The End of Economic Growth? Unintended Consequences of a Declining - PowerPoint PPT Presentation

The End of Economic Growth? Unintended Consequences of a Declining Population Chad Jones October 2020 0

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! 1

The Total Fertility Rate (Live Births per Woman) U.S. = 1.8 LIVE BIRTHS PER WOMAN H.I.C. = 1.7 China = 1.7 6 Germany = 1.6 Japan = 1.4 Italy = 1.3 5 Spain = 1.3 India 4 3 World U.S. 2 High income countries China 1950 1960 1970 1980 1990 2000 2010 2020 2

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 3

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

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 5

The Empty Planet Result 6

A Simplified Romer/AH/GH Model Y t = A σ t N t Production of goods (IRS) ˙ A t = α N t Production of ideas A t N t = N Constant population • Income per person: levels and growth y t ≡ Y t / N t = A σ t ˙ ˙ y t A t = σ = σα N y t A t • Exponential growth with a constant population ◦ But population growth means exploding growth? (Semi-endogenous fix) 7

Negative Population Growth in Romer/AH/GH Y t = A σ Production of goods (IRS) t N t ˙ A t = α N t Production of ideas A t N t = N 0 e − η t Exogenous population decline • Combining the 2nd and 3rd equations (note η > 0 ) ˙ A t = α N 0 e − η t A t • This equation is easily integrated... 8

The Empty Planet Result in Romer/GH/AH • The stock of knowledge A t is given by log A t = log A 0 + g A 0 1 − e − η t � � η where g A 0 is the initial growth rate of A • A t and y t ≡ Y t / N t converge to constant values A ∗ and y ∗ : � g A 0 � A ∗ = A 0 exp η � g y 0 � y ∗ = y 0 exp η • Empty Planet Result : Living standards stagnate as the population vanishes! 9

Semi-Endogenous Growth Y t = A σ t N t Production of goods (IRS) ˙ A t t A − β = α N λ Production of ideas t A t N t = N 0 e nt , n > 0 Exogenous population growth • Income per person: levels and growth t ∝ N λ/β y t = A σ A ∗ and t t g ∗ y = γ n , where γ ≡ λσ/β • Expanding Cosmos : Exponential income growth for growing population 10

Negative Population Growth in the Semi-Endogenous Setting Y t = A σ Production of goods (IRS) t N t ˙ A t = α N λ t A − β Production of ideas t A t N t = N 0 e − η t Exogenous population decline • Combining the 2nd and 3rd equations: ˙ A t 0 e − λη t A − β = α N λ t A t • Also easily integrated... 11

The Empty Planet in a Semi-Endogenous Framework • The stock of knowledge A t is given by 1 − e − λη t �� 1 /β � 1 + β g A 0 � A t = A 0 λη • Let γ ≡ λσ/β = overall degree of increasing returns to scale. • Both A t and income per person y t ≡ Y t / N t converge to constant values A ∗ and y ∗ : � 1 /β � 1 + β g A 0 A ∗ = A 0 λη � γ/λ � 1 + g y 0 y ∗ = y 0 γη 12

Numerical Example • Parameter values ◦ g y 0 = 2 % , η = 1 % ◦ β = 3 ⇒ γ = 1 / 3 (from BJVW) • How far away is the long-run stagnation level of income? y ∗ / y 0 Romer/AH/GH 7.4 Semi-endog 1.9 • The Empty Planet result occurs in both, but quantitative difference 13

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 14

Endogenous Fertility 15

The Economic Environment ℓ = time having kids instead of producing goods Y t = A σ Final output t ( 1 − ℓ t ) N t ˙ N t Population growth N t = n t = b ( ℓ t ) − δ b ( ℓ t ) = ¯ Fertility b ℓ t ˙ t A − β A t A t = N λ Ideas t � ∞ e − ρ t u ( c t , ˜ ˜ U 0 = N t ) dt , N t ≡ N t / N 0 Generation 0 utility 0 u ( c t , ˜ N t ) = log c t + ǫ log ˜ Flow utility N t c t = Y t / N t Consumption 16

