# Moral hazard and health insurance when treatment is preventive.

ABSTRACTWe consider a two-period model under moral hazard when treatment is preventive. In the second period, the treatment level under moral hazard is higher than that under no moral hazard. However, it may be lower than that under moral hazard when overinsurance is not allowed. In the first period, the treatment level is higher when treatment is preventive than when it is not. Treatment level is also higher as the discount factor increases. We demonstrate that a treatment increase following a coverage increase does not necessarily imply moral hazard. These findings imply that moral hazard is possibly overemphasized in the literature.

INTRODUCTION

In general, preventive health care is defined as health care that is consumed before illness occurs. Prevention is often classified into primary prevention and secondary prevention. Simply put, primary prevention aims to lower the probability of illness, whereas secondary prevention aims to reduce the severity of illness. Primary preventive care includes healthy diet and smoking cessation. Secondary preventive care, on the other hand, includes medical examinations and diagnostic screening. (1)

Traditionally, the insurance literature excludes the costs of preventive care from insurance coverage, since such preventive care is considered the consumer's choice, not a random event (e.g., see Zweifel and Breyer, 1997). Prevention is discussed mainly in the context of moral hazard. When the insurance benefit cannot be contingent on the level of prevention, the level of coverage for treatment will affect the consumer's selection of prevention level (Ehrlich and Becker, 1972; Shavell, 1979; Dionne, 1982; Kenkel, 2000; Winter, 2000, for a survey). As a result, optimal insurance coverage for treatment should take into account the effect of insurance on prevention level. Recently, a few studies have suggested that insuring preventive care costs may be optimal if, for example, it facilitates more efficient risk sharing or lower treatment costs (Barigozzi, 2004; Ellis and Manning, 2007; Newhouse, 2006).

Although the literature makes a distinction between prevention and treatment, there may not be a clear distinction in practice, as pointed out by Ellis and Manning (2007). For example, early screening through self-examination for certain conditions may lead to more sophisticated diagnostic follow-up examinations. Such diagnostic examination can be considered preventive care because it lowers the severity of the potential illness by detecting it at an early stage. However, it can also be regarded as part of treatment care if it detects a disease. Moreover, treatment care itself may work as preventive care with respect to future illness. For example, one of the aims of treatment for diabetes is to lower the risk of developing complications such as heart disease. Treatment for a stroke includes preventive care to lower the risk of future strokes.

In accordance with this observation, we will attempt in this article to understand the optimal level of treatment and insurance coverage when treatment care is also preventive. For this purpose, we set up a simple two-period model where the consumer may be sick in each period. Insurance for treatment is available in each period. Departing from the existing literature, we assume that treatment in the first period may affect the probability of being sick in the second period. Insurance coverage and treatment care level in the first period will be affected by their influence on second-period outcomes.

A related issue is the trade-off between moral hazard and risk sharing, as the need for insurance for prevention is often discussed in relation to the need for more generous insurance coverage (Ellis and Manning, 2007; Newhouse, 2006). Since Arrow (1963), Pauly (1968), and Zeckhauser (1970), the moral hazard problem has been one of the main economic issues in health insurance. On the one hand, health insurance increases the welfare level by rendering more efficient risk sharing. On the other hand, it may lower welfare by causing a moral hazard problem, that is, excessive utilization of health care. Optimal health insurance coverage is determined by balancing welfare gains and losses. In general, neither full insurance nor no insurance is optimal.

Several health economics papers report that the moral hazard problem in U.S. health insurance is not properly controlled (Feldstein, 1973; Feldstein and Friedman, 1977; Manning and Marquis, 1996; Pauly, 1974, 1986). The average U.S. coinsurance rate (costs shared by consumers) is known to be about 25 percent. However, several authors, for example, Manning and Marquis (1996), find that an optimal coinsurance rate is about 45 percent, although the efficiency loss with a 25 percent coinsurance rate does not seem to be large. From this viewpoint, efficiency would be improved by increasing the coinsurance rate.

Contrary to these findings, however, there is an increasing sentiment that the concern about moral hazard might be excessive. The fact that over 40 million Americans have no health insurance (U.S. Census Bureau, 2008) was an important consideration in the health-care reform of 2010 (see New York Times, 2010; Harrington, 2010a, 2010b, for an overview). In this regard, it is often argued that the current health insurance system overemphasizes the moral hazard problem, resulting in excessive cost sharing. From this viewpoint, efficiency would be improved by lowering the coinsurance rate.

One strand of the literature also suggests the possibility that the conventional approach might overemphasize the moral hazard problem. For example, Nyman (1999a, 1999b) argues that the conventional measure of total welfare loss may be exaggerated (for debates, see Blomqvist, 2001; Manning and Marquis, 2001). Newhouse (2006) also argues that lower cost sharing could result in increases in consumption of health services, which would eventually reduce total costs.

This article adds to the literature by investigating the trade-off between moral hazard and risk sharing when treatment is also preventive. The optimal treatment level in our model will be higher than that when treatment is not preventive. An important implication of our results is that insurance coverage may have to be more generous than when treatment is not preventive. Another interesting finding is that the treatment level may be lower under moral hazard than under no moral hazard if we do not allow for overinsurance. This result challenges the traditional belief that moral hazard is associated with overutilization of medical care. We also find that a coverage increase will improve the efficiency of the health insurance market if the original insurance contract is designed to ignore the preventive characteristics of treatment, or if consumers are shortsighted. We also show that an increase in treatment following a coverage increase does not necessarily imply moral hazard. These findings imply that the moral hazard problem may be overemphasized in literature.

