Advanced Microeconomics

Assumptions chapter 2

AssumptionsAssumptionsChapter 2AssumptionsAssumptions

Principal (P) - Agent (A)

Principal (P) - Agent (A)Principal (P) - Agent (A)We have a principal P (the employer, the bank) and an agent A (the worker, he has to sellProbability distribution of Return conditional on E§A can choose between two Effort levels: e can be either Low (e )LA can choose between two Effort levels: e can be either Low (e )financial products). Lor High (e )Principal (P) - Agent (A) (Figure 1)umptions H or High (e )A can choose between two effort levels: eL and eH, with eH>eL.H<Disutility: e eA can choose between two Effort levels: e can be either Low (e )L H L<Disutility: e eThe outcome of A’s behavior is the return (value of shares sold in one period) going to P,L HThe outcome of A’s behavior is the Return going to P, which is aor High (e )Principal (P) - Agent (A) HAssumptionsThe outcome of A’s behavior is the Return going to P, which is aewhich is a random variable R~ with possible values R2>R1.>random variable (R) with possible values: R R<Disutility: e e 2 1L H eA can choose between two Effort levels: e can be either Low (e ) >random variable (R) with possible values: R RLThe relationship between effort and returns is stochastic: 2 1The relationship between Effort and Return is stochastic (seeThe outcome of A’s behavior is the Return going to P, which is aor High (e )H The relationship between Effort and Return is stochastic (seePrincipal (P) - Agent (A)e-if effort is highFigure 1 ): >random variable (R) with possible values: R R" #<Disutility: e e 2 1L H Figure 1 ):A can choose between two Effort levels: e can be either Low (e )" #e LThe relationship between Effort and Return is stochastic (see= | =If e§ort is high: P R R e p ,The outcome of A’s behavior is the Return going to P, which is a2 H H Re R" #or High (e ) = | =If e§ort is high: P R R e p , 2 2H 2 H HFigure 1 ):e >" #random variable (R) with possible values: R R" #e P= | = −2 1P R R e 1 p <Disutility: e e L1 H H PL Hee" # " # H= | =If e§ort is high: P R R e p ,= | = −P R R e 1 pThe relationship between Effort and Return is stochastic (see2 H H1 H HThe outcome of A’s behavior is the Return going to P, which is a" # " # " #e e= | = = | = −If e§ort is low: P R R e p , P R R e 1 pFigure 1 ): 2 1L L L Le ee e e e" # L H= | = − >P R R e 1 p = | = = | = −random variable (R) with possible values: R RIf e§ort is low: P R R e p , P R R e 1 p1 2 1H H 2 1L L L LHigh e§ort makes higher outcome more likely than low e§ort does:" # " #e = | =If e§ort is high: P R R e p ,The relationship between Effort and Return is stochastic (see2 H H 1-P-if effort is low High e§ort makes higher outcome more likely than low e§ort does:e e" # 1-P H= | = = | = −If e§ort is low: P R R e p , P R R e 1 p>p p (stochastic dominance) L R R2 1L L L L" # " #H L 1 1Figure 1 ):e >p p (stochastic dominance)= | = − " #P R R e 1 p " # " #H Le e1 H HHigh e§ort makes higher outcome more likely than low e§ort does:| > |Implication: E R e E R e" # " # eH Le e= | =If e§ort is high: P R R e p ,| > |Implication: E R e E R e2 H He e H L>p p (stochastic dominance)= | = = | = −" #If e§ort is low: P R R e p , P R R e 1 p" # " #H L2 1L L L L P >PAngelo Baglioni () 2018 2 / 34e H Le e= | = −P R R e 1 p| > |Implication: E R e E R e1 H HHigh e§ort makes higher outcome more likely than low e§ort does:Angelo Baglioni () 2018 2 / 34H L" # " #>p p (stochastic dominance) e e= | = = | = −If e§ort is low: P R R e p , P R R e 1 p" # " #H L 2 1L L L LAngelo Baglioni () 2018 2 / 34e e| > |Implication: E R e E R eHigh effort makes higher outcome more likely than low effort does: pH>pL. DirectHigh e§ort makes higher outcome more likely than low e§ort does:H L Angelo Baglioni ()implication: >p p (stochastic dominance)" # " #H LAngelo Baglioni () 2018 2 / 34

Preferences: principale e principal

Preferences: Principale ePreferences: Principal | > |Implication: E R e E R eH LAngelo Baglioni () 2018 2 / 34

Asymmetric information: Hidden action A can observe both outcome R~ and action e.P can observe R~ but he cannot observe the action e taken by A.

