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 typeL # 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.