Moral Hazard in Teams
|
|
- Harold Paul
- 6 years ago
- Views:
Transcription
1 Moral Hazard in Teams Ram Singh Department of Economics September 23, 2009 Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
2 Outline 1 Moral Hazard in Teams: Model 2 Unobservable Individual Output 3 First Best without Budget Breaker 4 Risk-Averse Teams Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
3 Model I Moral Hazard in Teams: Model Many agents; at least two agents Effort on the part of each agent affects the output; Effort is not observable or contractible; Cost of effort by an agent is private Risk-neutral parties Example Firm as Team and Profit as Output; Cooperative (farm) as Team and Produce or profit as Output; Sales-persons as Team and sales as Outputs; Advocate as Team and Judicial judgement as Output Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
4 Model II Moral Hazard in Teams: Model General Model: Holmstrom (1982, BJE) n Agents; n 2 e = (e 1,..., e n ) Output Q = (e 1,..., e n ), { (q1,..., q Q = n ) R n, or; Q R,. Agents are weakly risk-averse. Team/partnership Contract: w(q) = (w 1 (Q),..., w n (Q)) where w i (Q) = s i (Q) is the output sharing rule such that s i 0. Typically, we have wi (Q) = s i (Q) = Q. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
5 Unobservable Individual Output Unobservable Individual Output I Simple Model: Q = Q(e 1,..., e n ) R is scalar deterministic output Q is increasing and concave; for all i, j, Q e i > 0, 2 Q e 2 i < 0, 2 Q e i e j 0, Matrix of second derivatives Q ij is Negative Definite Agent is risk neutral in wealth; u i (w i, e i ) = u i (w i ) ψ(e i ) = w i ψ(e i ) and ψ(e i ) is increasing and convex. w i (Q) = s i (Q) is continuously differentiable and ( Q)[ w i (Q) = s i (Q) = Q] Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
6 Unobservable Individual Output Unobservable Individual Output II The first best is solution to s.t. max e 1,...,e n {Q(e 1,..., e n ) ψ i (e i )} ( Q)[ w i (Q) = Q] (1) Let e i = (e 1,..., e i 1, e i+1,..., e n ). Therefore, the first best effort ei solves the following foc Q(e i,e i ) e i = ψ (e i ), for every i = 1,..., n. That is, for every i = 1,..., n Q(e i, e i ) e i = ψ (e i ) (2) Let e = (e 1,..., e i,..., e n) solve system 2. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
7 Unobservable Individual Output Is First Best Achievable? I In SB, e is not contractible but Q is Given e i = (e 1,..., e i 1, e i+1,..., e n ), agent i solves max e i {w i (Q(e i, e i ) ψ(e i )}. Therefore, a (Nash) equilibrium is characterized by the following n equations for every i = 1,..., n. dw i (Q(e i, e i )) Q(e i, e i ) = ψ (e i ), (3) dq e i Now e = (e 1,..., e i,..., e n) can solve (2) iff for every, we have ( i {1,..., n})[ dw i(q(e i, e i )) dq = 1] (4) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
8 Unobservable Individual Output Is First Best Achievable? II But from 1, we have dw i (Q(ei, e i )) = 1 (5) dq 4 and 5 give us a contradiction. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
9 Unobservable Individual Output First Best with Budget Breaker I Consider the following contract: BB demands an upfront payment of z i and appropriate the output; and pays ( Q)[w i (Q) = Q] to each agent Under this contract it is easy to see that BB and each agent is a residual claimant on the entire output; and e = (e 1,..., e i,..., e n) is a N.E. Is such a contract feasible? Yes, if for all i Q(e 1,..., e n) ψ i (e i ) z i, i.e., nq(e 1,..., e n) ψ i (e i ) z i (6) and zi + Q(e 1,..., e n) nq(e 1,..., e n) (7) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
10 Unobservable Individual Output First Best with Budget Breaker II That is, if which is clearly true. Q(e 1,..., e n) ψ i (e i ) > 0, Is e = (e 1,..., e i,..., e n) a unique N.E.? Suppose e i = (0,..., 0). Agent i solves max e i {Q(0,.., e i,..., 0) ψ(e i )}. > ψ(0) e i, the agent i will choose a positive effort. Now 0 implies that other agents will also increase their effort. If Assuming Q(0,..,0,...,0) e i 2 Q e i e j e = (e 1,..., e i,..., e n) a unique optimizer, iteration will continue till they reach e. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
11 Unobservable Individual Output First Best without BB I Consider{ the following Mirrlees Contract: bi 0, if Q = Q w i (Q) = ; k i, if Q Q where b. i 0 and k i < 0 BB pays b i if output Q = Q, where b i ψ i (e i ); and imposes penalty of k i if Q Q Can choose b i = Q Do not need external intervention in equilibrium Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
12 Unobservable Individual Output First Best without BB II Under this contract it is easy to see that e = (e 1,..., e i,..., e n) is a N.E. Multiple equilibria: Let ê i solve Now if holds (0,..., 0) is a N.E. If for some i, Q(0,..., ê i,..., 0) = Q(e 1,..., e i,..., e n) b i ψ i (ê i ) k i (8) b i ψ i (ê i ) > k i (9) there exist N.E. (ẽ 1,..., ẽ i,..., ẽ n ) such that for some j, ẽ j < e j Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
