Optimal Income Taxation Theory and Principles of Fairness” Marc Fleurbaey Woodrow Wilson School of Public and International Affairs

FALL 2019 NEW YORK UNIVERSITY SCHOOL OF LAW

“Optimal Income Taxation Theory and Principles of Fairness” Marc Fleurbaey Woodrow Wilson School of Public and International Affairs

November 5, 2019 Vanderbilt Hall – 202 Time: 4:00 – 5:50 p.m. Week 10

SCHEDULE FOR FALL 2019 NYU TAX POLICY COLLOQUIUM (All sessions meet from 4:00-5:50 pm in Vanderbilt 202, NYU Law School)

1. Tuesday, September 3 – Lily Batchelder, NYU Law School.

2. Tuesday, September 10 – Eric Zwick, University of Chicago Booth School of Business.

3. Tuesday, September 17 – Diane Schanzenbach, Northwestern University School of Education and Social Policy.

4. Tuesday, September 24 – Li Liu, International Monetary Fund.

5. Tuesday, October 1 – Daniel Shaviro, NYU Law School.

6. Tuesday, October 8 – Katherine Pratt, Loyola Law School Los Angeles.

7. Tuesday, October 15 – Zachary Liscow, Yale Law School.

8. Tuesday, October 22 – Diane Ring, Boston College Law School.

9. Tuesday, October 29 – John Friedman, Brown University Economics Department.

10. Tuesday, November 5 – Marc Fleurbaey, Princeton University, Woodrow Wilson School.

11. Tuesday, November 12 – Stacie LaPlante, University of Wisconsin School of Business.

12. Tuesday, November 19 – Joseph Bankman, Stanford Law School.

13. Tuesday, November 26 – Deborah Paul, Wachtell, Lipton, Rosen, and Katz.

14. Tuesday, December 3 – Joshua Blank, University of California at Irvine Law School.

Journal of Economic Literature 2018, 56(3), 1029–1079 https://doi.org/10.1257/jel.20171238

Optimal Income Taxation Theory and Principles of Fairness†

Marc Fleurbaey and François Maniquet*

The achievements and limitations of the classical theory of optimal labor-income taxation based on social welfare functions are now well known. Even though utilitarianism still dominates public economics, recent interest has arisen for broadening the normative approach and making room for fairness principles such as desert or responsibility. Fairness principles sometimes provide immediate recommendations about the relative weights to assign to various income ranges, but in general require a careful choice of utility representations embodying the relevant interpersonal comparisons. The main message of this paper is that the traditional tool of welfare economics, the social welfare function framework, is flexible enough to incorporate many approaches, from egalitarianism to libertarianism. ( JEL D63, H21, H24, J24)

1. Introduction benevolent planner who only observes the pretax labor income of agents whose wages he theory of optimal income taxation differ, but whose preferences are identical. Thas reached maturity and excellent The classical framework has been extended reviews of the field are available (Boadway in the last two decades in several direc- 2012, Mankiw, Weinzierl, and Yagan 2009, tions. On the one hand, many contributions Piketty and Saez 2013b, Salanié 2011). In have relaxed assumptions in order to take the classical framework initiated by Mirrlees account, for instance, of multidimensional (1971), the theory studies the maximization heterogeneity among agents (e.g., Mirrlees of a utilitarian social welfare function by a 1976, Saez 2001, Choné and Laroque 2010), or to cast the problem in a dynamic frame- * Fleurbaey: Princeton University. Maniquet: CORE, work (e.g., Golosov, Kocherlakota, and Université catholique de Louvain. This paper has benefited Tsyvinski 2003). On the other hand, several from conversations with R. Boadway, S. Coate, E. Saez, and S. Stantcheva, from reactions of participants at the Taxation authors have recently questioned the ethical Theory Conference (Cologne 2014) and the QMUL con- foundations of the classical framework and ference in honor of John Roemer, and comments and sug- the utilitarian social objective it relies upon. gestions by P. Pestieau, six referees, and the Editor, Steven Durlauf. François Maniquet’s work was supported by the The latter issue is the topic of this paper. We European Research Council under the European Union’s review the different reasons to endorse or to Seventh Framework Programme (FP7/2007-2013)/ depart from the classical utilitarian objective, ERC n° 269831. † Go to https://doi.org/10.1257/jel.20171238 to visit the and we discuss what policy conclusions the article page and view author disclosure statement(s). different alternative approaches can deliver.

1029 1030 Journal of Economic Literature, Vol. LVI (September 2018)

Unease with the classical utilitarian frame- theory neglects the diverse normative cri- work is due to internal and external limita- teria with which, as extensive evidence has tions. First, most of the corpus of optimal tax shown, most people evaluate policy” (2012, theory assumes that individuals have identical p. 1). Similarly, Piketty and Saez (2013b) preferences, which is unrealistic. But defin- emphasize “the limitations of the standard ing the utilitarian social welfare function for utilitarian approach” and argue: “While heterogenous utilities requires scaling the many recent contributions use general utilities and there is no commonly admit- Pareto weights1 to avoid the strong assump- ted recipe for this. For instance, Boadway tions of the standard utilitarian approach, we (2012) lists the “heterogeneity of individual feel that the Pareto weight approach is too utility functions” (p. 30) as one of the big general to deliver practical policy prescrip- challenges for optimal tax theory (along with tions in most cases. Hence, we think that it issues of government commitment, political is important to make progress both on nor- economy, and behavioral phenomena). “The mative theories of justice stating how social assumption of identical utility functions is welfare weights should be set and on positive made more for analytical simplicity than for analysis of how individual views and beliefs realism. It also finesses one of the key issues about redistribution are formed” (p. 393). in applied normative analysis … which is Sheffrin (2013) also argues that folk notions how to make interpersonal comparisons of of fairness are ignored in the economic the- welfare” (pp. 30–31). ory of taxation. Among the considerations The problem of interpersonal compari- that are missed by the classical approach, sons of welfare cannot indeed be assumed according to these authors, one finds: the away for the sake of analytical simplicity. libertarian view that the distribution of earn- When individuals have the same utility func- ings may deserve some respect; the principle tion, the main ethical question that has to be that income inequalities due to differences settled is the degree of inequality aversion, in preferences or effort are not as prob- over which it is not too difficult to perform a lematic as inequalities due to differences in sensitivity analysis spanning the various pos- qualifications or social background; and the sible value judgments (from utilitarianism to idea that tagging2 on the basis of statistical maximin). This is what optimal tax theory has discrimination violates horizontal equity. done very well. In contrast, when individual The nonutilitarian principles recently preferences differ, interpersonal compari- revived in the tax literature appear to find sons involve much more difficult questions, support in the empirical literature studying which, in philosophy (Rawls 1982) as well people’s values and opinions about redistri- as in folk justice (Gaertner and Schokkaert bution. For instance, Konow (2001) finds 2012), are generally addressed in terms of support for an “accountability principle” that fair allocation of resources or opportuni- recommends apportioning rewards to chosen ties—considerations that move the analysis out of the utilitarian frame. These considerations connect to the sec- 1 A (constrained or unconstrained) Pareto-efficient allo- ond source of dissatisfaction with the classi- cation is an extremum on the Pareto frontier for a weighted sum of utilities (and a maximum when the feasible utility cal framework. It concerns the gap between set is convex). These weights are called Pareto weights or, the normative underpinnings of the theory in some specific contexts, Negishi weights (Negishi 1972). and the relevant fairness values that seem 2 Tagging (Akerlof 1978) makes the tax paid by an agent depend on a characteristic that is ethically irrelevant but important in income redistribution. For statistically correlated to some ethically relevant variable, instance, Weinzierl writes that “conventional such as the agent’s skill. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1031 efforts and to neutralize the impact of exter- proposed to deal with this difficulty by relying nal circumstances. Gaertner and Schokkaert on weights applied to earning levels directly, (2012) review the empirical social choice and this technical solution to the multidi- literature in which various notions of desert mensional screening problem is made more appear relevant in opinion surveys, and, fol- attractive if, as Saez and Stantcheva argue, lowing an experimental approach, Cappelen the weights on earnings can be directly et al. (2007) and Cappelin, Sørensen, and derived from ethical principles. Tungodden (2010) also find a similar diversity One of our main points, in this paper, is of egalitarian, libertarian, and meritocratic that the classical social welfare function attitudes.3 framework is more flexible than commonly The attempts to make optimal taxation thought, and can accommodate a very large theory sensitive to nonutilitarian ethical val- set of nonutilitarian values. More specifi- ues, such as fairness values, have sometimes cally, fairness concepts can help solve the led scholars to abandon not just the utili- interpersonal comparison difficulties that tarian criterion but even the classical social the utilitarian approach faces when agents welfare function framework. For instance, have different preferences by providing a certain libertarian tradition has inspired useful selections of suitable individual util- the equal sacrifice approach (Young 1987), ity indexes. We devote a significant part of which involves looking at departures from the paper to illustrating this message in the market incomes rather than the final distri- case of the libertarian approach and the bution of well-being. The idea of abandoning resource-egalitarian approach, both of which the social welfare function has been recently can be formalized with money-metric utili- systematized by Saez and Stantcheva (2016). ties that incorporate desert and fairness prin- They propose to apply marginal social wel- ciples into the objective of the planner. Not fare weights not to unobservable utility all approaches retaining the social welfare indices but directly to observed earning lev- function framework are equally successful, els. The weight at each earning level depends however. As argued by Piketty and Saez, the on the characteristics of the subpopulation of Pareto-weight approach, for instance, is less agents earning that level, and can be inspired able than other approaches to deliver recom- by fairness principles, including libertarian mendations consistent with the fairness val- notions of desert. In this fashion, Saez and ues one may wish to capture. Stantcheva are able to retrieve some of the Note that our defense of the social wel- tax results inspired by fairness principles and fare function could even be understood as to find new results for different principles. a defense of the utilitarian approach, for an It should be recalled here that when mul- ecumenical notion of utilitarianism that is tidimensional heterogeneity is introduced flexible about the degree of inequality aver- in the taxation problem, there are techni- sion and the definition of individual utility. cal difficulties in identifying the efficient Generally, however, utilitarianism is under- incentive-compatible allocations, due to stood as a narrower class of social objectives, the bunching of agents of different types at with a zero or low aversion to inequality and the same income levels. Saez (2001, 2002) with a definition of utility that relies on subjective satisfaction or happiness. In this paper, we mostly confine our 3 If one assumes that the prevailing taxes are optimal attention to the taxation of earnings and the with respect to social preferences, it may even appear that they reject the Pareto principle (see Bourguignon and economic literature. There is, of course, Spadaro 2012 for an application to France). an important literature on the taxation of 1032 Journal of Economic Literature, Vol. LVI (September 2018)

commodities, capital, and inheritance, and the latter is chosen, reward individual talent we will briefly allude to it at the end. There is or fight inequalities due to unequal skills; iii) also a literature in philosophy (e.g., Murphy insofar as the latter is chosen, prioritize equal and Nagel 2002) and in law (e.g., Zelenak tax treatment for identical skills or compen- 2006) that discusses the ethical foundations sation for unequal skills; and iv) insofar as the for taxing income and other possible tax latter is chosen, favor individuals with low or bases. It is indeed worth emphasizing that high aversion to work. This methodology is ethical principles may be relevant not only meant to be applicable by practitioners who to the design of the income tax, but also to want to be in control of the ethical under- the selection of the tax base. In particular, pinnings of their choices of utility functions a controversy (reviewed in Zelenak 2006) without having to go through arcane axio- rages on the pros and cons of seeking to tax matics, and in section 9, we show how the individuals’ earning potential rather than various approaches can be practically used to actual earnings, and connects to familiar dis- study tax reforms and to find the optimal tax. cussions in economics about first-best redis- We conclude in section 10. tribution. We will briefly refer to these issues when discussing first-best allocations under 2. Achievements of the Classical Approach various approaches. In the following sections, we begin with a Optimal taxation theory studies how to brief description of the main achievements design tax systems that maximize social of the classical Mirrlees approach (sec- welfare. Let us begin by defining the main tion 2). We then discuss the various possi- ingredients of the theory formally. As ble interpretations of the concept of utility, announced, we focus in this paper on the tax- viewed as a proxy for well-being (section 3) ation of earnings. There are two goods, labor and examine utilitarianism as an aggregator and consumption, and n agents. A bundle of well-being levels (section 4). We review for individual i N {1, … , n} is a pair ∈ = various fairness approaches to optimal tax- z ( , c ), where is labor and c con- i = ℓi i ℓi i ation in section 5: the libertarian approach, sumption. The agents’ consumption set X is Roemer’s equality of opportunity, and the defined by the conditions0 1 and ≤ ℓi ≤ resource-egalitarian approach, as well as the c 0. The restriction of labor to an inter- i ≥ luck-desert approach of Saez and Stantcheva val is not always made in the tax literature, (2016). In that section, we also discuss but it will play a role in our analysis of some Kaplow and Shavell’s (2001) challenge to approaches. fairness principles. In the section following The individuals have two characteristics, that, we examine why it is difficult to incor- their personal utility function over the con- porate fairness principles in a weighted sumption set and their personal productivity. utilitarian social welfare function (section For every agent i N, the utility function ∈ 6). Next, we analyze the derivation of fair U : X represents preferences over i → ℝ optimal tax and the usefulness of Saez and labor and consumption. We assume that Stantcheva’s approach in terms of marginal individual utility functions are continu- social welfare weights (section 7). In section ous, quasi-concave, nonincreasing in , and ℓ 8, we provide a simple methodology for link- increasing in c. ing the construction of utility functions with The marginal productivity of labor is four distinct ethical choices: i) rely on sub- assumed to be fixed, as with a constant jective utility or only on fairness principles returns to scale technology. Agent i’s earn- involving ordinal preferences; ii) insofar as ing ability is measured by her productivity Fleurbaey and Maniquet: Optimal Income Taxation Theory 1033 or hourly wage, denoted w i , and is mea- the heterogeneity of utilities over (y, c) can sured in consumption units, so that w 0 be described by a single parameter like i ≥ is agent i’s production when working 1 w : V y, c U y w , c . Second, the ℓi = i i( ) = 0( / i ) and y w denotes the agent’s earnings social planner is utilitarian, which means that i = i ℓi (pretax income). Let V (y, c) U y w , c the social ordering is defined as maximizing i = i( / i ) denote the utility function derived from the sum of utility levels: Ui and representing i’s preferences over earnings–consumption bundles. (1) Vi ( y i, c i) . An allocation is a collection of bundles ∑i z (z 1, … , z n). A tax function T : A more general social welfare function = ℝ+ → ℝ delineates the budget constraint c y T(y), involving transformations of individual utili- = − which, in terms of labor and consumption, ties has also been considered, but if the social reads c w T w for all individuals welfare function is additively separable, this ≤ i ℓ − ( i ℓ) i N. An allocation is incentive compatible is just equivalent to considering various non- ∈ if every agent maximizes his utility in his bud- linear rescalings of U0 . get set, or equivalently, if the self-selection The questions that have been addressed constraint is satisfied: for all i, j N, in the optimal tax literature deal with the ∈ first-best implications of social welfare V y , c V y , c or y w . 4 i( i i) ≥ i( j j) j > i maximization, design of second-best tax schemes, and social welfare evaluation of An allocation is feasible if T y G, ∑i ( i) ≥ tax reforms. The literature has, in particular, where G is an exogenous requirement focused on deriving different formulas for of government expenditures, or equivalently, the optimal tax rates in the second-best con- c y G. text. These formulas show how marginal tax ∑i i ≤ ∑i i − The problem of optimal taxation is to rates depend on the elasticity of labor supply, evaluate tax functions and seek the best one the distribution of productivity levels and the under the feasibility and the self-selection shape of the U 0 function (which determines constraints. Since Mirrlees (1971), the eval- the social marginal value of consumption uation of T is derived from a consequential- that the social planner assigns to the differ- ist evaluation of the allocation(s) z that T ent types of agents). generates when every individual makes his It must be emphasized here that the choice in his personal budget determined by tax literature has not focused only on w i and T. The evaluation of allocations has the selection of the best tax, but has also to be made with a social-ordering function, explored how to describe the set of efficient which, for every particular population profile incentive-compatible allocations, i.e., the ( ( U1 , … , Un ) , ( w 1, … , w n) ) , defines a spe- efficient taxes in the second-best context cific ordering (i.e., a complete transitive rela- (e.g., Stiglitz 1982, 1987, Guesnerie 1995). tion) on the set of allocations X n. This investigation about efficiency has often The classical theory of labor income tax- relied on the weighted variant of the utilitar- ation has been initially developed under ian social welfare function, which has proved two main assumptions. First, the individuals have different productivity levels, but they 4 In the first-best, only the feasibility constraint applies, all have the same preferences over labor– whereas in the second-best, the self-selection constraint consumption bundles, represented by a also applies. The first-best is attainable if the govern- single utility function: for all i N: U U ment knows individual characteristics, whereas in the ∈ i = 0 second-best, the government only observes earnings (and or, at least, as shown in Mirrlees (1976, 1986), knows the statistical distribution of characteristics). 1034 Journal of Economic Literature, Vol. LVI (September 2018) to be a versatile tool, and is discussed in more The weighted-income approach offers a details in section 6. valuable solution to the technical difficulties In the last fifteen years, the theory has of optimal tax theory in the presence of het- been enlarged to consider the more realis- erogenous preferences.5 Nonetheless, the tic case in which agents differ both in their question of how to make interpersonal util- preferences and their wages. Introducing ity comparisons and, more specifically, how additional dimensions of heterogene- to compare high-productivity/high-aversion- ity makes it considerably more difficult to to-work agents and low-productivity/high- derive formulas for the optimal tax rates. willingness-to-work agents remains complex. First, the objective of the planner is much This is where fairness considerations can more difficult to define, as it requires one help, as recently advocated by many authors. to compare agents with the same produc- To prepare the background for such devel- tivity but different preferences. Second, the opments, in the next section we go back to taxation of each income interval influences the fundamental question of the meaning of high-productivity/high-aversion-to-work utility and its use in optimal tax theory. agents and low-productivity/low-aversion- to-work agents. Determining how much to 3. What Is Utility? tax such an interval of incomes is more difficult than when all agents have the same The objective of optimal taxation theory is preferences because in the latter case, richer to go beyond the Pareto principle and select agents also have higher productivity. among second-best allocations the ones that Solutions have been found for particu- are better justified from a normative point of lar models (see, e.g., Boadway et al. 2002, view. This requires social evaluation criteria Jacquet and Van de Gaer 2011, Choné that involve cardinality and or comparability / and Laroque 2010). A general solution has judgments about individual well-being. Such also been proposed by Saez (2001, 2002), judgments are embedded in the utilities that recently refined and extended in Jacquet enter the computation of social welfare. and Lehmann (2014). Saez’s approach con- There are two main views on utilities. sists of modifying the way the objective According to the first view, which domi- of the planner is defined. It is no longer a nates in the utilitarian tradition, utilities function of agents’ utilities, but a function are empirical objects that only need to be of agents’ incomes. All agents earning the measured and can be used as the inputs of a same income, whatever their productivity, social welfare function, the only ethical issue receive the same weight, and the objective being the degree of inequality aversion in the of the planner is defined in terms of the function. According to the second view, util- relative weights that are assigned to sets of ities themselves, not just the social welfare people earning different incomes. More details about this approach are provided in 5 However, it is primarily a “first-order” approach that section 7. One should also mention that the does not deal with bunching. Multidimensional heteroge- case of linear taxation is simpler, as shown neity is addressed in Saez’s (2001) main text, but not in in Mirrlees (1976) and further studied by the formal proof of the tax formula. A full formal proof is provided by Jacquet and Lehmann (2014) for separable Sandmo (1993), showing that the key issue utility functions and assuming smooth allocations with no determining whether the optimal tax is pro- bunching (they adopt Wilson’s 1993 method of classifying gressive or regressive is how preference the population into preference types, with single-crossing being satisfied across skills within each type). In its full for leisure affects the marginal utility of generality, multidimensional screening remains largely an consumption. unsolved problem (Rochet and Stole 2003). Fleurbaey and Maniquet: Optimal Income Taxation Theory 1035

