Topics in Theory

Problem Set 2

1. The market for pizza is perfectly competitive and demand is given by 𝑄𝐷 = 8 − 1

2 𝑃. Initially, all

the sellers have the same marginal cost 𝑀𝐶 = 10. Suppose one seller reduced her marginal cost

to $8 through some non-drastic innovation. Call this person the innovator.

a. What is the innovator’s profit prior to the innovation?

b. What is the innovator’s profit after the innovation?

c. What is the additional profit generated by the innovation?

2. The market for pizza is a monopoly and demand is given by 𝑄𝐷 = 8 − 1

2 𝑃. Initially, the

monopolist’s marginal cost 𝑀𝐶 = 10. Suppose the monopolist reduced her marginal cost to $8

through some non-drastic innovation.

a. What is the monopolist’s profit prior to the innovation?

b. What is the monopolist’s profit after the innovation?

c. What is the additional profit after the innovation?

d. Compared to the results in Question 1, does the monopolist have more or less incentive to

innovate than a seller from a competitive market? Why?

3. The market for pizza is a monopoly and demand is given by 𝑄𝐷 = 8 − 1

2 𝑃. Initially, the

monopolist’s marginal cost 𝑀𝐶 = 10. Suppose there is another firm (entrant) who is about to

enter the market. The monopolist can choose to acquire a technology that will lower her

marginal cost to $8. If she does so, her competitor will remain out. If she doesn’t do so, the

entrant will acquire that technology (with a lower marginal cost 𝑀𝐶 = 8) and enter the market.

Suppose the two firms will compete over quantities (Cournot Competition).

Don’t use any decimals in this question.

a. What is the monopolist’s profit if she doesn’t innovate?

b. What is the monopolist’s profit if she innovates?

c. What is the additional profit generated by the innovation for the monopolist?

4. Suppose two firms compete over quantities. Inverse demand is given by 𝑃 = 60 − 3𝑄, where

𝑄 = 𝑄1 + 𝑄2. At first, two firms have identical marginal cost 𝑀𝐶 = 12. The game has two

stages. In the first stage, (only) firm 1 can lower its marginal cost by 𝑘1 by investing 4𝑘1 2 in R&D.

In the second stage, two firms compete over quantities.

Don’t use any decimals in this question.

a. Solve the second stage equilibrium quantities and price at a function of 𝑘1.

b. What is firm 1’s optimal 𝑘1?

5. Suppose two firms compete over prices. Demand is given by

𝑄1 = 60 − 3𝑃1 + 2𝑃2

𝑄2 = 60 − 3𝑃2 + 2𝑃1

At first, two firms have the same marginal cost 𝑀𝐶 = 6.

In the first stage, (only) firm 1 can lower her marginal cost by 𝑘1 by investing in 2𝑘1 2 in R&D.

In the second stage, two firms compete over prices (Bertrand Competition)

Don’t use any decimals in this question.

a. Solve the second stage equilibrium prices and quantities as a function of 𝑘1.

b. What is firm 1’s optimal 𝑘1?