This article proceeds as follows. "The Model" section outlines the model. The "Second-Period Problem" section solves the second-period problem, and the "First-Period Problem" section solves the first-period problem. The "Discussion" section discusses the implications of our results for the health insurance debate. Finally, the "Conclusion" presents our conclusion.

THE MODEL

We consider a two-period model in which a consumer faces a random health loss in each period. Each period is denoted by t = 1, 2. The consumer is an expected utility maximizer with an endowment wealth of W in each period. The von Neumann-Morgenstern utility is denoted by U(x). The discount factor for period 2 is denoted as [beta]. There is an uncertainty regarding health status in each period. This uncertainty is described by two states regarding loss occurrence, denoted by [s.sub.t], in period t: the no-loss state ([s.sub.t] = 0), and the loss state ([s.sub.t] = 1). The state regarding loss occurrence can be observed by both the consumer and the insurer. The loss state can occur with probability [p.sub.t] in period t. The consumer suffers health loss [D.sub.t] in the loss state in period t. We assume that [D.sub.t] is a discrete random variable and can be observed only by the consumer. The probability of [D.sub.t] is denoted by q([D.sub.t]). In notations, [D.sub.t] [member of] [DELTA] [equivalent to] {[D.sup.1], ..., [D.sup.K]; F(x)} for some integer K > 0, where [DELTA] represents a loss distribution and F(x) is a probability distribution function.

Given health loss [D.sub.t], the consumer may receive medical treatment of cost [x.sub.t] = [x.sub.t]([D.sub.t]), which enhances health level by [H.sub.t]([x.sub.t]). The cost [x.sub.t] is public information observable by the insurer as well as the consumer. We assume that [H'.sub.t]([x.sub.t]) > 0 and [H.sub.t]"([x.sub.t]) < 0. Health level and health loss are expressed in monetary terms. We assume that [p.sub.t] is affected by the loss occurrence and treatment in period 1. When no loss occurs in period 1, [p.sub.2] = [p.sub.2](N). Given that a loss occurs in period 1, [p.sub.2] = [p.sub.2]([x.sub.1]) with [p'.sub.2] < 0 and [p".sub.2] > 0. That is, treatment in period 1 is for (primary) prevention. It is assumed that [p.sub.2](N) < [p.sub.2]([x.sub.1]) for any [x.sub.1].

The consumer may purchase health insurance for treatment before the realization of health status in each period. Given that [D.sub.t] is not observed by the insurer, the contract is based only on the treatment costs, [x.sub.t]. For simplicity, we focus on a short-term linear coinsurance contract, where [a.sub.t] is the insurance rate paid by the insurer. The indemnity It is thus given by [a.sub.t][x.sub.t]. The insurance premium is denoted by [Q.sub.t]. We assume that the premium is actuarially fair, that is, [Q.sub.t] = [p.sub.t]E([I.sub.t]) = [p.sub.t]E([a.sub.t][x.sub.t]). An insurance contract in period t will be denoted interchangeably by either {[Q.sub.t], {[I.sub.t]}} or {[Q.sub.t], {[a.sub.t]}}. We also assume that the indemnity payment is made immediately after treatment in each period.

When the consumer purchases an insurance contract {[Q.sub.t], {[I.sub.t]}}, her expected utility in period t is expressed as follows:

[EV.sub.t] = (1 - [p.sub.t])U(W - [Q.sub.t]) + [p.sub.t]EU(W - [Q.sub.t] - [D.sub.t] + [H.sub.t] ([x.sub.t]) - [x.sub.t] + [I.sub.t]). (1)

Let us denote [W.sub.ts] as wealth in state s in period t: [W.sub.t0] = W - [Q.sub.t] and [W.sub.t1] = W - [Q.sub.t] - [D.sub.t] + [H.sub.t]([x.sub.t]) - [x.sub.t] + [I.sub.t]. Let us also denote [U.sub.ts] for U([W.sub.ts]).

The overall ex ante expected utility in period I is denoted as

EU = [EV.sub.1] + [beta][EV.sub.2]. (2)

The consumer's problem is to select {[x.sub.1](x), [a.sub.1](x); [x.sub.2](x); [a.sub.2](x)} to maximize her expected utility, given the moral hazard problem. A moral hazard problem occurs because the indemnity is determined by treatment [x.sub.t], which the consumer selects in the loss state, given the insurance contract. Now, let us solve the problem via backward induction from the second-period problem.

SECOND-PERIOD PROBLEM

In period 2, the consumer selects treatment [x.sub.2] to maximize the ex post utility after a health loss occurs, leading to moral hazard. (2) Given [D.sub.2] and {[Q.sub.2], {[a.sub.2]}}, an optimal [x.sub.2](x) solves the following problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assuming an interior solution, this condition can be expressed as

[U'.sub.21] ([H'.sub.2] + [a.sub.2] - 1) = 0.

Under the assumption that U' > 0, this condition is equivalent to

[H'.sub.2] + [a.sub.2] - 1 = 0.

On the other hand, the consumer selects insurance coverage [a.sub.2] to maximize expected utility. Therefore, given [p.sub.2], the second-period problem can be stated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

To solve this problem, let us denote [[lambda].sub.2] and [mu][D.sub.2] for the Lagrange multipliers attached to constraints. For simplicity, let us suppress the time subscript 2 in this section, unless stated otherwise. Now the Lagrangian can be expressed as follows:

L = (1 - p)[U.sub.0] +p[EU.sub.1] + [lambda](Q - pE(ax)) + [summation][[mu].sub.D](H' + a - 1). (4)

First-order conditions are

[L.sub.x(D)] = pq[U'.sub.1] (H' + a - 1) - [lambda]pqa + [[mu].sub.D]H" = 0, for each D.

