EC 205 – Macroeconomics I

Macroeconomics I

Chapter 8 & 9:

Economic Growth

Why growth matters

In 2000, real GDP per capita in the United States was

more than fifty times that in Ethiopia.

Over the period 1975-2003, real GDP per capita in China

grew at a rate of 7.6% annually, while, in Argentina, real

GDP per capita grew at a rate of only 0.1%.

76 times slower!

From 1960 to 2000, the fastest growing country in the

world was Taiwan, which grew at 6%.

The slowest growing country was Zambia which grew at -

1.8%. That is, people in Zambia were markedly worse off

in 2000 than they were in 1960.

Why growth matters

The theory of economic growth seeks to address these

issues and provide explanations.

“Is there some action a government of India could take

that would lead the Indian economy to grow like

Indonesia's or Egypt's? If so, what exactly? If not, what is it

about the nature of India that makes it so? The

consequences for human welfare involved in questions

like these are simply staggering: Once one starts to think

about them, it is hard to think about anything else.”

- Robert Lucas Jr., Nobel Laureate (1995)

links to prepared graphs @ Gapminder.org

notes: circle size is proportional to population size,

color of circle indicates continent, press “play” on

bottom to see the cross section graph evolve over time

click here for one-page instruction guide

Income per capita and

Life expectancy

Infant mortality

Malaria deaths per 100,000

Adult literacy

Cell phone users per 100 people

Why growth matters

Anything that effects the long-run rate of economic

growth – even by a tiny amount – will have huge

effects on living standards in the long run.

1,081.4%243.7%85.4%

624.5%169.2%64.0%

2.5%

2.0%

…100 years…50 years…25 years

increase in standard

of living after…

annual

growth rate of

income per

capita

Why growth matters

If the annual growth rate of U.S. real GDP per

capita had been just one-tenth of one percent

higher from 2000–2010, the average person

would have earned $2,782 more during the

decade.

If the annual growth rate of Turkish real GDP

had been 1% higher between 1999 – 2007,

Turkey would have generated an additional

~1000TL of income per person measured in

1998 prices during the period.

The lessons of growth theory

…can make a positive difference in the lives of

hundreds of millions of people.

These lessons help us

understand why poor

countries are poor

design policies that

can help them grow

learn how our own

growth rate is affected

by shocks and our

government’s policies

Kaldor’s Stylized Facts

In 1957, Nicholas Kaldor documented some key

facts on economic growth by empirical

investigation.

Kaldor did not claim that any of the variables he

examines would be constant at all times; they

tend to be constant when averaging the data

over long periods of time.

Kaldor’s Stylized Facts

Six key facts:

Output per worker grows at a roughly constant rate that

does not diminish over time.

Capital per worker grows over time.

The rate of return to capital is constant.

The capital/output ratio is roughly constant.

The share of capital and labor in net income are nearly

constant.

Growth rate of output per worker differs substantially

across countries.

The Solow model

due to Robert Solow,

won Nobel Prize for contributions to

the study of economic growth

a major paradigm:

widely used in policy making

benchmark against which most

recent growth theories are compared

looks at the determinants of economic growth

and the standard of living in the long run

The Solow Model

How Solow model is different from

Chapter 3’s model

1. K is no longer fixed (exogenous), it is

endogenized:

investment causes it to grow,

depreciation causes it to shrink & whole

discussion is over the behavior of the evolution of

the capital stock

2. L is no longer fixed:

population growth causes it to grow

3. the consumption function is simpler

How Solow model is different from

Chapter 3’s model

4. no G or T

(only to simplify presentation;

we can still do fiscal policy experiments)

5. cosmetic differences

Representative Firm

Notation: Y denotes output, L denotes labor, K denotes

capital, and A denotes a constant measure of productivity

(also known as the Solow-residual)

Solow model assumes a country’s output is produced by a

representative neoclassical production function.

A function is called a neoclassical production function iff

it satisfies the following four properties:

1. Constant returns to scale (CRS or CRTS)

2. Positive and diminishing marginal products of capital and labor

3. Essentiality of inputs

4. Inada (limiting conditions)

Constant Returns to Scale

Let the production function be of the form Y=F(K,L)

A function satisfies CRTS in inputs if for all λ>0

F(λK,λL)=λF(K,L)

If both inputs double, so does output/production

Does the Cobb-Douglas production function satisfy

CRTS?

