Journal of Public Economics 118 (2014) 128–142

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Journal of Public Economics

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The promise of positive optimal taxation: normative diversity and a rolefor equal sacrifice☆

Matthew Weinzierl277 Morgan Hall, Harvard Business School, Boston MA 02163, United StatesNBER, Cambridge MA 02138, United States

☆ A number of technical details, footnotes, contextuaomitted from this published version can be found in thepaper and its Appendix, Weinzierl (2014), at the author'sincorporate and build on material from Weinzierl (2012we Redistribute so Much but Tag so Little?” I am graBierbrauer, Kim Clausing, Raj Chetty, Mihir Desai, Rafael dFleischer, John Friedman, Alex Gelber, Mikhail Golosov, CKaplow, Wojciech Kopczuk, Camille Landais, BenjaminYoram Margalioth, Joe Mazor, Jean-Baptiste Michaud, JeRaskolnikov, Meg Rithmire, Julio Rotemberg, EmmanuSamuelson, Florian Scheuer, Eytan Sheshinski, Ali ShouriTrannoy, Aleh Tsyvinski, David Weisbach, Glen Weyl, Johseveral anonymous referees for their helpful discussions.

E-mail address: mweinzierl@hbs.edu.1 Harsanyi (1953) provides the classic discussion of the

this context.

http://dx.doi.org/10.1016/j.jpubeco.2014.06.0120047-2727/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 10 September 2013Received in revised form 17 June 2014Accepted 18 June 2014Available online 1 July 2014

Keywords:Optimal taxationTaggingEqual sacrifice

A prominent assumption in modern optimal tax research is that the objective of taxation is Utilitarian. I presentnew survey evidence thatmost people reject this assumption's implications for several prominent features of taxpolicy, instead preferring tax policies based at least in part on a classic alternative objective: the principle of EqualSacrifice. I generalize the standard model to accommodate this preference for a mixed objective, proposing amethod by which to make disparate criteria commensurable while respecting Pareto efficiency. Then, I showthat optimal policy in this generalized model, calibrated to the survey evidence and U.S. microdata, is capableof quantitatively matching several features of existing tax policy that are incompatible in the conventionalmodel but widely endorsed in the survey and reality, including the coexistence of substantial redistributionand limited tagging. Together, these findings demonstrate the potential of a positive theory of optimal taxation.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Modern tax theorists have a workhorse model. Created by Mirrlees(1971) more than four decades ago, that model has been used tostudy countless aspects of tax policy. It provides the benchmark guide-lines against which policy proposals are often judged, and its recom-mendations form the basis of prominent policy advice.

When this standardmodel has been used to generate quantitative les-sons for policy, theorists commonly have imposed a strong assumption:the objective of tax policy is Utilitarian, either in its simplest form as asum of individual utilities or in a more general form as the sum of aconcave transformation of individual utilities. Mirrlees (1971) himselfintroduced this assumption with little explanation, but virtually alloptimal tax research in the last four decades has adopted it 1. To the

l information, and clarificationsworking paper version of thiswebsite. Portions of this paper), which was entitled “Why doteful to Alan Auerbach, Felixi Tella, Amy Finkelstein, Victor

aroline Hoxby, Bas Jacobs, LouisB. Lockwood, Greg Mankiw,

ff Miron, Dina Pomeranz, Alexel Saez, Bernard Salanie, Larrydeh, Stefanie Stantcheva, Alainn Weymark, Danny Yagan, and

applicability of Utilitarianism in

extent that this assumption has been relaxed, it has usually been toallow for a more redistributive normative criterion, such as theRawlsian priority on the least advantaged. The conventional case forUtilitarianism is usually traced to Harsanyi (1953, 1955).

Some theorists have taken a more agnostic approach by examiningonly whether policies are optimal given some set of weights on individ-uals' welfares; that is, Pareto efficient. An open question in that ap-proach is what weights to use when choosing between a wide rangeof Pareto-efficient policy options; in practice, Utilitarian (or Rawlsian)weights are typically the default assumption.2 The relatively little atten-tion paid to the Utilitarian assumption and its alternatives, as opposedto its policy implications, is especially surprising given that optimaltax theory is one of few forthrightly normative fields in economicresearch.

The first contribution of this paper is to present evidence of widedisagreement with this core assumption, at least in the United States. Idesign and implement a novel survey in which respondents are askedto choose between sets of feasible and incentive compatible tax policiesfor a society with the income distribution of the current United States.First, I ask them to choose between two policies: one based on the stan-dard (simple sum) Utilitarian criterion and the other based on the prin-ciple of Equal Sacrifice, a less redistributive and historically prominentalternative criterion for optimal tax design. In that case, nearly 60%of re-spondents prefer the Equal Sacrifice alternative over the conventionalUtilitarian objective. Disagreement with the conventional Utilitarian

2 See, e.g., Stiglitz, 1987; Werning, 2007; Rothschild and Scheuer, 2012, and Saez andStantcheva, 2014. In addition, specific normative limitations of the conventional modelhave been addressed directly (see Section 1.4 and Weinzierl, 2014).

129M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

assumption is even more striking when I give respondents a range ofchoices, including options that are based in part on Utilitarianism andin part on Equal Sacrifice. I find that 81% of individuals prefer policiesother than the pure Utilitarian or Rawlsian policies, and nearly halfmost prefer policies based on a combination of Utilitarianism andEqual Sacrifice.3 Of course, these responses may be due to a variety offactors other than an affinity for Equal Sacrifice, so I use additional ques-tions in the survey to test for more direct evidence on the relevance ofEqual Sacrifice. When asked explicitly how “sacrifice” from payingtaxes should be distributed, respondents prefer a distribution betweenthat implied by Utilitarianism and Equal Sacrifice. And the more enthu-siastic a respondent is about Equal Sacrifice, the more likely he or she isto reject tagging, the taxation of personal characteristics that is a featureof Utilitarian-optimal tax policy but that is rejected by Equal Sacrifice.

This evidence is admittedly far from definitive. The survey respon-dents are not a random representative sample of Americans, andmany variations in the survey's design, framing, and implementationare possible and could have large effects (see McCaffery and Baron,2004, for example). Nevertheless, the results are robust across subsam-ples, and the survey is designed to guard against a number of potentialdesign concerns. In the end, the survey evidence suggests that a numberof features of tax policy implied by the Utilitarian objective of conven-tional theory may not be, in the terminology of Diamond and Saez(2011), “socially acceptable.” That is, a large majority of individuals ap-pear to place substantial value on an alternative normative principle –

Equal Sacrifice – that rejects some of the conventional objective's policyimplications.

While my finding of a preference for a mixed objective is foreign tothe optimal tax literature, it is consistent with a large body of existingresearch showing that most individuals are not normative purists. Inthat research, whether individuals are asked to evaluate incomedistributions, answer conceptual questions, or participate in allocationgames, few appear to use a single normative criterion. As Scott et al.(2001) write: “Experimental research reveals that distributive justicejudgments usually involve several distinct allocation principles.”

How should we respond to this evidence? One possible response isto ignore it. We may decide that a normative theory ought to chooseits objective based on philosophical reasoning regardless of popularopinion. An alternative approach is to incorporate as much evidenceas possible on the way the agents included in these models thinkabout these very same issues. In their important synthesis of “empiricalsocial choice” research, Gaertner and Schokkaert (2012) make a strongargument for the value of eliciting public attitudes toward such issues.In optimal tax research, incorporating key aspects of reality into theconventional model has been a hallmark of major contributions suchas Diamond (1998), Saez (2001, 2002), Golosov and Tsyvinski (2006),and Farhi and Werning (2010), and often these efforts have improvedthe match between the theory's recommendations and real-world pol-icy. Diamond and Saez (2011) suggest a similar effort with regard to thenormative aspects of themodel, advocating a requirement of “social ac-ceptability” under which real-world normative beliefs would constrainthe set of relevant policy results.4 My paper falls in this tradition andproposes that we go one step further: it gathers formal evidence aboutpeople's views and interprets that evidence asmotivation for construct-ing a positive optimal tax theory. The broad aim of this positive optimaltaxation project is, then, to pursue empirically-supported generaliza-tions of the standard optimal tax model to better match the way inwhich real societies appear to evaluate tax policy. Specifically, thispaper's survey evidence, and a large body of prior work, suggests thatwe generalize the standard model to include a mixed policy objective.

3 A note on terminology: from this point on I will use “Utilitarian” to refer to the simplesum of individual utilities, not themore general version inwhich transformations of thoseutilities are made prior to aggregation.

4 Gaertner and Schokkaert (2012) provide a lucid and insightful discussion of the rela-tionship between normative and positive analyses of social preferences.

The second main contribution of this paper is to formally develop ageneralized model that can be used for positive optimal tax analysis.The generalized model combines multiple normative criteria into asingle policy objective while retaining both Pareto efficiency and the re-mainder of the familiar formal apparatus of conventional optimal taxtheory. In this way, I am following up on a suggestion made morethan three decades ago by Martin Feldstein (1976), that “optimal taxdesign involves a balancing of conflicting criteria.” This generalizationof the standard theory requires addressing long-standing concernsabout commensurability of different normative criteria. In keepingwith the survey evidence, I develop in depth the specific case of anobjective that combines Utilitarianism and Equal Sacrifice.

A complementary approach to generalizing the conventionalmodel's objective – part of the Pareto-efficient optimal tax approachmentioned above – can be found in contemporaneous research bySaez and Stantcheva (2014). They focus on the role of marginal socialwelfare weights in the aggregation of a given tax reform's effects on in-dividuals. By allowing these weights to take any non-negative values,they include the possibility that theymay be based in part on normativecriteria other than Utilitarianism.5 Their approach and this paper's canbe seen as two sides of the same coin: one might translate a mixedobjective function into a profile of marginal social welfare weights orvice versa. Each approach has applications for which it is more naturallysuited, and both contribute toward the broader goal of constructing apositive theory of optimal taxation.

One attractive feature of this paper's approach is that it requires aclear statement of each component of the set of criteria by which policyis judged. This requirement acts as a second test of the theory (in addi-tion to its ability tomatch observed policy features), in that criteria lack-ing intellectual coherence can be rejected andwe can avoid the risk thatfully-flexible welfare weights lose any explanatory power. More gener-ally, under the Pareto-efficient optimal tax approach, assumptions onthe welfare weights are often made in the interests of deriving morepowerful results. One way to interpret my contribution in the contextof that approach is that I look for evidence on the normative criteriathat seem to hold in reality and that, therefore, might inform the valuesof those weights that society would endorse. Specifically, I am able touse the principle of Equal Sacrifice as a disciplined way to give weightto a point on the Pareto frontier that appears to matter to the publicbut has been largely ignored by modern tax theory.

