Uncertain Bequest Needs and Long-Term Insurance Contracts 1

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1 Uncertain Bequest Needs and Long-Term Insurance Contracts 1 Wenan Fei (Hartford Life Insurance) Caude Fuet (Université du Québec à Montréa and CIRPEE) Harris Schesinger (University of Aabama) Apri 22, The authors thank Michae Hoy for hepfu comments on a draft of this paper.

2 Abstract We examine how ong-term ife insurance contracts can be designed to incorporate uncertain future bequest needs. An individua who buys a ife insurance contract eary in ife is often uncertain about the future nancia needs of his or her famiy, in the event of an untimey death. Ideay, the individua woud ike to insure the risk of having high future bequest needs; but since bequest motives are typicay unveri abe, a contract directy insuring these needs is not feasibe. We derive two equivaent ong-term ife insurance contracts that are incentive compatibe and achieve a higher wefare eve than the naïve strategy of deaying the purchase of insurance unti after one s bequest needs are known. We aso examine the wefare e ects of such contracts and we show how third-party nancia products, athough bene cia to the individua in the short run, can be wefare decreasing over one s ifetime Keywords: Asymmetric information, Bequest needs, Life insurance JEL Cassi cation: D82, D91, G22

3 1 Introduction Peope purchase ife insurance to protect their dependents against nancia osses caused by their deaths. In the ife insurance market, most contracts extend many years into the future. Such prevaence of ong-term contracts is party due to the premium risk or "insurabiity risk." In particuar, a person s heath status may deteriorate, which makes short-term ife insurance no onger a ordabe in the future. In the extreme, a person s heath coud deteriorate to such an extent that no ife insurance is avaiabe. A ongterm insurance contract with a front-oaded premium schedue, in a certain sense, aso provides insurance against this "insurabiity risk." However, even without such insurabiity risk, a ong-term contract can be bene cia. In particuar, we show how such arrangements can improve wefare by partiay insuring the risk of having a high bequest demand in the future. Athough it may be advantageous from an insurabiity standpoint to arrange for ife insurance eary, the need for ife insurance many years ater depends on the future demographic structure of the househod and may not be known in advance. The impact of one s death often depends on the - nancia condition of other famiy members, as we the future preferences of these famiy members, as examined by Lewis (1989). Moreover, as shown by Bernheim (1991), there is a demand by breadwinners to hod part of their assets in a soey bequeathabe form, as opposed a form that coud aso be used for current consumption if one is aive. Absent any insurabiity risk, it woud at rst appear to be optima to purchase ife insurance contracts ater in ife, when bequest needs are better known. Another possibiity is to purchase short-term contracts and to adjust the insurance eve as needed at a ater date, as in Poborn et a. (2006). If the status of one s heath is private information, this runs into the probem of 1

4 renewabiity risk. 1 However, even without the insurabiity risk, a short-term purchasing strategy for ife insurance is not optima. Intuitivey, athough deaying the purchase of ife insurance can hep individuas to determine the appropriate eve of insurance, in concordance with their known bequest demand, one must sti pay the extra insurance premium if one s demand turns out to be high. That is, one must pan for the possibiity of needing to spend more on insurance premia in the future. Note that this form of "premium risk" has nothing to do with the insurabiity risk. Here the risk is on the budget required to nance the required amount of ife insurance; not on whether or not the premium rate is higher. In this paper, we consider the design of a ong-term ife insurance contract that aso can hep to mitigate the risk of possiby having a high bequest need in the future. Our mode is simiar to that of Poborn et a. (2006), except that we do not consider the insurabiity risk. With no insurabiity risk, but with a risk of demand type, the insurance premium per unit of coverage shoud not change for short-term contracts. Hence one can aways buy more ife insurance ater at the same price. Poborn et. a. (2006) aso mention the case with no insurabiity risk, but they concude that there is no bene t to purchasing insurance earier, since they require zero-pro t insurance pricing within each period. 2 However, a ong-term contract can aso hep to mitigate the risk of bequest type. Athough this risk introduces no price risk per se, it does require that individuas with a high-bequest demand spend a higher share of their weath on ife insurance. Thus, a high-bequest demand eads to ess consumption 1 See Pauy et a. (1995). However, if this change in insurabiity is observabe, it might be possibe, at east in theory, to insure it directy in a manner simiar to Cochrane (1995). For commitment probems associated with ong-term contracting when changes in insurabiity are unobservabe risk see Hende and Lizzeri (2003). 2 Sheshinski (2007) derives simiar resuts in mode with annutities, rather than ife insurance. 2

