# Income, risk aversion, and the demand for insurance.

I. 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.

II. 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

[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:

[d.sup.2]EU/d[gamma.sup.2] = (1 - p)[lambda.sup.2][U".sub.1] + [p(1 - [lambda]).sup.2][U".sup.0]

<0 [SOC] (4') where the subscripts 0 and 1 refer to the income levels in states 0 and 1 at which the first and second derivatives are evaluated. The assumption of global risk aversion insures the SOC is satisfied. Rewriting the FOC, we obtain the usual marginal conditions:

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 full insurance. Here [gamma.sup.*] = L, optimal gross insurance coverage ([gamma.sup.*]) is equal to the potential loss. Or we have the equivalent statement [s.sup.*] = L - [pi][s.sup.*], optimal net insurance coverage ([s.sup.*]) is equal to the potential loss less the premium on the net coverage. From this point on in our analysis, the demand for insurance will refer to the demand for net coverage.

As Ehrlich-Becker showed, the demand curve for net insurance is everywhere downward sloping as in Figure 1. The corresponding comparative statics result is:

[Mathematical Expression Omitted]

In general, [character no conversion][s.sup.*]/[character no conversion][pi] will not be constant with respect to [pi] or, as we will show, with respect to [I.sup.E.[(5)

The price at which the amount of insurance demanded would be zero ([pi.sup.0]), the price intercept of the demand curve, is defined implicitly in the FOC (equation (5)) when the right-hand side is evaluated at the initial income endowments:

[pi.sup.0] = p U'([I.sup.E] - L)/(1 - p)U'([I.sup.E). (7)

In order to evaluate the effect of a change in the income endowment ([I.sup.E] on the demand for insurance ([s.sup.*]), we need to note two things. First, an increase in income increases the initial endowments in both states by increasing [I.sup.E]. Secondly, we wish to have a general specification of how the potential loss (L) might be affected by a change in income. The loss function allows us to examine cases where the loss may depend on the value of an underlying asset owned by the consumer.(6) We allow the loss to be a function of the initial endowment of income, L = L([I.sup.E]), where 0 [less than or equal to] dL/d[I.sup.E] = [L.sub.I] [less than or equal to] 1. While the upper bound on [L.sub.I] may seem extreme, it will be instructive as a limiting case.

Using our loss function specification, the FOC (equation (3)) can be written as:

dEU/ds = 0 = - [pi](i - p)U'([I.sup.E] - [pi]s]) + pU'([I.sup.E] - L([I.sup.E] + s). (8)

The comparative-static result for the effect of a change in endowed income on the net demand for insurance is given by:

[Mathematical Expression Omitted]

When the price of insurance is actuarially fair, equation [9] states that the demand for insurance will increase with initial income ([I.sup.E]) if [L.sub.I] > 0. The increase in the demand for net insurance will be [ds.sup.*] = (1 - p)[L.sub.I]d[I.sup.E]. This simply confirms that, if an increase in income leads to an increase in the loss, the demand for insurance at an actuarially fair price will increase since the consumer fully insures against the loss at this price. This is independent of how the consumer's degree of risk aversion changes with income. When the price of insurance is not actuarially fair, and the consumer is not fully insured, the risk aversion measures do not obligingly cancel out.

The denominator of the right-hand side of equation (9) is negative, so (9) wifl assume the same sign as:

X = [R.sub.AI] + [R.sub.AO]([L.sub.I - 1) (10) where, as before, the subscripts 0 and 1 refer to the income in the respective states at which the measure of absolute risk aversion is evaluated and X is the numerator from [9] multiplied by - 1.

Differentiation of equation (7) with respect to [I.sub.E] and simplification results in:

[Mathematical Expression Omitted]

where the 0 and 1 subscripts here refer to the initial endowment points [I.sup.E] - L and [I.sup.E].

The sign of[11] will be the same as the term in brackets which we can write as:

RA([I.sup.E]) + ([L.sub.I] -1) [R.sub.A]([I.sup.E] - L) = [R.sub.AI] + ([L.sub.I] - 1)

[R.sub.AO]. (12)

This is the same expression as[10] evaluated at the price intercept point. When evaluated at this point, [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] > (<)0 implies [Character no Conversion][s.sup.*]/[Character no Conversion] [I.sup.E] > (<)0.

III. Results

* We first take the simplest case where the potential loss is independent of the initial income, [L.sub.I] = 0. At an actuarially fair price [R.sub.A1] = [R.sub.A1] and equation (10) states that a change in initial income will not change the amount of insurance demanded. This is a standard result that a riskaverse consumer will fully insure at an actuarially fair price and, since the loss is independent of the consumer's initial income, a change in initial income will leave the demand for insurance unchanged.

At the price [pi.sup.0], with [L.sub.I] = 0, equation[12] shows that the sign of [Character no Conversion] [pi.sup.0]/[Character no Conversion][I.sup.E] will be determined by the sign of RA I - RAO. Thus, the price intercept of the demand curve will increase with initial income under increasing absolute risk aversion (IARA). It will decrease under decreasing absolute risk aversion (DARA). Constant absolute risk aversion (CARA) would leave the price intercept unchanged as initial income varied. Thus, for example, with DARA, as [I.sup.E] increases, the demand curve for insurance rotates counterclockwise through the full insurance point (F) as is illustrated in Figure 2.(7) With IARA, the demand curve would rotate clockwise through the full insurance point as [I.sup.E] increased. With CARA, the demand curve would be unaffected by a change in [I.sup.E]. These results for the case where [L.sub.I] = 0 are summarized in Table II.

