economic growth: malthus and solowfaculty.missouri.edu/~hedlunda/teaching/2013b/6 - growth... ·...

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?


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.


Introduction

The evolution of GDP per capita across countries, 1820 - 2000:


Introduction

The evolution of GDP per capita across countries, 1000 - 2000:


Introduction

North Korea vs. South Korea:


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.


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.


Can Money Buy Happiness?


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


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.


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.


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


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 .


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.


The Effects of Technological Progress

The effects of an increase in z in the short run and long run:


The Effects of Population Control

The effects of introducing population control:


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.


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 .


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.


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)


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∗.


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 .


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.


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.


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.


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.


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.


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.


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?



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.


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


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.


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.


economic growth: malthus and solowfaculty.missouri.edu/~hedlunda/teaching/2013b/6 - growth... ·...

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