Modern Principles:
Macroeconomics
Tyler Cowen
and Alex Tabarrok
Chapter 7
Growth, Capital Accumulation,
and the Economics of Ideas:
Catching Up vs. the Cutting
Edge
Copyright © 2010 Worth Publishers • Modern Principles: Macroeconomics • Cowen/Tabarrok
Introduction
• In 2006:
 China: GDP per capita grew by 10%
 United States: GDP per capita grew by 2.3%
• United States has never grown as fast as
•
the Chinese economy is growing today.
Why is China growing more rapidly than the
U.S.?
 Is there something wrong with the U.S.
economy?
 Do the Chinese have a magical potion for
growth?
Slide 2 of 73
Introduction
• There are two types of growth
 Catch-up growth
• Takes advantage of ideas, technologies, or
methods of management already in existence
 Cutting-edge growth
• Primarily about developing new ideas
• China is growing much faster than the U.S.
because:
 The U.S. economy is on the cutting edge.
 The Chinese economy is catching up.
12.3
Slide 3 of 73
Introduction
• What do we learn in this chapter?
 A model based on capital accumulation.
• Explains catch-up growth.
• Allows us to answer the following questions:
 Why China is growing faster than the U.S.
 Why the losers of WWII grew much faster
than the winner.
• How poor and rich countries can converge in
income over time.
 About cutting-edge growth and the economics
of ideas.
12.4
Slide 4 of 73
The Solow Model and Catch-Up Growth
• Robert Solow – Nobel Prize in Economics
• Total Output, Y, of an economy depends on:
 Physical capital: K
 Human capital: education x Labor = eL
 Ideas: A
• This can be expressed as the following
“production function”:
Y  F(A,K, eL)
12.5
Slide 5 of 73
The Solow Model and Catch-Up Growth
• For now, ignore changes in ideas, education, and
labor so that A, e, and L are constant. The
production function becomes:
Y  F(K)
•
•
MPK: marginal product of capital
 The additional output resulting from using an additional
unit of capital.
 As more capital is accumulated, the MPK gets smaller
and smaller.
We draw a particular production function in the
next slide where:
Y K
Slide 6 of 73
The Solow Model and Catch-Up Growth
•
The “Iron Logic” of Diminishing Returns
Output, Y
Y K
3.2
3
3.2  3.0
MPK 
 0.2
10  9
MPK 
1
1 0
1
1 0
Conclusion: as more
capital is added,
MPK declines.
Capital, K
0 1 2 3 4 5 6 7 8 9 10 11 12
12.7
Slide 7 of 73
The Solow Model and Catch-Up Growth
•
Growth in China and the United States
 The “iron logic of diminishing returns”
largely explains why…
• The Chinese economy is able to grow so
rapidly.
 It turned toward markets which increased
incentives.
 The capital stock was low
 The MPK was high.
• China will not be able to achieve these high
growth rates indefinitely.
12.8
Slide 8 of 73
The Solow Model and Catch-Up Growth
• Why Bombing a Country Can Raise Its
•
Growth Rate.
Also explained by the “iron law”…
 Much of the capital stock was destroyed
during WWII. Therefore the MPK was high.
 Following the war, both Germany and Japan
were able to achieve much higher growth
rates than the U.S. as they “caught up”.
• Check out the following table.
12.9
Slide 9 of 73
The Solow Model and Catch-Up Growth
Conclusions:
1. Catch-up growth (Germany, Japan) is much greater
than cutting-edge growth (U.S.)
2. Eventually the catch-up growth slows down.
Slide 10 of 73
The Solow Model and Catch-Up Growth
• Capital Growth Equals Investment Minus
•
Depreciation
 Capital is output that is saved and
invested.
 Let g be the fraction of output that is
invested in new capital.
The next figure shows how output is divided
between consumption and investment when
g = 0.3.
12.11
Slide 11 of 73
The Solow Model and Catch-Up Growth
• Capital Growth Equals Investment Minus Depreciation
Output, Y
20
When K = 100, Output = 10
Y K
15
10
Consumption = (1- 0.3) x 10 = 7
5
Investment = 0.3∙Y
3
2
Investment = (0.3) x 10 = 3
Capital, K
0
0
100
200
300
400
12.12
Slide 12 of 73
The Solow Model and Catch-Up Growth
• Capital Growth Equals Investment Minus
Depreciation (cont.).
