Altruism, matching, and nonmarket insurance.

Chami, Ralph ; Fischer, Jeffrey H.

I. INTRODUCTION

Nonmarket transactions are commonplace, even in situations in which market substitutes are available. The provision of insurance is one such case in which both market and nonmarket products coexist. Spousal support and income pooling, bequests and other income transfers designed to smooth consumption across generations, government social welfare schemes, and financial support from family and friends are all cases in which individuals or groups provide nonmarket coinsurance. The decision to enter into nonmarket insurance agreements, while individually rational, may affect equilibrium market insurance contracts such that overall welfare fails.

In many nonmarket situations, altruism plays an important role both in motivating the structure of the nonmarket arrangement - who insures whom - and in influencing the results of that arrangement.(1) This paper examines coinsurance arrangements between altruistically linked agents and highlights the role of altruism in creating the incentive to coinsure and the benefits from coinsurance. We focus on the effects of altruistically motivated nonmarket transfer on the market provision of insurance and on the welfare effects of this market-nonmarket interaction.

The provision of private insurance affects not only the individual participants but also equilibrium in the insurance market through the aggregate effects of nonmarket arrangements. One aim of this paper is to highlight the market-non-market interaction within the literature on altruism. Existing work on altruism and income transfers examines these nonmarket institutions taking the structure of the market as given. However, while a single pair of individuals correctly regard themselves as price-takers in the insurance market, the aggregate effect of nonmarket transfers does influence the market payout/premium ratio. We demonstrate that, through its effect on effort, nonmarket insurance lowers the market payout for a given insurance premium if the degree of altruism is sufficiently low. As partners care more about one another, nonmarket transfers increase the insurance payout; this positive externality on the market is socially beneficial.

Our model is one in which the insurance market cannot observe the care an individual takes to avoid an accident and, as a result of the moral hazard between the market insurer and the insured, individuals can obtain only incomplete insurance. Incomplete insurance creates an incentive for individuals to obtain additional insurance, and hence agents are willing to enter into coinsurance arrangements.

In the case in which coinsurance partners cannot observe the level of care one another take to avoid an accident, there is a second moral hazard problem, between the partners. Arnott and Stiglitz [1991] show that nonmarket insurance serves only to crowd out market insurance. Individuals view themselves as price-takers in the insurance market, but the aggregate effect of many such individuals is to decrease the amount of market insurance offered and shift risk from risk-neutral insurers to risk-averse individuals. In the absence of altruism, nonmarket insurance is unambiguously welfare decreasing.

Allowing individuals to choose coinsurance partners with whom there is mutual altruism has several important consequences. First, individuals prefer to pair with partners with whom there is mutual altruism (Proposition 1). That altruism plays a role in matching coinsurers is not surprising in light of the nature of non-market insurance. All the examples above, from spousal support to welfare schemes, involve a degree of altruism among the participants.(2) While altruism does not provide the sole motivation for nonmarket insurance - even non-linked agents are constrained in the amount of insurance they can purchase and thus are better off by coinsuring - altruism motivates the choice of partner and increases the effort partners take to avoid an accident.

Recent work on bequests and other voluntary income transfers recognizes the linkage between altruism and transfers and how these linkages may be used to modify the recipient's behavior.(3) Our approach draws on this literature. The difference between these papers and the present paper is that previous work has examined altruism in the context of full information (perfect observability) of the actions of the agent receiving the transfer.(4) Secondly, these papers all focus on a single institution - the family. Previous work has not examined the impact of transfers on an external market, such as the insurance or labor market. In contrast, we model a market outside the family, in addition to the family setting, which allows for spill-overs between effects in the insurance market and effects in the household in the presence of imperfect information (in the form of unobservable actions).(5) It is the imperfect information, which does not allow the parties to contract directly on effort, which creates the moral hazard.

A second, and perhaps more surprising, result of altruism in nonmarket insurance is that higher altruism does not necessarily translate into higher social welfare. Though effort increases with altruism (Proposition 2) which, all things equal, increases welfare, higher altruism makes one less willing to impose costs on one's partner - say, a spouse. This desire to insulate a spouse from one's own mistakes could lead to a decline in the equilibrium market transfer in the event of an accident.

We also show that, in contrast to the result in Arnott and Stiglitz [1991] with nonaltruistic partners, nonmarket insurance is not unambiguously welfare decreasing when there are altruistic partners. Without altruism, the sole effect of nonmarket insurance is to crowd out the insurance market. With altruism, the crowding-out effect is offset by higher levels of care because spouses do not want to impose costs on one another. If effort increases sufficiently, the good effect of higher effort can outweigh the effect of crowding out the insurance market.

A special case of the above result is when each agent values the utility of his spouse as much as his own. This case, which we call "balanced altruism," has the same effect as the Arnott and Stiglitz case in which effort is observable, although the reasons are different.(6) The moral hazard with respect to the insurance market also creates an externality between spouses; when one is in an accident, the other suffers. At the same time, the nonmarket participants suffer from the same disincentive to provide the socially optimal amount of effort to avoid an accident, since the spouse bears some of the costs of an accident. Thus there is moral hazard within the family as well as in the insurance market. Whereas in the effort-observable case spouses have the ability to contract around the externality by specifying a level of effort required before transferring income, spouses with balanced altruism completely internalize the externality and hence eliminate one source of moral hazard. The resulting equilibrium is a constrained optimum (Proposition 4). The result is still second-best because even in this case the nonmarket insurance leads to lower market transfers in the event of an accident, shifting risk onto the risk-averse consumers.

We develop a model in which spouses purchase insurance from a competitive market, then agree to coinsure one another in the event of an accident to one. We then solve for the subgame perfect equilibrium combination of insurance premium, insurance payout, nonmarket payout, and effort level as a function of the damage from an accident and the degree of altruism between the spouses.(7) Holding constant the market price of insurance, we derive some useful comparative statics and investigate the effect of altruism on effort and the nonmarket transfer. Then, recognizing that the price of insurance will change to maintain a zero-profit constraint in the insurance market, we solve the problem from the viewpoint of a social planner and derive conditions under which nonmarket insurance is welfare enhancing and conditions under which such coinsurance reduces welfare. We find that an altruistic agent who weights the utility of his partner less than his own will always transfer more than the socially desired amount (Proposition 5). We then provide some results in the case where altruism is even stronger than balanced altruism; that is, when an agent weights the utility of his partner more than his own; we refer to this case as "self-eclipsing altruism." With self-eclipsing altruism we find that nonmarket insurance provides a positive externality, so that the socially optimal amount of insurance is more than the amount for which individuals contract (Proposition 8). Section VI discusses some potential applications of the model. The final section provides a discussion of the results and some intuition about the case of unreciprocated altruism, in which the partners have different degrees of altruism toward one another.

II. THE MODEL

Individuals are endowed with wealth w, from which they gain utility. However, accidents create disutility to individuals, which we represent as a loss of income d. To reduce the probability of an accident, agents can increase the care taken to avoid an accident. Effort also creates disutility, but higher effort yields a lower accident probability: define

[p.sup.i] = p([e.sup.i]); p[prime] [less than] 0, p[double prime] [greater than] 0 [for every] [e.sup.i]

as the probability of an accident for agent i taking care [e.sup.i]. We assume that [p.sup.i] is a strictly convex, continuous function.

A competitive insurance market provides partial insurance to individuals. Full insurance is not optimal because effort levels are unobservable to the market. The resulting moral hazard problem - agents have partial control over the division of income between the agent and the company, but the parties cannot contract directly on the level of effort - makes risk-sharing optimal, as in Holmstrom [1979] and Shavell [1979]. The company charges a premium [Beta] and pays out an amount [Alpha] net of the premium in the event of an accident.

In the absence of nonmarket insurance, individual i receives expected utility of

(1) E[U.sup.i] = u(w - [Beta])[1 - p([e.sup.i])] + u(w - d + [Alpha])p([e.sup.i]) - [e.sup.i]

with u([center dot]) a continuously differentiable function where u[prime] [greater than] 0 and u[double prime] [less than] 0.

