Life and Growth
Charles I. Jones
Stanford GSB
Life and Growth – p.1/36
R. Posner (2004) Catastrophe: Risk and Response
“Certain events quite within the realm of possibility, such as a major
asteroid collision, global bioterrorism, abrupt global warming —
even certain lab accidents— could have unimaginably terrible
consequences up to and including the extinction of the human
race... I am not a Green, an alarmist, an apocalyptic visionary, a
catastrophist, a Chicken Little, a Luddite, an anticapitalist, or even a
pessimist. But... I have come to believe that what I shall be calling
the ‘catastrophic risks’ are real and growing...”
Life and Growth – p.2/36
Should we switch on the Large Hadron Collider?
• Physicists have considered the possibility that colliding
particles together at energies not seen since the Big Bang
could cause a major disaster (mini black hole, strangelets).
• Conclude that the probability is tiny.
• But how large does it have to be before we would not take
the risk?
• As economic growth makes us richer, should our decision
change?
Life and Growth – p.3/36
Growth involves costs as well as benefits
• Benefits | Costs
◦ Nuclear power | Nuclear holocaust
◦ Biotechnology | Bioterror
◦ Nanotechnology | Nano-weapons
◦ Coal power | Global warming
◦ Internal combustion engine | Pollution
◦ Radium, thalidomide, lead paint, asbestos
• Technologies (new pharmaceuticals, medical equipment,
airbags, pollution scrubbers) can also save lives
How do considerations of life and death affect the theory of
economic growth and technological change?
Life and Growth – p.4/36
The “Russian Roulette” Model
New ideas raise consumption,
but a tiny probability of a disaster...
Life and Growth – p.5/36
Simple Model
• Single agent born at the start of each period
• Endowed with stock of ideas ⇒ consumption c, utilty u(c)
• Only decision: to research or not to research
◦ Research:
With (high) probability 1 − π , get a new idea that
raises consumption by growth rate ḡ .
But, with small probability π , disaster kills the agent.
◦ Stop: Consumption stays at c, no disaster.
Life and Growth – p.6/36
• Expected utility for the two options:
U Research = (1 − π)u(c1 ) + π · 0 = (1 − π)u(c1 ), c1 = c(1 + ḡ)
U Stop
= u(c)
• Taking a first-order Taylor expansion around u(c), agent
undertakes research if
(1 − π)u′ (c)ḡc > πu(c)
• Rearranging:
ḡ >
u(c)
π
· ′
1 − π u (c)c
Life and Growth – p.7/36
Three Cases
• Consider CRRA utility:
c1−γ
u(c) = ū +
1−γ
ū is a key parameter
• Three cases:
◦ 0<γ<1
◦ γ>1
◦ log utility (γ = 1)
Life and Growth – p.8/36
Case 1:
0<γ<1
u(c)
π
· ′
ḡ >
1 − π u (c)c
• The value of life relative to consumption:
1
u(c)
γ−1
= ūc
+
.
′
u (c)c
1−γ
• ū not important, so set ū = 0
⇒
u(c)
= 1/(1 − γ)
′
u (c)c
Exponential growth, with rare disasters.
Life and Growth – p.9/36
Case 2:
γ>1
• Notice that we’ve implicitly normalized the utility from death
to be zero (in writing the lifetime expected utility function)
◦ So flow utility must be positive for consumer to prefer life
• But γ > 1 implies u(c) negative if ū = 0:
c1−γ
u(c) = ū +
1−γ
Example: γ = 2 implies u(c) = −1/c.
• Therefore ū > 0 is required in this case.
Life and Growth – p.10/36
Case 2:
γ > 1 (continued)
u(c)
π
· ′
ḡ >
1 − π u (c)c
• With γ > 1, the value of life rises relative to consumption!
1
u(c)
γ−1
= ūc
+
.
′
u (c)c
1−γ
Eventually, people are rich enough that the risk to life of
Russian Roulette is too great and growth ceases.
Life and Growth – p.11/36
The Research Decision when γ
>1
Utility
ū
(1 − π)ū
U research = (1 − π)u(c(1 + ḡ))
U stop = u(c)
Continue research
0
Stop research
c∗
Consumption, c
Life and Growth – p.12/36
Case 3: log utility (γ
= 1)
• Flow utility is unbounded in this case
• But the value of life relative to consumption still rises
u(c)/u′ (c)c = ū + log c
• So growth eventually ceases in this case as well.
Life and Growth – p.13/36
Microfoundations
in a Growth Model
Hall and Jones (2007) meet
Acemoglu (Direction TechChg)
Life and Growth – p.14/36
Production
R At 1/α α
R Bt 1/α α
Ct = ( 0 xit di) , Ht = ( 0 zit di)
RC (labor)
RC (scientists, pop)
Lct + Lht ≤ Lt , Lct ≡
Flow util.
