Question

Consider the following production function: ? = ? ?? ?? ? , where Y denotes output, and K denotes capital input, L denotes and labor input, and D denotes land input. There is a firm that uses this technology to produce output by choosing the quantity of each input a. Under what conditions does this production function exhibit constant returns to scale? b. Suppose the firm is perfectly competitive so it takes input prices as given (r is the rental rate for capital, w is the wage rate, and q is the rental rate for land). Set up the firm’s profit maximization and solve for the optimal quantity of each input.

Answer 1

Production function:

a)

For this production function exhibit constant returns to scale, increasing all inputs by t times should increase by t times as well.

b)

Supposing the firm is perfectly competitive, price of output = p (exogenous parameter. price- taker).

Total cost for producing Y units of output = wL + rK + qD

Profits are given by:

Firms' optimization problem:

Using first order conditions for optimization, we get:

Dividing (1) by (2), we get:

Dividing (2) by (3), we get:

Substituting (4) and (5) in (1), we get:

Similarly substituting back in (4) and (5), we get:

and