Y=F(K,L)=AKαL1-α

Positive & Diminishing Marginal Products

A production function F(K,L) displays positive and

diminishing marginal products in capital and labor iff

FK (K,L) > 0 & FL(K,L) > 0

FKK (K,L) < 0 & FLL(K,L) < 0

Additional labor and capital increase production, only at a

decreasing rate

Both the first and the second units of labor increase

production, but the first additional unit of labor contributes

more than the second for the same constant level of

capital

Does the Cobb-Douglas production function satisfy

positive and diminishing marginal products?

Essentiality of Inputs

A production function F(K,L) satisfies essentiality of

inputs property iff

F (K,0) = 0

F (0,L) = 0

Both inputs are essential for production

Does the Cobb-Douglas production function satisfy

positive and diminishing marginal products?

Inada Conditions

A production function F(K,L) satisfies Inada conditions iff

&

&

In the absence of capital or labor, the incremental

increase of the absent input boosts output substantially

(from zero to some positive amount)

As we increase one of the inputs so much while keeping

the other constant, after a point (infinite units to be exact)

the additional unit of the vastly abundant input does not

increase production at all.

Does the Cobb-Douglas production function satisfy

positive and diminishing marginal products?

limK®0

FK(K,L) = ¥

limL®0

FL(K,L) = ¥

limK®¥

FK(K,L) = 0

limL®¥

FL(K,L) = 0

Solow Model Assumptions

Output is produced with a neoclassical production

function

We are interested in not aggregate but per-capita

measures, since the latter is a better conventional

measure of standard of living, hence our focus.

We assume that everybody works, hence the number

of workers is equal to the number of citizens of a

country. Therefore L is both the number of workers,

AND the number of citizens,

Thus y=Y/L is the income/output per capita.

Population is constant so that

(… to be relaxed later)

L

t= L

t+1= ... = L

The production function

In aggregate terms: Y = F (K, L)

Define: y = Y/L = output per worker

k = K/L = capital per worker

Assume constant returns to scale:

zY = F (zK, zL ) for any z > 0

Pick z = 1/L. Then

Y/L = F (K/L, 1)

y = F (k, 1)

y = f(k) where f(k) = F(k, 1)

Properties of the Per-Capita Production

Function

MPK of F( ) is equal to marginal product of per-capita capital

As per-capita capital increases, so does per-capita output, yet

only at a diminishing rate

Capital is essential in production

Initial per-capita capital increase boosts production from zero

units to a positive amount, and when there is too much per-

capita capital, additional units of capital do not contribute

¶Y

¶K=

¶F(K ,L)

¶K=

df (k)

dk= f '(k)

f '(k) > 0& f ''(k) < 0

f (0) = 0

limk®0

f '(k) = ¥&limk®¥

f '(k) = 0

The production function

Output per

worker, y

Capital per

worker, k

1

MPK = f(k +1) – f(k)

f(k) (with A2 >A1 )

f(k) (with A3 <A1 )

f(k) (with A1 )

The Basics of the Solow Model

Households own the factor of production Kt & Lt, hence

the national output Yt =F(Kt, Lt)

Households consume a fraction of income and save

the rest

s = the saving rate,

the fraction of income that is saved

(s is an exogenous parameter)

(1-s) as marginal propensity to consume

Yt =F(Kt, Lt)=Ct + St

The Rule of Motion for Capital

Capital depreciates at a constant rate δ so that in period

t, δKt units of capital becomes useless

(1-δ) Kt carried over to the next period, t+1

Savings are used to finance investment and

investment contributes to the stock of capital

Next period’s capital increases by It=St=sYt=sF(Kt,

Lt)

The connection between Kt & Kt+1 then

Kt+1 =(1-δ) Kt +It=(1-δ) Kt + sYt=(1-δ) Kt + sF(Kt, Lt)

Kt+1 =(1-δ) Kt + sF(Kt, Lt) is the rule of motion for

capital

The Rule of Motion for Per-Capita

Capital

Kt+1 =(1-δ) Kt + sF(Kt, Lt)

