Slides presented at a seminar at Bonn University

Slides from Bonn presentation

1. A mechanism-design approach to property rights
Piotr Dworczak
?
Ellen Muir
(Northwestern; GRAPE) (Harvard)
November 29, 2023
Micro Theory Seminar, University of Bonn
?
Co-funded by the European Union (ERC, IMD-101040122). Views and opinions expressed are those of the authors
only and do not necessarily reect those of the European Union or the European Research Council.

2. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretradeoffsin the design of property rights:Investment incentives;Efciency of reallocation;Market power and distribution of surplus.Example: How to optimally design spectrum licenses?

3. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretradeoffsin the design of property rights:Investment incentives;Efciency of reallocation;Market power and distribution of surplus.Example: How to optimally design spectrum licenses?

4. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretradeoffsin the design of property rights:Investment incentives;Efciency of reallocation;Market power and distribution of surplus.Example: How to optimally design spectrum licenses?

5. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretradeoffsin the design of property rights:Investment incentives;Efciency of reallocation;Market power and distribution of surplus.Example: How to optimally design spectrum licenses?

6. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretradeoffsin the design of property rights:Investment incentives;Efciency of reallocation;Market power and distribution of surplus.Example: How to optimally design spectrum licenses?

7. Motivation
Assignment of property rightsmatters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
There aretradeoffsin the design of property rights:Investment incentives;Efciency of reallocation;Market power and distribution of surplus.Example: How to optimally design spectrum licenses?

8. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

9. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

10. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

11. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

12. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

13. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

14. Key feature of our framework
Our paper: Property rightsdetermine the holder's outside
options(with regard to the underlying good) in 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 propose a framework in which we characterize theoptimal
property rightusing a mechanism-design approach.

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34. To summarize
Fundamental friction: Designer cannot commit to future trading
mechanisms, resulting in
ex-post inefciency (mechanism may be suboptimal);
hold-up problem (inefcient investment decisions).Designerchooses property rightsto alleviate these frictions.Question:What is the optimal set of rights?Main result:The optimal property right is simple but exible,
often featuring anoption to own.

35. To summarize
Fundamental friction: Designer cannot commit to future trading
mechanisms, resulting in
ex-post inefciency (mechanism may be suboptimal);
hold-up problem (inefcient investment decisions).Designerchooses property rightsto alleviate these frictions.Question:What is the optimal set of rights?Main result:The optimal property right is simple but exible,
often featuring anoption to own.

36. To summarize
Fundamental friction: Designer cannot commit to future trading
mechanisms, resulting in
ex-post inefciency (mechanism may be suboptimal);
hold-up problem (inefcient investment decisions).Designerchooses property rightsto alleviate these frictions.Question:What is the optimal set of rights?Main result:The optimal property right is simple but exible,
often featuring anoption to own.

37. To summarize
Fundamental friction: Designer cannot commit to future trading
mechanisms, resulting in
ex-post inefciency (mechanism may be suboptimal);
hold-up problem (inefcient investment decisions).Designerchooses property rightsto alleviate these frictions.Question:What is the optimal set of rights?Main result:The optimal property right is simple but exible,
often featuring anoption to own.

38. To summarize
Fundamental friction: Designer cannot commit to future trading
mechanisms, resulting in
ex-post inefciency (mechanism may be suboptimal);
hold-up problem (inefcient investment decisions).Designerchooses property rightsto alleviate these frictions.Question:What is the optimal set of rights?Main result:The optimal property right is simple but exible,
often featuring anoption to own.

39. To summarize
Fundamental friction: Designer cannot commit to future trading
mechanisms, resulting in
ex-post inefciency (mechanism may be suboptimal);
hold-up problem (inefcient investment decisions).Designerchooses property rightsto alleviate these frictions.Question:What is the optimal set of rights?Main result:The optimal property right is simple but exible,
often featuring anoption to own.

40. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

41. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

42. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

43. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

44. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

45. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

46. Property rights in economics
Assignment of property rights matters, particularly in the
presence of transaction costs (Coase, 1960; Williamson, 1979).
Two large literatures:
Property rights affectefciencyunder private information:Myerson and Satterthwaite (1983);
Cramton, Gibbons and Klemperer (1987), ...Property rights affectinvestment incentives:Grossman and Hart (1986), Hart and Moore (1990), Aghion,
Dewatripont and Rey (1994), Hermalin and Katz (1993), ...

47. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

48. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

49. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

50. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

51. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

52. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

53. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

54. Marginal contribution
We study both effects in a setting with one-sided private
information and exogenous distribution of bargaining power.
To the best of our knowledge, our paper is rst tocharacterize
the optimal property right from a non-parametric class
(in a setting where the rst best cannot in general be achieved).
Attractive properties of option-to-own contracts were studied:
Liability rulesraise second-best efciency in bargaining.-
Options to own/ put optionsmay induce efcient
investments in hold-up problems.
- ¨oldeke and Schmidt (1995, 1998), Maskin and Tirole
(1999b)
We provide anoptimality foundation.

55. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

56. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

57. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

58. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

59. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

60. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

61. Marginal contribution
The setting allows us to study market-design applications:
optimal license design;
regulating a private rental market;optimal patent policy.On the technical side:We provide a novel solution method for mechanism design
withtype-dependent outside options(Lewis and
Sappington, 1988, Jullien, 2000) based on an extension of
theironingtechnique (Myerson, 1981).
We show how to optimize over the outside-option function.

62. Other related papers
Allocation mechanisms versus efcient investment:
Rogerson (1992), Bergemann and V¨alim¨aki (2002), Milgrom
(2017), Hateld, Kojima, and Kominers (2019), Gershkov,
Moldovanu, Strack, and Zhang (2021), Akbarpour, Kominers, Li,
Li, and Milgrom (2023), ...
Related techniques:Kleiner, Moldovanu, and Strack (2021),
Loertscher and Muir (2022), Kang (2023), Akbarpour
R
Dworczak
R
Kominers (2023),...
Spectrum license design:Posner and Weyl (2017), Milgrom,
Weyl and Zhang (2017), Weyl and Zhang (2017), ...
Optimal patent design:Wright (1983), Klemperer (1990),
Gilbert and Shapiro (1990), Gallini (1992), Kremer (1998),
Hopenhayn, Llobet and Mitchell (2006), Weyl and Tirole (2012)...

63. Model
Model

64. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

65. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

66. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

67. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

68. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

69. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

70. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

71. Model
There is anagent, aprincipal, and adesigner;
At timet=0:The designer designs a contract (set of rights) that the agent
holds.
At timet=1:The agent decides whether to invest at (sunk) costc>0;At timet=2:The agent's (privately observed) typeand a public state!
are drawn from a joint distribution that depends on whether
the agent invested or not.
The principal chooses a mechanism, and trade happens.

72. Model
Timet=2continued:
Principal chooses a trading mechanism(x!();t!()), where
x2[0;1]denotes an allocation, andt2Rdenotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.
Principal maximizesV(; !)x+t, where >0.Agent's utility isxt.

73. Model
Timet=2continued:
Principal chooses a trading mechanism(x!();t!()), where
x2[0;1]denotes an allocation, andt2Rdenotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.
Principal maximizesV(; !)x+t, where >0.Agent's utility isxt.

74. Model
Timet=2continued:
Principal chooses a trading mechanism(x!();t!()), where
x2[0;1]denotes an allocation, andt2Rdenotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.
Principal maximizesV(; !)x+t, where >0.Agent's utility isxt.

75. Model
Timet=2continued:
Principal chooses a trading mechanism(x!();t!()), where
x2[0;1]denotes an allocation, andt2Rdenotes a transfer;
Mechanism is chosen subject to IC and IR constraints, and must
respect the rights that the agent holds.
Principal maximizesV(; !)x+t, where >0.Agent's utility isxt.

76. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

77. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

78. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

79. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

80. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

81. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

82. Model
Agent's problem att=1
The agent decides whether to pay the costc>0 to invest.If the agent invests:!G,F!.Otherwise:!G,F
!, withF!
FOSD
F
!, for every!.
In thenon-contractible case, the mechanism att=2 and the
designer's contract do not depend on the investment decision.
In thecontractible case, the mechanism att=2 and the
designer's contract are contingent on the investment decision.

83. Model
Designer's problem att=0
Designer chooses a contract that is a menu of “rights”
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary (Mis compact).
The agent can “execute” any one of these rights att=2.Designer maximizesV
?
(; !)x+
?
t, where
?
0.
Assume: Investment is preferred by the designer to no investment;
investment can be induced by some contract; but it is not induced if
the agent holds no rights.

