3. Public Goods (15 points) Suppose that Strandsville has decided to add several public parks to its town. Strandsville is rather isolated, however (hence the name), and the town's two residents are the only ones that will be able to use the park system. The two residents, Anne (denoted A) and Ben (denoted B), have identical preferences:

Ui(Xi , G) = ln(Xi) + 2 ln(G); i = A, B

where G is the total number of public parks, where G is total number of public parks, and Xi is person i's private consumption. The total number of parks is determined by the sum of contributions from Anne and Ben, G = GA + GB. Both Anne and Ben earn an income of $24, and pG = pX = $1.

(a) Write down Anne's (private) optimization problem. (1 point)

(b) How many parks would Anne choose to build? How many total parks would be built in the private optimum? (4 points)

(c) Assume that the social planner has additive, equally weighted preferences over both people (in other words, she maximizes U = UA + UB). Set up the social planner's optimization problem. What is the socially optimal number of parks? (4 points)

(d) In two sentences or less: is the result you found in (3c) different than what you found in (3b)? Why? (1 point)

(e) Next, let's think about how the government can actually implement the social optimum. Suppose that, in an effort to improve outdoor activity in Strandsville, the federal government wants to optimally increase the number of parks compared to the private optimum. The government proposes two types of grants. For each grant, calculate the number of parks paid for by each person (Anne and Ben) as well as the total number of parks.

(i) A block grant of $4 to each person (Anne and Ben). (1 point)

(ii) A matching grant to each person (Anne and Ben) that supplements parks spending at a rate of 1 to 1 (so if Anne spends $2 on parks, the government allocates an additional $2 for parks). (2 points)

(f) What is the cost to the government for each grant? Which grant (if any) lets the government reach the social optimum in (3c) in the cheapest way? (1 points)

(g) Which of the grants do Anne and Ben prefer? (1 point)