1. MICROECONOMICS Principles and Analysis Frank Cowell
Prerequisites
Almost essential
Risk
Moral Hazard
MICROECONOMICS
Principles and Analysis
Frank Cowell
Note: the detail in slides marked “ * ” can only be seen if you run the slideshow
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

2. The moral hazard problem
A key aspect of hidden information
Information relates to actions
hidden action by one party affects probability of favourable/unfavourable outcomes
for hidden information about personal characteristics
see Adverse Selection
see also Signalling
However similar issues arise in setting up the economic problem
Set-up based on model of trade under uncertainty
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

3. Overview Information: hidden-actions model Moral Hazard The basics
A simplified model
The general model
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

4. Key concepts Contract: Wage schedule: Events:
An agreement to provide specified service
in exchange for specified payment
Type of contract will depend on information available
Wage schedule:
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
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

5. Strategic foundation A version of a Bayesian game Two main players
Alf is the Agent
Bill is the Boss (the Principal)
An additional player
nature is “player 0”
chooses a state of the world
Bill does not observe what this is
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

6. Principal-and-Agent: extensive-form game
“Nature” chooses a state of the world
Probabilities are common knowledge
Principal offers contract, not knowing state of world
Agent chooses whether to accept contract p
1-p
[RED]
[BLUE]
Bill
Bill
[NO]
[OFFER]
[NO]
[OFFER]
Alf
Alf
[low]
[high]
[low]
[high]
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

7. Extension of trading model
Start with trading model under uncertainty
there are two states-of-the world
so exactly two possible events
probabilities of the two events are common knowledge
Assume:
a single physical good
consumption in each state-of-the-world is a distinct “contingent good”
two traders Alf, Bill
CE in Edgeworth box determined as usual:
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
July 2015
Frank Cowell: Moral Hazard

8. Trade: p common knowledge
Certainty line for Alf
xRED b
pRED
– ____
pBLUE Ob
Alf's indifference curves
Certainty line for Bill
xBLUE a
Bill's indifference curves
Endowment point
CE prices + allocation •
Alf's reservation utility
If Bill can exploit Alf
pRED
– ____
pBLUE • •
xBLUE b Oa
xRED a
July 2015
Frank Cowell: Moral Hazard

9. Outcomes of trading model
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
July 2015
Frank Cowell: Moral Hazard

10. Trade: Bill is risk neutral
Certainty line for Alf
xRED b
pRED
– ____
pBLUE Ob
Alf's indifference curves
Certainty line for Bill
xBLUE a
Bill's indifference curves
Endowment point
CE prices + allocation •
Alf's reservation utility
If Bill can exploit Alf • •
xBLUE b Oa
xRED a
July 2015
Frank Cowell: Moral Hazard

11. Outcomes of trading model (2)
Minor modification yields clear-cut results
Risk-neutral Bill bears all the risk
So Alf is on his certainty line
Also if Bill has discriminatory monopoly power
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 hazard
July 2015
Frank Cowell: Moral Hazard

12. Overview Lessons from the 2x2 case Moral Hazard The basics
A simplified model
The general model
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

13. Outline of the problem Bill employs Alf to do a job of work
The outcome to Bill (the product) depends on
a chance element
the effort put in by Alf
Alf's effort affects probability of chance element
high effort – high probability of favourable outcome
low effort – low probability of favourable outcome
The issues are:
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
July 2015
Frank Cowell: Moral Hazard

14. Simple version – the approach
Start with simple case
two unknown events
two levels of effort
Build on the trading model
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
July 2015
Frank Cowell: Moral Hazard

15. Power: Principal and Agent
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)
July 2015
Frank Cowell: Moral Hazard

16. The 22 case: basics A single good
Amount of output q is a random variable
Two possible outcomes
failure q – _
success q
Probability of success is common knowledge:
given by p(z)
z is the effort supplied by the agent
The Agent chooses either
low effort z
high effort z
July 2015
Frank Cowell: Moral Hazard

17. The 22 case: motivation Agent's utility: Agent is risk averse
consumption of the single good xa ()
the effort put in, z ()
given vNM preferences utility is E ua(xa, z)
Agent is risk averse
ua(·, ·) is strictly concave in its first argument
Principal consumes all output not consumed by Agent
xb = q – xa
(In the simple model) Principal is risk neutral
utility is E q – xa
Can interpret this in the trading diagram
July 2015
Frank Cowell: Moral Hazard

