1. The document discusses a framework for determining the optimal allocation of goods when redistributive concerns are present. It considers allocating heterogeneous goods to agents who differ in observable characteristics, willingness to pay, and unobserved social welfare weights. 2. The designer aims to maximize a weighted sum of revenue and agents' utilities, where the weights reflect redistributive preferences. Non-market mechanisms may be optimal when observable information is imprecise or when certain groups have high welfare weights but monetary transfers are infeasible. 3. The optimal mechanism depends on how labels and willingness to pay statistically correlate with unobserved welfare weights, allowing the designer to estimate agents' relative "need".

1. Redistributive Allocation Mechanisms
(with applications to Covid-19 vaccines and energy pricing in Europe)
Mohammad Akbarpour r
O Piotr Dworczak r
O Scott Duke Kominers*
(Stanford) (Northwestern & GRAPE) (Harvard)
Feb 2, 2023
Saieh Family Fellows in Economics Seminar
*Authors’ names are in certified random order. Research funded by ERC grant IMD-101040122.

2. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing

3. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care

4. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food

5. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access

6. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits

7. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits
...

8. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits
...
Application #1: Covid-19 vaccines

9. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits
...
Application #1: Covid-19 vaccines
Application #2: Electrical energy to European consumers

10. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits
...
Application #1: Covid-19 vaccines
Application #2: Electrical energy to European consumers

11. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits
...
Application #1: Covid-19 vaccines
Application #2: Electrical energy to European consumers
Why not use a market mechanism when we know that it leads to
efficient outcomes?

12. Motivation
Many goods and services are allocated using non-market mechanisms:
Housing
Health care
Food
Road access
National park permits
...
Application #1: Covid-19 vaccines
Application #2: Electrical energy to European consumers
Why not use a market mechanism when we know that it leads to
efficient outcomes?
Common answer: Redistributive concerns

13. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.

14. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.
Designer allocates a set of goods of heterogeneous quality to agents
differing in their observed and unobserved characteristics.

15. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.
Designer allocates a set of goods of heterogeneous quality to agents
differing in their observed and unobserved characteristics.
We derive the optimal mechanism under an objective that reflects
redistributive concerns via welfare weights.

16. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.
Designer allocates a set of goods of heterogeneous quality to agents
differing in their observed and unobserved characteristics.
We derive the optimal mechanism under an objective that reflects
redistributive concerns via welfare weights.
It may be optimal to use a non-market mechanism in two broad
cases:

17. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.
Designer allocates a set of goods of heterogeneous quality to agents
differing in their observed and unobserved characteristics.
We derive the optimal mechanism under an objective that reflects
redistributive concerns via welfare weights.
It may be optimal to use a non-market mechanism in two broad
cases:
1 When observable information is not very precise and willingness to pay
is not aligned with the social objective function (primarily, due to
redistributive preferences).

18. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.
Designer allocates a set of goods of heterogeneous quality to agents
differing in their observed and unobserved characteristics.
We derive the optimal mechanism under an objective that reflects
redistributive concerns via welfare weights.
It may be optimal to use a non-market mechanism in two broad
cases:
1 When observable information is not very precise and willingness to pay
is not aligned with the social objective function (primarily, due to
redistributive preferences).
2 When observables identify groups of agents with high welfare weights
but monetary transfers to these groups are infeasible or costly.

19. Preview
This paper: A market-design framework for determining the optimal
allocation under redistributive concerns.
Designer allocates a set of goods of heterogeneous quality to agents
differing in their observed and unobserved characteristics.
We derive the optimal mechanism under an objective that reflects
redistributive concerns via welfare weights.
It may be optimal to use a non-market mechanism in two broad
cases:
1 When observable information is not very precise and willingness to pay
is not aligned with the social objective function (primarily, due to
redistributive preferences).
2 When observables identify groups of agents with high welfare weights
but monetary transfers to these groups are infeasible or costly.
We cast light on the structure of the optimal mechanism.

20. Literature
Price control, public provision of goods: Weitzman (1977), Nichols and
Zeckhauser (1982), Blackorby and Donaldson (1988), Besley and Coate (1991),
Viscusi, Harrington, and Vernon (2005), Gahvari and Mattos (2007), Bulow and
Klemperer (2012), Condorelli (2013), Dworczak r
O Kominers r
O Akbarpour (2021)
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), Olszewski
and Siegel (2019), ...
Methods (generalized ironing): Muir and Loertscher (2020), Ashlagi, Monachou,
and Nikzad (2020), Kleiner, Moldovanu, and Strack (2020), Z. Kang (2020a), ...
New threads in mechanism design with redistribution motives: Z. Kang
(2020b), M. Kang and Zheng (2020, 2022), Reuter and Groh (2021), Fan, Chen,
and Tang (2021), Pai and Strack (2022), Z. Kang (2022)...

