Redistribution through Markets

Inequality-aware Market Design
Piotr Dworczak
based on joint work with
Mohammad Akbarpour and Scott Duke Kominers
August 1, 2022
GRAPE Seminar

Motivation
Market designers often respond to participants’ wealth inequalities in
markets they control:

Motivation
▶ Housing market (rent control + public housing)

Motivation
▶ Health care

Motivation
▶ Health care
▶ Food

Motivation
▶ Health care
▶ Food
▶ Road access

Motivation
▶ Health care
▶ Food
▶ Road access
▶ (Iranian) kidney market

Motivation
▶ Health care
▶ Food
▶ Road access
▶ Covid-19 vaccines

Motivation
▶ Health care
▶ Food
▶ Road access
▶ ...

Motivation
▶ Health care
▶ Food
▶ Road access
▶ ...
At odds with classical intuitions (welfare theorems)?

Motivation
▶ Health care
▶ Food
▶ Road access
▶ ...
▶ Welfare theorems ignore private information...

Motivation
▶ Health care
▶ Food
▶ Road access
▶ ...
Equity-efficiency trade-off in public finance...

Motivation
▶ Health care
▶ Food
▶ Road access
▶ ...
▶ ... but doesn’t delve into the details of market design.

Motivation
▶ Health care
▶ Food
▶ Road access
▶ ...
▶ ... but doesn’t delve into the details of market design.
Inequality-aware Market Design: How to design a market
“optimally” when there is inequality among participants?

Wealth vs Market Design
Key consequences of wealth effects for market design:

Wealth may influence individuals’ preferences:

Wealth inequality may influence social preferences:

Maximizing welfare in the canonical quasi-linear framework in
mechanism design:

mechanism design:
Individuals have quasi-linear utilities

mechanism design:
Everyone “values” money the same

Our approach:
blank
Agents differ in their marginal utility of money

Our approach:
blank
The designer attaches different Pareto weights to individuals

The Pareto frontier under IC and IR constraints

Related literature (incomplete)
Rationing vs. market mechanism: Weitzman (1977); Condorelli
(2013); Kang and Zheng (2021, 2022); Reuter and Groh (2021);
Kang (2022a, b)
Auctions with budget-constrained bidders: Che and Gale (1998),
Fernandez and Gali (1999); Che, Gale, and Kim (2012); Pai and
Vohra (2014); Kotowski (2017)
Optimal taxation: Diamond and Mirrlees (1971); Atkinson and
Stiglitz (1976); Piketty and Saez (2013); Scheuer (2014), Saez and
Stantcheva (2016, 2017); Scheuer and Werning (2017)
Minimum wage and labor taxation: Allen (1987), Guesnerie and
Roberts (1987), Boadway and Cuff (2001), Lee and Saez (2012),
Cahuc and Laroque (2014)
Market design for fairness: Hylland and Zeckhauser (1979);
Bogomolnaia and Moulin (2001); Budish (2011)

Framework
There are two goods: K and M.
Good K is indivisible and agents demand one unit, good M is
divisible and agents can hold any amount of it.
There is a unit mass of sellers (owners of good K), and a mass µ > 0
of buyers (non-owners of good K).
Each agent is characterized by (her ownership type and) a
two-dimensional type (vK, vM );
If (xK, xM ) denote the holdings of the two goods, then each agent’s
utility is given by
vK · xK + vM · xM .
(vK, vM ) ∼ Fj(vK, vM ) on side j of the market.
The designer maximizes expected total value with vK and vM
interpreted as social values.

Framework – some observations
Individual preferences depend only on the rate of substitution:
r =
vK
vM
.

r =
vK
vM
.
It will turn out that this implies that it is wlog to consider mechanisms
whose outcomes only depend on r.

r =
vK
vM
.
whose outcomes only depend on r. Let:
r ∼ Gj with positive density gj on [rj, r̄j].

r =
vK
vM
.
Social preferences can also be expressed as a function of r:
µEB
[xB
K · vK + xB
M · vM ] + ES
[xS
K · vK + xS
M · vM ].

r =
vK
vM
.
µEB

vM

xB
K ·
vK
vM
+ xB
M

+ ES

vM

xS
K ·
vK
vM
+ xS
M

.

