Robust bounds on optimal tax progressivity
Anmol Bhandaria

Jaroslav Borovičkab

Yuki Yaoca University of Minnesota and NBERb New York University and NBER
c University of Kent

November 5, 2023
Preliminary and incomplete.

Abstract
We study the problem of a robust planner who designs an optimal taxation scheme fora heterogeneous population in presence of uncertainty about the shape of the distribution of underlying types. Low-income workers are well insured under the optimal
scheme, and so concerns about the left tail of the type distribution are negligible. On
the other hand, the planner fears misspecification of the right tail of the type distribution emerging from budgetary concerns. Even when the tail of the distribution is
Pareto, arbitrarily small misspecification concerns lead to zero marginal taxes at the
top. A quantitatively calibrated model shows that a plausible degree of uncertainty
leads to an optimal tax scheme with substantially reduced marginal tax rates for highincome earners and a peak marginal tax rate much lower than in the model without
uncertainty.

Bhandari acknowledges support from the Heller–Hurwicz Economic Institute.

1Introduction

It is well-known from the theory of optimal taxation (Mirrlees (1971)) that the shape of themarginal tax curve crucially depends on the shape of the underlying labor productivities
and labor supply preferences.1 Despite increasingly available surveys and administrative
micro-data on income, determining the joint distribution of skills and preferences that result in a specific distribution of reported income remains a complex task. The problem ismore severe in the tails, where sample sizes are extremely small and the data suffer from avariety of measurement issues.2
In this paper, we study optimal taxation of labor income when the government acknowledges uncertainty about the underlying distribution of skills and preferences. We
find that these concerns generally lead to lower progressivity. More strikingly, we findthat the celebrated Diamond (1998) and Saez (2001)’s high top marginal tax rate result isoverturned. By introducing a minimal degree of uncertainty to the Diamond–Saez setting,we prove that the top marginal tax approaches zero at a rate that is bounded away fromzero and independent of structural parameters.
Methodologically, we build on Hansen and Sargent (2008)’s notion of “robustness”
which aims to construct policies that work well not just for a single distribution but acrossa set of distributions. The concerns about the shape of the type distribution manifest as amin-max game between a government that chooses a nonlinear tax schedule to maximize
a given welfare criterion and its alter ego that adversely perturbs the joint distribution ofskills and preferences away from a given benchmark distribution subject to a penalty. Motivated by measurement concerns, we use a penalty that scales in a statistical measure of
distance between the two distributions, i.e., relative entropy. The min-max problem allows
the government to design a tax schedule that is robust with respect to a set of distributions
that are hard to distinguish from each other using an available finite data sample.
While our work has elements in common with public finance and Hansen–Sargent robustness literature, there are several distinct differences. Similar to the classic Mirrlees
problem, we examine a static environment where individuals vary in skills and preferences, and they supply labor given a non-linear tax schedule. The government devises
this schedule to maximize a social welfare criterion while adhering to a budget constraint.
Our approach differs in the explicit modeling of the uncertainty the government faces
regarding the type-distribution. Instead of assuming certainty as in the traditional Mir1 See, for example, Golosov, Troshkin, and Tsyvinski (2016).
2 For instance, even if earnings could be measured well, say in tax records, in order to map earnings toskills one needs to take a stand on income that is not reported, as well as what part of the reported income
is unrelated to labor supply. The problem gets even more difficult if we want to jointly estimate productivity
and labor supply elasticities for such individuals.

1rlees framework, our government uses the min-max formulation discussed previously.3 If
we set the penalty in the min-max problem to be large enough, our problem converges
to the standard Mirrlees problem in which the planner perfectly knows the shape of thetype distribution—a feature that gives us a convenient point of departure. We label the
problem involving an infinite penalty as the “baseline problem”, and we refer to the type
distribution in this baseline as the “benchmark” distribution. Similarly, we use terms like“robust problem” and “worst-case distribution” to denote the planner’s problem with afinite penalty and the adverse distribution chosen by the minimizing agent, respectively.
Compared to the robustness or more generally the literature on ambiguity4 , we focus
on a different type of uncertainty. While the existing literature primarily deals with the distribution of alternative potential outcomes or states of nature a decision-maker might faceunder uncertainty, our government is concerned about the uncertainty about the shapeof the cross-sectional distribution. Households who have private information about theirtypes of skills and preference face no uncertainty. Moreover, our formulation expands
upon the typically used one-parameter penalty specification, allowing us to express varying degrees of uncertainty in different segments of the underlying distribution. For example, we can formally articulate when the government is more uncertain about the extreme
ends (“tails”) of the distribution than the central part.
Our formalization allows us to examine the optimal nonlinear tax schedule using standard mechanism design techniques and use the revelation principle. This enables us toequivalently recast the (inner) problem of choosing a nonlinear tax schedule equivalently
as selecting an optimal allocation, subject to truth-telling constraints. To compare our results with existing literature, we begin with a scenario where the type-distribution is onedimensional (skills), and the penalty is scalar. We then broaden our scope to more complex
situations. For this simplified setting, Diamond–Saez show that in the baseline problem,the marginal tax rate for a particular skill level is determined by the so-called ABC formula. This formula consists of three components: A) a term dependent on labor supply
elasticities, B) a term tied to the hazard rate at a given skill level, and C) a term that relates
to the shape of the distribution above that skill level. These three factors strike a balance
between the efficiency costs of labor supply distortion caused by the marginal tax imposed
at a given skill level, and the benefits derived from redistributing additional income collected from workers above that skill level in a non-distortionary fashion.
3 There are a few papers that bring statistical concerns to an optimal taxation problem. See Lockwood, Sial,
and Weinzierl (2021) and Chang and Wu (2023). Our work differs from these studies in modeling statisticalconcerns over an infinite dimensional object—the entire type distribution—using a non-Bayesian robust control approach, while they model uncertainty over a finite set of parameters that are handled with Bayesian
techniques. Those papers found that parametric uncertainty in their environment leads to a more progressive
tax system while we find the optimal tax schedule to be less progressive relative to the baseline.
4 See, for example, Adam and Marcet (2011), Hansen and Sargent (2012), Karantounias (2013), Bhandari,
Borovička, and Ho (2022), and Hansen and Miao (2022).

2

We illustrate that in our context, the marginal tax rate adheres to a modified Diamond–Saez ABC formula. The difference lies in the replacement of the hazard rates and distributions with the worst-case distribution, which is determined as part of the min-max problem. This ex-post Bayesian representation helps us analyze the differences in the optimal
tax schedule with and without concerns for misspecification of the type-distributions bybreaking it down into two logical components: 1) what influences the shape of the worstcase distribution, and 2) how the tax schedule is pinned down by the Diamond–Saez ABC
formula given the shape of the worst-case distribution. Since the worst-case distribution
is an endogenous object that relies on the shape of the tax function, the solution to therobust optimal tax problem is a fixed-point that simultaneously characterizes the optimal
tax schedule and the worst-case distribution.
Our first theoretical result is the analysis of the top tax rate in the case where the benchmark distribution has a Pareto tail. In the baseline, when the penalty is infinite and theplanner is therefore certain that the benchmark distribution is the correct type distribution,
the Diamond–Saez ABC formula shows that the top tax rate approaches a positive finite
value that depends on the labor supply elasticity and the Pareto tail parameter. This tax
rate is quantitatively quite large—around 70–75% for reasonable elasticity and tail parameters. Intuitively, the thick tail of the productivity distribution means that the government
can always collect sufficient revenues from the right of any given threshold productivity
to offset the cost of distortions due to an increase in the marginal tax rate at that threshold.
For similar reasons, for distributions with bounded support or, more generally, with thintails, the top tax rate approaches zero.
Now consider the problem faced by a government which is concerned that the underlying bechmark Pareto distribution is misspecified. We first show that the ratio of densitiesof the worst-case distribution and the benchmark Pareto distribution has an exponential
tilting expression familiar from the robustness literature. The worst-case density shifts
mass away from worker types who are valuable to the government but the magnitude ofthis reweighting is disciplined by the entropy penalization of the statistical discrepancy
between the benchmark and the worst-case distributions. In our context, an individual’svalue to the government is determined by two components: first, their utilitarian contribution to the welfare objective function, which depends on the individiual’s indirect utility under the optimal allocation, and second, their contribution to easing the government
budget constraint, which hinges on the net tax revenue the government collects from that
individual weighted by the marginal social value of a unit of consumption.
Moving mass away from the right tail of the productivity distribution is costly to theplanner and hence desired by the adverse player in the min-max problem because eachhigh-productivity individual generates substantial tax revenue that a redistributive gov-

3

ernment can use to provide transfers to low-skilled individuals. While tax revenues raised
from a particular productivity type rely on the product of productivity level and the massof agents at that productivity level and decrease as we move further into the right tail, theentropy cost of shifting mass from the right tail diminishes much more rapidly because it
scales only with the density. Consequently, in the minimization part of the min-max problem, gains from skimming the density in the right tail grow as we move further into thetail, regardless of the penalty value. We demonstrate that the optimal top tax rate gradually reduces to zero, with the worst-case distribution approaching a distribution with an
exponentially decaying density, or a thin tail, in spite of the benchmark distribution beinga Pareto that has a polynomial decay in the tail, or, more generally, any other fat-tailed distribution. Thus, the mechanism formalizes the practical concern of policymakers to deal
with welfare-relevant aspects of tail behavior that are very hard to detect.
The asymptotic top tax rate is necessarily a statement about limits, and may not be
relevant if marginal tax rates approach the zero limit very slowly. To investigate that, westudy the elasticity of the marginal tax rate with respect to income. We show that in thelimit as productivity becomes large, the worst-case distortion is dominated by planner’s
concerns about the amount of tax revenue raised from those types. The slower the decay inthe marginal tax rate, the higher the revenue collected from the very productive individuals. But higher revenue means these individuals are more valuable to the government, andthe adverse agent in the min-max problem has stronger incentives to shift the worst-casedensity away from them. As their mass becomes smaller, the redistributive gains fromtaxing high-productivity individuals decline, and the marginal tax rate dictated by the
Diamond–Saez formula falls faster. We show that this fixed point is resolved in a uniquevalue such that the elasticity of the marginal tax rate with respect to income equals minus
one half. Remarkably, this asymptotic decay rate of the marginal tax rate is independentof any primitive parameters, such as preferences of the household or the government or
those characterizing the distribution of skills.
To examine the complete tax schedule rather than just the top tax rate, we resort to anumerical solution. The benchmark distribution is calibrated as an exponentially modified
Gaussian (EMG) distribution for the logarithm of productivity. We rely on the parameters
reported by Heathcote and Tsujiyama (2021), who estimated them using data from theSurvey of Consumer Finances. Although the top tax rate is not influenced by the specificvalue of the penalty parameter that controls the degree of misspecification concerns about
the shape of the productivity distribution, the overall shape of the tax function is. As
suggested in the literature (see Anderson et al. (2003)), we calibrate the penalty parameterto target a specific level of detection error probability. This probability measures the failure
rate of a likelihood ratio test in distinguishing two alternative distributions (in our case,the worst-case and benchmark distributions), given a specific sample size of draws from4the distribution. The detection error probability ranges from zero, when the distributions
are easily distinguishable, to one half, when it is impossible to differentiate between the
distributions. We use the sample size from the Survey of Consumer Finances and target a
detection error probability of 10% to calibrate our penalty parameter, a number frequently
used in the literature.
We evaluate our findings against the baseline scenario, in which the penalty parameteris infinite. We observe significant effects of the presence of misspecification concerns on
the marginal tax rate. For instance, households earning more than $500,000 face a baseline tax rate exceeding 70% in the absence of misspecification concerns, but under our
preferred calibration, the optimal tax rate peaks at 55% for earnings around $250,000. Thisrate falls to 40% for households with incomes of $5 million, ultimately diminishing to zero,
as suggested by our theory. These lower tax rates result in an approximately 2% increase in
output but also lead to about a 7% reduction in transfers to the lowest-income households.
When targeting lower detection error probabilities, the maximum marginal tax rates candrop as low as 30%, leading to a 10% surge in aggregate output and a substantial 50% cut
in transfers.
Finally, we consider a case in which the government is uncertain not just about the
distribution of labor productivity but the joint distribution of labor productivity and labor
supply elasticities. The theoretical tools to analyze taxation with multidimensional types isin a nascent stage, and we therefore only study this case numerically in a Ramsey fashion
via a splines-based approximation of the nonlinear tax function.5 Similar dynamics areat play in this scenario, and the worst-case distribution not only slims down the right tailin the skill dimension but also endogenously strengthens the correlation between labor
supply elasticity and labor productivity. As observed earlier, mass is shifted away fromtypes who contribute significantly to tax revenues. Consequently, the optimal tax rates thatemerge are lower than those in the baseline model, which disregards uncertainty abouttypes.
The rest of the paper is structured as follows. Section 2 describes the simple modelwith one-dimensional types and scalar penalty. Section 3 contains our main theoreticalresults about the top marginal tax rate and the rate of convergence. Section 4 uses a calibrated economy to study the full tax schedule. Section 5 extends the environment totype-dependent penalty, and Section 6 to multi-dimensional types. Section 7 concludes.
5 See Golosov and Krasikov (2023) for a theoretical approach to a two-dimensional Mirrlees setting applied
to the taxation of couples.

