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Economic Growth: Malthus and Solow
Economics 3307 - Intermediate Macroeconomics
Aaron Hedlund
Baylor University
Fall 2013
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 1 / 35
Introduction
Two questions:1 What explains differences in growth within a country over time?
2 What explains differences in growth between countries?
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 2 / 35
Introduction
Differences across time:I Japanese boy born in 1880 had a life expectancy of 35 years, today 81
years.
I An American worked 61 hours per week in 1870, today 34.
Differences across countries:I Average American income is 5 times larger than Mexican’s, 14 times
larger than Indian’s, and 35 times larger than African’s (using PPP).
I Out of 6.4 billion people, 0.8 do not have access to enough food, 1 tosafe drinking water, and 2.4 to sanitation.
I Life expectancy in rich countries is 77 years, 67 years in middle incomecountries, and 53 years in poor countries.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 3 / 35
Introduction
The evolution of GDP per capita across countries, 1820 - 2000:
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 4 / 35
Introduction
The evolution of GDP per capita across countries, 1000 - 2000:
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 5 / 35
Introduction
North Korea vs. South Korea:
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 6 / 35
Growth Facts
Kaldor’s stylized facts of economic growth:1 Real GDP per worker y = Y
N and capital per worker k = KN grow over
time at relatively constant and positive rates.
2 They grow at similar rates, so the capital-output ratio KY is
approximately constant over time.
3 The real return to capital and the real interest rate are relativelyconstant over time.
4 The capital and labor shares are roughly constant over time.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 7 / 35
Growth Facts
More growth facts:1 Pre-1800: constant per capita income across time and countries.
2 Post-1800: sustained growth in rich countries (2% in U.S. since 1900).
3 Across countries, high investment ⇔ high standard of living and highpopulation growth ⇔ low standard of living.
4 Divergence of per capita incomes from 1800 - 1950.
5 From 1960 - 2000, no relationship between output levels and outputgrowth across countries.
6 Richer countries more alike in growth rates than are poor countries.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 8 / 35
Can Money Buy Happiness?
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 9 / 35
Economic Growth and Welfare
“The consequences for human welfare involved in questions [of economicgrowth] are simply staggering: Once one starts to think about them, it is
hard to think about anything else.” - Robert Lucas
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 10 / 35
Economic Growth and Welfare
Growth is far more important to welfare than business cycles becauseof compounding.
Suppose constant growth rate of g : 1 + g = ytyt−1⇒ yt = (1 + g)ty0.
How long does it take for GDP to double?
2y = (1 + g)ty ⇒ t =ln 2
ln(1 + g)≈ 70/(g × 100%)
“Rule of 70.” Example: t = 35 years with 2% growth, t = 23 yearswith 3% growth, and t = 70 years with 1% growth.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 11 / 35
Malthusian Growth
Robert Malthus, An Essay on the Principle of Population, 1798.
Idea: technological advances for producing food ⇒ higher populationgrowth ⇒ lower average consumption until only subsistence.
The English Economy 1275 - 1800
No long run increase in standardof living without populationgrowth limits.
A good description of economichistory prior to the IndustrialRevolution.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 12 / 35
Malthusian Growth: Ingredients of the Model
Output Yt = zF (Lt ,Nt) where Lt is land and Nt is labor.I Interpret Yt as (perishable) food. No savings or investment.
I Fixed supply of land: Lt = L for all t.
I No government spending: Gt = 0.
I Inelastic labor supply: Nt = total labor = population.
Nt+1 = Nt + Birthst − Deathst = Nt(1 + birth ratet − death ratet)
Assume birth rate is an increasing function of CtNt
and death rate is a
decreasing function of CtNt
. Thus,
Nt+1
Nt= g
(Ct
Nt
)with g ′ > 0
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 13 / 35
Equilibrium and Steady State of the Malthus Model
Goods market clearing: Ct = Yt = zF (L,Nt)⇒ Nt+1
Nt= g
(zF (L,Nt)
Nt
).
