On the efficiency of equal sacrifice income tax schedules

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European Economic Review

European Economic Review 70 (2014) 399–418

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On the efficiency of equal sacrifice income tax schedules$

Carlos E. da Costa n, Thiago PereiraFGV-EPGE, Brazil

a r t i c l e i n f o

Article history:Received 2 July 2013Accepted 10 June 2014Available online 30 June 2014

JEL classification:H21D63

Keywords:EfficiencyEqual sacrifice

x.doi.org/10.1016/j.euroecorev.2014.06.00821/& 2014 Elsevier B.V. All rights reserved.

paper has been circulated under the title “

esponding author at: Fundação Getulio Var5 21 37995493; fax: + 55 21 25538821.ail addresses: [email protected] (C.E. daung (1990) has shown that most tax scheduacrifice principle while survey evidence gath that dominates the normative tax literatuprinciple to a Utilitarian metric to guide poung (1988) has shown that any method of apincomes; (ii) an increase in the tax total leadafter-tax incomes, and; (iv) the ordering of ttility function.

a b s t r a c t

In an economy which primitives are exactly those in Mirrlees (1971), we investigate theefficiency of tax schedules derived under the equal sacrifice principle. For a givenexogenous government consumption level we assess whether there is an alternative taxschedule that raises more revenue while delivering less utility to no one. For our preferredparametrizations, we find that inefficiency only arises at the top of the income distribu-tion for marginal tax rates well above the ones we currently observe in most countries.We also recover the implicit marginal social weights associated with the equal sacrificeschedule and find them not to be monotonic in types for the environments we study.

& 2014 Elsevier B.V. All rights reserved.

0. Introduction

Mirrlees (1971) has defined the standard for normative income tax analysis: the maximization of a social welfarefunctional subject to incentive and resource constraints. Despite its indisputable methodological advantages, the consensusregarding this procedure has obscured the fact that Welfarism need not reflect a society's actual view of what a just taxsystem is. Alternative non-welfarist views of distributive justice may capture more accurately the ideas that underlie notonly the political debate but also the actual policy making.

In this paper, we study a non-welfarist approach which has played a central role in the policy debate for most of the 19thand the early 20th century: equal sacrifice. This principle, aptly described in the words of John Stuart Mill: “…whateversacrifices the government requires should be made to bear as nearly as possible with the same pressure upon all” — see Mill(1844), appears to still be playing an important role in public debate as well as actual policy making.1 Not only does theequal sacrifice principle appeal to people's perception of fairness but also income taxes derived from it possess very sensibleproperties that may explain its presence in public debate.2

Sacrifice and Efficiency of the Income Tax Schedule.” First Version: July, 2010.gas, Escola de Pós-graduação em Economia Praia de Botafogo, 190 - 1105 Rio de Janeiro/RJ, Brazil.

Costa), [email protected] (T. Pereira).les that prevailed in the US for the period 1957–1987 may be rationalized by direct applications of thehered by Weinzierl (2012b) suggests that public opinion does not strictly adhere to the welfaristre. In fact, three-fifths of survey respondents in Weinzierl's (2012b) work prefer the use of the equallicy.portioning taxes such that (i) the way that taxpayers split a given tax total depends only on their owns everyone to pay more; (iii) every incremental increase in tax is apportioned according to taxpayers'axpayers by pre-tax income and after-tax income is the same as that of an equal sacrifice schedule for

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C.E. da Costa, T. Pereira / European Economic Review 70 (2014) 399–418400

Opposing these appealing features of equal sacrifice there are two potential drawbacks of applying this principle. First,the equal sacrifice principle need not lead to Pareto efficient schedules. Second it does not allow for purely redistributivetaxation. In this paper, we address both issues by (i) asking whether equal sacrifice leads to inefficient schedules under ‘realworld’ circumstances, and; (ii) verifying if these schedules are redistributive in the sense of being locally rationalized by aconcave social welfare function.3

Efficiency may not be addressed using the framework of the early equal sacrifice literature, since it does not explicitlymodel behavioral responses to taxation. We shall, then, derive equal sacrifice schedules in a Mirrlees' (1971) environment,i.e., an economy inhabited by a continuum of individuals with identical preferences defined over consumption and labor,who only differ with respect to their labor market productivity, w, which is private information. The efficiency question boilsdown, in this case, to whether welfare losses due to the informational structure of the problem are minimal when taxes arebased on the equal sacrifice principle.

Let Tð�Þ be an income tax schedule derived under the equal sacrifice principle. Associated with this schedule is anequilibrium utility profile vð�Þ, where v(w) is the utility attained by an individual with productivity w. We ask whether thereis an alternative tax schedule that generates at least as much revenue and which induces a utility profile vnð�Þ such thatvnðwÞZvðwÞ, 8w, with strict inequality for a subset of positive measure of individuals.

The first step toward our goal is to derive the minimum equal sacrifice allocation, i.e., an incentive compatible allocationwhich generates excess resources that are sufficient to finance the government consumption needs while imposing anidentical utility loss on all individuals. We use a truthful direct mechanism to find the incentive compatible equal sacrificeallocation. This is the same approach used by Berliant and Gouveia (1993), which, to the best of our knowledge, was the firstwork to explicitly take into account labor supply responses in an equal sacrifice based tax problem.4 From the allocationwe recover the equal sacrifice schedule, Tð�Þ, using the taxation principle. Finally, we check whether this schedule satisfies anefficiency condition which we derive using Werning's (2007) approach.

Throughout the paper, we adopt a separable specification for preferences. Separability is necessary for us to applyWerning's (2007) methodology for assessing efficiency. If not only preferences but also their specific utility representation isseparable, labor supply is shown to be independent of the level of sacrifice. It is only through reduced consumption, i.e.,through total taxes paid, that sacrifice is imposed on agents. The invariance of taxable income with respect to the level ofsacrifice, therefore, rationalizes the abstraction from labor supply responses in the early literature thus justifying ourdiscussion of Young's (1990) findings.

For commonly used parametric distributions of skills, regions of inefficiencies for the equal sacrifice schedule areintervals of the form ½ya;1Þ, with ya representing the lowest level of income for which the marginal tax rate is ‘too high’.ya is usually large and associated with very high marginal tax rates. In fact, only a very small share of the population hasincome in the region of inefficiency.

As for the other typical criticism of equal sacrifice, namely, that pure redistributive taxation is not allowed, we recall that,despite the fact that under equal sacrifice either everyone pays positive taxes or no-one does, there is a sense in whichredistribution may still be taking place. For large enough Government expenditures, an efficient tax system may prescribepositive taxes for the worse off individuals even if the associated social welfare function is concave. We ask whether this isthe case for the equal sacrifice schedule.

To answer this question, we devise a procedure to extract the marginal social welfare weights — see Diamond and Mirrlees(1971a, 1971b), Saez and Stantcheva (2013) — associated with the equilibrium allocations. We first relate these weights to theLagrange multipliers of the Pareto problem from which Werning's (2007) efficiency bounds are derived. Then, we show how toexpress them as a function of observed variables only.5 Once we extract the marginal weights, we ask whether they aredecreasing in agents' types, i.e., if the social welfare function that rationalizes the tax system is concave. We find this not to be thecase: the very poor receive too little weight under the equal sacrifice principle. Although these weights do increase asGovernment consumption increases, we get an inverted u-shaped pattern of weights in all our exercises.

Finally, we consider non-separable representations for the separable preferences. Non-separability means that societyperceives the sacrifice imposed by reduced consumption as being dependent on the level of effort one is making. Invarianceof labor supply is lost, but the efficiency tests remain valid and we are still able to characterize the marginal social welfareweights associated to the equal sacrifice schedule. Being able to handle non-separability is also important if one wants todisentangle the role of preferences from that of the society's perception of sacrifice in both the derivation of an equalsacrifice schedule and in the assessment of its efficiency.

We explore two different parametrizations of preferences; one associated with positive and other with negative cross-sectional elasticities of taxable income. By varying the utility representations associated with each parametrization ofpreferences, we are able to generate a rich variation in the shape of equal sacrifice schedules as well as in the pattern

3 From Kaplow and Shavell (2001) we know that any non-welfarist criterion will eventually lead to violations of the Pareto principle. Our question iswhether this is the case for empirically relevant circumstances.

4 Although Berliant and Gouveia (1993) raise the issue of efficiency, they do not address it formally. Indeed, while declaring that “One of the aspects ofthe model we still need to clarify are its welfare properties” and suggesting that inefficiency should result since, “The condition of a zero marginal tax rateat the top ability level, emphasized in Sadka (1976) and Seade (1977), is not generally satisfied”, they never produce a systematic discussion of the issue.