function, are normative indexes that need to Another version of the argument involves be constructed. adaptation. If declaring a high well-being The first view has serious weaknesses. level only reveals one’s ability to adapt to One can distinguish two main approaches objectively poor conditions, it does not jus- that adopt this view. In the first approach, tify a policy failing to address these poor con- utilities refer to subjective self-assessments ditions. Decancq, Fleurbaey, and Schokkaert of well-being. After a long period in which it (2015) emphasize that subjective well-being was believed that only preferences revealed data, by involving heterogenous aspiration by behavior were a suitable source of data levels, produce interpersonal comparisons about preferences, the use of surveys in that may disagree with the comparisons which people express their perceptions has made by the concerned individuals them- increased. This method has been popular- selves: a highly ambitious high achiever may ized in the last two decades by the economics have a better situation than someone else, as of happiness. It builds on answers to sur- unanimously evaluated by these individuals, vey questions like “Taken all together, how and yet have a lower satisfaction level.7 would you say things are these days? Would Philosophers have suggested the replace- you say that you are very happy, pretty happy, ment of utilities with other arguments. or not too happy?” There are many versions Dworkin (1981b), in particular, clearly advo- of this question. A variant relies on answers cates taking the bundles of resources that are to questionnaires that request the respon- assigned to agents as the appropriate argu- dent to decompose their time into a list of ment of a theory of justice. As we will see activities and, for each of them, to list and in section 5, some fairness approaches offer evaluate the positive and negative feelings ways to implement an ideal of equality of associated with it.6 resources. All these approaches are contemporary This rejection of utilities as an argument implementations of ideas that have long of a theory of justice by philosophers seems been salient in philosophy and economics. to echo a similar rejection by people when Criticisms of these approaches are also well they are asked to assess allocations. Yaari known. The most important, in the context and Bar-Hillel (1984) have initiated a lit- of this paper, comes from political philoso- erature, based on survey questionnaires, phy and was raised by Dworkin (1981a,b), dedicated to understanding the ethical Rawls (1982), and Sen (1985). It says that principles that guide people’s view on just subjective well-being is not a legitimate allocations. Summarizing that literature, argument of a theory of justice. One aspect Gaertner and Schokkaert (2012) write that of the criticism is the “expensive taste” argu- “the welfarist framework is not sufficient to ment. If declaring a lower well-being level capture all the intuitions of the respondents. only reveals a lower subjective disposition to […] Respondents distinguish between needs transform consumption into satisfaction, due and tastes and discount subjective beliefs to to a higher level of aspiration, it does not call a large extent. In general, intuitions about for compensation. distributive justice seem to depend on the context in which the problem is formulated” 6 Among many references, see in particular Clark, (pp. 94–5). The same authors also report the Frijters, and Shields (2008), Diener, Helliwell, and Kahneman (2010), Di Tella and McCulloch (2006), Dolan and White (2007), Graham (2009), Kahneman, Diener, 7 A much more extensive discussion of the subjec- and Schwarz (1999), Kahneman and Krueger (2006), and tive well-being approach can be found in Fleurbaey and Van Praag and Ferrer-i-Carbonell (2008). Blanchet (2013, chapter 5). 1036 Journal of Economic Literature, Vol. LVI (September 2018) fact that “issues of responsibility and account- Our conclusion on the literature on empir- ability, of acquired rights and claims, of asym- ical measures of well-being is not, however, metry between dividing harms and benefits, that they should be rejected. There are are highly relevant to understand real-world authors who are not convinced by the objec- opinions” (pp. 137–38). As we will see in sections against such measures. Some are con- tion 5, these questions are at the heart of the vinced hedonists and believe that individuals fairness approaches to optimal taxation. who pursue other goals than happiness are The second main approach that embraces mistaken (Layard 2005, Dolan 2014). Some the idea that utilities are empirical objects are more cautiously hoping that these mea- that only need to be measured refers to sures are good proxies of well-being and pro- choices under uncertainty. It is well known vide meaningful interpersonal comparisons that rational preferences can be represented on average (Clark 2016). Our review of the by von Neumann–Morgenstern (vNM) util- criticisms is meant to substantiate one point: ity functions, and such utility functions can adopting such measures cannot be done be given a cardinal meaning, provided one without relying on ethical assumptions. They assumes that risk aversion is a direct trans- are not neutral and ready-to-use measures.8 lation of preference intensity. This is the Once one acknowledges that the choice of assumption that Vickrey (1945) suggested, a particular utility measure is always strongly and Harsanyi (1976) and Mirrlees (1982) value laden, it is a small step to accept the endorsed it as well. second view and treat utilities as normative There are two main criticisms of this constructs. This second view was defended approach. The first criticism opposes the in particular by Atkinson (1995), and it is assumption that risk aversion is a measure of probably the dominant view among optimal intensity of preferences (Roemer 2008). This tax theorists. In the context of uniform pref- criticism rejects the view that vNM utility erences, Atkinson himself did not advocate functions can be given an ethically appealing relying on subjective well-being measures cardinal interpretation. and instead proposed to choose the least Even if one accepts the cardinal interpre- concave utility representation of the prefer- tation, though, vNM utility functions them- ences of the agents, and then to aggregate selves do not provide the comparability that them with a more or less inequality-averse one needs to build a social criterion, and this aggregator, reflecting the ethical preferences is especially relevant when one deals with het- of the social planner. Note that adopting a erogenous preferences. There have been pro- unique utility function when individuals posals to calibrate the vNM functions, e.g., by have identical preferences guarantees that letting all individual functions take the same interpersonal comparisons will align with the values (0 and 1) at particular points (Hildreth comparisons made by the individuals them- 1953, Dhillon and Mertens 1999, Adler 2012, selves—unlike subjective well-being levels Sprumont 2013). But it is debatable whether based on heterogenous aspiration levels. this makes the interpersonal comparisons Atkinson’s calibration is no longer appli- compelling. Like subjective well-being data, cable when agents have heterogenous they are vulnerable to the phenomenon that preferences. The least concave utility func- individuals with identical ordinal preferences tions of the individuals are then defined only but different risk attitudes may be ranked by up to a scaling factor, and are therefore not the calibrated vNM functions against their own assessment of their relative situations 8 The same point was hammered in Robbins’s (1935 (even in riskless contexts). [1937]) and Bergson’s (1938) classical texts. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1037 directly comparable. Additional assumptions an arbitrarily large loss to a minority. To put are needed to perform adequate interper- it differently, utilitarianism is unable to guar- sonal comparisons. antee a safety net to all agents, because an One of the main ideas that we would like increase in utility of a well-off agent may to defend in this paper is that the second offset a decrease in utility of a miserable view offers valuable ways to accommodate agent, independently of how low the utility interesting ethical principles about equality of this agent is, and a small utility gain for and redistribution. It is especially in the case many well-off individuals may outweigh a of heterogenous preferences that introduc- large utility loss for the miserable one. This ing ethical values may be particularly help- issue, however, is substantially alleviated in ful to guide interpersonal comparisons, and inequality-averse variants of utilitarianism. therefore the selection of utility represen- The second issue was identified by Mirrlees tations. As we will see, it turns out that the (1974). He proved that in a first-best world, traditional concept of money-metric utilities this social welfare function leads to the fol- is convenient and surprisingly versatile for lowing surprising result: if all agents have the incorporation of libertarian, as well as the same preferences, the high-ability agents resource-egalitarian, values. As a matter of end up enjoying lower satisfaction levels than fact, even in the case of identical preferences, the low-ability ones (assuming that leisure is money-metric utilities, based on libertarian a normal good). This is in sharp contrast with or egalitarian values may help select particu- what popular ethical views recommend. One lar scalings of utilities that are not obviously such view is that differences in productive found when one starts from the least concave abilities do not justify differences in out- representation and wonders about finding a comes. This calls for equalizing satisfaction suitable concave transform. Money-metric levels among agents with the same pref- utilities have not been used much in taxa- erences. Another popular view holds that tion theory, but have been quite common in agents own their ability at least partially, so other parts of the literature (such as the mea- that the high-ability agents should reach a surement of living standards across or within higher satisfaction level. different types of households).9 This issue is related to the common criticism that there is no place in utilitarianism for entitlements and rights. In his critical 4. Utilitarianism and Just Outcomes discussion of the Mirrlees Review, Feldstein The social criterion that is classically used (2012) points out the implicit assumption in optimal taxation theory is the utilitarian that redistribution is limited only by dis- social welfare function that adds up utili- incentive effects. How to accommodate ties. Several issues have been raised in the libertarian ideas in taxation theory will be literature about this criterion, in relation to discussed in the next section. As we will see certain notions of fairness inspired by egali- in section 9.2, however, certain approaches tarianism, libertarianism, or combinations of based on nonutilitarian ethical principles both. recognizing individual rights can also penal- The first issue was pointed out by Rawls ize some or all of the talented individuals in (1971). Utilitarianism is able to produce the the first-best allocations. undesirable outcome that a majority imposes The third issue is emphasized by Piketty and Saez (2013b) and echoes the liberal 9 See the reviews of this literature in Lewbel (1997) and egalitarian literature initiated by Rawls and Chiappori (2016). Dworkin. Maximizing a sum of utilities, even 1038 Journal of Economic Literature, Vol. LVI (September 2018) weighted,10 cannot produce the desirable expense of the individuals who were already properties that i) utility should be equalized worse off in the absence of tagging. when all agents have the same preferences, Let us assume, for instance, that skill is an objective which we will refer to later as the positively correlated to height. Because compensation objective,11 and ii) laissez-faire height is observable, and because tall people should prevail when all agents have the same should, on average, pay more tax than small productivity level, an objective which we agents (as utilitarianism justifies redistribut- will refer to as the laissez-faire objective. ing from richer to poorer agents), the optimal By “laissez-faire,” we mean the imposition tax scheme would consist first in redistribut- of a poll tax T y G n on all individ- ing a lump-sum amount of money from the ( ) = / uals, which is equivalent to the absence of tall agents to the small ones and, second, in redistribution. optimal second-best tax schemes among the The double goal of equalizing utilities small agents and among the tall agents. It is among agents having the same preferences clear that the small unskilled agents will ben- and not redistributing among agents having efit from the tagging, and it is unfortunate the same productivity is at the heart of the that the tall unskilled will suffer as a result. resource-egalitarian approach, and is also It is very hard to reject tagging when the closely connected to the equality of opportu- objective is maximizing a social welfare func- nity approach, both of which are reviewed in tion that does not put an absolute priority the next section. on respecting horizontal equity. However, The fourth issue is related to tagging the problem is partly alleviated if the crite- (Mankiw and Weinzierl 2010). It is clear that rion is strongly egalitarian, such as a maximin tagging represents an additional tool in the criterion. Whenever such a social criterion maximization of a social objective, because it increases, by definition this cannot harm uses relevant correlations between observed the worst off. Concretely, with tagging, the and unobserved individual characteristics unskilled, whether small or tall, will end to better target redistribution. Tagging, up enjoying the same increased satisfaction though, has been criticized on the ground level. But violations of horizontal equity that it leads to violations of the basic princi- will still typically happen higher up in the ple of equal treatment of equals. Indeed, if distribution. two agents that are identical in all ethically Observe how the maximin criterion, relevant dimensions but differ with respect applied to any utility indexes, escapes three to the dimension along which people are of the four shortcomings that we just men- tagged, they may be treated differently and tioned. It protects the worst off against sac- end up in unequal positions. rifices imposed on them to the benefit of If the social criterion is utilitarian, the sum well-off individuals. It produces equality in of the utilities will necessarily increase with the first best and therefore does not penalize tagging, but at the cost of these ethically the talented. And it implies respecting hori- dubious inequalities of treatment between zontal equity among the worst off. However, individuals. Moreover, these extra inequal- the maximin criterion has also been severely ities due to tagging may operate at the criticized for granting an absolute priority to the worst off, with devastating consequences for the better off in some circumstances 10 This holds true unless the weights are endogeneized (Arrow 1973, Harsanyi 1976). It is indeed in a very complex way—see section 6. 11 The word compensation reflects the goal of eliminat- striking when, for instance, the maximin ing the inequalities due to all causes other than preferences. requires choosing a steady state in a growth Fleurbaey and Maniquet: Optimal Income Taxation Theory 1039 model in which there exist constant-growth that lower their utility should receive less paths in which every generation except the resources if their marginal utility is also first one is better off than at the maximin lower, and this is the opposite of what typical allocation. But a main point of this paper is fairness criteria would recommend.12 that the choice of the utility index is crucial for this sort of issue. As we will see in the 5. Fairness Approaches to Optimal next section, the maximin is actually compat- Taxation ible with a very low degree of redistribution, and even with the most libertarian approach, In this section, we review the main fairness when rights and entitlements are embedded approaches to optimal taxation. By fairness, in the utility index. we refer to approaches that impose ethical Many authors consider that the utilitar- (typically libertarian or egalitarian) require- ian approach is strongly supported by the ments on objects other than ordinary utili- veil-of-ignorance argument (Harsanyi 1953) ties. Before we begin this review, though, and similar arguments such as Harsanyi’s we need to clarify the relationship between (1955) aggregation theorem. In the context our notion of fairness and the Pareto prin- of income taxation, one can view the utili- ciple. This can be usefully discussed with tarian criterion as reflecting the ex ante per- Kaplow and Shavell’s critique of fairness in spective of an individual who could end up the background. at any position in the distribution of wages 5.1 Kaplow and Shavell on Fairness (Varian 1980 applies this idea to risk bearing on returns to savings). This makes the utili- Kaplow and Shavell (2001, 2002) have tarian criterion particularly suitable to incor- argued that any continuous and non-welfarist porate the insurance perspective that makes method of policy assessment violates the many citizens support redistribution. Pareto principle, defined as stipulating that Whether the utilitarian criterion is actu- an allocation is strictly better than another ally appropriate in the context of risk, and if everyone is better off in it. In their work, whether the veil-of-ignorance idea is a good welfarism is defined by the principle of guide to think about redistribution, are two Pareto indifference, according to which soci- hotly debated topics in the literature. On ety should be indifferent between two allo- the first issue, there are critics who point cations if all agents are indifferent between to the lack of attention of utilitarianism to them.13 Clearly, the Pareto principle, under a inequalities, both ex ante (Diamond 1967) suitable continuity condition, implies Pareto and ex post (Broome 1991). On the second indifference. Therefore, under the same issue, Dworkin (2000) has developed a the- continuity condition, a violation of Pareto ory of justice around the idea of mimicking indifference implies incompatibility with the what an insurance market would produce Pareto principle. if individuals’ characteristics were insurable from behind a veil of ignorance. This idea 12 A recent survey on the first issue can be found in has been criticized for failing to incorporate Mongin and Pivato (2016). A long discussion of the second sufficient attention to inequalities, due to issue is in Fleurbaey (2008). 13 In the theory of social choice (e.g., d’Aspremont and the fact that individuals, ex ante, may want Gevers 2002), welfarism usually combines Pareto indiffer- to plan to consume less resources in states ence with an independence principle, and is defined as the with lower marginal utility (e.g., Roemer property that not only is the social ordering over allocations derived from a social ordering over utilities, but in 1996). Translated into redistribution, this addition, the social ordering over utilities remains the same means that individuals born with disabilities independently of the profile of utilities of the population. 1040 Journal of Economic Literature, Vol. LVI (September 2018)

