Anticipated expansions of life expectancy · Anticipated expansions of life expectancy and their long-run growth e ects3 A.O. Belyakov1;2 A.N. Kurbatsky1 K. Prettner3 1Lomonosov Moscow

Anticipated expansions of life expectancyand their long-run growth effects3

A.O. Belyakov1,2 A.N. Kurbatsky1 K. Prettner3

1Lomonosov Moscow State University2Central Economic Mathematical Institute of the Russian Academy of Sciences

3University of Hohenheim

XXI April International Academic Conference onEconomic and Social Development.

24 April 2020

3Kurbatskiy A. N. received funding from the Russian Foundation for BasicResearch (grant number 18-010-01169).Belyakov A. O. received funding from the Russian Science Foundation (grantnumber 19-11-00223).

Endogenous growth model

Paul M. Romer (1990)Endogenous Technological ChangeJournal of Political Economy 98(5), Part 2: S71-S102

Fixed number of Ramsay consumers with discountedpreferences

maxC(·)

∞∫0

e−ρtU(C (t))dt, u(C ) =C 1−θ

1− θ

Endogenous growth model with OLG

Klaus Prettner (2013)Population aging and endogenous economic growthJournal of Population Economics 26(2), pp 811–834

Klaus Prettner, Timo Trimborn (2017)Demographic Change and R&D-based Economic GrowthEconomica 84, pp 667–681

Agnieszka Gehringer, Klaus Prettner (2017)Longevity and technological changeMacroeconomic Dynamics

Blanchard (1985) overlapping generationswith “perpetual youth” (constant mortality rate µ)

maxC(·)

∞∫τ

e−(ρ+µ)tU(C (t))dt, u(C ) = lnC

Endogenous growth model with OLG

Main theoretical prediction

Intuitively, a decrease in the rate of mortality implies thathouseholds live longer and therefore they discount the future lessheavily. As a consequence, aggregate savings rise, exertingdownward pressure on the long-term market interest rate. Sincethe expected profits of R&D investments are discounted with themarket interest rate, the profitability of R&D rises. This impliesthat more resources are devoted to R&D activities with a positiveimpact upon technological progress and productivity growth.

Main extention

We assume that fertility and mortality rates are time dependent ingeneral equilibrium.

Demographic dynamics with “rejuvenation” I

Mortality equation for cohort born at τ

∂

∂tn(τ, t) = −µ(t) n(τ, t), ∀t ≥ τ,

Renewal equation

n(t, t) =

∫ t

−∞n(τ, t)β(t)dτ, ∀t ≥ 0,

Initial conditions

n(τ, 0) = n0(τ), ∀τ < 0,

µ(t) and β(t) are time-dependent mortality and fertility coefficients

Demographic dynamics with “rejuvenation” II

Number of agents

N(t) :=

∫ t

−∞n(τ, t)dτ

Population dynamics

N(t)

N(t)= β(t)− µ(t), N(0) =

∫ 0

−∞n0(τ)dτ,

Age profile

n(τ, t) = e−∫ tτ µ(θ)dθN(τ)β(τ).

Population aging: shock or not?

When both mortality and fertility decrease at the same value, whenthe difference β(t)− µ(t) is unchanged so is the dynamics of N,while the age profile n is shifting to older ages. This is calledpopulation aging – the issue for many industrialized countries.

Assumtion that fertility and mortality are time dependent ingeneral equilibrium model, allows to study the effect of predictedpopulation aging on economic growth.

Agent problem I

We assume that an agent born at time τ maximizes her discountedlife-time utility with respect to c(τ, t) for t ∈ [τ,∞)

u(τ) ≡∫ ∞τ

e−ρ(t−τ)−∫ tτ µ(θ)dθ log[c(τ, t)] dt,

where c(τ, t) denotes consumption at time t of the agent alongwith her children.

Agent problem II

Each agent from cohort τ is endowed at time t with one unit oflabor, which she inelastically supplies on the labor market to earnthe going wage rate w(t). We also assume that individuals areinsured against the risk of dying with positive assets by a fair lifeinsurance company that redistributes wealth of individuals whodied amongst those who are still alive. Therefore the real rate ofreturn r(t) on assets a(τ, t) is augmented by the mortality rateµ(t). Taking final goods as numeraire, the wealth constraint ofindividuals belonging to cohort τ reads

∂

∂ta(τ, t) = [r(t) + µ(t)] a(τ, t) + w(t)− c(τ, t). (1)

Agent problem III

Utility of representative consumer born at time τ∫ ∞τ

e−ρ(t−τ)−∫ tτ µ(θ)dθ log[c(τ, t)] dt → max

c>0,

c(τ, t) – consumption.

Asset dynamics

∂

∂ta(τ, t) = [r(t) + µ(t)] a(τ, t) + w(t)− c(τ, t),

w(t) – wage, r(t) – real interest rate.

No-Bequest and No-Ponzi-Game conditions

a(τ, τ) = 0, limt→∞

e−∫ tτ [r(θ)+µ(θ)]dθa(τ, t) ≥ 0.

Agent problem solution I

Note that the solution to the optimization problem has to obey theno-bequest condition and the no-Ponzi game condition

a(τ, τ) = 0, limt→∞

R(τ, t) a(τ, t) ≥ 0, (2)

where we define the discounting factor for convenience as

R(τ, t) := e−∫ tτ [r(θ)+µ(θ)] dθ. (3)

The latter condition in (2) results in transversality condition

limt→∞

R(τ, t) a(τ, t) = 0, (4)

that implies that the solution of (1) has the form

a(τ, t) =1

R(τ, t)

∫ ∞t

R(τ, z) [c(τ, z)− w(z)] dz . (5)

Agent problem solution II

Assuming convergence of the following improper integral

h(τ) :=

∫ ∞τ

R(τ, z)w(z)dz , (6)

we introduce life-time human wealth h(τ), that equals discountedlife-time consumption∫ ∞

τR(τ, z) c(τ, z)dz = h(τ), (7)

due to the no-bequest condition a(τ, τ) = 0 in (2). Notice that,since the wage income w(t) · 1 does not depend on date of birth τ ,according to definitions (3) and (6) assets (??) of each agent bornat τ

a(τ, t) =

∫ ∞t

R(t, z) c(τ, z)dz − h(t),

Agent problem solution III

are her discounted remaining consumption minus the humanwealth of a newly born agent, h(t).Maximization w.r.t. c(τ, t) is equivalent to the maximization ofthe following functional∫ ∞

τe−ρ(t−τ)−

∫ tτ µ(θ) dθ log[c(τ, t)] dt → max

c(τ,·), (8)

subject to (7). Optimal consumption

c(τ, t) =e−ρ(t−τ)+

∫ tτ r(θ) dθ∫∞

τ e−ρ(s−τ)−∫ sτ µ(θ) dθ ds

h(τ) = e−ρ(t−z)+∫ tz r(θ) dθc(τ, z)

(9)is proportional to human wealth h(τ).The last equality yields

c(τ, t)

∫ ∞t

e−ρ(z−t)−∫ zt µ(θ) dθ dz = a(τ, t) + h(t). (10)

Optimal consumption and asset profiles

Consumption and assets

c(τ, t) = e−ρ(t−τ)+∫ tτ r(θ) dθσ(τ) h(τ), a(τ, t) =

c(τ, t)

σ(t)− h(t).

Human wealth

h(τ) :=

∫ ∞τ

R(τ, z)w(z)dz , R(τ, t) := e−∫ tτ [r(θ)+µ(θ)]dθ.

Notation

σ(τ) :=1∫∞

τ e−ρ(z−τ)−∫ zτ µ(θ) dθ dz

.

if µ = const, then σ = ρ+ µ.

Aggregation

Aggregated consumption and assets

A(t) :=

t∫−∞

a(τ, t) n(τ, t) dτ, C (t) :=

t∫−∞

c(τ, t) n(τ, t)dτ.

A(t) = r(t)A(t) + w(t)N(t)− C (t), (11)

C (t) = [r(t)− ρ+ β(t)− µ(t)]C (t)− β(t)σ(t)A(t). (12)

Final good sector with perfect competition

Production function

Y (t) = [LY (t)]1−α∫ Q(t)

0[x(t, q)]α dq, (13)

LY – labor used in final goods production, Q(t) – technologicalfrontier, x(t, q) is the amount of intermediate good q ∈ (0,Q(t)].

Wage rate paid in the final goods sector

wY (t) = (1− α)Y (t)

LY (t), (14)

Prices paid for intermediate inputs

p(t, q) = α [LY (t)]1−α [x(t, q)]α−1 , (15)

Intermediate goods sector with monopolistic competition

Linear one-to-one production function

x(t, q) = k(t, q).

