Illiquid life annuities

DOIhttp://doi.org/10.1111/jpet.12253
Published date01 April 2018
AuthorHippolyte d'Albis,Johanna Etner
Date01 April 2018
Received: 25 August 2016 Accepted: 5 February2016
DOI: 10.1111/jpet.12253
ARTICLE
Illiquid life annuities
Hippolyte d’Albis1JohannaEtner2
1ParisSchool of Economics, CNRS
2EconomiX,CNRS and University Paris Nanterre
Weare grateful to Alexis Direr,Jean-Pierre
Drugeon,Ben Heijdra, and Jochen Mierau for
insightfulcomments. We are also grateful for
thevaluable remarks and suggestions made
bya referee and for the financial support from
theEuropean Research Council (ERC Stg Grant
283953)and Netspar. This research has been
conductedas part of the project Labex MME-DII
(ANR11-LBX-0023-01).The usual disclaimer
applies.
Hippolyted’Albis, Paris School of Economics,
CNRS,48 Boulevard Jourdan, F75014 Paris,
France(hdalbis@psemail.eu).
JohannaEtner, EconomiX, CNRS, and University
ParisNanterre, 200 Avenue de la République,
F92001Nanterre, France (johanna.etner@u-
paris10.fr).
In this paper,we consider illiquid life annuity contracts and show that
they maybe preferred to those illustrated by Yaari. In an overlapping
generations economy, liquid life annuities are demanded only if the
equilibrium is dynamically inefficient. However, an equilibrium dis-
playing a positivedemand for illiquid life annuities is indeed efficient.
In this latter case, the welfare at steady state is larger if illiquid life
annuity contracts are available.
1INTRODUCTION
In this paper, we challenge the common thought that the life annuity contract proposed byYaari in his seminal 1965
paper is optimal. We indeed show, in a standard neoclassical framework,that another contract, which actually resem-
bles more the contracts offered by annuity providers,may be preferred by rational individuals.
The economic theory of annuities has been strongly influenced by Yaari (1965), a paper that has studied optimal
demand for annuities in a life-cycle model with or without bequest motives. The financial asset that is named annuity
by Yaari has positive returns if the bearer is alive and zero if he is not. Annuities are nevertheless demanded since
their returns are larger than the one yielded by risk-free bonds. The difference between the two yields is the annuity
premium, which is said to be fair when it equals the inverse of the survival probability. Importantly, as the individual
ages, the premium increases. This characterization of an annuity has been quite influential and has lead to numerous
studies (see among others Davidoff,Brown, & Diamond, 2005; Sheshinski, 2008).
However, there are many types of annuity contracts (Cannon & Tonks, 2008). Their features are quite different
from Yaari’s annuities. For instance, the premium is age-independent. The individual purchases some annuities dur-
ing youth and, after a given age—let’s say post-retirement—he periodically receives a fixed amount as long as he sur-
vives. Another feature is that the contract is irreversible.Once payments have begun, one can not recover the amount
invested. An implicit assumption in Yaari is that agents, upon survival, receive the capital and the interests of their
annuity.This means that they are in position to renegotiate their contract at each period, hence the premium increases
as the individual ages.
Journal of Public Economic Theory.2018;20:277–297. wileyonlinelibrary.com/journal/jpet c
2017 Wiley Periodicals,Inc. 277
278 D’ALBISAND ETNER
In this paper, we propose a standard framework in which the individual has the choice between two types of life
annuity contracts. The first one, which we name here flexible, is the one proposed by Yaari (1965). The second one,
which we name illiquid, is irreversible and proposes age-independent returns. In both cases, we suppose that the annu-
ity premium is such that annuity providers make no profit. Illiquid annuities havebeen introduced in life-cycle models
byHorneff, Maurer, and Stamos (2008) and Peijnenburg, Nijman, and Werker(2016) in order to discuss the issue of low
demand for annuities. Our purpose is to study analytically the equilibrium and welfare consequences of the existence
of such contracts.
First, we analyze the consumer’s optimal decisions overthe life cycle under uncertain lifetime. We consider a setting
in which the individual ages, which more precisely means that survival probabilities decrease with age. We therefore
depart from the two-period life-cycle setting, which would not have allowed us to make a clear distinction between
increasing and fixedreturns. We show that illiquid annuities are preferred to flexible ones if the expected returns of the
first are sufficiently greater than those of the second. This is the consequence of an arbitrage between more flexibility
and more returns. We are aware that partial equilibrium analysis may bias the evaluation of the efficiency of annuity
markets (Heijdra& Mierau, 2012) and, therefore, we move to a general equilibrium analysis.
Second, we study an overlapping generations economy with neoclassical production (Diamond, 1965), in which
returns of both contracts are determined at the equilibrium. We show that illiquid annuities are preferred when the
equilibrium is dynamically efficient, whereas flexible annuities are preferred when it is inefficient. This result is based
on the fact that illiquid annuities represent a transfer from one generation to the next generation.When the popula-
tion growth rate is relatively low, which is the case when the equilibrium is efficient, this transfer is inexpensiveand
the investmentis profitable. We then discuss the optimality of both annuity contracts. In particular, for the dynamically
efficient equilibrium, the welfare at steady state is larger if illiquid life annuity contracts are available.
Theefficiency of transfer from the old to the young has been shown in overlapping generations models with acciden-
tal bequests by Pecchenino and Pollard (1997), Feigenbaum, Gahramanov, and Tang(2013), and Heijdra, Mierau, and
Reijnders (2014). In equilibrium, partial annuitization may dominate full annuitization strategiesas theyimply an unin-
tended transfer to the next generationand an increase in savings. Our results differ from theirs for two reasons. First,
we assume full annuitization and compare annuity contracts. Second, we show that illiquid annuities are preferred at
both the individual level and the long-run equilibrium.
Finally,to test the robustness of our results, we propose three extensions of our model by considering successively
a background risk, a bequest motive, and a subjective evaluationof survival probabilities. We also discuss our result by
introducing an alternative annuity pricing where redistribution between cohorts is forbidden.
2INDIVIDUAL BEHAVIOR
2.1 Demographics
We consider an overlappinggenerations model in which agents live a finite yet uncertain length of time. They live for a
maximum of three periods, also called ages, which are denoted i={0,1,2}.The probability of being alive at age i, con-
ditional on survival until age i1, is denoted pi. Survival probabilities at each age are constant over time, but decrease
with age.
Let Ni,t be the number of agents of age iat time t. At each time t=0,1,2,N0,t, identical agents are born. Thus,
the number of agents of age 1 born at time tis N1,t+1=p1N0,t and the number of agents of age 2 born at time
tis N2,t+2=p1p2N0,t. Finally, we assume that the number of agents of age 0 grows at a constant rate, denoted n,
with n>1:
N0,t =(1+n)N0,t1.(1)

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