Income, risk aversion, and the demand for insurance.

• Author: Cleeton, David L.

1. Introduction

   The expected utility hypothesis predicts that, when the price of insurance is actuarially fair to the consumer, a risk-averse consumer will choose to fully insure against a potential loss. The only role that income can play in affecting the amount of insurance demanded at the actuarially fair price is to affect the size of the potential loss. This result is independent of the consumer's degree of risk aversion or how it varies with income. However, at a price of insurance above or below the actuarially fair level the consumer's degree of risk aversion and its relation to the consumer's income level must be considered if we are to usefully describe consumer behavior.(1)

   Mossin [10] showed, for a consumer with declining absolute risk aversion and at a price of insurance above the actuarially fair level but below the price at which no insurance will be purchased, the optimal amount of insurance demanded against a loss of a given size is inversely related to the consumer's income. Chesney and Louberge [2] noted that Mossin's result does not take into account the empirically plausible situation in which higher income consumers may have greater potential losses to insure against. They present results concerning the relationship between risk aversion and the level and composition of income (in terms of the proportion of income subject to loss) and the maximum premium the consumer would be willing to pay for full insurance coverage. Since their results are in terms of the maximum willingness to pay for full insurance, or the ratio of this amount to insurable income, the situation they analyze has direct relevance only to the special case where the consumer is faced with a choice of purchasing full insurance or no insurance. The case more likely to be empirically observed in insurance markets is one where the consumer can purchase varying amounts of insurance, including zero or full, at a given price. Essentially, we attack the same problem as Chesney and Louberge, i.e., a description of the consumer's insurance purchasing behavior when the amount of the potential loss is a function of the consumer's income level.(2) However, it is a useful task to recast the theoretical analysis in terms of the consumer's demand curve for insurance so that it is more amenable to empirical investigation and intuitively more understandable.

   That the consumer's expected utility function yields a downward sloping demand curve for net insurance, i.e., the insurance pay-out less the premium due in the state in which the loss occurs, has been shown by Smith [14] and Ehrlich and Becker [5].(3) Extending the state preference model developed by Ehrlich and Becker, we develop a model showing how the degree of risk aversion of the consumer, the specification of the loss, and the price of insurance interact with income to affect the net demand for insurance.(4) We show that a change in the consumer's income has two effects on the consumer's demand curve for insurance. When the size of the potential loss is a positive function of income, an increase in income causes an outward shift in the demand curve. However, except in the case where the consumer's risk preferences have the attribute of constant absolute risk aversion, the increase in income also causes a rotation of the demand curve. The rotation will be counterclockwise if the consumer has declining absolute risk aversion and clockwise if the consumer has increasing absolute risk aversion. All of these results hold obversely for a decrease in income. We are able to state the conditions determining the sign of the income effect in a form that is directly relevant to empirical investigations of consumers' risk preferences and demands for insurable assets. We also show that correct specification of a demand function for insurance must contain a price-income interaction term. Indeed there may not be independent price or income effects, depending on the specific form of the utility function, but there must always be a price-income interaction term in the demand function except when the consumer has constant absolute risk aversion. We also include a section which clearly summarizes the relationships among the demand, income, and loss elasticities. The Mossin and Chesney and Louberge results can be shown to be subsumed in our more general analysis.

2. The Model

   Using the Ehrlich-becker framework for analyzing the demand for insurance, we consider the two state decision problem faced by a risk-averse consumer. We derive a number of comparative static results. We then use these results to analyze the effects of a change in income on the demand for insurance and indicate how the results depend on the way in which a change in income affects the consumer's degree of risk aversion and the size of the potential loss. We assume the utility function is state independent and exhibits risk aversion. We formulate the problem in terms of the choice of insurance coverage on both a gross and net basis. While the gross coverage problem may be more familiar to the reader, the net coverage problem is more tractable analytically and will be the focus of our discussion. The basic outline of the model follows the setup found in Ehrlich and Becker [5]. The consumer maximizes expected utility (EU) through the choice of the level of net (gross) insurance coverage, s (y) or:

   MAXEU = (1 - p)U([I.sub.1]) + pU([I.sub.0) (1)

   s

   [I.sub.1] = [I.sup.E] - [pi]s [I.sub.0] = [I.sup.E] - L + s (2) MAXEU = ( 1 - p)U([I.sub.1]) + pU([I.sub.0) (1') [gamma]

   [I.sub.1] = [I.sup.E] - [lambda][gamma] [I.sub.0] = [I.sup.E] - L + [gamma](1 - [lambda]). (2')

   The first and second-order conditions for the maximization problems are:

   dEU/ds = - (1 - p)[pi] [U'.sub.1] + p[U'.sub.0] = 0 [FOC] (3) [d.sup.2]EU/[ds.sup.2] = D = (1 - p) [pi.sup.2] [U".sub.1] + p[U".sub.0]

   (1 - p)[pi]/p = [U'.sub.0]/[U'.sub.1] (5) (1 - p)[lambda]/p(1 - [lambda]) = [U'.sub.0]/[U'.sub.1]. (5') With [pi] = [lambda]/(1 - [lambda]) and s = [gamma](1 - [lambda]), (5) and (5') are identical conditions. Inspecting (5), we see that if the consumer price were actuarially fair, [pi] = p/(1 - p) = [lambda]/(1 - [lambda]), the consumer would choose to insure up to the point where income is equalized between the two states, [I.sub.1] = [I.sub.0]. We will refer to this as...