Pop growth
R At
0
Sat + Sbt ≤ St ,
xit di, Lht ≡
St + Lt ≤ Nt
R Bt
0
zit di
δt = h−β
t , ht ≡ Ht /Nt
Mortality
Utility
λ Bφ
Ḃt = b̄Sbt
t
λ Aφ ,
Ȧt = āSat
t
Ideas
U=
R∞
0
e−ρt u(ct )Mt dt,
u(ct ) = ū +
c1−γ
t
1−γ ,
Ṁt = −δt Mt
ct ≡ Ct /Nt
Ṅt = n̄Nt
Life and Growth – p.15/36
Allocating Resources
• 14 unknowns, 11 equations (not counting utility)
◦ Ct , Ht , ct , ht , At , Bt , xit , zit , Sat , Sbt , St , Lt , Nt , δt
• Three allocative decisions to be made
◦ (st ) Scientists: Sat = st St
◦ (ℓt ) Workers: Lct = ℓt Lt
◦ (σt ) People: St = σt Nt
• Rule of Thumb allocation: st = s̄, ℓt = ℓ̄, and σt = σ̄
Life and Growth – p.16/36
BGP under the Rule of Thumb
P ROPOSITION 1: As t → ∞, there exists an asymptotic
balanced growth path such that growth is given by
∗
gA
=
∗
gB
λn̄
=
1−φ
δ∗ = 0
gc∗
=
gh∗
=
∗
αgA
=
∗
αgB
αλn̄
.
= ḡ ≡
1−φ
Life and Growth – p.17/36
The Optimal Allocation
max U =
{st ,ℓt ,σt }
Z
∞
Mt u(ct )e−ρt dt s.t.
0
ct = Aαt ℓt (1 − σt )
ht = Btα (1 − ℓt )(1 − σt )
Ȧt = āsλt σtλ Ntλ Aφt
Ḃt = b̄(1 − st )λ σtλ Ntλ Btφ
Ṁt = −δt Mt ,
δt = h−β
t
Life and Growth – p.18/36
Hamiltonian
• In solving, useful to define
H = Mt u(ct ) + pat āsλt σtλ Ntλ Aφt + pbt b̄(1 − st )λ σtλ Ntλ Btφ
−vt δt Mt
• Co-state variables:
◦ pat : shadow value of a consumption idea
◦ pbt : shadow value of a life-saving idea
◦ vt : shadow value of an extra person
Life and Growth – p.19/36
Optimal Growth with γ
>1+β
P ROPOSITION 2: Assume γ > 1 + β . There is an asymptotic
balanced growth path such that ℓt and st both fall to zero at
constant exponential rates, and
gs∗
=
gℓ∗
−ḡ(γ − 1 − β)
<0
=
αλ
1 + (γ − 1)(1 + 1−φ )
∗
λ(n̄
+
g
s)
∗
,
gA =
1−φ
∗
gB
gδ∗ = −βḡ,
∗
gc∗ = αgA
+ gℓ∗ = ḡ ·
λn̄
∗
> gA
=
1−φ
gh∗ = ḡ
αλ
1−φ )
αλ
1)(1 + 1−φ
)
1 + β(1 +
1 + (γ −
Life and Growth – p.20/36
Intuition
Ȧt = āsλt σtλ Ntλ Aφt and Ḃt = b̄(1 − st )λ σtλ Ntλ Btφ .
• 1 − st → 1, but st falls exponentially, slowing growth in At .
• Why? The FOC for allocating ℓt is
1 − ℓt
δt vt
=β ′
= δt ṽt
ℓt
u (ct )ct
where vt is the shadow value of a life, from the Hamiltonian.
• Numerator is extra lives that can be saved, denominator is
extra consumption that can be produced
• Race!
Life and Growth – p.21/36
Optimal Growth with γ
<1+β
P ROPOSITION 3: Assume 1 < γ < 1 + β . There is an asymptotic
balanced growth path such that ℓ̃t ≡ 1 − ℓt and s̃t ≡ 1 − st both
fall to zero at constant exponential rates, and
λn̄
∗
,
gA =
1−φ
∗
λ(n̄
+
g
∗
∗
s̃ )
gB =
< gA
1−φ
gc∗ = ḡ,
gδ∗ = −βgh∗ .
−ḡ(β + 1 − γ)
=
=
<0
αλ
1 + β(1 + 1−φ )
!
αλ
1 + (γ − 1)(1 + 1−φ )
∗
∗
∗
gh = gc ·
<
g
.
c
αλ
1 + β(1 + 1−φ )
gs̃∗
gℓ̃∗
Life and Growth – p.22/36
Optimal Growth with γ
=1+β
P ROPOSITION 4: Assume 1 < γ = 1 + β . There is an asymptotic
balanced growth path such that ℓt and st settle down to
constants strictly between 0 and 1, and
∗
gA
=
∗
gB
gc∗ = gh∗ = ḡ,
λn̄
=
1−φ
gδ∗ = −βḡ.