 Depreciation: amount of capital that wears out
each period
 Let d be the fraction of capital that wears out
each period. This is called the depreciation rate
so that:
depreciation
δ
K
 The next diagram shows that the amount of
depreciation depends on the capital stock.
12.13
Slide 13 of 73
The Solow Model and Catch-Up Growth
• Capital Depreciation Depends on the Amount of Capital
Depreciation
8
Depreciation = 0.02∙K
6
4
42
Slope 
200  100
2
0
Capital, K
0
100
200
300
400
12.14
Slide 14 of 73
The Solow Model and Catch-Up Growth
•
Capital Alone Cannot be the Key to Economic
Growth
 Again, the “iron logic of diminishing returns”
explains this insight. Let’s see how this works.
 As capital increases,
• depreciation increases at a constant rate = d.
• output increases at a diminishing rate.
• Because investment is a constant fraction of
output, at some point depreciation will equal
investment.
 The capital stock will stop growing.
 Output will stop growing.
12.15
Slide 15 of 73
The Solow Model and Catch-Up Growth
• Capital Increases or Decreases Until Investment = Depreciation
GDP, Y
8
Depreciation = 0.02∙K
At K = 400, Inv. < Dep. → ↓ K
6
Investment = 0.3∙Y
4.5
4
At K = 100,
Inv. > Dep.
→↑K
3
2
0
0
100
200 225
300
Result:
equilibrium
at K = 225
Y = 4.5
inv. = dep. =4.5
400 Capital, K
12.16
Slide 16 of 73
The Solow Model and Catch-Up Growth
• Capital Increases or Decreases Until Investment = Depreciation
Check the Math
• At K = 100, Y =√100 = 10
• Depreciation = 0.02x100 = 2
• Investment = 0.3x10 = 3
•Investment > Depreciation
Result: K and Y grow.
Check the Math
• At K = 400, Y =√400 = 20
• Depreciation = 0.02x400 = 8
• Investment = 0.3x20 = 6
•Investment < Depreciation
Result: K and Y decrease.
Check the Math
• At K = 225, Y =√225 =15
• Depreciation = 0.02x225 =
4.5
• Investment = 0.3x15 = 4.5
• Investment = Depreciation
Result:
1. Investment = Depreciation
2. K and Y are constant.
This is a steady state.
12.17
Slide 17 of 73
The Solow Model and Catch-Up Growth
•
Capital Alone Cannot be the Key to Economic
Growth (cont.)
 The logic of diminishing returns means that
eventually capital and output will cease
growing.
 Therefore, other factors must be responsible
for long-run economic growth.
 Consider:
• Human capital: knowledge, skills,
experience
• Technological knowledge: better ideas
12.18
Slide 18 of 73
The Solow Model and Catch-Up Growth
12.19
Slide 19 of 73
The Solow Model and Catch-Up Growth
•
Better Ideas Drive Long-Run Economic Growth
 Human Capital
• Like capital, it is subject to diminishing
returns and it depreciates.
• Logic of diminishing returns also applies to
human capital.
• Conclusion: Human capital also cannot
drive long-run economic growth.
 What about technological knowledge?
12.20
Slide 20 of 73
The Solow Model and Catch-Up Growth
•
Better Ideas Drive Long-Run Economic Growth
(cont.)
 Technological knowledge
• A way of getting more output from the same
input (an increase in productivity).
• We can include technological knowledge in
our model by letting A stand for ideas that
increase productivity. Therefore, let the
production function be:
YA K
12.21
Slide 21 of 73
The Solow Model and Catch-Up Growth
12.22
Slide 22 of 73
The Solow Model and Catch-Up Growth
• An Increase in A Increases Output Holding
K Constant (cont.)
 Conclusion:
• Technological knowledge or more generally
better ideas are the key to long-run
economic growth.
• Solow estimated that better ideas are
responsible for ¾ of our increased standard
of living.
12.23
Slide 23 of 73
CHECK YOURSELF
 What happens to the marginal
product of capital as more capital is
added?
 Why does capital depreciate? What
happens to the total amount of capital
depreciation as the capital stock
increases?
12.24
Slide 24 of 73
The Solow Model – Details and Further Lessons
• Let’s review what we know now:
 If Investment > Depreciation → K and Y grow.
 If Investment < Depreciation → K and Y fall.
 If Investment = Depreciation → K and Y are
constant.
• Two important conclusions
 Steady state equilibrium occurs when
investment equals depreciation.
 When K is in steady state equilibrium, Y is in
steady state equilibrium.
 These results are illustrated in the next two
diagrams.
