ECON 101 UCLA Marginal Cost Constant the Demand Function & Probability Worksheet

2 Problem 2

Firm 1 and rm 2 compete with each other by setting the price. For each rm i, its demand

depends on its own price pi and the price of its opponent p??i. The demand function can be written

as

qi = 20 ?? pi +

1

2

p??i:

The marginal cost is constant at $4 for both rms. There is no xed cost.

Firm 1 and rm 2 set their prices every year. In each year, there is probability 1 ?? that this

market dies and their relationship ends. Assume no discounting. Answer the following questions.

1. Suppose = 0 (this is a one-shot simutaneous game), compute the equilibrium prices, quan-

tities and pro ts for both rms.

2. Suppose = 0. If the two rms work together and collusively choose a price pC to maximize

the joint pro t, what is pC?

3. Consider the collusion where they both set the price as pC and split the joint pro t evenly.

For what values of will their collusion be sustainable by using a Grim Trigger” strategy?

4. For what values of will their collusion be sustainable by using a prefect Tit-for-Tat”

strategy?

3 Problem 3

Firm 1 and rm 2 compete with each other by choosing quantities. The market demand is given

by

P(Q) =

(

300 ?? Q; if Q < 300

0; otherwise

;

where Q = q1 + q2. Firm 1 has a cost function C1(q1) = 40q1, and rm 2 has a cost function

C2(q2) = 50q2. Answer the following questions.

1. Assume the game lasts only one period. Compute the equilibrium price, quantities and pro ts

for both rms.

2. If rm 1 becomes the monopolist on this market, what quantities will rm 1 choose to produce?

Denote this quantity as QM.

3. One possible strategy is that each rm produces QM

2 . Would the resulting outcome be better

for both rms (Pareto improvement)? Explain why this is not the equilibrium in the one-

period game.

4. Assume this game is in nitely repeated and the interest rate in this economy is r. For what

values of r the strategy in (3) is sustainable by using a Grim Trigger” strategy?