However, suppose agents i and j enter into an agreement to coinsure one another in the event of an accident to one. That is, they agree to transfer an amount [Delta] to i (j) if i (j) has an accident but j (i) does not; the parties make no transfer if both i and j have or neither i nor j has an accident. Then expected utility E[U.sup.i] depends on j's actions as well, so (1) become

(2) E[U.sup.i] = u(w - [Beta])[1 - p([e.sup.i])][1 - p([e.sup.j])] + u(w - d + [Alpha])p([e.sup.i])p([e.sup.j]) + u(w - [Beta] - [Delta])[1 - p([e.sup.i])]p([e.sup.j]) + u(w - d + [Alpha] + [Delta])p([e.sup.i])[1 - p([e.sup.j])] - [e.sup.i] = [u.sub.0](1 - [p.sup.i])(1 - [p.sup.j]) + [u.sub.1][p.sup.i][p.sup.j] + [u.sub.2](1 - [p.sup.i])[p.sup.j] + [u.sub.3][p.sup.i](1 - [p.sup.j]) - [e.sup.i]

where [u.sub.0] [equivalent to] u(w - [Beta]) is i's utility if neither i nor j is in an accident, [u.sub.1] [equivalent to] u(w - d + [Alpha]) is i's utility if both i and j are in an accident, [u.sub.2] [equivalent to] u(w - [Beta] - [Delta]) is i's utility if j but not i is in an accident (i transfers [Delta] to j), and [u.sub.3] [equivalent to] u(w - d + [Alpha] + [Delta]) is i's utility if i but not j is in an accident (j transfers [Delta] to i). Clearly [u.sub.0] [greater than] [u.sub.2] and [u.sub.3] [greater than] [u.sub.1]. Incomplete insurance implies [u.sub.2] [greater than] [u.sub.3] and [u.sub.0] [greater than] [u.sub.1] (individuals prefer not to be in accidents), while risk aversion implies ([u.sub.3] - [u.sub.1]) [greater than] ([u.sub.0] - [u.sub.2]) (individuals prefer more certain income streams).(8)

Arnott and Stiglitz [1991, 185] show that expected utility rises with [Delta] around [Delta] = 0; compared with the market-only situation, agents prefer to enter into coinsurance agreements. The intuition is simple: incomplete insurance induces agents to seek further smoothing of consumption. They do so by transferring income in some good states and receiving income in some bad states of nature. Irrespective of the equilibrium [Delta], individuals are still incompletely insured - when both i and j are hurt, the only transfer comes from the insurance company - but are closer to full insurance than before. Since i and j are atomistic with respect to the insurance market, they take the market transfers as given, even though the market responds to the aggregate behavior of individuals. As risk-averse agents, i and j perceive themselves to be better off.

Now suppose that i and j are altruistic toward one another. That is, E[U.sup.j] is a component of i's utility. For concreteness, let i and j be spouses, with i as the husband and j as the wife. Let [[Lambda].sup.ij] [greater than or equal to] 0 be the altruism of i toward j. We define [[Lambda].sup.ij] = 1 as balanced altruism since in this case the husband is equally concerned with his wife's utility as his own. Similarly, we define [[Lambda].sup.ij] [less than] 1 as self-preferring altruism and [[Lambda].sup.ij] [greater than] 1 as self-eclipsing altruism. For the remainder of this section and the next we restrict [Lambda] to the interval [0,1]; section V considers the case of [Lambda] [greater than] 1. Now define

(3) [[Omega].sup.i] = [Omega]([e.sup.i]; [e.sup.j], [[Lambda].sup.i,j], [Alpha], [Beta], [Delta], w) [equivalent to] E[U.sup.i] + [[Lambda].sup.i,j] E[U.sup.j] = (1 + [[Lambda].sup.i,j])[[u.sub.0](1 - [p.sup.i])(1 - [p.sup.j]) + [u.sub.1][p.sup.i][p.sup.j]] + (1 - [p.sup.i])[p.sup.j]([u.sub.2] + [[Lambda].sup.i,j][u.sub.3]) + [p.sup.i](1 - [p.sup.j])([u.sub.3] + [[Lambda].sup.i,j][u.sub.2]) - [e.sup.i] - [[Lambda].sup.i,j][e.sup.j]

as the husband's expected utility; a similar expression defines the utility of the wife.

The structure of the game is as follows: market insurers set ([Alpha],[Beta]) to maximize profits subject to a zero-profit constraint (entry dissipates supercompetitive profits); spouses then agree on [Delta] to maximize expected utility given [Alpha] and [Beta]; then i and j unilaterally decide on [e.sup.i] and [e.sup.j], given [Alpha], [Beta], and [Delta]. Hence [Delta] = [Delta] ([Alpha], [Beta]; [[Lambda].sup.ij]) and [e.sup.i] = [e.sup.i] ([Alpha], [Beta], [Delta]; [[Lambda].sup.i,j]).

A natural question at this point is: Does altruism make nonmarket insurance better or worse for the participants? Proposition 1 shows that altruism improves expected utility in a symmetric equilibrium. The proofs are given in the appendix.

PROPOSITION 1. Assume that [[Lambda].sup.i,j] = [[Lambda].sup.j,i] = [Lambda]. Given market insurance prices and payouts, expected utility increases with altruism.

Proposition 1 confirms the intuition that nonmarket arrangements are more useful in altruistic settings. The force of the proposition is that each agent prefers to match up with the agent for whom he or she has the strongest degree of mutual altruism.(9) This result provides part of the motivation for examining altruistic relationships.

It is important to bear in mind that partners view themselves as atomistic and hence ignore the effects their actions have on the insurance market. In particular, they perceive [Alpha] and [Beta] to be fixed, even though the collective action of the various partners does have an effect on the insurance market. Thus in Proposition 1 individuals perceive that matching with an altruistic partner is utility enhancing. Similarly, partners enter into nonmarket insurance arrangements because they perceive such arrangements to be beneficial. Whether or not the arrangements are indeed beneficial depends on the reaction of the insurance market. Section IV investigates circumstances under which nonmarket insurance is ultimately beneficial.

To simplify the analysis, we employ three standing assumptions throughout the paper:

1. altruism is reciprocated: [[Lambda].sup.i,j] = [[Lambda].sup.j,i] = [Lambda];

2. all spouses have the same degree of altruism; and

3. the probability of an accident is bounded by 1/2: [p.sup.i]([e.sup.i]) [less than] 1/2 [for every] [e.sup.i].

Assumption (1) allows us to exploit properties of the symmetric equilibrium to resolve questions that remain unresolved in the asymmetric case without more structure on the utility function. We discuss relaxing this assumption in the final section. Assumption (2) allows us to consider a representative pair of agents i and j and still solve for the market equilibrium. Assumption (3) is the "normal" accident probability.

The Optimal Level of Effort

In the final stage of the problem, when [Alpha] and [Beta] are fixed and the husband and wife have agreed upon [Delta], the remaining decision is to determine the level of effort each will take to avoid an accident.

The first-order condition for i with respect to [e.sup.i] is

(4) [Delta][Omega] / [Delta][e.sup.i] = {(1 + [Lambda])[-[u.sub.0](1 - p) + [u.sub.1]p] - ([u.sub.2] + [Lambda][u.sub.3])p + ([u.sub.3] + [Lambda][u.sub.2])(1 - p)}p[prime] - 1 = 0,

bearing in mind that p is a function of [e.sup.i] and p[prime] is evaluated at [e.sup.i].(10) The term [e.sup.*] = e([Alpha], [Beta], [Delta]; [Lambda]) is the level of effort which implicitly solves (4) and, by symmetry, is the solution to j's problem as well.

