Faculty of Business and Law School of Accounting, Economics and Finance

ECONOMICS SERIES

SWP 2012/1

Is It Really Good to Annuitize?

James Feigenbaum

and Emin Gahramanov

The working papers are a series of manuscripts in their draft form. Please do not quote without obtaining the author’s consent as these works are in their draft form. The views expressed in this paper are those of the author and not necessarily endorsed by the School or IBISWorld Pty Ltd.

Is It Really Good to Annuitize?∗

James Feigenbaum†

Utah State University

Emin Gahramanov

Deakin University

March 6, 2012

Abstract

Although rational consumers without bequest motives are better off investing exclusively with annu-

itized instruments in partial equilibrium, we demonstrate the welfare effect of annuitization is ambiguous

in general equilibrium on account of the pecuniary externality. Accidental bequests improve consump-

tion allocations by transferring capital mostly to young people rather than to the old, for whom the

present value of the transfer is much less. If households are not borrowing constrained in the rational

competitive equilibrium where they annuitize, there will exist a consumption/investment rule involving

nonannuitized investments that confers higher utility in general equilibrium while maintaining the same

equilibrium capital stock. Thus it may be that households eschew annuitization because society has

learned it is suboptimal. Regardless of the explanation for this behavior, policymakers should not take

steps to encourage more annuitization by the public.

JEL Classification: C61, D11, E21

Keywords: consumption, saving, coordination, learning, general equilibrium, pecuniary externality,

annuities puzzle, bequests, mortality risk, overlapping generations, optimal irrational behavior, Golden

Rule

∗The authors thank Frank Caliendo, Tom Davidoff, Jim Davies, Scott Findley, Hui He, Ian King, Val Lambson, Geng Li,Jochen Mierau, Moshe Milevsky, Svetlana Pashchenko, Eduardo Saucedo, Eytan Sheshinksi, and participants at seminars atthe Australasian Economic Theory Workshop; BYU; CEF Conference; Deakin University; the University of Hawai’i; the IFIDWorkshop; University of Pittsburgh; SABE Conference; Utah State; University of Sydney; and the Workshop on OptimalControl, Dynamic Games, and Nonlinear Dynamics for their views and comments. We also appreciate financial supportprovided by the TAER program at Deakin.†Corresponding author email: J.Feigen@aggiemail.usu.edu. Website: http://huntsman.usu.edu/jfeigenbaum/

1

Annuities, i.e. investment instruments that pay an income stream that terminates upon the owner’s

death, present a puzzle to economists. In their optimal format, annuities perfectly insure against longevity

risk by giving surviving investors, on top of the ordinary return to capital, a premium that increases with

the probability of dying.1 Deceased investors surrender their investment, and these assets are used to pay

the premiums of surviving investors. Without bequest motives, rational households are indifferent to the

disposal of their assets after death. Since annuities earn a higher return, such households should invest all

their wealth in annuities (Yaari (1965)). Even households with bequest motives ought to annuitize the wealth

intended to finance their own consumption. Nevertheless, private annuity investments account for only 1%

of total household wealth for households over age 65 in the United States,2 and only 6% of households in

this age range participate in private annuities (Pashchenko (2011)). Although many researchers believe the

near total rejection of annuities by the public can be explained if all relevant frictions are properly accounted

for, this stylized fact remains a diffi cult challenge for the rational-expectations paradigm.3 Moreover, even

if one finds that annuities products available to the public today are not attractive to rational agents, this

does not explain why the annuities market is so thin to begin with.

We consider the issue in a different light. Most of the literature on the annuities puzzle is motivated by

the presumption that people hurt themselves when they fail to annuitize. In the basic partial-equilibrium

model of Yaari (1965), this is certainly true from the perspective of a single individual, but is it true for

society as a whole in general equilibrium? If agents behave rationally, the answer is ambiguous, depending

on the parameterization of the model, though for our baseline calibration welfare will be higher if households

do not have access to annuities.4

If we expand our focus from individually rational behavioral rules to any behavioral rule consistent

with market-clearing constraints, the answer is more straightforward.5 If households are not borrowing

constrained in a rational competitive equilibrium where households have access to annuities, there will exist

a market-feasible consumption rule employing nonannuitized investments that confers higher lifetime utility

while maintaining the same capital stock. In our baseline calibration, the optimal consumption and saving

rule can nearly achieve the maximal welfare of the Golden Rule without extramarket transfers of consumption

These unintuitive results are a consequence of the pecuniary externality (McKean (1958), Prest and

Turvey (1965)), the property intrinsic to markets that people’s actions affect prices, which in turn affect

people’s behavior. Although this two-way causal relationship is the centerpiece of modern economics,

1 In practice, a financial intermediary will do the actual work of maintaining assets that finance the recipient’s consumptionstream. Frictions in the annuities market will reduce the effective return on annuities and the consequent consumption stream.See Sheshinski (2008) for a review of the general theory of annuities.

2This is according to the 2000 Health and Retirement Study (Johnson, Burman, and Kobes (2004)). Social Security anddefined benefit pension plans are not accounted for here, though they are effectively annuities, albeit suboptimal annuities thatdo not provide the intracohort risk-sharing benefits of financial annuities (Guo, Caliendo, and Hosseini (2012)). Transactioncosts and borrowing constraints may deter poor households from optimizing their portfolios, but this cannot explain why sofew wealthy households do so.

3Davidoff, Brown, and Diamond (2005) and Leung (2010) show that the management costs of annuitization would haveto be huge to prevent households from annuitizing a substantial portion of their wealth. See Pang and Warshawsky (2009),Lockwood (2009), and Pashchenko (2011) for estimates of how much annuitization will occur in lifecycle models with frictions.

4Heijdra, Mierau, and Reijnders (2010) have also found this in a two-period model.5There is no annuities puzzle if households are not required to be fully rational. See Brown (2007), Hu and Scott (2007),

and Milevsky and Young (2007).

2

economists generally assume that households take prices as given and ignore the effect their actions have

on prices. The Welfare Theorems prove this is an innocuous assumption in infinite-horizon, representative-

agent models for which Pareto optimality is equivalent to welfare maximization. However, this equivalence

does not hold for overlapping-generations models, even when competitive equilibria are Pareto optimal,

because there are multiple agents. Recent work (Feigenbaum and Caliendo (2010), Feigenbaum, Caliendo,

and Gahramanov (FCG) (2011)) has shown that different cohorts can coordinate their consumption and

saving behavior across generations to exploit the pecuniary externality and improve welfare for everyone in

the steady state without extramarket transfers of consumption.6 In the present context, we see this result

extends to portfolio allocation rules that apportion savings between annuities and nonannuitized investments.

Holding the received bequest fixed in a partial-equilibrium fashion, households do better investing only in

annuities. But if everyone follows this strategy, there will be no bequest to inherit. Thus households can

increase their welfare if they coordinate on a strategy of not annuitizing their wealth.

This behavior is irrational in that individual consumers could increase their welfare further by purchasing

annuities while enjoying the bequests left behind by their less selfish peers. Thus FCG (2011) term this

coordination “optimal irrational behavior”. In terms of game theory, markets in an overlapping-generations

setting induce a Prisoners Dilemma game for which the socially optimal outcome is not a Nash equilibrium.

What is remarkable about the application of optimal irrational behavior to annuities is that, whereas pre-

ceding papers have hypothesized that households might engage in socially optimal, yet irrational behavior

just as experimenters observe that people sometimes do cooperate in one-shot Prisoners Dilemma games,

here we actually have a real-world example where almost everyone behaves irrationally in a manner that

improves their welfare via macroeconomic mechanisms.

Bequests improve welfare by allowing better intertemporal consumption allocations. Both annuities and

bequests preserve capital by transferring the assets of deceased agents to living agents. However, annuities

primarily transfer this wealth to the elderly, who receive the largest insurance premiums. Bequests transfer

this capital across the whole population.7 A transfer received when young is discounted less in the household

budget constraint, so it purchases more consumption. The benefits of bequests could be further increased

if larger bequests are given to younger agents (Feigenbaum and Gahramanov (2011)).

Thus the mystery of why people do not annuitize can be accounted for by the fact that they are better

off as a whole, though not individually, if they abstain from annuitization. There is no friction or cognitive

error for policymakers to correct here. To the extent that real-world annuities are complicated financial

products, the widespread aversion to expending the time and effort necessary to understand them actually

protects us from a suboptimal world in which fewer people inherit bequests. While we do not observe people

following the best market-feasible consumption rule, which would bring us much closer to the Golden Rule,

people deviate from this rule primarily along the dimension of choosing how much to save, not in terms of

6Early generations will be hurt when this coordination is implemented, so the Pareto optimality of dynamically effi cientcompetitive equilibria is not violated.

7Here we focus on the case where bequests are spread uniformly across the whole population, but this is not essential as longas some assets are inherited by the young. Feigenbaum and Findley (2011) study the benefits of bequests in a model wherebequests are distributed across the surviving population more realistically with a greater portion going to the middle-agedchildren of deceased agents than goes to their grandchildren.

3

how they save.

With the retirement of the Baby Boom generation, there is great concern about how this surge in retirees

will convert their savings into consumption. Since annuities make this process easier than other investments,

many economists are now advocating for legislation that would facilitate the rollover of savings into annuities,

for example by making this the default option for the termination of 401(k) plans (Poterba, Venti, and Wise

(2011)). Our findings here demonstrate that such policies would be shortsighted. We should encourage

people to save more, but we should not interfere with their natural inclination to abstain from annuitization.