Divide both sides by Lt

Kt+1/Lt =(1-δ) Kt /Lt + sF(Kt,Lt) /Lt

Rearranging yields

Kt+1/Lt =(1-δ) kt + sf(kt)

Since we can rewrite this expression as

kt+1 =(1-δ) kt + sf(kt)

kt+1 =(1-δ) kt + sf(kt) is the rule of motion for per-capita

capital

L

t= L

t+1= ... = L

Output, consumption, and investment

Output per

worker, y

Capital per

worker, k

f(k)

sf(k)

k1

y1

i1

c1

Depreciation

Depreciation

per worker, δ k

Capital per

worker, k

δk

δ = the rate of depreciation

= the fraction of the capital stock

that wears out each period

1

δ

The Steady State

• If investment is just enough to cover

depreciation

[sf(k) = δk ],

then capital per worker will remain constant, i.e.

Δk = 0 or kt=kt+1=… = k*

• This one value of k, denoted k* is called the

steady state capital stock.

Alternative way of writing the rule of motion

Δkt+1 = s f(kt) – δkt where Δkt+1 =kt+1 – kt

The steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*

Moving toward the steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*

Δk = sf(k) - δ k

depreciation

Δk

k1

investment

Moving toward the steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*k1

Δk = sf(k) - δ k

Δk

k2

Moving toward the steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*

Δk = sf(k) - δ k

k2

investment

depreciation

Δk

Moving toward the steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*

Δk = sf(k) - δ k

Δk

k2

Moving toward the steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*

Δk = sf(k) - δ k

k2

Δk

k3

Moving toward the steady state

Investment

and

depreciation

Capital per

worker, k

sf(k)

δk

k*

Δk = sf(k) - δ k

k3

Summary:

As long as k < k*,

investment will exceed

depreciation,

and k will continue to

grow toward k*.

A numerical example

Production function (aggregate):

Y

t= F (K

t,L

t) = K

t´ L

t= K

t

1/2Lt

1/2

y

t= f (k

t) = k

t

1/2

Converting into-per capita terms

A numerical example, cont.

Assume:

s = 0.3

δ = 0.1

initial value of k = 4.0

Approaching the steady state:

A numerical example

Year k y c i δ k

Δ k

1 4.000 2.000 1.400 0.600 0.400 0.200

2 4.200 2.049 1.435 0.615 0.420 0.195

3 4.395 2.096 1.467 0.629 0.440 0.189

Assumptions: ; 0.3; 0.1; initial 4.0y k s k 4 4.584 2.141 1.499 0.642 0.458 0.184…10 5.602 2.367 1.657 0.710 0.560 0.150…25 7.351 2.706 1.894 0.812 0.732 0.080100 8.962 2.994 2.096 0.898 0.896 0.002

9.000 3.000 2.100 0.900 0.900 0.000¥

Example:

Solve for the steady state

40

Continue to assume

s = 0.3, = 0.1, and y = k 1/2

Imposing the steady state condition on the rule of

motion for per-capita-capital to solve for the steady-

state values of k, y, and c.

Example:

Solve for the steady state

41

eq'n of motion with s f k k k ( *) * 0

using assumed valuesk k0.3 * 0.1 *

*3 *

*

kk

k

Solve to get: k * 9 and y k * * 3

Finally, c s y * (1 ) * 0.7 3 2.1

An increase in the saving rate

Investment

and

depreciation

k

δk

s1 f(k)

*k1

An increase in the saving rate raises investment…

…causing k to grow toward a new steady state:

s2 f(k)

*k2

Prediction:

Higher s higher k*.

And since y = f(k) ,

higher k* higher y* .

Thus, the Solow model predicts that countries

with higher rates of saving and investment

will have higher levels of capital and income per

worker in the long run.

100

1,000

10,000

100,000

0 10 20 30 40 50

International evidence on investment rates and

income per person

Income per

person in

2009

(log scale)

Investment as percentage of output

(average 1961-2009)

The Golden Rule: Introduction

Different values of s lead to different steady states.

How do we know which is the “best” steady state?

