Problem 5 (20 points) Consider the following Cournot duopoly model. Firms 1 and 2 produce the same good. They simultaneously choose their output levels y, y2, and the resulting price is given by the inverse demand function p = 1 Y Y2. If firm 1 produces a (strictly) positive output level, it incurs a fixed cost k but no variable cost. Thus, its cost function is 0 if y = 0, k if y > 0. Firm 2 faces no cost at all; its cost function is C2 (Y2) = 0 for all y2 0. Show that (i) there is a unique Cournot-Nash equilibrium if the fixed cost k is below a lower threshold k or above an upper threshold k+ and (ii) there are two Cournot-Nash equilibria when k k kt. Compute the thresholds k, k and the equilibrium output levels explicitly. C (31) = {