84. Model
Designer's problem att=0
Designer chooses a contract that is a menu of “rights”
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary (Mis compact).
The agent can “execute” any one of these rights att=2.Designer maximizesV
?
(; !)x+
?
t, where
?
0.
Assume: Investment is preferred by the designer to no investment;
investment can be induced by some contract; but it is not induced if
the agent holds no rights.

85. Model
Designer's problem att=0
Designer chooses a contract that is a menu of “rights”
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary (Mis compact).
The agent can “execute” any one of these rights att=2.Designer maximizesV
?
(; !)x+
?
t, where
?
0.
Assume: Investment is preferred by the designer to no investment;
investment can be induced by some contract; but it is not induced if
the agent holds no rights.

86. Model
Designer's problem att=0
Designer chooses a contract that is a menu of “rights”
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary (Mis compact).
The agent can “execute” any one of these rights att=2.Designer maximizesV
?
(; !)x+
?
t, where
?
0.
Assume: Investment is preferred by the designer to no investment;
investment can be induced by some contract; but it is not induced if
the agent holds no rights.

87. Model
Designer's problem att=0
Designer chooses a contract that is a menu of “rights”
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary (Mis compact).
The agent can “execute” any one of these rights att=2.Designer maximizesV
?
(; !)x+
?
t, where
?
0.
Assume: Investment is preferred by the designer to no investment;
investment can be induced by some contract; but it is not induced if
the agent holds no rights.

88. Model
Designer's problem att=0
Designer chooses a contract that is a menu of “rights”
M=f(x
i;t
i)g
i2I;
wherex
i2[0;1],t
i2R, and setIis arbitrary (Mis compact).
The agent can “execute” any one of these rights att=2.
Designer maximizesV
?
(; !)x+
?
t, where
?
0.
Technicalities:[;

]is a compact subset ofR;VandV
?
are
continuous in(and measurable in!),Fhas a continuous positive
density on.

89. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

90. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

91. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

92. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

93. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

94. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

95. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

96. Comments about the model
A contract creates anoutside optionfor the agent: principal
must guarantee that the agent's type-utility from participating in
the mechanism is not lower than
max
i2I
fx
it
ig:
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, wherey2(0;1);Flexible property right:M=fs;p(s))g
s2[0;1].

97. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

98. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

99. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

100. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

101. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

102. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

103. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

104. Comments about the model
There are two frictions in the model:1.
principal (“ex-post inefciency”);
2. 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 designer and principal are
aligned, it is optimal to allocate no rights to the agent.
If the principal did not care about her revenue (=0), she would
“buy out” the agent's rights.

105. Analysis and results
Analysis and results

106. Main result
Theorem
There exists an optimal contract that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal contract is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.If there were no hold-up problem in the model,M
?
=f(1;p)g
would be always optimal.

107. Main result
Theorem
There exists an optimal contract that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal contract is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.If there were no hold-up problem in the model,M
?
=f(1;p)g
would be always optimal.

108. Main result
Theorem
There exists an optimal contract that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal contract is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.If there were no hold-up problem in the model,M
?
=f(1;p)g
would be always optimal.

109. Main result
Theorem
There exists an optimal contract that takes the form
M
?
=f(1;p);(y;p
0
)gfor some p;p
0
2Rand y2[0;1).
Comments:
The optimal contract is simple in that it consists ofat most two
types of rights.
One of the rights can be taken to be anoption to own.If there were no hold-up problem in the model,M
?
=f(1;p)g
would be always optimal.

110. Step 1: Outside option constraint
Lemma
A choice of contract M by the designer 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 by1.
The lemma follows immediately from the observation that, for anyM,
we can set
R() = maxf0;max
i2I
fx
it
igg:
The designermaximizes over type-dependent outside optionsfor
the agent.

111. Step 1: Outside option constraint
Lemma
A choice of contract M by the designer 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 by1.
The lemma follows immediately from the observation that, for anyM,
we can set
R() = maxf0;max
i2I
fx
it
igg:
The designermaximizes over type-dependent outside optionsfor
the agent.

112. Step 1: Outside option constraint
Lemma
A choice of contract M by the designer 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 by1.
The lemma follows immediately from the observation that, for anyM,
we can set
R() = maxf0;max
i2I
fx
it
igg:
The designermaximizes over type-dependent outside optionsfor
the agent.

