Exercise 2.9. Consider an industry where there are two firms, a large firm, Firm 1, and a small firm, Firm 1. The two firms produce identical products. Let x be the output of Firm 1 and y the output of Firm 2. Industry output is Q=x+y. The price Pat which each unit of output can be sold is determined by the inverse demand function P=130-100. For example, if Firm 1 produces 4 units and Firm 2 produces 2 units, then industry output is 6 and each unit is sold for P = 130-60 = $70. For each firm the cost of producing each q units of output is C(q)=10q+62.5. Each firm is only interested in its own profits. The profit of Firm 1 depends on both x and y and is given by ПI,(x, y) = x[130− (x+y)]-(10x+62.5) and similarly the profit function of revenue cost Firm 2 is given by П2(x, y) = y[130−(x+y)] −(10y+62.5). The two firms revenue cost play the following sequential game. First Firm 1 chooses its own output x and commits to it; then Firm 2, after having observed Firm 1's output, chooses its own output y, then the price is determined according to the demand function and the two firms collect their own profits. In what follows assume, for simplicity, that x can only be 6 or 6.5 units and y can only be 2.5 or 3 units. (a) Represent this situation as an extensive game with perfect information. (b) Solve the game using backward induction. (c) Write the strategic form associated with the perfect-information game. (d) Find the Nash equilibria of this game and verify that the backward- induction solutions are Nash equilibria.