5 This procedure connects Werning's (2007) efficiency bounds and Bourguignon and Spadaro's (2012) revealed social preferences methodologies forassessing efficiency.

C.E. da Costa, T. Pereira / European Economic Review 70 (2014) 399–418 401

of labor supply responses. We find that labor supply and marginal tax rates are considerably more sensitive to the level ofsacrifice when preferences are associated with positive cross-sectional elasticity of taxable income. As for efficiency, underno circumstance do we find inefficiency arising for income levels earned by more than 2% of individuals. Yet, we do findinefficiency arising at reasonably low income levels. This is due to strong labor supply responses.

The rest of the paper is organized as follows. Section 1 describes the economy. Implementable allocations are derived inSection 2, and used in Section 2.1 to derive the shape of equal sacrifice schedules for different parameterizations of preferences.Efficiency bounds are derived in Section 3 where we also show some back of the envelope calculations for the case of logpreferences. The first set of numeric results is in Section 4. In Section 5 we develop a procedure for extracting marginal socialweights and recover the weights for different assumptions regarding the environment. We extend the model to allow for non-separable utility in Section 6. Section 7 concludes. The appendix gathers the derivation of some of the main results.

1. The environment

The economy is inhabited by a continuum of measure one of individuals with identical preferences defined overconsumption, c, and effort, l. Preferences are represented by

Uðc; lÞ ¼ uðcÞ�hðlÞ;where u and h are smooth functions such that u0; �u″;h0;h″40, limc-0u0ðcÞ ¼1 and liml-1hðlÞ ¼1.

In most of what follows, we restrict our analysis to an iso-elastic specification for preferences:

hðlÞ ¼ lγ=γ

for γ41, and

u cð Þ ¼ c1�ρ�11�ρ

;

for ρ40, ρa1 and uðcÞ ¼ ln c when ρ¼1.6

Individuals differ from one another with respect to labor market productivity,wAW � Rþ þ , whereW is a closed interval.We assume that w is distributed according to F(w) with no mass points and associated density f, such that f ðwÞ40 for allwAW .

Technology is very simple: an individual with productivity w who makes effort l produces output y¼ lw, which isconverted one for one into one unit of consumption, c. y is measured in units of the consumption good. The economyis competitive so that each individual is paid his or her output. We, thus, refer to y as output and taxable income,interchangeably. As it turns, it is convenient to define choices over (c,y)-bundles instead of (c,l)-bundles, noting thatidentical preferences over (c,l), Uðc; lÞ ¼ uðcÞ�hðlÞ induce type-dependent preferences over (c,y), ~U ðc; y;wÞ ¼ uðcÞ�hðy=wÞ.

Following Mirrlees (1971), we assume that w is the private information; neither w nor l are observed separately.Finally, there is a government which must extract an exogenously given amount of resources, B, from the economy.An environment, E, is then defined as a triple ðU; F;BÞ where Uðc; lÞ ¼ uðcÞ�hðlÞ, Fð�Þ is the distribution of skills, and B the

Government's revenue requirement, along with its informational structure.An allocation is a mapping ðc; yÞ:W↦R2

þ that associates to each type, w, a consumption/output pair ðcðwÞ; yðwÞÞ. Let ΓðwÞdenote the set of choices available (budget sets) to an agent of productivity w. Each Γð�Þ induces an allocation, (c,y), through

ðcðwÞ; yðwÞÞAarg maxðc;yÞAΓðwÞ

fuðcÞ�hðy=wÞg

for all w.For an environment, E, the set of feasible allocations is defined by the economy's resource constraint:Z

W½yðwÞ�cðwÞ�f ðwÞ dwZB: ð1Þ

To induce an allocation satisfying (1), the Government designs a tax schedule defined as a mapping T :Rþ-R, fromoutput, y, to taxes, T(y). Associated with each tax schedule is a budget set ΓT � fðy; cÞ � R2

þ ; cry�TðyÞg. Under this taxschedule the maximum utility attained by an individual with productivity w is

vðwÞ �maxy

fuðy�TðyÞÞ�hðy=wÞg:

The sacrifice induced by the tax schedule on an individual of productivity w, s(w), is defined as

sðwÞ � v0ðwÞ�vðwÞ;where v0ðwÞ is the utility attained by type w individual when the budget set is the one associated with the chosen referencepoint. In all that follows we take as a reference point a ‘no-sacrifice world’ for which Γ0 � fðy; cÞ � R2

þ ; cryg. That is, the

6 In an equal sacrifice setting, iso-elastic preferences are associated with invariance with respect to rescaling among other appealing properties — seeYoung (1987, 1988).


no-sacrifice allocation is the allocation that results from

ðc0ðwÞ; y0ðwÞÞ � arg maxðc;yÞAΓ0

fuðcÞ�hðy=wÞg;

where ΓðwÞ ¼Γ0 � fðc; yÞ; cryg 8w, and v0ðwÞ ¼ uðc0ðwÞÞ�hðy0ðwÞ=wÞ.Equal sacrifice tax schedules are those which induce s(w) constant in w: sðwÞ ¼ s 8w. Naturally, the specific representation

for preferences that we use defines the standard by which sacrifice is to be measured.We have chosen as reference point a no-sacrifice world in which individuals pay no taxes and receive no transfers.

No-sacrifice ought not to be confused with no-Government. Instead, one should think of a fictitious state in which allGovernment activities necessary to sustain life in society are carried out without costs — see Musgrave (1985) for anexposition. Equal sacrifice is introduced as the principle which guides the procedure to define how society shares the costsof maintaining such Government activity when these costs are taken seriously into account. Because the no-Governmentworld in which social life would not be allowed to flourish is never formally discussed, one may get the impression thatsacrifice is introduced as the social cost for financing wasteful Government spending. We refrain from doing so since the factthat it is the very existence of a State that allows one to realize one's potential has often lead to the adoption of a ‘benefitdoctrine’ as a principle of distributive justice rather than the ‘ability to pay doctrine’ to which the equal sacrifice principle isassociated. This alternative non-welfarist view of justice appears to underlie President Barack Obama's call for the wealthyto pay more taxes on the grounds that others have “… helped to create this unbelievable American system that we have thatallowed you to thrive.”7

2. Incentive-compatible equal-sacrifice systems

The first step in our study is to find, for a given environment, E, the associated equal sacrifice tax schedule. Instead ofchoosing budget sets, we consider a direct mechanism and use the taxation principle to recover the associated tax schedule,as in Berliant and Gouveia (1993). The equivalence between a direct mechanism and the indirect mechanism represented bythe tax schedule is easy to verify — e.g., Guesnerie (1998).

The direct mechanism: We rely on the revelation principle to restrict ourselves to direct truthful mechanisms in which theplanner asks each individual his or her type, w, and uses the (possibly false) report w to assign a bundle ðcðwÞ; yðwÞÞ. Toguarantee truthful revelation, an allocation ðc; yÞ ¼ ðcðwÞ; yðwÞÞwAW must be such that

wAarg maxwAW

fuðcðwÞÞ�hðyðwÞ=wÞg; ð2Þ

for all wAW .Define vðwÞ � uðcðwÞÞ�hðyðwÞ=wÞ as the value attained in (2). The global incentive compatibility condition (2) is satisfied

if and only if the envelope condition

v0 wð Þ ¼ h0yðwÞw

� �yðwÞw2 ; ð3Þ

and the monotonicity condition

yðwÞ increasing in w; ð4Þ

are satisfied.Under the assumption that hð�Þ is strictly increasing and strictly convex, (3) and (4) lead to

yðwÞw

¼φ v0 wð Þwð Þ; ð5Þ

where φ is a strictly increasing function.Nowhere have we used the level of utility, only its variation. This is an interesting consequence of incentive compatibility

when preferences are separable: the cross-sectional variation in utility pins down the level of output produced by allindividuals.

Equal sacrifice, on the other hand, restricts ðcð�Þ; yð�ÞÞ to be such that v0ðwÞ�vðwÞ ¼ s 8w, and s, to be the minimum sfor which

RW ½yðwÞ�cðwÞ�f ðwÞ dwZB. Note that differentiability of v0 implies differentiability of v and v0ðwÞ ¼ v00ðwÞ 8w.

Since equal sacrifice is all about preserving ‘utility differences’, the cross-sectional variation of utility is invariant to the levelof sacrifice.

Combining incentive compatibility (5) and equal sacrifice, v0ðwÞ ¼ v00ðwÞ 8w, we have yðwÞ ¼ y0ðwÞ for all w. Everyoneproduces the exact same output they produce at the reference state!8 As a consequence sacrifice is all due to reduced

7 More explicit forms of non-wasted spending may be introduced by assuming that the Government produces a public good, z, equally valued acrossagents such that Uðc; l; zÞ ¼ uðcÞ�hðlÞþυðzÞ. All our results would go through with sþυðzÞ substituting for s in our formulae. The numeric implementationbecomes more involved since z is not monotonic in s.

