Question

Consider a firm with a production function of the following form:F (L, K) = L(1/4)K(1/2)

20

the corresponding marginal products are
MPL = 1 L(−3/4)K(1/2)

4 20MPK = 1 L(1/4)K(−1/2)

2 20
The cost of a unit of labour is the wage rate w and the cost of a unit of capital is the

rental rate r. The firm is free to adjust all factors of production. (19 points)
a. Does this production function exhibit decreasing returns, increasing returns, or

constant returns to scale?

b. Set up the cost minimization problem and derive the conditional input demand functions for the firm.

c. Provide an expression for the total cost curve for a given level of output q (the cost function should be a function of only the exogenous variables).

Answer 1

If the inputs are changed by a scale of and the output changes by exactly , less than or greater than , then we have constant, decreasing or increasing returns to scale respectively.

Here :

Hence if inputs are increased by , then output will increase by ^(3/4), that is lesser than . Hence decreasing returns to scale.

B) To set up the cost minimization problem of the firm subject to a given level of output say Q , we set up the following lagrangean problem:

These are the input demand functions for the firm.

C) The cost function will now be, wL + rK . We can put in the values of L and K derived above to find the cost function :

As required , C is a function of only the exogenous variables w,r,and exogenously given level of output Q.