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.