[TABULAR DATA OMITTED]

We now turn to the case where the potential loss is a positive function of the initial level of income, 0 < [L.sub.I] [less than or equal to] 1.(8) Let us begin by examining the limiting case where [L.sub.I] = 1. Equation (10) would simplify to: [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] has the same sign as X = [R.sub.A1], which is always positive. An increase in initial income would shift the demand curve for insurance outward at every price.(9)

Note again that this result does not depend on any assumption about the degree of risk aversion. This case provides an example where insurance can be said to be a non-inferior good, i.e., at all prices an increase in initial income leads to an increase in the demand for insurance regardless of the degree of consumer risk aversion. While the demand curve shifts outward regardless of the degree of risk aversion, the slope of the demand curve will be affected by the degree of risk aversion.

Rewriting equation (9) for the case where [L.sub.I] = 1 we obtain:

[Mathematical Expression Omitted]

At prices below the actuarially fair price, we know [I.sub.0] > [I.sub.1] and with DARA we would have [R.sub.A1] > [R.sub.AO]. Thus: if [pi] < p/(1 - p) then [R.sub.A0]/[R.sub.A1] < 1 and [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] > (1 - p). If [pi] > p/(1 -p) then [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] < (1 - p) and if [pi] = p /(1 - p) then [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] = (1 - p). As [I.sup.E] increases, the demand curve shifts outward by an amount which is increasingly larger the lower is the price. One can see, that under DARA and [L.sub.I] = 1, as [I.sup.E] increases, the demand curve shifts outward by (1 - p)[dI.sup.E] at the actuarially fair price and rotates counterclockwise about this new full insurance point as shown in Figure 3. With IARA, an increase in income shifts the demand curve out by the same amount at the actuarially fair price [[ds.sup.*] = (1 - p)[dI.sup.E] but the rotation is clockwise.

With DARA and 0 < [L.sub.1] < 1 the sign of [Character no Conversion][s.sup.*]/ [Character no Conversion][I.sup.E] depends on the specific values of [R.sub.A1], [R.sub.A0] and [L.sub.I]. Since [R.sub.A1] and [R.sub.A0] are both functions of [pi], the sign of [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] will depend on [pi] as well. One can think of the effect of an increase in initial income as causing the demand curve for insurance to rotate counterclockwise through a point which moves up the demand curve as [L.sub.I] increases. When [L.sub.I] equals zero, the rotation occurs at the full insurance point. As [L.sub.I] increases toward + 1, the point of rotation moves northwest along the demand curve and eventually the price intercept on the higher income demand curve increases above the price intercept on the lower income demand curve. Above the point of rotation [Character no Conversion][s.sup.*]/ [Character no Conversion][I.sup.E] < 0, at the point of rotation [Character no Conversion][s.sup.*]/[Character no Conversion][I.sub/E] = 0 and below the point of rotation [Character no Conversion][s.sup.*][Character no Conversion][I.sup.E] > 0. If we impose the assumptions of constant income elasticities of loss and risk aversion, it is possible to bring to bear some useful empirical evidence for the case where 0 < [L.sub.I] < 1. At the actuarially fair price, from equation (9), [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] = (1 - p)[L.sub.1]. Multiplication of both sides of this equation by [I.sup.E]/L, dividing both sides by (1 - p) and, since we are at the full-insurance point, the substitution of [s.sup.*] = (1 - p)L on the left-hand side obtains:

([ds.sup.*]s)([I.sup.E]/[dI.sup.E]) = [beta] = [eta]. (13)

That is, at the actuarially fair price, the income elasticity of demand for net insurance is equal to the income elasticity of the loss. To express the change in the price-intercept point due to a change in income in elasticity terms, we can multiply both sides of (12) by [I.sup.E]/[pi.sup.0] to obtain:

(d [pi.sup.0]/[pi.sup.0])/([dI.sup.E]/[I.sup.E]) = [element of] = [I.sup.E][(1 - [L.sub.I])

([U.sub.0"]/[U.sub.0"] - ([U.sub.1"]/[U.sub.1'] (14) where [element of] is the elasticity of [pi.sup.0] with respect to [I.sup.E].

Writing this in terms of the measures of absolute or relative risk aversion:

[element of] = [I.sup.E][[R.sub.A1] - (1- [L.sub.I][R.sub.A0]] = [R.sub.R1] -

[I.sup.E](1 - L.sub.I])

[R.sub.R0]/([I.sup.E] - L)] (15) where the subscripts 1 and 0 here refer to the initial endowments ([I.sup.E] and [I.sup.E] - L). We want to find the condition under which [Character no Conversion][pi.sup.0]/[Character no Conversion] [I.sup.E] > (<)0. These are the same as the conditions where [epsilon]> (<)0.

We can show; see the appendix:

[Mathematical Expression Omitted]

Where [phi.sub.a] is the arc elasticity of relative risk aversion with the percentage change in income measured as the potential loss divided by initial income and [eta] is the elasticity of the loss with respect to initial income, i.e., [phi.sub.a] = ([delta][R.sub.R]/L)([I.sub.0.sup.E]/[R.sub.RO]) and [eta] = (dL/[dI.sup.E])([I.sup.E]/L) = [L.sub.I]([I.sup.E]/L). [eta.sub.a] is the ratio of the percentage change in the measure of relative risk aversion when going from the initial endowment of income in the loss state to the initial endowment of income in the no-loss state divided by the percentage change in income between the loss and no-loss states.(10)

Three studies have produced estimates of the variation in relative risk aversion with respect to variation in wealth. Cohn, et al.,[4] concluded that there was declining relative risk aversion (fi,, < 0). Friend and Blume[7] found evidence of increasing relative risk aversion but concluded that "Corrections for various deficiencies of the theoretical model, the econometric analysis and the sample data would on balance tend to make the results closer to constant proportional risk aversion" ([eta.sub.a] = 0). Siegel and Hoban[13], using a total wealth measure that was more comprehensive than the earlier studies and with a sample that contained more variation in wealth, found evidence of increasing relative risk aversion (fi, > 0). A reasonable estimate for d), that can be derived from their study would be approximately 0.16.(11) Since Siegel and Hoban used a total wealth concept, and controlled for the effect of the ratio of marketable to non-marketable wealth, this estimate should, theoretically, apply to income as well as wealth.

