5.2 Optimal Nonlinear Schedule

5.3 Optimal Profile of Transfers

6.1 Tagging

We have assumed that depends only on earnings . In reality, the government can observe many other characteristics (denoted by vector ) also correlated with ability (and hence social welfare weights), such as gender, race, age, disability, family structure, height, etc. Hence, the government could set and use the characteristic as a “tag” in the tax system. There are two noteworthy theoretical results.

First, if characteristic is immutable then there should be full redistribution across groups with different . This can be seen as follows. Suppose is a binary 0–1 variable. If the average social marginal welfare weight for group 1 is higher than for group 0, a lump sum tax on group 0 funding a lump sum transfer on group 1 will increase total social welfare.

Second, if characteristic is not immutable, i.e., it can be manipulated through cheating,⁹¹ then it is still desirable to make taxes depend on (in addition to ). At the optimum however, the redistribution across the groups will not be complete. To see this, suppose again that is a binary 0–1 variable and that we start from a pure income tax . As is correlated with ability, the average social marginal welfare weight for group 1 is different from the one for group 0. Let us assume it is higher. In that case, a small lump sum transfer from group 0 to group 1 increases social welfare, absent any behavioral response. As is no longer immutable, this small transfer might induce some individuals to switch from group to group . However, because we start from a unified tax system, at the margin those who switch do not create any first order fiscal cost (nor any welfare cost through the standard envelope theorem argument).⁹²

Those points on tagging have been well known in the literature for decades following the analysis of Akerlof (1978) and Nichols and Zeckhauser (1982) for tagging disadvantaged groups for welfare benefits. It has received recent attention in Mankiw and Weinzierl (2010) and Weinzierl (2011) who use the examples of height and age respectively to argue that the standard utilitarian maximization framework fails to incorporate important elements of real tax policy design.

Indeed, in reality, actual tax systems depend on a very limited set of characteristics besides income. Those characteristics are primarily family structure (in particular the number of dependent children), disability status (for permanent and temporary disability programs). Hence, characteristics used reflect direct “need” (for example, the size of the household relative to income), or direct “ability-to-earn” (as is the case with disability status). To the best of our knowledge, the case for using indirect tags correlated with ability in the tax or transfer system has never been made in practice in the policy debate, implying that society does have a strong aversion for using indirect tags. We come back to this issue in Section 7 when we discuss the limits of utilitarianism.

6.2 Supplementary Commodity Taxation

The government can also implement differentiated commodity taxation in addition to nonlinear income taxes and transfers. The usual hypothesis is that commodity taxes have to be linear because of retrading (see e.g., Guesnerie, 1995, Chapter 1). The most common form of commodity taxation, value added taxes and general sales taxes, do display some variation in rates across goods, with exemptions for specific goods, such as food or housing. Such exemptions are in general justified on redistributive grounds. The government also imposes additional taxes on specific goods such as gasoline, tobacco, alcohol, airplane tickets, or motor vehicles.⁹³ Here, we want to analyze whether it is desirable to supplement the optimal nonlinear labor income tax with differentiated linear commodity taxation.

Consider a model with consumption goods with pre-tax prices . Individual derives utility from the consumption goods and earnings supply according to a utility function . The question we want to address is whether the government can increase social welfare using differentiated commodity taxation in addition to nonlinear optimal income tax on earnings . Naturally, adding fiscal tools cannot reduce social welfare. However, Atkinson and Stiglitz (1976) demonstrated the following.

Atkinson-Stiglitz Theorem. Commodity taxes cannot increase social welfare if utility functions are weakly separable in consumption goods vs. leisure and the subutility of consumption goods is the same across individuals, i.e., with the subutility function homogenous across individuals.

The original proof by Atkinson and Stiglitz (1976) was based on optimum conditions and not intuitive. Recently, Laroque (2005) and Kaplow (2006) have simultaneously and independently proposed a much simpler and intuitive proof that we present here.

Proof. The idea of the proof is that a tax system that includes both a nonlinear income tax and a vector of commodity taxes can be replaced by a pure income tax that keeps all individual utilities constant and raises at least as much tax revenue.

Let subject to be the indirect utility of consumption goods common to all individuals. Consider replacing with where is defined such that . Such a naturally exists (and is unique) as is strictly increasing in . This implies that for all . Hence, both the utility and the labor supply choice are unchanged for each individual .

By definition of an indirect utility, attaining utility of consumption at price costs at least . Let be the consumer choice of individual under the initial tax system . Individual attains utility when choosing . Hence . As , we have , i.e., the government collects more taxes with which completes the proof. QED.

Intuitively, with separability and homogeneity, conditional on earnings , the consumption choices do not provide any information on ability. Hence, differentiated commodity taxes create a tax distortion with no benefit and it is better to do all the redistribution with the individual nonlinear income tax. With the weaker linear income taxation tool, stronger assumptions on preferences, namely linear Engel curves uniform across individuals, are needed to obtain the commodity tax result (Deaton 1979).⁹⁴ Intuitively, in the linear tax case, unless Engel curves are linear, commodity taxation can be useful to “non-linearize” the tax system.

Heterogeneous Preferences.Saez (2002b) shows that the Atkinson-Stiglitz theorem can be naturally generalized to cases with heterogeneous preferences. No tax on commodity is desirable under three assumptions: (a) conditional on income , social marginal welfare weights are uncorrelated with the levels of consumption of good , (b) conditional on income , the behavioral elasticities of earnings are uncorrelated with the consumption of good , and (c) at any income level , the average individual variation in consumption of good with is identical to the cross-sectional variation in consumption of good with .