The Optimal Transfer

Given [Alpha] and [Beta] from the insurance market, and having inferred [e.sup.*] as a function of [Alpha], [Beta], and [Delta] from the final stage, the husband and wife must agree on a transfer [[Delta].sup.*]. In general, one could model how the parties determine [Delta] as a bargaining problem, or one could consider each spouse setting a (possibly different) transfer unilaterally. We assume the noncooperative approach, although the reciprocal nature of altruism yields a symmetric outcome which is also a point on the constrained utility frontier.

Differentiating [[Omega].sup.i] with respect to [Delta] gives us

(5) [Mathematical Expression Omitted].

If [[Delta].sup.*], the optimal transfer, is positive then [[Delta].sup.*] sets the expression (5) equal to zero. We now show that [Delta][[Omega].sup.i]/[Delta][Delta] [greater than] 0 at [Delta] = 0 so that the spouses agree on a positive transfer.

LEMMA 1.

(6) [Delta][e.sup.i] / [Delta][Delta] = { - [([u.sub.2][prime] - [Lambda][u.sub.3][prime])p + ([u.sub.3][prime] - [Lambda][u.sub.2][prime])(1 - p)]p[prime]} / {(1 + [Lambda])([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])[(p[prime]).sup.2] + p[double prime]/p[prime]}.

This expression is negative for [Lambda] [element of] [0,1).

LEMMA 2. When [Delta] = 0, [Delta] [[Omega].sup.i] / [Delta][Delta] [greater than] 0.

Increasing [Delta] lowers effort as long as agents are self-preferring; that is, they care more about themselves than about their partners. Equation (6) is the formal expression of the intuition that an increase in the nonmarket transfer reduces the expected cost of an accident, so agents take fewer precautions. However, reducing effort increases an agent's own utility at the expense of the agent's partner. As [Lambda] increases from 0 to 1, [Delta] [e.sup.i]/[Delta] [Delta] [approaches] 0, so altruism ameliorates the negative effect of altruism on effort. As [Lambda] increases beyond 1, agents care more about the utility of their partners than they care about their own, so effort actually increases with [Delta], as we show in section V.

Transfers affect utility through equation (7):

(7) [Delta] [[Omega].sup.i]/[Delta] [Delta] = (1 + [Lambda])p(1 - p)([u.sub.3][prime] - [u.sub.2][prime]) + (1 - [Lambda])[[u.sub.2] - [u.sub.3])p[prime] + 1][Delta][e.sup.j] / [Delta][Delta].

The term (1 + [Lambda])p(1 - p) ([u.sub.3][prime] - [u.sub.2][prime]) in (7) represents the utility-enhancing benefits of additional nonmarket insurance on i (and j, discounted by [Lambda]), which occurs with frequency p(1 - p). The term (1- [Lambda]) [([u.sub.2] - [u.sub.3]) p[prime] + 1] [Delta] [e.sup.j]/[Delta] [Delta] is the effect of moral hazard: as [Delta] increases, j exploits i more through her decrease in effort. As [Lambda] increases, however, i becomes less concerned with this effect and more concerned with the first effect, so higher altruism increases the beneficial effect of nonmarket insurance. Around [Delta] = 0 the term inside the brackets in (7) disappears, leaving a positive expression. Thus in the absence of nonmarket insurance, individuals value additional insurance and, in the absence of complete market insurance because of moral hazard, are willing to provide nonmarket insurance to one another.

Equilibrium in the Insurance Market

Families are price-takers in the insurance market, so decisions about [e.sup.i], [e.sup.j], and [Delta] are made given the insurance premium [Beta] and the insurance payout [Alpha]. However, collectively, nonmarket insurance affects the premium-payout combination insurance firms can profitably offer by affecting the effort level and hence the accident probability of each individual in the economy.

Firms are constrained to obtain zero profits in equilibrium, so [Alpha] and [Beta] must satisfy

-[Alpha] p([e.sup.*]) + [Beta][1 - p([e.sup.*])] = 0

or [[Alpha].sup.*] = {[1 - p([e.sup.*])]/p([e.sup.*])}[Beta]

for an arbitrary [Beta]. As long as [Delta] [e.sup.i] / [Delta] [Delta] [not equal to] 0, the introduction of nonmarket insurance changes [e.sup.*]. From Lemma 1, [Delta] [e.sup.i] / [Delta] [Delta] [less than] 0, so p([e.sup.*]) increases and hence [[Alpha].sup.*] falls. Because families do not take into account the externality they impose on insurance companies by entering into nonmarket insurance arrangements, nonmarket insurance crowds out market insurance. As Arnott and Stiglitz [1991] show, this crowding out is welfare reducing because insurance risk shifts from risk-neutral firms to risk-averse families.

Ill. EFFECTS OF ALTRUISM

This section examines the effects that changes in the degree of altruism have on effort, nonmarket transfers, and insurance payments.

The Effect of Altruism on Effort

From the first-order condition for effort, equation (4):

(8) [Delta][e.sup.*] / [Delta][Lambda] = { - p[prime][[u.sub.2] - [u.sub.0])(1 - p) + ([u.sub.1] - [u.sub.3])p]} / {p[double prime] / p[prime] + (1 + [Lambda])([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])[(p[prime]).sup.2]}.

Lemma 2 shows that the denominator of (8) is negative (risk aversion implies that ([u.sub.3] - [u.sub.1]) [greater than] ([u.sub.0] - [u.sub.2])). For [Delta] [greater than] 0, [u.sub.0] [greater than] [u.sub.2] and [u.sub.3] [greater than] [u.sub.1] so the numerator of (8) is also negative, which makes the entire expression in (8) positive. Hence we have the following proposition:

PROPOSITION 2. Holding constant market and nonmarket transfers, effort increases with altruism.

The Effect of Altruism on Nonmarket Transfers

Differentiating the first-order condition for [Delta] (equation (7)) with respect to [Delta] and [Lambda] gives us the effect of altruism on nonmarket transfers:

(9) [Delta] [[Delta].sup.*] / [Delta][Lambda] = { - [([u[prime].sub.3] - [u[prime].sub.2])[A.sub.1] + ([Delta][e.sup.*] / [Delta][Delta])][A.sub.2]} / {p(1 - p)(1 + [Lambda])([u.sub.3][double prime] + [u.sub.2][double prime]) - (1 - [Lambda])p[prime]([u.sub.2][prime] + [u.sub.3][prime])[Delta][e.sup.*] / [Delta][Delta]},

where

[A.sub.1] [equivalent to] p(1 - p) + (1 - 2p)(1 + [Lambda])p[prime][Delta][e.sup.*] / [Delta][Lambda]

and

[A.sub.2] [equivalent to] (1 - [Lambda])p[double prime]([u.sub.2] - [u.sub.3])[Delta][e.sup.*] / [Delta][Lambda] - [([u.sub.2] - [u.sub.3])p[prime] + 1].

From equation (6), [Delta][e.sup.*]/[Delta][Delta] [less than] 0, and [u.sub.2][double prime] [less than] 0 and [u.sub.3][double prime] [less than] 0 so the denominator of (9) is negative. [A.sub.1] reflects the direct effect of altruism on transfers, while [A.sub.2] reflects the indirect effect of altruism through the effort of j.

PROPOSITION 3. For [Lambda] sufficiently close to 1, [Delta][[Delta].sup.*]/[Delta][Lambda] [greater than] 0.

Intuitively, Proposition 3 says that if altruism does not affect effort directly ([Delta] [e.sup.*]/[Delta] [Lambda] = 0), then increasing altruism has no effect on the probability a spouse will receive [Delta] (nor does altruism affect the rate of change of the accident probability for the same reason). This leaves only two effects of higher transfers: the increased utility the wife receives by helping out her husband when he is injured and she is not [([u.sub.3][prime] - [u.sub.2][prime])p(1 - p)]; and the increased utility an additional unit of [Delta] yields through the change in effort ([Delta] [e.sup.*]/[Delta][Delta][([u.sub.2] - [u.sub.3])p[prime] + 1]). Since both effects increase the marginal utility schedule for [Delta], increasing altruism also increases [Delta].

