Download Chapter 5

Chapter 5
The Solow
Growth Model
By Charles I. Jones
• Additions / differences with the model:
– Capital stock is no longer exogenous.
– Capital stock is now “endogenized.”
– The accumulation of capital is a possible
engine of long-run economic growth.
Media Slides Created By
Dave Brown
Penn State University
5.1 Introduction
• In this chapter, we learn:
– How capital accumulates over time.
– How diminishing MPK explains differences
in growth rates across countries.
– The principle of transition dynamics.
– The limitations of capital accumulation, and
how it leaves a significant part of economic
growth unexplained.
5.2 Setting Up the Model
Production
• Start with the previous production model
– Add an equation describing the accumulation
of capital over time.
• The production function:
– Cobb-Douglas
– Constant returns to scale in capital and labor
– Exponent of one-third on K
• Variables are time subscripted (t).
• Output can be used for consumption or
investment.
• The Solow Growth Model:
– Builds on the production model by adding
a theory of capital accumulation
– Was developed in the mid-1950s by
Robert Solow of MIT
– Was the basis for the Nobel Prize he
received in 1987
Consumption
Investment
Output
• This is called a resource constraint.
– Assuming no imports or exports
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Capital Accumulation
• Goods invested for the future
determines the accumulation of capital.
• Capital accumulation equation:
Next year’s
capital
This year’s
capital
Investment
Depreciation
rate
Case Study: An Example of Capital
Accumulation
• To understand capital accumulation, we
must assume the economy begins with
a certain amount of capital, K0.
• Suppose:
– The initial amount of capital is 1,000
bushels of corn.
– The depreciation rate is 0.10.
• Depreciation rate
– The amount of capital that wears out each
period
– Mathematically must be between 0 and 1 in
this setting
– Often viewed as approximately 10 percent
• Change in capital stock defined as
• Thus:
• The change in the stock of capital is
investment subtracted by the capital
that depreciates in production.
Labor
• To keep things simple, labor demand
and supply not included
• The amount of labor in the economy is
given exogenously at a constant level.
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Investment
• Farmers eat a fraction of output and
invest the rest.
Fraction
Invested
• Therefore:
– Consumption is the share of output we
don’t invest.
Case Study: Some Questions about
the Solow Model
• Differences between Solow model and
production model in previous chapter:
– Dynamics of capital accumulation added
– Left out capital and labor markets, along
with their prices
• Why include the investment share but
not the consumption share?
– No need to—it would be redundant
– Preserve five equations and five
unknowns
• Stock
– A quantity that survives from period to
period.
• tractor, house, factory
• Flow
– A quantity that lasts a single period
• meals consumed, withdrawal from ATM
• A change in stock is a flow of
investment.
5.3 Prices and the Real Interest Rate
• If we added equations for the wage and
rental price, the following would occur:
– The MPL and the MPK would pin them.
– Omitting them changes nothing.
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• The real interest rate
- The amount a person can earn by saving
one unit of output for a year
- Or, the amount a person must pay to
borrow one unit of output for a year
- Measured in constant dollars, not in
nominal dollars
• Saving
– The difference between income and
consumption
5.4 Solving the Solow Model
• The model needs to be solved at every
point in time, which cannot be done
algebraically.
• Two ways to make progress
– Show a graphical solution
– Solve the model in the long run
• We can start by combining equations to
go as far as we can with algebra.
• Combine the investment allocation and
capital accumulation equation.
Depreciation
Investment
– Is equal to investment
A unit of investment becomes a unit
of capital
•
• Substitute the fixed amount of labor into
the production function.
- The return on saving must equal the
rental price of capital.
• Thus:
• We have reduced the system into two
equations and two unknowns (Yt, Kt).
- The real interest rate equals the
rental price of capital which equals
the MPK.
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• The Solow Diagram
– Plots the two terms that govern the change
in the capital stock
– New investment looks like the production
functions previously graphed but scaled
down by the investment rate.
• Notes about the dynamics of the model:
– When not in the steady state, the
economy exhibits a movement of
capital toward the steady state.
