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Chapter 7
Appendix:
The Solow Growth
Model
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7A The Solow Growth Model
Appendix Outline
20A.1
20A.2
20A.3
20A.4
20A.5
20A.6
The Three Building Blocks of the Solow Model
Steady-State Equilibrium in the Solow Model
Determinants of GDP
Dynamic Equilibrium in the Solow Model
Sources of Growth in the Solow Model
Calculating Average (Compound) Growth Rates
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7A The Solow Growth Model
Key Ideas
1.
2.
3.
There are three building blocks of the Solow
growth model.
The Solow growth model can be solved for a
steady-state equilibrium.
In the Solow growth model, increases in the
saving rate, human capital, and technology
increase the level of real GDP.
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7A The Solow Growth Model
Key Ideas
4.
5.
6.
In the Solow growth model, the steady-state
equilibrium is dynamic.
In the Solow growth model, sustained
economic growth can be achieved only with
increases in technology.
The compound growth formula is used to
calculate average annual growth rates.
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7A.1 The Three Building Blocks of the Solow Model
1. The aggregate production function—the first
block of the Solow model—determines the
level of real GDP:
Y = A× F ( K,H )
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7A.1 The Three Building Blocks of the Solow Model
2. An equation for physical capital
accumulation:
K now = K last year  K depreciated  I
where K is the stock of capital, and I is the
flow of new investment.
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7A.1 The Three Building Blocks of the Solow Model
Assuming a constant depreciation rate, the
physical capital accumulation becomes:
K now = K last year  (Depreciation rate × K last year  I )
K now = (1  d ) × K last year  I
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7A.1 The Three Building Blocks of the Solow Model
3. Saving by households:
I = saving = s  Y
I = s  Y = s  A × F (K , H )
where S is the constant saving rate.
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7A.1 The Three Building Blocks of the Solow Model
Total output Y is divided between C and I:
Exhibit 7A.1 Aggregate Income and Aggregate Saving
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7A.2 Steady-State Equilibrium in the Solow Model
Steady-state equilibrium
An economic equilibrium in which the
physical capital stock remains constant over
time:
K now = K last year = K
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7A.2 Steady-State Equilibrium in the Solow Model
A steady-state equilibrium occurs when new
investment is equal to depreciation:
I = depreciation
sxY=dxK
s x A x F (K,L) = d × K
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7A.2 Steady-State Equilibrium in the Solow Model
Exhibit 7A.2 Steady-State Equilibrium in the Solow Model
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7A.3 Determinants of GDP
An increase in either the saving rate, s, or the
stock of human capital, H, will increase the
steady-state level of GDP.
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7A.3 Determinants of GDP
Exhibit 7A.3 The Impact of the Saving Rate on the Steady-State
Equilibrium
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7A.3 Determinants of GDP
Exhibit 7A.4 Change in the Steady-State Equilibrium Resulting from an
Increase in the Human Capital of Workers
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7A.4 Dynamic Equilibrium in the Solow Model
A dynamic equilibrium traces the behavior of
the economy over time.
Suppose the economy begins with a physical
capital stock of K0 < K*.
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7A.4 Dynamic Equilibrium in the Solow Model
Exhibit 7A.5 Dynamic Equilibrium in the Solow Model
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7A.5 Sources of Growth in the Solow Model
Increases in the saving rate, s, is not a source of
sustained growth in real GDP.
Why? Increases in saving shift the investment
curve up and thus provide an increase in the level
of GDP.
Remember that we are looking for sources of
sustained growth.
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7A.5 Sources of Growth in the Solow Model
Exhibit 7A.6 Three Economies with Different Saving Rates in the Solow
Model
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7A.5 Sources of Growth in the Solow Model
Technological progress is a source of sustained
growth in real GDP.
Why? An increase in technology, A, raises
productivity, thus allowing physical and human
capital to produce more output.
As a result, technology progress (or constant
growth in technology) will lead to sustained
increases or growth in real GDP.
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7A.5 Sources of Growth in the Solow Model
Exhibit 7A.7 Sustained Growth Driven by Technological Change
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7A.5 Sources of Growth in the Solow Model
Another prediction of the Solow growth model is
that the ratio of the physical capital stock to GDP
should be constant through time.
At steady-state, investment = depreciation:
s Y = d  K
K
s
=
Y
d
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7A.5 Sources of Growth in the Solow Model
Exhibit 7A.8 The Ratio of Physical Capital Stock to GDP in the
United States
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7A.5 Sources of Growth in the Solow Model
Catch-up growth is driven by the accumulation of
physical and human capital.
Catch-up growth leads to increases in the level of
real GDP.
Although it can dramatically raise the level of
GDP, catch-up growth is not a source of sustained
growth in real GDP.
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7A.6 Calculating Average (Compound) Growth Rates
Compound growth is the phenomenon whereby
growth builds on growth.
Alternatively, compound growth is the earning of
interest on interest.
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7A.6 Calculating Average (Compound) Growth Rates
Example: For a money market account that earns
a 2% return (1 + 0.02), what is the return after one
year?
return2015 = principle2014 × (1 + 0.02)
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7A.6 Calculating Average (Compound) Growth Rates
Example: For a money market account that earns
a 2% return (1 + 0.02), what is the return after two
years?
return2015 = principle2014 × (1 + 0.02) × (1 + 0.02)
= principle2014 × (1 + 0.02)2
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7A.6 Calculating Average (Compound) Growth Rates
Example: For a money market account that earns
a 2% return (1 + 0.02), what is the return after 50
years?
return2064 = principle2014 × (1 + 0.02)50
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7A.6 Calculating Average (Compound) Growth Rates
Compound growth formula:
returnt+n = principlet × (1 + g)n
where t = starting year
g = growth rate
n = number of years
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7A.6 Calculating Average (Compound) Growth Rates
We can rewrite the compound growth equation
for the average annual growth rate, g:
returnt  n
n
(1  g ) =
principlet
1/ n
 returnt  n 
(1  g ) = 

 principlet 
1/ n
 returnt  n 
g= 

 principlet 
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1
7A.6 Calculating Average (Compound) Growth Rates
Example: U.S. GDP growth from 1960 to 2010:
GDP2010 = GDP2010 × (1 + g)50
41,365 = 15,398 × (1 + g)50
41,365
= (1  g )50  1
15,398
g = 2.68641/50 – 1 = 0.020 (1 + g)50
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