Topics in Microeconomics ECON 5210 ( Part II - Contract Theory )
Tapas Kundu & Tore Nilssen
University of Oslo
Spring Semester 2009
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 1 / 28 Private information on
I on what the agent does (hidden actions) I on who the agent is (hidden information) Information and sequence of actions
I Adverse Selection models :
F Screening - The un-informed party (UP), who is imperfectly informed about the informed party (IP), moves …rst F Signaling - Same informational situation as before, but IP moves …rst
I Moral Hazard models : UP, who is imperfectly informed about IP’s action, moves …rst
Introduction Introduction
Strategic interaction between privately informed agents
I Inadequacy of general equilibrium set up
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 2 / 28 Information and sequence of actions
I Adverse Selection models :
F Screening - The un-informed party (UP), who is imperfectly informed about the informed party (IP), moves …rst F Signaling - Same informational situation as before, but IP moves …rst
I Moral Hazard models : UP, who is imperfectly informed about IP’s action, moves …rst
Introduction Introduction
Strategic interaction between privately informed agents
I Inadequacy of general equilibrium set up Private information on
I on what the agent does (hidden actions) I on who the agent is (hidden information)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 2 / 28 Introduction Introduction
Strategic interaction between privately informed agents
I Inadequacy of general equilibrium set up Private information on
I on what the agent does (hidden actions) I on who the agent is (hidden information) Information and sequence of actions
I Adverse Selection models :
F Screening - The un-informed party (UP), who is imperfectly informed about the informed party (IP), moves …rst F Signaling - Same informational situation as before, but IP moves …rst
I Moral Hazard models : UP, who is imperfectly informed about IP’s action, moves …rst
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 2 / 28 Introduction Overview Introduction Overview of the course
Bilateral contracting with private information (class 1 & 2)
I Adverse selection I Moral hazard I Multidimensional informational problems Contracting under multilateral asymmetry (class 3) Repeated Contracting (class 4 & 5)
I Dynamic moral hazard, Dynamic adverse selection Incomplete contracts (class 6)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 3 / 28 Adverse Selection Discrete types Adverse Selection Bilateral setting with discrete types
B buys a good from S B gets v (q) T S gets = T cq H with probability 1 Assume = < with probability L H L Assume that S sets the terms of contract; B has reservation utility u.
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 4 / 28 B receives reservation utility i v (qi ) Ti = u ! Optimal qi solves i v 0 (qi ) = c.
Adverse Selection Discrete types Bilateral setting with discrete types First-best Outcome
S knows B’stype S solves max Ti cqi Ti ;qi
subject to i v (qi ) Ti u (IR)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 5 / 28 Optimal qi solves i v 0 (qi ) = c.
Adverse Selection Discrete types Bilateral setting with discrete types First-best Outcome
S knows B’stype S solves max Ti cqi Ti ;qi
subject to i v (qi ) Ti u (IR) B receives reservation utility i v (qi ) Ti = u !
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 5 / 28 Adverse Selection Discrete types Bilateral setting with discrete types First-best Outcome
S knows B’stype S solves max Ti cqi Ti ;qi
subject to i v (qi ) Ti u (IR) B receives reservation utility i v (qi ) Ti = u ! Optimal qi solves i v 0 (qi ) = c.
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 5 / 28 and qi = arg max i v (q) T (q) q
Adverse Selection Discrete types Bilateral setting with discrete types Second-best Outcome: Non linear pricing
S cannot observe B’stype, so S has to o¤er a set of choices T (q), independent of B’stype The optimization problem
max [T (qL) cqL] + (1 )[T (qH ) cqH ] Ti ;qi
subject to i v (qi ) Ti > 0 (IR)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 6 / 28 Adverse Selection Discrete types Bilateral setting with discrete types Second-best Outcome: Non linear pricing
S cannot observe B’stype, so S has to o¤er a set of choices T (q), independent of B’stype The optimization problem
max [T (qL) cqL] + (1 )[T (qH ) cqH ] Ti ;qi
subject to i v (qi ) Ti > 0 (IR) and qi = arg max i v (q) T (q) q
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 6 / 28 Adverse Selection Discrete types Revelation Principle
Without loss of generality, can restrict ourselves to the pair of optimal choices made by two types of buyers [T (qL) ; qL] ; [T (qH ) ; qH ] f g However, they have to be incentive compatible.