r =
vK
vM
.
µEB

EB
[vM |r]

xB
K · r + xB
M

+ ES

ES
[vM |r]

xS
K · r + xS
M

.

r =
vK
vM
.
µEB


EB
[vM |r]
| {z }
λB(r)

xB
K · r + xB
M

| {z }
UB(r)


 + ES


ES
[vM |r]
| {z }
λS(r)

xS
K · r + xS
M

| {z }
US(r)


 .

r =
vK
vM
.
r ≡ vK/vM ∼ Gj with positive density gj on [rj, r̄j].
TS(Λ) = µ
Z r̄B
rB
λB(r)UB(r)dGB(r) +
Z r̄S
rS
λS(r)US(r)dGS(r).

Framework
A key assumption:
λj(r) = E

vM |
vK
vM
= r

is non-increasing.

Framework
A key assumption:
λj(r) = E

vM |
vK
vM
= r

is non-increasing.
Example: What’s your willingness to pay for an Uber ride to the airport?
Agent 1: $100

= vK
vM

Agent 2: $10

= vK
vM

Framework
A key assumption:
λj(r) = E

vM |
vK
vM
= r

is non-increasing.
Agent 1: $100

= vK
vM

Agent 2: $10

= vK
vM

In our model, the designer attaches a higher weight to Agent 2.

Framework
A key assumption:
λj(r) = E

vM |
vK
vM
= r

is non-increasing.
Agent 1: $100

= vK
vM

Agent 2: $10

= vK
vM

Idea: Willingness to pay can be used to make a statistical inference about
the marginal value for money; The designer attaches a higher Pareto
weight to agents with lower willingness to pay (that are “poor in
expectation”)

Framework
A key assumption:
λj(r) = E

vM |
vK
vM
= r

is non-increasing.
Example: What’s your WTP for a Tommy Emmanuel’s concert ticket?
Agent 1: $100

= vK
vM

Agent 2: $10

= vK
vM

expectation”)

Framework
A key assumption:
λj(r) = E

vM |
vK
vM
= r

is non-increasing.
Example: What’s your willingness to pay for life-saving medicine?
Agent 1: $100

= vK
vM

Agent 2: $10

= vK
vM

expectation”)

Framework – Mechanisms
The designer chooses a trading mechanism subject to 4 constraints:
Incentive Compatibility
Individual Rationality
Market Clearing
Budget Balance

Simple mechanisms
Simple mechanisms in a simplified setting

Simple mechanisms
Assumptions:
The rate of substitution r is uniformly distributed on both sides of
the market.
The designer uses a single-price or two-price mechanism.

Simple mechanisms
Assumptions:
The rate of substitution r is uniformly distributed on both sides of
the market.
The designer uses a single-price or two-price mechanism.
We will first solve one-sided problems (for a fixed volume of trade Q and
revenue/budget R), and then link the two sides.

Simple mechanisms
Single-price mechanisms

Seller-side optimality

Single-price mechanisms – seller side
Problem: Choose a price pS to maximize seller welfare while buying
quantity Q and not exceeding a budget of R (where R QG−1
S (Q)).

S (Q)).
max
pS≥G−1
S (Q)
(
Q
GS(pS)
Z pS
rS
λS(r)(pS − r)dGS(r) + ΛS(R − pSQ)
)
,
where ΛS = ES[vM ].

S (Q)).
max
pS≥G−1
S (Q)
(
Q
GS(pS)
Z pS
rS
λS(r)(pS − r)dGS(r) + ΛS(R − pSQ)
)
,
where ΛS = ES[vM ].
We will refer to the price pC
S = G−1
S (Q) as the competitive price;
Any price pS pC
S leads to rationing.

Low seller-side inequality ⇐⇒ λS(rS) ≤ 2ΛS.

Proposition
When seller-side inequality is low, it is optimal to choose pS = pC
S .

Proposition
S .
When seller-side inequality is high, there exists Q̄ ∈ [0, 1) such that
Rationing at a price pS pC
S is optimal when Q Q̄;
Setting pS = pC
S is optimal when Q ≥ Q̄.

Proposition
S .
When seller-side inequality is high, there exists Q̄ ∈ [0, 1) such that
Rationing at a price pS pC
S is optimal when Q Q̄;
Setting pS = pC
S is optimal when Q ≥ Q̄.
This is in fact the optimal mechanism overall!

