Consider an oligopolistic market with two firms (1 and 2). Production costs are given by Ci(qi) = 30qi (for i = 1, 2) and the market demand function is: P = 180 Q Where P is the market price paid by consumers and Q is the total quantity sold. Firms compete in price. As they sell a homogeneous product, consumers always buy from the cheapest firm. If firms set the same price, they will split the market evenly. (a) Find the equilibrium quantities, prices and profits if firms are not subject to a production capacity constraint. [5 marks] (b) Explain why the solution to this problem is called the Bertrand Paradox? What are two important assumptions of the model that lead to this result? [5 marks] (c) Firm 3 enters this market with a production cost C3(q3) = 20q3 . We now have an oligopolistic market with three firms (1,2,3). The market demand and the cost functions of firms 1 and 2 remain unchanged. Assume that (i) if two firms charge the same price and have the same marginal cost, they share the market evenly, while (ii) if they charge the same price and don't have the same marginal cost, consumers buy from the more efficient firm. Find the equilibrium quantities, prices and profits if firms are not subject to a production capacity constraint. [5 marks] (d) Assume that each firm i (i = 1, 2, 3) has a capacity constraint of ki = 30 (i.e. each firm cannot produce more than 30 units of the good) and that the consumers behave according to the efficient rationing rule. Show that setting a price equal to 90 is the new Nash equilibrium price. What are the equilibrium quantities and profits? [15 marks] (e) Suppose that, due to an economic recession, the market demand shifts to P = 150 Q. However, firms are not able to adjust their capacity. What is the new Nash equilibrium price? What are the equilibrium quantities and profits? [5 marks] (f) Suppose that the market demand is P = 150 Q and firms now have the possibility to adjust their capacity. The game between the three firms becomes a two-stage game: in stage 1, firms set their production capacity simultaneously; in stage 2, firms set prices simultaneously. As before, we assume that there is efficient rationing. What will be the equilibrium prices, quantities and profits in this new game? [10 marks] (g) Compare the profits in parts (e) and (f).