The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are m individuals in the economy and that each individual is characterised by a skill level, ai, which indicates his or her ability to earn income. Without loss of generality suppose also that individuals are ordered by increasing ability. Pre-tax income itself is determined by skill level and effort, ci, which may or may not be sensitive to taxation. That is, . Suppose also that the utility cost of effort is given by ψ(c), ψ′> 0, ψ′′ <>ψ(0) = 0. Finally, the government wishes to choose a schedule of income taxes and transfers, T(I), which maximises social welfare subject to a government budget constraint satisfying
(where R is the amount needed to finance public goods).
a. Suppose that each individual’s income is unaffected by effort and that each person’s utility is given by
Show that maximisation of a CES social welfare function requires perfect equality of income no matter what the precise form of that function. (Note: For some individuals T () may be negative.)
b. Suppose now that individuals’ incomes are affected by effort. Show that the results of part (a) still hold if the government-based income taxation on ai
rather than on .
c. In general show that if income taxation is based on observed income, this will affect the level of effort individuals undertake.
d. Characterisation of the optimal tax structure when income is affected by effort is difficult and often counterintuitive. Diamond shows that the optimal marginal rate schedule may be U-shaped, with the highest rates for both low- and high-income people. He shows that the optimal top marginal rate is given by
where ki
(0 ≤ki≤ 1) is the top income person’s relative weight in the social welfare function and eL,w
is the elasticity of labour supply with respect to the after-tax wage rate. Try a few simulations of possible values for these two parameters, and describe what the top marginal rate should be. Give an intuitive discussion of these results.