Welfare Econ seminar

1. Ordeals and Redistribution
Piotr Dworczak*
(Northwestern University & GRAPE)
October 23, 2023
Based on research with Frank Yang and Mohammad Akbarpour
Normative Economics & Economic Policy webinar
*Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reflect those of the European Union or the European Research Council.

2. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?

3. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?
“Redistribution through markets” is very common:

4. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?
“Redistribution through markets” is very common:
rent control, public housing, food stamps, free road access, Covid-19
vaccines, legal services, in-kind provision of food items, controlling the
prices of energy, ...

5. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?
“Redistribution through markets” is very common:
rent control, public housing, food stamps, free road access, Covid-19
vaccines, legal services, in-kind provision of food items, controlling the
prices of energy, ...
Given some redistributive preferences (welfare weights), what
is the best way to structure such policies under imperfect
information of the planner?

6. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?
“Redistribution through markets” is very common:
rent control, public housing, food stamps, free road access, Covid-19
vaccines, legal services, in-kind provision of food items, controlling the
prices of energy, ...
Given some redistributive preferences (welfare weights), what
is the best way to structure such policies under imperfect
information of the planner?
D r
⃝
Kominers r
⃝
Akbarpour (2021), A r
⃝
D r
⃝
K (2023), A r
⃝
Budish r
⃝
D
r
⃝
K (2023), Tokarski r
⃝
K r
⃝
A r
⃝
D (2023)

7. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?
“Redistribution through markets” is very common:
rent control, public housing, food stamps, free road access, Covid-19
vaccines, legal services, in-kind provision of food items, controlling the
prices of energy, ...
Given some redistributive preferences (welfare weights), what
is the best way to structure such policies under imperfect
information of the planner?
D r
⃝
Kominers r
⃝
Akbarpour (2021), A r
⃝
D r
⃝
K (2023), A r
⃝
Budish r
⃝
D
r
⃝
K (2023), Tokarski r
⃝
K r
⃝
A r
⃝
D (2023)
Ongoing effort: Interaction of IMD with optimal taxation

8. Inequality-aware Market Design
Inequality-aware Market Design (IMD): How to design
individual markets in the presence of inequality?
“Redistribution through markets” is very common:
rent control, public housing, food stamps, free road access, Covid-19
vaccines, legal services, in-kind provision of food items, controlling the
prices of energy, ...
Given some redistributive preferences (welfare weights), what
is the best way to structure such policies under imperfect
information of the planner?
D r
⃝
Kominers r
⃝
Akbarpour (2021), A r
⃝
D r
⃝
K (2023), A r
⃝
Budish r
⃝
D
r
⃝
K (2023), Tokarski r
⃝
K r
⃝
A r
⃝
D (2023)
Ongoing effort: Interaction of IMD with optimal taxation
Doligalski, D, Krysta, and Tokarski (2023)

9. Ordeals and Redistribution
Today, we will focus on a particular distortion—ordeals.

10. Ordeals and Redistribution
Today, we will focus on a particular distortion—ordeals.
Ordeal: activity that is costly for the agent performing it but does
not directly benefit anyone. Examples:

11. Ordeals and Redistribution
Today, we will focus on a particular distortion—ordeals.
Ordeal: activity that is costly for the agent performing it but does
not directly benefit anyone. Examples:
Waiting time
Travel distance
Filling out forms
...

12. Ordeals and Redistribution
Today, we will focus on a particular distortion—ordeals.
Ordeal: activity that is costly for the agent performing it but does
not directly benefit anyone. Examples:
Waiting time
Deshpande Li (2019), Rose (2021), Zeckhauser (2021)
Travel distance
Alatas et al. (2016), Hernandez Seira (2022)
Filling out forms
Deshpande Li (2019), Finkelstein Notowidigdo (2019)
...

