Presentation from the MGTA event in Rome

Optimal Redistribution
via Income Taxation and Market Design
Mohammad Akbarpour (Stanford University)
Paweł Doligalski (University of Bristol)
Piotr Dworczak?
(Northwestern University; GRAPE)
Scott Duke Kominers (Harvard University; a16z)
June 19, 2024
Summer School, Markets and Governments: a Theoretical Appraisal
?
Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the European Research Council.

Inequality-aware Market Design
Policymakers frequently distort allocation of goods and services in
markets, often motivated by redistributive and fairness concerns:
Housing (rent control; public housing; LIHTC in the US)

Health care (public health care in Europe, Medicare and Medicaid in the US)

Food (food stamps in the US; in-kind food transfer programs (rice, wheat,...)
in developing countries)

Energy (electricity in virtually all European countries at the moment;
kerosene in India; energy subsidies in Latin America;... )

Covid-19 vaccines (“communist” solution within countries versus “wild west
capitalism” across countries)

Road access

Road access
Legal services

Road access
Legal services
...

Road access
Legal services
...
Policymakers didn’t take Econ 101...?

Running assumptions throughout:
I Welfare Theorem holds.
Social planner is benevolent and competent.
Social planner has redistributive preferences.

Conventional wisdom #1:
II Welfare Theorem implies that markets are a superior
allocation mechanism even under redistributive preferences.

Caveat: II WT does not account for private information!

Inequality-aware Market Design (IMD):
How to design individual markets in the presence of
socioeconomic inequality and private information?

Inequality-aware Market Design (IMD):
How to design individual markets in the presence of
socioeconomic inequality and private information?
Main takeaway: Market distortions (taxes, subsidies, inefficient
rationing) can be part of optimal mechanisms when the market
designer has sufficiently strong redistributive preferences (and
does not have access to other tools to effect redistribution).

Inequality-aware Market Design—Example
There are 3 agents: Ann, Bob, and Claire.

There are 2 homogeneous goods (e.g., houses) to allocate.

Agents have the following utilities u for a house:
Ann: 1/4, Bob: 3/4, Claire: 1.

However, these agents also differ in wealth w:
Ann: 8, Bob: 1, Claire: 1.

All agents have the same utility function
log(w) + u x;
where x is the probability of having a house.

All agents have the same utility function
log(w) + u x;
where x is the probability of having a house.
Social planner aims to maximize the sum of utilities.

value for good value for money willingness to pay
Ann 1/4 1/8
Bob 3/4 1
Claire 1 1

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Alternative interpretation of the problem:
Normalize the values for money in agent’s utilities to 1 (the usual
quasi-linear model);
Think of values for money as (unobserved) welfare weights.

val. good val. money WTP utility
Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Mechanism #1: Market equilibrium (+ lump-sum transfers)

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Market-clearing price: 1
House owners: Ann Claire

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Lump-sum transfer: 2=3

Ann 1/4 1/8 2 1=4 1=8 1=3
Bob 3/4 1 3/4
Claire 1 1 1

Ann 1/4 1/8 2 5=24
Bob 3/4 1 3/4 0 + 1 2=3
Claire 1 1 1

Ann 1/4 1/8 2 5=24
Bob 3/4 1 3/4 2=3
Claire 1 1 1 1 1 1=3

Ann 1/4 1/8 2 5=24
Bob 3/4 1 3/4 2=3
Claire 1 1 1 2=3
Total utility: 37=24

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Mechanism #2: Random allocation

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Price: 0
House owners: 66% Ann 66% Bob 66% Claire

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Price: 0
Lump-sum transfer: 0

Ann 1/4 1/8 2 2=3 1=4
Bob 3/4 1 3/4
Claire 1 1 1
Price: 0

Ann 1/4 1/8 2 1=6
Bob 3/4 1 3/4 2=3 3=4
Claire 1 1 1
Price: 0

Ann 1/4 1/8 2 1=6
Bob 3/4 1 3/4 1=2
Claire 1 1 1 2=3 1
Price: 0

Ann 1/4 1/8 2 1=6
Bob 3/4 1 3/4 1=2
Claire 1 1 1 2=3
Price: 0
Total utility: 32=24 ( 37=24)

Ann 1/4 1/8 2 1=6
Bob 3/4 1 3/4 1=2
Claire 1 1 1 2=3
Price: 0
Among all mechanisms charging a single price p, p = 1 is optimal.

