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2021 | OriginalPaper | Buchkapitel

3. Insurance Demand I: Decisions Under Risk Without Diversification Possibilities

verfasst von : Peter Zweifel, Roland Eisen, David L. Eckles

Erschienen in: Insurance Economics

Verlag: Springer International Publishing

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Abstract

Throughout this chapter, economic agents are assumed to have at their disposal two instruments of risk management only, viz., purchasing insurance coverage or exerting preventive effort. The possibility of coping with uncertainty through diversification of assets is, therefore, neglected.

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Vorheriges Kapitel Risk: Measurement, Perception, and Management

Nächstes Kapitel Insurance Demand II: Nontraditional Approaches to Decisions Under Risk

We use IB throughout the text to represent the consumer of insurance. The consumer is also often referred to as the policyholder.

In fact, preventive effort constitutes an action influencing probabilities of occurrence; see Sect. 3.5.

See method No. 1 for constructing the risk-utility function in Sect. 2.2.2.3.

The notation with brackets means that a function is evaluated at a particular value of the argument, e.g., the risk-utility function \(\upsilon (c)\) at a particular value \(c_{ij}\).

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 \(\upsilon (W)\). Therefore, the convexity of the indifference curve reflects risk aversion (specifically, \(R_{A}\)). For a proof, see, e.g., Eisen (1979b, 44) or Zweifel et al. (2009, Chap. 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.

\(W_s^*=W_u^*\) in Fig. 3.3 requires that the ransom (i.e., the horizontal distance between the two risk-utility functions) be compensated.

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.

Doherty (1985, 451) in addition distinguishes the franchise, where the insurer pays the full indemnity without deduction if it exceeds the deductible.

The relation between the elasticity of expenditure on insurance \(P=pW_{I}\) w.r.t. wealth, denoted by 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 \(e(P,Y)>1\).

This is the so-called zero-utility principle of premium calculation (see Sect. 7.1.3).

Conditional and nonconditional probabilities are related as follows. According to the Bayes theorem, the conditional probability is given by \(\pi _{N|L}=\pi _{N,L}/\pi _L\). Solving for \(\pi _{N,L}\), one obtains \(\pi _{N,L}=\pi _{N/L}\cdot \pi _L\).

The formula for a conditional probability reads, \(\pi _{N|L}=\pi _{N,L}/\pi _L\) implying \(\pi _{N,L}=\pi _{N|L}\pi _L=\pi _{L|N}\pi _N\). Substitution yields \(\pi _{N|L}=(\pi _{L|N}/\pi _L)\cdot \pi _N\). Therefore, \(\pi _{N|L}>\pi _N\) if \(\pi _{L|N}>\pi _L\), i.e., L occurs with greater probability if N happens as well. However, this is the consequence of L and N being positively correlated.

If one simplifies by setting average and marginal cost of prevention equal to 1 [such that \(C'(V) = C(V) = 1\)], then the optimum condition (3.53) becomes \((1-\pi )\upsilon '[2]/\pi \upsilon '[1]=L'[V^*]+1\). This condition is equivalent to maximizing expected utility if both the marginal utility of wealth and the marginal productivity of measures designed to reduce loss are decreasing [see Ehrlich (1972, 634)]. In the present context and referring to Fig. 3.7, indifference curves must be convex from, and the transformation curve TN concave to, the origin.

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Titel: Insurance Demand I: Decisions Under Risk Without Diversification Possibilities
verfasst von: Peter Zweifel
Roland Eisen
David L. Eckles
Verlag: Springer International Publishing
Buch: Insurance Economics
Print ISBN: 978-3-030-80389-6

Electronic ISBN: 978-3-030-80390-2

Copyright-Jahr: 2021
DOI: https://doi.org/10.1007/978-3-030-80390-2_3