The paper is organized as follows. Section 1 describes the basic model that will be used throughout

the paper. Section 2 compares what happens in rational competitive equilibria if markets are complete so

households have access to annuities or if markets are incomplete so households cannot invest in annuities.

Section 3 then establishes the result that a market-feasible consumption/investment rule that uses uninsured

investments will exist that gives higher utility in general equilibrium than the rational competitive equilibrium

with annuities. In Section 4, we derive the optimal consumption-saving rule both for a regime where only

annuities are used and a regime where only uninsured investments are used. Section 5 concludes.

1 The General Model

We consider a continuous-time overlapping generations model that generalizes Feigenbaum and Caliendo

(2010) to allow for an uncertain lifetime. This embeds Regime D of Yaari (1965) within a general-equilibrium

context as in Hansen and Imrohoroglu (2008). At each instant, a continuum of agents of unit measure is

born. Their lifespan is stochastic with a maximum value of T .8 Let Q(t) denote the probability of surviving

until age t, which is a strictly positive, strictly decreasing, C1 function. Denote the flow of consumption at

age t by c(t). An agent values allocations of consumption over his lifetime by the utility function

U =

∫ T

0

Q(t) exp(−ρt)u(c(t))dt, (1)

where u(c) is the period utility function and ρ is the discount rate. We restrict attention to the constant

relative risk aversion (CRRA) family

u(c; γ) =

{1

1−γ c1−γ γ 6= 1

ln c γ = 1, (2)

where γ > 0 is the risk aversion coeffi cient and, more relevantly, the reciprocal of the elasticity of intertem-

poral substitution. For all γ > 0, marginal utility is

u′(c; γ) = c−γ . (3)

8For our calculations in Section 4 the assumption of a maximum lifespan is not innocuous.

4

Note that this is the standard preference model for a finitely lived household. We could include non-

standard features such as hyperbolic discounting, but many such features are already known to overturn

the Yaari (1965) result and discourage agents from annuitizing even in partial equilibrium. Here we have

a preference model in which rational households definitely would annuitize, and we can analyze whether

annuitization increases or decreases welfare in general equilibrium.

At age t, the agent is endowed with e(t) effi ciency units of labor, which he supplies inelastically to the

market in return for the real wage w per effi ciency unit. Thus labor income at t is we(t). The consumer

allocates this income between consumption and saving. He has a choice of two saving instruments: annuities

and risk-free bonds. The flow of bonds b(t) earns a fixed net return r. The flow of annuities a(t) earns the

return r plus an insurance premium h(t) equal to the hazard rate of dying

h(t) = −d lnQ(t)

dt> 0. (4)

We assume for now that deceased agents bequeath their unannuitized wealth assets to everyone currently

alive.9 Let B denote this constant bequest. Then the consumer must choose the consumption path c(t)

subject to the budget constraint

c(t) +db(t)

dt+da(t)

dt= we(t) + rb(t) + (r + h(t))a(t) +B, (5)

the boundary conditions

b(0) = b(T ) = 0 (6)

a(0) = a(T ) = 0, (7)

and borrowing constraints

a(t), b(t) ≥ 0 ∀t ∈ [0, T ]. (8)

The latter are necessary to prevent households from accruing infinite wealth by exploiting the arbitrage

opportunity that arises because of the higher return paid to annuities. For much of the paper we will

consider cases where households either invest exclusively in annuities or exclusively in uninsured bonds, the

latter situation being closer to empirical reality. In either of these two restrictive cases, we dispense with

the borrowing constraints.

To complete the model, we give the economy a Cobb-Douglas production technology

F (K,N) = KαN1−α (9)

of labor N and capital K. The latter depreciates at the rate δ > 0. Labor is supplied inelastically by

households, so the aggregate supply is

N =

∫ T

0

Q(t)e(t)dt. (10)

9Feigenbaum and Gahramanov (2011) explore what happens if we loosen this assumption.

5

The supply of capital equals the aggregate of household investments

K =

∫ T

0

Q(t)(b(t) + a(t))dt. (11)

Firms behave competitively, so factor prices must satisfy the profit-maximizing conditions

w = w(K) ≡ (1− α)

(K

N

)α(12)

r = r(K) ≡ α(K

N

)α−1− δ. (13)

Finally, in equilibrium we must also have the bequest satisfy the inheritance-flow balance equation

B

∫ T

0

Q(t)dt =

∫ T

0

Q(t)h(t)b(t)dt, (14)

where the righthand side is the value of nonannuitized wealth belonging to recently deceased agents.

In the following, we are going to explore different consumption rules and compare their welfare in equi-

librium. Since we do not restrict our attention to consumption rules that maximize individual utility, we

need to generalize the usual definition of a market equilibrium to encompass a larger class of consumption

rules. Following FCG (2011), we define a generalized (steady-state) market equilibrium as a consumption

rule c(t), a demand for bonds b(t), a demand for annuities a(t), a capital stock K > 0, a bequest B ≥ 0,

a real wage w, and an interest rate r such that (i) the consumption rule c(t) and asset demands b(t) and

a(t) satisfy the budget constraint (5), boundary conditions (6)-(7), and borrowing constraints (8) given B,

r, and w; (ii) the capital stock K is the aggregate of the consumers’asset demands, satisfying (11); (iii) the

factor prices w and r satisfy the profit-maximizing conditions (12) and (13) given K; and (iv) the bequest B

satisfies the inheritance-flow balance equation (14) given b(t).10 A consumption/investment rule that is part

of a generalized market equilibrium is called market-feasible. A generalized market equilibrium differs from

the usual notion of a competitive equilibrium in that we only require c(t), b(t), and a(t) be affordable and

do not confine attention to consumption allocations that maximize the utility (1) given B, r, and w, subject

to the budget and borrowing constraints. A generalized steady-state market equilibrium that satisfies this

additional optimization condition constitutes a rational competitive (steady-state) equilibrium (RCE).

2 Rational Competitive Equilibria

First let us review what happens under the standard lifecycle approach. We consider separately what

happens if markets are complete and consumers can sell claims contingent on their survival, and what

10Here we abstract from population or technological growth, though it is straightforward to generalize this concept to allowfor a balanced-growth market equilibrium.

6

happens if consumers have no access to annuities markets.

2.1 Complete Markets

A rational household proceeds by maximizing (1) subject to the constraints (5)-(8) for a given set of

factor prices r and w and the bequest B. This problem has the Lagrangian density

Ld = Q(t) exp(−ρt)u(c(t)) + ρa(t)a(t) + ρb(t)b(t) (15)

+λ(t)

[we(t) + rb(t) + (r + h(t))a(t) +B − c(t)− da(t)

dt− db(t)

dt

].

The Euler-Lagrange equations for this problem are

∂Ld

∂c(t)= Q(t) exp(−ρt)u′(c(t))− λ(t) = 0 (16)

∂Ld

∂b(t)− d

dt

∂Ld

∂(db(t)/dt)= λ(t)r + ρb(t) +

dλ(t)

dt= 0 (17)

∂Ld

∂a(t)− d

dt

∂Ld

∂(da(t)/dt)= λ(t)(r + h(t)) + ρa(t) +

dλ(t)

dt= 0. (18)

Together the last two conditions imply

λ(t)h(t) + ρa(t) = ρb(t).

Since we have assumed h(t) > 0 for all t, the Kuhn-Tucker conditions for (8) imply the corresponding

Lagrange multipliers must satisfy ρb(t) > ρa(t) ≥ 0 for all t. Thus a rational household will never invest

in bonds (Yaari (1965)) in this environment without intrinsic bequest motives. This is purely an arbitrage

result. The household does not actually care whether it is insuring against mortality risk or not. It invests

in annuities solely because annuities pay a higher return, which the household earns by selling off the rights

to its annuitized wealth in the event of its death.

Note that we introduce the borrowing constraints in Section 1 to eliminate arbitrage opportunities and

not because we were particularly concerned about the debt of the dead (Feigenbaum (2008), Yaari (1965)).

Here and in the following, when only one of the two assets is held in positive quantities—in this case the

annuities—we ignore the borrowing constraint on that asset, which tremendously simplifies the computation.

After this simplification, Eq. (18) reduces to

d lnλ(t)

dt= −(r + h(t)). (19)

This has the well-known solution

λ(t) = λ0Q(t)

Q(0)exp(−rt),

7

where λ0 is an integration constant. Meanwhile, (16) implies

c(t) =

(λ(t)

Q(t)

)−1/γexp

(−ργt

). (20)

If we define

c0 =

(λ0Q(0)

)−1/γ, (21)

the lifecycle consumption profile will be

c(t) = c0 exp

(r(K)− ρ

γt

). (22)

After setting b(t) = 0 for all t and B = 0, we can rewrite the budget constraint (5) as

d

dt(Q(t) exp(−rt)a(t)) = Q(t) exp(−rt)

[−(r + h(t))a(t) +

da(t)

dt

]= Q(t) exp(−rt) [we(t)− c(t)] . (23)

Upon integrating (23), the borrowing conditions (7) determine c0:

c0 = w(K)

∫ T0Q(t)e(t) exp(−r(K)t)dt∫ T

0Q(t) exp

((1−γ)r(K)−ρ

γ t)dt. (24)

By investing in annuities, the household eliminates any effect of mortality risk on the shape of its con-

sumption profile. The survivor function only enters (22) to the extent that the expected present value of

the household’s labor endowment and consumption stream are weighted by Q(t) in (24). It is in this sense

that we say that annuities allow the household to insure against mortality risk. Conditional on being able

to consume, the path of consumption is independent of the realization of the time of death.

