Question

4. Suppose you are named the new Secretary of Economy of the United States of America. You are interested in investing in a project with great returns to the economy, so you are advised to focus on early childhood education investments, since it is shown they increase productivity and earnings in the economy in the long run. In order to fulfill your project, you need to construct a new school in each county, using two factors of production: capital (K) and labor (L). Let the rental rate of capital ber and the wage bew. Output(a new school)(q)is produced according to the following production function: F(K,L) = (K1 4 +L1 4)4

(a) Are capital and labor complementary factors of production? Prove you answer. (2 points)

(b) Now, suppose the production function is: F(K,L) = K1 2L1 2

Does labor have a diminishing marginal product? Prove your answer. (2 points)

(c) Write down the cost minimization problem for the function specified in part (a). What are the optimal levels for capital and labor that minimize the cost of the program? What is the cost function? (8 points)

i. L(q) =

ii. K(q) =

iii. C(q) =

(d) Solve for the marginal cost function using your result from the previous part.

Answer 1

(a) The given production function is:

For the factors to be perfect complements, the firm must hire the two factors in a constant ratio which does not depend on the wage and rental rate.

At equilibrium, marginal rate of technical substitution is equal to the ratio of factor prices:

Since the amount of capital and labor employed depend on the wage and rental rate, they are not perfect complements to the firm.

(b) The given production function is:

The marginal products of the two factors are obtained by differentiating the production function with respect to the factor.

It is clear from the above equations that as the quantity of factor increases, the marginal product decreases. Hence, both labor and capital exhibit diminishing marginal product.

(c) Cost of a firm is given by:

Using the equilibrium condition found in part (a):

Substituting this value in the production function:

Substituting the value into the demand for capital:

Cost is given by:

(d) Marginal cost is computed by differentiating the cost function with respect to quantity: