Suppose it's still the case that the consumer's utility function is now log$-$linear:

U $($c$,$ l$)=$ log$($c$)+$ $(1$ l$)$

where c is consumption, l is labor supply and is a relative importance of leisure.

Now suppose that a labor income tax reduces the household's take$-$home income:

c $=(1$ $)$wl $+$ T

$2$

where is the labor income tax rate $(0<$ $<1)$ and w is still the wage rate.

You may also assume that $0<$ l $<1,$ such that the consumer works a positive

number of hours, but not all hours in the day.

$1.$ Solve the optimization problem and describe the optimal consumption

$($c$)$ that the household want and labor supply $($l$)$ as a function of $,$ $,$ w

and T$.$ Call your solutions c$($$,$ T $)$ and l$($$,$ T $).$ We'll imagine all of the

other parameters are xed, except for the ones that pertain to taxes and

transfers.