Download Eco 300 Intermediate Micro

Eco 300 Intermediate Micro
Instructor: Amalia Jerison
Office Hours: T 12:00-1:00, Th 12:00-1:00,
and by appointment
BA 127A, [email protected]
A. Jerison (BA 127A)
Eco 300
Spring 2010
1 / 11
1. The necessary condition for a firm with different plants to be
maximizing its profit is that marginal costs are equal in all the
plants that produce. This does not mean that average costs have
to be the same. The two plants could have different cost functions
which lead to different average costs, even though their marginal
costs are the same.
For instance, suppose that plant A has the cost function
CA (qA ) = 27 + 2qA for all qA ≥ 0. Let plant B have the cost
function CB (qB ) = 10 + qB2 for all qB ≥ 0.
Then M CA (qA ) = 2, M CB (qB ) = 2qB , ACA (qA ) = 27/qA + 2 and
ACB (qB ) = 10/qB + qB .
A. Jerison (BA 127A)
Eco 300
Spring 2010
2 / 11
Setting M CA (qA ) equal to M CB (qB ), we get qB = 1. Then setting
ACA (qA ) equal to 3 × ACB (qB ), we get
27/qA + 2 = 10/qB + qB = 10 + 1 = 11, so 27/qA = 9, and qA = 3.
This shows that it is possible to have the average costs of plant A
be 3 times the average costs of plant B, yet the marginal costs of
the two plants are equal. So the firm owning these two plants can
be profit maximizing.
A. Jerison (BA 127A)
Eco 300
Spring 2010
3 / 11
2. a. q = 3L + 2K. When we multiply L and K by a positive
constant c, we get q = 3cL + 2cK = c(3L + 2K), which is c times
the original q. So this function has constant returns to scale.
The marginal product of labor is 3, so the marginal product of
labor doesn’t change as L is changed. Similarly, the marginal
product of capital is 2, so the marginal product of capital doesn’t
change as K is changed.
A. Jerison (BA 127A)
Eco 300
Spring 2010
4 / 11
b. q = (2L + 2K)1/2 . Given a positive constant c > 1,
q(cL, cK) = (2cL + 2cK)1/2 = c1/2 (2L + 2K)1/2 < c(2L + 2K)1/2 .
So this function has decreasing returns to scale.
The marginal product of labor is
(1/2)(2)(2L + 2K)−1/2 = (2L + 2K)−1/2 , which is decreasing as L
increases. Similarly the marginal product of capital is
(2L + 2K)−1/2 , which is decreasing as K increases.
A. Jerison (BA 127A)
Eco 300
Spring 2010
5 / 11
c. q = 3LK 2 . q(cL, cK) = 3cL(cK)2 = c3 (3LK 2 ) > c(3LK 2 ) for
c > 1. So the function has increasing returns to scale.
The marginal product of labor is 3K 2 which does not change with
L. The marginal product of capital is 6LK, which increases with
K.
A. Jerison (BA 127A)
Eco 300
Spring 2010
6 / 11
d. q = L1/2 K 1/2 . q(2L, 2K) = 21/2 L1/2 21/2 K 1/2 = 2L1/2 K 1/2 . So
this function has constant returns to scale.
The marginal product of labor is (1/2)L−1/2 K 1/2 , which decreases
with L. Similarly, the marginal product of capital decreases with
K.
A. Jerison (BA 127A)
Eco 300
Spring 2010
7 / 11
e. q = 4L1/2 + 4K. q(2L, 2K) = 4(21/2 )L1/2 + 8K < 8L1/2 + 8K.
So this function has decreasing returns to scale.
The marginal product of labor is 2L−1/2 , which decreases with L.
The marginal product of capital is 4, which is constant in K.
A. Jerison (BA 127A)
Eco 300
Spring 2010
8 / 11
3. The equation of the isoquant corresponding to an output of 1
chair is L + K = 4. The equation of an isocost line is
30L + 15K = C. So the slope of the isoquant is −1 and the slope
of an isocost line is −2. Because of these different slopes
everywhere, the solution to the cost-minimization problem must
be a corner solution.
We have to find the lowest-cost isocost line that touches the
isoquant. The lowest-cost point of intersection of the two lines is
where L = 0, K = 4. This makes sense because labor and capital
are equally productive, but capital costs less than labor. So the
firm should use only capital in its production.
The graphical illustration of the current production and optimal
production is below.
A. Jerison (BA 127A)
Eco 300
Spring 2010
9 / 11
K(hours)
Optimal
Current production
L (hours)
A. Jerison (BA 127A)
Eco 300
Spring 2010
10 / 11
4. Explain why the short run supply curve in a competitive
market might slope upward while the graph of the long run
supply in the same market is horizontal.
In the long run, firms can enter and exit the market, which they
cannot do in the short run (once a firm has decided to exit a
market, it takes quite a long time to do so, and once a potential
firm has decided to enter, it also takes some time to get set up).
When there is a positive demand shock, the short run market
response leads to an increase in price. However, in the long run,
this price increase is counteracted by an increase in short run
supply due to the entrance of more firms. This causes the long
run supply curve to be horizontal in a constant cost industry, as
nothing can cause an increase (or decrease) in price in the long
run.
A. Jerison (BA 127A)
Eco 300
Spring 2010
11 / 11