Redistributive Allocation Mechanisms

Redistributive Allocation Mechanisms
(with an application to the potential European energy crisis)
Mohammad Akbarpour r
O Piotr Dworczak r
O Scott Duke Kominers*
(Stanford) (GRAPE & Northwestern) (Harvard)
22 November, 2022
European Central Bank, invited speaker seminar
Research sponsored by ERC grant IMD-101040122.
*Authors’ names are in certified random order.

Motivation
Many goods and services are allocated using non-market mechanisms,
even though monetary transfers are feasible:
Housing

Motivation
Housing
Health care

Motivation
Housing
Health care
Food

Motivation
Housing
Health care
Food
Road access

Motivation
Housing
Health care
Food
Road access
...

Motivation
Housing
Health care
Food
Road access
...
Follow-up paper: Covid-19 vaccines

Motivation
Housing
Health care
Food
Road access
...
Follow-up paper: Covid-19 vaccines
Focus today: Energy this winter in Europe

Motivation—European energy crisis
Source: Sgaravatti, G., S. Tagliapietra, G. Zachmann (2021) ‘National policies to shield consumers from rising energy prices’

Motivation
Classical economic perspective: Distorting prices will lead to
inefficiency (I WT); if you want to redistribute, do so via lump-sum
payments (II WT)

Motivation
payments (II WT)
▶ echoed in The Economist on 08/09/2022 and IMF WP/22/152

Motivation
payments (II WT)
But: Welfare theorems ignore the issue of private information!

Motivation
payments (II WT)
Governments do have some information... but far from perfect in
practice:

Motivation
payments (II WT)
practice:
▶ Detailed financial situation?

Motivation
payments (II WT)
practice:
▶ Job market opportunities?

Motivation
payments (II WT)
practice:
▶ Exposure to interest rate hikes?

Motivation
payments (II WT)
practice:
▶ Energy efficiency of housing?

Motivation
payments (II WT)
practice:
▶ Heating method?

Motivation
payments (II WT)
practice:
▶ Heating method?
▶ Family size?

Inequality-aware Market Design
Our approach: A market-design framework for determining the optimal
allocation under redistributive concerns
(Market-design perspective: the designer only controls a single market)

Designer allocates a set of goods to agents differing in their observed
and unobserved characteristics.

We derive the optimal mechanism under IC and IR constraints and
an objective that reflects redistributive concerns via welfare weights.

Some key take-aways:

▶ Non-market mechanisms can be optimal!

▶ If lump-sum payments are available, uniform price caps are never
optimal;

optimal;
▶ Instead, price caps should only be applied up to some threshold
consumption level;

optimal;
▶ Instead, price caps should only be applied up to some threshold
consumption level;
▶ Uniform price caps can be optimal when lump-sum payments are
costly;

Literature
Weitzman (1977): First paper to point out that random allocation
might be preferred to market pricing when agents’ “needs” are not
well expressed by willingness to pay.
Condorelli (2013): A full mechanism-design approach to Weitzman’s
problem
▶ On a technical level: We add heterogeneous quality of objects,
continuum of goods and agents, groups of agents with the same
observable characteristics, and designer’s preferences over revenue.
▶ On a conceptual level: We focus on a particular objective function
(weighted sum of utilities and revenue) and connect market
circumstances with practical conclusions about the optimal design.
Dworczak r
O Kominers r
O Akbarpour (2021): two-sided market
but “maximally simplified” (no heterogeneous quality, no observable
information, no restrictions on lump-sum transfers)

Literature
Price control, public provision of goods: Nichols and Zeckhauser (1982), Besley
and Coate (1991), Blackorby and Donaldson (1988), Viscusi, Harrington, and
Vernon (2005), Gahvari and Mattos (2007), Currie and Gahvari (2008)
Optimal taxation: Diamond and Mirrlees (1971), Atkinson and Stiglitz (1976),
Akerlof (1978), Saez and Stantcheva (2016)
Auctions with budget-constrained bidders: Che and Gale (1998), Fernandez and
Gali (1999); Che, Gale, and Kim (2012); Pai and Vohra (2014); Kotowski (2017)
Costly-screening/ money-burning literature: McAfee and McMillan (1992),
Damiano and Li (2007), Hartline and Roughgarden (2008), Hoppe, Moldovanu,
and Sela (2009), Condorelli (2012), Chakravarty and Kaplan (2013)
Methods (generalized ironing): Muir and Loertscher (2021), Ashlagi, Monachou,
and Nikzad (2021), Kleiner, Moldovanu, and Strack (2021), Z. Kang (2020a).
New threads in mechanism design with redistribution motives: Z. Kang
(2020b), M. Kang and Zheng (2020, 2021), Reuter and Groh (2021)

Model
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)

Model
Comments for the energy context:
Quality normalized to 0 or 1, interpreted as quantity.
Then, q = 1 can be interpreted as the “normal consumption level.”
Instead of distribution of quality F, we just have total quantity S
available: F(q) = 1{q≥S}.
Throughout, assume that S < 1.
To the extent that some households have a higher “normal
consumption level,” we can (intuitively) fold it into the WTP r
r · qnormal = rqnormal
| {z }
r̃
· 1

Model
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)] .