H v (qH ) T (qH ) H v (qL) T (qL) and
Lv (qL) T (qL) Lv (qH ) T (qH )
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 7 / 28 Adverse Selection Discrete types The optimization problem
max [T (qL) cqL] + (1 )[T (qH ) cqH ] Ti ;qi
subject to
IRL : Lv (qL) TL 0 IRH : H v (qH ) TH 0 ICL : Lv (qL) T (qL) Lv (qH ) T (qH ) ICH : H v (qH ) T (qH ) H v (qL) T (qL)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 8 / 28 ICH is binding
If not, increase T (qH )
H v (qH ) T (qH ) > H v (qL) T (qL) > Lv (qL) T (qL) = 0 qH > qL (add ICs)
ICL and IRH can be ignored.(ICH + qH > qL ICL) f g !
Adverse Selection Discrete types Solving the model
IRL is binding
If not, increase T (qL) and T (qH ) by the same amount
H v (qH ) T (qH ) > H v (qL) T (qL) > Lv (qL) T (qL)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 9 / 28 qH > qL (add ICs)
ICL and IRH can be ignored.(ICH + qH > qL ICL) f g !
Adverse Selection Discrete types Solving the model
IRL is binding
If not, increase T (qL) and T (qH ) by the same amount
H v (qH ) T (qH ) > H v (qL) T (qL) > Lv (qL) T (qL) ICH is binding
If not, increase T (qH )
H v (qH ) T (qH ) > H v (qL) T (qL) > Lv (qL) T (qL) = 0
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 9 / 28 Adverse Selection Discrete types Solving the model
IRL is binding
If not, increase T (qL) and T (qH ) by the same amount
H v (qH ) T (qH ) > H v (qL) T (qL) > Lv (qL) T (qL) ICH is binding
If not, increase T (qH )
H v (qH ) T (qH ) > H v (qL) T (qL) > Lv (qL) T (qL) = 0 qH > qL (add ICs)
ICL and IRH can be ignored.(ICH + qH > qL ICL) f g !
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 9 / 28 Adverse Selection Discrete types Solving the model
1 max (Lv (qL) cqL) (H L) v (qL) + (1 )[H v (qH ) cqH ] qH ;qL " virtual surplus from type L #
H v 0 (qH ) = c v (q ) = c > c L 0 L 1 H L 1 L If S increases qL, it makes the [T (qL) ; qL] package more alluring to type H. To prevent type H from choosing [T (qL) ; qL], he must therefore reduces T (qH ) Positive surplus for type H
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 10 / 28 Every type but the lowest receives a positive informational rent. An agent of type H gets an informational rent as he can always pretend his type as L Every type but the lowest is indi¤erent between his contract and that of immediately lower type
Adverse Selection Discrete types Optimal allocation
Every type but the highest gets sub-e¢ cient allocation
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 11 / 28 Every type but the lowest is indi¤erent between his contract and that of immediately lower type
Adverse Selection Discrete types Optimal allocation
Every type but the highest gets sub-e¢ cient allocation Every type but the lowest receives a positive informational rent. An agent of type H gets an informational rent as he can always pretend his type as L
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 11 / 28 Adverse Selection Discrete types Optimal allocation
Every type but the highest gets sub-e¢ cient allocation Every type but the lowest receives a positive informational rent. An agent of type H gets an informational rent as he can always pretend his type as L Every type but the lowest is indi¤erent between his contract and that of immediately lower type
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 11 / 28 Adverse Selection Discrete types Application
Credit market: A bank facing borrowers with di¤erent risk pro…le (Stiglitz and Weiss 81) Insurance market: An insurer providing insurance for agents with di¤erent levels of risk (Stiglitz 77) Trade o¤ between allocative e¢ ciency and redistributive taxation (Mirrless 86) Regulating a natural monopolist(Baron and Myerson 82, La¤ont and Tirole 86)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 12 / 28 Adverse Selection Discrete types Regulating a natural monopolist - La¤ont Tirole 82 Think about an indivisible project run by a …rm for the government. Firm’scost c = e ; c is observable, but not or e e 2 costly e¤ort (e) = 2 Given a subsidy of amount s, the …rm gets s c (e) Government wants to have the project done at minimum cost First best: If it had known , e = c will be observable. Then the government can reimburse the e¤ort cost (s = (e)), and asks the …rm to produce e¤ort e that minimizes total cost
e = arg max s + c = arg max e + (e) e e 0 (e) = 1 ) E¤ort level is independent of type, but actual cost is higher for ine¢ cient type. Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 13 / 28 Adverse Selection Discrete types Regulation under Adverse Selection
Suppose Pr( = L) =
Government o¤ers a cost-contingent contract [sL; cL] ; [sH ; cH ] f g min [ (sL eL) + (1 )(sH eH )] sL (eL) 0 sH (eH ) 0 sL (eL) sH (eH ) sH (eH ) sL (eL + )
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 14 / 28 Adverse Selection Discrete types Regulation under Adverse Selection
Note that H is the ine¢ cient type here.
