Chapter 5
The Solow
Growth Model
By Charles I. Jones
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.
• 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
• 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.
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.
Consumption
Investment
Output
• This is called a resource constraint.
– Assuming no imports or exports
Capital Accumulation
• Goods invested for the future
determines the accumulation of capital.
• Capital accumulation equation:
Next
year’s
capital
This year’s Investment
capital
Depreciation
rate
• 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.
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.
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.
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.
• 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
– Is equal to investment
A unit of investment becomes a unit
of capital
•
- The return on saving must equal the
rental price of capital.
• Thus:
- The real interest rate equals the
rental price of capital which equals
the MPK.
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
• Substitute the fixed amount of labor into
the production function.
• We have reduced the system into two
equations and two unknowns (Yt, Kt).
• 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.
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.
• 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.
Solving Mathematically for the
Steady State
• In the steady state, investment equals
depreciation.
• Sub into the production function
• Solve for K*
• 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*.
• Higher steady-state production
– Caused by higher productivity and
investment rate
• Lower steady-state production
– Caused by faster depreciation
• 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.
5.5 Looking at Data through the Lens of
the Solow Model
The Capital-Output Ratio
• 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:
From Chapter 4
See figure 5.3 (next slide)
• We find that the factor of 108 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.
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:
– 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.
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
• Economists can experiment with the
model by changing parameter values.
An Increase in the 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
• 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**.
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 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**.
Experiments on Your Own
• Try experimenting with all the
parameters in the model:
– Figure out which curve (if either) shifts.
– Follow the transition dynamics of the Solow
model.
– Analyze steady-state values of capital (K*),
output (Y*), and output per person (y*).
Case Study: Wars and Economic
Recovery
• Hiroshima and Nagasaki
– Returned close to their original economic
position in just a few decades
• Vietnam
– In both villages that were bombed or left
untouched, poverty, literacy, and
consumption were similar 30 years after
the war.
• Implications of Solow growth model?
5.9 The Principle of Transition
Dynamics
• If an economy is below steady state
– It will grow.
• If an economy is above steady state.
– Its growth rate will be negative.
• When graphing this, a ratio scale is used.
– 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
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 10 percent of U.S. income
to 75 percent
• Philippines
– 1.6 percent per year
– Stayed at 10 percent of U.S. income
• Transition dynamics predicts
– South Korea must have been far below its
steady state.
– Philippines is already at steady state.
• 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
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 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
• 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
– 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.
• 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
• 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.
Additional Figures for Worked
Exercises
This concludes the Lecture
Slide Set for Chapter 5
Macroeconomics
Third Edition
by
Charles I. Jones
W. W. Norton & Company
Independent Publishers Since 1923
Additional Solow graph examples
from previous editions of slides
The Solow Diagram graphs these two
pieces together, with Kt on the x-axis:
Investment,
Depreciation
At this point,
dKt = sYt, so
Capital, Kt
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,
Kt increases from K0 to K1 > K0
K0
K1
Capital, Kt
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,
Kt decreases from K0 to K1 < K0
Capital, Kt
K1 K0
We call this the process of transition dynamics:
Transitioning from any Kt toward the economy’s
steady-state K*, where ∆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 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*
• So we also have
dynamics toward a
steady-state level of
income, Y*
K*
Capital, Kt