Assumption (a) is clearly necessary and might fail when earnings is no longer a sufficient statistic for measuring welfare. For example, if some individuals face high uninsured medical expenses due to poor health, then this assumption would not hold, and it would be desirable to subsidize health expenditures.⁹⁵ However, when heterogeneity in consumption reflects heterogeneity in preferences and not in need, assumption (a) is a natural assumption.

Assumption (b) is a technical assumption required to ensure that consumption of specific goods is not a tag for low responsiveness of labor supply to taxation. For example, if consumers of luxury cars happened to have much lower labor supply elasticities than average, it would become efficient to tax luxury cars as a way to indirectly tax more the earnings of those less responsive individuals. In practice, too little is known about the heterogeneity in labor supply across individuals to exploit such possibilities. Hence, assumption (b) is also a natural assumption.

Assumption (c) is the critical assumption. When it fails, the thought experiment to decide on whether commodity ought to be taxed is the following. Suppose high ability individuals are forced to work less and earn only as much as lower ability individuals. In that scenario, if higher ability individuals consume more of good than lower ability individuals, then taxing good is desirable. This can happen for two reasons. First, high ability people may have a relatively higher taste for good (independently of income) in which case taxing good is a form of indirect tagging of high ability. Second, good is positively related to leisure, i.e., consumption of good increases when leisure increases keeping after-tax income constant. This suggests taxing more holiday-related expenses and subsidizing work-related expenses such as child care.

In general the Atkinson-Stiglitz assumption is a good starting place for most goods. This implies that lower or zero VAT rates on some goods for redistribution purposes is inefficient (in addition to being administratively burdensome). Under those assumptions, eliminating such preferential rates and replacing them with a more redistributive income tax and transfer system would increase social welfare.⁹⁶

6.3 In-Kind Transfers

As we discussed in Section 3, the largest transfer programs are in-kind rather than cash. OECD countries in general provide universal public health care benefits and public education. They also often provide in-kind housing or nutrition benefits on a means-tested basis.

As is well known, from a rational individual perspective, if the in-kind benefit is tradable, it is equivalent to cash. Most in-kind benefits however are not tradable. In that case, recipients may be forced to overconsume the good provided in-kind and would instead prefer to receive the cash equivalent value of the in-kind transfer. Therefore, from a narrow rational individual perspective, cash transfers dominate in-kind transfers. From a social perspective, three broad lines of justification have been provided in favor of in-kind benefits.⁹⁷

6.4 Family Taxation

In practice, the treatment of families raises important issues. Any tax and transfer system must make a choice on how to treat singles vs. married households and how to make taxes and transfers depend on the number of children. There is relatively little normative work on those questions, in large part because the standard utilitarian framework is not successful at capturing the key trade offs. Kaplow (2008), Chapter 8 provides a detailed review.

Couples. Any income tax system needs to decide how to treat couples vs. single individuals. As couples typically share resources, welfare is best measured by family income rather than individual income. There are two main treatments of the family in actual tax (or transfer) systems. (a) The individual system where every person is taxed separately based on her individual income. In that case, couples are treated as two separate individuals. As a result, an individual system does not impose any tax or subsidy on marriage as tax liability is independent of living arrangements. At the same time, it taxes in the same way a person married to a wealthy spouse vs. a person married to a spouse with no income. (b) The family system where the income tax is based on total family income, i.e., the sum of the income of both spouses in case of married couples. The family system can naturally modulate the tax burden based on total family resources, which best measures welfare under complete sharing within families. However and as a result, a family tax system with progressive tax brackets cannot be neutral with respect to living arrangements, creating either a marriage tax or a marriage subsidy. Under progressive taxation, if the tax brackets for married couples are the same as for individuals, the family system typically creates a marriage tax. If the tax brackets for married couple are twice as wide as for individuals, the family system typically creates a marriage subsidy.⁹⁹

Hence and as is well known, it is impossible to have a tax system that simultaneously meets three desirable properties: (1) the tax burden is based on family income, (2) the tax system is marriage neutral, and (3) the tax system is progressive (i.e., the tax system is not strictly linear). Although those properties clearly matter in the public debate, it is not possible to formalize their trade off within the traditional utilitarian framework as the utilitarian principle cannot put a weight on the marriage neutrality principle.

If marriage responds strongly to any tax penalty or subsidy, it is better to reduce the marriage penalty/subsidy and move toward an individualized system. This issue might be particularly important in countries (such as Scandinavian countries for example), where many couples cohabit without being formally married and as it is difficult (and intrusive) for the government to observe (and monitor) cohabitation status.

Traditionally, the labor supply of secondary earners—typically married women—has been found to be more elastic than the labor supply of primary earners—typically married men (see Blundell & MaCurdy, 1999 for a survey). Under the standard Ramsey taxation logic, this implies that it is more efficient to tax secondary earners less (Boskin & Sheshinski, 1983). If the tax system is progressive, this goal is naturally achieved under an individual-based system as secondary earners are taxed on their sole earnings. Note however that the difference in labor supply elasticities between primary and secondary earners has likely declined over time as more and more married women work (Blau & Kahn, 2007).

In practice, most OECD countries have switched from family based to individual-based income taxation. In contrast, transfer systems remain based on family income. It is therefore acceptable to the public that a spouse with modest earnings would face a low tax rate, no matter how high the earnings of her/his spouse are.¹⁰⁰ In contrast, it appears unacceptable to the public that a spouse with modest earnings should receive means-tested transfers if the earnings of his or her spouse are high. A potential explanation could be framing effects as direct transfers might be more salient than an equivalent reduction in taxes. Kleven, Kreiner, and Saez (2009b) offer a potential explanation in a standard utilitarian model with labor supply where they show that the optimal joint tax system is to have transfers for non-working spouses (or equivalently taxes on secondary earnings) that decrease with primary earnings. The intuition is the following. With concave utilities, the presence of secondary earnings make a bigger difference in welfare when primary earnings are low than when primary earnings are large. Hence, it is more valuable to compensate one earner couples (relative to two earner couples) when primary earnings are low. This translates into an implicit tax on secondary earnings that decreases with primary earnings. Such negative jointness in the tax system is approximately achieved by having family based means-tested transfers along with individually based income taxation.