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 17

A Competitive Equilibrium with Externalities • Representative generation takes w t as given and solves � ∞ e − ρ t u ( c t , ˜ max N t ) dt { ℓ t } 0 subject to ˙ N t = ( b ( ℓ t ) − δ ) N t c t = w t ( 1 − ℓ t ) • Equilibrium wage w t = MP L = A σ t • Rest of economic environment closes the equilibrium 18

Solving for the equilibrium • The Hamiltonian for this problem is H = u ( c t , ˜ N t ) + v t [ b ( ℓ t ) − δ ] N t where v t is the shadow value of another person. • Let V t ≡ v t N t = shadow value of the population • Equilibrium features constant fertility along transition path V t = ǫ ρ ≡ V ∗ eq = 1 − ρ 1 1 ℓ t = 1 − = 1 − b ǫ ≡ ℓ eq ¯ ¯ ¯ bV ∗ bV t eq 19

Discussion of the Equilibrium Allocation b − δ − ρ n eq = ¯ ǫ • We can choose parameter values so that n eq < 0 ◦ Constant, negative population growth in equilibrium • Remaining solution replicates the exogenous fertility analysis The Empty Planet result can arise in equilibrium 20

The Optimal Allocation 21

The Optimal Allocation • Choose fertility to maximize the welfare of a representative generation • Problem: � ∞ e − ρ t u ( c t , ˜ max N t ) dt { ℓ t } 0 subject to ˙ N t = ( b ( ℓ t ) − δ ) N t ˙ A t t A − β = N λ t A t c t = Y t / N t • Optimal allocation recognizes that offspring produce ideas 22

Solution • Hamiltonian: H = u ( c t , ˜ t A 1 − β N t ) + µ t N λ + v t ( b ( ℓ t ) − δ ) N t t µ t is the shadow value of an idea v t is the shadow value of another person • First order conditions 1 ℓ t = 1 − , where V t ≡ v t N t ¯ bV t � ˙ � ρ = ˙ µ t u c σ y t A t + 1 + µ t ( 1 − β ) µ t µ t A t A t � � ˙ ρ = ˙ v t + 1 ǫ A t + µ t λ + v t n t v t v t N t N t 23

Steady State Conditions • The social value of people in steady state is t = ǫ + λ z ∗ V ∗ sp = v ∗ t N ∗ ρ where z denotes the social value of new ideas: σ g ∗ z ∗ ≡ µ ∗ t ˙ A ∗ A t = ρ + β g ∗ A (Note: µ ∗ finite at finite c ∗ and A ∗ ) • If n ∗ sp > 0 , then we have an Expanding Cosmos steady state λ n ∗ sp g ∗ A = β sp , where γ ≡ λσ g ∗ y = γ n ∗ β 24

Steady State Knowledge Growth This kink gives rise to two regimes... 0 0 25

Key Features of the Equilibrium and Optimal Allocations • Fertility in both n = ¯ b ℓ − δ ℓ = 1 − 1 ¯ bV where V is the “utility value of people” (eqm vs optimal). Therefore b − δ − 1 n ( V ) = ¯ V • Equilibrium: value kids because we love them (only): V eqm = ǫ ρ ◦ We can support n < 0 as an equilibrium for some parameter values • Planner also values the ideas our kids will produce: V sp = ǫ + µ ˙ A ⇒ V ( n ) ρ 26

Optimal Steady State(s) • Two equations in two unknowns ( V , n )  � � γ 1 ǫ + if n > 0  1 + ρ ρ V ( n ) = λ n ǫ if n ≤ 0  ρ b − δ + 1 n ( V ) = ¯ b ℓ ( V ) − δ = ¯ V • We show the solution graphically 27

A Unique Steady State for the Optimal Allocation when n ∗ eq > 0 Faster growth makes people more valuable – more ideas Steady State Equilibrium 28

Multiple Steady State Solutions when n ∗ eq < 0 High Steady State (Expanding Cosmos) Middle Steady State Equilibrium = Low Steady State (Empty Planet) 29