This paper's approach has a number of limitations. Positive optimaltax theory as developed here is not a positive tax theory, i.e., I havenot modeled the political economy that translates the public's prefer-ences into policy. While recognizing that establishing such a link is es-sential for a full understanding of how any normative principles affectreal-world allocations, that task is outside the main objective of thispaper. At the same time, positive optimal tax theory is not normativeoptimal tax theory, and we may reject the implications of the former ifwe believe the public is – at any given time – subject to biases ormistak-en beliefs. It is because of this very real and important risk that I empha-size the search for recognizable, and at least arguably defensible,philosophical principles in the development of the model. Moreover, itis important to clarify that positive optimal tax theory is not a substitutefor traditional normative optimal tax theory based on consideredjudgments of what society's objective function ought to be.

The third contribution of this paper is to show that this generalizedmodel, when calibrated to this survey evidence, can reconcile a numberof features of tax policy that are incompatible in conventional theory butendorsed in the survey evidence aswell as in reality. In particular, I sim-ulate optimal policy using the survey respondents' most-preferred

5 Saez and Stantcheva also note that welfare weights could be derived from existingpolicies or survey evidence. Bourguignon and Spadaro (2012) take the former approachto calibrating thewelfareweights in a standardmodel, as do Spadaro et al. (2012); Bargainet al. (2011, 2013); Zoutman et al. (2013a, 2013b); Hendren (2014); and Lockwood andWeinzierl (2014).

130 M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

normative objective, which combines Utilitarianism and Equal Sacrifice,and U.S. microdata. That policy simultaneously rejects the use of height,gender, and race as tags; accepts the use of blindness as a tag, endorsinga quantitatively realistic blindness benefit; and provides redistributionthrough a progressive schedule of average income tax rates that closelyresembles actual policy. I also briefly note two other examples: itsubstantially reduces the extent of utility rank reversals in the first-best policy, and it implies top marginal tax rates lower than whatconventional theory would recommend and closer to reality.6

The results on tagging are important because they provide evidencethat Equal Sacrifice's explanatory power is due to more than simplybeing a “less redistributive” criterion than Utilitarianism. While the op-timal extent of both tagging and redistribution decrease when EqualSacrifice is given more weight, tagging is reduced much more dramati-cally. Intuitively, Equal Sacrifice strongly rejects horizontal inequity intaxes (conditional on income-earning ability) because a taxpayer's sac-rifice is determined by his or her income-earning ability only, whileEqual Sacrifice can accommodate a range of tax progressivity acrossability levels. Consistent with this indirect evidence, I find more directevidence for Equal Sacrifice's role in explaining limited tagging in thesurvey, where a greater share of survey respondentswho oppose heightand blindness tags prefers policies based in part on Equal Sacrifice.

Taken together, the survey results, theoretical analysis, and calibrat-ed simulations of this paper demonstrate the potential of a positive op-timal taxation research agenda. They show that we can rigorouslycapture empirical evidence onwhat tax policies individuals find accept-able and, as one might hope, use the resulting model to better under-stand how actual tax policy is and (arguably) ought to be designed.

The support that I find for policies based in part on Equal Sacrificemay seem surprising, but in fact it ought not to be. Though Equal Sacri-fice has played only a minor role in tax research since 1971, it was orig-inally proposed by no less a Utilitarian than John Stuart Mill, and itavoids a prominent critique of Utilitarianism put forward by JohnRawls (1971), among others. In the early years of modern optimal taxtheory, Martin Feldstein (1976) saw a connection between Equal Sacri-fice and Robert Nozick's (1974) Libertarianism, arguing that “…taxschedules that impose equal utility sacrifice have an appeal that is clear-ly lacking in the utilitarian framework.” The pioneering work of H.Peyton Young (1987, 1988, 1990, 1994) and Berliant and Gouveia(1993) showed that existing income tax rate schedules were consistentwith the Equal Sacrifice principle by itself. In a sense, it would be sur-prising if Equal Sacrifice did not feature at least somewhat in theviews of many, especially in the United States.

The paper proceeds as follows. Section 1 reports the new survey ev-idence on normative preferences and discusses similar findings in priorwork. Section 2 generalizes the standardmodel to allow for amixed ob-jective, discusses Equal Sacrifice as an alternative to Utilitarianism, andapplies the model to the case of these two criteria. Section 3 showsthat the parameterizations of that model most preferred by survey re-spondents imply policies that resolve several disparities between con-ventional theory and real-world policy, especially the puzzle of limitedtagging. Section 4 concludes, andWeinzierl (2014) contains supportingmaterial.

7 The recent paper by Kuziemko et al. (2013) also presents respondents with realisticpolicy choices.

8 See Lockwood andWeinzierl (2014) on how greater preference heterogeneity lowersoptimal redistribution in the standard model. To the extent that heterogeneity in prefer-

2. New results on empirical normative preferences

In this section, I describe the design and results of a novel surveyeliciting normative preferences over realistic tax policies. I also providea range of robustness checks, all of which confirm the main findings:few individuals prefer the pure Utilitarian criterion standard in conven-tional optimal tax theory or the commonly-used Rawlsian alternative,and a plurality of individuals prefers tax policies reflecting a mixed nor-mative objective including both Utilitarianism and the classic

6 Weinzierl (2014) containsmuchmoredetail on these applications than is shownhere.

alternative criterion of Equal Sacrifice. Despite those checks, of coursethese results are far from definitive. Future research could exploremany variations on the surveys I perform, including changes to theway the data are presented, the design of the survey itself, and thechoice of the respondent sample.

One important ambiguity in a subset of the survey results is that theinfluence of the Equal Sacrifice criterion on the policy choices facing sur-vey respondents is by nomeans unique to Equal Sacrifice. Respondentsendorsing the less redistributive policies in the survey may be motivat-ed by a variety of factors other than an affinity for Equal Sacrifice. Toclarify, this paper is not intended to show that Equal Sacrifice influencespreferences to the exclusion of other, similar factors. Nevertheless, toaddress this ambiguity and bolster the case that Equal Sacrifice doesmatter for preferences, later in the paper I show the results of surveyquestions that explicitly gauge support for Equal Sacrifice, and I showthat Equal Sacrifice can have explanatory power along a dimension ofpolicy – tagging – not directly tied to the extent of redistribution(in Section 3).

This paper's survey makes a methodological contribution to empiri-cal research on tax preferences by having respondents face a task thatmimics the conventional social planner's optimal tax problem: that is,policies are constrained by both feasibility (in the context of govern-ment spending) and incentive compatibility. This innovation overmost prior work7 allows me to use the evidence on participant prefer-ences to calibrate a fully-specified optimal policy model.

2.1. Survey design

The survey, shown in full in Appendix toWeinzierl (2014), has threeparts. The first part tests whether respondents understand and can per-form simple calculations related to the concepts of before-tax income,after-tax income, and average tax rates. It defines each of these terms,shows a graphical illustration of them that parallels the figures used inthe remainder of the survey, and then asks four multiple-choice ques-tions to test comprehension. The third part of the survey asks respon-dents about their opinions on aspects of tax policy, political views, andpersonal traits, including economic status. To address any concernsthat these characteristics matter for the results, I examine my findings'robustness across all subgroups.

The second part is the centerpiece of the survey. Respondents areshown a graphical gross income distribution divided into eight typesof households (based on CBO data as discussed below). These types rep-resent the four lower quintiles and a division of the top quintile into thenext 10, 5, 4, and 1 percentiles. Wemight worry that respondents attri-bute some of the variation in incomes to differences in preferences forwhich, in the influential terminology of Fleurbaey and Maniquet(2006), individuals ought to be held “responsible” and not taxed or sub-sidized. I try to minimize that risk by clarifying in the survey text thatdifferences in earnings are not due to effort as follows8: “If there wereno taxes, these households would all work equally hard. But, type 2would earn more than type 1, type 3 would earn more than type 2,and so on.”

Respondents are then put in the position of objective policymakersfacing a constrained optimal tax problem. They are told “You are giventhe chance to choose taxes for this society. Please think of yourself asa policymaker for this society.” They are given information about theconstraints affecting their choices, as the survey states the requiredlevel of exogenous government spending (i.e., feasibility) and empha-sizes that households' labor supplies include responses to tax policy

ences exists in reality, the survey text to the contrary would be expected to push respon-dents toward a more redistributive policy than they would endorse in reality.

131M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

(i.e., incentive compatibility). Respondents are reminded that taxesmayserve a variety of purposes, from funding public goods to redistributingbefore-tax income. 9

In a series of choices, respondents rank sets of tax policies. For eachpolicy option, the survey displays two overlapping income distributions(see Fig. 1 for an example). The pretax distribution is shown as emptyoutlined columns while the after tax distribution is shown as filled-incolumns. The average tax rate for each household type is shown in atext box above their columns. Respondents are asked to rank the poli-cies from “best” to “worst” by clicking on numbered radio buttons. Byusing the general terms “best” and “worst” without further definingthe criteria by which tax policies ought to be judged, the survey leavesthe respondent free to use his or her own definition of optimality.

The surveywas listed in November, 2012 as an available task to up to400 members of the Amazon Mechanical Turk worker population fromthe United States who demonstrated good past performance on tasks.10

The title of the task was “Wewant your opinions on tax policy”, the de-scription was “Rank possible tax policies and give us your opinions ontaxes,” and the survey requestor was identified as “TaxSurvey.” Respon-dents had up to 30 min to complete the survey, and they were asked toenter their MTurk identification number aswell as a completion code atthe end of the survey for verification purposes. The respondentscompleted the survey in an average of 13 min and 6 s. They were paid$2.00 for the task, implying an average hourly rate of $9.16.

11 Formally, using notation specified below, these policies are as follows. To generate theRawlsian policy option, the planner solves the problem

maxci� ;y

i�f gI

i¼1∈ F∩Iℂf g

U c1� ; y1�=w

1� �

;

where F is defined in Eq. (2); Iℂ is defined in Eq. (3); and U(c1, y1 / w1) is defined inEq. (12). The Rawlsian policy is the most redistributive policy option. The poll tax policyis defined as follows:

ci� ¼ yi�−1IG for all i∈ 1;…; If g;

where households maximize utility subject to this constraint. The poll tax is the least re-distributive policy option (it is, in fact, regressive).12 An alternative approach would be to give the respondents control over a continuouspolicy lever that would trace out the entire range of redistribution (for instance, αES). Thatalternative has two drawbacks, however. First, communicating themeaning of that policylever would be difficult without influencing the respondents' answers. Second, we arelikely interested not merely in the respondents' ideal points but in their attitudes towardoptions along the entire range. Those would be difficult to elicit with this alternativeapproach.13 Note that the surveywas designed tominimize the risk that aversion to choosing end-

2.2. Results

Respondents' first rankings provide straightforward evidence thatthe Utilitarian criterion is less popular than the conventional model im-plies. Fig. 1 shows the two policies respondents rank, labeled A and B. InSection 2, I provide the details of how I calculated these (and all other)policy options in the survey. For now, note that option A reflects a con-ventional, pure Utilitarian objective for tax policy, while option B re-flects an objective based entirely on the principle of Equal Sacrifice(which sets the utility cost of taxation equal for all individuals). PolicyA is redistributive, while B is not.

