Annuities and Individual Welfare

By THOMAS DAVIDOFF,JEFFREY R. BROWN, AND PETER A. DIAMOND*

Advancing annuity demand theory, we present sufﬁcient conditions for the optimality of full annuitization under market completeness which are substantially less restrictive than those used by Menahem E. Yaari (1965). We examine demand with market incompleteness, ﬁnding that positive annuitization remains optimal widely, but complete annuitization does not. How uninsured medical expenses affect demand for illiquid annuities depends critically on the timing of the risk. A new set of calculations with optimal consumption trajectories very different from available annuity income streams still shows a preference for considerable annuitization, suggesting that limited annuity purchases are plausibly due to psychological or behavioral biases. (JEL D11, D91, E21, H55, J14, J26)

Since the seminal contribution of Yaari maintain most of these conditions. In particular, (1965) on the theory of a life-cycle consumer the literature has universally retained expected with an unknown date of death, annuities have utility and additive separability, the latter played a central role in economic theory. His dubbed “not a very happy assumption” by widely cited result is that certain consumers Yaari. While Yaari (1965) and B. Douglas should fully annuitize all of their savings. How- Bernheim (1987a, b) provide intuitive explana- ever, these consumers were assumed to satisfy tions of why the Yaari result may not depend on several very restrictive assumptions: they were these strict assumptions, the generality of this von Neumann-Morgenstern expected utility result has not been formally shown in the maximizers with intertemporally separable util- literature. ity; they faced no uncertainty other than time of The first contribution of this paper is to death; they had no bequest motive; and the present sufficient conditions substantially annuities available for purchase were actuari- weaker than those imposed by Yaari, under ally fair. While the subsequent literature on which full annuitization is optimal. The heart of annuities has occasionally relaxed one or two of the argument can be seen by comparing a one- these assumptions, the “industry standard” is to year bank certificate of deposit (CD), paying an interest rate r to a security that pays a higher interest rate at the end of the year conditional on * Davidoff: Haas School of Business, UC Berkeley, living, but pays nothing if you die before year- Berkeley, CA 94720 (e-mail: [email protected]); Brown: Department of Finance, University of Illinois at end. If you attach no value to wealth after death, Urbana-Champaign, 340 Wohlers Hall, MC-706, Cham- then the second, annuitized, alternative is a paign, IL 61820, and National Bureau of Economic Re- dominant asset. This simple comparison of oth- search (e-mail: [email protected]); Diamond: MIT erwise matching assets, when articulated in a Department of Economics, 50 Memorial Drive, Cambridge, MA 02142 (e-mail: [email protected]). We thank Amy setting that confirms its relevance, lies behind Finkelstein, Kevin Lang, and two anonymous referees for the Yaari result. The dominance comparison of helpful suggestions. Davidoff and Diamond are grateful for matching assets is not sensitive to their financial financial support from the Center for Retirement Research details, but does rely on the identical liquidity in (CRR) at Boston College. The research reported herein is an the annuitized and nonannuitized assets. extension of work supported by the CRR pursuant to a grant from the U.S. Social Security Administration funded as part This paper explores the implications of this of the Retirement Research Consortium. The opinions and comparison for asset demand in different set- conclusions are solely those of the authors and should not be tings, including both Arrow-Debreu complete construed as representing the opinions or policy of the markets and incomplete market settings. In the Social Security Administration or any agency of the federal government, or the Center for Retirement Research at Bos- Arrow-Debreu complete market setting, suffi- ton College. Diamond is grateful for financial support from cient conditions for full annuitization to be op- the National Science Foundation Award SES-0239 080. timal require that consumers have no bequest 1573 1574 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005 motive and that annuities pay a rate of return to sumption and the annuity trajectory, we allow a surviving investors, net of administrative costs, person’s utility to depend on how present con- that is greater than the return on conventional sumption compares to a standard of living to assets of matching financial risk. Thus, when which the individual has become accustomed, markets are complete, full annuitization is opti- which is itself a function of past consumption. mal without assuming exponential discounting, We model this “internal habit” as in Diamond the expected utility axioms, intertemporal sep- and James A. Mirrlees (2000).1 arability, or actuarially fair annuities. We also These results imply that annuities are quite relate the size of the welfare gain to the trajec- valuable to utility maximizing consumers, even tory of optimal consumption. under conditions that result in a very unfavor- A second contribution is to examine annuity able mismatch between available annuity in- demand in some incomplete market settings. If come streams and one’s desired consumption some desired consumption paths are not avail- path. Thus, while incomplete markets, when able when all wealth is annuitized in an incom- combined with preferences for consumption plete annuity market, full annuitization may no paths that deviate substantially from those of- longer be optimal. For example, if an individual fered by current annuity products, can certainly desires a steeply downward-sloping consump- explain the lack of full annuitization, it is diffi- tion path, but only constant real annuities are cult to explain the near universal lack of any available, then full annuitization is no longer annuitization outside of Social Security Insur- optimal. Thus, we explore conditions for the ance and defined-benefit pension plans, at least optimum to include partial annuitization. We at the higher end of the wealth distribution also consider partial annuitization with a be- where Social Security is a small part of one’s quest motive. portfolio and SSI is not relevant. This finding is Another example involves the relative liquid- strongly reminiscent of the literature on life ity of annuities and bonds in the absence of insurance, which is a closely related product.2 insurance for medical expenditure shocks. The The literature on life insurance has documented effect on annuity demand depends on the timing a severe mismatch between the life insurance of the risk. An uninsurable risk early in life may holdings of most households and their underly- reduce the value of annuities if it is not possible ing financial vulnerabilities (see, e.g., Bernheim to sell or borrow against future payments of the et al., 2003; Alan J. Auerbach and Laurence J. fixed annuity stream, but it is possible to do so Kotlikoff, 1987; Auerbach and Kotlikoff, 1991). with bonds. In contrast, loss of insurability of a These papers, taken together, are suggestive of shock occurring later in life may increase annu- psychological or behavioral considerations at ity purchases as a substitute for medical insur- play in the market for life-contingent products ance that has an inherent annuity character. that have not yet been incorporated into stan- Thus, an annuity can be a better substitute than dard economic models. a bond for long-term-care insurance. The focus of this paper is on the properties of Most practical questions about annuitization annuity demand in different market settings. We (e.g., the appropriate role of annuities in public do not address the more complex equilibrium pension systems) are concerned with partial an- question of what determines the set of annuity nuitization. The general theory itself is insuffi- products in the market. In light of the value of cient to answer questions about the optimal annuities to consumers in standard models, we fraction of annuitized wealth, and thus a large think examination of equilibrium would have to simulation literature has developed. Our third contribution is to extend the simulation literature by creating a new “stress test” of annuity 1 Different models of intertemporal dependence in utility valuation. We do so by generating optimal con- are discussed in, for example, James Dusenberry, 1949; sumption trajectories that differ substantially Andrew B. Abel, 1990; George M. Constantinides, 1990; from what is offered by a fixed real annuity Angus Deaton, 1991; John Y. Campbell and John H. Co- contract, and by showing that even under these chrane, 1999; Campbell, 1999; and Francesco Gomes and Alexander Michaelides, 2003. highly unfavorable conditions, the majority of 2 As discussed by Yaari (1965) and Bernheim (1991), the wealth is still optimally annuitized. To generate purchase of a pure life insurance policy can be viewed as the the unfavorable match between optimal con- selling of an annuity. VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1575 include the supply response to demand behavior A. The Optimality of Full Annuitization in a that is not consistent with standard utility max- Two-Period Model with No Aggregate imization. We do not develop such a behavioral Uncertainty theory of annuity demand, but rather clarify the mismatch between observed demand behavior Analysis of intertemporal choice is greatly and the value of annuitization in a standard simplified if resource allocation decisions are utility maximization setting. made completely and all at once, that is, without The paper proceeds as follows: In Section I, additional, later trades. Consumers will be will- we provide a general set of sufficient conditions ing to commit to a fixed plan of expenditures if, under which full annuitization is optimal under at the start of time, they are able to trade goods complete markets. Section II discusses why full across all time and all states of nature, as is annuitization may no longer be optimal with standard in the complete-market Arrow-Debreu incomplete markets, paying particular attention model. to the role of illiquidity of annuities. Section III Yaari considered annuitization in a continu- briefly discusses a bequest motive. In Section ous time setting where consumers are uncertain IV, we report simulation results reflecting the only about the date of death. Some results, quantitative importance of market incomplete- however, can be seen more simply by dividing ness with habit formation, showing that even time into two discrete periods: the present, pe- when the liquidity constraints on annuities are riod 1, when the consumer is definitely alive, binding, individuals still prefer a high level of and period 2, when the consumer is alive with annuitization. Section V concludes. probability 1 Ϫ q.4 ϭ By writing U U(c1, c2), we allow for a very I. Complete Markets general formulation of utility in a two-period setting with the assumptions that there is no Much of the focus of the annuities literature bequest motive and that only survival to period has been an attempt to reconcile Yaari’s (1965) 2 is uncertain. Lifetime utility is defined over “full annuitization” result with the empirical first-period consumption, c1, and consumption fact that few people voluntarily annuitize any of in the event that the consumer is alive in period 3 their private savings. This issue is of theoreti- 2, c2. We drop the requirement of intertemporal cal interest because it bears upon the issue of separability, allowing for the possibility that the how to model consumer behavior in the pres- utility from second-period consumption may ence of uncertainty. It is also of policy interest depend on the level of first-period consumption. because of the shift in the United States from Additionally, this formulation does not require defined-benefit plans, which typically pay out as that preferences satisfy the axioms for U to be an annuity, to defined contribution plans, which an expected value. rarely offer retirees, directly, the opportunity to The optimal consumption decision and the annuitize. Annuitization is also important in the welfare evaluation of annuities can be deter- debate about publicly provided defined contri- mined using a dual approach: minimizing ex- bution plans. penditures subject to attaining at least a given This section derives a general set of condi- level of utility. We measure expenditures in tions under which full annuitization is optimal, units of first-period consumption. Assume that relaxing many of the assumptions in the original there are two securities available. The first is a Yaari formulation. We begin with a simple two- bond that returns RB units of consumption in period model, and then show formally the gen- period 2, whether the consumer is alive or not, eralization to many periods and many states. in exchange for each unit of the consumption good in period 1. The second is an annuity that returns RA in period 2 if the consumer is alive, and nothing otherwise. 3 This assertion is consistent with the large market for what are called variable annuities, since these insurance products do not include a commitment to annuitize accu- mulations (nor does there appear to be much voluntary 4 A two-period model with a single consumer good in annuitization). See, for example, Brown and Mark J. War- each dated event precludes trade after the first period, but in shawsky (2001). a complete market setting, this is irrelevant. 1576 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005

