solow model best explain

8/19/2019 Solow Model Best Explain

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Lecture 7:History and Causes of Growth

1

September 22, 2015

Prof. Wyatt Brooks


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Causes of Growth

Before: Looked at special cases and tried toqualitatively understand causes of growth

Now evaluate causes of growth quantitatively

Motivating question: How important is history? That is, if a country has always been poor, are they at a

disadvantage relative to one that has been rich?


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Solow Growth Model

Robert Solow Macroeconomist

Professor at MIT

Nobel Prize (1987)

Important Contributions:

Developed a macroeconomic

model that allows for a

decomposition of GDP into

factors of production (capital,labor, productivity)

Showed that capital

accumulation is relatively

unimportant for growth Contrasts with the thinking of

the majority of economists

before (Smith, Marx, etc.)


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Macroeconomic Models Models: A theoretical construct designed to represent a

complex system. Economists use these models to predict the effects of

policy, such as:

If taxes are raised, what will happen to

unemployment?

Who will gain and lose from a free tradeagreement?

What happens to unemployment if the FederalReserve increases the supply of money in theeconomy?

What policies can increase growth in developing

countries?


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Key Features of a Macroeconomic Model

Consumers: Represent households who supplylabor, make investments and consume

Firms: Represent all businesses who use factors ofproduction (labor, capital, land, etc.) to produce

output

Equilibrium: The outcome of the model.

A prediction about how firms and consumers interactthrough markets


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Solow Growth Model Observation: Richer countries have more capital

(more machines, factories, etc.)

Is this the cause or the result of their greater income?

Two possibilities considered:

Countries have more capital because they save agreater part of their income

Countries have more capital because the return oninvesting in capital is higher

The whole model is beyond the scope of this class, sowe will consider a greatly simplified version


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Simplified Solow Growth ModelConsumers:

Consume a constant fraction of GDP and ownall the capital in the economy

Not modeling:

Unemployment (everyone always works)

Lifecycle (no children, students or retirees)

Within-country income inequality

Consumers described by one equation:

I = s Y

where s , a number between 0 and 1, is thefraction of output that gets invested.


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Simplified Solow Growth ModelFirms:

Use the capital to produce output

Not modeling:

Labor markets (searching for workers) Finance (borrowing to take on projects)

Executive compensation

Firms described by one equation:

Y = A K0.3

where Y is GDP, A is productivity

and K is the capital stock


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Simplified Solow Growth Model

Equilibrium: All output is used either in investment orconsumption (no trade, no government):

Y = C + I

How the stock of capital changes over time:

K’ = I + (1- δ)K where K’ is the capital stock next year,

K is the capital stock this year,I is investment this year, and

δ is the depreciation rate


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Simplified Solow Growth Model

So the entire model is described by fourequations:

Households: I = s Y

Firms: Y = A K0.3

Capital Accumulation: K’ = I + (1- δ)K GDP: Y = C + I

Rearranging terms:

I = s Y = s A K0.3 K’ = I + (1- δ)K = s A K0.3 + (1- δ)K


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How does the capital stockchange over time?

K’

K

K’= K

How are capital this year, and

capital next year related?


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

The equation above tells you

how much capital there will benext year


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Suppose the economy starts

with some low capital level K0

K0


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Then the equation says that

next year’s capital stock will

be K1

K0

K1


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Using the red 45 degree line

as a reference, we can find

K1 on the horizontal axis.

K0

K1

K1


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Then we can find K2

K0

K1

K1

K2


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Repeating these steps, we

can find the capital stock in

any future year

K0

K1

K1

K2


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K



any future year

K0

K1

K1

K2

K2


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K



any future year

K0

K1

K1

K2

K2

K3


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K



any future year

K0

K1

K1

K2

K2

K3

K3


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K



any future year

K0

K1

K1

K2

K2

K3

K3

K4


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K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Notice that the capital stock

is approaching the point

where the two lines meet

K0

K1

K1

K2

K2

K3

….

K10

K10

….


