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Problem Set 6 1. (10 points) In Cambridge, shoppers can buy apples from two sources: a local orchard, and a store that ships apples from out of state. The orchard can produce up to 50 apples per day at a constant marginal cost of 25c per apple. The store can supply any remaining apples demanded, at a constant marginal cost of 75c per unit. When apples cost 754 per apple, the residents of Cambridge buy 150 apples in a day. (a) (5 points) Draw the marginal cost curve of apple production in Cambridge. (b) (5 points) Assume that the city of Cambridge sets the price of apples within its borders. What price should it set, and should the price vary depending on where you purchase your apples? Problem 1 by MIT OpenCourse Ware. 2. (20 points) Tariffs are usually imposed in order to decrease imports, but they don't always have the same effect. Please draw graphs that demonstrate how shifts in the domestic supply curve for a product subject to a tariff could result in the following scenarios. (a) (10 points) No imports at all of that product. (b) (10 points) The country becoming an exporter of that product. Problem 2 by MIT OpenCourse Ware. 3. (35 points) (Suggestion: It may be helpful to read section 9.6 before doing this question. ) Moldavia is a small country that currently trades freely in the world barley market. Demand and supply for barley in Moldavia is governed by the following schedules: Demand: Q =4 -P Supply: Q' =P The world price of barley is $1/bushel. (a) (12 points) Calculate the free trade equilibrium price and quantity of barley in Moldavia. How many bushels do they import or export? On a well-labeled graph, depict this equilibrium situation, and shade the gains from trade relative to the autarkic (no trade) equilibrium in Moldavia. (b) (12 points) The Prime Minister of Moldavia, sympathetic as always, believes he can help those hurt by free trade in barley relative to the situation in autarky. He taxes the party that has benefited from free trade (either consumers or producers) an amount per bushel that is the difference between the autarkic price of barley and the free trade price. Furthermore, he rebates the entire government revenue of the tax back to the party harmed by free trade (again, either consumers or producers). In a new, well-labeled diagram, show this post-tax equilibrium situation. Calculate and show: . The new equilibrium price and quantity of barley in Moldavia . Changes in the quantity of imports or exports . The amount of revenue collected by the Prime Minister Who pays the larger burden of this tax, consumers or producers in Moldavia? Why? (c) (11 points) Are the free trade losers better off or worse off after the rebate than they were under autarky? Why? On your diagram from (b), shade the DWL (if any) of this tax rebate policy, relative to the free trade equilibrium you found in (a). 4. Problem removed due to copyright restrictions. This content is presented in audio form in the Solution Video for Problem Set 6, Problem 4.Questions Here we will study some properties of a demand system that is often used in oligopoly theory, and then make a start toward that theory. The new conceptual issues are in Ques- tion 1. The other two questions are mostly mechanical computations. Try to do one yourself, and then read and keep the rest for reference. Please send me an e-mail if you find typos. Question 1 - Demand Functions There are three goods, labelled 0, 1, and 2. The quantities are denoted by To, 21, and 12. All prices are measured in units of good 0, so the prices are denoted by 1, p1, and p2- The utility function is u(To, [1, T2) = To + and + 02 2 - 3 [61 (21)" + 2kx1 22 + 62(x2)"] and the budget constraint is To + Pidi + P242 = 1, (a) What are the first-order necessary conditions (FONCs) for the maximization? (You will find it easier to substitute out 2, than to do Lagrange.) (b) What are the second-order necessary conditions (SONCs) and sufficient conditions (SOSCs) for the maximization? (Here for once we need the full set of conditions you learned in your math courses.) (c) Solve the FONCs to express each of pi and p, as a function of (21, 12). These are the "inverse demand functions". (d) Solve the FONCs to express each of r, and r2 as a function of (p1, p2). These are the "direct demand functions". Under what conditions on (a1, a2, b1, b2, k) are the two goods 1 and 2 substitutes? When are they complements? And should we be using the Hicksian definition of substitutes/complements or the Marshallian one? (e) Take the function linking p1 to (21, 12) in (c). Holding 12 fixed, calculate the slope of the inverse demand curve in numerical value, namely -Op1/021. Then take function linking In to (Pi, p2) in (d). Holding p2 fixed, calculate the numerical value of the slope of the demand curve as it would look in the conventional diagram with p, on the vertical axis and 1 on the horizontal axis. Which of the two slopes you calculated is larger in numerical value? Can you think of an economic intuition for this? Question 2 - Quantity-Setting (Cournot) Duopoly Now suppose each of the goods 1 and 2 is produced by a single firm. Each firm chooses its quantity. Each has a constant per unit (average = marginal) cost, c for firm 1 and c2 for firm 2. The profit of firm 1 is I1 = (P1 - q) I1 . (a) Using the inverse demand function in Question 1(c), express this as a function of (1, 12). (b) If firm 1 thinks that firm 2 is choosing some particular quantity 12, what is the condition for the choice of z1 to maximize firm I's profit? (c) If firm 2 likewise thinks that firm 1 is choosing some particular quantity 21, what is the condition for the choice of z, to maximize firm 2's profit? (d) If each firm chooses its own quantity to maximize its own profit, regarding the quantity of the other as fixed, then the outcome is called the Cournot equilibrium of this duopoly. Take the above conditions for the two firms' maximization, and solve the two equations simultaneously to obtain the quantities in Cournot equilibrium in terms of the given algebraic constants (parameters) of the problem: (a1, (2, b1, b2, k, c1, C2).Question 3 - Price-Setting (Bertrand) Duopoly Again suppose each of the goods 1 and 2 is produced by a single firm. But now each firm chooses its price. Each has a constant per unit (average = marginal) cost, c, for firm 1 and cz for firm 2. Write the direct demand function as 01 = 01 - PIPI +*P2, $2 = 02+ KP1 - B2p2. Firm I's profit is MI=(mi- a)= =(p -a) (1 - Bpitop2). (a) If firm 1 thinks that firm 2 is choosing some particular price p2, what is the condition for the choice of p, to maximize firm I's profit? (b) If firm 2 likewise thinks that firm 1 is choosing some particular price pi, what is the condition for the choice of p, to maximize firm 2's profit? (c) If each firm chooses its own price to maximize its own profit, regarding the price of the other as fixed, then the outcome is called the Bertrand equilibrium of this duopoly. Take the above conditions for the two firms' maximization, and solve the two equations simultaneously to obtain the prices in Bertrand equilibrium in terms of the given algebraic constants (parameters) of the problem: (01, 02, 81, 82, k, (1, C2)
