debt, deficits, and age-specific mortality

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Review of Economic Dynamics 6 (2003) 300–312

www.elsevier.com/locate/red

Debt, deficits, and age-specific mortality

Hamid Faruqee

International Monetary Fund, Research Department, 10-548 Washington, DC 20431, USA

Received 6 March 2002; revised 20 September 2002

Abstract

This paper develops an overlapping agents model with age-specific mortality rates. The anframework also nests Blanchard’s (1985, Journal of Political Economy 82, 1095–1117) “peryouth” model as a special, though perhaps not realistic, case. With age specific mortality ratesis “fleeting.” Using standard hyperbolic functions, the model with fleeting youth is able to clreplicate the empirical relation between age and mortality. The steady-state model is used to ethe comparative implications of deficit finance, and age-specific mortality is shown to alter thRicardian properties of the model. 2003 Published by Elsevier Science (USA).

JEL classification: E21; E27; E62; H31

Keywords: Ricardian equivalence; Government debt; Saving

1. Introduction

In the fifteen years since Blanchard’s (1985) influential paper was published, numpapers have been written utilizing his overlapping agents model to examine arange of economic issues. The framework—often referred to as the Blanchard–Yaarmodel—has proven to be a valuable analytical tool for researchers with its straightfoaggregation properties. But this desirable feature of the model also highlights onemain drawbacks. The tractability of the model derives in part from the assumptiothe probability of death is constant and the same for all agents regardless of theThis constant hazard rate assumption has the peculiar feature that the planningor expected time until death for an agent is the same at age 20 or 120. Blan

E-mail address: [email protected].

1094-2025/03$ – see front matter 2003 Published by Elsevier Science (USA).doi:10.1016/S1094-2025(02)00020-0

H. Faruqee / Review of Economic Dynamics 6 (2003) 300–312 301

cle”

age-. Thecase.l butwell-own tolns of

do989)re ofh

n tofinite

el toright-erlyissuesrrects

talityerboliciricalmetersn, theal and

abilitye

ynastic

himself recognized this limitation of the simple model that lacked any “life-cydimension.

This paper derives a generalized version of Blanchard’s celebrated model withspecific mortality, incorporating the fact that mortality rates eventually rise with ageframework thus introduces a life-cycle dimension absent from the fixed mortality rateWith a rising probability of death over an agent’s lifetime, youth is no longer perpetua“fleeting.” Since this probability eventually reaches unity, the framework also places adefined age limit or upper bound on lifespans. This broader framework can also be shnest Blanchard’s perpetual youth assumption as a special case.1 By nesting both perpetuaand fleeting youth, the steady-state framework can isolate the long-run implicatiomortality rates for economic behavior.

Specifically, it is a well-known result that the mortality rate or finite horizonsnot matter for debt non-neutrality in the Blanchard model. As shown by Weil (1and Buiter (1988), it is the birth rate not the death rate that underlies the failuRicardian equivalence in the model. This result, however, isspecial to the perpetual youtassumption. Adding age-specific mortality, the distribution of mortality rates is showalter the real effects of public debt and identifies the circumstances under whichhorizons indeed matter.

The introduction of realistic age-specific mortality rates also allows the modgenerate a more realistic age distribution. With perpetual youth, a significanttail to the distribution results, implying an overstatement of the number of eldagents in the economy. This makes analysis of demographic and distributionalproblematic. With fleeting youth, having mortality rates the eventually increase cothis distortion.

With the use of standard hyperbolic functions, the model with age-specific moris shown to be generally tractable, at least for steady-state analysis. The hypspecification is also shown to be sufficiently flexible to closely capture the emprelation between age and mortality rates. Based on Gomperty’s Law, the paraentering the mortality function are estimated. Using the estimated mortality functiocomparative implications of deficit finance are then simulated in the cases of perpetufleeting youth.