When [Delta] [e.sup.*] / [Delta][Lambda] [greater than] 0, increased altruism somewhat dampens the effect of [Delta] through decreased effort. Since the husband knows the wife will transfer more to him than if her level of altruism were lower, he can exploit her altruism by reducing effort on the margin.(11)

Altruism reinforces some traits that were present without altruism and reduces other incentives. For example, equation (7) gives the first-order condition for the optimal transfer. The first term, (1 + [Lambda]) p (1 - p) ([u.sub.3][prime] - [u.sub.2][prime]), represents the direct effect of [Delta] on utility, holding constant effort and hence the probability of an accident. Full insurance would equate the marginal utility in each state, so [u.sub.3][prime] = [u.sub.2][prime] if and only if there is full insurance. Increasing [Delta] brings spouses closer to full insurance, holding constant effort (i.e., suppressing moral hazard effects). A higher degree of altruism increases the wife's concern for her husband's lack of full insurance, so increasing [Delta] yields an additional amount [Lambda] p (1 - p) ([u.sub.3][prime] - [u.sub.2][prime]) to the wife's utility.

At the same time, more altruism leaves the wife vulnerable to more exploitation as the husband reduces [e.sup.*] - this is the moral hazard between partners. Reducing [e.sup.i] imposes a loss of ([u.sub.2] - [u.sub.3]) p[prime] (the decrease in utility associated with a greater accident probability), but yields a gain of 1 multiplied by the change in effort - the gain in utility through lower effort. However, this gain in utility to i is at the expense of j's utility, because the wife bears the costs of a higher probability that her husband is in an accident. To the extent that i values E[U.sup.j] by an amount [Lambda], i's utility decreases by [Lambda] [([u.sub.2] - [u.sub.3]) p[prime] + 1] as effort drops in response to an increase in [Delta]. Since [Lambda] [less than or equal to] 1, the own effect outweighs the externality one spouse imposes on the other. As altruism increases, the wife is less concerned with the fact that her husband is taking advantage of nonmarket insurance to reduce his effort. Altruism mitigates the externality that nonmarket insurance causes. With balanced altruism, the externality from reducing effort causes i exactly as much disutility from the decrease in E[U.sup.j] as she gains from the increase in E[U.sup.i]. Thus transfers are neutral with respect to their indirect effect on utility, through effort, when there is balanced altruism.

The Effect of Altruism on Market Insurance

As we noted above, spouses make nonmarket insurance decisions given the premium-payout combination offered by the insurance market. Collectively, though, the effects of nonmarket insurance on effort force a reaction from insurance firms.

Recall that the equilibrium market payment is given by

[[Alpha].sup.*] = [(1 - p([e.sup.*]))[Beta]] / [p([e.sup.*])].

As the degree of altruism increases, [[Alpha].sup.*] changes according to

[Delta] [Alpha].sup.*]/[Delta] [Lambda] = -[p[prime] / p([e.sup.*])][[Delta][e.sup.*] / [Delta] [Lambda]]

[Beta]{[(1 - p)/p] + 1} [greater than] 0.

Increasing altruism increases effort, lowering the probability of an accident. Payouts to the insured decline, increasing profits in the insurance market. Positive profits induce competition in the payout rate. Ultimately, the ratio [Alpha]/[Beta] increases.

IV. WELFARE IMPLICATIONS OF NONMARKET INSURANCE

The previous section was concerned with various consequences of altruism: on effort, nonmarket transfers, and market transfers. Each of the comparative statics told a piece of the story of the overall effect that altruism has on welfare. In this section we bring together these elements in order to assess the net effect of altruism and nonmarket insurance.

Arnott and Stiglitz [1991] note that, in the absence of altruism, nonmarket insurance unambiguously decreases welfare when effort is unobservable. Nonmarket insurance lowers effort below the market-only level, which creates an uninternalized externality. In addition, risk shifts from the insurance company to risk-averse partners (because [Alpha]/[Beta] falls in reponse to lower effort). Both effects make the nonmarket insurance equilibrium Pareto-inferior to that with market insurance only. A social planner would choose the corner solution [Delta] = 0.

Altruism affects this equilibrium in several ways, not all of which work in the same direction. Altruism increases effort (Proposition 2), which reduces the consequences of moral hazard within the family and in the insurance market, increasing welfare. At the same time, altruism increases nonmarket transfers, which shifts proportionately more risk to the family, exacerbating one of the welfare-reducing aspects of nonmarket insurance.(12)

Does altruism, then, increase or decrease overall welfare? Consider a social planner who can impose the desired transfer [Delta], while recognizing that spouses will still set effort to maximize their utility. The planner's problem is to maximize [[Omega].sup.i] over [Delta] subject to the zero-profit constraint for insurers. That is, the planner solves

[Mathematical Expression Omitted]

subject to

-[Alpha]p([e.sup.*]) + [Beta](1 - p([e.sup.*])) = 0.

The function [e.sup.*] = [e.sup.*]([Alpha], [Beta], [[Delta].sup.P]; [Lambda]) is the solution to each partner's maximization problem for effort, where [[Delta].sup.P] is the transfer as determined by the planner.

The first-order condition for this problem is

(10) [Delta][[Omega].sup.i] / [Delta][Delta] = (1 + [Lambda])

{[ - [u[prime].sub.1] - [u[prime].sub.3](1 - p) / p]

[Beta]p[prime][Delta][e.sup.*]/[Delta][Delta] + p(1 - p)[u.sub.3][prime] - [u.sub.2][prime])]} + (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1][Delta][e.sup.*] / [Delta][Delta] = 0.

The first term in brackets in (10) is negative, and represents the effect the transfer has on risk sharing (recall that (1 - p)/p = [Alpha]/[Beta]) and moral hazard between the market insurer and the insured. This term is unique to the planner's problem; the remaining terms are also part of the individual's problem in determining [Delta]. The second term, p(1 - p) ([u.sub.3][prime] - [u.sub.2][prime]), is the effect of more complete insurance on i's own utility and is positive. The remaining term is negative and reflects the moral hazard within the family. This is the externality from nonmarket insurance.

Comparing (10) with (7), one can immediately see that for [Lambda] [less than or equal to] 1 the optimal [Delta] cannot be greater than the (decentralized) equilibrium transfer, [[Delta].sup.*]. With balanced altruism, we can say more:

PROPOSITION 4. When spouses have balanced altruism ([Lambda] = 1), the equilibrium ([[Alpha].sup.*], [[Delta].sup.*], [e.sup.*]) is a constrained optimum.

COROLLARY 1. When [Lambda] = 1, [[Delta].sup.*] = (d - [Alpha] - [Beta])/2.

Recall that [u.sub.2][prime] = [u.sub.3][prime] is also the condition for full insurance when only one spouse is in an accident. The sum of [[Alpha].sup.*] and [[Delta].sup.*] delivers the same utility of consumption whether i or j is insured. The market still provides less than full insurance, leaving utility in the event of an injury to both spouses lower than for other states of nature. When spouses have balanced altruism, Proposition 4 says, full insurance (in the sense above) is both desired by the spouses and optimal. To see why this is so, note that balanced altruism means that i weights j's utility of consumption and effort as much as his own. An accident to j is like an accident to i, from i's perspective, so i wants to equalize his marginal utility with that of j across states of nature. At the same time, i no longer wants to reduce effort on the margin and depend on j to transfer resources to i; the loss in j's utility results in an equal loss to i. The net result is that full insurance is now utility maximizing. And because balanced altruism eliminates the desire to take suboptimal care to avoid an accident, full insurance is also efficient.