13. Ordeals and Redistribution
Today, we will focus on a particular distortion—ordeals.
Ordeal: activity that is costly for the agent performing it but does
not directly benefit anyone. Examples:
Waiting time
Deshpande Li (2019), Rose (2021), Zeckhauser (2021)
Travel distance
Alatas et al. (2016), Hernandez Seira (2022)
Filling out forms
Deshpande Li (2019), Finkelstein Notowidigdo (2019)
...
Nichols and Zeckhauser (1982): Ordeals as costly screening
devices.

14. Ordeals and Redistribution
Today, we will focus on a particular distortion—ordeals.
Ordeal: activity that is costly for the agent performing it but does
not directly benefit anyone. Examples:
Waiting time
Deshpande Li (2019), Rose (2021), Zeckhauser (2021)
Travel distance
Alatas et al. (2016), Hernandez Seira (2022)
Filling out forms
Deshpande Li (2019), Finkelstein Notowidigdo (2019)
...
Nichols and Zeckhauser (1982): Ordeals as costly screening
devices.
Welfare properties of ordeals are not very well understood.

15. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.

16. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.

17. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.

18. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?

19. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:

20. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:
Direct cash subsidies in the US in Covid-19 pandemic;

21. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:
Direct cash subsidies in the US in Covid-19 pandemic;
Indonesia’s Conditional Cash Transfer Program.

22. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:
Direct cash subsidies in the US in Covid-19 pandemic;
Indonesia’s Conditional Cash Transfer Program.
Should we use an ordeal?

23. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:
Direct cash subsidies in the US in Covid-19 pandemic;
Indonesia’s Conditional Cash Transfer Program.
Should we use an ordeal?
“Equity-efficiency trade-off in quasi-linear environments”

24. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:
Direct cash subsidies in the US in Covid-19 pandemic;
Indonesia’s Conditional Cash Transfer Program.
Should we use an ordeal?
“Equity-efficiency trade-off in quasi-linear environments”
If yes, which ordeal to use?

25. Ordeals and Redistribution
Ordeals are particularly useful when allocating money.
Governments often engage in direct in-cash redistribution.
Of course, governments use verifiable information to target.
Should we also screen for unobserved characteristics?
Two examples for contrast:
Direct cash subsidies in the US in Covid-19 pandemic;
Indonesia’s Conditional Cash Transfer Program.
Should we use an ordeal?
“Equity-efficiency trade-off in quasi-linear environments”
If yes, which ordeal to use?
“Comparison of Screening Devices” (with F. Yang and M. Akbarpour)

26. Note on literature
It is well known that ordeals can improve targeting:
McAfee McMillan (1992), Hartline Rouchgarden (2008), Condorelli
(2012), Chakravarty Kaplan (2013), Yang (2023), ...
Most of the theory work focused on efficiency as design goal
(allocation of goods other than money!)
Moreover, all of these papers fix a screening device.
Classical references:
Weitzman (1977), Nichols Zeckhauser (1982), Besley Coate (1992),
Saez and Stantcheva (2016), ...
Other papers on IMD:
Condorelli (2013), Z. Kang (2020), M. Kang and Zheng (2020), Pai and
Strack (2022), ...

27. Paper #1
“Equity-efficiency trade-off in quasi-linear environments”

28. 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).

29. 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.

30. 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”)?

31. 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.

32. 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.
Main result: A condition for optimality of using an ordeal.

33. 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.
Main result: A condition for optimality of using an ordeal.
Simple intuition.

34. Model
Model

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

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, as in
Saez and Stantcheva, 2016)

37. 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, as in
Saez and Stantcheva, 2016)
The designer knows the distribution and maximizes total value.

38. 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, as in
Saez and Stantcheva, 2016)
The designer knows the distribution and maximizes total value.
Wlog, designer has no additional information.

39. 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, as in
Saez and Stantcheva, 2016)
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.

40. 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, as in
Saez and Stantcheva, 2016)
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”).

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”).

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].

43. 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.

44. 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.

45. 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.

46. 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.

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

48. 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.

49. 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.

50. 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.

51. Analysis
Analysis

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

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

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

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

56. Analysis
Suppose that the designer offers an additional payment t0 for an
ordeal y0 0.
Agents with k t0=y0 accept.
Let
E
h
vjc
v
= k
i
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)):

57. 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():

58. 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.

