The Solow Growth Model
MACROECONOMICS
Charles I. Jones
© 2008 by W. W. Norton & Company. All rights reserved
5.1 Introduction
 In this chapter, we learn:
 how capital accumulates over time, helping us understand
economic growth.
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 the role of the diminishing marginal product of capital in
explaining differences in growth rates across countries.
 the principle of transition dynamics: the farther below its
steady state a country is, the faster the country will grow.
 the limitations of capital accumulation, and how it leaves a
significant part of economic growth unexplained.
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The Solow growth model is the starting point to
determine why growth differs across similar
countries
 it builds on the production model by adding a
theory of capital accumulation
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 developed in the mid-1950s by Robert Solow of
MIT
 the basis for the Nobel Prize he received in 1987
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The Solow growth model
 capital stock is no longer exogenous
 capital stock is “endogenized”: converted from an
exogenous to an endogenous variable.
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 the accumulation of capital as a possible engine of
long-run economic growth
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5.2 Setting Up the Model
Start with the production model from chapter 4 and add an
equation describing the accumulation of capital over time.
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Production
 The production function:
 is Cobb-Douglas
 has constant returns to scale in capital and labor
 has an exponent of one-third on capital
 Variables are time subscripted as they may potentially
change over time
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 Output can be used for either consumption (Ct) or
investment (It)
 A resource constraint describes how an economy can
use its resources
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Capital Accumulation
 capital accumulation equation: the capital stock next
year equals the sum of the capital started with this
year plus the amount of investment undertaken this
year minus depreciation
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 Depreciation is the amount of capital that wears
out each period
 the depreciation rate is viewed as approximately
10 percent
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Thus the change in the capital stock is investment
less depreciation
represents the change in the capital stock between
today, period t, and next year, period t+1
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Labor
the amount of labor in the economy is given
exogenously at a constant level
Investment
the amount of investment in the economy is
equal to a constant investment rate times total
output
remember that total output is used for either
consumption or investment
therefore, consumption equals output times the
quantity one minus the investment rate
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The Model Summarized
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5.3 Prices and the Real Interest Rate
 If we added equations for the wage and rental price,
the MPL and the MPK would pin them down,
respectively -- omitting them changes nothing.
the real interest rate is the amount a person can earn
by saving one unit of output for a year
or equivalently, the amount a person must pay to
borrow one unit of output for a year
 measured in constant dollars, not in nominal dollars
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 saving is the difference between income and
consumption
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 Saving equals investment:
 a unit of saving is a unit of investment, which
becomes a unit of capital: therefore the return on
saving must equal the rental price of capital
 the real interest rate in an economy is equal to the
rental price of capital, which is equal to the
marginal product of capital
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5.4 Solving the Solow Model
 To solve the model, write the endogenous variables as
functions of the parameters of the model and graphically
show what the solution looks like and solve the model in
the long run.
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 combine the investment allocation equation with the
capital accumulation equation
(change in capital)
(net investment)
 net investment is investment minus depreciation
 substitute the supply of labor into the production function:
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 We now have reduced our system of five
equations and five unknowns to two equations and
two unknowns:
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 The key equations of the Solow Model are these:
The production function
And the capital accumulation equation
 How do we solve this model?
We graph it, separating the two parts of the capital
accumulation equation into two graph elements:
saving = investment and depreciation
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The Solow Diagram graphs these two
pieces together, with Kt on the x-axis:
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Investment,
Depreciation
At this point,
dKt = sYt, so
Capital, Kt
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Figure 5.1: The Solow Diagram
Investment, depreciation
Depreciation: d K
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Investment: s Y
Net investment
K0
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K*
Capital, K
15
Using the Solow Diagram
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 the amount of investment is greater than the amount
of depreciation, the capital stock will increase
 the capital stock will rise until investment equals
depreciation: this point, the change in capital is equal
to 0, and absent any shocks, the capital stock will
stay at this value of capital forever
 the point where investment equals depreciation is
called the steady state
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Suppose the economy starts at this K0:
•We see that the red line is above
Investment,
Depreciation
the green at K0:
•Saving = investment is greater
than depreciation
•So ∆Kt > 0 because
•Then since ∆Kt > 0,
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Kt increases from K0 to K1 > K0
K0
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K1
Capital, Kt
17
Now imagine if we start at a K0 here:
Investment,
Depreciation
•At K0, the green line is above the
red line
•Saving = investment is now less
than depreciation
•So ∆Kt < 0 because
•Then since ∆Kt < 0,
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Kt decreases from K0 to K1 < K0
Capital, Kt
K1 K0
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We call this the process of transition dynamics:
Transitioning from any Kt toward the economy’s
steady-state K*, where ∆Kt = 0
Investment,
Depreciation
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No matter where
we start, we’ll
transition to K*!