21. Model
Model

22. 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)

23. 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)] .

24. 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.

25. 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.

26. 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.
Designer’s objective becomes:
E(i,r) [αTi(r) + λi(r)Ui(r)] .

27. Comments about the model
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.

28. Comments about the model
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.

29. Comments about the model
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;

30. Comments about the model
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: Labels could capture family status, ethnicity, race,
educational background, neighborhood etc.

31. Comments about the model
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: $50,000
Agent B: $1,000
Naturally, we may presume that λi(r) is decreasing in r.

32. Comments about the model
Correlation of WTP with welfare weights:
Many labels, i captures low income, and we elicit information
about willingness to pay of two patients for a treatment:
Agent A: $50,000
Agent B: $1,000
Naturally, we may presume that λi(r) is (“less”) decreasing in r.

33. Comments about the model
Correlation of WTP with welfare weights:
Suppose there is just one label I = {i}, but we elicit information
about willingness to pay of two consumers for a concert ticket:
Agent A: $50
Agent B: $0
Naturally, we may presume that λi(r) is nearly constant in r.

34. Comments about the model
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.”

35. Comments about the model
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.

36. Comments about the model
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”.

37. Comments about the model
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.

38. Comments about the model
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.
When α = 0, we are in the money-burning/costly-screening case.

39. Derivation of Optimal Mechanism
Derivation of Optimal Mechanism

40. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;

41. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i. Details

42. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i. Details

43. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i. Details
Observation: Only expected quality, Qi(r), matters for payoffs.

44. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i. Details
Observation: Only expected quality, Qi(r), matters for payoffs.
Result: Within a group, agents are partitioned into intervals according to
WTP, with either the “market” or “non-market” allocation in each interval

45. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i. Details
Observation: Only expected quality, Qi(r), matters for payoffs.
Result: 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];

46. Derivation of Optimal Mechanism
1 Objects are allocated “across” groups: F is split into I cdfs F⋆
i ;
2 Objects are allocated “within” groups: For each label i, the objects
F⋆
i are allocated optimally to agents in group i. Details
Observation: Only expected quality, Qi(r), matters for payoffs.
Result: 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].

47. Economic Implications
Economic Implications

48. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.

49. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).

50. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
Pareto optimality with perfectly transferable utility:
Vi(r) = r

51. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
Pareto optimality with perfectly transferable utility:
Vi(r) = r
text

52. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
If Gi(r) is a distribution of WTP with positive density gi(r) on [ri, r̄i] in
group i, then:
Vi(r) =
1 − Gi(r)
gi(r)
+
r −
1 − Gi(r)
gi(r)
text

53. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
If Gi(r) is a distribution of WTP with positive density gi(r) on [ri, r̄i] in
group i, then:
Vi(r) =
1 − Gi(r)
gi(r)
| {z }
utility
+
r −
1 − Gi(r)
gi(r)
| {z }
revenue

54. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
If Gi(r) is a distribution of WTP with positive density gi(r) on [ri, r̄i] in
group i, then:
Vi(r) = Λi(r) ·
1 − Gi(r)
gi(r)
| {z }
utility
+α ·
r −
1 − Gi(r)
gi(r)
| {z }
revenue

55. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
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) = E[ λ | r̃ ≥ r, i].

56. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
Assuming differentiability, non-market mechanism is optimal when
Λ′
i(r)
1 − Gi(r)
gi(r)
+ α + (α − Λi(r))
d
dr
1 − Gi(r)
gi(r)
0.

57. Economic Implications
Let Vi(r) be the expected per-unit-of-quality social value from allocating a
good to an agent with WTP r in group i in an incentive-compatible
mechanism.
Proposition 1 (WTP-revealed inequality )
Suppose that lump-sum payments can be made to group i (α ≥ λ̄i).
Then, the market allocation in group i is optimal if and only if Vi(r) is
non-decreasing (and, otherwise, some agents within group i receive a
random allocation).
Assuming differentiability, non-market mechanism is optimal when
Λ′
i(r)
1 − Gi(r)
gi(r)
+ α + (α − Λi(r))
d
dr
1 − Gi(r)
gi(r)
0.
Some non-market allocation is optimal if willingness to pay is strongly
(negatively) correlated with the unobserved social welfare weights.