12.25
Slide 25 of 73
The Solow Model – Details and Further Lessons
•
in steady
stateequilibrium,
equilibrium, Y
Y is in steady state equilibrium.
WhenWhen
K is K
in issteady
state
is in steady state equilibrium.
Output, Y
8
Depreciation = 0.02∙K
6
Investment = 0.3∙Y
4.5
4
The Steady State K is found
where investment = Depreciation
3
2
0
Capital, K
0
100
200 225
300
400
12.26
Slide 26 of 73
The Solow Model – Details and Further Lessons
•
When K is in steady state equilibrium, Y is in steady state equilibrium.
Output, Y
20
Steady state output
Y K
15
Depreciation = 0.02∙K
10
Investment 0.3 K
5
Steady state capital stock
0
100
200 225 300
400
Capital, K
12.27 Slide 27 of 73
CHECK YOURSELF
 What happens when the capital
stock is 400?
 What is investment?
 What is depreciation?
 What happens to output?
12.28
Slide 28 of 73
The Solow Model – Details and Further Lessons
• Solow Model and an Increase in the
Investment Rate
 What happens when g, the fraction of output
that is saved and invested increases?
•↑g↑K↑Y
 Conclusion: an increase in the investment rate
increases a country’s steady state level of
GDP.
 We show this result in the next diagram.
12.29
Slide 29 of 73
The Solow Model – Details and Further Lessons
•
An Increase in the Investment Rate Increases Steady State Output
Output, Y
20
Y K
15
Depreciation = 0.02∙K
10
Inv.  0.4 K
Inv. 0.3 K
5
Capital, K
0
100
200 225
300
400
Slide 30 of 73
The Solow Model – Details and Further Lessons
• An Increase in the Investment Rate
Increases Steady State Output (cont.)
 The results presented in the previous diagram
predict that:
• An increase in investment rate, g, causes
output to increase.
• Because labor is held constant, output per
capita also increases.
 An important test of our model:
• Are its predictions consistent with real world
data?
• The next figure suggests that they are.
12.31
Slide 31 of 73
The Solow Model – Details and Further Lessons
Slide 32 of 73
The Solow Model – Details and Further Lessons
• An Increase in the Investment Rate
Increases Steady State Output (cont.)
 An Important Idea
• An increase in the investment rate = ↑ steady
state level of output.
• As the economy moves from the lower to the
higher steady state output = ↑ growth rate of
output
• This higher growth rate is temporary.
 Conclusion: ↑investment rate = ↑ steady state
level of output but not its long-run growth
rate.
 These points are illustrated in following case
study of South Korea.
12.33
Slide 33 of 73
The Solow Model – Details and Further Lessons
• The Case of South Korea
 In 1950, South Korea was poorer than
Nigeria.
 1950s: the investment rate was < 10%.
 1970s: Investment rate more than
doubled.
 1990s: Investment rate increased to over
35%.
 South Korea’s GDP increased rapidly.
 As GDP reached Western levels, the
growth rate has slowed.
12.34
Slide 34 of 73
The Solow Model – Details and Further Lessons
• What Determines High Investment Rates?
 Incentives which include
• Low real interest rates
• Low marginal tax rates
 Institutions which include
• Honest government
• Secure property rights
 One of the reasons that the investment
rate increased in South Korea is that
capitalists believed that their investments
would be protected.
• Effective financial intermediaries
12.35
Slide 35 of 73
The Solow Model – Details and Further Lessons
• The Solow Model and Conditional
Convergence
 Conditional Convergence: Among countries
with similar steady state levels of output, poorer
countries grow faster than richer countries.
 The Solow model predicts that a country will
grow faster the farther its capital stock is below
its steady state value.
• Conclusion: Conditional convergence is a
prediction of the Solow model.
 The next figure presents evidence of
convergence.
Slide 36 of 73
The Solow Model – Details and Further Lessons
12.37
Slide 37 of 73
The Solow Model – Details and Further Lessons
• From Catching Up to Cutting Edge
•
•
 Several predictions of Solow model are consistent
with the evidence.
• Countries with higher investment rates have
higher GDP per capita.
• Countries grow faster the farther their capital
stock is from the steady state level.
One prediction is NOT consistent with the
evidence:
 Steady state: Long-run growth = 0
What explains the observed long-run growth?
 Answer: Better ideas
12.38
Slide 38 of 73
The Solow Model – Details and Further Lessons
• Solow and the Economics of Ideas in
One diagram
 Generation of ideas results in long-run economic
growth.