Tentatively, let us conclude, for the sake of argument, that the weight of this evidence would support a value of [eta.sub.a] in the neighborhood of the range 0 to 0.16.(12) Thus, using the condition set out in equation[16], we should expect that those insurable losses whose income elasticity is greater than 1 could be expected to exhibit a positive income elasticity of demand for insurance at all insurance prices. For those insurable losses with an income elasticity less than 0. 84 we might expect that the sign of the income effect on the amount of insurance demanded would depend on the price of insurance at which it was measured.

IV. A Simple Example

In spite of its apparent usefulness, there are few published examples of econometric attempts at estimating the consumer's insurance demand function. Where such attempts have been published, most specifications of the insurance demand function have been on an ad hoc basis[11]. Because of the lack of empirical work on the demand for insurance, attempts at estimating the costs and benefits of specific government interventions in insurance markets, when they have been done, employ restrictive and untested assumptions in order to make the task manageable [6; 1].

One important application of our analysis of the relation between income, risk aversion and the demand for insurance is its implication for the econometric specification of an insurance demand function. The rotation of the demand curve in response to a change in income, which is theoretically to be expected, except in the case where the consumer's risk preferences exhibit CARA, requires the inclusion of a price-income interaction term in the demand equation to be estimated. The Bernoulli utility function, U = 1nI, illustrates this. The expected utility function across alternative states is:

EU = p 1n[I.sub.0] + (1 - p) 1n [I.sub.1] (17)

[I.sub.0] = [I.sup.E] - L([I.sup.E]) + s [I.sup.1] = [I.sub.E]- [pi] s (18)

Solving the FOC for the amount of net insurance demanded we obtain:

[s.sup.*] = p([I.supE]/[pi]) - (1- p)[I.sup.E] - L(I.sup.E])]. (19)

In this case, there is no independent price effect since [ds.sup.*]/d[pi] = [pI.sup.E]/[pi.sup.2]. Note also that the probabilities of alternative states interact with the income, price and potential loss variables. This is an implication of the expected utility model, not the specific utility function used.

What is needed then is a specification of the insurance demand function of sufficient generality to allow for the kinds of interactions we have shown are logically necessary if we are to rely on, or provide a test of, the expected utility model of consumer behavior under uncertainty.

V. Summary

We have detailed the relationship between the net demand for insurance and changes in the consumer's initial endowment of income. This complex relationship depends upon the consumer's degree of risk aversion and how it changes with income and the degree to which the loss itself depends upon income. Only in a very restricted set of cases can insurance be categorized as a normal good, i.e., the demand for insurance curve shifts outward at all prices as income increases. More generally, whether the demand increases or decrease with income and over what range of prices has been shown to be related with the income elasticity of demand through income elasticity measures of risk aversion and loss.

References

[1.] Chemick, Howard, Martin Holmer and Daniel Weinberg, "Tax Policy Toward Health Insurance and the Demand for Medical Services." Journal of Health Economics, March 1987, 1-25. [2.] Chesney, Marc and Henri Louberge, "Risk Aversion and the Composition of Wealth in the Demand for Full Insurance Coverage." Schweizerische Zeitschrift Volkswirtschaft und Statistik, September 1986, 359-69. [3.] Cleeton, David, "The Medical Uninsured: A Case of Market Failure?" Public Finance Quarterly, January 1989, 55-83. [4.] Cohn, Richard, Wilbur Lewellen, Ronald Lease, and Gary Schlarbaum, "Individual Investor Risk Aversion and Investment Portfolio Composition." The Journal of Finance, May 1975, 20. [5.] Ehrlich, Isaac and Gary Becker, "Market Insurance, Self-Insurance, and Self-Protection." Journal of Political Economy, July/August 1972, 623-48. [6.] Feldstein, Martin and Bernard Friedman, "Tax Subsidies, the Rational Demand for Insurance and the Health Care Crises.- Journal of Public Economics, April 1977, 155-78. [7.] Friend, Irwin and Marshall Blume, "The Demand For Risky Assets." American Economic Review, December 1975, 900-22. [8.] Hoy, Michael and Arthur Robson, "Insurance as a Giffen Good." Economics Letters, Vol. 8, No. 1, 1981, 47-51. [9.] Kaplow, Louis, "Income Tax Deductions for Losses as Insurance." American Economic Review, September 1992,1013-17. [10.] Mossin, Jan, "Aspects of Rational Insurance Purchasing." Journal of Political Economy, July/August 1968, 553-68. [11.] Phelps, Charles. The Demand for Health Insurance: A Theoretical and Empirical Investigation. Rand Corporation, Publ. R-1054-OEO, 1973. [12.] Schlesinger, Harris and Neil Doherty, "Incomplete Markets for Insurance: An Overview." Journal of Risk September 1985, 402-23. [13.] Siegel, Frederick, and James Hoban Jr., "Relative Risk Aversion Revisited." The Review of Economics and Skutics, August 1982, 481-7. [14.] Smith, Vernon, "Optimal Insurance Coverage." Journal of Political Economy, January/February 1968, 68-77.

