Abstract
This chapter surveys the theory of optimal insurance contracts under moral hazard. Moral hazard leads to insurance contracts that offer less than full coverage of losses. What form does the optimal insurance contract take in sharing risk between the insurer and the individual: a deductible or coinsurance of some kind? What are the factors that influence the design of the contract? Posed in the most general way, the problem is identical to the hidden-action principal–agent problem. The insurance context provides structure that allows more specific implications for contract design. This chapter reviews the static models of optimal insurance under ex ante and ex post moral hazard as well as the implications of repeated contracting.
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Notes
- 1.
In an ideal marriage, costs imposed on the spouse are internalized in an individual’s own utility function. Love solves the moral hazard problem.
- 2.
It has been suggested that the etymology of the term moral hazard may involve a second historical use of the term moral (Dembe and Boden 2000). Daniel Bernoulli (1738) in his resolution of the St. Petersburg paradox first posed by Nicholas Bernoulli in 1714 applied a theory that he referred to as “the theory of moral value.” Moral referred to the subjective or psychological value placed on the gain in an individual’s wealth. The moral expectation was distinguished from the mathematical expectation. Today we call Bernoulli’s moral value the Bernoulli utility or Von-Neumann Morgenstern utility of wealth. Bernoulli’s use of the term “moral”as meaning subjective or personal value is consistent with the usage of moral in the eighteenth and nineteenth centuries as meaning in accordance with the customers or norms of human conduct, rather than ethical as in current English usage. As Dembe and Boden note at p. 261, “The classical eighteenth-century mathematical analysis of subjective utility in risk-bearing situations can thus be considered as essentially value-neutral, despite being couched in the language of moral values and expectations.”It is tempting for an economist, who considers maximizing behavior under incomplete contracts simply to be rational behavior, to trace the use of the term moral hazard in economics to the essentially value-neutral language of moral expectation in the eighteenth century. Dembe and Boden place considerable weight on this possibility, although Arrow (1963) is quite explicit in citing the prior insurance literature, in introducing the moral hazard terminology to the economics literature.
- 3.
Models of insurance markets with asymmetric information are reviewed by Georges Dionne, Neil Doherty, and Nathalie Fombaron in a chapter in this handbook.
- 4.
Ma and Riordan (2002) adopt a general assumption on preferences that accommodates both non-pecuniary and pecuniary costs of effort.
- 5.
The assumptions of concavity of the distribution function F and the MLRP for the random loss x correspond to the assumptions in a conventional principal–agent model of the convexity of the distribution function and an MLRP with the opposite inequality. Here, x is a loss; in the conventional principal–agent problem, x is profit.
- 6.
Jewitt (1988) provides sufficient conditions for the first-order approach beyond the restrictive convexity condition and including the observation by the principal of multiple relevant statistics.
- 7.
This decomposition parallels (7) in Shavell (1979a), which considers the simpler insurance problem with only one possible value for the loss if an accident does occur.
- 8.
To elaborate on the proof of this inequality, note that ∫ fdx = 1 ⇒ \(\int f_{a}\mathrm{d}x = 0 \Rightarrow E(f_{a}/f) =\int (f_{a}/f)f\mathrm{d}x = 0\) so that \(\int xf_{a}\mathrm{d}x =\int x(f_{a}/f)f\mathrm{d}x = E[x {\ast} (f_{a}/f)] = \mathrm{cov}(x,f_{a}/f) <0\), by MLRP. Turning to the second term, \(\widetilde{{a}}^{{\prime}}> 0\) since less coverage leads to more effort. The third term is positive since u ′ is positive.
- 9.
This type of distribution can easily result from an exogenous uncertainty that has a bimodal distribution.
- 10.
The same set of assumptions can be used to justify the first-order approach in Shavell (1979a). Shavell adopts an assumption that the costs of care are pecuniary (i.e., a reduction in wealth) rather than purely non-pecuniary, as we assume here.
- 11.
An alternative theory supporting the optimality of deductibles in insurance contracts is costly state verification (Townsend 1979 and Gale and Hellwig 1985). This is a theory that endogenizes the extent of asymmetry in information, rather than taking it as given as in the basic principal–agent approach. Insurers cannot always costlessly observe the loss that an individual has incurred. If the loss (the “state”) can be verified only at a cost, then the optimal insurance policy will call for coverage only when the claimed loss exceeds a specific level. In other words, a deductible is optimal when the state can be verified only at a cost. The theory involves essentially a reinterpretation of the Townsend and Gale–Hellwig corporate-finance models in terms of insurance contracts.
- 12.
The standard insurance policy against baggage delay (as of 2012) allows the insured individual to claim up to 500 dollars to purchase clothing if baggages are delayed by more than 4 hours.
- 13.
With the substitution of the agent’s first-order condition for the ICC, the Lagrangian is
$$\displaystyle\begin{array}{rcl} \mathcal{L}& =& [1 - (p - a)]u(w - (p - a)I) {}\\ & & \quad + (p - a)E_{x}u(w - (p - a)I - x + I) {}\\ & & \quad - v(a) +\lambda [u(w - (p - a)I) {}\\ & & \quad - E_{x}u(w - (p - a)I - x + I) - {v}^{{\prime}}(a)] {}\\ \end{array}$$ - 14.
An alternative model is offered in Blomqvist (1977).
- 15.
This is a close approximation to the author’s preferences.
- 16.
The full model in Ma and Riordan allows for a pecuniary loss al as well as the utility loss.
- 17.
As Arrow (1963, p. 960) explained:
By certifying to the necessity of given treatment or the lack thereof, the physician acts as a controlling agent on behalf of the insurance companies. Needless to say, it is a far from perfect check; the physicians themselves are not under any control and it may be convenient for them or pleasing to their patients to prescribe more expensive medication,private nurses, more frequent treatments, and other marginal variations of care.
- 18.
For brevity, I omit the detailed development of the model and the proof of this statement. The intuition is clear: the optimality conditions for a long-term contract can be reduced to two sets of conditions: (1) a Borch condition on the optimal smoothing of consumption across states in each time period and (2) an optimal smoothing condition over time on the realized consumption in period 1 and the conditionally expected consumption in period 2. A sequence of short-run insurance contracts meets the first condition. The second condition is met by the individual’s optimal savings decision when the individual faces the same interest rate as the insurer.
- 19.
Note that the tax-shelter aspects of life insurance are somewhat constrained, at least in the USA, to prevent the avoidance of inheritance tax. In flexible-premium policies, large deposits of premium could cause the contract to be considered a “modified endowment contract” by the Internal Revenue Service (IRS), which would involve a tax liability, negating many of the tax advantages associated with life insurance.
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Acknowledgements
I am grateful to Kairong Xiao for excellent research assistance, to two reviewers for very helpful comments and to the Social Sciences and Humanities Research Council of Canada for financial support.
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Winter, R.A. (2013). Optimal Insurance Contracts Under Moral Hazard. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_9
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