At this value of K,
dKt = sYt, so
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K*
Capital, Kt
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 when not in steady state, the economy obeys
transition dynamics or in other words, the
movement of capital toward a steady state
 notice that when depreciation is greater than
investment, the economy converges to the same
steady state as above
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 at the rest point of the economy, all endogenous
variables are steady
 transition dynamics take the economy from its
initial level of capital to the steady state
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Output and Consumption in the
Solow Diagram
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 using the production function, it is evident that as K
moves to its steady state by transition dynamics,
output will also move to its corresponding steady
state by transition dynamics
 note that consumption is the difference between
output and investment
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We can see what happens to output, Y, and
thus to growth if we rescale the vertical axis:
• Saving = investment and
Investment,
Depreciation, Income
depreciation now appear
here
• Now output can be
Y*
graphed in the space
above in the graph
• We still have transition
dynamics toward K*
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• So we also have
dynamics toward a
steady-state level of
income, Y*
K*
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Capital, Kt
22
Figure 5.2: The Solow Diagram with
Output
Investment, depreciation,
and output
Output: Y
Y*
Consumption
Depreciation: d K
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Y0
Investment: s Y
K0
CHAPTER 5 The Solow Growth Model
K*
Capital, K
23
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Solving Mathematically for the Steady
State
 in the steady state, investment equals depreciation. If
we evaluate this equation at the steady-state level of
capital, we can solve mathematically for it
the steady-state level of capital is positively related
with the investment rate, the size of the workforce
and the productivity of the economy
the steady-state level of capital is negatively
correlated with the depreciation rate
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What determines the steady state?
 We can solve mathematically for K* and Y* in the
steady state, and doing so will help us understand the
model better
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 In the steady state:
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If we know K*, then we can find Y* using the
production function:
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 This equation also tells us about income per
capita, y, in the steady state:
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 notice that the exponent on the productivity
parameter is greater than in the chapter 4 model:
this results because a higher productivity
parameter raises output as in the production
model.
however, higher productivity also implies the
economy accumulates additional capital.
the level of the capital stock itself depends on
productivity
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5.5 Looking at Data through the Lens of
the Solow Model
The Capital-Output Ratio
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 the capital to output ratio is given by the ratio of the
investment rate to the depreciation rate:
 while investment rates vary across countries, it is assumed
that the depreciation rate is relatively constant
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Empirically, countries with higher investment
rates have higher capital to output ratios:
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Differences in Y/L
 the Solow model gives more weight to TFP in explaining
per capita output than the production model does
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 Just like we did before with the simple model of
production, we can use this formula to understand why
some countries are so much richer
 take the ratio of y* for a rich country to y* for a poor
country, and assume the depreciation rate is the same
across countries:
45 =
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18
x
2.5
31
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45 =
18
x
2.5
 Now we find that the factor of 45 that separates
rich and poor country’s income per capita is
decomposable into:
A 103/2 = 18-fold difference in this productivity
ratio term
A (30/5)1/2 = 61/2 = 2.5-fold difference in this
investment rate ratio
 In the Solow Model, productivity accounts for
18/20 = 90% of differences!
