Midterm Exam 2 – Answer Key
ECO 301 – Summer 2009
Instructor: Dr. Michael Malcolm
Instructions: You can use any written materials you would like and a calculator in
completing this exam.
Statement of academic honesty:
This exam entirely reflects my own work. I have not received assistance from
anyone or given assistance to anyone in completing this exam.
Signature: _________________________________
Name:
_____________________________________
They may be able to earn long-run profit. What drives profit to zero in the long run is entry of new firms. In this case, the fact that the government issues only 5000 licenses creates a barrier to entry. This barrier to entry can potentially allow existing firms to earn profit, without that profit being eaten away by newly entering firms.
Problem 2
a. Costs are minimized when extra output per dollar spent on capital is equal to extra output per dollar spend on labor:
100 40
20 20 5 2
MPK MPL
r = w
= ⇒ =
b. The firm will reduce its capital usage with a corresponding increase in usage of labor. Problem 3
By not gambling, Gale has $200,000 for sure. By taking the bet, Gale will have $250,000 with probability pwhen Obama wins, but $150,000 with probability 1−p. Comparing her expected utility from gambling with her expected utility from not gambling, she will take the gamble if:
(
)
(
)(
)
250, 000 1 150, 000 200, 000 500 1 387.2983 447.2136 500 387.2983 387.2983 447.2136
0.5316
p p
p p
p p
p
+ − ≥
+ − ≥
+ − ≥
⇒ ≥
a. The individual works 2000−N1 hours during the first year, so his first-year income is
(
)
1 2000 1
w −N . Similarly, his second-year income is w2
(
2000−N2)
. Equating the present value of consumption with the present value of income:(
)
2(
2)
2
1 1 1
2000 2000
1 1
w N
C
C w N
r r
−
+ = − +
+ +
b. Multiply through by 1+r to get the budget constraint in terms of future value:
(
)
(
)(
)
(
)
1 1 2 1 2000 1 1 2 2000 2
C + +r C =w −N + +r w −N
c. With perfect complements, all terms must be equal: C1 =C2 =N1 =N2. So we can simply substitute this into the budget constraint:
(
)
2(
1)
1
1 1 1
2000 2000
1 1
w C
C
C w C
r r
−
+ = − +
+ +
Solving this equation for C1 gives the answer. Problem 5
a. No – Since the marginal rates of substitution are not equal, there is a potential for mutually beneficial trades.
( )
2 2 0.5 2 2 2 0.5 A Anna AU F FC C
MRS
U C F F
∂ ∂ = = = = = ∂ ∂
( )
2 0.5 0.5 2 2 2 0.5 ABob
A
U F C C
MRS
U C FC F
∂ ∂
= = = = =
∂ ∂
b. No – Notice that both have utility of 0 with this allocation. Any trade that gave both of them some of each good will raise both of their utilities above 0.
a. Profit is given by:
( )
(
)
22 2 * 1 90 2 4 1 90 2 4 TR TC P q C q
q q q
q q q
Π = −
= −
= − −
= − −
Maximizing profit: 1
90 4 0
2 20
q q
q q
∂Π = − − =
∂ ⇒ =
Substituting back into the demand curve gives P=90 2− q=50 b. At the profit-maximizing level of output:
1 10 2 dTC MC q dq = = =
Since marginal cost is only $10 anyway, there is no reason to consider obtaining hats from the outside supplier, which cost $12 each. Thus, the firm will continue to sell 20 hats, producing all of them itself and obtaining none from the outside supplier.
c. If you solve the same problem as in (a), you get the optimal level of output at q=36. At this point, MC =18, and so it makes sense to obtain hats from the outside supplier at $12 each. The relevant marginal cost is now MC=12. The firm will set output where:
162 4 12 37.5
MR MC
q q
=
− = ⇒ =
Now, it will produce its own hats up until the point where the marginal cost exceeds $12: 1
12 24 2q> ⇒ >q
a. We solve for the amount of labor that the firm needs to produce q units of output when capital is fixed at 10 units.
(
)
210
10 10
q L
L q L q
= +
= − ⇒ = −
Thus, short-run costs are:
(
)
2( )
(
)
21 10 8 10 80 10
SRC=wL+rK = q− + = + q−
b. Marginal cost is:
(
)
2 10 2 20
dSRC
MC q q
dq
= = − = −
d. The firm chooses labor and capital to minimize costs subject to producing q units of output:
min wL+rK s t. . L+ =K q
Substituting w=1 and r=8, the Lagrangian is:
(
)
8
L= +L K+λ q− L−K
First-order conditions:
1/ 2 1
1 0 2 2
2
L
L L
L L
λ
λ − λ
∂ = − = ⇒ = ⇒ =
∂
8 0 8
L
K λ λ
∂ = − = ⇒ = ∂
Equating:
2 8
4 16
L
L L
=
= ⇒ =
Substitute back into the constraint to find K:
16 4
L K q
K q K q
+ =
+ = ⇒ = −
Thus, costs are:
( ) (
)
1 16 8 4 16 8 32 8 16
LRC=wL+rK = + q− = + q− = q−
e. Long-run average cost is:
8 16 16 8
LRC q
LRAC
q q q
−
= = = −
f. Notice that LRAC rises as output rises, which is the definition of diseconomies of scale. g. The firm starts by hiring 16 workers and then increases output by expanding only capital
In the short-run, the firm produces where P=MC. The MC function was derived in part (b). 10 2 20
2 30 15
q
q q
= −
= ⇒ =
Extra Credit 2