The results of this first choice are strikingly at odds with the conven-tional model's assumed objective. The share of respondents preferringthe Utilitarian policy A is 42%, with a standard error of 2%. In otherwords, nearly three-fifths of respondents prefer the pure Equal Sacrificepolicy B to policy A.

Respondents are then asked, over the course of two questions, to ranka wider range of seven policy options. Table 1 summarizes these choices,decreasing in redistributiveness from left to right. For each option, itshows the average tax rates levied on each household. In the second col-umn of the table, before-tax incomes in the no-tax scenario are shown.

The middle five policies in Table 1 combine Utilitarianism and EqualSacrifice, using a range of values for the weight on Equal Sacrifice, αES,discussed below. The value αES = 0.00 yields the conventional Utilitar-ian policy, while αES=1.00 yields the Equal Sacrifice policy. In between

these polar values, three values generate intermediate policies: αES ¼0:03; 0:10; 0:20n o

.

I also generate the two “endpoint” policies shown in Table 1. Theleft-most policy, C, is a “Rawlsian” policy that maximizes the utility of

9 I ran a follow-up survey, requested by a referee, that explicitly explained that transferswere made in kind. Though a greater percentage of the survey respondents self-identifiedas politically left-leaning, the main results highlighted belowwere unchanged. In particu-lar, amajority of respondents preferred policy B to policy A, and in the seven-policy choicethree-quarters preferred a policy other than Utilitarian or Rawlsian, with a plurality pre-ferring one of the three intermediate policies.10 Specifically, only respondents registered as in the United Stateswhose work had beenaccepted on 95% of previous tasks could take the survey. Horton et al. (2011) study the useof online labor markets, and specifically of Mechanical Turk, and find: “Online experi-ments, we show, can be just as valid—both internally and externally—as laboratory andfield experiments, while often requiring far less money and time to design and conduct.”

the lowest-ability household. The right-most policy is a “poll tax” thatsplits the financial cost of government spending G evenly acrosshouseholds.11. Adding these endpoint policies yields two benefits.First, offering the Rawlsian and poll tax options addresses a potentialframing problemwith presenting respondents with only the set of pol-icies along the Utilitarian-Equal Sacrifice spectrum. To the extent thatindividuals shy away from options that seem “extreme,” having policiesA and B as endpoints could bias us toward finding support for a mixedobjective. Adding the Rawlsian and poll tax options as the endpointson the redistributive spectrum may alleviate this concern. Second, themost commondeviation from simple-sumUtilitarianism in convention-al optimal tax theory is a generalized Utilitarianism under which theplanner takes a concave transformation of utilities before summingthem. The Rawlsian option is often included as an extreme version ofthis generalization. By including a Rawlsian policy as a choice, we cangauge the empirical support for this prominent criterion.

Respondents first compare option A to three additional options, twoof which are less redistributive than A, namely D and E, while C is moreredistributive. Respondents then compare option B from the first choiceto three additional options, two of which aremore redistributive than B,namely D and G, while F is less redistributive. Both choices can be seenin the survey as reproduced inWeinzierl (2014). Policy D is included inboth sets of four-option rankings so thatwe can infer respondents' pref-erences across the full range of seven policies.12 Note that the redistrib-utive spectrum is masked in both the alphabetical policy labels and thephysical placement of policieswithin the four-policy rankings (visible inWeinzierl 2014).

Fig. 2 shows each of these seven policy options and the share ofrespondents who placed them in their top-ranked or second-rankedgroup of policies.

Fig. 2 reveals twomain results from these rankings: one, support forthe conventional Utilitarian assumption and the Rawlsian alternative islow; two, a plurality of respondents prefer a mixed objective.13 Thepurely Utilitarian policy (A) makes up only 10% of the top-rankedchoices and 11% of the second-ranked choices.14 For the Rawlsian alter-native, these figures are 9% and 10%. Together, then, policies at least as

points is generating the observed preference for the intermediate policies D, E, and G. Eachof the two scenarios in which respondents chose between four policies had policy D as anendpoint option. In contrast, policies A and B were intermediate policies in each of theirfour-option scenarios.14 These classifications are made as follows. Using option D, which was included in bothof the four-option policy choices, we can create a weak ranking of all seven policy optionsfor each respondent. The top-ranked group includes any policy strictly dominated by noother policy. Less than 6% of respondents had more than one top-ranked policy. Thesecond-ranked group includes any policy strictly dominated by only policies in the top-ranked group. For example, if a respondent ranks option D as their first choice in bothfour-option choices, they will rank two policies second, each of which is therefore placedin the “second-ranked” group for that respondent.

Tax system A Tax system B

Fig. 1. The choice between the utilitarian policy A and the equal sacrifice policy B.

132 M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

redistributive as the conventional Utilitarianismmake up less than 20%of themost-preferred policies in this survey, the same share claimed bythe pure Equal Sacrifice policy B. In contrast, nearly half – 48% – of thetop-ranked policies were one of the three (E, D, and G) that correspondto a mixed normative criterion. These mixed policies also dominate thesecond-ranked preferences of respondents, making up 56% of thosechoices.

Together, these results sharply contradict the normative assump-tions that dominate modern optimal tax research. Respondents give lit-tle support to using the conventional Utilitarian criterion as the optimalpolicy benchmark. They also appear to disagreewith themost common-ly used alternative to pure Utilitarianism – a more concave socialwelfare function – as respondents are less enthusiastic about theRawlsian policy than any other option except (perhaps) the poll tax.Instead, empirical normative preferences appear to favor the use of amixed objective with some weight on a less redistributive criterionsuch as Equal Sacrifice.

More direct evidence of a preference for an objective that combinesUtilitarianism with Equal Sacrifice in particular is revealed when re-spondents are asked explicitly about the optimal distribution of sacrificein a tax system:

The responses to this question are as follows: 33% choose the first

Table 1Features of the tax systems among which respondents choose.

HH type No-tax earnings Tax system:

C A E D G B F

Average tax rates (in percent)

1 $6,205 −895 −731 −504 −345 −260 14 972 $24,314 −119 −94 −43 −8 11 14 333 $43,961 −14 −9 16 16 16 14 184 $70,254 30 28 22 19 17 14 125 $99,114 48 44 28 22 19 14 86 $127,252 56 51 33 25 21 14 67 $177,199 68 60 39 29 23 14 58 $476,167 81 79 59 45 35 12 2

option; 48% choose the second; and 19% choose the third option.15 Asthese responses demonstrate, the preference for mixed objectives thatwas apparent in respondents' choices over tax systems is echoed bytheir stated preferences over the distribution of sacrifice from the taxsystem. The conventional Utilitarian policy is most consistent with thethird option in this question, though it would in fact recommend amore redistributive option in which the poor received a net benefitfrom the tax system. The pure Equal Sacrifice policy is most consistentwith the first option. Fewer survey respondents show enthusiasm forthe conventional Utilitarian outcome than for the Equal Sacrificealternative, while the largest percentage of respondents prefers the in-termediate option. In other words, more than three-quarters of respon-dents choose policies reflecting some weight on Equal Sacrifice, thesame share as in the choices over tax policies as shown in Fig. 2.

15 In the follow-up survey, the order of possible answers was randomized at the requestof a referee. The results were 37, 39, and 24%. Though slightly less enthusiasm for themid-dle answer was apparent, that option still was chosen by the most respondents, and thelessons are the same as those drawn from the initial results.

Finally, Fig. 3 shows a degree of consistency that suggests the surveyis accurately eliciting respondents' policy preferences. It shows thesepreferences according to individuals' views on Equal Sacrifice. Policy op-tion B, based purely on Equal Sacrifice, claimsmore than twice the shareof the top rankings among those who state a preference for equal sacri-fice than among thosewhoprefer distributing sacrifice less equally. Thispattern holds despite that connection never beingmade apparent in thesurvey. Similarly, the Utilitarian and Rawlsian policies are supportedmore by those who prefer to have the poor bear no sacrifice, andintermediate policies are supported more by those who prefer theintermediate distribution of sacrifice.

2.3. Robustness

Here, I analyze the data along several dimensions to check therobustness of these results.

2.3.1. Respondents' understandingIf respondents fail to understand the questions being asked, we

might worry that their answers poorly reflect their true preferences.Two sets of observations offer reassurance on this point.

First, respondents appear to understand the economic concepts usedin the survey. The survey begins with definitions of the concepts ofbefore-tax income, after-tax income and the average tax rate. Itthen asks respondents to: 1) use before-tax income and taxes paid tocalculate after-tax income; 2) use before-tax income and taxes paid tocalculate a (positive) average tax rate; 3) use before-tax income andtaxes paid to calculate a (negative) average tax rate; 4) calculate the av-erage of three before-tax incomes. These questions test comprehension

Weight on equal sacrifice: αESs

Rawls 0.00 0:03 0.10 0.20 1.00 Polltax

Fig. 2. Respondent preferences across a range of policy options.Fig. 3. Policy preferences by view on equal sacrifice (top choices only).

16 Respondents with fewer correct answers were more supportive of both extremes othe policy distribution, especially the poll tax. As with those who answered at least threcorrectly, a substantial majority (more than two-thirds) of top choices for this groupwerfor a policy other than the Utilitarian or Rawlsian options.17 See Table 229 of the Statistical Abstract of the United States: 2012, published by thU.S. Census Bureau.

133M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

and the ability to work with the concepts, as well as numeracy. Theresults show that respondent understanding was very high, with 75%of respondents correctly answering all four questions and 87% answer-ing at least three correctly.16

Second, the pattern of rankings by most respondents suggests thatthey understood the choices they were making. If a respondent reportssingle-peaked preferences across the five policy options along the spec-trum between pure Utilitarianism and Equal Sacrifice, wemight be confi-dent in his or her understanding of the relationship among policies, not tomention the respondent's rationality. In fact, 68% of respondents exhibit-ed single-peaked preferences across these five policy choices. Important-ly, that result does not imply that 32% of respondents were makingirrational choices—someone may prefer policies that commit fully toone normative criterion or another, or multiple-peaked preferences maysuggest other influences on respondents' normative reasoning that arenot captured in this two-component objective function.