ϭ An actuarially fair annuity would yield RA nity to annuitize. Consider the minimization Ϫ RB/(1 q). Adverse selection and higher trans- under the further constraint of an upper bound action costs for paying annuities than for paying on purchases of annuities, A Յ A៮ . We know that bonds may drive returns below this level. How- utility-maximizing consumers will take advan- ever, because any consumer will have a positive tage of an arbitrage-like opportunity to annu- probability of dying between now and any fu- itize as long as bond holdings are positive and ture period, thereby relieving borrowers’ obli- can therefore be used to ﬁnance the purchase. Ͼ gation, we make the weak assumption that RA With no annuities available, bond holdings will 5 RB. be positive if second-period consumption is If we denote by A savings in the form of positive, as is ensured by the plausible condition annuities and by B savings in the form of bonds, that zero consumption is extremely bad: and if there is no other income in period 2 (e.g., ϭ ϩ the individual is retired), then c2 RAA RBB, ASSUMPTION 1: and expenditures for lifetime consumption are ϭ ϩ ϩ simply E c1 A B. Thus, the expenditure lim 3 ѨU/Ѩc ϭ ϱ : t ϭ 1, 2. minimization problem can be written as: ct 0 t

ϩ ϩ ͑ ϩ ͒ Ն ៮ (1) min c1 A B s.t. U c1 , RA A RB B U. Allowing consumers previously unable to annu- c1 , A, B itize any wealth to place a small amount of their We further impose the constraint that B Ն 0, savings into annuities (increasing A from zero) i.e., that the individual not be permitted to die in leaves second-period consumption unchanged. Ͼ debt. Otherwise with RA RB, purchasing an- (Since the cost of the marginal second-period nuities and selling bonds in equal numbers consumption is unchanged, so, too, is the opti- would cost nothing and yield positive consump- mal level of consumption in both periods.) tion when alive in period 2, but leave a debt if Thus, a small increase in A៮ from zero reduces dead, leaving lenders with expected financial the cost of achieving a given level of utility by Ϫ Ͻ losses in total. 1 (RA)/(RB) 0. This is the welfare gain from This setup leads immediately to the optimal- increasing the limit on available annuities for ity of full annuitization. If B Ͼ 0, then one is an optimizing consumer with positive bond able to reduce expenditures, while holding the holdings.6 consumption vector fixed, by selling RA/RB of If the upper bound constraint on available the bond and purchasing one unit of the annuity annuities is large enough that bond holdings are Ͼ (noting that RA RB). Thus, the solution to this zero, then the price of marginal second-period ϭ ៮ expenditure minimization problem is to set B consumption (up to A) falls from 1/(RB)to 0, i.e., to annuitize one’s wealth fully. The in- 1/(RA). With a fall in the cost of marginal second- tuition is that allowing individuals to substitute period consumption, its compensated level will annuities for conventional assets yields an rise. Thus, the welfare gain from unlimited an- arbitrage-like gain when the individual places nuity availability is made up of two parts. One no value on wealth when not alive. Such a gain part is the savings while financing the same enhances welfare independent of assumptions consumption bundle as when there is no annu- about preferences beyond the lack of utility itization, and the second is the savings from from a bequest. Nor must annuities be actuari- adapting the consumption bundle to the change ally fair. Indeed, all that is required is for con- in prices. We can measure the welfare gain in sumers to have no bequest motive and for the going from no annuities to potentially unlimited payouts from the annuity to exceed that of con- annuities from the expenditure function defined ventional assets for the survivor. over the price vector (1, p2) and the utility level Equation (1) also indicates an approach to U៮ . Integrating the derivative of the expenditure evaluating the gain from an increased opportu- function evaluated at the two prices 1/RA and 1/RB:

5 Ͻ Ͻ Ϫ That RB RA RB/(1 q) is supported empirically by Olivia S. Mitchell et al. (1999). If the ﬁrst inequality were 6 This point is made in the context of time-separable violated, annuities would be dominated by bonds. preferences by Bernheim (1987b). VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1577

͉ Ϫ ͉ ϭ ͑ ៮ ͒ ϭ ϩ (2) E A៮ ϭ 0 E A៮ ϭϱ E 1, 1/RB , U (5) c3 RB3 B3 RA3 A3 .

