Optimal reimbursement health insurance: An application of profit functions and Frischian demands

JOURNAL OF ECONOMIC THEORY 51, 403422 (1990)

Optimal Reimbursement Health Insurance: An Application of Profit Functions

and Frischian Demands*

TIMOTHY BESLEY

Princeton University, Princeton, Neus Jersey 08544

Received January 9, 1989; revised September 15, 1989

This paper develops a model of reimbursement health insurance which uses a consumer’s Frischian demands and their associated profit function. We contrast reimbursement insurance, which covers expenditure fluctuations on a particular subset of goods, with optimal insurance. A restriction on preferences in their profit function representation is shown to yield the fully optimal allocation for an economy with individual risks as described by Malinvaud. Journal of Economic Literature Classification Numbers: 022, 9 12, 92 1. 6 1990 Academic Press. Inc.

1. INTRODUCTION

This paper analyzes reimbursement health insurance, i.e., insurance which reimburses the insured for expenditures incurred in the pursuit of health. In exploring the optimality properties of such insurance, we emphasize the applicability of the theory of demands at a constant marginal utility of income which were extensively analyzed by Frisch [9]. To date most use of this theory has been in the study of intertemporal problems where utility functions are additive over time (see, for example, Browning, Deaton, and Irish [7]). There has been little explicit recognition of the fact that it may also be applied to insurance problems. The basic intuition for doing so is clear. As Malinvaud [14] demonstrated, in an economy with individual risks the optimal allocation of risk bearing has an agent setting his marginal utility of income constant across states of nature. Hence if an agent has optimal insurance, the relevant demands are his Frischian demands for commodities. It is this fact which we exploit in the ensuing analysis.

* I thank Martin Browning, Christopher Gilbert, Ian Jewitt, David Newbery, and seminar participants in Oxford, Bristol, and at the European meeting of the Econometric Society in Copenhagen for comments, suggestions, and encouragement at various stages. I am especially grateful to Terence Gorman for his help and guidance. This paper is based on part of Chap. 2 of my Oxford D. Phil thesis. I alone am responsible for errors.

403 OO22-0531/90 $3.00

Copyright 0 1990 by Academic Press, Inc All rights 01 reproduction m any form reserved.

404 TIMOTHY BESLEY

Malinvaud demonstrated that a complete set of Arrow-Debreu markets for individual risks allowed trade at personalized prices and required that there be a commodity yielding an individualized payoff in every state of the world for each consumer. Reimbursement insurance is a much cruder device than this. It can be thought of as a security whose payoff is tied to an agent’s health state via expenditures upon health care. While the demand for all goods may vary with one’s health state, such insurance stabilizes only one’s expenditures upon a subset of these, viz., expenditures on health care. Reimbursement insurance contracts do not do many things which one might have thought that a fully optimal insurance contract ought to. First, they do not offer compensation for the fact that illnesses are unpleasant, involving pain and suffering. Second, they do not offer reimbursement for changes in some parts of one’s consumption pattern despite this being directly related to the illness; for example, expenditures required in travelling to a health facility.

In this paper we show how restrictions upon the structure of Frischian demands permits reimbursement insurance to attain the optimal allocation of risk bearing described by Malinvaud. The restriction is clearly inter- pretable and is more general than additive separability of the direct utility function between health care and other goods. Along the way, we look at some aspects of the demand for health care and health insurance. In particular, we show how the demand for health insurance is related to health being a necessity in a Frischian sense.

Much of our analysis uses a model in which the demand for health care is nested in a demand for health. This notion was first suggested by Grossman [ 111. It is useful for us, not only as a heuristic device, but also since we wish to make precise the idea that there is a subset of goods which reimbursement insurance focuses upon. In our model these will be the goods which are direct inputs into the production of health. However, for the reader who is not content with this restriction, we offer a generalization which does not rely upon it.

The structure of this paper is as follows. In the next section we lay out the model together with its principal assumptions. Section 3 considers different kinds of insurance contracts. In Section 4 we examine the optimal insurance contract and in Section 5 we consider the reimbursement insurance contract. Section 6 compares these two arrangements and gives a necessary and sufficient condition for their coincidence in terms of Frischian demands. The implications of this for the consumer’s profit function are then derived. In Section 7, we offer a generalization of the results which does not rely upon invoking a demand for health. We also consider the interpretation of moral hazard in the model. Section 8 concludes.

OPTIMAL REIMBURSEMENT HEALTH INSURANCE 405

2. THE MODEL

Consider a single consumer who faces prices determined in perfectly competitive markets. There are two types of good which we refer to as “health goods” and “non-health goods,” denoting them respectively by XER, and ZER,. We also introduce a good “health”: y E R+. The consumer’s utility function is defined on non-health goods and health:

u = U(z, y). (2.1)

Assumption 1. U(z, y): [wM+, + lR+ is strictly concave and strictly increasing.