2. Model

To generalize the Blanchard model, we assume agents face a time-varying probof deathp(s, t), wheres is a generation index andt is the time index and their differenct− s represents age. The number of survivors from a cohort born at times at timet is givenby

n(s, t)= n(s, s)e− ∫ ts p(s,v)dv = bN(s)e− ∫ t

s p(s,v)dv, (1)

1 See Faruqee and Laxton (2000) for a discussion of Blanchard model from both life-cycle and dperspectives.

302 H. Faruqee / Review of Economic Dynamics 6 (2003) 300–312

us

lation

ghted)

l youth

d to

e age-erties

ecausekers

Faruqee-varying

whereb is the (fixed) birth rate—equal to the relative size of the newest cohortn(s, s) as aproportion of the contemporaneous total populationN(s).2 From Eq. (1), the instantaneochange in a particular cohort’s size is given byn(s, t) = −p(s, t)n(s, t), where a dotrepresents the derivative with respect to timet .

Using the fact that the total population is the sum of all existing cohorts, the popugrowth rate is then given by

N(t)

N(t)= n= b− p, (2)

wheren is the rate of population growth andp is theaggregate mortality rate defined bythe following adding-up condition:

p =t∫

−∞p(s, t)

n(s, t)

N(t)ds. (3)

This expression simply states that the overall death rate must equal the (weiaverage of age-specific death rates.3

The Blanchard case can be nested within these general conditions. The perpetuaassumption is the special case with a static probability of death—i.e., wherep(s, t) is equalto a fixedp for all s andt . Furthermore, assuming a stationary population normalizeunity (i.e.,n= 0 andN = 1) as we henceforth do, the birth rateb equals the death ratep. Inthat case, one exactly obtains Blanchard’s survivors formulan(s, t)= p e−p(t−s) from (1).4

3. Mortality function

If agents are taken to represent individuals, however, mortality rates should bdependent. Moreover, the mortality function should possess the following limit prop(for a givens):

limt→s

p(s, t)= p0> 0, (4)

limt→∞p(s, t)= 1. (5)

2 See Buiter Buiter (1988) for the extension of Blanchard model to the case of population growth. Bthe focus here is on theadult population, note that the “birth” rate denotes the arrival of new adults or wor(taken to be of age 20).

3 Birth rates, aggregate death rates, and population growth are assumed constant throughout. See(2001) for the case of time-varying birth and death rates and demographic dynamics. In the case of timerates, individual mortality rates would becohort-specific and not justage-specific.

4 With no population growth, the following adding-up condition forp(s, t) must be satisfied:

1

p=

t∫−∞

e− ∫ ts p(s,v)dv ds.


d thatnd or

e andate iste rises

le is

thent”cialg

thisrtalityn fitting

seholdsealing.

Fig. 1. Mortality function and Gomperty’s Law.

These Inada-type conditions require that the initial mortality rate be non-zero, anthe probability of death should eventually reach unity, reflecting a finite upper bouage limit.5

The function should also be able to replicate the empirical relation between agmortality. Following Blanchard (1985), Gomperty’s Law suggests that the death rroughly constant and near zero between age 20 and 40; beyond that, the mortality rain the following fashion:p50 = 1 percent,p60 = 3 percent,p80 = 16 percent,p100 = 67percent. Figure 1 shows these data points (as open circles).

A functional form that can satisfy all these conditions and still remain fairly tractabthe hyperbolic tangent function:

p(s, t)= c0 + c1[tanh(t − s − k)], c0 = p1 + p0

2, c1 = p1 − p0

2, (6)

where the parametersc0 andc1 are determined by the initial and terminal values ofmortality rate function—i.e.,p0 andp1, andk is a parameter representing the “mid-poiage—i.e., wherep(s, t) = c0.6 Blanchard’s (1985) perpetual youth model is the specase wherep0 = p1 = p, implying thatc0 = p andc1 = 0. Otherwise, the case of fleetinyouth has 0<p0<p1 = 1, implying thatc0 + c1 = 1.

