A farmer in a developing country has a production function y = qf(x) where
y is the amount of corn produced,
x ≥ 0 is the amount of labor input on his
farm and q > 0 is the quality of the land. Assume that f is twice continuously
differentiable, f'(x) > 0, f'(x) < 0 for all x ≥ 0 and satisfies f(0) = 0. The unit
prices of corn and labor are both $1 per unit.
(a) Assume that the farmer lives for one period only. The quality of the land is
known and equals q1 > 0. Formulate the farmer's profit maximization problem and obtain the first order condition for the profit maximizing amount of labor used.
(b) Suppose instead that the farmer lives for two periods and wishes to maximize
the sum of profits. His production function for output is yt = qtf(xt) where
xt is the amount of labor used and qt is the quality of land in periods t = 1; 2.
In the first period, the quality is q1 as above. However, if he uses x1 amount of
labor in the first period, the quality of his land in the second period becomes
q2 = h(x1) < q1, where h'(x1) < 0 for all x1 ≥ 0. (Intuitively, the harder
he works the land in the first period, the lower its quality in the second).
Obtain the first order conditions for the amount of labor he uses in each of
the two periods. How does the farmer's use of labor in the first period of (b)
compare with situation in (a)? Provide an economic justification for your