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Labor Markets
Chapter 16
1.
A baker can sell all the donuts that she chooses at price P and purchases equipment and labor
at competitive prices v and w, respectively.
Her production function is given by q=K
2
L
0.5
.
a)
Show whether labor has diminishing marginal productivity for this production
function.
b)
Show whether the function has diminishing returns to scale.
c)
Now suppose that her capital equipment usage is fixed at 43 in the short run.
Find the
marginal product of labor and the value of marginal product of labor.
d)
Find the baker’s shortrun input demand function for labor.
e)
Find the baker’s shortrun supply curve (Q as a function of P and W).
Answer:
a)
MP
L
=.5 K
2
L
0.5
, so MP
LL
=.25K
2
L
1.5
<0 & we have diminising returns
b)
Consider a new q’ when inputs are doubled.
Then q’=(2K)
2
(2L)
0.5
=2
2.5
q.
Output more
than doubles when we double inputs, so we have increasing returns to scale.
c)
MP
L
=924.5L
0.5
and VMP
L
=924.5PL
0.5
d)
w=VMP
L
=924.5PL
0.5
so L=(924.5P/w)
2
e)
q=1849(924.5P/w)
2.
Suppose a firm locates in an isolated community and produces widgets.
The community has
few employers and the supply of specialized workers needed by the widget firm is given by
L=40w280.
The firm’s production function is q=3.125L0.0025L
2
.
The firm sells its
product in a competitive output market at price 8.
a)
Find the point of diminishing marginal returns for labor.
b)
Find the marginal expense of labor function and the value of marginal product
function.
c)
What level of L should the firm choose to maximize profits?
What wage should the
firm pay workers?
d)
Suppose that the local community adapts a minimum wage law.
At what wage rate
will workers be displaced from their jobs at the widget factory?
Explain.
Answer:
a)
MP
LL
=.005, so we have diminishing marginal productivity over all ranges
b)
inverse labor supply=w=7+.025L, wage bill=wL=7L+.025L
2
.
ME=7+.05L,
VMP=8(3.125.005L)=25.04L
c)
ME=VMP implies 7+.05L=25.04L, L=200, w=12
d)
The firm will continue using workers as long as VMP of labor exceeds the minimum
wage.
In most cases, this will increase employment if the labor market is a
monopsony.
If the minimum wage is set above where ME for labor equals VMP for
labor, then the firm actually uses less labor after minimum wage legislation is enacted.
This occurs at 17 for this problem.
With miniumum wages above this wage, the firm
reduces employment.
3.
Jack owns an apple orchard and can sell all the apples that he chooses at price P.
He
purchases equipment(K) and labor(L) at competitive prices v and w, respectively, and has a
production function given by q=KL
0.50
, where q is bushels of apples.
a)
Find the marginal product of labor and show whether Jack has diminishing marginal
productivity for labor.
Show whether the production process has increasing returns to
scale.
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Suppose that K=300 and is fixed in the short run (assume K=300 for the remainder of
this problem).
Find the marginal expense for labor, the value of marginal product of
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 Winter '08
 Buddin
 Microeconomics

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