1. Repeated Cournot: Suppose two firms exist in a market with demand given by P = 24 - Q where Q is the total quantity produced by both firms given by Q = q1 + 92. Each firm has zero marginal cost. Assume that firms engage in simultaneous quantity competition. (a) Find the Nash Equilibrium quantity of the stage game (not repeated). (b) What is the monopoly quantity in this market. (c) Assume that the firms repeat this simultaneous quantity competition infinitely, and have common discount factor . Construct a grim trigger strategy that en- sures half the monopoly quantity is produced by each firm every period. (d) Find the necessary discount factor that sustains this strategy as an SPNE. 2. Repeated Stackelberg: Suppose the demand and marginal costs remain the same as in 2, however now the firms engage in Stackelberg competition, i.e. firm 1 chooses q1 first and then firm 2 observing q1 chooses q2. (a) Find the Nash Equilibrium quantity of the stage game (not repeated, i.e. the one interaction Stackelberg from week 6). Assume that the firms repeat this sequential quantity competition infinitely, and have common discount factor o. This means every stage is itself composed of two periods- firm 1 chooses q1 then firm 2 chooses q2, profits realize after these two sub-periods, the firms observe each other's quantity decisions and then the game repeats. We will construct a grim trigger strategy to sustain the monopoly quan- tity. Let the monopoly quantity be Q" and the stackelberg quantities from the previous question be (qi , 92 ). You can replace these by the actual numbers, I just did not want to give away the answers in this question.For firm 1, if the history is (qi, Q -q1) then play q1 = q1. Otherwise play q1 = q1. For firm 2, if the history is (qi, Q~ -q1), and firm 1 has played of in the current round play q2 = Q" - q1. Otherwise choose q2 = 12 - 2, i.e. firm 2 myopically best responds. The interpretation of this strategy is that the firms split monopoly profits and revert to stackelberg equilibrium if there is any deviation. The details of the split are parameterized by qf. The higher of the larger firm 1's share of the profit. We may need an unequal split because firm 1 and firm 2 are not symmetric (firm 1 chooses quantity first). We will try to find of such that this candidate equilibrium requires the smallest o to be sustained, i.e. requires the least patience from the players. (b) If firm 1 were to deviate what quantity would he choose and what would be his profits from such a deviation (remember that the strategy includes firm 2 best responding to any such deviation before firm 1 realizes any profits from it). (c) What is the condition on of so that firm 1 does not want to deviate. Does this depend on d?(d) Now consider rm 2's incentives. If rm 2 deviates, they should play a best response to g? If they deviate, in the rest of the following periods they will get second mover Stackelberg payos. I am telling you that the minimum required discount factor for rm 2 to have no incentive to deviate is increasing in qf. This is intuitive, rm 2 nds deviation less appealing if it is given a larger share of the monopoly quantity in the cooperation phase {but involves a slightly ugly calculation]. Given this information, and your answers to the previous questions, and the fact that rm 1 and 2 share a common discount factor, what is the {33' that makes cooperation sustainable under the largest range of discount factors. Does such a qf from the previous part involve a split that is favorable to rm 1, rm 2, or even. 1Why might rm 1 be easier to keep in check in the repeated Stackelberg despite having a more favorable punishment payo compared to rm 2. Calculate the discount factor required to sustain cooperation under the of from your previous answer. lCompare the required discount factor to sustain cooperation in Stackelberg in the previous question vs. that in lCouruot. 1What is the intuition for why this comparison arises?I