This document summarizes Piotr Dworczak's presentation on the equity-efficiency trade-off in redistributing cash. Governments often engage in direct cash redistribution through programs like unemployment insurance. Dworczak studies how to allocate cash to individuals who privately value money differently. With no information, the optimal mechanism is a lump-sum transfer. However, if individuals can complete costly "ordeals," targeting can improve at the cost of efficiency. Dworczak's model finds an ordeal mechanism is better when the expected value of money for the lowest-cost individual exceeds the average by over a factor of two, providing a simple condition seen in other works.

1. Equity-efficiency trade-off in
quasi-linear environments
Piotr Dworczak
(Northwestern University & Group for Research in Applied Economics)
January 8, 2023
NAWM 2023, New Orleans

2. Motivation #1
Governments often engage in direct in-cash redistribution.

3. Motivation #1
Governments often engage in direct in-cash redistribution.
Classical example: Unemployment insurance.

4. Motivation #1
Governments often engage in direct in-cash redistribution.
Classical example: Unemployment insurance.
Two other examples for contrast:

5. Motivation #1
Governments often engage in direct in-cash redistribution.
Classical example: Unemployment insurance.
Two other examples for contrast:
Direct cash subsidies in the US during the Covid-19
pandemic.

6. Motivation #1
Governments often engage in direct in-cash redistribution.
Classical example: Unemployment insurance.
Two other examples for contrast:
Direct cash subsidies in the US during the Covid-19
pandemic.
Indonesia’s Conditional Cash Transfer Program (see Alatas
et al., 2016)

7. Motivation #1
Governments often engage in direct in-cash redistribution.
Classical example: Unemployment insurance.
Two other examples for contrast:
Direct cash subsidies in the US during the Covid-19
pandemic.
Indonesia’s Conditional Cash Transfer Program (see Alatas
et al., 2016)
Of course, governments use verifiable information to target.

8. Motivation #1
Governments often engage in direct in-cash redistribution.
Classical example: Unemployment insurance.
Two other examples for contrast:
Direct cash subsidies in the US during the Covid-19
pandemic.
Indonesia’s Conditional Cash Transfer Program (see Alatas
et al., 2016)
Of course, governments use verifiable information to target.
But should we also screen for unobserved characteristics?

9. Motivation #2
Equity-efficiency trade-off at the core of (public) economics.
Classical references: Diamond and Mirrlees (1971), Atkinson
and Stiglitz (1976), ...
In the context of allocating goods: Weitzman (1977), Condorelli
(2013), ...
Ongoing research agenda with Mohammad Akbarpour
and Scott Kominers: Inequality-aware Market Design.
How to design markets in the presence of socioeconomic
inequality among participants?

10. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).

11. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.

12. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.

13. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.
Versions of this condition (with “2”) keep coming up:

14. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.
Versions of this condition (with “2”) keep coming up:
Z. Kang (2020);

15. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.
Versions of this condition (with “2”) keep coming up:
Z. Kang (2020);
M. Kang and Zheng (2020);

16. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.
Versions of this condition (with “2”) keep coming up:
Z. Kang (2020);
M. Kang and Zheng (2020);
Pai and Strack (2022);

17. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.
Versions of this condition (with “2”) keep coming up:
Z. Kang (2020);
M. Kang and Zheng (2020);
Pai and Strack (2022);
ADK (2022).

18. Motivation #2
Key idea: Private information of market participants + social
welfare weights (as in Saez and Stantcheva, 2016).
DKA: Use rationing (price control) in the optimal market design
when inequality is large enough.
Exact condition: Ration when the expected welfare weight
conditional on the lowest willingness to pay for the good exceeds
the average welfare weight by more than a factor of two.
Versions of this condition (with “2”) keep coming up:
Z. Kang (2020);
M. Kang and Zheng (2020);
Pai and Strack (2022);
ADK (2022).
Why “2”?

19. What I do
I study a “pure” redistribution problem: How to allocate cash to
agents differing in their (privately-observed!) marginal values for
money (welfare weights).