Health is produced using health goods as inputs. The health production function,

Y = gb, R, (2.2)

also depends upon 8, the consumer’s pre-treatment health state.

Assumption 2. g(x, 0): IR,, 1 --) R + is increasing in each element of x, decreasing in 8, and quasi-homothetic, i.e., has a cost function that can be written in the form

O,R Y) = min { P.X I g(x, e) a Y} = U(P, e) Y + b(p, e). x (2.3)

We make separate assumptions governing the properties of (2.3):

Assumption 3. The functions a(p, 0) and b(p, 0) are concave and twice continuously differentiable in p, and continuously differentiable in 8.

The function a(p, 0) is the marginal cost of health, while b(p, 0) is the fixed cost. Invoking assumption 2, both will be increasing in 0. Quasi- homotheticity is crucial for what follows. It is convenient mainly because it allows us to model the consumer as bugeting in two stages; at stage one, health and non-health goods are chosen, while at stage two, the choice of health goods is made. The convenience is more than analytical. It enables us to refer to the demandfor health coherently, rather than just the demand for health care.’ Furthermore it ties in with the fact that the second stage

1 The conditions under which this kind of two stage budgeting is possible are somewhat restrictive and are discussed in detail in Deaton and Muellbauer [8]. Quasi-homothetic separability, which is chosen here, is one such case.

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of budgeting is most often undertaken by physicians2 The indirect utility function associated with (2.1) is

V(~P, e), 4, I- HP, 0)) = max { U(z, Y) I a(p, 0) Y + WP, 0) + q.z G I>, .I’, :

(2.4)

where Z is lump sum income.

Assumption 4. V(.): RM+2 -+ R + is twice continuously differentiable.

Assumption 3 guarantees that it is also differentiable in p, while it is concave in income from assumption 1. For many purposes, we can treat a(p, 0) like a price argument in the indirect utility function and we shall refer to it as the “price of health.” Using Roy’s identity in this vein yields

13 V( .)/au .Y=Y(a(P, O), 43 W(P3 O))= -aV(~),aw’

where we have defined w(p, 0) := Z-b(p, 0). Equation (2.5) gives the (Marshallian) demand function for health. Differentiating (2.4) with respect to the price of health good i and using (2.5) yields

aV(.)/aPi WP, e) Wp, 0) x,= - aV(.,/aw

= ___ Y(~P, 01, 4, W(P, 0)) + ___ aPi aPi

(2.6)

which is the demand for health good i. Quasi-homothetic separability of health goods from non-health goods implies that this depends upon the price of non-health goods and income only via that demand for health. In the sequel, we use y to denote the amount of health and y to denote health as a function of the other parameters. This reflects an important distinc- tion: e(p, 8, y) is an increasing function of 8 (by assumption 2), while e(p, 8, y) need not be because of the dependence of y on 8. This notational device makes it unnecessary to list the arguments of the function y in every instance. Differentiating (2.4) to obtain the dependence of utility on the pre-treatment health state 8 yields

(2.7)

with arguments of functions omitted. It is clear that (2.7) is negative, which accords with intuition; i.e., the worse is the pre-treatment health state, the lower is utility.

* This paper sweeps a whole array of issues concerning the role of physicians in the demand for medical care under the carpet. These are undoubtedly important but would detract from the focus of the present paper.

OPTIMAL REIMBURSEMENTHEALTH INSURANCE 407

The uncertainty which motivates insurance is modeled here by supposing that 0 is a random variable. We assume 0 to be distributed on an interval of the real line, [& 61, with distribution function F(6). Hence expected utility is

Since all of the integrals in this paper are defined on the same interval, we omit the limits of integration hereafter.

The risk associated with variations in 0 is partly like a price risk, since a(p, 8) is a price argument in the indirect utility function, and partly like an income risk, since the dependence of b(p, 0) on 0 is like income risk.3 It is the variation in a(,~, 0) which gives a special character to health risk and which prevents straightforward applications from income risk analysis to the analysis of health insurance. Comparing the model presented here with that in Arrow [l], the main difference is that we relate variations in the consumer’s wealth to changes in health expenditures. However, in another sense, Arrow’s model is more general since the direct effect of 8 on utility is not restricted to entering via a price argument.