The demand for annuities is also determined by integrating (23):

a(t) =

∫ t

0

Q(s)

Q(t)exp(r(K)(t− s))[w(K)e(s)− c(s)]ds. (25)

The equilibrium capital stock Kannrce is then determined by the market-clearing condition

K =

∫ T

0

Q(t)a(t)dt. (26)

Note that in the special case where r(K) = 0 the lifecycle consumption profile (22) corresponds to the profile

of the Golden Rule allocation described in Appendix A. However, this will only happen for special choices

of the exogenous parameters.

8

2.2 Incomplete Markets

In the absence of annuities markets, the budget constraint becomes

db(t)

dt= we(t) + rb(t) +B − c(t), (27)

and the capital stock is simply the aggregate of nonannuitized bonds:

K =

∫ T

0

Q(t)b(t)dt. (28)

Now a rational household maximizes (1) for a given B, r, and w subject to the constraint (27) and boundary

conditions (6). In this case, the Lagrangian density for the household’s problem is

Lincd = Q(t) exp(−ρt)u(c(t)) + λ(t)

[we(t) + rb(t) +B − c(t)− db(t)

dt

]. (29)

The Euler-Lagrange equations for this problem are (16) and

∂Lincd∂b(t)

− d

dt

∂Lincd∂(db(t)/dt)

= λ(t)r +dλ(t)

dt= 0. (30)

The latter has the solution

λ(t) = λ0 exp(−rt). (31)

Substituting this into (20), we now get the lifecycle consumption profile

c(t) = c0 exp

(r(K)− ρ

γt

)(Q(t)

Q(0)

)1/γ, (32)

where again c0 is defined by (21). In this case, the budget constraint (27) can be rewritten

d

dt(exp(−rt)b(t)) = exp(−rt) [we(t) +B − c(t)] . (33)

The boundary conditions (6) then imply the lifetime budget constraint∫ T

0

exp(−r(K)t) [w(K)e(t) +B − c(t)] dt = 0 (34)

and thereby determine c0:

c0 =

∫ T0

exp(−r(K)t) [w(K)e(t) +B] dt∫ T0

(Q(t)Q(0)

)1/γexp

((1−γ)r(K)−ρ

γ t)dt

. (35)

Notice the difference between the lifecycle consumption profile with complete markets given by (22) and (24)

9

and the consumption profile with incomplete markets given by (32) and (35). Under complete markets, the

consumption profile only depends on the survivor function through c0. Mortality risk has no effect on the

shape of the consumption profile, only on the level of consumption. In contrast, under incomplete markets,

consumption is proportional to Q(t)1/γ , but the present value of expected income, i.e. the numerator in

(35) is not directly dependent on the survivor function.11 With incomplete markets mortality risk does

affect the shape of the consumption profile (Bullard and Feigenbaum (2007), Feigenbaum (2008), Hansen

and Imrohoroglu (2008)).

Integrating Eq. (33) gives us the demand for bonds

b(t) =

∫ t

0

exp(r(K)(t− s))[w(K)e(s) +B − c(s)]ds. (36)

An equilibrium is computed by solving the two remaining conditions (28) and (14) for the bequest Bbeqrce and

the capital stock Kbeqrce . The possibility for ineffi ciency stemming from the absence of life-insurance markets

can be demonstrated by the fact that when r(Kbeqrce) = 0 the lifecycle consumption profile (32) cannot equal

the consumption profile of the Golden Rule allocation in Appendix A, for the latter does not depend on Q(t).

Nevertheless, there exist calibrations of the model in which expected utility is higher in the equilibrium with

incomplete markets than it is in the equilibrium with complete markets, though both expected utilities must

then be less than the Golden Rule expected utility.

2.3 Calibration and Numerical Results

For our baseline calibration, we set the share of capital to α = 0.3375, the consumption to output ratio

C/Y = 0.75, and the capital to output ratio K/Y = 3.0, all common values from the literature. These

targets determine α and δ = 0.083. This leaves two parameters: the elasticity of intertemporal substitution

γ−1 and the discount rate ρ. Since these preference parameters are not separately identified by steady-state

behavior, we consider three common values of γ from the literature: 0.5, 1, and 3. For each choice of γ,

we set ρ so the rational competitive equilibrium with incomplete markets satisfies K/Y = 3. We use the

model with incomplete markets rather than complete markets to set the calibration since most people do

not annuitize. Thus the incomplete-markets model is more relevant empirically.

We assume households live for T = 75 years for a lifespan that corresponds to real ages of 25 to 100.

The survivor function Q(t) is taken from Feigenbaum (2008). For t < TR = 40, the endowment profile e(t)

is proportional to the income profile of Gourinchas and Parker (2002). For t ≥ TR, we assume consumers

retire and e(t) = 0.

Representative macroeconomic variables describing the rational competitive equilibria, both with com-

plete (COM) markets, so all investments are annuitized, and with incomplete (INC) markets, so no invest-

ments are annuitized, are presented in Table 1 for the three choices of γ.

The corresponding lifecycle consumption profiles for the incomplete markets equilibria are plotted in Fig.

1. To see the effect of annuitization without changing macroeconomic observables (as opposed to keeping

11 In equilibrium, the numerator will depend on the survivor function since the equilibrium bequest depends on Q(t).

10

γ ρ K/Y U ∆EV fBCOM INC COM INC

0.5 0.0240 3.24 3.00 64.238 63.906 −1.03% 5.24%1 0.0290 3.18 2.99 5.816 5.8444 0.11% 6.89%3 0.0533 2.77 3.00 −5.530 −4.908 6.15% 9.20%

Table 1: Macroeconomic observables for rational competitive equilibria, both with complete markets, whereonly annuities are employed, and with incomplete markets, where only uninsured investments are used.Three calibrations are considered with representative values of γ.

the structural parameters constant), we also plot the lifecycle consumption profile under complete markets

for a separate calibration in which K/Y = 3. This consumption profile is the same for all choices of γ since

(r − ρ)/γ is the same for each of these equilibria, and γ and ρ do not appear elsewhere in (22).12

For γ = 0.5, our a priori intuition that consumers should be better offwith complete markets is borne out.

However, for γ = 1 and γ = 3, that is not the case. In these calibrations, utility is higher when consumers

are forced to leave bequests. Since it is diffi cult to interpret the economic significance of differences in utility,

Table 1 also reports the equivalent variation ∆EV such that∫ T

0

Q(t) exp(−ρt)u(cINC(t))dt =

∫ T

0

Q(t) exp(−ρt)u((1 + ∆EV )cCOM (t))dt, (37)

i.e. the fraction by which consumption would have to be augmented at all instants in the complete markets

equilibrium to achieve the same lifetime utility as in the incomplete markets equilibrium. For γ = 1, the

change in utility from complete markets to incomplete markets is only slightly positive. However, for γ = 3,

it is huge. Consumption would have to be augmented by 6% in the complete markets equilibrium to achieve

the same utility as would arise without annuities. For comparison, Vidangos (2008) estimates the benefit

of eliminating idiosyncratic risk to be only 1-2%.

Why is annuitization welfare-improving only for low values of γ? Annuitization has two competing

effects on utility. First, as can be seen in Fig. 1, annuitization allows for smoother consumption streams.

The household will never run out of assets no matter how long it survives. Thus consumption drops by

only 35% from the beginning of life to the maximal lifespan for the complete markets profile in Fig. 1.

In contrast, those households that survive to age 100 see their consumption drop by 85% or more with

incomplete markets. This is the advantage of annuitization identified by Yaari (1965).

The second effect is that annuitization eliminates the possibility of receiving a bequest. The fraction of

lifetime wealth accounted for by accidental bequests in the incomplete markets equilibrium

fB =B∫ T0

exp(−rt)dt∫ T0

exp(−rt)[we(t) +B]dt(38)

is reported in the last column of Table 1.

12For all other exercises in this paper, we hold the structural parameters constant when comparing between complete andincomplete markets, rather than the macroeconomic observables.

11

Figure 1: Rational competitive equilibrium lifecycle consumption profiles for the baseline calibration withγ = 0.5, 1, 3 for both complete and incomplete markets. ρ is chosen so K/Y = 3 in all graphs.

12

Between Fig. 1 and Table 1, we see that as γ increases, two things happen. First, as the elasticity of

intertemporal substitution decreases, the household does a better job of smoothing its consumption on its

own with incomplete markets, so the smoothing benefits of annuitization become less important. Second,

because households run down their assets more slowly, they leave a larger estate when they die. Thus

the fraction of lifetime wealth contributed by the bequest in the incomplete markets equilibrium increases.

Together these two effects imply that utility will be higher under incomplete markets for suffi ciently high

γ. Since most estimates of intertemporal elasticity put γ somewhere between 1 and 3, this suggests that

households are actually better off because most do not annuitize.