5 than a ow-bequest demand, if an individua does not die eary. We show ong-term contracts can partiay hedge this future consumption risk. This is accompished by e ectivey transferring some weath in future states where one s bequest needs are ow to states for which bequest needs are higher. Since bequest needs are not ikey to be easiy veri abe, the contract cannot just pay a transfer to anyone who caims to have high bequests needs. Hence, the ong-term contract is written with particuar options, and the exercise of these options occurs via sef seection. Long-term contracting aso aows for any zero-pro t condition to impemented over the duration of the contract, rather than within each time period. In the next section, we set up the basic mode. We then examine a rst-best word in which bequest type is veri abe. We examine the optima insurance contract, which aso provides protection again the risk of having a high bequest need in this setting. Next, we derive two equivaent ongterm ife insurance contracts for the case where bequest type is unveri abe. These contracts are incentive compatibe and achieve a higher wefare eve than the naïve strategy of deaying the purchase of insurance unti after one s bequest needs are known. These second-best contracts are aso compared to the rst-best case. We concude by expaining how some reativey new third party nancia products, especiay so-caed "ife settement" contracts, can upset this ong-term contract arrangement. 2 The Mode We deveop a simpe three-period mode of ife-insurance purchases when individuas are uncertain about their bequest preferences. A person with initia weath w 0 at date t = 0 earns of his preferences for bequest at date t = 1. The individua faces a probabiity q of death at date t = 2. With probabiity 1 q, the individua ives to consume another period. To keep 3

6 the mode simpe and to focus on bequest needs, q is non-random and, thus, there is no insurabiity risk. For simiar reasons, we further assume that the interest rate for borrowing or ending is zero. Denote by w d and w the individua s na weath in the states of death and surviva respectivey. Let i refer to the individua s type with respect to preferences for bequest at t = 1. The expected utiity of na weath is then qv i (w d ) + (1 q)u(w ); where v i (w d ) is the utiity of eaving weath w d to dependents at t = 2 and u(w ) is the utiity of weath w in the state of iving. Both functions are increasing and stricty concave. Moreover, v 0 i(w) > u 0 (w) for a w, impying a demand for ife insurance. Taken together, v i (w d ) and u(w ) can be viewed as a state-dependent vaue function for the utiity derived from the optima consumption and savings strategies, given the individua s weath in each state at the beginning of date t = 2 and taking impicity into account the future abor income that a surviving individua woud earn. 3 Bequest needs are initiay uncertain. At t = 0, the individua does not know his bequest utiity function, which can be either v B () with probabiity or v A () with probabiity 1. We assume that vb 0 (w) > v0 A (w) for a w, so that type B is the high-bequest type. An individua s type, once earned, is private information, but insurers know the proportion of types in the popuation. Any amount of ife insurance coverage can be purchased at any time before t = 2. Let L be the death bene t purchased. 4 The ife 3 In this setting, we can view w 0 as the present vaue of ifetime earnings for this person, whom for simpicity we wi consider as the soe "bread winner" for the famiy. person dies, then famiy ifetime income wi become ess. If the The state-dependent utiity v captures this income oss. Obviousy, we are simpifying the basic insurance decision to a great extent. For exampe, we do not consider that future income might be risky, nor do we consider intermediate consumption. See, for exampe, Campbe (1980). For a survey of many of these theoretica ife insurance issues, see Vieneuve (2000). 4 We ignore any savings component buit into many ife insurance contracts. In this 4