Intuition: There are three effects associated with raising the price pS
above the competitive level pC
S :
1 Allocative efficiency is reduced ↓

S :
2 The mechanism uses up more money leaving a smaller amount
R − pSQ to be redistributed as a lump-sum transfer to all sellers ↓

S :
3 Sellers that sell receive a higher price ↑

S :
Consider the net redistribution effect 2+3:
−ΛS

S :
−ΛS + ES
[λS(r)|r ≤ pS]

S :
−ΛS + ES
[λS(r)|r ≤ pS] ≥ 0 ↑

S :
−ΛS + ES
[λS(r)|r ≤ pS] ≥ 0 ↑
This net effect must be stronger than the negative effect 1 to justify
rationing:
High seller-side inequality;
Low volume of trade: Q Q̄.

Buyer-side optimality

Single-price mechanisms – buyer side
Problem: Choose a price pB to maximize buyer welfare while selling
quantity Q and raising a revenue of R (where R QG−1
B (1 − Q)).

B (1 − Q)).
max
pB≤G−1
B (1−Q)

Q
1 − GB(pB)
Z r̄B
pB
λB(r)(r − pB)dGB(r) + ΛB(pBQ − R)

.
where ΛB = EB[vM ].

B (1 − Q)).
max
pB≤G−1
B (1−Q)

Q
1 − GB(pB)
Z r̄B
pB
λB(r)(r − pB)dGB(r) + ΛB(pBQ − R)

.
where ΛB = EB[vM ].
We will refer to the price pC
B = G−1
B (1 − Q) as the competitive price;
Any price pB pC
B leads to rationing.

Proposition
Regardless of buyer-side inequality, it is optimal to set pB = pC
B.
That is, the competitive mechanism is always optimal.

Intuition: There are three effects associated with lowering the price pB
below the competitive level pC
B:

B:
2 The mechanism generates less revenue pBQ − R that can be
redistributed as a lump-sum transfer to all buyers ↓

B:
3 Buyers that buy pay a lower price ↑

B:
−ΛB

B:
−ΛB + EB
[λB(r)|r ≥ pB]

B:
−ΛB + EB
[λB(r)|r ≥ pB] ≤ 0 ↓

B:
−ΛB + EB
[λB(r)|r ≥ pB] ≤ 0 ↓
The net redistribution effect is negative!

Two-price mechanisms

A designer can now post two prices, pH
j and pL
j , for each side j of the
market.

A designer can now post two prices, pH
j and pL
j , for each side j of the
market.
Traders trade with probability one at the less attractive price (higher
for buyers, lower for sellers), and trade with some interior probability
δ at the more attractive price.

Seller-side optimality

Two-price mechanisms – seller side
Nothing changes!

Buyer-side optimality

Two-price mechanisms – buyer side
max
pH
B ≥pL
B, δ





δ
Z rδ
pL
B
λB(r)(r − pL
B)dGB(r) +
Z r̄B
rδ
λB(r)(r − pH
B )dGB(r)
+ΛB pL
Bδ(GB(rδ) − GB(pL
B)) + pH
B (1 − GB(rδ)) − R






subject to the market-clearing and revenue-target constraints
1 − δGB(pL
B) − (1 − δ)GB(rδ) = Q,
pL
B)) + pH
B (1 − GB(rδ)) ≥ R,
where rδ is the type indifferent between the high and the low price.

max
pH
B ≥pL
B, δ





δ
Z rδ
pL
B
λB(r)(r − pL
B)dGB(r) +
Z r̄B
rδ
λB(r)(r − pH
B )dGB(r)
+ΛB pL
B)) + pH
B (1 − GB(rδ)) − R






subject to the market-clearing and revenue-target constraints
1 − δGB(pL
B) − (1 − δ)GB(rδ) = Q,
pL
B)) + pH
B (1 − GB(rδ)) ≥ R,
where rδ is the type indifferent between the high and the low price.
We say that there is rationing at the lower price pL
B if δ 1 and
GB(rδ) GB(pL
B), i.e., if a non-zero measure of buyers choose the lottery.

Low buyer-side inequality ⇐⇒ λB(rB) ≤ 2ΛB.

Proposition
When buyer-side inequality is low, it is optimal not to offer the low price
pL
B and to choose pH
B = pC
B.