The bequest effect is a general-equilibrium effect that is a consequence of the pecuniary externality. The

bequest B that households receive in the incomplete-markets equilibrium is accidental and determined by

the equilibrium conditions. The capital stock K that arises under the two equilibria is also determined by

equilibrium conditions. However, they are not determined separately. There is an interaction between the

capital stock and the bequest since a bequest allows the household to save and possibly increase K. The

bequest adds to income directly, but it can also affect income indirectly through its effect on the capital

stock.13 Does utility increase when we shut down annuitization because of the direct effect of bequests or

the indirect effect via the capital stock?

To assess this question we report equivalent variations relative to the complete markets general equilibrium

in various partial equilibria for γ = 1 and γ = 3. These are reported in Table 2. The first and second columns

report the equivalent variation from the complete and incomplete markets general equilibria respectively.

The third and fourth columns show the equivalent variation for partial equilibria respectively under complete

and incomplete markets but with the factor prices from the general equilibrium of the other market regime.

For the incomplete markets partial equilibrium, the bequest is the same as in the incomplete markets general

equilibrium. Finally, in the fifth column we consider what happens in a quasi-general equilibrium of the

incomplete markets model with factor prices from the complete markets general equilibrium but with the

equilibrating bequest that satisfies the balance equation (14). All equivalent variations are computed starting

from the complete-markets general equilibrium.

For the γ = 1 case, the capital stock is actually higher in the complete markets general equilibrium than

it is in the incomplete markets general equilibrium. Thus factor prices are more favorable with complete

markets since, holding the market structure constant, utility increases as we switch from the incomplete-

markets prices to the complete-markets prices. In this case the higher utility under incomplete markets

must be due to the direct effect of the bequest, and this is supported by the numbers in Table 2. If we go

partway from the complete markets general equilibrium to the incomplete markets general equilibrium by

changing the factor prices to their incomplete market values while maintaining annuitization, utility decreases

by the equivalent of one percent of consumption. If, instead, we change from annuitization to uninsured

investments while maintaining the complete markets factor prices, utility increases by the equivalent of 1.5%

of consumption. Thus the direct effect of having accidental bequests has a larger effect on lifetime utility

than the change in factor prices.

13Heijdra, Mierau, and Reijnders (2010) focus on the capital channel.

13

γ COM GE INC GE Annuities PE (INC GE Prices) Bequest PE (COM GE Prices) Bequest Quasi GE1 0.00% 0.11% −0.89% 1.50% −0.04%3 0.00% 6.15% 2.45% 4.69% 5.16%

Table 2: Equivalent variations relative to the complete markets general equilibrium for various partialequilibria under the γ = 1 and γ = 3 calibrations.

The size of the bequest is also important. In comparing the quasi-general equilibrium to the complete

markets equilibrium, we are keeping factor prices the same and looking at what happens with the smoothing

effects of annuitization vs what happens where there is no smoothing and a smaller bequest. The bequest

constitutes 5.6% of lifetime wealth in the quasi-general equilibrium as opposed to 6.9% in the full general

equilibrium. The smoothing benefits of annuitization slightly outweigh the benefit of this smaller bequest.

For the γ = 3 case, the capital stock is higher under incomplete markets so this is more consistent with

the story that the bequest improves utility by increasing the capital stock. Nevertheless we still see that

the direct effect of the bequest is larger than the indirect effect via the capital stock. When we change

factor prices to their incomplete market prices while preserving the use of annuities, utility increases by

the equivalent of 2.5% of complete-markets consumption. If, on the other hand, we change from annuity

investments to uninsured investments while preserving the complete markets factor prices, utility increases

by the equivalent of 4.7% of complete-markets consumption. Thus the larger increase comes through the

switch from annuities to bequests than from more favorable factor prices.

To further establish that accidental bequests improve welfare because they deliver the income that comes

from redistributing the assets of the dead at earlier ages than the mortality premium on annuities would, we

can see that a positive time value of money is important to get this improvement with rational agents. Let

us consider another set of partial equilibria where factor prices are the same both in the annuities regime and

in the bequest regime. Specifically, we consider the case where r = 0, so income received at different points

in the lifecycle is valued the same. In the bequest regime we set B to solve the bequest-balance equation

(14) as in the quasi-general equilibrium of Table 2.14 For the γ = 0.5, γ = 1, and γ = 3 calibrations we

find that utility is always higher in the annuities regime. The equivalent variation of the bequest partial

equilibrium relative to the annuities partial equilibrium is between −1% and −2% for all three calibrations.

3 Market-Feasible Consumption Rules

The results of the previous section demonstrate that annuitization will not be good for the economy as

a whole if the elasticity of intertemporal substitution is suffi ciently low. However, this finding contributes

nothing to the annuities puzzle since fully rational households still ought to completely annuitize their

savings if they can. For our next exercise we take a more evolutionary perspective. Instead of assuming

14 Ideally we would like to compare between partial equilibria while keeping the present value of income the same. However,this is problematic since present value is computed differently in the two regimes. By equilibrating B in the bequest regime,we ensure that the assets of the dead are fully recouped in both regimes.

14

that households individually maximize their utility, we follow FCG (2011) and assume that households may

adopt any market-feasible consumption rule, i.e. any rule that satisfies the instantaneous budget constraint

and clears markets in the aggregate. Suppose there exists a market-feasible consumption rule that confers

higher lifetime utility than the individually rational consumption rule. If individuals adopt consumption

rules taught to them by society rather than through individually rational cognition, then society may learn

to inculcate this higher-utility consumption rule.

To characterize the optimal market-feasible consumption rule, it is convenient to introduce a social

planner. This social planner maximizes (1) subject to the constraints (5), (8), (11), and (14) and the

boundary conditions (6)-(7). This imposes more constraints than a rational household would, as described

in Section 2, since the market-clearing conditions are imposed during the optimization instead of as an

equilibrium condition that must be satisfied afterwards. This new problem also differs from the problem

faced by a standard Pareto social planner who allocates goods to maximize (1) subject only to the constraint

that the economy is capable of producing all the goods allocated. The solution to that problem is what we

refer to as the Golden Rule allocation and is described in Appendix A.15

The Lagrangian for our social planner is

Ls =

∫ T

0

{Q(t) [exp(−ρt)u(c(t)) + µ(a(t) + b(t)) + ξ(h(t)b(t)−B)]

+ρa(t)a(t) + ρb(t)b(t)

+ λ(t)[w(K)e(t) + r(K)b(t) + (r(K) + h(t))a(t) +B − c(t)} dt

−∫ T

0

λ(t)

[da(t)

dt+db(t)

dt

]dt− µK, (39)

and the corresponding Lagrangian density is

Ls = Q(t)[exp(−ρt)u(c(t)) + µ(a(t) + b(t)) + ξ(h(t)b(t)−B)]

+ρa(t)a(t) + ρb(t)b(t)−µK

T+ λ(t)[w(K)e(t) + r(K)b(t)]

+λ(t)

[(r(K) + h(t))a(t) +B − c(t)− da(t)

dt− db(t)

dt

]. (40)

The Euler-Lagrange equations are

∂Ls∂c(t)

= Q(t) exp(−ρt)u′(c(t))− λ(t) = 0 (41)

∂Ls∂b(t)

− d

dt

∂Ls∂(db(t)/dt)

= Q(t)[µ+ ξh (t)] + ρb(t) + λ(t)r(K) +dλ(t)

dt= 0 (42)

15Here we assume the social planner has a zero generational discount rate, as Ramsey (1928) argued it should be, so wecan focus on steady-state equilibria. FCG (2011) show in a dynamic model that similar results would accrue with a positivegenerational discount rate.

15

∂Ls∂a(t)

− d

dt

∂Ls∂(da(t)/dt)

= Q(t)µ+ ρa(t) + λ(t)(r(K) + h(t)) +dλ(t)

dt= 0 (43)

∂Ls∂K

= −µ+

∫ T

0

λ(t)[w′(K)e(t) + r′(K)(a(t) + b(t))]dt = 0. (44)

Assuming the bequest must be nonnegative, the first-order condition for the bequest is

∂Ls∂B

=

∫ T

0

[λ(t)− ξQ(t)]dt ≤ 0, (45)

where inequality holds only if B = 0.

As in Feigenbaum and Caliendo (2010), we next focus our attention on finding the optimal market-feasible

consumption rule consistent with a given choice of (B,K) ∈ R+ ×R++. We refer to this as Subproblem

(B,K), which is governed by the Bellman equation

V (B,K) = maxc(t),b(t),a(t)

∫ T

0

Q(t) exp(−ρt)u(c(t))dt (46)

subject to the constraints

c(t) +db(t)

dt+da(t)

dt= w(K)e(t) + r(K)b(t) + (r(K) + h(t))a(t) +B

b(t), a(t) ≥ 0 ∀t ∈ [0, T ]

K =

∫ T

0

Q(t)(b(t) + a(t))dt (47)

B

∫ T

0

Q(t)dt =

∫ T

0

Q(t)h(t)b(t)dt, (48)

and the boundary conditions

b(0) = b(T ) = 0

a(0) = a(T ) = 0.