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Mechanism #3: Allocate 1st unit by lottery, sell 2nd unit in market

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1
Market-clearing price for 2nd unit: 1
(Ann’s indifference: 1=4 1=8 1 = 1=2 1=4)
House owners: Ann 50% Claire 50% Bob

Ann 1/4 1/8 2
Bob 3/4 1 3/4
Claire 1 1 1

Ann 1/4 1/8 2 1=4 1=8 2=3
Bob 3/4 1 3/4
Claire 1 1 1

Ann 1/4 1/8 2 4=24
Bob 3/4 1 3/4 1=2 3=4 + 1 1=3
Claire 1 1 1

Ann 1/4 1/8 2 4=24
Bob 3/4 1 3/4 17=24
Claire 1 1 1 1=2 1 1 1=3

Ann 1/4 1/8 2 4=24
Bob 3/4 1 3/4 17=24
Claire 1 1 1 20=24
Total utility: 41=24 ( 37=24)

Ann 1/4 1/8 2 4=24
Bob 3/4 1 3/4 17=24
Claire 1 1 1 20=24
This is in fact the optimal mechanism! (subject to IC constraints)

val. money WTP utility—market utility—rationing
Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20

Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:
Values for money (welfare weights) are negatively correlated with
willingness to pay.

Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:
willingness to pay.
Market behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.

Ann 1/8 2 5 4
Bob 1 3/4 16 17
Claire 1 1 16 20
Intuition:
willingness to pay.
Market behavior can be used for screening: Choosing the lottery
reveals a higher-than-average value for money.
The designer can then increase the utility of agents with higher
value for money by reducing the price of the rationed option.

Inequality-aware Market Design—Papers
DKA (2021):
Two-sided market with buyers and sellers for an indivisible good
Rationing used if significant inequality within a side of the market
Taxes/ subsidies used if inequality across the two sides of the market
ADK (2023):
One-sided allocation problem with goods differing in quality, observable
information, and flexible preferences over revenue
A Budish DK (2023):
Application to allocation of Covid-19 vaccines (interaction of redistributive
preferences with allocative externalities)
Tokarski KAD (2023): Application to energy pricing in Europe
D (2023), Yang DA (2024): Optimality of using costly screening (e.g., queuing)

IMD and Income Taxation
The Atkinson-Stiglitz theorem implies that redistribution should
be accomplished entirely via income taxation.

Caveat: Atkinson-Stiglitz theorem assumes that the only
heterogeneity in the population is in ability to generate income.

Main questions for this project:

Should markets be optimally distorted when there is also
heterogeneity in tastes for goods?

Should markets be optimally distorted when there is also
heterogeneity in tastes for goods?
What is the interaction between IMD and income taxation?

Result #1: Assuming separable utility functions and multidimensional
heterogeneity, if
1. utility functions feature no income effects;
2. taste type and ability type are statistically independent;
3. welfare weights depend only on the ability type,

heterogeneity, if
then the conclusion of the Atkinson-Stiglitz theorem holds.

heterogeneity, if
Result #2: If any of the assumptions 1, 2, or 3 are relaxed, then
distortions in markets may be optimal.

heterogeneity, if
We provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.

heterogeneity, if
We provide analytical solutions in a linear model with a single
good, binary ability type, and continuous taste type.
Distortions in markets are “generically” optimal.

