Question;Public GoodsRio de Janiero would like to improve the citys water supply. The new system will dependon the construction of a number of waste water treatment plants. Assume the city has tworesidents: Gisele and Neymar. Rio will fund the new water system solely from the individualcontributions of these residents. Each of the two residents has a utility function over privategoods (xi) and water quality (Q), of the form:U (xi, Q) = ln(xi5) + ln(Q5)The water quality reects the total capacity of the citys water system, Q, and is the sum ofthe number of treatment plants paid for by Gisele and Neymar: Q = qG + qN. Gisele has anincome of $100 and Neymar has an income of $80. Both the private good and the treatmentplants have a price of $1. Note: For any fraction or decimal answers, you may choose toround or keep as is.(a) How many treatment plants will be built if the government does not intervene? Howmany 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 anddecides to provide 30 treatment plants in addition to what Gisele and Neymar choose toprovide on their own. Gisele and Neymar each pay a $15 lump-sum tax to pay for the 30plants. What is the new total number of waste water treatment plants? How does youranswer compare to (a)? Did Rio achieve the social optimum with this plan? Why or whynot?(d) Suppose instead, starting from the situation in part (a), the World Bank pays for 30water treatment plants. What is the new total number of plants? How many are providedby Gisele? By Neymar? How does this compare to the level of provision in (c)? If there areany similarities or dierences, explain why.(e) Propose a mechanism the government could use to achieve the social optimum.
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