In addition to satisfying conditions (4) and (5) in the case of fleeting youth,functional form can also be shown to reproduce the empirical relation between moand age. Using non-linear least squares, structural time-series estimates of (6) showGomperty’s Law are as follows:

p(s, t)= 0.505+ 0.495

[tanh

(t − s − 74.2

16.6

)],

5 An alternative view of agents in the model, as discussed in Blanchard (1985), is that of dynastic houand wherep is the probability that the family line ends. In that case, a common fixed hazard rate is more app

6 See Appendix for a quick primer on hyperbolic functions.


sed tometerstriction

Law.hard’s

th a

r thein thetes in

e in

rticalIn theumbernd or

n show

R2 = 0.999, D.W.= 2.74. (7)

In the empirical specification, phase-shift and scaling parameters are both uimprove the fit. Corrected standard errors (not reported) indicate that all the paraare significant at the 1 percent level. These non-linear estimates also impose the resthat limt→∞p(s, t)= 1—i.e.,c0 + c1 = 1.

Figure 1 plots the fit of the empirical equation (7) that reproduces Gomperty’sIn a sense, the age-specific mortality rate model combines the features of Blancprobabilistic or uncertain lifetimes framework and a traditional life-cycle model wiknown time of death (i.e., maximum age).

4. Age distributions

For a given aggregate mortality (and birth) rate, the distributional implications fopopulation of the two models can be compared. Specifically, the common death rateperpetual youth case is set equal to the weighted-average of individual mortality rathe fleeting youth case.7 This allows the initial size of incoming cohorts to be the samboth settings, as well as the aggregate size of both populations.

The age distributions, however, are quite different as shown in Fig. 2. The veaxis shows the relative size (in percent of the total population) of each age group.Blanchard case, a sizable tail in the age distribution exists, tending to overstate the nof old-aged persons in the economy. With age-specific mortality, a finite upper-boumaximum age is quickly reached, truncating the right tail of the distribution.

Fig. 2. Age distribution.

7 Using the adding-up condition and the estimated mortality function based on Gomperty’s Law, one cathat the implied aggregate mortality ratep is 2.5 percent.


creterd’s

heel with

er’s

ability

in theumedtinctcould

ouldis latter

ual

am of

ofes.riginalalso be

To further highlight these age-distribution differences, consider the following conexample. With a world (adult) population of approximately 4 billion people, Blanchaconstant mortality rate would predict over 430million persons or nearly 11 percent of tpopulation to be age 110 years or older; for the same average death rate, the modage-specific mortality rates would predict less than 5 individuals in that age group.

4.1. Consumer’s problem

With log utility and assuming that utility is zero after the agent dies, the consumproblem can be formulated as maximizing the following expected utility function:

Et

{ ∞∫t

ln c(s, v)e−θ(v−t )dv}

=∞∫t

ln c(s, v)e− ∫ vt {θ+p(s,z)}dzdv, (8)

where θ is the rate of time preference. The mortality functionp(s, t) introduces anadditional discounting term that is now age-specific, based on the dynamic probthat the agent dies in some future period.

The treatment of the annuities or insurance market becomes more complicatedcase of age-specific mortality rates. Specifically, the zero profit condition for firms assin Blanchard (1985) and drawing on Yaari (1965) could be interpreted in two disways here. If firms cannot use the fact that hazard rates are age-specific, theystill balance their books based on population averages for mortality rates.8 If insurancefirms could discriminate (by age), however, they could make net paymentsp(s, t)w(s, t)

to individual agents in exchange for all their wealth contingent upon death. This wallow the insurance company to balance its payments and receipts by age group. Thscheme is adopted here:9

w(s, t)= [r(t)+ p(s, t)]w(s, t)+ y(s, t)− τ (s, t)− c(s, t), (9)

wherer is the interest rate,y − τ is disposable labor income, andc is consumption.Solving the consumer’s problem yields the following optimal solution for individ

consumption:

c(s, t)= φ(s, t)[w(s, t)+ h(s, t)], (10)

whereφ(s, t) is the marginal propensity to consume out of wealth, andh(s, t) is definedin the conventional way as human wealth or the present value of the future stre