Literature Review
Suboptimality of goods market distortions:
Diamond and Mirrlees (1971), Atkinson and Stiglitz (1976), Gauthier and
Laroque (2009), Doligalski et al. (2023), ...
Taste heterogeneity and failure of Atkinson-Stiglitz theorem:
Cremer and Gavhari (1995, 1998, 2002), Saez (2002), Kaplow (2008),
Golosov et al. (2013), Gauthier and Henriet (2018), Scheuer and Slemrod
(2020), Hellwig and Werquin (2023), Ferey et al. (2023), ...
Multi-dimensional heterogeneity in public finance:
Cremer et al. (2003), Blomquist and Christiansen (2008), Diamond and
Spinnewijn (2011), Piketty and Saez (2013), Saez and Stantcheva (2018),
Golosov and Krasikov (2023), Bierbrauer et al. (2023), ...
IMD precursors: Spence (1977), Weitzman (1977), Nichols and
Zeckhauser (1982), Besley and Coate(1991), Condorelli (2013), ...
Related mechanism-design papers: Jullien (2000), Che et al. (2013),
Fiat et al. (2016), Li (2021), Dworczak and Muir (2024), ...

Talk outline
1. Model
2. Result #1: Atkinson-Stiglitz theorem
3. Result #2: Optimality of goods market distortions
Optimal design with a simple income effect
Optimal design with welfare weights depending on taste
Optimal design under correlation of ability and taste

Model
We introduce a general model to prove the AS result; we make
simplifying assumptions later to solve the model when the AS
result fails.

Model
result fails.
There is a unit mass of agents characterized by two-dimensional
types: (t; ), where t is “taste” and is “ability.”

Model
result fails.
Social planner allocates:

Model
result fails.
numeraire consumption good c 2 R,

Model
result fails.
vector of goods x 2 X RL
+,

Model
result fails.
+,
earnings (pre-tax) y 2 R.

Model
result fails.
+,
earnings (pre-tax) y 2 R.
Each agent’s utility function is
U((c; x; y); (t; )) = u(c) + v(x; t) w(y; );
where the function u; v; w are non-decreasing.

Model
Types (t; ) have a joint distribution F(t; ) in the population.

Model
Goods in x can be produced at fixed marginal costs k 2 RL
+
(in terms of the numeraire).

Model
+
The planner must respect the resource constraint:
Z
[y(t; ) c(t; ) k x(t; )] dF(t; ) B;
where B could be positive (government expenditure) or negative
(outside source of revenue).

Model
+
The planner must respect the resource constraint:
Z
[y(t; ) c(t; ) k x(t; )] dF(t; ) B;
where B could be positive (government expenditure) or negative
(outside source of revenue).
The planner uses social welfare weights (t; ) 0, with
average weight normalized to 1.

Planner’s problem
Choose an allocation a = (c; x; y) to maximize
Z
(t; ) U(a(t; ); (t; )) dF(t; )
subject to incentive-compatibility constraints
U(a(t; ); (t; )) U(a(t0; 0); (t; )); 8(t; ); (t0; 0);
and the resource constraint
Z
[y(t; ) c(t; ) k x(t; )] dF(t; ) B:

Atkinson-Stiglitz result

Given the additive separability, the efficient allocation of goods is
x?
(t; ) = argmaxx fu(c̃(t; ) k x) + v(x; t)g, where c̃ denotes
post-tax earnings (or total consumption expenditures).

Given the additive separability, the efficient allocation of goods is
x?
(t; ) = argmaxx fu(c̃(t; ) k x) + v(x; t)g, where c̃ denotes
post-tax earnings (or total consumption expenditures).
Theorem
Suppose that:
1. There are no income effects: u(c) = c; c 2 R;
2. Welfare weights depend only on the ability type:
(t; ) = ¯
(); 8(t; );
3. Ability and taste types are statistically independent:
F(t; ) = Ft (t)F(); 8(t; ), where Ft and F denote the marginal
distributions.
Then, the optimal mechanism induces the efficient choice of x, and
can be implemented with an income tax alone.

Proof outline:
1. Without loss of generality to look at direct mechanisms.

Proof outline:
2. Consider the problem for a fixed taste type t, dropping the IC
constraints of reporting t truthfully.