The importance of Proposition 4 and Corollary 1 is that the solution to the nonmarket insurance problem when effort is unobservable is the same as the solution to the problem when effort is observable to the partners but partners are not altruistic toward one another. See Arnott and Stiglitz [1991, 186-87]. When effort is observable to the nonmarket partners (but not to the market insurer), partners can contract directly on effort, so they specify the optimal effort level and agree to provide full insurance if the partner in an accident has exerted this level of effort. This result is the "peer monitoring" effort of Arnott and Stiglitz's title: observable effort eliminates the inability to contract on effort, thereby eliminating the moral hazard problem between the partners. Since effort is set optimally, the solution is a constrained optimum. Even though the moral hazard between the insurers and the insured remains, it has no effect because monitoring creates a situation in which nonmarket insurance partners behave in exactly the same way the insurance market would have them behave if effort were observable and insurers could contract on effort.

Exactly the same result is achieved with balanced altruism, but for a different reason. The moral hazard between spouses is not eliminated with balanced altruism, since the spouses still cannot contract directly on effort, but each spouse completely internalizes this source of moral hazard. Because the within-family moral hazard no longer has an effect on effort, the effect of moral hazard between the market insurer and the insured is also reduced. Thus balanced altruism Pareto-dominates no altruism.

Bear in mind that the solution, even with balanced altruism, is still second best. That is, [[Delta].sup.*] = (d - [Alpha] - [Beta])/2 and [e.sup.*] = e([[Delta].sup.*; [Lambda] = 1) is still inferior to [[Delta].sup.*] = 0, [[Alpha].sup.*] = d - [Beta], and the same level of effort. Insurers cannot enforce [e.sup.*], so full market insurance is unprofitable. Without some nonmarket insurance to create the correct incentives for spouses, each takes insufficient care. Hence nonmarket insurance, though it inefficiently shifts risk, is socially desirable.(13)

When [Lambda] [less than] 1 but is sufficiently close to 1, the equilibrium transfer provides something less than full insurance, but the internalizing of the within-family moral hazard, though less than complete, still overcomes the suboptimal sharing of risk. Altruism is still beneficial, and the net effect of nonmarket insurance remains beneficial.(14)

Equations (7) and (10) allow us to compare the equilibrium amount of nonmarket insurance with the socially optimal amount. Proposition 5 summarizes this result.

PROPOSITION 5. When [Lambda] [less than] 1, the socially optimal nonmarket transfer, [[Delta].sup.P], is less than the equilibrium nonmarket transfer, [[Delta].sup.*].

Because nonmarket insurance reduces effort, partners overprovide such insurance, relative to the socially desired level. Because [[Delta].sup.P] = 0 when [Lambda] = 0, it is likely that a corner solution for the planner continues to be the case for [Lambda] close to 0. As [Lambda] [approaches] 1, the moral hazard within the family disappears, causing [[Delta].sup.P] to converge to [[Delta].sup.*] at [Lambda] = 1.

The next section examines the equilibrium and optimal transfers when [Lambda] [greater than] 1.

V. SELF-ECLIPSING ALTRUISM

When the degree of altruism is greater than 1, we enter the region of "self-eclipsing altruism," where the utility of one's spouse matters more than one's own. This section highlights the major differences between the case of [Lambda] [less than or equal to] 1 and [Lambda] [greater than] 1.

Proposition 6 notes that nonmarket transfers actually serve to increase effort when [Lambda] [greater than] 1.

PROPOSITION 6. When [Lambda] [greater than] 1, [Delta][e.sup.i] / [Delta][Delta] [greater than] 0.

With less than balanced altruism, insurance, including nonmarket insurance, creates a disincentive to exert effort to avoid an accident. This disincentive is a negative externality on the insurance market, since spouses fail to take into account the effect of their collective actions on the insurance payout.

In contrast, with self-eclipsing altruism, spouses are more concerned with the costs they impose on others than the costs they impose on themselves. Additional nonmarket transfers benefit an agent only in the event of an accident. To reduce the cost of transfers on his spouse, the agent increases effort to avoid an accident. This behavior creates a positive externality on the insurance market, as we will demonstrate below.

Despite the higher effort as the result of nonmarket insurance, the equilibrium transfer is lower for [Lambda] [greater than] 1 than for balanced altruism ([Lambda] = 1). Recall that balanced altruism leads to a transfer that fully insures the spouse when added to the market insurance payment. Proposition 7 makes this explicit.

PROPOSITION 7. When [Lambda] [greater than] 1, the equilibrium nonmarket transfer is less than the nonmarket transfer when [Lambda] = 1.

Why do nonmarket transfers decline as altruism increases toward balanced altruism? The key is the second term of (7), which represents the utility agent i receives from the effort of his spouse, agent j. When [Lambda] [greater than] 1, i receives disutility from additional effort by j. Spouses care enough about one another that they prefer a reduction in the spouse's effort (which imposes costs on the other spouse through a higher probability of exercising the nonmarket payment) despite the higher costs. Since effort increases in the nonmarket transfer, the only way to accomplish this goal is to reduce the transfer below that for balanced altruism. What might seem like bizarre behavior - reducing the nonmarket transfer below full insurance - is necessary because the transfer is the only method by which one spouse can affect the effort of the other, since spouses cannot observe nor.contract on effort itself.

Proposition 8 considers the welfare effects of nonmarket insurance when spouses exhibit self-eclipsing altruism.

PROPOSITION 8. When [Lambda] [greater than] 1, the socially optimal nonmarket transfer, [[Delta].sup.P], is greater than the equilibrium nonmarket transfer, [[Delta].sup.*].

Proposition 8 shows that individuals underprovide nonmarket insurance when [Lambda] [greater than] 1, whereas Arnott and Stiglitz [1991] showed that individuals overprovide nonmarket insurance for [Lambda] = 0. One way of looking at Proposition 8 is that an individual prefers to lower the nonmarket transfer below that of balanced altruism because of the effect that transfers have on the effort of his spouse. Individuals fail to take into account the externality that nonmarket transfers impose on the insurance market. In this case the externality is positive: higher nonmarket transfers yield higher effort, which feeds back through the insurance market in the form of lower premia for a given insurance payment. Hence a social planner prefers nonmarket transfers that are higher than the level under balanced altruism.(15) Both spouses would be better off, even after the market adjusts, if each could commit to a higher transfer and the same level of effort.

VI. EXAMPLES OF NONMARKET INSURANCE ARRANGEMENTS

Individuals engage in a variety of nonmarket insurance arrangements. This section provides examples of some of these arrangements.

Fully Reciprocated Altruism

Spouses implicitly agree on the additional amount of work one would undertake should the other lose his or her job.(16) For example, should the husband be laid off, a wife may move from part-time to full-time status at her job, or work more overtime hours. Each spouse privately chooses a level of on-the-job shirking, with a higher level of shirking associated with a higher probability of being fired. This implicit, nonmarket agreement supplements any market-based or government-instituted unemployment compensation. Spouses with a higher degree of altruism toward one another may be more likely to internalize the other's loss in utility if a job separation occurs, and hence may choose a lower level of shirking.

A second example similar to the first is that of spouses choosing a level of labor supply should the other become unable to work, whether through an on-the-job injury or a disabling medical condition (e.g., a heart attack or stroke). Each spouse privately chooses a level of risk behavior - an amount of exercise, a choice of diet, or risk-taking behavior on the job - conditional on his or her belief about the implicit agreement.

Not Fully Reciprocated Altruism

Other examples may arise in situations in which altruism is not fully reciprocated; that is, [[Lambda].sup.ij] [not equal to] [[Lambda].sup.ji]. While this paper does not explicitly model such situations, the intuition in these cases is similar to the case in which altruism is fully reciprocated.

Parents and their children often have implicit agreements on a level of financial support should the need arise. If a child loses his job and needs to (temporarily, one hopes) move back with his parents, the level of such support may depend on the degree of altruism the parents feel toward the child. Similarly, if an elderly parent needs supervision, the child may take in the parent, or provide some level of support for such care. Conditional on expectations about the level of support that will be provided if the need arises, the child chooses the amount of effort he expends in education and job training, his level of savings, and the amount of on-the-job shirking. A parent chooses her level of savings and degree of health care, both of which affect the probability she will require assistance from her child.