8 Firms could pay all age groups at the same average rate, payingpw(s, t) in return for the estate at the timedeath, provided that the common payout ratep equaled a wealth-weighted average of individual mortality rat

9 We operate under the assumption of price discrimination to further distinguish the model from the oassumptions used in Blanchard (1985). Derivations of the model under the common pricing rule couldderived (see previous footnote).


tant—

tewly

tractingh

udgetr, are

mancific—lth

2).

disposable labor income.10,11 The inverse of marginal propensity to consume∆(s, t) ≡φ(s, t)−1 evolves according to the following equation:

∆(s, t)≡ [θ + p(s, t)]∆(s, t)− 1. (11)

In the case of perpetual youth, the marginal propensity to consume would be consi.e.,∆(s, t)= 0, and identical for all agents:φ = θ +p.

The dynamics for aggregate consumption can be written generally as:

C(t)= pc(t, t)+ (r(t)− θ)C(t)−

t∫−∞

p(s, t)c(s, t)n(s, t)ds, (12)

where uppercase variables denote economy-wide aggregates.12 Equation (12) shows thathe evolution of total consumption depends on the additional consumption of narriving agents, the sum of the consumption changes of existing agents, and subout the consumption of those agents that die each period.13 In the simple perpetual youtcase, this last term would be given bypC(t).

The dynamics of aggregate financial and human wealth are given by

W (t)= r(t)W(t)+ Y (t)− T (t)−C(t), (13)

H (t)= ph(t, t)+ r(t)H(t)− [Y (t)− T (t)]. (14)

The dynamics of financial wealth are identical to the Blanchard case, reflecting a bconstraint for the economy as a whole. The dynamics of human wealth, howevedifferent and now include the marginal contribution of newly arriving agents’ huwealth to the economy-wide aggregate. If disposable labor income was not age-spei.e., y(s, t) − τ (s, t) = Y (t) − T (t) and youth was perpetual, individual human wea

10 Solving the maximization problem, one can show that with log utility:

φ(s, t)−1 =∞∫t

e− ∫ vt {θ+p(s,z)}dz dv.

This expression can be shown to have a closed-form solution given the functional form forp(s, t) . If p(s, t)= p,thenφ(s, t) is constant(= θ + p) as in Blanchard (1985).

11 Integrating forward the agent’s dynamic budget constraint (9), we haveh(s, t) given by

h(s, t)=∞∫t

{y(s, v)− τ (s, v)}e− ∫ v

t {r(z)+p(s,z)}dz dv.

12 Note that aggregate variables denoted with uppercase are defined as follows:

C(t)≡t∫

−∞c(s, t)n(s, t)ds.

Differentiating this expression and using the Keynes–Ramsey rule for optimal consumption, yields Eq. (113 Note that since agents are born without wealth—i.e.,w(t, t) = 0, initial consumption is given byc(t, t) =φ(t, t)h(t, t).


e

sity togregate

egatever, an

-

eas

haviorsolvedsuresnt ages.e,

ate

age

(under

would not be age-specific—i.e.,h(t, t) = H(t); then, Eq. (14) would reduce to thBlanchard case.

Based on individual consumption behavior and the fact that the marginal propenconsume is the same across age groups under perpetual youth, the solution for agconsumption in the Blanchard case is well known:

C(t)= (θ + p)[W(t)+H(t)]. (15)

Differentiating this expression yields the dynamics for consumption:

C(t)= (r(t)− θ − p)

C(t)+ p(θ +p)H(t). (16)

In the age-specific mortality case, obtaining a closed-form solution for aggrconsumption is somewhat more complicated. Using the mean-value theorem, howeanalytical solution can be expressed as follows:

C(t)= c(ξ, t)n(ξ, t)Λ, (17)

whereξ is the index representing the mean-value cohort in total consumption, andΛ isrelated to the finite range of ages in the population with fleeting youth.14 Using the meanvalue theorem again and Eq. (12), the law of motion for consumption is given by

C(t)= (r(t)− θ)C(t)− p(ζ, t)c(ζ, t)n(ζ, t)Λ+ pφ(t, t)h(t, t), (18)

whereζ is the mean-value cohort for the total consumption of agents who die, wherξis the mean-value cohort for the consumption of those still alive.15

5. Steady-state consumption behavior

Using the behavioral equations derived above, the consumption and saving beunder perpetual and fleeting youth can be compared. The model with fleeting isusing the fact that an individual’s steady-state time profile for key behavioral meacan also be used to describe the cross-sectional pattern across agents of differeSpecifically, at very old ages—i.e., asp(s, t) → 1, the marginal propensity to consumconverges to a well-defined upper bound:φ(s, t) → θ + 1; human wealth, meanwhilefalls to following lower bound:h(s, t)→ (Y − T )/(r + 1), where bars denote steady-stvalues. Using these terminal conditions and the laws of motion forφ andh, one can thensolve backwards for these values for younger agents.16

14 Using the mean-value theorem, we can then write:

c(ξ, t)n(ξ, t)Λ=t∫

t−Λc(s, t)n(s, t)ds, for ξ ∈ (t −Λ, t).

15 It should be noted that theΛ terms in (17) and (18) neednot be the same as they refer to the relevantranges for two different measures:c(s, t)n(s, t) andp(s, t)c(s, t)n(s, t). Becausep(s, t) ∈ (0,1), it follows thatthe age range is generally smaller in the second instance.

16 As a consistency check, one can also solve the integral calculus problem directly to obtain (say)φ(s, t)

directly. With θ = 3.5%, the model suggests a marginal propensity to consume of 5% for young agents


ratesaving

vidualbsencerizonsteady

oups,ter timeg rates

s needtor areumer the

, leaving

e thefactth and

with then

f5).

Fig. 3. Saving rates (in percent of total income).

Assuming the same rate of time preference and interest rate, individual savingin steady state implied by optimal consumption plans are shown in Fig. 3. The srate is defined as the rate of change in financial wealth. With perpetual youth, indisaving rate are always positive and the same across all age groups. Again, in the aof any life-cycle features, this result in not surprising. Agents face a fixed planning hoindependent of their age and choose to save at a fixed rate over time. This allowswealth accumulation and rising consumption over an individual’s lifetime.17

With fleeting youth, a clear life-cycle pattern of saving emerges. Across age gryoung and middle-age agents tend to be net savers, while older agents facing shorhorizons (higher discount rates) tend to dissave to some extent. Eventually, savinsettle down near zero.18

Interestingly, initial saving rates across models are the same in Fig. 3, though thinot always be the case. The probability of death and, hence, the effective discount faclower initially with the fleeting youth. Consequently, the marginal propensity to consis comparatively lower initially, but human wealth is higher for the same reason. Fointerest rate and time preference assumed here, these opposing effects just cancelinitial consumption rates to be essentially the same in the two cases.

These comparative implications should be taken as only partial results sincequilibrium interest rate will generallynot be the same across the two models. Thethat saving rates are uniformly lower for a given interest rate suggests less weal

age 30), rising to 10% by age 60, and 25% by age 80. These direct calculations are indeed consistentsaving rates in Fig. 3. (Details of the integration solution inMathematica© are available from the author uporequest.)

17 Recall that although individual consumption is continuously rising—withr > θ , aggregate consumptiongrowth is zero in steady state through population turnover, as discussed in Blanchard (1985).

18 These values could imply small positive saving rates near the end of life, depending on the value or − θ .In stock terms, agents maintain some financial wealth until death, similar to Davies (1981) and Abel (198


l with

age-imingnt ratesit.

the

nment

e initial

d debt

tock isstateormlyibriumck andology

teadywill

l stock.eficite theoral

own in

capitalteadyest

capital accumulation, and thus a higher long-run equilibrium interest rate in a modefleeting youth.