5

2

Model

The economy is populated by a continuum of workers indexed by their productivity z distributed according to density f (z) with a continuous support [z, z̄] ⊆ R+ where z̄ may
be infinite. Productivity types are private information of the worker. A worker with productivity z supplying labor n produces income y = zn. The worker solves the utilitymaximization problem
s.t. c = zn − T (zn)max U (c, n)
c,nwhere U (c, n) is a strictly concave and differentiable utility function representing worker’s
preferences over consumption and hours worked, and T (y) is the tax levied on income
y. Taking the tax function as given, worker’s optimal choice of labor supply yields theconditionUc (c, n) 1 − T ′ (zn) z + Un (c, n) = 0.

(1)

Denote the optimal choice of consumption and labor C (z; T ) and N (z; T ), respectively, theresulting output Y (z; T ) = zN (z; T ), and the associated indirect utility function U (z; T ).
For the rest of the paper, we drop T as an explicit argument of functions C(·), N (·), Y (·),
and U (·).
The government is in charge of choosing the tax schedule T as a function of observed
income y. Taxes are levied for redistribution purposes and to pay for an exogenous government expenditure B. The government welfare objective is given by
E [ψU ] =Z z̄z

ψ (z) U (z) f (z) dz

where ψ (z) is a Negishi (1960) welfare weighting function that satisfies E [ψ] = 1. For
example, ψ (z) = 1 implies a utilitarian planner, while ψ (z) = δz (z) / f (z) where δz (z)is the Dirac delta function yields the Rawlsian welfare criterion. In the absence of model
misspecification concerns, the government chooses the tax schedule so as to maximize thewelfare objective subject to the budget constraint
E [ T (Y )] ≥ B.
We study the optimal taxation problem in a situation when the government is concernedthat the underlying distribution of productivity types f (z) is misspecified. In the spirit ofHansen and Sargent (2001a,b), the government contemplates a set of alternative type distributions fe (z) that are statistically close to the ‘benchmark’ distribution f (z). We denotem (z) = fe (z) / f (z) the likelihood ratio between the benchmark and the alternative dise [·] the expectation operator under the distribution fe (z). By construction,
tribution, and E

6E [m] = 1. For any integrable function X (z), the Radon–Nikodým theorem implies
e [X] =EZ z̄z

X (z) fe (z) dz =Z z̄z

X (z) m (z) f (z) dz = E [mX ] .

The degree of statistical distinguishability of the two distributions f (z) and fe (z) is represented by their relative entropy
E [m log m] =Z z̄zm (z) log m (z) f (z) dz.

The relative entropy is nonnegative, and is equal to zero if an only if m ≡ 1 with probability
one. Alternative distributions fe (z) that are statistically easier to distinguish from f (z)yields a larger relative entropy.
The government desires to choose a tax function that would perform well across the setof alternative type distributions fe (z) that are statistically not too distinct from the benchmark distribution f (z). This leads to the maxmin problem
max min E [mψU ] + θE [m log m]Tm
E[m]=1s.t. E [mT (Y )] ≥ B.

(2)e [ψU ] evaluated
The first term in the objective function is equal to the government welfare E
under the alternative distribution fe (z). The second term is an entropy penalty that penalizes distributions with a large statistical distance from the benchmark distribution. The
degree of penalization is controlled by the parameter θ. The budget constraint dictates thate [ T (Y )] ≥ B.
the tax function must balance the budget under the alternative distribution, E
The government problem (2) leads to an optimal tax function that is robust to misspecifications of the type distribution that adversely affect the government objective. The problem can be interpreted as a two-player game in which the government faces an malevolent
nature that chooses alternative distributions with the most adverse welfare consequences
for the contemplated tax function.
Since fe (z) = m (z) f (z), the likelihood ratio m (z) plays the role of a weighting function that over- or underweighs the alternative distribution relative to the benchmark. The
government problem implies that the malevolent nature exploits both the direct welfare
impact as well as the budgetary consequences of adversely chosen distributions. On the
one hand, it desires to impose a high m (z) for types with low welfare impact ψ (z) U (z)to lower E [mψU ]. On the other hand, it strives for adverse budgetary consequences by
underweighing types for whom T (Y (z)) (net tax payers) is positive while, vice versa,
overweighing those for whom T (Y (z)) is negative (net tax recipients), so that the budget
constraint E [mT (Y )] ≥ B binds more strongly. At the same time, alternative adverse dis7

tributions fe (z) chosen by nature cannot be too distinct from the benchmark so as not toincur a large penalty θE [m log m].
In line with the literature, we call the distribution fe (z) that solves the minimax problem in (2) the worst-case distribution. A larger value of θ implies that the worst-case distribution is statistically closer to the benchmark. As θ → ∞, the entropy penalty becomesprohibitive, model misspecification concerns vanish, and we obtain fe ≡ f .

2.1

Mirrleesian formulation

In order to solve for the optimal tax function, we follow Mirrlees (1971), and characterize
the constrained optimal allocation under the restriction that the tax function can only depend on worker’s income. As usual, we cast the problem as a mechanism design problem,
focusing on incentive-compatible mechanisms in which workers truthfully reveal theirtypes.
The government offers to workers a menu of allocations (c (z) , y (z)) indexed by z.
Worker of type z chooses a reporting strategy ẑ that entitles to consumption c (ẑ) in exchange for providing output y (ẑ) that requires labor input y (ẑ) /z. The reporting strategy
therefore solves the announcement problemy (ẑ)
max U c (ẑ) ,
ẑ
z


.

Incentive-compatibility requires that the optimal report satisfies ẑ = z. The first-ordernecessary condition evaluated at ẑ = z yields

Uc

y (z)
c (z) ,z′c (z) + Uny (z)c (z) ,z

y′ (z)= 0.
z(3)

Totally differentiating the utility function with respect to z at the allocations (c (z) , y (z))
and plugging in the optimal reporting strategy condition derived in (3), we obtainy (z) y (z)dU= −Un c (z) ,
.
dz
z
z2

(4)

This is a condition on the utility gradient the menu (c (z) , y (z)) has to satisfy to be locally incentive-compatible (IC). When this condition holds, the worker has no incentives
to misrepresent the true type by an infinitesimal deviation. We assume that the functionυ (c, n) = −n

Un (c, n)Uc (c, n)(5)

is increasing in n for each fixed c. This implies a single-crossing property under which
8the local incentive-compatibility constraint also implies global incentive compatibility, andallocations (c (z) , y (z)) that satisfy incentive compatibility are also strictly increasing in z.
Since the IC constraint is type-by-type and does not depend on the underlying distribution, the model misspecification concern on the side of the planner does not alter its
form. We can therefore rely on the convenient Hamiltonian formulation of the allocation
problem.
The planner is solving
Z z̄



y (z)
max minψ (z) U c (z) ,
m
c,y
z
z


m (z) f (z) dz + θZ z̄
zm (z) log m (z) f (z) dz(6)subject to the IC constraint (4) and the budget constraint
Z z̄z(y (z) − c (z)) m (z) f (z) dz ≥ B.

(7)

We provide a solution to this problem using a Hamiltonian formulation that yields a modification of the Diamond (1998) and Saez (2001) elasticity formula for the marginal tax rate.
Treating U as the state variable, λ as its co-state, and y and m as control variables, we form
the constrained Hamiltonian
H (U , y, m, λ) = ψ (z) U (z) m (z) f (z) + θm (z) log m (z) f (z) − χm (z) f (z)y (z) y (z)−λ (z) Un c (z) ,+ µ [y (z) − c (z)] m (z) f (z) .
z
z2

(8)

Here, χ and µ are multipliers on the constraints E [m] = 1 and (7), respectively, and c (z) isdefined implicitly from the definition of the utility function as c (z) = C (U (z) , y (z)).
We derive a general characterization of the problem in detail in Appendix A.1. Here we
provide the analysis of a special case, in which we assume that preferences are quasilinearwith isoelastic labor disutility
U (c, n) = u (c) − v (n) = c −n 1+ γ.
1+γ(9)The first-order condition with respect to m (z) in (8) together with the restriction E [m] = 1
yields a characterization of the worst-case distortion in the form of an exponential tilting
formula
exp − 1θ [ψ (z) U (z) + µT (y (z))],
m (z) = R z̄

1
exp
−
ψ
z
U
z
+
µT
y
z
f
ζ
dζ
[
(
)
(
)
(
(
))]
(
)
θ
z

(10)

where T (y (z)) = y (z) − c (z) represents the effective tax the allocation imposes on worker
of type z. The remaining optimality conditions then imply the formula for the marginal

9

tax

e (z) − Fe (z) 1 − Fe (z)T ′ (y (z))Ψ
= (1 + γ )′1 − T (y (z))
1 − Fe (z)z fe (z)(11)

e (z) iswhere Fe (z) is the cumulative distribution function of the worst-case density, and Ψplanner’s cumulative welfare weight

Fe (z) =e (z) =
ΨZ z
zfe (ζ ) dζ =

Z z
z

R z̄
z

Z z
z

m (ζ ) f (ζ ) dζψ (ζ ) fe (ζ )dζ.
ψ (ξ ) fe (ξ ) dξ

The lump sum portion of the tax T (y (z)) imposed on the least productive worker is thendetermined so as the whole tax scheme raises sufficient revenue to satisfy the government
budget constraint
Z z̄
z

T (y (z)) dz = B.

The optimal marginal tax formula is analogous to that of Diamond (1998) and Saez (2001),
except that now, it depends on the endogenously determined distribution fe (z) represented by the distortion m (z) in (10). The first term on the right-hand side of (11) captures
distortionary effects of taxation on the labor supply, indicating that marginal taxes should
be lower when the inverse of the labor supply elasticity γ is low. The second term represents the desire for redistribution, and is bounded above by one. Marginal taxes will bee (z) > Fe (z), indicating a planner that puts higher welfare weights
strictly positive when Ψon lower worker types in the first-order stochastic dominance sense. Finally, the third termis determined by the shape of the tail of the type distribution, and it represents the tradeoffthat an increase in the marginal tax T ′ (y (z)) causes at a particular z. This marginal tax
has an adverse distortionary effect on the labor supply of all workers with type exactly atz, leading to a total output loss z fe (z), while generating the benefit of raising extra revenue
in lump sum fashion from all workers with type above z, whose mass is 1 − Fe (z).
The form of the distortion (10) reveals that the model misspecification concerns of therobust planner have a redistributive and a budgetary component. The numerator of theexpression for m (z) = fe (z) / f (z) in (10) indicates that the robust planner underweighs
worker types who, under the optimal tax policy, receive allocations with high weighted
utility ψ (z) U (z) or those with high net contributions to the planner’s budget, µT (y (z)).
The Lagrange multiplier µ converts the tax revenue to utility units under the government
welfare function.
The planner uses the tax policy to maximize the social welfare function. The worst-case10

distribution putting more weight on types with a low ψ (z) U (z) and less weight on types
with a high ψ (z) U (z) reflects the concern that the chosen tax function achieves lower gove [ψU ] than that measured under the benchmark model, E [ψU ]. Since
ernment welfare E
insurance is not perfect
At the same time, the government needs tax revenue to achieve the desired redistribution and tax collection goal. Underweighing worker types who deliver high tax revenueµT (y (z)) and overweighing those who deliver low tax revenue µT (y (z)) reflects concerns that worker types who contribute substantially to the budget are less abundant thanunder the benchmark model, making it more challenging to achieve the desired redistribution.
The parameter θ controls the entropy penalty in the planner’s problem (2) and hence
the degree of model misspecification concerns. A small value of θ reflects more substantial
concerns, which leads to stronger exponential tilting in (10). As θ → ∞, model misspecification concerns vanish, and the worst-case density fe (z) approaches the benchmark model
density f (z) in the statistical sense expressed by the entropy penalty E [m log m].
Importantly, the tax function T (y (z)) and the worst-case distortion m (z) are determined jointly as an outcome of the minimax problem (2). Given the tax function, theworst-case density delivers the lowest possible penalized objective in (2), and vice versa,
taking the worst-case density as given, the tax function maximizes the planner’s welfare.
The solution is a saddle point in the objective function that constitutes an equilibrium in a
two-person game between the benevolent government and the malevolent nature.

3

Optimal marginal tax rates at the top

In this section, we provide an analytical characterization of the asymptotic behavior of the
tax rate in (11) in the presence of model misspefication concerns.
When the planner is utilitarian with ψ (z) ≡ 1, then, due to the quasilinear form ofpreferences in (9), the motive for redistribution is absent. Equivalent to the case without
misspecification concerns, we obtain that marginal taxes are zero, T ′ (y (z)) = 0. Redistributive concerns are therefore induced by a decreasing welfare weighting function ψ (z).
Here we focus on the case in which there exists a ẑ such that ψ (z) = 0 for all z ≥ ẑ.
The planner hence puts a zero welfare weight on the right tail of the worker distribution.
e (z) = 1 for z ≥ ẑ, and the tax formula becomes
In this case, ΨT ′ (y (z))
1 − Fe (z)
=
1
+
γ
(
)
1 − T ′ (y (z))
z fe (z)11(12)

Since the second term in the tax formula (11) is bounded above by one, the current caseyields the highest possible tax rate in the tail of the type distribution across all alternative
welfare functions. The worst-case distortion for z ≥ ẑ then becomes µ
m (z) = m̄ exp − T (y (z)) ,θ(13)where m̄ is a normalization constant that assures E [m] = 1, corresponding to the reciprocalof the denominator in (10). The misspecification concerns in the right tail of the distribution
therefore do not involve the welfare of high-type agents, and only reflect the concerns
about the budgetary consequences of not having sufficiently many high-type workers who
contribute to the budget.
In Appendix C, we provide the analysis of the more general case that relaxes the assumption of quasilinear preferences in (9) and also treats more general welfare weighting
functions so that the welfare concern term ψ (z) U (z) in (10) is not zero in the right tail.
It turns out that in most cases, the budgetary concern µT (y (z)) dominates, and when it
does not, the additional welfare concern further reinforces our results.

