Problem 1 (20 points) Consider an industry composed of two firms producing the same good. Let y1 be the output of firm 1 and y2 the output of firm 2. For simplicity, assume that the price p of the good does not change with the output produced by the firms; this could be a reasonable assumption if the good is a luxury good. The total cost of producing y1 in firm 1 is C1(y1); the total cost of producing y2 in firm 2 is C2(y2). The two functions C1, C2 are increasing and convex. Firm 1 is more efficient in the sense that C1(y) 0. (1) The firms behave as a cartel and jointly choose y1, y2 to maximize the industry's total profit 7 (y1, y2) = p x (y1 + y2) - C1(y1) - C2(y2). Show that the solution to that maximization problem need not entail that firm 1 produces more than firm 2. More precisely: exhibit two increasing and convex cost functions C1, C2 satisfying (1) for which the output levels y1, 32 maximizing 7(y1, y2) are such that y1