Download ECON460: Answer Key to Problem Set 1

ECON460: Answer Key to Problem Set 1
Instructions: In answering the following questions, do not restrict yourself to finding answers that are natural
numbers. Instead, provide decimal answers where you compute them to be so, even though it may lack some
intuition.
1. Consider the following total cost function for an individual firm: C(q) = 10 + q + 14 q 2
a. At what level of output is average cost at its lowest? Also draw a graph showing firm average cost and
marginal cost curves.
Answer: AC = 10q −1 + 1 + 41 q
d(AC)
= −10q −2 + 14 = 0
dq
=⇒ q 2 = 40 =⇒ q = 6.32
b. Consider a short-run perfectly competitive equilibrium with 2 identical firms, each with the above cost
function. What is the equation for the industry supply curve?
Answer: Individual firm inverse supply function is given by
p = 1 + 12 qi , qi ≥ 0, where 0 is the minimum of the AVC function.
Individual firm supply curve is then given by
qi = 2p − 2, p ≥ 1, where 1 is the price corresponding the minimum of the AVC function.
The combined supply curve from two identical firms is Q = q1 + q2
Q = 4p − 4, p ≥ 1
Since we draw the inverse supply curve, this is given by
p = 14 Q + 1, Q ≥ 0.
c. The industry demand curve is estimated to be Q = 100 − p.
If there are two firms operating in the market, what is the equation for the residual demand curve for each
firm?
Answer: q1d = Qd − q2s = (100 − p) − (2p − 2) = 102 − 3p, p ≥ 1
d. Solve for the short-run perfectly competitive equilibrium levels of price and output in which there are two
firms.What are profits of each firm in this short-run equilibrium?
Answer: need to find the intersection of the industry demand curve and the 2-firm supply curve above:
Q = 100 − p = 4p − 4
=⇒ p∗ = 20.8,
Q∗ = 79.2
Profits:
q ∗ = 79.2/2 = 39.6
πi = REV EN U E − COST S = 20.8 × 39.6 − 10 + 39.6 +
1
4
× 39.62 = 382.04
2. What would a monopoly’s marginal revenue be if it chose a point on the demand curve where the price elasticity
of demand equals -1? Why would it never be optimal to choose such a point, given positive marginal costs?
Would the monopoly rather produce less or more?
Answer: Using the Lerner index, the monopoly’s marginal revenue would be zero. If its marginal costs are
positive, this implies that the monopoly earns less on the marginal unit of output than it cost to produce that
unit. Profits would therefore rise if it reduced its output.
3. Solve for the Nash equilibria of the following normal-form game. (Ignore mixed strategies —- if you don’t know
what this means, never mind).
Strategy A
player 1
Strategy B
Strategy C
Strategy A
(1,3)
(6,6)
(1,2)
Player 2
Strategy B Strategy C
(-1,3)
(0,1)
(0,7)
(-1,1)
(1,-1)
(1,2)
Answer: The only Nash Equilibrium is (C,C) where both players choose strategy C.
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4. An industry consists of three firms with identical costs C(q) = 18q + q 2 . Market demand is Q = 150 − p.
a. Write down the profit function for firm 1.
b. Solve for the Cournot best-response function for firm 1.
c. Compute the Cournot equilibrium level of total market output.
d. Compute equilibrium prices and profits for each firm.
Answer:
π1 = P q1 − C(q1 ) = [150 − (Q−1 + q1 )] q1 − 18q1 − q12 , where Q−1 = q2 + q3
∂π1
= 132 − Q−1 − 4q1 = 0
b.
∂q1
=⇒ 4q1 = 132 − Q−1
1
=⇒ q1BR = 33 − Q−1
4
c.
Since all firms are identical, the best-response functions are symmetric and in equilibrium all firms will
choose the same levels of output. Hence, each firm will choose q ∗ to satisty :
1
q1∗ = 33 − (2q1∗ )
4
=⇒ q1∗ = 22
and
Q∗ = 3q1∗ = 66
a.
d.
p = 150 − Q∗ = $84
π1∗ = π2∗ = π2∗ = $968
5. As a recent graduate you have landed a job in production management for Universal Clones, Inc. You are
responsible for the entire company on weekends.
Quantity
500
501
Average Total Cost
200
201
Your current level of production is 500 units. All 500 units have been ordered by your regular customers. The
phone rings. Its a new customer who wants to buy 1 unit of your product. This means you would have to
increase production to 501 units. Your new customer offers you $450 to produce the extra unit.
a. Should you accept this offer?
b. What is the net change in the firms profit?
Answer:
a. T C(500units) = 500 ∗ 200 = $100, 000
T C(501units) = 501 ∗ 201 = $100, 701
M C = 100, 000 − 100, 701 = $701
M R = $450
Since the marginal revenue is less than the marginal cost of production you should not accept the offer.
b. 701 − 450 = −$251 = Net change in profits
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