In the event that there is no market-feasible choice of c(t), b(t), and a(t) that satisfies these constraints, we

define V (B,K) = −∞.Note that ifKrce is the equilibrium capital stock in the general model of Section 1, including the borrowing

constraints, then the rational consumption rule with full annuitization will be the solution to Subproblem

(0,Krce). The problem of a rational household imposes fewer constraints than Subproblem (0,Krce) while

maximizing the same objective so it must earn utility Urce ≥ V (0,Krce). Yet since the rational consumption

rule satisfies the market-clearing conditions (47) with K = Krce and (48) with B = 0, it meets all the

constraints of Subproblem (B,K), so V (0,Krce) ≥ Urce. Thus the rational consumption rule with full

16

annuitization is the optimal market-feasible consumption rule consistent with (0,Krce).

The Lagrangian for Subproblem (B,K) remains (39), only now we view B and K as exogenous. The

Envelope Theorem and (44)-(45) then give us the derivatives of the value function:

∂V (B,K)

∂K=

∫ T

0

λ(t)[w′(K)e(t) + r′(K)(a(t) + b(t))]dt− µ (49)

∂V (B,K)

∂B=

∫ T

0

[λ(t)− ξQ(t)]dt. (50)

Note that the first-order condition for c(t), (41), is unchanged from the first-order condition (16) for the

rational competitive equilibrium. On the other hand, if we combine the Euler-Lagrange equations (42)-(43)

for b(t) and a(t), we now get

ρb(t)− ρa(t) = h(t)[λ(t)− ξQ(t)]. (51)

For the rational competitive equilibrium, we only had h(t)λ(t) on the righthand side. This is the additional

return, measured in utility, that can be earned immediately at t by investing an extra unit in annuities

as opposed to bonds. This must be positive on account of (41) and our assumptions about the survivor

function. Consequently, a rational household would never invest in uninsured bonds. Our social planner,

in contrast, must also consider the countervailing term −ξh(t)Q(t). The factor h(t)Q(t) is the additional

bequest produced by investing an extra unit in bonds as opposed to annuities. This is weighted by ξ, the

shadow price of bequests, to determine the return in utility that is obtained by this investment. Thus

the social planner would advise households to invest in bonds rather than annuities if ξQ(t) > λ(t). This

situation, of course, can only arise if ξ is positive, but from (45) we see that at the optimal (B,K) the

shadow price must satisfy

ξ ≥∫ T0λ(t)dt∫ T

0Q(t)dt

> 0. (52)

The first inequality can only be strict if B = 0. For an interior solution, the shadow price at the optimal

(B,K) is the sum of marginal utilities across the entire lifespan, since a uniform bequest provides a constant

income at each t, divided by the population the bequest is spread over.

One might naively conclude that (45) automatically implies that a rational competitive equilibrium must

be suboptimal since ξ presumably ought to be zero for the rational competitive equilibrium whereas the

fraction in (52) must be strictly positive. However, ξ does not appear in the equations of motion (41) and

(43) that carry over from the rational competitive model with just annuities. These equations revert to their

rational competitive equilibrium counterparts (16) and (18) when µ = 0, but it is not necessary to set ξ = 0.

Rather, when B = 0, ξ is determined so that ρb(t) ≥ 0 for all t. Indeed, the shadow price of bequests ought

not to be zero when B = 0 precisely because there is a benefit to having a bequest. As an example, when

it is possible to obtain the Golden Rule as a rational competitive equilibrium, λ(t) is proportional to Q(t)

for all t, and ξ > 0 is the constant of proportionality. In this special case the effective return from bequests

or annuities is the same for all t.

17

We can establish the following theoretical result that generalizes the main proposition from FCG (2011).

Proposition 1 Assuming households have access to annuities, in a rational competitive equilibrium that

does not conform to the Golden Rule and in which households are only borrowing constrained on a subset of

[0, T ] of measure zero, there will exist a market-feasible consumption rule involving nonannuitized investments

that produces higher lifetime utility U while maintaining the same capital stock Krce.

Proof. The rational consumption rule will be the solution to Subproblem (0,Krce), where Krce is the

equilibrium capital stock with full annuitization. Let a(t) be the lifecycle profile of annuities under this rule.

Define Sa = {t ∈ [0, T ] : a(t) > 0} and Sb = {t ∈ Sa : ρb(t) > 0}. Since ρa(t) = 0 ≤ ρb(t) for all t ∈ Sa, wehave λ(t) ≥ ξQ(t) for all t ∈ Sa. Furthermore, for t ∈ Sb we have λ(t) > ξQ(t), and for t ∈ Sa − Sb we haveλ(t) = ξQ(t). By assumption [0, T ]− Sa has measure zero, so (50) implies

∂V (0,Krce)

∂B=

∫ T

0

[λ(t)− ξQ(t)]dt =

∫Sa

[λ(t)− ξQ(t)]dt =

∫Sb

[λ(t)− ξQ(t)]dt.

If Sb has positive measure, ∂V (0,Krce)/∂B > 0. In that case, for some ε > 0, V (ε,Krce) > V (0,Krce). Thus

a market-feasible consumption rule with B = ε must exist that produces higher utility while maintaining the

same capital stock. The only exception occurs when Sb has measure zero, in which case ∂V (0,Krce)/∂B = 0.

Then (41) implies that

c(t) = ξ−1/γ exp

(−ργt

)(53)

for all but a set of measure zero of t. Since we are only considering smooth consumption functions, (53)

must hold for all t ∈ [0, T ], and this is the Golden Rule consumption profile.

The caveat in this proposition that households are never borrowing constrained must be emphasized here.

In all three calibrations of the model in 2.3, households hold negative amounts of annuities or bonds for small

t, so households would be borrowing constrained if we enforced the borrowing constraints in these examples.

While it is true that a rational competitive equilibrium will necessarily involve only annuities, Proposition 1

does not necessarily imply that a social planner can improve upon any non-Golden-Rule rational competitive

equilibrium. This is because it does not apply to such equilibria if both borrowing constraints bind on a set

of positive measure. The proposition would permit ∂V (0,K)/∂B ≤ 0 for some K > 0 if ρa(t) > ρb(t) > 0

for all t in the subset of positive measure that satisfy λ(t) < ξQ(t), though we have not found an example

where this occurs.

4 Optimal Irrational Behavior

A natural question to ask at this point is what is the best market-feasible (BMF) consumption rule.

This is what FCG (2011) refer to as the optimal irrational behavior.16 We obtain this by finding (B,K) ∈16This behavior is irrational in the sense that it requires actions contrary to the solution of an individual household’s

optimization problem. Households could get higher utility by deviating from the optimal irrational rule and following the fully

18

R+ ×R++ that maximizes V (B,K). Computationally, this problem is facilitated by our knowledge of the

first-order partial derivatives.

Note that, unless the Golden Rule allocation coincides with the complete-markets rational competitive

equilibrium, there will be infinitely many market-feasible consumption rules that confer higher utility than

the rational consumption rule. The best market-feasible rule is of interest primarily because it puts an

upper bound on how much utility could be obtained within markets. As we will see below, the best rule is

no more consistent with empirical behavior than the rational rule that requires full annuitization. There

may be nonmaterial costs that prevent adoption of the best market-feasible rule. For example, sticking to

the best market-feasible rule will often require tremendous self control since it may defer a great deal of

consumption till the end of life.

Computing equilibria in which both borrowing constraints may possibly bind is extremely diffi cult and

beyond the scope of the present paper. Instead, we consider what happens if we shut down either the

annuities market or the uninsured bond market, in which case we can relax the two borrowing constraints

without creating arbitrage opportunities.

4.1 With Annuities

Analogous to our procedure in 3, we define as Subproblem K the problem of choosing, for a given K > 0,

c(t) and a(t) to maximize

Vann(K) = maxc(t),a(t)

∫ T

0

Q(t) exp(−ρt)u(c(t))dt

subject to

c(t) +da(t)

dt= w(K)e(t) + (r(K) + h(t))a(t), (54)

(26), and (7). The Lagrangian for this problem is

Lanns =

∫ T

0

Q(t) [exp(−ρt)u(c(t)) + µa(t)] dt− µK (55)

+

∫ T

0

λ(t)

[w(K)e(t) + (r(K) + h(t))a(t)− c(t)− da(t)

dt

]dt,

and the associated Lagrange density is

Lanns = Q(t)[exp(−ρt)u(c(t)) + µa(t)] (56)

+λ(t)

[w(K)e(t) + (r(K) + h(t))a(t)− c(t)− da(t)

dt

]− µK

T.

rational rule. However, if all households do this, they will all end up worse off. Some have suggested that “optimal irrational”behavior might better be characterized as hyperrational behavior.