59. 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.

60. 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.

61. 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.
Intuitively: Using an ordeal becomes optimal when redistributive
preferences are strong enough and targeting effectiveness is high.

62. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.

63. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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.

64. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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:

65. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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);

66. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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);

67. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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);

68. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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 (2023).

69. Why 2?
Similar condition arises in papers studying optimal market design
under redistributive preferences.
DKA (2021): When designing a goods market, rationing may be
optimal 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 (2023).
Why “2”?

70. 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.

71. 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).

72. 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-cost agent at 2 or more.

73. Why 2?

74. Why 2?

75. Why 2?

76. Why 2?

77. Why 2?

78. Why 2?

79. Why 2?

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

81. 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)

82. 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)

83. 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 (??).

84. 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.

85. 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.

86. 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.

87. 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.

88. 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.

89. Concluding remarks
Concluding remarks

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

91. 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.

92. 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.
The condition is still sufficient without quasi-linearity in money!

93. 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.
The condition is still sufficient without quasi-linearity in money!
Importance of the assumption k = 0 ! policy implications.

94. 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.
The condition is still sufficient without quasi-linearity in money!
Importance of the assumption k = 0 ! policy implications.
Results silent about which ordeal to use.

95. Paper #2
“Comparison of Screening Devices”
(with Frank Yang and Mohammad Akbarpour)

96. Motivation
Many allocations use costly screening devices:
Waiting time;
Filling out forms;
Travel distance ;
...
For example:
Imposing an ordeal of traveling to a registration site improves
targeting in Indonesia’s Conditional Cash Transfer program
(Alatas et al. 2016)
Level of congestion at Social Security Administration field offices
affects targeting of disability programs (Deshpande Li 2019)
This paper: What makes one screening device better than another?

97. Illustrative Example
Illustrative Example

98. Illustrative Example
Allocate mass 2=3 of money to a unit
mass of agents, each receiving 1.
Types: p(oor), m(iddle class), r(ich)
Welfare weights: v = 3, 1, and 0
kA kB v
p 1 2 3
m 1 2 1
r 1 1 0
Screening devices A and B: kA and kB are per-unit-of-difficulty costs.

99. Illustrative Example
Allocate mass 2=3 of money to a unit
mass of agents, each receiving 1.
Types: p(oor), m(iddle class), r(ich)
Welfare weights: v = 3, 1, and 0
kA kB v
p 1 2 3
m 1 2 1
r 1 1 0
Screening devices A and B: kA and kB are per-unit-of-difficulty costs.
Screening device A achieves perfect targeting...

100. Illustrative Example
Allocate mass 2=3 of money to a unit
mass of agents, each receiving 1.
Types: p(oor), m(iddle class), r(ich)
Welfare weights: v = 3, 1, and 0
kA kB v
p 1 2 3
m 1 2 1
r 1 1 0
Screening devices A and B: kA and kB are per-unit-of-difficulty costs.
Screening device A achieves perfect targeting... but screening
device B is better than A!

101. Illustrative Example
Allocate mass 2=3 of money to a unit
mass of agents, each receiving 1.
Types: p(oor), m(iddle class), r(ich)
Welfare weights: v = 3, 1, and 0
kA kB v
p 1 2 3
m 1 2 1
r 1 1 0
Screening devices A and B: kA and kB are per-unit-of-difficulty costs.
Screening device A achieves perfect targeting... but screening
device B is better than A!
Under A, “market-clearing” difficulty = 1 unit.
Total welfare = 1
3 (3 (2) + 1 ) = 7
3
Under B, “market-clearing” difficulty = 1
2 unit.
Total welfare = 1
3 (3 (1 1
2) + 0 0) = 1 1
2

102. Illustrative Example
Allocate mass 2=3 of money to a unit
mass of agents, each receiving 1.
Types: p(oor), m(iddle class), r(ich)
Welfare weights: v = 3, 1, and 0
kA kB v
p 1 2 3
m 1 2 1
r 1 1 0
Screening devices A and B: kA and kB are per-unit-of-difficulty costs.
Screening device A achieves perfect targeting... but screening
device B is better than A!
Screening device B is worse at targeting but provides higher
rents to agents conditional on allocation.
kB has lower correlation with v but it has higher dispersion.