As we noted in the introduction, it will be useful to focus on Frischian demands. In this context, it is natural to take the profit function representation of consumer preferences, suggested in Gorman [lo]. In the present model this is

where r is the price of utility (the reciprocal of the consumer’s marginal utility of income). Properties of profit functions in general are well known (see McFadden [ 131) while Browning [6] considers applications to consumer theory in some detail,

Assumption 5. I)( .): IR, + * + Iw + is twice continuously differentiable.

Appendix 1 discusses further properties of profit functions and Frischian demands which prove useful in the sequel.

3. TYPES OF INSURANCE CONTRACT

In this paper we focus on two types of health insurance contract: reimbursement insurance and optimal insurance. Optimality is used in the

3 This makes the analysis somewhat analogous to that of commodity price risks in Newbery and Stiglitz [15].

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sense of Arrow [ 11, and refers to an actuarially fair compensation schedule which depends upon 8, and which maximizes the consumer’s expected utility. By contrast, a reimbursement insurance contract makes compensation depend upon the amount which the insured spends on health goods. Such a contract stabilizes an agent’s expenditures on health care, and when actuarially fair, the premium charged is mean health expenditure.

We shall consider Nash equilibrium insurance contracts between the insurance company and the insured; i.e., each party to the contract treats the choices made by the other party as given in making its own choices. Such contracts have been looked at in the context of insurance problems by Rothschild and Stiglitz [ 161 and Shave1 [ 171.

4. OPTIMAL INSURANCE

We shall denote an optimal insurance schedule by K(8). It is chosen to maximize expected utility subject to the constraint of actuarial fairness. Hence our program is

subject to

(4.1)

(4.2)

Denoting the Lagrange multiplier on (4.2) by I, the first order condition is

K~&(p, e),q, W(P, 0) + We)) = 1 each 0. (4.3)

The second order condition is guaranteed by assumption 1. Equation (4.3) states the familiar condition that the marginal utility of income is kept constant: the condition whose optimality is emphasized in Malinvaud [ 141. In contrast to Arrow [ 11, we have not bounded the values that may be taken by N(8). He argues for a lower bound of zero on the grounds that an insurance company cannot demand payments from an agent over and above the premium which has already been paid. We do not impose a bound here since at present we are only interested in optimality, unconstrained by considerations of practicality.

We cannot say a priori whether the optimal insurance schedule is increasing in the pre-treatment health state, 0. To see this, differentiate Eq. (4.2) to yield

dK wmm ae= -aV,.(.)/aw’ (4.4)

OPTIMALREIMBURSEMENTHEALTH INSURANCE

Differentiating Roy’s identity ( I’, = -y I’,.) yields

which when substituted into (4.4) and rearranged yields

409

(4.5)

(4.6)

where

Rr +! and ay w - -. M’ rzawy

The first part of (4.6) represents the effect of a change in the marginal cost of health on the marginal utility of income and is ambiguous, while the second represents the effect of an increase in the fixed costs of health. The latter is always positive since a rise in fixed costs reduces wealth, increasing the marginal utility of income. A deterioration of the consumer’s pre- treatment health state may actually reduce the compensation paid to the insured. This occurs since the marginal utility of income might fall when 6’ rises. It is not particularly surprising that this is ambiguous. It is unclear intuitively whether as one’s pre-treatment health state worsens, the marginal value of income should increase or not. There are two opposing effects; one may be able to do less with income if one is ill, making income less valuable at the margin. On the other hand, one has lost purchasing capacity through having to pay for healty care. If R> q then N(B) is increasing in the pre-treatment health state. This is equivalent to health being a necessity in a Frischian sense (see Besley [4]). This is discussed in greater detail below and in Appendix 1.

5. REIMBURSEMENT INSURANCE

Under a reimbursement insurance contract, the insured is reimbursed some fraction, a, of his expenditures on health care. The insured chooses this fraction which we refer to as the “reimbursement rate.” We consider actuarially fair insurance contracts; i.e., the premium equals expected health care expenditures. Our solution concept is Nash equilibrium. The insurance company chooses the premium, n, and a compensation schedule, h(8). In equilibrium, h(0) equals actual health care expenditures since it must reimburse the insured. The program for a Nash equilibrium reimbursement insurance contract is

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with

7c = h(B) dF(B) s (5.2)

and

h(e) = 4P, 8, Y(dA 0 4, a& 0) + aW(@) - 71))) for all BE [e, 61.

(5.3)

The first equation represents the insured’s optimization problem, the second determines the insurance premium by the principle of actuarial fairness, and the third is the requirement that the payment made by the insurance company equal the insured’s expenditure on health care.

One aspect of the equilibrium that we have constructed is that the consumer does not take account of the dependence of the payoff schedule on ~1. It is here that the Nash assumption bites. It is especially crucial that we have ruled out any recognition by the insured that variations in LY act like changes in the price of health care at the margin. This is dealt with in another paper: Besley [3].