(1.) We consider price ranges below an actuarially fair level for two reasons. First, a complete analysis of the demand function for insurance should examine all prices at which a positive demand exists. Second, the price variable in our model is the consumer price which may diverge from the producer price in distorted insurance markets. For example, in health, life, and accident insurance markets the consumer may have the ability to "pay" for insurance coverage through tax exempted payroll deductions. Such tax subsidies effectively lower the consumer price of insurance and may even in the presence of loading on the part of the supplier produce consumer prices for insurance which are below an actuarially fair level. Other features of the tax system such as medical expense and casualty loss deductions serve as partial stop-loss insurance for consumers. While these effects do not directly alter the consumer price they may alter the form of the effective self insurance package. See footnote 2 for information concerning self insurance and refer to Cleeton [3] and Kaplow [9] for details of distortionary effects in insurance markets from publicly provided stop-loss coverage.

(2.) We do not consider the possibility that the size of the potential loss may also be a function of the consumer's price of insurance. There are two situations in which such a dependency might be important to take account of explicitly in specifying the consumer's insurance demand function. First, the optimal amount of an insurable asset the consumer chooses to purchase may be determined jointly with the optimal amount of insurance. For example, if residential housing could not be insured against fire and storm damage except at a very high cost, the proportion of consumers' asset portfolios accounted for by home ownership would almost certainly be lower. A second situation in which the size of the potential loss may be determined by the price of insurance faced by the consumer is where there are opportunities for the consumer to expend resources in order to limit the size of the loss, i.e., opportunities for loss reduction or self insurance. Although the possibility of these situations does not alter our analysis of die role of,income and risk aversion, they may introduce their own complexities into the specification of the consumer's insurance demand function. We have worked out the model in the situation of self insurance and can report that the effects of changes in income on the demand for net insurance are identical to those we model in this paper, given a corresponding adjustment is made to die price intervals over which the effects are analyzed. This last statement also applies to another situation we omit from consideration: the possibility that the consumer can affect the probability of loss, i.e., loss prevention or self protection. [5]. (3.) Hoy and Robinson [8] showed that under certain conditions the demand for gross insurance, i.e., the pay-out in the loss state without consideration of the premium that the consumer might have to pay in that state, could behave as a Giffen good. However, they also showed that the conditions necessary for this to occur are empirically implausible. (4.) The incomplete insurance market problem in the context of a portfolio model is not addressed in our analysis. (See Schlesinger and Doherty [12] for a statement of the problem.) Such considerations would alter the demand for net insurance in terms of magnitude at alternative prices. However, the signs of the effects we analyze, over the correctly adjusted price intervals, would remain unaltered. (5.) Only where the marginal utility of income is a constant will the value [character no conversion] [s.sup.*]/[character no conversion] [pi] constant, i.e., the demand curve is linear. However, in this case the consumer is risk neutral and the demand curve is a horizontal line at the actuarially fair level of [pi]. (6.) The loss function may represent the replacement expenditure or the expenditure to repair the asset in the event of a decrease in its value due to an insurable event. The loss function may also be used to describe the way in which increases in income are accompanied by an increase in the proportion of the consumer's portfolio comprising risky versus safe assets [2]. (7.) This result is consistent with the result found in Chesney and Loubergd [2, 36] which states, under the assumptions of DARA and a loss independent of initial income, the willingness to pay for full insurance coverage will decrease with an increase in initial income. The maximum premium the consumer is willing to pay for full insurance is approximated by the area under the demand curve. In Figure 2 the increase in income reduces this area from [pi.sup.0i]F[s.sup.*]0 to [pi.sup.of] 0. The areas under the demand curves drawn in Figure 2 over estimate the consumer's true willingness to pay because as the consumer moves down the demand curve there is an income effect that raises the consumer's level of utility. Chesney and Louberge correctly measure the consumer's willingness to pay as that amount that keeps the consumer at the same level of utility as they were before purchasing insurance. (8.) This is identical to the case considered by Chesney and Loubergd [2] where the percentage of total wealth subject to loss is changed as income changes. (9.) This includes an upward shift in the price intercept [pi.sup.0]. This can be seen by simplifying (11) when [L.sub.I]= 1 to obtain: [Character no Conversion][pi.sup.0]/[Character no Conversion][I.sup.E] = [pi.suo.0][R.sub.A1] > 0 (10.) The relationship between [eta] and the income elasticity of absolute risk aversion, [tau], is [tau] + [eta] = 1. Thus the consumer's risk preferences exhibit increasing, constant or decreasing absolute risk aversion as [tau] > (=)(<)O, which is equivalent to [eta] > (+)(<)1. The conditions for [epsilon] > (<) in terms of the arc income elasticity of absolute risk aversion ([tau.sub.a]) and [sigma] are: [epsilon] > (<)0 as [tau.sub.a] + [sigma] > (<)0. (11). The Siegel and Hoban [13] study indicates that in moving from a total wealth class of over $200,000 to a total wealth class of from $1 to $10,000 the measure of relative risk aversion declines by 15%. When calculating the percentage change in wealth as (10,000-200,000)/200,000 = 95%, one yields an underestimate (since the average level of wealth in the highest wealth class was surely above $200,000 and that in the lowest wealth class was surely below $10,000). Thus, the estimate [eta.sub.a] = .15/.95 = .16 is an overestimate of the true value. (12.) Siegel and Hoban [13] showed that by restricting their sample to higher wealth households and defining wealth narrowly they were able to produce results compatible with the two earlier studies. However since our estimate of [eta.sub.a] derived from their study is an overestimate (see previous footnote) we entertain the possibility that the true value is within the indicated range.

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.

