Suppose two identical firms produce homogeneous products and that they are the o

Question

Suppose two identical firms produce homogeneous products and that they are the only firm in the market. Let the two firms face identical cost functions, c(y) = 10y + 0.5y2. Assume that the market demand function is given by P = 110-Y.

(1) Suppose that two firms compete in Bertrand fashion. That is, two firms choose their price simultaneously and independently. Write down the normal form of the game.

(2) Find the two firms' best response mapping.

(3) Prove that there does not exist Nash equilibrium in the Bertrand game. What will happen in the industry if the two firms play the Bertrand game?

Explanation / Answer

1)

In this game, firm seeks to undercut its rival by lowering price. Price war between these two continue until price becomes equal to the marginal cost of production.

P =MC

2)

C = 10Y +0.5Y^2

on differentiating TC

MC = 10 +Y

Best Resonse; P =MC

10+Y = 110 -Y

2Y = 100

Y = 100/2

= 50

P = 110- Y

= 110 - 50

= 60

3)

Each firm tries to reduce the price until it becomes equal to the marginal cost of production. Hence, there exists only one pure Nash equilibrium. Here, Nash equilibrium exists here.

if two firms play Bertrand game, it will get converted into perfectly competitive market. Profit will disappear.

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