We are now in a position to analyze some properties of the solution to the program described by (5.1)-(5.3). The first order condition for the choice of a is

s V,,(.)(h(e)-n)dF(e) =o

i

>o a=1

aE IX, 11 (5.4) <o a = 0.

Differentiating with respect to a a second time yields

I v,,, (. )(h(e) - ~)2 dqe), (5.5)

which, using Assumption 1, is negative. Hence, the second order condition holds at an interior optimum. Substituting (5.2) into (5.4) we have that, at an interior solution,

s ~,(.)(e(p,e,~)-7~)d~(e)=o. (5.6)

Using the fact that n is expected health expenditures, Eq. (5.6) has a convenient interpretation, It says that the optimal choice of a at an interior solution sets the covariance between the marginal utility of income and health expenditures to zero. Contrasting this with Eq. (4.3) which defined


the optimal insurance schedule, we have a weaker requirement in (5.6), i.e., (4.3) implies (5.6) but not vice versa. This covariance has proved to be important in previous analyses of reimbursement insurance. Arrow [Z] finds it so in his analysis of the welfare effects of changes in health co-insurance rates. Phelps [12] also finds it to be relevant in a related context.

Our next task is to find conditions under which an equilibrium exists with O<a< 1. Define

w U(P, e) =-z y

as the compensated demand elasticity for health and

as the share of the marginal cost of health in total expenditure.

THEOREM 1. There exists a unique c1 that solves program (5.1)-(5.3) at an interior point (i.e., CI E (0, 1)) if:

(i) R > q for all 0 E [Q, 61 and

(ii) R&/(1--j?p?)<qfor all t3E [e, 6-J.

For the proof of this result, the reader is referred to Appendix 2. There remains, however, the issue of interpretation. It is useful to express the conditions of the theorem in terms of the Firschian demand elasticities, whence they become

(i’) p < 1 for all 8 E [fl, 61 and

(ii’) d+p>O for all eE [Q, 61,

where 4 is the elasticity of demand for health with respect to a, with the marginal utility of income held fixed, and ,n is the elasticity of demand for health with respect to a change in the reciprocal of the marginal utility of income, i.e., the price of utility. These elasticities can be derived directly from the profit function, as Appendix 1 shows. To interpret these conditions it is useful to outline the proof of Theorem 1 verbally. Our assumptions guarantee that the covariance between the marginal utility of income and health expenditures is a continuous function of a. Therefore, if at a = 0 it is positive and at a = 1 it is negative, then by the intermediate value theorem there exists a value of c( for which the covariance is zero. Moreover, by Assumption 1 (strict concavity of the utility function) the covariance is strictly decreasing in a, making the value of a which satisfies this unique.

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Condition (i) ensures that when CI =0 (there is no insurance), the covariance in (5.6) is positive. The condition that p< 1 can be interpreted as saying that health is a necessity. This is nut according to the traditional definition of necessity which rests on the behavior of the budget share as a function of total expenditures. The definition we require describes a good as a necessity if the amount of money needed to stay at a fixed marginal utility of income increases with a rise in the price of a good. If to keep an agent’s marginal utility of income constant requires drawings from his stock of assets, then necessities, in the sense required here, are those goods he is prepared to run down his store of assets to consume in the face of a price rise. It does seem reasonable to suppose this of health care and it implies that there will be a demand for reimbursement health insurance. This condition is just that required in the last section to have the optimal insurance schedule increasing in the pre-treatment health state.

The second condition required can be interpreted better after exploiting the linear homogeneity of the profit function to yield

(5.7)

where xj (defined in Appendix 1) is the cross-elasticity of demand between the jth non-health good, zj, and the price of health, a(p, e), at a fixed marginal utility of income. Equation (5.7) says that non-health goods defined as a Hicks aggregate should be a Frisch complement with health. This condition is instrumental in proving that the covariance in (5.6) is negative at c( = 1 and hence in showing the desirability of some co-insurance. The explanation is as follows. When health and non-health goods are Frisch complements, then a worse pre-treatment health state leads to a fall in non-health goods consumption when the marginal utility of income is held fixed. Hence expenditures previously made upon non- health goods are “freed” for spending on health care. If reimbursement were complete, then the individual would have more expenditures in total than would be required to keep his marginal utility of income fixed and would thereby be prepared to take some co-insurance.