13 Unobservable Individual Output Problematic Features I Remark Under Holmstrom scheme, the payoff of the BB is w BB = z i + Q(e) Q(e) = z i (n 1)Q(e), i.e., dw BB dq = (n 1) < 0 Remark Note the results do not depend on output being stochastic or Risk aversion of agents BB want the scheme to fail BB may collude with one of the agents Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
14 Unobservable Individual Output Problematic Features II A side contract between BB and an agent gives back original problem Agents may collude to borrow Q and game with BB Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
15 First Best without Budget Breaker Deterministic Output and Finite Effort Space I Legros and Matthews (1993) Let Three agents, i = 1, 2, 3 Q = Q(e 1, e 2, e 3 ) e i {0, 1}, i = 1, 2, 3 ψ i (e i ) = ψ i (1) > ψ i (0) > 0, i = 1, 2, 3 The FB solves max{q(e 1, e 2, e 3 ) ψ i (e i )} Let (e1, e 2, e 3 ) = (1, 1, 1) Q i = Q(0, e i ), where e i = (1, 1) Q 1 Q 2 Q 3, a generic feature Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
16 First Best without Budget Breaker Deterministic Output and Finite Effort Space II Consider the following contract wi if Q = Q ; Q w i (Q) = 2 + δ if Q Q & Q Q i ; k i, if Q = Q i. δ 0. where w i = w i (Q ) ψ i > 0 and This contract implements the FB. However, if Q 1 = Q 2 = Q 3 the FB cannot be implemented. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
17 First Best without Budget Breaker Approximating FB with Deterministic Output I Legros and Matthews (1993) Let Two agents, i = 1, 2 Q = Q(e 1, e 2 ) = e 1 + e 2 e i [0, + ), i = 1, 2 ψ i = ψ i (e i ) = e2 i 2, i = 1, 2 The FB solves max{q(e 1, e 2 ) e1,e 2 ψ i (e i )} = max e 1,e 2 {e 1 + e 2 e2 1 2 e2 2 2 } Clearly (e1, e 2 ) = (1, 1) Consider the following contract Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
18 First Best without Budget Breaker Approximating FB with Deterministic Output II If Q 1 { w 1 (Q) = w 2 (Q) = (Q 1)2 2 and ; Q w 1 (Q). { w1 (Q) = Q + k and ; If Q < 1 w 2 (Q) = k. Proposition Under the above contract if agent acts as principal, then ((ɛ, 1 ɛ), (0, 1)) is a N.E. in which the first agent plays e = 0 and e = 1 with probability ɛ and 1 ɛ, respectively; and agent two plays e = 1 with probability one. Proof: Given e 2 = 1 opted by 2, agent 1 solves, max{w 1 (e 1 + 1) e2 1 e 1 2 } = max { e2 1 e 1 2 e2 1 2 } = 0 Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
19 First Best without Budget Breaker Approximating FB with Deterministic Output III i.e., all effort levels are equally good. So, (ɛ, 1 ɛ) is a best response for agent 1. Note agent 2 will never opt for e 2 > 1. Given that agent 1 opts for (ɛ, 1 ɛ), a choice of e 2 = 1 gives agent 2, (1 ɛ)[2 1 2 ] + ɛ[1 0] 1 2 = 1 ɛ 2. In contrast, when e 2 < 1 agent 2 s payoff is, (1 ɛ)[1 + e 2 e2 2 2 ] ɛk e e 2 e 2 2 ɛk, which is uniquely maximized at e 2 = 1 2. At e 2 1 2, agent 2 s payoff is e 2 = 1 is the best response for 2, if 5 4 ɛk k ɛ Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
20 Risk-Averse Team I Risk-Averse Teams Q = Q(e 1,..., e n ) R is scalar deterministic output Q is increasing and concave; for all i, j, Q e i > 0, 2 Q e 2 i < 0, 2 Q e i e j 0, Matrix of second derivatives Q ij is Negative Definite Agents are risk-averse in wealth; ũ i (w i, s i (Q), e i ) = u i (w i, s i (Q)) ψ i (e i ) = e r i s i (Q) ψ i (e i ) and ψ i (e i ) is increasing and convex. ( Q)[ s i (Q) = Q] Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
21 Risk-Averse Team II Risk-Averse Teams The First Best: max { ũ i (s i (Q), e i )}, i.e., e 1,...,e i,...,e n;s i s.t. max { [u i (s i (Q)) ψ i (e i )]} e 1,...,e i,...,e n;s i ( Q)[ s i (Q) = Q] Let e = (e 1,..., e i,..., e n) along with a sharing scheme s (Q) be the unique F.B. profile in this context. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
22 Risk-Averse Teams Risk-Averse Team III Remark For a sharing scheme s i (Q) and a profile of efforts (e 1,..., e i,..., e n ) (e 1,..., e i,..., e n), the following holds: There exists a sharing scheme s (Q) such that ( i)[e(s i, e i ) E(s i, e i )] (10) ( j)[e(s j, e j ) > E(s j, e j )] (11) If a sharing scheme ŝ i (Q) induces e = (e 1,..., e i,..., e n) as a N.E., then for any sharing scheme s i (Q) that induces (e 1,..., e i,..., e n ), the following cannot hold ( i)[e(s i, e i ) E(ŝ i, e i )] (12) ( j)[e(s j, e j ) > E(ŝ j, e j )] (13) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
23 Risk-Averse Team IV Risk-Averse Teams If a sharing contract does not induce e = (e 1,..., e i,..., e n) as a N.E., it cannot be F.B. Therefore, a P.O. sharing scheme will necessarily induce e = (e 1,..., e i,..., e n) as a N.E. We know that if agents are risk neutral, i.e., if u(x) = x, then no BB sharing scheme can induce e = (e 1,..., e i,..., e n) as a N.E. Can a BB sharing scheme can induce e = (e 1,..., e i,..., e n) as a N.E. if agents are risk-averse? Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