[L.sub.a(x)] = p[E.sup.r]([U'.sub.1]x) - [lambda]p[E.sup.r](x) + [[summation].sup.r] [[mu].sub.D] = 0, for each x, where [E.sup.r](x) and [[summation].sup.r](x) are calculated over D with the same x.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

No Moral Hazard Case

As a reference, let us first find the first-best outcome, assuming no moral hazard. The no moral hazard case corresponds to [[mu].sub.D] = 0 in (4). Note also that loss size is contractible.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Solving (6), we have the following results:

Lemma 1: [First-best outcome]

(A) Treatment is determined to maximize treatment efficiency.

(B) Insurance coverage increases in health loss.

(C) Overinsurance is optimal, when D > H([x.sup.*]).

Proof: From [L.sub.a(D,x)] = 0 and [L.sub.Q] = 0, we have that [lambda] = [U'.sub.1] for any (x, D), and [U'.sub.0] = [U'.sub.1], which implies that - D + H(x) - (1 - a(D))x = 0 for all (x, D). With [L.sub.x(D)] = 0, we have that the optimal H'([x.sup.*](D)) = 1 for all D. In sum, the following two conditions characterize the optimal solution:

H'([x.sup.*]) = 1,

[a.sup.*](x) = D - H([x.sup.*])/[x.sup.*] + 1. (7)

The first condition in (7) implies that treatment is determined such that the marginal benefit of treatment equals the marginal cost. The second condition implies that the risk is fully removed. In sum, the first-best solution achieves cost efficiency in treatment and a complete risk hedge, which proves (A). The second condition also implies that the insurance coverage increases in health loss, which proves (B). On the other hand, [a.sup.*] is greater than 1, exhibiting overinsurance, if D > H([x.sup.*]), which proves (C). Q.E.D.

Technically, [a.sup.*] can be negative if H([x.sup.*]) is very high. Since this case is not interesting, we will assume [a.sup.*] > 0 throughout this article. As it is common that a medical treatment can at best restore the health status before the loss, our main concern is with the case where D > H([x.sup.*]) (e.g., see Barigozzi, 2004; Ellis and Manning, 2007). Also for analytic convenience, we will make the following assumption throughout this article:

Assumption (A1): For any health loss D, D > H([x.sup.*]).

On the other hand, it is often the case that overinsurance is not allowed for diverse reasons, as commonly assumed in the literature: 0 [less than or equal to] a [less than or equal to] 1. (3) For comparison, let us call the solution in such a case the constrained first-best solution, to distinguish it from the first-best one. Under assumption (A1), the two solutions will differ from each other. Let us add the superscript ** for the constrained first-best solution. The solution exhibits the following properties:

Lemma 2: [Constrained first-best outcome]:

The constrained first-best outcome has the following properties:

(A) Insurance coverage increases in loss size.

(B) The treatment is equal to the first-best treatment for partial coverage, and is greater than the first-best treatment for full coverage.

Proof: For interior coverage (0 < a < 1), [lambda] = [U'.sub.1] and H' = 1 from [L.sub.a(D,x)] = 0 and [L.sub.x(D)] = 0. Thus, [x.sup.**](D) = [x.sup.*]. This implies that wealth levels are the same for interior coverage: -D + H([x.sup.*]) - (1 - a)[x.sup.*] = -K for some fixed K > 0, where K = W - Q - [U'.sub.1.sup.-1]([lambda]). Thus, [a.sup.**](D) = D-H([x.sup.*])-K/[x.sup.*] + 1. For D [less than or equal to] H([x.sup.*]) + K - [x.sup.*], we have [a.sup.**](D) = 0, implying that H' = 1, and [lambda] [greater than or equal to] [U'.sub.1]. Thus, [x.sup.**](D) = [x.sup.*]. For D [greater than or equal to] H([x.sup.*]) + K, we have [a.sup.**](D) = 1, implying that [U'.sub.1]H' = [lambda] [less than or equal to] [U'.sub.1]. Thus, [x.sup.**](D) [greater than or equal to] [x.sup.*] from H' [less than or equal to] 1. From these results, the lemma is obtained. Q.E.D.

When coverage is partial, the treatment level is first-best. When the coverage is full, however, it is greater than the first-best level. In such a case, the treatment level satisfies the following condition:

H'(x[(D).sup.**]) = [[lambda].sup.**]/[U.sup.**'.sub.1] < 1, (8)

where [[bar.U].sup.**'.sub.1] is evaluated at the optimal solution given D.

Although insurance coverage is full, the risk is not fully hedged, since the health status is not fully restored. In order to compensate for the unrecovered health loss, the consumer selects a high level of treatment.

Moral Hazard Case

Now, let us return to the moral hazard case. The solution will solve the first-order conditions in (5). From (5), we obtain the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Let us consider the following technical assumption.

Assumption (A2): H'"(x)/H"(x) x [greater than or equal to] 1, at x = x(a).

- H'"(x)/H"(x) x can be called relative prudence measure, given that - H'"(x)/H"(x) is absolute prudence measure for H(x). Thus, the condition can be expressed as the relative prudence measure for H(x) is greater than 1 at x(a). This condition is satisfied by functions often used in economics and finance: [square root of x], log x, [x.sup.1-[gamma]]/1 - [gamma] (with [gamma] > 0, [not equal to] 1), and _[e.sup.-[gamma]x] (with [gamma] > 0 for x [greater than or equal to] 1/[gamma]). With (A2) we can show the following result.

