1. Moral Hazard and State Dependent Utility with Loss Reduction
2014, July
Moral Hazard and State Dependent Utility with Loss Reduction
Jimin Hong & S. Hun Seog
Business School, Seoul National University

2. Table of Contents ► Introduction - Motivation ► Literature Review
► Model Description: State Dependent Utility & Loss Reduction
► Basic Model : without Moral Hazard
► The Model with Moral Hazard
► Two Periods Model – Extension
► Applications
► Conclusion

3. I. Introduction Moral Hazard with Loss Reduction
►Definition : Principal cannot observe the agent’s actions, so agent would have an incentive to under-invest
in order to reduce loss
State Dependent Utility
►If the loss occurs, then the utility function itself changes.
That is, that the monetary compensation cannot substitute the
loss perfectly.

4. I. Introduction Motivation ► Focus on the loss reduction effort
∙ The probability of natural disaster outbreaks such as typhoon,
earthquake and hurricane cannot be controlled by human effort.
∙ People can only care to minimize the loss size.
► Health insurance problem
∙ Ellis & Manning (2007), Barigozzi (2004) argue that loss reduction is
a combination of treatment & prevention

5. II. Literature Review State Independent Utility Shavell (1979)
- The optimal insurance contract should contain a deductible
under moral hazard
Ehrlich and Becker (1972) & Barigozzi (2004)
- The effort of the policyholder’s can be divided into
“Loss Prevention(Primary prevention)” and “Loss
Reduction”(Secondary prevention).
Winter (2000)
- The case of loss prevention (self-protection) and moral hazard
: the optimal form of insurance is full insurance above a
deductible
- The loss reduction and moral hazard case
: the optimal insurance involves full coverage up to a limit and
partial insurance above the limit

6. II. Literature Review State Dependent Utility Cook and Graham (1977)
- The loss of irreplaceable commodity decreases the
policyholders’ extra utility along with the monetary loss
- The marginal utility may also be changed depending on the
state whether an accident occurs or not.
Shioshansi (1982),Schlesinger (1984)
- The model for optimal insurance contract for irreplaceable
commodities without moral hazard
Huang and Tzeng(2006)
- The optimal insurance contract includes deductible and
coinsurance above the deductible endogenously.
Dionne (1982)
- focused on the loss prevention effort under moral hazard
using state dependent utility.

7. III. Model Description Assumptions one period & two states model
- S: state variable
- the loss state (S=1) and the no loss state (S=0)
utility function of the policyholder :
-twice differentiable and strictly concave
W : Income, Q: Premium, I(x): Indemnity
The policyholders’ cost function for the effort is
&

8. III. Model Description Assumptions
homogeneous policyholders – no adverse selection
competitive insurance market
- zero profit condition for insurer
the probability of loss occurrence : p
- p is not controlled by effort
loss size: x & effort : e, x~f(x;e)
Monotonic Likelihood Ratio Condition(Milgrom, 1981)
-This condition is that given is decreasing in x.

9. IV. Basic Model : without Moral Hazard
Benchmark Case-state independent utility
Suppose that the loss size x has the density function f(x).
For simplicity, use subscript S & N where S: loss, N:no loss 
FOC:
⇒ State independent case: full insurance is optimal
Optimal amount of effort
⇒ MB(effort)=MC(effort))

10. IV. Basic Model : without Moral Hazard
State-dependent utility uN uN A w w w

11. IV. Basic Model : without Moral Hazard
State-dependent utility
Lemma 1. (No moral hazard case) Under a state dependent utility, the optimal coverage is
(1) full insurance when
(2) full insurance if over-insurance is not allowed, and over-insurance is optimal if possible when
partial insurance when
Lemma 2. (No moral hazard case) Under a state dependent utility, the optimal effort
(1)is equivalent to a state independent utility case when
(2)is greater than that of state independent utility case when
(3)is less than that of state independent utility case when

12. V. Model : with Moral Hazard
Insureds maximize utility under state dependent utility
Lagrangian is

13. V. Model : with Moral Hazard
Rearranging the first order condition, we have
Lemma 3. Under a state dependent utility,
- Lemma 3 is consistent with Holstromm’s result
- The insurer also designs the contract to induce more effort
from the insured under a state dependent utility.