113. Step 1: Outside option constraint
Fixing!(and dropping it from the notation), the principal solves:
max
x:![0;1];u0
Z


W()x()du
s.t.xis non-decreasing;
U()u+
Z


x()dR();82;
where
W()(V() +B())f();
and
B() =
1F()
f()
:
This is a problem considered by Jullien (2000).

114. Step 1: Outside option constraint
Fixing!(and dropping it from the notation), the principal solves:
max
x:![0;1];u0
Z


W()x()du
s.t.xis non-decreasing;
U()u+
Z


x()dR();82;
where
W()(V() +B())f();
and
B() =
1F()
f()
:
This is a problem considered by Jullien (2000).

115. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

116. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

117. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

118. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

119. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

120. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

121. Step 2: Ironing Jullien (2000)
Naively, the principal wants to setx() =1 whenW()>0, and
setx()to be as low as possible (givenR) whenW()<0.
There arethree complications:this may violate monotonicity ofx();x()cannot be minimized point-wise, because allocation to
typeaffects the utility of higher types;
the principal may want to use the transfer to satisfy the
outside-option constraint.
These complications can be addressed by adapting theironing
techniquefrom Myerson (1981).
Dene
W() =
Z

W()d;W=co(W):

122. Step 2: Ironing Jullien (2000)

123. Step 2: Ironing Jullien (2000)

124. Step 2: Ironing Jullien (2000)

125. Step 2: Ironing Jullien (2000)

126. Step 2: Ironing Jullien (2000)

127. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

128. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

129. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()dU()R()u
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

130. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

131. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

132. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

133. Step 2: Ironing Jullien (2000)
Why does ironing work?
The outside option constraint is:
u+
Z


x()du
0
+
Z


x0()d:
This is (almost) asecond-order stochastic dominance
constraint ifxandx0are 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
R
Dworczak
R
Kominers (2023)

134. Step 3: Linearity of the designer's problem
We now go back tot=0.
Lemma (Linearity)
The designer's problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
Both the designer's objective and the agent's
investment-obedience constraint are linear inx
?
.

135. Step 3: Linearity of the designer's problem
We now go back tot=0.
Lemma (Linearity)
The designer's problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
Both the designer's objective and the agent's
investment-obedience constraint are linear inx
?
.

136. Step 3: Linearity of the designer's problem
We now go back tot=0.
Lemma (Linearity)
The designer's problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
Both the designer's objective and the agent's
investment-obedience constraint are linear inx
?
.

137. Step 3: Linearity of the designer's problem
We now go back tot=0.
Lemma (Linearity)
The designer's problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
Both the designer's objective and the agent's
investment-obedience constraint are linear inx
?
.

138. Step 3: Linearity of the designer's problem
We now go back tot=0.
Lemma (Linearity)
The designer's problem of choosing the optimal contract M is linear in
the outside option R().
By the above construction, the optimal allocation rulex
?
()for
the principal dependslinearlyonR().
Key intuition: The set of types at which the outside-option
constraint binds is pinned down by the distribution of agent's type
and principal's preferences—it does not depend onRitself!
Both the designer's objective and the agent's
investment-obedience constraint are linear inx
?
.

139. Step 4: Wrapping up
Lemma
The designer'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.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution such that the probability
distribution has support of size at most two (e.g., Fuchs and
Skrzypacz, 2015; Bergemann et al., 2018; Le Treust and Tomala,
2019; Doval and Skreta, 2022; Kang, 2023).
This gives us two items in the optimal menu.

140. Step 4: Wrapping up
Lemma
The designer'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.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution such that the probability
distribution has support of size at most two (e.g., Fuchs and
Skrzypacz, 2015; Bergemann et al., 2018; Le Treust and Tomala,
2019; Doval and Skreta, 2022; Kang, 2023).
This gives us two items in the optimal menu.

141. Step 4: Wrapping up
Lemma
The designer'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.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution such that the probability
distribution has support of size at most two (e.g., Fuchs and
Skrzypacz, 2015; Bergemann et al., 2018; Le Treust and Tomala,
2019; Doval and Skreta, 2022; Kang, 2023).
This gives us two items in the optimal menu.