18 I do not show the results for the Poll Tax because it received little support and is toodistinct from the Equal Sacrifice policy to be grouped with it.19 In a follow-up replication of the survey, requested by a referee, I also ask respondentsto self-report race, using the General Social Survey categories: white, black, other. The re-

2.3.2. Robustness across demographic groupsNatural concerns in any survey of this kind are whether the results

are driven by particular demographic groups and whether economicstatus is systematically related to respondents' preferences. To examinethese concerns, I ask the respondents to report their gender, age, educa-tion, and economic status when young and when an adult. Summarydemographic data for the 381 respondents who successfully completedthe survey is provided in Table 2.

Though not meant to be a representative sample, this group exhibitssubstantial variation in (self-reported) personal characteristics and back-grounds. The self-reported distribution of respondents across householdtypes when they were children matches the overall U.S. income distribu-tion remarkably well, with 36% reporting being from the bottom twoquintiles, 45% from the next two quintiles, and 19% from the top quintile.The respondent population also appears to be (or to expect to be) up-wardly mobile, with only about one-quarter of those who report theirchildhood household most resembled one of the two lowest-earninghousehold types reporting that their household at age 40 was also oneof those types. Consistent with that fact, the respondents were generallymore well-educated than the population aged 18–65 in the UnitedStates, where approximately 30% of adults are college graduates.17

fee

e

Table 2 also shows that there are few large differences in preferredpolicies across demographic groups. In the table, I group policies intothree groups: Utilitarian or Rawlsian (C or A); Mixed (E, D, or G); andEqual Sacrifice (B).18 Respondents with more education and higher eco-nomic status tend to be more supportive of Mixed policies. However, alarge majority of each demographic group prefers policies other thanthe Utilitarian or Rawlsian options, and a plurality of all but one groupprefers policies that result from a mixed normative criterion that com-bines Utilitarianism and Equal Sacrifice. The exception is those withlower status as adults, where a slight plurality (though less than one-third) prefers the more redistributive policies.19 With this exception,bothmainfindings from the full survey apply across demographic groups.

2.3.3. Robustness across political viewsAmajor conceptual question raised by this paper's results is how in-

dividuals' preferences are aggregated in a political system. Though Ilargely set that question aside, we can analyze the survey results totest whether the paper's main conclusions are likely to be sensitive tothe details of that aggregation. For example, if we found that individualsof only a particular political perspective were driving the results, wemight discount their relevance. To address these concerns, I ask the re-spondents to self-classify at three points on the (U.S.) political spectrumwith regard to economic issues: 1) Left-leaning, or Liberal; 2) Centrist,or Moderate; 3) Right-leaning, or Conservative. I also ask them to clas-sify themselves as (strongly or somewhat) supportive of or opposedto Libertarianism (which is left undefined in the survey). Table 3shows the distribution of responses. A plurality of the respondents,44%, self-classifies as left-leaning.20 Support for Libertarianism in thissample is consistent with the magnitudes for the U.S. population citedby Boaz and Kirby (2007).

Table 3 also shows that bothmain findings from the full survey char-acterize respondents across a wide range of political opinions.

sults for “white” and “other” respondents conform to the two main lessons highlighted inthe paper. For “black” respondents, less than 10% of the sample, a majority prefers policiesother than the Utilitarian or Rawlsian, consistent with the overall respondent group. Incontrastwith the overall group, however, relatively fewblack respondents prefer the com-bination policies, withmore than 40%preferring the Equal Sacrifice or Poll Tax policies andjust under 40% preferring the Utilitarian or Rawlsian policies.20 Wemight expect this group to bemore supportive of redistributive policy than a sam-ple centered on the “centrist” position.

Table 2Preferences across policy groups by demographic trait.

Tax policy group: Rawls orutilitarian(C or A)

Mixed(E, D, or G)

Equalsacrifice(B)

Share ofrespondents

GenderMale 0.16 0.48 0.23 0.56Female 0.24 0.47 0.16 0.44

Age18–25 0.16 0.52 0.17 0.3026–40 0.23 0.49 0.19 0.4841–65 0.18 0.40 0.25 0.21

EducationHigh school grad 0.21 0.36 0.29 0.13Some college 0.20 0.48 0.18 0.35College grad 0.19 0.52 0.18 0.51

Status when childTypes 1–2 (lower) 0.22 0.41 0.22 0.36Types 3–4 (middle) 0.19 0.50 0.19 0.45Types 5–8 (higher) 0.16 0.55 0.17 0.19

Status when adultTypes 1–2 (lower) 0.31 0.28 0.24 0.16Types 3–4 (middle) 0.19 0.50 0.19 0.51Types 5–8 (higher) 0.15 0.55 0.18 0.33

Table 3Preferences across policy groups by political views.

Tax policy group: Rawls orutilitarian(C or A)

Mixed(E, D, or G)

Equalsacrifice(B)

Share ofrespondents

Political positionLeft-leaning 0.19 0.57 0.11 0.42Centrist 0.21 0.52 0.18 0.30Right-leaning 0.18 0.29 0.37 0.22

View on libertarianismSupport 0.17 0.42 0.25 0.46Oppose 0.19 0.57 0.15 0.27

134 M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

Asmight be expected, right-leaning and Libertarian respondents aremore likely to favor less redistributive policies.21 However, across allgroups, and even among those who self-classify as left-leaning orliberal, a large majority of respondents prefer policies other than thosereflecting conventional objectives, and a plurality prefer a mixednormative framework with some (or all) weight on Equal Sacrifice.

2.4. Relation to existing evidence on normative preferences

This paper's survey evidence and the large body of prior empiricalwork on normative preferences share a commonmain conclusion: indi-viduals use and prefer a mixed normative criterion. Elsewhere, I discussthe related research in detail, but summary statements from studiesrepresenting three research designs in that literature illustrate themain point.22 Frohlich et al. (1987) use surveys in which participantsare asked to rank different distributions of resources, much as in thispaper, and find that “…subjects preferred a compromise. This impliesthat individuals treat choice between principles as involving marginaldecisions. Principles aremuch like economic goods inasmuch as individualsare willing to trade off between them [italics in the original].” Feldmanand Zaller (1992) ask a large group of Americans open-ended questionson distributive justice and write: “Most people are internally conflictedabout exactly what kind of welfare system they want…Ambivalencewith respect to socialwelfare policy ismore pronounced amongwelfareliberals…They end up acknowledging the values of economic individu-alism even as they try to justify their liberal preferences.” Engelmannand Strobel (2004) use allocation games among individuals to elicitvalues and conclude: “a combination of efficiency concerns, maximinpreferences, and selfishness can rationalize most of the data.”

This prior work is only indirectly, not directly, supportive of the roleof Equal Sacrifice as a factor in normative preferences. As far as I amaware, this paper is the first attempt to elicit (or infer) attitudes towardthat principle, though support for the Libertarian viewpoint that somehave linked to Equal Sacrifice has been found by a number of re-searchers (e.g., Cappelen et al., 2011; Boaz and Kirby, 2007; Frohlichet al., 2004; Konow, 2003).23

21 Support for the poll tax (policy F, not shown in the table) is generally low, but asmight be expected it is higher among those who identify as on the “right” (i.e., 16% oftop choices vs. 11% for the rest of the sample) and among those who support Libertar-ianism (i.e., 16% vs. 9%).22 See Weinzierl (2014).23 Gaertner and Schokkaert (2012) discuss somework related to equal losses in “claims”problems, where they find support for proportional monetary losses: see their Section 4.2.

An alternative normative perspective that has received substantialsupport in empirical work, such as in Fong (2001) and Gaertner andSchokkaert (2012), emphasizes individual responsibility for differencesin effort aswell as compensation for differences in ability. Fleurbaey andManiquet (2006) provide an influential theory of optimal taxation sen-sitive to this distinction, and Lockwood and Weinzierl (2012) discusshow incorporating a simple version of that distinction into the standardmodel affects optimal policy. While the framework of this paper is notdesigned to include that alternative perspective directly, our conclu-sions here are not in conflict with those of that literature, and a synthe-sis of them (along with others) may prove fruitful.

3. Generalizing the optimal tax model for multiple objectives

The survey results and related literature presented in the previoussection suggest two lessons for a positive theory of optimal taxation:first, the conventional optimal tax model's assumption of a Utilitarianobjective is counterfactually narrow; second, an accurate positiveoptimal tax theory must be able to accommodate multiple normativeobjectives simultaneously. In this section, I generalize the conventionalmodel to allow for this normative diversity, retainingmuch of the stan-dard theory's (familiar) formal apparatus. I then develop the details ofthat model for the case of the two main normative criteria used in thesurvey: Utilitarianismand Equal Sacrifice. Finally, I show the parameter-izations of the model that correspond to the policies offered to surveyrespondents.

3.1. The general model with multiple criteria

Appealing as it may be to generalize the normative objective in theoptimal tax model, there is a methodological obstacle: many plausiblenormative criteria evaluate outcomes inways that are not directly com-mensurable. For example, Utilitarianism ranks all possible allocations,but Equal Sacrifice yields only a most-preferred outcome and fails torank alternative allocations. To obtain a ranking of allocations thatreflects the judgments of both criteria therefore requires a translationof Equal Sacrifice into a more complete form. This case is an exampleof a more general problem with capturing unconventional principlesin a framework amenable to economic analysis.24

This paper ensures commensurability by representing the prioritiesof each normative criterion with a loss function that depends on

24 For example, Utilitarianism has a consequentialist (i.e., welfarist) criterion, namelymaximal aggregate utility, that ranks all possible allocations based exclusively on the util-ity levels of the individuals in society. In contrast, some normative frameworks stress themoral relevance of concerns such as freedom, rights, and rules, rather than the ends em-phasized by Utilitarianism. These frameworks are often referred to as deontological, anda long-standing concern inmoral philosophy iswhether the judgments of consequentalistand deontological frameworks can be compared. See Sen (1982).

135M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

deviations of the actual allocation of resources from each criterion's op-timal allocation. Of course, specifying these loss functions is a matter ofjudgment, and some may object to their use altogether. In the end, theappeal of my analysis will depend on how closely the optimal alloca-tions and loss functions I use align with the priorities of the normativecriteria. An important feature of this approach is that these loss func-tions can be specified in a way that respects Pareto efficiency, as theexamples below illustrate, avoiding the problem with non-welfaristcriteria noted by Kaplow and Shavell (2001).

In other words, one interpretation of this paper's contribution isas providing a basis, in the form of a specific alternative normative crite-rion, for including in the objective for policy some affinity for a pointalong the Pareto-efficient frontier that is far from the conventional Util-itarian or Rawlsian points. This interpretation relates to the distinctionmade earlier between this paper's approach and one in which weallow for any point on the Pareto frontier to be chosen, regardless ofwhether it is connected to any normative principle. It may be useful todraw an analogy to howRawls' theory has been reduced, in convention-al optimal tax analyses, to a simple maximin objective. While thisreductivism no doubt betrays many fundamental aspects of Rawls'framework, focusing on the maximin point along the Pareto-efficientfrontier is commonly justified by appeals to his work. Similarly, whileEqual Sacrifice as a principle may not correspond perfectly to anypoint on that frontier, it provides an intuition for considering onepoint that conventional criteria do not.

