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: [math]
Supply: [math]
Where [math] is wage, [math] is the quantity supplied or demanded of labour, and [math] are some constants.

If [math] is the tax rate (so if the income tax rate is 25% then r=.25) then:
[math]
Where [math] corresponds to the labour supply curve under the income tax. So a worker will receive [math] but will have to pay [math] to the government in the form of income tax.

If [math] and [math] are the equilibrium wage and quantity of labour supplied with an income tax (ei. quantity demanded equals quantity supplied), and [math], then the total tax is: [math].

To simplify the analysis we can choose the units of the quantity of labour and the wage so that [math].

Find the tax rate as a function of [math]
[math]

Find the revenue maximizing tax rate ([math]):
[math]