Children. Most tax and transfer systems offer tax reductions for children or increases in benefits for children. The rationale for such transfers is simply that, conditional on income , families with more children are more in need of transfers and have less ability to pay taxes. The interesting question that arises is how the net transfer (additional child benefits or reduction in taxes) per additional child should vary with income . On the one hand, the need for children related transfers is highest for families with very small incomes. On the other hand, the cost of children is higher for families with higher incomes particularly when parents work and need to purchase childcare.

Actual tax and transfers do seem to take both considerations into account. Means-tested transfers tend to offer child benefits that are phased-out with earnings. Income taxes tend to offer child benefits that increase with income for two reasons. First, the lowest income earners do not have taxable income and hence do not benefit from child-related tax reductions. Second, child-related tax reductions are typically a fixed deduction from taxable income which is more valuable in upper income tax brackets. Hence, the level of child benefits tends to be U-shaped as a function of earnings. Two important qualifications should be made.

First, as mentioned in Section 5.3.3, a number of countries have introduced in-work benefits that are tied to work and presence of children. This tends to make child benefits less decreasing with income at the low income end. In the United States, because of the large EITC and child tax credits and small traditional means-tested transfers, the benefit per child is actually increasing with family earnings at the bottom. Second, another large child benefit often subsidized or government provided is pre-school child care (infant child care, kindergarten starting at age 2 or 3, etc.). Such child care benefits are quantitatively large and most valuable when both parents work or for single working parents. Hence, economically, they are a form of in-kind in-work benefit which also promotes labor force participation (see OECD, 2006, chap. 4, Figure 4.1, p.129 for an empirical analysis). It is perhaps not a coincidence that cash in-work benefits for children are highest in the US and the UK, countries which provide minimal child care public benefits. Understanding in that context whether a cash transfer or an in-kind child care benefit is preferable is an interesting research question that has received little attention.

Child-related benefits raise two additional interesting issues.

First, families do not take decisions as a single unit (Chiappori, 1988). Interestingly, in the case of children, cash transfers to mothers (or grandmothers) have larger impacts on children’s consumption than transfers to fathers. This has been shown in the UK context (Lundberg, Pollak, & Wales, 1997) when the administration of child tax benefits was changed from a reduction in tax withholdings of parents (often the father) to a direct check to the mother. Similar effects have been documented in the case of cash benefits for the elderly in South Africa (Duflo, 2003). This evidence suggests that in-kind benefits (such as child care or pre-school) might be preferable if the goal is to ensure that resources go toward children. As mentioned above, primary education is again the most important example of in-kind benefits designed so that children benefit regardless of how caring parents are.

Second, child benefits might promote fertility. A large empirical literature has found that child benefits have sometimes positive but in general quite modest effects on fertility (see Gauthier, 2007 for a survey). There can be externalities (both positive and negative) associated with children. For example, there can be congestion effects (such as global warming) associated with larger populations. Alternatively, declines in populations can have adverse effects on sustainability of pay-as-you-go pension arrangements. Such externalities should be factored into discussions of optimal child benefits.

6.5 Relative Income Concerns

Economists have long been interested in the possibility that individuals care not only about their absolute income but also their income relative to others. Recently, substantial evidence coming from observational studies (e.g., Luttmer, 2005), lab experiments (e.g., Fehr & Schmidt, 1999), and field experiments (Card, Mas, Moretti, & Saez 2012), provide support for relative income effects. A number of optimal tax studies have incorporated relative income in the analysis (Boskin & Sheshinski, 1978 analyze the linear income tax case and Oswald, 1983 and Tuomala, 1990, Chapter 8 consider the nonlinear income tax case). Those studies find that in general relative income concerns tend to increase optimal tax rates. Relative income effects can be modeled in a number of ways. The simplest way, which we consider here, is to posit that individual utility also depends on the utility of others.¹⁰¹

Relative income concerns affect optimal tax analysis in two ways. First, it changes the social marginal welfare weights as a decrease in the utility of others has a direct effect on one’s utility (keeping one’s work and income situation constant), creating externalities. In our view, the simplest way to capture this effect is to consider that those externalities affect the social welfare weights. If a decrease in a person’s income increases others’ utility, then the social welfare weight on this person ought to be reduced by this external effect. Whether such externalities should be factored in the social welfare function is a deep and difficult question. Surely, hurting somebody with higher taxes for the sole satisfaction of envy seems morally wrong, Hence, social welfare weights should not be allowed to be negative for anybody no matter how strong the envy effects. At the same, it seems to us that relative income concerns are a much more powerful and realistic way to justify social welfare weights decreasing with income than standard utilitarianism with concave utility of consumption.

Second, relative income concerns affect labor supply decisions. For example, if utility functions are such that with average consumption in the economy, then a proportional tax on consumption affects and equally and hence has no impact on labor supply. This might be a simple explanation for why labor supply is relatively inelastic with respect to secular increases in wage rates over the long-term process of economic growth (Ramey and Francis, 2009).¹⁰² This labor supply channel effect is fully captured by the behavioral response elasticity and hence does not change the optimal tax formulas.

