Question

Suppose two firms are in Bertrand Competition. Both firms have a specific marginal cost of c = 5.

a.) Show that ( p∗1 , p∗2 ) = (5,5) is the unique Nash equilibrium.

b.) Now, suppose that the firms can only set their prices to the nearest cent (i.e., their pric- ing decision is now discrete instead of continuous). There are now three Nash equilibria:

(p∗1, p∗2)= (5,5) from before, and two more. Describe the two new Nash equilibria.

c.) Among the three Nash equilibria in b.), which are strict Nash equilibria, and which are not strict Nash equilibria?

Answer 1

A.The unique Nash equilibrium is mutual defection, for eoutcome that is worse for both players than mutual cooperation.in unique Nash equilibrium players are flawless in execution,players have sufficient intelligence to deduce the solution ,players know the planned equilibrium strategy of all of the other players.a Nash equilibrium is pair p*1,p*2) of prices such that p*1 is a best response to p*2 and p*2 is a best response to p*1 in this equilibrium (p1,p2)=(5,5) is anash equilibrium if one firm charges the prices to 5 then the other firm can do no better than charge the price 5 also .next we show that no other pair(p1,p2)is a Nash equilibrium, ad follows If pi<c for either i=1 or I=2 then the profit of the firm whose price is lowest is negative,and this firm can increase its profit by rising its price to c.if pi=c and pj>c then firm is better off increasing its price slightly making it profit positive rather than zero.if pi>c and pj>c suppose that pi>or equal pj then firm I can increase its profit by lowering pi to slightly below pj if D (pj)>0 (I.e.if pj<infinite) and to p D(PJ)=0 (I.e.if PJ>or equal infinite). In conclusion when unit cost of production is a constant c, the same for both firms and demand is linear,betrands game has a unique Nash equilibrium.