142. Step 4: Wrapping up
Lemma
The designer'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.
Using the representationR()u
0
+
R


x0()d, we can
optimize overx0(treated as a distribution) andu
0
.
There exists an optimal solution such that the probability
distribution has support of size at most two (e.g., Fuchs and
Skrzypacz, 2015; Bergemann et al., 2018; Le Treust and Tomala,
2019; Doval and Skreta, 2022; Kang, 2023).
This gives us two items in the optimal menu.

143. Corollaries of the proof
With no constraint, the problem is to maximize a linear objective
=)solution is an extreme point
=)option to ownis optimal.
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

144. Corollaries of the proof
With no constraint, the problem is to maximize a linear objective
=)solution is an extreme point
=)option to ownis optimal.
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

145. Corollaries of the proof
With no constraint, the problem is to maximize a linear objective
=)solution is an extreme point
=)option to ownis optimal.
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

146. Corollaries of the proof
With no constraint, the problem is to maximize a linear objective
=)solution is an extreme point
=)option to ownis optimal.
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

147. Corollaries of the proof
With no constraint, the problem is to maximize a linear objective
=)solution is an extreme point
=)option to ownis optimal.
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

148. Corollaries of the proof
With no constraint, the problem is to maximize a linear objective
=)solution is an extreme point
=)option to ownis optimal.
Easy extension toKlinear constraints
=)We need at mostK+1 options in the optimal menu.
Extension to continuous investment:Suppose agent choosese2[0;1]at strictly convex cost
c(e), andeis the probability thatis drawn fromF!.
Satisfying FOC at the target investment levele
?
is sufcient
for obedience.
FOC is linear inR() =)we need two options in the
optimal menu.

149. The monotone case
Assumethat:
Buyer and seller virtual surpluses aremonotone:
B()
1F()
f()
and
S()+
F()
f()
are non-decreasing in;
Both thedesigner's and the principal's objectivefunctions
V
?
(; !)andV(; !)arenon-decreasingin the agent's type.

150. The monotone case

151. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal 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 principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

152. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal 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 principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

153. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal 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 principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

154. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal 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 principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

155. The monotone case
Lemma (The monotone case)
For any outside option function R(), the principal 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 principal wants to allocate the good to the agent
anyway, so the outside-option constraint isslack;
For
?
!, the principal prefers to “buy out” the rights using a
monetary payment, so the constraint is again slack.

156. The monotone case
Suppose that the cost of investmentcis sufciently high.
Proposition
When the investment decision of the agent is contractible, the
designer optimally chooses a menu of the form
M
?
=f(1;p);(0;p
0
)gfor some p2Rand p
0
2R+.
When the investment decision of the agent is not contractible, the
designer optimally chooses a menu of the form M
?
=f(1;p);(y;0)g
for some p2Rand y2[0;1).
When investment is contractible,offering a cash paymentfor
investment is effective.
When investment is not contractible, the designer can incentivize
investment only byshifting more rents to higher types.

157. The monotone case
Suppose that the cost of investmentcis sufciently high.
Proposition
When the investment decision of the agent is contractible, the
designer optimally chooses a menu of the form
M
?
=f(1;p);(0;p
0
)gfor some p2Rand p
0
2R+.
When the investment decision of the agent is not contractible, the
designer optimally chooses a menu of the form M
?
=f(1;p);(y;0)g
for some p2Rand y2[0;1).
When investment is contractible,offering a cash paymentfor
investment is effective.
When investment is not contractible, the designer can incentivize
investment only byshifting more rents to higher types.

158. The monotone case
Suppose that the cost of investmentcis sufciently high.
Proposition
When the investment decision of the agent is contractible, the
designer optimally chooses a menu of the form
M
?
=f(1;p);(0;p
0
)gfor some p2Rand p
0
2R+.
When the investment decision of the agent is not contractible, the
designer optimally chooses a menu of the form M
?
=f(1;p);(y;0)g
for some p2Rand y2[0;1).
When investment is contractible,offering a cash paymentfor
investment is effective.
When investment is not contractible, the designer can incentivize
investment only byshifting more rents to higher types.

159. The monotone case
Suppose that the cost of investmentcis sufciently high.
Proposition
When the investment decision of the agent is contractible, the
designer optimally chooses a menu of the form
M
?
=f(1;p);(0;p
0
)gfor some p2Rand p
0
2R+.
When the investment decision of the agent is not contractible, the
designer optimally chooses a menu of the form M
?
=f(1;p);(y;0)g
for some p2Rand y2[0;1).
When investment is contractible,offering a cash paymentfor
investment is effective.
When investment is not contractible, the designer can incentivize
investment only byshifting more rents to higher types.