7 insurance premium is assumed to be actuariay fair, so the premium for the amount of coverage L is ql. As a preiminary step, we examine an individua s demand for ife insurance when coverage is purchased at t = 1, after the individua has earned his type. We then show that, from the perspective of t = 0, the individua woud ike to insure against the risk of being a high-bequest type. However, insurance against such a risk cannot be bought directy, since one s type is unveri abe. 3 Bequest type is veri abe Here we consider two insurance strategies. The rst is simpy to wait unti bequest type is known before buying insurance. Even in a word with no insurabiity risk, the individua has a risk as to how much the tota expenditure on insurance wi be. In an idea word, where bequest type is veri abe, this risk can be insured. The naïve strategy The simpest strategy for buying ife insurance is to wait unti t = 1 and to purchase coverage after earning one s type. It is usefu to characterize the demand for coverage as a function of some arbitrary weath w at date t = 1. Obviousy, if nothing has been done before this date, then w = w 0. For an individua with bequest type i and weath w at date t = 1, the ife-insurance objective is to max L i qv i (w ql i + L i ) + (1 q)u(w ql i ); i = A; B: sense, we can regard L as the pure death-protection bene t that is paid in the event of an eary death at date t = 2. More simpy, we can view the insurance as a type of term ife insurance product that ony pays a bene t if death is at date t = 2. 5

8 The optima coverage L i (w) satis es the rst-order condition v 0 i(w ql i (w) + L i (w)) u 0 (w ql i (w)) = 0; i = A; B: (1) Risk aversion ensures that the second-order condition is satis ed. It is easiy checked that L B (w) > L A (w), i.e., B is indeed the high-bequest type. Substituting for the optima amount of coverage yieds the date 1 optima expected utiity V i (w) qv i (w ql i (w) + L i (w)) + (1 q)u(w ql i (w)); i = A; B: Here V i (w) is the vaue function for a person of type i at date t = 1, who has weath w at that date. Viewed from date t = 0 and treating bequest type as a random variabe, V i (w) is a state-dependent utiity function exhibiting risk aversion in each state of the word. To see this, appy the enveope theorem and use (1) to obtain V 0 i (w) = u 0 (w ql i (w)); i = A; B: (2) Since L B (w) > L A (w), it foows that V 0 B (w) > V 0 A (w). second time yieds Di erentiating a The sign foows from V 00 i (w) = u 00 (w ql i (w)) (1 ql 0 i (w)) < 0: 1 ql 0 i (w) = v 00 i (1 q)v 00 i + qu00 > 0; (3) where the expression is obtained by tota di erentiation of (1). From (3), it is aso easiy veri ed that 1 ql 0 i (w) + L 0 i (w) = u 00 (1 q)v 00 i + qu00 > 0: (4) Thus, bequest and net weath in the surviva state are norma goods, i.e., w i w ql i (w) and w d i = w ql i (w) + L i (w) are stricty increasing in the date 1 weath w. 6

9 Insurance against bequest type An individua who decides to wait unti date t = 1 to purchase ife insurance knows that he wi purchase either L A (w 0) or L B (w 0), depending on his bequest needs. At date t = 0, his expected utiity is therefore (1 )V A (w 0 )+ V B (w 0 ). Since V 0 B (w 0) > V 0 A (w 0), transferring weath at a fair price from the ow to the high margina utiity state increases expected utiity. di erenty, the individua woud ike to insure against the risk of being a high-bequest type. Put Suppose for now, contrary to our earier assumption, that bequest types are veri abe. A contract coud then be written at date t = 0 that pays some amount Q at date t = 1 if the person turns out to be type B. The fair premium for such a contract is Q paid at date t = 0. The date 1 weath is now either w A = w 0 Q or w B = w 0 Q+Q depending on the individua s reaized bequest type, where (1 )w A + w B = w 0. It is a simpe dynamic programming probem to maximize the expectation of the vaue function max Q (1 )V A(w 0 Q) + V B (w 0 Q + Q): The optima Q satis es the rst-order condition V 0 B(w 0 Q + Q ) V 0 A(w 0 Q ) = 0: (5) It foows triviay that Q > 0, so that w B > w 0 > w A. The ife insurance purchased is then L A (w A ) if needs are ow and L B (w B ) if they are high. The possibiity of insuring against bequest needs yieds a soution characterized by where u 0 (w A) = u 0 (w B) = v 0 A(w d A ) = v 0 B(w d B ); (6) w i = w i ql i (w i ), w d i = w i ql i (w i ) + L i (w i ); i = A; B: 7