Our results suggest the following sufficient condition for the consumer not to insure:

(a) b(p, 0) is independent of tI and

(b) the indirect utility function can be written in the form

V~P, 04, W(P)) = G(~P, 0 q) + ff(q, W(P)). (5.8)

In this case p = 1 (health is neither a luxury nor a necessity) and the

OPTIMAL REIMBURSEMENTHEALTHINSURANCE 413

marginal utility of income is independent of the pre-treatment health state, 0, when c( = 0. When these hold, variations in 0 do not affect an agent’s marginal utility of income at all; however, the restrictions are very strong.

6. WHEN IS REIMBURSEMENT HEALTH INSURANCE OPTIMAL?

In this section we derive a restriction for preferences and the health care technology which makes the reimbursement contract equivalent to the optimal contract of Section 4. We deal first with the technological restriction that a(p, 0) be independent of 8. This “works” since we are then back in what Arrow [ 1, p. 2161 calls “the simple cash equivalent model,” i.e., all variations in the pre-treatment health state affect the consumer just like a change in his income, since all changes in health expenditures are induced by changes in the fixed cost of health. In this case, stabilizing health care expenditures stabilizes an agent’s marginal utility of income and is thereby equivalent to optimal insurance. More interesting, however, is the case where a(p, 0) depends upon 8:

assumption 6. &(p, (3)/&I > 0 for all 0 E [e, 61.

THEOREM 2. Reimbursement insurance is optimal insurance, i.e., a = 1 implies that V, (.) is constant, if and only if

1 Xj=O for all eE [e, 61. (6.1)

ProojI The sufficiency of (6.1) is contained in the proof of Theorem 1; we therefore concentrate on necessity. When a = 1, the price of utility, r, is determined from

I+ e(p, 4 y) - n - b(p, 0) = r$,(a(p, ‘3, q, r) - ICl(a(p, 0 q, r). (6.2)

Using the fact that e(p, 8, y) = a(p, e)( - $,) + b(p, 0) and exploiting the linear homogeneity of the prolit function we have

(6.3)

Differentiating (6.3), regarding it as a relationship which determines r, we derive that

which holds, given Assumption 6, only if (6.1) holds. i

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The intuitive interpretation of this condition is straightforward. When (6.1) holds aggregate expenditures on non-health goods are independent of the pre-treatment health state at a fixed marginal utility of income. Hence a marginal deterioration in the consumer’s pre-treatment health state, which leads to a change in health expenditures and which is entirely covered by insurance, requires no change in the budget allocated to non- health expenditures. Non-health goods expenditure is in a sense independent of expenditures on health care and therefore of the insurance payment received.

We now consider the structure of preferences which yields condition (6.1).

THEOREM 3. The consumer’s profit function satisfies (6.1) if and only if it can be written in the form

@(ah 0 4, r) = dq, r) + 4(4p, e), 4, r) (6.5 1

with cp( ., .) homogeneous of degree one, t(. , q, ‘) homogeneous of degree one, and r(a(p, 0), . , r) homogeneous of degree zero.

Proof: Sufficiency can be proved by checking, using Hotelling’s lemma, that (6.5) yields expenditures upon non-health goods independent of 0. To show necessity use the equivalence of (6.1), by linear homogeneity of the profit function, to

r %+a %=, ar aa ’

which after using Euler’s Theorem for homogeneous functions yields

(6.6)

with <,(.) homogeneous of degree zero in (a, r). Integrating (6.6) yields (6.4) for some function (p(q, r). Imposing linear homogeneity on the function as whole implies that <( .) will be linearly homogeneous in (a, r) as required. 4

The profit function in (6.1) encompasses two interesting special cases. The first arises when <( .) does not depend upon q at all. In this case, the restriction on preferences given in (6.5) can be stated in terms of the direct utility function. It implies that

U(z, y)=4z)+4y), (6.7)

i.e., that the function is additive in non-health goods and health. The


second interesting special case is that in which 5(.) does not depend upon r when

H&4 e), 4, r) = p(q) 41-5 0) (6.8)

with p( .) homogeneous of degree zero. It is easy to check that in this case 4 = p = 0. The cost function associated with (6.8) is easily derived using the fact that cost and profit functions are convex conjugates (see Browning [6] for further details) whence

C(~P, 0 9, u) = max {ru - IL(a(P, e),q, r)}, I (6.9)

yielding

C(4P, 0 4, u) = wq, u) - 4P3 0) P(4). (6.10)

where Y( ., U) is homogeneous of degree one. Since p(q) is homogeneous of degree zero, aggregate expenditures on non-health goods are independent of the pre-treatment health state at a given utility level. This can be checked by differentiating (6.10) with respect to q,, which yields, via Shepard’s lemma, that

