$6.$ Recall our rational criminal example using Table $13-3.$

TABLE $13-3$ The Expected Utility of Crime

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline & Baseline & $\backslash$begin$\{$tabular$\}\{$l$\}$

Higher $\backslash \backslash$

Probability of Prison

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

Longer $\backslash \backslash$

Prison $\backslash \backslash$

Term

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

Less $\backslash \backslash$

Loot

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

Higher $\backslash \backslash$

Income

$\backslash$end$\{$tabular$\}$ & Lower Probability of Prison $\backslash \backslash$

$\backslash$hline Lawful income $(\backslash$$$)$ & $100$ & $100$ & $100$ & $100$ & $400$ & $100\backslash \backslash$

$\backslash$hline Loot $(\backslash$$$)$ & $44$ & $44$ & $44$ & $21$ & $44$ & $44\backslash \backslash$

$\backslash$hline Probability of prison & $0\mathrm{.}5$ & $0\mathrm{.}75$ & $0\mathrm{.}5$ & $0\mathrm{.}5$ & $0\mathrm{.}5$ & $0\mathrm{.}25\backslash \backslash$

$\backslash$hline Prison time & $0\mathrm{.}36$ & $0\mathrm{.}36$ & $0\mathrm{.}51$ & $0\mathrm{.}36$ & $0\mathrm{.}36$ & $0\mathrm{.}36\backslash \backslash$

$\backslash$hline Lawful utility $\backslash (=(\backslash$text $\{$ lawful income $\})^\{1/2\}\backslash )$ & $10$ & $10$ & $10$ & $10$ & $20$ & $10\backslash \backslash$

$\backslash$hline $\backslash$multicolumn$\{7\}\{|$l$|\}\{$Utility from successful crime$\}\backslash \backslash$

$\backslash$hline Net income $\backslash (=\backslash )$ Legal income $+$ Loot & $144$ & $144$ & $144$ & $121$ & $444$ & $144\backslash \backslash$

$\backslash$hline Utility $\backslash (=(\backslash$text $\{$ Net income $\})^\{1/2\}\backslash )$ & $12$ & $12$ & $12$ & $11$ & $21$ & $12\backslash \backslash$

$\backslash$hline $\backslash$multicolumn$\{7\}\{|$l$|\}\{$Utility from failed crime$\}\backslash \backslash$

$\backslash$hline Prison cost $\backslash (=\backslash )$ prison time $\backslash (\backslash$cdot $\backslash )$ legal income & $36$ & $36$ & $51$ & $36$ & $144$ & $36\backslash \backslash$

$\backslash$hline Net income $\backslash (=\backslash )$ Legal income $-$ prison cost & $64$ & $64$ & $49$ & $64$ & $256$ & $64\backslash \backslash$

$\backslash$hline Utility $\backslash (=(\backslash$text $\{$ Net income $\})^\{1/2\}\backslash )$ & $8$ & $8$ & $7$ & $8$ & $16$ & $8\backslash \backslash$

$\backslash$hline Expected utility from crime $($utils$)$ & $10$ & $9$ & $9\mathrm{.}5$ & $9\mathrm{.}5$ & $18\mathrm{.}5$ & $11\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

Consider the higher income example. $($Income $=\backslash$$$400)$

A$)$ Calculate the amount of loot needed for Expected Utility from Crime to equal the Lawful Utility. Show your work.

B$)$ Now keep loot at $44,$ but calculate the amount of prison time needed to make the Expected Utility from Crime $=$ Lawful Utility. Show your work.