58. Economic Implications
Proposition 2 (Label-revealed inequality )
Suppose that lump-sum payments to group i are costly or infeasible, that
is, λ̄i α.

59. Economic Implications
Proposition 2 (Label-revealed inequality )
Suppose that lump-sum payments to group i are costly or infeasible, that
is, λ̄i α.
Then, if the good is universally desired (ri 0), there exists r⋆
i ri such
that the optimal allocation is random at a price of 0 for all types r ≤ r⋆
i .

60. Economic Implications
Proposition 2 (Label-revealed inequality )
Suppose that lump-sum payments to group i are costly or infeasible, that
is, λ̄i α.
Then, if the good is universally desired (ri 0), there exists r⋆
i ri such
that the optimal allocation is random at a price of 0 for all types r ≤ r⋆
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.

61. Economic Implications
Proposition 2 (Label-revealed inequality )
Suppose that lump-sum payments to group i are costly or infeasible, that
is, λ̄i α.
Then, if the good is universally desired (ri 0), there exists r⋆
i ri such
that the optimal allocation is random at a price of 0 for all types r ≤ r⋆
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.

62. Economic Implications
Proposition 2 (Label-revealed inequality )
Suppose that lump-sum payments to group i are costly or infeasible, that
is, λ̄i α.
Then, if the good is universally desired (ri 0), there exists r⋆
i ri such
that the optimal allocation is random at a price of 0 for all types r ≤ r⋆
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.
Intuition: A random allocation for free enables the designer to lower
prices for all agents.

63. Economic Implications
Proposition 3 (Optimality of free provision)
A necessary condition for a fully random allocation to be optimal within
group i is that
αr̄i ≤
Z r̄i
ri
rλi(r)dGi(r).
This condition becomes sufficient if Vi(r) is quasi-convex.

64. Economic Implications
Proposition 3 (Optimality of free provision)
A necessary condition for a fully random allocation to be optimal within
group i is that
αr̄i ≤
Z r̄i
ri
rλi(r)dGi(r).
This condition becomes sufficient if Vi(r) is quasi-convex.
Key take-away: Condition implies that α λ̄i, so it is never optimal to
provide the good for free if a lump-sum payment to group i is feasible.

65. Economic Implications
Proposition 4 (Across-group allocation )
Suppose that there are two groups. Then, (up to relabeling of the groups):

66. Economic Implications
Proposition 4 (Across-group allocation )
Suppose that there are two groups. Then, (up to relabeling of the groups):
1 If fully random allocation is optimal within both groups, then
max(supp(F⋆
1 )) ≤ min(supp(F⋆
2 )) in an optimal mechanism;

67. Economic Implications
Proposition 4 (Across-group allocation )
Suppose that there are two groups. Then, (up to relabeling of the groups):
1 If fully random allocation is optimal within both groups, then
max(supp(F⋆
1 )) ≤ min(supp(F⋆
2 )) in an optimal mechanism;
2 If the market allocation is optimal within both groups, then there
could be non-trivial overlaps in supp(F⋆
1 ) and supp(F⋆
2 );

68. Economic Implications
Proposition 4 (Across-group allocation )
Suppose that there are two groups. Then, (up to relabeling of the groups):
1 If fully random allocation is optimal within both groups, then
max(supp(F⋆
1 )) ≤ min(supp(F⋆
2 )) in an optimal mechanism;
2 If the market allocation is optimal within both groups, then there
could be non-trivial overlaps in supp(F⋆
1 ) and supp(F⋆
2 ); if the good is
not universally desired and r̄1 = r̄2, then supp(F⋆
1 ) = supp(F⋆
2 ).

69. Economic Implications
Proposition 4 (Across-group allocation )
Suppose that there are two groups. Then, (up to relabeling of the groups):
1 If fully random allocation is optimal within both groups, then
max(supp(F⋆
1 )) ≤ min(supp(F⋆
2 )) in an optimal mechanism;
2 If the market allocation is optimal within both groups, then there
could be non-trivial overlaps in supp(F⋆
1 ) and supp(F⋆
2 ); if the good is
not universally desired and r̄1 = r̄2, then supp(F⋆
1 ) = supp(F⋆
2 ).
3 If the market allocation is optimal for group 1, and
fully random allocation is optimal for group 2,
then group 2 receives intermediate quality: For some q ≤ q̄,
supp(F⋆
2 ) = [q, q̄] ∩ supp(F),
supp(F⋆
1 ) = ([0, q] ∪ [q̄, 1]) ∩ supp(F).

70. Conclusions
Conclusions

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

72. Conclusions
When to use in-kind redistribution? (random allocation of a range of
quality levels at a below-market price)
Roughly: When observable variables uncover inequality in the unobserved
welfare weights.