CRTS F ⇒ zF (L,Nt)Nt
= zF(
LNt, 1)⇒ Nt+1 = g
(zF(
LNt, 1))
Nt .
Unique steady state where N∗ = g(zF(
LN∗ , 1
))N∗.
When Nt < N∗, populationincreases: Nt+1 > Nt .
When Nt > N∗, populationdecreases: Nt+1 < Nt .
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 14 / 35
Determinants of Living Standards in the Malthus Model
Let yt ≡ YtNt
, lt ≡ LNt
, and ct ≡ CtNt
. Then yt = zf (lt) is theper-worker production function.
Equilibrium: ct = zf (lt) and Nt+1
Nt= g(ct).
In steady state, Nt+1
Nt= 1⇒ g(c∗) = 1 and c∗ = zf (l∗).
Consumption per-worker determined solely by g , implying thattechnological change has no impact on long run living standards.
Malthus’ solution: state-mandated population control.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 15 / 35
The Effects of Technological Progress
The effects of an increase in z in the short run and long run:
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 16 / 35
The Effects of Population Control
The effects of introducing population control:
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 17 / 35
Long Run Growth and the Solow Model
Malthus accurate prior to 1800 because of agricultural economy.
Main reasons for stagnation in the Malthus model: no accumulationof production inputs other than labor.
1 No accumulation of other production inputs ⇒ always cursed bydiminishing returns in the long run.
2 Assumes population growth increases in consumption per worker. Inreality, death rates decrease but so do birth rates.
Introducing capital raises prospects for long run growth because morecapital ⇒ more output ⇒ more capital. Capital begets capital.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 18 / 35
Ingredients of the Solow Model
The Representative Firm:
I CRTS production Yt = zF (Kt ,Nt)⇒ Yt
Nt= zF (Kt ,Nt)
Nt= zF
(Kt
Nt, 1)
.
F Firm profits are πt = zF (Kt ,Nt) − wtNt − rtKt .
I Let yt = Yt
Nt, kt = Kt
Nt, and f (kt) = F
(Kt
Nt, 1)
. Then yt = zf (kt).
Households:I Exogenous population growth Nt+1 = (1 + n)Nt with inelastic
household labor supply nst = 1⇒ Nst = Nt .
I Households own the capital and rent it to firms.
I Exogenous savings rate s ⇒ ct = (1− s)(wt + rtkst + πt
Nt) and
it = s(wt + rtkst + πt
Nt), where wt + rtk
st + πt
Ntis household income.
I Capital accumulation: K st+1 = (1− d)Ntk
st + Nt it = (1− d)K s
t + It .
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 19 / 35
Competitive Equilibrium of the Solow Model
A competitive equilibrium is prices {wt , rt}∞t=0 and allocations{Nd
t ,Kdt }∞t=0 and (ks0 , {ct , it}∞t=0) s.t.
1 Consumers satisfy their budget constraint: ct + it = wt + rtkst + πt
Nt.
2 Ndt and K d
t maximize firm profits πt = zF (Kt ,Nt)− wtNt − rtKt .
3 The labor market clears: Ndt = Nt .
4 The capital market clears: K dt = K s
t = Ntkst .
5 The goods market clears: Ntct︸︷︷︸Ct
+ Nt it︸︷︷︸It
= zF (K dt ,N
dt ).
Taking prices as given, households “choose” how much to consume ctand how much to invest it in new capital.
Taking prices as given, firms choose Ndt and Kd
t to maximize profits.
The prices adjust to clear each of the markets.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 20 / 35
Competitive Equilibrium, Cont’d
There is no consumer optimization because we did not specifypreferences. However, the budget constraint must be satisfied.
