Redistributive Market Design

Redistributive Market Design
Mohammad Akbarpour
(Stanford)
r
O Piotr Dworczak
(Northwestern)
r
O Scott Duke Kominers
(Harvard & a16z crypto)
June 28, 2022
Tutorial – ACM Conference on Economics and Computation
(Authors’ names are in ce r
Otified random order.)

Introduction: A “market design” approach to redistribution
presented by Scott Duke Kominers

Overarching Question
How should we design marketplaces in the presence of inequality?

Framing assumption: designer regulates/controls one market.

Price controls, subsidies, and various forms of in-kind distribution are
abundant!
I public housing (/rent control)
I health care
I food (e.g., food stamps)
I road access (opposition to congestion pricing)
I vaccines(!)

abundant!
I health care
I vaccines(!)
Are these sorts of policies a good idea?

abundant!
I health care
I vaccines(!)
Are these sorts of policies a good idea?
Knee-jerk economics answer: NO!

Standard Economic Intuition – Revisited

Thought Experiment
Consider a frictionless buyer/seller market.

Thought Experiment
As sellers become (systematically) poorer than buyers, the
competitive-equilibrium price decreases.

Thought Experiment
However, when sellers are poorer, the designer would like them to
have more money on the margin, all else equal.

Thought Experiment
However, when sellers are poorer, the designer would like them to
have more money on the margin, all else equal.
⇒ Past some point, competitive-equilibrium pricing will not be socially
optimal(!).

Market Pricing vs. Rationing (Weitzman, 1977)
Two-dimensional type model:
= “value for the good”
λ = “value for money”

utility = A − Bλp +


Compare welfare under (optimal) rationing price to market-clearing price:
Key dynamic: Trade-off between value for the good and value for
money in determining willingness to pay.
Those with high λ have less “ability” to pay at any given price.


Compare welfare under (optimal) rationing price to market-clearing price:
“The price system has greater comparative effectiveness in sorting
out the deficit commodity and in supplying it to those who need
it most when wants are more widely dispersed or when the soci-
ety is relatively egalitarian in its income distribution. Conversely,
rationing is more effective as needs for the deficit commodity are
more uniform or as there is greater income inequality.”

“Economists sometimes maintain or imply that the market system
is a superior mechanism for distributing resources. After all, the
argument goes, consider any other allocation. There is always
some corresponding way of combining the price system with a
specific lump-sum transfer arrangement which will make everyone
better off (or at least no worse). That is true enough in principle,
but not typically very useful for policy prescriptions, because the
necessary compensation is practically never paid.”

“[Moreover, t]here is a class of commodities whose just distribution
is sometimes viewed as a desirable end in itself, independent of
how society may be allocating its other resources. While it is
always somewhat arbitrary where the line should be drawn, such
‘natural right goods’ as basic food and shelter, security, legal aid,
military service, medical assistance, education, justice, or even
many others are frequently deemed to be sufficiently vital in some
sense to give them a special status.”

When/can this work?

When/can this work?
Inferring an agent’s level of “need” from their market behavior.
I (So when does that work?)

When/can this work?
Important Medical Treatment
A: WTP = $30,000
B: WTP = $5,000
⇒ B is probably “cash-poor”

When/can this work?
Important Medical Treatment
A: WTP = $30,000
B: WTP = $5,000
Bottle of Milk
A: WTP = $3
B: WTP = $0.50
⇒ B probably doesn’t like milk

When/can this work?
Important Medical Treatment 4
A: WTP = $30,000
B: WTP = $5,000
Bottle of Milk 6
A: WTP = $3
B: WTP = $0.50
⇒ B probably doesn’t like milk

When/can this work?
Our ability to do redistributive market design depends on the market
context—in particular, the correlation between agent behavior (and
potentially other observables) and “need”(/our welfare goals).

Part I: Can price controls ever be optimal?
Presented by: Mohammad Akbarpour

An old question
The question of “are prices the best way to allocate resources?” is a
classic economic question.

An old question
Weitzman (1977) asked a simple version of this question:
There are X goods and you want to allocate them among N people.
Each individual has a need v and a marginal value for cash λ.
Which one is better: market mechanism or free random allocation?

An old question
Main insight: It all depends on V ar(v)
V ar(λ) . If this is large enough, market
mechanism is better, and if not, random allocation is better.
Why?

