APPLICATIONS OF DUALITY THEORY TO AGRICULTURE

Duality Applications

linear duality is a generalization of conven-
tional duality in the sense that it allows both
the objective and constraint functions to be
non-linear.

The farm-household optimizing problem
can be seen as one of maximization of a non-
linear utility function subject to a non-linear
budget constraint. This constraint is non-
linear because an important proportion of the
farm-household income is given by the farm
returns which is a non-linear function of
household labor (which also appears in the
household’s utility function as leisure) and
fixed factors of production.

Since the seminal work by Sen, a number
of studies have analyzed the neoclassical
model of the farm-household with reference
to developing countries. In contrast with the
conventional models of the firm and of the
household, the farm-household model em-
phasizes the interdependences between util-
ity maximization and profit maximization de-
cisions which arise mainly as a consequence
of the existence of endogeneous prices of
labour and non-traded goods (Hymer and
Resnick). It is argued that family labour and
some goods which are only produced to satis-
fy the family’s own consumption necessities
are traded entirely within the farm-
household complex and hence that their
shadow prices are endogeneous and are de-
pendent on the farm production technology,
household preferences and prices of traded
consumption goods and outputs. These en-
dogeneous shadow prices are the main link-
ages between production and consumption
decisions.

A number of studies [Sen, Hymer and
Resnick, Rarnum and Squire] have been con-
cerned with obtaining empirically testable
predictions from the farm-household model
emphasizing agricultural output supply re-
sponses to price changes. Unfortunately, the
farm-household model does not a priori pro-
vide any definite predictions with respect to
output supply responses. Barnum and Squire
have analyzed alternative assumptions con-
sidering situations where more than one
output can be produced, the existence of
non-traded goods, etc. These authors have
concluded that each of the models analyzed
are consistent with positive or negative out-
put responses. This implies that if the atten-
tion is focused only on observed output sup-
ply responses it is not in general possible to
empirically verify the validity of the farm-
household model.

In this section we show how the use of
duality may help in deriving certain ex-
pressions which allow one to empirically test
the theory of the farm-household model. We
also illustrate the use of duality in deriving an
econometric framework appropriate to em-
pirically test the validity of the model by
estimating the farm-household behavioural
equations in a non ad-hoc manner. That is, to
explicitly derive the estimating equations
from the theoretical model, thus fully pre-
serving the connections between the theoret-
ical model and the estimating equations.

We first consider a variant of the farm-
household model which is a straight forward
generalization of the model used by Sen.
This model assumes no off-farm employ-
ment, that all outputs and inputs have ex-
Ogeneous prices with the only exception
being labour whose (shadow) price is
endogeneously determined within the farm-
household complex. It is also assumed that
the household maximizes a well-defined utili-
ty function which is a function of leisure and
the consumption of a vector of market-
purchased goods. Thus, the utility miximiza-
tion problem is

(6) Max U(H-L, X) :

H-L,X

(i) pX≤τr(q; T, L) +у

(ii) II⅛H-L⅛O. X⅛0
where U (.) is the household utility function,
H is total number of hours which household
members have available for work and leisure,
p is a vector of N market-purchased con-
sumption good prices, X is a vector of N
consumption goods, L is number of hours of
work, ττ (.) is a conditional variable profit

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