CSIO-IO

1. A mechanism-design approach to property rights
Piotr Dworczak?
Ellen Muir
(Northwestern; GRAPE) (Harvard)
October 7, 2023
20th CSIO/TSE Conference on Industrial Organization
?
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. Motivation
Vast literature in mechanism design studies the problem of
optimal allocation of economic resources.

3. Motivation
Vast literature in mechanism design studies the problem of
optimal allocation of economic resources.
Less is known about how to design the contractual rights
relating to the control of the resource (“property rights”).

4. Motivation
Vast literature in mechanism design studies the problem of
optimal allocation of economic resources.
Less is known about how to design the contractual rights
relating to the control of the resource (“property rights”).
Example:

5. Motivation
Vast literature in mechanism design studies the problem of
optimal allocation of economic resources.
Less is known about how to design the contractual rights
relating to the control of the resource (“property rights”).
Example:
Selling an apple...

6. Motivation
Vast literature in mechanism design studies the problem of
optimal allocation of economic resources.
Less is known about how to design the contractual rights
relating to the control of the resource (“property rights”).
Example:
Selling an apple...
... versus selling a spectrum license.

7. Motivation
Vast literature in mechanism design studies the problem of
optimal allocation of economic resources.
Less is known about how to design the contractual rights
relating to the control of the resource (“property rights”).
Example:
Selling an apple...
... versus selling a spectrum license.
We provide a framework for studying optimal design of
contractual rights.

8. Key features of our framework
In our framework, property rights determine the holder’s
outside options (with regard to the underlying good) in
economic interactions:

9. Key features of our framework
In our framework, property rights determine the holder’s
outside options (with regard to the underlying good) in
economic interactions:
Full property right: Holder can always keep the good;

10. Key features of our framework
In our framework, property rights determine the holder’s
outside options (with regard to the underlying good) in
economic interactions:
Full property right: Holder can always keep the good;
No rights: Outside option is (normalized to) zero;

11. Key features of our framework
In our framework, property rights determine the holder’s
outside options (with regard to the underlying good) in
economic interactions:
Full property right: Holder can always keep the good;
No rights: Outside option is (normalized to) zero;
Renewable lease: Holder can keep the good at a price;

12. Key features of our framework
In our framework, property rights determine the holder’s
outside options (with regard to the underlying good) in
economic interactions:
Full property right: Holder can always keep the good;
No rights: Outside option is (normalized to) zero;
Renewable lease: Holder can keep the good at a price;
...

13. Key features of our framework
In our framework, property rights determine the holder’s
outside options (with regard to the underlying good) in
economic interactions:
Full property right: Holder can always keep the good;
No rights: Outside option is (normalized to) zero;
Renewable lease: Holder can keep the good at a price;
...
We model the economic interaction as the holder of the rights
participating in a trading mechanism.

14. Key features of our framework
Fundamental friction: Social planner cannot commit to future
trading mechanisms, resulting in

15. Key features of our framework
Fundamental friction: Social planner cannot commit to future
trading mechanisms, resulting in
ex-post inefficiency (mechanism may be suboptimal);

16. Key features of our framework
Fundamental friction: Social planner cannot commit to future
trading mechanisms, resulting in
ex-post inefficiency (mechanism may be suboptimal);
hold-up problem (inefficient investment decisions).

17. Key features of our framework
Fundamental friction: Social planner cannot commit to future
trading mechanisms, resulting in
ex-post inefficiency (mechanism may be suboptimal);
hold-up problem (inefficient investment decisions).
Planner chooses property rights to alleviate these frictions.

18. Key features of our framework
Fundamental friction: Social planner cannot commit to future
trading mechanisms, resulting in
ex-post inefficiency (mechanism may be suboptimal);
hold-up problem (inefficient investment decisions).
Planner chooses property rights to alleviate these frictions.
Question: What is the optimal set of rights?

19. Key features of our framework
Fundamental friction: Social planner cannot commit to future
trading mechanisms, resulting in
ex-post inefficiency (mechanism may be suboptimal);
hold-up problem (inefficient investment decisions).
Planner chooses property rights to alleviate these frictions.
Question: What is the optimal set of rights?
Main result: The optimal property right is simple but flexible,
often featuring an option to own.

20. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.

21. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.

22. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:

23. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:

24. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:
Myerson and Satterthwaite (1983);

25. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:
Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987);

26. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:
Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987);
Matouschek (2004), Che (2006), Segal and Whinston (2016).

27. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:
Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987);
Matouschek (2004), Che (2006), Segal and Whinston (2016).
Property rights affect investment incentives:

28. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:
Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987);
Matouschek (2004), Che (2006), Segal and Whinston (2016).
Property rights affect investment incentives:
Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Maskin and Tirole (1999), ...

29. Property rights in economics
Coase (1960): Any assignment of property rights leads to
efficient outcomes as long as there are transaction costs.
Williamson (1979): Transaction costs take center stage in
economics.
Two large literatures:
Property rights affect efficiency under private information:
Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987);
Matouschek (2004), Che (2006), Segal and Whinston (2016).
Property rights affect investment incentives:
Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Maskin and Tirole (1999), ...
Hart (1995), Nöldeke and Schmidt (1995, 1998).