– 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.
Output and Consumption in the Solow
Diagram
• As K moves to its steady state by
transition dynamics, output will also
move to its steady state.
• Consumption can also be seen in the
diagram since it is the difference
between output and investment.
Using the Solow Diagram
• If the amount of investment is greater
than the amount of depreciation:
– The capital stock will increase until
investment equals depreciation.
• here, the change in capital is equal to 0
• the capital stock will stay at this value of capital
forever
• this is called the steady state
• If depreciation is greater than
investment, the economy converges to
the same steady state as above.
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Solving Mathematically for the
Steady State
• In the steady state, investment equals
depreciation.
• Sub into the production function
• Finally, divide both sides of the last
equation by labor to get output per
person (y) in the steady state.
• Note the exponent on productivity is
different here (3/2) than in the
production model (1).
– Higher productivity has additional effects
in the Solow model by leading the
economy to accumulate more capital.
• Solve for K*
5.5 Looking at Data through the Lens of
the Solow Model
The Capital-Output Ratio
• The steady-state level of capital is
– Positively related with the
• investment rate
• the size of the workforce
• the productivity of the economy
– Negatively correlated with
• the depreciation rate
• Plug K* into the production function to
get Y*.
• Plug in our solved value of K*.
• Recall the steady state.
• The capital to output ratio is the ratio of the
investment rate to the depreciation rate:
• Investment rates vary across countries.
• It is assumed that the depreciation rate is
relatively constant.
Differences in Y/L
• The Solow model gives more weight to TFP in
explaining per capita output than the
production model.
• We can use this formula to understand why
some countries are so much richer.
• Take the ratio of y* for two countries and
assume the depreciation rate is the same:
• Higher steady-state production
– Caused by higher productivity and
investment rate
• Lower steady-state production
– Caused by faster depreciation
From Chapter 4
See figure 5.3 (next slide)
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5.7 Economic Growth in the
Solow Model
• Important result: there is no long-run
economic growth in the Solow model.
• In the steady state, growth stops, and
all of the following are constant:
– Output
– Capital
– Output per person
– Consumption per person
• Empirically, however, economies
appear to continue to grow over time.
– Thus, we see a drawback of the model.
• According to the model:
• We find that the factor of 66 that
separates rich and poor countries’
income per capita is decomposable:
– TFP differences
– Investment differences
5.6 Understanding the Steady State
• The economy reachs a steady state because
investment has diminishing returns.
– The rate at which production and investment
rise is smaller as the capital stock is larger.
• Also, a constant fraction of the capital stock
depreciates every period.
– Depreciation is not diminishing as capital
increases.
• Eventually, net investment is zero.
– The economy rests in steady state.
– Capital accumulation is not the engine of
long-run economic growth.
– After we reach the steady state, there is no
long-run growth in output.
– Saving and investment
• are beneficial in the short-run
• do not sustain long-run growth due to
diminishing returns
Meanwhile, Back on the Family Farm
• Harvest starts with a small stock of
seed.
– Grows larger each year, for a time
– Settles down to a constant level
• Diminishing returns
– A fixed number of farmers cannot harvest
huge amounts of corn.
– Growth eventually stops.
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Case Study: Population Growth in the
Solow Model
• Can growth in the labor force lead to
overall economic growth?
– It can in the aggregate.
– It can’t in output per person.
• The presence of diminishing returns
leads capital per person and output per
person to approach the steady state.
– This occurs even with more workers.
5.8 Some Economic
Experiments
• The Solow model:
– Does not explain long-run economic
growth
– Does help to explain some differences
across countries
Case 1. An Increase in Saving/Investment Rate
• Suppose the investment rate increases
permanently for exogenous reasons.
– The investment curve
• rotates upward
– The depreciation curve
• remains unchanged
– The capital stock
• increases by transition dynamics to
reach the new steady state
• this happens because investment
exceeds depreciation
– The new steady state
• is located to the right
• investment exceeds depreciation
An Increase in the Investment Rate
– The capital stock
• increases by transition dynamics to
reach the new steady state
• this happens because investment
exceeds depreciation
– The new steady state
• is located to the right
• investment exceeds depreciation
• Economists can experiment with the
model by changing parameter values.
