HW XXXXXXXXXXPublic Goods XXXXXXXXXXThere are ?? members of a community. Each has an endowment ?? of wealth, which can be spent on a public good ????, or a private good ???? = ?? − ????, at the same...

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HW 4 (+24) Public Goods 1. (+12) There are ?? members of a community. Each has an endowment ?? of wealth, which can be spent on a public good ????, or a private good ???? = ?? − ????, at the same price ???? = ???? = 1. Utility is then given by ??(????,??) = ??????, where ?? = ∑ ????????=1 is the total amount of public good produced by the community. a. (+6) If all other citizens produce some level ???? = ?? of public good, what will an individual want to produce in response? Find a level ??∗ of public good such that, if all others produce ??∗, an individual wishes to produce the same amount ??∗ in response. For ??∗ = ????∗, what level ??∗ = ∑ ??(????∗,??∗)????=1 of (utilitarian) social welfare is obtained in that case? b. (+4) Now suppose that production decisions are governed by a social planner, who seeks to maximize total social welfare ???? = ∑ ??(??????,??????)????=1 , by instructing each citizen to contribute (the same number) ???? units of public good, and consume any remaining wealth ?????? = ?? − ????. Find ????. What level ???? of social welfare is obtained in that case? c. (+2) Your answers above may depend on the population size ??. Find and interpret lim ??→∞ ????(??) ??∗(??) to determine how welfare differs in centralized and decentralized systems as the community grows large. 2. (+12) The purpose of this exercise is to develop an intuition for the meaning of Tiebout’s assumption that the optimal club size is small. Suppose that individuals benefit from consuming both a private good ?? and a club good ??. However, the value of the club good diminishes quadratically with the number of other consumers (because of crowding); let an individual club member’s utility be given by the following: ??(??,??,??) = ?? � ?? ??2 � Each individual has wealth ?? with which to purchase ?? and ?? (at prices ???? = ???? = 1) but ?? can only be produced at all if a fixed cost ?? is paid. If individuals form a club to produce ??, they cannot exclude club-members from consuming ??, but can exclude non-club members. Since all individuals are identical, each pays the same club membership fee ?? = ??+?? ?? if they belong to the club, and spend the rest of their wealth on ??. Let ??∗ denote the number of club members and ??∗ denote the public good quantity that an individual prefers. Since individuals are identical, they unanimously prefer this club size and club good production level. a. (+2) Intuitively, how should the optimal number ??∗ of club members that an individual wants in his or her club change if the fixed cost ?? of club good production increases? b. (+8) Now find ??∗ and ??∗ explicitly in terms of ?? and ??. (Hint: substitute the budget constraints into the objective function to eliminate ?? and ??.) c. (+2) How do ??∗ and ??∗ change with ??? Does this match your conjecture above? HW 4 (+24) Public Goods
Answered Same DayJan 18, 2022

Answer To: HW XXXXXXXXXXPublic Goods XXXXXXXXXXThere are ?? members of a community. Each has an endowment ?? of...

Komalavalli answered on Jan 19 2022
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