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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 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 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?
. (+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 XXXXXXXXXXYour 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.
XXXXXXXXXXThe 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
e 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 ??,
ut 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
est 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?
. (+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 XXXXXXXXXXPublic Goods
Answered Same Day Jan 18, 2022

Solution

Komalavalli answered on Jan 19 2022
121 Votes
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