Thus, the key formal innovation in this paper's generalization of thestandard model is that the social planner minimizes a “social loss func-tion” that is the weighted sum of these criterion-specific losses. Theweight on a given criterion's loss represents the force that criterionexerts on society's moral evaluations. The social planner is thereforeinterpreted as an authority using a diverse normative criterion that isthe product of an (unspecified) political process.

This loss-minimization approach to combining disparate norma-tive criteria appears to be consistent with the “consequentialevaluation” of Amartya Sen (2000). Sen does not specify how thesecriteria ought to be combined, but a suggestive passage indicatesthat my approach of social loss minimization may not be far off themark: “…rights-inclusive objectives in a system of consequentialevaluation can accommodate certain rights the fulfillment of whichwould be excellent but not guaranteed, and we can still try to mini-mize the shortfall.”

In most other respects, the model economy in this paper is iden-tical to that considered in standard modern optimal tax models.Individuals differ in their innate ability to earn income, denoted wi

for types i ∈ {1, 2, …, I}, with the proportion of the population withability i denoted pi such that ∑ i = 1

I pi = 1. Individuals derive utilityfrom consumption c and disutility from exerting labor effort y/w toearn income y. Denote the interpersonally-comparable utility functionU(c, y/w).

A planner chooses allocations {c∗i , y∗i }i = 1I to minimize social loss

subject to feasibility and incentive compatibility constraints. Formally,the planner's problem is:

Problem 1. Social planner's problem (general case)

minci�; y

i�

n oI

i¼1∈ F∩Iℂf g

L ¼Xϕ∈Φ

αϕLϕ ciϕ; yiϕ

n oI

i¼1; ci�; y

i�

n oI

i¼1

� �; ð1Þ

where the criterion-specific loss functionsLϕ for each criterion ϕ in thesetΦ are defined below; and Fdenotes the set of feasible allocations forthe economy:

F ¼ ci; yin oI

i¼1:XIi¼1

pi yi−ci� �

≥G

( ); ð2Þ

where G is exogenous, required government spending on public goods;and Iℂ denotes the set of incentive compatible allocations:

Iℂ ¼ ci; yin oI

i¼1: U ci; yi=wi� �

≥U cj; yj=wi

� �for all i; j∈ 1;2;…; If g

n o:

ð3Þ

The weights {αϕ}ϕ ∈ Φ applied to each loss function represent theimportance of each normative criterion in society's evaluations ofpolicy. A number of models of the policymaking process could be usedto generate such weights, but incorporating a convincing model of thepolitical economy of policymaking is beyond this paper's scope.

The losses to which these weights apply are calculated using twocomponents that, together, capture the priorities of each normativecriterion.

First, each criterion generates a preferred, economically-feasibleallocation of consumption and income across types, which I label the“ϕ-optimal feasible allocation.” To identify these allocations, start byassuming that each normative criterion ϕ ∈ Φ implies a (possibly in-complete) preference relationv ϕ on the set F, so that we say allocation

ci1; yi1

n oI

i¼1∈F is weakly preferred under criterion ϕ to allocation

ci2; yi2

n oI

i¼1∈F if

ci1; yi1

n oI

i¼1vϕ ci2; y

i2

n oI

i¼1:

Givenv ϕ, the strict preference relation ≻ ϕ is defined as usual. For

any ci1; yi1

n oI

i¼1; ci2; y

i2

n oI

i¼1∈F,

ci1; yi1

n oI

i¼1≻ϕ ci2; y

i2

n oI

i¼1⇔ ci1; y

i1

n oI

i¼1vϕ ci2; y

i2

n oI

i¼1but not ci2; y

i2

n oI

i¼1vϕ ci1; y

i1

n oI

i¼1:

These preference relations allow the identification of the ϕ-optimalfeasible allocations, which I denote {cϕi , yϕi }i = 1

I , and formally define asfollows.

Definition 1. An ϕ-optimal feasible allocation {cϕi , yϕi }i = 1I is any alloca-

tion in the set F for which there is no other allocation {ci, yi}i = 1I in the

set F such that: {ci, yi}i = 1I ≻ ϕ{cϕi , yϕi }i = 1

I .

These ϕ-optimal feasible allocations provide a key link across nor-mative criteria. Note that no incentive compatibility constraints are im-posed when defining the ϕ-optimal feasible allocations, so that theyequal each criterion's “first-best” allocation in this context (i.e., whenability is observable).

Second, each criterion's priorities are represented by a loss functionthatmeasures the costs of deviations from the criterion'smost preferredallocation. I denote these loss functions Lϕ({cϕi , yϕi }i = 1

I , {c∗i , y∗i}i = 1I )

The loss functions {Lϕ}ϕ ∈ Φ that I use in this paper satisfy thefollowing three conditions. The first two are straightforward. Thethird, Pareto Efficiency, may be more controversial among politicalphilosophers but is generally viewed as a reasonable requirement inthe optimal taxation literature.

Remark 1. For all ϕ ∈ Φ, the loss function Lϕ(x, y) satisfies:

1. Ordinality: For any ci1; yi1

n oI

i¼1; ci2; y

i2

n oI

i¼1∈F,

Lϕ ciϕ; yiϕ

n oI

i¼1; ci1; y

i1

n oI

i¼1

� �≤Lϕ ciϕ; y

iϕ

n oI

i¼1; ci2; y

i2

n oI

i¼1

� �⇔ ci1; y

i1

n oI

i¼1vϕ ci2; y

i2

n oI

i¼1;

so that the loss from one allocation is no greater than that fromanother to which it is weakly preferred under criterion ϕ;

136 M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

2. Normalization: Lϕ({cϕi , yϕi }i = 1I , {cϕi , yϕi }i = 1

I ) = 0, so that the loss iszero when the equilibrium allocation equals the ϕ-optimal feasibleallocation.

3. Weak Pareto efficiency:

U ci1; yi1=w

i� �

≥U ci2; yi2=w

i� �

for all i∈ 1;2;…If g⇒Lϕ ciϕ; y

iϕ

n oI

i¼1; ci1; y

i1

n oI

i¼1

� �≤Lϕ ciϕ; y

iϕ

n oI

i¼1; ci2; y

i2

n oI

i¼1

� �;

ð4Þ

which can be converted into Strong Pareto Efficiency if desired.25

In words, Weak Pareto Efficiency as defined here says that if allindividuals do at least as well under allocation 1 as they do under allo-cation 2, the loss from allocation 1 cannot be greater than the lossfrom allocation 2. This conditionwill prevent the planner from rejectingPareto-improving allocations. It is too weak, however, to guarantee thatthe planner will avoid Pareto-inefficient allocations—for that, StrongPareto Efficiency is required.

Below, I apply this general approach to the case of the two maincriteria between which I have respondents to the survey choose: theconventional Utilitarian criterion and the principle of Equal Sacrifice.

3.2. Equal Sacrifice as an alternative to Utilitarianism

First, I provide a brief discussion of why Equal Sacrifice is a naturalchoice as an alternative to Utilitarianism.26 John Stuart Mill (1871)was the most famous proponent of Equal Sacrifice, and his argumentfor it is worth quoting at length.

“For what reason ought equality to be the rule in matters oftaxation? For the reason, that it ought to be so in all affairs ofgovernment…Equality of taxation, therefore, as a maxim of politics,means equality of sacrifice. It means apportioning the contributionof each person towards the expenses of government so that he shallfeel neither more nor less inconvenience from his share of thepayment than every other person experiences from his.”

To Mill (1871), the appeal of Equal Sacrifice was simple: it treats allindividuals equally. This argument for Equal Sacrifice was endorsedby other influential thinkers, including Alfred Marshall and HenrySidgwick, the latter of whom claimed it was the obviously equitableprinciple—assuming that the existing distribution of wealth is acceptedas just or not unjust.27

Utilitarianism, in contrast, is willing to trade the losses of some forgreater gains of others, a willingness that thinkers as diverse as JohnRawls and Robert Nozick have seen as a serious failing. The specificcontext inwhich this concern has been seen asmost forceful is “endow-ment” taxation, where individuals would be taxed on their potential toearn income rather than their actual earned income. Of course, endow-ment taxation is exactly the preferred policy of the conventionalUtilitarian optimal tax model.28 While Rawls and Nozick take fromtheir critiques very different lessons, they share a similar target:

25 Namely, U(c1i , y1i /wi) ≥ U(c2i , y2

i /wi) for all i ∈ {1, 2, …, I} and U ci01 ; y

i01=w

i0� �

N

U ci02 ; y

i02=w

i0� �

for some i′ ∈ {1, 2, …, I} ⇒ Lϕ({cϕi , yϕi }i = 1I, {c1i , y1i }i = 1

I ) b Lϕ({cϕi , yϕi }i = 1I ,

{c2i , y2i }i = 1I ).

26 Weinzierl (2014) includes an expanded version of this discussion.27 In addition to the work of H. Peyton Young and Berliant and Gouveia mentioned ear-lier, Yaari (1988), Ok (1995), Mitra and Ok (1996), D'Antoni (1999), and Moyes (2003)helped establish conditions on the progressivity of taxes designed in accordance withEqual Sacrifice and argue for the centrality of that principle. Lambert and Naughton(2009) is a recent contribution that reviews much of this literature.28 Legal scholars have extensively analyzed this issuewith endowment (ability) taxationunder the heading of “talent slavery,” the heavy taxation of those with high ability thatforces them to work exceptionally hard or at an occupation they dislike. See, for instance,Hasen (2007), Markovits (2003), Rakowski (2000), Shaviro (2002), Stark (2005), Sugin(2011), and Zelenak (2006).

Utilitarianism's potential to violate individual liberty due to its accep-tance of unequal treatment. This critique of Utilitarianism makes clearwhy Mill's Equal Sacrifice, with its emphasis on equal treatment of allindividuals, is a natural alternative normative criterion.

While a priority on equal treatmentmay be of paramount concern toonly a small minority of individuals, this paper's evidence and that citedin Section 2.4 suggests that it has at least some appeal to most. Formany, that appeal is linked to an affinity for a non-welfarist normativeperspective such as Libertarianism.29 As Feldstein (1976) noted priorto linking Nozick's logic to Equal Sacrifice: “Those who are fullypersuaded by Nozick will thus completely redefine the problem ofoptimal taxation. Others will reject Nozick completely…Many will bepersuaded that the entitlement principle limits the desirable degree ofredistribution.”