20. What I do
I study a “pure” redistribution problem: How to allocate cash to
agents differing in their (privately-observed!) marginal values for
money (welfare weights).
Absent additional information, the only incentive-compatible
mechanism is a lump-sum transfer.

21. What I do
I study a “pure” redistribution problem: How to allocate cash to
agents differing in their (privately-observed!) marginal values for
money (welfare weights).
Absent additional information, the only incentive-compatible
mechanism is a lump-sum transfer.
But what if the designer can ask agents to burn utility (an
“ordeal mechanism”)?

22. What I do
I study a “pure” redistribution problem: How to allocate cash to
agents differing in their (privately-observed!) marginal values for
money (welfare weights).
Absent additional information, the only incentive-compatible
mechanism is a lump-sum transfer.
But what if the designer can ask agents to burn utility (an
“ordeal mechanism”)?
Trade-off: Better targeting but lower efficiency.

23. What I do
I study a “pure” redistribution problem: How to allocate cash to
agents differing in their (privately-observed!) marginal values for
money (welfare weights).
Absent additional information, the only incentive-compatible
mechanism is a lump-sum transfer.
But what if the designer can ask agents to burn utility (an
“ordeal mechanism”)?
Trade-off: Better targeting but lower efficiency.
An ordeal mechanism is better than lump-sum transfer when the
expected value for money conditional on the lowest cost for the
ordeal exceeds the average value by more than a factor of two.

24. What I do
I study a “pure” redistribution problem: How to allocate cash to
agents differing in their (privately-observed!) marginal values for
money (welfare weights).
Absent additional information, the only incentive-compatible
mechanism is a lump-sum transfer.
But what if the designer can ask agents to burn utility (an
“ordeal mechanism”)?
Trade-off: Better targeting but lower efficiency.
An ordeal mechanism is better than lump-sum transfer when the
expected value for money conditional on the lowest cost for the
ordeal exceeds the average value by more than a factor of two.
Simple robust intuition.

25. Note on literature
It is well known that ordeals can improve targeting:

26. Note on literature
It is well known that ordeals can improve targeting:
Hartline and Rouchgarden (2008), Condorelli (2012),
Chakravarty and Kaplan (2013), ...

27. Note on literature
It is well known that ordeals can improve targeting:
Hartline and Rouchgarden (2008), Condorelli (2012),
Chakravarty and Kaplan (2013), ...
Alatas et al (2016), Rose (2021), Zeckhauser (2021), ...

28. Note on literature
It is well known that ordeals can improve targeting:
Hartline and Rouchgarden (2008), Condorelli (2012),
Chakravarty and Kaplan (2013), ...
Alatas et al (2016), Rose (2021), Zeckhauser (2021), ...
Most of the theory work focused on efficiency as design goal
(allocation of goods other than money!)

29. Note on literature
It is well known that ordeals can improve targeting:
Hartline and Rouchgarden (2008), Condorelli (2012),
Chakravarty and Kaplan (2013), ...
Alatas et al (2016), Rose (2021), Zeckhauser (2021), ...
Most of the theory work focused on efficiency as design goal
(allocation of goods other than money!)
To the best of my knowledge, the condition is new.

30. Outline
1. Model
2. Main result + intuition
3. Mechanism design approach
4. Concluding remarks

31. Model
Designer has a budget B of money that she allocates to a unit
mass of agents.

32. Model
Designer has a budget B of money that she allocates to a unit
mass of agents.
Each agent has a “marginal value for money” v that is the
agent’s private information (this is just a welfare weight!)

33. Model
Designer has a budget B of money that she allocates to a unit
mass of agents.
Each agent has a “marginal value for money” v that is the
agent’s private information (this is just a welfare weight!)
The designer knows the distribution and maximizes total value.

34. Model
Designer has a budget B of money that she allocates to a unit
mass of agents.
Each agent has a “marginal value for money” v that is the
agent’s private information (this is just a welfare weight!)
The designer knows the distribution and maximizes total value.
Wlog, designer has no additional information.