The “best” steady state has the highest possible

consumption per person: c* = (1–s) f(k*).

An increase in s

leads to higher k* and y*, which raises c*

reduces consumption’s share of income (1–s),

which lowers c*.

So, how do we find the s and k* that maximize c*?

The Golden Rule capital stock

the Golden Rule level of capital,

the steady state value of k

that maximizes consumption.

*

goldk

To find it, first express c* in terms of k*:

c* = y* - i*

= f (k*) - i*

= f (k*) - δ k*

In the steady state:

i* = δ k*

because Δk = 0.

Then, graph

f(k*) and δk*,

look for the

point where

the gap between

them is biggest.

The Golden Rule capital stock

steady state

output and

depreciation

steady-state capital per

worker, k*

f(k*)

δ k*

*

goldk

*

goldc

* *

gold goldi k* *( )gold goldy f k

The Golden Rule capital stock

c* = f(k*) - δ k*

is biggest where the

slope of the

production function

equals

the slope of the

depreciation line:

steady-state capital per

worker, k*

f(k*)

δ k*

*

goldk

*

goldc

MPK = δ

The transition to the

Golden Rule steady state

The economy does NOT have a tendency to

move toward the Golden Rule steady state.

Achieving the Golden Rule requires that

policymakers adjust s.

This adjustment leads to a new steady state with

higher consumption.

But what happens to consumption

during the transition to the Golden Rule?

Starting with too much capital

then increasing c*

requires a fall in s.

In the transition to

the Golden Rule,

consumption is

higher at all points

in time.

If goldk k* *

timet0

c

i

y

Starting with too little capital

then increasing c*

requires an

increase in s.

Future generations

enjoy higher

consumption,

but the current

one experiences

an initial drop

in consumption.

If goldk k* *

timet0

c

i

y

Population growth

Assume the population and labor force grow

at rate n (exogenous):

Lt+1=(1+n) Lt , Lt+2=(1+n) Lt+1 ,…

EX: Suppose L1 = 1,000 in year 1 and the population is

growing at 2% per year (n = 0.02).

Then ΔL2 = n L1 = 0.02 × 1,000 = 20,

so L2 = 1,020 in year 2.

Ln

L

Solow Model with Population Growth

Kt+1 =(1-δ) Kt + sF(Kt, Lt)

Divide both sides by Lt

Kt+1/Lt =(1-δ) Kt /Lt + sF(Kt,Lt) /Lt

Rearranging yields

Kt+1/Lt =(1-δ) kt + sf(kt)

Since for any t we can rewrite this

expression as

kt+1(1+n) =(1-δ) kt + sf(kt)

Lt=

Lt+1

(1+ n)

Solow Model with Population Growth

We cannot use the former diagram to look at the effects of

population growth on the steady state variables

Instead we’ll define a draw a diagram with kt & kt+1 on the

axis

Let k

t+1= g(k

t) =

sf (kt)+ (1-d )k

t

1+ n

45

k

t+1

k

t

g(k

t) =

sf (kt)+ (1-d )k

t

1+ n

Steady-State

Solow Model with Population Growth

k1 k2 k3k*

k2

k3

k*

k4

45

k

t+1

k

t

g(kt) =

sf (kt) + (1-d )k

t

1+ n1

g(kt) =

sf (kt) + (1-d )k

t

1+ n2

n

2> n

1

k *

1

Solow Model with Population Growth

k *

2

k *

2

k *

1

45

g(k

t) =

s1f (k

t)+ (1-d

1)k

t

1+ n

g '(k

t) =

s1f (k

t)+ (1-d

2)k

t

1+ n

s

2> s

1&d

2> d

1

g(k

t) =

s2f (k

t)+ (1-d

1)k

t

1+ n

k*

s1

,d1

k*

s1

,d1

k*

s1

,d2

k*

s1

,d2

k*

s2,d

1

k*

s2,d

1

k

t+1

k

t

Solow Model with Population Growth

Prediction:

Higher n lower k*.

And since y = f(k) ,

lower k* lower y*.

Thus, the Solow model predicts that countries

with higher population growth rates will have

lower levels of capital and income per worker in

the long run.

