310, ECO Fall 2008 Problem Set 5: Equilibrium Analysis Due in class on November 18 Question 1
In this problem we will consider the sh industry on the island of San Serife. For this purpose we will aggregate all the other goods into one composite, and measure it in units of the islands currency Arial. Thus the price of the other good is 1. (a) There are 160 consumers, each with the utility function U (x, y ) = 10x 5x2 + y, where x is the consumption of sh and y the consumption of the other good. Let P denote the price of sh. Find one consumers demand function (x expressed as function of P ). Be careful about boundary solutions with x = 0, but ignore boundary solutions with y = 0, i.e., assume that each consumer has suciently high income. Find the market demand for sh. Show this in a graph with P on the vertical axis and the market quantity Q on the horizontal axis. (b) The shing industry consists of several rms. Each rm, to produce and sell a positive amount of sh, must get a boat and hire a market stall. The cost per period of owning a boat (the interest on the money tied up) is 3 Arials, and the cost of hiring a market stall is 1 Arial. In the long run, neither of these costs is xed. In the short run, the cost of the boat is xed because boats have no alternative use, whereas the cost of the market stall is avoidable if the rm chooses zero output because the same stall can be used to sell other things. In addition to these costs, the rm has to pay q 2 to produce q units of output. Write down expressions for each rms long-run total cost (C), long-run average cost (AC), short-run variable cost (SVC), short-run average variable cost (SAVC), and marginal cost (MC), in each case as functions of q . Find the values of q that minimize AC and SAVC, and the minimum values of these two average costs. Find the equation for the rms short-run supply curve. Draw rough sketches of AC, SAVC, MC, and the supply curve. (c) In the long run, there is free entry and exit of shing rms. What is the industrys supply? long-run (d) Suppose the industry is initially in long-run equilibrium. Putting together the market demand curve you found in part (a) and the industry supply curve you found in part (c), nd the long-run equilibrium price. How many rms operate in this equilibrium? What is the prot of each rm? What is the consumer surplus? 1
(e) Now suppose the government levies a tax of 2.5 Arials per unit of sh. Find the new short-run equilibrium with the same number of rms as in the original long-run equilibrium. What is the price paid by the consumers? What is the price received by the rms? What is the prot of each rm? Will any rm want to stop production in the short-run? What is the aggregate loss of consumer surplus? How much tax revenue does the government collect? What is the dead-weight loss? (f) Will rms want to exit in the long run with the tax? In this new long run equilibrium, what is the price paid by the consumers? What is the price received by the rms? How many rms are active? What is the prot of each rm? What is the aggregate loss of consumer surplus? How much tax revenue does the government collect? What is the dead-weight loss?
Question 2
Ann has an endowment of 20 units of good x and 5 units of good y . Bob has an endowment of 10 units of good x and 5 units of good y . Consider the following two cases: Case 1 Anns utility function is UA (xA , yA ) = xA yA and Bobs utility function UB (xB , yB ) = xB yB . Case 2 Anns utility function is UA (xA , yA ) = min(xA , yA ) and Bobs utility function UB (xB , yB ) = xB + yB . For each case, answer the following questions algebraically but illustrate your answers in the Edgeworth box. (a) Describe the set of ecient allocations in this economy. (b) Describe the set of allocations that Pareto-improve (make both individuals weakly better o and one of them strictly better o) on the endowment allocation. (c) Describe the contract curve. (d) Find all competitive equilibrium price vectors and allocations for this economy.
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