Lemma 3: x'(a)/x(a) is increasing in a where x'(a) = dx(a)/da, if and only if(A2) holds.

Proof: From H' + a - 1 = 0, we have that x(a) = G(1 - a) and x'(a) = -G'(1 - a), where G(x) = [H.sup.'-1](x). Then, we have x'(a)/x(a) = - G'(1-a)/G)1 - a).

d (x'(a)/x(a)) / da = - -G"(1 - a)G(1 - a) G'[(1 - a).sup.2]/G[(1 - a).sup.2].

Thus, x'(a)/x(a) is increasing if and only if - G"(1 - a)G(1 - a) + G'[(1 - a).sup.2] [less than or equal to] 0.

Using the differentiation rules regarding an inverse function, this condition can be further arranged into - H'"(x)/H" (x) [greater than or equal to] 1, at x = x(a). (4) Q.E.D.

Lemma 3 is reminiscent of the monotone likelihood ratio property (MLRP) in the standard moral hazard model. (5) Similar to the MLRP, Lemma 3 leads to the monotone relation between the health loss and the coverage as shown in Proposition 1. Now, we can characterize the solutions as follows (let us add superscript m for the solution):

Proposition 1: Second-Period Outcome

(A) Insurance coverage is partial.

(B) Insurance coverage is increasing in health loss under (A2).

(C) The treatment is greater than the first-best.

Proof:

(A) Full (or over-) insurance is not optimal for any D, because [a.sup.m] [greater than or equal to] 1 implies H' [less than or equal to] 0, an infinite or infeasible treatment.

(B) Suppose [D.sub.1] > [D.sub.2]. We need to show that [a.sub.1] > [a.sub.2], where [a.sub.i] = [a.sup.m]([D.sub.i]). On the contrary, let us assume that [a.sub.1] [less than or equal to] [a.sub.2]. Then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [x.sub.i] = [x.sup.m]([D.sub.i]). We also have U'([W.sub.1]) [greater than or equal to] U'([W.sub.2]), since [W.sub.1] [less than or equal to] [W.sub.2] where [W.sub.i] = W - Q - [D.sub.i] + H([x.sub.i]) - (1 - [a.sub.i])[x.sub.i]. Now, let us simultaneously increase [a.sub.1] by [da.sub.1] and decrease [a.sub.2] by [da.sub.2], keeping the premium the same. This implies that [q.sub.1]([x.sub.1] + [ax'.sub.1])[da.sub.1] = [q.sub.2]([x.sub.2] + [a.sub.2] [x'.sub.2])[da.sub.2] from Q = pE(ax).

That is, [da.sub.2] = [q.sub.1]/[q.sub.2] [x.sub.1] + [a.sub.1][x'.sub.1]/[x.sub.2] + [a.sub.2] [x'.sub.2] [da.sub.1].

Now, the effect of this change will lead to a change in expected utility as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The last inequality comes from the fact that U'([W.sub.1]) > U'([W.sub.2]), and 1+[a.sub.1][x'.sub.1]/[x.sub.1]/1 + [a.sub.2] [x'.sub.2]/[x.sub.2] [less than or equal to] 1 under assumptions of [a.sub.1] [less than or equal to] [a.sub.2] and Lemma 3.

(C) Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [x.sup.m](D) [greater than or equal to] [x.sup.*]. Equality holds only if [a.sup.m] = 0. Q.E.D.

The results of Proposition 1 are typical of those in the moral hazard literature. Let us provide intuitive explanations as follows. For part (A), full coverage is not desirable under moral hazard, since higher risk sharing lowers moral hazard costs. Furthermore, part (B) states that a higher loss leads to higher coverage under (A2). For this, note that assumption (A2) and Lemma 3 implies that the effect of coverage on treatment is greater for higher coverage. Since higher coverage and treatment level lead to a higher wealth level, a higher loss needs to be associated with higher coverage in order to prevent a risk increase. This result implies that there is one-to-one mapping between D and [a.sup.m], so that [a.sup.m] can be considered a function of D, even though D is not directly observable.

Part (C) states that moral hazard leads to a higher treatment level than first-best treatment. With a lower level of coverage than first-best, the consumer is exposed to a high risk of health loss. To compensate for the health loss, the consumer will select a higher level of treatment than first-best.

Let us comment further on this proposition. First, note that the Lagrange multiplier [[mu].sub.D] is negative (see 9). This result implies that the insurer wants the consumer to reduce the treatment level, which confirms the moral hazard problem. Second, the proposition does not say that the treatment is greater than the constrained first-best, or [x.sup.m](D) [greater than or equal to] x[(D).sup.**] for all D. When [a.sup.**](D) < 1, [x.sup.m](D) [greater than or equal to] x[(D).sup.**] = [x.sup.*]. However, for D such that [a.sup.**](D) = 1, the relative sizes between [x.sup.m](D) and x[(D).sup.**] are ambiguous. For this, let us define b by H'([x.sup.**]) = 1 - b, given D. Given that H'([x.sup.**]) < 1, b > 0. Thus, [x.sup.m](D) > x[(D).sup.**] if and only if [a.sup.m](D) > b.

From (8), b = 1 - [[lambda].sup.**]/[U.sub.1.sup.**'] = 1 - [EV.sup.**']/[U.sub.1.sup.**'].