1/RB Ϫ E͑1, 1/R , U៮ ͒ ϭ Ϫ͵ c ͑p ͒ dp If our assumption that the return on annuities A 2 2 2 exceeds that of bonds holds, period by period, 1/R A then our full optimization result extends trivi- where c2(p2) is compensated demand arising ally. Note that the standard definition of an from minimization of expenditures equal to Arrow security distinguishes between states ϩ c1 c2p2, subject to the utility constraint with- when an individual is alive and when he or she out a distinction between asset types. Consum- is not alive. That is, a standard Arrow security is ers who save more (have larger second-period an “Arrow annuity.” We have set up what we consumption) benefit more from the ability to call “Arrow bonds” (here B2 and B3)bycom- annuitize completely. bining matched pairs of events that differ in whether the consumer’s death has occurred. B. The Optimality of Full Annuitization with This representative of a standard bond is what Many Periods and Many States becomes a dominated asset once one can separate the Arrow bond into two separate Arrow While a two-period model with no uncer- securities, and the consumer can choose to pur- tainty other than length of life has a clarity that chase only one of them. In a setting with addi- derives from its simplicity, real consumers face tional sources of uncertainty, the combination a more complicated decision setting. In partic- of a matched pair of events to depict a nonan- ular, they face many periods of potential con- nuitized asset is straightforward when the death sumption, and each period may have several of the particular consumer is independent of possible states of nature. For example, a 65- other events, and unimportant in determining year-old consumer has some probability of sur- equilibrium. In some settings, there may not be viving to be a healthy and active 80-year-old, such a decomposition of a bond because the some chance of finding herself sick and in a recognition of important differences between nursing home at age 80, and some chance of not different events is strongly related to the sur- being alive at all at age 80. Moreover, returns on vival of the individual.8 some assets are stochastic. In this section, we To take the next logical step, and assuming show that the optimality of complete annuitiza- we can decompose Arrow bonds, we continue tion survives subdivision of the aggregated fu- to treat c1 as a scalar and interpret c2, B2, and A2 ture defined by c2 into many future periods and as vectors with entries corresponding to arbi- states, as long as markets are complete. trarily many future periods (t Յ T), within ar- ␻ Յ ⍀ A simple subdivision would be to add a third bitrarily many states of nature ( ). RA2 period, while continuing with the assumption of (RB2) is then a matrix with columns correspond- no other uncertainty. In keeping with the com- ing to annuities (bonds) and rows corresponding plete market setting, we have bonds and annu- to payouts by period and state of nature. Thus, ities that pay out separately in period 2 with rates the assumption of no aggregate uncertainty can RB2 and RA2, and period 3 with rates RB3 and be dropped. Multiple states of nature might re- 7 RA3. That is, defining bonds and annuities fer to uncertainty about aggregate issues such as purchased in period 1 with the appropriate output, or individual specific issues beyond subscript: mortality such as health. To extend the analysis, we assume that the consumer is sufficiently ϭ ϩ ϩ ϩ ϩ (3) E c1 A2 A3 B2 B3 “small” (and with death uncorrelated with other events) that for each state of nature where the ϭ ϩ (4) c2 RB2 B2 RA2 A2

