Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

310 INPUT PRICING AND EMPLOYMENT [CHAP. 13
lethargic employers to institute more productive techniques. However, the imposition of a minimum wage
also tends to reduce the level of employment. Therefore, while those remaining employed are better off,
others find themselves jobless. Training programs for the unemployed might help them find jobs.
However, this is not easy to accomplish. The United States has had a minimum wage since 1938. In April
1991, the minimum wage was raised to $4.25 per hour.
(c) The ability of unions to increase wages is a controversial subject. Union labor does receive wages that are 20%
higher than nonunion labor wages in the United States today. However, unionized industries are generally
large-scale industries that employ more skilled labor and that paid higher wages even before unionization.
On the other hand, comparison of wage differences between unionized and nonunionized labor may lead to
underestimating the effectiveness of unions in raising wages because nonunionized firms may more or less
match union wages in order to retain their workers and to keep unions out. Most economists who have
studied this question have tentatively concluded that unions in the United States have increased the wages
of their members by about 10% to 15%.
PRICE AND EMPLOYMENT OF INPUTS
13.32 Let P and Q equal the commodity price and output, w and r the wage rate of labor and the rental price of
capital, and L and K the amounts of labor and capital used in production by a firm which is a perfect
competitor in the product and input markets. Derive, using calculus, the condition for the amount of
labor and capital that the firm should use in order to maximize its total profits.
Total profit (p) is
p ¼ TR TC
¼ PQ wL rK
Since Q ¼ f (L, K), we can rewrite the profit function as
p ¼ Pf (L, K) wL rK
Taking the partial derivative of p with respect to L and K and setting them equal to zero, we get
@p
@L ¼ P @f
@L
@p
@K ¼ P @f
@K
Since P ¼ MR, we can rewrite the above equations as
Dividing the first equation by the second, we get
w ¼ 0
r ¼ 0
(MP L )(MR) ¼ MRP ¼ w
(MP K )(MR) ¼ MRP ¼ r
Cross multiplying, we have
MP L
MP K
¼ w r
MP L
w
¼ MP K
r
13.33 Suppose that the production function of a firm is Q ¼ 100L 0.5 K 0.5 and that K ¼ 100, P ¼ $1, w ¼ $30,
and r ¼ $40. Determine (a) the quantity of labor that the firm should hire in order to maximize its total
profits and (b) the maximum profit of this firm.
(a)
Substituting K ¼ 100 into the production function we get
Q ¼ 100L 0:5 100 0:5
¼ 1000L 0:5