Under moral hazard, from (9),

[a.sup.m](D) = [x.sup.m](D)H"([x.sup.m](D)) (1 - [U.sup.m']/[EV.sup.m']), where [U.sup.m'.sub.1] is evaluated at D under moral hazard.

Let us define R[(D).sup.**] = [EV.sup.**']/[U.sub.1.sup.**'], and R[(D).sup.m] [EV.sup.m']/[U.sup.m'.sub.1]. Note that R[(D).sup.**], [R.sup.m](D) < 1. With these notations, we have the following result:

Corollary 1: For D such that [a.sup.**](D) < 1, [x.sup.m](D) [greater than or equal to] x[(D).sup.**] = [x.sup.*].

For D such that [a.sup.**](D) = 1,

[x.sup.m](D) [greater than or equal to] x[(D).sup.**] if and only if [x.sup.m](D)H"([x.sup.m](D)) (1 - 1/R[(D).sup.m]0 [greater than or equal to] 1 - R[(D).sup.**]. (10)

Proof: See text above. Q.E.D.

Although (10) is not of an explicit form, the following inferences can be made. First, as R[(D).sup.**] is close to 1, [x.sup.m](D) is likely to be greater than x[(D).sup.**]. Intuitively, R[(D).sup.**] will be close to 1 when the risk contribution from D, under no moral hazard, is well hedged. In such a case, x[(D).sup.**] will not be great. Thus, [x.sup.m](D) is likely to be greater than x[(D).sup.**]. On the other hand, R[(D).sup.m] will be close to 1 when the risk contribution from D, under moral hazard, is well hedged. From (10), as R[(D).sup.m] is close to 1, [x.sup.m](D) is likely to be smaller than x[(D).sup.**]. Note that the treatment level is likely to be low and the risk hedge level is likely to be high if the moral hazard problem is not severe. Thus, high R[(D).sup.m] is likely to be associated with low [x.sup.m](D).

The above observation may appear to contradict the conventional belief that moral hazard leads to a higher treatment level. However, recall that [x.sup.m](D) > [x.sup.*]. The only factor to affect the difference between [x.sup.*] and [x.sup.**] is the coverage constraint. In the no moral hazard case with the coverage constraint, risk is not fully hedged due to the coverage constraint. As a result, a higher level of treatment is needed to compensate for the incomplete risk hedge. If the risk hedge benefit is high enough, then the treatment level can be higher than that under moral hazard. In the Appendix, we present an example of a constant absolute risk aversion (CARA) utility to show that [x.sup.m](D) is greater than x[(D).sup.**] when the risk hedge benefit is high. The example shows that x[(D).sup.**] tends to be greater than [x.sup.m](D) when risk aversion is greater, or when the damage to health is greater.

FIRST-PERIOD PROBLEM

In the first period, the consumer selects treatment [x.sub.1] and coverage [a.sub.1]. It is assumed that there is no savings in the model. This problem is similar to the second-period problem, except that the consumer now needs to take into consideration the effect on second-period utility. As in period 2, the consumer selects treatment [x.sub.1] after the health loss occurs. However, the objective function includes not only the ex post utility, but also the discounted expected utility of period 2. That is, an optimal [x.sub.1] ([D.sub.1]) solves

Max U(W - [Q.sub.1] - [D.sub.1] + [H.sub.1]([x.sub.1]) - [x.sub.1] + [a.sub.1][x.sub.1]) + [beta]E([V.sup.m.sub.2]|[x.sub.1]), given [Q.sub.1] and [a.sub.1]([x.sub.1]),

where E([V.sup.m.sub.2]|[x.sub.1]) is the expected utility in period 2, which is optimally determined given treatment [x.sub.1] under the loss state in period 1, as described in the previous section. The first-order condition for an interior solution is

[U'.sub.11]([H'.sub.1] + [a.sub.1] - 1) + [beta]dE ([V.sup.m.sub.2]|[x.sub.1])/d[x.sub.1] = 0,

Note that dE[V.sup.m.sub.2]/d[x.sub.1] is positive, since higher [x.sub.1] leads to lower [p.sub.2], which in turn leads to the higher expected utility of period 2.

Now, the problem can be stated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Note that E([V.sup.m.sub.2]|N) denotes the expected utility in period 2 given no loss occurrence in period 1, and that [U.sub.10] = U(W - [Q.sub.1]), and [U.sub.11] = U(W - [Q.sub.1] - [D.sub.1] + [H.sub.1]([x.sub.1]) - [x.sub.1] + [a.sub.1][x.sub.1]). To solve this problem, let us denote [[lambda].sub.1] and [[mu].sub.1] for the Lagrange multipliers attached to the constraints. For simplicity, let us suppress the time subscript I in this section, unless stated otherwise. Now, the Lagrangian can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The first-order conditions are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Let us add superscript "1m" for the solution. The solution to (13) has the following properties:

Proposition 2: First-period outcome

(A) Insurance coverage is partial.

(B) Insurance coverage increases in health loss under (A2).

(C) Treatment is greater under the preventive treatment case than under the no preventive treatment case.

Proof: (A) and (B) Although there is an additional term [beta]dE([V.sup.m.sub.2]/d[x.sub.1])/d[x.sub.1], the logic of the proof of Proposition 1 (A) and (B) can be employed without difficulty to show that full or zero coverage cannot be optimal for [x.sup.1m] to be an interior solution. (C) Let us first consider the first-order conditions under no preventive effect. These first-order conditions can be obtained from (13) by noting that dE[V.sub.2.sup.m]/dx = 0.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

Although the problem is the same as in the previous section, (14) differs from (5) since the incentive constraint is written differently: p[U'.sub.1](H' + a - 1) = 0, instead of H' + a - 1 = 0. The reason for this difference is to make a clear comparison between the preventive case and the no preventive case.