A final example involves the degree of mutual support in small communities. Community members implicitly agree on the amount of financial assistance or other support (providing meals, transportation, companionship) the community will provide to a member who loses a job, becomes ill, or shows a need for assistance. Community members then choose actions that affect the likelihood of losing a job or becoming ill. As with the previous example, there is no reason to assume that the degree of altruism among all community members, or between parents and their children, is the same; nevertheless, to the extent that altruism does exist, this ameliorates the moral hazard problem that arises in such situations.

VII. DISCUSSION AND CONCLUSIONS

Altruism is inextricably linked to various nonmarket insurance arrangements, so it is only natural that we consider the two together. We examine one aspect of altruism - nonmarket insurance transfers which supplement market insurance - and the effect of these transfers on the insurance market equilibrium and, ultimately, on consumer welfare. The literature on altruistically motivated transfers, such as bequests, is concerned with the interaction only among agents outside the context of a market. By explicitly bringing the insurance market into this literature, we show conditions under which altruism is socially harmful and other conditions under which altruism is beneficial.

By itself, nonmarket insurance is a dysfunctional institution, created as a private response to a market imperfection - the inability of individuals to purchase full insurance - but ultimately harmful because of the way it reduces care to avoid accidents and suboptimally shifts risk. The cause of the market imperfection is the inability of the market insurer to specify a level of effort that the insured must take.

Altruism, however, causes nonmarket partners to take greater care to avoid accidents because neither wants to impose costs on the other. The externality that altruism creates offsets the externality that the inability to contract on effort creates. At the same time, altruism encourages higher levels of nonmarket transfers, reducing the positive effect altruism has on effort and shifting still more risk to the nonmarket partners.

We show that altruism causes one partner to internalize at least some of the moral hazard within the family. If the degree of altruism is high enough, the partners internalize enough of the within-family moral hazard to increase effort to the point where the net effect of nonmarket insurance is beneficial. In the case of balanced altruism, the resulting equilibrium yields a constrained efficient level of effort and transfers.

When partners exhibit self-eclipsing altruism - weighting their spouse's utility more than their own - we show that nonmarket transfers increase the effort each spouse undertakes to avoid an accident. Nevertheless, the equilibrium transfer decreases relative to the case of balanced altruism. Because self-eclipsing altruism creates a positive externality with respect to the insurance market, the social optimum involves a higher level of nonmarket insurance than individuals are willing to supply.

Throughout the paper we have employed the assumption that altruism is reciprocated between partners. How does this assumption affect the results?

Whether altruism is reciprocated or not, agent i, given [e.sup.j], picks effort independently of the degree of altruism j feels toward i. However, j's degree of altruism affects [e.sup.j] and hence affects [e.sup.i] through [e.sup.j]; recall that each partner picks effort given his belief over the effort level of the spouse. We believe that the result that effort is increasing in [Lambda] carries through to the case in which altruism is not reciprocated. If [[Lambda].sup.i] [greater than] [[Lambda].sup.j], then [e.sup.j] will be less than if [[Lambda].sup.i] = [[Lambda].sup.j]. Since i's utility depends in part on the probability that i is in an accident when j is not, lowering [e.sup.j] has the effect of raising [e.sup.i] for a given transfer: i finds himself in a state of nature in which he is uninsured by j (because j is in an accident) more often than if altruism were reciprocated, so i's response is to raise [e.sup.i] to protect himself. By having a lower degree of altruism, j reduces the incentive for i to rely on transfers from j. Nevertheless, we believe this consequence of unreciprocated altruism changes the results of the present paper in a quantitative, rather than qualitative, way.

A second consequence of unreciprocated altruism is that the nonmarket transfers, if set unilaterally, will be different between i and j. Agent j gains less utility from i's utility than i does from j, so one would expect [[Delta].sup.i] [greater than] [[Delta].sup.j]. Both spouses still desire nonmarket insurance, and both would be willing to provide some to the other (recall that i and j perceive nonmarket insurance to be beneficial even when [[Lambda].sup.i] = [[Lambda].sup.j] = 0), but they no longer agree on the magnitude of the transfer. Again, however, the effect is quantitative rather than qualitative.

We have focused on a single type of nonmarket activity - coinsurance. Other kinds of activity may have no market substitutes and so are provided solely by nonmarket institutions; for example, Bernheim, Shleifer, and Summers [1985] point to filial affection as something which has no market counterpart. Another class of activities could be done outside the market but are more efficiently conducted in a market setting. Economies of scale, or comparative advantage, coupled with transactional costs associated with coordinating these activities, are two reasons that market production might be more efficient than nonmarket production. In each case the more effective method (market or nonmarket provision) crowds out the less effective. Even in these cases, however, informational asymmetries can lead to second-best outcomes. It remains an open and interesting question as to whether nonmarket provision of goods and services when no market substitute is available is, in general, efficient.(17)

Finally, whatever the effectiveness of various nonmarket institutions, altruism appears to be a device which allows altruistically linked individuals to internalize nonmarket externalities. In the debate over whether the family or the individual is the appropriate decision-making unit, research must take into account the role of altruism in the process. With enough altruism among family members, the decisions of the individual and the family converge.

APPENDIX

PROOF OF PROPOSITION 1. Hold [Alpha] and [Beta] constant. Then

[Mathematical Expression Omitted]

The first term is simply E[U.sup.j] [greater than] 0. The variables [e.sup.i] and [e.sup.j] are determined in equilibrium by maximizing [[Omega].sup.i] and [[Omega].sup.j], so [Delta][[Omega].sup.i] / [Delta][e.sup.i] = [Delta][[Omega].sup.j] / [Delta][e.sup.j] = 0. In a symmetric equilibrium, [Delta] also maximizes [[Omega].sup.i] and [[Omega].sup.j], so [Delta][[Omega].sup.i]/[Delta][Delta] = [Delta][[Omega].sup.j] / [Delta][Delta] = 0. Then

[Mathematical Expression Omitted].

Proof of Lemma. 1. From the first-order condition (4) above, [Delta][[Omega].sup.i]/[Delta][e.sup.i] = 0. Taking the differential of (4) with respect to [e.sup.i] and [Delta] and manipulating the resulting expression,

(11) [Delta][e.sup.i]/[Delta][Delta] = -{[([u.sub.2][prime] - [Lambda][u.sub.3][prime])p

+ ([u.sub.3][prime] - [Lambda][u.sub.2][prime])(1 - p)]p[prime]}/{[[Delta].sub.1] + [[Delta].sub.2]}

where

[[Delta].sub.1] = [(1 + [Lambda])([u.sub.0] + [u.sub.1]) - ([u.sub.2] + [Lambda][u.sub.3]) - ([u.sub.3] + [Lambda][u.sub.2])][(p[prime]).sup.2]

and [[Delta].sub.2] = {(1 + [Lambda])[-[u.sub.0](1 - p) + [u.sub.1]p]

- ([u.sub.2] + [Lambda][u.sub.3])p + ([u.sub.3] + [Lambda][u.sub.2])(1 - p)}p[double prime].

Substituting equation (4) into (11) and manipulating yields

(12) [Delta][e.sup.i]/[Delta][Delta] = { - [([u.sub.2][prime] - [Lambda][u.sub.3][prime])p

+ ([u.sub.3][prime] - [Lambda][u.sub.2][prime])(1 - p)]p[prime]}/{(1 + [Lambda])([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])[(p[prime]).sup.2] + (p[double prime]/p[prime])}.

Risk aversion implies ([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3]) [less than] 0 [ILLUSTRATION FOR FIGURE 1 OMITTED], while p[double prime] [greater than] 0 and p[prime] [less than] 0, so the denominator of (12) is negative. To sign the numerator of (12), note that [Lambda] [element of] [0,1]. If [Lambda] = 0, the numerator is

-[[u.sub.2][prime]p + [u.sub.3][prime](1 - p)]p[prime] [greater than] 0.