8 If the death of a consumer occurs only with other large changes, then there is no way to construct an Arrow bond that differs from an Arrow annuity only in the death of a 7 In keeping with the complete market setting, there is single consumer, for example, if an individual will die in no arrival of asymmetric information about future life some future period if and only if an influenza epidemic expectancy. occurs. 1578 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005 consumer is alive, there exists a state where the COROLLARY 1: In a complete-market Arrow- consumer is dead and the equilibrium relative Debreu equilibrium, a consumer without a be- prices are otherwise identical. Completeness of quest motive annuitizes all savings. markets still has Arrow bonds that represent the combination of two Arrow securities. Note, Thus, with complete markets, the result of however, that the ability to construct such bonds full annuitization extends to many periods, the is not necessary for the result that a consumer presence of aggregate uncertainty, actuarially without a bequest motive does not purchase unfair but positive annuity premiums, and consumption when not alive in the complete intertemporally dependent utility that need market setting. not satisfy the expected utility axioms. This Annuities paying in only one dated event are generalization of Yaari holds, so long as mar- contrary to conventional life annuities that pay kets are complete. Thus, if the puzzle of why so out in every year until death. With complete few individuals voluntarily annuitize is to be markets, however, separate annuities with pay- solved within a rational, life-cycle framework, outs in each year can be combined to create we have shown that the answer does not lie in such conventional annuities. the specification of the utility function, per se, It is clear that the analysis of the two-period (beyond the issue of a bequest motive). Below, model extends to this setting, provided we we briefly consider the role of annuitization maintain the standard Arrow-Debreu market with a bequest motive. The results do demon- structure and assumptions that do not allow an strate, however, the importance of market com- individual to die in debt, since the consumer can pleteness, an issue that we turn to next. purchase a combination of annuities with a structure of benefits across time and states as II. Incomplete Markets desired. In addition to the description of the optimum, the formula for the gain from allow- With complete markets, a higher yield of an ing more annuitization holds for state-by-state Arrow annuity over the matching Arrow bond, increases in the level of allowed level of annu- dated event by dated event, leads directly to the itization. Moreover, by choosing any particular result that a consumer without a bequest motive price path from the prices inherent in bonds to fully annuitizes. But markets are not complete. the prices inherent in annuities, we can measure There are two forms of incompleteness that we the gain in going from no annuitization to full consider. The first is that the set of annuities is annuitization. This parallels the evaluation of highly limited, relative to the set of nonannu- the price changes brought about by a lumpy in- itized securities that exist.9 The second is the vestment (see Diamond and Daniel L. McFadden, incompleteness of the securities market. 1974). Hence, with complete markets, preferences matter only through optimal consumption; this A. Incomplete Annuity Markets fact may clarify, for example, the unimportance of additive separability to the result of complete Most real world annuity markets require that annuitization. a consumer purchase a particular time path of Stating the result more formally, the full an- payouts, thereby combining in a single security nuitization result is a corollary of the following: a particular “compound” combination of Arrow annuities. Privately purchased immediate life annuities are usually fixed in nominal terms, or THEOREM 1: If there is no future trade once offer a predetermined nominal slope such as a portfolio decisions have been made, if there is 5-percent increase per year. Variable annuities no bequest motive, and if there is a set of link the payout to the performance of a partic- annuities with payouts per unit of investment that dominate the payouts of some subset of bonds that are held, then a welfare gain is 9 Explaining why this is the case would take us into the available by selling the subset of bonds and industrial organization of insurance supply, which would necessarily make use of consumer understanding and per- replacing the bonds with the annuities, so long ceptions of insurance. These issues are well beyond the as this transfer does not lead to negative wealth scope of this paper. Rather, we analyze rational annuity in any state. demand in different market settings that are taken as given. VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1579 ular underlying portfolio of assets, and combine plete bonds and a single available annuity and Arrow securities in that way. CREF annuities no opportunity for trade after the initial con- are also participating, which means that the tracting. The minimization problem is now payout also varies with the actual mortality ex- ϩ ϩ ϩ perience for the class of investors. To explore (6) min : c1 B2 B3 A issues raised by such restrictions, we restrict our c1 , A, B analysis to a single kind of annuity—a constant real annuity—although we state some of our (7) results in a more general vocabulary. We exam- ͑ ϩ ϩ ͒ Ն ៮ ine the demand for such an annuity, distinguish- s.t. : U c1 , RB2 B2 RA2 A, RB3 B3 RA3 A U ing whether all trade must occur at a single time Ն Ն or whether it is possible also to purchase bonds (8) B2 0, B3 0. later. Given our return assumption and positive con- Trade Occurs All at Once.—For the case of a sumption whenever alive, then, we have an ar- conventional annuity, we must revise the supe- bitrage-like dominance of the annuity over the rior return condition for Arrow annuities that matching combination of bonds, as long as this Ͼ @ ␻ RAt␻ RBt␻ t . An appropriate formulation trade is feasible. Thus, we can conclude that for a compound security is that it costs less than some annuitization is optimal and that the opti- purchasing the same consumption vector using mum has zero bonds in at least one dated event. bonds. Define by ഞ a row vector of ones with The logic extends to a setting with more dates length equal to the number of states of nature and states. We would not get complete annuiti- occurring in the annuity, and so distinguished zation, however, if the consumption pattern by bonds. Let the set of bonds continue to be with complete annuitization were worth chang- represented by a vector with elements corre- ing by purchasing a bond. That is, purchasing a sponding to the columns of the matrix of returns bond would be worthwhile if it raised utility by RB, and let RA be a vector of annuity payouts more than the decline from decreased first- multiplying the scalar A to define state-by-state period consumption. Thus, denoting partial de- payouts. Then the cost of the bonds exceeds the rivatives of the utility function with subscripts, cost of the annuity under the following there will be positive bond holdings if we sat- assumption: isfy either of the conditions: ͑ ͒ ASSUMPTION 2: For any annuitized asset A (9) U1 c1 , RA2 A, RA3 A and any collection of conventional assets B, ϭ f Ͻ ഞ Ͻ ͑ ͒ RAA RBB A B. RB2 U2 c1 , RA2 A, RA3 A

For example, if there is an annuity that costs or one unit of first-period consumption per unit and pays R per unit of annuity in the second A2 ͑ ͒ (10) U1 c1 , RA2 A, RA3 A period and RA3 per unit of annuity in the third period, then we would have 1 Ͻ R /R ϩ A2 B2 Ͻ ͑ ͒ 10 RB3 U3 c1 , RA2 A, RA3 A . RA3/RB3. By linearity of expenditures, this implies that any consumption vector that may be purchased strictly through annuities is less ex- By our return assumption, we cannot satisfy pensive when financed through annuities than both of these conditions at the same time, but when purchased by a set of bonds with match- we might satisfy one of them. ing payoffs. Consider a three-period model, with com- Additional Trading Opportunities.—The previous analysis stayed with the setting of a single time to trade. If there are additional trading 10 The right-hand side represents the required invest- opportunities, the incompleteness of annuity ments in two Arrow bonds which cost one unit each in the markets may mean that such opportunities are first period to replicate the annuity payout. taken. Staying within the setting of perfectly 1580 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005 predicted future prices and assuming that prices which is inconsistent with the FOC for positive for the same commodity purchased at different holdings of both A and B2 (and some annuiti- dates are all consistent, we can see that repeated zation is part of the optimum). In the commonly bond purchase may increase the degree of an- used model of intertemporally additive prefer- nuitization. If desired consumption occurs later ences with identical period utility functions, a than consumption of all of the annuitized ben- constant discount rate and a constant interest efit, then the ability to save by purchasing rate, a sufficient condition for full initial annu- bonds, rather than consuming all of the annuity itization in a constant real annuity, is thus ␦(1 ϩ payment, means that annuitization is made more r) Ն 1. If the available annuity (a constant real attractive. benefit) provides consumption later than an in- Returning to the three-period model with dividual wants, then the assumed illiquidity of only mortality uncertainty, we can write this by an annuity limits its attraction. We consider denoting saving at the end of the second period illiquidity below. by Z (Z Ն 0). We assume that the return on savings between the second and third periods, Z, B. Incomplete Securities and Annuity ϭ is consistent with the other bond returns (RZ Markets: The Role of Liquidity RB3/RB2). The minimization is now: A widely recognized basis for incomplete ϩ ϩ ϩ (11) min c1 B2 B3 A annuitization is that there may be an expendi- c1 , A, B, Z ture need in the future that cannot be insured. This might be an individual need, like a medical ͑ ϩ Ϫ (12) s.t.:U c1 , RB2 B2 RA2 A Z, RB3 B3 expense that is not insurable, or an aggregate event such as unexpected inflation, which low- ϩ ϩ ͑ ͒ Ն ៮ RA3 A RB3 /RB2 Z) U. ers the real value of nominal annuity payments in the absence of real annuities. With incom- The restriction of not dying in debt is the plete markets, the arbitrage-like dominance ar- nonnegativity of wealth (including the value of gument used above will no longer hold if bonds future payments) if A equals zero: are liquid while annuities are not.11 We assume total illiquidity of annuities without exploring ϩ ͑ ͒ Ն (13) RB2 B2 RB2 /RB3 RB3 B3 0 the possible arrival of asymmetric information about life expectancy, which would naturally ϩ ͑ ͒ Ն (14) RB3 B3 RB3 /RB2 Z 0. reduce the liquidity of annuities far more than the liquidity of nontraded bonds, such as certif- Dissaving after full annuitization (if possible) icates of deposit. would not be attractive if: We show, however, that the presence of uninsured risks may add to or subtract from the ͑ ͒ (15) RB2 U2 c1 , RA2 A, RA3 A optimal fraction of savings annuitized, depending on the nature of the risk. We illustrate the Յ ͑ ͒ RB3 U3 c1 , RA2 A, RA3 A . role of illiquidity by examining two cases: uninsured medical expenditures, and the arrival of Under Assumption 2, (15) is now sufficient asymmetric information combined with inferior for the result of full annuitization of initial sav- annuity returns. ings. To see this, note that Assumption 2 im- Ͻ ϩ plies that RB3 RA2RZ RA3. Thus, holdings Annuities and Medical Expenditures.—For of B3 are dominated by the annuity, with the concreteness, let us start with the two-period second-period return fully saved. Positive hold- model above, where the only uncertainty is ings of B2 are ruled out by Assumption 2 and length of life. To this model, let us add a risk of condition (15), since they imply: a necessary medical expense which we consider