IR is binding for H
IC is binding for L
min (1 )( (eH ) eH ) + ( (eL) eL + (eH ) (eH )) 0 (eL) = 1; 0 (eH ) (1 ) 0 (eH ) = 0 No distortion in eL; Allocative e¢ ciency for the e¢ cient type, but comes at an informational rent But, eH < e
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 15 / 28 Consider preference of type L.
I Since IRL is binding, and if L is risk averse, S can only charge something less than TL < TL given a lottery over q with mean qL. I Such a scheme will only be pro…table if S can extract su¢ ciently more from H
Consider preference of H
I ICH is binding for the deterministic contract. As L is getting a worse deal than [TL; qL], S can possibly charge a little more from H Gain over type H exceeds loss over type L if type H is more risk averse (Maskin and Riley 84)
Adverse Selection Discrete types Stochastic Contract
B faces a lottery instead of a …xed allocation
Assume the optimal deterministic allocation [TL; qL] ; [TH ; qH ] and both types are risk averse f g
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 16 / 28 Consider preference of H
I ICH is binding for the deterministic contract. As L is getting a worse deal than [TL; qL], S can possibly charge a little more from H Gain over type H exceeds loss over type L if type H is more risk averse (Maskin and Riley 84)
Adverse Selection Discrete types Stochastic Contract
B faces a lottery instead of a …xed allocation
Assume the optimal deterministic allocation [TL; qL] ; [TH ; qH ] and both types are risk averse f g
Consider preference of type L.
I Since IRL is binding, and if L is risk averse, S can only charge something less than TL < TL given a lottery over q with mean qL. I Such a scheme will only be pro…table if S can extract su¢ ciently more from H
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 16 / 28 Gain over type H exceeds loss over type L if type H is more risk averse (Maskin and Riley 84)
Adverse Selection Discrete types Stochastic Contract
B faces a lottery instead of a …xed allocation
Assume the optimal deterministic allocation [TL; qL] ; [TH ; qH ] and both types are risk averse f g
Consider preference of type L.
I Since IRL is binding, and if L is risk averse, S can only charge something less than TL < TL given a lottery over q with mean qL. I Such a scheme will only be pro…table if S can extract su¢ ciently more from H
Consider preference of H
I ICH is binding for the deterministic contract. As L is getting a worse deal than [TL; qL], S can possibly charge a little more from H
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 16 / 28 Adverse Selection Discrete types Stochastic Contract
B faces a lottery instead of a …xed allocation
Assume the optimal deterministic allocation [TL; qL] ; [TH ; qH ] and both types are risk averse f g
Consider preference of type L.
I Since IRL is binding, and if L is risk averse, S can only charge something less than TL < TL given a lottery over q with mean qL. I Such a scheme will only be pro…table if S can extract su¢ ciently more from H
Consider preference of H
I ICH is binding for the deterministic contract. As L is getting a worse deal than [TL; qL], S can possibly charge a little more from H Gain over type H exceeds loss over type L if type H is more risk averse (Maskin and Riley 84)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 16 / 28 Note that IR is satis…ed as long as v (q ()) T () 0 What about IC constraints?
Adverse Selection Continuous types Continuous types
max [T () cq ()]dF () q();T () subject to R IR : v (q ()) T () 0 IC : v (q ()) T () v q T b b
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 17 / 28 What about IC constraints?
Adverse Selection Continuous types Continuous types
max [T () cq ()]dF () q();T () subject to R IR : v (q ()) T () 0 IC : v (q ()) T () v q T Note that IR is satis…ed as long asv (q ()) T () 0 b b
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 17 / 28 Adverse Selection Continuous types Continuous types
max [T () cq ()]dF () q();T () subject to R IR : v (q ()) T () 0 IC : v (q ()) T () v q T Note that IR is satis…ed as long asv (q ()) T () 0 b b What about IC constraints?