103. Framework
Framework

104. Costly Screening—framework
A screening device D is a conditional distribution
D : v ! ∆(R++):

105. Costly Screening—framework
A screening device D is a conditional distribution
D : v ! ∆(R++):
Interpretation: D(v) is the distribution of per-unit-of-difficulty
costs k for agents with welfare weight v.

106. Costly Screening—framework
A screening device D is a conditional distribution
D : v ! ∆(R++):
Interpretation: D(v) is the distribution of per-unit-of-difficulty
costs k for agents with welfare weight v.
This allows us to vary the screening device while holding fixed
designer’s redistributive preferences.

107. Costly Screening—framework
A screening device D is a conditional distribution
D : v ! ∆(R++):
Interpretation: D(v) is the distribution of per-unit-of-difficulty
costs k for agents with welfare weight v.
This allows us to vary the screening device while holding fixed
designer’s redistributive preferences.
Given D, denote by F the cumulative distribution function of k.

108. Costly Screening—framework
A screening device D is a conditional distribution
D : v ! ∆(R++):
Interpretation: D(v) is the distribution of per-unit-of-difficulty
costs k for agents with welfare weight v.
This allows us to vary the screening device while holding fixed
designer’s redistributive preferences.
Given D, denote by F the cumulative distribution function of k.
Screening device identified by the joint distribution of (v; k).

109. Costly Screening—framework
A screening device D is a conditional distribution
D : v ! ∆(R++):
Interpretation: D(v) is the distribution of per-unit-of-difficulty
costs k for agents with welfare weight v.
This allows us to vary the screening device while holding fixed
designer’s redistributive preferences.
Given D, denote by F the cumulative distribution function of k.
Screening device identified by the joint distribution of (v; k).
Normalize the maximal quantity of money received by agent to 1;
budget B 2 [0; 1]; no restrictions on difficulty y 0.

110. Costly Screening—framework
In the paper:
Allocation of goods with heterogeneous quality q 2 [0; 1].

111. Costly Screening—framework
In the paper:
Allocation of goods with heterogeneous quality q 2 [0; 1].
Each agent has a private value w for the good and welfare
weight v.

112. Costly Screening—framework
In the paper:
Allocation of goods with heterogeneous quality q 2 [0; 1].
Each agent has a private value w for the good and welfare
weight v.
Screening device D maps (v; w) into distribution of costs k.

113. Costly Screening—framework
In the paper:
Allocation of goods with heterogeneous quality q 2 [0; 1].
Each agent has a private value w for the good and welfare
weight v.
Screening device D maps (v; w) into distribution of costs k.
Utility is wq ky, contribution to social welfare is v(wq ky).

114. Costly Screening—framework
In the paper:
Allocation of goods with heterogeneous quality q 2 [0; 1].
Each agent has a private value w for the good and welfare
weight v.
Screening device D maps (v; w) into distribution of costs k.
Utility is wq ky, contribution to social welfare is v(wq ky).
Marginal rate of substitution: r = w=k.

115. Costly Screening—framework
In the paper:
Allocation of goods with heterogeneous quality q 2 [0; 1].
Each agent has a private value w for the good and welfare
weight v.
Screening device D maps (v; w) into distribution of costs k.
Utility is wq ky, contribution to social welfare is v(wq ky).
Marginal rate of substitution: r = w=k.
All results apply if we replace the distribution of inverse costs 1=k
with the distribution of the marginal rate of substitution.

116. Designing an ideal screening device?
Allocate supply B of money to agents with welfare weights distributed
uniformly: v U[0;1].

117. Designing an ideal screening device?
Allocate supply B of money to agents with welfare weights distributed
uniformly: v U[0;1].
The following screening device achieves the first best: k = 1fv1 Bg.