19

The Euler-Lagrange equations for Subproblem K are (41) for c(t), which is unchanged from Section 2, and

∂Lanns

∂a(t)− d

dt

∂Lanns

∂(da(t)/dt)= µQ(t) + λ(t)[r(K) + h(t)] +

dλ(t)

dt= 0. (57)

Meanwhile the Envelope Theorem gives us

V ′ann(K) =∂Lanns

∂K= −µ+

∫ T

0

λ(t)[w′(K)e(t) + r′(K)a(t)]dt = 0. (58)

The differential equation for λ, (57), has the solution

λ(t) =

(

λ0Q(0) exp(−r(K)t)− µ

r(K) [1− exp(−r(K)t)])Q(t) r(K) 6= 0(

λ0Q(0) − µt

)Q(t) r(K) = 0

, (59)

where λ0 is an integration constant. Let us define

χ =µQ(0)

λ0= µcγ0 , (60)

recalling our definition of c0, (21). Converting marginal utilities to consumption via (20), we obtain the

lifecycle consumption profile

c(t) =

c0 exp(r(K)−ρ

γ t)(

1− χr(K) [exp(r(K)t)− 1]

)− 1γ

r(K) 6= 0

c0 exp(− ργ t)

(1− χt)−1γ r(K) = 0

. (61)

Note that this is exactly the same consumption profile as was obtained by Feigenbaum and Caliendo (2010)

without any mortality risk, although c0 is determined by the instantaneous budget constraint (23) that we

used in Section 2.1 and so does depend on the survivor function:

c0 =

∫ T0

exp(−r(K)t)Q(t)e(t)dt∫ T0

exp((1−γ)r(K)−ρ

γ t)Q(t)

(1− χ

r(K) [exp(r(K)t)− 1])− 1

γ

dt

w(K) (62)

for r(K) 6= 0 and

c0 =

∫ T0Q(t)e(t)dt∫ T

0exp(− ρ

γ t)Q(t) (1− χt)−1γ dt

w(K) (63)

for r(K) = 0. Likewise, the asset demand profile is computed using (25).

The only remaining variable that needs to be determined is χ. This is obtained by solving the capital

constraint (26). Suppose r(K) 6= 0. Then we must have

χ

r(K)[exp(r(K)t)− 1] ≤ 1 (64)

20

for all t ∈ [0, T ] in order for c(t) to be defined for all t.17 The lefthand side of (64) is increasing in t if

χ ≥ 0 and decreasing otherwise. Since the condition is always satisfied for t = 0, a necessary and suffi cient

condition for the consumption profile to be defined is

χ ≤ r(K)

exp(r(K)T )− 1. (65)

This imposes an upper bound on the set of χ we must search over to solve Eq. (26). We can then compute

Vann(K). We find the best market-feasible capital stock K∗ann by maximizing Vann(K).

If we set χ = 0, (61) and (62) revert to the solutions, (22) and (24), for a rational household with annuities

given factor prices r(K) and w(K). Thus χ can be interpreted as a measure of how much the consumption

profile that solves Subproblem K deviates from the a rational consumption profile. Since the rational

competitive equilibrium solves Subproblem Kannrce , it must be the case that Vann(K∗ann) ≥ Vann(Kann

rce ).

We refer the reader to Feigenbaum and Caliendo (2010) for a characterization of the properties of the best

market-feasible consumption profile with annuities, which aside from the dependence of c0 on the survivor

function are the same as in a model without mortality risk.

4.2 Without Annuities

In the absence of annuities markets, we define Subproblem (B,K) for (B,K) ∈ R+ ×R++ to be

Vbeq(B,K) = maxc(t),b(t)

∫ T

0

Q(t) exp(−ρt)u(c(t))dt

subject to (27), (28), (14), and (6). The Lagrangian for the problem with a time-varying hazard rate18 is

Lbeqs =

∫ T

0

Q(t) [exp(−ρt)u(c(t)) + µb(t) + ξ (h(t)b(t)−B)] dt (66)

+

∫ T

0

λ(t)

[w(K)e(t) + r(K)b(t) +B − c(t)− db(t)

dt

]dt− µK.

The Lagrange density is

Lbeqs = Q(t) [exp(−ρt)u(c(t)) + µb(t) + ξ (h(t)b(t)−B)] (67)

+λ(t)

[w(K)e(t) + r(K)b(t) +B − c(t)− db(t)

dt

]− µK

T.

17For r(K) = 0, it is necessary and suffi cient that χT ≤ 1.18The problem with a constant hazard rate has to be treated differently since the bequest balance equation implies that B

is proportional to K, so the constraint qualification does not hold. Since the empirical hazard rate is not constant, we do notconsider this special case here.

21

The Euler-Lagrange equation (41) for c(t) is unchanged from Section 4. The remaining equation is

∂Lbeqs∂b(t)

− d

dt

∂Lbeqs∂(db(t)/dt)

= λ(t)r(K) + [µ+ ξh(t)]Q(t) +dλ(t)

dt= 0. (68)

The Envelope Theorem then gives the partial derivatives of Vbeq:

∂Vbeq(B,K)

∂K=∂Lbeqs∂K

=

∫ T

0

λ(t)[w′(K)e(t) + r′(K)b(t)]dt− µ = 0. (69)

∂Vbeq(B,K)

∂B=∂Lbeqs∂B

= −ξ∫ T

0

Q(t)dt+

∫ T

0

λ(t)dt = 0. (70)

For this model, the solution to the λ equation (68) can be written

λ(t) = exp(−r(K)t)(λ0 − µG(t)− ξH(t)), (71)

where we define

G(t) =

∫ t

0

Q(s) exp(r(K)s)ds (72)

and

H(t) =

∫ t

0

Q(s)h(s) exp(r(K)s)ds. (73)

If we again use the notation c0, defined by (21), and we introduce new Lagrange multipliers

ν =µ

λ0(74)

ζ =ξ

λ0, (75)

we obtain from (20) the lifecycle consumption profile

c(t) = c0 exp

(r(K)− ρ

γt

)(Q(t)

Q(0)

)1/γ[1− νG(t)− ζH(t)]

−1/γ. (76)

The intertemporal budget constraint is the same as in Section 2.2, when we considered the rational com-

petitive equilibrium without annuities markets. Thus the demand for bonds b(t) is given by (36) and the

boundary conditions (6) determine c0:

c0 =

∫ T0

exp(−r(K)t)[w(K)e(t) +B]dt∫ T0

exp((1−γ)r(K)−ρ

γ t)(

Q(t)Q(0)

)1/γ[1− νG(t)− ζH(t)]

−1/γdt

. (77)

Not surprisingly, when ν = ζ = 0, (76) and (77) reduce to the solution (32) and (35) for a rational

household in a partial equilibrium with factor prices r(K) and w(K). This special case must then be the

22

solution of Subproblem (Bbeqrce ,Kbeqrce), which is therefore nested within the set of optimal market-feasible

rules for each (B,K). Consequently, generalizing the result from Section 4.1, we have Vbeq(B∗beq,K∗beq) ≥

Vbeq(Bbeqrce ,K

beqrce), where B∗beq is the best market-feasible bequest and K

∗beq is the best market-feasible capital

stock, both with annuities markets closed.

Using (70) we can isolate the value of ξ at the optimal B, which, assuming B > 0, is

ξ =

∫ T0λ(t)dt∫ T

0Q(t)dt

> 0. (78)

Inserting (71) into (78), we obtain

ξ

∫ T

0

Q(t)dt =

∫ T

0

exp(−r(K)t)(λ0 − µG(t)− ξH(t))dt.

Using (74) and (75), this simplifies to

ζ

∫ T

0

Q(t)dt =

∫ T

0

exp(−r(K)t)(1− νG(t)− ζH(t))dt,

which we can solve for ζ:

ζ(K, ν) =

∫ T0

exp(−r(K)t)(1− νG(t))dt∫ T0

[Q(t) + exp(−r(K)t)H(t)] dt. (79)

Note that ν and K are the only endogenous variables in (79). Having isolated ζ, we can solve for the optimal

equilibrium bequest B given K and ν:

B(K, ν) = w(K)

∫ T0

exp(−r(K)t)[Z(K, ν)−H(t)]e(t)dt∫ T0

[Q(t) + (H(t)− Z(K, ν)) exp(−r(K)t)]dt, (80)

where

Z(K, ν) =

∫ T0H(t) exp

(r(K)(1−γ)−ρ

γ t)(

Q(t)Q(0)

)1/γ[1− νG(t)− ζ(K, ν)H(t)]

−1/γdt∫ T

0exp

((1−γ)r(K)−ρ

γ t)(

Q(t)Q(0)

)1/γ[1− νG(t)− ζ(K, ν)H(t)]

−1/γdt

. (81)

These are derived in Appendix B.

Define ν(K) to be the ν that satisfies the capital constraint (28) given the capital K and B = B(K, ν).

Then we can define

Vbeq(K) = Vbeq(B(K, ν(K)),K),

and we have reduced the problem of finding the best market-feasible behavior to a one dimensional opti-

mization of Vbeq(K) with respect to K. The primary diffi culty of computing Vbeq(K) is then to find the

Lagrange multiplier ν(K). Analogous to Section 4.1, we can restrict attention to ν that satisfy

νG(t) + ζ(K, ν)H(t) ≤ 1 ∀t ∈ [0, T ]. (82)

23

Since G and H are both strictly increasing in t, for the case where ν, ζ(K, ν) ≥ 0, a necessary and suffi cient

condition for (82) to hold is that ν satisfy

νG(T ) + ζ(K, ν)H(T ) ≤ 1. (83)

What does the best market-feasible consumption profile look like? Using (20), we can write the growth

rate of consumption as a function of the decay rate of the instantaneous marginal utility λ(t):

d ln c(t)

dt=

1

γ

[−d lnλ(t)

dt− ρ− h(t)

]. (84)

From (71), (72), and (73), we obtain the decay rate of λ:

−d lnλ(t)

dt= r(K) + (µ+ ξh(t))

Q(t)

λ(t). (85)

Plugging this into (84), we derive

d ln c(t)

dt=

1

γ

[r(K)− ρ− h(t) + (µ+ ξh(t))

Q(t)

λ(t)

]. (86)

In the special case where the Lagrange multipliers µ and ξ vanish, (86) reduces to the growth rate of

consumption for an individually rational household. Ignoring the pecuniary externality, a rational household

will increase (decrease) its consumption when the interest rate is greater (less) than the effective discount

rate ρ + h(t), inclusive of mortality risk. This arises because a unit of consumption deferred at t will earn

the household 1 + rdt units of consumption at t + dt, but a unit of consumption at t + dt is only worth

1− (ρ+ h(t))dt at t+ dt.

