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
characterized by a skill level, \(a_{p}\), which indicates his or her ability to
earn income. Without loss of generality suppose also that individuals are
ordered by increasing ability. Pretax income itself is determined by skill
level and effort, \(c_{b}\) which may or may not be sensitive to taxation. That
is, \(I_{i}=I\left(a_{b} c_{i}\right) .\) Suppose also that the utility cost of
effort is given by \(\psi(c), \psi^{\prime}>0, \psi^{\prime \prime}<0,
\psi(0)=0 .\) Finally, the government wishes to choose a schedule of income
taxes and transfers, \(T(I),\) which maximizes social welfare subject to a
government budget constraint satisfying \(\sum_{i=1}^{m} T\left(I_{i}\right)=R\)
(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
\(u_{i}=u_{i}\left[I_{i}-T\left(I_{i}\right)-\right.\) \(\psi(c)]\). Show that
maximization of a CES social welfare function requires perfect equality of
income no matter what the precise form of that function. (Note: for some
individuals \(T\left(I_{i}\right)\) 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
\(a_{i}\) rather than on \(I_{i}\)
c. In general show that if income taxation is based on observed income, this
will affect the level of effort individuals undertake
d. Characterization of the optimal tax structure when income is affected by
effort is difficult and often counterintuitive. Diamond \(^{25}\) shows that the
optimal marginal rate schedule may be U-shaped, with the highest rates for
both low- and highincome people. He shows that the optimal top rate marginal
rate is given by
\\[
T^{\prime}\left(I_{\max }\right)=\frac{\left(1+e_{L,
w}\right)\left(1-k_{i}\right)}{2 c_{L, w}+\left(1+e_{L,
w}\right)\left(1-k_{i}\right)}
\\]
where \(k,\left(0 \leq k_{l} \leq 1\right)\) is the top income person's relative
weight in the social welfare function and \(e_{L w}\) is the elasticity of labor
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.