14. V. Model : with Moral Hazard
Proposition 1.
(Moral hazard case) Under a state dependent utility, the optimal coverage

15. V. Model : with Moral Hazard
Proposition 1.
Winter remarks (Moral Hazard & State Independent Utility)
“the optimal coverage involves the full coverage of small losses and partial insurance for high losses”
If under state dependent utility, we obtain the same result as Winter’s
The utility with no loss is equal to the expected utility with loss because the penalty is only given to the loss size regardless of loss occurrence.
- Insurer cannot infer the effort level exactly & the effort
affects the loss size  insurer penalizes insured along the
loss size to induce the effort as much as possible.

16. V. Model : with Moral Hazard
Proposition 1.
If the state dependency is larger then full insurance is possible when
- policyholders need more indemnity than the state
independent case to smooth the consumption between two
states.
When, and state dependency is sufficiently larger, optimal insurance is a deductible up to a limit and coinsurance above the limit
- Policyholders may require lower indemnity than the case with state
independent utility.

17. VI. Effort Comparison Proposition 2.
(Moral hazard case) Under a state dependent utility, if the coverage is equal to the coverage with a state independent utility for all coverage level, then the optimal effort

18. VI. Effort Comparison Proposition 2.
-the moral hazard problem is still significant problem under the state dependent utility with loss reduction
- the concern about the moral hazard problem might be excessive or should be higher depending on the marginal utility of income.
- The effort amount in proposition 2 is less than the lemma 2.

19. VII. Two periods Model - Extension
additional assumptions
- state : St , effort: et t=1 or 2
- indemnity:
- joint density:
Model

20. VII. Two periods Model - Extension
Model-continued

21. VII. Two periods Model - Extension
Model-continued

22. VII. Two periods Model - Extension
Lemma 4

23. VII. Two periods Model - Extension
Lemma 4
Proposition 3
Proposition 3. (Moral hazard in Two Period Case) Under a state dependent utility, if the coverage of the second period when loss doesn’t occur in the first period is equal to the coverage with a loss in the first period for all coverage level, then the optimal amount of an effort at time 2 with loss at time 1
- state : St , effort: et t=1 or 2
- indemnity:
- joint density:

24. VIII. Applications Debt Contract
enterprise with two states denoted by S
(S = 1) : the profit is lower than the debt with probability p
financial distress cost occurs
(S = 0) : the profit is higher than the debt with probability 1-p
In a low profit state, the profit π given effort e follows the distribution f(π,e)
the profit follows the distribution g(π,e) in a high profit state.
Debt principal : I, interest rate: r
C(e): cost of effort to increase the profit
V(π): financial distress cost & convex function
v(π, S=0)= v(π, S=1)- A(π), A(π)>0
-B(π): debt repayment schedule, nondecreasing.

25. VIII. Applications
Debt Contract

26. VIII. Applications
Result 1

27. VIII. Applications Wage Contract
An employee with two states denoted by S
(S = 1) : the profit is lower than the limit value π* with probability p
(S = 0) : the profit is higher than the limit value π* with probability 1-p
Salary: S(π), Reservation utility :
In a low profit state, the profit π given effort e follows the distribution f(π,e)
the profit follows the distribution g(π,e) in a high profit state.
Suppose that
:sentiment value (reputation…)

28. VIII. Applications
Wage Contract

29. VIII. Applications
Result 2

30. VIII. Conclusions
According to comparison with marginal utility of income between loss state and no loss state, the optimal insurance coverage and effort are changed.
-If marginal utility of income in the loss state is larger than that of income in the no loss state, then the optimal insurance includes full insurance.
-If the relation between the marginal utilities is reverse, then optimal insurance includes a deductible up to a limit and coinsurance above the limit. 30

31. VIII. Conclusions
If the indemnity level is equal to the indemnity under a state independent utility, the effort is higher than that of a state independent utility when marginal utility of income in the loss state is greater than that of income in the no loss state.
-then the moral hazard problem is less severe for all indemnity level.
The moral hazard is more severe when marginal utility of income in the loss state is lower than that of income in the no loss state. 31

32. Thank you! 32