Lopez
Duality Applications
to support in agriculture than simple cost
minimization because of risk related prob-
lems which are mainly related to the variabil-
ity of output yields and price rather than to
costs of production.
The factor demands estimated using a pro-
fit function framework allow one to measure
input substitution and output scale effects of
factor price changes. Additionally, one can
measure the cross effects of output price
changes on factor demands and vice versa as
well as output supply responses and their
cross price effects. A major advantage of the
profit function framework is that it allows the
estimation of multi-output technologies in a
much simpler way than a cost function or a
transformation function. The profit function,
π, is defined by
(5) ττ (p,w⅛K)≡{Max py-wx:F(y,x;K) = O}
y,w
where y is a vector of M outputs, x is a vector
of N variable inputs, K is a vector of S fixed
inputs, F(∙) is a continuous, concave transfor-
mation function, p, w are vectors of M output
prices and N input prices.
It has been shown that the profit function
ττ (.) is non-decreasing in p, non-increasing in
w, linear homogeneous and convex in p and
w. Moreover, its Hessian matrix with respect
to p and w is symmetric. As in the case of the
cost function, knowledge of these properties
has allowed to develop suitable functional
specifications which permit to test, verify or
impose the above properties. The factor de-
mands and output supply equations are de-
rived from the specified profit function by
simple differentiation with respect to input
prices and output prices, respectively (Hotel-
ling’s lemma). Furthermore, the shadow
price of fixed resource Ki is the derivative of
π (■) with respect to Ki.
Most applications of profit functions to ag-
riculture have assumed a single output tech-
nology. The earlier works used very simple
and restrictive specifications for profit func-
tions. Among these one may mention the
studies by Lau and Yotopoulos of 1972 and
Yotopoulos, Lau and Lin of 1976 who used a
Cobb-Douglas specification for a single out-
put restricted profit function. They estimated
output supply and input demand responses
using data from India and Taiwan, respec-
tively.
More recent studies have used flexible
functional form specifications for the profit
function. Binswanger and Evenson tried
various single output flexible form specifi-
cations using Indian data including the gen-
eralized Ieontief, translog and the quadratic
normalized function. They found that, in
general, the results obtained using the trans-
log specification were less compatible with
the restrictions implied by economic theory
than the other two forms. An undesirable
feature of the specifications used by Binswan-
ger and Evenson for both the generalized
Ieontief and normalized quadratic forms is
that the shadow prices of fixed resources are
implicitly assumed constant independent of
the level of fixed resources.
Another recent application of the profit
function approach is the study by Sidhu and
Baanante to analyze input demand and wheat
supply in the Punjab region of India. They
used a normalized restricted translog profit
function considering wheat output, three
variable inputs (labor, fertilizers and animal
power) and seven fixed factors (machinery
and equipments, land, various soil nutrients,
schooling and irrigation area). They obtained
estimates for the elasticities of wheat supply
responses as well as for the three variable
factor demands. They showed that the Cobb-
Douglas profit function specification is not
supported by the data, and that the symmet-
ry restrictions are not rejected. They ob-
tained a wheat supply elasticity of 0.6 and,
surprisingly, they found that the output price
effect is more powerful in affecting demand
for labor, fertilizer and animal power than
their respective prices.
As indicated before, the vast majority of
the profit function applications to agriculture
assume a single output technology. Since
agricultural production is carried on in farm
units which typically produce several out-
puts, this implies that either there exist no
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