160. The monotone case
Suppose that the cost of investmentcis sufciently high.
Proposition
When the investment decision of the agent is contractible, the
designer optimally chooses a menu of the form
M
?
=f(1;p);(0;p
0
)gfor some p2Rand p
0
2R+.
When the investment decision of the agent is not contractible, the
designer optimally chooses a menu of the form M
?
=f(1;p);(y;0)g
for some p2Rand y2[0;1).
When investment is contractible,offering a cash paymentfor
investment is effective.
When investment is not contractible, the designer can incentivize
investment only byshifting more rents to higher types.

161. Applications
Applications

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

163. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

164. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

165. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

166. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

167. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

168. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

169. Application #1: Dynamic resource allocation
In the monotone case, when investment iscontractible, the
optimal license is a cash payment plus an option to own.
If investment isnon-contractible, the optimal license is a partial
property right plus an option to own.
Suppose thatF!is uniform on[0;1], andF
!is an atom at 0.The optimal property right as a function of value for revenue:=
?
=1:option to own(1;p)
(pmakes the investment-obedience constraint bind);
=1;
?
=0:partial property right(0;y)
(ymakes the investment-obedience constraint bind);
=
?
=0:no right
(consistent with Rogerson (1992))

170. Application #2: Regulating a private rental market
Players:
Agent: Tenant
Designer: Regulator
Principal: Rental company
Agent invests in the property determining her valuefor staying.
The state!is the seller's outside option (market rental price).
The seller wants to maximize revenue:
V(; !)x+t=!x+t.
The designer wants to maximize efciency:
V
?
(; !)x+
?
t= (!)x:

171. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0:
Suppose that!is known (and lies in a certain range).The optimal contract is a renewable lease with price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:

172. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0:
Suppose that!is known (and lies in a certain range).The optimal contract is a renewable lease with price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:

173. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0:
Suppose that!is known (and lies in a certain range).The optimal contract is a renewable lease with price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:

174. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0:
Suppose that!is known (and lies in a certain range).The optimal contract is a renewable lease with price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:

175. Application #2: Regulating a private rental market
Suppose that investment increases the value by>0:
Suppose that!is known (and lies in a certain range).The optimal contract is a renewable lease with price
p
?
=!;
whereis the Lagrange multiplier on the investment constraint.
If investment constraint is slack,p
?
=!, so the designer “forces”
the seller to use a VCG mechanism.
If!is random, then (assuming interior solution)
p
?
=E[!j
?
!p
?

?
!]:

176. Application #3: Vaccine development
Players:
Agent: Pharmaceutical company
Designer: Health agency
Principal: Health agency
Company develops a vaccine; the marginal cost of production
conditional on investment isk
e
F(=k);
Allocationxis 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=
?
.

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

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).

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

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).

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

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).

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

?
;

k
o
:
If the designer cares about revenue as much as the principal,
she sets a price equal to!,as ifshe maximized efciency.
For
?
close enough to 0, the optimal contract is a guaranteed
sale (for all producer types).

181. Application #3: Vaccine development
If!is stochastic, then the optimal price satises
p
?
= min
(
E

!j!2[!
p
?;!p
?]


?
;

k
)
;
where[!
p
?;!p
?]is the interval of!'s for which the choice ofp
?
changes the second-period mechanism.
If the realized!is high (health crisis is severe), the principal will
offer a price better thanp
?
.
If the realized!is low (health crisis is mild), the principal will
offer a price lower thanp
?
and compensate the producer with an
additional cash payment (on top of the payment for investment).

182. Application #3: Vaccine development
If!is stochastic, then the optimal price satises
p
?
= min
(
E

!j!2[!
p
?;!p
?]


?
;

k
)
;
where[!
p
?;!p
?]is the interval of!'s for which the choice ofp
?
changes the second-period mechanism.
If the realized!is high (health crisis is severe), the principal will
offer a price better thanp
?
.
If the realized!is low (health crisis is mild), the principal will
offer a price lower thanp
?
and compensate the producer with an
additional cash payment (on top of the payment for investment).

183. Application #3: Vaccine development
If!is stochastic, then the optimal price satises
p
?
= min
(
E

!j!2[!
p
?;!p
?]