24 Risk-Averse Team V Risk-Averse Teams Consider the following BB Scapegoat sharing contract: If Q = Q(e ), then s i (Q) = b i, where b i s are such that b i = Q(e ); If Q > Q(e ), then s i (Q) = b i + Q Q(e ) n If Q < Q(e ), choose one agent j randomly and fix shares such that s j (Q) = w j ( i j) s i (Q) = b i + b j + w j + Q Q(e ) n 1 Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
25 Risk-Averse Team VI Risk-Averse Teams Remark Note when Q < Q(e ), n s i (Q) = s j (Q) + i=1 n s i (Q) = w j + i j n [bi + b j + w j + Q Q(e ) ] = Q. n 1 i j Therefore, the above contract meets the BB constraint. Suppose, e i = e i, i.e., all agents apart from i have opted for FB effort. If i opts for ei, his payoff is u i (bi ) ψ i (ei ). If he opts for some e i > ei, his payoff is u i (bi + Q Q(e ) ) ψ i (e i ). n Since e is P.O. profile, u i (bi + Q Q(e ) ) ψ i (e i ) > u i (bi ) ψ i (ei ) n Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
26 Risk-Averse Team VII Risk-Averse Teams cannot hold. Now, if i opts for some e i < ei, his share { wi with probability 1 s i (Q) = n ; bi + z i with probability 1 n n, where z i is a random variable. For each j i, probability of z i = b i + b j +w j +Q Q(e ) for some e i < ei, his payoff is n 1 is 1 n 1. Therefore, if i opts n 1 n Eu i(b i + z i ) + 1 n u( w i) ψ i (e i ) (14) n 1 n n [ 1 n 1 u i(bi + z i )] + 1 n u( w i) ψ i (e i ) (15) i j Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
27 Risk-Averse Teams Risk-Averse Team VIII For e i < e i, agent i s payoff function is concave. Let ê i uniquely solve in region e i < e i. Now let Y i = u i (b i ) ψ i (e i ) [ n 1 n Eu i(b i + z i ) + 1 n u( w i)] ψ i (ê i ) (16) Clearly, if Y i > 0, e i is a unique best response for agent i. Now, using envelop theorem Moreover, concavity of u i implies dy i dw i = 1 n u i > 0 (17) d 2 Y i dw 2 i = 1 n u i > 0 (18) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
28 Risk-Averse Team IX Risk-Averse Teams That is Y i is increasing in w i at an increasing rate. So, there exits w i such that for all w i w i, Y i > 0. That is, for all w i w i, ei is a unique best response to e i. Therefore, Proposition If w i is sufficiently large for all i, then e = (e1,..., e i,..., en) is a N.E. Proposition If r i is sufficiently large for all i, then e = (e 1,..., e i,..., e n) is a N.E. Proof: Rewriting as Y i = u i (b i ) ψ i (e i ) [ n 1 n Eu i(b i + z i ) + 1 n u( w i) ψ i (ê i )] Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
29 Risk-Averse Team X Risk-Averse Teams Y i = u i (bi ) ψ i (ei ) [ n 1 n n [ 1 n 1 u i(bi + z i )] + 1 n u( w i) ψ i (ê i )] i j i.e., as Y i = e r i b i ψ i (e i ) + 1 n ( i j e r i {b i + 1 n 1 [b j +w j Q(e )+Q(ê i,e i )]} ) (19) + 1 n er i w i + ψ i (ê i ) Note as r i goes up, the first and the third terms approach zero. The second term is unaffected and the fifth one is bounded by ψ i (0) and ψ i (e i ). But, the fourth term exploded towards infinity. Therefore, for sufficiently large r i, Y i > 0 holds. Again, e = (e 1,..., e i,..., e n) is a N.E. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
30 Risk-Averse Teams Scapegoats Versus Massacres When agents are identical, the scapegoat contract is: Q n if Q Q(e ); Q+w n 1 s i (Q) = n 1 with probability n if Q < Q(e ); w with probability 1 n if Q < Q(e ). When agents are identical, the massacre contract is: Q n if Q Q(e ); s i (Q) = Q + (n 1)w with probability 1 n if Q < Q(e ); w with probability n 1 n if Q < Q(e ). Reference: Rasmusen (1984, RJE) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30
Moral Hazard: Characterization of SB
Moral Hazard: Characterization of SB Ram Singh Department of Economics March 2, 2015 Ram Singh (Delhi School of Economics) Moral Hazard March 2, 2015 1 / 19 Characterization of Second Best Contracts I
More informationLinear Contracts. Ram Singh. February 23, Department of Economics. Ram Singh (Delhi School of Economics) Moral Hazard February 23, / 22
Ram Singh Department of Economics February 23, 2015 Ram Singh (Delhi School of Economics) Moral Hazard February 23, 2015 1 / 22 SB: Linear Contracts I Linear Contracts Assumptions: q(e, ɛ) = e + ɛ, where
More informationRelative Performance Evaluation
Relative Performance Evaluation Ram Singh Department of Economics March, 205 Ram Singh (Delhi School of Economics) Moral Hazard March, 205 / 3 Model I Multiple Agents: Relative Performance Evaluation Relative
More information1 Moral Hazard: Multiple Agents 1.1 Moral Hazard in a Team
1 Moral Hazard: Multiple Agents 1.1 Moral Hazard in a Team Multiple agents (firm?) Partnership: Q jointly affected Individual q i s. (tournaments) Common shocks, cooperations, collusion, monitor- ing.
