Abstract
This chapter presents the basic theoretical model of insurance demand in a one-period expected-utility setting. Models of coinsurance and of deductible insurance are examined along with their comparative statics with respect to changes in wealth, prices and attitudes towards risk. The single risk model is then extended to account for multiple risks such as insolvency risk and background risk. It is shown how only a subset of the basic results of the single-risk model is robust enough to extend to models with multiple risks.
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Notes
- 1.
Technically an “indemnity” reimburses an individual for out-of-pocket losses. I will use this terminology to represent generically any payment from the insurance company. For some types of losses, most notably life insurance, the payment is not actually indemnifying out-of-pocket losses, but rather is a specified fixed payment.
- 2.
See Segal and Spivak (1990). Although extensions to the case where u is not everywhere differentiable are not difficult, they are not examined here. See Schlesinger (1997) for some basic results.
- 3.
Obviously real-world costs include more than just the indemnity itself, plus even competitive insurers earn a “normal return” on their risk. Thus, we do not really expect \(c\left [I(x)\right ] = 0\). That the zero-profit case is labeled “perfect competition” is likely due to the seminal article by Rothschild and Stiglitz (1976). We also note, however, that real-world markets allow for the insurer to invest premium income, which is omitted here. Thus, zero-costs might not be a bad approximation for our purpose of developing a simple model. The terminology “fair premium” is taken from the game-theory literature, since such a premium in return for the random payoff \(I(\tilde{x})\) represents a “fair bet” for the insurer.
- 4.
- 5.
If we wish to strengthen the claims in Proposition 1 to hold for every possible starting wealth level and every possible random loss distribution, then DARA, CARA and IARA would also be necessary.
- 6.
If the support of \(\tilde{x}\) is [0, L], it may be useful to define W ≡ W 0 + L. If the loss exposure is unchanged, an increase in W can be viewed as an increase in W 0. More realistically, an increase in W will consist of increases in both W 0 and L.
- 7.
A necessary and sufficient condition for insurance not to be Giffen is given by Briys et al. (1989).
- 8.
See the essay by Gollier (2013) in this Handbook for a detailed analysis of the optimality of deductibles.
- 9.
Meyer and Ormiston (1999) make a strong case for using \(E[I(\tilde{x})]\), although its often much simpler to use D as an inverse proxy for insurance demand.
- 10.
Leibniz rule states that \(\frac{\mathrm{d}} {\mathrm{d}t}\int \limits _{a(t)}^{b(t)}H(x,t)\mathrm{d}x = H(b,t)\) \({b}^{{\prime}}(t) - H(a,t){a}^{{\prime}}(t) +\int \limits _{ a(t)}^{b(t)}\frac{\partial H} {\partial t} \mathrm{d}x.\)
- 11.
- 12.
The problematic issue deals with differentiability of the objective function. The details can be found in Schlesinger (2006).
- 13.
- 14.
Although not modeled in this manner, the possibility of a probationary period is examined by Eeckhoudt et al. (1988), who endogenize the length of probation.
- 15.
In a two-state (loss vs. no loss) model, there is no distinction between coinsurance and deductibles. A coinsurance rate α is identical to a deductible level of \(D = (1-\alpha )L\).
- 16.
Note that if there is no default risk with q = 1, then \({u}^{{\prime}}(Y _{1}) = {u}^{{\prime}}(Y _{2})\) implying that α ∗ = 1, as we already know from Mossin’s Theorem. Also, if insurance in default pays for most of the claim (as opposed to none of the claim), it is possible for full coverage or even more-than-full coverage to be optimal. See Doherty and Schlesinger (1990) and Mahul and Wright (2007).
- 17.
This is easiest to see by noting that − u ′ is an affine transformation of u.
- 18.
For CARA, v(y) = ku(y) and for quadratic utility \(v(y) = u(y) + c\), where \(k = E[\exp (r\tilde{\varepsilon })] > 0\), r denotes the level of risk aversion, and \(c = -t\mathrm{var}\left (\tilde{\varepsilon }\right )\) for some t > 0. Gollier and Schlesinger (2003) show that these are the only two forms of utility for which v represents preferences identical to u.
- 19.
- 20.
In stating that two random variables are equal, we mean that they each yield the same value in every state of nature, not simply that they have equal distributions.
- 21.
- 22.
Losses \(\tilde{x}\) are positively expectation dependent on \(\tilde{\varepsilon }\) if \(E(\tilde{x}\vert \tilde{\varepsilon } \leq k) \leq E(\tilde{x})\,\,\,\forall k\). In a certain sense, a smaller value of \(\tilde{\varepsilon }\) implies that expected losses will be smaller. Negative expectation dependence simply reverses the second inequality in the definition. It should be noted that Hong et al. (2011) do not consider the case of a positive premium loading. Thus, their theorem only extends one part of Mossin’s Theorem. See also the article by Dana and Scarsini (2007), which uses similar dependence structures to examine the optimal contractual form of insurance.
- 23.
To the best of my knowledge, Tibiletti (1995) also introduces the use of copulas into insurance models. Copulas allow one to describe the joint distribution of \((\tilde{\varepsilon },\tilde{x})\) as a joint distribution function of the marginal distributions of \(\tilde{\varepsilon }\) and \(\tilde{x}\), which is a type of normalization procedure. This allows one to both simplify and generalize the relationship between H and G. The use of particular functional forms for the copula allows one to parameterize the degree of statistical association between \(\tilde{x}\) and\(\tilde{\varepsilon }\). See Frees and Valdez (1998) for a survey of the use of copulas.
- 24.
The fact that detrimental changes in the background risk \(\tilde{\varepsilon }\) do not necessarily lead to higher insurance purchases under simple risk aversion is examined by Eeckhoudt et al. (1996), for the case where the deterioration can be measured by first- or second-degree stochastic dominance. Keenan et al. (2008) extend the analysis to consider deteriorations via background risks that either reduce expected utility or increase expected marginal utility.
- 25.
This follows easily using Jensen’s inequality, since marginal utility is convex under prudence.
- 26.
- 27.
Actually, this result depends on the differentiability of the von Neumann–Morgenstern utility function. “Kinks” in the utility function can lead to violations of Mossin’s result. See, for example, Eeckhoudt et al. (1997).
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Schlesinger, H. (2013). The Theory of Insurance Demand. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_7
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