Explain what happens to the economy according
to the Solow model when some fundamentals
change (Increase/Decrease):
• Case 1. Saving/Investment rate
• Case 2. Depreciation rate
• Case 3. Capital
• Case 4. Productivity
Present the results using the following:
1. Solow diagram;
2. Dynamics over time graph;
3. In words.
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• 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 steadystate level Y* to the new steady state Y**.
Case 2. A Rise in the Depreciation Rate
• Suppose the depreciation rate is
exogenously shocked to a higher rate.
– The depreciation curve
• rotates upward
– The investment curve
• remains unchanged
– The capital stock
• declines by transition dynamics until it
reaches the new steady state
• this happens because depreciation
exceeds investment
– The new steady state
• is located to the left
What can we say about the rate of growth during
the transition to the new steady state, K** and Y**?
•After s has
Investment,
Depreciation,
Income
increased to s’, we
are still at K*
temporarily
Y**
•∆K = sY – dK is the
vertical distance
between red and
green and is now
very large
Y*
• 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**.
•So growth is rapid!
•As K increases,
etc.
K*
K**
that distance
shrinks, growth
slackens
Capital, Kt
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The graph of Yt against time shows a discontinuous
jump in Yt brought on by the destruction of K t
Yt
In the old steady
state before the
calamity, Yt is
constant at Y*
Growth slackens as
we return to the new
steady state at Y*
Y*
At time t0, the destruction
reduces Kt, and Yt falls
immediately to Y’ < Y*
Y’
Growth, shown by the slope,
accelerates during recovery
t0
Case 3. Destruction of capital
•
•
Why would we be interested in this application? What are
examples of destruction of capital?
Important (but tragic) world events involve capital destruction:
Time, t
Case 4. Increase in Productivity
•
•
Try to solve on your own.
Will be reviewed in class.
– Warfare destroys productive capital — World War II resulted in the virtual
obliteration of German and Japanese industrial capacity. And 9/11
resulted in the destruction of a lot of physical capital. (In both cases, the
value of human lives lost surely dwarfed the value of lost capital.)
– Natural disasters do too — One recent example in this country is
Hurricane Katrina in 2005, which destroyed capital and killed people in
New Orleans and along the Gulf Coast
•
How do we think about the destruction of capital in the Solow
Model? What does it imply about initial effects and recovery?
Destruction of capital per se does not change
any fundamentals like s or d; it just removes K
•Prior to the
Investment,
Depreciation,
Income
destruction, we are
in steady state at K*
and Y*
Y*
•The destruction
moves us from K*
to K’ < K* as capital
is destroyed
Y’
Destruction
•But since the
fundamentals
haven’t changed,
the steady state is
still at K* and Y*,
and we transition
back toward it
K’
K*
5.9 The Principle of Transition
Dynamics
• 1. Direction of movement
– the economy will always move toward
the steady state.
• 2. Speed of movement
– the farther from the study state, the
faster the growth
Capital, Kt
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The Principle of Transition
Dynamics
• If an economy is below steady state
– It will grow.
The Solow Diagram graphs these two
pieces together, with Kt on the x-axis
Investment,
Depreciation
• If an economy is above steady state.
– Its growth rate will be negative.
So what?
• When graphing this, a ratio scale is used.
At this point, dKt
= sYt, so
– Allows us to see that output changes more
rapidly if we are further from the steady state
– As the steady state is approached, growth
shrinks to zero.