Proof outline:
3. Since there is no taste heterogeneity and we have (additive)
separability, the standard Atkinson-Stiglitz result holds (formally,
we use the approach from Doligalski et al., 2023).

Proof outline:
4. Define individual expenditure on all goods by
c̃(t; ) = c(t; ) + k x?
(t) (note: x?
does not depend on ).

Proof outline:
4. Define individual expenditure on all goods by
c̃(t; ) = c(t; ) + k x?
(t) (note: x?
does not depend on ).
5. Key step: (c̃; y) that solves the relaxed problem for fixed t does
not depend on t.

Proof outline continued:
5. (c̃; y) does not depend on t because:

By assumption 2, the distribution of is the same for all t;

By assumption 3, the welfare weights do not vary with t;

By assumption 1, t only affects the “split” of total
consumption between numeraire and other goods.

6. The mechanism (c̃(); y(); x?
(t)) is IC for the full problem:

Total expenditure and labor choices are incentive-compatible
by construction (they do not depend on t)

Total expenditure and labor choices are incentive-compatible
by construction (they do not depend on t)
Expenditure on goods x is incentive-compatible because
each type t obtains the efficient amount.

Comments:
Assumption 1 is crucial—in particular, we have used the fact that
c can be potentially negative.

Comments:
Assumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.

Comments:
Assumptions 2 and 3 similar in spirit to the conditions derived by
Saez (2002) using the perturbation approach.
If we add a participation constraint, and assume that the planner
maximizes revenue (or is Rawlsian), the result breaks down.

Optimality of market distortions
Optimality of market distortions

Framework
To solve the model without Assumptions 1–3, we make further
simplifying assumptions.

Framework
There is a single indivisible good, x 2 [0; 1] (can be interpreted
as bounded quality); earnings y 2 [0; ȳ].

Framework
Ability type is binary: 2 fL; Hg, where L 0, and H 1.

Framework
Let i be the mass of ability types i, and Fi(t) the CDF of taste t
conditional on ability type i, supported on [0; t̄], for i 2 fL; Hg.

Framework
Let i be the mass of ability types i, and Fi(t) the CDF of taste t
conditional on ability type i, supported on [0; t̄], for i 2 fL; Hg.
Utility function is piece-wise linear:
(
c + t x 1
y c c;
1 c c;
where c is a “subsistence” level.

Framework
Comments about the framework:
Willingness to pay is
WTP =





t c c;
2 [0; t] c = c;
0 c c:

Framework
WTP =





t c c;
2 [0; t] c = c;
0 c c:
So efficiency requires
x?
(t; c) =





1 c c and t k;
2 [0; 1] c = c and t k;
0 c c or t k:

Framework
WTP =





t c c;
2 [0; t] c = c;
0 c c:
So efficiency requires
x?
(t; c) =





1 c c and t k;
2 [0; 1] c = c and t k;
0 c c or t k:
The framework is a “best-case” scenario for income taxation.

Results
We show that in the optimal mechanism:
Provision of labor is always efficient.
Details

Results
The allocation rules for low- and high-ability agents are steps
functions with few steps (one step for low-ability agents).
Details

Results
Rough idea of the proof:
Details

Results
Low-ability agents cannot mimic high-ability agents;
Details

Results
High-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
Details

Results
High-ability agents face an endogenous outside-option
constraint (from mimicking low-ability agents);
Such constraints can be handled by adapting Myerson’s
ironing technique.
Details

Case # 1: Income effect

Suppose that Assumptions 2 and 3 from the AS result hold:

FL(t) = FH(t) = F(t), for all t;

H(t) = 0 and LL(t) = 1, for all t.

Suppose that buyer and seller virtual surpluses
t (1 F(t))=f(t) and t + F(t)=f(t) are non-decreasing.

Suppose that buyer and seller virtual surpluses
t (1 F(t))=f(t) and t + F(t)=f(t) are non-decreasing.
Finally, assume that
H

1
1
H

B + c + k;
=) income effect has bite.