30. Marginal contribution
Our framework studies both effects in a simple setting.

31. Marginal contribution
Our framework studies both effects in a simple setting.
To the best of our knowledge, our paper is first to characterize
the optimal property right from a non-parametric class (in a
setting where the first best cannot in general be achieved).

32. Marginal contribution
Our framework studies both effects in a simple setting.
To the best of our knowledge, our paper is first to characterize
the optimal property right from a non-parametric class (in a
setting where the first best cannot in general be achieved).
The attractive properties of an option-to-own contract have been
studied—our work provides an optimality foundation.

33. Marginal contribution
Our framework studies both effects in a simple setting.
To the best of our knowledge, our paper is first to characterize
the optimal property right from a non-parametric class (in a
setting where the first best cannot in general be achieved).
The attractive properties of an option-to-own contract have been
studied—our work provides an optimality foundation.
On the technical side:

34. Marginal contribution
Our framework studies both effects in a simple setting.
To the best of our knowledge, our paper is first to characterize
the optimal property right from a non-parametric class (in a
setting where the first best cannot in general be achieved).
The attractive properties of an option-to-own contract have been
studied—our work provides an optimality foundation.
On the technical side:
We provide a novel solution method for mechanism design
with type-dependent outside options (Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
the ironing technique (Myerson, 1981).

35. Marginal contribution
Our framework studies both effects in a simple setting.
To the best of our knowledge, our paper is first to characterize
the optimal property right from a non-parametric class (in a
setting where the first best cannot in general be achieved).
The attractive properties of an option-to-own contract have been
studied—our work provides an optimality foundation.
On the technical side:
We provide a novel solution method for mechanism design
with type-dependent outside options (Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
the ironing technique (Myerson, 1981).
We show how to optimize over the outside-option function.

36. Other related papers
Allocation mechanisms versus efficient investment:
Rogerson (1992), Bergemann and Välimäki (2002), Milgrom
(2017), Hatfield, Kojima, and Kominers (2019), Akbarpour,
Kominers, Li, Li, and Milgrom (2023), ...
Related techniques: Kleiner, Moldovanu, and Strack (2021),
Loertscher and Muir (2022), Kang (2023), Akbarpour ®
Dworczak ® Kominers (2023),...
Spectrum license design: Milgrom, Weyl and Zhang (2017),
Weyl and Zhang (2017), ...
Optimal patent design: Matutes, Regibeau and Rockett (1996),
Wright (1999), Hopenhayn, Llobet and Mitchell (2006), ...

37. Model
Model

38. Model
There is an agent, a designer, and a social planner;

39. Model
There is an agent, a designer, and a social planner;
At time t = 0:

40. Model
There is an agent, a designer, and a social planner;
At time t = 0:
The planner designs a contract (set of rights) that the agent
holds.

41. Model
There is an agent, a designer, and a social planner;
At time t = 0:
The planner designs a contract (set of rights) that the agent
holds.
At time t = 1:

42. Model
There is an agent, a designer, and a social planner;
At time t = 0:
The planner designs a contract (set of rights) that the agent
holds.
At time t = 1:
The agent decides whether to invest at (sunk) cost c 0;

43. Model
There is an agent, a designer, and a social planner;
At time t = 0:
The planner designs a contract (set of rights) that the agent
holds.
At time t = 1:
The agent decides whether to invest at (sunk) cost c 0;
At time t = 2:

44. Model
There is an agent, a designer, and a social planner;
At time t = 0:
The planner designs a contract (set of rights) that the agent
holds.
At time t = 1:
The agent decides whether to invest at (sunk) cost c 0;
At time t = 2:
The agent’s (privately observed) type and a public state !
are drawn from a joint distribution that depends on whether
the agent invested or not.

45. Model
There is an agent, a designer, and a social planner;
At time t = 0:
The planner designs a contract (set of rights) that the agent
holds.
At time t = 1:
The agent decides whether to invest at (sunk) cost c 0;
At time t = 2:
The agent’s (privately observed) type and a public state !
are drawn from a joint distribution that depends on whether
the agent invested or not.
The designer chooses a mechanism, and trade happens.

46. Model
Time t = 2 continued:
Designer chooses a trading mechanism (x!(); t!()), where
x 2 [0; 1] denotes an allocation, and t 2 R denotes a transfer;

47. Model
Time t = 2 continued:
Designer chooses a trading mechanism (x!(); t!()), where
x 2 [0; 1] denotes an allocation, and t 2 R denotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.

48. Model
Time t = 2 continued:
Designer chooses a trading mechanism (x!(); t!()), where
x 2 [0; 1] denotes an allocation, and t 2 R denotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.
Designer maximizes V(; !) x + t, where 0.

49. Model
Time t = 2 continued:
Designer chooses a trading mechanism (x!(); t!()), where
x 2 [0; 1] denotes an allocation, and t 2 R denotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.
Designer maximizes V(; !) x + t, where 0.
Agent’s utility is x t.

50. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.

51. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.
If the agent invests:

52. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.
If the agent invests:
! G, F!.

53. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.
If the agent invests:
! G, F!.
Otherwise:

54. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.
If the agent invests:
! G, F!.
Otherwise:
! G, F!, with F! FOSD F!, for every !.

55. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.
If the agent invests:
! G, F!.
Otherwise:
! G, F!, with F! FOSD F!, for every !.
In the non-contractible case, the mechanism at t = 2 and the
planner’s contract do not depend on the investment decision.

56. Model
Agent’s problem at t = 1
The agent decides whether to pay the cost c 0 to invest.
If the agent invests:
! G, F!.
Otherwise:
! G, F!, with F! FOSD F!, for every !.
In the non-contractible case, the mechanism at t = 2 and the
planner’s contract do not depend on the investment decision.
In the contractible case, the mechanism at t = 2 and the
planner’s contract are contingent on the investment decision.

57. Model
Social planner’s problem at t = 0
Social planner chooses a contract that is a menu of “rights”
M = f(xi; ti)gi2I;
where xi 2 [0; 1], ti 2 R, and set I is arbitrary (M is compact).

58. Model
Social planner’s problem at t = 0
Social planner chooses a contract that is a menu of “rights”
M = f(xi; ti)gi2I;
where xi 2 [0; 1], ti 2 R, and set I is arbitrary (M is compact).
The agent can “execute” any one of these rights at t = 2.

59. Model
Social planner’s problem at t = 0
Social planner chooses a contract that is a menu of “rights”
M = f(xi; ti)gi2I;
where xi 2 [0; 1], ti 2 R, and set I is arbitrary (M is compact).
The agent can “execute” any one of these rights at t = 2.
Social planner maximizes V?
(; !)x + ?
t, where ?
0.

60. Model
Social planner’s problem at t = 0
Social planner chooses a contract that is a menu of “rights”
M = f(xi; ti)gi2I;
where xi 2 [0; 1], ti 2 R, and set I is arbitrary (M is compact).
The agent can “execute” any one of these rights at t = 2.
Social planner maximizes V?
(; !)x + ?
t, where ?
0.

61. Model
Social planner’s problem at t = 0
Social planner chooses a contract that is a menu of “rights”
M = f(xi; ti)gi2I;
where xi 2 [0; 1], ti 2 R, and set I is arbitrary (M is compact).
The agent can “execute” any one of these rights at t = 2.
Social planner maximizes V?
(; !)x + ?
t, where ?
0.
Assume: Investment is preferred by the social planner to no
investment; investment can be induced by some contract; but it is not
induced if the agent holds no rights.

62. Model
Social planner’s problem at t = 0
Social planner chooses a contract that is a menu of “rights”
M = f(xi; ti)gi2I;
where xi 2 [0; 1], ti 2 R, and set I is arbitrary (M is compact).
The agent can “execute” any one of these rights at t = 2.
Social planner maximizes V?
(; !)x + ?
t, where ?
0.
Technicalities: Θ [; ¯
] is a compact subset of R; V and V?
are
continuous in (and measurable in !), F has a continuous positive
density on Θ.

63. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:

64. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:

65. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:
Property right: M = f(x = 1; t = 0)g;

66. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:
Property right: M = f(x = 1; t = 0)g;
Cash payment: M = f(0; p)g;

67. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:
Property right: M = f(x = 1; t = 0)g;
Cash payment: M = f(0; p)g;
Property right with a resale option: M = f(1;0); (0; p)g;

68. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:
Property right: M = f(x = 1; t = 0)g;
Cash payment: M = f(0; p)g;
Property right with a resale option: M = f(1;0); (0; p)g;
Renewable lease/ option to own: M = f(1; p)g;

69. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:
Property right: M = f(x = 1; t = 0)g;
Cash payment: M = f(0; p)g;
Property right with a resale option: M = f(1;0); (0; p)g;
Renewable lease/ option to own: M = f(1; p)g;
Partial property right: M = f(y; 0)g, where y 2 (0; 1);

70. Comments about the model
A contract creates an outside option for the agent: Designer
must guarantee that the agent’s type- utility from participating in
the mechanism is not lower than
max
i2I
fxi tig:
The framework captures many conventional rights:
Property right: M = f(x = 1; t = 0)g;
Cash payment: M = f(0; p)g;
Property right with a resale option: M = f(1;0); (0; p)g;
Renewable lease/ option to own: M = f(1; p)g;
Partial property right: M = f(y; 0)g, where y 2 (0; 1);
Flexible property right: M = fs; p(s))gs2[0; 1].

71. Comments about the model
There are two frictions in the model:

72. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);

73. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);
2. Non-contractible investment decision (“hold-up problem”);

74. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);
2. Non-contractible investment decision (“hold-up problem”);
Additionally, we limited the power of property rights by assuming
that they cannot be state-contingent (“incomplete contracts”);

75. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);
2. Non-contractible investment decision (“hold-up problem”);
Additionally, we limited the power of property rights by assuming
that they cannot be state-contingent (“incomplete contracts”);
We can “switch off” some frictions and provide tighter results.

76. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);
2. Non-contractible investment decision (“hold-up problem”);
Additionally, we limited the power of property rights by assuming
that they cannot be state-contingent (“incomplete contracts”);
We can “switch off” some frictions and provide tighter results.
Uninteresting cases:

77. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);
2. Non-contractible investment decision (“hold-up problem”);
Additionally, we limited the power of property rights by assuming
that they cannot be state-contingent (“incomplete contracts”);
We can “switch off” some frictions and provide tighter results.
Uninteresting cases:
If there is no investment and the planner and designer are
aligned, it is optimal to allocate no rights to the agent.

78. Comments about the model
There are two frictions in the model:
1. Divergence of preferences between the planner and
designer (“ex-post inefficiency”);
2. Non-contractible investment decision (“hold-up problem”);
Additionally, we limited the power of property rights by assuming
that they cannot be state-contingent (“incomplete contracts”);
We can “switch off” some frictions and provide tighter results.
Uninteresting cases:
If there is no investment and the planner and designer are
aligned, it is optimal to allocate no rights to the agent.
If the designer did not care about her revenue ( = 0), she would
“buy out” the agent’s rights.

79. Application #1: Dynamic resource allocation
Players:
Agent: Firm
Planner: Regulator
Designer: Regulator
Agent invests in infrastructure determining value ;
State ! represents the value for an alternative use;
The planner maximizes a combination of efficiency and revenue:
V?
(; !)x + ?
t = ( !)x + ?
t:
The designer might put more weight on revenue:
V(; !)x + t = ( !)x + t; where ?
.

80. Application #2: Regulating a private rental market
Players:
Agent: Tenant
Planner: Regulator
Designer: Rental company
Agent invests in the property determining her value for staying.
The state ! is the seller’s outside option (market rental price).
The seller wants to maximize revenue:
V(; !)x + t = !x + t.
The planner wants to maximize efficiency:
V?
(; !)x + ?
t = ( !)x:

81. Application #3: Vaccine development
Players:
Agent: Pharmaceutical company
Planner: Health agency
Designer: Health agency
Company develops a vaccine; the marginal cost of production
conditional on investment is k F̃ ( = k);
Allocation x is the number of units purchased by the health
agency, with 1 being the mass of the patient population;
Health agency maximizes the total value of allocation,
V?
(; !)x = V(; !)x = !x,
where ! measures the severity of the health crisis.

82. Application #4: Patent policy
Players:
Agent: Firm
Planner: Regulator
Designer: Patent agency
Firm invests in a new technology; if investment is made, the firm
can produce at constant marginal cost k F̃.
Conditional on x = 1, the firm provides a monopoly quantity to
maximize profits; conditional on x = 0, the firm competes in a
competitive market. Market demand is D(p) = 1 p.
The regulator and the patent agency maximize consumer
surplus with weight ! (but attach a positive weight to revenue).

83. Application #5: Supply chain contracting
Players:
Agent: Small supplier
Planner: Large producer
Designer: Large producer
Producer buys some amount x of inputs from the supplier.
The supplier must invest at time t = 1 in relationship-specific
technology to produce the inputs at marginal cost c .
Through the interaction with the supplier, the large firm can learn
the supplier’s costs: The state ! is a noisy signal of .
Producer maximizes profits having a constant marginal value 1
for each unit of the input: V!() = 1 and = 1:
Producer proposes a contract: V?
!() = 1 and ?
= 1.

84. Analysis and results
Analysis and results

85. Main result
Theorem
There exists an optimal contract that takes the form
M?
= f(1;p);(y;p0)g for some p;p0 2 R and y 2 [0;1).

86. Main result
Theorem
There exists an optimal contract that takes the form
M?
= f(1;p);(y;p0)g for some p;p0 2 R and y 2 [0;1).
Comments:
The optimal contract is simple in that it consists of at most two
types of rights.

87. Main result
Theorem
There exists an optimal contract that takes the form
M?
= f(1;p);(y;p0)g for some p;p0 2 R and y 2 [0;1).
Comments:
The optimal contract is simple in that it consists of at most two
types of rights.
One of the rights can be taken to be an option to own.

88. Main result
Theorem
There exists an optimal contract that takes the form
M?
= f(1;p);(y;p0)g for some p;p0 2 R and y 2 [0;1).
Comments:
The optimal contract is simple in that it consists of at most two
types of rights.
One of the rights can be taken to be an option to own.
If there were no hold-up problem in the model, M?
= f(1; p)g
would be always optimal.

89. Step 1: Outside option constraint
Lemma
A choice of contract M by the social planner is equivalent to choosing
an outside option function R() for the agent in the second-period
mechanism, where R() is non-negative, non-decreasing, convex,
and has slope bounded above by 1.

90. Step 1: Outside option constraint
Lemma
A choice of contract M by the social planner is equivalent to choosing
an outside option function R() for the agent in the second-period
mechanism, where R() is non-negative, non-decreasing, convex,
and has slope bounded above by 1.
The lemma follows immediately from the observation that, for any M,
we can set
R() = maxf0; max
i2I
fxi tigg:

91. Step 1: Outside option constraint
Lemma
A choice of contract M by the social planner is equivalent to choosing
an outside option function R() for the agent in the second-period
mechanism, where R() is non-negative, non-decreasing, convex,
and has slope bounded above by 1.
The lemma follows immediately from the observation that, for any M,
we can set
R() = maxf0; max
i2I
fxi tigg:
As a result, the planner maximizes over type-dependent outside
options for the agent.