Proposition
pL
B and to choose pH
B = pC
B.
When buyer-side inequality is high, there exists Q ∈ (0, 1] such that
Rationing at the low price is optimal when Q Q;
Setting pH
B = pC
B (and not offering the low price pL
B) is optimal for
Q ≤ Q.

Proposition
pL
B and to choose pH
B = pC
B.
When buyer-side inequality is high, there exists Q ∈ (0, 1] such that
Rationing at the low price is optimal when Q Q;
Setting pH
B = pC
B (and not offering the low price pL
B) is optimal for
Q ≤ Q.
This is in fact the optimal mechanism overall!

Intuition: This time rationing (a discounted price) is offered to buyers
with relatively low types r ∈ [pL
B, rδ], so that it is possible that the net
redistribution effect is positive:
−ΛB

−ΛB + EB
[λB(r)| r ∈ [pL
B, rδ]]

−ΛB + EB
[λB(r)| r ∈ [pL
B, rδ]] ≥ 0

−ΛB + EB
[λB(r)| r ∈ [pL
B, rδ]] ≥ 0 ↑

−ΛB + EB
[λB(r)| r ∈ [pL
B, rδ]] ≥ 0 ↑
This net effect must be stronger than the negative effect on allocative
efficiency to justify rationing:
High buyer-side inequality;
High volume of trade: Q Q.

Cross-side optimality

Proposition
When same-side inequality is low on both sides of the market, it is optimal
to set pB ≥ pS such that the market clears, GS(pS) = µ(1 − GB(pB)),
and redistribute the resulting revenue as a lump-sum payment to the side
of the market j ∈ {B, S} with higher average value for money Λj.

Proposition
For example, when there is no same-side inequality:
pB − pS =

ΛS − ΛB
ΛS

1 − GB(pB)
gB(pB)
.

Proposition
For example, when there is no same-side inequality:
pB − pS =

ΛS − ΛB
ΛS

1 − GB(pB)
gB(pB)
.
We will refer to the situation ΛS ̸= ΛB as cross-side inequality.

Proposition
Competitive equilibrium, defined by pB = pS = pCE, where
GS(pCE
) = µ(1 − GB(pCE
)),
is optimal only when there is no cross-side inequality (ΛS = ΛB).

Sellers:
Proposition
When seller-side inequality is high and ΛS ≥ ΛB, if µ is low enough, then
it is optimal to ration the sellers by setting a single price above the
competitive-equilibrium level.

Sellers:
Proposition
Intuition:
Low µ guarantees a low enough volume of trade;

Sellers:
Proposition
Intuition:
Low µ guarantees a low enough volume of trade;
ΛS ≥ ΛB guarantees that the budget constraint does not alter the optimal
mechanism for the seller side.

Buyers:
Proposition
If seller-side inequality is low and rB = 0, then the optimal mechanism
does not ration the buyers (regardless of buyer-side inequality).

Buyers:
Proposition
Rationing the buyers is suboptimal for any µ!

Proposition
Intuition:
Under these conditions, it is never optimal to have a high volume of
trade in the mechanism;

Proposition
Intuition:
High volume of trade and rB = 0 imply that the revenue generated
by the mechanism is small;

Proposition
Intuition:
Rationing = providing the good at a low price to “poor” agents, is
ineffective when rB = 0;

Proposition
Intuition:
Rationing = providing the good at a low price to “poor” agents, is
ineffective when rB = 0;
A better idea: Limit volume, raise revenue, and help the “poor”
buyers by giving a lump-sum transfer.

Proposition
If there is high buyer-side inequality, ΛB ≥ ΛS, and
rB − r̄S ≥
1
2
(r̄B − rS),
then there is some ϵ 0 such that it is optimal to ration the buyers for
any µ ∈ (1, 1 + ϵ).

Proposition
rB − r̄S ≥
1
2
(r̄B − rS),
any µ ∈ (1, 1 + ϵ).
Intuition:
Because there are large gains from trade, any optimal mechanism
must maximize volume of trade;

Proposition
rB − r̄S ≥
1
2
(r̄B − rS),
any µ ∈ (1, 1 + ϵ).
Intuition:
Because there are slightly more buyers than sellers in the market,
almost all buyers buy;

Proposition
rB − r̄S ≥
1
2
(r̄B − rS),
any µ ∈ (1, 1 + ϵ).
Intuition:
Because there are slightly more buyers than sellers in the market,
almost all buyers buy;
Thus, we have high volume + high buyer-side inequality =⇒ buyer
rationing.