More informationModule 8: Multi-Agent Models of Moral Hazard
Module 8: Multi-Agent Models of Moral Hazard Information Economics (Ec 515) George Georgiadis Types of models: 1. No relation among agents. an many agents make contracting easier? 2. Agents shocks are
More informationWhat happens when there are many agents? Threre are two problems:
Moral Hazard in Teams What happens when there are many agents? Threre are two problems: i) If many agents produce a joint output x, how does one assign the output? There is a free rider problem here as
More informationHidden information. Principal s payoff: π (e) w,
Hidden information Section 14.C. in MWG We still consider a setting with information asymmetries between the principal and agent. However, the effort is now perfectly observable. What is unobservable?
More informationMicroeconomic Theory (501b) Problem Set 10. Auctions and Moral Hazard Suggested Solution: Tibor Heumann
Dirk Bergemann Department of Economics Yale University Microeconomic Theory (50b) Problem Set 0. Auctions and Moral Hazard Suggested Solution: Tibor Heumann 4/5/4 This problem set is due on Tuesday, 4//4..
More informationGame Theory Correlated equilibrium 1
Game Theory Correlated equilibrium 1 Christoph Schottmüller University of Copenhagen 1 License: CC Attribution ShareAlike 4.0 1 / 17 Correlated equilibrium I Example (correlated equilibrium 1) L R U 5,1
More informationMechanism Design: Basic Concepts
Advanced Microeconomic Theory: Economics 521b Spring 2011 Juuso Välimäki Mechanism Design: Basic Concepts The setup is similar to that of a Bayesian game. The ingredients are: 1. Set of players, i {1,
More informationMoral hazard in teams
Division of the Humanities and Social Sciences Moral hazard in teams KC Border November 2004 These notes are based on the first part of Moral hazard in teams by Bengt Holmström [1], and fills in the gaps
More informationMarket Failure: Externalities
Market Failure: Externalities Ram Singh Lecture 21 November 10, 2015 Ram Singh: (DSE) Externality November 10, 2015 1 / 18 Questions What is externality? What is implication of externality for efficiency
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Adverse Selection We have now completed our basic analysis of the adverse selection model This model has been applied and extended in literally thousands of ways
More informationA New Class of Non Existence Examples for the Moral Hazard Problem
A New Class of Non Existence Examples for the Moral Hazard Problem Sofia Moroni and Jeroen Swinkels April, 23 Abstract We provide a class of counter-examples to existence in a simple moral hazard problem
More informationMoral Hazard: Hidden Action
Moral Hazard: Hidden Action Part of these Notes were taken (almost literally) from Rasmusen, 2007 UIB Course 2013-14 (UIB) MH-Hidden Actions Course 2013-14 1 / 29 A Principal-agent Model. The Production
More informationThis is designed for one 75-minute lecture using Games and Information. October 3, 2006
This is designed for one 75-minute lecture using Games and Information. October 3, 2006 1 7 Moral Hazard: Hidden Actions PRINCIPAL-AGENT MODELS The principal (or uninformed player) is the player who has
More informationAdverse selection, signaling & screening
, signaling & screening Applications of game theory 2 Department of Economics, University of Oslo ECON5200 Fall 2009 Seller Buyer Situation 1: Symmetric info One market 1 2 prob high quality 1 2 prob high
More informationGame Theory. Monika Köppl-Turyna. Winter 2017/2018. Institute for Analytical Economics Vienna University of Economics and Business
Monika Köppl-Turyna Institute for Analytical Economics Vienna University of Economics and Business Winter 2017/2018 Static Games of Incomplete Information Introduction So far we assumed that payoff functions
More informationSimple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) Jonathan Heathcote. updated, March The household s problem X
Simple Consumption / Savings Problems (based on Ljungqvist & Sargent, Ch 16, 17) subject to for all t Jonathan Heathcote updated, March 2006 1. The household s problem max E β t u (c t ) t=0 c t + a t+1
More informationMarket Equilibrium and the Core
Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange
More informationTeoria das organizações e contratos
Teoria das organizações e contratos Chapter 6: Adverse Selection with two types Mestrado Profissional em Economia 3 o trimestre 2015 EESP (FGV) Teoria das organizações e contratos 3 o trimestre 2015 1
More informationCombinatorial Agency of Threshold Functions
Combinatorial Agency of Threshold Functions Shaili Jain 1 and David C. Parkes 2 1 Yale University, New Haven, CT shaili.jain@yale.edu 2 Harvard University, Cambridge, MA parkes@eecs.harvard.edu Abstract.
