$1.(40)$ Farmer John has $\backslash (\backslash$bar$\{$L$\}=24\backslash )$ hours per day to divide between farming cabbages $\backslash ($ L $\backslash )$ and leisure time $\backslash ($ R $\backslash ).$ Suppose that he can earn a wage of $\backslash ($ W$=6\backslash )$ per hour, which he can in turn spend on cabbages $\backslash ($ Q $\backslash ),$ priced at $\backslash ($ P$=1\backslash )$ each. As a consumer, his utility as a function of his consumption of cabbages and hours of leisure time is given by:

$\backslash [$

U$($Q$,$ R$)=\backslash$frac$\{$Q R$^\{2\}\}\{192\}$

$\backslash ]$

$($a$)(10)$ Write John's budget constraint. What is the "price" of leisure time and what is his "income" level?

$($b$)(10)$ Using the table, find John's utility maximizing demands for consumption and leisure, and the hours of labor he will supply.

$($c$)(10)$ Suppose that John's hourly wage may either rise to $\backslash ($ W$=9\backslash )$ or fall to $\backslash ($ W$=3\backslash ).$ Using the table, find how many hours of labor he will supply at each wage level.

$($d$)(10)$ Using the following table, plot John's labor supply curve. What is the price elasticity of labor supply?