In class, we developed a simple model with mug utility and money utility, and we derived implications for the reservation values of buyers, sellers, and choosers in endowment$-$effect experiments. This question asks you to think through several variants of that model. For all parts, assume that the person has wealth $w=$$$15,000,$ and that $($Total Utility$)=($Mug Utility$)+($Money Utility$).$

$($a$)$ Let's begin with the case studied in class: Suppose that money utility is $u_m(m)=m,$ and that mug utility is $u(c,r)=c+v(c-r)$ where

$v(x)=\{\begin{array}{ccc}{}_{\text{m}\text{u}\text{g}}**x& if& x0\\ {}_{\text{m}\text{u}\text{g}}**{}_{\text{m}\text{u}\text{g}}**x& if& x0.\end{array}$

$($i$)$ Derive the reservation values for buyers, sellers, and choosers as a function of $,{}_{\text{m}\text{u}\text{g}},$ and ${}_{\text{m}\text{u}\text{g}}.$

$($ii$)$ Discuss how and why the three types differ. If it helps, consider the special case when $=3,{}_{\text{m}\text{u}\text{g}}=0\mathrm{.}5$ and ${}_{\text{m}\text{u}\text{g}}=5.$

$($b$)$ Now, let's introduce loss aversion over money: Suppose that mug utility is as in part $($a$),$ but now suppose that money utility is $u_m(m,r_m)=m+v_m(m-r_m)$ where $r_m=w$ and

$v_m(x)=\{\begin{array}{ccc}{}_{\text{m}\text{o}\text{n}\text{e}\text{y}}**x& if& x0\\ {}_{\text{m}\text{o}\text{n}\text{e}\text{y}}**{}_{\text{m}\text{o}\text{n}\text{e}\text{y}}**x& if& x0.\end{array}$

$($i$)$ Derive the reservation values for buyers, sellers, and choosers as a function of $,{}_{\text{m}\text{u}\text{g}},{}_{\text{m}\text{u}\text{g}},{}_{\text{m}\text{o}\text{n}\text{e}\text{y}\text{,}}$ and ${}_{\text{m}\text{o}\text{n}\text{e}\text{y}}.$

$($ii$)$ Discuss how and why the three types differ. If it helps, consider the special case when $=3,{}_{\text{m}\text{u}\text{g}}=0\mathrm{.}5,{}_{\text{m}\text{u}\text{g}}=5,{}_{\text{m}\text{o}\text{n}\text{e}\text{y}}=0\mathrm{.}3,$ and ${}_{\text{m}\text{o}\text{n}\text{e}\text{y}}=4.$

$($c$)$ Next, instead of assuming loss aversion over money, let's assume diminishing marginal utility from money: Suppose again that mug utility is as in part $($a$),$ but now suppose that money utility is $u_m(m)=15,000**lnm.$ To simplify things, assume that $=3,{}_{\text{m}\text{u}\text{g}}=0\mathrm{.}5,{}_{\text{m}\text{u}\text{g}}=5.$

$7$

$($i$)$ Derive the reservation values for buyers, sellers, and choosers.

$($ii$)$ Discuss how and why the three types differ.

$($d$)$ Finally, let's keep diminishing marginal utility from money, but now eliminate loss aversion over mugs: Suppose that mug utility is $u(c)=3\mathrm{.}5c,$ and that money utility is $u_m(m)=15,000**lnm.$

$($i$)$ Derive the reservation values for buyers, sellers; and choosers.

$($ii$)$ Discuss how and why the three types differ.