3.1

Zero marginal taxes at the top

In the absence of model misspecification concerns, m (z) ≡ 1, which implies that the worstcase distribution in (12) corresponds to the exogenously specified benchmark model. The
limiting tax rate then depends on the shape of the benchmark distribution. When the
distribution f (z) is sufficiently thick-tailed, then the marginal tax rate determined (12) has
a strictly positive limit. For example, in the case of the Pareto distribution with shapeparameter α, we have (1 − F (z)) / (z f (z)) = α−1 . On the other hand, bounded or thintailed distributions, such as normal or lognormal, imply a zero limit. This has lead towidely differing policy prescriptions about the range of recommended marginal tax rates
the planner should impose on top incomes.6
When model misspecification concerns are present, we characterize the shape of theworst-case density µ
fe (z) = m (z) f (z) = m̄ exp − T (y (z)) f (z) .
θ(14)The distribution fe (z) remains continuous, which implies that limz→z Fe (z) = 0. Whenz is finite, the conclusion about the top marginal tax rate is the same as without model
6 For example, Diamond and Saez (2011) find a mid-range estimate for the top marginal tax of 73%, basedon labor supply elasticity γ−1 = 0.25 and a Pareto distribution of types z with shape parameter α = 1.875(under the given elasticity of labor supply, this translates to a Pareto distribution of incomes y (z) with shapeparameter αy = 1.5).

12

misspecification concerns, and limz→z T ′ (y (z)) = 0. To see this, notice that in order forthe top marginal tax rate in (12) to be different from zero, we need limz→z z fe (z) = 0. In
this case, we can apply L’Hôpital’s rule to obtain
1 − Fe (z)= lim −z→z z fe ( z )
z→z

lim

1
d log fe(z)1 + z dz

= 0,

where the conclusion follows from the fact that limz→z log fe (z) = −∞ for a finite z, whichmeans that the derivative must diverge to infinity as well.
We therefore focus on the more interesting case when z = ∞. It turns out that themarginal tax rate at the top still asymptotically converges to zero.
Assumption 1. There exists a ẑ such that the type density f (z) under the benchmark distribution
is continuously differentiable on [ẑ, z), and z f (z) is strictly decreasing on [ẑ, z), withlim
z→z

d log f (z)< −1,
d log zwith the limit possibly being −∞.
Theorem 3.1. Assume that worker’s preferences are given by the quasilinear form (9), the type
distribution satisfies Assumption 1 with z̄ = ∞, and θ < ∞. Then the marginal tax rate vanishes
to zero at the top:
lim T ′ (y (z)) = 0.

z→∞

We formally prove the theorem in Appendix B.1. The proof requires a technical treatment of the existence of the limit but conditional on its existence, the result is intuitive.
′ (yDenote Trat
rat ( z )) the opitmal marginal tax rate in the model without model misspecifi-

cation concerns, θ = ∞. Then
′ (y
Trat
1
1 − F (z)rat ( z ))= (1 + γ) lim= (1 + γ) lim d log f (z)
< ∞,
z→∞ −
z→∞ 1 − T ′ ( yrat ( z ))z→∞ z f ( z )
−1
rat

lim

d log zwhere the second equality follows from an application of L’Hôpital’s theorem, and the′ (yfinal inequality is implied by Assumption 1. This yields limz→∞ Trat
rat ( z )) < 1.

Further, the single-crossing property (5) implies that the optimal incentive compatible
scheme implies that output y (z) is strictly increasing in worker’s type z, and since themarginal tax is strictly positive, m (z) in (14) is strictly decreasing. This implies that, for
any z ≥ ẑ,1 − Fe (z)1 − F (z)<z f (z)z fe (z)′ (yand hence also limz→∞ T ′ (y (z)) ≤ limz→∞ Trat
rat ( z )).

13

The limiting tax rate under model misspecification cannot, however, be positive. If it
were converging to τ∞ > 0, then the output function y (z) implied by the optimal labor
choice (3) for the case of quasilinear preferences (9) would asymptotically behave as
1 1+ γ 1 1+ γ
y (z) = 1 − T ′ (y (z)) γ z γ ≈ τ∞γ z γ ,(15)and the worst-case distortion as


 µ

1 1+ γµγ
γ
m (z) = m̄ exp − T (y (z)) ≈ m̄ exp − (1 − τ∞ ) τ∞ z
.
θ
θ

(16)

In that case, an application of L’Hôpital’s rule to the tax formula (12) yields
1 − Fe (z)1+γT ′ (y (z))
=
lim
1
+
γ
= lim µ= 0, (17)()
d log f (z)
d
z→∞
z
→
∞
z→∞ 1 − T ′ ( y ( z ))ez f (z)z T (y (z)) −
−1
lim

θ

dz

d log zbecause the first term in the denominator diverges to ∞. This is a contradiction to theassumption that limz→∞ T ′ (y (z)) = τ∞ > 0, and hence the tax rate has to covnerge tozero.
The striking result is that the marginal tax rate at the top converges to zero irrespective of the degree of model misspecification concerns. Since we abstracted from welfare
concerns at the top of the income distribution, the planner only cares about the budgetaryconsequences associated with taxing top incomes. From this perspective, the robust planner is concerned that there are fewer high-productivity workers that can be taxed.
The form of the distortion in (13) indicates that the concerns grow proportionally withthe marginal social value of the tax revenue µT (y (z)) that the worker with a given productivity z contributes to the budget. The distortion m (z) therefore becomes more severe
as z increases, effectively generating a thinner tail in the worst-case distribution fe (z) in(14). This consequently implies a lower and vanishing optimal marginal tax as z → ∞,since the tradeoff of an increase in the marginal tax T ′ (y (z)) at z that compares the extra
benefit of taxing workers above z with the cost of distorting labor supply of workers at z
becomes less favorable under the worst-case distribution.
While we have shown the vanishing marginal tax rate result for the case of quasilinear
utility and no planner’s welfare concerns in the right tail of the type distribution, these results carry over to more general cases. In particular, we show in Appendix C that the result
also when utility from consumption is concave, and in the presence of welfare concerns at
the top of the type distribution when Ψ (z) only asymptotically converges to one.
The intuition for these generalizations is straightforward. These generalizations yield
the more general form of the distortion of the type distribution represented by expression

14

(10). The case of concave utility leads to a different marginal social value of public funds µ
and to a different optimal tax function trajectory T (y (z)) but since µ remains strictly positive, we only need to show that the tax revenue T (y (z)) under the optimal tax continues
to diverge to ∞ as z → ∞.
Adding welfare concerns corresponds to a nonzero ψ (z) function as z → ∞. Since U (z)is increasing in z, this can only lead to a more strongly decreasing distortion m (z) since
the budgetary concerns and welfare concerns of not having sufficiently many workers
with high types who contribute substantially both to the budget as well as to the welfare
objective reinforce each other.

3.2

Rate of decay of the tax rate

Theorem 3.1 shows that the marginal tax rates at the top vanish to zero irrespective of theunderlying type distribution. However, the theorem does not determine the rate of convergence. In quantitative applications, the rate of convergence matters because it sharpens
information about the practical importance of the asymptotic behavior for finite values of
productivity z.
In this subsection, we derive this rate of convergence. We first state it in the form of
a theorem, and then provide a derivation of the result that leads to a specific differential
equation that will be helpful in numerical implementation of the full characterization of
the optimal tax scheme.
In order to simplify the formula, strengthen Assumption 1 and assume that the tail ofthe benchmark density f (z) is given by a Pareto distribution with shape parameter α. Thisis the prototypical choice that yields strictly positive marginal taxes in absence of model
misspecification.
We characterize the following result directly in income space, treating y = y (z) as theendogenous income of each worker. Previous results imply that limz→∞ y (z) = ∞, so that
limits z → ∞ and y → ∞ are equivalent.
Theorem 3.2. Assume that worker’s preferences are given by the quasilinear form (9), underlying
productivity has a right tail that is Pareto distributed with shape parameter α, and θ < ∞. Thenthe limiting tax rate satisfies2µ ′T (y) y = γ.
y→∞ θ
lim

(18)

In particular, the limiting elasticity of the marginal tax with respect to income, if the limit exists, isequal to1
d log T ′ (y)=− .
y→∞d log y
2
lim

(19)

The proof of the theorem is provided in Appendix B.3. Expression (18) depends on
15

one endogenous object, the marginal social value of wealth, which needs to be determined
separately. The expression implies that the marginal tax rate has to decay to zero at rate
1
log T ′ (y) ≈ − log y,2which implies that the elasticity of the marginal tax rate with respect to income has a limit,
this limit has to be equal to − 21 .
In Section 3.4 and Appendix C, we again show how this result generalizes when werelax the assumptions of the theorem. Broadly speaking, the elasticity expression (19) remains robust but the rate of decay of the marginal tax rate to zero may be even faster
when, for example, planner’s welfare concerns for the top income earners are sufficiently
strong, or when the type distribution under the benchmark density is already sufficiently
thin-tailed to begin with.
Remarkably, the elasticity of the marginal tax rate with respect to income (19) does notdepend on any of the parameters of the model. In order to understand the result, assume
that for incomes y ≥ ŷ, the marginal tax rate T ′ (y) is sufficiently well approximated by aconstant elasticity function with elasticity ν. Taking a ȳ ≫ ŷ, we can write
T (ȳ) = T (ŷ) +

Z ȳ
ŷ

T ′ (y) dy ≈ T (ŷ) +

Z ȳ
ŷ

(1 + ν) yν dy = T (ŷ) − ŷ1+γ + ȳ1+ν .

This means that asymptotically, for high-productivity types z̄ with income ȳ = y (z̄), theworst-case distortion behaves as
 µ

 µ

e exp − ȳ1+νm (z̄) = m̄ exp − T (ȳ) ≈ m
θ
θ

(20)

e absorbs the contribution of terms T (ŷ) − ŷ1+γ . At the same time, the output funcwhere m
1+ γtion y (z) in (15) implies that as T ′ (y) → 0, we can approximate y (z) ≈ z γ . Applying
L’Hôpital’s rule as in (17), we obtain
T ′ (y (z))
1
= (1 + γ) lim µz→∞ 1 − T ′ ( y ( z ))z→∞ zT ′ y z y′ z − d log f (z) − 1( ( )) ( )
lim

θ

(21)

d log zThe denominator on the right-hand side is then dominated by the the first term,µ ′µ1 + γ 1+γ γ (ν+1)
zT (y (z)) y′ (z) ≈ (1 + ν)
z
.
θ
θ
γ
Comparing the elasticities of the left-hand and right-hand side of (21) with respect to the16type z yields
1+γ
1+γ
ν=−
( ν + 1) ,γ
γ
from which we obtain ν = − 21 .
Intuitively, expression (21) indicates that the optimal rate of decay of the tax rate balances two forces, the effect on the tax revenue T (y (z)) collected from high worker typesz, and the effect this tax revenue has on the worst-case distortion m (z). If the decay rate
was higher (a more negative ν), then the tax revenue T (y (z)) would grow more slowly asz → ∞. This would consequently diminish the budgetary concerns of the model misspecification in (20), the worst-case density fe (z) would be less distorted with a thicker tail, andthe optimal tax formula would indicate a more gradual decay of the tax rate. The chain ofarguments is reversed if the decay rate was lower (a less negative ν).
The elasticity choice ν = − 12 thus uniquely identifies the asymptote of the saddle point
between the maximization problem that seeks the optimal tax rate, and the minimization
problem that finds the model misspecification with the most adverse consequences for theplanner.

3.3

Phase diagram

We formalize the proof of Theorem 3.2 by deriving a differential equation for the marginal
tax rate T ′ (y). After differentiating the tax formula (12) with respect to the productivity z
and a sequence of algebraic manipulations, we obtain the differential equation −1 
2
1+γ+α ′
T ′′ (y) yµ ′1+γ+α ′
= − 2−−T (y)T (y) y − γ + γ
T (y) . (22)
1 − T ′ (y)
1+γ
θ
1+γ
This equation depends on z only implicitly through y (z), and we therefore can treat theequation directly as a function of income y. This is a first-order differential equation for
the unknown marginal tax T ′ (y), and the unique strictly positive solution is pinned downby the terminal condition limy→∞ T ′ (y) = 0. More detail concerning the analysis of thisdifferential equation is provided in Appendix B.2.
The left-hand side of the differential equation is the elasticity of the take-home ratewith respect to income−

T ′′ (y) yd log (1 − T ′ (y)).
=
1 − T ′ (y)d log yThis elasticity must converge to zero as y → ∞. In order for the right-hand side of (22)to converge to zero, the last bracket has to converge to zero. Since limy→∞ T ′ (y) = 0, we
obtain equation (18) in Theorem 3.2.