Profits of firm q

π(t, q) = p(t, q) k(t, q)− r(t) k(t, q)− δ k(t, q)

= α [LY (t)]1−α [k(t, q)]α − [r(t) + δ] k(t, q)→ maxk(t,q)

δ – depreciation rate of machines.

Capital and prices of intermediate good

k(t) :=

[α2

r(t) + δ

] 11−α

LY (t), p(t) :=r(t) + δ

α.

Final and intermediate goods sectors I

Optimal profit

π(t, q) = (1− α) p(t) k(t) = (1− α)α

[α2

r(t) + δ

] α1−α

LY (t).

(16)

Aggregation of capital

K (t) :=

Q(t)∫0

k(t, q)dq =

[α2

r(t) + δ

] 11−α

LY (t)Q(t).

Capital and labor take constant shares of final outcome

p(t)K (t) = αY (t), w(t)LY (t) = (1− α)Y (t).

Final and intermediate goods sector equations

Labor employed in production, price of intermediate goods,output, and interest rate

LY (t) =1− αα

K (t)

w(t)p(t), p(t) = α

[(1− α)

Q(t)

w(t)

] 1−αα

,

r(t) = αp(t)− δ, Y (t) = p(t)K (t)/α (17)

as functions of the capital K , the wage rate w , and thetechnological frontier Q.

Total operating profit of firms

Π(t) :=

Q(t)∫0

π(t, q) dq = (1− α) p(t)K (t) = αw(t) LY (t). (18)

R&D sector with perfect competition

Poduction function

Q(t) = λQ(t) LQ(t), (19)

LQ(t) – labor of scientists employed at time t in R&D sector.

Present value of the real profit flow firm q ∈ [0,Q(t)]

v(t, q) =

∫ ∞t

exp

[−∫ s

tr(θ) dθ

]π(s, q)ds. (20)

Zero-profit condition

v [t,Q(t)] Q(t) = w(t) LQ(t), (21)

R&D sector I

No-arbitrage condition (time-derivative of (20) with q = Q(t))

v [t,Q(t)]

v [t,Q(t)]+π[t,Q(t)]

v [t,Q(t)]= r(t), lim

T→∞e−

∫ Tt r(θ)dθv [T ,Q(T )] = 0,

(22)

Rate of capital gain (valuation gains of shares)

v [t,Q(t)]

v [t,Q(t)]=

w(t)

w(t)− Q(t)

Q(t). (23)

R&D sector II

Operating profit and price of patent, assuming LQ(t) > 0

v [t,Q(t)] =w(t) LQ(t)

Q(t), π[t,Q(t)] = α

w(t) LY (t)

Q(t). (24)

Profit rate (dividend payments per share)

π[t,Q(t)]

v [t,Q(t)]= α

LY (t)

LQ(t)

Q(t)

Q(t). (25)

No-arbitrage condition (22)

w(t)

w(t)− Q(t)

Q(t)+ α

LY (t)

LQ(t)

Q(t)

Q(t)= r(t),

R&D sector equations

Technological frontier dynamics

Q(t) = λQ(t) LQ(t),

No-arbitrage condition

w(t)

w(t)+ λ {α LY (t)− ϕ LQ(t)} = r(t). (26)

subject to

limT→∞

e−∫ Tt r(θ) dθw(T )

Q(T )= 0.

Balances

Labor market clearing

LQ(t) + LY (t) = N(t). (27)

Product market clearing

K (t) = Y (t)− C (t)− δK (t), K (0) = K0 > 0. (28)

Balance of asset flows

A(t) = K (t) + V (t), V (t) :=

Q(t)∫0

v(t, q)dq. (29)

No “bubbles” in asset market

Balance of assets

A(t) = K (t) + V (t) ⇒ A(t) = K (t) + V (t), (30)

44

V (t) = v [t,Q(t)]Q(t) =w(t) LQ(t)

Q(t)Q(t) =

w(t)

λ. (31)

We combine

A(t) = r(t)A(t) + w(t)N(t)− C(t),

K(t) = r(t)K(t) + Y (t)− C(t)− (r(t) + δ)K(t),

V (t) = r(t)V (t) + w(t) LQ(t)− Π(t),

with the use of Y (t) = p(t)K(t) + w(t)LY (t), r(t) + δ = αp(t) to get

A(t)− K(t)− V (t) = r(t) (A(t)− K(t)− V (t)) .

