A Duality Approach to Testing the Economic Behaviour of Dairy-Marketing Co-operatives: The Case of Ireland



Hotelling’s Lemma (Varian (1978)) gives us the processed output supply and variable input
demand equations for the cooperative firm as equations (4)(a) and (4)(b):

< c λi
k,m


~ = yc =yc(pyyym,k) (a)

,Py J

r -. -. -                                           141

- = -χc = χc(p W,m,k) (Z>)

<Py J

where,
χc= profit-maximising quantities of variable inputs demanded by the cooperative firm such that
χc < X.

On the face of it equations (3) - (4) appear quite different from the system in (1) - (2). However,
under certain additional assumptions the derivative relationships may be shown to be identical.
This is of course an implication of the well-known Le Chatelier-Samuelson principle or the
Envelope Theorem (Varian (1978)).

To see the correspondence between the system (1) - (2) and (3) - (4), consider the implied returns
to capital for the cooperative firm which are defined in equation (5) as:

(5)


k= kΛPy^w^k)-Prnm

where,
pm = price of milk paid to cooperative members.



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