10 We essentiay have a compete contingent caims market and equate margina utiity in a four possibe states of the word. This is achieved by combining two types of insurance products: one insures against a premature death and the other insures the uncertain bequest needs. Coverage against the risk of being the high bequest type, equivaenty the transfer of weath from state A to state B individuas is wb wa = Q = q(l B (w B) L A (w A)), the di erence in the ife insurance premia. We wi refer to this set of contracts as the rst-best soution. Comparison It is instructive to compare this rst-best soution with the naïve strategy used when preference risks are not insurabe. Using the rst-best strategy, weath in the surviva state is now equaized across bequest types. Moreover, because of the weath transfer and since bequests are norma goods, bequests are now arger in the high-bequest state and smaer in the ow-bequest state, i.e., w d B > w 0 ql B (w 0) + L B (w 0) and w d A < w 0 ql A (w 0) + L A (w 0). Thus, the possibiity of insuring against preference risks aows the bequest amount to more cosey re ect needs. -- Fig. 1 and 2 about here -- Figures 1 and 2 provide a state-space representation of the consumer s probem at date t = 1, when bequest type is known but one s date of death is sti uncertain. In gure 1 preference risks are not insured. The negativey soped straight ine is the budget constraint arising from the insurer s zero pro t condition, i.e., (1 q)wi + qwi d = w 0 : Indi erence curves (iso-expected-utiity) for both bequest types are shown. For bequest type i, the margina rate of substitution between weath in the 8

11 surviving state and weath in the death state is dw i qv 0 = i(w d ) (1 q)u 0 (w ) : dw d i For each type, the equiibrium occurs at the tangency point with the budget ine, impying v 0 i(w d i ) = u 0 (w i). Since type B has steeper indi erence curves, wb d > wd A and w B < w A, eading to equiibria such as the contingent caims E A and E B in gure 1. 5 Figure 2 iustrates the case where preference risks can be insured. The date 1 budget constraint is then (1 q)w i + qw d i = w i ; i = A; B: The equiibrium contingent caims in this case are E A and E B characterized by the condition wb = w A. Moreover, wd A is smaer and wd B arger than the corresponding amounts in the uninsured case. Of course, direct insurance against bequest type is not feasibe if one s bequest type is unveri abe. An individua purchasing such a poicy woud aways want to caim that he is the high bequest type in order to receive the indemnity Q. This is obvious from gure 2. Rather than staying at E A, a type-a individua is better o caiming he is B and moving to the higher budget ine. 4 Bequest type is unveri abe We now turn our attention to the case where bequest type is private information and show how we can improve upon the naïve strategy of waiting unti date t = 1 to purchase insurance. Note that it does not matter whether or not type is veri abe by the insurer to impement the naïve strategy. 5 See Karni (1985) for a genera treatment of modes using such state-dependent preferences. 9

12 Long-term ife insurance contracts are purchased at date t = 0, before individuas know their bequest preferences. Many extant ife insurance contracts often incude provisions that aow for changes to the contract at some future date, at the option of the insured. One such type of provision is an opting out opportunity: the insured can trade-in his poicy at a ater date at some pre-speci ed buy-back price. Aternativey, the contract can incude an option for the purchase of additiona coverage at some pre-speci ed rate. We show that such ong-term insurance contracts can improve the individua s wefare even though bequest types are non-veri abe. In particuar, a we designed poicy aows weath to e ectivey be transferred from type-a individuas to type-b individuas. Opting out contracts We consider a contract with a se back option. We de ne the contract by the tripet (P; L; K), where P denotes the premium paid at t = 0, L denotes the death bene t and K is the price at which the poicy can be traded in (i.e. sod back to the insurer) at date t = 1. With insurers earning zero pro t, if ony type-a individuas se back their poicies, such a contract wi e ectivey transfer the amount ql K from type-a individuas to type-b individuas at date t = 1. In essence, the insurer ses the origina coverage L at a subsidized price. The insurer nances this subsidy by buying back the poicy at an unfair price from the ow-bequest types, who then subsequenty purchase a ower eve of coverage. 6 Such an arrangement works if the foowing three incentive-compatibiity conditions are satis ed: 6 As we show beow in Proposition 2, the B-type contract provides more insurance than woud be desirabe with just a weath transfer. Thus, the B-types have no desire to purchase more insurance at date t = 1. Indeed, they woud ideay ike to purchase ess coverage at date t = 0, but are forced to over-insure in satisfying the constraint (7). 10