Principal is Risk NeutralPrincipal is Risk NeutralPreferences Principal: risk neutral.Principal is Utility function is e f( − )Utility function v R W , where:e f( − )Utility function v R W , where: eR is his ReturneR is his Return fwhere R~ is the return, W~ is the wage it pays to A (assuming W~ is aW is the Wage he pays to APreferences: AgentfW is the Wage he pays to A f efunction of R~, with W2>=W1). ≥(assuming that W is a function of R, with W 2f e ≥(assuming that W is a function of R, with W W )2 1v() is a linear utility function v(x)=a+b(x). From the properties of the linear function() ( ) = +v is a Linear utility function: v x a bx() ( ) = +v is a Linear utility function: v x a bx ( )] = +[ [Property of the linear function: E v x E a( )] = + = + ( )[ [ ]Property of the linear function: E v x E a bx a bE xSo P wants to maximise the Expected Value of his Net Return:e fHence P wants to maximize the Expected Value of his net return E(R~-W~).( − )E R W Agent is Risk AversePreferences Agent: risk averse.Agent is Utility function is f f( ) = ( ) −Utility function u W , e u W ef f f0 00( ) ( ) > ( ) <where u W Concave: u W 0 and u W 0where u() is concave u’(W~)>0 and u’’(W~)<0. Reservation level of utility: uAngelo Baglioni ()A has a reservation level of utility of u.Angelo Baglioni () 2018 5 / 34

Solving the P-A problem

Solving the P-A problemWe look for the optimal equilibrium contract, the best contract that the two parties candesign, given the constraints of the problem.

Bargaining structure is the following:-P makes a take-it-or-leave it offer to A-A can accept or refuse-so all bargaining power is assigned to P, but this does not alter the properties of eq.Angelo Baglioni ()Two step procedure: we find the optimal contract under Sym, the 1st best; then theoptimal with Asym, the 2nd best.Agency cost can be derived by comparing 1st best to 2nd best.

Symmetric information

A) Symmetric Information contractible.Given that P can observe both R~ and e, the action e isSymmetric info: Low e§ortFor each action, we must find the least costly contract for P, given the reservationconstraint of APrincipal’s problem:+ ( − )min p W 1 p W2 1L L( ) + ( − ) ( ) − ≥s.to p u W 1 p u W e u (PC)2 1L L L= + ( − ) − ( ) + ( − ) ( ) − −l [ ]L p W 1 p W p u W 1 p u W e u2 1 2 1L L L L LFOC:∂L 0= − ( ) =lpp u W 02L L∂W 2∂L 0= ( − ) − ( − ) ( ) =l1 p 1 p u W 0 We can derive ORS principle1L L∂W 1 We can derive ORS principle=lCan be 0 (PC slack)?= = ( − ) =lNo: with 0 FOC imply p 0 and 1 p 0, which cannotL Lbe >lSo it is 0 and PC binding FOC can be written as:FOC can be written as:0= ( )lu1 WFOC can be written as 20= ( )lu1 W0 2= ( )lu1 W 1Angelo Baglioni () 2018 9 / 340= ( )lu1 W 1 =which imply that W W (Fixed Wage): Optimal Risk1 2=which imply that W W (Fixed Wage): Optimal Risk1 2 −P bears the whole risk: R REquilibrium contract (1st best)Fixed Wage: Optimal Risk Sharing ORS,which imply that W1=W2 P bears the whole2 1−P bears the whole risk: R R2 1risk R2-R1. What determines the level of W ?