General Framework
General Framework

General Framework
A designer chooses a mechanism to allocate a unit mass of objects to
a unit mass of agents.
Each object has quality q ∈ [0, 1], q is distributed acc. to cdf F.
Each agent is characterized by a type (i, r, λ), where:
i is an observable label, i ∈ I (finite);
r is an unobserved willingness to pay (for quality), r ∈ R+;
λ is an unobserved social welfare weight, λ ∈ R+;
The type distribution is known to the designer.
If (i, r, λ) gets a good with quality q and pays t, her utility is qr − t,
while her contribution to social welfare is λ(qr − t)

General Framework
The designer has access to arbitrary (direct) allocation mechanisms
(Γ, T) where Γ(q|i, r, λ) is the probability that (i, r, λ) gets a good with
quality q or less, and T(i, r, λ) is the associated payment, subject to:
Feasibility: E(i,r,λ) [Γ(q|i, r, λ)] ≥ F(q), ∀q ∈ [0, 1];
IC constraint: Each agent (i, r, λ) reports (r, λ) truthfully;
IR constraint: U(i, r, λ) ≡ r
R
qdΓ(q|i, r, λ) − T(i, r, λ) ≥ 0;
Non-negative transfers: T(i, r, λ) ≥ 0, ∀(i, r, λ).
The designer maximizes, for some constant α ≥ 0, a weighted sum of
revenue and agents’ utilities:
E(i,r,λ) [αT(i, r, λ) + λU(i, r, λ)] .

Framework
The role of revenue:
The weight α on revenue captures the value of a dollar in the
designer’s budget, spent on the most valuable “cause.”

Framework
α = maxi λ̄i:
As if a lump-sum transfer to group i were allowed;

Framework
α = maxi λ̄i:
α = averagei λ̄i:
As if lump-sum transfers to all agents were allowed.

Framework
α = maxi λ̄i:
α λ̄i for all i:
There is an “outside cause.”

Framework
α = maxi λ̄i:
α λ̄i for all i:
There is an “outside cause.”
α λ̄i for some i:
Lump-sum payments to agents in group i are prohibited or costly.

Framework
Lemma
It is optimal for the designer to condition the allocation and transfers only
on agents’ labels i and WTP r.

Framework
Lemma
Let Vi(r) be the expected per-unit-of-quality social benefit from allocating
a good to an agent with WTP r in group i in an IC mechanism.

Framework
Lemma
Pareto optimality with perfectly transferable utility (first welfare theorem):
Vi(r) = r
text

Framework
Lemma
Let Gi(r) be a continuous distribution of WTP conditional on label i.
Vi(r) = r
text

Framework
Lemma
If the designer maximizes revenue a’la Myerson (1981):
Vi(r) = r −
1 − Gi(r)
gi(r)
text

Framework
Lemma
A useful decomposition:
Vi(r) = Λi(r)·
1 − Gi(r)
gi(r)
+ α·

r −
1 − Gi(r)
gi(r)

text

Framework
Lemma
A useful decomposition:
Vi(r) = Λi(r)·
1 − Gi(r)
gi(r)
| {z }
utility
+ α·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
text

Framework
Lemma
With redistributive preferences:
Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
where Λi(r) = Er̃∼Gi [ λ | r̃ ≥ r, i].

Framework
Economic idea:
The designer assesses the “need” of agents by estimating the unobserved
welfare weights based on the observable (label i) and elicitable (wtp r)
information:
Λi(r) = Er̃∼Gi [ λ | r̃ ≥ r, i].

Framework
Economic idea:
information:
Λi(r) = Er̃∼Gi [ λ | r̃ ≥ r, i].
Consequence:
The optimal allocation depends on the statistical correlation of labels
and willingness to pay with the unobserved social welfare weights.

Framework
Economic idea:
information:
Λi(r) = Er̃∼Gi [ λ | r̃ ≥ r, i].
Consequence:
The optimal allocation depends on the statistical correlation of labels
and willingness to pay with the unobserved social welfare weights.
Correlation is likely to be negative (however, the direction and strength
depend on the market context!)