International evidence on population growth and

income per person

Income per

person in

2009

(log scale)

Population growth

(percent per year, average 1961-2009)

100

1,000

10,000

100,000

0 1 2 3 4 5

The Golden Rule with population

growth

To find the Golden Rule capital stock,

express c* in terms of k*, first rewrite impose the

steady state restrictions on the rule of motion:

which yields

Since c* = f(k*)-sf(k*)= f(k*)-(n+δ)k*

c* is maximized when

f’(k*)-(n+δ)=0 or MPK = δ + n

k* =

sf (k*) + (1-d )k *

1+ n (n +d )k* = sf (k*)

Growth empirics: Convergence

Solow model predicts that, other things equal,

“poor” countries (with lower Y/L and K/L) should

grow faster than “rich” ones.

If true, then the income gap between rich & poor

countries would shrink over time, causing living

standards to “converge.” (Also known as

“Absolute Convergence”)

In real world, many poor countries do NOT grow

faster than rich ones. Does this mean the Solow

model fails?

Growth empirics: Convergence

Solow model predicts that, other things equal,

“poor” countries (with lower Y/L and K/L) should

grow faster than “rich” ones.

No, because “other things” aren’t equal.

In samples of countries with

similar savings & pop. growth rates,

income gaps shrink about 2% per year.

In larger samples, after controlling for differences

in saving, pop. growth, and human capital,

incomes converge by about 2% per year.

Growth empirics: Convergence

What the Solow model really predicts is

conditional convergence - countries converge

to their own steady states, which are determined

by saving, population growth, and education.

This prediction seems to be true in the real

world.

Growth empirics: Convergence

Not controlling for

other things:

Controlling for

other things:

Average annual growth rates, 1970-89

closedopen

Growth empirics:

Production efficiency and free trade

Since Adam Smith, economists have argued that

free trade can increase production efficiency and

living standards.

Research by Sachs & Warner:

0.7%4.5%developing nations

0.7%2.3%developed nations

Growth empirics:

Production efficiency and free trade

To determine causation, Frankel and Romer

exploit geographic differences among countries:

Some nations trade less because they are farther

from other nations, or landlocked.

Such geographical differences are correlated with

trade but not with other determinants of income.

Hence, they can be used to isolate the impact of

trade on income.

Findings: increasing trade/GDP by 2% causes

GDP per capita to rise 1%, other things equal.

Some Strengths and Weaknesses of the

Solow Model

The Solow-model predicts that

Countries converge to their steady-state levels of per-

capita capital and output in the long-run, and if they have

different saving rates, depreciation rates, or population

growth rates, or productivity levels, the steady-states may

differ.

Poor countries (countries with lower levels of per-capita

capital and output) grow faster.

Countries with high saving rates, low depreciation rates,

high productivity, and low population growth reach higher

levels of steady states.

Solow model somewhat successfully addresses the some

of the stylized facts a-la Kaldor.

Solow Model & Data

Some Strengths and Weaknesses of the

Solow Model

The model has clear weaknesses, as well

Naïve modeling of consumption limits the reliability of

predictions

It does not explain why different countries have different

saving and productivity rates

It focuses on investment and capital, while the much more

important factor of TFP is still unexplained

The model does not provide a theory of sustained long-run

economic growth

The role of institutions, cultures, resources, … are not

ignored

The dissatisfaction gave rise to endogenous growth

models

Alternative perspectives on population

growth

The Malthusian Model (1798)

Predicts population growth will outstrip the

Earth’s ability to produce food, leading to the

impoverishment of humanity.

Since Malthus, world population has increased

sixfold, yet living standards are higher than ever.

Malthus neglected the effects of technological

progress.

Chapter Summary

1. The Solow growth model shows that,

in the long run, a country’s standard of living

depends:

positively on its saving rate

negatively on its population growth rate

2. An increase in the saving rate leads to:

higher output in the long run

faster growth temporarily

but not faster steady-state growth

Chapter Summary

3. If the economy has more capital than the

Golden Rule level, then reducing saving will

increase consumption at all points in time,

making all generations better off.

If the economy has less capital than the

Golden Rule level, then increasing saving will

increase consumption for future generations, but

reduce consumption for the present generation.