Comments on “Robust bounds on optimal tax progressivity”

Tomoyuki Nakajima
June 7, 2024
University of Tokyo

Very interesting and important paper!
Mirrleesian optimal taxation problem, where the planner is uncertain about the distribution ofproductivity of workers.
robust control approach
Such uncertainty reduces the optimal tax rate at the top (to zero).
Quantitative results illustrate how the tax function is affected.

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Mirrleesian taxation problem

Optimal taxation problem
a continuum of workers indexed by their productivity z ∈ R+f : R+ → R+ is the probability density function of z.

Worker’s utility maximization problem:

max U(c, n)
c ,ns.t.

c = zn − T(zn)

Solution: c = c(z), n = n(z), y(z) = zn(z).

Optimal taxation problem:

Z ∞maxT(·)

ψ(z)U (c(z), n(z)) f(z) dz
Z0 ∞s.t.

y(z) − c(z) f(z) dz ≥ B.

0

where ψ(z) is the Pareto weights withZ ∞ψ(z)f(z) dz = 102/10

ABC formulaquasi-linear utility:

U(c, n) = c −n1+γ1+γABC formula:T′ (y(z))Ψ(z) − F(z) 1 − F(z)= (1 + γ)
1 − T′ (y(z))
1 − F (z )zf(z)whereZ z
F (z ) =f(ζ) dζZ0 z
Ψ(z) =

ψ(ζ)f(ζ) dζ.
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Marginal tax rate at the top
If the productivity distribution is bounded, then the optimal marginal tax rate at the top is zero′

T (y(z̄)) = 0

This remains true as long as the distribution is thin-tailed (e.g. normal, log-normal, etc),

lim T′ (y(z)) → 0z→∞This result is overturned if the distribution is fat-tailed. For instance, with a Pareto with parameter α,

1 − F (z )1=zf(z)αThus, if Φ(z) = 1 for large z,T′ (y(z))
1= (1 + γ)
1 − T′ (y(z))αAccording to Diamond and Saez (2011), γ = 4 and α = 1.875, andlim T′ (y(z)) ≈ 73%z→∞

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Robust taxation

Robust taxation problem
The government is uncertain about the distribution of productivities.
It designs the tax function so as to perform well under the “worst-case distribution.”
The worst-case distribution is chosen endogenously based on a relative entropy penalization.

Robust taxation problem:

Z ∞max minT(·)

m(·)

Z ∞
m(z) log m(z)f(z) dz

ψ(z)U (c(z), n(z)) m(z)f(z) dz + θ
Z0 ∞s.t.

0y(z) − c(z) f(z) dz ≥ B,Z0 ∞m(z)f(z) dz = 1.
0This reduces to the standard Mirrleesian program if θ = ∞.

The worst-case distortion:

exp − θ1 [ψ(z)U(z) + µT(y(z))]
exp − θ1 [ψ(z)U(z) + µT(y(z))] f(ζ) dζ ′zm(z) = R z̄

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ABC formulaModiﬁed ABC formula:e z) − eT′ (y(z))Ψ(F (z ) 1 − eF (z )= (1 + γ)′e
1 − T (y(z))
1 − F (z )zef(z)
whereeF (z ) =Z zf̃(ζ) dζ =0e z) =Ψ(Z zZ z0m(ζ)f(ζ) dζ
0

ψ(ζ)f̃(ζ)dζ.
R∞
ψ(ξ)ef(ξ) dξ
0If ψ(z) = 0 for large z, thenT′ (y(z))
1−eF (z )= (1 + γ)′
1 − T (y(z))
zef(z)

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Marginal tax rate at the top

Theorem 3.1: If θ < ∞, thenlim T′ (y(z)) = 0z→∞

The worst-case distribution f̃ puts lower weights on high types.
Even though the objective distribution f is Pareto, the worst case distribution f̃ is thin-tailed, and thus themarginal tax rate at the top is zero.

Extensions:concave separable preferences
welfare concerns at the top
power divergence functions

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Quantitative resultqe (0.99)

12.56811.949

9.601

7.765

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

010−3

10−2

10−1

100

101

102

103104zFigure 3: Optimal marginal tax schedules for alternative levels of misspecification concerns. Thedashed 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

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Comments

Comment 1

What is the gain of adopting a robust policy in this problem?
How much welfare would be lost when the productivity distribution is indeed given by theworst-case distribution?

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Comment 2

Other kinds of uncertainty or fear of misspeciﬁcation might also be relevant.
Indeed, some of those might lead to an increase in the progressiveness of optimal taxation.
misspeciﬁcation regarding the social welfare function
inequality aversion/relative income concerns

misspeciﬁcation regarding the wage determination
rent seeking activity

Is there a way to choose the kinds of misspeciﬁcation the government should focus on?

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