17

1

T ′ (y)

0.5

+

−

0

+

−0.5

T ′ (y) solution
other paths satisfying ODE (23)isoclines h(y, T ′ (y)) = 0

−1
100

102

101

103

y
Figure 1: Phase diagram for differential equation (23). The dashed solid lines correspond to isoclines h(y, T ′ (y)) = 0, and the sign of h(y, T ′ (y)) in between these lines is depicted in the boxes.
The magenta line with circles corresponds to the unique strictly positive solution that satisfies theterminal condition. Green dashed lines are other trajectories that satisfy (23). The parameters areµ = 1, θ = 1, α = 1.5, γ = 2.

Let us denote

 

1−τ
1 + γ + α −1 µ 2
1+γ+αh (y, τ ) =2−ττ y−γ+γ
τ .
y
1+γ
θ
1+γ
Then the differential equation (22) can be rewritten as
T ′′ (y) = h y, T ′ (y) .

(23)

In Figure 1, we plot the phase diagram for this differential equation. The differential equakljklrwjerlwjer wlejr
tion exogenously fixes the marginal social value of resources µ, which must be determined
separately and jointly with the lump-sum tax on the lowest worker type T (y (z)) so thatthe planner’s budget constraint holds. We reiterate that this characterization holds for theright tail of the type distribution that has a Pareto density and for which the planner has nowelfare concerns. This solution for the right tail can then be combined with that for the restof the type distribution that possibly has a different shape and for which welfare concerns
are nonzero, using the general expression for the marginal tax rate given by (10)–(11). The
marginal social value of resources µ connects the solutions.
The black dashed lines are the isoclines for which the slope of the marginal tax curve18

is equal to zero. Since the solution must satisfy the terminal condition limy→∞ T ′ (y) = 0
and must be strictly positive, these isoclines bound the solution into the positive part of
the region denoted with the minus sign.
The solution is depicted with the magenta line with circles. Taking this trajectory fromthe perspective of an initial value problem, the solution constitutes an unstable saddle
path. Starting from any other initial condition, the trajectories satisfying equation (23)converge to one of two stable saddle paths visible in the graph, so they either converge toone, or become negative. This also verifies that the terminal condition limy→∞ T ′ (y) = 0
pins down a unique strictly positive solution. In addition, this solution for the marginal
tax rate must be strictly declining.

3.4

Generalizations

The preceding analysis studies the case when workers have quasilinear preferences andthe planner has no welfare concerns for high-type workers in the right tail of the productivity distributions. In this subsection, we briefly discuss generalizations of these results,
with detailed calculations provided in Appendix C. The central insight is that the marginal
tax converging to zero at an exponential rate equal to (at least) − 12 is a robust result that
holds in a range of extensions.
3.4.1

Concave separable preferences

We first consider the case of general separable isoelastic preferences of the form
U (c, n) =c 1− ρ
n 1+ γ
−ψ
1−ρ
1+γ

(24)

where ρ is is the inverse of the consumption elasticity. We still focus on the right tailof the type distribution for which we assume no welfare concerns on the planner’s side,ψ (z) = 0. In this case, the optimality conditions for the Hamiltonian (8) yield the optimal
tax formula in the form
T ′ (y (z))
1 − Fe (z)
=
1
+
γ
,
(
)
1 − T ′ (y (z))
z fe (z)where
fe (z) = κ −1 (c (z))ρ m (z) f (z)

19

(25)

is now the inverse marginal utility weighted density under the worst-case model, withnormalization constant
κ=

Z z̄
z

e [cρ ] ,(c (ζ ))ρ m (ζ ) f (ζ ) dζ = E

and c (z) = y (z) − T (y (z)). The function Fe (z) is the corresponding cumulative distribution function associated with density fe (z) defined in (25).
Theorem 3.3. Assume that worker’s preferences are given by the separable form (24), underlying
productivity has a right tail that is Pareto distributed, and θ < ∞. Then the marginal tax ratevanishes at the top,
lim T ′ (y) = 0.

y→∞More specifically, the limiting tax rate satisfies
lim

2
µ ′
e [cρ ] ,T (y) y = (γ + ρ) E

y→∞ θ

(26)

and hence the limiting elasticity of the marginal tax with respect to income is equal tod log T ′ (y)
1
=− .
y→∞
d log y2limThis theorem generalizes Theorems 3.1 and 3.2 to the case of concave marginal utilityof consumption. Setting ρ = 0 recovers the special quasilinear case. As in the quasilinear
case, the zero limiting marginal tax is preserved in cases when the density that characterizes the right tail of the distribution is thinner than Pareto, even though the decay rate maythen be faster than − 21 .
Intuitively, decreasing marginal utility from consumption effectively reduces the elasticity of labor supply with respect to the productivity. But since the zero limiting marginal
tax result does not depend on the labor supply elasticity to start with, it is also robust withrespect to the introduction of more general separable utility form (24). The only differe [cρ ]ence is expression (26) that adjusts for the consumption elasticity, with the last term E
converting goods to their marginal social value units.
3.4.2

Welfare concerns at the top

When the welfare function does not assign zero weights ψ (z) to top productivity types,
the determination of the limiting tax must also take into account the distortions under
the worst-case model that are due to welfare concerns. To illustrate the consequences, weconsider here the case of a utilitarian planner with ψ (z) ≡ 1. The worst-case distortion20

then takes the form1m (z) = m̄ exp − [U (z) + µT (y (z))] ,θ(27)

and the marginal tax formula can be expressed ase (z) − Fe (z)T ′ (y (z))
Ψ
=
1
+
γ
,
(
)
1 − T ′ (y (z))
z fe (z)(28)

where fe (z) is the inverse marginal utility weighted worst-case density (25), Fe (z) the corree (z) spesponding cumulative distribution function, and the cumulative welfare weight Ψcializes to
e (z) =
ΨZ z
zfe (ζ ) dζ.

e (z), we have Ψe (z) > Fe (z) whenever ρ > 0, which reflects theComparing Fe (z) and Ψ
redistributive motives of the utilitarian planner. When ρ = 0, the redistributive motive ise (z) = Fe (z).
absent, and Ψ
The worst-case distortion m (z) in (27) now combines the contributions of welfare andbudgetary concerns. The distortion from the utility term U (z) reflects the concern that
there are fewer high-type workers in the distribution, which directly adversely affects theplanner’s objective function. Since both U (z) and T (y (z)) are strictly increasing in z, bothconcerns imply a strictly decreasing m (z). Which of the two terms dominates depends onthe curvature of the utility function.
Theorem 3.4. Assume that worker’s preferences are given by the separable form (24), underlying
productivity has a right tail that is Pareto distributed, the planner is utilitarian with ψ (z) ≡ 1, thecurvature of the utility function is ρ > 0, and θ < ∞. Thenlim T ′ (y) = 0,

y→∞

and



d log T ′ (y)1lim= min − , ρ − 1 .
y→∞d log y2
The theorem shows that the direct welfare concern dominates when the preferences are

sufficiently elastic, ρ < 12 . In this case, the limited curvature of the utility function implies
that the utility term U (z) grows faster than the tax revenue that determines the budgetaryconcern. However, the decay rate − 21 derived for the benchmark model constitutes theslowest rate of decay we can anticipate.

21

3.4.3

Power divergence functions

The objective function (6) of the robust planner penalizes deviations from the benchmark
using an entropy penalty, also known as the Kullback–Leibler divergence. While entropy isa natural penalty choice from a statistical perspective, the tendency toward lower progressivity holds for more general divergence functions. Here, we consider the Cressie and Read
(1984) class of power divergence functions analyzed, for example, in Almeida and Garcia(2017) or Borovička et al. (2016). The class of divergences is given by Eη (m) = E ϕη (m)with
m 1+ η − 1
ϕη (m) =,η (1 + η )where m (z) = fe (z) / f (z) and η ∈ R. Divergences Eη (m) for η ∈ {−1, 0} are constructedby appropriate limiting arguments, yielding the entropy E0 (m) = E [m log m] in the limit
as η → 0. Relative to entropy, power divergences Eη (m) for η > 0 penalize relatively
more the deviations in the left tail of the distribution, while for η < 0 they penalize morestrongly the right tail.
Appendix C.3 provides more detail on power divergences. It also shows that replacingθ E0 (m) with θ Eη (m) in the planner’s objective function (6) leads, in the case when utilitarian concerns are absent in the right tail of the productivity distribution, to the same
optimal tax formula (12) but the worst-case distortion (13) is now given bym (z) =hηθ(χ − µT (y (z)))

i η1

(29)

where χ is the Lagrange multiplier on the constraint E [m] = 1. As in the entropy case,the worst-case distortion is decreasing because the expression for the optimal marginal tax
implies that the marginal tax is positive.
Specializing to the case of quasilinear utility and Pareto-distributed productivity underthe benchmark distribution, the differential equation for the optimal marginal tax (22) now
takes the form#
 −1 "
2T ′′ (y) yµ [ T ′ (y)] y
1+γ+α ′
1+γ+α ′
−
T (y)
−
−γ+γ
T (y) .
= − 2−
1 − T ′ (y)1+γλ µT (y) − χ
1+γThis leads to the following result.
Theorem 3.5. Assume that worker’s preferences are given by the quasilinear form (9), the type
distribution satisfies Assumption 1 with z̄ = ∞, the divergence penalty in the planner’s problem isθ Eη (m), and θ < ∞. Then the optimal marginal tax rate T ′ (y (z)) for an agent with type z under
the robust planner is lower than under the planner without model misspecification concerns.

22Moreover, assume in addition that the underlying productivity has a right tail that is Pareto
distributed with shape parameter α. When η ≥ 0, then the marginal tax rate at the top satisfies
lim T ′ (y) = 0.

(30)

y→∞

When η < 0, then the marginal tax rate at the top is given bylim T ′ (y) = τη =

y→∞

1+γ
1+γ+e
α

with eα = α−

1+γ 1
> α.
γ η

(31)

Expressions (30) and (31) show that the asymptotic top marginal tax is continuous in
the parameter η that indexes the power divergences. As η ↗ 0, it also converges to theentropy case with zero asymptotic marginal tax.
Details of the derivation are provided in Appendix C.3. To provide intuition, consider
first the case η > 0. In this case, the worst-case distortion formula (29) implies that χ −µT (y) > 0. This means that T (y) must be bounded from above, and since the marginal
tax T ′ (y) has to be positive, it must converge to zero.
When η < 0, expression (29) implies that χ − µT (y) < 0, so T (y) can be unbounded.
Conjecturing that the limiting marginal tax τη ∈ (0, 1), an asymptotic approximation analogous to that in (15) and (16) implies that the worst-case distortion (29) asymptotically behaves as a power function of z. This means that the worst-case density fe (z) = m (z) f (z)behaves asymptotically as a Pareto density, with an adjusted shape parameter e
α. Thisshape parameter becomes arbitrarily large as η ↗ 0, when the divergence function approaches entropy, and, in this case, the asymptotic marginal tax rate approaches zero.

4

Quantitative application

In Section 3, we provided a theoretical characterization of the tail behavior of the optimal
tax function when the planner is concerned about misspecification of the type distribution. We now focus on a quantitative evaluation of the whole tax function. Specifically, we
are interested in a plausible calibration of the magnitude of the misspecification concerns,
and implications for the relative distortions across the type distribution. We envision thatthe planner has available survey evidence in the form of a random sample of I observations of workers’ productivities {zi }iI=1 , and ask how informative this sample is about theunderlying productivity distribution.

23

4.1

Quantifying model misspecification concerns

In order to quantify the magnitude of model misspecification concerns, we use detection
error probabilities proposed by Anderson et al. (2003). Consider a random sample of inII
dependent draws ziB i=1 from the benchmark distribution f (z), and a sample ziA i=1
drawn from the alternative distribution fe (z; θ ) determined as the worst-case distribution
I
under parameter θ. We evaluate the probability that the random sample ziB i=1 is assigned a higher likelihood under the alternative distribution than under the correct benchmark distribution,
I

P



I





∑ log fe ziB ; θ > ∑ log f ziB

i =1



!

I

=P

i =1





∑ log m ziB ; θ > 0

!
,

i =1


I
and, conversely, the probability that the random sample ziA i=1 is assigned a higher likelihood under the benchmark distribution than under the correct alternative distribution,
IP∑ log f



ziA



i =1

I

> ∑ log feziA ; θ



!

I

=P

i =1

∑ log m

ziA ; θ



!

<0 .

i =1

The detection error probability is then defined as the average of the two probabilities
above,1d (θ, I ) =2

IP∑ log m

ziB ; θ



!

>0 +P

i =1

I

∑ log m

ziA ; θ



!!