 Let’s see how this works:
• We begin at steady state equilibrium.
• New ideas → ↑A → ↑Output at every level of K
• ↑ Output → ↑Investment → Investment >
Depreciation →↑ K→ ↑ Output (movement
along new production function).
• As ideas continue to grow, output continues to
grow.
12.39
Slide 39 of 73
The Solow Model – Details and Further Lessons
• Solow and the Economics of Ideas in One diagram
(cont.)
Output, Y
Effect of ↑A from 1 to 1.5
c
33.7
Output ↑
b
Better
Ideas
15
Y  (1.5) K
a
Y  (1) K
Depreciation = 0.02∙K
Investment 0.3(1.5) K
Investment 0.3(1) K
225
506
Capital, K
Slide 40 of 73
CHECK YOURSELF
What happens to investment and
depreciation at the steady state level of
capital?
In Figure 7.9, how much is consumed in
the old steady state? How much is
consumed in the new steady state?
Do countries grow faster if they are far
below their steady state or if they are close?
Do countries with higher investment rates
have a lower or higher GDP per capita?
12.41
Slide 41 of 73
Growing on the Cutting Edge: The Economics of Ideas
• The United States and other developed
regions such as Japan and Western Europe
are on the cutting edge of economic growth.
• In order to keep on growing these countries
must develop new ideas to increase the
productivity of capital and labor.
• Conclusion: The economics of ideas
becomes the key to growth on the cutting
edge.
12.42
Slide 42 of 73
Growing on the Cutting Edge: The Economics of Ideas
•
The Economics of Ideas
1. Ideas for increasing output are primarily
researched, developed, and
implemented by profit-seeking firms.
2. Spillovers mean that ideas are
underprovided.
3. Government has a role in improving the
production of ideas.
4. The larger the market, the greater the
incentive to research and develop new
ideas.
12.43
Slide 43 of 73
Growing on the Cutting Edge: The Economics of Ideas
1. Research and Development Is Investment
for Profit.
 keys to increasing technological knowledge:
• Incentives
• Institutions that encourage investment in
physical and human capital and R&D.
 70% of scientists and engineers in the U.S. work
for private firms.
 Profits provide incentive to invest in R&D
• Implication: Property rights, honest
government, political stability, a dependable
legal system, and competitive open markets
help drive the generation of technological
knowledge.
Slide 44 of 73
Growing on the Cutting Edge: The Economics of Ideas
1. Research and Development Is Investment
for Profit (cont.).
 Not just the number of scientists and engineers
that are important
• All kinds of people come up with new ideas.
• Business culture and institutions are also
important.
 Institutions that are especially important:
• Commercial settings that help innovators to
connect with capitalists
• Intellectual property rights
• A high-quality education system
12.45
Slide 45 of 73
Growing on the Cutting Edge: The Economics of Ideas
1.Research and Development is Investment
for Profit (cont.).
 A commercial setting that helps innovators
connect with capitalists.
• Ideas without financial backers are sterile.
• The U.S. is good at connecting innovators
with businessmen and venture capitalists.
• American culture supports entrepreneurs:
 People like Apple CEO Steve Jobs are
lauded in the popular media.
 Contrast this to the treatment of 18th
century British entrepreneur John Kay.
Slide 46 of 73
Growing on the Cutting Edge: The Economics of Ideas
John Kay (1704-1780) invented the
“flying shuttle” used in cotton
weaving, the single most important
invention launching the industrial
revolution. Kay, however, was not
rewarded for his efforts. His house
was destroyed by “machine breakers,” who
were afraid that his invention would put them
out of a job. Kay was forced to flee to France
where he died a poor man.
12.47
Slide 47 of 73
Growing on the Cutting Edge: The Economics of Ideas
• Institutions that are especially important
 Intellectual property rights
• New processes, products, and methods can be
copied by competitors.
 World’s first MP3 player was the Eiger Labs
MPMan introduced in 1998.
 Copied by other firms and Eiger Labs lost out
in the competition.
 Patents
• Grant temporary monopoly.
• Can slow down spread of technology.
• Trade-off between creating incentives to
research and develop new products and
avoiding too much monopoly power = one of
trickiest in economic policy
12.48
Slide 48 of 73
Growing on the Cutting Edge: The Economics of Ideas
• Institutions that are especially important
(cont.)
 A high-quality education system
• Important at all levels of education.
• Creates necessary talent.
• Universities generate basic and
applied research.