8 This result was first proved by Berliant and Gouveia (1993) — see their Proposition 4.


consumption

s¼ uðy0ðwÞÞ�uðy0ðwÞ�Tðy0ðwÞÞÞ: ð6Þ

2.1. The shape of equal sacrifice tax schedules

Let ξð�Þ ¼ u�1. Then, using yðwÞ ¼ y0ðwÞ; 8w and TðyðwÞÞ ¼ yðwÞ�ξðuðyðwÞÞ�sÞ one recovers Tð�Þ only using s and yð�Þ. Tð�Þ,therefore, inherits the properties of uð�Þ. In particular, if u is a smooth function such that u0ðxÞ40 for all xARþ , then T″ willbe a well defined continuous function.

Define the retention function, RðyÞ ¼ y�TðyÞ. Then, differentiate (6) with respect to y to obtain u0ðyðwÞÞ ¼ u0ðRðyðwÞÞÞR0ðyðwÞÞ.9

If uð�Þ is iso-elastic, marginal and average retention rates are connected through

RðyðwÞÞyðwÞ ¼ R0ðyðwÞÞ1=ρ: ð7Þ

Moreover, letting rðwÞ ¼ R0ðyðwÞÞ, then, for any s,

r wð Þ ¼ ðyðwÞ1�ρ�sð1�ρÞÞρ=ð1�ρÞ

yðwÞρ ; ð8Þ

provided that ρa1. If ρ¼1, rðwÞ ¼ r¼ expf�sg. In particular, limw-1rðwÞ ¼ 0 when ρ41.Note that s contains all the information necessary to define r(w). As for s, first note that because y(w) is invariant to s,

ϱ sð Þ ¼RW fyðwÞ�ξðuðyðwÞÞ�sÞgf ðwÞ dwR

WyðwÞf ðwÞ dw : ð9Þ

defines a strictly increasing function ϱð�Þ linking Government revenues as a share of GDP to the level of sacrifice. Theminimum equal sacrifice allocation by choosing s such that ϱðsÞRWyðwÞf ðwÞ dw¼ B.

3. Efficient tax schedules

Let ðcðwÞ; yðwÞÞ be the allocation induced by Tð�Þ, and v(w) the associated utility profile. A tax schedule, Tð�Þ, is efficient ifand only if there is no alternative tax schedule that induces an allocation ð~cðwÞ; ~yðwÞÞ such that ~vðwÞZvðwÞ for all w andwhich raises more revenue.

Werning (2007) writes down and solves this problem by substituting the necessary envelope condition for the incentivecompatibility constraints. Because equal sacrifice schedules induce a taxable income function, y(w), which is a strictlyincreasing function of w, the envelope condition is also sufficient for the allocation to be incentive compatible in our setting.Hence, the main assumption under which Werning's (2007) procedure is valid holds here. We combine Werning's (2007)Proposition 4 with the properties of equal sacrifice schedules under separable and iso-elastic preferences to obtainProposition 1.

Proposition 1. For separable and iso-elastic preferences, an equal sacrifice labor income tax schedule, Tð�Þ, is efficient if and onlyif marginal retention rates, r(w), are such that, for all w,

ηρrðwÞ2�1=ρ� γþα wð Þ�1� �

r wð Þþα wð Þrηγγ�1þρ γþ1

� �� ; ð10Þ

with αðwÞ ¼ �d ln f =d ln w, and η¼ γ=ðγþρ�1Þ.

Proof. See Appendix A.1.

For any r(w) the left-hand side of (10) is increasing in αðwÞ. It is then easy to check that (10) is satisfied for any rðwÞ40 ifαðwÞo ð1�ρþγðρþ1ÞÞ=ðγþρ�1Þ. That is, for low values of αðwÞ, any positive marginal tax rate below 100% is efficient.Conversely, when ρ41 the following corollary of Proposition 1 guarantees that inefficiency always arises if the right tail ofthe distribution of skills is not too thick.

Corollary 1. Let α ¼ lim supw-1αðwÞ, defined over the extended real numbers, and assume that ρ41. Then, if

α4 ð1�ρþγðρþ1ÞÞ=ðγþρ�1Þ; ð11Þthe equal sacrifice income tax schedule exhibits inefficiencies.

9 Since R0ðyÞrRðyÞ=y is both necessary and sufficient for average taxes to be increasing, when the tax schedule is smooth, one immediately connectsrisk aversion and progressivity: an equal sacrifice schedule is progressive if and only if the coefficient of relative risk aversion of the chosen utility functionis greater than one. Samuelson (1947) derived this result disregarding incentives and assuming that utility depended only on consumption. For the specialcase of separable preferences, taxable income is invariant to the level of sacrifice and Samuelson (1947) result remains valid.


Proof. From (8) we get limw-1rðwÞ ¼ 0 for ρ41. Then, as r approaches zero, the two first terms in the left-hand side of (10)approach 0. Inequality (10) is therefore violated for large enough w if (11) holds. □

To illustrate the efficiency test, consider an environment with uð�Þ ¼ lnð�Þ and a Pareto distribution of skills, FðwÞ ¼1�ðw=wÞα�1, with support ½w;1Þ, w40, and associated density f ðwÞ ¼ κw�α, α41, where κ ¼ ðα�1Þwα�1.

When ρ¼1, the equal sacrifice schedule is such that rðwÞ ¼ r.10 Inequality (10), therefore, simplifies to

r wð Þ ¼ rZα�2

αþγ�2: ð12Þ

If αo2, (12) is satisfied for any value of r, so we focus on α42. This is also the range deemed more relevant from anempirical perspective.11 Saez (2001) considers the following values of α for the US economy: 2.5, 3 and 3.5; Werning (2007),α¼ 3. In all cases, condition (12) has a bite.

To complete the environment's description only γ remains to be defined. The elasticity of taxable income, ϵ¼d ln y=d ln r, is of little use in this case since it is identically zero for all γ: income and substitution effects exactly offset.Of course, ϵ¼0 is compatible with compensated and income elasticities being both low or both high, depending on γ. Thus,we could choose γ to match the compensated (or the income) elasticity of taxable income. Alternatively, we may use theFrisch elasticity of labor supply, ϵf ¼ γ=ðγ�1Þ, for which reliable estimates are available.

With linear taxes, ϱðsÞ ¼ 1�r. Table 1 displays, for each combination of εf and φwithin the range used by Saez (2001), themaximum share of GDP that may be raised by the Government with an equal sacrifice schedule without violating (12).

4. Sacrifice and efficiency in practice

In this section we use the tests derived in Section 3 to analyze equal sacrifice schedules associated with an environmentthat approximates the US economy. Recall that given our preference choices, an environment is fully described by thepreference parameters, γ and ρ, and the distribution of skills, F(w).

Let us start with γ and ρ. In the absence of non-labor income, these two parameters determine both the uncompensatedand the compensated elasticities of taxable income. Hence, ρ and γ may be chosen in such a way as to guarantee thatthe values for these two statistics generated by the model are in an empirically relevant range. Alternatively, one can useYoung's (1990) procedure to recover ρ from the empirical effective tax schedule under the assumption that the equalsacrifice principle does rationalize these schedules. γ is then chosen to match a single statistic, e.g., the cross-sectionalelasticity of taxable income, ε.

The two procedures make explicit the connection between preferences and the society's views of sacrifice. We haverenounced to the possibility of disentangling preferences from views of sacrifice in order to benefit from the separablespecification of utility. Putting it differently, by fixing the parameters of the utility function we commit to a specific view onhow consumption changes translate into utility changes, i.e., a specific perception of sacrifice, an issue we shall reconsider inSection 6.

As for F(w), we must define it consistently with our choices of ρ and γ. Let

yðwÞ ¼ arg maxy

fuðy� T ðyÞÞ�hðy=wÞg;

where T ð�Þ is the actual income tax schedule of the economy we study. Assume that yð�Þ has an inverse, i.e., there is ωð�Þ suchthat w¼ωðyðwÞÞ.12 Using the observed distribution of taxable income, G(y), we find the distribution of w, F(w) throughGðyÞ ¼ FðωðyÞÞ. For separable, iso-elastic preferences, and differentiable Tð�Þ, we have

w¼ yðwÞðγþρ�1=γÞð1�T 0ðyðwÞÞÞðρ�1Þ=γ 1�T 0ðyðwÞÞ1�TðyðwÞÞ=yðwÞ

� ��ρ=γ

: ð13Þ

We use the approximation of effective tax schedules due to Gouveia and Strauss (1994):

y�TðyÞ ¼ yð1�bÞþb½y�νþκ��1=ν; ð14Þwhere the parameters b, ν and κ, are estimated from the US actual effective tex schedule by Guner et al. (Forthcoming).