<0

.

i =1

The detection error probability expresses the chance that the likelihood ratio leads to theerrorneous conclusion about which of the two distributions generated the random sample.
The construction implies that 0 ≤ d (θ, I ) ≤ 21 , achieving the upper bound when f (z)and fe (z; θ ) are identical. The function d (θ, I ) is increasing in θ, reflecting the fact that ahigher entropy penalty θ implies more similar benchmark and worst-case distributions. In
other words, more severe model misspecification concerns captured by a lower value of θlead the planner to consider as plausible a worst-case distribution fe (z; θ ) that is statistically more distinct from the benchmark distribution f (z), leading to a smaller detection
error probability d (θ, I ). The detection error probability is also decreasing in the sample
size I, as long as f (z) and fe (z; θ ) are statistically distinguishable.
In order to calibrate θ and the magnitude of the misspecification concerns, we fix aplausible level for the detection error probability d (θ, I ) = d.¯ We fix the sample size I tomatch the sample size of a given survey, and then infer θ d,¯ I from the implicit equation d θ d,¯ I , I = d.¯24A more informative survey with a larger sample size I will lead to a higher value θ d,¯ I
for a given fixed d,¯ reflecting the fact that larger sample sizes achieve the same detection
error probabilities for distributions that are statistically more alike.

4.2

Model calibration

We base our benchmark model calibration on the results in Heathcote and Tsujiyama(2021), who use labor income data from the Survey of Consumer Finances (SCF) to infer
the productivity distribution. Heathcote and Tsujiyama (2021) argue that the SCF provides substantially more information about the right tail of the productivity distribution
than other household surveys like the Current Population Survey (CPS).
They also show that the labor income distribution in SCF is very well approximatedby using the exponentially modified Gaussian (EGM) distribution for the logarithm of
the productivity x = log z. The EGM distribution describes the sum of a normal and an
exponential random variable, with density given by
α α2 (2µ+ασ2 −2x)
e
erfc
2
Z ∞
2
2
√
e−t dt.
π x

f x ( x; µ, σ, α) =erfc ( x ) =µ + ασ2 − x√
2σThis distribution implies that productivity z = exp ( x ) has support (0, ∞), the left tail follows the log-normal distribution with parameters µ, σ, and the right tail is asymptotically
Pareto distributed with shape parameter α. We can therefore invoke the theoretical results
derived in Section 3 for the case of the Pareto-distributed right tail.7 As in Heathcote andTsujiyama (2021), we choose α = 2.2 and σ2 = 0.142. The first moments of the logarithm
and the level of the productivity distribution are given by
E [log z] = µ +

1
α



1
α
E [z] = exp µ + σ2.
2
α−1

We set µ to normalize the average productivity to E [z] = 1.
We further assume a utilitarian planner endowed with a concave separable utility function (24), with parameters ρ = 1 implying logarithmic utility from consumption, ψ = 1,
and γ = 2. In the benchmark model, we set B = 0, implying that the government only
taxes for redistributive purposes.
In order calibrate θ, we target the detection error probability of about 10%. For example,
Hansen and Sargent (2010) argue that d¯ = 0.2 is a conservative choice, reflecting that f (z)7 Truncating the Gaussian component of the distribution fx ( x ) for a sufficiently high value of x so that the
tail has an exact Pareto distribution is quantitatively inconsequential.

25

and fe (z; θ ) are sufficiently hard to distinguish. In general, the literature uses value in therange between 5% and 20%. We choose θ = 20, which yields d¯ = 0.09 for a sample sizeof I = 3, 500, which corresponds to the number of households in the SCF survey. In what
follows, we also provide comparative statics exercises with respect to alternative values ofthe parameter θ.

4.3

Asymptotic behavior of the marginal tax rate

We start with the following result that characterizes the marginal tax rate for the left andright tail of the productivity distribution.
Lemma 4.1. When the logarithm of productivity z follows the exponentially modified Gaussianwith parameters (µ, σ, α), the following results hold.
1. In the rational model (θ = ∞), the asymptotic marginal tax rate is given bylim T ′ (y) =y→∞( γ + ρ ) (1 + γ ).
α ( γ + ρ ) + γ (1 + γ )2. Under model misspecification concerns (θ < ∞),lim T ′ (y) = 0.

(32)

z→∞

d log T ′ (y)y→∞d log y1= min − , ρ − 1
2

lim


(33)

3. For the left tail of the productivity distribution,
lim T ′ (y) = 0,y →0irrespective of the value of θ.
A sketch of the proof is provided in Appendix C. The first result evaluates the tax
formula (28) that reflects planner’s utilitarian concerns for the workers in the right tailof the productivity distribution, and the concave shape of the utility function. The limitaligns with the case when the tail is exactly Pareto distributed, since the contribution of the
log-normal component in the right-tail vanishes. The second result restates results shownin Theorem 3.4.
Finally, the third result follows from the log-normal shape of the productivity distribution at zero. To understand the reason why the marginal tax rate is zero irrespective of
the presence of misspecification concerns, we note that because the marginal utility of consumption is infinite at zero consumption, the planner optimally provides a finite, strictly
26

2

θ=∞
θ = 100
θ = 20θ = 10
θ=1

1−20
log fe(z)

fe(z)

1.5

0

0.5
0
10−1

−40
−60
−80

100
z

−100
10−310−210−1 100 101 102 103 104z

101

Figure 2: Worst-case distributions fe(z) for alternative levels of misspecification concerns given byθ. The case θ = ∞ corresponds to the rational benchmark for which fe(z) = f (z).

positive transfer T (z) to workers with zero productivity z = 0. Such a transfer has infinite
marginal social value against a finite social marginal cost of resources. Consequently, U (z)is finite, and the distortion m (z) in (27) is bounded and bounded away from zero in theneighborhood of z = 0. Since, as we show in the appendix, the limiting tax for the left tailis zero in the rational model, a perturbation of the tax rate formula (28) by a finite m (z) will
not alter the zero limiting marginal tax rate in the model with misspecification concerns.

4.4

Worst-case distributions and marginal tax rates

Figure 2 plots the worst-case productivity distributions fe (z) across different choices ofthe parameter θ. The orange line with an infinite entropy penalty θ = ∞ corresponds to
the rational case for which fe (z) = f (z), with subsequent lines representing increasing
misspecification concerns as θ decreases.
The left panel shows the distribution in the proximity of the mean under the benchmark
model, which is equal to E [z] = 1, while the right panel focuses on the broad range of
productivities z and plots the density in logarithms to highlight distortions in the tail. In
the log-log plot in the right panel, the straight orange line in the right tail reflects the Pareto
shape of the right tail under the benchmark distribution f (z).
As is apparent from the right graph, even modest levels of misspecification concerns
lead to sharp distortions in the right tail of the distribution. This is in line with theoretical results from Section 3 that show that an arbitrarily small amount of misspecification
concerns leads to a thin-tailed worst-case distribution. At the same time, the center of the
distribution gets noticeably distorted only as misspecification concerns become substan-

27

quantiles \ θ
qe (0.01)
qe (0.05)
qe (0.25)
qe (0.50)
qe (0.75)
qe (0.95)
qe (0.99)

∞
0.291
0.394
0.629
0.919
1.490
4.348
12.568

100
0.290
0.393
0.628
0.916
1.478
4.241
11.949

20
0.289
0.391
0.622
0.902
1.434
3.826
9.601

10
0.287
0.388
0.614
0.885
1.384
3.438
7.765

1
0.252
0.252
0.500
0.675
0.933
1.597
2.460

Table 1: Quantiles of the distribution of productivity z under the worst-case distribution for alternative values of θ. The case θ = ∞ corresponds to the rational case.

0.8

T ′ (y(z))

0.6

θ=∞
θ = 100
θ = 20θ = 10
θ=1

0.4

0.2

0
10−3

10−2

10−1

100

101

102

103

104

z
Figure 3: Optimal marginal tax schedules for alternative levels of misspecification concerns. The
dashed line corresponds to the limiting marginal tax rate for the rational case.

tial (θ = 1), and the left tail remains essentially undistorted. These conclusions are also
confirmed in Table 1 that tabulates the quantiles of the productivity z under the alternative
worst-case distributions fe (z).
Figure 3 shows the optimal marginal tax rate schedule for alternative levels of themodel misspecification concerns. The orange line represents the marginal tax rate for therational case. In line with the literature, since the underlying productivity distribution exhibits a Pareto tail, the asymptotic tax rate limz→∞ T ′ (y (z)) is positive and quantitatively
large, at 71.4%.8 The tax rate asymptotes to zero as z → 0, in line with Lemma 4.1.
When misspecification concerns are present, the shape of the optimal tax schedules
looks notably different. While for wage levels around the mean (E [z] = 1), the optimal
marginal tax looks similar to that under the rational case, it starts departing quickly for
8 In Heathcote and Tsujiyama (2021), the computed marginal tax rates in the right tail asymptote to zero
because they truncate the distribution and focus on numerical solutions for the truncated case.

28

0.6

θ=∞
θ = 100
θ = 20θ = 10θ=1

d log T ′ (y)/d log y

0.4
0.2
0

−0.2
−0.4
−0.6
10−3

10−2

10−1

100

101

102

103

104

y
Figure 4: Growth rates of optimal marginal tax schedules, d log T ′ (y)/d log y, for alternative levelsof misspecification concerns. The dashed black line corresponds to the theoretical limit, equal to−1/2.

higher wage levels. For the benchmark case of θ = 20, the marginal tax peaks at 58.5% forwages equal to 7.5 the average wage, and starts declining thereafter.
Theoretical results from Section 3 show that not only should marginal tax rates decline
to zero as z → ∞, but the asymptotic rate of decline is also pinned down. From Lemma 4.1,
given that ρ = 1, we have that the rate of decline should converge to − 21 . Figure 4 verifies
this result numerically. The orange line, representing the rational case, is above zero andconverges to zero, reflecting an increasing marginal tax rate schedule converging to a positive limiting tax rate. On the other hand, across all levels of misspecification concerns, thedecay rate indeed asymptotically converges to the theoretically predicted value.

4.5

Insurance provision and budgetary concerns

The decision problem of the robust planner trades off utilitarian and budgetary concerns,
reflected in the worst-case distortion1
m (z) = m̄ exp − [U (z) + µT (y (z))] .
θ


(34)

On the one hand, the planner is concerned that there are more agents in the left tail of the
productivity distribution, who receive low utility U (z) and generate low (in fact negative)
net tax revenue T (y (z)). The planner can diminish the utilitarian concerns by providing
more insurance, thus raising U (z) and lowering m (z). This insurance comes in the form

29

4

θ=∞
θ = 100
θ = 20θ = 10θ=1

2−20
log m(z)

m(z)

3

0

1
0
10−2

−40
−60
−80

10−1

100
z

101

−100
10−310−210−1 100 101 102 103 104z

102

Figure 5: The likelihood ratio m(z) = fe(z)/ f (z) representing the distortion of the worst-case distribution relative to the benchmark distribution, plotted for alternative levels of misspecification
concerns given by θ. The case θ = ∞ corresponds to the rational benchmark for which m(z) = 1.

of transfers, and hence at the cost of a lower net tax revenue T (y (z)). On the margin,
the optimal tax schedule designed by the planner trades off a unit of tax revenue at themarginal social value µ against a unit of consumption transferred to the low-productivity
agent at marginal value U ′ (z).
Any transfers provided to low-productivity workers must come from those in upperparts of the productivity distribution. As shown in Section 3, when the utility function
is sufficiently concave (ρ > 21 ), budgetary concerns dominate the shape of the distortion(34) in the right tail of the productivity distribution. Following the tax formula (28), theplanner chooses the marginal tax rate for a particular z as a result of a tradeoff between
the tax distortion imposed on the fe (z) agents at z against the social benefit of raising lump
sum revenue from the (marginal utility weighted) mass of agents with productivities abovez. Since the misspecification concern increases with z, the planner fears that the mass ofagents above z is lower relative to agents residing exactly at z who are distorted by themarginal tax at z, and hence opts for lower marginal tax rates in the tail.
This desire to lower marginal taxes at the top combined with concerns about the higher
prevalence of low-productivity and lower prevalence of high-productivity workers increases the marginal social value of a unit of tax revenue µ, pushing toward lower overall
redistribution.
Figure 5 represents these implications quantitatively by plotting the shape of the distortion m (z) = fe (z) / f (z). The plots reveal that in the left tail of the productivity distribution, the utilitarian and budgetary concerns reflected in the shape of m (z) are minimal
for the benchmark choice θ = 20. Without any redistribution scheme, limz→0 U (z) = −∞,

30

1.2

104

1

103
102

0.6
0.4
0.2

c(y)

n(z)

0.8
θ=∞
θ = 100
θ = 20θ = 10
θ=1

101
100
10−1
10−2
10−3 −3 −2 −1 0
10 10 10 10 101 102 103 104y0
10−310−210−1 100 101 102 103 104z

Figure 6: Labor supply under optimal tax schedules (left panel), and consumption as a function ofincome (right panel) under alternative levels of misspecification concerns.