Comments about the model
Lemma: It is without loss of optimality for the mechanism designer to
only elicit information about r through the mechanism—allocation and
transfers depend on (i, r) but not on λ directly.

This implies that the designer forms expectations over the unobserved
social welfare weights:
λi(r) ≡ E(i,r,λ) [λ| i, r] ;
we call these functions Pareto weights, and assume they are continuous.

This implies that the designer forms expectations over the unobserved
social welfare weights:
λi(r) ≡ E(i,r,λ) [λ| i, r] ;
we call these functions Pareto weights, and assume they are continuous.
Designer’s objective becomes:
E(i,r) [αTi(r) + λi(r)Ui(r)] .

λi(r) ≡ E(i,r,λ) [λ| i, r]
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) ≡ E(i,r,λ) [λ| i, r]
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.
Consequence:
The optimal allocation depends on the statistical correlation of labels
and willingness to pay with the unobserved social welfare weights.

Correlation of labels with welfare weights:
If label i captures income brackets, then we may naturally think that
E[λ| i] ≡ λ̄i ≥ λ̄j ≡ E[λ| j]
if i corresponds to a lower income bracket than j;

Correlation of labels with welfare weights:
If label i captures income brackets, then we may naturally think that
E[λ| i] ≡ λ̄i ≥ λ̄j ≡ E[λ| j]
if i corresponds to a lower income bracket than j;
Other possibilities (varies by country): Labels could capture family
size, energy efficiency of housing, heating method, exposure to
interest rate hikes.

Correlation of WTP with welfare weights:
Suppose there is just one label I = {i}, but we elicit information
about willingness to pay of two patients for a treatment:
Agent A: e50,000
Agent B: e1,000
Naturally, we may presume that λi(r) is decreasing in r.

Many labels, i captures low income, and we elicit information
about willingness to pay of two patients for a treatment:
Agent A: e 50,000
Agent B: e1,000
Naturally, we may presume that λi(r) is (“less”) decreasing in r.

Suppose there is just one label I = {i}, but we elicit information
about WTP of two consumers for maintaining “normal” energy
consumption:
Agent A: $10,000
Agent B: $5,000
Naturally, we may presume that λi(r) is decreasing in r.

Many labels, i captures family size and energy efficiency, but we
elicit information about WTP of two consumers for maintaining
“normal” energy consumption:
Agent A: $10,000
Agent B: $5,000
Naturally, we may presume that λi(r) is (“more”) decreasing in r.

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.”

When α = maxi λ̄i, then it is as if a lump-sum transfer to group i
were allowed; when α = averagei λ̄i, then it is as if lump-sum
transfers to all agents were allowed.

When α > λ̄i for all i, there is an “outside cause”.

When α < λ̄i for some i, lump-sum payments to agents in group
i are prohibited or costly (insurance motive as in Gadenne et al.,
2022).

When α < λ̄i for some i, lump-sum payments to agents in group
i are prohibited or costly (insurance motive as in Gadenne et al.,
2022).
When α = 0, we are in the money-burning/costly-screening case.

Universally desired goods

Let Gi(r) be the cdf of willingness to pay conditional on label i, and let
gi(r) be its continuous positive density on [ri, r̄i].

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.

Examples:
“Essential” goods: housing, energy, food, basic health care;

Examples:
“Essential” goods: housing, energy, food, basic health care;
“Stores of value”: shares, currencies, commodities

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. Details

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

i ;
F⋆
Result: Within a group, agents are partitioned into intervals according to
WTP, with either the “market” or “non-market” allocation in each interval

i ;
F⋆
Market allocation: assortative matching between WTP and quality
Qi(r) = (F⋆
i )−1
(Gi(r)), ∀r ∈ [a, b];

i ;
F⋆
Market allocation: assortative matching between WTP and quality
Qi(r) = (F⋆
i )−1
(Gi(r)), ∀r ∈ [a, b];
Non-market allocation: random matching (rationing!) between
WTP and quality
Qi(r) = q̄, ∀r ∈ [a, b].

Derivation of Optimal Mechanism: within-group allocation

Derivation of Optimal Mechanism: across-group allocation

Economic Implications

Proposition 1 (WTP-revealed inequality )
Suppose that α ≥ λ̄i (lump-sum payments are available).

Then, it is optimal to ration agents with willingness to pay in some
(non-degenerate) interval if and only if the function αJi(r) + Λi(r)hi(r) is
not non-decreasing, where
hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].

hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].
Assuming differentiability, the condition becomes
Λ′
i(r)hi(r) + α + (Λi(r) − α)h′
i(r) < 0.

hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].
Assuming differentiability, the condition becomes
Λ′
i(r)hi(r) + α + (Λi(r) − α)h′
i(r) < 0.
Some non-market allocation is optimal if willingness to pay is strongly
(negatively) correlated with the unobserved social welfare weights.