As an illustration, let us go back to the optimal top tax rate analysis from Section 5.1 with a small variation in the top tax rate. The key difference in the analysis is that the reduction in welfare for top bracket earners would now have a positive externality on the utility of lower income individuals. As long as this external effect is weakly separable from labor supply choices, i.e., where is the standard utility function and is the vector of utilities of all other (non ) individuals, the individual earnings decisions are not affected by the external effect. The external effect is proportional to the direct welfare effect on top bracket earners and the strength of the externality. Therefore, the external effect simply reduces the social marginal value of consumption of top bracket earners from to . The optimal tax formula retains the same form as before .

In sum, we think that relative income concerns are a useful way to interpret and justify optimal tax analysis and can be incorporated within standard optimal tax analysis.

6.6 Other Extensions

Endogenous Wages. The standard assumption in optimal labor income tax theory is that pre-tax wage rates are exogenous, i.e., that there is perfect substitutability between skills in production. Interestingly, in the discrete occupational models we have introduced in Section 5.2.2, this assumption can be relaxed without affecting the general optimal tax formula (12). To see this, consider a general production function of the consumption good with constant returns to scale.¹⁰³ In that case, wages are set by marginal product . The maximization of the government can be rewritten as choosing to maximize

Note that any explicit reference to wages has disappeared from this maximization problem and the first order condition with respect to immediately leads to the same optimal tax formula (12).

The intuition in a basic two skill model is the following. Suppose an increase in high skill taxes leads to a reduction in high skill labor supply and hence an increase in high skill wages (and a decrease in low skill wages) through demand effects. Because of the absence of profits, those demand effects are a pure transfer from low to high skill workers. Therefore, the government can readjust the tax on high and low skills to offset those demand effects on the net consumption levels at no net fiscal cost, leaving the optimal tax formula unchanged.¹⁰⁴

Theoretically, this result arises because the discrete occupational model is effectively mathematically identical to a Diamond and Mirrlees (1971), optimal commodity tax model where each occupation is a specific good taxed at a specific rate. As is well known from Diamond and Mirrlees (1971), optimal Ramsey tax formulas depend solely on consumers’ demand and do not depend on production functions. This generates two important additional consequences. First, the production efficiency result of Diamond and Mirrlees (1971) carries over to the discrete occupational choice model, implying that distortions in the production process or tariffs (in the case of an open economy) are not desirable. Second, in an extended model with many consumption goods, the theorem of Atkinson and Stiglitz (1976) also carries over to the discrete occupational choice model. Namely, differentiated commodity taxation is not desirable to supplement optimal nonlinear earnings taxation under the standard separability assumption presented above. Those results are formally proven in Saez (2004b). They stand in sharp contrast to results obtained in the Stiglitz (1982) discrete model with endogenous wages where it is shown that the optimal tax formulas are affected by endogenous wages (Stiglitz, 1982), and where the production efficiency theorem and the Atkinson-Stiglitz theorem do not carry over (Naito, 1999). Saez (2004b) argues that the occupational model best captures the long-term when individuals choose their occupations while the Stiglitz (1982) model captures a short-term situation where individuals have fixed skills and only adjust hours of work.

Workfare, Take-Up Costs, and Screening. Workfare can be defined as requiring transfer beneficiaries to work, typically for a public project. In its extreme form, the work required has no productive value. In that case, workfare is similar to imposing an ordeal, such as time consuming take-up costs, on welfare beneficiaries. The literature has focused primarily on such “useless workfare requirements.” Besley and Coate (1992) show that, if the government cares about poverty measured by net-income rather than individual utilities, it can be optimal to impose workfare. In their model, workfare screens away higher wage individuals who have a higher opportunity cost of time.¹⁰⁵

Cuff (2000) shows, in a standard Stiglitz (1982) two-type discrete model that a useless workfare program is never desirable with a standard welfarist objective. Interestingly, Cuff (2000)then extends the analysis to include heterogeneity in tastes for work (in addition to the standard wage rate heterogeneity). When there are lazy vs. hard working low skill workers and when society does not like to redistribute toward lazy low skill workers, workfare can become desirable. This is because work requirements are more costly to lazy types than hard working types.

In practice, finding ordeals which hurt more the undeserving beneficiaries than the deserving beneficiaries seems difficult. In particular, if society feels that welfare is too generous, it is more efficient to cut benefits directly rather than impose ordeals. Both reduce welfare benefits (and hence the incentives to become a recipient), but at least direct cuts save on government spending.

Screening mechanisms that also impose costs on recipients, (e.g., filing out forms, medical tests, etc.) can be desirable when they are successful in screening deserving recipients (e.g., the truly disabled) vs. undeserving recipients (e.g., those faking disability). Diamond and Sheshinski (1995) propose an analysis along those lines in the case of disability insurance (see also the chapter by Chetty and Finkelstein in this volume for more details on optimal social insurance). The key difference with useless workfare or ordeals is that such screening is directly designed at separating deserving vs. undeserving recipients. It is very unlikely that blanket ordeals can achieve this. Today, data driven screening (i.e., checking administrative databases for potential earnings, etc.) are far more powerful and efficient than direct in person screening (and a lot less intrusive for recipients).

Minimum Wages. The minimum wage is another policy tool that can be used for redistribution toward low skill workers. At the same time minimum wages can create unemployment among low skill workers, creating a trade off between equity and efficiency. A small literature has examined the desirability of minimum wages in addition to optimal taxes and transfers in the standard competitive labor market with endogenous wage rates (as in the model discussed above).¹⁰⁶

Lee and Saez (2012) use the occupational model of Section 5.3.2 with endogenous wages and prove two results. First, they show that a binding minimum wage is desirable under the strong assumption that unemployment induced by the minimum wage hits the lowest surplus workers first. The intuition for this result is simple and can be understood using Figure 8. Suppose a minimum wage is set at level and that transfers to low-skilled workers earning are increased. The presence of the minimum wage at rations low skill work and effectively prevents the labor supply responses from taking place. Some non-workers would like to work and earn but cannot find jobs because those jobs are rationed by the minimum wage. Therefore, the minimum wage enhances the ability of the government to redistribute (via an EITC type benefit) toward low skill workers.

