ) + C (Y2 ) > C (Y )
where C (Y1 ) is the cost of producing Y1 alone and C (Y2 ) is
the cost of producing Y2 alone. The economies of scope
[ SCN (Y ) ] is then defined as
SC N (Y ) =
C (Y1 ) + C (Y2 ) − C (Y )
C (Y )
4. Both the economies of scope [ SCN (Y ) ] and product-specific
economies ( Si ) can be combined to give an overall measure of the
returns to scale for an individual firm:
α S (Y ) + (1 − α1 ) S 2 (Y )
S N (Y ) = 1 1
1 − SC N (Y )
C. Results
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 EstateLoans
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
4