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5.6 Understanding the Steady State
 the economy will settle in a steady state because the
investment curve
has diminishing returns
 however, the rate at which production and
investment rise is smaller as the capital stock is larger
 a constant fraction of the capital stock depreciates
every period, which implies depreciation is not
diminishing as capital increases
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 In summary, as capital increases, diminishing returns
implies that production and investment increase by
less and less, but depreciation increases by the same
amount .
 Eventually, net investment is zero and the economy
rests in steady state.
There are diminishing returns to capital: less Yt per
additional Kt
That means new investment is also diminishing: less
sYt = It
But depreciation is NOT diminishing; it’s a constant
share of Kt
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5.7 Economic Growth in the Solow
Model
 there is no long-run economic growth in the Solow
model
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 in the steady state: output, capital, output per person,
and consumption per person are all constant and
growth stops
both constant
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 empirically, economies appear to continue to
grow over time
thus capital accumulation is not the engine of
long-run economic growth
 saving and investment are beneficial in the shortrun, but diminishing returns to capital do not
sustain long-run growth
 in other words, after we reach the steady state,
there is no long-run growth in Yt (unless Lt or A
increases)
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5.8 Some Economic Experiments
 while the Solow model does not explain long-run economic
growth, it does help to explain some differences across
countries
 economists can experiment with the model by changing
parameter values
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An Increase in the Investment Rate
 the investment rate increases permanently for exogenous
reasons
 the investment curve rotates upward, but the deprecation
line remains unchanged
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Figure 5.4: An Increase in the
Investment Rate
Investment, depreciation
Depreciation: d K
New investment
exceeds depreciation
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Old investment: s Y
K*
CHAPTER 5 The Solow Growth Model
K**
Capital, K
38
 the economy is now below its new steady state and
the capital stock and output will increase over time by
transition dynamics
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 the long run, steady-state capital and steady-state
output are higher
 What happens to output in response to this increase in
the investment rate?
the rise in investment leads capital to accumulate over
time
this higher capital causes output to rise as well
output increases from its initial steady-state level Y*
to the new steady state Y**
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Figure 5.5: The Behavior of Output
Following an Increase in s
Investment, depreciation, and output
Depreciation: d K
Output: Y
Y**
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Y*
New
investment:
s ‘Y
Old
investment:
s Y
K*
K**
(a) The Solow diagram with output.
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Capital, K
40
Figure 5.5: The Behavior of Output
Following an Increase in s (cont.)
Output, Y
(ratio scale)
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Y**
Y*
2000
2020
2040
2060
2080
2100
Time, t
(b) Output over time.
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A Rise in the Depreciation Rate
 the depreciation rate is exogenously shocked to a higher
rate
 the depreciation curve rotates upward and the investment
curve remains unchanged
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 the new steady state is located to the left: this means that
depreciation exceeds investment
 the capital stock declines by transition dynamics until it
reaches the new steady state
 note that output declines rapidly at first but less rapidly as it
converges to the new steady state
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Figure 5.6: A Rise in the Depreciation
Rate
Investment, depreciation
New
depreciation:
d ‘K
Old
depreciation:
dK
Depreciation
exceeds
investment
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Investment: s Y
K**
CHAPTER 5 The Solow Growth Model
K*
Capital, K
43
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 What happens to output in response to this increase
in the depreciation rate?
the decline in capital reduces output
output declines rapidly at first, and then gradually
settles down at its new, lower steady-state level Y**
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Figure 5.7: The Behavior of Output
Following an Increase in d
Investment, depreciation,
and output
New depreciation: d‘K
Output: Y
Y*
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Y**
Investment:
s Y
Old depreciation: d K
K**
K*
Capital, K
(a) The Solow diagram with output.
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Figure 5.7: The Behavior of Output
Following an Increase in d (cont.)
Output, Y
(ratio scale)
Y*
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Y**
2000
2020
2040
2060
(b) Output over time.
CHAPTER 5 The Solow Growth Model
2080
2100
Time, t
46
Experiments on Your Own
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 Try experimenting with all the parameters in the
model:
1. Figure out which curve (if either) shifts.
2. Follow the transition dynamics of the Solow model.
3. Analyze the steady-state values of capital, output,
and output per person.