118. Designing an ideal screening device?
Allocate supply B of money to agents with welfare weights distributed
uniformly: v U[0;1].
The following screening device achieves the first best: k = 1fv1 Bg.
Not practical:
cannot freely design screening devices;
sensitive to the available supply level.

119. Comparing Screening Devices
Screening device D1 dominates screening device D2 if
OPT(D1;B) OPT(D2;B)
for every supply level B 2 [0;1], where
OPT(D;B) is the social welfare induced by the optimal
allocation mechanism given screening device D and budget B.

120. Comparing Screening Devices
Screening device D1 dominates screening device D2 if
OPT(D1;B) OPT(D2;B)
for every supply level B 2 [0;1], where
OPT(D;B) is the social welfare induced by the optimal
allocation mechanism given screening device D and budget B.
Question: When does one screening device dominate another?

121. Comparing Screening Devices
Screening device D1 dominates screening device D2 if
OPT(D1;B) OPT(D2;B)
for every supply level B 2 [0;1], where
OPT(D;B) is the social welfare induced by the optimal
allocation mechanism given screening device D and budget B.
Question: When does one screening device dominate another?
Remark: While we focus on optimal mechanisms, our results also
hold with “market mechanisms” (demand = supply).

122. Main Results
Main Results

123. Characterization of Dominance
For any screening device D, define its welfare index at parameter s:
W(D;s) :=
Z s
0
1
F 1(t)
F 1(s)
| {z }
Rent Provision
E
h
v j k = F 1
(t)
i
| {z }
Targeting Effectiveness
dt:

124. Characterization of Dominance
For any screening device D, define its welfare index at parameter s:
W(D;s) :=
Z s
0
1
F 1(t)
F 1(s)
| {z }
Rent Provision
E
h
v j k = F 1
(t)
i
| {z }
Targeting Effectiveness
dt:
Rent provision:
utility net of screening costs for an agent whose cost is in the
bottom t quantile, given supply s;
measured in units of money.

125. Characterization of Dominance
For any screening device D, define its welfare index at parameter s:
W(D;s) :=
Z s
0
1
F 1(t)
F 1(s)
| {z }
Rent Provision
E
h
v j k = F 1
(t)
i
| {z }
Targeting Effectiveness
dt:
Rent provision:
utility net of screening costs for an agent whose cost is in the
bottom t quantile, given supply s;
measured in units of money.
Targeting effectiveness:
converts monetary units to social welfare units;
uses information revealed through agent’s cost k.

126. Characterization of Dominance
For any screening device D, define its welfare index at parameter s:
W(D;s) :=
Z s
0
1
F 1(t)
F 1(s)
| {z }
Rent Provision
E
h
v j k = F 1
(t)
i
| {z }
Targeting Effectiveness
dt:
Theorem 1
For any two screening devices D1 and D2, if
W(D1;s) W(D2;s); for all s 2 [0;1]; (?)
then screening device D1 dominates screening device D2.

127. Characterization of Dominance
For any screening device D, define its welfare index at parameter s:
W(D;s) :=
Z s
0
1
F 1(t)
F 1(s)
| {z }
Rent Provision
E
h
v j k = F 1
(t)
i
| {z }
Targeting Effectiveness
dt:
Theorem 1
For any two screening devices D1 and D2, if
W(D1;s) W(D2;s); for all s 2 [0;1]; (?)
then screening device D1 dominates screening device D2.
Remark: A weakening of (?) is both necessary and sufficient

128. Math Break: Two Notions of Spread
Dispersive order: For two random variables X and Y with CDFs
FX and FY , X disp Y if
F 1
X (t) F 1
X (s) F 1
Y (t) F 1
Y (s); for all 0 s t 1:

129. Math Break: Two Notions of Spread
Dispersive order: For two random variables X and Y with CDFs
FX and FY , X disp Y if
F 1
X (t) F 1
X (s) F 1
Y (t) F 1
Y (s); for all 0 s t 1:
Majorization order: For two functions h1 and h2 on [0;1], h2 maj h1
if
Z s
0
h2(t)dt
Z s
0
h1(t)dt; for all s 2 [0; 1], with equality at s = 1.
If h1;h2 are non-decreasing, then this is the same as
h2(U) mps h1(U) ;
where mps is mean-preserving spread and U is a uniform random
variable.