?
;

k
)
;
where[!
p
?;!p
?]is the interval of!'s for which the choice ofp
?
changes the second-period mechanism.
If the realized!is high (health crisis is severe), the principal will
offer a price better thanp
?
.
If the realized!is low (health crisis is mild), the principal will
offer a price lower thanp
?
and compensate the producer with an
additional cash payment (on top of the payment for investment).

184. Application #4: Patent policy
Players:
Agent: Firm
Designer: Regulator
Principal: Patent agency
Firm invests in a new technology; if investment is made, the rm
can produce at constant marginal costk
~
F.
Conditional onx=1, the rm provides a monopoly quantity to
maximize prots; conditional onx=0, the rm competes in a
competitive market. Market demand isD(p) =1p.
The regulator and the patent agency maximize consumer
surplus with weight!(but attach a positive weight to revenue).

185. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal contract is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.

186. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal contract is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.

187. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal contract is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.

188. Application #4: Patent policy
This setting maps into our framework with=
1
4
(1k)
2
and
V
?
(;!) =V(;!) =
3
2
!:
Perfect competition is more benecial when marginal cost is low.
Suppose that the density ofis non-decreasing, and the weight
on revenueis small enough.
Then, an optimal contract is to offer full patent protection to the
invention (x=1) with exogenous probabilityy2(0;1]at no
payment.

189. Application #5: Supply chain contracting
Players:
Agent: Small supplier
Designer: Large producer
Principal: Large producer
Producer buys some amountxof inputs from the supplier.
The supplier must invest at timet=1 in relationship-specic
technology to produce the inputs at marginal costc .
Through the interaction with the supplier, the large rm can learn
the supplier's costs: The state!is a noisy signal of.
Producer maximizes prots having a constant marginal value 1
for each unit of the input:V!() =1 and=1:
Producer proposes a contract:V
?
!() =1 and
?
=1.

190. Application #5: Supply chain contracting
Suppose investment by the small supplier isnot contractible.
The producer will in general commit to atwo-price scheme,
committing to buy up toyunits at some pricep
H, and any
number of units at some lower pricep
L.
If investment by the small supplier iscontractible, assuming the
cost of investment is high enough, the producer will offer an
upfront paymentfor setting up production and aguaranteed
purchase pricefor any number of units.

191. Application #5: Supply chain contracting
Suppose investment by the small supplier isnot contractible.
The producer will in general commit to atwo-price scheme,
committing to buy up toyunits at some pricep
H, and any
number of units at some lower pricep
L.
If investment by the small supplier iscontractible, assuming the
cost of investment is high enough, the producer will offer an
upfront paymentfor setting up production and aguaranteed
purchase pricefor any number of units.

192. Application #5: Supply chain contracting
Suppose investment by the small supplier isnot contractible.
The producer will in general commit to atwo-price scheme,
committing to buy up toyunits at some pricep
H, and any
number of units at some lower pricep
L.
If investment by the small supplier iscontractible, assuming the
cost of investment is high enough, the producer will offer an
upfront paymentfor setting up production and aguaranteed
purchase pricefor any number of units.

193. Summary and future steps
Summary and future steps

194. Summary
We introduced a simple but exible framework for analyzing
optimal design of (property) rights.
Optimally designed rightspartially restore commitmentto
future trading mechanisms.
The optimal right often features anoption to own.

195. Summary
We introduced a simple but exible framework for analyzing
optimal design of (property) rights.
Optimally designed rightspartially restore commitmentto
future trading mechanisms.
The optimal right often features anoption to own.

196. Summary
We introduced a simple but exible framework for analyzing
optimal design of (property) rights.
Optimally designed rightspartially restore commitmentto
future trading mechanisms.
The optimal right often features anoption to own.

197. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

198. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

199. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

200. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

201. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

202. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?

203. Future steps
Maskin-Tirole (1999) critique of incomplete contracts: Can the
designer do better byconditioning rights on the statein an
incentive-compatible way?
Consider designing a menu of menusM=fM
jg
jsuch that the
principal selects a menuM
jbased on realized state and the
agent chooses an option fromM
j.
This can help!
What if both parties (att=2) haveprivate information and
bargaining poweris not exogenous?
Allocate rights to both
parties?
What if the designer can prohibit the usage of certain
allocation-transfer pairs by the principal?
Optimalallocation of contracts att=0 to multiple agents?