However, the (B,K) that maximizes Vbeq must have that the shadow price of capital is

µ =

∫ T

0

λ(t)[w′(K)e(t) + r′(K)b(t)]dt (87)

from (69) while the shadow price of bequests ξ is given by (78). Thus we can rewrite (86) as

d ln c(t)

dt=

1

γ

[r(K)− ρ− h(t) +

∫ T

0

Q(t)

λ(t)

λ(t′)

Q(t′)Q(t′)[w′(K)e(t′) + r′(K)b(t′)]dt′ + h(t)

∫ T

0

Q(t)

λ(t)

λ(t′)

Q(t′)dt′

].

(88)

The second term, involving w′(K) and r′(K), also appeared in the consumption growth equation without

mortality risk in Feigenbaum and Caliendo (2010). This term accounts for the fact that, by deferring a unit

of consumption from t to dt, the household increases the capital stock by dt. Thus expected income at age

t′ will change by Q(t′)[w′(K)e(t′) + r′(K)b(t′)] at every moment t′ of the lifecycle. Since marginal utility

is λ(t)/Q(t), the factor (λ(t′)/Q(t′))/(λ(t)/Q(t)) then converts the utility gain at t′ into a corresponding

utility at t. The third term, involving h(t), accounts for the fact that there is a probability h(t)dt that the

24

household will die between t and t + dt. In that event, the extra saving will be spread across the whole

surviving population. Again the utility gain at t′ must be weighted by (λ(t′)/Q(t′))/(λ(t)/Q(t)) to obtain

a time-t measure of this utility gain. Overall these contributions from the effect of deferring consumption

on K and B introduce a term inversely proportional to marginal utility in (86). The best market-feasible

consumption profile must deviate from the standard Lifecycle/Permanent-Income Hypothesis profile of a

rational household in order for it to be consistent in equilibrium with a capital stock K and a bequest B.

It deviates most when the deviation will have the smallest impact on utility, i.e. when marginal utility is

lowest.

Now let us see how the consumption growth rate evolves over the lifecycle since this will determine the

shape of the consumption profile. In Appendix C we derive

d2 ln c(t)

dt2=

1

γ

[(ξQ(t)

λ(t)− 1

)dh(t)

dt+ (µ+ ξh(t))

Q(t)

λ(t)

(r(K)− h(t) + (µ+ ξh(t))

Q(t)

λ(t)

)]. (89)

Let us suppose that the hazard rate of dying h(t) is strictly increasing with age as it is empirically for adults.

For an individually rational household, µ = ξ = 0 and (89) reduces to(d2 ln c(t)

dt2

)RCE

= − 1

γ

dh(t)

dt< 0. (90)

Thus the consumption profile for a rational household will be strictly concave. Another interesting special

case is when h(t) = 0 as in Feigenbaum and Caliendo (2010). Then (89) simplifies to

d2 ln c(t)

dt2

∣∣∣∣h(t)=0

=µ

γ

Q(t)

λ(t)

(r(K) + µ

Q(t)

λ(t)

).

For r(K)µ ≥ 0, which will happen if the economy is dynamically effi cient and the shadow price of capital

is positive, the consumption profile will be strictly convex. For the general case, the sign of (89) depends

on both the rate of change of h(t), whether the interest rate is bigger or smaller than the hazard rate, and

how ξQ(t)/λ(t) compares to 1. Conceivably we could obtain best market-feasible consumption profiles with

many different shapes depending upon the parameters.19

4.3 Numerical Results

Now we compute the best market-feasible equilibria, both for the case where households only annuitize

and where households never annuitize, for the same calibration as we used in 2.3 and compare them to

the corresponding rational competitive equilibrium. Table 3 reports macroeconomic observables for these

equilibria for the γ = 1 calibration, and Fig. 2 displays the corresponding lifecycle consumption profiles.

Table 4 and Fig. 3 provide the same information for the γ = 0.5 calibration, and Table 5 and Fig. 4 for the

19The empirical consumption profile is hump-shaped (Gourinchas and Parker (2002)). According to (90), a rational householdwill also have a hump-shaped consumption profile. However, Feigenbaum (2008) has shown it is diffi cult to find parameters forthe present model with bequests that give a rational competitive equilibrium consumption profile that matches empirical data.

25

Regime K B U ∆EV K/Y C/Y fB

RCE Annuity 270 0.000 5.816 0.00% 3.18 0.735 0.000RCE Bequest 246 0.082 5.844 0.11% 2.99 0.751 0.069BMF Annuity 398 0.000 6.525 2.78% 4.11 0.657 0.000BMF Bequest 394 0.262 7.981 8.72% 4.08 0.660 0.226Golden Rule 388 N/A 8.066 9.08% 4.05 0.662 N/A

Table 3: Macroeconomic observables under the baseline calibration with γ = 1 for rational competitiveequilibria and best market-feasible equilibria both under annuities and bequests as well as for the GoldenRule allocation.

γ = 3 calibration.

We focus on the γ = 1 calibration. We have already seen that the rational competitive equilibrium

with accidental bequests confers slightly higher lifetime utility than the rational competitive equilibrium

with annuities. The best market-feasible equilibrium with annuities gives higher utility than both of the

rational competitive equilibria, providing the equivalent of a 3% increase in consumption in the rational

competitive equilibrium with annuities. However, consistent with our finding in Section 3, the best market-

feasible equilibrium with accidental bequests does substantially better, providing the equivalent of a 9%

increase in consumption relative to the rational equilibrium with annuities. Indeed, there is not much room

for improvement over the best market-feasible equilibrium with bequests. The Golden Rule allocation,

which confers the highest utility of any feasible allocation, only gives a 0.36% bigger increase in equivalent

consumption.

Both of the best market-feasible equilibria have capital stocks higher than the Golden Rule capital stock,

so they are both dynamically ineffi cient. Thus there would be a market failure if households adopt the best

market-feasible rule either with annuities or accidental bequests. The government could achieve the Golden

Rule allocation by implementing wealth transfers, possibly even with a Pareto optimal transition. However,

in the equilibrium with bequests, the possible gain from this effort would be negligible.

Looking at Fig. 2, one might be concerned about how the BMF equilibrium with bequests is achieving

such high utility. Both with annuities and accidental bequests, consumption shoots up as t → T . This

spike in terminal consumption was also observed in Feigenbaum and Caliendo (2010), though it is blatantly

counterfactual. Is all the utility coming from a gigantic consumption binge enjoyed by the small fraction of

the population that survives to the maximum lifespan? In fact only 0.025% of lifetime utility comes from

the last year of life, so the answer is no. The spike is an artifact of the assumption that everyone dies at T .20

Since there is no bequest motive, everyone intends to die broke, yet the household keeps a huge amount of

savings that must be liquidated before it dies. We could eliminate the spike if we introduce a small terminal

bequest motive so a household that survives to T can pass on a portion of its wealth to the next generation.

The unimportance of the spike for utility can also be seen by the fact that the BMF consumption profile

with bequests nearly coincides with the Golden Rule allocation up until age 90. Since the BMF profile is

20Since the behavior of the model at large t depends so crucially on T , it is not clear how to compute the model with anunbounded lifespan.

26

Figure 2: Lifecycle consumption profiles for rational competitive equilibria (RCE) and best market-feasible(BMF) equilibria both with annuities and with bequests and for the golden-rule allocation in the baselinecalibration with γ = 1.

27

Regime K B U ∆EV K/Y C/Y fB

RCE Annuity 278 0.000 64.24 0.00% 3.24 0.730 0.000RCE Bequest 247 0.061 63.90 −1.03% 3.00 0.750 0.052BMF Annuity 409 0.000 65.06 2.58% 4.19 0.651 0.000BMF Bequest 400 0.394 68.06 12.25% 4.12 0.656 0.307Golden Rule 388 N/A 68.30 13.06% 4.05 0.662 N/A

Table 4: Macroeconomic observables under the calibration with γ = 0.5 for rational competitive equilib-ria and best market-feasible equilibria both under annuities and bequests as well as for the Golden Ruleallocation.

Regime K B U ∆EV K/Y C/Y fB

RCE Annuity 219 0.000 −5.530 0.00% 2.77 0.769 0.000RCE Bequest 246 0.112 −4.908 6.15% 3.00 0.750 0.092BMF Annuity 402 0.000 −4.821 7.10% 4.14 0.655 0.000BMF Bequest 390 0.171 −4.457 11.39% 4.06 0.661 0.159Golden Rule 388 N/A −4.455 11.41% 4.05 0.662 N/A

Table 5: Macroeconomic observables under the calibration with γ = 3 for rational competitive equilibria andbest market-feasible equilibria both under annuities and bequests as well as for the Golden Rule allocation.

higher than the Golden Rule allocation for ages 90 and above yet the Golden Rule allocation produces higher

utility, the bulk of the utility must be coming from earlier in the lifecycle.