Equilibrium contract (1st best)

Equilibrium contract (1st best)What determines the level of W ?∗ ( )] − =[PC: u W e e uWhat determines the level of W? L L ∗ (∗The same procedure applies to High e§ort, leading to W( )] − =[PC: u W e e uEquilibrium contract (1st best) L L ∗determined by: ( )The same procedure applies to High e§ort, leading to W e Hand the same applies for High eff: ∗ ( )] − =[PC: u W e e udetermined by: H H∗∗ ( )The same procedure applies to High e§ort, leading to W e ∗ ∗( )] − =[PC: u W e e u > ( ) > ( )Of course, from e e we get W e W e .HH H H L H Ldetermined by: ∗ ∗ ∗> ( ) > ( )Of course, from e e we get W e W e . [ (Which contract will P propose? Either high e§ort, W eH L H LOf course eH>eL, so we get W*(eH)>W*(eL). H∗ ( )] − =[PC: u W e e u ∗H H ( )] ∗e§ort, W e ? [ ( )] [Which contract will P propose? Either high e§ort, W e or lowWhich contract will P propose? [high effort, W*(eH)] or [low effort, W*(eL)]?L H∗ ∗ Angelo Baglioni ()> ( ) > ( )Of course, from e e we get W e W e . ∗H L H L ( )]e§ort, W e ?P will choose the one that maximises his own Net expectedP will choose the one that maximizes his own net expected return as we’ve said:Angelo Baglioni ()L ! $∗[ ( )] [Which contract will P propose? Either high e§ort, W e or lowe f ∗H − ( )P will choose the one that maximises his own Net expected return:E R W e! $∗ ( )]e§ort, W e ? e fL ∗− ( )E R W e Two alternative cases:! $ ! $P will choose the one that maximises his own Net expected return:! $ e e∗ ∗Two alternative cases: | − ( ) > | − ( )(A) If E R e W e E R e W e , the! $ ! $H H L Le f ∗− ( )E R W eTwo alternative cases: e e∗ ∗ ∗| − ( ) > | − ( )(A) If E R e W e E R e W e , then the 1st( )][best Equilibrium Contract is: high e§ort, W eH H L L H! $ ! $Two alternative cases:! $ ! $ ∗ ( )][best Equilibrium Contract is: high e§ort, W ee e∗ ∗| − ( ) < | − ( )H(B) If E R e W e E R e W e , thee e ! $ ! $∗ ∗ H H L L| − ( ) > | − ( )(A) If E R e W e E R e W e , then the 1stH H L L e e∗ ∗ ∗| − ( ) < | − ( )(B) If E R e W e E R e W e , then the 1st( )][best Equilibrium Contract is: low e§ort, W eH H L L∗ L( )][best Equilibrium Contract is: high e§ort, W e H! $ ! $ ∗ ( )][best Equilibrium Contract is: low e§ort, W e Le e∗ ∗| − ( ) < | − ( )(B) If E R e W e E R e W e , then the 1stAngelo Baglioni ()H H L L∗ ( )][best Equilibrium Contract is: low e§ort, W eAngelo Baglioni () 2018 11 / 34LAngelo Baglioni () 2018 11 / 34

Asymmetric information

Asymmetric informationB) Asymmetric Informationcannot observeP can observe R~, but he the action e taken by A.eP can observe R, but he Cannot Observe the action e taken byLow effort.Let us now focus on P proposes the 1st best contract, [low effort, W*(eL)].A.Does it work? Let us focus on Low e§ort. P proposes the 1st best contract:Yes, because ∗∗ ( )][ low e§ort, W e . Does it work?LL ∗ ∗∗ ∗( )] − > ( )] −[ [Yes, because u W e e u W e e , so A will notL L L HL L L Hdeviate.so A will not deviate. ∗∗ ( )Fixed wage contract (W e : the same as under Symmetric Info)LLFixed wage contract ( the same as under symmetric info) can be used when P wants A tocan be used when P wants A to take the Least Costly action forNo Moral Hazard Here.take the least costly action for A:A: No moral hazard here!High effort.Let us now focus on P proposes the 1st best contract, [high effort, W*(eH)].Let us focus on High e§ort. P proposes the 1st best contract:Does it work? ∗∗ ( )][ high e§ort, W e . Does it work?HHNo, because ∗ ∗∗ ∗( )] − < ( )] −[ [No, because u W e e u W e e , so A will deviate:H H H LH H H LMoral Hazard!P cannot apply any penalty for deviation, because he does notMoral Hazard.so A will deviate: observe A’s e§ort (action is not contractible).any penaltyP cannot apply for deviation, because he does not observe A’s effort(remember action e is not contractible).Angelo Baglioni () 2018 12 / 34Angelo Baglioni () 2018 12 / 34So P will not propose 1st best contract with high effort.