13 (a) type-a individuas choose to se back their poicy at t = 1 and buy a new short-term poicy on the spot market at actuariay fair prices 7 : V A (w 0 P + K) qv A (w 0 P + L) + (1 q)u(w 0 P ): (7) (b) type-b individuas prefer keeping their poicy at t = 1: qv B (w 0 P + L) + (1 q)u(w 0 P ) V B (w 0 P + K): (8) (c) From the perspective of date t = 0, the arrangement dominates the strategy of waiting unti t = 1 to buy insurance: U (1 )V A (w 0 ) + V B (w 0 ); (9) where U denotes the expected utiity provided by the ong-term contract U (1 )V A (w 0 P +K)+ [qv B (w 0 P + L) + (1 q)u(w 0 P )] : (10) In addition, the contract must yied a non-negative pro t: P ql + (1 )K: (11) It is easiy seen that the set of contracts that satisfy the above constraints is not empty. In particuar, consider the contract de ned by L = L B (w 0) and P = K = ql B (w 0), where L B (w 0) is the optima death bene t for type B under the naïve strategy. The non-negative pro t condition and (8) are then satis ed as equaities, and (7) is satis ed as a strict inequaity. Ceary, this arrangement yieds the same outcome as the naïve strategy described in gure 1, impying that (9) is then satis ed as an equaity. 7 We make the usua assumption that an individua chooses the action designed for him when he is just indi erent between two courses of action. 11

14 In a competitive market, insurers are ed to o er the best contract subject to pro ts being non negative. The equiibrium contract is therefore the one that maximizes U de ned as in (10) subject to the non-negative pro t condition and the incentive compatibiity conditions. Since it is a maximum, the optima contract is at east as good as the naïve strategy, i.e., the constraint (9) is triviay satis ed. Aso, given K P, it is easiy checked that (9) impies (8). Thus, the ony reevant constraints are (7), which is the opting out condition for type A, and the non-negative pro t condition (11). Second-best arrangement Under the above arrangement, type A s weath at date t = 1, after exercising his option to se back his poicy, is w A = w 0 P + K. This type then purchases the optima death bene t L A (w A) in the date 1 market. This yieds the na contingent weath eves w A = w A ql A (w A) and w d A = w A ql A (w A) + L A (w A). Type B does not opt out and thus the na contingent weath aocation foows directy from the ong-term insurance contract, i.e., w B = w 0 P and w d B = w 0 P + L. The date 1 vaue of this aocation is w B = qw d B + (1 q)w B = w 0 P + ql: The impied weath transfer from type-a to type-b individuas is therefore w B w A = ql K. Written in terms of w A, wb and wd B, the second-best arrangement soves the foowing program: max U = (1 )V A (w A ) + qv B (wb) d + (1 q)u(wb) w A ;wb wd B subject to V A (w A ) qv A (w d B) + (1 q)u(w B) (12) 12

15 and (1 )w A + qw d B + (1 q)w B w0 : (13) The rst inequaity is type A s incentive compatibiity constraint; the second foows from the insurer s non-negative pro t condition. It is straightforward to characterize the main features of the soution to the above probem. The resource constraint (13) is obviousy binding. We show rst that the sef-seection condition (12) must be binding as we. Suppose, to the contrary, that the optima soution maximizes U subject to (13) ony. This is readiy seen to yied a soution VA(w 0 A ) = vb(w 0 B) d = u 0 (wb): Substituting from (1) and (2), we then have u 0 (wa) = va(w 0 A) d = vb(w 0 B) d = u 0 (wb); which corresponds to the rst-best aocation represented in gure 2. However, as is cear from the gure, type A stricty prefers EB to E A, impying that type A woud not opt out, i.e., (12) is not satis ed. Secondy, the naïve strategy is not a soution. As discussed above, (12) hods as a strict inequaity under the naïve strategy. Since this condition must bind, the naïve strategy does not sove the probem. However, since it nevertheess satis es the constraints, it must be the case be that the individua is stricty better o under the ong-term contract. The foowing proposition summarizes these resuts. Proposition 1 Long-term opting-out contracts make individuas stricty better o than the naïve strategy, but they remain second-best compared to the (compete-information) case where bequest needs are directy insurabe. 13