6. Steady-state effects of government debt

From a policy perspective, an important question is whether the introduction ofspecific mortality rates affects the impact of public debt. Changes in debt—i.e., the tof taxes—have real effects in the standard model because public and private discoudiffer. Thus, national saving could fall in response to an increase in the budget defic

To examine the debt implications of age-specific mortality, we first introducegovernment sector in the standard way:

D(t)=G(t)− T (t)+ r(t)D(t), (19)

whereD is the stock of outstanding government debt,G is public expenditures, andTis (lump-sum) taxes. Financial wealth held by the private sector consists of goverbonds and the capital stock:W =D +K.

For convenience, government expenditures are assumed to be fixed at zero. In thsteady state, taxes and debt are also assumed to be zero (i.e.,D = T = 0). Otherwise, fromthe government’s budget constraint, the following long-run relation between taxes anholds:

T

D= r . (20)

The initial steady state in the perpetual youth case is one where the capital sequal to unity.19 From the economy’s budget constraint, this implies that steadyconsumption is also unity. In the case of fleeting youth, because saving is uniflower for a given interest rate, there is less capital accumulation and a higher equilreal interest rate than in the perpetual youth case. Consequently, the capital stoconsumption are less than unity with fleeting youth, given identical taste and technparameters.

Now consider a unit shock to government debt. From Eq. (20), taxes in the new sstate will be positive, equal to the new equilibrium interest rate. The interest ratealso be higher as the increase in public debt crowds out to some extent the capitaBecause public and private discount rates differ, a shift in the timing of taxes (i.e., dfinance) will affect national saving and investment as agents do not fully internalizfuture tax implications of higher debt that follow from the government’s intertempbudget constraint. The extent to which debt has real effects in the two models is shTable 1.

19 With a standard Cobb–Douglas production function, a unit capital (and labor) stock requires thatshare be equal toθ +p(θ +p), which we assume for convenience. In the perpetual youth case, the initial sstate real interest rate will also equalθ + p(θ + p). In the fleeting youth case, however, the equilibrium interrate (capital stock) will be higher (lower).


in thean-an theiscountunt ratetality.ality or

other, when

re noaringInrdian

rrtalityment

n-valuee, therate.of totalinallygmentsard’st rates

(1990).that the

Table 1Steady-state effects of government debt (change in capital stock frominitial values)

Model θ = 1% θ = 3.5% θ = 10%

Perpetual youth −7.4% −4.2% −3.5%Fleeting youth −20.1% −10.1% −7.0%

As seen in the table, the real effects of government debt are significantly highercase of fleeting youth.20 The intuition for this result can be seen from Eq. (17). The mevalue cohort in consumption has a higher discount rate (and higher mortality rate) thpopulation-wide average, which represents the wedge between public and private drates in the perpetual youth case. Consequently, the wedge between the public discoand the effective private discount rate in consumption is larger with age-specific morA larger wedge between discount rates induces a larger departure from debt neutrRicardian equivalence.

Apart from these quantitative results, it is useful to compare these basic findings tostudies. In a paper extending the Blanchard model to include population growth (i.e.birth and death rates could differ), Buiter (1988) showed that birth ratesnot death rateswere the reason behind debt non-neutrality in the case of perpetual youth.21 The intuition isas follows. New taxpayers arriving in the future allowed current individuals—who shaintergenerational connection—to enjoy the benefits of lower taxes today without beall of the future tax burden. Thelevel of the mortality rate, however, did not matter.other words, finite horizons were not, in any way, the source of the failure of Ricaequivalence in Blanchard model.