92. Step 1: Outside option constraint
Fixing ! (and dropping it from the notation), the designer solves:
max
x:Θ![0;1]; u0
Z
W()x()d u
s.t. x is non-decreasing;
U() u +
Z
x() d R(); 8 2 Θ;
where
W() (V() + B()) f();
and
B() =
1 F()
f()
:

93. Step 1: Outside option constraint
Fixing ! (and dropping it from the notation), the designer solves:
max
x:Θ![0;1]; u0
Z
W()x()d u
s.t. x is non-decreasing;
U() u +
Z
x() d R(); 8 2 Θ;
where
W() (V() + B()) f();
and
B() =
1 F()
f()
:
This is the problem considered by Jullien (2000).

94. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.

95. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.
There are three complications:

96. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.
There are three complications:
this may violate monotonicity of x();

97. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.
There are three complications:
this may violate monotonicity of x();
x() cannot be minimized point-wise, because allocation to
type affects the utility of higher types;

98. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.
There are three complications:
this may violate monotonicity of x();
x() cannot be minimized point-wise, because allocation to
type affects the utility of higher types;
the designer may want to use the transfer to satisfy the
outside-option constraint.

99. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.
There are three complications:
this may violate monotonicity of x();
x() cannot be minimized point-wise, because allocation to
type affects the utility of higher types;
the designer may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting the ironing
technique from Myerson (1981).

100. Step 2: Ironing Jullien (2000)
Naively, the designer wants to set x() = 1 when W() 0, and
set x() to be as low as possible (given R) when W() 0.
There are three complications:
this may violate monotonicity of x();
x() cannot be minimized point-wise, because allocation to
type affects the utility of higher types;
the designer may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting the ironing
technique from Myerson (1981).
Define
W() =
Z ¯
W()d; W = co(W):

101. Step 2: Ironing Jullien (2000)

102. Step 2: Ironing Jullien (2000)

103. Step 2: Ironing Jullien (2000)

104. Step 2: Ironing Jullien (2000)

105. Step 2: Ironing Jullien (2000)

106. Step 2: Ironing Jullien (2000)
Why does ironing work?

107. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u +
Z
x()d U() R() u0 +
Z
x0()d:

108. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u +
Z
x()d U() R() u0 +
Z
x0()d:

109. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u +
Z
x()d u0 +
Z
x0()d:

110. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u +
Z
x()d u0 +
Z
x0()d:
This is (almost) a second-order stochastic dominance
constraint if x and x0 are treated as CDFs.

111. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u +
Z
x()d u0 +
Z
x0()d:
This is (almost) a second-order stochastic dominance
constraint if x and x0 are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.

112. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u +
Z
x()d u0 +
Z
x0()d:
This is (almost) a second-order stochastic dominance
constraint if x and x0 are treated as CDFs.
Ironing corresponds to taking a mean-preserving spread of the
distribution—it preserves the outside-option constraint.
Related work: Kleiner, Moldovanu, and Strack (2021), Loertscher
and Muir (2022), Akbarpour ® Dworczak ® Kominers (2023)

113. Step 3: Linearity of the planner’s problem
We now go back to t = 0.

114. Step 3: Linearity of the planner’s problem
We now go back to t = 0.
Lemma (Linearity)
The planner’s problem of choosing the optimal contract M is linear in
the outside option R().

115. Step 3: Linearity of the planner’s problem
We now go back to t = 0.
Lemma (Linearity)
The planner’s problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rule x?
() for
the designer depends linearly on R().

116. Step 3: Linearity of the planner’s problem
We now go back to t = 0.
Lemma (Linearity)
The planner’s problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rule x?
() for
the designer depends linearly on R().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent’s type
and designer’s preferences—it does not depend on R itself!

117. Step 3: Linearity of the planner’s problem
We now go back to t = 0.
Lemma (Linearity)
The planner’s problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rule x?
() for
the designer depends linearly on R().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent’s type
and designer’s preferences—it does not depend on R itself!
Both the planner’s objective and the agent’s
investment-obedience constraint are linear in x?
.

118. Step 4: Wrapping up
Lemma
The planner’s problem of choosing the optimal contract M is
equivalent to choosing a probability distribution and a scalar to
maximize a linear objective subject to single linear constraint.

119. Step 4: Wrapping up
Lemma
The planner’s problem of choosing the optimal contract M is
equivalent to choosing a probability distribution and a scalar to
maximize a linear objective subject to single linear constraint.
By standard results, there exists an optimal solution such that
the probability distribution has support of size at most two.

120. Step 4: Wrapping up
Lemma
The planner’s problem of choosing the optimal contract M is
equivalent to choosing a probability distribution and a scalar to
maximize a linear objective subject to single linear constraint.
By standard results, there exists an optimal solution such that
the probability distribution has support of size at most two.
This gives us two items in the optimal menu.

