$(20pt)$ The cities $of$ Rize and Artvin are trying $to$ decide how much they want $to$

contribute $to$ a joint fire department $to$ provide fire protection services for both cities.

The two cities have identical preferences: $u(x_i,f)=log{x}_i+3logf,$ where $x_{i}is$ city

$i\text{'}s$ private consumption, and $f=f_R+f_{A}is$ the sum $of$ total contributions $to$ the fire

department. Rize contributes $f_R$ and Artvin contributes $f_{A}to$ the joint fire department.

Assume that $p_x=p_f=1$ and that $(for$ simplicity$)$ each city has $a100TL$ budget for

both goods.

$(a)$ What are the two characteristics $of$ a pure public good?

$(b)Do$ fire departments $\frac{?}{?}$ fire protection services fit these characteristics $of$ a pure

public good? Explain why $or$ why not.

$(c)$ Determine the private equilibrium level $of$ contributions for each city, $f_R$ and $f_A,$

$as$ well $as$ the total level $of$ fire protection services, $f_{\text{e}\text{q}\text{m}}.$

$(d)$ Determine the Pareto efficient level for the joint fire department, $f_{\text{e}\text{f}\text{f}}.$

$(e)$ Why does the socially optimal quantity $of$ fire services $in(d)$ differ from the level

$in(c)?$

$(f)$ Ministry $of$ finance $is$ aware $of$ the difference $in$ private contributions versus the

Pareto efficient level $of$ fire protection for the two cities, and $he$ proposes $a(f_{\text{e}\text{f}\text{f}}-f_{\text{e}\text{q}\text{m}})$

$TL$ block grant for each city $to$ spend $on$ fire protection services. Will the new market

equilibrium level $of$ fire protection $be$ the same $as$ efficient level?

$(g)$ Now instead $of$ a block grant, a matching grant $of25\%of$ any contributions $by$

Rize and Artvin will $be$ added $tof.$ Determine the private contribution levels $f_R$ and

$f_A$ for each city under this program. $Is$ the optimum level $in(d)$ achieved?