Second, when labor supply responses are along the extensive margin only, which is the empirically relevant case, the co-existence of a minimum wage with a positive tax rate on low-skilled work is always (second-best) Pareto inefficient. A Pareto improving policy consists of reducing the pre-tax minimum wage while keeping constant the post-tax minimum wage by increasing transfers to low-skilled workers, and financing this reform by increasing taxes on higher paid workers. Importantly, this result is true whether or not rationing induced by the minimum wage is efficient or not. This result can also rationalize policies adopted in many OECD countries in recent decades that have decreased the minimum wage while reducing the implicit tax on low skill work through a combination of reduced payroll taxes for low skill workers and in-work benefits of the EITC type for low skill workers.

Optimal Transfers in Recessions. In practice, some transfers (such as unemployment insurance in the United States) can be made more generous during recessions. Traditionally, optimal policy over the business cycle has been analyzed in the macro-economics literature rather than the public economics literature.¹⁰⁷ The macro-economics literature, however, rarely focuses on distributional issues. There are three channels through which recessions can affect the calculus of optimal transfers for those out-of-work.

First, recessions are a time of high unemployment where people want to work but cannot find jobs. This suggests that employment is limited by demand effects rather than the supply effects of the traditional optimal tax analysis. As a result, in recessions, unemployment is likely to be less sensitive to supply-side changes in search efforts and job search is likely to generate a negative externality on other job seekers in the queue. Landais, Michaillat, and Saez (2010) capture this effect in a search model where job rationing arises in recessions and show that unemployment insurance should be more generous during recessions. Crépon, Esther, Marc, Roland and Philippe (in press), using a large scale job placement aid randomized experiment in France, show that indeed there are negative externalities of job placement aid on other job seekers and that those externalities are larger when unemployment is high.

Second, in recessions, the ability to smooth consumption might be reduced, as the long-term unemployed might exhaust their buffer stock savings and might face credit constraints. This implies that the gap in social marginal utility of consumption between workers and non-workers might grow during recessions, further increasing the value of redistributing from workers to the unemployed (Chetty, 2008).

Third and related, individuals are less likely to be responsible for their unemployment status in a recession than in an expansion. In an expansion when jobs are easy to find, long unemployment spells are more likely to be due to low search efforts than in a recession when jobs are difficult to find even with large search efforts. If society wants to redistributive toward the hard-searching unemployed—i.e., those who would not have found jobs even absent unemployment benefits—then it seems desirable to have time limited benefits during good times combined with expanded benefit durations in bad times. We will come back to such non-utilitarian social preferences in Section 7.

Education Policy. Education plays a critical role in generating labor market skills. All advanced economies provide free public education at the K-12 level and heavily subsidize higher education. As we have seen earlier, there is a strong rationale for providing K-12 public education to correct potential parenting failures. For higher education, the presence of credit constraints might lead to suboptimal educational levels, providing a strong rationale for government provision of loans (see e.g., Lochner and Monge, 2011).¹⁰⁸ However, governments in advanced economies not only provide loans but also direct subsidies to higher education. Direct subsidies could be justified by “behavioral considerations” if a significant fraction of young adults are not able to make wise educational choices on their own—due for example to informational or self-control issues.

A small literature in optimal taxation has examined the desirability of education subsidies in fully rational models. Higher education subsidies encourage skill acquisition but tend to benefit more the relatively skilled and hence are likely regressive. Absent any ability to observe educational choices, the total elasticity of earnings with respect to net-of-tax rates is due to both labor supply and education choices. If education choices are elastic, the corresponding optimal income tax should incorporate the full elasticity and not solely the labor supply elasticity. This naturally leads to lower optimal tax rates than those calibrated using solely the labor supply elasticity. Diamond and Mirrlees (Unpublished) develop this point, which they call the “Le Chatelier” principle.¹⁰⁹

Suppose now that the government can observe educational choices and hence directly subsidize (or tax) them in addition to using income-based taxes and transfers. In that context, redistributive taxes and transfers discourage both labor supply and education investments as they reduce the net rewards from higher education. Bovenberg and Jacobs (2005) consider such a model and show that combining educational subsidies with redistributive income-based taxation is optimal—consistent with real policies.

In the simplest version of their model, education increases the wage rate (with increasing and concave and being innate ability) at a cost . Individuals choose and to maximize utility subject to where is the income tax rate, the subsidy rate on education expenses , and the demogrant. In this simple model, is an intermediate good that does not directly enter the utility function which depends solely on and . The education choice is given by the first order condition . Hence, education is pure cost of production and individuals should be taxed on their earnings net of education costs . This implies that should be set exactly equal to .

7.1 Issues with the Welfarist Approach

All our analysis so far has followed the standard welfarist approach whereby the government objective is to maximize a weighted sum of individual utilities (or an increasing transformation of utilities). As we saw, all optimal tax formulas can be expressed in terms of the social marginal welfare weights attached to each individual which measure the social value of an extra dollar of consumption to each individual.

In standard optimal tax analysis, the utilitarian case (maximizing the unweighted sum of individual utilities) is by far the most widely used. In that case, social welfare weights are proportional to the marginal utility of consumption. As we have seen, this criterion generates a number of predictions at odds with actual tax systems and with people’s intuitive sense of redistributive justice.

First, if individuals do not respond to taxes, i.e., if pre-tax incomes are fixed, and individual utilities are concave, then utilitarianism recommends a 100% tax, and full redistribution. In reality, even absent behavioral responses, many and perhaps even most people would still object to confiscatory taxation on the grounds that people deserve to keep part of the income they have created.