and consequently limz→0 m (z) = ∞, as the utilitarian concerns about the low-productivity
workers dominate. However, the optimal tax schedule insures the low-productivity workers sufficiently, leading to bounded and quantitatively modest distortions of the left tail.
Only when the misspecification concerns are substantial (θ = 1), the planner’s concerns
about insufficient tax revenue to insure low-productivity workers start increasing morenotably.
On the other hand, in the right tail of the productivity distribution, the concerns aboutloss of tax revenue from high-productivity workers are severe. The planner, concerned
about the distortion of the labor supply at the top, lowers marginal tax rates asymptotically to zero, yet the marginal tax rate declines sufficiently slowly to make the tax revenue
T (y (z)) from each high-productivity worker grow without bound as z → ∞. Consequently, the planner’s concerns that these high-productivity workers are less prevalentthan assumed under the benchmark distribution also grow without bound, leading tolimz→∞ m (z) = 0.
The left panel in Figure 6 depicts the labor supply under the optimal tax schedule foralternative levels of the misspecification concerns. The lower marginal tax rates when
misspecification concerns are present generally increase labor supply. At the same time,
the right panel shows that low productivity households who produce little output are wellinsured by the optimal tax scheme.
Table 2 summarizes these insights in the form of moments under the benchmark andworst-case distributions. The benchmark mean E [z] is identical and normalized to one
across parameterizations since the distribution f (z) is exogenously specified by the calie [z] decrease as misspecbrated EGM distribution in Section 4.2. The worst case means E
31moments \ θ
E [z]
e [z]
E
E [y]
e [y]
E
µ
T0
maxy T ′ (y) (%)
arg maxy T ′ (y)E [T ]e [T ]
E

∞
1.000
1.000
0.824
0.824
1.220
−0.311
71.4
∞
0.000
0.000

100
1.000
0.986
0.828
0.815
1.229
−0.30764.5
15.541
0.008
0.000

20
1.000
0.951
0.838
0.792
1.261
−0.292
58.6
6.955
0.023
0.000

10
1.000
0.918
0.848
0.770
1.297
−0.278
55.0
4.609
0.037
0.000

1
1.000
0.657
0.918
0.579
1.726
−0.178
35.2
1.992
0.110
0.000

Table 2: Moments under the benchmark distribution f (z) and the worst-case distribution fe(z)under optimal tax schedules for alternative values of θ.

ification concerns increase, reflecting shifts toward more adversely slanted productivity
distributions subjectively perceived by the robust planner.
Objective means of the income distribution E [y (z)] = E [zn (z)] increase with increasing misspecification concerns, as lower marginal taxes increase labor supply, plotted in theleft panel of Figure 6. Nevertheless, the mean of the income distribution under the worst
e [y], declines.
case distribution, E
As mentioned above, since increases in misspecification concerns are manifested inincreases in the marginal social value of tax funds µ, reflecting planner’s fear about more
severe scarcity of tax revenue.
Finally, the bottom part of Table 2 reports statistics for the tax schedule. The lump sum
transfer to the lowest productivity worker, T0 = T (y (z)), expressed as a share of average
income E [y], is roughly 33–37%, and rather stable across the a range of the values of theparameter θ. Only when misspecifications concerns and the fear of lack of tax revenue
worsen severely, the transfer is reduced more substantially.
While the lump sum transfer to the lowest productivity worker is rather insensitive tothe degree of misspecification concerns for a range of values of θ, the peak marginal tax ratechanges substantially. For the case without misspecification concerns, the top marginal tax
rate asymptotes at 71.4%, for the benchmark calibration θ = 20, the marginal tax rate
peaks at 58.6% at an income arg maxy T ′ (y) corresponding to 8.3 of the average income,and starts declining thereafter.
Since the planner’s optimization problem involves running a balanced budget underthe endogenously determined subjective distribution, and we chose B = 0, this implies
e [ T ] = 0 for all choices of θ. However, the tax revenue under the benchmark distributionEgenerally differs from zero. Since the worst-case distribution is pessimistically biased,
then, compared to the benchmark distribution, the planner underestimates the amount32

I
θ = 100
θ = 20θ = 10
θ=1

100
0.462
0.432
0.348
0.001

500
0.452
0.298
0.172
0.000

1000
0.437
0.247
0.092
0.000

3500
0.377
0.090
0.007
0.000

10000
0.304
0.010
0.000
0.000

Table 3: Detection error probabilities for models with alternative choices of misspecification concerns parameterized by θ, and for alternative sizes of the random samples I.

of tax revenue the given tax schedule raises under that benchmark distribution. For the
preferred choice θ = 20, the extra surplus generated by the tax policy is about 2.7% of totallabor income in the economy.
If we interpret the benchmark distribution as the true productivity distribution under the data-generating measure, then the optimal tax policy of the robust planner indeed
generates this surplus. This raises the question of what the planner does with the surplus
resources. Our model is static so it does not speak to intertemporal tradeoffs but it is plausible to envision a dynamic extension of this model in which the planner also manages
the accumulated debt or assets over time. Hansen and Sargent (2012, 2015), Kwon andMiao (2017), Ferriere and Karantounias (2019), or Karantounias (2023) are important contributions in this direction that continue optimal dynamic policies in representative agent
frameworks in which the planner is ambiguous about the stochastic path of the aggregate
economy. Introducing dynamic debt management into our framework faces novel challenges but is a natural way of moving forward.

4.6

Detection error probabilities

We calibrated the baseline magniftude of the misspecification concerns to achieve a detection error probability d¯ = 0.09 when the benchmark and worst-case distributions arecompared using random draws of samples with size I = 3500, corresponding to the sizeof the SCF sample. In Table 3 we compare the detection error probabilities across alternative values of θ and I. In line with the theoretical construction outlined in Section 4.1, thedetection error probabilities decline in θ and I.
For θ = 100, the misspecification concerns are negligible and the two distributions fe (z)and f (z) statistically close to each other, so that the detection error probabilities are close
to their upper bound of 0.5, at which the two distributions are indistinguishable. On theother hand, for θ = 1, the worst-case distribution is extremely distinct from the benchmark,and even a small random sample can convincingly distinguish the two distributions.

33

5Type-dependent model misspecification concerns

In the baseline model, we used a constant parameter θ to express the magnitude of theuncertainty about the underlying productivity distribution. However, surveys and otherevidence can provide differential amount of information about the distribution in different
parts of its support. For example, the government may be less confident about both the left
and right tails of the distribution of the productivity distribution, compared to its center,
due to lower response rates in the survey, undersampling, or issues with measurement.
To incorporate this idea, we extend the baseline model with a type dependent penalty
parameter θ (z). We postulate a penalty function of the form

E (m, θ ) = E [θ (m log m − m + 1)] =Z z̄zθ (z) (m (z) log m (z) − m (z) + 1) f (z) dz, (35)where θ (z) is a strictly positive function. The minimax problem (2) augmented with thetype-dependent penalty parameter θ (z) now reads
min max E [mψU ] + E [θ (m log m − m + 1)]m
E[m]=1T

s.t. E [mT (Y )] ≥ B.

(36)

Key properties and implications of the state dependent penalty are summarized in thefollowing lemma.
Lemma 5.1. The entropy penalty E (m, θ ) in (35) satisfies the following properties.
1. When θ (z) = θ is constant, E (m, θ ) reduces to the entropy penalty used in the baselinemodel

E (m, θ ) = θE [m log m] .
2. For any given strictly positive function θ (z), the entropy penalty satisfies E (m, θ ) ≥ 0, andis minimized for m = 1.
3. Denote µ the Lagrange multiplier on the budget constraint E [mT (Y )] ≥ B, and v (z) =ψ (z) U (z) + µT (y (z)). Then, for a z and z′ for which θ (z) = θ (z′ ), if v (z) > v (z′ ) thenm (z) < m (z′ ) is decreasing in v (z) is constant across all worker types, When ψ (z) U (z)is constant across worker types,
4. If v (z) is constant across all worker types, then there the minimizing probability measure is
not distorted, m (z) = 1.
Proof of Lemma 5.1. When θ (z) = θ, then definition (35) impliesE m, θ = θE [m log m] + θ (1 − E [m]) = θE [m log m]34

since E [m] = 1, showing property 1. To show 2., notice thatm (z) log m (z) ≥ m (z) − 1
with equality if and only if m (z) = 1. Premultiplying by θ (z), and integrating over z
yieldsZ z̄z

θ (z) (m (z) ln m (z) − m (z) + 1) f (z) dz = E (m, θ ) ≥ 0,and equality is attained if and only if m (z) = 1 almost everywhere on the support of the
distribution of z.
Taking the first-order condition with respect to m (z) in (36) yields
ψ (z) U (z) + µT (y (z)) − χ + θ (z) log m = 0,
where χ is the Lagrange multiplier on the constraint E [m] = 1, and µ is the multiplier on
the budget constraint. This can be reorganized as1m (z) = exp −(ψ (z) U (z) + µT (y (z)) − χ) .
θ (z)

(37)

The multiplier χ can be found from the restriction E [m] = 1, by solving the implicit equation

Z z̄



1
exp −
(ψ (z) U (z) + µT (y (z)) − χ)
θ (z)
z


f (z) dz = 1.

Equation (37) directly implies property 3. Finally, when v (z) = v̄ is constant, then theunique solution to (37) is equal to χ = v̄, and m (z) = 1, confirming property 4.
The function θ (z) can now be calibrated to capture heterogenous precision of information about the shape of the productivity distribution in different parts of its support,
assigning a smaller penalty to the parts of the support where misspecification concerns arestronger.
[TO BE COMPLETED]

6

Multidimensional uncertainty[TO BE COMPLETED]35

7

Conclusion

The design of optimal tax schedules crucially depends on assumptions about the underlying distribution of types of taxed individuals. These types are typically unobservable, andmeasurement of observable quantities like earnings hinges upon surveys with often small
samples, especially in the tails that matter most.
In this paper, we tackle this uncertainty about the underlying type distribution by
studying optimal taxation in the presence of concerns that the underlying distribution
is misspecified. The robust approach we employ allows us to avoid making parametricassumptions about the structure of uncertainty, and instead focus on nonparametric misspecifications of the underlying type distribution that the planner fears to be most consequential for the welfare function.
In stark contrast to the existing literature, we find that marginal taxes should optimallydecline to zero, even when the underlying benchmark distribution has fat tails, like in the
Pareto case. Misspecification concerns do not only have a theoretical asymptotic impact,
they decrease marginal tax rates of high income individuals in quantitatively importantways, against a modest decrease in redistribution toward low-productivity workers.
We calibrate the plausible degree of misspecification concerns using detection error
probabilities that capture the ability to distinguish alternative distributions using sampled
data, with sample sizes corresponding to those in available surveys. While doing so, we
also provide a theoretical extension of entropy penalties widely used in the literature toallow for type-dependent penalization that captures differential amount of available information about the shape of the underlying distribution in different parts of the support of
the type distribution.
Our main theoretical results and the benchmark quantitative application involve a
static model of labor income taxation in the presence of a one-dimensional unobserved
type. This sharply theoretically and quantitatively analyzed problem opens up avenues
toward applications in which uncertainty about the underlying type distribution can beeven more pressing. One such example that we provide involves multidimensional unobserved distribution of worker types in which the planner lacks information on the joint
distribution of productivity and labor supply elasticity.
Other important directions are left for follow-up work. A natural candidate involves adynamic extension where optimal taxation of heterogeneous workers is combined with an
optimal debt management problem. Another prominent application in which uncertainty
about the underlying distribution is substantial involves taxation of wealth. Insights developed in this work will help analyze such cases as well.

36

Appendix
A

Derivations of optimal tax formulas

A.1

Derivation of the tax formulamax min
m

c,y

Z z̄



Z z̄
y (z)
ψ (z) U c (z) ,m (z) f (z) dz + θm (z) log m (z) f (z) dzzzz

subject to the IC constraintdU
y (z) y (z).
= −Un c (z) ,
dz
z
z2

(38)

and the budget constraint
Z z̄z(y (z) − c (z)) m (z) f (z) dz ≥ B.