130. Dispersion and Correlation
Let V(t) := E[v j k = F 1(t)] be the posterior mean of v when k is
equal to its quantile t.
Theorem 2
For any two screening devices D1 and D2, if
(i) log(k2) disp log(k1),
(ii) V2 maj V1,
then screening device D1 dominates screening device D2.

131. Dispersion and Correlation
Let V(t) := E[v j k = F 1(t)] be the posterior mean of v when k is
equal to its quantile t.
Theorem 2
For any two screening devices D1 and D2, if
(i) log(k2) disp log(k1),
(ii) V2 maj V1,
then screening device D1 dominates screening device D2.
Condition (i): marginal distribution of k
Condition (ii): correlation of v and k

132. Dispersion and Correlation
Let V(t) := E[v j k = F 1(t)] be the posterior mean of v when k is
equal to its quantile t.
Theorem 2
For any two screening devices D1 and D2, if
(i) log(k2) disp log(k1),
(ii) V2 maj V1,
then screening device D1 dominates screening device D2.
Condition (i): marginal distribution of k
Condition (ii): correlation of v and k

133. Dispersion and Correlation
Let V(t) := E[v j k = F 1(t)] be the posterior mean of v when k is
equal to its quantile t.
Theorem 2
For any two screening devices D1 and D2, if
(i) log(k2) disp log(k1),
(ii) V2 maj V1,
then screening device D1 dominates screening device D2.
Condition (i): marginal distribution of k
Condition (ii): correlation of v and k
Dispersion at log level for k () Rent provision
Negative correlation of (v;k) () Targeting effectiveness

134. Economic Implications
Economic Implications

135. Limits of Costly Screening
Proposition 1
For any 0, there exists a screening device D such that, for any
budget B, D achieves welfare that is within of the first-best welfare.

136. Limits of Costly Screening
Proposition 1
For any 0, there exists a screening device D such that, for any
budget B, D achieves welfare that is within of the first-best welfare.
D approximates the first-best robustly for all budget levels!

137. Limits of Costly Screening
Proposition 1
For any 0, there exists a screening device D such that, for any
budget B, D achieves welfare that is within of the first-best welfare.
D approximates the first-best robustly for all budget levels!
Proof sketch:

138. Limits of Costly Screening
Proposition 1
For any 0, there exists a screening device D such that, for any
budget B, D achieves welfare that is within of the first-best welfare.
D approximates the first-best robustly for all budget levels!
Proof sketch:
Perfect targeting: make k a strictly decreasing function of v.
Increasingly dispersed: make log(k) increasingly dispersed as
decreases.

139. Limits of Costly Screening
Proposition 2
For any device D and any 0, there exists D such that:
(i) D has better targeting effectiveness than D: V maj V;
(ii) For any budget B, D achieves social welfare that is less than
under the market-clearing rule.

140. Limits of Costly Screening
Proposition 2
For any device D and any 0, there exists D such that:
(i) D has better targeting effectiveness than D: V maj V;
(ii) For any budget B, D achieves social welfare that is less than
under the market-clearing rule.
Proof sketch:
D can be made to achieve perfect targeting;
But if log(k) is increasingly compressed then we have arbitrarily
low welfare.

141. Importance of Dispersion
Proposition 3
Consider any screening devices D; D1; D2; D3 such that
k1 = k ∆; k2 = min(k;∆0); k3 = ∆00k;
for some constants ∆;∆0; ∆00 0. Then
D1 dominates D;
D dominates D2;
D is equivalent to D3.