Derivation of Optimal Mechanism

1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;

i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i.

i ;
F⋆
Brief recap of Myerson’s ironing:
Start with virtual surplus: V (r) = r − 1−G(r)
g(r) ;

i ;
F⋆
g(r) ;
Integrate (in the quantile space): Ψ(t) :=
R 1
t V (G−1(x))dx;

i ;
F⋆
g(r) ;
R 1
t V (G−1(x))dx;
Concavify the function Ψ;

i ;
F⋆
g(r) ;
R 1
t V (G−1(x))dx;
Differentiate Ψ to get the “ironed” virtual surplus V̄ (r);

i ;
F⋆
g(r) ;
R 1
t V (G−1(x))dx;
▶ Ironed virtual surplus is negative =⇒ do not allocate;

i ;
F⋆
g(r) ;
R 1
t V (G−1(x))dx;
▶ Ironed virtual surplus is constant =⇒ randomize (const. allocation);

i ;
F⋆
g(r) ;
R 1
t V (G−1(x))dx;
▶ Ironed virtual surplus is constant =⇒ randomize (const. allocation);
▶ Ironed virtual surplus is increasing =⇒ auction (increasing allocation).

i ;
F⋆
Observation: Only expected quality, Qi(r), matters for payoffs.

i ;
F⋆
Within a group, agents are partitioned into intervals according to WTP,
with either the “market” or “non-market” allocation in each interval:
Market allocation: assortative matching between WTP and quality
Qi(r) = (F⋆
i )−1
(Gi(r)), ∀r ∈ [a, b];

i ;
F⋆
Within a group, agents are partitioned into intervals according to WTP,
with either the “market” or “non-market” allocation in each interval:
Market allocation: assortative matching between WTP and quality
Qi(r) = (F⋆
i )−1
(Gi(r)), ∀r ∈ [a, b];
Non-market allocation: random matching between WTP and quality
Qi(r) = q̄, ∀r ∈ [a, b].
Details

Derivation of Optimal Mechanism: within-group allocation

Derivation of Optimal Mechanism: across-group allocation

Economic Implications

Assume first that the designer can give label-contingent lump-sum
transfers.

Assume first that the designer can give label-contingent lump-sum
transfers.
Proposition (WTP-revealed inequality )
Suppose that α ≥ λ̄i. Then, it is optimal to provide a random allocation
to agents with willingness to pay in some (non-degenerate) interval if and
only if the function
Λi(r) ·
1 − Gi(r)
gi(r)
+ α ·

r −
1 − Gi(r)
gi(r)

is not increasing.

Assuming differentiability, the condition for using a non-market mechanism
around type r in group i becomes
Λ′
i(r)
1 − Gi(r)
gi(r)
+ α + (Λi(r) − α)
d
dr

1 − Gi(r)
gi(r)

0,
where recall that Λi(r) = Er̃∼Gi [ λ | r̃ ≥ r, i].

Λ′
i(r)
1 − Gi(r)
gi(r)
+ α + (Λi(r) − α)
d
dr

1 − Gi(r)
gi(r)

0,
Thus, non-market allocation is optimal if willingness to pay is strongly
(negatively) correlated with the unobserved social welfare weights.

Λ′
i(r)
1 − Gi(r)
gi(r)
+ α + (Λi(r) − α)
d
dr

1 − Gi(r)
gi(r)

0,
Thus, non-market allocation is optimal if willingness to pay is strongly
(negatively) correlated with the unobserved social welfare weights.
This is more likely to hold when:
The designer has strong redistributive preferences (dispersion in λ’s);
The good being allocated is relatively expensive and everyone needs it;
The label i is not very informative about λ.

What if the designer cannot give a lump-sum transfer to group i?

We call the good universally desired (for group i) if ri 0.

Interpretation: A vast majority of agents have a willingness to pay that is
bounded away from zero.

Interpretation: A vast majority of agents have a willingness to pay that is
bounded away from zero.
Main example:
“Essential” goods: housing, food, basic health care;

Proposition (Label-revealed inequality )
Suppose that
the average Pareto weight λ̄i for group i is strictly larger than the
weight on revenue α;
the good is universally desired (ri 0).
Then, there exists r⋆
i ri such that the optimal allocation is random at a
price of 0 for all types r ≤ r⋆
i .

