Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Risk
Frank Cowell: Microeconomics
June 2004
Moral Hazard
MICROECONOMICS
Principles and Analysis
Frank Cowell
The moral hazard problem
Frank Cowell: Microeconomics




Jump to
“Adverse
selection”
Jump to
“Signalling”
A key aspect of hidden information
Information relates to actions.
Hidden action by one party affects probability of
favourable/unfavourable outcomes.




Hidden information about personal characteristics is
dealt with...
... under “adverse selection.”
... under “signalling.”
However similar issues arise in setting up the
economic problem.
Set-up based on model of trade under uncertainty.
Overview...
Moral Hazard
Frank Cowell: Microeconomics
The basics
Information:
hidden-actions
model
A simplified
model
The general
model
Key concepts
Frank Cowell: Microeconomics

Contract:

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Wage schedule:

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
An agreement to provide specified service…
…in exchange for specified payment
Type of contract will depend on information available.
Set-up involving a menu of contracts
The Principal draws up the menu
Allows selection by the Agent
Again the type of wage schedule will depend on information
available
Events:



Assume that events consist of single states-of-the-world
Distribution of these is common knowledge
But distribution may be conditional on the Agent’s effort
Strategic foundation
Frank Cowell: Microeconomics
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A version of a Bayesian game.
Two main players


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An additional player

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
Alf is the Agent.
Bill is the Boss (the Principal)
Nature is “player 0”
Chooses a state of the world
Bill does not observe what this is...
Principal-and-Agent: extensiveform game
Frank Cowell: Microeconomics
 "Nature" chooses a state of the world
 Probabilities are common knowledge
0
p
[NO]
 Principal may offer a contract, not
knowing the type
1-p
[RED]
[BLUE]
Bill
Bill
[OFFER]
[OFFER]
[NO]
Alf
[low]
 Agent chooses whether to accept
contract
Alf
[high]
[low]
[high]
Extension of trading model
Frank Cowell: Microeconomics

Start with trading model under uncertainty

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Assume:

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A single physical good…
…so consumption in each state-of-the world is a distinct
“contingent good”.
Two traders Alf, Bill
CE in Edgeworth box determined as usual:



There are two states-of-the world
So exactly two possible events
Probabilities of the two events are common knowledge
Draw a common tangent through the endowment point.
Gives equilibrium prices and allocation
But what happens in noncompetitive world?

Suppose Bill can completely exploit Alf
Trade: p common knowledge
Frank Cowell: Microeconomics
b
xRED
pRED
– ____
pBLUE
a
xBLUE
 Certainty line for Alf
b
 Alf's indifference
curves
O
 Certainty line for Bill
 Bill's indifference curves
 Endowment point
 CE prices + allocation
 Alf's reservation utility
 If Bill can exploit Alf...
pRED
– ____
pBLUE
•
•
Oa
•
b
xBLUE
a
xRED
Outcomes of trading model
Frank Cowell: Microeconomics

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
CE solution as usual potentially yields gains to
both parties
Exploitative solution puts Alf on reservation
indifference curve
Under CE or full-exploitation there is risk sharing


Exact share depends on risk aversion of the two parties.
What would happen if Bill, say, were risk neutral?


Retain assumption that p is common knowledge
We just need to alter the b-indifference curves
The special
case
Trade: Bill is risk neutral
Frank Cowell: Microeconomics
b
xRED
pRED
– ____
pBLUE
a
xBLUE
•
 Certainty line for Alf
b
 Alf's indifference
curves
O
 Certainty line for Bill
 Bill's indifference curves
 Endowment point
 CE prices + allocation
 Alf's reservation utility
 If Bill can exploit Alf...
•
Oa
•
b
xBLUE
a
xRED
Outcomes of trading model (2)
Frank Cowell: Microeconomics
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Minor modification yields clear-cut results
Risk-neutral Bill bears all the risk

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Also if Bill has discriminatory monopoly power

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So Alf is on his certainty line
Bill provides Alf with full insurance
But gets all the gains from trade for himself
This forms the basis for the elementary model of
moral hazad.
Overview...
Moral Hazard
Frank Cowell: Microeconomics
The basics
Lessons from the
2x2 case
A simplified
model
The general
model
Outline of the problem
Frank Cowell: Microeconomics
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Bill employs Alf to do a job of work
The outcome to Bill (the product) depends on

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
Alf's effort affects probability of chance element.