18. Low effort b Ob Ob a Oa xRED xBLUE
pRED
– ____
pBLUE
xRED b
Certainty line for Alf (Agent)
Alf's indifference curves Ob Ob
xBLUE a
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 risk-neutral and Alf risk averse then outcome is on Alf's certainty line ua
Switch to high effort
xBLUE b Oa
xRED a
July 2015
Frank Cowell: Moral Hazard

19. *High effort b Ob Ob Ob a Oa xRED xBLUE
Alf: Cert.line and indiff curves
pRED
– ____
pBLUE
Bill: Cert. line and indiff curves Ob Ob Ob
xBLUE a
Endowment point
Alf's reservation utility
High effort tilts ICs, shifts equilibrium outcome
Contrast with low effort
Combine to get menu of contracts
xBLUE b Oa
xRED a
July 2015
Frank Cowell: Moral Hazard

20. Full information: max problem
Agent's consumption is determined by the wage
Principal chooses a wage schedule
w = w(z)
subject to the participation constraint:
E ua(w,z)  ua
So, problem is choose w(·) to maximise
E q – w + l[E ua(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
July 2015
Frank Cowell: Moral Hazard

21. Full-information contractsq –
xRED b
Alf's low-effort ICs Ob
Bills ICs
xBLUE a
Alf's high-effort ICs
Bills ICs
Low-effort contract
High-effort contract – q –
w(z) –
w(z)
xBLUE b Oa –
w(z) –
w(z)
xRED a
July 2015
Frank Cowell: Moral Hazard

22. Full-information contracts: summary
Schedule of contracts for high and low effort
effort is verifiable
Contract specifies payment in each state-of-the-world
State-of-the-world is costlessly and accurately observable
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
July 2015
Frank Cowell: Moral Hazard

23. Second best: principles
Utility functions
as before
Wage schedule
effort is unobservable
cannot condition wage on effort or on the state-of-the-world
but resulting output is observable
you can condition wage on output
Participation constraint
essentially as before
(but we'll have another look)
New incentive-compatibility constraint
cannot observe effort
agent must get the utility level attainable under low effort
Maths formulation
July 2015
Frank Cowell: Moral Hazard

24. Participation constraint
Principal can condition the wage on the observed output:
_ _
wage w if output is q
wage w if output is q
Agent will choose high or low effort
this determines the probability of getting high output
so the probability of getting a high wage
Let's assume he would choose high effort
(check this out in next slide)
To ensure that Agent doesn't reject the contract
must get the utility available elsewhere:
_ _ _ _ _
p(z) ua(w, z) + [1 – p(z)] ua(w, z)  ua
July 2015
Frank Cowell: Moral Hazard

25. Incentive-compatibility constraint
Assume that Agent will actually participate
_ _
wage w if output is q
wage w if output is q
Agent will choose high or low effort
To ensure that high effort is chosen, set wages so that:
_ _ _ _ _
p(z) ua(w, z) + [1 – p(z)] ua(w, z) 
p(z) ua(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
July 2015
Frank Cowell: Moral Hazard

26. Second-best contracts
xRED b
Alf's low-effort ICs Ob
Bills ICs
xBLUE a
Alf's high-effort ICs
Bills ICs
Full-information contracts ua
Participation constraint
Incentive-compatibility constr.
Bill’s second-best feasible set
Second-best contract – w
Contract maximises Bill’s utility over second-best feasible set
xBLUE b Oa w –
xRED a
July 2015
Frank Cowell: Moral Hazard

27. Simplified model: summary
Participation constraint
set of contingent consumptions giving Alf his reservation utility
if effort is observable get one such constraint for each effort level
Incentive compatibility constraint
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
July 2015
Frank Cowell: Moral Hazard

28. Overview Extending the “first-order” approach Moral Hazard The basics
A simplified model
The general model
July 2015
Frank Cowell: Moral Hazard
Frank Cowell: Moral Hazard

29. General model: introduction
Retain assumption that it is a two-person contest
same essential roles for Principal and Agent
allow for greater range of choice for Agent
allow for different preferences for Principal
Again deal with full-information case first
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
July 2015
Frank Cowell: Moral Hazard

30. Model components: output and effort
Production depends on effort z and state of the world w:
q = f(z,w)
w  W
Effort can be anything from “zero” to “full”
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)
July 2015
Frank Cowell: Moral Hazard

31. Effect of effort f(q, z) q – q q –
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 –
July 2015
Frank Cowell: Moral Hazard