12.49
Slide 49 of 73
Growing on the Cutting Edge: The Economics of Ideas
2. Spillovers, and Why There Aren’t Enough
Good Ideas
 Ideas are non-rivalrous.
 Ideas can be used simultaneously.
• Use of an idea by one individual does not mean
less of the idea available to someone else.
 The spillover or diffusion of new ideas generates
widespread economic growth.
 Implication: Spillovers mean that the generator of
the idea doesn’t get all of the benefits.
• Result: Too few ideas are produced.
• Let’s see why.
Slide 50 of 73
Growing on the Cutting Edge: The Economics of Ideas
2. Spillovers, and Why There Aren’t Enough
Good Ideas (cont.)




Optimal social investment in R&D occurs where:
MSB = MSC
Optimal private investment occurs where:
MPB = MPC
With spillover benefits: MSB = MPB + spillovers
and MSC = MSB
Conclusion:
Optimal Private
Investment in R&D

<
Optimal Social
Investment in R&D
Implication: Spillovers result in too little
investment in research and development.
12.51
Slide 51 of 73
Growing on the Cutting Edge: The Economics of Ideas
• Spillovers Mean Too Little Investment in Research and Development
$
Spillover benefits
IP = optimal private investment in R&D
IS =optimal social investment in R&D
MPC = MSC
MPB = MPC
MSB = MSC
Assumes there
MSB are no spillover
costs
MPB
IP
IS
Quantity of R&D
Slide 52 of 73
Growing on the Cutting Edge: The Economics of Ideas
3. Government’s Role in the Production of New
Ideas
 Ideas in mathematics, physics, and molecular
biology have many applications so spillovers
can be large.
• Problem: Even if the social benefits are large,
the private benefits can be small.
• Solution: Subsidize the production of new
ideas or give tax breaks for R&D
expenditures.
 Both shift the MC of R&D curve down → ↑
R&D investment.
12.53
Slide 53 of 73
Growing on the Cutting Edge: The Economics of Ideas
3. Government’s Role in the Production of New
Ideas (cont.)
 Large spillovers to basic science suggest a role
for government subsidies to universities.
• Especially those parts of the universities that
produce innovations and the basic science
behind those innovations.
• Universities produce scientists
 Most of the 1.3 million scientists were
trained in government subsidized
universities.
12.54
Slide 54 of 73
Growing on the Cutting Edge: The Economics of Ideas
4. Market Size and Research and
Development
 Innovations like pharmaceuticals, new
computer chips, software, and chemicals
require large R&D expenditures.
 Companies will avoid investing in innovations
with small potential markets.
 Larger markets mean increased rewards (thus
incentives) for R&D.
 As the world market grows companies will
increase their R&D investments.
12.55
Slide 55 of 73
CHECK YOURSELF
 What would happen to the incentive to produce
new ideas if all countries imposed high tax rates
on imports?
 What are spillovers and how do they affect the
production of ideas?
 Some economists have proposed that the
government offer large cash prizes for the
discovery of cures for diseases like malaria that
affect people in developing countries. What
economic reasons might there be to support a
prize for malaria research rather than, say,
cancer research?
12.56
Slide 56 of 73
The Future of Economic Growth
• Over the last 10,000 years per capita world
GDP has been growing.




Dawn of civilization to about 1500: growth = 0%
AD 1500 – 1760: growth = 0.08%
Growth doubled in next 100 years.
Increased even further during the 19th and 20th
centuries.
 Today: world wide growth of per capita GDP =
2.2%
12.57
Slide 57 of 73
The Future of Economic Growth
• Economic growth can be even faster. How?
 The following framework helps us think about
this.
A (ideas) = Population x Incentives x Ideas/Hour
 Population
• ↑population → ↑ number of people with new
ideas
 Much of the world is poor; thousands of
potentially great scientists are laboring in menial
jobs.
 As the world gets richer → ↑ production of ideas
→ everyone benefits
12.58
Slide 58 of 73
The Future of Economic Growth
• Economic growth can be even faster. How?
(cont.)
• A (ideas) = Population x Incentives x Ideas/Hour
 Incentives
• Appear to be increasing
 Consumers are richer
 Markets are expanding due to trade
 World wide improvement in institutions
 Property rights
 Honest government
 Political stability
12.59
 Dependable legal system
Slide 59 of 73
The Future of Economic Growth
• Economic growth can be even faster. How?
(cont.)