Second and related, views on taxes and redistribution seem largely shaped by views on whether the income generating process is fair and whether individual incomes are deserved or not. The public tends to dislike the redistribution of fairly earned income through one’s effort but is in favor of redistributing income earned unfairly or due to pure luck (see Piketty, 1995 for a theoretical model and Alesina & Giuliano, 2011, chap. 4 for a recent survey). Such distinctions are irrelevant for utilitarianism.

Third, as we have seen in Section 6.1 on tagging, under utilitarianism, optimal taxes should depend on all observable characteristics which are correlated with intrinsic earning ability. In practice, taxes and transfers use very few of the potentially available tags. Society seems to have horizontal equity concerns and using tags to achieve indirect redistribution is hence perceived to be unfair.

Fourth, perceptions about recipients seem to matter a great deal for the public views on transfers. Most people support transfers for people really unable to work, such as the truly disabled but most people dislike transfers to people able to work and who would work absent transfers. In the standard model, behavioral responses matter for optimal taxes only through their effects on the government budget. In reality, the presence of behavioral responses also colors the public perceptions on how deserving transfer beneficiaries are.

7.2 Alternatives

A number of alternatives to welfarism have been proposed in the literature.

Pareto Principle. First, let us recall that the standard utilitarian criterion can be easily extended, as we have seen, by considering a weighted sum of individual utilities (instead of a simple sum). Those positive weights are called Pareto weights. By changing those weights, we can describe the set of all second-best Pareto efficient tax equilibria. It seems natural that any “optimal tax system” should be at least second-best Pareto efficient, i.e., no feasible tax reform can improve the welfare of everybody. Hence, the Pareto principle imposes a reasonable but weak condition on tax optima. Indeed, optimal tax analysis was particularly interested in finding properties that hold true for all such second-best optima.¹¹⁰ Those properties are relatively few, an example being the Atkinson and Stiglitz theorem. Hence, considering arbitrary weights is not going to be enough to obtain definite conclusions in general. Hence, it is necessary to be able to put more structure on those Pareto weights so that we can select among the wide set of second-best Pareto optimal tax systems.

All the examples of alternatives to utilitarianism we describe next show that any criterion leads to a specific set of marginal social welfare weights.

Rawlsian Criterion. In the Rawlsian criterion, Pareto weights are concentrated solely on the most disadvantaged person in the economy. This amounts to maximizing the utility of the person with the minimum utility, hence this criterion is also called the maxi-min objective. A judgment needs to be made as to who is the most disadvantaged person. In models with homogeneous preferences and heterogeneous skills, the most disadvantaged person is naturally the person with the lowest skill and hence the lowest earnings. This criterion has the appealing feature that, once society agrees on who is the most disadvantaged person, the optimum is independent of the cardinal choice for individual utilities. The key weakness of this criterion is that it concentrates all social welfare on the most disadvantaged and hence represents extreme redistributive tastes. Intuitively, it seems clear that the political process will put weight on a broader set of voters than solely the most disadvantaged. Hence, the Rawlsian principle makes sense politically only if the most disadvantaged form a majority of the population. This is not a realistic assumption in the case of redistribution of labor income.¹¹¹ For example, we have seen in Section 4.1 that a standard median voter outcome puts all the weight on the median voter preferences.

Libertarianism and Benefits Principle. At the other extreme, libertarians argue that the government should not do any redistribution through taxes and transfers. Therefore, taxes should be set according to the benefits received from government spending, individual by individual. This is known as the benefits principle of taxation. Any redistribution over and above benefits is seen as unjust confiscation of individual incomes. Such a principle can be formally captured by assuming that social marginal welfare weights are identical across individuals (in the situation where taxes correspond to benefits). In that case, additional redistribution does not add to social welfare.¹¹² While some voters may hold libertarian views, as we discussed in Section 2.1, all OECD countries do accomplish very substantial redistribution across individuals, and hence depart very significantly from the benefits principle of taxation. This shows that the benefits principle cannot by itself account for actual tax systems.

Principles of Responsibility and Compensation. The general idea is that individuals should be compensated for circumstances affecting their welfare over which they have no control, such as their family background or disability at birth. This is the principle of compensation. In contrast, individuals should be held responsible for circumstances which they control such as how many hours they work. Hence, no redistribution should take place based on such choices. This is the principle of responsibility. These principles are presented and discussed in detail in Kolm (1996), Roemer (1998), Fleurbaey (2008), Fleurbaey and Maniquet (2011).

An example often presented in the literature is that of individuals differing by their wage rate which they do not control (for example because it is due to exogenous ability), and by their taste for leisure (some people prefer goods consumption, some people prefer leisure consumption). By the principle of compensation, it is fair to redistribute from high wage to low wage individuals. By the principle of responsibility, it is unfair to redistribute from goods lovers toward leisure lovers. When there is only one dimension of heterogeneity, those principles are easy to apply. For example, if individuals differ only according to their wage rate (and not in their tastes), then the principle of compensation boils down to a Rawlsian criterion whereby the tax and transfer system should provide as much compensation as possible to the lowest wage people. In terms of welfarism, social marginal welfare weights are fully concentrated on the lowest wage person. If individuals differ solely in taste for work, the principle of responsibility calls for no redistribution at all because everybody has the same time endowment that they can divide between work and leisure based on their relative tastes for goods consumption vs. leisure consumption. It would be unfair to redistribute based on tastes.¹¹³ The standard welfarist approach cannot easily obtain this meaningful result, except through a renormalization of Pareto weights so that social marginal utilities of consumption are the same across individuals (absent transfers).¹¹⁴

