Question

Consider the following variant of the Bertrand Model of Duopoly. Suppose there are two firms producing the same good and they simultaneously set prices for their product. If firm i sets a price pi and firm j sets a price pj, the total quantity demanded for firm i’s product is given by:

qi= 10–pi+ ½ pj

Each firm produces exactly the qi demanded by the market. Both firms have the same marginal cost of production: c=4. For example, if a firm produces 5 units it has to incur a cost of 20.

What is the profit function of each firm? What is the best response function? What is the Nash equilibrium?

Answer 1

Total quantity demanded for firm 1’s product is given by:q1= 10 –p1+ 0.5p2 and that of firm 2 is q2 = 10 - p2 + 0.5p1. There is a same marginal cost of production: c=4.

What is the profit function of each firm?

It is given by π1 = p1q1 - cq1 = p1(10 –p1+ 0.5p2) - 4p1 and since the game is symmetric, π2 = p2q2 - cq2 = p2(10 –p2+ 0.5p1) - 4p2

What is the best response function?

It is found when marginal profit is zero

10 - 2p1 + 0.5p2 = 4 and 10 - 2p2 + 0.5p1 = 4

Best response functions are

p1 = 3 + 0.25p2 and p2 = 3 + 0.25p1

Use the value of p1 and place it in the second

p2 = 3 + 0.25*(3 + 0.25p2)

p2 = 3 + 0.75 + 0.0625p2

0.375p2 = 3.75 and so p2 = 4 and p1 = 4

What is the Nash equilibrium?

It is that both firm charge the same price of 4, equal to marginal cost.