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The End of Economic Growth?
Unintended Consequences of a Declining Population
Chad Jones
October 2020
0
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Key Role of Population
• People ⇒ ideas ⇒ economic growth
◦ Romer (1990), Aghion-Howitt (1992), Grossman-Helpman
◦ Jones (1995), Kortum (1997), Segerstrom (1998)
◦ And most idea-driven growth models
• The future of global population?
◦ Conventional view: stabilize at 8 or 10 billion
• Bricker and Ibbotson’s Empty Planet (2019)
◦ Maybe the future is negative population growth
◦ High income countries already have fertility below replacement!
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The Total Fertility Rate (Live Births per Woman)
1950 1960 1970 1980 1990 2000 2010 2020
2
3
4
5
6
World
India
China
U.S.
High income
countries
LIVE BIRTHS PER WOMAN
U.S. = 1.8
H.I.C. = 1.7
China = 1.7
Germany = 1.6
Japan = 1.4
Italy = 1.3
Spain = 1.3
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What happens to economic growth if population growth is negative?
• Exogenous population decline
◦ Empty Planet Result: Living standards stagnate as population vanishes!
◦ Contrast with standard Expanding Cosmos result: exponential growth for an
exponentially growing population
• Endogenous fertility
◦ Parameterize so that the equilibrium features negative population growth
◦ A planner who prefers Expanding Cosmos can get trapped in an Empty Planet
– if society delays implementing the optimal allocation
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Literature Review
• Many models of fertility and growth (but not n < 0)
◦ Too many papers to fit on this slide!
• Falling population growth and declining dynamism
◦ Krugman (1979) and Melitz (2003) are semi-endogenous growth models
◦ Karahan-Pugsley-Sahin (2019), Hopenhayn-Neira-Singhania (2019), Engbom
(2019), Peters-Walsh (2019)
• Negative population growth
◦ Feyrer-Sacerdote-Stern (2008) and changing status of women
◦ Christiaans (2011), Sasaki-Hoshida (2017), Sasaki (2019a,b) consider capital,
land, and CES
◦ Detroit? Or world in 25,000 BCE?4
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Outline
• Exogenous negative population growth
◦ In Romer / Aghion-Howitt / Grossman-Helpman
◦ In semi-endogenous growth framework
• Endogenous fertility
◦ Competitive equilibrium with negative population growth
◦ Optimal allocation
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The Empty Planet Result
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A Simplified Romer/AH/GH Model
Production of goods (IRS) Yt = Aσt Nt
Production of ideasAt
At= αNt
Constant population Nt = N
• Income per person: levels and growth
yt ≡ Yt/Nt = Aσt
yt
yt= σ
At
At= σαN
• Exponential growth with a constant population
◦ But population growth means exploding growth? (Semi-endogenous fix)
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Negative Population Growth in Romer/AH/GH
Production of goods (IRS) Yt = Aσt Nt
Production of ideasAt
At= αNt
Exogenous population decline Nt = N0e−ηt
• Combining the 2nd and 3rd equations (note η > 0)
At
At= αN0e−ηt
• This equation is easily integrated...
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The Empty Planet Result in Romer/GH/AH
• The stock of knowledge At is given by
logAt = logA0 +gA0
η
(
1 − e−ηt)
where gA0 is the initial growth rate of A
• At and yt ≡ Yt/Nt converge to constant values A∗ and y∗:
A∗ = A0 exp
(
gA0
η
)
y∗ = y0 exp
(
gy0
η
)
• Empty Planet Result: Living standards stagnate as the population vanishes!
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Semi-Endogenous Growth
Production of goods (IRS) Yt = Aσt Nt
Production of ideasAt
At= αNλ
t A−βt
Exogenous population growth Nt = N0ent, n > 0
• Income per person: levels and growth
yt = Aσt and A∗
t ∝ Nλ/βt
g∗y = γn, where γ ≡ λσ/β
• Expanding Cosmos: Exponential income growth for growing population
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Negative Population Growth in the Semi-Endogenous Setting
Production of goods (IRS) Yt = Aσt Nt
Production of ideasAt
At= αNλ
t A−βt
Exogenous population decline Nt = N0e−ηt
• Combining the 2nd and 3rd equations:
At
At= αNλ
0 e−ληtA−βt
• Also easily integrated...
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The Empty Planet in a Semi-Endogenous Framework
• The stock of knowledge At is given by
At = A0
(
1 +βgA0
λη
(
1 − e−ληt)
)1/β
• Let γ ≡ λσ/β = overall degree of increasing returns to scale.