The γ = 0.5 calibration produces consumption profiles that are qualitatively similar to the γ = 1 calibra-

tion, although consumption varies more across the lifecycle since the elasticity of intertemporal substitution

is higher. The spike in terminal consumption for the BMF allocations is also more pronounced. Note that

while the rational competitive equilibrium with annuities produces higher utility than the rational competi-

tive equilibrium with accidental bequests, this is reversed for the BMF equilibria. The best market-feasible

equilibrium with annuities only gives the equivalent of a 2.6% increase in RCE consumption with annuities.

The BMF equilibrium with accidental bequests gives the equivalent of a 12% increase in consumption, which

again nearly matches the Golden Rule utility.

For the γ = 3 calibration, we get the same ranking of the different allocations in terms of utility with

only a 0.02% difference in equivalent RCE consumption (with annuities) between the BMF equilibrium with

accidental bequests and the maximum feasible utility of the Golden Rule allocation.21 Since the elasticity

of intertemporal substitution is lower, the consumption profiles are all much smoother. In particular, the

spike in terminal consumption for the BMF equilibrium with accidental bequests is softened to the degree

that c(T ) < c(0), although the spike for the BMF equilibrium with annuities remains quite pronounced.

5 Conclusion21For γ = 3, Feigenbaum and Gahramanov (2010) show that the Golden Rule utility can be achieved by a BMF equilibrium

if the bequest is not spread uniformly across the surviving population but is allowed to depend on age.

28

Figure 3: Lifecycle consumption profiles for rational competitive equilibria (RCE) and best market-feasible(BMF) equilibria both with annuities and with bequests and for the golden-rule allocation in the baselinecalibration with γ = 0.5.

29

Figure 4: Lifecycle consumption profiles for rational competitive equilibria (RCE) and best market-feasible(BMF) equilibria both with annuities and with bequests and for the golden-rule allocation in the baselinecalibration with γ = 3.

30

We find that if households with access to annuities markets are not borrowing constrained in a rational

competitive equilibrium, there will exist a market-feasible consumption rule involving nonannuitized invest-

ments that results in higher lifetime utility. We also find that for the range of parameterizations of a CRRA

utility function that most macroeconomists would consider plausible, a general equilibrium in which house-

holds are disallowed from investing in annuities usually produces higher utility than a general equilibrium

where households annuitize. Both these results stem from the fact that accidental bequests transfer the

wealth of recently deceased households to more young survivors than the insurance premiums from annuities

would, and a gift of income to a young agent has a bigger impact on lifetime wealth and utility than the same

gift to an older agent. Although evolutionary pressures will typically engender a distribution of behaviors

rather than the best market-feasible (or individually rational) behavior (Feigenbaum (2011)), these findings

suggest that there will be strong evolutionary pressures encouraging society to foster a tendency to eschew

annuities.

We have not computed transition paths here,22 though FCG (2011) offers guidance regarding transition

costs. If the economy were to start in a rational competitive equilibrium with complete markets, there

would be no Pareto-improving path to an equilibrium without annuities. Early cohorts would suffer because

they would not benefit from the higher returns they could earn on annuities. Once cohorts start to earn

bequests, though, the benefit of this additional wealth received early in the lifecycle will soon dominate the

utility loss from not annuitizing.

However, the complete-markets rational competitive equilibrium is not the situation existing in reality

now. Though households have access to annuities, most do not make use of them. Somehow they have

coordinated upon a rule that they should not annuitize. If households were to begin following the advice

of most economists to annuitize, there would be short-term gains as households enjoy higher returns on

their savings. But later generations would be hurt as they stop receiving accidental bequests. In the long

run, everyone would be worse off. So we would argue that policymakers should not implement measures

intended to encourage annuitization. Consumers should be encouraged to save more, but they should do so

via uninsured investments.

A The Golden-Rule Allocation with Mortality Risk

The golden-rule allocation solves the problem of a social planner who maximizes lifetime utility∫ T

0

Q(t) exp(−ρt)u(c(t))dt

subject only to the feasibility constraint∫ T

0

Q(t)c(t)dt+ δK = KαN1−α (91)

22Heijdra, Mierau, and Reijnders (2010) do compute transition paths for the two-period model.

31

that aggregate consumption and any investment necessary to replace depreciated capital equals the societal

output. This problem has the Lagrangian

Lgr =

∫ T

0

Q(t) exp(−ρt)u(c(t))dt+ λ

[KαN1−α −

∫ T

0

Q(t)c(t)dt− δK]

and associated Lagrange density

Lgr = Q(t)[exp(−ρt)u(c(t))− λc(t) +λ

T

[KαN1−α − δK

].

The first-order conditions are∂Lgr∂c(t)

= Q(t)[exp(−ρt)u′(c(t))− λ] = 0 (92)

∂Lgr∂K

= λ

[α

(K

N

)α−1− δ]

= 0. (93)

Since, if there were markets, the interest rate would be given by Eq. (13), Eq. (93) implies

r(K) = 0.

Integrating (92), we obtain

c(t) = c0 exp

(−ρtγ

), (94)

where the constraint (91) determines c0.

B The Optimal Market-Feasible Bequest

Inserting (36) into the bequest-balance equation (14), we obtain

B

∫ T

0

Q(t)dt =

∫ T

0

Q(t)h(t)

∫ t

0

exp(r(K)(t− s))[w(K)e(s) +B − c(s)]dsdt.

32

We can switch the order of the integrals in the double integral to obtain

B

∫ T

0

Q(t)dt =

∫ T

0

∫ T

s

Q(t)h(t) exp(r(K)(t− s))[w(K)e(s) +B − c(s)]dtds

=

∫ T

0

[H(T )−H(t)] exp(−r(K)t)[w(K)e(t) +B − c(t)]dt

= H(T )

∫ T

0

exp(−r(K)t)[w(K)e(t) +B − c(t)]dt−∫ T

0

H(t) exp(−r(K)t)[w(K)e(t) +B − c(t)]dt

= −∫ T

0

H(t) exp(−r(K)t)[w(K)e(t) +B − c(t)]dt,

where we have used the lifetime budget constraint (34) in the last step. From (77) and (76), this becomes

B

∫ T

0

[Q(t) +H(t) exp(−r(K)t)]dt = −w(K)

∫ T

0

H(t) exp(−r(K)t)e(t)dt

+

∫ T

0

H(t) exp

(r(K)(1− γ)− ρ

γt

)(Q(t)

Q(0)

)1/γ[1− νG(t)− ζ(K, ν)H(t)]

−1/γdt

×∫ T0

exp(−r(K)t)[w(K)e(t) +B]dt∫ T0

exp((1−γ)r(K)−ρ

γ t)(

Q(t)Q(0)

)1/γ[1− νG(t)− ζ(K, ν)H(t)]

−1/γdt

.

If we define

Z(K, ν) =

∫ T0H(t) exp

(r(K)(1−γ)−ρ

γ t)(

Q(t)Q(0)

)1/γ[1− νG(t)− ζ(K, ν)H(t)]

−1/γdt∫ T

0exp

((1−γ)r(K)−ρ

γ t)(

Q(t)Q(0)

)1/γ[1− νG(t)− ζ(K, ν)H(t)]

−1/γdt

,

we have

B

∫ T

0

[Q(t)+H(t) exp(−r(K)t)]dt = −w(K)

∫ T

0

H(t) exp(−r(K)t)e(t)dt+Z(K, ν)

∫ T

0

exp(−r(K)t)[w(K)e(t)+B]dt.

Solving this for B, we obtain

B(K, ν) = w(K)

∫ T0

exp(−r(K)t)[Z(K, ν)−H(t)]e(t)dt∫ T0

[Q(t) + (H(t)− Z(K, ν)) exp(−r(K)t)]dt.

C Rate of Change of Consumption Growth

33

We can rewrite (86) entirely in terms of c(t):

d ln c(t)

dt=

1

γ[r(K)− ρ− h(t) + (µ+ ξh(t))c(t)γ exp(ρt)] .

Differentiating this with respect to time, we obtain

d2 ln c(t)

dt2=

1

γ

[(ξc(t)γ exp(ρt)− 1)

dh(t)

dt+ ρ(µ+ ξh(t))c(t)γ exp(ρt)

+ γ(µ+ ξh(t))c(t)γ exp(ρt)d ln c(t)

dt

]

d2 ln c(t)

dt=

1

γ

[(ξc(t)γ exp(ρt)− 1)

dh(t)

dt+ ρ(µ+ ξh(t))c(t)γ exp(ρt)

+ (µ+ ξh(t))c(t)γ exp(ρt)

(r(K)− ρ− h(t) + (µ+ ξh(t))

Q(t)

λ(t)

)]

d2 ln c(t)

dt2=

1

γ(ξc(t)γ exp(ρt)− 1)

dh(t)

dt

+1

γ[(µ+ ξh(t))c(t)γ exp(ρt) (r(K)− h(t) + (µ+ ξh(t))c(t)γ exp(ρt))] .

To ease interpretation we reexpress the righthand side in terms of λ:

d2 ln c(t)

dt2=

1

γ

[(ξQ(t)

λ(t)− 1

)dh(t)

dt+ (µ+ ξh(t))

Q(t)

λ(t)

(r(K)− h(t) + (µ+ ξh(t))

Q(t)

λ(t)

)].

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36