Assume the gizmo industry is monopolistically competitive, with economies of scale. Total market size (S) in the Home country is a fixed 5 million gizmos, and each firm has a fixed cost (FC) of $10 million and a constant marginal cost of $100. For every $1 a firm charges below the average price (AvgP) of its competitors, its market share increases by 0.5%. Thus, the firm’s demand curve can be written as Q = S×(1/N + B×(AvgP – P)), where B = 0.005. a) How do we solve for the firm’s price? First, solve the firm’s demand curve for P = AvgP + [1/(B×N) – Q/(B×S)], multiply by Q to get the firm’s revenue, and then take the derivative with respect to Q to solve for MR = P – Q/(B×S). Or, you could read my notes on the model math. Then set this equal to the marginal cost of $100, replace Q/S with 1/N, and solve for P as a function of N. Using the numbers given above, solve for P as a function of N. Graph this. b) The firm’s average cost equals its average fixed cost plus its average variable cost, or AC = FC/Q + MC. Because this is falling with Q, we conclude the firm has economies of scale. Now replace Q with S/N, and solve for AC as a function of N. Using the numbers given above, solve for AC as a function of N. Graph this. c) Under monopolistic competition, firms will enter the market if P > AC, and exit if P < AC. Solve for the equilibrium number of firms by setting P = AC. Remember that if the optimal number is not an integer, you should round down. Using the numbers above, solve for the optimal number N of firms, the firm's market price P, and the average cost AC. Graph this. What is the output Q of the average firm? d) For the typical firm, calculate profit = Q(P-AC), and total industry profits as S(P-AC). I suggest you use Excel or a good calculator so you don’t make rounding errors. If profit is nonzero, how is this possible? e) Now, assume that the Home country opens up to trade with a neighboring country with a similar industry, with an identical cost structure. The foreign country’s total market size is 3 million gizmos, but other things remain the same. Solve for the optimal number of firms in the foreign market at autarky, the foreign autarky price, and the foreign average cost. Graph this. What is the output of the average foreign firm, and the profit of the typical foreign firm? f) Now, combine the autarkies into one market with a quantity-demanded of 8 million gizmos. Solve for the new free trade price, the equilibrium number of firms, and the average cost. g) What is the equilibrium output of the average firm under free trade, and what is the profit of the typical firm under free trade? Calculate the change in producer surplus as the difference in total industry profits between autarky and free trade. h) Calculate the change in consumer surplus in each country as the total quantity-demanded times the change between the autarky and free-trade prices. What is the total consumer gain from trade? How does this compare to the change in total producer surplus? i) Assume that in the short run, firms in both countries are unable to adjust output and do not shut down, but prices change in response to the increased competition. Solve for the short-run price if the combined number of firms stays the same as in autarky, and then solve for the losses of the typical home firm and the typical foreign firm. How many firms must shut down overall to reach the new equilibrium?