• A (ideas) = Population x Incentives x Ideas/Hour
 Ideas per Hour
• New ideas do not experience diminishing
returns.
• Two reasons why this is so.
1. Many ideas make creating new ideas
easier.
2. The field of ideas that can be explored is
so large that diminishing returns may not
set in for a very long time.
12.60
Slide 60 of 73
The Future of Economic Growth
• Recap: Economic growth might be even
faster in the future than it has been in the
past.
 There are more scientists and engineers in the
world than ever before, and their numbers are
also increasing as a percentage of the
population.
 Incentives are increasing due to growing
markets resulting from
• Increasing trade
• Increasing wealth in developing countries
 Better institutions and more secure property
rights are spreading throughout the world.
12.61
Slide 61 of 73
Takeaway
• As K accumulates, the MPK declines until
investment = depreciation, and growth stops.
• The Solow model tells us three things about
economic growth:
 Countries that have higher investment rates will
be wealthier.
 Growth will be faster the further away a
country’s capital stock is from its steady state
value.
 Capital accumulation cannot explain long-run
economic growth.
12.62
Slide 62 of 73
Takeaway
• New ideas are the driving force behind
long-run economic growth.
 Ideas are non-rivalrous which means there
are spillover benefits.
 Spillover benefits means that the originator of
the new idea will not receive all of the
benefits.
 In order to achieve the optimal number of
ideas government can support production of
new ideas…
• By protecting intellectual property.
• By subsidizing production of new ideas.
12.63
Slide 63 of 73
Takeaway
• There is a trade-off between providing
appropriate incentives to produce new ideas
and providing appropriate incentives to share
new ideas.
• The larger the size of the market, the greater
the incentive to invest in R&D.
• More people and wealthier countries increase
the number of people devoted to the
production of new ideas.
• The increased wealth of many developing
nations, the move to freer trade, and the
spread of better institutions all encourage the
future of economic growth.
12.64
Slide 64 of 73
Modern Principles:
Macroeconomics
Tyler Cowen
and Alex Tabarrok
Chapter 7 Appendix:
Excellent Growth
Copyright © 2010 Worth Publishers • Modern Principles: Macroeconomics • Cowen/Tabbarrok
Appendix
• Excellent Growth
 Using a spreadsheet, you can easily explore
the Solow model and duplicate all the graphs.
First, calculate the
increasing capital stock
using the formula in A3 and
let the spreadsheet do
the rest.
Note: Clicking on the lower right
corner of a cell and dragging it
down will duplicate the formula
in the lower cells.
12.66
Slide 66 of 73
Appendix
• Excellent Growth (cont.)
Second, calculate output,
Y, using the formula:
Y K
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Appendix
• Excellent Growth (cont.)
Third, graphs can be created using the data generated
In the steps one through three.
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Appendix
• Excellent Growth (cont.)
Lastly, you can experiment with different investment
shares in E2 or the depreciation rates in F2.
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Appendix
• The Mathematics of Economic Growth
•
•
along the Transition Path
Objective: To see how economic growth
varies along the transition path to a new
steady state equilibrium.
We will do two things:
 Outline the mathematics.
 Use a spreadsheet to visualize our
results.
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Appendix
• The Mathematics
Recall
1
2
Investm ent  gY  γ K  γK (e.g., 0.3 K )
Depreciation  dK (e.g., 0.02 K )
thus
1
2
ΔK  Investm ent- Depreciation  gK  dK
T hegrowth rateof thecapitalstockis given by
1
2
ΔK
gK
dK
g
 Growth rateof K 

 1 d
K
K
K
K2
Implication :
g
If
K
g
K
1
2
1
2
By plotting these two
expressions
separately on a graph,
we can see how the
steady state changes
with the values of the
investment rate and
depreciation rate.
 d  Growth rateof K is positive
 d  Growth rateof K is negative
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Appendix
• The Mathematics
d, g/K1/2
0.08
0.07
0.06
Difference is the growth rate of the
capital stock. The bigger the difference
the faster K grows.
0.05
0.04
0.03
d = 0.02
0.02
0.01
0.4/K1/2
400
Capital, K
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Appendix
• The Spreadsheet
Plotting Y against time shows the transition to steady
state
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Appendix
• The Spreadsheet
Output, Y
16.00
14.00
12.00
10.00
8.00
Output, Y
6.00
4.00
2.00
0.00
0
100
200
300
400
500
600
Time
Result: The transition to steady state proceeds at a
decreasing rate. As K approaches 400 growth slows down.
Slide 74 of 73