Fig. 2 displays the distribution of skills recovered from the Panel Study of Income Dynamic (PSID) labor income data,which we refer to as the ‘empirical’ distribution of skills. We then adjust a parametric (Log-normal) distribution to theempirical distribution of skills — see Fig. 2. The preference parameters used to recover the distribution of skill displayed inthe figure are ρ¼1.5 and γ¼3.5. This corresponds to cross-sectional unconditional and conditional elasticities of laborsupply of �0.125, and 0.25, respectively. We concentrate on ρ41 since this is needed to generate progressive equal sacrificeschedules.

10 With log utility and a linear tax yðwÞ ¼w. Therefore, the associated distribution of income is also Pareto, GðyÞ ¼ 1�ðy=yÞφ�1, and αðwÞ ¼�d ln f =d ln w¼ �d ln gðyÞ=d ln y¼φ. This is a commonly used specification for the distribution of income — e.g., Diamond (1998), Saez (2001), andWerning (2007).

11 Recall that a Pareto distribution does not have a finite mean if αo2, and does not have a finite variance if αo3.12 This rules out concave kinks in the actual tax schedule. For yðwÞ to be a function, rather than a correspondence, we rule out convex kinks, as well.

Table 1The table displays for each combination of the decay parameter, φ, and the Frisch elasticity parameter, εf, the maximum share of GDP that may be efficientlyfinanced with an equal sacrifice schedule.

α εf (%)

0.05 0.3 0.4 0.5 2

2.5 98 90 88 86 753 95 81 78 75 603.5 93 74 70 67 50

Fig. 1. The two top figures display marginal and average tax rates as a function of taxable income for three different levels of sacrifice. In all figures, solidline refers to the lowest level of sacrifice and dotted line to the highest. The bottom left figure displays Virtual Income defined as ð1�τÞy�TðyÞ. The bottomright figure shows how the elasticity of taxable income d ln y=d lnð1�τÞ varies with taxable income.


The 90–50 and the 50–10 wage ratios of skills distribution generated by our procedure are 2.90 and 2.29, respectively.The same ratios in the data are 2.32 and 2.07.13 Our procedure seems to generate a distribution of skills which has a greaterdispersion than the distribution of wages. It is important to recall that productivity and wages are not exactly the same thingsince we collapse individuals entire lives into a single period. All in all, the Log-normal distribution matches reasonably wellall deciles of the empirical distribution, but fails to approximate the data for the top percentiles. The right tail of the incomeand skill distributions are known to be better approximated by a Pareto distribution.

The level of sacrifice is chosen to raise 25% of GDP in taxes. Fig. 3 displays rð�Þ, the retention rate, and rð�Þ, the minimumretention rate compatible with efficiency, as a function of taxable income, y. Because ρ41, the marginal retention rate isdecreasing in y. For the Log-normal distribution, αðwÞ is non-decreasing and the region of inefficiency is an interval ½wa;1Þ.Inefficiency arises only when the marginal tax rate reaches 68% for an individual who earns US$ 655k annually.

13 To generate model data which is comparable to actual wage data we made two changes in our procedures. We first introduced an additionalparameter, χ, in the dis-utility of labor, which is now χlγ instead of lγ=γ. We then truncated the distribution of wages at 70% of the minimum wage. Notethat these changes are only used in order to compare the wage distributions. They have no impact on any of the procedures in the paper. Data is fromCongressional Budget Office (2011).

1 0 1 2 3 4 5 6x 1014

0

1

2

3

4

5

6

7

8 x 10 15

Skill

f(w)

Log normal: Skill density 90%

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 1015

0

1

2

3

4

5

6

7

8 x 10 15

Skill

f(w)

Log Normal: Skill density 99%

Fig. 2. The figure displays the distribution of skills we obtain by using PSID data for the year 1993, and approximating the tax systemwith the Gouveia andStrauss (1994) functional form. We also adjust a Log-normal distribution to the empirical one.

Fig. 3. This figure displays the marginal retention rates, r, associated with each level of taxable income as well as the minimummarginal retention rate, r ð�Þ,for which the tax system is efficient. It is assumed that the distribution of skills is Log-normal, ρ¼1.5, γ¼3.5, and two levels of tax revenues are considered:20% — low G — and 30% — high G — of GDP.


To highlight the role of the skill distribution's upper tail, we show in Fig. 4 the lower bounds for marginal retention ratesfor both the Log-normal and the Pareto distributions. The level of sacrifice is the same in the two examples, chosen to raise25% of GDP under the Log-normal distribution. It is apparent that inefficiency arises earlier in the case of a Log-normaldistribution. Indeed, the difference between the bounds of a Pareto and of a Log-normal distribution is still significant for αas high as 3.5, as the figure clearly shows. Inefficiency only arises for the Pareto distribution when annual income reaches US$ 1354k and r¼77.2%. The contrasting results for the Log-normal and the Pareto distributions are hardly surprising. Theymirror the current debate on the efficient taxation of high earners.14

Next, we ask how these numbers change if the Government is to raise 30% instead of 25% of GDP in taxes. For a Log-normal distribution inefficiency arises when r¼71.2% corresponding to US$ 504k in taxable income. For a Pareto distributionwith α¼3.5 the corresponding figures are r¼77.2% and US$ 769k.

Young's (1990) equal sacrifice schedules: Young (1990) has taken real world distributions of before and after tax incomesand has shown that one could find a common (and empirically sound) utility function that equalizes the utility loss of allindividuals, and such that this loss was minimal to finance the government revenue requirements. He asked whether the

14 Diamond (1998) first emphasized how the results regarding optimal taxes may dramatically change from those prescribed by Mirrlees (1971) whenone substitutes the Pareto distribution for the Log-normal.

Fig. 4. This figure displays the marginal retention rate, r, associated with each level of taxable income as well as the minimum marginal retention rate forwhich the tax system is efficient. Both a Log-normal and a Pareto distribution of skills, with α¼3.5 are considered. The preference parameters are ρ¼1.5,γ¼3.5, and the level of sacrifice is chosen to generate tax revenues of 20% of GDP for the Log-normal distribution.

Table 2The table displays the critical retention rates, r , and the associated labor income earnings in thousands of US$ for the equal sacrifice schedules adjusted byYoung (1990) to the years 1957, 1967 and 1977. In the case of a Log-normal distribution adjusted to the current distribution of skills, total revenues raised asa percentage of GDP, ϱðsÞ, are also calculated. No such figure is calculated for the Pareto distribution.

Year rho Log-normal Pareto (α¼3.5)

1957 1967 1977 1957 1967 1977(ρ¼ 1:610) (ρ¼ 1:519) (ρ¼ 1:718) (ρ¼ 1:610) (ρ¼ 1:519) (ρ¼ 1:718)

r ð%Þ 22.9 33.2 22.9 18.5 21.9 14.5US$ k 305 426 406 424 858 684ϱðsÞ ð%Þ 34.9 27.8 26.1 – – –


effective tax rates which prevailed in the US during the 1957–1987 period could be rationalized by schedules derived underthe equal sacrifice principle. We now evaluate the equal sacrifice schedules adjusted by Young (1990) for the US economy.15

Young (1990) considered constant relative risk aversion preferences and estimated the parameters ρ from the data usingthis assumption. Since we borrow the curvature parameter ρ from Young (1990) we adopt the alternative procedure forpinning down the utility function parameters. We use Young's (1990) values for ρ, along with the corresponding levelof sacrifices, and vary γ to hold the elasticity of taxable income constant at �0.125. In Table 2, we report the critical value ofrð�Þ and the associated critical level of income for each value of ρ (and associated level of sacrifice) used in Young's (1990)work. Provided that αðwÞ is weakly increasing, we need not use the entire distribution of skills to apply Proposition 1. Weare, therefore, able to also use Pareto distribution, which is supposed to fit the upper portion of the distribution of skillsbetter than the Log-normal. For the Log-normal distributions we also report how much revenue would be raised for eachlevel of sacrifice if the distribution of skills was the one we have extracted from current data. Ideally we would use thecorresponding period's distribution of skills. Instead, we rely on scale invariance to obtain a rough estimate of whether thesevalues are compatible with actual Government consumption values. Results are displayed in Table 2. Because the Paretodistribution only matches the top of the distribution we refrain from making any statement regarding total revenues raised.

