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Annuities and Individual Welfare Annuities and Individual Welfare By THOMAS DAVIDOFF,JEFFREY R. BROWN, AND PETER A. DIAMOND* Advancing annuity demand theory, we present sufficient conditions for the optimal- ity 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 de- mand 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 litera- ture 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.
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