Question

A monopolist (Firm 1) may be a low-cost type, with constant marginal cost of production 10, or a high-cost type, with constant marginal cost of 20, with probabilities p and 1-p respectively. It has no fixed cost. Only the monopolist knows his type. The game has two stages. In stage one, a potential entrant (Firm 2) with constant marginal cost of 15 decides whether to enter the market. Entry requires a fixed investment of $100. If Firm 2 enters the market, it learns what Firm 1’s type is, and both engage in Bertrand competition in homogeneous products in the second stage. Consumer demand is 100 units regardless of the price.

(a)What is the Nash Equilibrium of the second-stage game if Firm 2 enters? Solve the game for each type of Firm 1.

(B)Argue that Firm 2 would not enter if it believes Firm 1 is certainly low cost but would enter if it believes Firm 1 is certainly high cost.

(C)What is the value of p under which Firm 2 will decide to enter?

Answer 1

Since firms are in Bertrand competition, they will charge prices equal to their marginal costs.

In this game, there are two players, Firm 1 and Firm 2. Further, Firm 1 is of two types: low cost and high cost. If firm 2 enters when firm 1 isn't producing, its payoff would be 15*100-100= 1400 (total revenue generated minus fixed cost to enter). If it doesn't, then it gets nothing. If there is a low cost firm and  it decides to participate in market, then it's payoff will be 100*10= 1000 and firm 2 will get nothing because it has a higher MC. Firm 1 gets nothing if it doesn't produce.

The table form is shown below:

	Firm 2 enters	Firm 2 does not enter
Low cost firm 1 produces	1000, -100	1000, 0
LC firm 1 does not produce	0, 1400	0,0

Similar table for a High cost firm 1 with firm 2 game: Here firm 1 will gain nothing because its MC is higher than firm2. Firm 2 will end up attracting all the consumers:

	Firm 2 enters	Firm 2 does not enter
firm 1 produces	0, 1400	2000, 0
firm 1 does not produce	0, 1400	0,0

If Firm 1 is low cost, then Nash equilibrium is (Firm 1 produce, Firm 2 does not enter)

If Firm 1 is high cost, then Nash equilibrium (Firm 1 produce/does not produce, firm 2 enters)

b) If Firm 2 believes that Firm 1 is high cost, then it does not matter whether Firm 1 decides to produce or not, because 2 will end taking all of the market. But if it is certain that firm 1 is low cost, then it would incur a loss of $100 upon entering. It is better off if it doesnt enter.

c) Firm 2 will enter iff expected MC of firm 1 is equal to or below Firm 2 MC:

p*10+(1-p)*20<15

5<10p: p> 0.5

Hence probability should be equal to or more than 0.5