Mill himself provides a telling example of exactly this form of mixednormative reasoning, writing approvingly of both Equal Sacrifice andminimal total sacrifice (which is similar to the Utilitarian criterion):

“As a government ought tomake no distinction of persons or classesin the strength of their claims on it, whatever sacrifices it requiresfrom them should be made to bear as nearly as possible with thesame pressure upon all, which, it must be observed, is the mode bywhich least sacrifice is occasioned on the whole.”

Mill is incorrect, as many others have noted, in the assertion thatEqual Sacrifice implies minimized total sacrifice. But this mistake re-veals that, for Mill, both equal and minimized total sacrifice were prin-ciples he believed appealing and likely to be accepted by his readers.This paper is built on the idea that Mill's split normative intuition ismore the rule than the exception.30

3.3. A two-criterion case: Utilitarianism and Equal Sacrifice

In this section, I apply the general approach fromabove to the case ofthe two main criteria used in the survey of Section 1.

3.3.1. ϕ-optimal feasible allocationsThe first step in this application is to define the preference relations

that determine the ϕ-optimal feasible allocations. The preference rela-tion for Utilitarianism is familiar from the conventional optimal taxliterature: allocations are preferred that generate a greater sum ofindividual utilities. Formally, v Util is defined by:

ci1; yi1

n oI

i¼1vUtil ci2; y

i2

n oI

i¼1⇔XIi¼1

piU ci1; yi1=w

i� �

≥XIi¼1

piU ci2; yi2=w

i� �

:

ð5Þ

The Utilitarian-optimal feasible allocation is therefore:

ciUtil; yiUtil

n oI

i¼1∈F :

XIi¼1

piU ciUtil; yiUtil=w

i� �

≥XIi¼1

piU ci; yi=wi� �

;

for all possible ci; yin oI

i¼1∈F.

The preference relation for the principle of Equal Sacrifice requiresmore discussion. The key question is from what starting point is eachindividual's sacrifice to be calculated? Though one could defend a num-ber of choices for that starting point, one natural option is the allocationthat would obtain absent any government intervention, i.e., the no-tax

29 Murphy and Nagel (2002) have argued: “If (and only if) [libertarianism] is the theoryof distributive justice we accept, the principle of equal sacrifice does make sense.”30 Mill (1871) also wrote: “An income not exceeding 50 l. should not be taxed at all, ei-ther directly or by taxes on necessaries,” again illustrating how his affinity for Equal Sacri-ficewas tempered byUtilitarian (or even Rawlsian) intuitions aswell. I thank a referee forbringing this example to my attention.

31 An alternative approachwould be to use a common loss function for all criteria. Whilethis has the seeming advantage of consistency, it in fact would lead to pathologies. For ex-ample, if fullweightwas put on theUtilitarian criterion, but the loss function usedwas notthe same as expression (8), the model would yield a different ranking of policies than theconventional Utilitarian model.32 This property is consistent with the classic “loss aversion” of Kahneman and Tversky(1979). However, equal sacrifice is not consistentwith thediminishing sensitivity to lossesthat is part of classic prospect theory.

137M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

allocation. In particular, the allocation with no taxation is the preferredallocation of the Libertarian framework with which the principle ofequal sacrifice has been linked. As Murphy and Nagel (2002) haveargued: “The implication for tax policy of rights-based libertarianism inits pure or absolute form is that no compulsory taxation is legitimate…”

For clarity, Iwill refer to the allocationwith no taxation as the laissez-faireallocation and formally define it as follows.

Definition 2. The laissez-faire allocation, cilf ; yilf

n oI

i¼1∈F, where G= 0,

satisfies the following conditions (where Ux(c, y/w) denotes the partialderivative of individual utility with respect to x):

1. Ucilfcilf ; y

ilf =w

i� �

¼ Uyilfcilf ; y

ilf =w

i� �

=wi

2. cilf ¼ yilf :

These conditions are simply that each individual maximizesutility and there are no interpersonal transfers. In the statement ofthe definition, I clarify that G = 0, as this is the allocation with nogovernment.

Awell-known conceptual issuewith the idea of the laissez-faire allo-cation is that any economy is, in reality, inseparable from the govern-ment and state institutions that taxes fund. The laissez-faire allocationis, therefore, not well-defined, because G = 0 implies a very differenteconomy than that the status quo. Without a well-defined startingpoint, calculating “sacrifice” is impossible. In formal terms, if G N 0 isrequired for the status quo economy to function, the laissez-faireallocation is not in the feasible set F.

Fortunately, though I am not aware of this being recognized before,the Equal Sacrifice principle provides a naturalway to convert the infea-sible hypothetical laissez-faire allocation into a feasible one. Considerthe following thought experiment. Suppose that the public goods neces-sary to support the current economy are sustained without any cost tothe economy, so that G = 0 but the status quo economic system is fea-sible. According to Equal Sacrifice, the (no tax) laissez-faire outcome inthis scenario is surely optimal, as it satisfies Equal Sacrifice with thesmallest possible uniform sacrifice – that is, zero – for all individuals.Now, suppose that sustaining those public goods is costly, so thatG N 0. The Equal Sacrifice principle implies that the cost of the publicgoods will be distributed across individuals such that the utility loss isidentical (and as small as possible) for all.

Formally, defineESas the set of all feasible allocations that satisfy theprinciple of Equal Sacrifice relative to the laissez-faire allocation:

ES ¼(

ci; yin oI

i¼1∈F : U cilf ; y

ilf =w

i� �

−U ci; yi=wi� �

¼ U cjlf ; yjlf =w

j� �

−U cj; yj=wj

� �for all i; j∈ 1;2;…; If g

):

ð6Þ

The Equal Sacrifice preference relation, denoted v ES, indicates thatone allocation in ES is preferred to another if it generates a smalleruniform sacrifice:

ci1; yi1

n oI

i¼1vES ci2; y

i2

n oI

i¼1⇔U cilf ; y

ilf =w

i� �

−U ci1; yi1=w

i� �

≤U cilf ; yilf =w

i� �

−U ci2; yi2=w

i� �

;

ð7Þ

for ci1; yi1

n oI

i¼1; ci2; y

i2

n oI

i¼1∈ ES and for any i ∈ {1, 2, …, I}.

Consequently, the Equal Sacrifice-optimal feasible allocation is thatwhich achieves the smallest equal sacrifice while funding G. Formally,

we define {cESi , yESi }i = 1I as follows:

ciES; yiES

n oI

i¼1∈ES : U cilf ; y

ilf =w

i� �

−U ciES; yiES=w

i� �

≤U cilf ; yilf =w

i� �

−U ci; yi=wi� �

;

for any i ∈ {1, 2, …, I} and for all possible ci; yin oI

i¼1∈ES.

Once we have specified the ϕ-optimal feasible allocations, the nextstep is to specify the loss functions for the planner.

3.3.2. Loss functionsThe Utilitarian loss function LUtil is:

LUtil ciUtil; yiUtil

n oi; ci�; y

i�

n oi

� �¼XIi¼1

pi U ciUtil; yiUtil=w

i� �

−U ci�; yi�=w

i� �h i

:

ð8Þ

In words, it is the sum of individuals' utility losses from having theequilibrium allocation {c∗i , y∗i}i deviate from theUtilitarian-optimal feasi-ble allocation. This loss function has the appealing property that itdirectly adopts the cardinal welfare comparisons underlying theUtilitarian preference relation and, thus, the conventional optimal taxmodel.31 Note that it converts the familiar goal of aggregate utilitymaximization into aggregate sacrifice minimization.

Unlike Utilitarianism, the Equal Sacrifice criterion does not rank allo-cations that deviate from its preferred allocation. As far as I am aware,no previous work has studied how to obtain a complete ranking ofallocations based on Equal Sacrifice. While my approach is, therefore,by necessity somewhat speculative, I design the Equal Sacrifice lossfunction to reflect the priorities of that principle. In words, these prior-ities are simple: deviations from equal sacrifice are costly, even if theyreduce the aggregate level of sacrifice, and outcomes with less sacrificefor some and no more for all are preferred (i.e., Pareto efficiency). Ofcourse, future research may discover alternative specifications thatprove more useful. The goal of this paper is to propose one reasonableway, not the definitive way, to capture the priorities of the EqualSacrifice principle.

I will assume an Equal Sacrifice loss functionLESwith three features:first, deviations of individual utility below the Equal Sacrifice-optimalfeasible allocation are costly but deviations above the Equal Sacrifice-optimal feasible allocation yield little or no offsetting benefits32;second, losses increase more than proportionally with the sizeof the deviation of individual utility below the Equal Sacrifice-optimal feasible allocation; third, gains are concave in the size ofthe deviation of individual utility above the Equal Sacrifice-optimalfeasible allocation.

I formalize these properties as follows:

LES ciES; yiES

n oi; ci�; y

i�

n oiÞ ¼

XIi¼1

piV U ciES; yiES=w

i� �

;U ci�; yi�=w

i� �� �

;

ð9Þ

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 21.4

1.42

1.44

1.46

1.48

1.5

1.52

1.54

1.56

1.58

Utility for low type

Util

ity fo

r hig

h ty

pe

Utilitarian optimum (not incentive compatible)

Utilitarian optimum

ES optimum

Planner optimum

Utility Possibilities FrontierUtilitarian indifference curvePlanner indifference curveEqual Sacrifice indifference curve

Fig. 4. The utility possibilities frontier and indifference curves in the two-type example.

138 M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

where

V UiES;U

i�

� �¼ f− δ Ui

�−UiES

� �h iθif Ui

ESbUi�

λ UiES−Ui

�� �h iρ

if UiES≥Ui

�

;

for scalars δ≥0; λNδ; θ∈ð0;1�; ρN1f g:

ð10Þ

Consistent with the first property, the loss function inexpressions (9) and (10) applies weights δ and λ, where 0 ≤δ b λ, to deviations of individual utility above and below the EqualSacrifice-optimal feasible allocation.

The kink at the Equal Sacrifice-optimal feasible allocation implied byδ b λ, and thus the asymmetric punishment of downward deviationsfrom that allocation, rejects the Utilitarian idea that the distribution ofutility across individuals is irrelevant. Though nondifferentiability is tech-nically inconvenient, it is conceptually important to capture the EqualSacrifice criterion's priorities. The reason is that the alternative – a smoothloss function at the Equal Sacrifice-optimal feasible allocation – implieslocal indifference to symmetric deviations from that allocation. That indif-ference is in direct conflict with the principle's priority on equal sacrifice.

The assumption that δ ≥ 0 respects Weak Pareto Efficiency asdiscussed above (δ N 0 would respect Strong Pareto Efficiency).Consistent with the second and third properties, the parameters ρ N 1and θ ∈ (0, 1] imply losses that increase more than proportionallywith deviations below and gains that increase (weakly) less than pro-portionally for deviations above the Equal Sacrifice-optimal feasible al-location. As noted above, I do not mean to claim that these functionalforms or these assumptions on its parameter values are the only possi-ble choices, but rather to construct a plausible representation of theEqual Sacrifice principle for analysis.