16 Leves of coverage We next examine how the eves of coverage di er under the various insurance arrangements. The Lagrangian of the second-best program is L = (1 )V A (w A ) + qv B (w d B) + (1 q)u(w B) + V A (w A ) qv A (w d B) (1 q)u(w B) + w 0 (1 )w A qw d B + (1 q)w B ; with positive mutipiers and. Together with (12) and (13) hoding as equaities, the soution satis es the A = (1 + )V 0 A(w A ) (1 ) = B = q v 0 B(w d B) v 0 A(w d B) = 0; (15) = (1 q) ( ) u 0 (w B) = 0: (16) Denote the soution by ( bw A ; bw B d ; bw B ). The date-1 vaue of type B s aocation is bw B q bw B d +(1 q) bw B. Type A s aocation is bwd A and bw A satisfying v 0 A( bw d A) = u 0 ( bw A) = V 0 A( bw A ): (17) An iustration is given in gure 3. We derive two resuts. First, the weath transfer from type-a to type-b individuas in the ong-term arrangement is smaer than in the rst-best contract. Secondy, the second-best contract provides a greater death bene t than type B woud wish if he coud purchase freey on the basis of his contractuay de ned date 1 weath. -- Fig. 3 about here -- 14

17 Weath transfer. We show that the subsidy from type-a to type-b individuas is ower in the second-best soution vis-à-vis the rst-best one: bw B bw A < wb wa. Suppose, to the contrary, that bw B bw A w B w A: (18) From the zero-pro t condition, the state weath eves satisfy (1 ) bw A + bw B = (1 )w A + w B = w 0 : Hence, (18) impies bw A wa and bw B wb. A s date-1 budget ine in the second-best arrangement is then beow the rst-best one represented in gure 2, whie B s budget ine woud be above the one in gure 2. A s aocation satis es (17). Since bequest and surviva weath are norma goods, (18) therefore impies bw A w A and bwd A wd A. Consider now B s aocation. This is given by the intersection of A s indi erence curve through ( bw d A ; bw A ) and B s budget ine. Obviousy, the foregoing impies bwd B > wd B. Combining both these resuts yieds v 0 A( bw d A) v 0 A(w d A ) = v 0 B(w d B ) > v 0 B( bw d B), (19) where the equaity foows from the optimaity conditions for a rst best. We now turn to the restrictions imposed by the rst-order conditions. From (14) and (15) it foows that 1 + vb( 0 bw B) d = V 0 1 A( bw A ) + v 0 A( bw B): d (20) Substituting for V 0 A ( bw A) = v 0 A ( bwd A ) from (17), (20) impies v0 B ( bwd B ) > v0 A ( bwd A ) which contradicts (19). The weath transfer must therefore be stricty smaer in the second-best arrangement. Distortion. Here we show that the B-type is forced to "overinsure," which can be interpreted as a type of signaing cost in the second-best setting. This distortion is represented by a point such as b E B in gure 3. As 15

18 drawn, the ong-term contract provides a arger bequest (and correspondingy smaer surviva weath) than type B woud wish to purchase vountariy on the basis of the post-transfer weath eve bw B. In other words, L > L B ( bw B) or equivaenty bw d B > bw B ql B ( bw B) + L B ( bw B). This is a necessary feature of the second-best arrangement. The intuition is that this distortion faciitates the transfer of weath from state A to state B, by making it more costy for type A not to opt out of the initia contract. To see this more formay, substituting from (15) and (16) yieds v 0 B( bw d B) u 0 ( bw B) = v0 A( bw d B) u 0 ( bw B) : No distortion woud require that the eft-hand side is zero, which in turn impies va 0 ( bwd B ) = v0 B ( bwd B ), a contradiction. Moreover, B s indi erence curve through E b B cannot be steeper than the fair-odds ine. Otherwise, a pair (wb d ; w B ) coud be chosen on the same fair-odds ine beow A s indi erence curve through E b B that satis es A s incentive compatibiity constraint and is stricty preferred by B. Hence, we must have L arger than L B ( bw B) as caimed. The next proposition summarizes our resuts. Proposition 2 Under the second-best ong-term contract (i) the weath transfer between type A and type B is smaer than the rst-best transfer and (ii) high-bequest types are over-insured reative to the coverage they woud ike to have at the contractuay de ned weath eve. The foregoing resuts impy that type-a s bequest and surviva weath are greater than in the rst best, whie type-b s surviva weath is smaer. However, how type-b s bequest compares with the rst-best eve is ambiguous. There are two opposing e ects so to speak. On the hand, because of the distortion, B s equiibrium bequest is arger than he woud wish at the weath 16

