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Ec 106 Problem Set #2Oligopoly: practice problemsReturn to the 2-firm case. Assume each firm produces with C(q) = cq, and market demand curveis p = a − bQ.1. Cournot:• (***) Solve f or the Cournot Nash equilibrium quantities, prices, and profits for the twofirms. Call these q∗, p∗, π∗.• (***) What if these two firms formed a cartel and maximized joint profits? Solve for theresulting quantities, prices, and profits; call these qj, pj, πj.• What if firm 2 cheats when firm 1 sets q1= qj? What are the resulting quantities,prices, and profits?• What does this have to do with the p risoner’s dilemma?• (Think about) What if firms play the Cournot game for two consecutive periods? Whatare the chances that a cartel could survive, and how could this happen? What if theyplay for ten periods? What if they play forever? (Hint: how do we solve multi-periodgames?)2. (***) Bertrand: derive the Bertrand nash equilibrium prices, quantities, and profits. Callthese qb, pb, πb.3. (***) Stackelberg: If firm 1 is the Stackelberg leader, what are the resulting quantities, prices,and profits (qs1, q2s), (ps1, ps2), (πs1, πs2).4. Rank th e quantities, prices, and profits comp uted in the problems marked (***).5. Consider the following game tree (see figure 1)(a) List all of player 1’s strategies(b) List all of player 2’s strategies(c) What are the Nash equilibria of this game? Show why.(d) What are the subgame perfect equilibria of this game? Show why.6. Construct a “Nash reversion”-type subgame-perfect equilibrium to the infinitely repeatedBertrand (price-setting) game. Assu me there are two identical firms, each p roducing at constantmarginal cost c. The market demand curve is p = a − bQ.Ec 106 Problem Set #2Figure 1: Game tree for question 5PLAYER 1PLAYER 2PLAYER 2PLAYER


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CALTECH EC 106 - Problem Set 2

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