“j= yjyi(4, u)-a(P, e, P,(q), (6.11)

from which

C q,i'j=C 9jyj(4, u)-a(P7 e)C q.jPj(q) i i i

= wq, u). (6.12)

In this case the demand for health is independent of both prices and income; i.e., the amount of health demanded after treatment is not affected either by one’s initial health condition or by the cost of health care and we are in essence back to a simple cash equivalent model. This is confirmed by looking at the indirect utility function obtained by inverting (6.10),

b(~, e), 4, a4 0)) = v(q, ah 0) -4p, 0) dd), (6.13)

from which it is clear that the uncertainty associated with variations in the pre-treatment health state acts like income variation. It is therefore unsur- prising that we get the optimal insurance result for the reasons that we gave above when discussing the case where u(p, 0) is independent of 0.

In the “general” case given by (6.5) we have neither an additive direct utility function, nor do we rule out income and substitution effects in the demand for health. This vindicates our approach based upon the profit

416 TIMOTHY BESLEY

function representation for preferences and Frichian demands, since when more convential representations of preferences are used the restriction implied by (6.5) has no visible implication.

7. EXTENSIONS AND GENERALIZATIONS

In this section, we shall present some extensions to the results presented above. To begin with, consider a world in which we cannot speak at all of a demand for health underpinning the demand for health care; i.e., the conditions required for two stage budgeting are not fulfilled. We continue to refer to % as the pre-treatment health state, but we allow it to enter the indirect utility function quite generally:

(7.1)

Suppose once again that reimbursement insurance covers only a subset of the goods which will continue to be those with price vector p. Assume furthermore that all goods have a common insurance rate. In general, this will not be optimal: see Besley [S] for details. The condition required in Theorems 1 and 2 generalizes readily to this case. The requirement is that the profit function be of the form

(7.2)

with &J, 9, ., Y) homogeneous of degree zero. This “works,” since expenditures upon non-health goods are independent of 9, in aggregate, when the marginal utility of income is held fixed. Put otherwise, the crucial property is that health goods expenditures not depend upon the level of prices for non-health goods. It is more general than the condition in Theorem 3, since the prices of health goods enter the function y(.). The demand for health model allowed the pre-treatment health state to enter preferences only through an aggregator function which includes prices of health goods. While the same basic flavor is revealed, proceeding using this model precludes one from interpreting the conditions for insurance demand in any particularly convenient way.

The reader will also have noted that the results presented here generalize to two types of multidimensional situation. The first is where there are many facets to the pre-treatment health state (9 is a vector) and the other is where there are multidimensional insurance contracts with a scalar pre-treatment health state. Both are quite straightforward, whether or not one maintains a demand for health framework. We shall not, however, go into them in detail here.

More interesting is to consider the meaning and importance of moral


hazard in a model such as this, an issue which we have judiciously avoided to date. It would seem natural to suppose that moral hazard is absent in the model that we have analyzed so far, since we have shown that there are equilibria in reimbursement insurance markets which can achieve the first best. Moral hazard is normally thought of as something which precludes the attainment of the first best in insurance markets. Note also that in the model of this paper, reimbursement insurance may achieve the first best despite the fact that insurance increases health care expenditures. The Marshallian demands for health care, which are those in the absence of insurance, differ from the Frischian demands, which are those with an optimal allocation of resources across health states. Some are inclined to argue that health insurance inclines agents to excessive demands for health care. While there are senses in which this is valid, the statement requires care since some increase in health expenditures from the pre-insurance condition is compatible with a first best outcome. To check whether demands are “excessive” the valid comparison is not with the Marshallian demands but with the Frischian demands.

A simple illustration of this point is available by introducing an element into the analysis which means that reimbursement insurance cannot attain the first best. Suppose that the insurance schedule h(8) is seen to depend upon the rate of insurance since once insured, the agent faces a subsidy at rate c( on every unit of health care consumed. Our Nash equilibrium defined above precluded this and hence made the first best available (at least in principle). Remaining within the confines of actuarially fair insurance contracts, the problem faced by the insured is

V((l-cc)a(p,0),q,Z--(l-cr)b(p,8)-ctn:)dF(B)

with X= e((l-u)p,8,y)dF(B). s (7.3)

It is easy to check (see Besley [3] for details) that if health is a necessity in the Frischian sense then the solution satisfies c1* = 1 and the consumer has a marginal utility of income which is independent of his pre-treatment state of health and equal to V,.(O, q, I- CUC’), where rc” is the premium when IX is one. This is interesting, since it is apparent that the relevant demands are once again Frischian demands for health care. However, the set of demands in force may differ from those under a fully optimal allocation since there is a distortion created by reimbursement insurance in this case. Let r” denote the price of utility in the fully optimal situation and Y’ when health insurance creates a distortion; then substantiating the claim that health care expenditures are excessive requires comparing x,(0, q, r”)

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with xi@, 0, q, r”). This shows how the Frischian demands for health are a useful benchmark for some questions which are of interest in the analysis of health insurance problems outside the first best.

