Conference 5Intermediate Microeconomics - Fall 201801/10/2018Problem 1: Price competition and MCC clausesConsider a price situation among duopolies offering differentiated products. The two firms canchoose between setting the monopoly (pm) or the competition price (pc) for their product, withpc< pm. The normal form game with resulting payoffs is the following:pmpcpm(4, 4) (1, 5)pc(5, 1) (2, 2)Note that this is a simplified version of the Bertrand competition in prices seen in class.The dominant strategy is pcand the Nash equilibrium is (pc, pc).1. Find firms’ dominant strategies in this game and deduce its Nash equilibrium.Firms’ dominant strategies are to set price pc. Therefore the game’s Nash equilibrium is(pc, pc)Now, a possible way to change the game is to introduce Meet-the-Competition clauses (MCC).MCC is a contractual agreement giving the company the option of retaining a customer bymatching the price offered by competition. For instance if firm 1 charges pmand firm 2 pc, thenwithout MCC firm 1 would lose its customers to firm 2. But with MCC firm 1 can change itsprice to pcand keep its customers.We are now going to consider a game where the two firms can adopt MCC. Each companyhas now four actions: charge the competitive price (pc) with or without MCC or charge themonopoly price (pm) with or without MCC.2. Fill in the firms’ new payoffs in the table below.pmpcpm-MCC pc-MCCpm(4, 4) (1, 5)pc(5, 1) (2, 2)pm-MCCpc-MCC1pmpcpm-MCC pc-MCCpm(4, 4) (1, 5) (4, 4) (1, 5)pc(5, 1) (2, 2) (2, 2) (2, 2)pm-MCC (4, 4) (2, 2) (4, 4) (2, 2)pc-MCC (5, 1) (2, 2) (2, 2) (2, 2)3. Find the Nash equilibrium by iteratively eliminating dominated strategies.pm-MCC dominates pm. Remove strategy pm, and the game simplifies to:pcpm-MCC pc-MCCpc(2, 2) (2, 2) (2, 2)pm-MCC (2, 2) (4, 4) (2, 2)pc-MCC (2, 2) (2, 2) (2, 2)Now, observe that in this game pm-MCC is a dominant action. Thus both firms adoptMCC and price high at pm, i.e. they get profits at monopoly level.Problem 2: Three players game. Consider the three players game with the payoffs givenbelow. (Player 1 chooses one of the two rows, player 2 one of the two columns and player 3 oneof the three tables.)L RT (0, 0, 3) (0, 0, 0)B (1, 0, 0) (0, 0, 0)AL RT (2, 2, 2) (0, 0, 0)B (0, 0, 0) (2, 2, 2)BL RT (0, 0, 0) (0, 0, 0)B (0, 1, 0) (0, 0, 3)C1. Find the four pure strategy Nash equilibriaL RT (0, 0, 3) (0, 0, 0)B (1, 0, 0) (0, 0, 0)AL RT (2, 2, 2) (0, 0, 0)B (0, 0, 0) (2, 2, 2)BL RT (0, 0, 0) (0, 0, 0)B (0, 1, 0) (0, 0, 3)CThe pure strategy equilibria are{(T, R, A), (B, L, A), (T, R, C), (B, L, C)}2. Show there is a mixed stragegy equilibrium in which player 3 always chooses B and player1 (resp. player 2) mix between T and B (resp. L and R ) with equal probabilities.We show it is an equilibrium by eximining deviations’ possibilities and showing that theyare not profitable.2Player 3 : knowing that players 1 and 2 play T and L with probability12each, player 3has no incentive to deviate and play A or C. Since the situations (T, L), (T, R), (B, L)and (B, R) are all equally likely to arise with probablity14each. Therefore player 3gets expected payoff34by playing A or C and 1 by playing B.Player 2 : knowing that player 3 plays B and player 1 plays T with probability12,player 2 gets expected payoff 1 by playing L with probablity12. If player 2 deviatesshe can only do so within game B, and putting any weights above12on L or R wil sether expected payoff below 1. Therefore there is no profitable deviation for player 2.Player 1 : has the same reasoning as player 2.Problem 3: More price competitionTwo firms produce heterogenous goods and compete in price. The demand and cost functionsfor each firm are as follows:q1= 2 − 2p1+ p2and C1(q1) = q1+ q21q2= 2 − 2p2+ p1and C2(q2) = q2+ q221. What problem do firms have to solve to maximize their profit?∀i ∈ {1, 2}, firm i solves:maxpipiqi− Ci(qi)= maxpipiqi− qi− q2i= maxpipi(2 − 2pi+ p−i) − (2 − 2pi+ p−i) − (2 − 2pi+ p−i)2= maxpi(pi− 1)(2 − 2pi+ p−i) − (2 − 2pi+ p−i)2= maxpi(pi− 1 − 2 + 2pi− p−i)(2 − 2pi+ p−i)= maxpi(−3 + 3pi− p−i)(2 − 2pi+ p−i)2. Express firms’ Best Reponse functionsTo find the Best Response functions, we need to solve the firm’s maximization problem.The First Order Conditions is:3(2 − 2pi+ p−i) − 2(−3 + 3pi− p−i) = 0⇔ 6 − 6pi+ 3p−i+ 6 − 6pi+ 2pi= 0⇔ −12pi+ 5pi+ 12 = 0⇔ pi= 1 +512p−iTherefore firm i’s best response to firm −i setting its price at p−iis to set its own price to1 +512p−i, as long as this quantity is above zero. Mathematically we write:BRi(pi) = max{1 +512p−i, 0}33. What are the equilibrium prices? (Hint: we are looking for p1and p2such that firms playtheir Best Response functions)Draw BR1and BR2on a graph to get an idea of how they look like:0 2 4 6 8 100246810p1p2Equilibrium prices (p∗1, p∗2) are given by the lines’ intersection. To solve analytically we set:(BR1(p∗1) = p∗2BR2(p∗2) = p∗1Let us work on the second equation:p∗1= 1 +512p∗2⇔ p∗2=125(p∗1− 1)If we equate this result with the second equation, we get that:1 +512p∗1=125(p∗1− 1)⇔112(12 + 5p∗1) =125(p∗1− 1)⇔560(12 + 5p∗1) =14460(p∗1− 1)⇔ 60 + 25p∗1= 144p∗1− 144⇔ 204 = 119p∗1⇔ p∗1=1274Now we can find p∗2easily:p∗2= 1 +512p∗1= 1 +512127=7 + 57=127Therefore the equilibrium prices are