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Solutions to Problem Set 1

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ECON 3340
Andrea Podhorsky

Chapter 3 1. (a) The Pareto criterion cannot satisfy completeness, the axiom that a desirable choice mechanism should provide ranking between any two choices. Equivalent to unanimity, the Pareto criterion says A is preferred to B if everyone prefers A; B is preferred to A if everyone prefers B. Of course, in social choice generally some people prefer A and some B. In these cases the Pareto criterion fails to provide a ranking. (b) Plurality rule voting (the single winner is the person with the most votes, with no requirment that the winner have a majority of votes) fails the independence of irrelevant alternatives axiom. For instance, if there are 35 voters, if candidates X, Y and Z run for election, candidate X gets 10 votes, Y gets 12 votes and Z gets 13 votes. But if Z does not run, if Z and X have similar platforms (ie. They are both liberal), then suppose X gets 20 votes and Y gets 15 votes. Since Z splits the votes between X and Y, the preference for X over Y depends on whether Z is running for election. (c) Majority rule voting cannot provide transitivity. It is possible that a simple majority vote between two alternatives can produce A preferred to B, and B preferred to C, but C preferred to A. This is a non-transitivity. For instance (where “>” means “is preferred to”) Voter 1: A>B>C; Voter 2: B>C>A; Voter 3: C>A>B. Now according to a majority vote: A vs. B: A>B B vs. C: B>C C vs. A: C>A. [Alternatively, a vote could be held to choose among 3 or more choices directly. Of course, there is no guarantee one choice would receive a majority. This is a violation of completeness.] (d) Pulling an alternative out of a hat has few desirable properties as a choice mechanism. The most serious of its shortcomings is the failure even to provide unanimity. Everyone may prefer a to b, but if b is drawn from the lint, it is "preferred" by this meaningless criterion. It also would fail to provide transitivity. 2. A B (a) No No (b) No Yes (c) No No (a) The Pareto criterion requires that an action only be undertaken when it makes at least one person better off, while making no one else worse off. If anyone would vote against a proposition, it is not a Pareto improvement. Plan A is not attractive to the two upwind residents: their willingness-to-pay is $1,000 each, the plan calls for a contribution of $7,000 each. Since they would be made worse off by plan A, it fails the Pareto criterion. 1 Plan B would also fail on this basis. Here, the downwind resident is charged with paying for the full amount of the smoke-guzzler. The upwind residents are better off, but the downwind resident is asked to pay an amount 6000$ greater than his willingness to pay which makes him worse off. (b) By similar reasoning, plan A would fail to earn a majority vote, but pl an B would achieve a 2 of 3 majorities. For all the reasons outlined above, the downwind residents would vote against plan A (defeating the initiative), and for plan B (carrying the initiative). (c) A good starting point is to recognize that the sum of the willingnesses-to-pay would not finance the "smoke-guzzler": if each resident was charged exactly their maximum willingness - to-pay, $17, 000 could be raised, while the project costs $21,000. Under these circumstances, no financing scheme exists, even with compensation that would produce unanimous support. For plan A, the upwind residents would have to be compensated $6,000 each to be in favor of the plan, but the sum of the downwind resident's tax and this necessary compensation ($7,000 + $12,000 = $19,000) is greater than his willingness -to-pay ($15,000). Similarly, the upwind residents could compensate the downwind resident $1,000 each, but this is not enough to make up the difference between his willingness -to-pay ($15,000) and the cost of the project ($21,000). 3. It will not pass a majority vote since 60% of residents are not willing to pay $2 each. It will also not pass the Pareto criterion since 60% of residents will object to the head tax. It is a good idea, however, according to the compensation p rinciple since the gains to the wealthy exceed the losses to poor and hence, with compensation, there can be a Pareto improvement. (400,000 x $98 > 600,000 x $1.) 4. Z and R. Chapter 4 1. (a) Under perfect competition, P=MC è 50 – Q = 10 or Q = 40 and P = 10. (b) Unregulated monopolist, MR = 50 – 2Q, MC = MR è 50 – 2Q = 10 or Q = 20 and P = 50 – 20 = 30. (c) In part a): Consumer surplus = ½ x 40 x 40 = 800, Net surplus = 800 – (15 x 40) = 200 In part b): CS = ½ x 20 x 20 = 200, Net surplus = 200 – (15 x 20) = -100. 2. (a) D must be south and west of points Z and R but strictly north of S. b) E is on the vertical axis, south of Z. 3. Yes, this makes sense, the marginal cost of producing another unit of wine is increasing as wine increases. 4. See class notes. 5. Since the MRS > MRT, to move toward the efficient quantitiy of wine and garbage disposal, more wine and less garbage disposal is needed. 2 6. This will be inefficient if marginal control costs are not equalized across all polluters. Otherwise we can reallocate pollution (ie. let high cost polluter pollute one more unit and low cost polluter pollute one less unit) such that aggregate pollution is unchanged although control costs are lower. Chapter 5 2. 2 (a) We are given that "noise costs " are N = 1/d and transportation costs are T = 2d (1$/ Km, each way). Daily total costs are give
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