8. CONCLUSIONS

This paper has shown how demands at a constant marginal utility of income, Frischian demands, are relevant to the analysis of reimbursement health insurance. Since the optimal allocation of risk bearing in a large economy with individual risks has individuals holding their marginal utility of income fixed across states of nature, such demands are of interest. They are estimated in empirical studies, although most often in models of intertemporal optimization. We have shown here how they might also be relevant to the analysis of health insurance and how appropriate structure can imply the optimality of reimbursement insurance. Even when the conditions for optimality do not hold, Frischian demands remain interesting as the appropriate measure against which claims of excessive health care demands may be assessed. This makes Frischian demands and profit functions a worthwhile focus for the kind of problem that we have considered here.

APPENDIX 1: PROFIT FUNCTIONS AND FRISCHIAN DEMANDS

This appendix presents some results on Frischian demands which prove useful in the text. A useful source of further results is Browning [6]. The consumer’s profit function defined by (2.10) is a linearly homogeneous and convex function of its price arguments. Hotelling’s lemma yields that

(All)

which gives the demand functions and the utility level with the marginal utility of income held fixed. It is useful to define the following elasticities for health:

4 = - azt,b/aa2 a/(a+/aa), the own price elasticity of demand for health which is negative given the convexity of the profit function.

p= - a*$/(aa ar) r/(&b/LJa), the elasticity of demand for health with respect to a change in the price ,of utility.

x, = - a’$/(aa aqj) qjj(a$/da), the cross price elasticity of demand between health and the price of non-health good j.


Since the profit function is linearly homogeneous

4+P+c x,=0 (A1.2)

and by differentiating the identity

(A1.3)

with respect to a and w, we obtain

4 = 4~ + Bv*/R) (A1.4)

and

PS1 as Rzq, (A1.5)

where

RE -!+f!, 3 w af a

‘I’GY’ E= -m and

n

The superscript ‘c’ is used to denote the compensated (Slutsky) demand derivative. Equation (A1.4) gives a useful link between the Frischian demand elasticities and the more conventional Marshallian ones, while Eq. (Al.5) links the Frischian elasticity p with the coefficient of relative risk aversion and the income elasticity of demand for health. To interpret this elasticity, in particular whether it is greater or less than one, we consider the compensation function

N&J, Q4, r) = rll/,(&-h e), 4, r) - I~/(&J, e), 9, r) (A1.6)

which can be interpreted as giving the amount of expenditure required to stay at a fixed marginal utility of income. Differentiating with respect to a, we obtain

am -=4,-$(I/, aa

=B(l -P). (A1.7)

Hence whether p is greater than or less than unity determines whether (A1.7) is positive or negative or, in words, whether an agent needs to spend more or less to stay at a fixed marginal utility of income. This can be thought of as giving a definition of luxury and necessity. Further discussion can be found in Besley [4].

420 TIMOTHY BESLEY

APPENDIX 2: PROOF OF THEOREM 1

Continuity of the covariance between the marginal utility of income and health expenditures as a function of u is guaranteed by assumption 4. We are looking for a point at which this covariance is zero when 0 < CI < 1. In fact we show that under the conditions of Theorem 1, the covariance is positive at c( = 1 and negative c1= 0. The proof is via three lemmas. First, however, it is useful to state the Tchebychev inequality, without proof:

J r(0) o(6) S(B) $0 as z(6) 0,

where l w(0) dF(8) = 0, o(e) is increasing in 8 and z(e) is positive. Take w(0) = e(p, 8, y) - n; then since insurance is actuarially fair, it

satisfies the first condition of the Tchebychev inequality. Lemma 3 below shows that w(e) is also increasing. We take t(e) = I’,,( .) and show that when (i) holds it is an increasing function of 8 at o( = 0 and when (ii) holds it is decreasing in 0 at c( = 1.

LEMMA 1. If R > r] for all 0 E [& 61, then V,. (.) is increasing in 0 at u = 0.

Prooj Differentiating V,( .) with respect to 8 at CI = 0 yields Eq. (4.6) in the text, which can be rearranged to yield

dV 6Jb L= V,, $ (R-q)- I’,., as ae

(A2.1)

which is positive if R > 4. 1

LEMMA 2. Zf RE/( 1 -/Iv) < q for all 0 E [f3, 61, then V,( .) is decreasing in 8 at CL= 1.