More informationFundamental Theorems of Welfare Economics
Fundamental Theorems of Welfare Economics Ram Singh Lecture 6 September 29, 2015 Ram Singh: (DSE) General Equilibrium Analysis September 29, 2015 1 / 14 First Fundamental Theorem The First Fundamental
More informationEC476 Contracts and Organizations, Part III: Lecture 2
EC476 Contracts and Organizations, Part III: Lecture 2 Leonardo Felli 32L.G.06 19 January 2015 Moral Hazard: Consider the contractual relationship between two agents (a principal and an agent) The principal
More informationIntroduction to General Equilibrium: Framework.
Introduction to General Equilibrium: Framework. Economy: I consumers, i = 1,...I. J firms, j = 1,...J. L goods, l = 1,...L Initial Endowment of good l in the economy: ω l 0, l = 1,...L. Consumer i : preferences
More informationLecture Notes on Solving Moral-Hazard Problems Using the Dantzig-Wolfe Algorithm
Lecture Notes on Solving Moral-Hazard Problems Using the Dantzig-Wolfe Algorithm Edward Simpson Prescott Prepared for ICE 05, July 2005 1 Outline 1. Why compute? Answer quantitative questions Analyze difficult
More informationNTU IO (I) : Auction Theory and Mechanism Design II Groves Mechanism and AGV Mechansim. u i (x, t i, θ i ) = V i (x, θ i ) + t i,
Meng-Yu Liang NTU O : Auction Theory and Mechanism Design Groves Mechanism and AGV Mechansim + 1 players. Types are drawn from independent distribution P i on [θ i, θ i ] with strictly positive and differentiable
More informationAdvanced Microeconomics
Advanced Microeconomics ECON5200 - Fall 2012 Introduction What you have done: - consumers maximize their utility subject to budget constraints and firms maximize their profits given technology and market
More informationA Solution to the Problem of Externalities When Agents Are Well-Informed
A Solution to the Problem of Externalities When Agents Are Well-Informed Hal R. Varian. The American Economic Review, Vol. 84, No. 5 (Dec., 1994), pp. 1278-1293 Introduction There is a unilateral externality
More informationGame Theory, Information, Incentives
Game Theory, Information, Incentives Ronald Wendner Department of Economics Graz University, Austria Course # 320.501: Analytical Methods (part 6) The Moral Hazard Problem Moral hazard as a problem of
More informationGame Theory. Professor Peter Cramton Economics 300
Game Theory Professor Peter Cramton Economics 300 Definition Game theory is the study of mathematical models of conflict and cooperation between intelligent and rational decision makers. Rational: each
More informationGeneral Equilibrium with Production
General Equilibrium with Production Ram Singh Microeconomic Theory Lecture 11 Ram Singh: (DSE) General Equilibrium: Production Lecture 11 1 / 24 Producer Firms I There are N individuals; i = 1,..., N There
More informationSome Notes on Moral Hazard
Some Notes on Moral Hazard John Morgan University of California at Berkeley Preliminaries Up until this point, we have been concerned mainly with the problem of private information on the part of the agent,
More informationStatic (or Simultaneous- Move) Games of Complete Information
Static (or Simultaneous- Move) Games of Complete Information Introduction to Games Normal (or Strategic) Form Representation Teoria dos Jogos - Filomena Garcia 1 Outline of Static Games of Complete Information
More information1. Linear Incentive Schemes
ECO 317 Economics of Uncertainty Fall Term 2009 Slides to accompany 20. Incentives for Effort - One-Dimensional Cases 1. Linear Incentive Schemes Agent s effort x, principal s outcome y. Agent paid w.
More informationGeneral idea. Firms can use competition between agents for. We mainly focus on incentives. 1 incentive and. 2 selection purposes 3 / 101
3 Tournaments 3.1 Motivation General idea Firms can use competition between agents for 1 incentive and 2 selection purposes We mainly focus on incentives 3 / 101 Main characteristics Agents fulll similar
More informationGame Theory -- Lecture 4. Patrick Loiseau EURECOM Fall 2016
Game Theory -- Lecture 4 Patrick Loiseau EURECOM Fall 2016 1 Lecture 2-3 recap Proved existence of pure strategy Nash equilibrium in games with compact convex action sets and continuous concave utilities
More informationAssortative Matching in Two-sided Continuum Economies
Assortative Matching in Two-sided Continuum Economies Patrick Legros and Andrew Newman February 2006 (revised March 2007) Abstract We consider two-sided markets with a continuum of agents and a finite
More informationMoral Hazard. EC202 Lectures XV & XVI. Francesco Nava. February London School of Economics. Nava (LSE) EC202 Lectures XV & XVI Feb / 19
Moral Hazard EC202 Lectures XV & XVI Francesco Nava London School of Economics February 2011 Nava (LSE) EC202 Lectures XV & XVI Feb 2011 1 / 19 Summary Hidden Action Problem aka: 1 Moral Hazard Problem
More informationOnline Appendix for Sourcing from Suppliers with Financial Constraints and Performance Risk
Online Appendix for Sourcing from Suppliers with Financial Constraints and Performance Ris Christopher S. Tang S. Alex Yang Jing Wu Appendix A: Proofs Proof of Lemma 1. In a centralized chain, the system
More informationOn the Informed Principal Model with Common Values
On the Informed Principal Model with Common Values Anastasios Dosis ESSEC Business School and THEMA École Polytechnique/CREST, 3/10/2018 Anastasios Dosis (ESSEC and THEMA) Informed Principal with Common
More informationOnline Appendix for Dynamic Procurement under Uncertainty: Optimal Design and Implications for Incomplete Contracts
Online Appendix for Dynamic Procurement under Uncertainty: Optimal Design and Implications for Incomplete Contracts By Malin Arve and David Martimort I. Concavity and Implementability Conditions In this
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More informationLecture Notes - Dynamic Moral Hazard
Lecture Notes - Dynamic Moral Hazard Simon Board and Moritz Meyer-ter-Vehn October 23, 2012 1 Dynamic Moral Hazard E ects Consumption smoothing Statistical inference More strategies Renegotiation Non-separable