(39)

Treating U as the state variable, λ as its co-state, and y and m as control variables, we can form
the constrained Hamiltonian
H (U , y, m, λ)= ψ (z) U (z) m (z) f (z) + θm (z) log m (z) f (z) − κm (z) f (z)y (z) y (z)−λ (z) Un c (z) ,+ µ [y (z) − c (z)] m (z) f (z) .
zz2Here, κ and µ are multipliers on the constraints E [m] = 1 and (7), respectively, and c (z) is defined
implicitly from the definition of the utility function


y (z)
U (z) = U c (z) ,z
as c (z) = C (U (z) , y (z)).
The optimality condition with respect to output choice is
0

=




y (z) 1Hy = µ 1 − Cy (U (z) , y (z)) m (z) f (z) − λ (z) Un C (U (z) , y (z)) ,
(40)
z
z2




 
y (z)
y (z) 1 y (z)−λ (z) Unc C (U (z) , y (z)) ,Cy (U (z) , y (z)) + Unn C (U (z) , y (z)) ,
z
z
z
z2




with respect to the distortion is
0 = Hm = ψ (z) U (z) f (z) + θ [log m (z) + 1] f (z) − κ f (z) + µ [y (z) − C (U (z) , y (z))] f (z)and the costate dynamics restriction yields
dλ (z)dz= − HU = [µCU (U (z) , y (z)) − ψ (z)] m (z) f (z) +y (z) y (z)+λ (z) Unc C (U (z) , y (z)) ,CU (U (z) , y (z)) .
z
z2

37

(41)

The transversality condition isλ (z̄) U (z̄) = 0
and since the problem is unrestricted at the left end of the type distribution (the choice U (z) is
unrestricted), we also have λ (z) = 0.
The condition (41) can be integrated up over the whole range of types (z, z̄):Z z̄zdλ (z)
dz
dz

Z z̄

Z z̄

= λ (z̄) − λ (z) = 0 = µ
CU (U (z) , y (z)) m (z) f (z) dz −
ψ (z) m (z) f (z) dz
z
z


Z z̄
y (z) y (z)+CU (U (z) , y (z)) dz
λ (z) Unc C (U (z) , y (z)) ,zz2z

which can be solved for the Lagrange multiplier µ


R z̄
µ=

R z̄
y(z)
z ψ ( z ) m ( z ) f ( z ) dz − z λ ( z ) Unc C (U ( z ) , y ( z )) , z
R z̄
z CU (U ( z ) , y ( z )) m ( z ) f ( z ) dzy(z)CU (U (z) , y (z)) dz
z2

(42)

This Lagrange multiplier can be interpreted as the marginal social value of public funds to theplanner.
With an expression for µ at hand, we can solve for λ (z) forward by integrating (41) on (z, z̄).
For that purpose, denote the terms in (41) as follows:= µCU (U (z) , y (z)) f (z) − ψ (z) m (z) f (z)y (z) y (z)HU,N (z) = Unc C (U (z) , y (z)) ,CU (U (z) , y (z)) .
zz2
HU,S (z)The term HU,N (z) is only present when preferences U (c, n) are non-separable. We can then rewrite(41) asλ′ (z) = HU,S (z) + λ (z) HU,N (z) .
This equation has the solution
λ (z) = −

Z z̄
z


 Z ζ
HU,S (ζ ) exp −HU,N (ξ ) dξ dζ.
z

To simplify notation, let us simplify the arguments of functions above, and write, for example,


y (z)
Unc (z) = Unc C (U (z) , y (z)) ,.
z

38

Further, we utilize the following notational simplifications:
1 + εuεc
1 − T ′ (y (z)) z

Unn + wUcn
Un
Un (z)−Uc (z)1
Uc (z)
1 Un (z)−z Uc (z)= 1+n=

CU (U (z) , y (z))=

Cy (U (z) , y (z))=

The first equation follows from the definition of uncompensated and compensated labor supply
elasticities, εu and εc , respectively. The second equation is the worker’s individual optimality condition for the choice of labor supply and consumption given a particular tax schedule. The last two
follow from the definition of the implicit function C (U (z) , y (z)).
With this notation and substitutions, we can reorganize the optimality condition (40) as


 ′


λ (z)
Unn (z) + w (z) Unc (z)′−µ T (y (z)) f (z) =Uc (z) 1 − T (y (z)) 1 + n (z)z
Un (z)
which then yields

1 + εu (z)λ (z)Uc (z)T ′ (y (z))
=
−
.
′
c
1 − T (y (z))ε (z) zm (z) f (z) µ(43)

To obtain the Lagrange multiplier µ, simplify equation (42) to obtain

R z̄ hµ=zin ( ζ )U ( ζ )λ(ζ )m (ζ ) f (ζ ) dζψ (ζ ) − zm(ζ ) f (ζ ) U (ncc ζ)
R z̄ 1z U (ζ ) m ( ζ ) f ( ζ ) dζcFinally, the expression for λ (z) can be simplified as


HU,S (z)=HU,N (z)

=

λ (z)

=


µ
− ψ (z) m (z) f (z)Uc (z)n (z) Unc (z)zUc (z)
 Z ζ

Z z̄
−
HU,S (ζ ) exp −HU,N (ξ ) dξ dζ.
zz

For the special case when the utility function is quasilinear of the form (9), we obtain Uc (z) = 1,
HU,N (z) = 0, and
µ=

Z z̄
z

ψ (ζ ) m (z) f (ζ ) dζ,

yielding
λ (z) = −Z z̄z[µ − ψ (ζ )] m (ζ ) f (ζ ) dζ.

The tax formula (43) then yields (11) when we notice that (1 + ε u ) /ε c = 1 + γ.

39

B

Proofs for the baseline modelThroughout this section, we restrict our attention to the analysis of the baseline model, analyzed in
Sections 3.1–3.3. In particular, we assume the quasilinear utility function
U (c, n) = c −n 1+ γ,1+γan unbounded type space, z̄ = ∞, and restrict our attention to the characterization of the optimale (ẑ) =marginal tax on the interval [ẑ, z̄) on which the planner has no utilitarian concerns, Ψ (ẑ) = Ψ
1, and for which Assumption 1 holds. In the proof of Theorem 3.2, we additionally assume that theshape of the type distribution for z ≥ ẑ under the benchmark model is proportional to the Pareto
distribution.

B.1

Proof of Theorem 3.1

Before proving Theorem 3.1, we start with three preliminary lemmas.
Lemma B.1. Optimal output chosen by individual workers under a given tax scheme T (y) satisfies:y′ (z) =y(z)z.
T ′′ (y(z))γ + 1−T ′ (y(z)) y (z)(1 + γ )(44)

Proof. Under the given tax scheme T (y), the optimality condition (1) of a worker of type z for the
special case of quasilinear preferences (9) implies labor supply n (z) given by
1 − T ′ (zn (z)) z = n (z)γ
which can be rewritten as

 1 1+ γ
y (z) = 1 − T ′ (y (z)) γ z γ .

Differentiating with respect to z:
y′ (z)

 1− γ 1+ γ
1 1
1+γ
1
1 − T ′ (y (z)) γ z γ T ′′ (y (z)) y′ (z) +1 − T ′ (y (z)) γ z γγγ′′
1 T (y (z))
1 + γ y (z)−y (z) y′ (z) +γ 1 − T ′ (y (z))
γ
z

= −
=

yields
1 T ′′ (y (z))
1 + γ y (z)1+
y (z) y′ (z) =γ 1 − T ′ (y (z))γ
zand hencey′ (z) =(1 + γ )
γ+

y(z)
z

T ′′ (y(z))
y1− T ′ (y(z))

40

(z)

.

(45)

Lemma B.2. Let f (z) and fei (z), i = 1, 2 be density functions related by fei (z) = mi (z) f (z) where mi (z)are strictly positive functions representing changes of measure, and m1 (z) /m2 (z) is strictly decreasing.
Then
1 − Fe1 (z)
1 − Fe2 (z)<.
z fe1 (z)z fe2 (z)Proof. The expression can be rewritten as
1 − Fe1 (z)z fe1 (z)R z m1 ( ζ )Rz=z m1 ( ζ ) f ( ζ ) dζzm1 (z) f (z)R z m2 ( ζ )<z m2 (z) f ( ζ ) dζz f (z)z m1 (z) f ( ζ ) dζ

=

(46)

z f (z)Rz=z m2 ( ζ ) f ( ζ ) dζzm2 (z) f (z)=1 − Fe2 (z),z fe2 (z)(47)

where the inequality comes from the fact that for any ζ > z,
m1 ( ζ )
m (z)
< 1
.
m2 ( ζ )m2 ( z )Lemma B.3. Define
e (z) =ϕThene′ (z) =ϕ

1 − Fe (z)1T ′ (y (z)).
=
1 + γ 1 − T ′ (y (z))
z fe (z)d 1 − Fe (z)11 µ ′f ′ (z)′e− T (y (z)) y (z) +.
= − − ϕ (z)dz z fe (z)
z
z
θ
f (z)

Proof. By direct computation:d 1 − Fe (z)dz z fe (z)− fe (z) z fe (z) − 1 − Fe (z)fe (z) + z fe′ (z)=2z fe (z)"#
1 1 − Fe (z) 1fe′ (z)
1
1
d
= − −
+
= − − ϕe (z)+log fe (z) .
z
z
z
z
dz
z fe (z)fe (z)

Using fe (z) from expression (14), we obtain the last line of the lemma.
Proof of Theorem 3.1. We restrict attention to z ≥ ẑ for which the planner’s welfare weight is zero,ψ (z) = 0, and for which Assumption 1 holds. In this case, the tax formula (12) is given by
T ′ (y (z))
1 − Fe (z)= (1 + γ )= (1 + γ) ϕe (z) .
′1 − T (y (z))
z fe (z)(48)

The single-crossing property (5) implies that y (z) is strictly increasing in z. Since the tax rate
T ′ (y (z)) is strictly positive, the tax function T (y (z)) strictly increases in z. This means that m (z)in (14) is strictly decreasing, and, by Assumption 1, fe (z) is also strictly decreasing for sufficiently

41large z. Lemma B.2 then implies that
1 − Fe (z)1 − F (z)<,ez f (z)z f (z)(49)

and hence for all z > ẑ, the marginal tax rate T ′ (y (z)) must be strictly lower than the marginal tax′ (yrate Trat
rat ( z )) in the model without model misspecification.
An application of L’Hôpital’s rule to the tax rate under the model without model misspecification implies′ ( y ( z ))Trat
1 − F (z)1
= (1 + γ) lim= lim d log f (z)<∞′z→∞ z f (z )z→∞z→∞ 1 − T ( y ( z ))
rat
−
−1

lim

d log z′ ( y ( z )) < 1, and combined with T ′ ( y ( z )) < T ′ ( yby Assumption 1. This yields limz→∞ Trat
rat ( z ))rat′
implied by (49), we obtain that for sufficiently large z, T (y (z)) must be bounded away from one,e (z) is bounded,T ′ (y (z)) < 1 − ε̄ T , and ϕe (z) =
ϕ

1 1 − ε̄ T
1T ′ (y (z))<= Kϕ .
′
1 + γ 1 − T (y (z))
1 + γ ε̄ TConsequently, the optimal allocation formula (45) implies that limz→∞ y (z) = ∞.
e (z)From the optimal tax formula (48), we obtain 1 − T ′ (y (z)) = 1 + (1 + γ) ϕ
we can rewrite expression (45) ase (z)y ( z ) = 1 + (1 + γ ) ϕ

− γ1

 −1

, and then

1+ γ

z γ .

Differentiating this expression with respect to z yieldsy′ (z)= −=1γ(1 + γ) ϕe′ (z)e (z)1 + (1 + γ ) ϕ

1+γ
γ

1+ γ



1+ γ
γ

1
e (z)
1 + (1 + γ ) ϕz γ + 1+γ γ z

1
γ



1+γ
γ

1

 γ1 z1γe (z)1 + (1 + γ ) ϕ


f′ z
µ
e (z) 2 + γ − zT ′ (y (z)) y′ (z) + z ( )
2+ϕ
,
θ
f (z)

e′ (z). Since y′ (z) > 0, the last bracket
where the second line uses Lemma B.3 to substitute in for ϕmust be strictly positive.
We now show that limz→∞ T ′ (y (z)) = 0. Assume that it is not. Then there exists ε T > 0 suchthat for an arbitrarily large z̄ there exists a z > z̄ such that T ′ (y (z)) > ε T . Take such a z for which
T ′ (y (z)) > ε T . Since f ′ (z) < 0, we have
f ′ (z)µµ ′′e
0 < 2 + ϕ (z) 2 + γ − zT (y (z)) y (z) + z< 2 + Kϕ 2 + γ − zy′ (z) ε T ,
θ
f (z)
θ

42

which yields a bound on zy′ (z):θ 
2 + γ + 2Kϕ−1 = Ky .
εT µzy′ (z) <(50)

Since, from the result in Lemma B.1, we have that

(1 + γ ) y ( z )zy′ (z) =T ′′ (y(z))γ + 1−T ′ (y(z)) y (z)< Ky ,

we can derive a restriction on T ′′ (y (z)):
T ′′ (y (z)) > 1 − T ′ (y (z))

 (1 + γ ) y ( z ) − K y γ(1 + γ ) y ( z ) − K y γ> ε̄ T.
Ky y ( z )Ky y ( z )(51)

Recall that we can find an arbitrarily large z for which this inequality holds. Since limz→∞ y (z) =∞, we can find such a z that is sufficiently large to satisfy (1 + γ) y (z) > Kγ, denote it ž. ThenT ′′ (y (ž)) > 0, and, consequently, T ′ (y (z)) is increasing at ž. Hence the inequality T ′ (y (z)) > ε Tholds also for z in the right neighborhood of ž, so that the lower bound on T ′′ (y (z)) given in (51)also hold for z to the right of ž, and this argument can then be extended for any z ≥ ž. This thenimplieslim T ′ (y (z))

z→∞

= T ′ (y (ž)) +

Z ∞
y(ž)

> T ′ (y (ž)) + ε̄ T

T ′′ (ξ ) dξ > T ′ (y (ž)) +

Z ∞
y(ž)

ε̄ T

(1 + γ ) ξ − K y γdξKy ξ(1 + γ) y (ž) − Ky γ Z ∞dξ = ∞,
Ky y (ž)y(ž)

which contradicts the bound T ′ (y (z)) < 1 − ε̄ T . Therefore, the marginal tax must converge to zero,lim T ′ (y (z)) = 0.

z→∞

Is is worth noting that misspecification concerns enter the proof by way of a finite bound on zy′ (z)in (50). In the absence of misspecification concerns, θ = ∞, so that Ky = ∞ in (50), and there does
not exist a y (z) for which the right-hand side in (51) is positive, implying we cannot guarantee astrictly positive lower bound on T ′′ (y (z)).