Suppose that
i .
Interpretation: If the designer would like to redistribute to group i but
cannot give agents in that group a direct cash transfer, then it is optimal
to provide the lowest qualities of universally desired goods for free.

Suppose that
i .
Interpretation: If the designer would like to redistribute to group i but
cannot give agents in that group a direct cash transfer, then it is optimal
to provide the lowest qualities of universally desired goods for free.
Note: There might still be a price gradient for higher qualities.

Suppose that
i .
“Wrong” intuition: A random allocation for free increases the welfare of
agents with lowest willingness to pay

Suppose that
i .
“Wrong” intuition: A random allocation for free increases the welfare of
agents with lowest willingness to pay
Correct Intuition: A random allocation for free enables the designer to
lower prices for all agents Picture

Conclusions
When to use in-kind redistribution? (random allocation of a range of
quality levels at a below-market price)

Conclusions
Roughly: When observable variables uncover inequality in the unobserved
welfare weights.

Conclusions
welfare weights.
1 Label-revealed inequality:

Conclusions
welfare weights.
1 Label-revealed inequality: When the average Pareto weight on
group i exceeds the weight on revenue, it is optimal to use at least
some in-kind redistribution for universally desired goods.

Conclusions
welfare weights.
▶ Qualified support for programs allocating essential goods to agents
satisfying verifiable eligibility criteria (if cash transfers are not feasible)

Conclusions
welfare weights.
2 WTP-revealed inequality: When the welfare weights are strongly
and negatively correlated with willingness to pay.

Conclusions
welfare weights.
2 WTP-revealed inequality: When the welfare weights are strongly
and negatively correlated with willingness to pay.
▶ Provision of a subsidized lower-quality option in addition to a
higher-quality option priced by the market (e.g., public health care)

Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)

Conclusions
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.

Conclusions
1 Revenue maximization:

Conclusions
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal:

Conclusions
▶ Allocation of goods to corporations (oil leases, spectrum licenses etc.)

Conclusions
▶ Whenever lump-sum transfers to the target population are feasible
(e.g. tax credits for renters in the housing market).

Conclusions
2 Efficiency maximization: When the welfare weights are not strongly
correlated with willingness to pay:

Conclusions
▶ Small dispersion in welfare weights to begin with;

Conclusions
▶ Small dispersion in welfare weights to begin with;
▶ Large dispersion in welfare weights but little correlation with WTP
(affordable goods for which WTP depends heavily on taste).

Extensions and Future Work

Allocative externalities

Allocative externalities =⇒ vaccine allocation (ADK + Eric Budish)

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.
Pricing versus Rationing

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.
Pricing versus Rationing versus Queuing (ADK + Shengwu Li)

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.
vK · xK + vM · xM − vQ · xQ

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.
Other on-going projects:

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.
▶ Application to congestion pricing;

Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+ α ·

r −
1 − Gi(r)
gi(r)

| {z }
revenue
+ E [Tex|i, r]
| {z }
externality
.
▶ Application to congestion pricing;
▶ Comparison of screening devices.

Interaction of IMD with conventional redistributive tools (income
taxation):

taxation):
▶ Should redistribution through markets still be used? When?

taxation):
▶ Does IMD have a greater scope in developed or developing countries?

taxation):
The effects of concave utility functions:

taxation):
▶ Risk aversion (force against rationing)

taxation):
▶ Income effects (force against using constant welfare weights)

taxation):
The effects of aftermarkets:

taxation):
▶ Is rationing even feasible?

taxation):
▶ How do aftermarkets modify the optimal mechanism?

taxation):
▶ How do aftermarkets modify the optimal mechanism?
... Much more work remains to be done:)

Intuition for factor 2
Why “2” in the definition of same-side inequality?

Why rationing becomes optimal when λj(rj) 2Λj?

Consider the seller side and let rS = 0.

Consider the seller side and let rS = 0.
We will consider the welfare effects of raising the price from p to p + ϵ and
rationing, for some initial small price p ≈ 0.

Competitive mechanism with price p

Gain: G0 =
R p
0 λS(r)(p − r)dGS(r);

Gain: G0 ≈ gS(0)λS(0)
R p
0 (p − r)dr;

R p
0 (p − r)dr;
Cost: C0 = ΛS · pGS(p).

