APPLICATIONS OF DUALITY THEORY TO AGRICULTURE



December 1982


Western Journal of Agricultural Economics

function, q is an exogeneous price vector of
the S net outputs produced by the farm,
using the convention of representing output
prices by positive quantities and purchased
input prices by negative quantities, T is a
fixed factor of production, say land an y is
non-labour income net of fixed obligations.

Constraint (i) in (6) indicates that the total
expenditures on consumption goods cannot
be greater than the income associatd with the
net farm returns to labour and owned fixed
resources represented by the profit (function)
plus the net non-labour income which may
include government transfers, asset income,
etc.

The conditional variable profit function τr
(q;T,L) is defined as follows:

(7) ττ(q; T, L)≡ Max {qQ : (Q; T, L)ετ}
Q

where Q is a vector of S net outputs including
M outputs and S-M inputs, and τ is the
production possibilities set which is assumed
to be a compact, non-empty, convex set.

It is easy to verify that τr (q; T,L)∙ is non-
negative, continuous linearly homogeneous
and convex in q, nondecreasing and concave
in T and L for fixed q.

Notice that this specification allows for a
rather general production technology and, in
contrast with the two crop model of Barnum
and Squire, for example, it allows for the
existence of economies (or diseconomies) of
joint production. It also allows for the ex-
istence of several purchased inputs. Also
note that ττ (∙) is a variable profit function
conditional on a given level of work L, which
is jointly determined as an equilibrium level
obtained from the labour supply schedule
associated with the household’s preferences
for leisure and the demand for labour sched-
ule associated with the production side rep-
resented by the variable profit function.

If problem (6) is defined locally for the
compact subset M, then we can define an
indirect utility function associated with such
a problem in the following manner:

360


(8) G(p, q, T, y)≡ Max {U(H-L, X) :

H-L,X


(i) pX-ττ(q; T, L)≤y


(ii) (H-L, X)ε∕ and (p, q, T, Y)εP}

where the attention is restricted the set of
utility levels M≡{μ: μ≤μ≤μ} which implies
that the corresponding commodity space ʃ
and parameter space P are compact, non-
empty sets.

Epstein has shown the existence of a duali-
ty relationship in the context of a more gen-
eral non-linear model of which (8) is a special
case. Epstein’s results imply that an indirect
utility function G(∙) exists and, moreover,
that there exists a one-to-one duality relation
between the function G(∙) and U(∙) for a given
function ττ (∙). The function G(∙) is non-
increasing in p, non-decreasing in q,T and y
and homogeneous of degree zero in p,q and
y. Moreover, minimization of G(∙) subject to
the budget constraint allows to retrieve a
function U* with identical behavioural impli-
cations as U and, from the first order con-
ditions of this minimization problem, one
obtains a relationship between the indirect
utility function and the farm-household be-
havioural equations:


(ɪ) Qi


(ii) Xj


/...ʌ дтт

(llɪ) =

∖ /


0G∕aqi
∂G∕∂y


i=l,-,S


<9G∕<⅛j
dG/ду


∂G∕∂T
dG/ду


j = l,-,N


Notice that the net output supply equations
(9i) are dependent on the structural prop-
erties of both the production technology and
household’s preferences. Moreover, output
supply responses are affected not only by the
level of net output prices, but also by the
price level of those commodities consumed
by the houseshold as well as by the house-




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