1. There are three individuals and each has an endowment of 40 units of the private good. Let U1 = α1√x + y1, U2 = α2√x + y2, U3 = α3 ln (x + 1) + y3. The production of 1 unit of X requires δ units of Y as input. Express the Samuelson condition as simply as you can, then further simplify the condition for the case δ = α3.
2. Utility function of agent i is Ui = αix + yi for i = 1, 2, · · · , n. The production of x units of the public good requires 1/2 x2 units of the private good as input. An allocation is determined by having each i report a benefit parameter βi, producing β1 + β2 + · · · + βn units of the public good and collecting a total of 1/2(β1 + β2 + · · · + βn)2 units of the private goods from households.(Note that we use αi to denote individual i's true benefit parameter, with βi representing the reported benefit parameter.) Suppose each individual i's consumption of the private good is reduced by 1/2βi(β1 +β2 +· · ·+ βn) units. Suppose that α2 + α3 +· · ·+ αn = 2. What value of β1, person 1's reported benefit parameter, maximizes U1, given 0 < α1 < 1 and the fact that the benefit parameters reported by the others sum to 2?
3. Consider the case of three individuals, Soren, Rosie and Edie, and two public projects F and G. Table below gives the benefit derived by each person from each project.
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Soren
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Rosie
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Edie
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Project F
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10
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20
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30
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Project G
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15
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10
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40
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(a) What project would be undertaken if the pivotal mechanism were employed and individuals reported their benefits truthfully? Calculate surtax for each individual.
(b) Show the none of the three individuals has an incentive to misrepresent his or her benefit in this setting. What does it mean to say that truthful revelation of preference is a dominant strategy in this case?
(c) What is the size of the government budget surplus associated with your answer?