19 eve bw B. On the other hand, his weath eve is ower than the rst-best wb. When conditiona weath eves do not di er too much from the rst best, it may therefore be that B eaves a arger bequest. In this case, bequest by both types are greater than in the rst best. It is interesting to note that at date 1, an individua who turns out to be type A wi be better o with the second-best contract than with the rst-best one. This is to be expected, since the subsidy is ower under the secondbest arrangement. In other words, at date 0 the individua woud prefer the extra protection a orded by a arger subsidy. However, an individua who eventuay turns out to be of a ow-bequest type wi be happier if the subsidy is smaer when date 1 arrives. 8 Opting in contracts An aternative to the opting out arrangement is to o er a contract with an option to purchase additiona coverage at date t = 1. Such an "opt-in" contract is de ned by the vector (P ; L; S; k) where P is the premium paid at t = 0 for coverage L and S is the optiona additiona coverage that the individua can purchase at date t = 1 for an additiona premium k. If P > ql and k < qs, then weath wi be transferred from type-a individuas to type-b individuas, provided of course ony type B exercises the option to purchase additiona coverage. Here, the origina coverage L is sod at an unfair premium. The non-negative pro t constraint under this opting-in arrangement is P ql + (qs k): (21) The above contract is equivaent to seing both types an initia contract with death bene t (L + S) for a premium of (P + k). The type-b individua "opts in" by maintaining this package at date t = 1. The type-a individua 8 This is an often overooked artifact of most adverse-seection modes. 17

20 refuses to "opt in" at date t = 1 by obtaining a refund of the extra premium k, and reducing the death tota bene t by an amount S. 9 As before, e ectivey ony the A-type s incentive compatibiity constraint matters, with the type-a individua being just indi erent between opting in or not opting in. De ning P P + k; L L + S and K ql + k; (22) it is easiy seen that the non-negative pro t constraint (21) is equivaent to the previous non-negative pro t constraint (11), and it wi be satis ed once again with an equaity at the optimum. We have one additiona requirement here, namey that a type-a individua who woud receive a refund ql+k at date t = 1 woud purchase an optima eve of insurance coverage L A (w 0 P +ql) = L, if insurance woud be avaiabe at a fair price. More formay, from the rstorder condition for L A, this requires that v 0 A(w 0 P + ql ql A + L A) = u 0 (w 0 P + ql ql A) (23) be satis ed when L A (w 0 P + ql) = L. It then foows in a straightforward manner that the same tripe (P; L; K) is optima, which together with (22) and (23) determine the parameters for the optima opt-in contract: (P ; L; S; k). Thus, the opting-in and opting-out arrangements are e ectivey identica These opting in contracts have the property that the premium oading is coected in the rst period, which is simiar to the types of contracts that one observes for renewabe insurance contracts when there is an insurabiity risk. Of course, the design in those modes is for quite a di erent purpose; namey, to cope with asymmetry about one s risk type. 10 One can aso verify this directy by writing out the incentive compatibiity constraints and then nding the optima (P ; L; S; k) directy, which together with (23) shows the equivaence of the two types of contract arrangements. 18

