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 q first and then firm 2 observing q, chooses 43- (a) Find the Nash Equilibrium quantity of the stage game (not repeated, Le. the one interaction Stackelberg from week 6). Assume that the firms repeat this sequential quantity competition infinitely, and have common discount factor s. This means every stage is itself composed of two periods- firm 1 chooses q then firm 2 chooses 92. 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 QU and the stackelberg quantities from the previous question be (of, q?). 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 (q;. Q" -of) then play 9 = of. Otherwise play q = dj. For firm 2, if the history is (q;. Q" - of), and firm 1 has played of in the current round play ga = Q - qi. Otherwise choose or = 12 - 4, 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 of. 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 I were to deviate what quantity would be 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 6? (d) Now consider firm 2's incentives. If firm 2 deviates, they should play a best response to of. If they deviate, in the rest of the following periods they will get second mover Stackelberg payoffs. I am telling you that the minimum required discount factor for firm 2 to have no incentive to deviate is increasing in of. This is intuitive, firm 2 finds 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 firm 1 and 2 share a common discount factor, what is the of that makes cooperation sustainable under the largest range of discount factors. (e) Dors such a of from the previous part involve a split that is favorable to firm 1, firm 2, or even. Why might firm 1 be easier to keep in check in the repeated Stackelberg despite having a more favorable punishment payoff compared to firm 2