II. 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] < 0 [SOC] (4) dEU/d[gamma] = - (1 - p) [lambda][U'.sub.1] + P(1 - [lambda]) [U'.sub.0] + 0 [FOC] (3')

[d.sup.2]EU/d[gamma.sup.2] = (1 - p)[lambda.sup.2][U".sub.1] + [p(1 - [lambda]).sup.2][U".sup.0]

<0 [SOC] (4') where the subscripts 0 and 1 refer to the income levels in states 0 and 1 at which the first and second derivatives are evaluated. The assumption of global risk aversion insures the SOC is satisfied. Rewriting the FOC, we obtain the usual marginal conditions:

(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 full insurance. Here [gamma.sup.*] = L, optimal gross insurance coverage ([gamma.sup.*]) is equal to the potential loss. Or we have the equivalent statement [s.sup.*] = L - [pi][s.sup.*], optimal net insurance coverage ([s.sup.*]) is equal to the potential loss less the premium on the net coverage. From this point on in our analysis, the demand for insurance will refer to the demand for net coverage.

As Ehrlich-Becker showed, the demand curve for net insurance is everywhere downward sloping as in Figure 1. The corresponding comparative statics result is:

[Mathematical Expression Omitted]

In general, [character no conversion][s.sup.*]/[character no conversion][pi] will not be constant with respect to [pi] or, as we will show, with respect to [I.sup.E.[(5)

The price at which the amount of insurance demanded would be zero ([pi.sup.0]), the price intercept of the demand curve, is defined implicitly in the FOC (equation (5)) when the right-hand side is evaluated at the initial income endowments:

[pi.sup.0] = p U'([I.sup.E] - L)/(1 - p)U'([I.sup.E). (7)

In order to evaluate the effect of a change in the income endowment ([I.sup.E] on the demand for insurance ([s.sup.*]), we need to note two things. First, an increase in income increases the initial endowments in both states by increasing [I.sup.E]. Secondly, we wish to have a general specification of how the potential loss (L) might be affected by a change in income. The loss function allows us to examine cases where the loss may depend on the value of an underlying asset owned by the consumer.(6) We allow the loss to be a function of the initial endowment of income, L = L([I.sup.E]), where 0 [less than or equal to] dL/d[I.sup.E] = [L.sub.I] [less than or equal to] 1. While the upper bound on [L.sub.I] may seem extreme, it will be instructive as a limiting case.

Using our loss function specification, the FOC (equation (3)) can be written as:

dEU/ds = 0 = - [pi](i - p)U'([I.sup.E] - [pi]s]) + pU'([I.sup.E] - L([I.sup.E] + s). (8)

The comparative-static result for the effect of a change in endowed income on the net demand for insurance is given by:

[Mathematical Expression Omitted]

When the price of insurance is actuarially fair, equation [9] states that the demand for insurance will increase with initial income ([I.sup.E]) if [L.sub.I] > 0. The increase in the demand for net insurance will be [ds.sup.*] = (1 - p)[L.sub.I]d[I.sup.E]. This simply confirms that, if an increase in income leads to an increase in the loss, the demand for insurance at an actuarially fair price will increase since the consumer fully insures against the loss at this price. This is independent of how the consumer's degree of risk aversion changes with income. When the price of insurance is not actuarially fair, and the consumer is not fully insured, the risk aversion measures do not obligingly cancel out.

The denominator of the right-hand side of equation (9) is negative, so (9) wifl assume the same sign as:

X = [R.sub.AI] + [R.sub.AO]([L.sub.I - 1) (10) where, as before, the subscripts 0 and 1 refer to the income in the respective states at which the measure of absolute risk aversion is evaluated and X is the numerator from [9] multiplied by - 1.

Differentiation of equation (7) with respect to [I.sub.E] and simplification results in:

[Mathematical Expression Omitted]

where the 0 and 1 subscripts here refer to the initial endowment points [I.sup.E] - L and [I.sup.E].

The sign of[11] will be the same as the term in brackets which we can write as:

RA([I.sup.E]) + ([L.sub.I] -1) [R.sub.A]([I.sup.E] - L) = [R.sub.AI] + ([L.sub.I] - 1)

[R.sub.AO]. (12)

This is the same expression as[10] evaluated at the price intercept point. When evaluated at this point, [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] > (<)0 implies [Character no Conversion][s.sup.*]/[Character no Conversion] [I.sup.E] > (<)0.

III. Results

* We first take the simplest case where the potential loss is independent of the initial income, [L.sub.I] = 0. At an actuarially fair price [R.sub.A1] = [R.sub.A1] and equation (10) states that a change in initial income will not change the amount of insurance demanded. This is a standard result that a riskaverse consumer will fully insure at an actuarially fair price and, since the loss is independent of the consumer's initial income, a change in initial income will leave the demand for insurance unchanged.

At the price [pi.sup.0], with [L.sub.I] = 0, equation[12] shows that the sign of [Character no Conversion] [pi.sup.0]/[Character no Conversion][I.sup.E] will be determined by the sign of RA I - RAO. Thus, the price intercept of the demand curve will increase with initial income under increasing absolute risk aversion (IARA). It will decrease under decreasing absolute risk aversion (DARA). Constant absolute risk aversion (CARA) would leave the price intercept unchanged as initial income varied. Thus, for example, with DARA, as [I.sup.E] increases, the demand curve for insurance rotates counterclockwise through the full insurance point (F) as is illustrated in Figure 2.(7) With IARA, the demand curve would rotate clockwise through the full insurance point as [I.sup.E] increased. With CARA, the demand curve would be unaffected by a change in [I.sup.E]. These results for the case where [L.sub.I] = 0 are summarized in Table II.