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High effort – high probability of favourable outcome
Low effort – low probability of favourable outcome
The issues are:




A chance element
The effort put in by Alf
Does Bill find it worth while to pay Alf for high effort?
Is it possible to monitor whether high effort is provided?
If not, how can Bill best construct the contract?
Deal with the problem in stages
Simple version – the approach
Frank Cowell: Microeconomics

Start with simple case

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Build on the trading model

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Two unknown events
Two levels of effort
Principal and Agent are the two traders
But Principal (Bill) has all the power
Agent (Alf) has the option of accepting/rejecting the
contract offered.
Then move on to general model


Continuum of unknown events.
Agent has general choice of effort level
Power: Principal and Agent
Frank Cowell: Microeconomics

Because Bill has power:
Can set the terms of the contract
 ...constrained by the Alf’s option to refuse
 Can drive Alf down to reservation utility


If the effort supplied is observable:
Contract can be conditioned on effort: w(z)
 Get all the insights from the trading model


Otherwise:

Have to condition on output: w(q)
The 22 case: basics
Frank Cowell: Microeconomics

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A single good
Amount of output q is a random variable
Two possible outcomes

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
Probability of success is common knowledge:

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Failure q
–_
Success q
given by p(z)
z is the effort supplied by the agent
The Agent chooses either


Low effort z
_
High effort z
The 22 case: motivation
Frank Cowell: Microeconomics

The Agent's utility derives from

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The Agent is risk averse


xb = q – xa
(In the simple model) Principal is risk neutral

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ua(•, •) is strictly concave in its first argument
The Principal consumes all output not consumed
by Agent


consumption of the single good xa ()
the effort put in, z ()
Given vNM preferences utility is Eua(xa, z) .
Utility is Eq – xa
Can interpret this in the trading diagram
Low effort
Frank Cowell: Microeconomics
b
xRED
a
xBLUE
pRED
– ____
pBLUE
 Certainty line for Alf (Agent)
b
 Alf's
curves
Ob indifference
O
 Certainty line for Bill
 Bill's indifference curves
 Endowment point
 Alf's reservation utility
If Bill exploits Alf
then outcome is on
reservation IC, ua
If Bill is riskneutral and Alf risk
averse then
outcome is on Alf's
certainty line.
ua
b
xBLUE
Oa
a
xRED
Switch to high
effort
High effort
Frank Cowell: Microeconomics
b
xRED
a
xBLUE
pRED
– ____
pBLUE
 Certainty line and indifference
curves
forb Alf
O
Obb O
 Certainty line and indifference
curves for Bill
 Endowment point
 Alf's reservation utility
High effort tilts
the ICs, shifts
the equilibrium
outcome.
Contrast with
low effort
b
xBLUE
Oa
a
xRED
Combine to get
menu of
contracts
Full information: max problem
Frank Cowell: Microeconomics
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The Agent's consumption is determined by the wage paid.
The Principal chooses a wage schedule...

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...subject to the participation constraint:
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Eua(w,z)  ua.
So, problem is choose w(•) to maximise


w = w(z)
Eq – w + l[Eua(w,z) – ua]
Equivalently



_
Find w(z) that maximise p(z) q + [1
_ – p(z)] q – w(z)...
... for the two cases z = z and z = z.
Choose the one that gives higher expected payoff to Principal
Full-information contracts
Frank Cowell: Microeconomics
–
q
b
xRED
a
xBLUE
Ob
q
–
–
w(z)
w(z)
–
b
xBLUE
Oa
–
w(z)
w(z)
–
a
xRED
 Alf's low-effort ICs
 Bills ICs
 Alf's high-effort ICs
 Bills ICs
 Low-effort contract
 High-effort contract
Full-information contracts: summary
Frank Cowell: Microeconomics

Schedule of contracts for high and low effort

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Contract specifies payment in each state-of-the-world
State-of-the-world is costlessly and accurately observable

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Effort is verifiable
Equivalent to effort being costlessly and accurately observable
Alf (agent) is forced on to reservation utility level
Efficient risk allocation



Bill is risk neutral
Alf is risk averse
Bill bears all the risk
Second best: principles
Frank Cowell: Microeconomics

Utility functions

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Wage schedule





Because effort is unobservable…
...cannot condition wage on effort or on the state-of-the-world.
But resulting output is observable...
... so you can condition wage on output
Participation constraint

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
As before
Essentially as before
(but we'll have another look)
New incentive-compatibility constraint

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Cannot observe effort
Agent must get the utility level attainable under low effort
Maths
formulation
Participation constraint
Frank Cowell: Microeconomics

The Principal can condition the wage on the observed
output:

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Agent will choose high or low effort.