3.3.3. Planner's problemWith the loss functions defined by expressions (8), (9) and (10), the

planner in this case chooses {c∗i , y∗i}i = 1I to solve the following problem.

Problem 2. Social planner's problem (specific case)

minci�; y

i�

n oI

i¼1∈ F∩Iℂf g

αUtil

XIi¼1

pi U ciUtil; yiUtil=w

i� �

−U ci�; yi�=w

i� �h i

þ αES

XIi¼1

piV U ciES; yiES=w

i� �

;U ci�; yi�=w

i� �� �

8>>><>>>:

9>>>=>>>;;

ð11Þ

where

αUtil þ αES ¼ 1;

V(⋅) is defined in (10), F is defined in (2), and Iℂ is defined in (3).

This planner's problem is equivalent to the conventional approach ifαES = 0.

To illustrate the effect of positive αES on optimal policy, I simulate asimple model with two types of workers and show how this form ofnormative diversity affects the well-being of individuals in theeconomy.

3.3.4. Example with two typesIndividual income-earning ability is eitherw1= 10 orw2= 50, each

of whichmakes up half the population, so p1= p2 = 0.5. The individualutility function is

U ci; yi=wi� �

¼ci� �1−γ−1

1−γ− 1

σyi

wi

!σ

;

where γ=1.5,σ=3. The Equal Sacrifice loss function's parameters are

δ=0.5, λ=20,ρ=2.0, θ=1.0, and the social loss function'sweight onthe Equal Sacrifice loss function is αES = 0.20. Government spending Gis set to zero.

This simple example is most useful for showing the effect of such amixed objective on the allocation of utility across individuals. Fig. 4plots the utility of the high-ability individual against that of the low-ability individual. The bold solid line shows the utility possibilities fron-tier (UPF): that is, the highest incentive-compatible, feasible utility forthe low-ability individual given a utility level for the high-ability indi-vidual. The thin solid and dotted lines are the indifference curves pass-ing through the ϕ-optimal feasible (but not necessarily incentivecompatible) allocations for the Utilitarian and Equal Sacrifice criteria.The dashed line is the indifference curve for the planner that chooses(by tangency with the UPF) the optimal allocation for the economy.Also shown are the optimal feasible and incentive-compatible alloca-tions chosen by each criterion.

Fig. 4 shows how the Equal Sacrifice loss function, LES, differs fromthe Utilitarian, LUtil. To remain indifferent while moving away from itsoptimal allocation, L ES requires a greater gain for the low-abilityindividual in exchange for a given loss for the high-ability individual.Moreover,LES increasesmore than proportionallywith these deviations,whileLUtil is linear. The impact of incorporating this loss function in theplanner's decisions is as expected: the planner compromises betweenthe competing normative criteria, implementing some redistribu-tion but stopping well short of what a Utilitarian would choose. Byvarying αUtil, we can shift the planner's chosen allocation alongthe UPF.

3.4. Generating the survey's policy options from the model

The set of policy options presented to respondents in the survey ofSection 1 was generated using this section's generalized optimal taxmodel. Here, I describe the calibration of the model to data on the U.S.income distribution and the parameterizations of themodel that gener-ate those policies.

For each policy objective I simulate a constrained planner's prob-lem as in expression (11), calibrated to data on the U.S. income dis-tribution from 2006 as calculated by the Congressional BudgetOffice. In particular, I take the gross labor income distribution as cal-culated (by the CBO) into eight bins: the bottom four quintiles andthe next 10, 5, 4, and 1 percentiles. The CBO also provides taxes paid

139M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

for these households,33 so I use the utility function specified below toback out the earnings ability implied by the households' pre-taxearnings and tax payments. Then, I calculate the earnings each house-hold would choose if there were no taxation, again using the individualutility function defined below. This calculation yields the distributionpresented to respondents as the baseline “no tax” income distribution.The distribution of ability for the model, where wi denotes the abilityand pi denotes the population proportion of type i, is as follows.

Ability distribution

3

pop

wi

3 Theaymenf incomroach y

3.10

CBO provts. For sime for eachield very

12.16

ides incomplicity, Itype of hosimilar re

21.98

e taxes, wmultiply tousehold tosults in ter

35.13

hich are dtal persongeneratems of the

49.56

ue to bothal incomelabor incomdistributio

63.63

labor andtaxes paide taxes. V

n of abilitie

88.59

capital inby the la

ariations os.

238.08

pi 0.20 0.20 0.20 0.20 0.10 0.05 0.04 0.01

All model parameters other than αES and αUtil retain the same valuesacross simulations. These parameters, and the underlying formal struc-ture of the problem, are never disclosed to respondents. I assume thefollowing parameter values.

Parameter values

ρ

θ δ λ 1σ−1 φ

cobon t

G

2.0

1.0 0 10 0.10 12−σ 8.26

34 As noted by Berliant and Gouveia (1993), among others, Equal Sacrifice endorses pro-gressivity if, in the notation of this paper, utility is separable across consumption and lei-sure and γ N 1. Mill's (1871) writings suggest that he thought logarithmic utility was anatural specification.35 See Weinzierl (2014) for numerical results on this point.36 Detail on the applications in this section, including analytical results on the optimalextent of tagging under Equal Sacrifice, can be found in Weinzierl (2014).37 This statement assumes that there are no differences in preferences or elasticitiesacross tagged groups, which may reinforce or weaken the case for a given tag.38 Aside from blindness (analyzed below), disability and the presence of children are thedominant tags used (age is not properly thought of as a tag, as all individuals pass through

The utility function for all households is

U ci; yi=wi� �

¼ ln ci� �

−φσ

yi

wi

!σ

: ð12Þ

The values of ρ, θ, δ, and λ determine the shape of the Equal Sacrificeloss function. The parameter σ controls the elasticity of labor supply,while φ is a taste shifter used only to normalize labor effort. The valueof G is chosen so that government expenditure as a share of equilibriumtotal output roughly matches that in the United States. Several of theseparameter values deserve additional comment.

First, assuming δ = 0 implies that deviations of individual utilityabove the Equal Sacrifice-optimal feasible allocation generate no gainsaccording to the Equal Sacrifice criterion. This is the strictest version ofthe Equal Sacrifice loss function, in that it rejects redistribution even ifit generates enormous gains for some as long as it generates any lossesfor others. To the extent that respondents are, in reality, sympathetic toa more moderate version of Equal Sacrifice, this assumption biases thesurvey toward support for more Utilitarian policies.

Second, the implied Frisch labor supply elasticity 1σ−1 is low in these

parameterizations, at 0.10. That is below most mainstream estimates,though not for prime-aged heads of households. Lower labor supplyelasticities will reduce the efficiency costs of redistributive policies,increasing their appeal. Therefore, our survey results are likely to bebiased toward Utilitarianism due to this choice.

Third, assuming logarithmic utility of consumption has two implica-tions. If it underestimates the concavity of that subutility function, thesimulations generate policies with less income redistribution thanwhat a more realistic calibration would produce. As with the other as-sumptions above, this bias would tend to increase the reported supportfor Utilitarianism in the survey, as the survey results show that mostpeople prefer less redistributive policies than the purely Utilitarianone. Log utility of consumption also means that average tax rates areflat under the Equal Sacrifice criterion, and a flat tax may have someappeal to respondents due to its simplicity. Working against this, if wewere to use a more concave form of utility, Equal Sacrifice would yield

me taxr sharehis ap-

progressive average taxes, making it appear closer to the most-preferred policies in the survey. 34

Related to the choices of both 1σ−1 and the concavity of the utility of

consumption (which I will later parameterize with the coefficient ofrelative risk aversion γ) is the question of whether the Utilitarian andEqual Sacrifice criteria become quantitatively indistinguishable asredistribution becomes more costly ( 1

σ−1 increases) and less rewarding(γ decreases). It turns out that the answer is “no.” The clearest exampleis that Utilitarianism endorses negative average tax rates at lowincomes, while Equal Sacrifice does not because, by definition, EqualSacrifice requires all individuals to bear some sacrifice. More generally,the two policies remain quite distinct even for unconventionally highvalues of 1

σ−1 and low values of γ.35

Finally, the values of ρ, θ, and λ are necessarily chosen without anydirect empirical guidance. I do not mean to suggest, therefore, thatthese are necessarily the correct values, or that the results of thispaper are robust to their choice. They are chosen to represent a plausiblespecification of the Equal Sacrifice criterion, and the (indirect) test oftheir appropriateness is that they contribute to the explanatory powerof this paper's model for real-world policy.

4. Descriptive power of the positive optimal tax model

In this brief section I illustrate how the optimal tax model, as pro-posed and empirically estimated in this paper, is able to explain aspectsof existing policy that are difficult to reconcile in conventional theorybutwidely endorsed in reality. I focus especially on the puzzle of limitedtagging—the taxation of personal characteristics in addition to incomeintroduced to the literature in Akerlof (1978). I also briefly discuss theimplications of Equal Sacrifice for the topic of rank-reversals in thecase of optimal policy with full information.36

In themodern theory of optimal taxation, tagging is a free lunch, andawide variety of candidate tags exist. Any observable and largely inelas-tic characteristic across which the distribution of abilities differs oughtto affect tax schedules.37 In comparison, the role for tagging in moderntax policy is highly constrained. Some sizeable tagging does occur, butonly for tags that are virtually guaranteed to indicate that a taxpayerhas low income-earning ability.38 In this section, I use numerical simu-lations calibrated to theU.S. microdata to show that the combinations ofUtilitarian and Equal Sacrifice principles preferred by the survey respon-dents can quantitatively match the simultaneous rejection of most, butnot all, forms of tagging and the acceptance of substantial income redis-tribution in U.S. policy.

The importance of the tagging application to this paper is that itsuggests Equal Sacrifice has explanatory power beyond simply being a“less redistributive” criterion than Utilitarianism. I show that, whilethe optimal extent of both tagging and redistribution decrease whenEqual Sacrifice is given more weight, tagging is disproportionately dis-couraged. Intuitively, Equal Sacrifice rejects tagging because it causesdifferential sacrifice across people of the same underlying ability, notbecause it causes too much sacrifice by those with high ability. In an

all ages, subject tomortality differences—seeWeinzierl, 2011). To the extent that disabilitystatus implies zero earning ability, it by definition merits tagging. Future work could use-fully focus on showing whether the model can explain the substantial tagging on depen-dent children in existing policy. That task will require making judgments on the propermodeling and normative treatment of households.