These authors target the normative Surprisingly, however, a good deal of lib- theories that discriminate among Pareto- ertarianism can actually be embedded in the indifferent allocations on the basis of fairness choice of the utility functions, as we show in principles. A typical example, in the context this paper. Many of the fairness approaches of taxation, is the principle of horizontal that we review in this section can be formu- equity stipulating that it is better if people lated with a social welfare function and do with the same earnings obtain the same final satisfy the Pareto principle.15 consumption, which we have discussed ear- 5.2 Libertarianism lier in relation to tagging. Other examples include the socialist principle that consump- Maybe the most radical departure from tion should be proportional to labor, or the the utilitarian approach to taxation is the lib- libertarian principle according to which the ertarian view that, absent all kinds of mar- laissez-faire allocation is superior to any ket imperfections, the competitive income other allocation. These authors’ critique also distribution is just. The maximization of a aims at the theories of justice that recom- social welfare function is replaced by the mend gauging individual advantage not in application of an extended version of the terms of well-being, but in objective terms laissez-faire objective: earnings are fair if they such as resources or capabilities.14 reflect the natural differences among people. Kaplow and Shavell argue that fairness These differences come from differences in principles violating Pareto indifference talents, which are rewarded at their marginal are therefore harmful and they propose to productivity, and from differences in prefer- restrict considerations of fairness to inequal- ences, which are rewarded proportionally to ity aversion over well-being, which is per- labor times. Inequalities due to differences fectly compatible with the standard social in earning capacities are no longer consid- welfare approach. ered unjust. As Mankiw (2010) puts it, “each As mentioned in the introduction, in person’s income reflects the value of what he this paper, we highlight another way in which at least some fairness principles can 15 It is easy to be confused about whether such remain compatible with the Pareto princi- fairness-inspired social orderings should be called welfarist ple. Fairness principles can indeed guide or not, according to the social choice terminology intro- the selection of the individual utility func- duced in the previous footnote. Consider a social welfare ∗ ∗ ∗ function W(u 1 , … , u n ) for which u i is a particular rep- tions that serve to measure well-being and resentation of individual preferences that is selected out perform interpersonal comparisons. Insofar of fairness principles. Obviously, the social ordering over ∗ ∗ as a fairness principle helps picking utility ( u 1 , … , u n ) represented by W is welfarist because it does not change when the profile of utilities change. But sup- functions among the set of functions that pose that one rewrites the same social welfare as a function faithfully represent individual preferences, of empirically given utilities (u 1, … , u n) , e.g., happiness it does not clash with the Pareto principle. data, so that Not all fairness principles fall in this cate- W u ∗ , … , u ∗ W ∗ u , … , ∗ u , ( 1 n) = (φ1( 1) φn ( n)) gory, obviously, and the socialist and liber- ∗ where the i transformations bring the fairness consider- φ ∗ tarian principles mentioned two paragraphs ations and produce the relevant u i . It is then generally the earlier provide examples of non-Paretian case that the social ordering over ( u 1, … , u n) is not stable ∗ when ( u 1, … , u n) changes, because the transformations i approaches. φ∗ have to be changed. For instance, if the selection of u i ∗ depends only on individual ordinal preferences, u i does not change when u i changes without altering the preferences, therefore requiring ∗ to change. It is with refer- φi 14 See in particular the discussion in Kaplow (2008, ence to this observation that Fleurbaey, Tungodden, and chapter 13). Chang (2003) call such fair approaches non-welfarist. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1041 contributed to society’s production of goods progressive taxation, since the same absolute and services.” In a classical paper, Feldstein utility loss requires a greater tax for richer (1976) had earlier drawn attention to the lib- taxpayers when marginal utility is decreas- ertarian viewpoint and its implications for ing. A classical analysis of equal sacrifice was taxation, inspired by Nozick (1974). made by Young (1987). The libertarian view is in radical opposi- The benefits approach has been thor- tion to the pure redistributive objective of oughly studied in the fair allocation litera- labor income taxation, an objective that is ture (classical references are Kaneko 1977, shared by utilitarian ethics and the other Moulin 1987). In this literature, the distribu- fairness approaches that we discuss below. tion of resources is assumed to be fair prior In the presence of market imperfections to the production of public goods (which such as pollution, or public goods such as is in line with the libertarian principles of security, libertarianism is compatible with this section) and the question is to compare some taxation of incomes (as explained in the gain in well-being obtained by different Mankiw 2010).16 The issue of externalities people with different preferences over the has recently been tackled by Lockwood, private–public good trade-off. Lindahl equi- Nathanson, and Weyl (2017), who study the libria (which treat public goods like private Pigouvian taxes that would be needed to curb goods but with person-specific prices) have externalities attached to various professions, been criticized for being too indeterminate in the situation in which professions cannot and for failing to satisfy basic fairness condi- be taxed directly and one has to estimate the tions (at a Lindahl equilibrium, for instance, average externalities produced by the vari- identical individuals may end up facing dif- ous professions present at any given level of ferent Lindahl prices). Alternative propos- earnings. They actually show that being able als often rely on the equivalence approach, to directly tax or subsidize occupations would which consists, in the first-best context, in be much more effective than the income tax putting every individual in a situation that (even though the policy may still be imper- is equally good as a benchmark situation fect because externalities may vary a lot (such as enjoying a certain quantity of the within occupations). public goods for free). While the literature For the case of public goods, there are two initially focused on the first-best context, main schools of thought. One school advo- the second-best context has been addressed cates taxing according to the benefit received in Maniquet and Sprumont (2004, 2005, by the tax payers, the other proposes to min- 2010) and Fleurbaey and Maniquet imally distort the distribution by imposing (2011a). equal sacrifice to all. As shown in Weinzierl But this literature simply ignores earn- (2014a), the two approaches may sometimes ings, which are left out of the model, and recommend the same allocation. In general, focuses on the contributions to the public however, they diverge, with the equal sac- good as they relate to individual prefer- rifice approach being more likely to induce ences. Therefore, it is silent about the inter- action between income taxation and the funding of public goods. It would be worth 16 The absence of market imperfections is the typical assumption of the Mirrlees optimal tax approach, as the exploring how income tax can be used for wage rates are assumed to be equal to the productivity of funding public goods, under such benefit the agents. The Mirrlees framework has of course been criteria, when there is a known correlation extended to take market imperfection into account, such as public goods, or non-competitive labor markets (see between individual preferences about the reviews in Kaplow 2008 and Boadway 2012). private–public good trade-off and individual 1042 Journal of Economic Literature, Vol. LVI (September 2018)

earnings.17 In an interesting variant of the consists in suitably choosing the utility func- model, Weinzierl (2014a) considers the dif- tions representing agents’ preferences. ferent case in which the public goods affect A key concept here is the notion of individuals’ private productivity,18 so that money-metric utility, due to Samuelson income taxation then appears an even more (1974) and which, in this model (after nor- natural way to apportion contributions to malizing the price of consumption to 1), can benefits (preferences are assumed to be be defined as the value of the expenditure identical in his model). function for a reference wage rate w and the Libertarianism, either in the benefit or utility level U i ( z i) (note that in the following equal sacrifice variant, does receive some formula, as well as later on in the paper, we degree of popular support. In a recent sur- use t to denote lump-sum transfers): vey bearing directly on the ranking of tax schemes, Weinzierl (2014b), for instance, m i( w, z i) min{t | ( , c) X, finds that no less than 33 percent of the = ∈ ℝ ∃ ℓ ∈ respondents favor tax schemes in which even c t w , the poorest people contribute to the funding = + ℓ of governmental expenditures (20 percent Ui ( , c) Ui ( z i) . favoring an equal-sacrifice kind of taxation ℓ ≥ } scheme and 13 percent favoring a poll tax). The remaining majority of respondents still To put it differently, m i( w, z i) is the lump-sum favor redistribution toward the poorest, but transfer (negative if it is a tax) that leaves in a way that is less generous than what the agent i indifferent between consuming z i or utilitarian or the maximin in utility social receiving that lump-sum amount and being welfare functions would recommend. free to work at wage w. An important feature The main point we want to make here is of this definition is that wagew plays the role that the libertarian ethics we just described of a parameter of the money-metric index, can still be adapted to the social welfare and need not be the actual wage of agent i. function framework, at least to some extent. A second feature is that, once w is fixed, the More precisely, we can show that it is pos- function m i( w, ⋅ ) is a numerical representa- sible to construct a social welfare function tion of i’s preferences.19 For further refer- that achieves the laissez-faire allocation in ence, let the set the absence of market imperfections, and that proposes a benefit-based allocation in , c X | c m w, z w {(ℓ ) ∈ = i( i) + ℓ} the case that a fixed amount of government expenditures has to be collected or public be called i’s “implicit budget” (at reference goods have to be funded. The central issue wage w ). This is the budget with lump-sum transfer and wage rate w that would enable i to obtain utility Ui ( z i) . Note that z i need not 17 The classical literature has focused on the validity belong to that budget. of the Samuelson rule for public goods in the presence Consider the following social ordering. It of income taxation (e.g., Christiansen 1981 and Boadway applies an inequality-averse social welfare and Keen 1993) and on the marginal cost of public funds (e.g., Gahvari 2006). In a discrete model (finite number of function W to individual well-being indexes individual types, discrete public good), Bierbrauer (2009) introduces a double heterogeneity in skills and preferences for the public good, but retains a utilitarian objective. 19 This is an immediate consequence of the fact that, for 18 Christiansen (1981) also studies a special case in any fixed w, the family of sets { , c X | c t w } is (ℓ ) ∈ ≤ + ℓ which the public good is an input. nested and continuously increasing in t. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1043 defined as the value of themoney-metric Let us come back to the laissez-faire utility function at the personal wage rate w i. recommendation and study under which This social ordering is then represented by conditions it could come out of the maximi- the function zation of a weighted sum of subjective utilities, for a suitable choice of weights. The W m w , z . laissez-faire allocation being first-best effi- (( i( i i))i N) ∈ cient, it is clear (under mild assumptions) Assuming that G 0, the laissez-faire allo- that it maximizes a weighted sum of utilities, = cation achieves m w, z 0 for all i, and but the weights depend on the allocation i( i i) = any form of redistribution generates a nega- in a complex way. In particular, the weight tive m i( w i, z i) for some i. Moreover, for every of an individual depends on characteristics feasible allocation the average m i( w i, z i) is (preferences, productivity) of other individ- non-positive. Therefore, given the inequal- uals. In contrast, the money-metric utility is ity aversion of W, the laissez-faire alloca- easy to compute and only depends on the tion is among the best feasible allocations. individual’s own characteristics. Plugged in Moreover, this ordering is intuitive because to any economy with an arbitrary profile, it considers that the worst off are those who and any inequality-averse social welfare are in a situation equivalent to suffering the function, it makes the laissez-faire allocation largest lump-sum tax in the population. one of the best. The limitations of weighted Note that the same laissez-faire conclu- utilitarianism will be further discussed in sion is obtained even when W is the maximin section 6. criterion To incorporate libertarian values into the social objective, Weinzierl (2014a,b) W m w , z min m w , z . proposes to take the first-best allocation as (( i( i i))i N) i i( i i) ∈ = a benchmark and maximize a sum of utilities computed in a way that incorporates a The maximin may require that nobody be cost of deviating from the benchmark. This taxed! This shows that a social ordering based is tantamount to defining social welfare by on the maximin aggregator can be compati- the (opposite of the) distance to the desired ble with a wide array of redistributive poli- first-best allocation. This is the most obvious cies. The choice of utility indexes is key. way to extend an allocation rule that selects Let us now look at the optimal allocation a particular allocation into a social order- according to this maximin ordering if a fixed ing that ranks all allocations, and can then budget G needs to be collected. Again, the be maximized in the second-best context. egalitarian objective must yield equality in Varian (1976) proposed something similar m i( w i, z i) , which is achieved when all agents for the incorporation of the no-envy crite- pay an identical tax G n and are then free to rion into a social welfare function. What we / choose their labor time, being paid at their suggest here is that the alternative method own wage. The same social ordering that jus- of incorporating the values that underlie the tifies thelaissez-faire in the pure redistribu- selection of the first-best goal into the indi- tive problem therefore recommends the poll vidual utility measures is a less obvious, but tax to finance public expenditures. Note that quite effective, option. this is obtained when public expenditures (or To see how flexible this alternative meth- their outputs) do not matter directly to indi- odology is, consider an extension of the vidual preferences. A different case is exam- model of this paper, in which there is a bun- ined below. dle of public goods g that enters individuals’ 1044 Journal of Economic Literature, Vol. LVI (September 2018) consumption bundles z , c , g . As in 5.3 Roemer’s Theory of Equality of i = (ℓi i ) Hammond (1994), pick a reference value Opportunity g and define the money-metric utility ̃ In contrast to Mankiw, Roemer’s (1993, 1998) theory of equality of opportunity is m w, z min t | , c X, against rewarding individuals for their nat- i( i) = { ∈ ℝ ∃(ℓ ) ∈ ural talents, but it retains an idea of des- c t w , ert. This theory is inspired by followers of = + ℓ Rawls and Dworkin in political philosophy, U , c, g U z . especially Arneson and Cohen (see Arneson i(ℓ ̃) ≥ i( i)} 1989, and Cohen 1989). It is also closely con- The choice of the reference g is convenient nected to the capability approach proposed ̃ in order to assess the benefit enjoyed by indi- by Sen (1985). The central idea of the the- viduals with different preferences. If a low ory is that one should provide individuals value of g is retained, then the individuals with equal or at least equivalent menus of ̃ with strong preference for the public goods options (called opportunities by some, capa- will be considered pro tanto advantaged by bilities by others). This idea naturally implies the extra quantity of g that is made available. that the sources of inequalities in individu- In contrast, if a value greater than the current als’ achievements should be divided into two g is retained as the reference, the individuals groups. The first group gathers individual with a strong preference for the public goods characteristics for which agents should not will be considered pro tanto disadvantaged, be held responsible. Such characteristics i.e., they will have a lower money-metric util- call for compensation, which means that the ity because it subtracts the larger amount of resulting inequalities in outcomes should consumption they would be willing to pay for be eliminated. They are called the “circum- more public goods. This shows that the “ben- stances” of the individuals, and define the efit” approach can be extended into a “harm” “type” of individuals. approach if one considers that the current The second group gathers the character- production of public goods is insufficient istics for which individuals should be held and harms those who are eager to have more responsible, typically because individuals public goods. control or choose them. They are called If one instead considers Weinzierl’s model “effort variables.” Agents should be held in which the public goods do not enter util- responsible for their effort, which means ity directly but only affect productivity, one that society should be indifferent to inequal- can introduce wage functions w i( g) and rely ities in agents’ outcomes that are caused by on the same notion of money-metric utility, such characteristics. This is a key innovation which, in this particular case, reads: in welfare economics, and it is in sharp contrast with utilitarianism and welfarism as a m w , z min t | , c X, whole, since the causes of individuals’ out- i( i i) = { ∈ ℝ ∃(ℓ ) ∈ comes play a key role in the evaluation of c t w g , individual situations. = + i( ̃)ℓ The social criterion that follows from U , c U z . these principles works as follows. Roemer i(ℓ ) ≥ i( i)} assumes that individual outcomes are car- We will illustrate the implications of this dinal and comparable. The set of agents is approach for optimal taxation in section 9. partitioned according to their “genuine” Fleurbaey and Maniquet: Optimal Income Taxation Theory 1045 effort—more will be said about this notion goal would then be that two agents having in the next paragraph. In each effort group, different skills but the same effort level the worst off are given absolute priority, and should also have the same outcome level. We social welfare is computed as the average are then left to define effort and outcome. value of outcome for the worst off of all effort If income is again retained as the relevant subgroups. In other words, social welfare is outcome, and individual effort is measured based on the maximin criterion within effort by the agent’s percentile in the distribution groups, reflecting the compensation ideal for of his skill group, then the goal becomes individuals with identical effort but unequal the maximization of the average income of circumstances; but the utilitarian criterion the unskilled agents if their distribution of is applied between effort groups, because income is first-order stochastically domi- there is no concern for inequalities linked to nated by the income distributions of all other differential effort. skill groups. This is reminiscent of Besley Roemer also advocates a particular way and Coate’s (1995) study of optimal taxation to measure genuine effort. He measures an under the goal of minimizing the poverty individual’s effort as the percentile of the rate. One worry about such an approach distribution of outcomes at which this indi- focusing on income is that it is unlikely to vidual stands in the subgroup sharing his cir- satisfy the Pareto principle.20 cumstances (i.e., the individual’s type). The Another possibility, more respectful of measurement of effort therefore requires individual preferences, would be to take util- partitioning the population by types, and ity as the outcome (assuming there is a com- measuring effort within each type by the rel- parable measure of utility). The approach ative ranks in the distribution of outcome. would then define effort as the relative rank Roemer et al. (2003) apply the approach to of an individual in the distribution of utility income taxation. The relevant achievement is in his type. If the distribution of utility for assumed to be income. Observe that income unskilled agents is dominated by the dis- is indeed a cardinal and comparable outcome. tribution of utility for the other types, then The set of circumstances is restricted to the the goal is to maximize the average utility level of education of the individuals’ parents. of the unskilled agents. This is similar to an The set of efforts is assumed to gather all approach followed by Boadway et al. (2002) the characteristics that generate variations in the special case in which there are two skill in how the influence of parents’ education levels and two preference types in the econ- is transformed into income. The tax systems omy. In that paper, preferences are assumed in ten countries are then compared in terms to be quasi-linear in leisure, which suggests of their ability to equalize the distribution of a natural cardinalization of the preferences. incomes across types. The optimal linear tax As Roemer assumes that the relevant is computed and the tax rate is compared to outcome is cardinal and comparable, his the average tax rate in the various countries. approach does not solve the difficult ques- One may think of many other application of how to construct utilities. In fact, tions of Roemer’s theory to optimal taxa- he explicitly recommends not to apply his tion. In particular, the set of circumstances approach to utilities and has restricted can be much larger than the parental edu- attention to cases in which the outcomes cation level. It would come closer to the naturally come in cardinal and comparable classical objective of optimal taxation theory to assume that the circumstances of an 20 When all individuals enjoy the same circumstances, agent include her skill. The compensation Roemer’s criterion implies maximizing total income. 1046 Journal of Economic Literature, Vol. LVI (September 2018) units, such as incomes. This severely limits ending up on the same indifference curve).21 the relevance of his approach for optimal This is clearly a pairwise (hence, stronger) taxation conceived as a tool for improving version of the compensation objective which, social welfare. But there does not seem to be as defined in section 4, dealt only with the a fundamental objection against seeking to case in which all the population has identical extend his approach to social welfare by tak- preferences. ing some relevant notion of well-being as the The approach also retains a responsibility outcome. objective, but not Roemer’s utilitarian objec- Responsibility, in Roemer’s approach, and tive. This principle is replaced with a pair- desert in the libertarian perspective, seem to wise version of the laissez-faire objective: follow the same objective, but they are quite there should be no redistribution between different. The difference is best seen if one agents having the same skill level, i.e., they thinks of an economy in which all agents have should be submitted to the same degree of the same circumstances, including the same redistribution.22 skills. The libertarian approach recommends Let us note that by combining the pair- that agents be rewarded according to their wise compensation objective with the pair- common level of skills, and all income differ- wise laissez-faire objective, this approach ences then come from different choices, for offers a solution to Piketty and Saez’s criti- which no correction is needed. Laissez-faire cism of utilitarianism (section 4). Indeed, is then considered fair. In contrast, for this restricted to economies in which all agents economy, Roemer’s equality of opportunity have the same preferences, the pairwise approach recommends indifference about compensation objective boils down to the inequalities in outcomes, meaning that the compensation objective. Restricted to econ- utilitarian criterion should be applied. As a omies in which all agents have the same consequence, the optimal policy has no rea- productive skill, the pairwise laissez-faire son to coincide with the laissez-faire (except objective boils down to recommending the when utilities are quasi-linear in consump- laissez-faire allocation. It is also worth noting tion). The Roemer approach is therefore that, according to Gaertner and Schokkaert compatible with income redistribution, even (2012), there is substantial empirical support in economies in which all agents have the for this combination of compensation and same earning capacities. laissez-faire.23 5.4 The Resource-Egalitarian Approach 21 It should be clear that this principle involves only The third fairness approach we survey purely ordinal preferences. Enjoying the same satisfac- offers a combination of the previous two tion level, indeed, means that these two agents should consume bundles they deem equivalent. Alternatively, approaches. This approach takes inspiration the requirement can be stated by reference to the (ordi- from the resource egalitarian philosophical nal) fairness concept of no-envy, introduced in the formal literature (Rawls, Dworkin) as well as the theory of fair allocation by Kolm (1999 [1972]) and Varian (1974): such agents should not envy each other. economic literature on fair allocation the- 22 In the literature, the pairwise laissez-faire objec- ory. It pursues the compensation objective, tive has been variably called the responsibility principle, under the assumption that the characteris- the natural reward principle, or the liberal reward principle (see Fleurbaey 2008 and Fleurbaey and Maniquet tics for which individuals are held responsi- 2011a,b). ble are their preferences. The compensation 23 However, they also obtain results that are not in principle then requires that agents having accordance with the theoretical literature. For instance, some respondents want to widen the market inequalities, the same preferences should also enjoy even when they are due to innate talent. A Nietzchean the same satisfaction level (in the sense of view on redistribution? Fleurbaey and Maniquet: Optimal Income Taxation Theory 1047

