Exercise 22.31. Consider the following one-period economy populated by a mass 1 of agents. A fraction λ of these agents are capitalists, each owning capital k. The remainder have only human capital, with human capital distribution μ(h). Output is produced in competitive markets, with aggregate production function Y = K1−αHα, where uppercase letters denote total supplies. Assume that factor markets are competitive and denote the market clearing rental price of capital by r and that of human capital by w.

(1) Suppose that agents vote over a linear income tax, τ . Because of tax distortions, total tax revenue is T ax = (τ − v (τ )) µ λrk + (1 − λ) w Z hdμ (h) ¶ where v (τ ) is strictly increasing and convex, with v (0) = v0 (0) = 0 and v0 (1) = ∞ (why are these conditions useful?). Tax revenues are redistributed lump sum. Find the ideal tax rate for each agent. Find conditions under which preferences are single peaked, and determine the equilibrium tax rate. How does the equilibrium tax rate change when k increases? How does it change when λ increases? Explain the intuition for these results. (2) Suppose now that agents vote over capital and labor income taxes, τ k and τh, with corresponding costs v (τ k) and v (τh), so that tax revenues are T ax = (τ k − v (τ k)) λrk + (τh − v (τh)) (1 − λ) w Z hdμ (h) Determine the most preferred tax rates for each agent. Suppose that λ < 1/2. Does a voting equilibrium exist? Explain. How does it change when λ increases? Explain why this would be different from the case with only one tax instrument? (3) In this model with two taxes, now suppose that agents first vote over the capital income tax, and then taking the capital income tax as given, they vote on the labor income tax. Does a voting equilibrium exist? Explain. If an equilibrium exists, how does the equilibrium tax rate change when k increases? How does it change when λ increases?