Since dE[V.sup.m.sub.2]/dx > 0, we should have H' + a - 1 < 0 under the preventive case. As a result, given a, the treatment level will be higher under the preventive case than under the no preventive case, ceteris paribus. Let us show that [x.sup.1m] [greater than or equal to] [x.sup.m] for all D. For this, suppose on the contrary that [x.sup.1m] < [x.sup.m] for some D. This implies that [a.sup.1m] should be lower than [a.sup.m]. Now, let us slightly increase [a.sup.1m] to [a.sup.0], so that [x.sup.1m] < [x.sup.0] < [x.sup.m], where [x.sup.0] is the treatment corresponding to [a.sup.0] under the preventive case. It is more intuitive to regard the consumer's problem as [max.sub.a](x)[(1 - p){[U.sub.0] + [beta]E([V.sub.2]| N)} + [max.sub.x](x) p{[U.sub.1] + [beta]E([V.sub.2]| x)}]. By definition, ([a.sup.m], [x.sup.m]) solves the same problem, assuming [beta]E([V.sub.2]| x) is fixed. Since ([a.sup.0], [x.sup.0]) can be chosen to be arbitrarily close to ([a.sup.m], [x.sup.m]), ([a.sup.0], [x.sub.0]) produces higher expected utility than ([a.sup.1m], [x.sup.1m]) does assuming [beta]E([V.sub.2]| x) is fixed. Moreover, [beta]E([V.sub.2]| x)is higher with [x.sup.0] than with [x.sup.1m], since [x.sup.0] > [x.sub.1m]. Thus, we arrive at a contradiction to the optimality of ([a.sup.1m], [x.sup.1m]). Q.E.D.

The intuitions of parts (A) and (B) are the same as in Proposition 1. Part (C) is also intuitive, since an increase in treatment has an additional positive effect on the future expected utility under the preventive case. By reasoning similar to that in part (C), we can show the more general fact that the optimal treatment will increase as [beta] increases. In other words, as the consumer becomes more concerned about the future, she will select a higher level of treatment. This result is also intuitive.

Corollary 2: A short-sighted (low [beta]) consumer selects a lower level of treatment than a far-sighted (high [beta]) consumer, ceteris paribus.

On the other hand, the relative amount of insurance coverage is not clearly determined. (6) However, an intuitive explanation can be derived from observation of the first-order conditions. Given D, let us denote x = x(a), [lambda] = [lambda](a), and [mu] = [mu](a) to be the solutions corresponding to a, ignoring [L.sub.a] = 0 in the first-order conditions. From the first-order conditions, we have [a.sup.1m] > [a.sup.m]. if [L.sub.a]|a.sup.m] = pq[U'.sub.1] x - [lambda]pqx + [[mu].sub.D][U".sub.1](H' + a - 1)x + [U'.sub.1]] > 0, where variables are evaluated at a = [a.sup.m]. The first term represents the benefit of the marginal coverage from the utility increase after the loss. The second term represents costs following the premium increase. These two terms can be referred to as the nonmoral hazard net benefit of marginal coverage. The third term measures the moral hazard costs resulting from increased coverage. Under the preventive case, the increase in coverage changes the costs with respect to the moral hazard, since the increase in treatment has preventive effects. The net effect should weigh the nonmoral hazard net benefit and the moral hazard costs. When the former is greater than the latter, then the optimal coverage [a.sup.1m] is greater than [a.sup.m], ceteris paribus. Intuitively, as the preventive effect is greater, allowing more treatment is preferred. As a result, insurance coverage will be greater. However, if the preventive effect is small, moral hazard costs may be greater than the nonmoral hazard net benefit. In this case, insurance coverage will be reduced.

DISCUSSION

In the second-period problem, we find that the treatment level is generally higher under moral hazard than under no moral hazard. However, when overinsurance is not allowed, the treatment level can be lower under moral hazard. This finding may contradict the conventional belief that moral hazard leads to a higher utilization of resources. The treatment level is high under no moral hazard, since it helps hedge the risk. With no coverage constraints, risk is hedged via overinsurance. With the coverage constraint, a high level of coverage may be needed to lower the risk exposure. A main cause for this outcome is that health damage is so high that it cannot be fully compensated for by an affordable range of treatment. This result is similar in spirit to what is postulated by Ellis and Manning (2007), who argue that uncompensated health costs lead to lower cost sharing.

It is important to note that although treatment under moral hazard is lower than that under no moral hazard, this does not mean that an increase in treatment level improves efficiency. That is because this treatment level is a second-best one.

In the first-period problem, we investigate the optimal treatment level, where treatment not only increases health level but also lowers the future possibility of health damage. The treatment level tends to be higher when treatment is preventive. We also find that the treatment level is higher as the discount factor becomes higher. These results have implications for the current debate around health insurance coverage in the United States (Gruber, 2008; Newhouse, 2006; Nyman, 1999b).

Suppose that the current coverage level is set without taking into account the preventive characteristics of treatment. For example, coverage is set at [a.sup.m]. If treatment were not preventive, treatment would be [x.sup.m]. Given that treatment is preventive, the consumer selects treatment x([a.sup.m]), which is higher than [x.sup.m]. It would appear that overtreatment exists. If the insurer considers the overtreatment excessive moral hazard, then it may seek to lower the coverage level to control the moral hazard problem. Lowering coverage will reduce the treatment level, as anticipated. It also obviously reduces the health-care costs spent in that period. However, this does not mean that efficiency is improved.