More informationAn Introduction to Moral Hazard in Continuous Time
An Introduction to Moral Hazard in Continuous Time Columbia University, NY Chairs Days: Insurance, Actuarial Science, Data and Models, June 12th, 2018 Outline 1 2 Intuition and verification 2BSDEs 3 Control
More informationPositive Models of Private Provision of Public Goods: A Static Model. (Bergstrom, Blume and Varian 1986)
Positive Models of Private Provision of Public Goods: A Static Model (Bergstrom, Blume and Varian 1986) Public goods will in general be under-supplied by voluntary contributions. Still, voluntary contributions
More informationArea I: Contract Theory Question (Econ 206)
Theory Field Exam Winter 2011 Instructions You must complete two of the three areas (the areas being (I) contract theory, (II) game theory, and (III) psychology & economics). Be sure to indicate clearly
More informationMechanism design and allocation algorithms for energy-network markets with piece-wise linear costs and quadratic externalities
1 / 45 Mechanism design and allocation algorithms for energy-network markets with piece-wise linear costs and quadratic externalities Alejandro Jofré 1 Center for Mathematical Modeling & DIM Universidad
More informationA note on the take-it-or-leave-it bargaining procedure with double moral hazard and risk neutrality
A note on the take-it-or-leave-it bargaining procedure with double moral hazard and risk neutrality A. Citanna HEC - Paris; and GSB - Columbia University, NY September 29, 2003 In this note we study a
More informationMinimum Wages and Excessive E ort Supply
Minimum Wages and Excessive E ort Supply Matthias Kräkel y Anja Schöttner z Abstract It is well-known that, in static models, minimum wages generate positive worker rents and, consequently, ine ciently
More informationPerfect Competition in Markets with Adverse Selection
Perfect Competition in Markets with Adverse Selection Eduardo Azevedo and Daniel Gottlieb (Wharton) Presented at Frontiers of Economic Theory & Computer Science at the Becker Friedman Institute August
More informationInformed Principal in Private-Value Environments
Informed Principal in Private-Value Environments Tymofiy Mylovanov Thomas Tröger University of Bonn June 21, 2008 1/28 Motivation 2/28 Motivation In most applications of mechanism design, the proposer
More informationONLINE ONLY APPENDIX. Endogenous matching approach
ONLINE ONLY APPENDIX Endogenous matching approach In addition with the respondable risk approach, we develop in this online appendix a complementary explanation regarding the trade-off between risk and
More information1 Web Appendix: Equilibrium outcome under collusion (multiple types-multiple contracts)
1 Web Appendix: Equilibrium outcome under collusion (multiple types-multiple contracts) We extend our setup by allowing more than two types of agent. The agent s type is now β {β 1, β 2,..., β N }, where
More informationPrincipal-Agent Games - Equilibria under Asymmetric Information -
Principal-Agent Games - Equilibria under Asymmetric Information - Ulrich Horst 1 Humboldt-Universität zu Berlin Department of Mathematics and School of Business and Economics Work in progress - Comments
More informationAuthority and Incentives in Organizations*
Authority and Incentives in Organizations* Matthias Kräkel, University of Bonn** Abstract The paper analyzes the choice of organizational structure as solution to the trade-off between controlling behavior
More informationBargaining, Contracts, and Theories of the Firm. Dr. Margaret Meyer Nuffield College
Bargaining, Contracts, and Theories of the Firm Dr. Margaret Meyer Nuffield College 2015 Course Overview 1. Bargaining 2. Hidden information and self-selection Optimal contracting with hidden information
More informationECONOMICS 001 Microeconomic Theory Summer Mid-semester Exam 2. There are two questions. Answer both. Marks are given in parentheses.
Microeconomic Theory Summer 206-7 Mid-semester Exam 2 There are two questions. Answer both. Marks are given in parentheses.. Consider the following 2 2 economy. The utility functions are: u (.) = x x 2
More informationPatience and Ultimatum in Bargaining
Patience and Ultimatum in Bargaining Björn Segendorff Department of Economics Stockholm School of Economics PO Box 6501 SE-113 83STOCKHOLM SWEDEN SSE/EFI Working Paper Series in Economics and Finance No
More informationGame Theory Review Questions
Game Theory Review Questions Sérgio O. Parreiras All Rights Reserved 2014 0.1 Repeated Games What is the difference between a sequence of actions and a strategy in a twicerepeated game? Express a strategy
More informationLEN model. And, the agent is risk averse with utility function for wealth w and personal cost of input c (a), a {a L,a H }
LEN model The LEN model is a performance evaluation frame for dealing with unbounded performance measures. In particular, LEN stands for Linear compensation, negative Exponential utility, and Normally
More informationOptimal Insurance of Search Risk
Optimal Insurance of Search Risk Mikhail Golosov Yale University and NBER Pricila Maziero University of Pennsylvania Guido Menzio University of Pennsylvania and NBER November 2011 Introduction Search and
More informationIn the Name of God. Sharif University of Technology. Microeconomics 1. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 1 44715 (1396-97 1 st term) - Group 1 Dr. S. Farshad Fatemi Chapter 10: Competitive Markets
More informationLecture Notes - Dynamic Moral Hazard
Lecture Notes - Dynamic Moral Hazard Simon Board and Moritz Meyer-ter-Vehn October 27, 2011 1 Marginal Cost of Providing Utility is Martingale (Rogerson 85) 1.1 Setup Two periods, no discounting Actions
More informationOrganization, Careers and Incentives
Organization, Careers and Incentives Chapter 4 Robert Gary-Bobo March 2018 1 / 31 Introduction Introduction A firm is a pyramid of opportunities (Alfred P. Sloan). Promotions can be used to create incentives.
