Consider a consumer whose preferences for consumption $($C$)$ and leisure

$($R$)$ are given by: U$($C$,$R$)=$alpha ln C$+(1-$alpha$)$ln R$.$ Suppose the consumer has T units of time that

can be allocated to labor $($L$)$ and leisure $($R$).$ Let p denote the price of one unit of consumption

and w denote the nominal wage rate.

$($a$)$ If M denotes consumer's non$-$labor income, write out the consumer's budget constraint.

$($b$)$ Write out the Lagragian associated with this problem.

$($c$)$ Derive the consumer's labor supply function.

$($d$)$ For what value of M will the consumer supply positive hours of labor? How does con$-$

sumer's labor supply decision vary with non$-$labor income, M$?$

$($e$)$ Now suppose alpha$=0\mathrm{.}5,$w$=$$$10,$T$=24.$ There are two types of workers: one type has

no non$-$labor income and the other type has M$=$$$100.$ What are their optimal labor

supply decisions based on your answer in part c$?$

$($f$)$ Suppose the firm wants everyone to work at least $8$ hours and it has a policy that it can

use to induce workers who are working less than $8$ hours to work $8$ hours. The firm will

subsidize every worker's wage at a $t per hour. What is the minimum subsidy the firm

needs to offer for this policy? What is the cost of the policy to the firm per worker type

if it needs to pay the same wage to both types of workers?