The tax analyzed in exercise 3.15 is a proportional wage tax. The U.S. federal income tax, however, is progressive. This means that the average tax rate one pays increases the more wage income is earned.
A: For instance, suppose the government exempts the first $500 of weekly earnings from taxation, then taxes the next $500 at 20% and any earnings beyond that at 40%. Suppose that you again have 60 hours of leisure per week and can earn $25 per hour.
(a) Graph your weekly budget constraint illustrating the tradeoffs between leisure and consumption.
(b) The marginal tax rate is defined as the tax rate you pay for the next dollar you earn, while the average tax rate is defined as your total tax payment divided by your before-tax income. What is your average and marginal tax rate if you choose to work 20 hours per week?
(c) How does your answer change if you work 30 hours? What if you work 40 hours?
(d) On a graph with before-tax weekly income on the horizontal axis and tax rates on the vertical, illustrate how average and marginal tax rates change as income goes up. Will the average tax rate ever reach the top marginal tax rate of 0.4?
e) Some have proposed that the U.S. should switch to a “flat tax”—a tax with one single marginal tax rate. Proponents of this tax reform typically also want some initial portion of income exempt from taxation. The flat tax therefore imposes two different marginal tax rates: a tax rate of zero for income up to some amount x per year, and a single rate t applied to any income earned above x per year. Is such a tax progressive?
B: Suppose more generally that the government does not tax income below x per week; that it taxes income at t for anything above x and below 2x, and it taxes additional income (beyond 2x) at 2t .
Let I denote income per week.
(a) Derive the average tax rate as a function of income and denote that function a(I , t ,x)—where I represents weekly income.
b) Derive the marginal tax rate function m(I , t ,x).