In this section, we illustrate the combina- social ordering that maximizes min i N U0 ( z i) tion of the compensation and the laissez-faire is then exactly the same as every member∈ of objectives by introducing a new class of social the reference-wage egalitarian-equivalent orderings. How to choose a particular order- class. Indeed, when preferences are iden- ing among this class, in relation to a variety tical, the ranking of individuals in terms of of ethical principles, is discussed in greater money-metric utilities is then the same as the detail in section 8. ranking in terms of utility U0 ( z i) , whatever The money-metric utility is the key tool, w , because m w , z is a numerical repre- ̃ i( ̃ i) here again. Consider the following social sentation of the same preferences as U 0 , for ordering. It applies the maximin criterion all w . The result that utilities are equalized in ̃ to individual well-being indexes, which are a first-best context then follows from the fact defined as the value of themoney-metric that the social ordering is a maximin. utility function at a common reference wage One may worry that when preferences are w . This social ordering is then represented identical but utility functions differ, picking ̃ by the function24 a common representation or a money-metric utility that only depends on ordinal prefer- min m w , z . ences ignores potentially relevant inequal- i N i( ̃ i) ∈ ities in utilities that come from unequal For the moment, let us only assume that capacities for enjoyment (see, e.g., Boadway the reference wage lies between the lowest 2012, p. 521). In order to examine this issue, and the largest wages observed in the popu- two possibilities must be considered. lation: w min w , max w . It, therefore, The first possibility is that the different ̃ ∈ [ i i i i] has to be a function of, at least, the profile of calibrations of satisfaction simply reflect that wages in the population. Letting this func- some individuals are more difficult to satisfy tion remain unspecified, we thus obtain a than others. This directly connects to the dis- class of social-ordering functions, rather cussion of “expensive tastes” and adaptation than a precise one. Let us call this the class in section 3. It can be argued that fairness of reference-wage egalitarian-equivalent25 is on the side of well-established approaches social-ordering functions. that ignore such differences in utilities. The first point we want to make here is The second possibility is that the capaci- that, in a first-best world, every member of ties for enjoyment reflect internal parameters this class of social-ordering functions satisfies (metabolism, health, mental health, etc.) that the combination of the compensation and matter to individuals and create real inequal- laissez-faire objectives discussed in section 4. ities. This means that the model is incom- Consider the case in which all preferences plete and such internal parameters have to are identical. Pick any common represen- be made explicit, together with individual tation of the agents’ preferences, U 0 . The preferences over them. The bundles can then be denoted z , , where denotes ( i θi) θi the internal parameters. Note that individu- 24 The well-being index m w , z is the money-metric als then have three sources of heterogene- i( ̃ i) utility discussed in Preston and Walker (1999, p. 346) for ity in such an extended model (preferences, this same model. 26 25 The idea of “egalitarian-equivalence” is due to Pazner wages, and internal parameters). The and Schmeidler (1978). The expression refers to a social criterion that seeks to achieve an allocation that is Pareto indifferent to an egalitarian allocation. In the case at hand, 26 Such a triple-heterogeneity model is studied by the egalitarian allocation grants all individuals an equal Valletta (2014). The problem of compensating for unequal budget with a lump-sum tax and a wage rate equal to w . internal characteristics is developed at length in a related ̃ 1048 Journal of Economic Literature, Vol. LVI (September 2018) extension involves defining money-metric (i.e., seek to eliminate all inequalities utilities between pairs of individuals who differ only in their productivities) is fully satisfied. m w , , z min t | , c X, This compensation effort actually goes i( ̃ θ̃ i) = { ∈ ℝ ∃(ℓ ) ∈ beyond the case of individuals with identical c t w , preferences. It also applies to cases in which = + ̃ ℓ one agent’s indifference curve in X lies every- U , c, U z , . where above another agent’s. In such cases, i(ℓ θ̃) ≥ i( i θi)} the money-metric utility of the individual at In conclusion, either way, the worry can be the lower indifference curve is necessarily suitably addressed. We continue the discus- lower than the other’s. Consequently, the sion, now, under the assumption that agents maximin objective recommends to transfer only differ in productivity and preferences. goods from the former to the latter agent. Let us come back to the combination of Such a transfer reduces the “inequality in the compensation and laissez-faire objec- indifference curves” between these two tives and check the laissez-faire side of the agents. picture. When all productivities are equal, The same pairwise extension holds for w must necessarily equal the common wage, the laissez-faire property, but in a more ̃ and equality of m w , z is achieved by the modest form.28 It applies to individuals i( ̃ i) laissez-faire allocation (with a poll tax G n), having the same wage, but only when their / which is also efficient and incentive com- common wage is equal to w . For this spe- ̃ patible, and therefore maximizes the lowest cial case w w w , the counterpart i = j = ̃ m w , z under the relevant constraints of of the laissez-faire ideal is that the equality i( ̃ i) feasibility and incentive compatibility. m w , z m w , z is achieved in the first i( ̃ i) = j( ̃ j) Admittedly, the two cases of identi- best. This means that the two individuals are cal preferences and equal productivi- just as satisfied as they would be by receiv- ties are very special, and it is important to ing an identical lump-sum transfer or paying check if a social-ordering function in the an identical lump-sum tax and working at reference-wage egalitarian-equivalent class their common wage. For individuals with a behaves well in other cases. The main obser- common wage w w that does not differ i = j vation is that the compensation property too much from w , this laissez-faire property ̃ for identical preferences indeed applies to will only hold approximately, i.e., the equality pairs of individuals. Reducing inequalities m w , z m w , z will be approxi- i( i i) = j( j j) between two individuals sharing the same mately satisfied. preferences is always deemed acceptable Therefore, one sees that a reference-wage for a reference-wage egalitarian-equivalent egalitarian-equivalent social-ordering func- social ordering, and is even deemed a strict tion consistently seeks to reduce inequal- improvement if one considers the leximin ities in indifference curves (whether they variant of such a social ordering.27 In other belong to the same preferences or not), and words, the pairwise compensation principle in a milder form seeks to avoid allocations that depart too much from the laissez-faire when redistribution is not needed. The literature (Fleurbaey 2008, Fleurbaey and Maniquet 2011b, 2012). 27 The leximin extends the maximin lexicographically by 28 It is impossible to satisfy the two pairwise properties considering the very worst-off, then the second worst-off, simultaneously (for details, see Fleurbaey and Maniquet and so on. 2011a,b). Fleurbaey and Maniquet: Optimal Income Taxation Theory 1049