Life and Growth – p.23/36
Empirical Evidence
Life and Growth – p.24/36
Empirical Evidence
• γ − 1 versus β
◦ γ > 1 is the “normal” case
◦ Evidence from health “production functions” suggests β
is relatively small
• Trends in R&D: Toward health
• Quantifying the “growth slowdown”
Life and Growth – p.25/36
Evidence on β (Recall: δt
= h−β
t )
• Plausible upper bound compares trends in mortality to
trends in health spending
◦ Attributes all decline in mortality to real health spending
◦ Minimal quality adjustment reinforces “upper bound”
view for β
• Numbers for 1960 – 2007
◦ Age-adjusted mortality rates fell at 1.2% per year
◦ CPI-deflated health spending grew at 4.1% per year
⇒ β upper bound ≈
1.2
4.1
≈ .3
• Hall and Jones (2007) more careful analysis along these
lines finds age-specific estimates of between .10 and .25.
Life and Growth – p.26/36
Evidence on γ
• Risk aversion evidence suggests γ > 1
◦ Asset pricing (Lucas 1994), Labor supply (Chetty 2006)
• Intertemporal substitution elasticity 1/γ
◦ Traditional view is EIS<1 ⇒ γ > 1 (Hall 1988)
◦ Careful micro work supporting this view: Attanasio and
Weber (1995), Barsky et al (1997), Guvenen (2006), Hall
(2009)
◦ Other recent work finds some evidence for EIS>1
⇒ γ < 1 (Vissing-Jorgensen and Attanasio 2003,
Gruber 2006)
• Mixed evidence.
Life and Growth – p.27/36
Evidence on the Value of Life
• Same force as in Hall and Jones (2007) on health spending
◦ Consumption runs into sharply diminishing returns: u′ (c)
◦ While life becomes increasingly valuable: u(c)
• Evidence on value of life?
◦ Nearly all is cross sectional
◦ Costa and Kahn (2004), Hammitt, Liu, and Liu (2000)
• Other evidence? Safety standards?
Life and Growth – p.28/36
The Changing Composition of U.S. R&D Spending
Health Share of R&D (percent)
30
CMMS + NSF Recent
25
20
NIH estimates
15
CMMS + NSF IRIS
10
5
0
1960
Non−commercial health research (CMMS)
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Year
Life and Growth – p.29/36
The Changing Composition of OECD R&D Spending
Health Share of R&D (percent)
22
20
18
16
U.S.
14
12
10
OECD
8
6
4
2
0
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
Year
Life and Growth – p.30/36
Fraction of Patents for Medical Eq. or Pharma
Percent
18
16
U.S. only
14
12
10
Total
8
6
4
2
1960
1965
1970
1975
1980
1985
1990
1995
2000
Year
Source: Jeffrey Clemens
Life and Growth – p.31/36
An Income Effect in Health Spending
Health spending (percent of GDP)
18
16
United States
14
12
France
10
Germany
Japan
8
U.K.
6
4
2
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Year
Life and Growth – p.32/36
Health and Consumption
Real quantity per person (1950=100)
1600
Health (pce)
4.67%
800
Health (official)
3.10%
400
2.05%
1.84%
200
Non−health consumption (pce ,official)
100
1950
1960
1970
1980
1990
2000
2010
Year
Life and Growth – p.33/36
The Growth Drag: Ratio of gc to gh
αλ
1−φ
0.50
1.00
2.00
β = .25
γ = 1.5
γ=2
0.79
0.75
0.70
0.55
0.50
0.44
β = .10
γ = 1.5
γ=2
0.66
0.60
0.52
0.46
0.40
0.33
Life and Growth – p.34/36
A Future Slowdown?
• Calibration to past growth suggests
αλ
1−φ
< 2, so that ḡ < 2%.
◦ Therefore growth in h must slowdown significantly from
its 4 + % rate.
◦ And gc ≈ 1 gh < 1% suggests a slowdown of
2
consumption growth as well.
• Intuition: h has been growing much faster than its steady
state rate because of the rising share of research devoted to
life-saving technologies.
Life and Growth – p.35/36
Conclusions
• Including “life and death” considerations in growth models
can have first order consequences.
• For a large class of preferences, safety is a luxury good.
◦ Diminishing returns to consumption on any given day
means that additional days of life become increasingly
valuable.
◦ R&D may tilt toward life-saving technologies and away
from standard consumption goods.
◦ Consumption growth may be substantially slower than
what is feasible, possibly even slowing all the way to
zero.
Life and Growth – p.36/36