(16) R U Ͻ R U ϩ ͑R /R ͒R U B2 2 A2 2 A3 B3 B2 2 11 More generally, it is well known that life-cycle con- Յ ϩ sumers may be unwilling to invest in illiquid assets when RA2 U2 RA3 U3 , they face stochastic cash needs (Ming Huang, 2003). VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1581 separately for each period. Assume that this early redemption penalty, but annuities cannot expenditure enters into the budget constraint as be sold. The analysis would be similar with a required expenditure, but does not enter into liquid annuities that had a larger penalty for the utility function. Assume further that the early redemption or the ability to reduce spend- occurrence of illness has no effect on life ing and add to bond holdings (at a lower interest expectancy. rate) after a realization of no health risk. Then If it is possible to fully insure future medical the problem becomes: expenses on an actuarially fair basis, then the ϩ ϩ ϩ optimal plan is full medical insurance and full (19) min c1 A B I annuitization. Thus, removing the ability to in- c1 , A, B, I, Z sure medical expenses, while continuing to as- ͑ Ϫ ͒ ͑ ϩ ͒ sume that annuity benefits can be purchased (20) s.t. 1 m U c1 , RA A RB B separately period by period, can only raise the ϩ ͑ Ϫ ϩ ␤ ϩ ␣ demand for bonds and may do so if the medical mU c1 M I B Z, risk occurs in period 1, but not if it occurs in ϩ ͑ Ϫ ͒ Ն ៮ period 2. That is, the illiquidity of annuities may RA A RB B Z ) U be relevant if the risk occurs early in life, but not toward the end of life. For example, contrast where Z is the value of bonds withdrawn early ␣ ␣ Ͻ the cost of a hospitalization early in retirement and B ( B 1) is the fraction of value re- with the need for a nursing home toward the end ceived, net of the early withdrawal penalty. of life. The former calls for shifting expenses Since annuities still dominate any bonds that earlier and so increases the value of an asset that would never be cashed in, the level of bond permits such a change. Since medical expenses holdings would not exceed the amount cashed occur only for the living, annuities are a better in early, and we can rewrite the problem as substitute for nonexistent insurance for medical ϩ ϩ ϩ expenses later in life than are bonds. (21) min c1 A B I To the two-period problem considered in (1), c1 , A, B, I we add the risk of a first-period medical expense ͑ Ϫ ͒ ͑ ϩ ͒ of size, M, with probability m and insurance at (22) s.t. 1 m U c1 , RA A RB B cost I, paying a benefit of ␤ per dollar of insur- ϩ ͑ Ϫ ϩ ␤ ϩ ␣ ͒ Ն ៮ ance. If there is no additional trading after the mU c1 M I B B, RA A U. initial purchases, the problem is: In this case, it is possible to generate prefer- ϩ ϩ ϩ (17) min c1 A B I ences that have an optimum with some bonds, c1 , A, B, I provided there is a small difference between annuity and bond returns, a small early with- ͑ Ϫ ͒ ͑ ϩ ͒ (18) s.t. 1 m U c1 , RA A RB B drawal penalty for bonds, and sufficiently actuarially unfair medical insurance pricing. ϩ ͑ Ϫ ϩ ␤ ϩ ͒ Ն ៮ mU c1 M I, RA A RB B U. We turn now to a medical risk that occurs only in the second period. With medical insur- In the case of no trading after the initial date, ance purchased at the start of period 1, expected Ϫ ϩ annuities continue to dominate bonds. If the utility can be written as (1 m)U(c1, RAA ϩ ϩ Ϫ ϩ ␤ insurance is actuarially fair, there will be com- RBB) mU(c1, RAA RBB M I). Thus, plete medical insurance (and so c1, A, and B are medical risk in the second period does not the same as if there were no risk and expendi- change the dominance of annuities over bonds, tures were reduced by mM). With less favorable whatever the pricing of medical insurance. In medical insurance, the presence of both risk and the absence of the arrival of information, there insurance generally affects the level of illiquid are two equivalent ways of organizing medical savings and thus the level of annuitization, but insurance. One is to purchase medical insurance all savings are annuitized since bonds are still at the start of period 1 (as with long-term care dominated. insurance). The other is to purchase an annuity To bring out the role of liquidity, assume that with a plan to purchase medical insurance bonds can be sold in the first period with an at the start of period 2, if alive. With both 1582 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005 formulations, a worsening of the pricing of quidity of annuities were an explanation for lack medical insurance will generally alter first-pe- of demand for illiquid annuities, Bernheim riod consumption, and thus total savings. In the (1987b) has pointed out that illiquidity is not a second formulation, this translates directly into complete explanation for the near absence of the demand for annuities. In addition, there annuity markets. Bernheim proposes the cre- would generally be a change in the amount of ation of annuities that are subject to cancellation medical insurance purchased in the second pe- at any time. In terms of the notation above, riod. In the first formulation, unlike the second, bonds would have a return if held to maturity, ␣ a change in the level of medical insurance RB, and a return is withdrawn early, B, with ␣ would also change the level of annuity pur- annuities characterized by RA and A. Even with chase, even if preferences were such that first- an early withdrawal option, we might plausibly Ͻ ␣ Ͼ ␣ period consumption did not change. For have RB RA and B A. This could be the example, going from fair medical insurance to outcome, since early withdrawal from an annu- no medical insurance would increase the spend- ity has implications for the cost of providing the ing on annuities by the full amount previously annuity, while this is less so in the case of a spent on medical insurance if preferences were bond (where withdrawal may just reflect avail- such that first-period consumption did not able alternative investments). change. That is, removing insurance that is ef- fectively annuitized can increase the demand for C. Inferior Returns to Annuities a standard annuity. Thus, we can conclude that the timing of the Another route to limited annuitization is if risk of medical expenses is key to understanding annuity pricing and the arrival of asymmetric the interaction between the availability of med- information imply an advantage to delayed an- ical insurance and the purchase of annuities. In nuitization. For example, Moshe A. Milevsky the absence of strong assumptions, it is thus and Virginia R. Young (2002) consider a cost to impossible to sign the effect of liquidity needs annuitization associated with illiquidity that on annuity demand. In one parameterization, renders the return on annuities essentially infe- Cassio M. Turra and Mitchell (2005) find that rior to the return on some bonds. In particular, the optimal fraction of wealth annuitized re- they show that it may be optimal to wait to mains large even when out-of-pocket (unin- annuitize wealth if returns on investment in the sured) medical expenditures are possible and future may exceed present returns. Similarly, if are associated with truncated lifetimes. These annuities purchased later in life are priced more simulations also show, however, that these un- favorably than those purchased earlier, deferral insured expenditures tend to reduce demand for of annuitization may be optimal.13 If, however, annuities below 100 percent of savings. there are utility gains to annuitizing at, say, age We have assumed an absence of a relation- 65, and yet it is optimal to delay annuitization, ship between medical expenses and life expect- this implies there are even larger utility gains to ancy. If a medical expense in period 1 implies a annuitizing later in life. Empirically, however, lower survival probability, then that would we do not observe households choosing to an- strengthen the value of liquid bonds. The next nuitize at later ages, suggesting that such an subsection briefly considers a role of the arrival explanation cannot fully explain the near ab- of information about life expectancy.12 sence of demand for private market annuities. Since we examine annuity demand in differ- Of course, if annuity returns, net of adminis- ent settings, rather than a model of equilibrium, trative costs, are inferior to those of conven- we do not explore the illiquidity of annuities tional assets, annuity demand can go to zero. (relative to bonds) that is present. Even if illi- Mitchell et al. (1999) show, however, that available pricing on annuities does not seem to be sufficient to render annuitization unattractive. 12 If the medical condition is observable to the provider, Recent work by Irena Dushi and Anthony Webb it may be that bundling insurance for the cash need with the annuity could improve pricing for both forms of insurance by eliminating adverse selection that would exist for either product individually, as has been proposed by Warshawsky 13 Michael Dennis (2004) also finds a gain to deferred et al. (forthcoming). annuitization for some price patterns. VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1583

