Download Problem Set 5 Answer Key

iniEconomics 101
Spring 2002
Section 5 – Alley
Problem Set #5
Due: Wednesday, May 1, 2002
1.
Assume that the manufacturing of biking socks is a perfectly competitive industry. The market demand for
D
biking socks is described by a linear demand function Q = 400 - 2 P . The inverse demand is
P = 200 -
1 D
Q . There are 30 manufacturers of biking socks. Each manufacturer has the same production
2
costs. These are described in the long-run total and marginal cost functions below.
LRTC(q) = 200 + 10q + 2q2
MC(q) = 10 + 4q.
a.
Show that an individual firm in this industry maximizes profit by producing q =
P - 10 1
= P - 2.5.
4
4
To find the profit maximizing output, a competitive firm will set P = MC.
P = 10 + 4q = MC
Now, just solve for q:
4q = P – 10
q = ¼ P – 10/4
This expression is the supply curve for a single competitive firm (above the min of ATC).
b.
Derive the industry supply curve and show that it is Q S = 7.5P - 75.
There are 30 identical firms in the market, each with the same supply function found in part a.
All we have to do is add up all of the individual supply functions to derive the market supply
function:
QS = 30q = 30( ¼ P – 2.5)
= 7.5P - 75
c.
Find the equilibrium market price by setting supply equal to demand. The answer is P = $50. Show your
work.
To find the equilibrium market price, just set QS = QD :
7.5P – 75 = 400 – 2P
Solve for P:
9.5P = 475
P = 50
d.
Find the aggregate quantity traded in equilibrium. The answer is Q = 300. Show your work.
To find the quantity traded, just plug the price that we found in part c into the supply or demand
function. I will use demand:
Q = 400 – 2(50)
= 300
e.
How much output does each firm produce? The answer is 10. Show your work.
You can do this part in two ways. First, we know that the total market output is 300. We have 30
identical firms producing that output, so each firm is producing 300/30 = 10.
Another way to answer this part is to plug the market price into the individual firm supply function
that we found in part a.
q = ¼ (50) – 10/4
= 12.5 – 2.5 = 10
2.
Consider a monopolist with the following demand, cost, and marginal cost functions were q is output:
q = 120  p
C(q) = 900  .5q 2
MC(q) = q
a.
What is the inverse demand function (What is the price as a function of output)?
Solve the demand function for P.
q = 120 – p
p = 120 - q
b.
Marginal revenue as a function of output is given by 120 - 2q. Find the level of output q m, that this firm
will choose as a monopolist (where marginal cost equals marginal revenue).
The monopolist will set MR = MC
120 – 2q = q
Solve for q:
3q = 120
q = 40
c.
What is the price pm at this level of output?
Plug the quantity from part b into the inverse demand function found in part a:
p = 120 - 40
p = 80
d.
What is marginal cost at this level of output?
MC = q, so MC = 40
e.
What is marginal revenue at this level of output?
MR = 120 – 2q
= 120 – 2(40) = 40
Parts d and e are just a check that MR actually does equal MC.
f.
What is the firm's profit at this level of output?
Profit = TR – TC
TR = pq = (80)(40) = 3200
TC = 900 + .5(402) = 1700
Profit = 3200 – 1700 = 1500
2. Firm 1 and Firm 2 are movie producers. Each has the option of producing a blockbuster
romance or a blockbuster suspense film. The payoff matrix displaying the payoffs for each
of the four possible strategy combinations (in thousands of dollars) is shown in the following
table. Each firm must make its choice without knowing the choice of its rival.
Firm 2
Romance
Suspense
Romance
Π1 = 300 Π2 = 300
Π1 = 1000 Π2 = 400
Suspense
Π1 = 400 Π2 = 1,000
Π1 = 750 Π2 = 750
Firm 1
Identify the Nash equilibrium or equilibria for this problem. Explain your answer.
There are two Nash Equilibria for this game. They are highlighted in yellow. First,
consider the box where firm one chooses romance and firm two chooses suspense. Neither
has an incentive to move given the other’s choice. If firm one changes to suspense, its profit
falls by 250. If firm 2 changes to romance, its profit falls by 100. Neither firm has an
incentive to change. Thus, the strategies (romance/suspense) is a Nash Equilibrium.
Now, consider the box where firm one chooses suspense and firm 2 chooses romance. If
firm 1 changes to romance, its profit falls by 100. If firm 2 changes to suspense, its profit
falls by 250. Again, neither firm has an incentive to change.