B.2

Analysis of the ODE for the optimal marginal tax rate

We now derive and analyze the differential equation (22) that characterizes the behavior of the
optimal marginal tax rate. This will provide intuition for the subsequent proof of Theorem 3.2 inthe next subseection. We restrict our attention to the case when the benchmark type distribution is
Pareto, as the typical case that leads to nonzero top marginal taxes in absence of model misspecification concerns.
Lemma B.4. When z is Pareto distributed with shape parameter α under the benchmark model, the worst-

43

case density satisfies
d
d
d
µ
1
log fe (z) =log m (z) +log f (z) = − T ′ (y (z)) y′ (z) − (α + 1) .
dz
dz
dz
θ
z
Proof. Since µ
= m̄ exp − T (y (z))θα
f (z) =
z α +1m (z)we have, by direct computation,dµ
1
log fe (z) = − T ′ (y (z)) y′ (z) − (α + 1) .
dz
θz

Proposition B.5. When the type distribution under the benchmark model is Pareto with shape parameter α,
the optimal marginal tax T ′ (y) obeys the differential equation− −1 
2
1+γ+α ′
T ′′ (y) yµ ′1+γ+α ′
=
−
2
−
T
y
T
y
y
−
γ
+
γ
T
y
.
(
)
(
)
(
)
1 − T ′ (y)
1+γ
θ
1+γ

(52)

Proof. We start with the optimal tax formula
1 − Fe (z)T ′ (y (z))= (1 + γ )= (1 + γ) ϕe (z) .
′1 − T (y (z))
z fe (z)Differentiating this formula with respect to z, and using Lemma B.3,



1 e
f ′ (z)
1 µ ′
′
=
1
+
γ
−
−
ϕ
z
−
T
y
z
y
z
+
.
(
)
(
)
(
(
))
(
)
z
z
θ
f (z)
(1 − T ′ (y (z)))2
T ′′ (y (z)) y′ (z)Combining y′ (z) terms, we have

′T ′′ (y (z))µ ′′e ( z ) −1 − 1 − z f ( z ) .
−
T
y
z
y
z
z
=
−
ϕ
(
(
))
(
)
T ′ (y (z)) (1 − T ′ (y (z)))θ
f (z)We can now use Lemma B.1 to substitute out y′ (z) and obtain
T ′′ (y (z))µ ′f ′ (z)(1 + γ ) y ( z )Tyz−= −ϕe (z)−1 − 1 − z.
(())′′ (y (z ))′′TT (y (z)) (1 − T (y (z)))θ
f (z)γ + 1−T ′ (y(z)) y (z)We can now multiply by the denominator of the compound fraction on the left-hand side, use the44e (z), and combine terms that contain T ′′ (y (z)):
expression for ϕf ′ (z)2 −γ − z f (z)1+γ

f ′ (z)γ − z f (z)2
T ′′ (y (z))µ ′T ′ (y (z))
y
z
=
T
y
z
y
z
−
γ
+
γ
T ′ (y (z)) .
(
)
(
(
))
(
)
1 − T ′ (y (z))θ
1+γFinally, for the case of the Pareto density,−zf ′ (z)= 1 + α.
f (z)Substituting this expression in, we notice that the resulting differential equation does not depend
explicitly on z. Since y (z) is strictly monotonic, we can drop the z argument and rewrite the equation as a differential equation for T (y), yielding the expression in the statement of the proposition.

We now study the phase diagram of the differential equation (52), which we can rewrite as
T ′′ (y) = −1 2
1 − T ′ (y)1+γ+α ′
1+γ+α ′
µ ′
2−
T (y)
T (y) y − γ + γ
T (y) .
y
1+γ
θ
1+γ

The resulting phase diagram is depicted in Figure 1.
Define the right-hand side of the above equation as a function h : [0, ∞) × (−∞, 1) → R: 
1 + γ + α −1 µ 2
1+γ+α
1−τ
2−
τ
τ y−γ+γτ .
h (y, τ ) =
y
1+γ
θ
1+γ
We study the function on (y, τ ) ∈ (0, ∞) × (−∞, 1). For simplicity, we assume that
1+γ+α
< 2,
1+γ
so that the first bracket in the definition of h (y, τ ) is never zero for τ ∈ (−∞, 1). This does notchange any conclusions about asymptotic behavior of the optimal tax.
For a given y ∈ (0, ∞), we first find the isoclines by solving for τ̄ (y) such that f (y, τ̄ (y)) = 0.
This τ̄ (y) solves the cubic equation(1 − τ̄ (y))


µ
1+γ+α
yτ̄ (y)2 + γ
τ̄ (y) − γ = 0θ
1+γwith three solutionsα−γ 1+1+γ+γ ±τ̄1,2 (y)=τ̄3 (y)

= 1,

r

γ+α
γ 1+1+γ

2+ 4 µθ γyµ2θywhere τ̄1 (y) denotes the root with the minus sign. The isoclines are depicted with the black dashed45

lines in Figure 1. Asymptotically, the differential equation has two steady stateslim τ̄1,2 (y)= 0lim τ̄3 (y)

= 1.

y→∞

y→∞

We can order the three isoclines asτ̄1 (y) < 0 < τ̄2 (y) < τ̄3 (y) ,and then, as depicted in the phase diagram,f (y, τ )> 0

τ < τ̄1 (y)f (y, τ )< 0

τ̄1 (y) < τ < τ̄2 (y)f (y, τ )> 0

τ̄2 (y) < τ < τ̄3 (y) .

The result from Theorem 3.1,
lim T ′ (y (z)) = lim T ′ (y) = 0,

z→∞

y→∞

is a transversality condition that pins down the unique optimal marginal tax function T ′ (y). This
optimal path is depicted in the phase diagram with the red solid line with bullet markers. It follows
from the proofs of theorems 3.1 and 3.2 that any other path for T ′ (y) that satisfies equation (52)either converges to one, or becomes negative for some sufficiently high y, both of which violate
conditions that the optimal marginal tax function has to satisfy.

B.3

Proof of Theorem 3.2

Using the insights from the phase diagram, we now turn to the proof of Theorem 3.2.
Proof of Theorem 3.2. We investigate the limiting behavior of the differential equation (52). The
left-hand side of this equation is the elasticity of take-home rate 1 − T ′ (y) with respect to income,or −1 
2d log (1 − T ′ (y))
1+γ+α ′µ ′1+γ+α ′
= − 2−T (y)T (y) y − γ + γ
T (y) .
d log y
1+γ
θ
1+γ

(53)

Since limy→∞ (1 − T ′ (y)) = 1, if the limit as y → ∞ of the left-hand side of the above equation
exists, it has to be zero. Assume for now that this limit exists. Since limy→∞ T ′ (y) = 0, the firstbracket on the right-hand side converges to a positive number as y → ∞. For the same reason,
the last term of the second bracket converges to zero as well. Hence the only way how the second
bracket converges to zero is when
lim

2
µ ′
T (y) y = γ.

y→∞ θ

(54)

In the remainder of the proof, we prove that the limit indeed exists. Assume it does not, so that46

there exists an ε such that for any y̌, there exists a y ≥ y̌ such that2µ ′T (y) y − γ > ε > 0.
θ(55)

Further, the analysis of the phase diagram implies that along the optimal path, T ′ (y) monotonicallydecreases to zero. This means that T ′′ (y) < 0, and the elasticity in (53) is strictly positive. In
addition, for any arbitrarily small ε τ , there exists a y̌τ such that 0 < T ′ (y) < ε τ for all y ≥ y̌τ .
2Assume first that inequality (55) holds as θ [ T ′ (y)] y − γ > ε. Then the elasticity in (53) can bebounded asµ −1 
d log (1 − T ′ (y))− T ′′ (y) y1+γ+α ′
1+γ+α ′=
< − 2−T (y)ε+γ
T (y) < 0,d log y1 − T ′ (y)1+γ1+γwhich is a contradiction with T ′′ (y) < 0, given that T ′ (y) > 0 and limy→∞ T ′ (y) = 0.
2

On the other hand, assume that inequality (55) holds as θ [ T ′ (y)] y − γ < −ε. Then the elasticity in (53) can be bounded asµd log (1 − T ′ (y))
d log y21 µ ′− T ′′ (y) y1+γ+α ′
>
−
T
y
y
−
γ
+
γ
T
y
(
)
(
)
1 − T ′ (y)2 θ1+γi γ1+γ+α2ε
γ1+γ+αε
1 hµ  ′T (y) y − γ −ετ > −ετ > ,
> −
2 θ
2 1+γ
2
2 1+γ
4

=

where the last inequality follows from the fact that ε τ can be taken to be arbitrarily small when we
restrict our attention to sufficiently large y ≥ y̌ ≥ y̌τ . We therefore obtain
T ′′ (y) y < −ε1 − T ′ (y) .
4

As a consequencei
2d hµ  ′T (y) y − γ
dy θ

=
<
=


2 i
µ h ′′
2T (y) T ′ (y) y + T ′ (y)
θ


2 i
µh ε
− 1 − T ′ (y) T ′ (y) + T ′ (y)
θ
2 h
i
ε ′
µ ′
ε 
T (y) − + 1 +T (y) ,
θ22which becomes negative for sufficiently large y because limy→∞ T ′ (y) = 0. Denote this y as y∗ . Thisµ2means that inequality θ [ T ′ (y)] y − γ < −ε continues to hold as y increases above y∗ . Integrating
up the inequalityεd log 1 − T ′ (y) > d log y
4for y ≥ y∗ yieldsε
1 − T ′ (y) − log 1 − T ′ (y∗ ) > (log y − log y∗ )4and hence ε/4y′
T (y) < 1 −1 − T ′ (y∗ ) .
y∗Since y → ∞, the right-hand side must ultimately become negative, which violates the restriction47

T ′ (y) > 0.
We have thus shown that condition (55), which is equivalent to a violation of equation (54),cannot hold simulatenously with other restrictions on the optimal marginal tax rate. Either themarginal tax rate function would have to become increasing, or become negative. From the perspective of the phase diagram in Figure 1, analyzed in Appendix B.2, if condition (55) holds fora sufficiently large y, it must be that the given T ′ (y) is either on a trajectory above the optimal
path that crosses the τ̄2 (y) isocline and converges to one, or becomes negative, crossing the τ̄1 (y)isocline.
Finally, equation (54) implies that the marginal tax rate has to decay as
1
log T ′ (y) = − log y + o (y)2where o (y) converges to a constant as y → ∞. Hence, differentiating this expression with respect
to log y, and taking the limit as y → ∞, this limit, if it exists, must be given by expression (19).

C

Generalizations of the baseline model

In this appendix, we provide details on the derivation of results from Section 3.4.

C.1

Concave separable preferences[TO BE COMPLETED]

C.2

Welfare concerns at the top[TO BE COMPLETED]

C.3

Power divergence functions

Cressie and Read (1984) study a class of power divergence functions 1+ η
m−1.
Eη (m) = E ϕη (m) = E
η (1 + η )The function ϕη (m) satisfies ϕη′′ (m) = mη −1 , so it is convex for m > 0. The functions for η ∈{−1, 0} can be constructed by taking appropriate limits using L’Hôpital’s rule. For η = 0, we
obtain the entropy

lim Eη (m) = lim

η →0

η →0

h
i
E e(1+η ) log m − 1
η (1 + η )

= lim

h
i
E e(1+η ) log m log m

η →0

48

1 + 2η

.
= E [m log m] = E0 (m) .

For η = −1, the same calculation yields the negative of the empirical likelihood.
lim Eη (m) = E [− log m] = E−1 (m) .

η →−1

The divergence Eη (m) is positive for all positive m, and uniquely minimized at m ≡ 1. To see this,observe that the minimization problem
min E ϕη (m)s.t. E [m] = 1mleads to first-order conditions
ϕη′ (m) =mη=χηwhere χ is the Lagrange multiplier on the constraint. Hence m must be constant, and thereforem ≡ 1.

C.3.1

Problem formulation

The characterization of the planner’s problem follows Appendix A. The planner solves
max min
c,y

m



Z z̄
y (z)
m ( z )1+ η − 1
ψ (z) U c (z) ,m (z) f (z) dz + θ
f (z) dzz
η (1 + η )
z
z

Z z̄

subject to the incentive constraint (38) and the budget constraint (39). Specializing to the case ofquasilinear utility (9) and restricting attention to the part of the space z ≥ ẑ for which ψ (z) = 0,
the associated Hamiltonian becomes
H (U , y, m, λ)=m ( z )1+ η − 1
λ (z)θ
f (z) − χm (z) f (z) +η (1 + η )z+µ [y (z) − C (U (z) , y (z))] m (z) f (z) .



y (z)
z

 1+ γThe optimality condition with respect to the probability distortion m (z) yields
0 = Hm =θm (z)η f (z) − χ f (z) + µ [y (z) − C (U (z) , y (z))] f (z) .
η(56)This yields the worst-case distortionm (z) =hηθ(χ − µT (y (z)))

i1
η

.

The remaining optimality conditions remain as in Appendix A, yielding the optimal tax formula
1 − Fe (z)T ′ (y (z))= (1 + γ ).
′1 − T (y (z))
z fe (z)49References
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