121. Step 4: Wrapping up
Lemma
The planner’s problem of choosing the optimal contract M is
equivalent to choosing a probability distribution and a scalar to
maximize a linear objective subject to single linear constraint.
By standard results, there exists an optimal solution such that
the probability distribution has support of size at most two.
This gives us two items in the optimal menu.
With no constraint, the problem is to maximize a linear objective
=) solution is an extreme point
=) option to own is optimal.

122. Step 4: Wrapping up
Lemma
The planner’s problem of choosing the optimal contract M is
equivalent to choosing a probability distribution and a scalar to
maximize a linear objective subject to single linear constraint.
By standard results, there exists an optimal solution such that
the probability distribution has support of size at most two.
This gives us two items in the optimal menu.
With no constraint, the problem is to maximize a linear objective
=) solution is an extreme point
=) option to own is optimal.
Easy extension to K linear constraints.

123. The monotone case
Assume that:
Buyer and seller virtual surpluses are monotone:
B()
1 F()
f()
and
S() +
F()
f()
are non-decreasing in ;
Both the planner’s and the designer’s objective functions
V?
(; !) and V(; !) are non-decreasing in the agent’s type .

124. The monotone case

125. The monotone case
Lemma (The monotone case)
For any outside option function R(), the mechanism designer in the
second period will choose an optimal mechanism in which the
outside-option constraint binds for types in the interval [!; ¯
!].

126. The monotone case
Lemma (The monotone case)
For any outside option function R(), the mechanism designer in the
second period will choose an optimal mechanism in which the
outside-option constraint binds for types in the interval [!; ¯
!].
Moreover, except for the case of a boundary solution,
V(!; !) + S(!) = 0;
V(¯
!; !) + B(¯
!) = 0:

127. The monotone case
Lemma (The monotone case)
For any outside option function R(), the mechanism designer in the
second period will choose an optimal mechanism in which the
outside-option constraint binds for types in the interval [!; ¯
!].
Moreover, except for the case of a boundary solution,
V(!; !) + S(!) = 0;
V(¯
!; !) + B(¯
!) = 0:
For ¯
!, the designer wants to allocate the good to the agent
anyway, so the outside-option constraint is slack;

128. The monotone case
Lemma (The monotone case)
For any outside option function R(), the mechanism designer in the
second period will choose an optimal mechanism in which the
outside-option constraint binds for types in the interval [!; ¯
!].
Moreover, except for the case of a boundary solution,
V(!; !) + S(!) = 0;
V(¯
!; !) + B(¯
!) = 0:
For ¯
!, the designer wants to allocate the good to the agent
anyway, so the outside-option constraint is slack;

129. The monotone case
Lemma (The monotone case)
For any outside option function R(), the mechanism designer in the
second period will choose an optimal mechanism in which the
outside-option constraint binds for types in the interval [!; ¯
!].
Moreover, except for the case of a boundary solution,
V(!; !) + S(!) = 0;
V(¯
!; !) + B(¯
!) = 0:
For ¯
!, the designer wants to allocate the good to the agent
anyway, so the outside-option constraint is slack;
For !, the designer prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

130. The monotone case
Suppose that the cost of investment c is sufficiently high.

131. The monotone case
Suppose that the cost of investment c is sufficiently high.
Proposition
When the investment decision of the agent is contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(0; p0)g
for some p 2 R and p0 2 R+.

132. The monotone case
Suppose that the cost of investment c is sufficiently high.
Proposition
When the investment decision of the agent is contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(0; p0)g
for some p 2 R and p0 2 R+.
When the investment decision of the agent is not contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(y;0)g
for some p 2 R and y 2 [0; 1).

133. The monotone case
Suppose that the cost of investment c is sufficiently high.
Proposition
When the investment decision of the agent is contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(0; p0)g
for some p 2 R and p0 2 R+.
When the investment decision of the agent is not contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(y;0)g
for some p 2 R and y 2 [0; 1).
When investment is contractible, offering a cash payment for
investment is effective.

134. The monotone case
Suppose that the cost of investment c is sufficiently high.
Proposition
When the investment decision of the agent is contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(0; p0)g
for some p 2 R and p0 2 R+.
When the investment decision of the agent is not contractible, the
planner optimally chooses a menu of the form M?
= f(1;p);(y;0)g
for some p 2 R and y 2 [0; 1).
When investment is contractible, offering a cash payment for
investment is effective.
When investment is not contractible, the planner can incentivize
investment only by shifting more rents to higher types.

135. Applications
Applications

136. Application #1: Dynamic resource allocation
Players:
Agent: Firm
Planner: Regulator
Designer: Regulator
Agent invests in infrastructure determining value ;
State ! represents the value for an alternative use;
The planner maximizes a combination of efficiency and revenue:
V?
(; !)x + ?
t = ( !)x + ?
t:
The designer might put more weight on revenue:
V(; !)x + t = ( !)x + t; where ?
.

137. Application #1: Dynamic resource allocation
In the monotone case when investment is not contractible, the
optimal license is a partial property right plus an option to own.

138. Application #1: Dynamic resource allocation
In the monotone case when investment is not contractible, the
optimal license is a partial property right plus an option to own.
Suppose that F! is uniform on [0; 1], and F! is an atom at 0.