35. Model
Designer has a budget B of money that she allocates to a unit
mass of agents.
Each agent has a “marginal value for money” v that is the
agent’s private information (this is just a welfare weight!)
The designer knows the distribution and maximizes total value.
Wlog, designer has no additional information.
The value of giving a lump-sum transfer is simply E[v] B.

36. Model
Designer has a budget B of money that she allocates to a unit
mass of agents.
Each agent has a “marginal value for money” v that is the
agent’s private information (this is just a welfare weight!)
The designer knows the distribution and maximizes total value.
Wlog, designer has no additional information.
The value of giving a lump-sum transfer is simply E[v] B.
I normalize E[v] = 1 (the “marginal value of public funds”).

37. Model
Suppose the designer can ask agents to “burn utility”:
Complete a task that is publicly observable, costly to the agent,
and socially wasteful (an “ordeal”).

38. Model
Suppose the designer can ask agents to “burn utility”:
Complete a task that is publicly observable, costly to the agent,
and socially wasteful (an “ordeal”).
The designer can choose a difficulty level y 2 [0; 1].

39. Model
Suppose the designer can ask agents to “burn utility”:
Complete a task that is publicly observable, costly to the agent,
and socially wasteful (an “ordeal”).
The designer can choose a difficulty level y 2 [0; 1].
Each agent has a privately-observed cost c.

40. Model
Suppose the designer can ask agents to “burn utility”:
Complete a task that is publicly observable, costly to the agent,
and socially wasteful (an “ordeal”).
The designer can choose a difficulty level y 2 [0; 1].
Each agent has a privately-observed cost c.
Utility of each agent is quasi-linear:
cy + vt;
when completing ordeal y and receiving a transfer t.

41. Model
Suppose the designer can ask agents to “burn utility”:
Complete a task that is publicly observable, costly to the agent,
and socially wasteful (an “ordeal”).
The designer can choose a difficulty level y 2 [0; 1].
Each agent has a privately-observed cost c.
Utility of each agent is quasi-linear:
cy + vt;
when completing ordeal y and receiving a transfer t.
Cost c measured in “social-utility units;” let k c=v be the cost
measured in money— only k matters for agents’ choices.

42. Model
Suppose the designer can ask agents to “burn utility”:
Complete a task that is publicly observable, costly to the agent,
and socially wasteful (an “ordeal”).
The designer can choose a difficulty level y 2 [0; 1].
Each agent has a privately-observed cost c.
Utility of each agent is quasi-linear:
cy + vt;
when completing ordeal y and receiving a transfer t.
Cost c measured in “social-utility units;” let k c=v be the cost
measured in money— only k matters for agents’ choices.
Assume k F with C1 density f on [k; k̄] with k = 0, f(k) 0.

43. Model
The designer now maximizes
E[ cy(k) + vt(k)]
subject to budget constraint and incentive-compatibility
(self-selection based on k).

44. Model
The designer now maximizes
E[ cy(k) + vt(k)]
subject to budget constraint and incentive-compatibility
(self-selection based on k).
Note: Any positive level of y is a pure social waste.

45. Model
The designer now maximizes
E[ cy(k) + vt(k)]
subject to budget constraint and incentive-compatibility
(self-selection based on k).
Note: Any positive level of y is a pure social waste.
If no redistributive preferences (all v = 1), then trivially lump-sum
payment is optimal.

46. Model
The designer now maximizes
E[ cy(k) + vt(k)]
subject to budget constraint and incentive-compatibility
(self-selection based on k).
Note: Any positive level of y is a pure social waste.
If no redistributive preferences (all v = 1), then trivially lump-sum
payment is optimal.
But in general, there is an equity-efficiency trade-off.

47. Analysis
Suppose that the designer offers an additional payment t0 for an
ordeal y0 0.

48. Analysis
Suppose that the designer offers an additional payment t0 for an
ordeal y0 0.
Agents with k t0=y0 accept.

49. Analysis
Suppose that the designer offers an additional payment t0 for an
ordeal y0 0.
Agents with k t0=y0 accept.
Let
E[vjc
v
= k]
be the conditional expectation of v conditional on the relative
cost being equal to k (assumed continuous).

