Chapter 5 - Solow Model for Growth

Chapter 5: Solow Growth Model
Ryan W. Herzog
Spring 2021
Ryan W. Herzog (GU) Solow Spring 2021 1 / 59

1 Introduction
2 Setting up the Model
3 Prices and the Real Interest Rate
4 Solving the Solow Model
5 Looking at Data through the Lens of the Solow Model
6 Understanding the Steady State
7 Economic Growth in the Solow Model
8 Some Economic Experiments
9 The Principle of Transition Dynamics
10 Strengths and Weaknesses of the Solow Model

Introduction
Learning Objectives
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.

Introduction
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

Introduction
Changes in the Solow 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.

Setup
Production
Start with the previous production model and 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
Yt = F(Kt, Lt) = AK
1/3
t L
2/3
t (1)

Setup
Output
We assume a closed economy so output can be used for either
consumption or investment
Ct + It = Yt (2)
This is called the resource constraint (how an economy can use its
resources).

Setup
Capital Accumulation
Goods invested for the future determines the accumulation of capital.
Capital accumulation equation:
Kt+1 = Kt + It − dKt (3)
where
Kt+1 is next year’s capital stock
Kt is this year’s capital stock
It is investment this year
d is the depreciation rate

Setup
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 (some examples assume
d = 0.07).

Setup
Evolution of Capital
Let ∆Kt+1 ≡ Kt+1 − Kt
Thus:
Kt+1 − Kt = It − dKt
or
∆Kt+1 = It − dKt (4)

Setup
An Example of Capital Accumulation
Assume the economy begins with an initial level of capital, k0 of
1,000 units:
Time, t Capital, Kt Investment, It Depreciation, dKt ∆Kt+1
0 1,000 200 100 100
1 1,100 200 110 90
2 1,190 200 119 81
3 1,271 200 127 73
4 1,344 200 134 66
5 1,410 200 141 59

Setup
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 (L = L).

Setup
Investment
In our single firm economy (i.e. farmers) we assume a constant
portion of the output is consumed and the rest is investment:
It = sYt (5)
Therefore:
Ct = (1 − s)Yt
You can think of s as a constant saving rate. In this model saving
equal investment, implicitly assuming a closed economy (in reality,
use the investment rate).

Setup
The Solow Model
The model has five endogenous variables (Yt, Kt, Lt, Ct, It)
The model has five equations:
Production function: Y = AK1/3
L2/3
Capital accumulation: ∆Kt+1 = It − dKt
Labor force: L = L
Resource constraint: Ct + It = Yt
Allocation of resources: It = sYt
Exogenous parameters: A, s, d, L, K0

Setup
Questions/Differeces in 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

Setup
Key Definitions
Stock: A quantity that survives from period to period.
Flow: A quantity that lasts a single period
A change in the stock of capital is the flow of investment.

Prices
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

Prices
Saving
The difference between income and consumption
Is equal to investment
Yt − Ct
| {z }
saving
= It
The closed economy assumptions forces all saving to flow into
domestic investment.
In an open economy saving could fund domestic or foreign
investment, plus foreign saving can fund domestic investment.

Prices
Return on Saving and MPK
A unit of investment becomes a unit of capital which means the
return on saving must equal the rental price of capital.
The real interest rate equals the rental price of capital which equals
the MPK.

Solving
Solving the Model
The model needs to be solved at every point in time, which cannot be
done algebraically.
Two ways to make progress
1 Show a graphical solution
2 Solve the model in the long run
We can start by combining equations to go as far as we can with
algebra.

Solving
Solving the Model
Let’s start by combining the investment allocation (It = sYt) with the
capital accumulation equation (∆Kt+1 = It − dKt)
∆Kt+1
| {z }
change in capital
= sYt − dKt
| {z }
net investment
(6)
We can interpret equation 6 as the change in capital equals
investment less depreciation.

Solving
Production
Since we assume the supply of labor is constant (L = L) we can
reduce the production function down to two unknowns (Yt, Kt).
Y = AK1/3
L
2/3
(7)
We can now combine our two equations by plugging in equation 7
into equation 6.
This would provide us with a single dynamic equation that describes
the evolution of the capital stock.

Solving
Creating the Solow Diagram
The Solow Diagram plots the two terms (sY and dK)
New investment looks like the production functions previously
graphed but scaled down by the investment rate.
sY = sAK1/3
L
2/3
Depreciation is constant so dK is linear with an intercept of 0 and
slope of d

Solving

Solving
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.
When they equal 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.