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 17 / 28 Adverse Selection Continuous types
Given the Spence-Mirrless Condition, which is going to be satis…ed in our scenario, @2u is of constant sign (> 0) @@q where u (; q; T ) = v (q) T ; ICs are equivalent of Monotonicity : dq() 0 d Local IC : v (q ()) dq() = T () for all ; 0 d 0 2
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 18 / 28 Adverse Selection Continuous types The Optimization Problem
max [T () cq ()]dF () q();T () subject to R IR : v (q ()) T () 0 Monotonicity : dq() 0 d LIC : v (q ()) dq() = T () for all ; 0 d 0 2
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 19 / 28 Next, no incentive to give the lowest type any positive surplus v (q ()) T () = 0 (all IRs are satis…ed) max [v (q ()) v (q (x)) dx cq ()]dF () q() R hR i
Adverse Selection Continuous types Mirrless’approach
De…ne the rent for type : W () = v (q ()) T () dW () As q and T will be chosen optimally, we have d = v (q ()) or W () = v (q (x)) dx (notice that I am using LICs here) R
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 20 / 28 Adverse Selection Continuous types Mirrless’approach
De…ne the rent for type : W () = v (q ()) T () dW () As q and T will be chosen optimally, we have d = v (q ()) or W () = v (q (x)) dx (notice that I am using LICs here) R Next, no incentive to give the lowest type any positive surplus v (q ()) T () = 0 (all IRs are satis…ed) max [v (q ()) v (q (x)) dx cq ()]dF () q() R hR i
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 20 / 28 But does it satisfy dq() 0? d f () It turns out that a su¢ cient condition is the hazard rate 1 F () to be increasing in
Adverse Selection Continuous types First order condition
Integrate by parts, and then look at the …rst order condition 1 F () v (q ()) = c f () 0 Suboptimalh i allocation for all types other than the highest type.
Let q () be the solution of the above problem.
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 21 / 28 Adverse Selection Continuous types First order condition
Integrate by parts, and then look at the …rst order condition 1 F () v (q ()) = c f () 0 Suboptimalh i allocation for all types other than the highest type.
Let q () be the solution of the above problem. But does it satisfy dq() 0? d f () It turns out that a su¢ cient condition is the hazard rate 1 F () to be increasing in
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 21 / 28 Adverse Selection Continuous types Bunching max [v (q ()) v (q (x)) dx cq ()]dF () q() h dq() i subject toR Monotonicity R: 0. Let q () be the solution. d
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 22 / 28 Without bundling, optimal price charged : P1 = P2 = 80 80 80 Expected pro…t = 2 + 2 = 80 Alternate strategy: sell good 1 and good 2 each at price P1 = P2 = 90 and o¤er the bundle at pb = 120 120 90 90 Expected pro…t = 2 + 4 + 4 = 105
Adverse Selection Multidimensional types Adverse selection with multidimensional types Consider a seller with some market power who sells at least two di¤erent goods. Example, a supermarket. Bundling can be a screening device (Adams and Yallen 76)
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 23 / 28 Alternate strategy: sell good 1 and good 2 each at price P1 = P2 = 90 and o¤er the bundle at pb = 120 120 90 90 Expected pro…t = 2 + 4 + 4 = 105
Adverse Selection Multidimensional types Adverse selection with multidimensional types Consider a seller with some market power who sells at least two di¤erent goods. Example, a supermarket. Bundling can be a screening device (Adams and Yallen 76)
Without bundling, optimal price charged : P1 = P2 = 80 80 80 Expected pro…t = 2 + 2 = 80
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 23 / 28 Adverse Selection Multidimensional types Adverse selection with multidimensional types Consider a seller with some market power who sells at least two di¤erent goods. Example, a supermarket. Bundling can be a screening device (Adams and Yallen 76)
Without bundling, optimal price charged : P1 = P2 = 80 80 80 Expected pro…t = 2 + 2 = 80 Alternate strategy: sell good 1 and good 2 each at price P1 = P2 = 90 and o¤er the bundle at pb = 120 Kundu & Nilssen (UiO) 120 90 90 Contract Theory Spring Semester 2009 23 / 28 Expected pro…t = 2 + 4 + 4 = 105 As buyer can buy the goods separately, we have Pb < P1 + P2
Let P1 and P2 be the monopoly prices.