Looking for the incentive compatible contract

Looking for the Incentive Compatible Contract. P looks for a contract that creates anincentive for A to exert high effort.Intuition: P should design a random wage W~ increasing in R~. Hence A knows that, bytaking high effort, he will make higher outcomes R~ and higher wage W~ more likely.P will look for the wage schedule that minimizes E(W~), given the Participation ConstraintPC and the Incentive Compatibility IC constraint.IC constraint: A gets more utility from high effort than from low effort.Principal’s problem:+ ( − )min p W 1 p W2 1H Hs.toGetting to the solution( ) + ( − ) ( ) − ≥p u W 1 p u W e u (PC)2 1H H H( ) + ( − ) ( ) − ≥ ( ) + ( − ) ( ) −p u W 1 p u W e p u W 1 p u W e2 1 2 1H H H L L L(IC) > >l µ=So it must be 0 and 0 (both PC and IC binding)L + ( − ) − ( ) + ( − ) ( ) − − −l [ ]p W 1 p W p u W 1 p u W e uFOC can be written as2 1 2 1h iH H H H H…[ ( ) + ( − ) ( ) − − ( ) − ( − ) ( ) + ]µ p u W 1 p u W e p u W 1 p u W ep1 = + −l µ 1 L2 1 2 1H H H L L L0 p( )u W(with both lambda and mu =0 ph=0 and 1-ph=0, which cannot be.h iH2FOC: ( − )1 p1 L= + −l µ 1with lambda=0 the second FOC becomes …, where LHS>0 and RHS<0.∂L 0 0 00 ( ) ( − )u W 1 p= − ( ) − ( ) − ( )] =lp µ [p u W p u W p u W 01 H2 2 2H H H L∂Wwith mu=0 u’(W1)=u’(W2), implying W2=W1=W, which leads to a violation of IC (because2 1∂L > > lFirst FOC: p p implies 0= ( − ) − ( − ) ( ) −l1 p 1 p u WH L 0 1H H ( )u W∂WIC becomes W-eH>=W-eL implying eL-eH>=0 which cannot be, eH>eL). )21 0 0 1− ) ( ) − ( − ) ( )] =µ [( 1 p u W 1 p u W 0( − ) < ( − ) < lSecond FOC: 1 p 1 p implies1 1H LH L… 0 ( )u W 1which can be written as:10 0So u’(W2)<u’(W1), which implies (since marginal utility u’(W1) is decreasing) W2>W1.( ) < < ( ) >Therefore: u W u W , which implies W W (given2 1 2 1lp − − − =lp µ [ ]p p 0H Equilibrium contract (2nd best)0H H L( )Bottom line: The Incentive Compatible Contract is a random wage W*~ Increasing inthat marginal utility u W is decreasing)0 ( )u W 2( − )1 pthe outcome R~:H − ( − ) − − ) − ( − )] =l µ [(1 p 1 p 1 p 0Bottom line: the Incentive Compatible Contract is a random wageH H L0 ( )u W 1f e∗ ∗ ∗> >W Increasing in the outcome R: W W (given R R ).2 12 1Angelo Baglioni () 2018 14 / 34

Which contract will P propose?

Which contract will P propose? fWhich contract will P propose? W*(eL) (inducing A to take low effort) or∗ ∗( )Either W e (inducing A to take low e§ort) or W (inducing A toLAngelo Baglioni () 2018 16 / 34(inducing A to take high effort)? take high e§ort)?P will choose the one that maximises his own Net expected return.Two alternative cases:" $ " $e f e∗ ∗&minu