50. Analysis
Suppose that the designer offers an additional payment t0 for an
ordeal y0 0.
Agents with k t0=y0 accept.
Let
E[vjc
v
= k]
be the conditional expectation of v conditional on the relative
cost being equal to k (assumed continuous).
E[vjk] measures targeting effectiveness.

51. Analysis
Suppose that the designer offers an additional payment t0 for an
ordeal y0 0.
Agents with k t0=y0 accept.
Let
E[vjc
v
= k]
be the conditional expectation of v conditional on the relative
cost being equal to k (assumed continuous).
E[vjk] measures targeting effectiveness.
Designer’s payoff from this “ordeal mechanism:”
Z t0=y0
0
E[vjk] (t0 ky0)f(k)dk + (B t0F(t0=y0)):

52. Analysis
We want to check whether the ordeal mechanism strictly
outperforms the lump-sum transfer for small t0 = y0:
Z
0
E[vjk]( k)f(k)dk F():

53. Analysis
We want to check whether the ordeal mechanism strictly
outperforms the lump-sum transfer for small t0 = y0:
Z
0
E[vjk]( k)f(k)dk F():
Enough to prove that the ratio of the RHS to LHS is above 1 in
the limit:
lim
!0
R
0 E[vjk]( k)f(k)dk
F()
(h)
= lim
!0
R
0 E[vjk]f(k)dk
f() + F()
(h)
=
E[vjk]
2
;
by L’Hôpital’s rule.

54. Analysis
We want to check whether the ordeal mechanism strictly
outperforms the lump-sum transfer for small t0 = y0:
Z
0
E[vjk]( k)f(k)dk F():
Enough to prove that the ratio of the RHS to LHS is above 1 in
the limit:
lim
!0
R
0 E[vjk]( k)f(k)dk
F()
(h)
= lim
!0
R
0 E[vjk]f(k)dk
f() + F()
(h)
=
E[vjk]
2
;
by L’Hôpital’s rule.
So ordeal is better than lump-sum when E[vjk] 2.

55. Analysis
Proposition
If the expected value for money conditional on the lowest relative cost
exceeds the value of public funds by more than a factor of two, i.e., if
E[vjk] 2; (?)
then the ordeal mechanism strictly outperforms giving a lump-sum
transfer for some positive 0.

56. Analysis
Proposition
If the expected value for money conditional on the lowest relative cost
exceeds the value of public funds by more than a factor of two, i.e., if
E[vjk] 2; (?)
then the ordeal mechanism strictly outperforms giving a lump-sum
transfer for some positive 0.
High-level intuition:
Better targeting requires sacrificing some surplus.

57. Analysis
Proposition
If the expected value for money conditional on the lowest relative cost
exceeds the value of public funds by more than a factor of two, i.e., if
E[vjk] 2; (?)
then the ordeal mechanism strictly outperforms giving a lump-sum
transfer for some positive 0.
High-level intuition:
Better targeting requires sacrificing some surplus.
For every dollar of public funds spent, only 1=2 of the dollar is
received by an agent (the other half is “burned” in screening).

58. Analysis
Proposition
If the expected value for money conditional on the lowest relative cost
exceeds the value of public funds by more than a factor of two, i.e., if
E[vjk] 2; (?)
then the ordeal mechanism strictly outperforms giving a lump-sum
transfer for some positive 0.
High-level intuition:
Better targeting requires sacrificing some surplus.
For every dollar of public funds spent, only 1=2 of the dollar is
received by an agent (the other half is “burned” in screening).
Thus, ordeal is better if the designer values each dollar given to
lowest-k (“poorest”?) agents at 2 or more.

