Consider two consumers with utility functions given by u1(C1, G) = C1 + ln(G) and u2(C2, G) = 2C2 + 5 ln(G) where G is the level of a public good and Ci is private consumption

Question 1.
1. Consider two consumers with utility functions given by u1(C1, G) = C1 +
ln(G) and u2(C2, G) = 2C2 + 5 ln(G) where G is the level of a public good
and Ci
is private consumption. Assume that prices of private consumption
and the public good are both 1, and that each consumer’s income is 10.
(a) What would be the socially efficient level of provision of the public
good?
(b) Suppose the consumers rely on a private solution, so that each contributes gi where g1 + g2 = G. Find consumer 1’s best response
function g1(g2), which is consumer 1’s optimal level of g1 as a function of consumer 2’s contribution g2. Similarly, find consumer 2’s
best response function g2(g1). Show these functions on a graph with
axes g1 and g2 (hint: don’t omit the portions of the best response
functions for which g1 = 0 and g2 = 0).
(c) At the one point where the best response functions intersect, how
much G is provided? Who provides it? Explain briefly the intuition
for why the private market under-provides G.
(d) Government decides to provide 0.5 unit of the public good. Now
consumers’ income remains the same, and G = g1 + g2 + 0.5. What
would be the socially efficient level of provision of the public good in
this case? What would the total provision be if the consumers rely
on a private solution? Does the government’s investment move G in
the private case closer to the socially efficient level?
The questions below are adapted from former midterms – hence the point values.
Question 2. (80 points). This question asks you to consider two problems of
public good provision, in two scenarios.
I. Each member of a society composed of n ≥ 2 individuals must decide whether
or not to contribute to a public good G. Each individual is endowed with 1 unit
of an indivisible good that can either be consumed individually or contributed
in its entirety to G.
If individual i consumes her unit, she derives utility Ui = θi where θi
is
known only to individual i. It is commonly known, however, that θi
is equally
likely to assume any value in the interval [1, 6], for all i, and that each realization
of θi
is independent across individuals.The public good G equals the sum of the
1
contributions: if m ≤ n individuals contribute, G = m. A level G of the public
good yields to every individual utility 4G. Hence individual i’s utility is given
by:
Ui = θiI + 4G
where I is an indicator function that equals 1 if i has not contributed her
endowment to the public good and 0 if she has.
a. (10 points). Consider individual i, deciding whether or not to contribute her
unit. (1) Does her utility maximizing choice depend on her expectation of
m? (2) On θi? What is her choice? (You can suppose that if indifferent,
i contributes.)
b. (5 points). Are individual choices efficient (i.e. do they maximize the sum
of utilities in society)? Why?
c. (10 points). In these confusing times, it is easy to forget that making decisions through voting can be desirable. Suppose that a tax τ ∈ [0, 1] on
each individual endowment is being debated. The tax proceeds will all be
transformed into the public good. Thus:
Ui = (1 − τ )θi + 4G
where G = nτ . (1) What is voter i’s preferred level of τ? (2) Is there
disagreement in the society? (3) Suppose the decision is made by majority
voting. Is the resulting level of τ efficient?
d. (5 points). Why do the answers to points a. and c. above differ–i.e. why
is the ideal tax rate different from the ideal individual contribution? Do
you see a general lesson about the power of voting?
II. Consider now a modified problem. As before, there is a society of n ≥
2 individuals, each endowed with a good that yields utility θi
if individually
consumed, where θi
is private information. As before, private endowments can
be transformed into a public good G, but in this second problem the public
good suffers from congestion: if its level is G, then the utility every society
member receives from it is G/n. No voluntary individual contributions are
requested, but a law is being debated. If it is imposed, all private endowments
will be expropriated and transformed into the public good: G = n. If not, each
individual consumes her endowment individually (the status quo). Thus:
Ui =
(
θi
if the law is not imposed
4G/n with G = n if the law is imposed
e. (5 points). Under what condition, would such a law maximize the sum of
utilities in this society?
f. (5 points). Individuals are thus asked to announce their private θi values.
The law will be applied if it maximizes the sum of utilities, taking the
announcements as truthful. Do you expect them to be truthful? Why?