money-metric utility m w , z makes satisfying the pairwise laissez-faire objective i( ̃ i) interpersonal comparisons of resources in (seeking to equalize transfers for all pairs an interestingly versatile way to achieve the of agents with identical wages) and slightly desired combination of the compensation less strong on compensation (elimination of and laissez-faire objectives: individuals with inequalities in indifference curves for each identical preferences are compared in terms pair of agents with identical preferences is of indifference curves (and this extends to obtained for a subset of preferences). The individuals with different preferences but basic idea underlying their construction, non-crossing indifference curves), and indi- along with one core example, is developed in viduals with the reference wage are com- section 8. Their implications for taxation are pared in terms of the lump-sum transfers studied in detail in Fleurbaey and Maniquet they receive when tax operates by lump-sum (2007, 2011a,c). transfers (of which the laissez-faire is a One might want to question the extreme degenerate case). form of egalitarianism that is adopted through That these two ways of making interper- the maximin approach. The literature on fair sonal comparisons can be performed by social orderings (see, in particular, Fleurbaey the same well-being indexes is quite nota- and Maniquet 2011a, chapter 3), echoing ble. The money-metric utility has generally earlier studies of the money-metric utility been considered in the profession as a mere (Blackorby and Donaldson 1988), shows convenience, although it was sometimes that mild egalitarian requirements (such as presented as more than that. For instance, convexity of the social ordering on X n, or a Deaton and Muellbauer (1980, p. 225) sug- Pigou–Dalton transfer principle applied to gest that the money-metric utility reflects the consumptions of individuals with iden- “the budget constraints to which the agents tical preferences and equal labor) can be are submitted.”29 The money-metric utility satisfied only with an absolute priority for respects individual preferences while using the worst off when the evaluation satisfies an objective measuring rod to compare indi- informational simplicity requirements. But vidual situations. This combination enables of course, it is always possible to weaken the it to respect individual preferences not only egalitarian requirements further in order to for intrapersonal comparisons, but also for obtain a less extreme priority for the worst interpersonal comparisons when individuals off.30 unambiguously agree about who has a bet- As we mentioned in the previous sub- ter situation because indifference curves do section, Roemer’s theory of equality of not cross. It also enables it to be sensitive to opportunity relies on a distinction between transfers in the case of individuals with the circumstances and effort. In the usual opti- reference wage. mal income tax model, individuals are char- There exist other social-ordering func- acterized by their preferences and their tions that are much stronger with respect to skills. That forces us to put the cut between circumstances and effort between these two characteristics. It would be possible, 29 See also Deaton (1980, p. 51): “I believe that practi- however, to enrich the model with other cal welfare measurement should be fundamentally based on opportunities rather than on their untestable conse- elements, such as the influence of fam- quences. No government is going to give special treatment ily background on preferences, and study to an individual who claims his extra sensibilities require special facilities, at least not without some objective evidence of why money means something different to him 30 See, e.g., Fleurbaey and Maniquet (2005) or than to anyone else.” (emphasis added) Fleurbaey and Tadenuma (2014). 1050 Journal of Economic Literature, Vol. LVI (September 2018) the compensation and laissez-faire objec- relative to net incomes, creating, if earnings tives with another cut. However, there is are disincentivized by taxation, a reinforcing an interesting difference between Roemer’s mechanism that can generate multiple equi- approach and this one. In Roemer’s perspec- libria. Low-tax equilibria incur a lower ran- tive, a family background influencing prefer- dom shock relative to net incomes, justifiying ences may be a genuine handicap in attaining the low tax, and the converse is true for the the relevant outcome (such as income). In high-tax equilibria. the approach described here, there is no As a matter of fact, the separation between comparable outcome and one cannot view luck and desert is at the core of the literature an influence on preferences as a handicap in that has just been reviewed in the previous the satisfaction of these preferences. Instead, subsections. The decomposition of gross such an influence must be viewed as distort- income into a deserved and an undeserved ing preferences and implying that the agent’s part can be analyzed easily using Roemer’s situation should be assessed with “ideal” approach or the resource-egalitarian preferences, i.e., preferences that would be approach. In both cases, interestingly, the free from the alleged influence. The fairness weights on various types of individuals are literature has been reluctant to follow this different from the intuitive ones proposed route because it means dropping the Pareto by Saez and Stantcheva. principle and considering that individual In Roemer’s approach, the random shock preferences are not fully respectable. This can be added to the circumstances, and if the is, however, a route familiar to the literature random shock is uncorrelated with earnings on behavioral phenomena such as myopia in each type, the partitioning of individuals and hyperbolic discounting (e.g., Choi et in terms of “genuine effort” is unchanged. al. 2003). The relevance of behavioral stud- Then, the worst type, when luck is included ies to these issues will be discussed in the in the description of type, gathers the indi- conclusion. viduals with bad luck and otherwise bad circumstances. Roemer’s criterion would 5.5 Luck and Desert then advocate maximizing their average Saez and Stantcheva (2016, Appendix outcome. B2), inspired in particular by Alesina and In the resource-egalitarian approach, Angeletos (2005), examine the case in which one would also treat the random shock as individual income has two components, a circumstance to be compensated, and the ordinary earnings and a random shock, would apply the laissez-faire principle only which for simplicity can be assumed to have to the pairs of individuals sharing the same a zero mean.31 Saez and Stantcheva show, in skill and shock. Interestingly, this does particular, that if only the individuals with not require any change to the index mea- net income below their earnings are given sure. Indeed, the money-metric utility an equal positive weight in the social objec- m w , z defined in the previous subsec- i( ̃ i) tive, income taxation may have to be greater tion displays the nice dual property that when the random shock has greater variance two individuals with the same preferences, but possibly different skills and or different / 31 In a classic paper, Varian (1980) studied the case of ex luck, should ideally be given final bundles ante identical individuals facing random returns on their on the same indifference curve, whereas savings, and showed how the optimal tax depends on the for individuals with wage equal to w , the distribution of shocks. Desert in his model can be viewed ̃ as the amount of savings, but all consumers being identical, ideal state is to give them lump-sum trans- this dimension of the problem vanishes. fers, canceling their inequalities in luck and Fleurbaey and Maniquet: Optimal Income Taxation Theory 1051 letting them work freely (with no further allocation, rather than any (distorted) tax). earnings. The advantage of these two approaches, compared to the intuitive weighting proposed 6. Can Weighting Utilities Yield Fair by Saez and Stantcheva, is that it provides Outcomes? a sensible ordering of individuals, which makes it possible to prioritize the very worst If the population comes with a given pro- off (if the maximin criterion is adopted) or to file of utility functions (U1 , … , Un ) , is it nec- give a positive weight to all individuals, but essary to replace such functions by indexes with decreasing priority according to their like m w , z before applying a social wel- i( ̃ i) position as measured by m w , z . fare function? Couldn’t one simply weight i( ̃ i) If one adopts the view that earned income these utility functions in a utilitarian sum? is fully deserved and only the random shock is Facing the problem of heterogeneous util- undeserved, then one can use another utility ities, the literature32 has indeed considered function, namely m i( w i , z i) , which we have weighting them in a utilitarian social welfare shown to be associated with the libertarian function i N i Ui z i . These i coefficients ∑ ∈ α ( ) α approach. As the random shocks are equiv- are the so-called Pareto weights referred to alent to redistribution between individuals, in the introduction (quoting Piketty and Saez any inequality-averse social welfare func- 2013b). It is true that, provided the utility tion applied to the distribution of m i( w i , z i) possibility set is convex, every (constrained) will seek to undo them via a compensating efficient allocation can be viewed as opti- redistribution. This approach is unlikely mal for such a weighted utilitarian function. to end up giving equal positive weight to One could therefore imagine seeking the those with a net income below their earn- weights that induce the same choice of tax αi ings because it will prioritize those with the function as, for instance, a reference-wage greater gap. egalitarian-equivalent social ordering. But There is another interesting difference this idea does not work well, as we now with the weights proposed by Saez and explain. Stantcheva. The weights they propose can Let us consider a second-best world in generate multiple equilibria because the rel- which only incomes are observable. Assume ative share of the shocks in income is endog- that the individuals with the lowest wage are enous to the tax. In particular, if the tax is sufficiently diverse in preferences so that 100 percent, as they note, the posttax earn- they span all the labor–consumption prefer- ings are null and all income is undeserved ences of the population. It is then possible (and has zero elasticity), implying that the to derive the conclusion that, at the socially optimal tax is indeed 100 percent. In con- best allocation, only them should be given a trast, a social welfare function with m i( w i , z i) positive weight. This is because, under the is unlikely to generate multiple equilibria incentive constraints, a less productive agent and be satisfied with a 100 percent tax. The necessarily faces a less favorable budget set reason is that it does not treat posttax earn- than a more productive agent. Therefore, ings as deserved when they are strongly dis- an agent with a high wage will always have torted by the tax. It treats the tax as being just a higher well-being index m i( w , z i) than an as bad as a negative shock when it reduces ̃ m i( w i , z i) . It therefore offers a defense of the principle that individuals deserve 32 See, in particular, Boadway et al. (2002), Kaplow to keep their earnings of the laissez-faire (2008), Choné and Laroque (2010). 1052 Journal of Economic Literature, Vol. LVI (September 2018) agent with the same preferences and the Following the simplifying method to deal lowest wage. The latter agent should then be with heterogenous preferences introduced given full priority over the former, because by Mirrlees (1976) and Brett and Weymark the objective is a maximin. (2003), they assume that individual behav- The question then becomes: for two ioral heterogeneity is one-dimensional. First, agents, say j and k, having different prefer- preferences are parameterized by a unidi- ences and the lowest wage, how should we mensional number i . Moreover, preferences determine and ? The key point is that θ αj αk and skill interact in such a way that all agents their value would have to depend on the with the same n w are behaviorally i = θi i whole profile of the population because indistinguishable and have the same utility this profile determines the set of feasible U y n, c at the same earning–consumption ( / ) and incentive-compatible allocations, and bundle ( y, c). therefore the exact bundles assigned to They rely on a weighted social welfare these agents in the optimal allocation. That function is, once we have identified the second-best optimal allocation for a reference-wage ∞ n U(n) f (n) dn, egalitarian-equivalent social ordering, it is ∫0 α possible to compute the corresponding s. α But it is impossible to guess what these where U(n) is a short notation for weights should be before computing the opti- U yn n, cn , the utility of agents i such ( ( )/ ( )) mal allocation. Therefore the Pareto-weights that n n. The marginal social value of i = methodology cannot help in finding the opti- consumption c n is g n , where g n ( ) αn ( ) ( ) mal allocation. U n c n is the marginal utility of = ∂ ( )/∂ ( ) The correct weights, moreover, would be consumption. They propose to make αn of limited use even if they could be guessed, inversely proportional to because the function i N iUi z i using ∑ ∈ α ( ) these weights is only good at selecting the E g LF w | n n , [ (θi ̅ ) i = ] best allocation. It cannot reliably be used to evaluate suboptimal allocations, for instance i.e., the average value, among the agents in the context of a reform in which both of (actual) parameter n, of g( · ) in the the pre-reform and the post-reform alloca- laissez-faire allocation of a hypothetical tions are suboptimal. The evaluation might economy in which all agents have the aver- then go against what the reference-wage age wage w̅ of the actual economy. When egalitarian-equivalent social ordering rec- the actual economy already has equal wages ommends for such suboptimal allocations, for all, the computation of such weights because the individuals’ relative marginal implies that g LF n is a constant in n αn ( ) utilities may be completely different between and laissez-faire is an optimal allocation. the optimal allocation and the suboptimal An alternative method would start from ones. As a result, new s would have to be α the individual weights, for each individ- computed for each new problem, that is, as a ual i, that would deliver the laissez-faire in function of the set of allocations among which the hypothetical economy with equalized the choice has to be made and, again, the LF wages: i 1 g ( i w̅ ) . One cannot actu- values of these s could only be ascertained α = / θ α ally use such weights at the individual level after the optimal allocation is identified. because is not observed. But, given that θi The recent interesting work of Lockwood the incentive-compatible allocations give the and Weinzerl (2012) illustrates this difficulty. same utility U (n) to agents with the same n , Fleurbaey and Maniquet: Optimal Income Taxation Theory 1053 the weighted utilitarian objective can be depend only on w, not on .34 In this case, θ written: within any subset of agents with the same wage, the social objective is indifferent to U ∞ U n f n dn, ∫i αi i = ∫0 αn ( ) ( ) redistribution by lump-sum transfers, and therefore admits equal lump-sum transfers for (which represent the “laissez-faire” ideal in this case) as optimal, although it does not E n n strictly prefer such equality to any other αn = [αi | i = ] distribution of lump-sum transfers with the E 1 g LF w | n n . same sum. = [ / (θi ̅ ) i = ] In conclusion, it appears that the replace- The evaluation of incentive-compatible allo- ment of arbitrary utility functions U i ( z i) by cations according to ∞ U n f n dn for suitable well-being indexes cannot gener- ∫0 αn ( ) ( ) such weights always coincides with the ally be mimicked by a weighting system. αn evaluation that would be made with the cor- Weighted utilitarianism is not an all-purpose rect individual weights .33 tool. Relying on it in order to incorpo- αi Whatever the precise formula for the rate the fairness principles that underlie weights, the evaluation of allocations in the reference-wage egalitarian-equivalent and actual economy is not geared toward the similar social-ordering functions is possible laissez-faire in a systematic way. In particu- only if the weights are specific to the allo- lar, the weighted objective may not pursue cation that is evaluated and depend on the the pairwise laissez-faire objective in the whole profile of the population. Far from actual economy, even for agents with average being the simplest amendment to classical wage. Take two agents, j and k, enjoying the welfare economics, weighted utilitarianism average wage w but different , . The sum seems an arduous detour compared to the ̅ θj θk direct adoption of well-being indexes such as U z g LF w U z g LF w the no-less-classical money-metric utility. j( j)/ (θj ̅ ) + k( j)/ (θk ̅ ) does not generally seek equal tax treatment 7. Weighting Incomes for these two agents in the actual economy when they are far from the laissez-faire bun- An important progress in optimal taxa- dles they would receive in the hypothetical tion theory has been recently accomplished economy. by Saez (2001, 2002) and followed up by Utilitarianism accepts the laissez-faire Saez and Stantcheva (2016). This progress as optimal (among other allocations, in the has been made possible by a shift of focus. first-best context) when individuals have In Saez’ formulation, social preferences quasi-linear preferences and identical wages. are represented by weights that are endog- Lockwood and Weinzierl (2015) make use enously assigned to incomes at the con- of this insight. The utilities are assumed to templated allocation, rather than to utility be quasi-linear and the weights are set to levels. The underlying rationale is illuminat- ingly simple. For a social welfare function U z , a marginal change T to the ∑i αi i( i) δ

33 Lockwood and Weinzierl’s weights are the harmonic 34 In the quasi-linear case, the method proposed by mean of the individual weights , in every n group, instead Lockwood and Weinzierl (2012) produces equal weights αi of the arithmetic mean. for all types. 1054 Journal of Economic Literature, Vol. LVI (September 2018) function T will induce a change in social wel- y depends on T so that the weights are, in fare equal, by the envelope theorem,35 to principle, endogenous and depend on the particular allocation under consideration. In ____ Ui 2 i ∂ T(y i) . this section, we show how the theory of fair ( ) −∑i α c δ ∂ i social orderings, by relying on specific utility This expression can be read as a sum over indices, can sometimes help compute these earning levels y of the change in tax T y weights. δ ( ) weighted by the total marginal social weight A simplification comes from the fact that, of the subpopulation earning the level y . At given that fair social orderings are of the a given allocation, one can take the weights maximin type, a positive weight will only be U c as given and focus on the assigned to the agents exhibiting the lowest βi = αi∂ i/∂ i weighted sum of T y over all levels of y. value of the well-being index. In a first-best δ ( ) This can be done to evaluate whether a context, of course, these values should be (small) reform is a social improvement, or equalized, in which case all agents have a whether the current allocation is optimal (in positive weight. In the second-best context of which case, no feasible reform yields a posi- optimal taxation theory, it is quite likely that it tive weighted sum). will be impossible to equalize well-being lev- The important question raised by Saez and els. In that case, the agents whose well-being Stantcheva (2016) is whether one can directly is bounded below by incentive compatibility guess what the weights should be, or whether constraints will receive a weight of zero. one has to deduce them from a social wel- In some cases, the objective of maximizing fare function and a formula like U the minimal value of some well-being index βi = αi∂ i c . They provide many examples covering enables us to completely determine the /∂ i tagging, desert, and fairness, and argue that weights that should be assigned to incomes. extending the approach to cases in which In this section, we present one such case, the weights are not related to an underlying obtained when the social objective consists of social welfare function enlarges the set of “maximinning” a particular well-being index possible ethical approaches applicable to tax- in the reference-wage egalitarian-equivalent ation. Here, we mainly focus on the issue of family presented above, m i ( w min , z i ), where computing weights that derive from a social w min min i N w i. That is, the well-being = ∈ welfare function, or at least a social ordering of an agent is measured by reference to the (satisfying transitivity and the Pareto princi- lump-sum tax that would leave him indif- ple). The idea of dropping the social welfare ferent between his actual bundle and freely function framework is discussed at the end choosing his labor time, should he be paid at of the section. the minimal wage. When relying on a social welfare function, Let us first examine a reform problem, in a difficulty is that the subpopulationearning which an arbitrary tax scheme prevails but is not optimal. A reform has to be designed, and the tax scheme can only be changed 35 The additional term marginally. How should we change it? _ Ui _ Ui The reasoning is illustrated in figure 1. i ∂ 1 T′ ( y i) dy i ∂ d i α ( c i ( − ) + i ℓ ) ∑i ∂ ∂ℓ It represents the pretax after-tax income / vanishes when either the first-order condition space. The 45º line represents the relation U U _ ∂ i 1 T y w _∂ i 0 c ( ′( i)) i between pretax and after-tax incomes in the ∂ i − + ∂ℓi = or the condition dy d 0 (obtained for corner absence of taxation. The tax scheme that i = ℓi = choices at 0 or 1) holds for all agents. we try to evaluate, T , is represented by the ℓi = ℓi = Fleurbaey and Maniquet: Optimal Income Taxation Theory 1055

c 45˚

y T y − ( )

c c*

0 * y min ′ y

Figure 1. Deriving the Income Weights: The Case of a Reform

corresponding function c y T(y) that index. It is convenient to do it in two steps. = − describes how after-tax incomes c depend on First, the agents with the lowest well-being pretax incomes y.36 index need to be identified in each produc- Evaluating a tax scheme requires identi- tivity subgroup. Let us begin with agents fying the agents with the lowest well-being whose productivity is equal to w min. Given our assumption that labor is bounded (0 i 1), we know that these agents’ 36 Note that T exhibits a decreasing marginal tax rate on ≤ ℓ ≤ earnings are in the [0, w min ] interval earnings in the [0, w min ] interval. As a result, the budget set of the low productivity agents is not convex. Nothing in the (remember that w i also stands for agent i’s reasoning here depends on that non-convexity. pretax income, would he work full time). 1056 Journal of Economic Literature, Vol. LVI (September 2018)

Let y ∗ [0, w ] be the pretax income Individual i ’s choice reveals that he weakly ∈ min 0 for which the tax T (y) is maximal over the prefers bundle (y ∗, c ∗ ) to all other possible [0, w min ] interval, or, equivalently, the dif- bundles affordable given the tax scheme, but ference c y (which is measured in the also, by construction of the tangent line seg- − graph as the vertical distance between the ment, to all affordable bundles in the budget ∗ y T(y) curve and the no-tax line) is mini- (y, c) 0, w min | c t y . The − { ∈ [ ] × ℝ+ ≤ + } mal. Graphically, y ∗ is found at the point of bundle (y ∗, c ∗ ) is the preferred one in both tangency between the curve representing budgets. Therefore, one has y T(y) and a line segment of slope 1, as − drawn on the figure.37 The intercept associ- V (y ∗, c ∗) ated with that line is denoted t ∗. i 0 Let us assume that some agents with min- max Vi (y, c) 0, w min | c y T(y) imal productivity, one of them having index = 0({ ∈ [ ] × ℝ+ ≤ − }) ∗ 38 i0 , happen to earn y . Given the tax scheme, this grants them an after-tax income max V (y, c) 0, w | c t ∗ y , i 0( { [ min] }) c ∗ y ∗ T( y ∗ ). = ∈ × ℝ+ ≤ + = − To formalize the argument, it is convenient to recall the duality relationship between the latter equality implying that indirect utility and money-metric utility, m (w , z ) t ∗. i 0 min i 0 = implying that the indirect utility derived For all other agents with w w , since i = min from the implicit budget (defined in sec- 39 tion 5.1) is equal to the actual utility level: ∗ (y, c) 0, w min | c t y { ∈ [ ] × ℝ+ ≤ + } Ui (z i) max Ui ( { ( , c) X | c m i (w , z i) w }) = ℓ ∈ ≤ ̃ + ̃ ℓ (y, c) 0, w min | c y T y ⊆ { ∈ [ ] × ℝ+ ≤ − ( )} max Vi (y, c) 0, w i | = ({ ∈ [ ] × ℝ+ one always has _y c m i(w , z i) w . ≤ ̃ + ̃ w i }) Vi y i, c i max Vi (y, c) 0, w min | ( ) = { ∈ [ ] × ℝ+