An old question
Main insight: It all depends on V ar(v)
V ar(λ) . If this is large enough, market
mechanism is better, and if not, random allocation is better.
Why?
We now ask this question in a more general way, before tackling it from a
mechanism design perspective.

Framework: Buying from sellers
There is an indivisible good K and a unit mass of sellers.
Each agent is characterized by a two-dimensional type (r, λ)
r is an unobserved willingness-to-receive and λ is an unobserved
social welfare weight.
If (r, λ) sells an object at price P, her utility will be P − r and her
contribution to social welfare will be λ(P − r).
G(r) is the distribution of r, known to the designer.
A designer wants to buy a quantity Q ≤ 1 of the good.
The designer’s budget is R, where R QG−1(Q). If spent less than
R, the rest will be redistributed as a lump-sum transfer.

Simplifying assumptions
The rate of substitution r is uniformly distributed on both sides of
the market.
In principle, an allocation mechanism can be fairly complex
I It can ask for agents types and offer them a (potentially infinite) menu
of prices and probabilities.
We consider a simple class of mechanisms where the designer uses a
single-price mechanism.
I Consequently, we can only elicit r. We then can estimate welfare
weights:
λ(r) = E [λ | r]

Simple price control – buying from sellers

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

max
pS≥G−1(Q)

Q
G(pS)
Z pS
r
λ(r)(pS − r)dG(r) + ΛS(R − pSQ)

,
where ΛS = E[λ].

max
pS≥G−1(Q)

Q
G(pS)
Z pS
r
λ(r)(pS − r)dG(r) + ΛS(R − pSQ)

,
where ΛS = E[λ].
We will refer to the price pC
S = G−1(Q) as the competitive price;
Any price pS pC
S leads to rationing.

Low seller-side inequality ⇐⇒ λ(r) ≤ 2ΛS.

Proposition 1
When seller-side inequality is low, it is optimal to choose pS = pC
S .

Proposition 1
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 1
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!

Economic trade-offs of simple price control
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 + E[λ(r)|r ≤ pS]

S :
−ΛS + E[λ(r)|r ≤ pS] ≥ 0 ↑

S :
−ΛS + E[λ(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.

Simple price controls: Selling to buyers

Framework: Selling to buyers
There is an indivisible good K and a unit mass of buyers.
Each agent is characterized by a two-dimensional type (r, λ)
r is an unobserved willingness-to-pay. and λ is an unobserved social
welfare weight.
If (r, λ) buys an object at price P, her utility will be r − P and her
contribution to social welfare will be λ(r − P).
G(r) is the distribution of r, known to the designer.
A designer wants to sell a quantity Q ≤ 1 of the good.
The designer wants to raise a revenue is R (R QG−1(1 − Q)). If
raised more than R, the rest will be redistributed as a lump-sum
transfer.

Simple price control – selling to buyers

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(1−Q)

Q
1 − G(pB)
Z r̄
pB
λ(r)(r − pB)dG(r) + ΛB(pBQ − R)

.
where ΛB = E[λ].

B (1 − Q)).
max
pB≤G−1(1−Q)

Q
1 − G(pB)
Z r̄
pB
λ(r)(r − pB)dG(r) + ΛB(pBQ − R)

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

Proposition 2
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 + E[λ(r)|r ≥ pB]

B:
−ΛB + E[λ(r)|r ≥ pB] ≤ 0 ↓

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

Two-price mechanisms

A designer can now post two prices, pH and pL, for sellers or buyers.

A designer can now post two prices, pH and pL, for sellers or buyers.
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
λ(r)(r − pL
B)dG(r) +
Z r̄
rδ
λB(r)(r − pH
B )dG(r)
+Λ pL
Bδ(G(rδ) − G(pL
B)) + pH
B (1 − G(rδ)) − R






subject to the market-clearing and revenue-target constraints
1 − δG(pL
B) − (1 − δ)G(rδ) = Q,
pL
Bδ(G(rδ) − G(pL
B)) + pH
B (1 − G(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
λ(r)(r − pL
B)dG(r) +
Z r̄
rδ
λB(r)(r − pH
B )dG(r)
+Λ pL
Bδ(G(rδ) − G(pL
B)) + pH
B (1 − G(rδ)) − R






subject to the market-clearing and revenue-target constraints
1 − δG(pL
B) − (1 − δ)G(rδ) = Q,
pL
Bδ(G(rδ) − G(pL
B)) + pH
B (1 − G(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
G(rδ) G(pL
B), i.e., if a non-zero measure of buyers choose the lottery.

