Session 3:
Why Do Economies Grow?
Inputs?
Chad Jones
Stanford GSB
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Levels and Growth Rates of Per Capita GDP
PER CAPITA GDP GROWTH (1960 TO 2017)
10%
8%
Botswana Malta South Korea
6% China Singapore
Romania Taiwan
Egypt
Thailand Malaysia Ireland
4% Indonesia Hong Kong
Japan
India Morocco Brazil
Ethiopia Spain Norway
Argentina
Pakistan Luxembourg
Nepal Iran Israel Netherlands
Mali Bolivia U.K.
2% Mozambique Uganda Bangladesh Fiji Mexico Uruguay
U.S. Switzerland
Honduras Ecuador New Zealand
Burkina Faso
Malawi Togo Kenya Jamaica South Africa
Haiti Senegal Venezuela Algeria
0% Burundi
Chad Zimbabwe Nigeria
Nicaragua Barbados
Niger Guinea
Central Afr. Rep.
-2% D.R. of the Congo
1/64 1/32 1/16 1/8 1/4 1/2 1
PER CAPITA GDP (RATIO SCALE, US=1)
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Outline of Today’s Class
• How do we understand sustained U.S. economic growth at 2% per year?
What about different growth rates across countries?
• The basics of the Solow Model
• The determination of the real interest rate
• The Solow Diagram
• Economic growth according to Solow
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South Korea and the Philippines
• In 1960, South Korea and the Philippines were quite similar
◦ Per capita GDP: Both around $1500
◦ Population: Both about 25 million, 1/2 working age
◦ Similar sectoral composition (industry, agriculture)
◦ College enrollment: S. Korea=5%, Philippines=13%
• Between 1960 and 2017, macroeconomic performance diverged sharply
◦ Growth: S. Korea=6.5%, Philippines=2.5%
◦ Per capita GDP in 2017:
S. Korea=$36,000, Philippines=$7,500
• Why?
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The Solow Growth Model
• Robert Solow in 1950s, Nobel prize in 1987
◦ Build on production function, endogenize capital
◦ Can the accumulation of capital — computers, machine tools,
factories — explain sustained economic growth, like U.S. 2%?
• Analogy: A Corn Farm
◦ Farm has a silo containing bushels of seed corn
◦ Farmers plant the seed, tend the crop, and harvest
◦ They eat 3/4ths of the harvest and save the remaining 1/4th in the
silo for next year’s planting
◦ Repeat.
• Key: each seed kernel produces ten ears of corn, each with hundreds of
kernels, so harvest grows
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The Economy of Solovia
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Production function Yt = ĀKt Lt
Note: ∆Kt+1 ≡ Kt+1 − Kt ,
Capital accumulation ∆Kt+1 = It − ¯
dKt so Kt+1 = Kt + It − ¯
dKt
Labor force Lt = L̄
Resource constraint C t + It = Y t
Allocation of resources It = s̄Yt
Unknowns: Yt , Kt , Lt , Ct , It
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Prices and the Real Interest Rate
• We’ve left prices — the wage and the rental price of capital — out of the
model.
◦ Just a simplification, could add them back easily
• Worth pausing to introduce the real interest rate.
• Definition: ???
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Prices and the Real Interest Rate
• We’ve left prices — the wage and the rental price of capital — out of the
model.
◦ Just a simplification, could add them back easily
• Worth pausing to introduce the real interest rate.
• Definition: the (real) amount someone can earn by saving one unit of
output for a year
• Measured in units of output, or constant dollars
• What is the real interest rate equal to in the Solow world?
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Prices and the Real Interest Rate
• We’ve left prices — the wage and the rental price of capital — out of the
model.
◦ Just a simplification, could add them back easily
• Worth pausing to introduce the real interest rate.
• Definition: the (real) amount someone can earn by saving one unit of
output for a year
• Measured in units of output, or constant dollars
• What is the real interest rate equal to in the Solow world?
The real interest rate equals the rental price of capital = return on
investment = marginal product of capital
◦ Saving equals investment in this Solow economy
◦ Save one unit ⇒ invest one unit ⇒ earn the marginal product of
capital, which equals the rental price
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Solving the Solow Model
• The equations can be combined and simplified significantly, yielding
∆K = s̄ Yt − ¯
dK
| {zt+1} | {z }t
change in capital net investment
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Yt = ĀKt L̄2/3
• These two equations govern the dynamics of the Solow model.
• Analyze in a Solow diagram
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The Solow Diagram
∆Kt+1 = s̄Yt − ¯
1/3
dKt Yt = ĀKt L̄2/3
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The Solow Diagram with Output and Consumption
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Looking at Data using the
Solow Model: Capital
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The Capital-Output Ratio
• The key addition of the Solow model is that it explains where the capital
stock comes from.
• Recall that in steady state s̄Y∗ = ¯
dK∗ , which implies
K∗ s̄
= ¯
Y∗ d
• Different countries have different investment rates (measured as
investment divided by GDP).