37 ∗ In this example, y is interior to the interval. It need c y T(y)} not be the case. If T is convex over [0, w min ], for instance, ≤ − so that the y T(y) curve is concave, then either y ∗ 0, − = in which case the analysis yields m (w , z ) T 0 , or i 0 min i 0 = − ( ) max Vi (y, c) 0, w min | y ∗ w , in which case the analysis yields m (w , z ) ≥ { ∈ [ ] × ℝ+ = min i 0 min i 0 T( w ). In both cases, the analysis presented here = − min goes through. c t ∗ y , 38 This assumption, actually, can be imposed without loss ≤ + } of generality. Indeed, if no agent earns that income, then the tax scheme is irrelevant at that income level, so that ∗ implying that m i (w min, z i) t . the tax amount can be decreased until it becomes relevant, ≥ that is, until one agent is indifferent between her bundle What about individuals for whom w w ? One more line segment is drawn and this new bundle. In that case, we can assume that this i > min agent actually chooses the new bundle. Consequently, the in the figure. It starts from the interceptt ∗ _ only assumption that is needed is that as soon as a group of and ends at consumption level c t ∗ agents are willing to earn some income level in the range = + [0, w min ], there is at least one agent among them who has w min, for y w′ . The slope of this line _ ∗ = _ ∗ the lowest productivity w min. is ( c t ) w′ . Using the equality c t 39 We let V( B) denote the image of the set B by the − / − w min, the slope can be expressed as function V. Therefore max V( B) is the maximal value = w w . obtained by V on set B. min/ ′ Fleurbaey and Maniquet: Optimal Income Taxation Theory 1057

If an agent i of productivity w w was T. It is the intercept b of the budget line i = ′ proposed such a budget, he would reach a c b w that is tangent from below = + min ℓ utility level to the budget curve induced by T in the [0, w min ] range of incomes. Formally, max Vi (y, c) 0, w′ ({ ∈ [ ] × ℝ+ | b min ( T(y)). w = y w min − c t ∗ _min y , ≤ ≤ + w′ }) Observe that all agents earning an income of y′ or less have a subsidy at least as great as b . implying that m (w , z ) t ∗. Now, as is Let us now consider the new tax function i min i = transparent from the figure, any such bud- T′ . It consists in applying a constant amount get is below the actual budget of this agents, of subsidy, b , to all income levels lower than since the function y T(y) is non-decreasing. y . Beyond y , T and T coincide. Compared − ′ ′ ′ As a result, the actual indirect utility cannot to T , the new tax scheme T ′ has two import- be lower, which translates into m i ( w min, z i ) ant features. t ∗, where z denotes the actual bundle First, the minimal well-being level remains ≥ i of agent i facing T . The inequality is strict if identical at b . The planner interested in max- y T(y) is increasing. Therefore, more gen- iminning m (w , z ) is, therefore, indiffer- − i min i erally, all agents having a wage greater than ent between T and T′ . w min have a strictly larger well-being index Second, compared to T , the new scheme than i if y T(y) is increasing. T allows the planner to obtain a budget sur- 0 − ′ In conclusion: assuming that y T(y) is plus. Indeed, all agents who earn more than − increasing, one should, at the status quo, y′ under T will continue to earn exactly the assign a positive weight to earning levels like same income under T′ . In particular, the y ∗, i.e., corresponding to maximum T(y) for change from T to T′ will not induce them to y w , and a zero weight to all other earn less than y , since the lower portion of ≤ min ′ incomes. the budget has become less attractive. Their By a simple extension of this reasoning, influence on the budget therefore remains the result of zero weight for y w also the same. > min holds under the optimal tax for these social Agents earning less than y′ under T are preferences. It is also a small additional step likely to change their labor time and, there- to show that, for the optimal tax, the marginal fore, their earning, but the key point is that rate of taxation should be zero on incomes they will move from an income at which they below w min. This gives us a precise formula received a subsidy of at least b to another for the optimal tax scheme. This case is also income at which they receive a subsidy of discussed in Saez and Stantcheva (2016), and at most b. That is, no taxpayer will pay less here we provide an intuitive explanation for tax under T′ than under T, and some of them the zero marginal tax result. will pay more. This proves that the planner The intuition for this result is illustrated will now run a budget surplus. By redistrib- in figure 2. Let us consider the tax function uting this surplus to all agents (which can be T. The marginal rates of taxation differ from done by slightly translating the budget curve zero for incomes below w min. We need to generated by T′ upwards), we can obtain a show that this cannot be optimal. new allocation that strictly Pareto dominates By the same reasoning as above, the previous one, thereby strictly increas- we can identify the level of the lowest ing its lowest well-being level above b. well-being index in the population under This proves that T cannot be optimal. As a 1058 Journal of Economic Literature, Vol. LVI (September 2018)

(1)

(T T ) = ′

y T y) − ( y T y) − ′(

0 y y min ′ max

Figure 2. The Optimal Tax Has a Zero Marginal Rate on Low Incomes

result, we need a tax scheme that is flat in the distribution of the earnings, and its density, [0, w ] range of incomes, in which all which we denote (y), F(y), and f (y) respec- min ϵ agents receive the same subsidy: The mar- tively. We obtain:40 ginal tax rate is equal to zero in this range. How should incomes be taxed above y w , T ( y) 0, ∀ ≤ min ′ = w min? As explained earlier, their weight is null and therefore the only objective of tax- ______T′ ( y) ______1 F(y) y w min, − . ing those incomes should be to maximize the ∀ ≥ 1 T′ ( y) = (y) yf (y) tax return so as to have as large a subsidy on − ϵ low incomes as possible (assuming, of course, that after-tax income remains an increasing function of pretax income). This is achieved 40 If the income effect is different from zero, then, the by applying the Saez (2001) formula with a second part of the formula needs to be replaced with weight of zero on all incomes above w , if ∞ y′ u (z) min _T′ ( y) _1 _ϵ _dz c ∗ y exp y 1 c z f (y′ ) dy′ , 1 T ( y) = (y) y f (y) [ ( − (z)) ] one assumes that the first-order approach − ′ ϵ ∫ ∫ ϵ on which the formula relies is valid. In the where u and c stand for the uncompensated and ϵ ϵ simple case of no income effect, the mar- compensated earning supply elasticity functions, respec- tively, and f ∗ is a modified density function. See Saez ginal taxation rates are a function of the elas- (2001) for the derivation of this formula, and Jacquet and ticity of the earning supply, the cumulative Lehmann (2014) for a similar formula. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1059

In conclusion, the optimal tax scheme that directly weights tax changes at the various should be implemented by an egalitarian earning levels is compatible with transitive planner interested in the m i ( w min, z i ) index and Paretian social preferences, and then of well-being consists of a zero marginal rate extendable to the study of nonlocal reforms, on incomes below w min and a marginal tax only if it relies on the classical framework of rate that follows Saez’ formula with zero the social welfare function. Many reforms to weights above w min. the tax code are not small. Therefore, out of There are other cases in which the deter- respect for rationality (transitivity) and indi- mination of the weights and optimal tax is vidual preferences (Pareto), the social wel- less simple. For the different social order- fare function framework seems to provide ing studied in Fleurbaey and Maniquet a safe, even if sometimes complex, toolbox. (2006), for instance, one can show that the Fortunately, as we will show in section 9, for marginal tax rate at a second-best alloca- an interesting and rather large set of social tion is non-positive on average over low preferences, there exist extremely simple incomes, but the lowest well-being level may criteria for the evaluation of reforms—cri- be attained by a subset of the low-income teria that make it possible not only to easily agents that is hard to identify, so that the identify what levels of earnings should be weights at the optimal allocation cannot be given a positive weight in the evaluation of easily determined. local reforms, but also make it very easy to In conclusion, this section shows that the assess nonlocal reforms. weights approach, though useful, does not always provide an easy shortcut, unfortu- 8. Ethical Selection of Utility Indexes nately, for the determination of the optimal tax. This is because the link between the fair- In the previous sections, we have empha- ness principles embodied in the social pref- sized the need to carefully select the utility erences and the weights on earnings is quite indexes (instead of just weighting them), and complex. Only in some cases involving the have hinted at the wide possibilities offered maximin criterion can one deduce that some by the relevant span of indexes. In this sec- levels of earnings should have zero weight. tion, we analyze this array of possibilities and When the social ordering is not a maximin, try to make the underlying ethical choices the determination of the optimal tax and the transparent, so that practitioners can easily associated marginal social welfare weights connect with the methodology of choos- for incomes is harder. ing utility indexes. The theory of fair social It may also be worth emphasizing that orderings has mostly relied on an axiomatic the social welfare function approach has approach. While this is useful to theorists been introduced by Bergson (1938) and who want to grasp the logical underpinnings Samuelson (1947) not out of a taste for ele- of the objects under study, practitioners seek gance, but because it is the only way to define a more direct reading of the meaning of the social preferences that are both transitive tools they use. Therefore, here, we ignore and Paretian.41 Therefore, a method that the formal aspect of axiomatics and focus on the intuitive meaning of the various features 41 This claim is formally true only if transitivity is log- on an index of well-being. Our analysis here ically strengthened into being representable (by a func- echoes the comprehensive review of indexes tion). Transitive Paretian social preferences may or may in Preston and Walker (1999), where the not be representable by a function. The social welfare function approach is here meant to include all such social underlying normative principles were not preferences. made explicit, and an analysis in Decoster 1060 Journal of Economic Literature, Vol. LVI (September 2018) and Haan (2015), where three main exam- than the labor–consumption bundle that is ples of indexes are discussed together with the focus of the taxation model. This is an their ethical meaning. important objection, and it calls for embed- ding the model into a larger space in which 8.1 Four Choices the relevant aspects of life that individuals We will focus on four ethical choices that care about are taken into account. Truly guide the selection of a well-being index in enough, this objection falls short of provid- the taxation model: ing a reason to take subjective utility at face value. Nonetheless, it does raise a serious 1. Does one trust subjective utility or rely question when the model is not enlarged, only on ordinal preferences? because an approach that only looks at ordi- 2. Does one seek to reduce inequalities due nal preference over labor–consumption to unequal skills or consider that individ- bundles may miss important dimensions of uals deserve their own productivity? inequality. 3. Does one prioritize the inequalities due It is commonly believed that no inter- to unequal skills or the inequalities of personal comparisons can be made on the tax treatment between equally skilled basis of individual ordinal noncomparable individuals? preferences. In the rest of this section, we 4. Does one want to pay special attention focus on indexes that are based on ordinal to the individuals with high or low aver- noncomparable preferences in order to show sion to work? the wide array of possibilities offered by this informational basis. While the first two questions are obvious 8.1.2 Does One Seek to Reduce Inequalities and relate to issues already discussed, the due to Unequal Skills or Consider last two are less obviously relevant and their That Individuals Deserve the Fruits of importance will be explained below. Their Talents? 8.1.1 Does One Trust Subjective Utility or We have seen in section 5 that the Rely Only on Ordinal Preferences? money-metric utilities m w , z , fed into an i( ̃ i) As explained in section 3, indexes that are inequality-averse social welfare function, do constructed on the basis of individuals’ ordi- seek to eliminate inequalities due to skills nal preferences are less vulnerable to expen- among individuals having the same prefer- sive tastes and the adaptation problem than ences, whereas the different money-metric subjective declarations of utility or satisfac- utilities m i( w i, z i) embody the libertarian tion. Moreover, subjective declarations may goal of letting the skilled individuals enjoy contradict the individuals’ own interpersonal their advantage. comparisons, and the latter are especially Interestingly, one could explore a com- compelling when indifference curves do promise view in which one would use the not cross. An individual may be on a higher indexes m w 1 w , z in order i(λ ̃ + ( − λ) i i) indifference curve but declare a lower satis- to let individuals enjoy their skills to some faction than another, simply because individ- extent (1 ) and limit inequalities due to − λ ual self-assessments rely on heterogeneous skills to the complementary extent ( ). λ standards. More importantly, one can also generalize It may be objected that utility takes account the indexes and observe that a money-metric not only of standards of satisfaction, but also utility really defines a budget, rather than just of other aspects of the individual’s situation a number. One can then choose what part of Fleurbaey and Maniquet: Optimal Income Taxation Theory 1061 the budget to consider for the comparison these implicit budgets, which is extreme42 across money-metric utilities. This, of course, because the implicit budgets for greater skills is of no consequence for money-metric util- are then dominated by the implicit budgets ities of the m w , z sort, because all the for lower skills (except at 1, where they i( ̃ i) ℓ = implicit budgets are parallel. But for the meet). m i( w i, z i) indexes, it matters a lot, because For ease of reference, we will call their slope in the , c space is w , so that “egalitarian-equivalent” the case in which (ℓ ) i the budget lines often cross. 1, i.e., w w , and “libertarian” the λ = ̃ i = ̃ A simple generalization of the m w , z case in which 0, i.e., w w , even i( i i) λ = ̃ i = i indexes, therefore, consists in picking a ref- though, when 0, a substantial amount ℓ ̃ > erence value for labor and evaluating how of redistribution can ensue (only when ℓ ̃ much one would consume with the implicit 0 does one have “pure” libertarianism). ℓ ̃ = budget if one worked hours. This is com- A further generalization (explored in ℓ ̃ puted as the index m w , z w . Such an Fleurbaey and Maniquet 1996 and Fleurbaey i( i i) + i ℓ ̃ index has been advocated at length in Kolm 2008) would not focus on a particular labor (2004). It is grounded in the idea that every- reference , and would instead pick a refer- ℓ ̃ one should consider as the fraction of one’s ence preference ordering and apply a cor- ℓ ̃ time due to the community, and can fully responding indirect utility function to the enjoy the benefits of the remainder. When various implicit budgets.43 In other words, 0, one obtains the standard libertarian budgets would be compared by the indiffer- ℓ ̃ = view that everyone fully owns one’s talents. ence curves of the reference preference rela- When this new index is equal across indi- tion that are tangent to the budgets. viduals, their implicit budgets cross at the There are, thus, two ways of introduc- point where . To sum up this discus- ing a redistributive attitude in the indexes ℓ = ℓ ̃ sion, we obtain the general index m w , z w , where w w 1 w : i( ̃ i i)+ ̃ iℓ ̃ ̃ i = λ ̃ +( −λ) i let 1 or let 1. A natural question λ → ℓ ̃ → m w , z w , here is whether taking one route or the other i( ̃ i i) + ̃ iℓ ̃ makes a difference. It does, and this is where where w w 1 w . the next question becomes relevant. ̃ i = λ ̃ + ( − λ) i The construction of the value of this index 8.1.3 Does One Prioritize the Inequalities for a general bundle z and utility function i due to Unequal Skills or the U is illustrated in figure 3 (slopes of budget i Inequalities of Tax Treatment between lines are noted between parentheses below Equally Skilled Individuals? the line). With such an index, there are two ways to seek redistribution across individuals Compare the properties of m w , z and i( ̃ i) with unequal skills. One way is to let the per- m w , z w . The former always consid- i( i i) + iℓ ̃ sonalized reference wage w be equal across ers that an individual on a dominated indif- ̃ i individuals ( 1), yielding the index ference curve is worse off, and therefore λ = m w , z w which is, up to a constant, i( ̃ i) + ̃ ℓ ̃ equivalent to m w , z . Another way to i( ̃ i) adopt a redistributive attitude is to pick a 42 See section 9.2 for further discussion. 43 The reference corresponds to the case in which the large value for , because one then seeks ℓ ̃ ℓ ̃ reference preferences always choose a labor time equal to make the implicit budgets for the skilled to whatever the actual productivity w . The general- ℓ ̃ i ization amounts to assuming that may itself depend on agents low compared to those of the less- ℓ ̃ w , in which case it can only be an increasing function ̃ i skilled agents. When ̃ 1, one seeks to of w , because the compensated labor supply cannot but ℓ = ̃ i equalize the full incomes corresponding to increase in the wage. 1062 Journal of Economic Literature, Vol. LVI (September 2018)

c i

′i

(i)

m ( , ) i i i + i ℓ

mi (i, i)