More informationIntroduction to Game Theory Lecture Note 2: Strategic-Form Games and Nash Equilibrium (2)
Introduction to Game Theory Lecture Note 2: Strategic-Form Games and Nash Equilibrium (2) Haifeng Huang University of California, Merced Best response functions: example In simple games we can examine
More informationAsymmetric Information in Economic Policy. Noah Williams
Asymmetric Information in Economic Policy Noah Williams University of Wisconsin - Madison Williams Econ 899 Asymmetric Information Risk-neutral moneylender. Borrow and lend at rate R = 1/β. Strictly risk-averse
More informationECON 2060 Contract Theory: Notes
ECON 2060 Contract Theory: Notes Richard Holden Harvard University Littauer 225 Cambridge MA 02138 rholden@harvard.edu September 6, 2016 Contents 1 Introduction 2 1.1 Situating Contract Theory.............................
More informationMoral Hazard: Part 2. April 16, 2018
Moral Hazard: Part 2 April 16, 2018 The basic model: A is risk neutral We now turn to the problem of moral hazard (asymmetric information), where A is risk neutral. When A is risk neutral, u (t) is linear.
More informationDiscussion Papers in Economics
Discussion Papers in Economics No. 10/11 A General Equilibrium Corporate Finance Theorem for Incomplete Markets: A Special Case By Pascal Stiefenhofer, University of York Department of Economics and Related
More informationLecture December 2009 Fall 2009 Scribe: R. Ring In this lecture we will talk about
0368.4170: Cryptography and Game Theory Ran Canetti and Alon Rosen Lecture 7 02 December 2009 Fall 2009 Scribe: R. Ring In this lecture we will talk about Two-Player zero-sum games (min-max theorem) Mixed
More informationIBM Research Report. Equilibrium in Prediction Markets with Buyers and Sellers
RJ10453 (A0910-003) October 1, 2009 Mathematics IBM Research Report Equilibrium in Prediction Markets with Buyers and Sellers Shipra Agrawal Department of Computer Science Stanford University Stanford,
More informationVariational inequality formulation of chance-constrained games
Variational inequality formulation of chance-constrained games Joint work with Vikas Singh from IIT Delhi Université Paris Sud XI Computational Management Science Conference Bergamo, Italy May, 2017 Outline
More informationGeorge Georgiadis. Joint work with Jakša Cvitanić (Caltech) Kellogg School of Management, Northwestern University
Achieving E ciency in Dynamic Contribution Games George Georgiadis Joint work with Jakša Cvitanić (Caltech) Kellogg School of Management, Northwestern University Cvitanić and Georgiadis E ciency in Dynamic
More informationThe Principal-Agent Problem
Andrew McLennan September 18, 2014 I. Introduction Economics 6030 Microeconomics B Second Semester Lecture 8 The Principal-Agent Problem A. In the principal-agent problem there is no asymmetric information
More informationFundamentals in Optimal Investments. Lecture I
Fundamentals in Optimal Investments Lecture I + 1 Portfolio choice Portfolio allocations and their ordering Performance indices Fundamentals in optimal portfolio choice Expected utility theory and its
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationAdvanced Microeconomic Theory. Chapter 6: Partial and General Equilibrium
Advanced Microeconomic Theory Chapter 6: Partial and General Equilibrium Outline Partial Equilibrium Analysis General Equilibrium Analysis Comparative Statics Welfare Analysis Advanced Microeconomic Theory
More informationMicroeconomics. 3. Information Economics
Microeconomics 3. Information Economics Alex Gershkov http://www.econ2.uni-bonn.de/gershkov/gershkov.htm 9. Januar 2008 1 / 19 1.c The model (Rothschild and Stiglitz 77) strictly risk-averse individual
More informationA Rothschild-Stiglitz approach to Bayesian persuasion
A Rothschild-Stiglitz approach to Bayesian persuasion Matthew Gentzkow and Emir Kamenica Stanford University and University of Chicago September 2015 Abstract Rothschild and Stiglitz (1970) introduce a
More informationSURPLUS SHARING WITH A TWO-STAGE MECHANISM. By Todd R. Kaplan and David Wettstein 1. Ben-Gurion University of the Negev, Israel. 1.