5. Efficiency and marginal social welfare weights

Notwithstanding its appealing properties, equal sacrifice has been criticized as a guiding principle for tax-benefit policiesdue to the absence of explicit redistributive motives. Indeed, everyone pays taxes if any amount of revenue must be raised,and no taxes are paid for purely redistributive reasons. Recall, however, that for high enough government consumption

15 Young (1990), for example, suggests but cannot explore the possibility that efficiency concerns may explain the poor fit of equal sacrifice schedulesat the high end of the distribution of income. In his words (Young, 1990, p. 264) “For high incomes, therefore, the departure from equal sacrifice may be dueto efficiency considerations…”


everyone may end up paying positive taxes even if the tax schedule results from the maximization of a concave socialwelfare function. We cannot rule out a priori the possibility that equal sacrifice schedules are redistributive in this sense ofthe word. This is what we investigate in this section.

We start by discussing an alternative formulation of our efficiency discussion due to Bourguignon and Spadaro (2012)which will be useful for dealing with the redistributive properties of equal sacrifice schedules.16

Let U be the range of Uð�Þ, and Ψ :U-R an arbitrary Bergson–Samuelson social welfare function. Then, define the Mirrlees'problem for environment E given Ψ as

maxZWΨ ðuðcðwÞÞ�hðyðwÞ=wÞÞf ðwÞ dw ð15Þ

subject toZWfyðwÞ�cðwÞgf ðwÞ dwZB ð16Þ

and

wAarg maxw

fuðcðwÞÞ�hðyðwÞ=wÞg: ð17Þ

We say that a tax schedule, Tð�Þ, is rationalizable at environment E if there is a social welfare functionΨ, increasing in v(w),such that the allocation that solves the Mirrlees problem at environment E for the social welfare function Ψ is induced byTð�Þ. We then say that the pair ðE;Ψ Þ rationalizes Tð�Þ.

The question we ask in this paper is, therefore, whether, for a given environment, E, and an equal sacrifice tax schedule,Tð�Þ, it is always the case that we may find a Paretian social welfare function Ψ such that the pair ðE;Ψ Þ rationalizes Tð�Þ.

We associate redistributive taxes with an underlying concave social welfare function. Our approach is local, in the sensethat we may only hope to recover the derivative of such function, Ψ 0ðvðwÞÞ, the marginal Pareto weights. Redistribution is,therefore, associated with decreasing marginal Pareto weights, i.e., we ask whether the equal sacrifice tax system may be(locally) rationalized by a concave social welfare function.

In Appendix A.2 we show how to use Werning's (2007) procedure to extract the marginal social weights — see Diamondand Mirrlees (1971a, 1971b), Saez (2001), Saez and Stantcheva (2013) — associated with the equal sacrifice schedules.Regions for which marginal weights become negative are those for which (10) is violated.

Our procedure to recover the marginal weights is simple. We note that the marginal weights are related to the Lagrangemultiplier associated with the promise keeping constraint (28) of the Pareto problem that defines the efficiency bounds —

Appendix A.2. It is then a matter of algebra to arrive at an expression for this Lagrange multiplier that only depends on observables.

Proposition 2. For an environment E and a schedule Tð�Þ with associated marginal retention rate function rð�Þ, let w40 be thelowest skill level in the support of the distribution of skills, Fð�Þ, with associated density, f ð�Þ. The social marginal utility of incomein the hands of a type-w person is

Ψ 0 v wð Þð ÞcðwÞ�ρ ¼ λ Υ wð Þ1�rðwÞγrðwÞ þ1

� ;

where

Υ wð Þ � �Φ wð Þ�d ln rðwÞd ln w

rðwÞ1�rðwÞþγ;

λ¼ZWcðwÞρf wð Þ dw� 1

γ�11�rðwÞrðwÞ f w

� �cðwÞρ

� �1

;

and

Φ wð Þ � �d ln f ðwÞd ln w

þ γ�1� �d ln yðwÞ

d ln w�1:

Proof. See Appendix A.2.

It is important to note that the proposition applies not only to equal sacrifice but also to any smooth tax schedule that inducesa monotone allocation. Moreover, although we have assumed that the utility function is separable and iso-elastic to derive theexpressions in Proposition 2, an analogous expression is easily derived for any separable preferences — see Section 6.

The marginal weight Ψ 0ðvðwÞÞ is the direct value placed by the planner on type-w's utility. It, therefore, reflects howutility differences affect social welfare at the margin. For a concave social welfare function, which we take as the main notionof equity concerns, Ψ 0ð�Þ is weakly decreasing in vð�Þ. The identification of Ψ 0ð�Þ is, of course, conditional on a specific

16 We shall relate it to the use of generalized social marginal welfare weights championed by Saez and Stantcheva (2013).

Fig. 5. The left panel displays marginal social welfare weights, Ψ 0ð�Þ, for two different levels of revenue as a percentage of GDP, 20% — low G, and 30% —

high G. In the right panel the associated marginal social values of income, Ψ 0ð�Þcð�Þ�ρ , are displayed. Both are normalized to integrate to one.


representation of preferences. Because the assessment of sacrifice does require the commitment to a specific representationof preferences, it is with respect to this representation that captured the notion of sacrifice that we identify Ψ 0ð�Þ.17

Of course, it is not only Ψ ð�Þ that defines the society's desire to redistribute income, but the curvature of u is also crucialin determining why a society may weight income in the hands of different people differently. Ψ 0ðvðwÞÞcðwÞ�ρ, whichDiamond and Mirrlees (1971a, 1971b) call social marginal utility of income in the hands of individual type w, and which Saezand Stantcheva (2013) call generalized social marginal welfare weight, is the term which captures this full motive for incomedistribution.

Fig. 5 displays both the marginal Pareto weights and the generalized social marginal welfare weights for two differentlevels of sacrifice, corresponding to the schedules derived for the Log-normal distribution in Section 4. The first thing to noteis that marginal Pareto weights (left panel) are not monotonic. They are close to zero for the very poor, increase for themiddle income and decline as income gets high enough. For a high level of Government spending it turns negative, thusreflecting inefficiency of the tax schedule for income levels within the range we explore.

It is apparent that the marginal social value of income increases for low income and decreases for high incomeindividuals as Government consumption increases. For high levels of Government consumption taxing lightly the poor maybe a fairly redistributive policy. Indeed, had we chosen to truncate the distribution to get a more realistic lower part of thedistribution, we would not observe these extremely low weights at the bottom. Even so, we do not believe that we wouldobtain a downward sloping curve for the levels of Government consumption we consider here.

Fig. 6 displays the effects of changes in the distribution of skills on marginal Pareto weights. The graphs in the leftcompare marginal retention rates and associated bounds, marginal social welfare weights and social marginal value ofincome for two economies that have the same Gini coefficient but differ with respect to average income. The same amountof consumption as a percentage of GDP namely 20% determines the level of sacrifice. Note that the same level of incomerepresents very different accumulated fractions of population when we change the distribution. The right column comparesthe same statistics for two economies with the same mean income but different Gini coefficients. The same commentregarding income and the accumulated fraction of population applies here. Note that high taxes on a highly egalitariansociety means that the wealthy are being distorted without the benefit of raising a lot of revenue.

6. Non-separability

So far, we have strongly relied on the assumption that not only the preferences but also the utility function used to assesssacrifice is additively separable. Underlying this assumption is a commitment to a specific view of how effort affects theutility value of consumption; namely dUc=dl¼ 0. The curvature of the relevant utility representation from preferences isultimately pinned down from choices through revealed preferences exactly as risk aversion is identified by labor supplychoices in Chetty's (2006) work. It is important to bear in mind that identification is accomplished through a commitment toany dUc=dl, not necessarily, dUc=dl¼ 0.

Therefore, if we want to disentangle preferences from the society's views on sacrifice, we must allow for differentassumptions regarding dUc=dl. In particular, we must relax dUc=dl¼ 0.

17 In Section 6 we vary the representation of preferences and show how this alters the way society perceives the mapping from consumption changesto sacrifice.


The first consequence of assuming Ucla0 is that invariance no longer characterizes equal sacrifice allocations. Indeed, bycombining incentive compatibility with equal sacrifice we obtain

yðwÞy0ðwÞ ¼

Ulðy0ðwÞ; y0ðwÞ=wÞUlðcðwÞ; yðwÞ=wÞ ;

which cannot be satisfied for yðwÞ ¼ y0ðwÞ, cðwÞay0ðwÞ unless Ul is independent of c. Still, by combining incentivecompatibility and equal sacrifice,

�Ul c wð Þ; yðwÞw

� �yðwÞw2 ¼ v00 wð Þ and U c wð Þ; yðwÞ

w

� �yðwÞw2 ¼ v0 wð Þ�s;

one may derive the minimum equal sacrifice allocation. This system of two equations in two unknowns may be easily solved(numerically at least) under mild regularity conditions. So, non-separability is not an issue for the generalization of ourmodel. The real problem is that the efficiency tests require separability of preferences.

