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Notes on Moral HazardSummary based on The Theory of Incentives, Laffont and Martimort (2014)

Typos and small mistakes may be present, and they are entirely mine.

2Overview

Basics of the model Incentive Feasible Contracts Complete information optimal contract Risk Neutrality and First Best Implementation TradeOff LLR extraction and Efficiency TradeOff Insurance and Efficiency Optimal Transfers and SB Effort Contract Theory at work: Sharecropping and Financial Contracts

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While adverse selection deals with a principal who delegates a task to an agent without knowing its type (efficient or inefficient),

Moral hazard deals with a principal who delegates a task to an agent without knowing if puts effort or not. Principal induces the agent to put effort because it obtains a higher production level.

Two solutions, the first under complete information, and second best under moral hazard. Some cases will deal with insurance and limited liability rent; moreover the risk aversion of the agent will play a specific role in determining optimal contract.

4Introduction

In Adverse Selection the UNCERTAINTY is EXOGENOUS ( nonobservability of the type is determined outside the model)

Moral Hazard the UNCERTAINTY is ENDOGENOUS ( non observability of effort determined within the model)

Principal delegate a task to the agent, BUT agent can take actions to affect value of trade or affect the production:

Agent PUT EFFORT e=1 creating utility for the principal (higher production) but disutility for agent;

Or Agent does NOT PUT EFFORT e=0 creating lower utility for P and no disutility for himself

Since the effort is not observable the actions cannot be contracted upon, we can only contract on the final levels of production.

5Principal-agent model: Moral Hazard

Examples. How safely you drive affects your insurance contract. How does a shareholder can control the manager in managing her

business?

Each project is associated to a particular stream of cash flows for Principal and transferable private benefit for the agent.

Moral Hazard : CONFLICT BETWEEN P and A objective functions Principal designs a contract based on observable results of agent =

Production levels.

6INCENTIVE FEASIBLE CONTRACTS

Principal implements: Incentive Schemes inducing positive and costly effort = IC Inducing the volountary participation of the Agent, even of the

inefficient agent with the higher cost = PC

Incentive Feasible Contracts are the ones satisfying PC AND IC.

Principal preferes the incentive schemes that minimizes his cost. In general we find two solutions, the First Best and the Second

Best.

7Risk Neutral Agent: RC, NO TC.

When the agent is RISK NEUTRAL NO TRANSACTION COSTS.

NO OBSERVABILITY OF EFFORT CREATES NO PROBLEMS ON EFFICIENCY OF TRADE. NO TRANSACTION COSTS. FB DEPENDS ON PRODUCTION LEVELS = IF IT IS HIGH, AGENT OBTAINS REWARD, IF IT IS LOW IT IS PENALIZED, WHICH IS ABLE TO BEAR SINCE IT IS RISK NEUTRAL.

AGENT IS RESIDUAL CLAIMANT FROM GAINS FROM TRADE AND TAKE EXPECTED GAINS BY EX ANTE LUM SUM TRANSFER TO THE PRINCIPAL.

8Risk Averse Agent

CONSTANT WAGE PROVIDES FULL INSURANCE BUT INDUCES NO EFFORT.

INDUCING EFFORT FOR THE PRINCIPAL ENTAILS BEARING SOME RISK.

AGENT SHOULD RECEIVE A RISK PREMIUM TO ACCEPT A RISKY CONTRACT.

HIGH EFFORT IS LESS IMPLEMENTED UNDER ASYMMETRIC INFO. THAN COMPLETE INFORMATION

9Basic Definitions

P DELEGATES TASK TO A. A CAN PUT EFFORT e=1 or not (e=0). This is not observable by P.

If A put effort faces the cost of putting effort is >0. Utility function of the agent is U : U = u(t) - Utility function is increasing and concave (positive marginal

utility u’>0 with diminishing returns u’’<0). The utility function can be expressed as h=1/u or h= u^-1

10Stochastic Distributionis common knowledge

THE PROBABILITY ASSOCIATED WITH HIGH PRODUCTION ( ARE: IF PUT POSITIVE EFFORT : IF PUT NO EFFORT :

Probability associated with low production (Q_): IF PUT POSITIVE EFFORT : IF PUT NO EFFORT :1-

Difference in probabilities:

11Principal’s Utility based on Stochastic Production

Effort improves production in sense of First order stochastic dominance. Probability of obtaining high production level conditioned on putting effort is higher than obtaining high production without putting effort.

V Principal Utility if agent puts effort

V Principal Utility if agents puts NO effort

12Incentive feasible contracts

In order to be incentive feasible, the contract must satisfy : 1. Incentive Compatibility Constraint = IC

The expected utility from putting effort given cost of put effort must be greater than not putting effort.