Adverse Selection Multidimensional types When is bundling optimal? McAfee, McMillan, and Whinston 89
A monopolist selling two di¤erent items, 1 and 2
Buyers’valuation (v1; v2) [v ; v 1] [v ; v 2] distributed as F (v1; v2) 2 1 2 Cumulative Distribution (and density): Hi (vi ) [and hi (vi )]
Cost of production: c1; c2
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 24 / 28 Let P1 and P2 be the monopoly prices.
Adverse Selection Multidimensional types When is bundling optimal? McAfee, McMillan, and Whinston 89
A monopolist selling two di¤erent items, 1 and 2
Buyers’valuation (v1; v2) [v ; v 1] [v ; v 2] distributed as F (v1; v2) 2 1 2 Cumulative Distribution (and density): Hi (vi ) [and hi (vi )]
Cost of production: c1; c2
As buyer can buy the goods separately, we have Pb < P1 + P2
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 24 / 28 Adverse Selection Multidimensional types When is bundling optimal? McAfee, McMillan, and Whinston 89
A monopolist selling two di¤erent items, 1 and 2
Buyers’valuation (v1; v2) [v ; v 1] [v ; v 2] distributed as F (v1; v2) 2 1 2 Cumulative Distribution (and density): Hi (vi ) [and hi (vi )]
Cost of production: c1; c2
As buyer can buy the goods separately, we have Pb < P1 + P2
Let P1 and P2 be the monopoly prices.
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 24 / 28 Consuming 1 only: v1 P and v2 P 1 2 Consuming 2 only: v2 P + " and v1 P " 2 1 Consuming the bundle: v1 + v2 P + P ; v2 P and v1 P " 1 2 2 1
Adverse Selection Multidimensional types
Consider an alternate o¤er: P1 = P1; P2 = P2 + "; Pb = P1 + P2
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 25 / 28 Consuming 2 only: v2 P + " and v1 P " 2 1 Consuming the bundle: v1 + v2 P + P ; v2 P and v1 P " 1 2 2 1
Adverse Selection Multidimensional types
Consider an alternate o¤er: P1 = P1; P2 = P2 + "; Pb = P1 + P2
Consuming 1 only: v1 P and v2 P 1 2
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 25 / 28 Consuming the bundle: v1 + v2 P + P ; v2 P and v1 P " 1 2 2 1
Adverse Selection Multidimensional types
Consider an alternate o¤er: P1 = P1; P2 = P2 + "; Pb = P1 + P2
Consuming 1 only: v1 P and v2 P 1 2 Consuming 2 only: v2 P + " and v1 P " 2 1
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 25 / 28 Adverse Selection Multidimensional types
Consider an alternate o¤er: P1 = P1; P2 = P2 + "; Pb = P1 + P2
Consuming 1 only: v1 P and v2 P 1 2 Consuming 2 only: v2 P + " and v1 P " 2 1 Consuming the bundle: v1 + v2 P + P ; v2 P and v1 P " 1 2 2 1
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 25 / 28 Adverse Selection Multidimensional types
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 26 / 28 Adverse Selection Multidimensional types Local incentive
Find whether the seller has incentive to increase P2 from P2
P1 " v 2 (") = (P2 + " c2) v P +" f (v1; v2) dv2 dv1 1 2 P1 v 2 + (P + P + c1 c2)R hR f (v1; v2i) dv2 dv1 1 2 P " P +P v1 1 2 1 If we assume valuationsR are drawnhR independently, theni the …rst order condition can be written as
H1 (P ) [1 H2 (P )] (P c2) h2 (P ) 1 f 2 2 2 g + (P c1)[1 H2 (P )] h1 (P ), which is always positive 1 2 1 If not independent, bundling is not always optimal. Armstrong and Rochet 99 provides a rigorous global analysis.
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 27 / 28 Adverse Selection references References and exercises
Adverse Selection
I Discrete types (BD Ch. 2.1) I Examples including regulation (BD Ch. 2.2) I Stochastic Contracts (BD Ch. 2.3.2) I Continuous types (BD Ch. 2.3.3) I Multidimensional types (Bundling) (BD Ch. 6.1.1.,6.1.2) Problems BD 2.2, BD 2.3
Kundu & Nilssen (UiO) Contract Theory Spring Semester 2009 28 / 28