• The principle of transition dynamics
– The farther below its steady state an
economy is, (in percentage terms)
• the faster the economy will grow
– The farther above its steady state
• the slower the economy will grow
– Allows us to understand why economies
grow at different rates
Capital, Kt
Suppose the economy starts at this K0:
•We see that the
red line is above
the green there:
Investment,
Depreciation
•Saving =
investment is
greater than
depreciation
•So ∆Kt > 0
because
•Then since ∆Kt >
0, Kt increases
from K0 to K1 > K0
K0
What we’ve learned so far
Capital, Kt
K1
Now imagine if we start at a K0 here
Investment,
Depreciation
•There, the green is
above the red
•Saving =
investment is now
less than
depreciation
• The key equations of the Solow Model are these:
– The production function
– And the capital accumulation equation
•So ∆Kt < 0 because
• 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
•Then since ∆Kt < 0,
Kt decreases from K0
to K1 < K0
K1 K0
Capital, Kt
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Transition dynamics: Transitioning from any Kt
toward the economy’s steady-state K*, ∆Kt = 0
Investment,
Depreciation
No matter where
we start, we’ll
transition to K*!
At this value of K,
dKt = sYt, so
K*
Capital, Kt
We can see what happens to output, Y, and
thus to growth if we rescale the vertical axis
•Saving = investment
Investment,
Depreciation,
Income
and depreciation now
appear here
•Now output can be
Y*
graphed in the space
above
•We still have
transition dynamics
toward K*,
•So we also have
dynamics toward a
steady-state level of
income, Y*
K*
Capital, Kt
Understanding Differences in Growth Rates
• Empirically, for OECD countries, transition
dynamics holds:
– Countries that were poor in 1960 grew quickly.
– Countries that were relatively rich grew slower.
• Looking at the world as whole, on average,
rich and poor countries grow at the same
rate.
– Two implications of this:
• most countries have already reached their
steady states
• countries are poor not because of a bad shock,
but because they have parameters that yield a
lower steady state
Case Study: South Korea and the Philippines
• South Korea
– 6 percent per year
– Increased from 15 percent of U.S. income
to 75 percent
• Philippines
– 1.7 percent per year
– Stayed at 15 percent of U.S. income
• Transition dynamics predicts
– South Korea must have been far below its
steady state.
– Philippines is already at steady state.
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• Assuming equal depreciation rates
• The long-run ratio of per capita
incomes depends on
– The ratio of productivities (TFP levels)
– The ratio of investment rates
• The weaknesses of the Solow Model:
– It focuses on investment and capital
• the much more important factor of TFP
is still unexplained
– It does not explain why different countries
have different investment and productivity
rates.
• a more complicated model could
endogenize the investment rate
– The model does not provide a theory of
sustained long-run economic growth.
Summary
• The starting point for the Solow model is
the production model.
• The Solow model
– Adds a theory of capital accumulation.
– Makes the capital stock an endogenous
variable
• The capital stock today
– Is the sum of past investments
– Consists of machines and buildings that
were bought over the last several decades
5.10 Strengths and Weaknesses
of the Solow Model
• The strengths of the Solow Model:
– It provides a theory that determines how
rich a country is in the long run.
• long run = steady state
– The principle of transition dynamics
• allows for an understanding of
differences in growth rates across
countries
• a country further from the steady state
will grow faster
• 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.
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• As the economy accumulates more
capital
– Depreciation rises one-for-one
– 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.
– The capital stock ceases to grow.
– Out capital stock ceases to grow.
– The economy settles down to a steady
state.
• The first major accomplishment of the
Solow model is that it provides a
successful theory of the determination
of capital.
– Predicts that the capital-output ratio is
equal to the investment-depreciation ratio
• Countries with high investment rates
– Should thus have high capital-output ratios
– This prediction holds up well in the data.
• In general, most poor countries have
– Low TFP levels
– 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
– It would grow rapidly
– This is due to the principle of transition
dynamics.
This concludes the Lecture
Slide Set for Chapter 5
Macroeconomics
Second Edition
by
Charles I. Jones
W. W. Norton & Company
Independent Publishers Since 1923
• The second major accomplishment of
the Solow model is the principle of
transition dynamics.
– The farther below its steady state an
economy is, the faster it will grow.
• Transition dynamics
– Cannot explain long-run growth
– Provide a nice theory of differences in
growth rates across countries.
• Increases in the investment rate or TFP
– Increase a country’s steady-state position
and growth for a number of years
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