• Both At and income per person yt ≡ Yt/Nt converge to constant values A∗ and y∗:
A∗ = A0
(
1 +βgA0
λη
)1/β
y∗ = y0
(
1 +gy0
γη
)γ/λ
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Numerical Example
• Parameter values
◦ gy0 = 2%, η = 1%
◦ β = 3 ⇒ γ = 1/3 (from BJVW)
• How far away is the long-run stagnation level of income?
y∗/y0
Romer/AH/GH 7.4
Semi-endog 1.9
• The Empty Planet result occurs in both, but quantitative difference
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First Key Result: The Empty Planet
• Fertility has trended down: 5, 4, 3, 2, and less in rich countries
◦ For a family, nothing special about “above 2” vs “below 2”
• But macroeconomics makes this distinction critical!
◦ Negative population growth may condemn us to stagnation on an Empty Planet
– Stagnating living standards for a population that vanishes
◦ Vs. the exponential growth in income and population of an Expanding Cosmos
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Endogenous Fertility
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The Economic Environment
ℓ = time having kids instead of producing goods
Final output Yt = Aσt (1 − ℓt)Nt
Population growth Nt
Nt= nt = b(ℓt)− δ
Fertility b(ℓt) = bℓt
Ideas At
At= Nλ
t A−βt
Generation 0 utility U0 =∫∞
0e−ρtu(ct, Nt)dt, Nt ≡ Nt/N0
Flow utility u(ct, Nt) = log ct + ǫ log Nt
Consumption ct = Yt/Nt
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Overview of Endogenous Fertility Setup
• All people generate ideas here
◦ Learning by doing vs separate R&D
• Equilibrium: ideas are an externality (simple)
◦ We have kids because we like them
◦ We ignore that they might create ideas that benefit everyone
◦ Planner will desire higher fertility
• This is a modeling choice — other results are possible
• Abstract from the demographic transition. Focus on where it settles
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A Competitive Equilibrium with Externalities
• Representative generation takes wt as given and solves
max{ℓt}
∫ ∞
0
e−ρtu(ct, Nt)dt
subject to
Nt = (b(ℓt)− δ)Nt
ct = wt(1 − ℓt)
• Equilibrium wage wt = MPL = Aσt
• Rest of economic environment closes the equilibrium
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Solving for the equilibrium
• The Hamiltonian for this problem is
H = u(ct, Nt) + vt[b(ℓt)− δ]Nt
where vt is the shadow value of another person.
• Let Vt ≡ vtNt = shadow value of the population
• Equilibrium features constant fertility along transition path
Vt =ǫ
ρ≡ V∗
eq
ℓt = 1 −1
bVt
= 1 −1
bV∗eq
= 1 −ρ
bǫ≡ ℓeq
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Discussion of the Equilibrium Allocation
neq = b − δ −ρ
ǫ
• We can choose parameter values so that neq < 0
◦ Constant, negative population growth in equilibrium
• Remaining solution replicates the exogenous fertility analysis
The Empty Planet result can arise in equilibrium
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The Optimal Allocation
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The Optimal Allocation
• Choose fertility to maximize the welfare of a representative generation
• Problem:
max{ℓt}
∫ ∞
0
e−ρtu(ct, Nt)dt
subject to
Nt = (b(ℓt)− δ)Nt
At
At= Nλ
t A−βt
ct = Yt/Nt
• Optimal allocation recognizes that offspring produce ideas
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Solution
• Hamiltonian:
H = u(ct, Nt) + µtNλt A1−β
t + vt(b(ℓt)− δ)Nt
µt is the shadow value of an idea
vt is the shadow value of another person
• First order conditions
ℓt = 1 −1
bVt
, where Vt ≡ vtNt
ρ =µt
µt+
1
µt
(
ucσyt
At+ µt(1 − β)
At
At
)
ρ =vt
vt+
1
vt
(
ǫ
Nt+ µtλ
At
Nt+ vtnt
)
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Steady State Conditions
• The social value of people in steady state is
V∗sp = v∗t N∗
t =ǫ+ λz∗
ρ
where z denotes the social value of new ideas:
z∗ ≡ µ∗t A∗
t =σg∗A
ρ+ βg∗A
(Note: µ∗ finite at finite c∗ and A∗)
• If n∗sp > 0, then we have an Expanding Cosmos steady state
g∗A =λn∗
sp
β
g∗y = γn∗sp, where γ ≡
λσ
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Steady State Knowledge Growth
00
This kink gives rise
to two regimes...
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Key Features of the Equilibrium and Optimal Allocations
• Fertility in bothn = bℓ− δ
ℓ = 1 −1
bV
where V is the “utility value of people” (eqm vs optimal). Therefore
n(V) = b − δ −1
V
• Equilibrium: value kids because we love them (only): Veqm = ǫρ
◦ We can support n < 0 as an equilibrium for some parameter values
• Planner also values the ideas our kids will produce: Vsp = ǫ+µAρ ⇒ V(n)
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Optimal Steady State(s)
• Two equations in two unknowns (V, n)
V(n) =
1
ρ
(
ǫ+ γ1+
ρλn
)
if n > 0
ǫρ if n ≤ 0
n(V) = bℓ(V)− δ = b − δ +1
V
• We show the solution graphically
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A Unique Steady State for the Optimal Allocation when n∗eq > 0
Steady State
Equilibrium
Faster growth makes peoplemore valuable – more ideas
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Multiple Steady State Solutions when n∗eq < 0
High Steady State
(Expanding Cosmos)
Middle Steady State
Equilibrium = Low Steady State
(Empty Planet)
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Parameter Values for Numerical Solution
Parameter/Moment Value Comment
σ 1 Normalization
λ 1 Duplication effect of ideas
β 1.25 BJVW
ρ .01 Standard value
δ 1% Death rate
neq -0.5% Suggested by Europe, Japan, U.S.
