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This is a question that came in one of the previous year's entrance test to a master's program in a reputed institution in my country. I am also attaching how I tackled the problem in each part of the question. I am stuck in part (c) and part (d). I didn't really take public economics as an undergrad hence I am facing a bit of tough spot here. Also the question seems weird as there is no private good.

Consider a city with $n$ people in which each person gets a welfare benefit of $U(G)$ from having $G$ hectares of public parks in the city. Suppose the cost of providing each hectare of parks is$\ 1,000,000 \space $ in local currency units. The net utility of person $i$ who contributes $c$ million for park provision is $U(G) - c$ when there are $G$ hectares of public parks in the city. $U(.)$ is strictly increasing and strictly concave, and $U'(G)$ approaches $\infty$ as $G$ approaches zero from the right.

(a) If a utilitarian social planner decides the level of park provision $G^*$ with the cost being financed by equal per capita taxes, what condition must $G^*$ satisfy [5 points]

My answer: Social Planner's Problem $$\max_{G}\space n[U(G)-c]$$ First Order Necessary condition for interior optimum $G^*$ is $U'(G^*)=0$. That is the marginal benefit by increasing another hectare of public park provision is zero for the representative individual. Also, since $U''(G)<0$ $\space \space (\because \space U(.) \space is \space concave)$ the above optimal solution is indeed a global maximum.

(b) Suppose each person voluntarily decides how much to contribute to a fund that will be used for park provision, taking all other persons' contributions as given. What condition must the resulting level of park provision $G_s$ satisfy? How does it compare to the optimum of the social planner? [5 points]

My answer: $$G = \sum c_i $$ Each agent solves

$$\max_{c_i} \space \space U\left(\sum _{j\ne i}c_j +c_i \right)-c_i $$

First order necessary condition for interior optimum is $$U'\left(\sum _{j\ne i}c_j +c_i \right)-1 =0$$ $$\implies U'\left(\sum _{j\ne i}c_j +c_i \right) = 1 \implies U'(G_s)=1$$

Since $U'(.)$ is decreasing $(\because U''(.)<0)$ hence $$U'(G^*)=0<1=U'(G_s) \implies G^*>G_s$$

Now here is the problem. Parts (c) and (d).

(c) Now suppose that the level of park provision is decided by voting. Each person votes for their own most preferred level of provision knowing that the cost will be equally shared among the residents of the city. The median voter's choice will be implemented. What condition must the resulting level of provision $G_m$ satisfy? How does it compare to the optimum $G^*$ of the social planner? [5 points]

My approach: We can always relabel the city residents in such a way that the preferred public provision level for resident $i$ , $G_i$ satisfies $G_1 \le G_2 \le ...\le G_i\le G_{i+1}\le ...G_{n}$

Then there are two cases: If $n$ is odd, $G_m = G_{\frac{n+1}{2}}$ will be implemented. If $n$ is even, $G_m = \frac{1}{2}\left(G_{\frac{n}{2}}+G_{\frac{n+1}{2}}\right)$ will be implemented.

I can't really compare this with $G^*$. I feel some kind of distribution should have been provided. Either a probability mass function or cdf.

(d) Now suppose instead that one-third of the voters each get a welfare benefit of $3U(G)$ from $G$ hectares of parks, while the remaining two-thirds each get a welfare benefit of $U(G)$ as before. Costs are as before. Now compare the social planners' solution with the one obtained under median voting. [5 points]

Again same problem as part (c).

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  • $\begingroup$ Your answer to part (a) is not correct. To provide $G$ hectares costs $G$ million. Each person would pay $G/n$ in tax. The objective function is thus $n[U(G)-G/n]=nU(G)-G$. (The assumption of equal per capita taxes is not actually needed given that the marginal disutility of expenditure is constant.) The first-order condition is $nU'(G)=1$ or $U'(G)=1/n$. Your answer would only be correct in the case where the public good is costless and there exists a value of $G$ such that $U'(G)=0$ (such a value of $G$ cannot exist given the assumptions on $U$). $\endgroup$
    – smcc
    Commented May 13 at 18:58

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For part (c), each person's net utility if the provision is $G$ would be

$$U(G)-\frac{G}{n}$$

The first-order condition is $U'(G)=1/n$, the same as for the social planner in (a). Thus everyone will vote for $G^*$ and therefore $G_m=G^*$.

Hopefully you can answer part (d) now.

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