59. Why 2?

60. Why 2?

61. Why 2?

62. Why 2?

63. Why 2?

64. Why 2?

65. Why 2?

66. Optimal mechanism
Is this intuition robust to choosing the optimal mechanism?

67. Optimal mechanism
Is this intuition robust to choosing the optimal mechanism?
max
y(k)2[0; 1]; t(k)0
Z k̄
k
E[vjk]( ky(k) + t(k))dF(k); (OBJ)
ky(k) + t(k) ky(k0) + t(k0); 8k; k0; (IC)
ky(k) + t(k) 0; 8k; (IR)
Z k̄
k
t(k)dF(k) = B: (B)

68. Optimal mechanism
Proposition
The optimal mechanism uses an ordeal (y is strictly positive for some
agents) if and only if
E[V(k)jk k0] 0 for some k0; (??)
where
V(k) =
E
h
vjc
v
k
i
1
F(k)
f(k)
k: (V)

69. Optimal mechanism
Proposition
The optimal mechanism uses an ordeal (y is strictly positive for some
agents) if and only if
E[V(k)jk k0] 0 for some k0; (??)
where
V(k) =
E
h
vjc
v
k
i
1
F(k)
f(k)
k: (V)
Condition (?) implies that V(k) 0 for small enough k; hence,
condition (?) implies condition (??).

70. Optimal mechanism
Proposition
The optimal mechanism uses an ordeal (y is strictly positive for some
agents) if and only if
E[V(k)jk k0] 0 for some k0; (??)
where
V(k) =
E
h
vjc
v
k
i
1
F(k)
f(k)
k: (V)
Condition (?) implies that V(k) 0 for small enough k; hence,
condition (?) implies condition (??).
Conversely, when V(k) is quasi-concave, condition (??) implies that
condition (?) must hold as a weak inequality.

71. Optimal mechanism
Proposition (continued)
When B is large enough, the optimal mechanism offers a single
payment k?
for completing the ordeal y = 1, and allocates the
remaining budget as a lump-sum payment.

72. Optimal mechanism
Proposition (continued)
When B is large enough, the optimal mechanism offers a single
payment k?
for completing the ordeal y = 1, and allocates the
remaining budget as a lump-sum payment.
Intuition:
Because I assumed k = 0, we have V(0) = 0.

73. Optimal mechanism
Proposition (continued)
When B is large enough, the optimal mechanism offers a single
payment k?
for completing the ordeal y = 1, and allocates the
remaining budget as a lump-sum payment.
Intuition:
Because I assumed k = 0, we have V(0) = 0.
Turns out that condition (?) is equivalent to V0(0) 0.

74. Optimal mechanism
Proposition (continued)
When B is large enough, the optimal mechanism offers a single
payment k?
for completing the ordeal y = 1, and allocates the
remaining budget as a lump-sum payment.
Intuition:
Because I assumed k = 0, we have V(0) = 0.
Turns out that condition (?) is equivalent to V0(0) 0.
This condition is (almost) necessary if V is quasi-concave.

75. Concluding remarks
Looked at a simple(st?) equity-efficiency problem.

76. Concluding remarks
Looked at a simple(st?) equity-efficiency problem.
Condition with factor 2 reflects a fundamental equity-efficiency
trade-off in quasi-linear environments.

77. Concluding remarks
Looked at a simple(st?) equity-efficiency problem.
Condition with factor 2 reflects a fundamental equity-efficiency
trade-off in quasi-linear environments.
Importance of the assumption k = 0—some conclusions about
which ordeals could work.

78. Concluding remarks
Looked at a simple(st?) equity-efficiency problem.
Condition with factor 2 reflects a fundamental equity-efficiency
trade-off in quasi-linear environments.
Importance of the assumption k = 0—some conclusions about
which ordeals could work.
Policy implications

79. Concluding remarks
Looked at a simple(st?) equity-efficiency problem.
Condition with factor 2 reflects a fundamental equity-efficiency
trade-off in quasi-linear environments.
Importance of the assumption k = 0—some conclusions about
which ordeals could work.
Policy implications
More papers coming up in the Inequality-aware Market Design
(IMD) agenda.