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5.9 The Principle of Transition
Dynamics
 when the depreciation rate and the investment rate
were shocked, output was plotted over time on a ratio
scale
ratio scale allows us to see that output changes more
rapidly the further we are from the steady state
as the steady state is approached, growth shrinks to
zero
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 the principle of transition dynamics says that the
farther below its steady state an economy is, in
percentage terms, the faster the economy will grow
 similarly, the farther above its steady state, in
percentage terms, the slower the economy will grow
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 this principle allows us to understand why economies
may grow at different rates at the same time
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Understanding Differences in Growth
Rates
 empirically, OECD countries that were relatively poor in
1960 grew quickly while countries that were relatively rich
grew slower
 if the OECD countries have the same steady state, then the
principle of transition dynamics predicts this
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 looking at the world as whole, on average, rich and poor
countries grow at the same rate
 two implications: (1) most countries have already reached
their steady states; and (2) countries are poor not because
of a bad shock, but because they have parameters that yield
a lower steady state
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CHAPTER 5 The Solow Growth Model
51
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CHAPTER 5 The Solow Growth Model
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5.10 Strengths and Weaknesses of
the Solow Model
 The strengths of the Solow model are:
1. It provides a theory that determines how rich a country is
in the long run.
2. The principle of transition dynamics allows for an
understanding of differences in growth rates across
countries.
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 The weaknesses of the Solow model are:
1. It focuses on investment and capital, while the much more
important factor of TFP is still unexplained.
2. It does not explain why different countries have different
investment and productivity rates.
3. The model does not provide a theory of sustained long-run
economic growth.
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Summary
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1. The starting point for the Solow model is the
production model of Chapter 4. To that framework,
the Solow model adds a theory of capital
accumulation. That is, it makes the capital stock an
endogenous variable.
2. The capital stock is the sum of past investments.
The capital stock today consists of machines and
buildings that were bought over the last several
decades.
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3. The goal of the Solow model is to deepen our
understanding of economic growth, but in this it’s
only partially successful. The fact that capital runs
into diminishing returns means that the model does
not lead to sustained economic growth. As the
economy accumulates more capital, depreciation
rises one-for-one, but output and therefore
investment rise less than one-for- one because of the
diminishing marginal product of capital. Eventually,
the new investment is only just sufficient to offset
depreciation, and the capital stock ceases to grow.
Output stops growing as well, and the economy
settles down to a steady state.
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4. There are two major accomplishments of the Solow
model. First, it provides a successful theory of the
determination of capital, by predicting that the
capital-output ratio is equal to the investmentdepreciation ratio. Countries with high investment
rates should thus have high capital-output ratios,
and this prediction holds up well in the data.
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5. The second major accomplishment of the Solow
model is the principle of transition dynamics, which
states that the farther below its steady state an
economy is, the faster it will grow. While the model
cannot explain long-run growth, the principle of
transition dynamics provides a nice theory of
differences in growth rates across countries.
Increases in the investment rate or total factor
productivity can increase a country’s steady-state
position and therefore increase growth, at least for a
number of years. These changes can be analyzed
with the help of the Solow diagram.
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6. In general, most poor countries have low TFP levels
and low investment rates, the two key determinants
of steady-state incomes. If a country maintained
good fundamentals but was poor because it had
received a bad shock, we would see it grow rapidly,
according to the principle of transition dynamics.
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Figure 5.10: Investment in South Korea
and the Philippines, 1950-2000
Investment rate (percent)
South Korea
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U.S.
Philippines
Year
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Figure 5.11: The Solow Diagram
Investment, depreciation,
and output
Output: Y
Y*
Depreciation: d K
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Y0
Investment: s Y
K0
CHAPTER 5 The Solow Growth Model
K*
Capital, K
60
Figure 5.12: Output Over Time, 2000-2100
Output, Y
(ratio scale)
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Y*
Y0
2000
2020
2040
2060
2080
2100
Time, t
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