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

CHAP. 13] INPUT PRICING AND EMPLOYMENT 311
Then we find the equation of the marginal product of labor (MP L ):
MP L ¼ @Q
0:5
¼ 500L
@L
To maximize profits the firm should hire labor until the MRP L ¼ w. Since P ¼ MR ¼ $1,
Thus
MRP L ¼ (MR)(MP L ) ¼ ($1)(500L 0:5 ) ¼ $50 ¼ w
L ¼ $500 2
¼ 100
$50
(b) With L ¼ 100 and K ¼ 100
Q ¼ 100(100) 0:5 (100) 0:5 ¼ 10,000
The total revenue and the total costs of the firm are
so that the total profit of the firm is
TR ¼ (P)(Q) ¼ ($1)(10,000) ¼ $10,000
TC ¼ wL þ rK ¼ $50(100) þ $40(100) ¼ $9000
p ¼ TR TC ¼ $10,000 $9000 ¼ $1000
This represents the maximum profits that the firm can earn.
13.34 A monopsonist hiring only labor faces the total cost function TC ¼ wL. Derive, using calculus, the
expression (a) for the marginal resource cost of labor (MRC L ) and (b) relating the MRC L , the wage
rate (w), and the wage elasticity of the supply of labor (e L ).
(a) MRC L ¼ d(TC L)
¼ w þ L dw
dL dL
(b)
Rearranging the above equation, we get
MRC L ¼ w 1 þ L
dw
w dL
Therefore,
MRC L ¼ w 1 þ 1 e L
Graphically, this means that the MRC L lies above the (positively sloped) S L curve (see Fig. 13-5). If the firm
were instead a perfect competitor in the labor market, e L ¼ 1 and MRC L ¼ w (i.e., the MRC L curve would
coincide with the horizontal S L curve faced by the firm at the given level of w).