139. Application #1: Dynamic resource allocation
In the monotone case when investment is not contractible, the
optimal license is a partial property right plus an option to own.
Suppose that F! is uniform on [0; 1], and F! is an atom at 0.
The optimal property right as a function of value for revenue:

140. Application #1: Dynamic resource allocation
In the monotone case when investment is not contractible, the
optimal license is a partial property right plus an option to own.
Suppose that F! is uniform on [0; 1], and F! is an atom at 0.
The optimal property right as a function of value for revenue:
= ?
= 1: option to own (1; p)
(p makes the investment-obedience constraint bind);

141. Application #1: Dynamic resource allocation
In the monotone case when investment is not contractible, the
optimal license is a partial property right plus an option to own.
Suppose that F! is uniform on [0; 1], and F! is an atom at 0.
The optimal property right as a function of value for revenue:
= ?
= 1: option to own (1; p)
(p makes the investment-obedience constraint bind);
= 1; ?
= 0: partial property right (0; y)
(y makes the investment-obedience constraint bind);

142. Application #1: Dynamic resource allocation
In the monotone case when investment is not contractible, the
optimal license is a partial property right plus an option to own.
Suppose that F! is uniform on [0; 1], and F! is an atom at 0.
The optimal property right as a function of value for revenue:
= ?
= 1: option to own (1; p)
(p makes the investment-obedience constraint bind);
= 1; ?
= 0: partial property right (0; y)
(y makes the investment-obedience constraint bind);
= ?
= 0: no right
(this is just Rogerson (1992)!)

143. Application #2: Regulating a private rental market
Players:
Agent: Tenant
Planner: Regulator
Designer: Rental company
Agent invests in the property determining her value for staying.
The state ! is the seller’s outside option (market rental price).
The seller wants to maximize revenue:
V(; !)x + t = !x + t.
The planner wants to maximize efficiency:
V?
(; !)x + ?
t = ( !)x:

144. Application #2: Regulating a private rental market
Suppose that without investment, value is uniform on [0; 1];
with investment, the value is drawn from [∆; 1 + ∆] instead.

145. Application #2: Regulating a private rental market
Suppose that without investment, value is uniform on [0; 1];
with investment, the value is drawn from [∆; 1 + ∆] instead.
Suppose that ! is known (and lies in a certain range).

146. Application #2: Regulating a private rental market
Suppose that without investment, value is uniform on [0; 1];
with investment, the value is drawn from [∆; 1 + ∆] instead.
Suppose that ! is known (and lies in a certain range).
The optimal contract is a renewable lease with price
p?
= ! ∆;
where is the Lagrange multiplier on the investment constraint.

147. Application #2: Regulating a private rental market
Suppose that without investment, value is uniform on [0; 1];
with investment, the value is drawn from [∆; 1 + ∆] instead.
Suppose that ! is known (and lies in a certain range).
The optimal contract is a renewable lease with price
p?
= ! ∆;
where is the Lagrange multiplier on the investment constraint.
If investment constraint is slack, p?
= !, so the planner “forces”
the seller to use a VCG mechanism.

148. Application #2: Regulating a private rental market
Suppose that without investment, value is uniform on [0; 1];
with investment, the value is drawn from [∆; 1 + ∆] instead.
Suppose that ! is known (and lies in a certain range).
The optimal contract is a renewable lease with price
p?
= ! ∆;
where is the Lagrange multiplier on the investment constraint.
If investment constraint is slack, p?
= !, so the planner “forces”
the seller to use a VCG mechanism.
If ! is random, then (assuming interior solution)
p?
= E[ ! j ?
! p?
?
!] ∆:

149. Application #3: Vaccine development
Players:
Agent: Pharmaceutical company
Planner: Health agency
Designer: Health agency
Company develops a vaccine; the marginal cost of production
conditional on investment is k F̃ ( = k);
Allocation x is the number of units purchased by the health
agency, with 1 being the mass of the patient population;
Health agency maximizes the total value of allocation,
V?
(; !)x = V(; !)x = !x,
where ! measures the severity of the health crisis.
Assume that 1 = ?
.

150. Application #3: Vaccine development
Assume that investment is observable and that ! is known.

151. Application #3: Vaccine development
Assume that investment is observable and that ! is known.
The optimal contract, for c high enough, is a cash payment for
the investment, plus a guaranteed sale price equal to
p?
= min
n !
?
; k̄
o
:

152. Application #3: Vaccine development
Assume that investment is observable and that ! is known.
The optimal contract, for c high enough, is a cash payment for
the investment, plus a guaranteed sale price equal to
p?
= min
n !
?
; k̄
o
:
If the planner cares about revenue as much as the designer, she
sets a price equal to !, as if she maximized efficiency.

153. Application #3: Vaccine development
Assume that investment is observable and that ! is known.
The optimal contract, for c high enough, is a cash payment for
the investment, plus a guaranteed sale price equal to
p?
= min
n !
?
; k̄
o
:
If the planner cares about revenue as much as the designer, she
sets a price equal to !, as if she maximized efficiency.
For ?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).