However, those two principles can conflict in situations where there is heterogeneity in both dimensions (skills and taste for leisure). Fleurbaey (2004) presents a simple example in a two skill, two levels of taste for leisure model showing that it is not possible to fulfill both the responsibility principle and the compensation principle at the same time. Therefore, some trade off needs to be made between the two principles. This trade-off needs to be specified through a social objective function. Fleurbaey (2008) reviews this literature and the many criteria that have been proposed.¹¹⁵

Equal Opportunity. One prominent example of how to trade-off the responsibility vs. the compensation principles is Roemer (1998) and Roemer et al. (2003) who propose an Equal Opportunity criterion. In the model of Roemer et al. (2003), individuals differ solely in their wage rate but the wage rate depends in part on family background and in part on merit (i.e., personal effort in getting an education, getting ahead, etc.). The model uses quasi-linear utility functions uniform across individuals. In the model, people are responsible for wage differences due to merit but not for wage differences due to family background. Suppose for simplicity there is a low and high family background. The distribution of wage rates is equal to and among those coming from low and high family backgrounds respectively. Assume that high family background provides an advantage so that stochastically dominates . The government wants to redistribute from high to low family backgrounds but does not want to redistribute across individuals with different wages within a family background group because their position within the group is due to merit. The government can only observe earnings and cannot observe family background (nor the wage rate). Hence, the government is limited to using a nonlinear income tax and cannot discriminate directly based on family background. Individuals choose to maximize their utility .

By assumption, two individuals in the same wage percentile within their family background group are equally deserving. Therefore, any discrepancy in the utility across family background conditional on wage percentile should be corrected. This can be captured by a local social welfare function at percentile given by where is the

th percentile wage rate in family background group , and the labor supply choice of the

th percentile wage person in group . Total social welfare is then obtained by summing across all percentiles. Hence, we have

Effectively, the social criterion is locally Rawlsian as it wants to redistribute across family background groups conditional on merit (percentile) to level the field as much as possible but does not value redistribution within a family background group (as utilities are quasi-linear).

Because high family background provides an advantage, we have . Hence the

th percentile individual in the high family background has a higher utility than the

th percentile individual in the low family background. As a result, total social welfare can be rewritten as:

This criterion is equivalent to a standard welfarist objective with the following social marginal welfare weights. The weights are equal to zero for those with high family background and equal and constant for those with low family background. Hence, the average social welfare weight at wage is simply , i.e., the relative fraction of individuals at wage coming from a low family background. Presumably, decreases with as it is harder to obtain (through merit) a high wage when coming from a low family background.

The standard Diamond (1998) optimal nonlinear tax theory of Section 5 applies in this case by simply substituting the standard welfarist weights by those weights. For example, the optimal top tax rate is given again by the simple formula where is the relative fraction of top earners coming from a low family background. If nobody coming from a low family background can make it to the top, then and the optimal top tax rate is set to maximize tax revenue.

Generalized Social Welfare Weights. A systematic approach recently proposed by Saez and Stantcheva (2013) is to consider generalized social marginal welfare weights that are ex-ante specified to fit justice principles. Those social marginal welfare weights reflect the relative value of marginal consumption that society places on each individual. Hence, they can be used to evaluate the aggregate social gain or loss created by any revenue neutral tax reform. A tax system is “optimal” if no small revenue neutral reform yields a net gain when adding gains and losses across individuals weighted using those generalized social marginal welfare weights. Importantly, the optimum no longer necessarily maximizes an ex-ante social objective function. Naturally, the optimal tax system that arises is second-best Pareto efficient as long as the social marginal welfare weights are specified to be non-negative.

This framework is therefore general and contains as special cases virtually all the situations we have discussed before. The use of suitable generalized social welfare weights can resolve many of the puzzles of the traditional utilitarian approach and account for existing tax policy debates and structures.

First, if generalized social marginal welfare weights depend positively on net taxes paid, in addition to net disposable income, the optimal tax rate is no longer 100% even absent behavioral responses.

Second, generalized social welfare weights can also capture the fact that society prefers taxes on income due to luck rather than taxes on income due to work. As shown in the example above from Roemer et al. (2003), the social welfare weights can be set to zero for those who have an undue advantage because of family background or income due to luck. Such “locally Rawlsian” weights capture the intuition that it is fair to redistribute along some dimensions but not others. When redistribution is deemed fair, it should be as large as possible as long as it benefits those deem Roemer et al. (2003), Piketty and Saez (2012a,2012b) also use such weights in the context of inheritance taxation where weights are set to zero for all those who receive positive inheritances. In the context of inheritance taxation, this yields relatively robust outcomes, due to the fact that the bottom half of the population generally receives close to zero inheritance. We suspect that this approach could be fruitfully extended to the optimal taxation of top labor incomes. For example, if individuals whose parents were in the bottom half of the income distribution have small probabilities to reach the top 1% of the earnings distribution, then this probability could be used as the welfare weight for the top 1%. One key advantage of this approach-based upon transition probabilities and mobility matrices is that it provides an objective, non-ideological basis upon which welfare evaluations can be made.

Third and related, generalized social welfare weights can capture horizontal equity concerns as well. Weights can be set to zero on anybody who benefits from a favorable treatment based on a policy that creates horizontal inequity (such as, for instance, shorter people in a tax system based on height). In that case, tax policies creating horizontal inequities will arise only if they benefit the group that is being discriminated against, i.e., taxing the tall more is desirable only if the tall end up better off in this new tax system as well. This drastically reduces the scope for using additional characteristics in the tax and transfer system, consistent with the rare use of tags in real policies.

Fourth, generalized social welfare weights can be made dependent on what individuals would have done absent taxes and transfers. For example, social welfare weights can be set to zero on “free loaders” who would have worked absent means-tested transfers. This sharply reduces the desirability of transfers when behavioral responses are large for fairness reasons (in addition to the standard budgetary reason).

