$($a$)$ Suppose that Johnny is an expected utility maximizer with $u(x)=-e^{-0\mathrm{.}001x},$ and has initial wealth is $$75,000.$ Derive how Johnny feels about the following bets:

$($i$)$ Johnny will accept $(-100,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>????$

$($ii$)$ Johnny will accept $(-200,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>????$

$($iii$)$ Johnny will accept $(-500,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>????$

$($iv$)$ Johnny will accept $(-750,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>????$

$($b$)$ Like Johnny, Tommy has initial wealth $$75,000.$ But unlike Johnny, Tommy evaluates gambles according to prospect theory with $(p)=p$ and

$v(x)=\{\begin{array}{ccc}x& if& x0\\ x& if& x0\end{array}$

If we observe that Tommy accepts $(-100,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>210,$ what can we conclude about Tommy's $?$

$($c$)$ Suppose that Tommy has the $$ that you found in part $($b$),$ and derive how he feels about the following bets:

$($i$)$ Tommy will accept $(-200,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>????$

$($ii$)$ Tommy will accept $(-500,\frac{1}{2}$;$x,\frac{1}{2})$ if and only if $x>????$