154. Application #3: Vaccine development
If ! is stochastic, then the optimal price satisfies
p?
= min
(
E
!j! 2 [!p? ; ¯
!p? ]
?
; k̄
)
;
where [!p? ; ¯
!p? ] is the interval of !’s for which the choice of p?
changes the second-period mechanism.

155. Application #3: Vaccine development
If ! is stochastic, then the optimal price satisfies
p?
= min
(
E
!j! 2 [!p? ; ¯
!p? ]
?
; k̄
)
;
where [!p? ; ¯
!p? ] is the interval of !’s for which the choice of p?
changes the second-period mechanism.
If the realized ! is high (health crisis is severe), the designer will
offer a price better than p?
.

156. Application #3: Vaccine development
If ! is stochastic, then the optimal price satisfies
p?
= min
(
E
!j! 2 [!p? ; ¯
!p? ]
?
; k̄
)
;
where [!p? ; ¯
!p? ] is the interval of !’s for which the choice of p?
changes the second-period mechanism.
If the realized ! is high (health crisis is severe), the designer will
offer a price better than p?
.
If the realized ! is low (health crisis is mild), the designer will
offer a price lower than p?
and compensate the producer with an
additional cash payment (on top of the payment for investment).

157. Application #4: Patent policy
Players:
Agent: Firm
Planner: Regulator
Designer: Patent agency
Firm invests in a new technology; if investment is made, the firm
can produce at constant marginal cost k F̃.
Conditional on x = 1, the firm provides a monopoly quantity to
maximize profits; conditional on x = 0, the firm competes in a
competitive market. Market demand is D(p) = 1 p.
The regulator and the patent agency maximize consumer
surplus with weight ! (but attach a positive weight to revenue).

158. Application #4: Patent policy
This setting maps into our framework with = 1
4 (1 k)2 and
V?
(; !) = V(; !) =
3
2
!:

159. Application #4: Patent policy
This setting maps into our framework with = 1
4 (1 k)2 and
V?
(; !) = V(; !) =
3
2
!:
Perfect competition is more beneficial when marginal cost is low.

160. Application #4: Patent policy
This setting maps into our framework with = 1
4 (1 k)2 and
V?
(; !) = V(; !) =
3
2
!:
Perfect competition is more beneficial when marginal cost is low.
Suppose that the density of is non-decreasing, and the weight
on revenue is small enough.

161. Application #4: Patent policy
This setting maps into our framework with = 1
4 (1 k)2 and
V?
(; !) = V(; !) =
3
2
!:
Perfect competition is more beneficial when marginal cost is low.
Suppose that the density of is non-decreasing, and the weight
on revenue is small enough.
Then, an optimal contract is to offer full patent protection to the
invention (x = 1) with exogenous probability y 2 (0; 1] at no
payment.

162. Application #5: Supply chain contracting
Players:
Agent: Small supplier
Planner: Large producer
Designer: Large producer
Producer buys some amount x of inputs from the supplier.
The supplier must invest at time t = 1 in relationship-specific
technology to produce the inputs at marginal cost c .
Through the interaction with the supplier, the large firm can learn
the supplier’s costs: The state ! is a noisy signal of .
Producer maximizes profits having a constant marginal value 1
for each unit of the input: V!() = 1 and = 1:
Producer proposes a contract: V?
!() = 1 and ?
= 1.

163. Application #5: Supply chain contracting
Suppose investment by the small supplier is not contractible.

164. Application #5: Supply chain contracting
Suppose investment by the small supplier is not contractible.
The producer will in general commit to a two-price scheme,
committing to buy up to y units at some price pH, and any
number of units at some lower price pL.

165. Application #5: Supply chain contracting
Suppose investment by the small supplier is not contractible.
The producer will in general commit to a two-price scheme,
committing to buy up to y units at some price pH, and any
number of units at some lower price pL.
If investment by the small supplier is contractible, assuming the
cost of investment is high enough, the producer will offer an
upfront payment for setting up production and a guaranteed
purchase price for any number of units.

166. Summary and future steps
Summary and future steps

167. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.

168. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.

169. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.

170. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.

171. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.
Future steps:

172. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.
Future steps:
Can the planner do better by designing a menu of menus:
M = fMjgj; such that the designer selects a menu Mj based on
realized state, and the agent chooses an option from Mj?

173. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.
Future steps:
Can the planner do better by designing a menu of menus:
M = fMjgj; such that the designer selects a menu Mj based on
realized state, and the agent chooses an option from Mj? Yes!

174. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.
Future steps:
Can the planner do better by designing a menu of menus:
M = fMjgj; such that the designer selects a menu Mj based on
realized state, and the agent chooses an option from Mj? Yes!
Symmetric bargaining power and “two-sided” optimal rights.

175. Summary and future steps
We introduced a simple but flexible framework for analyzing
optimal design of (property) rights.
Optimally designed rights partially restore commitment to future
trading mechanisms.
The optimal right often features an option to own.
Future steps:
Can the planner do better by designing a menu of menus:
M = fMjgj; such that the designer selects a menu Mj based on
realized state, and the agent chooses an option from Mj? Yes!
Symmetric bargaining power and “two-sided” optimal rights.
Optimal allocation of contracts at t = 0.