Theorem
There exists an optimal mechanism that can be implemented as
follows:
Income is taxed at a rate 1 1=H (high-ability agents work);

Theorem
follows:
Anyone can purchase either

Theorem
follows:
x̄L of the good at a unit price pL k; or

Theorem
follows:
x̄H units of the good at a unit price pH(1 x̄L=x̄H) + pL;
where pH lies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if x̄H = 1)

Theorem
follows:
x̄H units of the good at a unit price pH(1 x̄L=x̄H) + pL;
where pH lies between marginal cost k and the optimal
monopoly price (equal to optimal monopoly price if x̄H = 1)
Everyone receives a lump-sum transfer equal to c + pLx̄L.

Discussion:
Income tax extracts all surplus but is not distortionary.

Discussion:
Low-ability agents with sufficiently high t purchase the good at a
subsidized price subject to rationing.

Discussion:
Basic intuition:

Discussion:
Basic intuition:
Suppose the good is priced at k. Low-ability agents are
constrained and buy x 1 despite having high taste t.

Discussion:
Basic intuition:
Consider perturbing the price to p = k , for small 0.

Discussion:
Basic intuition:
There is a second-order loss in efficiency (marginal effect)...

Discussion:
Basic intuition:
... but there is a first-order gain in welfare since all types
t k can now buy more of the good (inframarginal effect)

Discussion:
Basic intuition:
... but there is a first-order gain in welfare since all types
t k can now buy more of the good (inframarginal effect)
High-ability agents can “top up” in the market but pay a unit price
higher than marginal cost.

Goods allocation in optimal mechanism

Numeraire consumption in optimal mechanism

Is the result driven by our stylized “income effect”?
Model with income effect very difficult to solve analytically.

We are working on two extensions:

Concave utility u(c) in a 2 2 model;

Optimality of distortion in the “same direction” in a model
with concave utility u(c).

Optimality of distortion in the “same direction” in a model
with concave utility u(c).
Numerically, results appear robust.

Goods allocation in optimal mechanism (u(c) =
pc)

Numeraire consumption in optimal mechanism (u(c) =
pc)

Goods allocation in optimal mechanism (multiple ability types)

Numeraire consumption in optimal mechanism (multiple ability types)

Labor supply in optimal mechanism (multiple ability types)

Intuition for the result under concave u:
Purchasing the good implies higher marginal utility for money
u0(c) relative to agents who do not buy.

Thus, the planner endogenously values giving more money to
agents who buy the good (higher type t).

Welfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).

Welfare improvement can therefore be accomplished by
subsidizing the purchase of the good (below marginal cost).
Helpful thought exercise: Think of the good being treatment for a
serious illness.

Case # 2: Welfare weights depend on taste type

u(c) = c; c 2 R.

u(c) = c; c 2 R.
Suppose that welfare weights can depend on the taste type.

u(c) = c; c 2 R.
Let Λi(t) be the average welfare weight on all types higher
than t, conditional on i 2 fL; Hg.

u(c) = c; c 2 R.
Let Λ(t) = LΛL(t) + HΛH(t):

u(c) = c; c 2 R.
Let h(t) be the inverse hazard rate, and J(t) = t h(t) be
the virtual surplus function associated with distribution F.

u(c) = c; c 2 R.
Assume that

u(c) = c; c 2 R.
Assume that
(Λi(t)h(t) + J(t) k) f(t) is non-decreasing whenever it is
negative, for i 2 fL; Hg.

u(c) = c; c 2 R.
Assume that
(Λi(t)h(t) + J(t) k) f(t) is non-decreasing whenever it is
negative, for i 2 fL; Hg.
ΛL(t) ΛH(t); for all t (weaker than L(t) H(t) for all t).

Theorem
There are two candidate optimal mechanism:
1. High-ability agents work efficiently at net wage of 1=H (with the
remaining surplus taxed away), and the good is provided at a
single price p?
given by
p?
= k + (1 Λ(p?
))h(p?
):
2. High-ability agents work efficiently at net wage w 1=H; and
the good is sold at a lower price to low-ability agents than to
high-ability agents.

