We show here that in a share-tenancy contract, under certainty, any combination of input levels which can be achieved by the landlord stipulating those levels can also be achieved by cost-sharing. An intuitive argument was provided in § 3.2; a formal proof is given in this appendix.
The problem where the landlord specifies the input levels is considered first -- we call it problem A. We adopt the same notation as in § 3.2, when we discussed the Cheung model, except that we generalize to include purchased inputs (we denote the inputs by the vector x and their prices by p) as well as the non-purchased input labour, l. The landlord's maximization problem is as follows.
Maximize n r F
r, l, n, x
subject to (1 - r)F - p.x ≧ w l
where output from area M/n if a tenant applies labour, 1, and inputs x is F(l, M/n, x).
Substituting from the constraint (it will bind at the optimum) we have, without constraint
Maximize n (F - p.x - w l)
l, n, x
Hence, differentiating with respect to xi we have ∂ F/∂ xi = pi; with respect to l, ∂ F/∂ l = w; and the rent per tenant (R/n) is F - p.x - w l, which is equal to M/n ∂ F/∂h (where ∂ F/∂h is the marginal product of land) since we impose constant returns to scale. Thus, when the landlord stipulates inputs, the marginal product of each input is equal to its price.
We consider now the problem where the tenant decides on inputs but the landlord offers to contribute C(x) when inputs are at level x. The landlord can choose the function C( ). We call his new maximization problem, B. The tenant's income is P (x, n. r, l, C ( )) = (1 - r) F - p.x + C (x) and he chooses x (but not l)
Maximize n(rF - C(x)) n, r, C( ), l
subject to P≧wl, and the constraint that the tenant chooses x to maximize P given n, r, C( ), l
Write R' = n(rF - C). The landlord's problem is Problem B'
R', n, l, C ( )
subject to (i) F - p. x - Rn ≧ wl
and (ii) the tenant chooses x to maximize his income.