This analysis shows, however, that thedistribution of mortality rates does matter fodebt non-neutrality. In particular, for a given average death rate, age-specific moaccording to Gomperty’s Law tends to increase the crowding-out effects of governdebt. The reason is that the effective private discount rate—represented by the meacohort—in consumption is higher with fleeting youth. Seen from another perspectivconsumption-weighted mortality rate is higher than the population-weighted mortalityOlder agents with higher than average discount rates tend to comprise a larger shareconsumption. Hence, finite horizons matter in this context as Blanchard had origposited, by increasing the wedge between public and private discount rates. This authe real effects of public debt, beyond the impact of a positive birth rate. In Blanchspecial case of perpetual youth, these distributional implications regarding discounvanish and, as shown by Buiter (1988), the role of finite horizons is eliminated.22

20 The small effect of government debt in the perpetual youth model is a well-known result; see EvansAdding age-earnings profiles, Faruqee et al. (1997) and Faruqee and Laxton (2000) show, however,departures from Ricardian equivalence can be much larger. See Barro (1974) for the classical case.

21 Weil (1989) derives a similar result in an infinite horizon model with population growth.22 It is straightforward to prove that the consumption-weighted mortality rate isidentical to the overall mortality

rate when youth is perpetual:C(t)−1 ∫ t−∞ pc(s, t)n(s, t)ds = p = ∫ t

−∞ pn(s, t)ds.


pecificicateeas ofws adeathreality.-cyclea flatt debt,s tendes thecing a-state

. All

erivections

pertiestions.

7. Conclusions

This paper extends Blanchard’s (1985) overlapping agents model to include age-smortality rates. Using hyperbolic functions to introduce mortality rates that replGomperty’s Law, the steady-state model with fleeting youth produces several ardeparture from the original model. The introduction of age-specific mortality allomore direct interpretation of agents as individuals, facing realistic probabilities ofat different ages. This also produces a demographic age structure more in line withMoreover, fleeting youth introduces a form of heterogeneity across agents and lifeimplications that were absent from the simpler model. Finally, the change (fromdistribution) in mortality rates increases the steady-state real effects of governmenas finite horizons now matter. Older agents with higher than average discount rateto comprise a larger share of total consumption. Age-specific mortality thus increaswedge between the effective private discount rate and the public discount rate, indularger departure from Ricardian equivalence. Future work could build on this steadyanalysis to explore the dynamic implications of age-specific mortality.

Acknowledgments

I would like to thank Alfredo Cuevas and Robin Brooks for helpful discussionsremaining errors are my own.

Appendix. Hyperbolic functions

The appendix presents a quick primer on the hyperbolic functions used to danalytically tractable solutions. The basic hyperbolic sine, cosine, and tangent funare defined as follows:

sinh(x)= 1

2(ex − e−x), (A.1)

cosh(x)= 1

2(ex + e−x), (A.2)

tanh(x)= sinh(x)

cosh(x). (A.3)

These functions have straightforward integrals and derivatives and possess prothat, in many instances, are broadly similar to their counterpart trigonometric funcSome useful results are as follows:

cosh2(x)− sinh2(x)= 1, (A.4)∫cosh(x)dx = sinh(x), (A.5)∫sinh(x)dx = cosh(x), (A.6)


ten as

nt-wayr

91.

293.my 89,

44.traints:

nomies,

, IMF

nomic

Fig. 4. Hyperbolic tangent function.∫tanh(x)dx = log

[cosh(x)

]. (A.7)

A more elaborate parameterization of the hyperbolic tangent function can be writfollows:

p(t)= c0 + c1[tanh

(t − ka

)], c0 = p0 +p1

2, c1 = p1 − p0

2. (A.8)

In Eq. (A.8),p0 = c0 − c1 andp1 = c0 + c1 represent the limit values of the functioas t → −∞ and t → ∞, respectively;a is a scaling parameter andk is a phase-shifparameter, marking the point in time when the value of the function is exactly halfbetween the limit values—i.e.,p(k)= c0. Figure 4 depicts the “step” function in (A.8) foa = 1, k > 0.

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