Competition Policy Problem Set

Question 1

Suppose that firm 1 and firm 2 produce imperfect substitutes. The demand for firm i ∈ {1, 2} is given by:

Qi(pi, p-i) = 30 - pi + 1/2p-i

Assume that both firms can produce their respective product at a marginal cost of $10. The
two firms compete in prices.

Solve for p1* and p2*
in the pure-strategy Nash equilibrium. What is the cross-price elasticity of demand for either product in the Nash equilibrium?

Answer 1

1. Each firm i maximizes:

π = pi *qi - ci
Sub Qi in: In this case:

Qi(pi, p-i) = 30 - pi + 1/2p-i
π = pi *qi - ci
π = pi *(30 - pi + 1/2p-i) - 10
π = pi *(30 - pi + 1/2p-i) - 10
π = (pi -10) *(30 - pi + 1/2p-i)
expand
πi=40pi - 300 - pi^2 + 1/2p-ipi - 5p-i
Compute the FOC
dπ/di = 40 - 2pi + 1/2p-i
Set the FOC = 0
0 = 40 - 2pi + 1/2p-i
Rearrange for Pi(P-i)
Pi(P-i) = 20 + 1/4p-i

2. Resolve Using the same steps for firm -i, it would be the same but switched since they have the same MC
   P-i(Pi) = 20 + 1/4pi
3. Solve for P 1
   Substitute Plug P2 into P1
   P1 = 80/3
   Then, P1 = P2 = 80/3
4. To find the cross price elasticity of demand find the demand for either products

Q1 = Q2 = 30 - pi + 1/2p-i
Q1 = Q2 = 30 - 80/3 + 1/2(80/3)
= 50/3

Cross Price Elasticity of Demand:

Ed Cross = dQi/dP-i * P-i/Qi
Ed Cross = 1/2 * 80/3 / 50/3
Ed Cross = 0.8

Question 2

Suppose that firm 1 and firm 2 produce imperfect substitutes. The demand for firm i ∈ {1, 2} is given by:

Qi(pi, p-i) = 30 - pi + 1/2p-i

Assume that both firms can produce their respective product at a marginal cost of $10. The
two firms compete in prices.

Conduct the SSNIP test. Do the two firms share the same market

Answer 2

Conducting the SSNIP:

A hypothetical monopolist would maximize:

max. (30 - p1 + 1/2p2)(p1 - 10) (30- p1, p2 p2+1/2p1) (p2 - 10)

2. Expand:
3. Take the FOC For P1 and P2
   dπ /dp1 = 40 - 2p1+ p2 - 5 = 0
   Solve for P1 or Pi
   dπ /dp2 = p1 - 5 + 20 - 2p2 = 0
   Solve for P2 of P-i
4. Substitute P2 INTO P1 to find Pm
5. Conduct the SSNIP Test
   (Pm* - Pi)/pi = 35 - 80/3 /80/3 = 0.3125
6. Compare to 0.05
   0.3125 > 0.05.

This is a small but significant price increase , the price increase is substantial enough for the hypothetical monopolist to profit from, raising concerns about market power and potential antitrust issues. Products of firm 1 and firm 2 belong to the same market.

Question 3

Determine the level of market concentration in the Nash equilibrium.

Info Needed
P* = 35 P* = 35
Qi(pi, p-i) = 30 - pi + 1/2p-i

Answer 3

The total market size is
Q1(P,P) + Q2(P, P)

1. Find the quantities

Q1(35,35) = 30 - 35 + 1/2(35) = 12.5
Q2(35,35) = 30 - 35 + 1/2(35) = 12.5

2. Calculate the market shares

Firm 1:
S1 = Q1/ Q1+Q2 = 12.5 / 12.5+12.5 = 0.5

Firm 2:
S2 = Q2/Q1 + Q2 = 12.5 / 12.5 + 12.5 = 0.5

HHI = 10,000((0.5)^2 + (0.5)^2)
HHI = 5000
-
DOJ/FTC classifies this as highly concentrated
- <1500 unconcentrated
- between 1500 and 2500 it is moderately concentrated

Question 4

Firm 1 and firm 2 argue that their marginal cost would go down to $4 if they merged.
Assuming that was true, should they be allowed to merge?

Background Info:

Qi(pi, p-i) = 30 - pi + 1/2p-i

Answer 4

max. (30 - pi + 1/2p-i) (pi - 4) + (30 p1, p2 - p-i + 1/2pi) (p-i - 4)

1. Expand this equation
   max. (32pi - 240 - pi^2 + p-ipi +
   p1, p2 32p-i - p-i)
2. Need to take 2 FOC
   dπ /dpi = 0
   Solve for Pi
   dπ /dp-i = 0
   Solve for P-i
3. Plug P-i into Pi, Rearrange to solve for Pi which is Pm*
4. Calculate the SSNIP

Pm* - Pi* / Pi* = 32 - 80/3 /80/3 = 0.2

0.2 > 0.05

The marginal cost reduction is not enough to offset the lack of competition, the marginal cost decrease is not substantial enough for the hypothetical monopolist to profit from.