32. Model components: preferences
Again the Agent's utility derives from
the wage paid, w ()
the effort put in, z ()
E ua(w, z)
ua(·, ·) is strictly concave in its first argument
Principal consumes output after wage is paid
but we allow for non-neutral risk preference
E ub(xb) = E ub(q – w)
ub(·) is concave
July 2015
Frank Cowell: Moral Hazard

33. Full information: optimisation
Alf’s participation constraint:
E ua(w,z)  ua
Bill sets the wage schedule
can be conditioned on the realisation of w
w = w(w)
To set up the maximand, also use
Bill’s utility function ub
production function f
Problem is then
choose w(·)
to max E ub(f(z,w))
subject to E ua(w(w),z)  ua
Lagrangian is
E ub(f(z,w) – w(w)) + l[E ua(w(w), z) – ua]
July 2015
Frank Cowell: Moral Hazard

34. Optimisation: outcomes
The Lagrangian is
E ub(xb) + l[E ua(xa,z) – ua]
where xa = w(w) ; xb = f(z,w) – w(w)
Each w(w) and z can be treated as control variables
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
E uxb(f(z,w) – w(w))fz(z,w) + lE uza(w(w), z) = 0
Combining we get
uxb(xb) / uxa(xa) = l
 uza(xa, z) 
E uxb(xb )fz(z,w) + E   uxb(xb) = 0
 uxa(xa, z) 
xa = w(w)
xb = f (z, w) – w(w)
July 2015
Frank Cowell: Moral Hazard

35. Full information: results
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) 
E uxb(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
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
July 2015
Frank Cowell: Moral Hazard

36. Full information: lessons
Principal fully exploits Agent
because Principal drives Agent down to reservation utility
follows from assumption that Principal has all the power
(no bargaining)
Efficient risk allocation
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
July 2015
Frank Cowell: Moral Hazard

37. Second-best: introduction
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
July 2015
Frank Cowell: Moral Hazard

38. Second-best: components
Objective function of Principal and of Agent are as before
Distribution of output f depends on effort z
probability density at output q is f(q, z)
Participation constraint for Agent still the same
modify it to allow for redefined distribution
Require also the incentive-compatibility constraint
builds on the (hidden) optimisation of effort by the Agent
Again use Lagrangian technique
assumes problem is “well-behaved”
this may not always be appropriate
July 2015
Frank Cowell: Moral Hazard

39. Second-best: problem Bill sets the wage schedule
cannot be conditioned on the realisation of w
but can be conditioned on observable output
w = w(q)
Bill knows that Alf must get at least “reservation utility” :
E ua(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) + E uza(w, z) = 0
this is the incentive-compatibility constraint
July 2015
Frank Cowell: Moral Hazard

40. Second-best: optimisation
Problem is then
choose w(·) to max E ub(q – w(q))
subject to E ua(w(q),z)  ua
and E (ua(w(q), z)bz) + E uza(w(q), z) = 0
Lagrangian is
E ub(q – w(q)) + l [E ua(w(q), z) – ua ]
+ m [E (ua(w(q), z)bz) + E uza(w(q), z) ]
l is the “price” on the participation constraint
m is the “price” on the incentive-compatibility constraint
Differentiate Lagrangian with respect to w(q)
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
July 2015
Frank Cowell: Moral Hazard

41. Second-best: FOCs Use a simplifying assumption: Lagrangian is
uxza(·,·) = 0
Lagrangian is
E ub(xb) + l[E ua(xa, z) – ua ] + m[ E (ua(xa, z)) / z ]
where
xa = w(q)
xb = q – w(q)
Differentiating with respect to w(q):
FOC1: – uxb(xb) + luxa(xa, z) + muxa(xa, z)bz = 0
Differentiating with respect to z:
FOC2: E ub(xb)bz+ m[ 2E (ua(xa, z)) / z2 ] = 0
July 2015
Frank Cowell: Moral Hazard

42. Second-best: results From FOC2: From FOC1: – Eub(xb)bz m = ———————
bz is +ve where xb is large
From FOC2:
– Eub(xb)bz
m = ———————
2E(ua(xa, z))/z2
m > 0
so the incentive-compatibility constraint is binding
From FOC1:
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
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)
2nd derivative is negative
July 2015
Frank Cowell: Moral Hazard

43. Principal-and-Agent: Summary
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)
incentive-compatibility constraint is also binding
Principal pays Agent more if output is high
Principal no longer insures Agent fully
July 2015
Frank Cowell: Moral Hazard