Thus, provided that there is a monotone transformation of U which is separable, i.e., Uðc; lÞ ¼ ςðuðcÞ�hðlÞÞ, for somestrictly increasing ςð�Þ; uð�Þ and hð�Þ as in Section 1, the efficiency tests remain valid.18 Assuming Uðc; lÞ ¼ ςðuðcÞ�hðlÞÞ forsome triple ðςð�Þ;uð�Þ;hð�ÞÞ is not without loss, of course. Still, this extension provides enough flexibility for us to disentanglethe effects of changes in preferences from those of changes in the metric of sacrifice. Moreover, we can easily adapt theprocedure in Appendix A.2 to recover the marginal social welfare weights for this class of utility functions.

Let ~v0ðwÞ ¼maxyfuðyÞ�hðy=wÞg, and v0ðwÞ ¼ ςð ~v0ðwÞÞ, then, assuming that ςð�Þ is differentiable with ς0ð�Þ40, it is possibleto solve for y(w) using a single equation:

ς0ð ~v0ðwÞÞ ~v 00ðwÞ

ς0ðς�1ðςð ~v0ðwÞÞ�sÞÞ ¼ h0yðwÞw

� �yðwÞw2 ; ð18Þ

where ς�1ð�Þ is the inverse of ςð�Þ.We will use the same preferences of previous sections, uðcÞ ¼ ðc1�ρ�1Þ=ð1�ρÞ and hðlÞ ¼ lγ=γ, and ςðvÞ ¼ v1�σ=ð1�σÞ.

Preferences in the consumption/leisure space are defined by ρ and γ while views on sacrifice are allowed to vary throughchanges in σ. When σ40, dUc=dl40 the more the one works, the greater the sacrifice induced by a given drop inconsumption. The opposite is true for σo0. The main consequence for equal sacrifice allocations is that labor supplydecreases with sacrifice when σ40 and increases when σo0. It is also possible to show that changes in labor supplyinduced by the equal sacrifice schedule are greater for more productive or less productive agents depending on whetherσ41 or σo1. An interesting borderline case is ςð�Þ ¼ lnð�Þ, which corresponds to equal proportional sacrifice. In this case,labor supply declines by the same proportion for all w. Fig. 8 displays the variation in labor supply for different values of σ.

The first set of results — our benchmark results — are displayed in Table 3. We consider two sets of preferenceparameters: ðρ¼ 1:5; γ ¼ 2:8Þ and ðρ¼ 0:8; γ ¼ 3:5Þ. Both parametrizations yield the same cross-sectional compensatedelasticity of taxable income, εc ¼ 0:3. In each line we report the results for a different σ, from σ ¼ �0:5 to σ¼3. Governmentconsumption in all exercises is equivalent to 25% of GDP. It is important to note that, except for the separable case, σ¼0, GDPis not constant across exercises.

Finally recall that the main reason for focusing on ρ41 up to this point was that this was necessary for tax schedules tobe progressive when σ¼0. Here, in contrast, we obtain progressive tax system with ρ¼0.8, both when σ¼1.5 and whenσ¼3 — see Fig. 7.

Tax schedules and labor supply display little variation with σ when ρ¼1.5, whereas both vary substantially when ρ¼0.8as Figs. 8 and 7 clearly show. The change in GDP induced by the tax schedule is the most salient consequence of suchdifferences. The variation in GDP is orders of magnitude greater for the parametrization with ρo1 as one sees in Table 3. Asa consequence, results regarding efficiency are far more sensitive to σ if ρ¼0.8 than if ρ¼1.5 (Fig. 9).

A cautionary remark is due. In all cases, less than 1% of the population earns more than the minimum level of income atwhich inefficiency arises. Since we are using a Log-normal distribution which does not adjust too well at the top of thedistribution, one should take the reported values with a grain of salt.

In Table 4 we assume γ¼1.5 in both parametrization. That is, we fix the Frisch elasticity of labor supply at εf ¼ 2, usingboth values for ρ: 0.8 and 1.5. We are, therefore, considering substantially higher compensated elasticities: εc ¼ 0:77 forρ¼0.8 and εc ¼ 0:5 for ρ¼1.5. Inefficiency arises for lower values of income and higher marginal retention rates. Again, forno value of σ do we find that more than 1% of earnings in the inefficiency region.

We finally assess efficiency when the government must raise 50% of GDP in taxes — Table 5. Recalling that there are notransfers, this is an extremely high number. For ρ¼1.5 we find that for all values of σ a little over 1% of the population facesinefficiently high marginal tax rates. For ρ¼0.8 we still find that less than 1% of the population is in the inefficiency region.

18 An important example is balanced growth path preferences, the specification of preferences most commonly used by macroeconomists.

Fig. 6. The left column compares marginal retention rates, bounds (top row), marginal social welfare weights (middle row) and marginal value of income(bottom row) for two economies that have the same Gini coefficient but different mean income. The right column compares the same things for twoeconomies with the same mean income but different Gini coefficients.


However, when σ¼3 this occurs for taxable incomes as low as US$ 89k. Despite this very low value, it is still the case thatless than 1% of individuals face inefficiently low retention rates. This is due to the fact that there is a large labor supplyresponse.19

19 We have chosen to hold revenues as a percentage of GDP constant. For σ¼0 this is equivalent to holding total revenues constant since GDP isinvariant to the level of sacrifice. However, when σa0, GDP varies with the level of sacrifice, and this variation need not be trivial. For ϱðsÞ ¼ 0:5, ρ¼0.8 andσ¼3, for example, GDP is 47% lower than in the no-sacrifice world.

Table 3For two distinct set of preference parameters, ðρ; γÞ, we vary σ, the parameter that captures the specific utility representation. r indicates the marginalretention rate at which inefficiency arises, US$ k, the taxable income in thousands of US dollars at which it occurs, and ΔY the percentage change in GDPinduced by the tax schedule. The level of sacrifice is chosen to raise 25% of GDP in taxes. We only report inefficiency if it arises for at least 0.01% ofindividuals.

ρ¼1.5, γ¼2.8 ρ¼0.8, γ¼3.5

US$ k r (%) ΔY (%) US$ k r (%) ΔY (%)

σ ¼ �0:5 288 40.4 0.07 σ ¼ �0:5 – – 3.10σ¼0 286 40.4 0.00 σ¼0 – – 0.00σ¼0.5 285 40.3 �0.07 σ¼0.5 – – �2.91σ¼1 284 40.2 �0.14 σ¼1 – – �5.77σ¼1.5 283 40.2 �0.20 σ¼1.5 – – �8.69σ¼3 279 40.0 �0.41 σ¼3 425 46.7 �19.16

Fig. 7. The figures display τðwÞ as a function of productivity for different values of σ. We have used ρ¼1.5 for the panel in the left and ρ¼0.8 for the panel inthe right.

Fig. 8. The figures display the percentage change in labor supply, l(w), for each level of productivity given different values of σ. We have used ρ¼1.5 for thepanel in the left and ρ¼0.8 for the panel in the right. In all cases, sacrifice was chosen to finance 25% of GDP.


7. Conclusion

We study the efficiency and distributive properties of equal sacrifice schedules in a Mirrlees' (1971) setting. Separabilityof preferences allows us to evaluate efficiency using the methodology developed by Werning (2007). It also makes possiblefor us to develop a simple procedure to extract the marginal social weights associated with each policy.

Our focus on equal sacrifice schedules reflects a broader increase in academic interest in alternative views of distributivejustice. This has led in recent years to new studies aimed at, on the one hand, better understanding the society's view ofwhat a just tax system is as in Weinzierl (2012a), and, on the other, on using this information to guide policy, as in Weinzierl(2012b) and Saez and Stantcheva (2013). Equal sacrifice itself has experienced a revival in the 1990s possibly due to thefindings of Young (1987, 1988, 1990). Works that followed allowed us to better understand the restrictions imposed on

Fig. 9. The figures display the marginal social weight, Ψ 0ðwÞ, for each level of productivity given different values of σ. We have used ρ¼1.5 for the panel inthe left and ρ¼0.8 for the panel in the right. For purely illustrative purposes, sacrifice was chosen to finance 50% of GDP.