21 5 Concuding Remarks This paper has shown how a ong-term insurance contract can be designed, within a competitive insurance market, to insure the uncertain future bequest needs of the individua. We derived two equivaent forms for this ong-term insurance contract: 11 (i) It provides a high eve of initia coverage at a subsidized (ow) price, with an option to se back the poicy at an unfair price (i.e. at a oss to the insured) or (ii) It provides a ow eve of initia coverage at an unfair (high) price, with an option to purchase additiona coverage at a subsidized (ow) price. The existence of such contracts in the market pace depends cruciay on the sef seection of types in exercising the various options. But some reativey new innovations in the nancia marketpace may have an untoward e ect on the deveopment of the ong-term contracts we propose. In particuar, the market for ife settements poses such an obstace. A ife settement contract essentiay o ers to "buy back" the ife insurance poicy of an individua. This is e ected via a third party paying cash to the insured, in exchange for being named the bene ciary of the ife insurance death bene t. Athough this seems to eiminate any bene t to the origina bene ciary, this wi not be the case. In particuar, under contract (i), the ow-bequest need individua wi opt to se the poicy to a ife settement broker, rather than back to the insurer, and receive more money for the poicy. The insured can then purchase insurance at a fair price, since there is no insurabiity risk. Under contract (ii), both bequest types might purchase 11 These two forms wi not be unique. For exampe, an intermediate eve of insurance coud be o ered with both "opt in" and "opt out" opportunities to achieve the same na weath eves. 19

22 the additiona extra insurance at the ow price, with the ow-bequest need type individua then immediatey seing back the extra coverage in the ife settement market for a pro t. The existence of such markets provides an aternative for the insured that is bene cia ex post (i.e., after signing the origina ong-term contract). Insurance companies had originay protested as these markets deveoped, caiming that they shoud have the excusive right to buy-back (i.e. "sette") contracts that they had written. But others disagree. For exampe, Doherty and Singer (2002) tout the bene ts of ife settement markets to the insurance consumer. Such anaysis might be incompete, however, in that it excudes the fact that ex ante (i.e. prior to earning one s bequest type) one woud prefer the onger term contracts described in this paper; and the ife settement market might precude such contracts from ever being o ered. Athough the ong-term contracts we describe in this paper give the insurer monopoy power ex post, a competitive market ex ante shoud ensure that insurers cannot earn undue monopoy rents. Obviousy, we simpi ed the setting of our anaysis by assuming away many compicating factors, such as the insurabiity risk. This aowed our focus to be on the bequest needs and the (non-random) probabiity of death. Integrating these resuts into more compex settings is di cut. Hopefuy, our paper takes a good rst step in this direction. References [1] Bernheim, D. A. (1991). How strong are bequest motives? Evidence based on estimates of the demand for ife insurance and annuities. Journa of Poitica Economy 99, [2] Campbe, R. A. (1980). Demand for ife insurance: An appication of 20

23 the economics of uncertainty. Journa of Finance 35, [3] Cochrane, J. H. (1995). Time-consistent heath insurance. Journa of Poitica Economy 103, [4] Doherty, N. A. and Singer, H. J. (2002). The bene t of a secondary market for ife insurance poicies, Wharton Financia Institutions Center Working Paper No [5] Hende, I. and Lizzeri, A. (2003). The roe of commitment in dynamic contracts: Evidence from ife insurance. Quartery Journa of Economics 118, [6] Karni, E., (1985). Decision Making Under Uncertainty: The Case of State-Dependent Utiity. Cambridge, MA: Harvard University Press. [7] Lewis, F. D. (1989). Dependents and the demand for ife insurance. American Economic Review 79, [8] Pauy, M. V.; Kunreuther, H.; and Hirth, R. (1995). Guaranteed renewabiity in insurance. Journa of Risk and Uncertainty 10, [9] Poborn, M. K., Hoy, M., & Sadanand, A. (2006). Advantageous e ects of reguatory adverse seection in the ife insurance market. Economic Journa 116, [10] Sheshinski, E. (2007). Optimum and risk-cass pricing of annuities. Economic Journa 117, [11] Vieneuve, B. (2000). Life insurance, in: G. Dionne, Editor, Handbook of Insurance, Boston: Kuwer Academic Pubishers. 21

24 w d w = w w 0 1 q A B w A E A w B E B d w A d w B w 0 q w d Figue 1: Purchase after type is known w d w = w * w B 1 q A B * w A 1 q * * A w B w = * E A * E B d w * A d w * B w d Figure 2: First-best insurance 22

25 w d w = w ˆ w B 1 q A B ˆ w A 1 q ŵ A Ê A ŵ B d ŵ A Ê B d ŵ B A B w d Figure 3: Second-best ong term contracts 23

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