Exercise Set 3.5

1. Consider a firm's production function Y = F(K,L), where Y is output, K is capital and L labor. Suppose F(K,L) = 10K^0.5L^0.5.

a. Obtain MP_K and MP_L.

b. Does the marginal product of capital rise, or fall, as capital increases, ceteris paribus?

c. Show that F₂₂ < 0. What is the economic interpretation of this derivative and its sign?

d. Suppose the firm produces 500 units of output.

1. Obtain the combinations of K and L such that output is 500. To do this, set Y = 500, and write K as a function of L. This is an isoquant.
2. Plot the isoquant, K vs. L.
   

2. A firm's production function is Q = 3L^4/3E^1/3, where Q is output, L is labor, and E energy (measured in gallons of gasoline). The price of labor (w) is $30 per worker; the price of energy (g) is $2 per gallon.

a. Suppose the firm wishes to produce 300 units of output. Set Q = 300, and obtain L as a function of E:

L(E) = __________________.

b. The firm's cost of production is: C = wL + gE. Substitute for input prices and L(E) to obtain C(E):

C(E) = __________________.

c. Obtain the first-order derivative of C(E).

d. Set C'(E) = 0. Solve for E, and call it E₀.

e. Obtain L₀ = L(E₀). Show that it is equal to 383.15.

f. Suppose the price of labor increases. Using a higher value of w in (b), obtain the new values of L₀ and E₀. How does the firm change its use of labor and energy as a result of the higher wage?