(2004), which builds upon previous work by than the value of a bond, evaluated at zero Mitchell et al. (1999) and Brown and James M. annuities: Poterba (2000), shows that a combination of ͑ Ϫ ͒ Ј͑ ͑ Ϫ ͒͒ low annuity returns, within couple-risk sharing (26) 1 q RB RAv RB RB B c2 that substitutes for a formal annuity market, and very high levels of pre-existing annuities can Ͻ RB RB ͑qvЈ͑RB RB B͒ render additionally fixed annuities unattractive ϩ͑ Ϫ ͒ Ј͑ ͑ Ϫ ͒͒ at any age. Even these results, however, suggest 1 q v RB RB B c2 ). that we should observe frequent annuitization Ͼ by surviving spouses, which we do not. With c2 0, this is violated for an actuarially Ϫ ϭ fair annuity (1 q)RA RB, and for annuities III. Bequests close enough to fair. If the annuity is actuarially fair, then Throughout the paper, we have maintained the assumption that there was no bequest mo- (27) tive. While a bequest motive reduces the de- Ј͑ ͑ ϩ Ϫ ͒͒ ϭ Ј͑ ͒ mand for annuities, it does not eliminate it in v RB RA A RB B c2 qv RB RB B general. To see this, consider a model with two ϩ ͑ Ϫ ͒ Ј͑ ͑ Ϫ ͒͒ periods and uncertain survival to the second 1 q v RB RB B c2 . period as the sole risk. Ignoring the role of inter ϭ vivos gifts, there can be a bequest at the end of This implies RAA c2, a point implicit in Yaari the first period or of the second. We model the (1965). That is, annuitization equals second- bequest motive as an additive term depending period consumption. Thus, with this approach on the two levels of bequests that might happen to bequests, the case for significant annuitiza- at the end of periods 1 and 2. In this case, we tion survives the presence of a bequest can express utility in terms of own consumption motive.14 and the two possible bequest levels. With death at the end of period two, the bequest is received IV. Simulations in period 3, involving additional interest. Using the dual formulation, we have: The theory shows that while full annuitization is not guaranteed unless every bond is ϩ ϩ (23) min : c1 A B dominated by some annuity, an unrealistic as- c1 , A, B, c2 sumption, it is also clear that partial annuitization is optimal in many settings. The theory ͑ ͒ (24) s.t. : U c1 , c2 itself is insufficient for answering practical policy questions that are often centered on what ϩ ͑ ͑ ϩ Ϫ ͒͒ Ն ៮ V RB B, RB RA A RB B c2 U. fraction of savings is optimally annuitized in different market settings and with different pref- If the utility of bequest is the expectation of a erences. Thus, a large simulation literature has concave value of the bequest, v, evaluated at the arisen to address this. We briefly review this start of period three (whether given then or literature, and then extend it by conducting a earlier), we have: “stress test” of the annuity valuation results by considering cases that are intentionally designed ͑ ͑ ϩ Ϫ ͒͒ (25) V RB B, RB RA A RB B c2 14 The use of a “guarantee” with annuities is common. ϭ qv͑RB RB B͒ This often involves a minimum number of payments, re- sulting in a lower monthly annuity for a given purchase ϩ͑1 Ϫ q͒ ͑R ͑R A ϩ R B Ϫ c ͒͒ price. We note that this generates a random bequest com- v B A B 2 pared with purchasing the same lower annuity without the guarantee and bequeathing (or giving before death) the reduced purchase price. In the case of fair annuities, the where q is the probability that the individual guarantee is a pure gamble and implausibly optimal. With dies before period 2. For there to be no annu- selection issues, the demand depends on pricing and the itization, the value of an annuity must be less guarantee may be part of addressing selection issues. 1584 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005 to create a more extreme mismatch between cases, however, the optimal level of annuitiza- desired consumption and the annuity income tion was generally high. stream than we would expect to see in practice. By showing that the optimal level of annuitiza- A. Relaxing Additive Separability: A “Stress tion remains high even in what we consider to Test” of Annuity Valuation be extreme cases, we conclude that the virtual absence of voluntary annuitization cannot be A key practical message from our theoretical easily explained by the standard set of reasons results is that the welfare gains from annuitiza- that come from the life-cycle framework. tion depend on preferences mainly through the Benjamin M. Friedman and Warshawsky consumption trajectory. Thus, if one is seeking (1990) and Mitchell et al. (1999) are examples to reconcile the theory with the empirical small- of the simulation literature that calculate utility ness of voluntary annuity markets, one ought to gains from annuities under alternative assump- be looking for situations in which there is a tions. These papers, like many others, consider severe mismatch between the desired consump- a retired individual whose preferences can be tion trajectory and the income path provided by represented by a utility function that exhibits a limited set of annuities in an incomplete mar- constant relative risk aversion, exponential dis- kets setting. counting, and intertemporal separability. They In this section, we extend the simulation lit- find that these consumers would accept a sub- erature by modeling preferences that lead to a stantial reduction in wealth in exchange for severe mismatch, with the goal of seeing access to actuarially fair annuity markets.15 whether such mismatches can plausibly explain Numerous subsequent papers have explored the small size of the market. We do this by the gains from annuitization under a wider exploring a class of models which has the real- range of assumptions, including: uncertainty istic property that preferences are not intertem- about future asset returns (Milevsky and Young, porally additive and has the feature that 2002); risk pooling within couples (Brown and different parameterizations, meant to reflect the Poterba, 2003); uninsured medical expenditures heterogeneity of household financial positions (Turra and Mitchell, 2005; Sven H. Sinclair and at retirement, lead to very different time shapes Kent A. Smetters, 2004); actuarially unfair of optimal consumption. prices due to mortality heterogeneity (Brown, We consider a 65-year-old male with survival 2003; Oded Palmon and Avia Spivak, 2001); probabilities taken from the U.S. Social Secu- and higher levels of pre-existing annuities rity Administration for the year 1999, modified (Dushi and Webb, 2004). In nearly every case, (to ease computation) so that death occurs for the annuity products considered in these papers certain by age 100. His utility function is: are constrained in some way (i.e., annuity markets are incomplete), the most common assump- 100 tion being that the annuity’s payment stream is ϭ ͸ ͑ ϩ ␦͒Ϫt͑ ͒1 Ϫ ␥ ͑ Ϫ ␥͒ (28) U 1 ct /st / 1 . fixed in real terms. Consistent with our theoret- t ϭ 65 ical findings, these papers find that there are substantial gains to some annuitization, but that When the parameter st is constant, this is the full annuitization is not always optimal. Our frequently modeled case of CRRA preferences. theoretical results provide a framework in Following this literature, we set the real interest which such results can be interpreted: when full rate and discount rate (␦) equal to 0.03, and annuitization is suboptimal, the settings were calculate the utility gains from annuitization. such that incomplete markets led to a mismatch We set ␥ ϭ 2 and find the consumption vector between desired consumption trajectories and that solves the expenditure minimization prob- available annuity income paths. Even in these lem numerically using standard optimization techniques.16 We begin simply by verifying that our model