ρ¼1.5, γ¼1.5 ρ¼0.8, γ¼1.5

US$ k r (%) ΔY (%) US$ k r (%) ΔY (%)

σ ¼ �0:5 144 49.6 0.07 σ ¼ �0:5 – – 3.37σ¼0 144 49.6 0.00 σ¼0 – – 0.00σ ¼ 0:5 143 49.4 �0.07 σ ¼ 0:5 – – �3.16σ¼1 142 49.3 �0.14 σ¼1 – – �6.22σ¼1.5 142 49.3 �0.21 σ¼ 1.5 523 68.1 �9.19σ¼3 141 49.0 �0.43 σ¼3 177 53. �18.73


ρ¼1.5, γ¼2.8 ρ¼0.8, γ¼3.5

US$ k r (%) ΔY (%) US$ k r (%) ΔY (%)

σ ¼ �0:5 129 22.2 0.19 σ ¼ �0:5 – – 7.93σ¼0 128 22.1 0.00 σ¼0 – – 0.00σ ¼ 0:5 128 22.1 �0.19 σ ¼ 0:5 – – �6.92σ¼1 127 22.0 �0.38 σ¼1 554 50.5 �13.52σ¼1.5 126 21.9 �0.57 σ¼1.5 253 42.8 �20.36σ¼3 125 21.7 �1.14 σ¼3 89 27.0 �47.23


observed tax schedules by the equal sacrifice principle — Mitra and Ok (1996) and Ok (1995) — as well as the consequencesof taking incentives into account explicitly — Berliant and Gouveia (1993).20

In our work we assume that preferences are separable and vary the preference parameters in a range that producessensible elasticities of taxable income. For the iso-elastic specification a coefficient of risk aversion that varies in the intervalbetween 1.5 and 1.7 provides a good fit with the data. This is also, according to Young (1990), what one needs to rationalizethe US income tax schedule using the equal sacrifice principle and a separable utility function.

20 Gouveia and Strauss's (1994) equal-sacrifice based approximation of the US schedule, widely used by macroeconomists: Conesa and Krueger (2006),Conesa et al. (2009), is another reminder that these findings have not remained unnoticed by academia at large.


Whereas separability of preferences is crucial for our procedure, separability of the utility function is not. Allowing fornon-separable representation of preferences endows us with an extra degree of freedom to play around with differentnotions of sacrifice without changing preferences. We show that different preferences may lead to substantially differentsensitivities of schedules and efficiency bounds to the level of sacrifice.

Finally, we devise a procedure for extracting the implicit marginal Pareto weights associated with any arbitrary taxschedules. We find that weights associated with equal sacrifice schedules are not monotone, which indicates that, evenlocally, the policy is not redistributive toward the poorest individuals. As the level of Government consumption increases,the implicit weights placed on the poorest individuals do increase. Yet, because we have allowed individuals to havearbitrarily low productivity we never obtain decreasing weights. We find it unlikely that we would do so for any reasonablelevel of Government consumption.

We have followed Mirrlees' (1971) tradition of collapsing agents' entire lives into a single period. Therefore, issuespertaining to investments both in human and physical capital and many other choices that involve dynamics cannot beaddressed here. Still, it is possible to imagine that these intertemporal choices might have important consequences for theassessment of efficiency. In particular, progressive schedules are associated with disincentives to accumulate human capital.Cross-sectional labor-supply elasticities may, in this case, grossly underestimate the overall behavioral effects of taxation —

e.g., Keane (2011b, 2011a). The other side of the coin is that we have not allowed for taxes to distort savings, which therecent new dynamic public finance literature has suggested that may play an important role in improving the overallefficiency of tax systems — e.g., Golosov et al. (2003), Farhi and Werning (2013), and Weinzierl (2011). Finally, differences inaccumulated wealth generate differences in incentives to make effort, thus representing an important additional dimensionof heterogeneity for the purpose of our investigation. The technical issues that arise with multidimensional screening arewell known and represent the main reason why we left this for future work.

Acknowledgments

da Costa thanks the hospitality of MIT and gratefully acknowledges financial support from CNPq. We thank Luis Braido,Tiago Berriel, Ricardo Cavalcanti, Bev Dahlby, Érica Diniz, Alexandre Sollaci and seminar participants at INSPER, EESP-FGV,UFRJ the 2011 PET Meeting, the 67th IIPF Meeting and the 2012 ESEM Meeting for their invaluable comments. We retain fullresponsibility for all remaining errors.

Appendix A. Mathematical appendix

A.1. Proofs

Lemma 1. For all s40, 0oro1.

Proof. Let u0ðwÞ ¼ uðc0ðwÞÞ and use the fact that utility differences are the same for all w to see that

ξ0ðu0ðwÞ�sÞξ0ðu0ðwÞÞ ¼ r wð Þ;

where ξ0ðuÞ is the marginal cost, measured in consumption units, of delivering utility u and rðwÞ ¼ R0ðyðwÞÞ. Note that ξ is anincreasing convex function of u which means that 0oτo1 for uðy0Þ�u4s40, where u ¼ limc-0uðcÞ. □

Proof of Proposition 1. Let Rð�Þ be a smooth retention function and y(w) the induced efficient labor income schedule.Assume that the associated marginal retention function rð�Þ is rðwÞ40 8w. Differentiating (6) and rearranging terms yield

u0ðy0ðwÞÞu0ðRðy0ðwÞÞÞ ¼ R0 y0 wð Þ� �

: ð19Þ

Differentiate (19) to obtain

R″ðy0ðwÞÞy0ðwÞR0ðy0ðwÞÞ � Ar y0 wð Þ� ��Ar R y0 wð Þ� �� R0ðy0ðwÞÞ

Rðy0ðwÞÞy0 wð Þ�

; ð20Þ

where ArðyÞ is the coefficient of relative risk aversion at income level y. For the case of CRRA preferences, ArðcÞ ¼ ρ for all c,and expression (20) reduces to

d ln R0ðyÞd ln y

y ¼ y0ðwÞ

¼ ρR0ðy0ðwÞÞRðy0ðwÞÞy0 wð Þ�1

� :

Recall that rðwÞ ¼ R0ðyðwÞÞ and note that

r0ðwÞrðwÞw¼ R″ðyðwÞÞ

R0ðyðwÞÞy wð Þy0ðwÞyðwÞw


or,

d ln rðwÞd ln w

¼ frðwÞ1�1=ρ�1gρη: ð21Þ

where we have used (7) and η¼ γ=ðγþρ�1Þ. Next, note that the equal sacrifice schedule induces a taxable income profile,yð�Þ, that is increasing in w. The first order conditions used by Werning (2007) to derive his results are therefore alsosufficient for the allocation to be implementable. Define21

Φ wð Þ � �d ln f ðwÞd ln w


d ln w�1: ð22Þ

Assume that ΦðwÞ4d ln rðwÞ=d ln w, then from (33) we have that a retention function R:Rþ-Rþ is efficient if and only if22

r wð ÞZ ΦðwÞ�γ

Φ wð Þ�d ln rðwÞd ln w

: ð23Þ

To save on notation, let αðwÞ ¼ �d ln f ðwÞ=d ln w. Then, for an equal sacrifice retention function associated with separableiso-elastic preferences we have

Φ wð Þ ¼ γ�1� � γ

γþρ�1þα wð Þ�1: ð24Þ

Substituting (24) into (23), and using (21) we get

r wð Þ γþα wð Þ�1� ��rðwÞ2�1=ρ ργ

γþρ�1Zα wð Þ�γ�1þρð1þγÞ

γþρ�1; ð25Þ

for all w.Condition (23) not always has a bite. We have assumed that ΦðwÞ4d ln rðwÞ=d ln w, which guarantees that the

denominator in (23) is positive. If, however ΦðwÞoγ, all one is requiring is for the marginal retention rate to be positive,or, equivalently, for marginal tax rates not to exceed 100%. On the other hand, if ΦðwÞ4d ln rðwÞ=d ln w, and d ln rðwÞ=d ln w4γ, then only with marginal retention rates that exceed 100% will the schedule be efficient. □

A.2. Marginal social welfare weights

Next, we show how the procedure devised in Werning (2007) can be used to extract the marginal social welfare weights,in the sense of Saez and Stantcheva (2013). We start by providing a sketch of the necessity of condition (23). A completeproof of both necessity and sufficiency is found in Werning (2007).

Deriving condition (23): Following Werning (2007), if a tax schedule is efficient the underlying allocation must solve thefollowing program:

maxyð�Þ;vð�Þ

ZW½yðwÞ�eðvðwÞ; yðwÞ;wÞ�f ðwÞ dw

s.t.,

v0 wð Þ ¼ yðwÞγwγþ1 ; ð26Þ

yðwÞ increasing; ð27Þand

vðwÞZvðwÞ 8w; ð28Þwhere eðvðwÞ; yðwÞ;wÞ is implicitly defined through

v wð Þ ¼ u e v wð Þ; y wð Þ;wð Þð Þ�yðwÞγγwγ

Disregarding the monotonicity constraint (26), we may write the LagrangianZW

y wð Þ�e v wð Þ; y wð Þ;wð Þ½ �f wð Þþμ wð Þ v0 wð Þ�yðwÞγwγþ1

� �þζ wð Þ v wð Þ�v wð Þ½ �

� dw;

21 The term d ln y=d ln w that appears in Eq. (22) is the cross-sectional elasticity of taxable income, i.e. the percentage change in taxable income whenwecompare individuals whose productivities differ by 1% for a given tax structure. This makes the application of (23) quite simple under the separable iso-elastic specification for preferences, since d ln y=d ln w¼ γ=ðγþρ�1Þ for all levels of sacrifice. The elasticity of taxable income proper, in contrast, is notinvariant to the level of sacrifice. See Fig. 1.

