One of the primary policy tools to address economic inequality is redistributive taxation. Redistributive taxation is, obviously, a very contested field of policy. There is a field of economics that aims to derive "optimal taxes," including optimal redistributive income taxes, inheritance taxes, etc. We will discuss some of the basic ideas of this field in the present chapter. A key reference for our discussion is Saez (2001).
Recall that we discussed the distributional impact of changes in prices on individuals' welfare in the previous chapter. What we will do next is very similar, with changing taxes taking the place of changing prices. Additional complications arise because we need to talk about government revenues, and about how to compare the welfare of different people.
There are many different kinds of taxes in practice, including value-added taxes, income taxes, wealth taxes, inheritance taxes, etc. The framework we discuss applies, in principle, to the analysis of all of these.
There are some general principles in common to the analysis of "optimal taxes" for different kinds of taxes:
Let us now state these principles in a more formal way. Suppose we are changing a tax parameter \(\alpha\), individual welfare for person \(i\) is given by \(v_i\), and government revenues are given by \(g\). A choice of \(\alpha\) is optimal if
$$ \partial_\alpha SWF = 0. $$Adding up all components of social welfare, and using the appropriate welfare weights, we get
The envelope theorem tells us that \(\partial_\alpha v_i\) can be calculated as the effect on the individual's budget constraint, holding behavior constant.
The effect on government revenue \(\partial_\alpha g\) has two components, the direct effect (holding behavior fixed), and the behavioral effect of individuals reacting to the policy change.
Let us go through these terms in a more specific context, where individuals choose their labor supply \(l\) and consumption \(x\) subject to a linear income tax \(t=\alpha + \beta \cdot l \cdot w\), where \(l\) denotes labor supply and \(w\) denotes the wage. Real income taxes are rarely linear, but this assumption allows us to considerably simplify our discussion. Different individuals have different utility functions and different wages. In generalization of the setup we considered in section 1, assume individuals solve
subject to the budget constraint
Note that the choice variables \(x_i\) and \(l_i\) are functions of prices \(p\), wages \(w_i\), and the tax parameters \(\alpha\) and \(\beta\). Realized utility, as before is given by
$$v_i = u_i(x_i, l_i).$$By exactly the same arguments as in CHAPTER 9, we get that the envelope theorem in this setting implies that the equivalent variation of marginally increasing \(\alpha\), and of marginally increasing \(\beta\), is given by
$$ \begin{align*} EV_\alpha &= -1\\ EV_\beta &= - w_i \cdot l_i. \end{align*} $$As an exercise, try to prove this, going step by step through the arguments of CHAPTER 9.
What about government revenues? Effects on these are given by the sum of a mechanical and a behavioral component,
$$ \begin{align*} \partial_\alpha g &= N + \beta \cdot \sum_i w_i \cdot \partial_\alpha l_i \\ \partial_\beta g &= \sum_i w_i \cdot l_i + \beta \cdot \sum_i w_i \cdot \partial_\beta l_i, \end{align*} $$where \(N\) is the number of people in the population. To simplify exposition, we shall assume that there are no effects of changing \(\alpha\) on labor supply, so that \(\partial_\alpha l_i =0\) and thus \(\partial_\alpha g = N\).
Now we have all terms that we need to calculate the marginal effect on social welfare of changing \(\alpha\) and \(\beta\):
$$ \begin{align*} \partial_\alpha SWF &= \sum_i (\lambda - \omega_i)\\ \partial_\beta SWF &= \sum_i (\lambda - \omega_i)\cdot w_i \cdot l_i + \lambda \cdot \beta \cdot \sum_i w_i \cdot \partial_\beta l_i. \end{align*} $$These expressions are obtained by simply adding up everyone's equivalent variation, weighted by \(\omega_i\), and the impact on government revenues, weighted by \(\lambda\).
At the optimal linear income tax, both of these expressions have to equal zero. This implies
$$ \begin{align*} \lambda &= E[ \omega_i]\\ \lambda \cdot \beta \cdot E[ w \cdot \partial_\beta l] &= \text{Cov}(\omega, w\cdot l), \end{align*} $$where \(E\) denotes the average across individuals, and \(\text{Cov}\) the covariance across individuals.
The first equation says that the value of an additional dollar for the government is the same as the average value of an additional dollar across the population. The second equation can be rewritten as
$$ \beta = \frac{\text{Cov}(\omega / \lambda, w\cdot l)}{ E[ w \cdot \partial_\beta l]}. $$This equation says that the marginal tax rate \(\beta\), that is the degree of redistribution,
Let us now turn to nonlinear income taxes, where we go through a simplified exposition of the arguments in Saez (2001). We will only consider how to set the top tax rate. In standard models, welfare weights ("the marginal welfare value of additional income") go to zero as income goes to infinity, relative to the welfare weights of people with average income. Put differently, an additional dollar for a billionaire is considered to be of much smaller value than an additional dollar for a poor person. If that is so, we want to set the top tax rate to maximize revenues, since the assumption implies
$$ \partial_\tau SWF = \lambda \cdot \partial_\tau g, $$where \(\tau\) is the top tax rate. This top tax rate applies to everyone above the income threshold \(\underline{y}\).
Assume, returning to CHAPTER 3, that top incomes follow a Pareto distribution with parameter \(\alpha\): $$ P(Y>y | Y \geq \underline{y}) = \left ( \underline{y} /y\right )^{\alpha}. $$
Assume further that the elasticity of taxable income with respect to the "net of tax" rate \(1-\tau\) is equal to \(\eta\) for those above the income threshold \(\underline{y}\), that is
Government revenues from taxes on top income receivers are equal to
$$ g(\tau) = \tau \cdot N \cdot\left (E[Y| Y \geq \underline{y} ] - \underline{y}\right ), $$where \(N\) is the number of individuals above the threshold. We have all terms that we need to calculate the effect of a change of \(\tau\) on government revenues, which is given by a sum of mechanical and behavioral effects:
Solving the first order condition \(\partial_\tau g = 0\) yields
or, after some algebra
$$ \tau = \frac{1}{1+\alpha \cdot \eta}. $$If we plug in the realistic parameter values \(\alpha = 2\) and \(\eta = .25\), this formula implies an optimal top tax rate of \(1/(1+0.5) = 67\%\). More generally, optimal top tax rates are larger (i) the more unequal the distribution of incomes is (small \(\alpha\)), and (ii) the less responsive taxable incomes are to changes in tax rates (less tax loopholes, better tax enforcement).