0 1 ℓ ℓ

Figure 3. The Construction of Well-Being Indices

gives clear priority to the compensation goal But inequalities due to unequal skills are of eliminating inequalities due to skills. In less of a priority for this index. Because contrast, as explained in section 5, it applies individuals with the same preferences may the pairwise laissez-faire objective (i.e., seek have implicit budgets that cross (when their equal lump-sum transfers) to individuals wages differ), their ranking according to the with equal skills only when their wage coin- m w , z w index may not always i( i i) + iℓ ̃ cides with w . The laissez-faire objective is coincide with the ranking of their indiffer- ̃ then less prominent. ence curves. This is illustrated in figure 4. In contrast, the latter index produces par- Coincidence is guaranteed only for individ- allel budget lines for individuals with the uals who have a strong preference (of an same wage, and therefore seeks to make the almost Leontief sort) for working exactly ℓ ̃ implicit budgets equal for such individuals. hours (or in the generalization using a refer- When redistribution is made by lump-sum ence preference relation, only for individuals transfers in the first-best context, it then with personal preferences identical to the dutifully gives the same lump-sum trans- reference preferences). fers to individuals with the same wage. In This shows that the choice between the the second-best context, it seeks to obtain egalitarian-equivalent m w , z and the i( ̃ i) a similar pattern for the implicit budgets. libertarian m w , z w is a choice i( i i) + iℓ ̃ Fleurbaey and Maniquet: Optimal Income Taxation Theory 1063

c =

m ( ) + ℓ ()

0 1 ℓ ℓ

Figure 4. U(z) U(z) whereas m(w, z) w m(w, z) w j j > k k k k k + k ℓ ̃ > j j j + j ℓ ̃

between compensating unequal skills and utility, one works with m w g , z . It is i( i( ̃) i) laissez-faire among individuals with identical then possible to be in a situation in which skills: pairwise compensation versus pairwise w g 1 w ; for instance, if w g i( ̃) = ( − λ) i i( ) laissez-faire. And, as explained in the previ- w h g and g g. Therefore, one = i ( ) ̃ < ous question, when 0, the m w , z sees that the introduction of the interme- ℓ ̃ → i( i i) + w leans toward laissez-faire tout court. diate and personalized reference wage w iℓ ̃ ̃ i Remember that in section 5.1 we showed 1 w may also be useful to capture = ( − λ) i that the libertarian approach, in the pres- the virtual wage the individual would have ence of public goods g , can be applied with under the reference quantity of public goods. money-metric utilities that incorporate a 8.1.4 Does One Want to Pay Special reference value g of the public goods. The ̃ Attention to the Individuals with High pure libertarian choice w w and 0 ̃ i = i ℓ ̃ = or Low Aversion to Work? is then compatible with some redistributive income taxation if there is a correlation When one gives priority to inequali- between m w , z , with z , c , g , ties in skills and wages, one cannot apply i( i i) i = (ℓi i ) and y i. In the special case in which pub- the laissez-faire principle to a great extent. lic goods affect productivity rather than One virtue of the laissez-faire principle is 1064 Journal of Economic Literature, Vol. LVI (September 2018) that it is neutral with respect to individual violate the participation constraint (this con- preferences. In a group of agents with straint stipulates that every i should never identical skills, it seeks to give them the prefer the (0, 0) bundle to his assigned same implicit budget, disregarding their labor–consumption bundle).45 One could, of preferences. course, add a participation constraint to the Such neutrality is necessarily lost, then, search for the optimal tax, but it seems pref- when one focuses on eliminating inequali- erable to make sure that the social objective ties due to skills, i.e., on the compensation itself guarantees that it will be satisfied by objective. That this is logically necessary has the optimal redistribution, whether in the been well established in the literature. It is first- or in the second-best context.46 due to the fact that one cannot at the same 8.2 Discussion time give the same budget to individuals having identical wages and give bundles on the The previous subsection has shown that, same indifference curves to individuals with even restricting attention to the class of identical preferences.44 indexes of the form When neutrality is lost, one has to decide what preferences to favor. This is where the m w , z w , i( ̃ i i) + ̃ iℓ ̃ choice of the parameter w in m w , z plays a ̃ i( ̃ i) role. With a low value for w , individuals who there is a large spectrum of possibilities. This ̃ are more averse to work tend to obtain lower calls for two remarks. First, only a small sub- implicit budgets than individuals who are less set of all the possible utility representations averse. And the contrary occurs with a high of given preferences are justified from a value for w . This is illustrated in figure 5. normative point of view. The properties dis- ̃ Therefore, with a social-ordering function cussed in this section provide a guide to the displaying a strong degree of inequality aver- relevant ethical choices. sion, the work-averse individuals are better Second, one may think that all second-best treated under a low w than under a high w . efficient allocations may turn out to be jus- ̃ ̃ If one considers, for instance, that work aver- tified by some appropriate choice ofw and ̃ i sion in this model may be partly due to low . This is definitely wrong, because at the ℓ ̃ job quality for the unskilled, a feature that laissez-faire allocation, the worst off for any is not well captured by this simple model, it of the social orderings in the class considered may be prudent, or charitable, to choose a here include agents with the lowest wage low value for w . w . Therefore, an allocation that penal- ̃ min Two additional considerations suggest izes all of the unskilled, compared to the that the choice of w w is worth con- laissez-faire, can never be socially optimal ̃ = min sidering seriously. First, it is endogenous to for any of these orderings. We conjecture the wage distribution and implies that w is that many other restrictions could be found. ̃ the common wage when all individuals have We close this section with a remark the same wage, which itself entails that the about a general impossibility to write the laissez-faire is an optimal allocation in this indexes discussed here as functions of a case. Second, and more specifically, it is the unique parameter gathering the prefer- only value in the [w min, w max] interval that ences and skill heterogeneities. Let us guarantees that redistribution will never

45 See Fleurbaey and Maniquet (2011a). 44 For details, see Fleurbaey and Maniquet (1996, 46 See Fleurbaey (2008) and Fleurbaey and Maniquet 2005). (2011a) for further discussion of the choice of w w . ̃ = min Fleurbaey and Maniquet: Optimal Income Taxation Theory 1065

( ) m ( ′ ′ m ( ′ ( )

m (

0 1 ℓ Figure 5. m(w, z) m(w, z) whereas m (w , z) m(w , z) j ̃ j > k ̃ k k ̃ ′ k > j ̃ ′ j

consider, for instance, the reference-wage on (once the common v is given). Then θi egalitarian-equivalent social ordering. Let us one computes assume that preferences are quasi-linear and __y i ∗ can be represented by the following utility m i w , z i c i v w ( i) ( ̃ ) = − (n i) − ̃ ℓ θ function: U ( , c ) c v , for some i ℓi i = i − (ℓi/θi) increasing and convex v function satisfying ∗ ( ) v ______ℓ θ i . v(0) 0. With this utility function, one has + = ( θi ) V (y , c ) c v y w , so that the i i i = i − ( i/( i θi)) parameter n w completely cap- It is transparent that this utility index cannot i = i θi tures the behavioral heterogeneity of the be written as a function of the n i parameter population. only. This implies that optimal tax analysis Let ∗ ( ) be the optimal labor time of cannot then be performed with a unidimen- ℓ θi agent i if her wage were the reference w . sional screening method, in general. ̃ The quasi-linearity assumption precisely Here, one may worry that the class of guarantees that it is fixed and only depends utility indexes described in this section 1066 Journal of Economic Literature, Vol. LVI (September 2018)

complicates, rather than enhances, the work 9.1 Assessing Feasible Taxes of tax analysts. The next section examines what can be done. The literature has mainly focused on the so-called “reform” problem (introduced by Feldstein 1976), which is the identification 9. Applications of Taxation Theory of marginal changes that improve the tax. In the previous section, we have spanned The (weighted or unweighted) utilitarian a wide set of possible social objectives from social welfare function lends itself well to this utilitarianism to libertarianism and intro- analysis, as shown in formula (2), which dis- duced a subclass of intermediate objec- plays weights for each local marginal change tives that involve money-metric utilities. As T y that simply aggregate the marginal δ ( ) already explained in the previous sections, social value of money for people earning y , defining the social objective is only the and can be derived from knowledge of the beginning of the analysis, and deriving policy statistical characteristics of the population. conclusions requires the additional work that The Saez–Stantcheva approach proposes to makes the bulk of tax theory. In this section, derive these weights directly from intuitions we briefly review methodologies and provide about how deserving the population earning illustrations. such or such level of income is. A systematic It is useful to distinguish two types of study of the reform problem, focusing on lin- applications. First, when a small set of tax ear taxes, can be found in Guesnerie (1995). options is offered, e.g., when a few possible Kaplow (2008) proposed to decompose reforms are considered, how can they be every policy project (not just a tax reform) ranked in order to select the best according into an efficiency component and a distri- to the chosen social objective? This is the butional component. The idea is that, under typical context of practical policy decisions. the separability assumptions that make the The political process is always full of polit- Atkinson and Stiglitz (1976) result valid,47 one ical constraints that limit the set of options can always decompose a change in allocation that can be considered. The second type due to a particular project into two steps: i) of application, which has attracted much the change in the reform under consideration, more attention in economic theory, is the accompanied by an extra (virtual) adjustment “utopian” context in which the whole set of in the income tax that is budget neutral and feasible taxes is considered and one can pick produces a new distribution of welfare that is the best. It is important to see that the two Pareto comparable to the initial distribution; contexts require different tools. Knowing and ii) removing the extra adjustment in the the optimal tax formula does not provide a income tax, which yields the final outcome of recipe for comparing two suboptimal taxes, the reform. The idea is that, if the interme- and conversely, a simple criterion for the diate allocation is Pareto superior to the ini- comparison of arbitrary taxes does not pro- tial allocation, there is a sort of Kaldor–Hicks vide the optimal formula. It is, in particular, argument in favor of the reform: the reform incorrect to draw lessons about reforms from could produce a Pareto-improving change the qualitative shape of the optimal tax. For if accompanied by a suitable reshuffling of instance, if one finds that the optimal tax isU the income tax. Of course, the distributional shaped in marginal tax rates (as in Diamond 1998), this does not mean that any reform 47 Individual preferences about commodities must be moving the tax formula toward a U shape is identical and independent of the quantity of labor. See an improvement. Laroque (2005) for details. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1067 impact of the second step should not be Incidentally, note that, for any arbitrary allo- forgotten, as it may be substantial in some cation, one can compute the money-metric reforms. In a similar fashion to the Kaldor– utility of an individual i in the ( y, c) space by ∗ ∗ Hicks approach, this type of analysis raises computing k such that B ( k , w i) is tangent two possibilities: either the compensation is to i’s indifference curve. ∗ made and the Pareto principle is sufficient Take the union of all the sets B (k , w), to evaluate the reform, or the compensation for w w , w , and construct the ∈ [ min max] is not made, and potential Pareto superior- function f (y) that espouses the upper bound ity is not a decisive argument in favor of the of this set. It is a matter of simple algebra to reform. compute that

The maximin criterion generally simplifies ∗ _y k ( w (1 ) w min) w ̃ for y w min the analysis of reforms by reducing the num- f (y) + λ ̃ + − λ ( min − ℓ ) ≤ . = ∗ {k w (1 ) y 1 for y w min ber of income levels with a positive weight to + (λ ̃ + − λ )( − ℓ ̃ ) ≥ a small set, possibly a singleton. In this case, The graph of this function is a very sim- a marginal reform is an improvement if it ple piecewise linear curve, with slope w w 1 0 for y w , increases income at this particular level. λ ̃ / min + ( − λ) ≥ ≤ min The combination of the maximin cri- and 1 1 0, 1 for y w .48 ( − λ)( − ℓ ̃ )∈ [ ] ≥ min terion with the money-metric utilities Note that f is always concave, since m w , z w discussed in the previous 1 1 w w 1 . i( ̃ i i) + ̃ iℓ ̃ ( − λ) ( − ℓ ̃ ) ≤ λ ̃ / min + ( − λ) section produces an extra simplification that The graph of the function f provides a makes it possible to evaluate all reforms, very simple graphical tool to evaluate taxes. marginal and non-marginal, in a way that Consider any given tax T and its induced allo- requires very little information about the cation, and compute k ∗ so that the graph of population characteristics. This is a direct f is tangent from below to the graph of generalization of the analysis made in sec- y Ty . By construction, there is a w such − ( ) tion 7 for the special case of 1 and that the curve y T(y) touches, without λ = − w w . Indeed, imagine an allocation crossing, the set Bk ∗ , w from above for ̃ = min ( ) (which may or may not be feasible by a tax) a level y 0. Assume that among the group for which the individual money-metric util- of individuals sharing this particular w, ∗ ities are all equal to a given k . Focus on there is one, i 0, whose income is y 0. the group of individuals sharing the same The money-metric utility of i 0 is necessarily ∗ w i. Equality in their utilities means that k —it cannot be greater, since i 0’s bundle their implicit budget (i.e., the budget with belongs to B (k ∗, w), and it cannot be lower, slope w that is tangent to their indifference since i ’s indifference curve is weakly above ̃ i 0 curve in , c coordinates) is, for all of them, y T(y), therefore weakly above B k ∗, w . (ℓ ) − ( ) the same set Moreover, this agent is necessarily among the worst off of the w -group, because the , c X | c k ∗ w . indifference curves of everyone in this group {(ℓ ) ∈ = + ̃ i(ℓ − ℓ ̃ )} are weakly above y T(y). A fortiori, for the ∗− In (y, c) coordinates, this corresponds to the agents whose B (k , w) is strictly below the set 48 When the set of values of w is discrete, the func- ∗ B(k , w i) ( y, c) [ 0, w i] | tion is slightly less simple above w min. The line of slope = { ∈ × ℝ+ 1 1 is replaced by a succession of lines of ( − λ)( − ℓ ̃ ) y slope 0 and 1, with the points f (w) being on the line of ∗ _ slope 1 1 . For details, see Fleurbaey and c k w i . ( − λ) ( − ℓ ̃ ) = + ̃ (w i − ℓ ̃ )} Maniquet (2011c). 1068 Journal of Economic Literature, Vol. LVI (September 2018) graph of y T y , their money-metric util- Third, the libertarian criterion, with 0 − ( ) λ = ity must be above k ∗. This means that i is and 0, has a unitary slope throughout 0 ℓ ̃ = among the worst off of the whole population, and identifies as the worst off those who pay and that k ∗ is the lowest utility. the most taxes. At the opposite extreme, If we assume that it is a good approxima- the egalitarian-equivalent ( 1) case λ = tion to consider that every group of agents with w 0 has a zero slope throughout, ̃ = with a given w is spread everywhere on implying a focus on the lowest consumption the budget curve y T y over the range level and seeking to maximize the minimum − ( ) y 0, w , one sees that computing k ∗ income support. Note that for 0, by ∈ [ ] λ > by seeking tangency between the graph choosing a high w there is no upper bound to ̃ of f and the graph of y T y provides the slope before w . In contrast, the slope − ( ) min a good approximation of the lowest util- above w min is always between zero and one. ity. Therefore, when k ∗, thus computed, is Let us briefly discuss the case of public greater for one tax than for another, social goods affecting productivity. For this case, welfare is greater with the corresponding we saw in the previous section that the lib- tax. ertarian approach was compatible with tak- The computation of k ∗ requires no knowl- ing m w , z for w 1 w if w g i( ̃ i i) ̃ i = ( − λ) i i( ) edge at all of the characteristics of the pop- w h g . The function f is then quite sim- = i ( ) ulation, except the value of w and the ple: it has slope 1 throughout. This min ( − λ) assumption that there is sufficient diversity pushes the optimal tax toward a proportional of preferences so that individuals within tax, which is reminiscent of Smith’s (1776) each wage group are spread on their com- intuition about the benefit tax, as recalled by mon budget curve. Weinzierl (2014a). Observe that the concavity of f means that This methodology can also be used, in a basic form of progressivity is built in to the the context of a particular reform, to find analysis. Of course, an optimal tax may still the subset of criteria (in the class of max- have declining marginal tax rates if efficiency imin criteria applied to utilities of the requires it, due to incentives. m w , z w form) that are compatible i( ̃ i i) + ̃ iℓ ̃ A few salient cases are worth describing. with this reform. Assuming that the reform First, for 1 (egalitarian-equivalent cri- is motivated by a consistent objective, one λ = terion) or 1 (focus on full incomes for can then retrieve the revealed preferences ℓ ̃ = the implicit budgets), the slope is nill after of the policy maker. For instance, Fleurbaey w min, which means that the comparison of (2008) examines the US 1996 welfare reform taxes bears only on the graph of y T y and finds that it is an improvement for a − ( ) below w min, reflecting an absolute priority reference-wage egalitarian-equivalent cri- granted to low incomes. terion (i.e., 1 ) only if w is sufficiently λ = ̃ Second, for 0 (priority to laissez-faire) high (meaning that hardworking agents are λ = or w w , the slope is one below w , favored enough by the decision maker). ̃ = min min implying that the goal is to have a zero mar- Figure 6 illustrates the assessment of taxes ginal tax on low incomes. The particular with the f function. The budget reflects the reference-wage egalitarian-equivalent cri- 2013 US income tax for a couple with two terion studied in section 7, with 1 and children.49 For an egalitarian-equivalent λ = w w , has a unit slope until w , and ̃ = min min zero beyond, meaning that the goal is to give 49 The data come from the OECD Tax Benefit a lump-sum transfer to low incomes, and to Calculator (http://www.oecd.org/els/benefits-and-wages- maximize it. models.htm) based on the OECD report on benefits and Fleurbaey and Maniquet: Optimal Income Taxation Theory 1069