ℓeq 1/8 Time spent raising children
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Implied Parameter Values and “Expanding Cosmos” Steady-State Results
Result Value Comment
b .040 neq = bℓeq − δ = −0.5%
ǫ .286 From equation for ℓeq
nsp 1.74% From equations for ℓsp and nsp
ℓsp 0.68 From equations for ℓsp and nsp
gspy = g
spA 1.39% Equals γnsp with σ = 1
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Transition Dynamics
• State variables: Nt and At
• Redefine “state-like” variables for transition dynamics solution: Nt and
xt ≡ Aβt /Nλ
t = “Knowledge per person”
• Why?
At
At=
Nλt
Aβt
=1
xt
Key insight: optimal fertility only depends on xt
• Note: x is the ratio of A and N, two stocks that are each good for welfare.
◦ So a bigger x is not necessarily welfare improving.
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Equilibrium Transition Dynamics
25 100 400 1600 6400
-0.5%
0%
0.5%
1.0%
1.5%
2.0%
2.5%
Equilibrium rateAsymptotic Low SS
(Empty Planet)
KNOWLEDGE PER PERSON, x
POPULATION GROWTH, n(x)
where x = Aβ
Nλ
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Optimal Population Growth
25 100 400 1600 6400
-0.5%
0%
0.5%
1.0%
1.5%
2.0%
2.5%
High Steady State
(Expanding Cosmos)
Middle Steady State
Equilibrium rate
Asymptotic Low SS
(Empty Planet)
KNOWLEDGE PER PERSON, x
POPULATION GROWTH, n(x)
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The Economics of Multiple SS’s and Transition Dynamics
• The High SS is saddle path stable as usual
◦ Equilbrium fertility depends on utility value of kids
◦ Planner also values the ideas the kids will produce ⇒ nspt > n
eqt
• Why is there a low SS?
◦ Diminishing returns to each input, including ideas
◦ As knowledge per person, x, goes to ∞, the “idea value” of an extra kid falls to
zero ⇒ nsp(x) → neq
• Why is the low SS stable?
◦ Since neq < 0, we also have nsp(x) < 0 for x sufficiently high
◦ With nsp(x) < 0, x = Aβ/Nλ rises over time
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What about the middle candidate steady state?
• Linearize the FOCs. Dynamic system has
◦ imaginary eigenvalues
◦ with positive real parts
• So the middle SS is an unstable spiral — a “Skiba point” (Skiba 1978)
• Numerical solution reveals what is going on...
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The Middle Steady State: Unstable Spiral Dynamics
KNOWLEDGE PER PERSON, x
POPULATION GROWTH, n(x)
What path is optimal?
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Population Growth Near the Middle Steady State
1000 1500 2000 2500 3000 3500-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Middle Steady State
50.79
60.86
69.85
78.01
83.74
87.19 69.12
77.75
83.74 87.30
83.67
87.19 77.73
KNOWLEDGE PER PERSON, x
POPULATION GROWTH, n(x) (percent)
Welfare in red
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Surprising Result
• The optimal allocation features two very different steady states
◦ One is an Expanding Cosmos
◦ One is the Empty Planet
• Start the economy with low x
◦ The equilibrium converges to the Empty Planet steady state
◦ If society adopts optimal policy soon, it goes to the Expanding Cosmos
But if society delays, even the optimal allocation converges to the Empty Planet
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Even the optimal allocation can get trapped
25 100 400 1600 6400
-0.5%
0%
0.5%
1.0%
1.5%
2.0%
2.5%
High Steady State
(Expanding Cosmos)
Middle Steady State
Equilibrium rate
Asymptotic Low SS
(Empty Planet)
KNOWLEDGE PER PERSON, x
POPULATION GROWTH, n(x)
If society delays, even the optimal allocation
converges to the Empty Planet
high x ⇒ high ideas per person
⇒ low µA
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Conclusion
• Fertility considerations may be more important than we thought:
◦ Negative population growth may condemn us to stagnation on an Empty Planet
◦ Vs. the exponential growth in income and population of an Expanding Cosmos
• This is not a prediction but rather a study of one force...
• Other possibilities, of course!
◦ Technology may affect fertility and mortality
◦ Evolution may favor groups with high fertility
◦ Can AI produce ideas, so people are not necessary?
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