22 This is Werning's (2007) Proposition 4 adapted to our setting.


where

v wð Þ ¼ u e v wð Þ; y wð Þ;wð Þð Þ�yðwÞγγwγ :

Integrating by parts,ZW

y wð Þ�e v wð Þ; y wð Þ;wð Þ½ �f wð Þ�μ0 wð Þv wð Þ�μ wð ÞyðwÞγwγþ1þζðwÞ½vðwÞ�vðwÞ� dw

� þμ wð Þv wð Þ�μ w

� �v w� �

First order conditions are

1�ey v wð Þ; y wð Þ;wð Þ� �f wð Þ ¼ μ wð ÞγyðwÞγ�1

wγþ1 ; ð29Þ

�evðvðwÞ; yðwÞ;wÞf ðwÞ ¼ μ0ðwÞ�ζðwÞ; ð30Þand μðwÞr0, limw-1μðwÞ ¼ 0 (μðwÞZ0, if F has a bounded support).

This implies

�evðvðwÞ; yðwÞ;wÞf ðwÞrμ0ðwÞ: ð31ÞTo save on notation, let evðwÞ � evðvðwÞ; yðwÞ;wÞ, and eyðwÞ � eyðvðwÞ; yðwÞ;wÞ. Then, assuming 14eyðwÞ ¼ rðwÞ40, (29) canbe written in logs:

lnð1�rðwÞÞþ ln f ðwÞ ¼ ln μþ ln γþðγ�1Þln yðwÞ�ðγþ1Þln w;

which implies

d lnð1�rðwÞÞd ln w

þd ln f ðwÞd ln w

¼ d ln μðwÞd ln w


d ln w� γþ1� �

: ð32Þ

Next note that

d ln μðwÞd ln w

¼ μ0ðwÞμðwÞwZ�evðwÞf ðwÞ

μðwÞ w¼ �γyðwÞγ�1evðwÞf ðwÞw1�eyðwÞ� �

f ðwÞwγþ1

� γ1�rðwÞev wð ÞyðwÞγ�1

wγ ¼ �γrðwÞ

1�rðwÞ:

Hence, using (32), we get

�d ln rðwÞd ln w

rðwÞ1�rðwÞþ

d ln f ðwÞd ln w

Z�γrðwÞ

1�rðwÞþ γ�1� �d ln yðwÞ

d ln w� γþ1� �

:

We may, then, write the efficiency condition for r(w) as

r wð Þ Φ wð Þ�d ln rðwÞd ln w

� �ZΦ wð Þ�γ; ð33Þ

where ΦðwÞ is as defined in (22). Assuming ΦðwÞ4d ln rðwÞ=d ln w, Eq. (23) is obtained.23

Proof of Proposition 2. Using the same definitions applied throughout the paper, the problem defined in Section 5 —

henceforth, primal program — may be written as

maxyð�Þ;vð�Þ

ZWΨ ðvðwÞÞf ðwÞ dw ð34Þ

s.t.,

v0 wð Þ ¼ h0yðwÞw

� �1w; ð35Þ

y(w) increasing, andZW½yðwÞ�eðvðwÞ; yðwÞ;wÞ�f ðwÞ dwZG: ð36Þ

The associated Lagrangian isZWfΨ v wð Þð Þþλ y wð Þ�e v wð Þ; y wð Þ;wð Þ�G½ �gf wð Þ dw

23 d ln rðwÞ=d ln w is positive (resp., negative) if marginal tax rates are increasing (resp., decreasing). If d ln rðwÞ=d ln w40, thenΦðwÞ40 suffices. Notealso that we have assumed 0orðwÞo1, which is always the case in our model. A reversed inequality results when rðwÞ41 using a similar derivation whichdoes not rely on taking logs.


�ZW

μ 0 wð Þv wð Þþμ wð Þh0 yðwÞw

� �1w

� dw

with transversality conditions, μðwÞr0 and limw-1μðwÞ ¼ 0.The first order conditions are

fΨ 0ðvðwÞÞ�λevðvðwÞ; yðwÞ;wÞgf ðwÞ ¼ μ 0ðwÞ; ð37Þand

λ 1�ey v wð Þ; y wð Þ;wð Þ� �f wð Þ ¼ μ wð Þh″ yðwÞ

w

� �yðwÞw2 : ð38Þ

Let ðvðwÞ; yðwÞÞ denote the allocation that solves the primal program above. Comparing (30) with (37), and (29) with (38)it is apparent that ðvðwÞ; yðwÞÞ solves (30) and (29) for

ζ wð Þ ¼Ψ 0ðvðwÞÞf ðwÞλ

; andμðwÞλ

¼ μ wð Þ:

Integrating (37) and applying the transversality conditions,ZWfΨ 0ðvðwÞÞ�λevðvðwÞ; yðwÞ;wÞgf ðwÞ dw¼

ZWμ 0ðwÞ dw¼ �μðwÞ;

lead to

λ¼RWΨ

0ðvðwÞÞf ðwÞ dwþμðwÞRWevðvðwÞ; yðwÞ;wÞf ðwÞ dw: ð39Þ

Next, note that

λ½1�eyðv w� �

; y w� �

;w�f w� �¼ μ w

� �h″

yðwÞw

� �yðwÞw2 ;

which allows us to write

λZWevðvðwÞ; yðwÞ;wÞf ðwÞ dw�

ZWΨ 0ðvðwÞÞf ðwÞ dw¼ μðwÞ

Adopting the normalizationRWΨ

0ðvðwÞÞf ðwÞ dw¼ 1, and after some straightforward algebra, we may re-write (39) as

λ¼ZWev v wð Þ; y wð Þ;wð Þf wð Þ dw� 1�rðwÞ

ðγ�1ÞrðwÞf w� �

ev v w� �

; y w� �

;w� �� 1

;

where we have used

h″ l w� �� lðwÞ

w¼ h″ðlðwÞÞlðwÞ

h0ðlðwÞÞh0ðlðwÞÞ

wevðvðwÞ; yðwÞ;wÞevðvðwÞ; yðwÞ;wÞ

¼ ðγ�1ÞrðwÞevðvðwÞ; yðwÞ;wÞ:

When μðwÞ ¼ 0, rðwÞ ¼ 1 and λ¼ fRWevðvðwÞ; yðwÞ;wÞf ðwÞ dwg�1. For our purposes, however, equal sacrifice will alwayslead to μðwÞ ¼ μðwÞ=λ40. In any case we may recover, from the data, using

ζ wð Þ ¼Ψ 0ðvðwÞÞf ðwÞλ

; ð40Þ

provided that we know μ0ðwÞ.The only issue is how to find an expression for μ0ðwÞ. For this we may use (29) which we re-write simply as

½1�rðwÞ�f ðwÞ ¼ μðwÞγyðwÞγ�1w�γ�1: ð41Þ

Differentiating (41) leads to

1�r wð Þ½ �f 0 wð Þ�r0 wð Þf wð Þ ¼ μ0 wð ÞγyðwÞγ�1

wγþ1 �μ wð Þγ γþ1� �yðwÞγ�1

wγþ2

þμ wð Þγ γ�1� �yðwÞγ�2

wγþ1 y0 wð Þ;


which may also be written as

Υ wð Þ 1�r wð Þ½ �f wð Þ ¼ μ0 wð ÞγyðwÞγ�1

wγ ;

where

Υ wð Þ � f 0ðwÞwf ðwÞ � r0ðwÞw

½1�rðwÞ�� γ�1� �y0ðwÞw

yðwÞ � γþ1� ��

:

We, then, obtain the following expression for μ0ðwÞ:

μ0 wð Þ ¼ Υ wð Þ1�rðwÞγrðwÞ ev v wð Þ; y wð Þ;wð Þf wð Þ;

which gives us

ζ wð Þ ¼ Υ wð Þ1�rðwÞγrðwÞ þ1

� ev v wð Þ; y wð Þ;wð Þf wð Þ:

Finally,

Ψ 0 v wð Þð Þ ¼ λ Υ wð Þ1�rðwÞγrðwÞ þ1

� ev v wð Þ; y wð Þ;wð Þ: □

Appendix B. Supplementary materials

Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.euroecorev.2014.06.008.

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On the efficiency of equal sacrifice income tax schedules

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