Naturally, the flexibility of generalized social weights begs the question of what social welfare weights ought to be and how they are formed. First, generalized welfare weights can be derived from social justice principles, leading to a normative theory of taxation. The most famous example is the Rawlsian theory where the generalized social marginal welfare weights are concentrated solely on the most disadvantaged members of society. As we discussed, “locally Rawlsian” weights as in Roemer (1998), Roemer et al. (2003), or Piketty and Saez (2012a,2012b) can also be normatively appealing to model preferences for redistribution based on some but not all characteristics. Second, generalized welfare weights could also be derived empirically, by estimating actual social preferences of the public, leading to a positive theory of taxation. There is indeed a small body of work trying to uncover perceptions of the public about various tax policies. Those approaches either start from the existing tax and transfers system and reverse engineer it to obtain the underlying social preferences (see e.g., Ahmad & Stern (1984) for commodity taxation and Bourguignon and Spadaro (2012) for nonlinear income taxation) or directly elicit preferences on various social issues in surveys (see e.g., Fong, 2001 and Frohlich & Oppenheimer, 1992). Social preferences of the public are shaped by beliefs about what drives disparities in individual economic outcomes (effort, luck, background, etc.) as in the model of Piketty (1995). In principle, economists can cast light on those mechanisms and hence enlighten public perceptions so as to move the debate back to higher level normative principles.

A.1 Formal Derivation of the Optimal Nonlinear Tax Rate

We specialize the Mirrlees (1971) model to the case with no income effects, as in Diamond (1998). All individuals have the same quasi-linear utility function where is disposable income and is labor supply with increasing and convex in . Individuals differ only in their skill level, denoted by , which measures their marginal productivity. Earnings are equal to . The population is normalized to one and the distribution of skills is , with density and support [0,∞). The government cannot observe skills and thus is restricted to setting taxes as a function only of earnings, . Individual chooses to maximize utility leading to first order condition .

Under a linearized income tax system with constant marginal tax rate , the labor supply function is implicitly defined by the equation . Hence and hence the elasticity of labor supply with respect to the net-of-tax rate is . As there are no income effects, this elasticity is both the compensated and the uncompensated elasticity.

Let , and denote the consumption, earnings, and utility level of an individual with skill . The government maximizes a social welfare function,

In the maximization program of the government, is regarded as the state variable, as the control variable, while is a function of and . Using the envelope theorem and the individual first order condition, the utility of individual satisfies .

Hence, the Hamiltonian is

where is the multiplier of the state variable. The first order condition with respect to is

The first order condition with respect to is

which can be integrated to yield where we have used the transversality condition . The other transversality condition yields , i.e., social marginal welfare weights average to one.

Using this equation for , and noting that , and that , we can rewrite the first order condition with respect to as:

(17)

where is the social marginal welfare weight on individual . This formula is derived in Diamond (1998).

Under a linearized income tax system with marginal tax rate , we have and hence . Therefore, denoting by the density of earnings at if the nonlinear tax were replaced by a linearized tax with marginal tax rate , we have and hence . Therefore, and we can rewrite Eq. (17) as

(18)

where is the average marginal social welfare weight on individuals above . Changing variables from to , we have where is the actual (not virtual) cumulative distribution of earnings. This establishes Eq. (11) in the main text. Note that the transversality condition implies that .

Equation (17) is particularly easy to use for numerical simulations calibrated to the actual income distribution. Using the specified utility function , the distribution is calibrated so that, using the actual tax system, the resulting earnings distribution match the actual earnings distribution. Once is obtained, formula (17) can be used iteratively until a fixed point tax system is found. See e.g., Brewer et al. (2010) for an application to the UK case.

A.2 Optimal Bottom Tax Rate in the Mirrlees Model

In the Mirrlees (1971) model, all individuals have the same utility function increasing in disposable income and decreasing in labor supply . Individuals differ only in their skill level, denoted by , which measures their marginal productivity. Earnings are equal to . The population is normalized to one and the distribution of skills is , with density , and support . The government cannot observe skills and thus is restricted to setting taxes as a function only of earnings, . Individual chooses to maximize utility leading to first order condition . Let , and denote the consumption, earnings, and utility level of an individual with skill . Note that and .

To have a fraction of non-workers, we assume that for all . As a result, all individuals with skill below defined as will not work and choose the corner solution and . Hence, the fraction non-working in the population is and naturally depends on both (substitution effects) and (income effects).

Using the envelope theorem, the utility of individual satisfies . Note that this equation remains true even for non-workers at the bottom as is constant with and hence for .

The government maximizes a social welfare function,

Following Mirrlees (1971), in the maximization program of the government, is regarded as the state variable, as the control variable, while is determined implicitly as a function of and from the equation . The Hamiltonian is

where is the multiplier of the state variable. As , the first order condition with respect to is

At , and this first order condition becomes

As , the first order condition with respect to is

For are constant with so that this equation simplifies to:

and can be integrated from to to yield

where we have used the transversality condition . Replacing this expression for into the first order condition for at yields

which can be rewritten as

(19)

where is the social marginal welfare weight on non-workers.¹¹⁶

Recall that which defines . Hence, the substitution effect of on (keeping constant) is such that . Hence, the elasticity of the fraction non-working with respect to is

which allows us to rewrite (19) as

exactly as in the discrete model formula (14) presented in the text.

Note that with quasi-linear iso-elastic preferences of the form , the individual first order condition is so that everybody with works. If there is a positive fraction of individuals with zero skill (and hence not working), the formula above applies with so that . Intuitively, the fraction of individuals affected by a change in is negligible relative to the number of non-workers so that behavioral responses are negligible and hence .