We also assume it is costly to raise taxes, so we introduce a general deadweight cost of taxation related to the tax rate. The greater the taxes, the greater are the costs. Economist Arthur Okun (1975) characterized these in terms of the metaphor of the “leaky bucket.” Redistributing income or assets is a leaky bucket in the sense that when income or assets are taken from someone, as they are transferred to someone else, part of what was taken dissipates, like water falling through the leaks in a bucket. The leaks are due to the costs of administering taxes and creating a bureaucracy and possibly also because of corruption and sheer incompetence. More important, however, is that greater taxes also distort the investment and labor supply incentives of asset holders and create distortions in the production process. For these reasons, the citizens, who form the majority in democracy, determine the level of taxation and redistribution by trading off the benefits from redistribution and the costs from distortions (i.e., the leaks in the bucket).
Economists often discuss these distortions in terms of the “Laffer Curve,” which is the relationship between the tax rate and the amount of tax revenues. The Laffer Curve is shaped like an inverted U. When tax rates are low, increasing the tax rate increases tax revenues. However, as tax rates increase, distortions become greater and eventually tax revenues reach a maximum. After this point, increases in the tax rate actually lead to decreases in tax revenues because the distortions created by taxation are so high.
In our model, these distortions are captured by an aggregate cost, coming out of the government budget constraint of C(τ)nῩ when the tax rate is r. Total income in the economy, nῩ, is included simply as a normalization. We adopt this normalization because we do not want the equilibrium tax rate to depend in an arbitrary way on the scale of the economy. For example, if we vary ny, we do not want equilibrium tax rates to rise simply because the costs of taxation are fixed while the benefits of taxation to voters increase. It seems likely that as nῩ increases, the costs of taxation also increase (e.g., the wages of tax inspectors increase), which is considered in this normalization. We assume that C : [0, 1] → R+, where C(0) = 0 so that there are no costs when there is no taxation; C’ (·) > 0 so that costs are increasing in the level of taxation; C” (·) > 0 so that these costs are strictly convex - that is, they increase faster as tax rates increase (thus ensuring that the second-order condition of the maximization problem is satisfied); and, finally, C’(0) = 0 and C’(1) = 1 so that an interior solution is ensured: the first says that marginal costs are small when the tax rate is low, and the second implies that costs increase rapidly at high levels of taxation. Together with the convexity assumption, both of these are plausible: they emphasize that the disincentive effects of taxation become substantial as tax rates become very high. Think, for example, of the incentives to work and to produce when there is a 100 percent tax rate on your earnings!
From this, it follows that the government budget constraint is as follows:
(4.4)
which uses the definition of average income above (4.3). This equation emphasizes that there are proportional income taxes and equal redistribution of the proceeds, so higher taxes are more redistributive. For example, a higher τ increases the lump-sum transfer and, because rich and poor agents receive the same transfer but pay taxes proportional to their incomes, richer agents bear a greater tax burden.
All individuals in this society maximize their consumption, which is equal to their post-tax income, denoted by(τ) for individual i at tax rate τ. Using the government budget constraint (4.4), we have that, when the tax rate is τ, the indirect utility of individual i and his post-tax income are
(4.5)
The indirect-utility function is conditioned only on one policy variable, τ, because we have eliminated the lump-sum transfer T by using (4.4). We also condition it on y’ because, for the remainder of the book, it is useful to keep this income explicit. Thus, we use the notation V(yi| τ) instead of Vi(τ).
More generally, individuals also make economic choices that depend on the policy variables. In this case, to construct V(yi| τ), we first need to solve for individual i’s optimal economic decisions given the values of the policy variables and then define the induced preferences over policies, given these optimally taken decisions (Persson and Tabellini 2000, pp. 19-21 ).
It is straightforward to derive each individual i’s ideal tax rate from this indirect-utility function. Recall that this is defined as the tax rate τi that maximizes V(yi| τ). Under the assumptions made about C(τ), V(yi | τ) is strictly concave and twice continuously differentiable. This tax rate can then be found simply from an unconstrained maximization problem, so we need to set the derivative of V(yiτ) with respect to τ equal to zero. In other words, τ’ needs to satisfy the first-order condition:
(4.6)
which we have written explicitly emphasizing complementary slackness (i.e., τiτ’ can be at a corner). In the rest of the book, we will not write such conditions out fully as long as this causes no confusion.
The assumption that C” (.) > 0 ensures that the second-order condition for maximization is satisfied and that (4.6) gives a maximum. More explicitly, the second-order condition (which is derived by differentiating (4.6) with respect to τ) is< 0, which is always true, given C”(·) > 0. This second-order condition also implies that V(y1 τ) is a strictly concave function, which is a sufficient condition for it to be single-peaked.