[TABULAR DATA OMITTED]

We now turn to the case where the potential loss is a positive function of the initial level of income, 0 < [L.sub.I] [less than or equal to] 1.(8) Let us begin by examining the limiting case where [L.sub.I] = 1. Equation (10) would simplify to: [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] has the same sign as X = [R.sub.A1], which is always positive. An increase in initial income would shift the demand curve for insurance outward at every price.(9)

Note again that this result does not depend on any assumption about the degree of risk aversion. This case provides an example where insurance can be said to be a non-inferior good, i.e., at all prices an increase in initial income leads to an increase in the demand for insurance regardless of the degree of consumer risk aversion. While the demand curve shifts outward regardless of the degree of risk aversion, the slope of the demand curve will be affected by the degree of risk aversion.

Rewriting equation (9) for the case where [L.sub.I] = 1 we obtain:

[Mathematical Expression Omitted]

At prices below the actuarially fair price, we know [I.sub.0] > [I.sub.1] and with DARA we would have [R.sub.A1] > [R.sub.AO]. Thus: if [pi] < p/(1 - p) then [R.sub.A0]/[R.sub.A1] < 1 and [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] > (1 - p). If [pi] > p/(1 -p) then [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] < (1 - p) and if [pi] = p /(1 - p) then [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] = (1 - p). As [I.sup.E] increases, the demand curve shifts outward by an amount which is increasingly larger the lower is the price. One can see, that under DARA and [L.sub.I] = 1, as [I.sup.E] increases, the demand curve shifts outward by (1 - p)[dI.sup.E] at the actuarially fair price and rotates counterclockwise about this new full insurance point as shown in Figure 3. With IARA, an increase in income shifts the demand curve out by the same amount at the actuarially fair price [[ds.sup.*] = (1 - p)[dI.sup.E] but the rotation is clockwise.

With DARA and 0 < [L.sub.1] < 1 the sign of [Character no Conversion][s.sup.*]/ [Character no Conversion][I.sup.E] depends on the specific values of [R.sub.A1], [R.sub.A0] and [L.sub.I]. Since [R.sub.A1] and [R.sub.A0] are both functions of [pi], the sign of [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] will depend on [pi] as well. One can think of the effect of an increase in initial income as causing the demand curve for insurance to rotate counterclockwise through a point which moves up the demand curve as [L.sub.I] increases. When [L.sub.I] equals zero, the rotation occurs at the full insurance point. As [L.sub.I] increases toward + 1, the point of rotation moves northwest along the demand curve and eventually the price intercept on the higher income demand curve increases above the price intercept on the lower income demand curve. Above the point of rotation [Character no Conversion][s.sup.*]/ [Character no Conversion][I.sup.E] < 0, at the point of rotation [Character no Conversion][s.sup.*]/[Character no Conversion][I.sub/E] = 0 and below the point of rotation [Character no Conversion][s.sup.*][Character no Conversion][I.sup.E] > 0. If we impose the assumptions of constant income elasticities of loss and risk aversion, it is possible to bring to bear some useful empirical evidence for the case where 0 < [L.sub.I] < 1. At the actuarially fair price, from equation (9), [Character no Conversion][s.sup.*]/[Character no Conversion][I.sup.E] = (1 - p)[L.sub.1]. Multiplication of both sides of this equation by [I.sup.E]/L, dividing both sides by (1 - p) and, since we are at the full-insurance point, the substitution of [s.sup.*] = (1 - p)L on the left-hand side obtains:

([ds.sup.*]s)([I.sup.E]/[dI.sup.E]) = [beta] = [eta]. (13)

That is, at the actuarially fair price, the income elasticity of demand for net insurance is equal to the income elasticity of the loss. To express the change in the price-intercept point due to a change in income in elasticity terms, we can multiply both sides of (12) by [I.sup.E]/[pi.sup.0] to obtain:

(d [pi.sup.0]/[pi.sup.0])/([dI.sup.E]/[I.sup.E]) = [element of] = [I.sup.E][(1 - [L.sub.I])

([U.sub.0"]/[U.sub.0"] - ([U.sub.1"]/[U.sub.1'] (14) where [element of] is the elasticity of [pi.sup.0] with respect to [I.sup.E].

Writing this in terms of the measures of absolute or relative risk aversion:

[element of] = [I.sup.E][[R.sub.A1] - (1- [L.sub.I][R.sub.A0]] = [R.sub.R1] -

[I.sup.E](1 - L.sub.I])

[R.sub.R0]/([I.sup.E] - L)] (15) where the subscripts 1 and 0 here refer to the initial endowments ([I.sup.E] and [I.sup.E] - L). We want to find the condition under which [Character no Conversion][pi.sup.0]/[Character no Conversion] [I.sup.E] > (<)0. These are the same as the conditions where [epsilon]> (<)0.

We can show; see the appendix:

[Mathematical Expression Omitted]

Where [phi.sub.a] is the arc elasticity of relative risk aversion with the percentage change in income measured as the potential loss divided by initial income and [eta] is the elasticity of the loss with respect to initial income, i.e., [phi.sub.a] = ([delta][R.sub.R]/L)([I.sub.0.sup.E]/[R.sub.RO]) and [eta] = (dL/[dI.sup.E])([I.sup.E]/L) = [L.sub.I]([I.sup.E]/L). [eta.sub.a] is the ratio of the percentage change in the measure of relative risk aversion when going from the initial endowment of income in the loss state to the initial endowment of income in the no-loss state divided by the percentage change in income between the loss and no-loss states.(10)

Three studies have produced estimates of the variation in relative risk aversion with respect to variation in wealth. Cohn, et al.,[4] concluded that there was declining relative risk aversion (fi,, < 0). Friend and Blume[7] found evidence of increasing relative risk aversion but concluded that "Corrections for various deficiencies of the theoretical model, the econometric analysis and the sample data would on balance tend to make the results closer to constant proportional risk aversion" ([eta.sub.a] = 0). Siegel and Hoban[13], using a total wealth measure that was more comprehensive than the earlier studies and with a sample that contained more variation in wealth, found evidence of increasing relative risk aversion (fi, > 0). A reasonable estimate for d), that can be derived from their study would be approximately 0.16.(11) Since Siegel and Hoban used a total wealth concept, and controlled for the effect of the ratio of marketable to non-marketable wealth, this estimate should, theoretically, apply to income as well as wealth.

