Consider Bertrand competition with homogeneous products. Two firms, 1 and 2, pro

Question

Consider Bertrand competition with homogeneous products. Two firms, 1 and 2, produce an identical product and compete by choosing price. Consumers buy from the firm with the lower price. If the prices are identical, however, assume all consumers buy from firm 1. Inverse demand for the product is given by Q = 80 P and each firm has a marginal cost of 10. Assume that the firms can only set integer prices, so if one wants to undercut the other by the smallest amount possible, the undercut must be at least a dollar.

(a) Write out the best response function of firm 1 for any price that firm 2 could choose.

(b) Write out the best response function of firm 2 for any price that firm 1 could choose.

(c) What is/are the Nash equilibrium/equilibria of this game?

Explanation / Answer

Inverse demand function of both firms is Q= 80 - P and MC =10
Demand function ; P =80 -Q

The best response function for each firm will be equal to:

Q1= (a-c-bQ2)/2b
Where Q1 and Q2 designate the quantities of output chosen by each firm, a and b are the intercept and slope from the demand function, (ie. P=a-bQ), and c represents marginal cost.
So, a = 80 , b = 1 and c= 10
a) Q1 = (80-10-1*Q2)/2*1 = 70- Q2/2
b) Q2 = (80-10-1*Q1)/2*1 = 70 - Q1/2
Solving part (a) and part(b) we get :-
From (a) :-
2Q1 = 140- Q2
From (b) :-
2Q2 = 140 - Q1
=> 2Q2 = 140 - 70 + Q2/2
=> 3Q2/2 = 70
=> Q2 = 140/3 = 46.67 units.
Similarly, Q1 = 70 - 23.33 = 46.67 units.
c) Nash equilibrium price will be :-
Q = 80-P
=>46.67 = 80 -P
=> P = 33.33 $
So, required quantilty is 46.67 units and price is 33.33$ at nash equilibrium.

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