We have written the first-order condition (4.6) in the Kuhn-Tucker form (Blume and Simon 1994, pp. 439-41) to allow for the fact that the preferred tax rate of agent i may be zero. In this case, we have a corner solution and the first-order condition does not hold as an equality. If τi > 0, then (4.6) says that the ideal tax rate of voter i has the property that its marginal cost to individual i is equal to its marginal benefit. The marginal cost is measured by yi, individual i’s own income, because an incremental increase in the tax rate leads to a decline in the individual i’s utility proportional to his income (consumption). The benefit, on the other hand, is (1 — C’(τi))Ῡ, which comes from the fact that with higher taxes, there will be more income redistribution. The term (1 —is the extra income redistribution, net of costs, generated by a small increase in the tax rate.
The conditions in (4.6) imply the intuitive result that rich people prefer lower tax rates and less redistribution than poor people. For a rich person, the ratiois higher than it would be for a poor person. This means that for (4.6) to hold, 1 — C‘(τi) must be higher, so that C’(τi) must be lower. Because C‘(τ’) is an increasing function (by the convexity of C(.)), this implies that the preferred tax rate must be lower. The model actually has a more specific prediction. For a person whose income is the same as the mean, (4.6) becomes 0 = — C‘(τ’), which implies that τ’ = 0 for such a person. Moreover, for any person with income yi >the Kuhn-Tucker conditions imply that there is a corner solution. Hence, people whose income is above average favor no income redistribution at all, whereas people with yifavor a strictly positive tax rate, which is why we use the Kuhn-Tucker formulation.
To derive these comparative static results more formally, let us assume τ‘> 0 and use the implicit function theorem (Blume and Simon 1994, p. 341) to write the optimal tax rate of individual i as a function of his own income, τ(yi). This satisfies (4.6). The implicit function theorem tells us that the derivative of this function, denoted τ’(yi), exists and is given by
Throughout the book, we appeal frequently to the implicit-function theorem to undertake comparative static analysis of the models we study. We undertake two types of comparative statics. First is the type we have just analyzed. Here, we use the conditions for an equilibrium to express a particular endogenous variable, such as the tax rate, as a function of the various exogenous variables or parameters of the model, such as the extent of inequality. Comparative statics then amounts to investigating the effect of changes in exogenous variables or parameters, such as inequality, on the value of the endogenous variable.
(When inequality is higher, does the tax rate increase?) We often use the answers to such questions not just to derive predictions for what would happen within one country if inequality increased but also to compare across countries: Would a country where inequality was higher have a higher tax rate than a country with lower inequality?
We also conduct a different type of comparative statics. In game-theoretic models, various types of behavior may be equilibria in different types of circumstances. For instance, in the repeated prisoner’s dilemma, cooperation forever may be an equilibrium if players value the future sufficiently. We derive conditions under which particular types of behavior - for instance, the creation of democracy - are an equilibrium. We then conduct comparative statics of these conditions to investigate which factors make democracy more or less likely to be created. When we do this, however, we are not directly investigating how a change in an exogenous variable (smoothly) changes the equilibrium value of an endogenous variable. Rather, we examine how changes in exogenous variables influence the “size of the parameter space” for which democracy is created. In essence, democracy can only be created in certain circumstances, and we want to know what makes such circumstances more likely.
We can now think of a game, the (Nash) equilibrium of which will determine the level of redistributive taxation. We can do this in the context of either a direct democracy or a representative democracy, but the most intuitive approach is the one we developed leading up to Proposition 4.2. This result implies that the equilibrium of the game will be for both political parties to propose the ideal point of the median voter, which will be the tax rate chosen in a democracy. The model has this prediction despite the fact that there is political conflict. Poor people would like high taxes and a lot of redistribution; rich people, those with greater than average income, are opposed to any redistribution. How can we aggregate these conflicting preferences? The MVT says that the outcome is the tax rate preferred by the median voter and, for most income distributions, the income of the median person is less than average income (i.e., yM
The comparative statics of this condition follow from the discussion of (4.6). If yM decreases relative tothen the median voter, who becomes poorer relative to the mean, prefers greater tax rates and more redistribution.
4.2 A Two-Group Model of Redistributive Politics
Although many of the results in this book follow from the previous model in which the income of each person is different, a useful simpler model is one in which there are just two income levels. Consider, therefore, a society consisting of two types of individuals: the rich with fixed income yr and the poor with income yp < yr. To economize on notation, total population is normalized to 1; a fraction 1 — δ > ½ of the agents is poor, with income yp; and the remaining fraction δ is rich with income yr. Mean income is denoted by y. Our focus is on distributional conflict, so it is important to parameterize inequality. To do so, we introduce the notation θ as the share of total income accruing to the rich; hence, we have:
(4.7)
Notice that an increase in θ represents an increase in inequality. Of course, we need yP< yr, which requires that:
As in the last subsection, the political system determines a nonnegative income-tax rate τ >_ 0, the proceeds of which are redistributed lump sum to all citizens. We assume that taxation is as costly as before and, from this, it follows that the government budget constraint is:
(4.8)
With a slight abuse of notation, we now use the superscript i to denote social classes as well as individuals so, for most of the discussion, we have i = p or r. Using the government budget constraint (4.8), we have that, when the tax rate is τ, the indirect utility of individual i and his post-tax income are:
(4.9)
As in the last subsection, all agents have single-peaked preferences and, because there are more poor agents than rich agents, the median voter is a poor agent. We can think of the model as constituting a game as in the previous subsection; democratic politics will then lead to the tax rate most preferred by the median voter: here, a poor agent. Notice that because they have the same utility functions and because of the restrictions on the form of tax policy (i.e., taxes and transfers are not person-specific), all poor agents have the same ideal point and vote for the same policy. Here, there is no need for coordination and no sort of collective-action problem (discussed in Chapter 5).