If [Lambda] = 1, the numerator is

([u.sub.3][prime] - [u.sub.2][prime])(1 - 2p)p[prime]

which is nonnegative. By inspection, if the numerator is positive when [Lambda] = 1, the numerator is positive for all [Lambda] [element of] (0, 1), since the expression is decreasing in [Lambda]. By symmetry, the right-hand side of (12) is also the expression for [Delta][e.sup.j]/[Delta][Delta]. When accidents are infrequent, additional transfers increase the moral hazard problem within the family and reduce effort.

Proof of Lemma 2. Differentiating (3) with respect to [e.sup.j],

(13) [Delta][[Omega].sup.i]/[Delta][e.sup.j] = [(1 + [Lambda])(-[u.sub.0](1 - p) + [u.sub.1]p)

+ (1 - p)([u.sub.2] + [Lambda][u.sub.3]) - p([u.sub.3] + [Lambda][u.sub.2])]p[prime] - [Lambda]

where p[prime] is now evaluated at [e.sup.j]. Using symmetry and the first-order condition (4),

(14) [Delta][[Omega].sup.i]/[Delta][e.sup.j] = [(1 + [Lambda])(-[u.sub.0](1 - p) + [u.sub.1]p)

- p([u.sub.2] + [Lambda][u.sub.3]) + (1 - p)([u.sub.3] + [Lambda][u.sub.2])]p[prime] - 1

+ (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1]

= [Delta][[Omega].sup.i]/[Delta][e.sup.i] + (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1]

= (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1]

since [Delta][[Omega].sup.i]/[Delta][e.sup.j] = 0 at e* and p([e.sup.i]) = p([e.sup.j]) = p(e*) = p by symmetry. Note that, with balanced altruism, [Delta][[Omega].sup.i]/[Delta][e.sup.j] = 0: the wife weights her husband's utility as much as her own.

Holding e constant we obtain

(15) [Mathematical Expression Omitted].

Combining the results in (15), (12), and (14),

(16) [Delta][[Omega].sup.i]/[Delta][Delta] = (1 + [Lambda])p(1 - p)([u.sub.3][prime] - [u.sub.2][prime])

+ (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1][Delta][e.sup.j]/[Delta][Delta].

Note that equation (16) is the same equation (7) following Lemma 2. At the competitive equilibrium without nonmarket insurance, [Delta] = 0. Then [u.sub.0] = [u.sub.2] and [u.sub.1] = [u.sub.3]. Re-evaluating (4),

(17) [Mathematical Expression Omitted].

But then ([u.sub.3] - [u.sub.2])p[prime] = 1 so

(1 - [Lambda])[([u.sub.2] [u.sub.3])p[prime] + 1] = 0

and (16) reduces to

(18) [Mathematical Expression Omitted].

Equation (18) indicates that individuals value nonmarket insurance, so [Delta]* [greater than] 0.(18)

To prove Proposition 3, we employ the following lemma:

LEMMA 3. Suppose [Lambda] = 1. Then [u.sub.2] = [u.sub.3].

Proof. When [Lambda] = 1, equation (16) becomes

[Delta][[Omega].sup.i]/[Delta][Delta] = 2p(1 - p)([u.sub.3][prime] - [u.sub.2][prime]) = 0.

Hence the equilibrium value of [Delta] must satisfy [u.sub.3][prime] = [u.sub.2][prime]. But the continuity of u([center dot]) requires that, for the derivatives to be equal, [u.sub.2] = [u.sub.3].

Proof of Proposition 3. At [Lambda] = 1, Lemma 3 shows that [u.sub.2] = [u.sub.3] so [u.sub.2][prime] = [u.sub.3][prime] and [u.sub.3][double prime] = [u.sub.2][double prime]. Then as [Lambda] approaches 1 from below, the denominator of (9) approaches 4 p(1- p)[u.sub.2][double prime] [less than] 0. [A.sub.2] approaches -1 so the numerator of (9) approaches [Delta]e*/[Delta][Delta], which is negative.

Hence [Delta][Delta]*/[Delta][Lambda] [greater than] 0 near [Lambda] = 1. From the continuity of u([center dot]) and p, [Delta][Delta]*/[Delta][Lambda] [greater than] 0 for k less than but sufficiently close to 1.

Proof of Proposition 4. The socially optimal solution for [Delta] reflects the zero-profit constraint of the firms. Thus [[Delta].sup.P] = [Delta]([Alpha],[Beta] [where] [Alpha] = {[1 - p(e)]/p(e)}[Beta]). Hence

(19) [Delta]e*/[Delta][[Delta].sup.P] = -{[(1 - p)[u.sub.3][prime] - [Lambda][u.sub.2][prime]) + p([u.sub.2][prime] - [Lambda][u.sub.3][prime])]p[prime]}/{[(1 + [Lambda])([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])][(p[prime]).sup.2] + (p[double prime]/p[prime]) + ([Beta][(p[prime]).sup.2]/[p.sup.2])[-(1 + [Lambda])p[u.sub.1][prime] + [u.sub.3][prime](p[Lambda] - (1 - p))]}.

At [Lambda] = 1, (19) becomes

[Delta]e*/[Delta][[Delta].sup.P] = -{[u.sub.2][prime]p - [u.sub.3][prime](1 - p)}/{2([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])p[prime] + [p[double prime]/[(p[prime]).sup.2]] - [([Beta]p[prime])/[p.sup.2]](2p[u.sub.1][prime] + (1 - p)[u.sub.3][prime])}.

Substituting (19) into (10) and rearranging, we have:

(20) [Delta][[Omega].sup.i]/[Delta][Delta] = [2([u.sub.3][prime] - [u.sub.2][prime])]/[Delta] [multiplied by] {[Beta][u.sub.1][prime]p[prime]p - [Beta][u.sub.3][prime]p[prime][[(1 - p).sup.3]/p] - p(1 - p)[p[double prime]/[(p[prime]).sup.2]] -2p(1 - p)p[prime]([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])}

where

[Delta] = -[2([u.sub.0] + [u.sub.1] - [u.sub.2] - [u.sub.3])p[prime] + [p[double prime]/[(p[prime]).sup.2]] - [([Beta]p[prime])/[p.sup.2]](2p[u.sub.1][prime] + (1 - p)[u.sub.3][prime])].

Note that in (20) the term in brackets that is multiplied by 2([u.sub.3][prime] - [u.sub.2][prime]) is negative, and [Delta] is negative, so (20) equals zero when [u.sub.2][prime] = [u.sub.3][prime].

Now examine the decentralized transfers, [Delta]*. From (15), when [Lambda] = 1, [Delta]* solves

[Delta][[Omega].sup.i]/[Delta][Delta] = 2p(1 - p)([u.sub.3][prime] - [u.sub.2][prime]) = 0.

Again, [u.sub.2][prime] = [u.sub.3][prime]. Hence [Delta]* is a constrained optimum.

Proof of Corollary 1. [u.sub.2][prime] = [u.sub.3][prime]. Then (w - [Beta] - [Delta]*) = (w - d + [Alpha] - [Delta]*). The result is immediate.

Proof of Proposition 5. Let [Delta]* be the [Delta] that solves the individual's maximization problem, and let [[Delta].sup.P] be the [Delta] that solves the social planner's maximization problem. From (9) and the continuity of [u.sub.i], i = 0, ..., 3, [Delta]* is continuous with respect to k. The same argument shows that [[Delta].sup.P] is also continuous in [Lambda]. From Arnott and Stiglitz [1991], [[Delta].sup.P] [less than] [Delta]* when [Lambda] = 0, while [[Delta].sup.P] = [Delta]* at [Lambda] = 1 from Lemma 3. Factoring out ([u.sub.3][prime] - [u.sub.2][prime]) from (10), we note that [[Delta].sup.P] = [Delta]* if and only if [u.sub.3][prime] = [u.sub.2][prime]. However, Proposition 4 shows that this occurs only when [Lambda] = 1. Then continuity of [[Delta].sup.P] and [Delta]* implies that [[Delta].sup.P] - [Delta]* [less than] 0 for all [Lambda] [less than] 1.

