In this question you will derive a simple Laffer curve, which relates tax revenue to the tax rate.

Consider two simple labour supply and labour demand curves:
Demand: $w = b - c L$
Supply: $w = a L$
Where $w$ is wage, $L$ is the quantity supplied or demanded of labour, and $a,b,c$ are some constants.

If $r$ is the tax rate (so if the income tax rate is 25% then r=.25) then:
$w_1 = \frac{1}{1-r}aL$
Where $w_1$ corresponds to the labour supply curve under the income tax. So a worker will receive $w_1$ but will have to pay $w_1 - w$ to the government in the form of income tax.

If $w^*_1$ and $L^*$ are the equilibrium wage and quantity of labour supplied with an income tax (ei. quantity demanded equals quantity supplied), and $w^* = w(L^*)$, then the total tax is: $TAX = (w^*_1 - w^*)L^*$.

To simplify the analysis we can choose the units of the quantity of labour and the wage so that $a,b,c=1$.

Find the tax rate as a function of $r$
$TAX =$

Find the revenue maximizing tax rate ($r^*$):
$r^* =$