Let this equilibrium tax rate be τp. We can find it by maximizing the post-tax income of a poor agent; that is, by choosing τ to maximize V(yPτ). The first-order condition for maximizing this indirect utility now gives:
(4.10)
because yp < y. Equation (4.10), therefore, implicitly defines the most preferred tax rate of a poor agent and the political equilibrium tax rate. For reasons identical to those in the previous subsection, it is immediate that preferences are single-peaked.
Now, using the definitions in (4.7), we can write the equation for τ ρ in a more convenient form:
(4.11)
where both sides of (4.11) are positive because θ > δ by the fact that the poor have less income than the rich.
Equation (4.11) is useful for comparative statics. Most important, consider an increase in θ, so that a smaller share of income accrues to the poor, or the gap between the rich and the poor widens. Because there is a plus sign in front of θ, the left side of (4.11 ) increases. Therefore, for (4.11 ) to hold, τ ρ must change so that the value of the right side increases as well. Because C” ( . ) > 0, when τ ρ increases, the derivative increases; therefore, for the right side to increase, τ ρ must increase. This establishes that greater inequality (higher θ) induces a higher tax rate, or, written mathematically using the implicit function theorem:
It is also the case that total (net) tax revenues as a proportion of national income increase when inequality rises. Total net tax revenues as a proportion of national income are:
Notice that d (τρ — C(τρ)) /dθ= (1 — C‘(τρ)).dτρ/dθ.We know that higher inequality leads to higher taxes; that is, dτρ/dθ> 0. Moreover, (4.11) implies that C’(rρ) = (θ — δ) /(1 — δ) < 1, so 1 - C’ (τρ) > 0, which then implies that d (τρ — C(τ ρ)) /dθ> 0. In other words, greater inequality leads to a higher proportion of net tax revenues in national income, as argued by Meltzer and Richard (1981) in the context of a slightly different model. In fact, it is straightforward to see that the burden of taxation on the rich is heavier when inequality is greater even if the tax rate is unchanged. Let us first define the burden of taxation as the net redistribution away from the rich at some tax rate τ. This is:
As inequality increases (i.e., θincreases), this burden increases, which simply reflects the fact that with constant average incomes, transfers are constant; and, as inequality increases, a greater fraction of tax revenues are collected from the rich. This observation implies that, even with unchanged tax rates, this burden increases and, therefore, with great inequality, the rich will be typically more opposed to taxation.
Finally, it is useful to conclude this subsection with a brief discussion of efficiency. In this model, taxes are purely redistributive and create distortionary costs as captured by the function C(τ ρ). Whether democracy is efficient depends on the criterion that one applies. If we adopted the Pareto criterion (Green, Mas-Colell, and Whinston 1995, p. 313), the political equilibrium allocation would be Pareto optimal because it is impossible to change the tax policy to make any individual better off without making the median voter worse off — because the democratic tax rate maximizes the utility of the median voter, any other tax rate must lower his utility.
However, in many cases, the Pareto criterion might be thought of as unsatisfactory because it implies that many possible situations cannot be distinguished from an efficiency point of view. An alternative approach is to propose a stronger definition of social welfare, such as a utilitarian social welfare function, and exa
mine if political equilibria coincide with allocations that maximize this function (Green, Mas-Colell, and Whinston 1995, pp. 825-31 ). The democratic political equilibrium here is inefficient compared to the utilitarian social optimum, which would involve no taxation. That taxation creates distortionary costs is a feature of most of the models we discuss throughout this book. In some sense, this is plausible because taxation creates disincentive effects, distorting the allocation of resources.
Its tendency to redistribute income with its potential distortions might suggest that democracy is inefficient relative to a regime that allocates political power to richer agents, who would choose less redistribution. Nevertheless, there are also plausible reasons in general for why greater redistribution might improve the allocation of resources. First, if we allowed people to get utility from public goods that were provided out of tax revenues, it is a standard result in median-voter models that the rich prefer too few public goods whereas the poor prefer too many (Persson and Tabellini 2000). In this case, depending on the shape of the income distribution, the level preferred by the poor may be closer to the social optimum, and democracy, giving political power to the poor, would improve the social efficiency of public goods provision.
Economic Origins of Dictatorship and Democracy Page 16
