1. Find the 4-firm concentration ratio and the total industry HHI for the following three industries.

The 4-firm concentration ratio for industry 1 is (40+15+10+5) = 70, for industry 2 is (40+30+15+5) = 90, and for industry 3 is (20+15+15+15) = 65. The HHI is the sum of squared market shares which equals 2100 for industry 1, 2800 for industry 2, and 1350 for industry 3.

2. In Hotelling's town, if all firms are required to charge the same fixed price, describe the equilibrium location of three firms. Explain your answer. Now describe the equilibrium for four firms.

There is no equilibrium for three firms, even if firms are allowed to locate "on top" of each other. There are three possibilities to consider:

1. If none of the firms occupy the same location, both the leftmost firm A, and the rightmost firm C, want to move as close to the middle firm B as possible because they then have to share the smallest segment of market with B. As a result B is hemmed in on both sides with no customers to sell to, and it therefore wants to move to the left of A, to the right of C or to the same location as either of them. This is therefore not an equilibrium.

2. If exactly two firms, say A and B, occupy the same location, then C wants to move as close as possible to them, on the side with the longest distance to the end of Main Street. In that way, C gets at least half the market, while A and B each get at most a quarter. Either A or B can then do better by moving right next to C again, on the side with the longest distance to the end of main street. This is therefore not an equilibrium.

3. Finally, if all three firms occupy the same location, they each get a third of the market. Any one of them can then do better by moving slightly to the side with the longest distance to the end of Main Street, thereby getting at least half the market, This is therefore not an equilibrium.

An equilibrium for four firms does exist. It has two firms located right next to each other at a quarter of the length of Main Street from one end, and the other two firms located right next to each other at the same distance from the other end. Given that the other three firms stay put, each firm is indifferent about either remaining in this location or moving closer towards the center.

3. All firms in a Cournot monopolistically competitive industry have the same cost function C=25+10q. Market demand is Q= 110-p.

A. Calculate the equilibrium price, firm output, total output and the number of firms in the industry.

B. How would these values change if a franchise tax of $75 were imposed on each firm?

C. What if a technical innovation were to reduce unit production cost to $5?

A. P=110-Q; AC=(25+10q)/q; Q=q1+(n-1)q2

profit 1 = Pq1-TC=(110-Q)q1-(25+10q1)=(110-(q1+(n-1)q2))q1-(25+10q1)

profit 1 = 110q1-q1²-(n-1)q1q2-25-10q1

d(profit1)/dq1=110-2q1-(n-1)q2-10=0

100-2q1-(n-1)q2=0

in equilibrium, q1=q2

100-2q1-(n-1)q1=0

100=q1(2+n-1)

q1=100/(n+1)

Q=nq1=100n/(n+1)

P=110-Q=110-100n/(n+1)=(10n+110)/(n+1)

firms will enter until profit = 0 or P=AC , so

P=AC

(10n+110)/(n+1)=25/q+10=25/(100/(n+1))+10

10n+110=(25/100)(n+1)²+10(n+1)

0=(25/100)(n+1)²-100=((5/10)(n+1)+10)((5/10)(n+1)-10)

(1/2)n-19/2=0

n=19

q1=5

Q=95

P=15

B. Franchise tax increase fixed costs from 25 to 100. Increase in fixed costs does not change the output or pricing decision. However P=AC now becomes

(10n+110)/(n+1)=100/(100/(n+1)) + 10

10n+110 = (n+1)² + 10(n+1)

0=(n+1)² - 100
0 = (n+1 - 10)(n+1 +10)
0=(n-9)
n=9
P=20
q=10; Q=90

C. Lower unit costs means that C=25+5q, so

profit 1 = Pq1-TC=(110-Q)q1-(25+5q1)=(110-(q1+(n-1)q2))q1-(25+5q1)

profit 1 = 110q1-q1²-(n-1)q1q2-25-5q1

d(profit1)/dq1=110-2q1-(n-1)q2-5=0

105-2q1-(n-1)q2=0

in equilibrium, q1=q2

105-2q1-(n-1)q1=0

105=q1(2+n-1)

q1=105/(n+1)

Q=nq1=105n/(n+1)

P=110-Q=110-105n/(n+1)=(5n+110)/(n+1)

firms will enter until profit = 0 or P=AC , so

P=AC

(5n+110)/(n+1)=25/q+10=25/(105/(n+1))+5
5n+110=(25/105)(n+1)²+5(n+1)

0=(25/105)(n+1)²-105=25(n+1)²-(105)²

0=(5(n+1)-105)(5(n+1)+105)

0=(5n+5-105)

n=20

P=10

Q=100; q=5