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Some Things to Notice

The further the economy starts below the steady

state level of capital, the faster the economyinitially grows

Mankiw refers to this as the “catch-up” effect

This is due to the effect of “diminishing returns”

The amount of extra output from eachadditional unit of capital goes down as thecapital stock gets larger

Growth slows over time until the capital stockreaches the steady state level


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Savings and Productivity

What happens if the savings rate of the countrychanges? Increase s from its initial level to a higher level


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Increase in the Savings Rate

K’

K

K’= K

K’ = s A K0.3 + (1- δ)K

Suppose the economy is in

a steady state with savings

rate s.

K*

K*


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K’

K

K’= K

K’ = s’ A K0.3 + (1- δ)K

Then the savings rate

increases to s’ .

K*

K*


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K’

K

K’= K

K’ = s’ A K0.3 + (1- δ)K

Now capital accumulates

according to the new

equation with the higher

savings rate

K1

K*


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K’

K

K’= K

K’ = s’ A K0.3 + (1- δ)K

And we proceed

exactly like before.

K1

K*


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K’

K

K’= K

K’ = s’ A K0.3 + (1- δ)K

Eventually a new,

higher steady state

capital stock is

reached.

K0

K0 K*

K*


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What happens if instead productivity is

increased?

Same thing.

Income goes up, so consumers have more to

invest, which increases the capital stock.

How are they different?

Higher savings: Decreases consumptiontoday, increases it (maybe) in the future

Higher productivity: Increases consumption both today and in the future


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0

20000

40000

60000

80000

100000

120000

-20 -10 0 10 20 30 40 50 60 70

G D P

p e r C a

p i t a

Savings Rate


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Back to what Solow found:

Savings rates (even historical) have littlerelationship to relative wealth

Apparently the wealth of countries that are

now rich is not because of long term savingsand investment per se

That is, clearly the fact that rich countries are

rich is partly because they have morecapital. BUT they have more capital because they have high productivity.


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Savings and Productivity This is an extremely important finding.

Suggests that a long history of capitalaccumulation is not necessary to be wealthy

If a country is able to increase its productivity,

capital will “catch up” quite quickly This shifted the emphasis in the study of

promoting development in low income countriesaway from trying to send them capital, and toward

trying to make their economies more efficient

How do you do that?

Perhaps the most important open question in

economics.


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More Direct: Growth Accounting Now take the basic Solow model and extend it to

include labor and human capital Decompose changes in GDP per capita using this

production function:

Y = A K 1/3

H 2/3

Here, H is human capital: h x L

h is average human capital

L is the number of workers

N is total population

= 1.5

0.5

ℎ


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Sources of Growth?

Growth through savings:

Increases in GDP driven by higher K/Y

Growth through productivity:

Increases in GDP driven by higher A

K/Y is roughly constant (as in Solow)

Other possibilities:

Labor: L/N Human capital: h


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USA Growth Accounting

50

100

200

400

1950 1960 1970 1980 1990 2000 2010

Y/N

K/Y^(1/2)

h

L/N

A^(3/2)


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China Growth Accounting

50

100

200

400

800

1600

3200

1950 1960 1970 1980 1990 2000 2010

Y/N

K/Y^(1/2)

h

L/N

A^(3/2)


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South Korea Growth Accounting

50

100

200

400

800

1600

3200

1960 1970 1980 1990 2000 2010

Y/N

K/Y^(1/2)

h

L/N

A^(3/2)


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Argentina Growth Accounting

50

100

200

400

1950 1960 1970 1980 1990 2000 2010

Y/N

K/Y^(1/2)

h

L/N

A^(3/2)


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Zimbabwe Growth Accounting

25

50

100

200

400

1960 1970 1980 1990 2000 2010

Y/N

K/Y^(1/2)

h

L/N

A^(3/2)


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Findings

In growing countries, growth is not driven by K/Y ,

it’s driven by A

In non-growing countries, big fluctuations in all ofthe factors

Why does A go up in growing countries?

Look to histories

Improvements in technology

Improvements in institutions

solow model best explain

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