Question

# Consider the following variant of the Bertrand Model of Duopoly. Suppose there are two firms producing...

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?

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.

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