Solving
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.

Solving
Output and Consumption
As K moves to its steady state by transition dynamics, output will
also move to its steady state.
Y = AK(1/3)
L
2/3
Consumption can also be seen in the diagram since it is the difference
between output and investment.
Ct = Yt − It

Solving
Solow Diagram with Output

Solving
Solving Mathematically for the Steady State
In the steady state, investment equals depreciation.
sY ∗
= dK∗
Substitute in the production function
sAK∗1/3
L
2/3
= dK∗

Solving
Solve for K∗
Solve for K∗
K∗
=

sA
d
3/2
L (8)
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

Solving
Solve for Y ∗
We can plug K∗ into the production function:
Y ∗
= AK∗1/3
L
2/3
Doing so yields:
Y ∗
=

s
d
1/2
A
3/2
L (9)
Higher steady-state production caused by higher productivity and
investment rate
Lower steady-state production caused by faster depreciation

Solving
Output per Worker (Person)
Finally, divide both sides of the last equation by labor to get output
per person (y) in the steady state.
y∗
≡

Y ∗
L∗

=

s
d
1/2
A
3/2
(10)
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.

Data
The Capital-Output Ratio
Recall the steady state, where ∆Kt+1 = 0 and sY ∗ = dK∗
The capital to output ratio is the ratio of the investment rate to the
depreciation rate:
K∗
Y ∗
=
s
d
(11)
Investment rates vary across countries.
It is assumed that the depreciation rate is relatively constant.

Data
Capital-output Ratio and Investment Rates

Data
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
| {z }
70
=

Arich
Apoor
3/2
| {z }
35
×

srich
spoor
1/2
| {z }
2
(12)

Data
Explaining Differences in TFP and investment
Investment rates in rich countries are approximately 25 to 30 percent.
Investment rates in poor countries are approximately 7 percent.
Then

srich
spoor
1/2
= (28/7)1/2 = 41/2 ≈ 2
We can then find the ratio of TFP as 70/2 ≈ 35

Steady State
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.

Growth
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

Growth
Economic Growth
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 but do not
sustain long-run growth due to diminishing returns

Growth
Adding Population Growth to The 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 even with more
workers

Experiments
An Increase in the Investment Rate
Suppose investment increases from s to s
0
, then...
The investment curve rotates upward
The depreciation curve remains unchanged.
The capital stock increases by transition dynamics to reach the new
steady state because investment exceeds depreciation
The new steady state is located to the right where s
0
Y = dK.

Experiments
An Increase in Investment

Experiments
Output
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 ∗∗.

Experiments
Changes in Output

Experiments
A Rise in the Depreciation Rate
Suppose depreciation rate is exogenously shocked to a higher rate
from d to d
0
, then...
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 where sY = d
0
K.

Experiments
A Higher Depreciation Rate

Experiments
Output in Response to a Change in Depreciation
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
Output in Response to a Change in Depreciation

Experiments
What Else Can Change?
We looked at changes to depreciation and investment rates. What
happens if...
Technology increase?
Labor decreases?
A country has a lower level of initial capital.

Transition
Transition Dynamics
If an economy is below steady state then economic growth will be
positive.
If an economy is above steady state then economic growth will be
negative.
When graphing this, a ratio scale is used. This 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.

Transition
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

Transition
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.
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.

Transition
Growth Convergence

Transition
Rich and Poor Countries

Transition
South Korea and Philippines
South Korea
Grew at 6 percent per year
Income increases from 15 percent to 75 percent of US income
Philippines
Grew at 1.7 percent per year
Stayed at 15 percent of U.S. income

Transition
Comparing US and Korea
Assuming equal depreciation rates
y∗
Korea
y∗
US
=

AKorea
AUS
3/2
×

sKorea
sUS
1/2
(13)
The long-run ratio of per capita incomes depends on the ratio of
productivities (TFP levels) and the ratio of investment rates

Transition
South Korea, Philippines, and US

Strengths and Weaknesses
Strengths
It provides a theory that determines how rich a country is in the long
run.
The long run occurs at the 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

Strengths and Weaknesses
Weaknesses
It focuses on investment and capital but 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.

Chapter 5 - Solow Model for Growth

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