142. Importance of Dispersion
Proposition 3
Consider any screening devices D; D1; D2; D3 such that
k1 = k ∆; k2 = min(k;∆0); k3 = ∆00k;
for some constants ∆;∆0; ∆00 0. Then
D1 dominates D;
D dominates D2;
D is equivalent to D3.
Even though both D1; D2; D3 reduce the level of costs,
uniformly decreasing costs is welfare-enhancing;
allowing delegation of the task is welfare-reducing.
multiplying costs by a constant is welfare-neutral.

143. Simple Regression Test
Proposition 4
If two screening devices D1 and D2 satisfy
log(k2) = log(k1) + ;
for some 2 [0;1] and random variable that is independent of
(v;k1), then D1 dominates D2.

144. Simple Regression Test
Proposition 4
If two screening devices D1 and D2 satisfy
log(k2) = log(k1) + ;
for some 2 [0;1] and random variable that is independent of
(v;k1), then D1 dominates D2.
D1 and D2 sort agents the same way up to idiosyncratic shocks:
Smaller means less dispersion on the log level;

145. Simple Regression Test
Proposition 4
If two screening devices D1 and D2 satisfy
log(k2) = log(k1) + ;
for some 2 [0;1] and random variable that is independent of
(v;k1), then D1 dominates D2.
D1 and D2 sort agents the same way up to idiosyncratic shocks:
Smaller means less dispersion on the log level;
introduces a trade-off between dispersion and targeting;

146. Simple Regression Test
Proposition 4
If two screening devices D1 and D2 satisfy
log(k2) = log(k1) + ;
for some 2 [0;1] and random variable that is independent of
(v;k1), then D1 dominates D2.
D1 and D2 sort agents the same way up to idiosyncratic shocks:
Smaller means less dispersion on the log level;
introduces a trade-off between dispersion and targeting;
Adding idiosyncratic noise makes screening device worse;

147. Simple Regression Test
Proposition 4
If two screening devices D1 and D2 satisfy
log(k2) = log(k1) + ;
for some 2 [0;1] and random variable that is independent of
(v;k1), then D1 dominates D2.
D1 and D2 sort agents the same way up to idiosyncratic shocks:
Smaller means less dispersion on the log level;
introduces a trade-off between dispersion and targeting;
Adding idiosyncratic noise makes screening device worse;
Intuitively, we want variation in log(k) across (not within)
groups of different v.

148. Combining Screening Devices
For any two screening devices D1 and D2, either:
Bundle two devices: Induce a new device D3 by
c3 = c1 + c2;
Offer a choice: Induce a new device D3 by
c3 = min
n
c1;c2
o
:

149. Combining Screening Devices
For any two screening devices D1 and D2, either:
Bundle two devices: Induce a new device D3 by
c3 = c1 + c2;
Offer a choice: Induce a new device D3 by
c3 = min
n
c1;c2
o
:
Proposition 5
If D1 and D2 satisfy c2 = (c1;), where
is independent of (v;c1);
0 d log (c;)
d log c 1 for all c, almost surely in ;
then D1 dominates D3.

150. Concluding remarks
Concluding remarks

151. Concluding Remarks
What makes a costly screening device better than another?
Two channels:

152. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;

153. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.

154. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:

155. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:
dispersion of the distribution of costs,

156. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:
dispersion of the distribution of costs,
correlation of costs and welfare weights.

157. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:
dispersion of the distribution of costs,
correlation of costs and welfare weights.
Future directions:

158. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:
dispersion of the distribution of costs,
correlation of costs and welfare weights.
Future directions:
“Useful” ordeals? (Kleven Kopczuk, 2011, Dréze, 2019)

159. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:
dispersion of the distribution of costs,
correlation of costs and welfare weights.
Future directions:
“Useful” ordeals? (Kleven Kopczuk, 2011, Dréze, 2019)
Ethical ramifications?

160. Concluding Remarks
What makes a costly screening device better than another?
Two channels:
rent provision;
targeting effectiveness.
Characterized by:
dispersion of the distribution of costs,
correlation of costs and welfare weights.
Future directions:
“Useful” ordeals? (Kleven Kopczuk, 2011, Dréze, 2019)
Ethical ramifications?
Interaction with other tools for redistribution? (Ooghe, 2021)