03 Jun Question Public Goods Rio de Janiero would like to improve the citys wa
Question
Public Goods
Rio de Janiero would like to improve the citys water supply. The new system will depend
on the construction of a number of waste water treatment plants. Assume the city has two
residents: Gisele and Neymar. Rio will fund the new water system solely from the individual
contributions of these residents. Each of the two residents has a utility function over private
goods (xi ) and water quality (Q), of the form:
U (xi ; Q) = ln(xi
5) + ln(Q
5)
The water quality reects the total capacity of the citys water system, Q, and is the sum of
the number of treatment plants paid for by Gisele and Neymar: Q = qG + qN . Gisele has an
income of $100 and Neymar has an income of $80. Both the private good and the treatment
plants have a price of $1. Note: For any fraction or decimal answers, you may choose to
round or keep as is.
(a) How many treatment plants will be built if the government does not intervene? How
many are paid for by Gisele? By Neymar?
(b) What is the socially optimal number of treatment plants? If your answer diers from
(a), explain why.
(c) Suppose the city of Rio de Janiero is not happy with the private equilibrium and
decides to provide 30 treatment plants in addition to what Gisele and Neymar choose to
provide on their own. Gisele and Neymar each pay a $15 lump-sum tax to pay for the 30
plants. What is the new total number of waste water treatment plants? How does your
answer compare to (a)? Did Rio achieve the social optimum with this plan? Why or why
not?
(d) Suppose instead, starting from the situation in part (a), the World Bank pays for 30
water treatment plants. What is the new total number of plants? How many are provided
by Gisele? By Neymar? How does this compare to the level of provision in (c)? If there are
any similarities or dierences, explain why.
(e) Propose a mechanism the government could use to achieve the social optimum.
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