Proposition 2 (Optimality of uniform rationing)
A necessary condition for uniform rationing (at a fixed low price) to be
optimal within group i is that
αr̄i ≤
Z r̄i
ri
rλi(r)dGi(r).
This condition becomes sufficient if αJi(r) + Λi(r)hi(r) is quasi-convex.

αr̄i ≤
Z r̄i
ri
rλi(r)dGi(r).
Key take-away: It is never optimal to use uniform rationing if a
lump-sum payment to group i is feasible.

αr̄i ≤
Z r̄i
ri
rλi(r)dGi(r).
Key take-away: It is never optimal to use uniform rationing if a
lump-sum payment to group i is feasible.
By an extension of the same argument: It is never optimal to have a
uniform price cap.

Key idea behind optimality of rationing:
Offering a reduced-price option with rationing allows to screen for
vulnerable households.

Each household chooses between a high per-unit price with no limit
on consumption or a lower per-unit price but up to a consumption
threshold q̄ < 1.

threshold q̄ < 1.
Selection will be based on WTP.

threshold q̄ < 1.
Key question: Do we get the “right” selection?

threshold q̄ < 1.
If yes, subsidize (by reducing the per-unit price) the group that
self-identified as being in need.

threshold q̄ < 1.
If yes, subsidize (by reducing the per-unit price) the group that
self-identified as being in need.
If no, then the rationing policy is pure allocative inefficiency.

The answer to the question largely depends on available labels!

Mathematically, we need strong negative correlation between WTP
and welfare weights (Λ′
i(r) < 0)

i(r) < 0)
This might hold when we control for confounding factors that wold
raise the WTP of poor households:

i(r) < 0)
▶ Large family size;

i(r) < 0)
▶ Energy inefficiency of housing;

i(r) < 0)
▶ Energy inefficiency of housing;
▶ Electricity-based heating.

Advantage of consumption-capped price subsidy over lump-sum
payments: Additional screening!

Market choices may reveal which households are really in need.

Then, clear why uniform price and quantity caps can’t work!

They distort economic efficiency without offering the screening
benefit.

benefit.
Thus, they are dominated by lump-sum payments (here, we agree
with the Economist and the IMF paper)

benefit.
Thus, they are dominated by lump-sum payments (here, we agree
with the Economist and the IMF paper)
What if lump-sum payments are costly? (administration costs,
insurance motive, behavioral aspects, etc.)

Proposition 3 (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 to ration (at
a price of 0) 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 small quantities 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 small quantities of universally desired goods for free.
Note: There might still be a price gradient for higher quantities (the exact
condition is the one we have seen before)

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? (provision of goods at a
below-market-clearing prices)

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 lump-sum payments are costly and
some groups can be identified as vulnerable, it is optimal to use some
in-kind redistribution for universally desired goods.

Conclusions
welfare weights.
▶ Energy context: Controlling electricity bills for poorest households may
be easier/ cheaper than direct cash transfers.

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

Conclusions
welfare weights.
2 WTP-revealed inequality: When welfare weights are strongly and
negatively correlated with willingness to pay conditional on labels.
▶ Energy context: A well-designed subsidy system can screen for most
vulnerable households, better than a cash transfer scheme could.

Conclusions
We proposed a model of Inequality-aware Market Design.

Conclusions
Our analysis uncovers the importance of three factors:

Conclusions
1 Correlation between the unobserved welfare weights and the
information that the designer can elicit or observe directly;

Conclusions
2 Importance of availability of lump-sum transfers;

Conclusions
3 Whether the good is universally desired or not.

Conclusions
More work is needed on market design with redistributive
concerns:

Conclusions
concerns:
1 There are hundreds (thousands?) of theory papers focusing on how to
maximize revenue or efficiency when allocating resources...

Conclusions
concerns:
2 ... but only a handful on how to achieve optimal redistribution.

Conclusions
concerns:
2 ... but only a handful on how to achieve optimal redistribution.
3 Much more work remains to be done... (many stylized assumptions in
our model!)

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) = αJi(r) + Λi(r)hi(r), where
hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].

max
Qi(r)
Z r̄i
ri
hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].
Feasible distribution of expected quality: CDF Φi such that
Fi ≻MPS Φi.

max
Qi(r)
Z r̄i
ri
hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].
Fi ≻MPS Φi.
Incentive-compatibility =⇒ Gi(r) = Φi(q)

max
Qi(r)
Z r̄i
ri
hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ r].
Fi ≻MPS Φi.
Incentive-compatibility =⇒ Qi(r) = Φ−1
i (Gi(r))

max
Qi(r)
Z r̄i
ri
hi(r) =
1 − Gi(r)
gi(r)
, Ji(r) = r − hi(r), Λi(r) = Ei
[λi(r̃) | r̃ ≥ 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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