E620 – Economics of Industry

Problem Set #6

Davis

Fall 2004KEY

  1. Collusion. Consider a market consisting of 4 sellers with a history of fierce competitive behavior. Each seller has unit costs of $10 per unit, and the inverse market demand curve is P = 100 –Q, where Q is the joint production of the 4 sellers.
  2. Assuming that the 4 sellers share equally the gains from conspiracy, how much would each of them earn in an efficient conspiracy?

In an efficient conspiracy, the sellers would price as a monopolist, and then divide up the profits. . A monopolist maximizes

=TR-TC

=(100 – Q)Q -10Q

’=100 – 2Q-10=0

Q=45 (or 11.25 per seller)

P=100 – 45 = 55

Profits = 452 = 2025, or $5056.25 per seller

  1. What is the deadweight cost of the conspiratorial behavior?

The DWL is the triangle bound by unit costs, the quantity produced, and the competitive outcome. In this case

DWL=½ (55 – 10)(90 – 45)

=1012.5

  1. Suppose that the sellers might each service the entire market on their own. What are the gains to cheating for any defector?

Using the monopoly price as a reference, any defector might earn essentially the entire monopoly profits ($2025), by posting a price just below the JPM level, and then assuming the entire market Thus, the gain from defection is 2025 – 506.25 = 1518.75

  1. Suppose now that the market consisted of just 2 sellers rather than 3, and that the 2 sellers again plan to divide the market evenly. How does changing the number of conspirators affect the efficient collusive price, quantity and DWL of collusion?

Changing the number of conspirators does not affect the aggregate monopoly price, quantity and profit level or the DWL of collusion.. Reducing the number of competitors means that the given surplus is divided. Thus, with 2 sellers we have

Q=45 (or 22.5 per seller)

P=100 – 45 = 55

Profits = 452 = 2025, or $1012.5 per seller

DWL=½ (55 – 10)(90 – 45)

=1012.5

Observe that reducing the number of competitors increases the profitability of collusion.

  1. How does changing the number of sellers affect the incentive to defect from a collusive arrangement? Be explicit

Reducing the number of sellers diminishes the incentive to defect from a collusive arrangement. For example, with 4 sellers, a defector gains 2025 – ¼ (2025) =1518.75

With 2 sellers a defector gains 2025 – ½ (2025) = 1012.5

  1. Consider further the 2 firm case. Suppose that the cost function for each firm is 1012.5 + 10q. How does the addition of the fixed cost affect the profitability of collusion? Do you think collusion in this case is more or less likely? How does the fixed cost affect the DWL of collusion?

With two sellers, and joint Fixed cost obligations of 2025, the combined firm earns profits of $0 in the collusive outcome. Thus, absent the arrangement, both firms would lose their fixed costs ($1012.5), so collusion yields 0 profits. Note nevertheless that relative to earnings in the competitive situation, collusion is equally profitable as in the no fixed cost situation.

I would suspect that collusion is more likely in this case, since the firms must collude to avoid going out of business

Collusion does not affect the DWL of collusion.

  1. Cournot Duopoly. Consider a duopoly with demand p = 120 – (q1+q2), mc = 30 Q=q1+q2. Suppose that the sellers are Cournotcompetitors.
  2. What is the Cournot Equilibriuim (price, quantity andearnings?)

1=(120 –(q1+q2))q1-30q1

2=(120 –(q1+q2))q2-30 q2

1/q1=120 –2q1- q2-30=0

2/q2=120 –q1- 2q2 -30=0

Solving

q1= 90 -q2/2

q2= 45 -q1/2

Thus

q1=q2=30

p=120 – 60=60

1=2=30(60-30) =900 per seller

  1. What are the Joint Profit Maximizing Outcomes (price, quantity and earnings)

Acting as a monopolist,

=(120 –Q)Q-30Q

’=120 –2Q-30=0

Q = 45 (22.5 per seller)

P= 75

=452=$2025 (or $1012.5 per seller)

  1. What are competitive equilibrium predictions?

P=30

Q=120 – 30=90

=0

  1. Illustrate a, b and c in a graph

  1. The Effects of Increasing the Number of Competitors on the Cournot Equilibrium. Consider the same demand and cost function as above, but suppose that instead of two sellers there were three. That is, p = 120 –(q1+q2+q3), mc = 30.
  2. Calculate the Cournot Equilibrium (price, quantity and earnings)

i=(120 –(q1+q2+ q3))qi-30qii==1,2,3

Consider firm 1 as a representative case.

1/q1=(120 –2q1- q2 - q3) i-30=0

Given the symmetry of the sellers, we can solve by equating the qi ‘s

90=4q1

Thus,

q1=22.5, and Q = 90

p=120-3(22.5) =52.5

and i=22.52=506.25

  1. Use an illustration to compare your result to part a in question 2.

  1. What do you think this result suggests about the relationship beween the number of sellers and the price/cost markup?

A smooth inverse relationship exists between the number of sellers and the price/cost markup.