Discussion:
In the first case, there is:

Discussion:
Subsidy for the good if the average welfare weight on agents
buying it, Λ(p?
), exceeds the average welfare weight 1.

Discussion:
buying it, Λ(p?
Tax on the good if the average welfare weight on agents
buying it, Λ(p?
), is below the average welfare weight 1.

Discussion:
buying it, Λ(p?
buying it, Λ(p?
In the second case:

Discussion:
buying it, Λ(p?
buying it, Λ(p?
In the second case:
The reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.

Discussion:
buying it, Λ(p?
buying it, Λ(p?
In the second case:
The reduced price of the good is available only to agents
who don’t work—the two instruments are bundled together.
Incentive compatibility is maintained by decreasing tax on
labor, so that high-ability agents get a strictly positive
surplus from working.

Case # 3: Correlation of taste and ability types

i(t) = ¯
i, for all t;

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.
Suppose that FL(t) 6= FH(t).

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.
Let Λ(t) be the average welfare weight on all types higher than t:
Λ(p) =
¯
L L(1 FL(p)) + ¯
H H(1 FH(p))
L(1 FL(p)) + H(1 FH(p))
:

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.
Λ(p) =
¯
L L(1 FL(p)) + ¯
H H(1 FH(p))
L(1 FL(p)) + H(1 FH(p))
:
Assume that

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.
Λ(p) =
¯
L L(1 FL(p)) + ¯
H H(1 FH(p))
L(1 FL(p)) + H(1 FH(p))
:
Assume that
t (1 ¯
H)hH(t) k

fH(t) is increasing when negative;

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.
Λ(p) =
¯
L L(1 FL(p)) + ¯
H H(1 FH(p))
L(1 FL(p)) + H(1 FH(p))
:
Assume that
t (1 ¯
H)hH(t) k

t (1 ¯
L)hL(t) k

fL(t) crosses zero once from below;

i(t) = ¯
i, for all t;
u(c) = c; c 2 R.
Λ(p) =
¯
L L(1 FL(p)) + ¯
H H(1 FH(p))
L(1 FL(p)) + H(1 FH(p))
:
Assume that
t (1 ¯
H)hH(t) k

t (1 ¯
L)hL(t) k

fL(t) crosses zero once from below;
(¯
L 1)hL(r) (¯
H 1)hH(r); for all t:

Theorem
There are two candidate optimal mechanism:
1. High-ability agents work efficiently at net wage of 1=H (with the
remaining surplus taxed away), and the good is provided at a
single price p?
given by
p?
= k + (1 Λ(p?
))h(p?
):
2. High-ability agents work efficiently at net wage w 1=H; and
the good is sold at a lower price to low-ability agents than to
high-ability agents.

Discussion: (recall that p?
= k + (1 Λ(p?
))h(p?
))
Conditional on buying the good, t p?
, the designer can infer
the likelihood of the agent being a high- or low-ability worker.

= k + (1 Λ(p?
))h(p?
))
Thus, the goods market can be used to redistribute more utility
from high- to low-ability agents.

= k + (1 Λ(p?
))h(p?
))
Assuming that ¯
H = 0, we have that
Λ(p) 1 FL(p)
L(1 FL(p)) + H(1 FH(p))
1 () FL(p) FH(p):

= k + (1 Λ(p?
))h(p?
))
Assuming that ¯
H = 0, we have that
Λ(p) 1 FL(p)
L(1 FL(p)) + H(1 FH(p))
1 () FL(p) FH(p):
If ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;

= k + (1 Λ(p?
))h(p?
))
Assuming that ¯
H = 0, we have that
Λ(p) 1 FL(p)
L(1 FL(p)) + H(1 FH(p))
1 () FL(p) FH(p):
If ability and taste are positively correlated, the good is taxed,
and revenue is redistributed lump-sum;
If ability and taste are negatively correlated, the good is
subsidized.