Walras’ law: (1) + (2) + (3) + (4) ⇒ (5). From (1),
ct + it = wt + rtkst +
πtNt⇒ Ntct + Nt it = Nt
(wt + rtk
st +
πtNt
)⇒ Ct + It = wtNt + rtK
st + πt
From (2), πt = zF (Kdt ,N
dt )− wtN
dt − rtK
dt
⇒ Ct + It = wtNt + rtKst +
(zF (Kd
t ,Ndt )− wtN
dt − rtK
dt
)From (3) and (4), Nt = Nd
t and K st = Kd
t ≡ Kt
⇒ Ct + It = wtNt + rtKt + zF (Kt ,Nt)− wtNt − rtKt
⇒ Ct + It = zF (Kt ,Nt)
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 21 / 35
Solving the Model
Investment it = s(wt + rtkst + πt
Nt)
⇒ Nt it = s(wtNt+rtKt+πt) = szF (Kt ,Nt)⇒ Kt+1 − (1− d)Kt︸ ︷︷ ︸
=Nt it=It
= szF (Kt ,Nt).
The dynamics of Kt and Nt are therefore
Kt+1 = (1− d)Kt + szF (Kt ,Nt)
Nt+1 = (1 + n)Nt
In per-worker terms,
kt+1 =1− d
1 + nkt +
szf (kt)
1 + n
Steady state szf (k∗) = (n + d)k∗.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 22 / 35
Analyzing the Steady State
An increase in s causes an increase in k∗ and y∗ but not always c∗.
The golden rule savings rate sgr maximizes steady stateconsumption c∗ = (1− sgr )zf (k∗gr ) = zf (k∗gr )− (n + d)k∗gr .
Optimality condition:(zf ′(k∗gr )− (n + d)
) dk∗gr
ds = 0⇒ MPK = n + d .
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 23 / 35
Analyzing the Steady State
An increase in n causes a decrease in k∗, y∗, and c∗.
An increase in z causes an increase in k∗, y∗, and c∗.I No limit to long run economic growth as long as z keeps rising.
I Fluctuations in z can also be used to study business cycles.
I 2008 - 2009 recession: credit disruptions reflected in TFP decrease.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 24 / 35
Simulated Effects of a Temporary TFP Drop
Calibrate a version of the Solow model and simulate a 5% drop in z .
zF (K ,N) = zK 0.36N0.64 ⇒ zf (k) = zk0.36, s = 0.14, d = 0.1,n = 0.01.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 25 / 35
The Solow Model with Continual Technological Progress
Production Yt = F (Kt ,AtNt) with labor augmenting technologicalprogress. Growth rates 1 + g = At+1
Atand 1 + n = Nt+1
Nt.
Let xt = XtAtNt
= xtAt
for Xt = Ct ,Kt ,Yt .
Rewrite Kt+1 = (1− d)Kt +
It︷ ︸︸ ︷sF (Kt ,AtNt) as
kt+1(1 + g)(1 + n) = (1− d)kt + f (kt)
Dividing by (1 + g)(1 + n) gives
kt+1 =1− d
(1 + g)(1 + n)kt +
s
(1 + g)(1 + n)f (kt)
Steady state k∗ implies long run balanced growth path.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 26 / 35
The Solow Model with Continual Technological Progress
Compute growth rate of per capita Xt as gx = Xt+1/Nt+1
Xt/Nt= xt+1
xt.
Steady state growth rates gy = gk = gc = g .
I To see this, note that kt = Kt
AtNt= kt
At⇒ kt = At kt . Therefore
kt+1
kt= At+1
At
kt+1
kt→ (1 + g) k∗
k∗ = 1 + g in the steady state.
Wage growth wt+1
wt= FN(Kt+1,At+1Nt+1)At+1
FN(Kt ,AtNt)At= FN(Kt+1/(At+1Nt+1),1)At+1
FN(Kt/(AtNt),1)At
= FN(kt+1,1)At+1
FN(kt ,1)At→ (1 + g)FN(k
∗,1)
FN(k∗,1)= 1 + g .
