Reference no: EM132414811

Problem: 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.

Required:

(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 answer.