73. Conclusions
When to use in-kind redistribution? (random allocation of a range of
quality levels at a below-market price)
Roughly: When observable variables uncover inequality in the unobserved
welfare weights.
1 Label-revealed inequality:

74. Conclusions
When to use in-kind redistribution? (random allocation of a range of
quality levels at a below-market price)
Roughly: When observable variables uncover inequality in the unobserved
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.

75. Conclusions
When to use in-kind redistribution? (random allocation of a range of
quality levels at a below-market price)
Roughly: When observable variables uncover inequality in the unobserved
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.

76. Conclusions
When to use in-kind redistribution? (random allocation of a range of
quality levels at a below-market price)
Roughly: When observable variables uncover inequality in the unobserved
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.
2 WTP-revealed inequality: When the welfare weights are strongly
and negatively correlated with willingness to pay.

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

78. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.

79. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization:

80. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal

81. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal
▶ Allocation of goods to corporations (oil leases, spectrum licenses etc.)

82. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal
▶ Allocation of goods to corporations (oil leases, spectrum licenses etc.)
▶ Whenever lump-sum transfers to the target population are feasible.

83. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal
▶ Allocation of goods to corporations (oil leases, spectrum licenses etc.)
▶ Whenever lump-sum transfers to the target population are feasible.
2 Efficiency maximization: When the welfare weights are not strongly
correlated with willingness to pay:

84. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal
▶ Allocation of goods to corporations (oil leases, spectrum licenses etc.)
▶ Whenever lump-sum transfers to the target population are feasible.
2 Efficiency maximization: When the welfare weights are not strongly
correlated with willingness to pay:
▶ Small dispersion in welfare weights to begin with;

85. Conclusions
When to use market mechanisms? (assortative matching of a range of
quality levels at market-clearing prices)
Roughly: When the motives to maximize revenue and allocative efficiency
dominate the redistributive concerns.
1 Revenue maximization: As long as the weight on revenue is above
the average Pareto weight, some assortative matching is optimal
▶ Allocation of goods to corporations (oil leases, spectrum licenses etc.)
▶ Whenever lump-sum transfers to the target population are feasible.
2 Efficiency maximization: When the welfare weights are not strongly
correlated with willingness to pay:
▶ 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).

86. Conclusions
We proposed a model of optimal object allocation under redistributive
concerns—Inequality-aware Market Design
Our analysis uncovers the importance of three factors:
1 Correlation between the unobserved welfare weights and the
information that the designer can elicit or observe directly;
2 The role of raising revenue and availability of lump-sum transfers;
3 Whether the good is universally desired or not.
Open questions:
1 Relaxing linearity assumptions (income effects, risk aversion);
2 Interaction with “conventional” public finance.
3 Additional tools (queuing, information provision, resale etc.)

87. Derivation of Optimal Mechanism: within-group allocation
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88. Derivation of Optimal Mechanism: within-group allocation
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89. Derivation of Optimal Mechanism: within-group allocation
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90. Derivation of Optimal Mechanism: within-group allocation
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91. Derivation of Optimal Mechanism: within-group allocation
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92. Derivation of Optimal Mechanism: within-group allocation
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93. Derivation of Optimal Mechanism: within-group allocation
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94. Derivation of Optimal Mechanism: across-group allocation
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95. Derivation of Optimal Mechanism: across-group allocation
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96. Derivation of Optimal Mechanism: across-group allocation
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97. Derivation of Optimal Mechanism: across-group allocation
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98. Derivation of Optimal Mechanism: across-group allocation
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99. Derivation of Optimal Mechanism: across-group allocation
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100. Derivation of Optimal Mechanism: across-group allocation
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101. Derivation of Optimal Mechanism: across-group allocation
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102. Derivation of Optimal Mechanism: across-group allocation
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103. Derivation of Optimal Mechanism: across-group allocation
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104. Derivation of Optimal Mechanism: across-group allocation
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105. Derivation of Optimal Mechanism: across-group allocation
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106. Derivation of Optimal Mechanism: across-group allocation
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107. Derivation of Optimal Mechanism: across-group allocation
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108. Derivation of Optimal Mechanism: across-group allocation
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109. Derivation of Optimal Mechanism: across-group allocation
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110. Derivation of Optimal Mechanism: across-group allocation
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111. Derivation of Optimal Mechanism: across-group allocation
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112. Derivation of Optimal Mechanism: across-group allocation
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113. Derivation of Optimal Mechanism: across-group allocation
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