Advanced Microeconomics

Equilibrium Contract

H L Lf *Equilibrium Contract is: W . "$ " $e f e*-* | < | - ( )(B) If E R W e E R e W e , then the 2st bestH L LWhich contract will P propose? f* *()( )Either W e (inducing A to take low e§ort) or W (inducing A toLP will chose again the one that maximized his own net expected return. Two alternativetake high e§ort)?cases:P will chose the one that maximises his own NET expected return.Two alternative cases:" "$ " $e f e*-* | > | - ( )(A) If E R W e E R e W e , then the 2nd bestH L Lf *Equilibrium Contract is: W . "$ " $e f e*-* | < | - ( )(B) If E R W e E R e W e , then the 2st bestH L L* ( )Equilibrium Contract is: W e .LComparing 1st best and 2nd best (see Figures 2,3,4)Comparing 1st best and 2nd best (see Figures 2,3,4)Assume case (A) holds both under SYM and ASYM.Comparing 1st best and 2nd best (see Figures 2,3,4)The 1st

best eq contract is [high effort, W*(eH)].

Assume Case (A) holds both under Simm. and Asymm. Information:Angelo Baglioni () 2018 17 / 34∗

The 2nd best eq contract is

Assume Case (A) holds both under Simm. and Asymm. Information:( )][The 1st best Equilibrium Contract is: high e§ort, W e HAssume Case (A) holds both under Simm. and Asymm. Information:∗f ( )][The 1st best Equilibrium Contract is: high e§ort, W e∗

Comparing 1st best and 2nd best (see Figures 2,3,4)The 2nd best Equilibrium Contract is: W H∗ ( )][The 1st best Equilibrium Contract is: high e§ort, W ef ∗ HThe 2nd best Equilibrium Contract is: W

PC holds in both cases, so A gets Reservation level of utility from f ∗

Comparing 1st best and 2nd best (see Figures 2,3,4)The 2nd best Equilibrium Contract is: WPC holds in both cases, so A gets Reservation level of utility from both contracts:PC holds in both cases, so A gets Reservation level of utility fromboth contracts: h #

$i h # $i f ∗ ∗= ( )][

Therefore: E u W u W eboth contracts: h # $iHf f From which, by applying the Jensen’s inequality, we get:∗ ∗ ∗

Assume Case (A) holds both under Simm. and Asymm. Information:> = ( )][u E W E u W u W eh # $i h # $iPC holds in both cases, so A gets Reservation level of utility fromfH∗ ∗( )] − = − =[u W e e u, E u W e uFrom which, by applying the Jensen’s inequality, we get:# $ H H Hf fh # $i h # $i∗ ∗ ∗ ∗> = ( )][u E W E u W u W e ( )][The 1st best Equilibrium Contract is: high e§ort, W eh # $iboth contracts: h # $i H Hf ∗ ∗ f f∗ ∗ ∗> ( )which implies E W W e # $> = ( )][u E W E u W u W efH ∗ ∗fFrom which, by applying Jensen’s inequality f∗ ∗ H= ( )][Therefore: E u W u W e∗( )] − = − =[u W e e u, E u W e uThe 2nd best Equilibrium Contract is: W# $ Hf# $H H

W ewhich lowers his NET Expected profit,> = ( )][u E W E u W u W ee e f# $ # $ # $H∗ ∗H| − ( ) | −going from E R e W e to E R e E Wf # $∗ ∗H H H# $ = ( )][Therefore: E u W u W ee e f∗ ∗H| − ( ) | −going from E R e W e to E R e E WAngelo Baglioni () 2018 18 / 34f ∗ ∗f H H H∗ ∗ = − ( )AGENCY COST paid by P: AC E W W e ,> ( )which implies E W W e HHFrom which, by applying the Jensen’s inequality, we get:AGENCY COST paid by P:Angelo Baglioni () 2018 18 / 34h # $i h # $i # $which lowers his NET Expected profit,# $ # $ # $Angelo Baglioni () 2018 18 / 34f f f∗ ∗ ∗∗ ∗> = ( )][u E W E u W u W e= − ( )AGENCY COST paid by P: AC E W W e ,H He e f∗ ∗| − ( ) | −# $going from E R e W e to E R e E WH H Hwhich lowers his NET Expected profit,f# $ # $ # $∗ ∗> ( )which implies E W W e He e f∗ ∗#

$| − ( ) | −going from E R e W e to E R e E WAngelo Baglioni () 2018 18 / 34H H Hfwhich lowers his net expected profit, going from ∗ ∗= − ( )AGENCY COST paid by P: AC E W W e ,HAngelo Baglioni () 2018 18 / 34which lowers his NET Expected profit,# $ # $ # $e e f∗ ∗| − ( ) | −going from E R e W e to E R e E WH H HAngelo Baglioni () 2018 18 / 34Is AC a redistribution from P to A?NO: going from 1st best to 2nd best, P is worse off but A is not better off.AC is a DEADWEIGHT LOSS: an extra payment made by P to keep A on the same level ofutility.Why the inefficiency? Two contrasting principles:-ORS: all risk should be given to P, so wage should be fixed W-IC: some risk has to be given to A, so wage has to be random W~AC is the extra payment requested by A to be compensated for risk: Risk Premium.In utility theory, W*(eH) is the Certainty Equivalent CE of W*~.Figure 3re 2 Angelo Baglioni () 2018 21 / 34Angelo Baglioni () 2018 20 /

1. Credit Markets
• Credit risk: risk of default, borrower not being able to repay in full the lenders
• Lender - borrower conflict. Assumptions
• Asymmetric information:
• Moral hazard: hidden action, opportunistic behavior of borrowers
• Adverse selection: hidden information, best potential borrowers withdraw from the market and lenders provide money only to bad borrowers
2. We focus on two players: Bank (B) and Firm (F)
3. What are the implications of the debt contract (with limited liability)?
4. Lender-Borrower Conflict
• To examine the impact of debt contract on incentives, we assume assumptions
• The principal we make are the following. We focus on two players, Bank B that both B and F are risk neutral and Firm F.
• To examine the impact of the debt contract on incentives we assume that Assume F can choose among different investment projects /risk neutral.
• Both B and F are management strategies

projects to which locate moneyfEach project has a random value: X , taking up either value X or X

Figure 1. Projects’ valueH Lreceived from B. Each project has a random value X~.> ( − )(where X X ) with probabilities p and 1 p respectively (seeH L H HsameAll projects have the expected value:Figure 1 )All projects have the same expected value:e( ) = + ( − )E X p X 1 p XH H H L = −Projects di§er for their risk, measuerd by d X XH Lrisk,Projects have a different measured by d=XH-XL.Players’ payo§s (Figure 2)F borrows an amount D to be repaid at the end of the period: D(1+r).Angelo Baglioni () 2018 3 / 46To have an interesting case, we assume that for all projects:XH>S(1+r)>XL. Otherwise: Angelo Baglioni () 2018 4 / 46-if D(1+r)>XH: F is never solvent-if D(1+r)<XL: F is never insolventPlayers’ payo§s (Figure 2) Players’ expected payo§Angelo Baglioni () 2018 6 / 46e( ) = ( + ) + ( − )Bank: E R p D 1 r 1 p XB

H_H Le( ) = − ( + )][Firm: E R p X D 1 rF H H ?

How do the two expected payo§s respond to a change in r?

Figure 4. Firm’s payo§

Bank’s expected payoff increases in r and

The same argument assuming a continuous X

decreases in d. The opposite for Firm’s expected

payoff.

The same ar