Provided that r(t) 6= 0 we have (30) due to implicit assumption in (20):

limt→∞

v(t, q) exp

[−∫ t

0

r(θ) dθ

]= 0.

Equations of general equilibrium I

We have 9 unknown functions Q, C , K , LY , LQ , A, w , r , p

Q(t)

Q(t)= λ LQ(t), Q(0) = Q0 > 0,

C(t)

C(t)= r(t)− ρ+ β(t)− µ(t)− β(t)σ(t)

A(t)

C(t),

K(t)

K(t)=

r(t) + δ(1− α2

)α2

− C(t)

K(t), K(0) = K0 > 0,

A(t)

A(t)= r(t) + N(t)

w(t)

A(t)− C(t)

A(t),

w(t)

w(t)= r(t)− λ [α LY (t)− ϕ LQ(t)] ,

LQ(t) = N(t)− LY (t), r(t) = α p(t)− δ,

LY (t) =1− αα

K(t)

w(t)p(t), p(t) =

[(1− α)

Q(t)

w(t)

] 1−αα

A(t) = K(t) +w(t)

λ

Equations of general equilibrium in intensive form

We have 3 unknown functions p, CK , K

w and 3 differential equations

p(t)

p(t)=

1− αα

([λ (1− α)

K (t)

w(t)− α

]p(t) + δ

)(32)

d

dtlog

(C (t)

K (t)

)=

C (t)

K (t)− 1− α2

αp(t)− ρ+ β(t)− µ(t)

− β(t)σ(t)C(t)K(t)

1 +1

λ K(t)w(t)

, (33)

d

dtlog

(K (t)

w(t)

)=

[λK (t)

w(t)+ 1

]1− α2

αp(t)− λN(t)− C (t)

K (t), (34)

Balanced Growth Path with constant mortality and replacement fertility µ = β

N = const, σ(t) ≡ ρ+ µ

Q(t)

Q(t)=

C (t)

C (t)=

K (t)

K (t)=

A(t)

A(t)=

w(t)

w(t)=

p(t)

p(t):= g .

Consider the case of ϕ = 1 and δ = 0:

λ (1− α)K (t)

w(t)− α = 0, ⇒ λ

K (t)

w(t)=

α

1− α

C (t)

K (t)− 1− α2

αp − ρ− β σ

C(t)K(t)

1 +1

λ K(t)w(t)

= 0,

[λK (t)

w(t)+ 1

]1− α2

αp − λN − C (t)

K (t)= 0,

We obtain the quadratic equation with one positive root

αC (t)

K (t)− (1− α)λN − ρ− β σ

α C(t)K(t)

= 0

C (t)

K (t)=ρ

2+

(1− α)λN

2+

√(ρ

2+

(1− α)λN

2

)2

+ β σ

Anticipated aging I

If abrupt change in mortality and fertility at time t∗

µ(t) = β(t) =

{µ1, t < t∗,µ2, t ≥ t∗,

is anticipated, then for all τ ≥ t∗

σ(τ) :=1∫∞

τ e−ρ(z−τ)−∫ zτ µ(θ) dθ dz

= µ2 + ρ,

while for all τ ≤ t∗ we have

σ(τ) =ρ+ µ1

1 + µ1−µ2ρ+µ2

e−(ρ+µ1)(t∗−τ)→ µ1 + ρ,

as τ → −∞.

The GDP growth rate

It follows from p(t)K (t) = αY (t) that the GDP growth rate has

the expression g(t) ≡ Y (t)Y (t) = K(t)

K(t) + p(t)p(t) , which with

r(t) = α p(t)− δ,

K (t)

K (t)=

r(t) + δ(1− α2

)α2

− C (t)

K (t),

andp(t)

p(t)=

1− αα

([λ (1− α)

K (t)

w(t)− α

]p(t) + δ

)yields

g(t) =

(λ (1− α)2

K (t)

w(t)+ 1− α + α2

)p(t)

α− C (t)

K (t)+

1− 2α

αδ.

Figure: Comparison of the GDP growth rates between unanticipatedaging and anticipated aging.

Conclusions

I Population aging, i.e., a decrease in β and µ, implies fastereconomic growth along the balanced growth path.

I Anticipated aging is associated with faster economic growthduring the transition to the long-run balanced growth pathand, hence, a higher level of per capita GDP in the long run.

I Overall, the accurate information of the population leads toeconomic gains that could be valuable for a society in thetransition to an older population.