2. Participation Constraint = PC (equal to left side above) 0 ensures that agents putting effort enters into the contract ( since it has

higher costs and could shutdown or refuse the contract).

13Complete Information Optimal Contract Differently from Asymmetric information, under complete information effort is

observable, there are no transaction costs. 0 L=)=0 Where lambda should be positive since PC is greater or equal than zero.

Solution: is equal to the inverse of utility function and the transfer is the same no matter state of nature.

Full insurance is obtained for the agent. This is the FIRST BEST SOLUTION which is not feasible under moral hazard.

Transfer is the FB COST t=h() and expected gain of effort is greater than fb cost of inducing effort.

14Risk Neutrality FB Implementation

We replace utility = transfer u(t)=t and inverse utility h(u) = u

Since transfer = ; is transfer with high production

And t*_ = ; the agent is rewarded if production is high;

Utility of agent is transfer minus disutility of effort. Principal induces effort because ensures to provide the second transfer instead of providing the first.

Under complete information FB delegation is costless for the principal.

Even under incomplete information if agent is risk neutral, since it can be punished for low production and rewarded from high production through transfers of FB.

PROVIDED INCENTIVE COMPATIBILITY IS BINDING

15Trade off LLR and EFFICIENCY - i

In inducing effort principal faces the trade off of limited liability extraction and efficiency. We introduce two additional constraints besides the above 4 PC and IC:

As the transfer should be above a certain level l: it reduces the set of incentive feasible allocations and prevent FB

even if agent is risk neutral.

When the red binds principal is limited in punishments to induce efforts, risk neutral agents does not have nough assets to cover the punishment if q_ realizes

16Trade off LLR and EFFICIENCY - ii

Proposition 4.2 with limited liability optimal contract inducing effort: For that is only IC and PC are binding. Optimal transfers are given by =

and t*_ = Agent has no exp. LLR : EUsb=0 For optimal transfers are and EUsb = LLC in bad state is binding = Punishments are limited to induce effort

With moral hazard and LLR tradeoff between inducing effort and giving up ex ante limited liability rent to the agent, so the principal chooses to induce a high effort from the agent less often.

17Trade off LLR and EFFICIENCY - iii

Principal expected utility is V SB when induces effort: And it is worth inducing effort is VSB is greater than V0:

= Cost of inducing effort is limited liability rent + disutility of effort. SB COST exceeds FB cost.

18Trade Off btw Insurance and Eff.

Second source of inefficiency is risk aversity of agent.

Kuhn tucker works only if concavity works, and to ensure it change variables. these new variables are levels of utility ex post agent obtains his status of

nature. IC and PC are now linear (since u and no more u(t)):

0

we replaced transfers.

19Optimal Transfers

Solve the lagrangian and obtain:

then I find mu and lambda strictly positive thus both are binding. Risk averse agent does not obtain full insurance anymore.

Jensen Inequality

20Optimal SB Effort

With Moral Hazard and Risk Aversion.

implied by Jensen’s Inequality;

With moral hazard and risk aversion tradeoff of inducing effort and insurance is solved by principal in inducing less often positive effort.

= E[h(u) ] and decompose it into variance to obtain Agency Cost AC= informational problem for principal: large when small.

21Sharecropping & linear sharing rule

Principal is landlord and agent is tenant. If the agent puts effort in harvesting the associated probability of high production increases.

Max IC PC. NO TRANSFER WITH LINEAR SHARING RULE.

linear sharing rule : P offers A a fixed share of prod. In a certain sense this is a insurance scheme because if produces low it is rewarded (even if reward is lower it is still positive) so it is inefficient. Max] s.t. ) and PC.

; ; Where Adding the fixed fee Beta equal to the expected utility of the agent s.t. efficiency is

restored.

22Financial Contracts & Credit Rationing When PRINCIPAL IS RISK AVERSE. Initial Investment: I. P has no cash. Return on project is V- with probability and V_ with inverse probability. S.T.

Change variables t=V-Z . Second Best cost: ; considering costs: : Iif only projects with positive value V>0 are financed investment must be low enough:

Under no moral hazard projects are financed: This is a form of credit rationing. Contract is debt contract no money left to borrower in

bad state of nature and residula being pocketed by lender in good state. No limited liability constraint is satisfied now t_=0 instead of being t_=.

23Source

Laffont, J. and Martimort, D.: The Theory of Incentives: The Principal Agent’s Model. (2014), pp. 145-180.