Question 5

Exercise 2
Firm 1 and Firm 2 produce a homogeneous good. The inverse market demand function is
PD(Q) = 6 −1/2Q, where Q = q1 + q2 is the aggregate market output.

Firm 1 produces with constant marginal cost of c1 = 1, firm 2 produces with constant marginal cost of c2 = 2. The twofirms are competing in quantities. There are no fixed cost.

Find the Nash equilibrium
(q1, q2)

Answer 5

• Competing in quantities add q unlike price competition to the profit max equation

1. Firm 1:
   π = pi *qi - ciq1
   π = (6 −1/2Q) *qi - ciq1
   Compute the FOC = 0
   dπ /dq1 = 6 - q1 - 1/2q2 - 1
   0 = 6 - q1 - 1/2q2 - 1
   Rearrange:
   q1(q2) = 5 - 1/2q2
2. Firm 2:
   π = pi *qi - ciq2
   π = (6 −1/2Q) *qi - 2q2
   Compute the FOC = 0
   dπ /dq2 = 4 - 1/2q1 - q2
   0 = 4 - 1/2q1 - q2
   Rearrange:

q2(q1) = 4 - 1/2q1

3. Plug q2(q1) into q1 and solve for q1
   Then substitute q1 into q2 to get the value of q2

Nash Equilibrium:
q1 = 2, q2 = 4

4. Find the Q
   Q = 4 + 2 = 6
5. Calculate the HHI Index

Find the market share:

s1 = q1 / q1 + q2 = 4/6 = 2/3

s2 = q2/q1+q2 = 2/6 = 1/3

HHI = 10,000 ((1/3)^2 + (2/3)^2)
HHI = 5555.55

6. Equilibirum Price
   PD(Q) = 6 −1/2Q
   PD(Q) = 6 −1/2(6) = 3

Question 6

Suppose firm 1 and firm 2 merged. How does the Herfindahl-Hirschmann index change?
Should this merger be blocked?

Background Info:
Firm 1 and Firm 2 produce a homogeneous good. The inverse market demand function is
PD(Q) = 6 −1/2Q, where Q = q1 + q2 is the aggregate market output.

Answer 6

1. The new profit maximization problem becomes:

max ( 6 - 1/2q)q - q
q

max ( 6q -1/2q^2) - q
q

5q - 1/2q^2

2. Taking the FOC

dπ /dq1 = 5 - q
0 = 5 - q
q = 5

3. Plug q = 5 to find the price

PD(Q) = 6 −1/2Q
PD(Q) = 6 −1/2(5)
= 3.5

4. Calculating HHI

Merged: HHI
Merged Share = q1 + q2/ q1 + q2 = 6/6 = 1

HHI = 10000(1^2) = 10000

The HHI index will increase from 5555.56 to 10,000. A higher HHI indicates less competition and that can lead to anti competitive behaviour, such as potential barriers to entry for new firms. As a result the merger should be blocked to prevent further concentration and preserve competition in this highly competitive market.

The equilibrium price is now 3.5 >3

Question 7

Suppose there was rm 3 with a marginal cost of $2. How do your answers to (a) and
(b) change?

Backgorund Information:
Background Info:
Firm 1 and Firm 2 produce a homogeneous good. The inverse market demand function is
PD(Q) = 6 −1/2Q, where Q = q1 + q2 is the aggregate market output.

Firm 1 produces with constant marginal cost of c1 = 1, firm 2 produces with constant marginal cost of c2 = 2. The two firms are competing in quantities. There are no fixed cost.

Answer 7

1. Calculate the profit for each firm separately
2. Remember π = p *q - ciq
3. Find the FOC for each firm, set that equal to 0
4. Solve for the best response functions of BR(1) BR(2) BR(2)

NOTE that at equilibirum q2 = q3 = q* replace that in BR(1) to solve and substitute back to get the value of q*

Solve all values:
q1(q2, q3) = 5 - 1/2(q2 + q3)
q2(q1, q3) = 4 - 1/2(q1 + q3)
q3(q1,q2) = 4 - 1/2(q1+q2)

q1 = 3.5, q2 = q3 = 1.5

Nash Equilibrium
(3.5, 1.5, 1.5)

5. Find HHI

q1 + q2 + q3 = 6.5
s1 = q1/6.5 = 3.5/6.5

HHI = 10,000(3.5/6.5^2 + 1.5/6.5^2 + 1.5/6.5^2)
HHI = 3964.49

Market concentration is higher in a) 5555.55 - 3964.49 = 1591. However the market price decreases relative to a) as the merged firm now produces a larger fraction of the output totalling 6.5 > 6.