Axiomatic Utility Theory under Risk pp 123-161 | Cite as

# Application to Insurance Economics

## Abstract

In the course of this thesis, a bulk of empirical studies have been presenteded which were carried out in terms of laboratory experiments. Although these studies provide many insights into individual decision making under risk, the overall evidence is not unambiguous. A possible response to this fact is to consider field data. The assessment of non-experimental field data is, however, quite difficult because individual preferences are not directly observable. Thus, there is a need for producing testable implications of non-expected utility theories which differ from those of expected utility theory (EU). The goal of this third part is to derive such implications in the context of insurance economics.^{1}

## Keywords

Risk Aversion Expected Utility Risk Averse Indifference Curve Insurance Demand## Preview

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## Notes

- 1.Some of the results presented in sections 3.3 and 3.4 already appeared in Schmidt (1996).Google Scholar
- 2.For further applications of non-expected utility theory to insurance economics cf. Konrad and Skaperdas (1993), Doherty and Eeckhoudt (1995), Diewert (1995), Karni (1995), and Schlee (1995).Google Scholar
- 3.Although Fréchet-differentiability implies second-order risk aversion the converse is not true. For instance, weighted utility also exhibits second-order risk aversion [cf. Segal and Spivak (1990, p. 119)] but is not Fréchet-differentiable.Google Scholar
- 4.Cf. Loomes and Segal (1994), p. 239.Google Scholar
- 5.Cf. Epstein and Zin (1990).Google Scholar
- 6.Cf. section 1.3.3.Google Scholar
- 7.Cf. section 1.4.2.3.Google Scholar
- 8.For convenience we make a slight change in notation in the third part. From now on,
*p*denotes a single probability and not, as in parts 1 and 2, a probability measure. This allows us to write simply*p*instead of the more cumbersome*p*(*x*).Google Scholar - 9.
- 10.Cf. section 1.4.2.2.Google Scholar
- 11.Cf. section 1.4.1.4. It is assumed that preferences satisfy axiom SM.Google Scholar
- 12.Cf. Gul (1991, p. 675). To be exact, strict risk aversion is also consistent with
*β =*0. In this case, however, TDA reduces to EU.Google Scholar - 13.Note that this definition of
*g*is admissible since*g*(0) = 0,*g*(1) = 1, and*g*is strictly increasing and strictly concave.Google Scholar - 14.Cf. Mossin (1968, pp. 556-559), Smith (1968, pp. 70-71), and Ehrlich and Becker (1972, pp. 625-627).Google Scholar
- 15.See, e.g. Ehrlich and Becker (1972).Google Scholar
- 16.Cf. Chiang (1984, p. 728).Google Scholar
- 17.Cf. Fuchs (1976) and Camerer (1989b).Google Scholar
- 18.This is because a smaller value of
*p*implies a longer interval [1 —*g*(*p*),*g*(*p*)]. See also Hogarth and Kunreuther (1989).Google Scholar - 19.Cf. Gui (1991, p. 676).Google Scholar
- 20.Cf. Hirshleifer and Riley (1979, p. 1386).Google Scholar
- 21.For comprehensive surveys of these problems, cf. Winter (1992) and Dionne and Doherty (1992), respectively.Google Scholar
- 22.Cf. Stiglitz (1983, p. 4).Google Scholar
- 23.This form of contract is generally called “forcing contract”. Cf. Rees (1987, p. 49).Google Scholar
- 24.Cf. section 1.3.3.Google Scholar
- 25.This assumption, in conjunction with
*p*′(*a*) < 0, allows us to use the first-order approach in the proof of Propositions 3.6–3.8. Cf. Rogerson (1985, pp. 1361-1362). Alternatively, one could also assume that the agent’s choice of*a*is unique.Google Scholar - 26.Note that, according to assumption (i), we have
*u*′_{A}> 0 and*u*′_{B}> 0, which implies λ > 0.Google Scholar - 27.Cf. Holmstrm (1979, p. 77).Google Scholar
- 28.Cf. section 1.4.3.1.Google Scholar
- 29.For a discussion of this assumption see Shavell (1979a, p. 60, note 14).Google Scholar
- 30.The argument is taken from Shavell (1979, pp. 59-60).Google Scholar
- 31.Cf. section 3.1.Google Scholar
- 32.
- 33.In order to derive this result, one must also assume that “the cost of taking care” is not too high.Google Scholar
- 34.For analysis in the expected utility framework cf. Spremann (1987, pp. 30-35).Google Scholar