INTERNATIONAL ECONOMIC REVIEW Vol. 41, No. 2, May 2000 SURPLUS SHARING WITH A TWO-STAGE MECHANISM By Todd R. Kaplan and David Wettstein 1 Ben-Gurion University of the Negev, Israel In this article we consider
More informationGame Theory Fall 2003
Game Theory Fall 2003 Problem Set 1 [1] In this problem (see FT Ex. 1.1) you are asked to play with arbitrary 2 2 games just to get used to the idea of equilibrium computation. Specifically, consider the
More informationModels of Wage Dynamics
Models of Wage Dynamics Toshihiko Mukoyama Department of Economics Concordia University and CIREQ mukoyama@alcor.concordia.ca December 13, 2005 1 Introduction This paper introduces four different models
More information(a) Output only takes on two values, so the wage will also take on two values: z(0) = 0 0 z(0) 0. max s(d)z { d. n { z 1 0 (n + d) 2.
Steve Pischke/Jin Li Labor Economics II Problem Set Answers. An Agency Problem (a) Output only takes on two values, so the wage will also take on two values: z( ) z 0 z The worker s problem: z(0) 0 0 z(0)
More informationLecture 1. Evolution of Market Concentration
Lecture 1 Evolution of Market Concentration Take a look at : Doraszelski and Pakes, A Framework for Applied Dynamic Analysis in IO, Handbook of I.O. Chapter. (see link at syllabus). Matt Shum s notes are
More informationThe Revenue Equivalence Theorem 1
John Nachbar Washington University May 2, 2017 The Revenue Equivalence Theorem 1 1 Introduction. The Revenue Equivalence Theorem gives conditions under which some very different auctions generate the same
More informationNotes on Recursive Utility. Consider the setting of consumption in infinite time under uncertainty as in
Notes on Recursive Utility Consider the setting of consumption in infinite time under uncertainty as in Section 1 (or Chapter 29, LeRoy & Werner, 2nd Ed.) Let u st be the continuation utility at s t. That
More informationA Theory of Financing Constraints and Firm Dynamics by Clementi and Hopenhayn - Quarterly Journal of Economics (2006)
A Theory of Financing Constraints and Firm Dynamics by Clementi and Hopenhayn - Quarterly Journal of Economics (2006) A Presentation for Corporate Finance 1 Graduate School of Economics December, 2009
More informationOptimal contract under adverse selection in a moral hazard model with a risk averse agent
Optimal contract under adverse selection in a moral hazard model with a risk averse agent Lionel Thomas CRESE Université de Franche-Comté, IUT Besanon Vesoul, 30 avenue de l Observatoire, BP1559, 25009
More informationWhat kind of bonus point system makes the rugby teams more offensive?
What kind of bonus point system makes the rugby teams more offensive? XIV Congreso Dr. Antonio Monteiro - 2017 Motivation Rugby Union is a game in constant evolution Experimental law variations every year
More informationNoncooperative Games, Couplings Constraints, and Partial Effi ciency
Noncooperative Games, Couplings Constraints, and Partial Effi ciency Sjur Didrik Flåm University of Bergen, Norway Background Customary Nash equilibrium has no coupling constraints. Here: coupling constraints
More informationReciprocity in the Principal Multiple Agent Model
WORKING PAPER NO. 314 Reciprocity in the Principal Multiple Agent Model Giuseppe De Marco and Giovanni Immordino May 2012 University of Naples Federico II University of Salerno Bocconi University, Milan
More informationWelfare Economics: Lecture 12
Welfare Economics: Lecture 12 Ram Singh Course 001 October 20, 2014 Ram Singh: (DSE) Welfare Economics October 20, 2014 1 / 16 Fair Vs Efficient Question 1 What is a fair allocation? 2 Is a fair allocation
More information5. Externalities and Public Goods. Externalities. Public Goods types. Public Goods
5. Externalities and Public Goods 5. Externalities and Public Goods Externalities Welfare properties of Walrasian Equilibria rely on the hidden assumption of private goods: the consumption of the good
More information5. Externalities and Public Goods
5. Externalities and Public Goods Welfare properties of Walrasian Equilibria rely on the hidden assumption of private goods: the consumption of the good by one person has no effect on other people s utility,
More informationGame Theory. Bargaining Theory. ordi Massó. International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB)
Game Theory Bargaining Theory J International Doctorate in Economic Analysis (IDEA) Universitat Autònoma de Barcelona (UAB) (International Game Theory: Doctorate Bargainingin Theory Economic Analysis (IDEA)
More information6.254 : Game Theory with Engineering Applications Lecture 7: Supermodular Games
6.254 : Game Theory with Engineering Applications Lecture 7: Asu Ozdaglar MIT February 25, 2010 1 Introduction Outline Uniqueness of a Pure Nash Equilibrium for Continuous Games Reading: Rosen J.B., Existence
More informationMathematical Methods and Economic Theory
Mathematical Methods and Economic Theory Anjan Mukherji Subrata Guha C 263944 OXTORD UNIVERSITY PRESS Contents Preface SECTION I 1 Introduction 3 1.1 The Objective 3 1.2 The Tools for Section I 4 2 Basic
More informationGame Theory and Economics of Contracts Lecture 5 Static Single-agent Moral Hazard Model
Game Theory and Economics of Contracts Lecture 5 Static Single-agent Moral Hazard Model Yu (Larry) Chen School of Economics, Nanjing University Fall 2015 Principal-Agent Relationship Principal-agent relationship
More informationArea I: Contract Theory Question (Econ 206)
Theory Field Exam Summer 2011 Instructions You must complete two of the four areas (the areas being (I) contract theory, (II) game theory A, (III) game theory B, and (IV) psychology & economics). Be sure
More information