lecture4-solow growth model

8/13/2019 Lecture4-Solow Growth Model

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Simplified Solow Growth Model

Consumers: Consume a constant fraction of GDP and own

all 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 the

fraction of output that gets invested.


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Simplified Solow Growth Model

Firms: 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 Kwhere Yis GDP,A is productivity

and Kis the capital stock


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Simplified Solow Growth ModelEquilibrium:

All output is used either in investment or

consumption (no trade, no government):

Y = C + I

How the stock of capital changes over time:

K = I + (1- )Kwhere Kis the capital stock next year,

Kis the capital stock this year,

Iis investment this year, and

is the depreciation rate


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Simplified Solow Growth ModelSo the entire model is described by four equations:

Households: I = s Y

Firms: Y = A K0.3

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

Rearranging terms:

I = s Y= s A K0.3

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

Then the equation says that

next years capital stock will

be K1

K0

K1


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K

K

K= K

K = s A K0.3+ (1- )K

Using the red 45 degree line

as a reference, we can find

K1on the horizontal axis.

K0

K1

K1


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K

K

K= K

K = s A K0.3+ (1- )K

Then we can find K2

K0

K1

K1

K2


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K

K

K= K

K = s A K0.3+ (1- )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- )K



any future year

K0

K1

K1

K2

K2


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K

K

K= K

K = s A K0.3+ (1- )K



any future year

K0

K1

K1

K2

K2

K3


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K

K

K= K

K = s A K0.3+ (1- )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- )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- )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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K

K

K= K

K = s A K0.3+ (1- )K

The point where the two

lines meet is the steady

state level of capital. Once

the economy is at this level,

the capital level does not

change.

K*

K*


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Some Things to Notice

The further the economy starts below the steadystate level of capital, the faster the economy

initially 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 each

additional unit of capital goes down as the

capital stock gets larger Growth slows over time until the capital stock

reaches the steady state level


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Further Remarks

This model can be made more complicated (andrealistic) by makingAgrow over time

The mechanics of the model are the same, except

instead of reaching a steady state, the capital

stock grows at the same rate as productivity

This is called a balanced growth path


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Savings and Productivity

What happens if the savings rate of the countrychanges?

Increase sfrom 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- )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- )K

Then the savings rate

increases to s.

K*

K*


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K

K

K= K

K = s A K0.3+ (1- )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- )K

And we proceed

exactly like before.

K1

K*


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K

K

K= K

K = s A K0.3+ (1- )K

Eventually a new,

higher steady state

capital stock is

reached.

K0

K0 K*

K*


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If the savings rate is increased, the economytransitions to a higher steady state capital level

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 consumption

today, increases it in the future

Higher productivity: Increases consumption

both today and in the future


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Back to what Solow found: Savings rates (even historical) have little

relationship to relative wealth

Apparently the wealth of countries that are now

rich is notbecause of long term savings andinvestmentper se

That is, clearly the fact that rich countries are

rich is partly because they have more capital.

BUT they have more capital becausethey have

high productivity.


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This is an extremely important finding. Suggests that a long history of capital accumulation 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 countries awayfrom trying

to send them capital, and towardtrying to make their

economies more efficient

How do you do that?

Perhaps the most important open question in social

science.

lecture4-solow growth model

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