15 Another strand of the literature examines the “probability of a shortfall” when a consumer tries to self-insure rather than using formal annuity markets. Milevsky (1998) 16 Results for other levels of risk aversion are available is a good example of this approach. in some of the papers listed above. VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1585

TABLE 1—SIMULATED UTILITY GAINS FROM ACCESS TO ANNUITIZATION

% of savings Habit Ratio of initial annuitized Welfare gain when Welfare gain when

adjustment habit s65 to Discount with real optimal % of annuity markets Case speed ␣ real annuity rate annuities savings annuitized complete (1) (2) (3) (4) (5) (6) (7) 1 n/a n/a 0.03 100 56 56 2 1 1 0.03 100 71 96 3 1 0.5 0.03 100 67 74 4 1 2 0.03 90 51 54 5 1 2 0.1 75 25 30

␥ Notes: Risk aversion is set at 2 throughout. Column 3 compares the habit level (s65 in equation (28)) to the annual payout of an actuarially fair annuity with equal payouts in years 66 through 100. Welfare gains in columns 6 and 7 are calculated as the amount of initial wealth that a 65-year-old male unable to annuitize any savings would have to be given to attain the same level of utility as if he were allowed to annuitize an optimal fraction of savings. Column 5 is the optimal fraction of savings placed in a real annuity, corresponding to the equivalent variation in column 6. Complete annuitization is optimal when the trajectory of payouts is unconstrained. (Figure 1 plots optimal consumption trajectories with no annuities, real annuities, and optimally designed annuities for each of the five sets of parameters.) matches the results of the previous literature in level relative to past consumption, that matters the case in which st is fixed at a value of one, for utility. For example, life in a studio apart- corresponding to the CRRA utility case. In this ment is surely more tolerable for someone used setting, which we denote as “Case 1” in Ta- to living in such circumstances than for some- ble 1, we find that it is optimal for an individual one who was forced by a negative income shock to fully annuitize all wealth. Moreover, with r ϭ to abandon a four-bedroom house. In choosing ␦, real annuities provide the optimal annuity how to allocate resources across periods, “habit stream. consumers” trade off immediate gratification A rough estimate of the magnitude of EVs from consumption not only against a lifetime can be obtained by observing the difference in budget constraint, but also against the effects of trajectories between the unconstrained (circles consumption early in life on the standard of and ϫ’s in Figure 1, Case 1) consumption plan living later in life. and the constrained real annuity (triangles in Following Diamond and Mirrlees (2000), we Figure 1) consumption plan. When optimal con- model the evolution of the habit as follows: sumption is sharply decreasing, the constraints ϭ ͑ ϩ ␣ ͒ ͑ ϩ ␣͒ bind consumption away from the optimal path. (29) st st Ϫ 1 ct Ϫ 1 / 1 . In these cases, the benefit of annuitization is relatively small because the sum of future con- ␣ is the parameter that governs the speed of sumption is relatively small and the gain is adjustment of the habit level. When ␣ is zero, offset further by the constraints. When optimal the habit is constant and we are back in the unconstrained (zero annuitization) consumption additively separable case. As ␣ approaches in- is hump shaped and less steeply decreasing, finity, present habit approaches last period’s there are greater benefits, and the constraints consumption. We select an intermediate speed impose fewer costs, so the net benefit to annu- of habit adjustment of the habit (␣ ϭ 1). Away itization is greater. from zero, we find that changes to ␣ make no What differentiates our more general setup substantive difference to our results, and there- ␣ from prior work is that we can vary st in equa- fore do not report results for a range of tion (1) so that the utility function exhibits an values.17 “internal habit,” which we can then adjust to When the habit evolves (␣ Ͼ 0), present create optimal consumption trajectories that dif- consumption increases the future standard of fer markedly from the usual CRRA case. The intuition behind our utility function, taken from Diamond and Mirrlees (2000), is that it is not 17 Additional results are available from the authors upon the level of present consumption, but rather the request. 1586 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005

