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

Hotelling’s Lemma gives us the processed output supply and input-demand functions consistent
with maximising the returns to capital as equations (2)(a) and (2)(b):

( A
к

= У = У(Р_у,™,к) (a)

< PyJ

л ʌ

-∖ = -_x = _x(_pwj_i) (b)
у w J

where,

x = vector of variable input quantities.

The path-breaking and enduring work on the economic behaviour of the dairy-marketing
cooperative was developed in a series of seminal papers in the early 1960s by Peter Helmberger
and colleagues (see Helmberger and Hoos (1962), Helmberger (1964) and Helmberger and Youde
(1966)). For convenience we will hereafter refer to this body of work as the theory of the
Cooperative-Managed Firm (CMF). According to this theory the objective function of the
cooperative firm is different to the price-taking profit-maximising firm. The CMF seeks to
maximise the price paid per unit of raw material supplied by its farmer members. The CMF is also
obliged to process all the raw material supplied by its member suppliers.

CMFs are divided into “restricted” and “open” membership forms. In the “restricted” form the
cooperative is constituted as a club and can decide to admit new members as it wishes and thus
adjust its raw material supply through its membership policy. However, it has to process all the
raw material supplied by its current membership. In the “open” co-operative there is little or no
restriction on entry that can be exercised by the cooperative itself. In other words the existing
membership cannot preclude new members from joining. Irish dairy cooperatives and we would

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