Tax Revenues and the Laffer curve: In this exercise, we will consider how the tax rate on wages relates to the amount of tax revenue collected.
A. As introduced in Section B, the Laffer curve depicts the relationship between the tax rate on the horizontal axis and tax revenues on the vertical. Because people’s decision on how much to work may be affected by the tax rate, deriving this relationship is not as straightforward as many think.
(a) Consider first the extreme case in which leisure and consumption are perfect complements. On a graph with leisure hours on the horizontal and consumption dollars on the vertical, illustrate how increases in the tax on wages affect the consumer’s optimal choice of leisure (and thus labor).
(b) Next, consider the less extreme case where a change in after-tax wages gives rise to substitution and wealth effects that exactly offset one another on the leisure axis. In which of these cases does tax revenue rise faster as the tax rate increases?
(c) On a graph with the tax rate (ranging from 0 to 1) on the horizontal and tax revenues on the vertical, how does this relationship differ for tastes in (a) and (b)?
(d) Now suppose that the substitution effect outweighs the wealth effect on the leisure axis as after-tax wages change. Illustrate this and determine how it changes the relationship between tax rates and tax revenue.
(e) Laffer suggested (and most economists agree) that the curve relating tax revenue (on the vertical axis) to tax rates (on the horizontal) is initially upward sloping but eventually slopes down —reaching the horizontal axis by the time the tax rate goes to 1. Which of the preferences we described in this problem can give rise to this shape?
(f) True or False: If leisure is a normal good, the Laffer curve can have an inverted U-shape only if leisure and consumption are (at least at some point) sufficiently substitutable such that the substitution effect (on leisure) outweighs the wealth effect (on leisure).
B. we derived a Laffer curve for the case where tastes were quasilinear in leisure. Now consider the case where tastes are Cobb-Douglas—taking the form u(c,ℓ) = cαℓ(1−α). Assume that a worker has 60 hours of weekly leisure endowment that he can sell in the labor market for wage w.
(a) Suppose the worker’s wages are taxed at a rate t. Derive his optimal leisure choice.
(b) For someone with these tastes, does the Laffer curve take the inverted U-shape. Why or why not? Which of the cases described in A does this represent?
(c) Now consider the more general CES function (αc−ρ + (1−α) ℓ−ρ) −1/ρ. Again derive the optimal leisure consumption.
(d) Does your answer simplify to what you would expect when ρ = 0?
(e) Determine the range of values of ρ such that leisure consumption increases with t .
(f) When ρ falls in the range you have just derived, what happens to leisure consumption as t approaches 1? What does this imply for the shape of the Laffer curve?
(g) Suppose α = 0.25, w = 20 and ρ = −0.5. Calculate the amount of leisure a worker would choose as a function of t. Then derive an expression for this worker’s Laffer curve and graph it.