Proof: Differentiating V,( .) with respect to 8 when tl is not zero yields

. (A2.2)

The use of a total derivative dy/dO follows from the fact that y is determined by

Y = y(a(p, a ~,w(P~ 0) + 444 0) Y + b(p, 0) - 4) (A2.3)


at the Nash equilibrium for health insurance. Differentiating (A2.3) with respect to tI yields

(A2.4)

where the superscript c denotes the compensated (Slutsky) demand derivative. An expression for aP’,/&z is available by differentiating (4.5) in the text with respect to a. Substituting this expression together with (A2.4) into (A2.2) and evaluating the result at c( = 1 yields (in elasticity form)

which is negative as required. 1

Combining conditions (if and (ii) of the theorem yields

E+flff< 1;

i.e., the Marshallian demand for health is price inelastic.

(A2.5)

(A2.6)

LEMMA 3. If E + /?q < 1 for all 8 E [fl, 61, then e(p, 8, y) is increasing in 8 for ctE [0, 11.

ProoJ Differentiating health expenditures with respect to 8, recognizing that y is determined by (A2.3), yields

de 1 -&I--E da l-Bq ab de= l-& ys+iqs+

(A2.7)

which is positive for any value of CI in the unit interval provided that the price elasticity of demand is less than one. 1

To complete the proof of the theorem we invoke the intermediate value theorem in conjunction with the Tchebychev inequality, and Lemmas 1 through 3, to establish the existence of a point at which the covariance is zero. The uniqueness of such a point follows from (5.5) which shows that the covariance is strictly decreasing in CI. 1

REFERENCES

1. K. J. ARROW, Optimal insurance and generalized deductibles, reprinted in “Collected Papers of Kenneth J. Arrow, Volume 3: Individual Choice under Uncertainty,” Blackwell, Oxford. 1986.

422 TIMOTHY BESLEY

2. K. J. ARROW, Welfare analysis of health co-insurance rates, in “The Role of Health Insurance in the Health Services Sector” (R. N. Rosett, Ed.), Nat]. Bur. Econ. Res. New York, 1973.

3. T. J. BESLEY, Publicly provided disaster insurance for health and the control of moral hazard, J. Public Econ. 39 (1989). 141-156.

4. T. J. BESLEY, A definition of luxury and necessity for cardinal utility functions, Econ. J. 99 (1989), 844-849.

5. T. J. BESLEY, Optimal reimbursement insurance and the theory of Ramsey taxation, J. Health Econ. 7 (1988), 321-336.

6. M. J. BROWNING, “Profit Function Representations for Consumers’ Preferences,” Bristol University Discussion Paper No. 82/125, 1982.

7. M. J. BROWNING, A. S. DEATON, AND M. IRISH, A profitable approach to labor supply and commodity demands over the life cycle, Econometrica 53 (1985), 503-543.

8. A. S. DEATON AND J. N. J. MUELLBAIXR, “Economics and Consumer Behavior.” Cambridge Univ. Press, Cambridge, UK, 1980.

9. R. FRISCH, A complete scheme for computing all direct and cross demand elasticities in a model with many sectors, Econometrica 27 (1959), 117-196.

10. W. M. GORMAN, Tricks with utility functions, in “Essays in Modern Economic Analysis” (M. J Artis and A. R. Nobay, Eds.), Macmillan & Co., London, 1976.

11. M. GROSSMAN, “The Demand for Health,” Natl. Bur. Econ. Res., New York, 1972. 12. C. E. PHELPS, The demand for reimbursement insurance, in “The Role of Health Insurance

in the Health Services Sector” (R. N. Rosett, Ed.), Natl. Bur. Econ. Res. New York, 1973. 13. D. MCFADDEN, Cost, revenue, and production functions, in “Production Economics:

A Dual Approach to Theory and Applications” (M. Fuss and D. McFadden, Eds.), North-Holland, Amsterdam, 1978.

14. E. MALINVAUD, The allocation of individual risks in large markets, J. Econ. Theory 4 (l972), 312-328.

15. D. M. G. NEWBERY AND J. E. STIGLITZ, “The Theory of Commodity Price Stabilization,” Oxford Univ. Press, Oxford, 1982.

16. M. ROTHSCHILD AND J. E. STIGLITZ, Equilibrium in competitive insurance markets: An essay on the economics of imperfect information, Quart. J. Econ. 87 (1976), 629-650.

17. S. SHAWLL, Moral hazard and insurance, Quart. J. Econ. 90 (1979). 541-562.

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Optimal reimbursement health insurance: An application of profit functions and Frischian demands