Low buyer-side inequality ⇐⇒ λ(r) ≤ 2ΛB.

Proposition 3
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 3
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 3
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 + E[λ(r)| r ∈ [pL
B, rδ]]

−ΛB + E[λ(r)| r ∈ [pL
B, rδ]] ≥ 0

−ΛB + E[λ(r)| r ∈ [pL
B, rδ]] ≥ 0 ↑

−ΛB + E[λ(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.

Part II: A mechanism-design approach
presented by Piotr Dworczak

Preview
So far, we have focused on conceptual issues, and illustrated them
through simple examples.

Preview
But now we would like to understand the problem in some generality
and derive the optimal mechanism.

Preview
We can adapt the classical ironing technique of Myerson (1981) to
solve for the optimal mechanism under arbitrary redistributive
preferences.

Preview
We can adapt the classical ironing technique of Myerson (1981) to
solve for the optimal mechanism under arbitrary redistributive
preferences.
This part builds on Myerson (1981), Condorelli (2013), and
Akbarpour r
O Dworczak r
O Kominers (2021).

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)

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 1
It is optimal for the designer to condition the allocation and transfers only
on agents’ labels i and WTP r.

Framework
Lemma 1
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 1
Pareto optimality with perfectly transferable utility (first welfare theorem):
Vi(r) = r
text

Framework
Lemma 1
Let Gi(r) be a continuous distribution of WTP conditional on label i.
Vi(r) = r
text

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

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

r −
1 − Gi(r)
gi(r)

text

Framework
Lemma 1
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 1
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;
I Ironed virtual surplus is negative =⇒ do not allocate;

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

i ;
F?
g(r) ;
R 1
t V (G−1(x))dx;
I Ironed virtual surplus is constant =⇒ randomize (const. allocation);
I 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 1 (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
Λ0
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].

Λ0
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.

Λ0
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 2 (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.
I 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.
I 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
I Allocation of goods to corporations (oil leases, spectrum licenses etc.)

Conclusions
I 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
I Small dispersion in welfare weights to begin with;

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

Coda: A “market design” approach to redistribution
presented by Scott Duke Kominers

Review
Micro “market design” approach to redistribution.

Review
It may be worth distorting a market’s allocative efficiency in
order to improve equity.

Review
I Gives a justification for in-kind redistribution and priority-based
allocation; complements macro approaches/PF.

Review
Driving force is how/much agents’ behavior reveals about
welfare weights.

Review
Driving force is how/much agents’ behavior reveals about
welfare weights.
I Market behavior can sometimes reveal information that would not
otherwise be observable, even to a government(!).

Far more being done:
M.Kang–Zheng (2022): endogenous buyer/seller sorting ( welfare
weight on types near the cut-off determines whether to ration or not)

Z.Y.Kang (2020a): availability of public good affects who buys
private good ( public option rationed, although value of that option
goes to “middle class” rather than “poorest”)

Z.Y.Kang (2020b): “exploit the correlation between each individual’s
consumption behavior and the amount of externality generated”

Allen–Rehbeck (2021): joint identification of vK and vM

Che–Gale–Kim (2013): budget-constrained agents

Fan–Chen–Tang (2021): heterogeneous values and bargaining power

Ashlagi–Monachou–Nikzad (2021): optimal dynamic rationing

Ashlagi–Monachou–Nikzad (2021): optimal dynamic rationing
Also, e.g., M.Kang–Zheng (2020); Reuter–Groh (2020); Doval–Skreta
(2021); Matsushima (2021); Muir–Loertscher (2022),. . .

Far more to think about:
queuing – sorting on value for money vs. value of time (in progress w/Li);
relaxing linearity assumptions (income effects, risk aversion);
general equilibrium effects;
considering ex post fairness;
impact on investment incentives;
interaction with conventional redistributive tools;
practical limitations and implementation challenges. . .

Far more to think about:
queuing – sorting on value for money vs. value of time (in progress w/Li);
relaxing linearity assumptions (income effects, risk aversion);
general equilibrium effects;
considering ex post fairness;
impact on investment incentives;
interaction with conventional redistributive tools;
practical limitations and implementation challenges. . . .
end{tutorial}

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
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];
I 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];
I Non-market allocation: random matching between WTP and quality
Qi(r) = q̄, ∀r ∈ [a, b].
Back

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