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
This determines the probability of getting high output
...and so the probability of getting a high wage.
Let's assume he would choose high effort


_
_
Pay wage w if output is q
Pay wage w if output is q
(check this out in next slide)
To ensure that Agent doesn't reject the contract...
...must get the utility available elsewhere:

_
_ _
_
_
a
a
p(z) u (w, z) + [1 – p(z)] u (w, z)  ua
Incentive-compatibility constraint
Frank Cowell: Microeconomics

Assume that the Agent will actually participate

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
Agent will choose high or low effort.
To ensure that high effort is chosen, set wages so the
following holds:


_
_
Pay wage w if output is q
Pay wage w if output is q
_
_ _
_
_
a
a
p(z) u (w, z) + [1 – p(z)] u (w, z) 
_
_
a
p(z) u (w, z) + [1 – p(z)] ua(w, z)
This condition determines a set of w-pairs


a set of contingent consumptions for Alf
must not reward Alf too highly if failure is observed
Second-best contracts
Frank Cowell: Microeconomics
 Alf's low-effort ICs
Ob Bills ICs
 Alf's high-effort ICs
 Bills ICs
 Full-information contracts
 Participation constraint
 Incentive-compatibility
constraint
 Bill’s second-best
feasible set
 Second-best contract
b
xRED
a
xBLUE
ua
–w
b
xBLUE
Oa
–
w
a
xRED
Contract
maximises Bill’s
utility over
second-best
feasible set
Simplified model: summary
Frank Cowell: Microeconomics

Participation constraint

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Incentive compatibility constraint

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
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Set of contingent consumptions giving Alf his reservation utility.
If effort is observable get one such constraint for each effort level
Relevant for second-best policy.
Set of contingent consumptions such that Alf prefers to provide
high effort.
Implemented by making wage payment contingent on output
Intersection of these two sets gives feasible set for Bill
Outcome depends on information regime


Observable effort: Bill bears all the risk
Moral hazard: Alf bears some risk
Overview...
Moral Hazard
Frank Cowell: Microeconomics
The basics
Extending the
“first-order”
approach
A simplified
model
The general
model
General model: introduction
Frank Cowell: Microeconomics

Retain assumption that it is a two-person contest.


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Again deal with full-information case first.

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Same roles for Principal and Agent. But…
Allow for greater range of choice for Agent
Allow for different preferences for Principal
Draw on lessons from 2×2 case
Same principles apply
Then introduce the possibility of unobserved
effort.


Needs some modification from 2×2 case
But similar principles emerge
Frank Cowell: Microeconomics
Model components: output and
effort

Production depends on effort z and state of the
world w:



Effort can be anything from “zero” to “full”


q = f(z,w)
wW
z  [0,1]
Output has a known frequency distribution




f(q, z)
Support is the interval [q, q]
Increasing effort biases distribution rightward
Define proportional effect of effort bz := fz(q, z)/f(q, z)
Effect of effort
Frank Cowell: Microeconomics
Support of the distribution
Output distribution: low
effort
f(q, z)
Output distribution: high
effort
Higher effort
biases frequency
distribution to
the right
q
–
q–
q
Model components: preferences
Frank Cowell: Microeconomics

Again the Agent's utility derives from





the wage paid, w ()
the effort put in, z ()
Eua(w, z) .
ua(•, •) is strictly concave in its first argument
The Principal consumes output after wage is paid



But we allow for non-neutral risk preference
Eub(xb) = Eub(q – w)
ub(•) is concave
Full information: optimisation
Frank Cowell: Microeconomics

Alf’s participation constraint:

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Bill sets the wage schedule.

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Bill’s utility function ub
production function f
Problem is then




Can be conditioned on the realisation of w
w = w(w)
To set up the maximand, also use


Eua(w,z)  ua.
choose w(•)
to max Eub(f(z,w))
subject to Eua(w(w),z)  ua.
Lagrangean is

Eub(f(z,w) – w(w)) + l[Eua(w(w), z) – ua]
Optimisation: outcomes
Frank Cowell: Microeconomics

The Lagrangean is

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Each w(w) and z can be treated as control variables



Eub(xb) + l[Eua(xa,z) – ua]
where xa = w(w) ; xb = f(z,w) – w(w)
Bill chooses w(w) .
Alf chooses z, knowing the wage schedule set by Bill.
First-order conditions are


– uxb(f(z,w) – w(w)) + luxa(w(w),z) = 0
Euxb(f(z,w) – w(w))fz(z,w) + lEuza(w(w), z) = 0
xb = f(z,w)
w(w)
 –Combining


a
x =
we get
w(w)
uxb(xb) / uxa(xa) = l
 uza(xa, z)

b
b
b
b
Eux (x )fz(z,w) + E   ux (x ) = 0
 uxa(xa, z)

Full information: results
Frank Cowell: Microeconomics

Result 1





uxb(xb) / uxa(xa) = l
Because uxa and uxb are positive l must be positive.
So participation constraint is binding
Ratio of MUs is the same (l) in all states of nature
Result 2






 uza(xa, z)