Budget under US tax 350 US tax Laissez-faire 1, ref. 0.76 300 λ = = min 0, ref. labor 0.4 λ = =

250 70

60 200 50

150 40 Net income 30 100 20

10 50 0 0 10 20 30 40 50 60 70 80 0 0 50 100 150 200 250 300 350 400 450 500 Earnings

Figure 6. Evaluating the US Income Tax

approach ( 1) with a value of w that is In fact, since the marginal tax rate is slightly λ = ̃ sufficiently low (at0.76 w min), the earnings above 0.4 for high incomes, the function that receive positive weight and correspond f shown on the figure actually crosses the to the lowest money-metric utility are zero budget curve around $650K. Therefore, and w min (set here at $25K). A lower value the worst off are actually the top incomes for w would single out the zero earnings, a for this particular f. The value of 0.439 ̃ ℓ ̃ = greater value would focus on the minimum (corresponding to marginal tax rate for high wage (working poor).50 For a libertarian incomes) is the threshold below which the criterion ( 0, with 0.4 on the fig- top incomes get the priority and above which λ = ℓ ̃ = ure), the focus is not just on the low incomes the priority shifts to lower incomes. (around $50K), but also the high incomes. 9.2 Optimal Tax wages (http://www.oecd.org/els/soc/benefits-and-wages- Let us very briefly discuss the first-best country-specific-information.htm) for the United States. context in which incentive constraints do We are grateful to Dirk Neumann for guiding us to these not apply. It was recalled in section 4 that data. 50 Note that the budget curve is not monotonic, due to utilitarianism penalizes the more produc- the phasing out of social assistance, and that the trough at tive individuals for given preferences. The $33K is also tangent to f. Our analysis is compatible with maximin criterion naturally avoids this prob- preferences failing to be monotonic in leisure, in which case some individuals could end up picking a point in the lem for utility indexes such that individu- non-monotonic part of the curve. als with identical preferences are given the 1070 Journal of Economic Literature, Vol. LVI (September 2018) same utility index. This property holds true optimal tax is an important theoretical ques- with the money-metric utilities m w , z tion, and provides valuable insights into the i( ̃ i i) + w if 1. In contrast, when 1, exact form of the information that is needed ̃ iℓ ̃ λ = λ < the utility index depends not just on the to compute the tax (sufficient statistics). individual’s preferences, but also on w i. The The typical tax formulas (as derived in first-best allocations equalize these utilities, Atkinson and Stiglitz 1980, Diamond 1998, which means that individual i is indiffer- Saez 2001, Jacquet and Lehmann 2014, ent to a budget with slope w and all such and Saez and Stantcheva 2016) contain two ̃ i budgets give the same consumption for . If endogenous terms on the right-hand side of ℓ ̃ 0, the budget lines cross and the lower the marginal tax equation: the average mar- ℓ ̃ > part of these budgets (below ) are therefore ginal social value of income above the level ℓ ̃ less favorable for the individuals with greater of earnings y , and the elasticity of earnings w i. It is then possible to see some pairs of with respect to the net income rate (one individuals with identical preferences in a minus the tax rate). For instance, the for- configuration such that the one with greater mula obtained by Jacquet and Lehmann is: wage in the pair is on a lower indifference curve. This will happen for individuals who T ′ ( y) 1 1 H(y) y + ∞[ g(z) (z) T ′ ( z)] h(z) dz (3) ______ ____ ______− 1 ______∫ + η , 1 T ′ y = (y) yh y − 1 H y are sufficiently averse to work so that their − ( ) ϵ ( ) ( − ( ) ) optimal choice of labor in these implicit budgets are below . When 1, which where y is the average elasticity of y with ℓ ̃ ℓ ̃ = ϵ( ) implies equalizing full incomes at the opti- respect to 1 T among agents earning y , h − ′ mum, this becomes a general pattern, as with and H are the PDF and CDF of the distri- utilitarianism. This has famously been called bution of y at the allocation, g(z) is the mar- the “slavery of the talented” by Dworkin ginal social value at earning level z , and z η( ) (1981b).51 In summary, with the m w , z is the average derivative of y with respect to i( ̃ i i) w class of utilities, if one wants to fully a lump-sum transfer among agents earning + ̃ iℓ ̃ avoid the risk of penalizing any individual z. In order to implement such formulas, it for being more productive, one has to pick is then common to assume a fixed distribu- either 1 or 0.52 tion of marginal social values (and or a fixed λ = ℓ ̃ = / In the second-best context, the computa- distribution of income) and a constant elas- tion of the optimal tax formula, or at least the ticity, and when possible an iteration can be marginal tax rates, has been the focus of the performed. literature since Mirrlees (1971). In practice, The purpose of this section is to illustrate it is always possible to find the optimal tax the various optimal taxes that are obtained with a sufficiently powerful computer for for a population calibrated to resemble any given sample describing the population the US population, and with quasi-linear 1 1 characteristics. However, understanding preferences represented by c a + /ε. − i ℓ the relation between the characteristics of The calibration is similar to Lockwood and the population, the social objective, and the Weinzierl’s (2015), with a few changes. First, a full-time work bound is introduced, which matters in the definition of money-metric 51 The prospect that such a tax might force the talented to work in order to pay their tax liability is discussed in the utilities. Second, the diversity of prefer- law literature (see Zelenak 2006 and Hasen 2007). ences is specified since some social objec- 52 In the case of identical preferences over consump- tives require knowing the joint distribution tion and leisure, it is enough to pick below the range of ℓ ̃ labor values the individuals would choose in their implicit of wages and preferences, even if behav- budgets. ioral heterogeneity is unidimensional, as Fleurbaey and Maniquet: Optimal Income Taxation Theory 1071

300 Ineq. aversion = 0.5 Ineq. aversion = 3 250 Ineq. aversion = in nity Laissez-faire

200

150 Net income 100

0 0 50 100 150 200 250 300 Earnings

Figure 7. Utilitarian Taxes (for 0.5, 3, ) ρ = +∞

explained at the end of the previous section.53 underlying formula (3) cannot be applied. With this population, many agents work Moreover, the maximin money-metric cri- full time, so that the first-order approach teria would provide weights that can vary abruptly with small changes in the allocation and therefore do not lend themselves to the 53 The population contains 300 types of households with use of this formula (except when the mar- wages w i distributed according to a lognormal distribution with parameters , 2.2, 0.6 so that the distri- ginal social value is zero, but for the lowest (μ σ) = ( ) bution of earnings fits the quintiles of the distribution of income). We therefore focus on the compu- pretax household income from the census for 2014 for a tation of an optimal piecewise linear tax.54 flat tax of 30 percent ($29k, 53k, 82k, 129k, and 230k for the 95th percentile). A household is considered in our Figure 7 displays the various taxes obtained application to be like 1.66 adults (census), so that for an with a utilitarian social welfare function individual minimum wage of $15,000 per year we use a _1 1 i u i −ρ for different values of . The household minimum of $25,000. The parameters a i of the 1 ∑ ρ 1 1 − ρ utility functions c a + /ε take three values so that the − i ℓ utilities adopted here are simply the inter- percentage of households working full time (more than 1 1 cept utilities c a + /ε, which can be 90 percent), part time, and less than 1 3, at the status i / − ℓ quo, is 45 percent, 43 percent, and 12 percent. We have interpreted as the amount of consumption introduced a greater variety of preference parameters for that i would accept in absence of work— the minimum-wage households so that their labor supply spreads over the whole interval. The parameter is set ε at 0.5 to match a reasonable estimate of the elasticity of part-time workers, and the average elasticity of the whole 54 There are sixteen tax brackets. The maximum income population at the status quo is 0.26. in the sample is 500. 1072 Journal of Economic Literature, Vol. LVI (September 2018)

Panel A. Budget under egalitarian-equivalent Panel B. Budget under libertarian optimal tax 1, ref. wage 0, 25, or 50 optimal tax 0, ref. labor 0, 0.2, 0.5 or 1 λ = = λ = =

250 250

200 200

150 150 Net income Net income 100 100

50 50

0 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 Earnings Earnings

Ref. wage 0 Ref. labor 1 = = Ref. wage ( 25) Ref. labor 0.5 = min = = Ref. wage 50 Ref. labor 0.2 = = Laissez-faire Ref. labor 0 ( Laissez-faire = = )

Figure 8. Optimal Egalitarian-Equivalent and Libertarian Taxes

these are the money-metric utilities for optimal tax as Roemer’s equal opportunity 1 and w 0. This choice of cardinal criterion that maximizes the average utility λ = ̃ = utilities is of course only one possibility among of the unskilled agents whose wage is w min many. As one can see on the figure, the mar- (assuming that intercept utilities are used in ginal tax rates go from an inverted U shape Roemer’s criterion, as in the above simula- to a declining curve, and then to a U-shaped tions of utilitarianism). pattern (Diamond 1998) as inequality aver- Figure 8, in the right graph, also shows sion increases. various taxes obtained with the libertarian Figure 8, on the left graph, shows vari- criteria, i.e., when w w and varies ̃ i = i ℓ ̃ ous taxes obtained with the reference-wage from 0 (pure libertarianism) to 1 (which is egalitarian-equivalent criteria, i.e., the most redistributive in this category and when w w for all agents. Given the ̃ i = ̃ coincides with the most redistributive utili- absence of income effects and the fact that tarian tax in figure 7).55 Note that the US tax only the situation of the income below w min matters for these criteria, the budget 55 curves are parallel for y w min. Note that For these criteria, the function f introduced in > the previous subsection has slope 1 until w and 1 in this model with quasi-linear preferences, min − ℓ ̃ beyond. The optimal tax for ̃ 0.2 produces a budget the criterion with w w has the same ℓ = ∗ ̃ = min that actually espouses the graph of f for a well-chosen k . Fleurbaey and Maniquet: Optimal Income Taxation Theory 1073 shown on figure 6 bears some resem- individual utility indexes are malleable tools blance with the libertarian tax one would that can incorporate many of the fairness obtain with 0.4, since it has a mar- considerations listed in the previous para- ℓ ̃ = ginal tax close to zero for low incomes, and graph. Perhaps the current weariness with then close to 40 percent (although around the social welfare function comes from an 30 percent for certain intermediate levels of exclusive focus on weighted variants of the earnings). utilitarian criterion. The utility-weighting For a value of between 0 and 1, one approach, indeed, is limited because the λ obtains optimal taxes that are intermediate weights cannot be transparently connected between those of the two sides of figure 8. to fairness ideas. Even the method of For 1 2, with w 0 and 0 (a weighting incomes directly, as proposed by λ = / ̃ = ℓ ̃ = case in which f has constant slope 1 2, and Saez and Stantcheva (2016), cannot always / may arise for a libertarian approach taxing be applied easily because identifying the lev- benefits from public goods, considering that els of income that deserve a positive weight half of everyone’s wage can be attributed to may sometimes require a detour by a suit- the benefits brought by public goods), in our ably defined social welfare function involving calibrated economy one obtains an optimal appropriate utility indexes. tax that makes y T y coincide with f y We have focused on the maximin crite- − ( ) ( ) and is a flat tax (with a marginal tax rate equal rion in a large part of the paper. As we have to 50 percent throughout). stressed, however, less extreme degrees of inequality aversion than the maximin can be studied, with the same set of possible utility 10. Conclusion functions as listed in section 8. The maximin In this paper, we have examined the con- is worth considering for two reasons. First, tribution that notions of fairness can make it is important to recognize that a distinc- to optimal income taxation theory. Recent tion must be made between redistribution interest in fairness principles capturing the and inequality aversion in the social welfare relevant differences between deserved and function. The maximin is compatible with undeserved income, as well as between cir- very little redistribution, even with the full cumstances and effort, the importance of libertarian approach, since a crucial element laissez-faire, and the problems with tagging, in deciding how much to redistribute is the makes it timely to connect public economics measure of individual advantage (the utility with the theory of fair allocation, which pro- function), which can take account of endow- vides useful concepts. ments, desert, and similar considerations. While some authors have argued for a rad- The second reason for paying attention to ical overhaul of taxation theory that would the maximin is that it makes the comparative throw out the welfare economics baby with evaluation of taxes easier, as shown in section the utilitarian bath water, we have pleaded 9.1, where a very simple graphical tool has for going beyond the conventional utilitarian been provided, which can be adapted to the criterion while retaining the social welfare whole array of utility functions presented in function and its arguments, the utility func- section 8. tions. Specifically, we have shown that the Two approaches have been highlighted, with all intermediate cases being possible. The egalitarian-equivalent approach focuses More generally, when the graph of f corresponds to an efficient budget, it is associated to the optimal tax (Fleurbaey on low incomes, and lets the decision maker and Maniquet 2011c). choose to prioritize the poor workers who 1074 Journal of Economic Literature, Vol. LVI (September 2018) work full time, or those who work less (or have access to less pleasant or more danger- not at all). The libertarian approach is neu- ous jobs, or because they have children or rel- tral among poor workers and, in its moderate atives needing their care at home, or because version, can accept a certain degree of redis- their health reduces their ability to do cer- tribution represented by a graphical rod fea- tain tasks. As noted in section 8, the worry turing a flat tax above low incomes. Unlike that greater work aversion may be explained the egalitarian-equivalent approach, it does by disadvantages can partly be addressed by pay attention to all incomes, and can declare answering the fourth question in the previ- the worst off to be among the top incomes, ous section in a particular way, by selecting as in the example provided for the US tax in a low reference wage in the construction of figure 6. the utility index. However, addressing these We hope that the last two sections will help issues completely and satisfactorily requires optimal tax theorists and applied analysts to adding the relevant features into the model, incorporate the relevant fairness consider- and, for applications, finding estimates of the ations of their choice easily, via a suitable distribution of characteristics in the relevant choice of the utility indexes. population. The analysis in this paper has been An important extension of the Mirrlees focused on the standard income tax model, model that has been developed recently and similar considerations can be applied to involves dynamic processes (see Golosov, related standard models of taxation or new Tsyvinski, and Werning 2006 for a detailed models incorporating additional features. It introduction to this literature). In the appears that both in theory and in popular dynamic optimal taxation literature, the discourse, different contexts often call for focus is on the ability of the tax system to different fairness principles. For instance, allow the economy to reach constrained effi- inheritance taxation raises issues about altru- cient allocations in the presence of individual ism toward descendants that do not appear and/or aggregate uncertainty about future in the Mirrlees model. The methodology laid productivity (see, e.g., Farhi and Werning out in this paper is, in principle, applicable 2013) or government expenditures (see, e.g., to a wide set of taxation problems, such as Kocherlakota 2005); the impossibility of the capital income taxation, commodity taxation, government to commit not to use the infor- or Pigovian taxation.56 mation revealed by agents’ past choices (see, Such extensions can also serve to address e.g., Berliant and Ledyard 2014); and the the worry that the preferences over labor and advantage of using age or history-dependent consumption that play an important role in taxation schemes (see, e.g., Weinzerl 2011). the Mirrlees model and in usual applications To the extent that they deal with efficiency, of optimal tax theory may be influenced by the results obtained in that literature are of factors for which the laissez-faire principle interest to all approaches that are consistent is not justified. Some workers may be more with efficiency, which includes libertarian and averse to work than others because they only resource-egalitarian approaches. The potential contribution of fairness approaches to dynamic optimal taxation questions should be, 56 See, e.g., Ambec and Ehlers (2016) on externalities and Fleurbaey (2006) on commodity taxes. The study of as in the static optimal taxation setting exam- inheritance taxation in Piketty and Saez (2013a) relies on ined in this paper, to help design the individ- the maximin criterion and could be extended to heteroge- ual utility indices that could be used when the neous utilities with the help of money-metric utilities. One also finds a brief analysis of capital income taxation with goal of the research is to go beyond efficiency money-metric utilities in Fleurbaey (2008). analysis and to study dynamic redistribution. Fleurbaey and Maniquet: Optimal Income Taxation Theory 1075

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