The document summarizes Chapter 5 of the Solow Growth Model. It introduces the Solow model, which uses a Cobb-Douglas production function to model how capital accumulates over time. It shows graphically and mathematically how the model reaches a steady state where growth stops. While the model does not explain long-term growth, it can be used to understand differences in output across countries and experiment with changes to parameters. The strengths and weaknesses of the Solow model are also discussed.

1. Chapter 5: The Solow Growth Model
Hikaru Saijo
University of California, Santa Cruz
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).
Yt = F (Kt , Lt) =
• Output can be used for consumption or investment.
= Yt
• This is called a resource constraint.
• Assumes no imports or exports

2. Capital Accumulation
• Goods invested for the future determines the
accumulation of capital.
• Capital accumulation equation:
Kt+1 =
• 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
d
̄ = 0.10
• Change in capital stock defined as
∆Kt+1 ≡ Kt+1 − Kt
• Thus:
∆Kt+1 =
• The change in the stock of capital is investment
subtracted by the capital that depreciates in production.
Labor

3. • To keep things simple, labor demand and supply not
included
• The amount of labor in the economy is given exogenously
at a constant level.
Lt = L
̄
Investment
• Farmers eat a fraction of output and invest the rest.
It = s̄ Yt
• Therefore:
Ct =
• Consumption is the share of output we don’t invest.
• Unknowns/endogenous variables:
1 Production function:
2 Capital accumulation:
∆Kt+1 =
3 Labor force: Lt = L
̄
4 Resource constraint:
Yt =

4. 5 Allocation of resources:
It =
• Parameters/exogenous variables: Ā, s̄ , d
̄ , L
̄ , K
̄ 0
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
Yt − Ct︸ ︷︷︸
saving
= It︸︷︷︸
investment

5. • 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.
∆Kt+1 =
• Substitute the fixed amount of labor into the production
function.
Yt =

6. • 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
s̄ Y d
̄ K
• New investment looks like the production functions
previously graphed but scaled down by the investment
rate.
s̄ Y = s̄ ĀK 1/3L
̄ 2/3
Drawing the Solow diagram
Using the Solow Diagram
• If the amount of is greater than
the amount of :
• 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 is greater than

7. , 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 capital 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.
Drawing the Solow diagram with output.
Solving Mathematically for the Steady State
• In the steady state, investment equals depreciation.

8. s̄ Y ∗ = d
̄ K ∗
• Sub in the production function
= d
̄ K ∗
Solving for K ∗
The steady-state level of capital is
• Positively related with
•
•
•
• Negatively related with
•
Solving for Y ∗
The steady-state level of output is
• Positively related with

9. •
•
•
• Negatively related with
•
• Finally, divide both sides of the last equation by labor to
get output per person (y) in the steady state.
y ∗ ≡ Y
∗
L∗
=
• 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.
By what proportion does per capita output change in the long
run in response to the following changes?
1 Saving rate decreases by 10%.

10. 2 The productivity level falls by 20%.
3 The capital stock increases by 50% as a result of foreign
investment.
5.5 Looking at Data through the Lens of the Solow
Model: The Capital-Output Ratio
• Recall the steady state.
s̄ Y ∗ = d
̄ K ∗
• The capital to output ratio is the ratio of the investment
rate to the depreciation rate:
K ∗
Y ∗
=
s̄
d
̄
• Investment rates vary across countries.
• It is assumed that the depreciation rate is relatively
constant.

11. 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:
y ∗ rich
y ∗ poor︸ ︷︷︸
64
=
(
Ā∗ rich
Ā∗ poor
)3/2
︸︷︷︸
32
×
(
s̄ ∗ rich
s̄ ∗ poor
)1/2
︸︷︷︸
2

12. 5.6 Understanding the Steady State
• The economy reaches 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

13. • 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
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

14. • Suppose the investment rate increases permanently for
exogenous reasons.
s̄ −→ s̄ ′
• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.
A Rise in the Depreciation Rate
• Suppose the depreciation rate is exogenously shocked to a
permanently higher rate.
d
̄ −→ d
̄ ′
• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.
A Rise in the Productivity