FIGURE 1. OPTIMAL CONSUMPTION TRAJECTORIES UNDER DIFFERENT ANNUITY AVAILABILITY Notes: The five cases correspond to the parameter values listed in Table 1. In each case, circles plot optimal consumption for a consumer who is unable to annuitize any wealth. Triangles plot optimal consumption when only a constant real annuity is available. ϫ’s plot optimal consumption when any consumption trajectory can be funded solely by annuities. The level of expenditures is the minimum to hold utility equal to the maximum utility when a constant real annuity is available and expenditures are equal to 100. VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1587 living. This increase in the standard of living without. The accuracy of this intuition hinges, reduces the level of utility and increases mar- however, on the level of the initial habit s65 ginal utility in the future. With later consump- relative to resources at retirement. If the initial tion, there are fewer future periods that are habit is small, then it is correct that the presence adversely affected by an increased standard of of a potentially increasing habit leads to de- living. ferred consumption and increased valuation of While some researchers have attempted to the annuity. However, if the initial habit is so measure parameters of alternative habit forma- large as to be unsustainable, so that the habit tion models, largely involving “external habit” level must fall over time, then the presence of formation in calibration exercises, there is, to the habit will push optimal consumption earlier, our knowledge, no empirical study that provides because of the desire to smooth the ratio of estimates of the initial standard s65 in our consumption to habit across time. If the habit model. This is not a major limitation, however, decreases with time, then smoothing likewise as our objective is not to carefully calibrate a requires that consumption decrease with time. realistic model of a representative lifetime op- Hence, a very high initial habit relative to re- timizing consumer, but rather to create optimal sources provides a mechanism beyond heavy consumption trajectories that create an inten- discounting whereby the deferred consumption tionally more severe mismatch with the avail- required by a constant real annuity may be very able annuity structure than we actually observe burdensome. in available consumption data, and to recognize To calibrate the model, and to build some the wide diversity in wealth at retirement rela- intuition, we first consider an individual who tive to lifetime income. Doing so allows us to has reached retirement with resources that, “stress test” the annuity valuations in order to when annuitized, are sufficient to satisfy exactly see if one can plausibly explain the paucity of their habit level of consumption for the rest of voluntary annuity purchases within a strictly their lifetime. In this scenario, labeled “Case 2” rational model. We do so by considering initial in Table 1, the individual enters retirement with standards of living that generate consumption a habit level s65 that is precisely equal to the paths that are both extremely friendly and ex- annual annuity value of their wealth.18 As extremely unfriendly to annuitization. pected, the presence of a potentially increasing The first four columns of Table 1 indicate the habit leads to deferred consumption, i.e., an case number and the relevant parameters that upward-sloping consumption path in the early are varied across cases. Column 5 reports the years (see Figure 1, Case 2). When only real fraction of wealth that is optimally placed in a annuities are available, the optimal allocation is constant real annuity instead of bonds. In col- to annuitize fully, because doing so provides the umn 6, we report the equivalent variation individual with the highest possible return with- (“EV”) associated with this optimal amount of out imposing any binding liquidity constraints. real annuitization. This is the increase in wealth While the utility gain is even higher than in our required to hold utility constant, while moving base case, the utility gains are higher still if the from having the optimal amount annuitized in a individual is able to purchase annuities in a real annuity to having all of wealth in bonds. complete market, and thus match exactly the Column 7 reports the gains from annuitization desired consumption trajectory. In this latter (again as an EV) for the case in which the case, the ability to choose an optimal annuity individual is permitted to choose an optimal trajectory is equivalent in utility terms to nearly payout trajectory, i.e., they are no longer con- doubling of nonannuitized wealth. strained to purchase a constant real annuity. For Reaching retirement with precisely the re- shorthand, we refer to this as the “complete sources required to maintain one’s past living markets” result, although, strictly speaking, all standard is not the empirical norm, however. that is necessary is that the subset of annuities Indeed, given the wide range of retirement re- desired by this individual be available. sources across the population, it would not Because an internal habit introduces a new cost to early consumption, one might expect that the fixed real annuity will be even more 18 Given our mortality assumptions, the annual annuity desirable with an internal habit (␣ Ͼ 0) than value of $100 of starting wealth is $8.59. 1588 THE AMERICAN ECONOMIC REVIEW DECEMBER 2005 make sense to consider a common level of pre- would optimally annuitize three-quarters of his retirement consumption relative to retirement wealth. In results not reported, we have found resource across the entire population. Michael that even when the individual’s habit is so high Hurd and Susann Rohwedder (2003) provide that retirement wealth can provide for an annu- new evidence on the distribution of the percent- ity that is only one-sixth of the initial habit age change in spending from pre- to post-retire- level, the optimal fraction of wealth invested in ment. They find a mean consumption drop of 13 a real annuity is still nearly two-thirds of initial to 14 percent. At the twentieth percentile, they retirement wealth. find a 30-percent decline in consumption, These simulations indicate that it is ex- whereas at the ninety-fifth percentile, they find a tremely difficult to generate optimal consump- 20-percent increase in consumption.19 We tion profiles such that the optimal fraction of choose to examine even more extreme points in wealth annuitized drops much below two-thirds the consumption distribution by evaluating the of initial wealth at retirement without appealing case in which the habit level of consumption is to many additional factors. This suggests that 50 percent and 200 percent of the amount that the absence of annuitization outside of Social can be sustained by one’s retirement wealth. Security and defined-benefit pensions cannot In Case 3 of Table 1 and Figure 1, the indi- easily be explained within a rational life-cycle vidual has sufficient wealth to purchase a real model, even with preferences that lead to a annuity stream that is double the initial habit severe mismatch between desired consumption level in retirement. As in Cases 1 and 2, it is still and available annuity paths. optimal to annuitize fully all wealth. Doing so in a constant real annuity generates a utility gain V. Conclusions and Future Directions that is equivalent to a 67-percent increase in nonannuitized wealth. The ability to match an- With complete markets, the result of com- nuities to the desired upward-sloping consump- plete annuitization survives the relaxation of tion trajectory is even more valuable. several standard, but restrictive, assumptions. More interesting for purposes of our “stress Utility need not satisfy the von Neumann– test” is to examine the case in which the indi- Morgenstern axioms and need not be additively vidual has only half of the level of wealth that separable. Further, annuities need not be actu- would be required to sustain his initial habit arially fair, but only must offer positive net consumption level. As can be seen in Case 4, premia over conventional assets. this leads the individual to want to sharply re- With incomplete markets, the full annuitiza- duce consumption early in retirement in order to tion result can break down when there is a rapidly bring down the level of habit to a more sufficient mismatch between the optimal con- sustainable level. As such, the optimal fraction sumption path and the income stream offered by of wealth held in a real annuity declines to 90 the annuity market. In the much-studied case of percent, as the individual uses the 10 percent of a world where only individual mortality is un- nonannuitized wealth to supplement consump- certain, we find that there may be considerable tion in the first ten years, and then consumes the individual heterogeneity in the value of annu- annuity for the remainder of life after bring the itization. Heterogeneity in annuity valuations is habit down to a lower level. driven by heterogeneity in the willingness to Pushing the “stress test” even further, Case 5 substitute late consumption for early consump- adds a high discount rate to the low wealth-to- tion. We find that even for preferences that habit ratio. By holding r fixed at 0.03 and rais- stretch the bounds of plausible impatience, a ing the discount rate to 0.10, the optimal large fraction of wealth is optimally placed in a consumption trajectory, as shown in Case 5, is constant real annuity. even more heavily front-loaded, and thus the In our simulations, we have retained the ab- mismatch with the real annuity income stream is stractions of no bequest motive, no risks other particularly severe. Even so, the individual than longevity, and no learning about health status or other liquidity concerns. Exploring the consequences of dropping these assumptions in 19 Hurd and Rohwedder (2003) do not report points in the context of nonseparable preferences and un- the distribution below the twentieth percentile. fair annuity pricing would be an important gen- VOL. 95 NO. 5 DAVIDOFF ET AL.: ANNUITIES AND INDIVIDUAL WELFARE 1589 eralization, but obtaining results will require ment: Testing the Pure Life Cycle Hypothe- strong assumptions on annuity returns, on the sis,” in Zvi Bodie, John B. Shoven, and nature of bequest preferences and liquidity David A. Wise, eds., Issues in pension eco- needs, and on the stochastic structure of bond nomics. NBER Project Report Series, Chi- returns. cago: University of Chicago Press, 1987a, pp. It is sometimes argued that the lack of annu- 237–74. ity purchase is evidence for a bequest motive. Bernheim, B. 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