140 M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

optimal policy that gives weight to both Equal Sacrifice and Utilitarian-ism, only those tags that provide sufficiently strong information aboutability, and therefore Utilitarian welfare gains, will be optimal.

4.1. Numerical results on optimal tagging and redistribution

First, I consider three prominent potential tags – height, gender,and race – that are not used in reality.39 Then, I consider blindness,one of the few personal characteristics explicitly tagged in the U.S.tax code.

For the optimal tax policy simulations in this section, I use thefollowing parameter values:

Parameter values

3

C4

frsesh4

αES

9 See Mankiw and Weinzierl (2009)remer, Gahvari, and Lozachmeur (2010 I set δ N 0 (rather than δ= 0) to caom consumption is 1

1−γ c1−γ−1� �

, wht σ = 3 (rather than σ = 11) to beare of the U.S. government expend1 See Weinzierl (2014) for more deta

ρ

; Alesin0); andpture aere I scloser

iture inil.

θ

a, IchinBlumkless stret γ N 1to maitotal i

δ

o, and Kin, Margaict versio(rathernstreamncome.

λ

arabarlioth,n of Eqthan γestim

γ

bounisand Sadual Sac= 1, lates. I

1σ−1

(2011ka (200rifice. Uog utiliset G t

G

0:00; 0:03; 0:10; 0:20; 1:00n o

2.0

1.0 0.01 10 1.5 0.5 20

A few of these values differ from those used to generate the policyoptions for the survey. While I chose values for the survey of δ, γ, andσ to increase the appeal of the conventional Utilitarian policy, here Ichoose values to maximize realism.40

Data on ability distributions by height, gender, and race come fromthe National Longitudinal Survey of Youth. Meanwages for each taggedgroup are shown in Table 4. For each of thefive values ofαES, Table 4 alsoreports measures of the optimal extent of tagging: the “extra” averagetax paid by or transfer made to the members of each tagged group asa share of their income when the planner can use tagging as comparedto when it cannot.

To gauge the progressivity of the optimal income tax under eachvalue of αES, Table 5 reports the average tax rate paid by the membersof each wage range.

The results in these tables show that the support expressed in thesurvey of Section 1 for objectives that include Equal Sacrifice can yieldan optimal tax policy that rejects most forms of tagging but pursuessubstantial income redistribution, as in reality. Note, in particular, thecase of αES = 0.20, the most-preferred policy in the survey. In thatcase, tagging is negligible but average tax rates (and top-income mar-ginal tax rates, not shown) closely mimic those in current U.S. policy.41

The U.S. tax code includes a special deduction or exemption for indi-vidualswith substantially impaired vision. To analyze optimal blindnesstagging, I use data from the Statistics of Income microdata of the U.S.Internal Revenue Service. Those claiming the blindness exemptionmake up 0.3% of the population, 79% of whom report zero income.

Table 6 shows the optimal extent of tagging in the conventional cal-ibration with αES = 0 and in the most-favored calibration in the surveyof Section 1,withαES= 0.20. All other parameters are as before (thoughG is adjusted to be a similar share of total income).

As with height, gender, and race, Table 6 shows that adding thisweight on Equal Sacrifice to the objective function substantially reducesthe optimal extent of tagging on blindness. Unlike those other tags,however, the optimal extent of tagging on blindness in the Utilitarianbenchmark is so great that even the dramatically reduced extent ofoptimal tagging is sizeable—namely, a 20 percent transfer to the blindon average. Using the data from the IRS, we can calculate mean incomefor the blind (including those with zero income) to be approximately$2350per year. A 16percent transfer to the blindon average is therefore

) and9)tilityty). Io the

equivalent to approximately $376, not far from the value of actual blind-ness deductions and exemptions in the mid-1980s.

4.2. Equal sacrifice, horizontal equity, and vertical equity

The popularity of the principle of horizontal equity – that “equalsought to be treated equally” – has long been used to explain the limiteduse of tagging. Boadway and Pestieau (2006) write: “Of course, such asystem may be resisted because, if the tagging characteristic has no di-rect utility consequences, a differentiated tax system violates the princi-ple of horizontal equity”. Similar statements are made by, e.g., Atkinsonand Stiglitz (1980), Auerbach and Hassett (1999), and King (1983).

Importantly, the analysis of this paper is consistent with these argu-ments stressing horizontal equity. Horizontal equity is implied by EqualSacrifice, as Equal Sacrifice values both it and “vertical” equity as part ofits overall priority on equal treatment. Thus, one possibility raised bythis paper is that the popular enthusiasm for horizontal equity comesout of an underlying affinity for Equal Sacrifice. This possibility is espe-cially interesting in light of the critiques, for example by Musgrave(1959) and Kaplow (2008), of the intellectual coherence of horizontalequity.

In fact, when respondents to the survey of Section 1 were given thelogic for height and blindness tags and askedwhether they supported oropposed them, the individuals who oppose tagging disproportionatelysupport Equal Sacrifice. In Weinzierl (2014), I show that this result isstatistically significant and holds when we control for gender, age,education, race, and income status when a child and adult.

Horizontal equity, however, has no implications for the “vertical”distribution of income, while the ability of Equal Sacrifice to implyempirically-realistic levels of redistribution along with limited taggingbolsters the case for its use. A particularly stark version of this argumentrelates to the issue of so-called rank reversals. It has been known sincethe analyses in Mirrlees (1971) and Mirrlees (1974) that an optimalUtilitarian tax policy in the case of full information generally induces anegative relationship between innate ability and the allocation of utilityacross individuals. This reversal of pre-tax and post-tax utility orderingshas generated considerable discomfort among optimal tax theorists(Saez and Stantcheva, 2014) and tax law scholars (see, for example,King, 1983; Zelenak, 2006). Horizontal equity raises no concerns aboutrank reversals. In contrast, simulations following the same process asabove for the U.S. income distribution show that mixed objective mostpreferred in the survey generates a nearly uniform utility distributionin the first-best, substantially limiting rank reversals. Intuitively, EqualSacrifice leaves the utility ordering of agents unchanged, even in thecase of full information, because its reluctance to redistribute does notdepend on the distortionary costs of taxation.

5. Conclusion

The optimal tax literature occupies a rare place in economic researchin which the normative assumptions of economists are given priority.The conventional use of Utilitarianism as the criterion for quantitativeanalyses of optimal policy is expedient, as it narrows the range ofmodels to consider. It may also be compelling, if we believe the Utilitar-ian criterion is the right one.

An alternative to the conventional approach is to use empirical evi-dence on normative preferences to develop a positive optimal tax theoryin which economists' normative intuitions are replaced by those thathold sway among voters and taxpayers in reality. Of course, a numberof classic questions arise about such an approach, such as: whose pref-erences matter for policymaking, how are individual preferencesaggregated, and what are the admissable normative criteria. Thispaper has not focused on these questions, which are important topicsfor future work. The conventional approach sidesteps these questions,but at the potential cost of relevance.

Table 4Extent of Tagging: extra tax (+) or transfer (−) rate, in percent.

αES

1 2 3 4 5 6 7 8 9 10 11 12Tall Med. Short Tall Tall Short Med. Med. Short Tall Med. ShortM M M M F M M F F F F FWhite White White NW White NW NW White White NW NW NW

0 10.5 8.1 6.3 1.6 −4.3 −5.5 −3.5 −11.7 −13.4 −17.7 −22.0 −23.40:03 4.5 3.6 3.1 1.5 −1.1 −1.5 −1.6 −5.0 −6.5 −8.4 −11.8 −12.70.10 1.8 1.3 1.3 0.9 −0.4 −0.5 −1.3 −2.0 −2.6 −2.8 −4.9 −5.20.20 0.8 0.5 0.5 0.7 −0.1 −0.1 −1.0 −0.9 −1.2 −0.8 −2.0 −2.71.00 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Mean wage 17.7 16.9 16.3 15.3 14.3 13.6 13.5 12.8 12.3 11.2 10.7 10.5

Table 5Extent of progressivity (average tax rates, in percent).

αES

Average wage rate in range

2.81 6.50 10.03 13.82 17.80 21.70 27.28 43.25 62.06 95.96

0 −396 −64 −5 17 27 32 38 50 52 530:03 −346 −51 1 18 25 30 35 47 50 520.10 −300 −39 3 18 23 27 31 43 47 500.20 −258 −29 7 17 22 24 28 40 44 471.00 −5 11 13 14 16 17 19 22 25 29

141M. Weinzierl / Journal of Public Economics 118 (2014) 128–142

In this paper, I make three contributions toward demonstrating thepromise of such a positive optimal tax theory.

First, I present novel survey evidence on the empirical normativepreferences of individuals in the United States. Using a fully-specifiedplanner's problem, I generate feasible and incentive-compatible tax pol-icies that are optimal according to a range of social objective functions,and I have respondents rank these policies. I find striking and robust re-sults: few respondents prefer the conventional Utilitarian policy or theRawlsian alternative, and a plurality (nearly half) prefers policies thatreflect a mixed objective that gives weight to both Utilitarianism andEqual Sacrifice. Additional questions in the survey provide more directevidence in support of these results. When asked explicitly how“sacrifice” from paying taxes should be distributed, respondents prefera distribution between that implied by Utilitarianism and Equal Sacri-fice. And the more enthusiastic a respondent is about Equal Sacrifice,the more likely he or she is to reject tagging, the taxation of personalcharacteristics that is a feature of Utilitarian-optimal tax policy butthat is rejected by Equal Sacrifice. This evidence is consistentwith a sub-stantial body of previouswork showing that the normative reasoning ofmost individuals draws on a diverse set of criteria.

Second, I generalize the conventional optimal tax model to accom-modate this evidence of a mixed objective for taxation. This generaliza-tion requires overcoming the challenge of combining disparate,sometimes incommensurable, criteria for optimality. I develop a meth-od bywhich any set of criteria can be integrated into a unified objectivethat respects Pareto efficiency, and I apply that method to the specificcase of two criteria at the heart of this paper: Utilitarianism and EqualSacrifice. More generally, this method provides a way to inform thechoice of welfare weights in the generalized Pareto-efficient approachto optimal taxation.

Third, I show that the empirically-preferred calibration of the gener-alized theory can explain some conventionally puzzling features of real-

Table 6Extent of tagging on blindness (extra tax or transfer rate, in percent).

αES Not blind Blind

0 0.07 −1300.20 0.01 −16

world tax policy. I focus on themodel's ability to explain the limited roleof tagging in policy that is otherwise quite redistributive. I simulate op-timal policywith the objective functions favored by survey respondents,calibrated to microdata from the United States. That policy rejects theuse of height, gender, and race as tags; it accepts the use of blindnessas a tag, endorsing a quantitatively realistic blindness benefit; and itprovides redistribution through a progressive schedule of averageincome tax rates that closely resembles actual policy.

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