15. • Suppose the productivity is exogenously shocked to a
permanently higher rate.
Ā −→ Ā′
• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.
A Destruction of Capital Stock
• Suppose the capital stock is exogenously shocked to a
lower level (due to, for example, a natural disaster).
K −→ K ′
• What happens to the economy (capital and output) over
time and in the long run?
• Use Solow diagram to answer the question.
5.9 The Principle of Transition Dynamics

16. • If an economy is below
• It will grow.
• If an economy is above
• 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 further below its steady state an economy is, (in
percentage terms)
• the the economy will grow
• The further above its steady state
• the 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:

17. • Countries that were poor in 1960 grew quickly.
• Countries that were relatively rich grew slower.
• For the world as a whole, on average, rich and poor
countries grow at the same rate.
• Two implications of this:
• Most countries (rich and poor) 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 (determinants of the steady state invest rates and
A).
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
• allows for an understanding of differences in growth
rates across countries

18. • 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
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
Chapter 4: A Model of Production
Hikaru Saijo
University of California, Santa Cruz
4.2 A Model of Production
• Vast oversimplifications of the real world in a model can
still allow it to provide important insights.

19. • Consider the following model
• Single, closed economy
• One consumption good
Setting Up the Model
• Inputs
• Labor (L)
• Capital (K )
• Production function
• Shows how much output (Y ) can be produced given any
number of inputs
• Production function:
Y = F (K, L) = ĀK
1
3L
2
3
• Output growth corresponds to changes in Y .
• There are three ways that Y can change:
• Capital stock (K ) changes.

20. • Labor force (L) changes.
• Ability to produce goods with given resources (K , L)
changes.
• Technological advances occur (changes in A).
• TFP is assumed to be exogenous in the Solow model.
• The Cobb-Douglas production function is the particular
production function that takes the form of
Y = KaL1−a
Assumed to be a = 1/3. Explained later.
• A production function exhibits constant returns to scale if
doubling each input exactly doubles output.
• Standard replication argument
• A firm can build an identical factory, hire identical
workers, double production stocks, and can exactly
double production.
If the sum of exponents in the inputs. . .
• sum to more than 1 −→ returns to scale
• sum to 1 −→ returns to scale
• sum to less than 1 −→ returns to scale

21. Returns to Scale Example
Are these production functions IRS, CRS, or DRS?
1 Y = K
1
3 L
1
3
2 Y = L1.1
3 Y = K1.5L0.9
Allocating Resources
max
K,L
Π = F (K, L) − rK −wL
• The rental rate and wage rate are taken as given under
perfect competition.
• For simplicity, the price of the output is normalized to
one.
• The marginal product of labor (MPL)
• The additional output that is produced when one unit of

22. labor is added, holding all other inputs constant.
• The marginal product of capital (MPK)
• The additional output that is produced when one unit of
capital is added, holding all other inputs constant.
MPL =
∂Y
∂L
=
MPK =
∂Y
∂K
=
If the production function has constant returns to scale in
capital and labor, it will exhibit decreasing returns to scale in
capital alone.
max
K,L
Π = F (K, L) − rK −wL
• The solution is to use the following hiring rules:

23. • Hire capital until the MPK = r
• Hire labor until MPL = w
Solving the Model: General Equilibrium
• The model has five endogenous variables:
• Output (Y )
• the amount of capital (K )
• the amount of labor (L)
• the wage (w)
• the rental price of capital (r)
• The model has five equations:
• The production function
• The rule for hiring capital
• The rule for hiring labor
• Supply equals the demand for capital
• Supply equals the demand for labor
• The parameters in the model:
• The productivity parameter
• The exogenous supplies of capital and labor
• Unknowns/endogenous variables:

24. 1 Production function:
2 Rule for hiring capital:
3 Rule for hiring labor:
4 Demand = supply for capital:
5 Demand = supply for labor:
• Parameters/exogenous variables:
• A solution to the model
• A new set of equations that express the five unknowns in
terms of the parameters and exogenous variables
• Called an equilibrium
• General equilibrium
•
Solution
to the model when more than a single market
clears

25. Solving the Model
1 Find a market clearing (equilibrium) capital and labor.
2 Plug in the equilibrium capital and labor to the
production function and get output.
3 Given equilibrium capital and labor, solve for prices.
Solving the Model
Solving the Model
Interpreting the