Two students live together $in$ a rented apartment. After a week, the apartment gets dirty

and they have $to$ decide who cleans $it,$ and how much. Cleaning $upis$ a public good.

The strategy $of$ each student $isto$ choose how many hours $to$ clean, $h_{1}0$ for student $1$

and $h_{2}0$ for student $2(any$ real number greater than $or$ equal than zero $is$ allowed$).$

$If$ a total $ofH=h_1+h_2$ hours $of$ cleaning are produced, then each student will enjoy $a$

benefit $of2\sqrt[\phantom{2}]{H}.$ Cleaning $iscostly.If$ student i exerts $h_i$ hours $of$ cleaning, $heor$ she suffers

a disutility $of{c}_i{h}_i,$ where $c_{i}is$ a constant. Thus, the utility functions $of$ student $1$ and $2$ are

given $by$:

$u_1(h_1,h_2)=2\sqrt[\phantom{2}]{H}-c_1{h}_1$

and

$u_2(h_1,h_2)=2\sqrt[\phantom{2}]{H}-c_2{h}_2$

Suppose that $f(x)=2\sqrt[\phantom{2}]{x}x{x}^{-\frac{1}{2}}{h}_1{h}_2{c}_1=1,c_2=1\mathrm{.}50.In$ particular, student $1$ has the lowest marginal cost $of$ cleaning.

$(Hint$: the derivative $of$ the function $f(x)=2\sqrt[\phantom{2}]{x}$ with respect $toxis{x}^{-\frac{1}{2}}.)$

$(a)[5$ points$]$ Suppose the students make their choices simultaneously. $Isit$ a Nash

equilibrium for each student $to$ choose zero hours $of$ cleaning? $Ifso,$ show $itis$ Nash. $If$ not,

explain why not.

$(b)[10$ points$]$ Suppose the students make their choices simultaneously. $Is$ there a Nash

equilibrium $in$ which the total number $of$ hours spent cleaning $is$ greater than zero? $Ifso,$ find

$it.If$ not, explain why not.

$(c)[10$ points$]$ Suppose now that the players move sequentially: Student $1$ first chooses $h_1,$

and then student $2,$ after observing student $1\text{'}s$ choice, chooses $h_2.$ Find the the backward

induction solution for the case where $c_1=1,c_2=1\mathrm{.}5.$