14. Consider a firm that produces output from capital and labor according to the

Question

14. Consider a firm that produces output from capital and labor according to the production function q =
10LK. The price of capital (r) is $120 per unit. The price of labor (w) is $40 per unit.
a. If capital is fixed at 10 in the short run, how much labor must be employed to produce 500 units of
output in the short run?
b. What is the total cost of producing 500 units of output in the short run? What is the short run
marginal cost when 500 units of output are produced?
c. If the price of labor increases from $40 per unit to $60 per unit, what happens to the total cost of
producing 500 units of output in the short run (up or down, and by how much)? What happens to
the short run marginal cost (up or down, and by how much)?
d. At the initial prices, which bundle of inputs (L, K) would enable the firm to produce 500 units of
output at the lowest cost in the long run?
e. What is the total cost of producing 500 units of output in the long run at the initial prices? What is
the long run marginal cost when 500 units are produced?
f. If the price of labor increases from $40 per unit to $60 per unit, what happens to the total cost of
producing 500 units of output in the long run (up or down, and by how much)? What happens to the
long run marginal cost?

Explanation / Answer

A firm produces according to the following production function: Q = K^.5L^.5 where Q = units of output, K = units of capital, and L = units of labor. Suppose that in the short run K = 100. Moreover, wage of labor is W = 5 and price of the product is P = 10. What are the optimal units of labor? Consider a firm with Cobb-Douglas production function Q = K^aL^(1-a). (Note that in this special case of a Cobb-Douglas production function, the exponents a + b = 1.) and budget constraint C = rK + wL We can set up a constrained maximization problem to calculate how many units of X and Y that this person will buy. maximize Q = K^aL^(1-a). subject to: C = rK + wL Suffice it to say that the results are: Cost-minimizing Factor Demand equations : K = aC/r L = (1-a)C/w (Note also a useful feature of this special Cobb-Douglas production function: the cost-minimizing firm spends fraction "a" of its budget on K, and fraction "1-a" of its budget on L.) Problem And Answer. Q = K^.5*Q^.5 K costs 100 each and L costs 5 each. Price of Product=10=P. This firm’s factor demand equations are: K = .5P/100 L = .5P/5 K=0.05 L=1.

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