Tentatively, let us conclude, for the sake of argument, that the weight of this evidence would support a value of [eta.sub.a] in the neighborhood of the range 0 to 0.16.(12) Thus, using the condition set out in equation[16], we should expect that those insurable losses whose income elasticity is greater than 1 could be expected to exhibit a positive income elasticity of demand for insurance at all insurance prices. For those insurable losses with an income elasticity less than 0. 84 we might expect that the sign of the income effect on the amount of insurance demanded would depend on the price of insurance at which it was measured.

IV. A Simple Example

In spite of its apparent usefulness, there are few published examples of econometric attempts at estimating the consumer's insurance demand function. Where such attempts have been published, most specifications of the insurance demand function have been on an ad hoc basis[11]. Because of the lack of empirical work on the demand for insurance, attempts at estimating the costs and benefits of specific government interventions in insurance markets, when they have been done, employ restrictive and untested assumptions in order to make the task manageable [6; 1].

One important application of our analysis of the relation between income, risk aversion and the demand for insurance is its implication for the econometric specification of an insurance demand function. The rotation of the demand curve in response to a change in income, which is theoretically to be expected, except in the case where the consumer's risk preferences exhibit CARA, requires the inclusion of a price-income interaction term in the demand equation to be estimated. The Bernoulli utility function, U = 1nI, illustrates this. The expected utility function across alternative states is:

EU = p 1n[I.sub.0] + (1 - p) 1n [I.sub.1] (17)

[I.sub.0] = [I.sup.E] - L([I.sup.E]) + s [I.sup.1] = [I.sub.E]- [pi] s (18)

Solving the FOC for the amount of net insurance demanded we obtain:

[s.sup.*] = p([I.supE]/[pi]) - (1- p)[I.sup.E] - L(I.sup.E])]. (19)

In this case, there is no independent price effect since [ds.sup.*]/d[pi] = [pI.sup.E]/[pi.sup.2]. Note also that the probabilities of alternative states interact with the income, price and potential loss variables. This is an implication of the expected utility model, not the specific utility function used.

What is needed then is a specification of the insurance demand function of sufficient generality to allow for the kinds of interactions we have shown are logically necessary if we are to rely on, or provide a test of, the expected utility model of consumer behavior under uncertainty.

V. Summary

We have detailed the relationship between the net demand for insurance and changes in the consumer's initial endowment of income. This complex relationship depends upon the consumer's degree of risk aversion and how it changes with income and the degree to which the loss itself depends upon income. Only in a very restricted set of cases can insurance be categorized as a normal good, i.e., the demand for insurance curve shifts outward at all prices as income increases. More generally, whether the demand increases or decrease with income and over what range of prices has been shown to be related with the income elasticity of demand through income elasticity measures of risk aversion and loss.

References

[1.] Chemick, Howard, Martin Holmer and Daniel Weinberg, "Tax Policy Toward Health Insurance and the Demand for Medical Services." Journal of Health Economics, March 1987, 1-25. [2.] Chesney, Marc and Henri Louberge, "Risk Aversion and the Composition of Wealth in the Demand for Full Insurance Coverage." Schweizerische Zeitschrift Volkswirtschaft und Statistik, September 1986, 359-69. [3.] Cleeton, David, "The Medical Uninsured: A Case of Market Failure?" Public Finance Quarterly, January 1989, 55-83. [4.] Cohn, Richard, Wilbur Lewellen, Ronald Lease, and Gary Schlarbaum, "Individual Investor Risk Aversion and Investment Portfolio Composition." The Journal of Finance, May 1975, 20. [5.] Ehrlich, Isaac and Gary Becker, "Market Insurance, Self-Insurance, and Self-Protection." Journal of Political Economy, July/August 1972, 623-48. [6.] Feldstein, Martin and Bernard Friedman, "Tax Subsidies, the Rational Demand for Insurance and the Health Care Crises.- Journal of Public Economics, April 1977, 155-78. [7.] Friend, Irwin and Marshall Blume, "The Demand For Risky Assets." American Economic Review, December 1975, 900-22. [8.] Hoy, Michael and Arthur Robson, "Insurance as a Giffen Good." Economics Letters, Vol. 8, No. 1, 1981, 47-51. [9.] Kaplow, Louis, "Income Tax Deductions for Losses as Insurance." American Economic Review, September 1992,1013-17. [10.] Mossin, Jan, "Aspects of Rational Insurance Purchasing." Journal of Political Economy, July/August 1968, 553-68. [11.] Phelps, Charles. The Demand for Health Insurance: A Theoretical and Empirical Investigation. Rand Corporation, Publ. R-1054-OEO, 1973. [12.] Schlesinger, Harris and Neil Doherty, "Incomplete Markets for Insurance: An Overview." Journal of Risk September 1985, 402-23. [13.] Siegel, Frederick, and James Hoban Jr., "Relative Risk Aversion Revisited." The Review of Economics and Skutics, August 1982, 481-7. [14.] Smith, Vernon, "Optimal Insurance Coverage." Journal of Political Economy, January/February 1968, 68-77.