Concluding Remarks
Concluding Remarks

Concluding Remarks
We investigated the optimal joint design of an income tax and
a market for goods.
Income taxation is used as the only redistributive instrument
under narrow assumptions.
Whenever these assumptions fail, redistribution through
markets can also be useful.
This is not to say that all market interventions motivated by
redistributive motives are justified...
... but this does mean that traditional theoretical foundations
opposing goods market distortions are insufficient.

Concluding Remarks
Future research directions:
Richer models of how Inequality-aware Market Design
interacts with optimal taxation.
Empirical applications.
Political economy.
Alternative approaches to measuring welfare and inequality.

Results
Lemma
There exists an optimal mechanism that features
efficient provision of labor (high-ability agents choose y = ȳ,
low-ability agents choose y = 0);

Results
Lemma
one-step allocation rule xL (possibly with rationing) for low-ability
agents;

Results
Lemma
agents;
at most a three-step allocation rule xH for high-ability agents.

Results
Lemma
agents;
at most a three-step allocation rule xH for high-ability agents.
If there is no income effect (c ! 1 or B ! 1), then both
allocation rules reach 1 at the highest step, and xH has at most two
steps.
Back

Results
Proof overview:
If high-ability agents do not work full time, increase their labor
supply y and consumption c to keep them indifferent:
Back

Results
Proof overview:
leaves utilities unchanged;
Back

Results
Proof overview:
relaxes the incentive-compatibility constraints;
Back

Results
Proof overview:
relaxes the resource constraint =) strict improvement.
Back

Results
Proof overview:
Use the envelope formula to solve for the consumption rule in
terms of the two allocation rules, a lump-sum payment, and an
extra monetary payment to high types.
Back

Results
Proof overview:
Lump-sum payment pinned down by the resource constraint.
Back

Results
Proof overview:
Lump-sum payment pinned down by the resource constraint.
The “subsistence constraint” (ci(t) c) binds only at the highest
type t̄ for each i 2 fL; Hg, and hence depends on xi(t̄).
Back

Results
Parameterize x̄L = xL(t̄), x̄H = xH(t̄), and assign Lagrange
multipliers L 0; H 0 to the two subsistence constraints.
Back

Results
Incentive-compatibility constraints:
Back

Results
No misreport of t conditional on true report of = L:
=) xL(t) is non-decreasing;
Back

Results
No misreport of t conditional on true report of = H:
=) xH(t) is non-decreasing;
Back

Results
No misreport of t and misreport of = L
() ruled out by the assumption that L 0.
Back

Results
No misreport of t and misreport of = L
() ruled out by the assumption that L 0.
No misreport of t and misreport of = H
() no misreport of = H but truthful report of t
() an outside option constraint.
Back

Results
An outside option constraint:
∆ +
Z t
0
xH()d UL(t) :=
Z t
0
xL()d; 8t 2 [0; t̄]:
Back

Results
∆ +
Z t
0
xH()d UL(t) :=
Z t
0
xL()d; 8t 2 [0; t̄]:
If we fix xL, the problem is analogous to linear mechanism design
with a type-dependent outside option constraint (Jullien, 2000)
Back

Results
∆ +
Z t
0
xH()d UL(t) :=
Z t
0
xL()d; 8t 2 [0; t̄]:
Dworczak and Muir (2024): We can solve this problem using an
adaptation of the ironing technique (Myerson, 1981).
=) The optimal (∆; xH) depend linearly on xL.
Back

Results
∆ +
Z t
0
xH()d UL(t) :=
Z t
0
xL()d; 8t 2 [0; t̄]:
The problem of optimizing over xL is linear with no constraints:
=) The optimal xL takes the form xL(t) = x̄L 1ftt?
L
g.
Back

Results
∆ +
Z t
0
xH()d UL(t) :=
Z t
0
xL()d; 8t 2 [0; t̄]:
The problem of optimizing over xL is linear with no constraints:
=) The optimal xL takes the form xL(t) = x̄L 1ftt?
L
g.
This implies a bound on the number of steps of xH(t).
Back

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