# Palanpur, the Economy of an Indian Village

By C. J. Bliss; N. H. Stern | Go to book overview

Appendix to Chapter 3

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.

Problem A

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)

Problem B

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'

Maximize R'

R', n, l, C ( )

subject to (i) F - p. x - Rnwl

and (ii) the tenant chooses x to maximize his income.

-88-

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Palanpur, the Economy of an Indian Village

• Title Page i
• Preface v
• Contents ix
• 1 - Introduction 1
• 2 - The Village 8
• Appendix to Chapter 2 34
• 3 - Theory and India 50
• Appendix to Chapter 3 88
• 4 - Markets in Palanpur 90
• 5 - Tenancy 124
• 6 - Output and Income 167
• Appendix to Chapter 6 211
• 7 - Wheat : The New Varieties and Practices in Palanpur 222
• 8 - Wheat: Productivity and Expectations 251
• 9 - Reflections on Economic Theory 299
• 10 - A Revisit and Some Speculations on the Future 314
• Glossary 327
• References 331
• Author Index 335
• Subject Index 336
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