(1.) We consider price ranges below an actuarially fair level for two reasons. First, a complete analysis of the demand function for insurance should examine all prices at which a positive demand exists. Second, the price variable in our model is the consumer price which may diverge from the producer price in distorted insurance markets. For example, in health, life, and accident insurance markets the consumer may have the ability to "pay" for insurance coverage through tax exempted payroll deductions. Such tax subsidies effectively lower the consumer price of insurance and may even in the presence of loading on the part of the supplier produce consumer prices for insurance which are below an actuarially fair level. Other features of the tax system such as medical expense and casualty loss deductions serve as partial stop-loss insurance for consumers. While these effects do not directly alter the consumer price they may alter the form of the effective self insurance package. See footnote 2 for information concerning self insurance and refer to Cleeton [3] and Kaplow [9] for details of distortionary effects in insurance markets from publicly provided stop-loss coverage.

(2.) We do not consider the possibility that the size of the potential loss may also be a function of the consumer's price of insurance. There are two situations in which such a dependency might be important to take account of explicitly in specifying the consumer's insurance demand function. First, the optimal amount of an insurable asset the consumer chooses to purchase may be determined jointly with the optimal amount of insurance. For example, if residential housing could not be insured against fire and storm damage except at a very high cost, the proportion of consumers' asset portfolios accounted for by home ownership would almost certainly be lower. A second situation in which the size of the potential loss may be determined by the price of insurance faced by the consumer is where there are opportunities for the consumer to expend resources in order to limit the size of the loss, i.e., opportunities for loss reduction or self insurance. Although the possibility of these situations does not alter our analysis of die role of,income and risk aversion, they may introduce their own complexities into the specification of the consumer's insurance demand function. We have worked out the model in the situation of self insurance and can report that the effects of changes in income on the demand for net insurance are identical to those we model in this paper, given a corresponding adjustment is made to die price intervals over which the effects are analyzed. This last statement also applies to another situation we omit from consideration: the possibility that the consumer can affect the probability of loss, i.e., loss prevention or self protection. [5]. (3.) Hoy and Robinson [8] showed that under certain conditions the demand for gross insurance, i.e., the pay-out in the loss state without consideration of the premium that the consumer might have to pay in that state, could behave as a Giffen good. However, they also showed that the conditions necessary for this to occur are empirically implausible. (4.) The incomplete insurance market problem in the context of a portfolio model is not addressed in our analysis. (See Schlesinger and Doherty [12] for a statement of the problem.) Such considerations would alter the demand for net insurance in terms of magnitude at alternative prices. However, the signs of the effects we analyze, over the correctly adjusted price intervals, would remain unaltered. (5.) Only where the marginal utility of income is a constant will the value [character no conversion] [s.sup.*]/[character no conversion] [pi] constant, i.e., the demand curve is linear. However, in this case the consumer is risk neutral and the demand curve is a horizontal line at the actuarially fair level of [pi]. (6.) The loss function may represent the replacement expenditure or the expenditure to repair the asset in the event of a decrease in its value due to an insurable event. The loss function may also be used to describe the way in which increases in income are accompanied by an increase in the proportion of the consumer's portfolio comprising risky versus safe assets [2]. (7.) This result is consistent with the result found in Chesney and Loubergd [2, 36] which states, under the assumptions of DARA and a loss independent of initial income, the willingness to pay for full insurance coverage will decrease with an increase in initial income. The maximum premium the consumer is willing to pay for full insurance is approximated by the area under the demand curve. In Figure 2 the increase in income reduces this area from [pi.sup.0i]F[s.sup.*]0 to [pi.sup.of] 0. The areas under the demand curves drawn in Figure 2 over estimate the consumer's true willingness to pay because as the consumer moves down the demand curve there is an income effect that raises the consumer's level of utility. Chesney and Louberge correctly measure the consumer's willingness to pay as that amount that keeps the consumer at the same level of utility as they were before purchasing insurance. (8.) This is identical to the case considered by Chesney and Loubergd [2] where the percentage of total wealth subject to loss is changed as income changes. (9.) This includes an upward shift in the price intercept [pi.sup.0]. This can be seen by simplifying (11) when [L.sub.I]= 1 to obtain: [Character no Conversion][pi.sup.0]/[Character no Conversion][I.sup.E] = [pi.suo.0][R.sub.A1] > 0 (10.) The relationship between [eta] and the income elasticity of absolute risk aversion, [tau], is [tau] + [eta] = 1. Thus the consumer's risk preferences exhibit increasing, constant or decreasing absolute risk aversion as [tau] > (=)(<)O, which is equivalent to [eta] > (+)(<)1. The conditions for [epsilon] > (<) in terms of the arc income elasticity of absolute risk aversion ([tau.sub.a]) and [sigma] are: [epsilon] > (<)0 as [tau.sub.a] + [sigma] > (<)0. (11). The Siegel and Hoban [13] study indicates that in moving from a total wealth class of over $200,000 to a total wealth class of from $1 to $10,000 the measure of relative risk aversion declines by 15%. When calculating the percentage change in wealth as (10,000-200,000)/200,000 = 95%, one yields an underestimate (since the average level of wealth in the highest wealth class was surely above $200,000 and that in the lowest wealth class was surely below $10,000). Thus, the estimate [eta.sub.a] = .15/.95 = .16 is an overestimate of the true value. (12.) Siegel and Hoban [13] showed that by restricting their sample to higher wealth households and defining wealth narrowly they were able to produce results compatible with the two earlier studies. However since our estimate of [eta.sub.a] derived from their study is an overestimate (see previous footnote) we entertain the possibility that the true value is within the indicated range.

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Author: | Zellner, B. Bruce |
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Publication: | Southern Economic Journal |

Date: | Jul 1, 1993 |

Words: | 5383 |

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