The effect of lowered coverage on efficiency depends on the relative sizes of [a.sup.1m] and [a.sup.m]. If [a.sup.1m] [greater than or equal to] [a.sup.m], then reducing coverage will lower efficiency. However, if [a.sup.1m] < [a.sup.m], then lowering coverage toward a will increase efficiency. That is, controlling moral hazard does not necessarily improve efficiency, even though it lowers short-term health-care costs. This result emphasizes the importance of the preventive characteristic of treatment. When treatment is preventive, the relevant health-care costs should be the long-term costs, which incorporate the effects of treatment on future health.

Corollary 3: Suppose that the preventive characteristic is ignored in health insurance design. Then, a coverage increase may improve efficiency.

Another case for increasing coverage may be found where the consumer is short-sighted. Suppose that the consumer thinks of [beta] as zero, whereas the true [beta] is in fact positive. For the coverage given, the consumer selects a lower level of treatment under [beta] = 0 than under [beta] > 0. For example, with [a.sup.1m], the consumer selects an x' that is less than [x.sup.1m]. Now, there is less treatment. If a benevolent social planner knows the true [beta] of the consumer, then it is possible to improve efficiency by increasing coverage, which leads to increased treatment.

Corollary 4: Suppose that consumers are more shortsighted than they should be. Then, a coverage increase may improve efficiency.

Finally, the discussion in the preceding section (specifically Corollary 1) has an interesting implication in the debates on moral hazard and health insurance coverage. As an illustration, suppose that there is no moral hazard problem. However, suppose further that health insurance is designed with the false assumption that a moral hazard problem exists. For example, let us assume that coverage is set at [a.sup.m]. With [a.sup.m], the consumer, under no moral hazard, will select, say, [x.sup.*]([a.sup.m]), not [x.sup.m]. Now, the increase in coverage from [a.sup.m] to [x.sup.*] may be followed by an increase in treatment, even though there is no moral hazard (see the Appendix). That is, the increase in treatment does not necessarily imply moral hazard. Moreover, if [x.sup.**] > [x.sup.*]([a.sup.m]), then the increase in coverage leading to an increase in treatment may improve efficiency. However, if the increase in treatment is interpreted as moral hazard, the insurer may not want to increase coverage, resulting in a failure to improve efficiency. (7) This observation points out the possibility that the moral hazard problem may be overemphasized.

This case is illustrated in the Appendix. In Table A1, we report [x.sup.*]([a.sup.m]) in a CARA utility example, showing that [x.sup.**] > [x.sup.*]([a.sup.m]) in many cases. As an example, let us consider the base case. From Table A1, we know that [a.sup.m] = 0.71, [a.sup.*] = 21.22, and [a.sup.**] = 1 in the base case. Figure A1 depicts treatment as a function of coverage under no moral hazard. In the range of low coverage, the increase in coverage will lead to an increase in treatment. Both treatment and coverage increase until (and beyond) full coverage. Since the constrained first-best outcome is obtained with full coverage, efficiency is improved by increasing coverage toward 1.

CONCLUSION

We consider a two-period model under moral hazard when treatment is also preventive. In the second period, we find the standard result that the treatment level under moral hazard is higher than that under no moral hazard. However, it may be lower than that under moral hazard if overinsurance is not allowed. In the first period, the treatment level is higher when treatment is preventive than when it is not. Treatment level is also higher as the discount factor increases. These results imply that an increase in insurance coverage may improve the efficiency of the health insurance market if the original insurance contract is designed ignoring the preventive characteristics of treatment, or if consumers are shortsighted. We also demonstrate that an increase in treatment following a coverage increase does not necessarily imply moral hazard. These findings point out that the moral hazard problem is possibly overemphasized in literature.

Even though this article points out the possibility of efficient prevention following an increase in coverage, the results should be understood with some caveats. First, there are different types of prevention in reality. In some prevention, only a small amount of medical spending may be needed. However, there can also be wasteful prevention, such as an expensive magnetic resonance imaging (MRI), which can detect a disease with high Type II error. It may be important to distinguish between different types of prevention, which will require future empirical research. Second, this article focuses on the demand side, ignoring the supply side. However, coverage for prevention may affect the medical supply decisions of doctors (McGuire, 2000). Third, a health loss may change the utility function whereas this article adopts the standard expected utility (see Dionne, 1982). Under the state dependent utility, the interaction between prevention and moral hazard will be different from this article.

DOI: 10.1111/j.1539-6975.2011.01459.x

APPENDIX

We provide a simple numerical example for the second-period problem. For expository simplicity, we consider a degenerate distribution of health loss; that is, D is fixed.

Assumptions and notations:

U(W) = - exp(-AW), where A is a positive constant.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

H(x) = h [square root of x], where h is a positive constant.

H'(x) = (1/2)[hx.sup.-1/2] = h/(2[square root of x]).

H"(x) = - (1/4)[hx.sup.-3/2] = -h/(4x [square root of x]).

Q = pax

EV = (1 - p)[U.sub.0] + p[U.sub.1].

Let us first find the outcomes under moral hazard. The program under moral hazard can be stated as follows:

Max EV (A1)

s.t. Q = pax,

H'(x) + a - 1 = 0.

From the second constraint, we have

a = 1 - H'(x) = 1 - h/(2[square root of x]). (A2)

Thus, Q = pax = p[1 - h/(2[square root of x])]x = p[x - h[square root of x]/2]. (A3)

We also have

Q' = dQ/dx -- p[1 - h/(4[square root of x])].