Rental rate growth rt+1
rt= FK (Kt+1,At+1Nt+1)
FN(Kt ,AtNt)→ FK (k
∗,1)
FK (k∗,1)= 1.
I Mathematical note: if F (aK , aN) = aF (K ,N) for all a, thenFK (aK , aN) = FK (K ,N) and FN(aK , aN) = FN(K ,N) for all a.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 27 / 35
Evaluating the Solow Model
Two difficulties:1 Difficulty evaluating cross-country measurements.
2 Limited time span of data. Are economies in steady state?
Two predictions of the Solow model:1 Higher savings rates s = It
Ytlead to higher living standards.
2 Higher population growth n = Nt+1
Ntleads to lower living standards.
The data show a positive correlation between ItYt
and YtNt
and a
negative correlation between Nt+1
Ntand Yt
Nt.
The Solow model matches the Kaldor facts well.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 28 / 35
Evaluating the Solow Model
Limitations of the Solow model:1 Savings and population growth rates are not exogenous.
2 No steady state growth unless z is continually increasing.
3 Technological progress not exogenous.
4 The model cannot account for the magnitude of developmentdifferences between countries.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 29 / 35
From Malthus to Solow
Pre-Industrial Revolution: land-intensive production with fixed supplyof land and decreasing return to labor.
Post-Industrial Revolution: constant returns to scale production withlabor and capital inputs.
What accounts for the transition from stagnation to steady growth?
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 30 / 35
From Malthus to Solow
Malthus to Solow (Hansen and Prescott, 2002):
Two technologies: Malthus technology and Solow technology.
YMt = AMtKφMtN
µMtL
1−φ−µMt and YSt = AStK
θStN
1−θSt
where LMt is in fixed supply and θ > φ.
Along the equilibrium growth path, only Malthus technology is usedin early stages of development because the Solow technology isunprofitable.
As TFP grows, firms adopt the Solow technology.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 31 / 35
From Malthus to Solow
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 32 / 35
Growth Accounting
Growth accounting decomposes economic growth into growth offactor inputs and TFP, where Y = zF (K ,N).
Cobb-Douglas a good fitfor U.S. data:
Y = zK 0.36N0.64
Generate Solowresiduals as follows:
zt =Yt
K 0.36t N0.64
t
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 33 / 35
Solow Residuals and the Productivity Slowdown/Recovery
Average Annual Growth Rates in the Solow Residual
Years Average Annual Growth Rate
1950 – 1960 1.42
1960 – 1970 1.61
1970 – 1980 0.50
1980 – 1990 1.05
1990 – 2000 1.36
2000 – 2007 0.76
Three common reasons for the slowdown:1 Measurement problems due to change in quality of goods/services
during shift from manufacturing to services.
2 Increases in relative price of energy. Old capital not energy efficient,became obsolete.
3 Disruption arising from costs of adopting new technology. Beginning ofIT revolution.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 34 / 35
A Growth Accounting Exercise
Average Annual Growth Rates
Years Y K N z
1950 – 1960 3.48 3.68 1.11 1.42
1960 – 1970 4.19 3.86 1.80 1.61
1970 – 1980 3.19 3.24 2.36 0.50
1980 – 1990 3.26 2.85 1.81 1.05
1990 – 2000 3.28 2.72 1.43 1.36
2000 – 2007 2.32 2.64 0.93 0.76
Growth rate for X between years m and n is gXmn =
(XnXm
) 1n−m − 1.
Cobb-Douglas Ym = zmKαmN
1−αm , Yn = znK
αn N
1−αn
⇒ YnYm
= znzm
(KnKm
)α (NnNm
)1−α⇒ 1 + g y
mn = (1 + g znm)(1 + gK
mn)α(1 + gNmn)1−α
Approximation g ymn ≈ g z
mn + αgKmn + (1− α)gN
mn.
Econ 3307 (Baylor University) Malthus and Solow Fall 2013 35 / 35