Proof of Proposition 6. From (6), increasing [Lambda] above 1 does not change the sign of the denominator, which remains negative. However, the numerator is negative for all [Lambda] [greater than] 1, so [Delta][e.sup.i]/[Delta][Delta] [greater than] 0.

Proof of Proposition 7. The equilibrium transfer sets equation (16) to zero. When [Lambda] [greater than] 1, ([u.sub.3][prime] - [u.sub.2][prime]) [greater than] 0. We show this by contradiction. Suppose ([u.sub.3][prime] - [u.sub.2][prime]) [less than] 0. Then ([u.sub.2] - [u.sub.3]) [less than] 0 so (1 + [Lambda])p (1 - p)([u.sub.3][prime] - [u.sub.2][prime]) [less than] 0 and (1 - [Lambda]) [([u.sub.2] - [u.sub.3])p[prime] + 1] [Delta][e.sup.j]/[Delta][Delta] [less than] 0 (where [Delta][e.sup.j]/[Delta][Delta] [greater than] 0 when [Lambda] .[greater than] 1 from Proposition 6). But then (16) cannot equal zero. Hence [u.sub.3][prime] [greater than] [u.sub.2][prime]. But [u.sub.3][prime] is decreasing in [Delta] and [u.sub.2][prime] is increasing in [Delta], so [Delta]* when [Lambda] [greater than] 1 is less than [Lambda]* when [Lambda] = 1.

Proof of Proposition 8. Let [[Delta].sup.P] be the transfer that solves the planner's problem, and let [Delta]* be the transfer that solves the individual's problem. From Proposition 6, [Delta]e*/[Delta][Delta] [greater than] 0 when [Lambda] [greater than] 1 so

(1 + [Lambda]){-[u.sub.1][prime] - [u.sub.3][prime][(1 - p)/p]}[Beta]p[prime]([Delta]e*/[Delta][Delta]) [greater than] 0.

At [Delta]*.

(1 + [Lambda])p(1 - p)([u.sub.3][prime] - [u.sub.2][prime])

+ (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1]([Delta]e*/[Delta]) = 0;

this is the first-order condition for the individual agent. Combining these expressions,

(1 + [Lambda])p(1 - p)([u.sub.3][prime] - [u.sub.2][prime])

+ (1 - [Lambda])[([u.sub.2] - [u.sub.3])p[prime] + 1][Delta]e*/[Delta])

+ (1 - [Lambda]){-[u.sub.1][prime] - [u.sub.3][prime][(1 - p)/p]}[Beta]p[prime]([Delta]e*/[Delta][Delta]) [greater than] 0.

However, this expression is the first-order condition for the social planner, which at [[Delta].sup.P] must be equal to zero. Hence, for the planner,

[Mathematical Expression Omitted].

This [[Delta].sup.P] [greater than] [Delta]*.

1. Stark [1989] shows how altruism leads to cooperation in the standard Prisoner's Dilemma game if the agents care "enough" about each other.

2. This is not to say that motivations other than altruism are not important. In the case of welfare, for example, one might argue that political gains play a role in determining the size of the allocation of the welfare pie. However, to the extent that taxpayers agree to transfer resources to welfare recipients out of concern for their fellow humans, altruism plays a role.

3. See Hirshleifer [1977]; Bernheim and Stark [1988]; Bergstrom [1989]; Bruce and Waldman [1990]; Becker [1991]; and others. Presenting an alternative view is Bernheim, Shleifer, and Summers [1985], in which parents use transfers to induce their children to provide various nonmarket services, such as affection. Transfers buy these services within the household. These transfers, at least on the margin, are nonaltruistic in nature. However, Bernheim, Shleifer, and Summers model a single market with full observability of actions.

4. Rasmusen [1989] refers to this as certain information.

5. See also Coate [1995] and Chami and Fischer [1996] for examples of altruism and the interaction between market and nonmarket institutions.

6. When effort is observable between partners, partners can make nonmarket transfers conditional on effort. Arnott and Stiglitz show that this "peer monitoring" effect disciplines individuals and creates a positive externality on the insurance market. In this case, nonmarket insurance is welfare enhancing.

7. In solving the model, we do not consider the strategies discussed in Bernheim and Stark [1988]. They note that "in a variety of situations, altruism entails exploitability, and therefore causes family members to behave in ways that leave all parties worse off" (p. 1035). Thus they show that altruists, for example, may overconsume in an early period to reduce the incentive for the beneficiary to overconsume in expectation of a large transfer in a later period (the Samaritan's Dilemma; see Bruce and Waldman [1990]). They also note that threats to enforce cooperative behavior by punishing defectors are less effective the higher is the degree of altruism. While recognizing the importance of this kind of strategic behavior, we do not consider cooperative solutions because they are unenforceable in a one-shot game of imperfect information. In our model, the lack of full insurance, along with the presence of altruism, helps ameliorate the Samaritan's Dilemma. The lack of full insurance implies that an accident imposes more costs than benefits on the victim, even with insurance, so the incentive to "overconsume" (in the sense of strategically reducing effort in order to raise the probability of an accident) is reduced. Similarly, altruism induces each spouse to internalize the externality imposed on the other. When altruism is balanced ([Lambda] = 1 in the model), the Samaritan's Dilemma disappears. For a discussion of this effect in the context of bequests, see Chami [1992], particularly the section on time-consistent contracts.

8. See Figure 1.

9. When altruism is not reciprocated, it does not necessarily follow that i wants to match with the partner for whom he feels the strongest. Bernheim and Stark [1988] show that when one partner cares more about the other, he is vulnerable to exploitation by his partner. To avoid this possibility, individuals are likely to pair with people of a similar degree of altruism. With a large number of possible matches, this symmetry is not unreasonable. See Chami [1992] for an analysis of one-sided altruism.

10. We suppress the superscript on [p.sup.i] in the symmetric equilibrium because [e.sup.i] = [e.sup.j] = [e.sup.*] so [p.sup.i] = [p.sup.j] = p.

11. The same kind of behavior is present in Bernheim and Stark [1988], though in a model without any informational asymmetry.

12. We also showed that [Alpha]/[Beta] increases, which benefits the insured. However, ceteris paribus, this is merely a transfer of wealth from insurers to the insured, a transfer which is neutral from a welfare standpoint.

13. The effort-observable case parallels this result, in the sense that nonmarket insurance is necessary as long as effort remains private information unavailable to the insurance firm.

14. This is a result of the continuity and monotonicity of the utility function and the fact that [Lambda] = 1 is a constrained optimum (Proposition 4).

15. The planner still chooses a [[Delta].sup.P] such that ([u.sub.3][prime] - [u.sub.2][prime]) [greater than] 0, so the social optimum involves less than full insurance. However, individuals perceive [Alpha] and [Beta] to be fixed. Given the belief that [Alpha] and [Beta] are the same for [Lambda] [greater than] 1 as for [Lambda] = 1, the planner prefers a higher nonmarket transfer than for [Lambda] = 1; the planner recognizes that the market insurance premium will adjust accordingly.

16. We would like to thank the coeditor for suggesting this example.

17. Chami [1992] deals with the provision of nonmarket transfers under asymmetric information and asymmetric altruism when an altruistic parent cares about the utility of a nonaltruistic child and cares about the level of effort the child incurs in achieving that utility. The Chami paper has no outside market for transfers per se, but does include outside income which controls the effectiveness of the nonmarket transfer as a motivational device. Transfers which are compensatory on the margin create a disincentive to increase effort in the labor market, an externality which the employer, since he cannot observe the transfer, cannot internalize through his wage contract.

18. Note that the sign of (18) is independent of the value of [Lambda], so this result does not depend on altruism.

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