Euxb(xb )fz(z,w) + E   uxb(xb)  = 0
 uxa(xa, z)

In each state Bill’s (the Principal’s) MU is used as a weight.
In the special case where Bill is risk-neutral...
...this weight is the same in all states. Then we have:
 uza(xa, z) 
E fz(z,w) = – E   
 uxa(xa, z) 
Expected MRT = Expected MRS for the Agent
Full information: lessons
Frank Cowell: Microeconomics

Principal fully exploits Agent




Efficient risk allocation



Because Principal drives Agent down to reservation utility
Follows from assumption that Principal has all the power
(No bargaining)
Take MRS between consumption in state-of-the-worlds w and w
MRSa = MRSb
Efficient allocation of effort


In the case where Principal is risk neutral...
Expected MRTSzx = Expected MRSzx
Second-best: introduction
Frank Cowell: Microeconomics



Now consider the case where effort z is unobserved
This is equivalent to assuming state-of-the-world w
unobserved
Can work with the distribution of output q:





Transformation of variables from w to q
Just use the production function q= f(z,w)
Clearly effort shifts the distribution of output
Use the expectation operator E over the distribution of output.
All model components can be expressed in terms of this
distribution
Second-best: components
Frank Cowell: Microeconomics


Objective function of Principal and of Agent are as before.
Distribution of output f depends on effort z.


Participation constraint for Agent still the same


Modify it to allow for redefined distribution
Require also the incentive-compatibility constraint


Probability density at output q is f(q, z)
Builds on the (hidden) optimisation of effort by the Agent
Again use Lagrangean technique


Assumes problem is “well-behaved”
This may not always be appropriate
Second-best: problem
Frank Cowell: Microeconomics

Bill sets the wage schedule.




Bill knows that Alf must get at least “reservation utility” :



Cannot be conditioned on the realisation of w
But can be conditioned on observable output
w = w(q)
Eua(w(q),z)  ua.
participation constraint
Also knows that Alf will choose z to maximise own utility



So Bill assumes (correctly) that the following FOC holds:
E(ua(w, z)bz) + Euza(w, z) = 0
This is the incentive-compatibility constraint.
Second-best: optimisation
Frank Cowell: Microeconomics

Problem is then




Lagrangean is
 Eub(q – w(q)) + l [Eua(w(q), z) – ua ]
+ m [E(ua(w(q), z)bz) + Euza(w(q), z) ]



l is the “price” on the participation constraint
m is the “price” on the incentive-compatibility constraint
Differentiate Lagrangean with respect to w(q) …


choose w(•) to max Eub(q – w(q))
subject to Eua(w(q),z)  ua.
and E(ua(w(q), z)bz) + Euza(w(q), z) = 0
each output level has its own specific wage level.
... and with respect to z.


Bill can effectively manipulate Alf’s choice of z ...
... subject to the incentive-compatibility constraint.
Second-best: FOCs
Frank Cowell: Microeconomics

Use a simplifying assumption:


Lagrangean is



Eub(xb) + l[Eua(xa, z) – ua ] + m[ E(ua(xa, z)) / z ]
where
 xa = w(q)
 xb = q – w(q)
Differentiating with respect to w(q):


uxza(•,•) = 0
FOC1: – uxb(xb) + luxa(xa, z) + muxa(xa, z)bz = 0
Differentiating with respect to z:

FOC2: Eub(xb)bz+ m[ 2E(ua(xa, z)) / z2 ] = 0
Second-best: results
Frank Cowell: Microeconomics

From




2E(ua(xa, z))/z2
m>0
So the incentive-compatibility constraint is binding
From FOC1:





–
m
bz is +ve where
FOC2:
xb is large
2nd derivative
Eub(xb)bz
= ——————— is negative
uxb(xb) / uxa(xa, z) = l + m bz
We know that bz < 0 for low q...
So if l = 0, this would imply LHS negative for low q (impossible)
Hence l > 0: the participation constraint is binding.
From FOC1:



Because uxb(xb) / uxa(xa, z) = l + m bz
Ratio of MUs > l if bz > 0; ratio of MUs < l if bz < 0
So a-consumption is high if q is high (where bz > 0).
Principal-and-Agent: Summary
Frank Cowell: Microeconomics

In full-information case:




participation constraint is binding
risk-neutral Principal would fully insure risk-averse Agent.
Fully efficient outcome
In second-best case:





(where the moral hazard problem arises)
participation constraint is binding
incentive-compatibility constraint is also binding
Principal pays Agent more if output is high
Principal no longer insures Agent fully.