2. Public Good and Contribution. Consider the following scheme for contributing to the provision of a public good, say the opening of a community library for a community of n families. Each family i decides a contribution gi towards a public good, which represents a number of books. The total quantity of public good is equal to G=8, is then the size of the library. Each family incurs an addition cost of y(g)/2 for contributing g,, equivalent to the cost of transporting the books to the collection point. Therefore, the utility function of family i is (81)² (81)² Ugig)=G-Y- 8-1- 2 2 (a' (b) (c) Consider the socially optimal solution. Define the social wel- fare V as the sum of the all the families' utility functions V=U(g;.g-;). Maximise V with respect to g, for i = 1,...,n. Using the first order condi- tions, determine the socially optimal level of contribution per family. gso.i for i=1,...,n and the socially optimal size of the library: Go How do you know that the solutions for go, are a maximum? (Hint: what conditions are needed for the solution of an optimisation to be a maximum) Consider now the simultaneous contribution to the public good by the families in the community. Find the Nash equilibrium in the con- tribution level, by having each family maximise their utility holding the contribution of others g, constant. What is the quantity contributed g? (d) Compare the contributions in the Nash equilibrium to the so- cial optimum quantity go. What intuition can you use to explain the difference? In particular, compute the ratio g;/8so. How does this vary with the number of families n? In which communities is the problem of underprovision of public goods more serious?
2. Public Good and Contribution. Consider the following scheme for contributing to the provision of a public good, say the opening of a community library for a community of n families. Each family i decides a contribution gi towards a public good, which represents a number of books. The total quantity of public good is equal to G=8, is then the size of the library. Each family incurs an addition cost of y(g)/2 for contributing g,, equivalent to the cost of transporting the books to the collection point. Therefore, the utility function of family i is (81)² (81)² Ugig)=G-Y- 8-1- 2 2 (a' (b) (c) Consider the socially optimal solution. Define the social wel- fare V as the sum of the all the families' utility functions V=U(g;.g-;). Maximise V with respect to g, for i = 1,...,n. Using the first order condi- tions, determine the socially optimal level of contribution per family. gso.i for i=1,...,n and the socially optimal size of the library: Go How do you know that the solutions for go, are a maximum? (Hint: what conditions are needed for the solution of an optimisation to be a maximum) Consider now the simultaneous contribution to the public good by the families in the community. Find the Nash equilibrium in the con- tribution level, by having each family maximise their utility holding the contribution of others g, constant. What is the quantity contributed g? (d) Compare the contributions in the Nash equilibrium to the so- cial optimum quantity go. What intuition can you use to explain the difference? In particular, compute the ratio g;/8so. How does this vary with the number of families n? In which communities is the problem of underprovision of public goods more serious?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:2. Public Good and Contribution.
Consider the following scheme for
contributing to the provision of a public good, say the opening of a community
library for a community of n families. Each family i decides a contribution g;
towards a public good, which represents a number of books. The total quantity
of public good is equal to G =E8; is then the size of the library. Each family
incurs an addition cost of r(g,)*/2 for contributing g;, equivalent to the cost of
transporting the books to the collection point. Therefore, the utility function of
family i is
(g.)
U{81»8-1)= G-y-
2
j=1
(g.)
2
(a'
Consider the socially optimal solution. Define the social wel-
fare V as the sum of the all the families' utility functions V = E U,(8j,g=j).
Maximise V with respect to g, for i = 1,...,n. Using the first order condi-
tions, determine the socially optimal level of contribution per family. goi
for i = 1, ..,n and the socially optimal size of the library: Go
(b)
How do you know that the solutions for gio are a maximum?
(Hint: what conditions are needed for the solution of an optimisation to
be a maximum)
| Consider now the simultaneous contribution to the public good
(c)
by the families in the community. Find the Nash equilibrium in the con-
tribution level, by having each family maximise their utility holding the
contribution of others g*, constant. What is the quantity contributed g;?
(d) -
cial optimum quantity gšo What intuition can you use to explain the
difference? In particular, compute the ratio g/g3o How does this vary
with the number of families n? In which communities is the problem of
underprovision of public goods more serious?
Compare the contributions in the Nash equilibrium to the so-
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