Academic journal article Journal of Risk and Insurance

Second-Degree Stochastic Dominance Decisions and Random Initial Wealth with Applications to the Economics of Insurance

Academic journal article Journal of Risk and Insurance

Second-Degree Stochastic Dominance Decisions and Random Initial Wealth with Applications to the Economics of Insurance

Article excerpt

Introduction

A weakness of many economic analyses relying on a second-degree stochastic dominance ordering is that the variable which appears as an argument of the utility function is typically not total wealth, as should be, but rather some concept of partial wealth. The proposition, for example, that a risk-averse agent, at constant mean, prefers to have a small rather than a large insurance deductible on the basis of the distribution of his or her insurable wealth is always vulnerable to the attack that it is the distribution of total wealth and not only insurable wealth which matters to the agent. It is not difficult to concoct an example where the result no longer holds if it is the distribution of total wealth which is assessed. Many other economic applications could be given. Incorporating random initial wealth in decision making under uncertainty is a major concern of a number of recent papers, and this article contributes to this growing literature.

In this context, it is of interest to identify as undemanding as possible sufficient conditions for decisions taken under the second-degree dominance rule applied to partial wealth to be invariant to the remaining wealth of the agent, called here initial wealth. The joint distribution of the partial wealth under consideration and the initial wealth is, of course, the key. It is known that if random prospect Y stochastically dominates random prospect X by the second degree, then all risk-averse agents will prefer to hold Y and some initial wealth Z rather than X and the same Z as long as the latter is statistically independent of both X and Y (Hadar and Russell, 1971, and Levy and Sarnat, 1971; see also Levy and Kroll, 1978, for extensions). This article provides more general conditions which include independence as a particular case for the above invariance of the stochastic dominance ordering to initial wealth to hold. Some restrictions are needed, which are particularly natural in an insurance context: essentially that there be no moral hazard and that the insurance contracts preclude overinsurance.

The next section provides the main result: the sufficient conditions are that X be more positively dependent on Z than Y is, in a well defined sense, and that either (X,Z) or (Y,Z) be positively dependent in a well defined sense. Then, we discuss the concept of dependence used here and show that, of the many possible such concepts, it is the most suitable one. Then, we apply the main results to the choice of an insurance deductible and to the choice of insurance coverage by a risk-averse agent in a situation where fair insurance is available. The conclusion relates our results to the recent economic literature on this subject and to the germane subject of characterizing the risk premia when the insurance is partial and/or initial wealth is random.

Main Results

Let in general X and Y be discrete random variables taking values x = ([x.sub.1],..., [x.sub.n]) and y = ([y.sub.1],..., [y.sub.n]) with a common set of probabilities [Mathematical Expression Omitted], pertaining to a set of relevant states of nature 1,..., n. Assume furthermore that x and y are similarly ordered, that is, ([x.sub.i] - [x.sub.i-1])([y.sub.i] - [y.sub.i-1]) [greater than or equal to] 0; i = 2,..., n. We assume without loss of generality x and y to be increasingly ordered; formally,

x, y [element of][D.sup.n] with [D.sup.n] = {w [element of] [R.sup.n] [where] [w.sub.1] [less than or equal to] ... [w.sub.n]}. Let [S.sub.n[Pi]] be the set of discrete random variables taking values [w.sub.1] [less than or equal to] ... [less than or equal to] [w.sub.n] with prescribed probabilities [Pi]. We thus assume that X,Y [element of] [S.sub.n[Pi]]. The economic interpretation of those restrictions is discussed below.

Let Z be a discrete random variable taking values z = ([z.sub.1],..., [z.sub.m]) with probabilities p = ([p.sub.1],..., [p.sub.m]). We assume without loss of generality that z [element of] [D. …

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