Question

...... Consider the standard Bertrand model of price competition. There are two firms that produce a homogenous good with the same constant marginal cost of c. a. Suppose that the rule for splitting up consumers when the prices are cgunl.assigns all consumers to firm when both firms charge the same price. Show that PP.)=(cc) is a Nash equilibrium and that no other pair of prices is a Nash equilibrium... ! Now, we assume that the Bertrand game in part a as the stage game in an infinitely repeated game. Consider the following trigger strategy for both firms: - in "Start to charge the monopoly price (p") at the first period and receive the half of monopoly profits (7"12). At the i period if no one deviates from monopoly pricing up to (1-1) period. charge the monopoly price(p"). Otherwise charge the Bertrand price as a punishment." What is the condition on ő for collusion to be possible? (Hint: you can assume the deviation price would be slightly undercutting the inonopoly price so that you can capture all of monopoly profits.)

consider the standard Bertrand model of price competition. There are two firms that produce a homogenous good with the same constant marginal cost of c.

a) Suppose that the rule for splitting up cunsumers when the prices are equal assigns all consumers to firm1 when both firms charge the same price. show that (p1,p2) =(c,c) is a Nash equilibrium and that no other pair of prices is a Nash equilibrium.

b) Now, we assume that the Bertrand game in part a) as the stage game in an infinitely repeated game. Consider the following trigger stragegy for both firms: " start to charge the monopoly price p at the first period and receive the half of monopoly profits . At the t period if no one deviates from monopoly pricing up to (t-1) period, charge the monopoly price p. otherwise charge the Bertrand price as a punishment."

What is the condition on discount factor (delta) for collusion to be possible?( Hint: you can assume the deviation price would be slightly undercutting the monopoly price so that you can captire all of monopoly profits?

Answer 1

6) given infort given information . Collusive Profit T = (when both Charge b=5") deviation Profit (when one of them Chaoge bab) o grim trigger strategy Leacts To T (ic NE com profit to each of them after deviation ) = 0. Venure Collusive value (Vc) = T +ST+8+. . . o I to o deviation Valu in an + 8 + = "TT + S - - - F+S (0) + 5*(0) 70 = m S da for collusion Is Pregjerable colusive Value Should be greater Than or equal to deviation value i.e. Dvd EF 1 er PPPP 185 1/4 87 - So Smin? 1/2.

- (a) To show that (bi, b2 )= (c, c) is a unique NE. it suffices To Show That you all other 8 price combinations there is atleast one from (1 or 2) That has an incentive To deviate. case-I. If bis P2 >c minn from I can increase its profit by Setting PE CCP2) ie from- I can increase its profit by charging & its frice sighly above c and sightly below p2. so given Case is Not a Nasnego Case-2. If by = P2 > c. In this case each firm has incentive to be increase its Profits by slightly undercutting the rival price. case-3 If !,> 62=0 fim-2 can increase its layoya by invecoing its price above c and below for kr. 3 — So confirming That bı= z=c is The ne of the game.