Competitive mechanism with price p + ϵ
R p+ϵ
0 (p + ϵ − r)dr;
Cost: C1 = ΛS · (p + ϵ)GS(p + ϵ).

Rationing at price p + ϵ
Gain: G1 ≈ gS(0)λS(0) p
p+ϵ
R p+ϵ
0 (p + ϵ − r)dr;
Cost: C1 = ΛS · (p + ϵ)GS(p).

Rationing at price p + ϵ
Change in gain: λS(0)
| {z }
value for money
GS(p)
| {z }
mass
1
2
ϵ
|{z}
per agent surplus
;
Change in cost: ΛS
|{z}
value for money
GS(p)
| {z }
mass
ϵ
|{z}
per agent cost
.

Derivation of the optimal “within-group” mechanism
The “within-group” problem:
Fixing a group of agents i, and a distribution of quality Fi available for
group i, what is the optimal way to allocate quality subject to IC and IR
constraints?

The designer’s problem: (Qi(r) denotes expected quality)
max
Qi(r), Ui(ri)
Z r̄i
ri
Vi(r)Qi(r)dGi(r) + (λ̄i − α)Ui(ri)

max
Qi(r)
Z r̄i
ri
Vi(r)Qi(r)dGi(r) + max{0, λ̄i − α}riQi(ri).

max
Qi(r)
Z r̄i
ri
In the above, Vi(r) = α(r − 1−Gi(r)
gi(r) ) + Λi(r)1−Gi(r)
gi(r) .

max
Qi(r)
Z r̄i
ri
gi(r) .
Feasible distribution of expected quality: CDF Φi such that
Fi ≻MPS Φi.

max
Qi(r)
Z r̄i
ri
gi(r) .
Fi ≻MPS Φi.
Incentive-compatibility =⇒ Gi(r) = Φi(q)

max
Qi(r)
Z r̄i
ri
gi(r) .
Fi ≻MPS Φi.
Incentive-compatibility =⇒ Qi(r) = Φ−1
i (Gi(r))

max
Qi(r)
Z r̄i
ri
gi(r) .
Φ−1
i ≻MPS F−1
i .
Incentive-compatibility =⇒ Qi(r) = Φ−1
i (Gi(r))

The designer’s problem is
max
Ψi
Z r̄i
ri
Vi(r)Ψi(Gi(r))dGi(r) + max{0, λ̄i − α}riΨi(0)
s.t. Ψi ≻MPS
F−1
i
Back

max
Ψi
Z 1
0
Vi(t)
z }| {
Z 1
t
Vi(G−1
i (x))dx + max{0, λ̄i − α}ri1{t=0}

dΨi(t)
s.t. Ψi ≻MPS
F−1
i
Back

max
Ψi
Z 1
0
Vi(t)
z }| {
Z 1
t
Vi(G−1
i (x))dx + max{0, λ̄i − α}ri1{t=0}

dΨi(t)
s.t. Ψi ≻MPS
F−1
i
Agents are partitioned into intervals according to WTP:
Back

max
Ψi
Z 1
0
Vi(t)
z }| {
Z 1
t
Vi(G−1
i (x))dx + max{0, λ̄i − α}ri1{t=0}

dΨi(t)
s.t. Ψi ≻MPS
F−1
i
▶ Market allocation: assortative matching between WTP and quality
Qi(r) = F−1
i (Gi(r)), ∀r ∈ [a, b];
Back

max
Ψi
Z 1
0
Vi(t)
z }| {
Z 1
t
Vi(G−1
i (x))dx + max{0, λ̄i − α}ri1{t=0}

dΨi(t)
s.t. Ψi ≻MPS
F−1
i
Qi(r) = F−1
i (Gi(r)), ∀r ∈ [a, b];
▶ Non-market allocation: random matching between WTP and quality
Qi(r) = q̄, ∀r ∈ [a, b].
Back

max
Ψi, F̃i
Z 1
0
Vi(t)
z }| {
Z 1
t
Vi(G−1
i (x))dx + max{0, λ̄i − α}ri1{t=0}

dΨi(t)
s.t. Ψi ≻MPS
F̃−1
i ≻FOSD
F−1
i
Qi(r) = F−1
i (Gi(r)), ∀r ∈ [a, b];
▶ Non-market allocation: random matching between WTP and quality
Qi(r) = q̄, ∀r ∈ [a, b].
Back

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