Abstract
Throughout this chapter, the economic agent is assumed to dispose of two instruments of risk management only, viz. purchasing insurance coverage or exerting preventive effort. The possibility of coping with uncertainty through a diversification of assets is therefore neglected. This alternative is available to enterprises and their owners and will be treated in Chap. 4. Section 3.1 refers to the risk utility function derived in Sect. 2.2.2 and presents the expected utility maximization hypothesis which is used to resolve the decision making problem under uncertainty. With this, the groundwork is laid for developing the basic model of insurance demand in Sect. 3.2. It predicts the choice of full coverage if the potential purchaser of insurance is charged the so-called fair premium, i.e. a premium that just covers the expected value of the loss insured.
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Notes
- 1.
In fact, preventive effort constitutes an action influencing probabilities of occurrence, see Sect. 3.5.
- 2.
See method No. 1 for constructing the risk utility function in Sect. 2.2.2.3.
- 3.
The notation with brackets means that a function is evaluated at a particular value of the argument, e.g. the risk utility function υ(c) at a particular value c = c ij .
- 4.
As to the curvature, the indifference curve is convex from the origin. It can be shown that (strict) convexity follows from (strict) concavity of the risk utility function υ(W). Therefore, the convexity of the indifference curve reflects risk aversion (specifically, R A ). For a proof, see e.g. Eisen (1979, 44) or Zweifel et al. (2009, Ch. 5).
- 5.
In conventional microeconomics, the precise slope of the indifference curve is not known. The additional information available here is due to the fact that the EU-function (3.8) is additive while conventional utility functions u(W 1, W 2) can be of any form.
- 6.
W s ∗ = W u ∗ in Fig. 3.3 requires that the ransom (i.e. the horizontal distance between the two risk utility functions) be compensated.
- 7.
In the jargon of insurance, the fair premium is often called “pure premium” or “risk premium”, in contradistinction to the definition in Sect. 2.3.2.
- 8.
Doherty (1985, 451) in addition distinguishes the franchise, where the insurer pays the full indemnity without deduction if it exceeds the deductible.
- 9.
The relation between the elasticity of expenditure on insurance P = pW I w.r.t. wealth e(P, W) and the income elasticity e(P, Y ) can be established as follows. By expansion, one obtains \(e(P,Y ) = \frac{\partial P} {\partial Y } \cdot \frac{Y } {P} = \frac{\partial P} {\partial W} \cdot \frac{\partial W} {\partial Y } \cdot \frac{W} {P} \cdot \frac{Y } {W} = e(P,W) \cdot e(W,Y )\). If outlay on insurance as a share of wealth is to be constant, it must be true that e(P, W) = 1. On the other hand, the fact that the concentration of wealth exceeds that of income implies e(W, Y ) > 1. In combination, these two statements result in e(P, Y ) > 1.
- 10.
This is the so-called zero-utility principle of premium calculation (see Sect. 6.1.3).
- 11.
Conditional and nonconditional probabilities are related as follows. According to the Bayes theorem, the conditional probability is given by \({\pi }_{N\vert L} = {\pi }_{N,L}/{\pi }_{L}\). Solving for π N, L , one obtains \({\pi }_{N,L} = {\pi }_{N/L} \cdot {\pi }_{L}\).
- 12.
The formula for a conditional probability reads, \({\pi }_{N\vert L} = {\pi }_{N,L}/{\pi }_{L}\) implying \({\pi }_{N,L} = {\pi }_{N\vert L}{\pi }_{L} = {\pi }_{L\vert N}{\pi }_{N}\). Substitution yields \({\pi }_{N\vert L} = ({\pi }_{L\vert N}/{\pi }_{L}) \cdot {\pi }_{N}\). Therefore, π N | L > π N if π L | N > π L , i.e. L occurs with greater probability if N happens as well. However, this is the consequence of L and N being positively correlated.
- 13.
If one simplifies by setting average and marginal cost of prevention equal to 1 [such that \(C(V ) = V = 1\)], then the optimum condition (3.52) becomes \((1 - \pi )\upsilon \prime[2]/\pi \upsilon \prime[1] = L\prime[{V }^{{_\ast}}] + 1\). This condition is equivalent to maximizing expected utility if the marginal utility both of wealth and the marginal productivity of measures designed to reduce loss are decreasing [see [see Ehrlich and Becker (1972, 634)]. In the present context and referring to Fig. 3.7, indifference curves must be convex and the transformation curve TN concave to the origin.
- 14.
The term “anomaly” is due to Kuhn (1962) who defines it as “a phenomenon that researchers are not prepared to encounter given the prevalent paradigm or in other words, it is the recognition that nature (in the natural sciences) has failed to satisfy the expectations generated by the paradigm” (p. 65f). In the present context, the Bernoulli principle constitutes a paradigm that has the main merit of “generating a precision of information and interaction between observation and theory that cannot be attained otherwise” [see Kuhn (1962)].
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Zweifel, P., Eisen, R. (2012). Insurance Demand I: Decisions Under Risk Without Diversification Possibilities. In: Insurance Economics. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20548-4_3
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DOI: https://doi.org/10.1007/978-3-642-20548-4_3
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