H(x) - (1 - a)x = h [square root of x] - h[square root of x]/2 = h [square root of x]/2.

d{H(x) - (1 - a)x}/dx -- h/(4[square root of x]).

Now, let us replace Q and a in EV with the earlier results.

EV = - [(1 - p)[e.sup.-A(W - Q)] + [pe.sup.-A(W - Q - D + H(x) - (1 - a)x)]] (A4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A5)

In sum, from (A2) and (A5), the solution under moral hazard is determined as follows: [x.sup.m] solves (A5) and [a.sup.m] = 1 - h/(2[square root of [x.sup.m]]).

Now, let us consider the case of no moral hazard. The first best outcome will solve the following program.

Max EV (A6)

s.t. Q = pax.

Let us replace Q with pax in EV and solve the first-order conditions. [partial derivative]EV/[partial derivative]x = 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A8)

From (A8) to (A7),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A9)

From (A8) and (A9), the solution under no moral hazard is determined as follows:

[x.sup.*] = [h.sup.2]/4, (A10)

[a.sup.*] = (4/[h.sup.2])D - 1. (A11)

Note that [a.sup.*] > 1 if D > [h.sup.2]/2.

Assuming that D > [h.sup.2]/2, coverage [a.sup.*] exhibits overinsurance. If we do not allow overinsurance, we have the following (constrained first best) outcome:

[a.sup.**] = 1. (A12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A13)

Table A1 provides numerical results for several parameter values. The table shows that [x.sup.m] becomes smaller than [x.sup.**] as risk aversion (A) becomes higher or as health damage (D) increases.

[FIGURE A1 OMITTED]

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(1) See Kenkel (2000) for a review of prevention in the economics of health care.

(2) This type of moral hazard is often called ex post moral hazard. On the other hand, moral hazard in the selection of loss preventive action is called ex ante moral hazard (e.g., see Shavell, 1979).

(3) Under no moral hazard, the rationales for disallowing overinsurance include adverse selection and institutional and social norms prohibiting gambling on losses.

(4) The differentiation rules are as follows: When y = f(x), and x = g(y) = [f.sup.-1](y), we have g'(y) = 1/f'(x), and g"(y) = -f"(x)/f'[(x).sup.3].

(5) In a standard moral hazard setting of Holmstrom (1979), the differentiable version of MLRP states that [f.sub.e](x:e)/f(x:e) is increasing in x, where e is effort and f(x: e) is the probability density function of output x, given e. This leads to the monotone relation between the output and the output sharing.

(6) Determining the relative sizes of coverage requires complex technical assumptions.

(7) See, for example, Gruber (2008), as well as Manning and Marquis (1996), for discussions of the deadweight loss due to the negative elasticity of health-care demand with respect to its price.

S. Hun Seog is with the Business School, Seoul National University. The author can be contacted via e-mail: seogsh@snu.ac.kr. The author would like to thank the participants in the Risk Theory Society meeting, the American Risk and Insurance Association meeting, and the Korean Insurance Academic Society meeting in 2009 for their comments. He gratefully acknowledges the support from the Institute of Management Research of Seoul National University. He would also like to thank Thi Nha Chau for her support.

TABLE A1 A Numerical Example Parameter Base Case Al A2 A3 B1 p 0.200 0.200 0.200 0.200 0.100 A 0.100 0.200 0.050 O.O05 U.100 h 3.000 3.000 3.000 3.000 3.000 D 50.000 50000 50.000 50.000 50.000 Outcome [x.sup.m] 26.812 29.519 15.985 3.523 59.795 [x.sup.**] 40.323 54.570 20.675 3.189 71.790 [x.sup.*] 2.250 2.250 2.250 2.250 2.250 [x.sup.*] ([a.sup.m]) 11.643 12.700 7.339 2.410 27.058 [a.sup.m] 0.710 0.724 0.625 0.201 0.806 [a.sup.**] 1.000 1.000 1.000 1.000 1.000 [a.sup.*] 21.222 21.222 21.222 21.222 21.222 Parameter B2 B3 C1 C2 C3 p 0.300 0.400 0.200 0.200 0.200 A 0.100 O.l00 0.100 0.100 0.100 h 3.000 3.000 1.000 5.000 10.000 D 50.000 50.000 50.000 50.000 50.000 Outcome [x.sup.m] 17.798 14.592 3.125 61.066 68.692 [x.sup.**] 22.068 13.252 5.842 46.858 25.000 [x.sup.*] 2.250 2.250 0.250 6.250 25.000 [x.sup.*] ([a.sup.m]) 7.319 5.497 1.348 27.478 40.094 [a.sup.m] 0.644 0.607 0.717 0.680 0.397 [a.sup.**] 1.000 1.000 1.000 1.000 1.000 [a.sup.*] 21.222 21.222 199.000 7.000 1.000 Parameter Dl D2 D3 p 0.200 0.200 0.200 A 0.100 0.100 0.100 h 3.000 3.000 3.000 D 10.000 30.000 70.000 Outcome [x.sup.m] 5.183 17.359 29.153 [x.sup.**] 4.104 18.778 52.789 [x.sup.*] 2.250 2.250 2.250 [x.sup.*] ([a.sup.m]) 3.002 7.932 12.547 [a.sup.m] 0.341 0.640 0.722 [a.sup.**] 1.000 1.000 1.000 [a.sup.*] 3.444 12.333 30.111

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Author: | Seog, S. Hun |
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Publication: | Journal of Risk and Insurance |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Dec 1, 2012 |

Words: | 8711 |

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