◦ The Solow framework predicts that these should show up as
different capital-output ratios (“capital intensities”).
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Explaining Capital in the Solow Model
Countries with high investment rates
have high capital-output ratios.
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Economic Growth
in the Solow Model
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Economic Growth in the Solow Model
• What is the growth rate of the economy in the long run according to the
Solow model?
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Economic Growth in the Solow Model
• What is the growth rate of the economy in the long run according to the
Solow model?
Zero! There is no long-run growth in Solow!!
• Why?
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Understanding the Steady State Result
• Why does the economy settle down to a steady state?
◦ s̄Y∗ = ¯
dK∗ — investment equals depreciation.
◦ The new investment we make is just enough to offset the
wear-and-tear that depreciates the existing capital
• Because investment — the s̄Y curve — exhibits diminishing returns.
◦ As we increase capital, output rises by a smaller and smaller
amount.
◦ But a constant fraction of capital depreciates.
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Understanding the Steady State Result
• Why does the economy settle down to a steady state?
◦ s̄Y∗ = ¯
dK∗ — investment equals depreciation.
◦ The new investment we make is just enough to offset the
wear-and-tear that depreciates the existing capital
• Because investment — the s̄Y curve — exhibits diminishing returns.
◦ As we increase capital, output rises by a smaller and smaller
amount.
◦ But a constant fraction of capital depreciates.
So diminishing returns to capital is at the heart of why growth eventually
ceases in the Solow model.
— A huge, disappointing failure...
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Growth and Transition Dynamics
• Despite this negative result on long-run growth, the Solow framework is
extraordinarily useful (see “Nobel Prize”)
• The reason is related to Transition Dynamics
• And this is most easily studied by considering an “experiment” in
Solovia:
Suppose Solovia begins in steady state in the year 2000 with an
investment rate of 10%. What happens if the investment rate then
increases permanently in the year 2020 to 15%?
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A Permanent Increase in the Investment Rate
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The Response of Output to a Rise in Investment
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The Principle of Transition Dynamics
• As suggested by the Solow Diagram and by the last example, a key
prediction of the Solow framework is
The Principle of Transition Dynamics: the farther below its steady state
an economy is, the faster it will grow; similarly, the farther above its
steady state, the faster it will decline (the slower it will grow).
Differences in growth rates across countries largely reflect
how near or far countries are from their steady state.
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Examples: The United States and China
• United States
◦ Growth at a constant rate of 2% per year for more than a century
• China
◦ Rapid growth at more than 8% per year during the last 25 years
• Think about Japan after World War II...
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Examples: The United States and China
• United States
◦ Growth at a constant rate of 2% per year for more than a century
◦ In some sense, this reflects the fact that the U.S. is close to its
steady state.
◦ We’ll understand this more after the next two lectures.
• China
◦ Rapid growth at more than 8% per year during the last 25 years
• Think about Japan after World War II...
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Examples: The United States and China
• United States
◦ Growth at a constant rate of 2% per year for more than a century
◦ In some sense, this reflects the fact that the U.S. is close to its
steady state.
◦ We’ll understand this more after the next two lectures.
• China
◦ Rapid growth at more than 8% per year during the last 25 years
◦ Suggests China is below its steady state and therefore growing
rapidly
◦ Why might this be true?
• Think about Japan after World War II...
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Per capita GDP in Several Countries
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Growth Rates in the OECD, 1960–2017
In the OECD, the poor countries grew most
rapidly and the rich countries grew most slowly.
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Growth Rates around the World, 1960–2017
PER CAPITA GDP GROWTH, 1960-2017
7% Malta
Botswana South Korea For the world as a whole, however, growth
6% China Singapore
rates are unrelated to initial GDP... Why?
Taiwan
5% Egypt
Ireland
Indonesia Malaysia
Thailand
4% Panama
Hong Kong
Japan
Ethiopia Brazil Norway
Portugal Spain
3% India Turkey
Sri Lanka
Austria
Italy
Finland Germany
Netherlands Luxembourg
Pakistan
Nepal Peru Iran Greece Israel Denmark
Mali Colombia
Gabon BelgiumFrance Switzerland
Sweden Australia
2% Mozambique
Jordan Costa RicaMexico
Uruguay U.K. United States
Fiji Ecuador New Zealand
1% Côte dIvoire Kenya South Africa
Malawi Togo
BeninGhana Algeria
Jamaica Venezuela
Burundi Nicaragua
0% Chad
Nigeria
Zimbabwe Barbados
Guinea
-1% Niger
Central Afr. Rep.
-2% D.R. of the Congo
0 4,000 8,000 12,000 16,000 20,000
PER CAPITA GDP, 1960 (IN 2011 DOLLARS)
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Investment in South Korea and the Philippines
INVESTMENT RATE (PERCENT)
50
South Korea
40
30
U.S.
20
Philippines
10
0
1950 1960 1970 1980 1990 2000 2010 2020
YEAR
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Can investment be too high?
• What investment rate would maximize steady state consumption?
◦ s = 0 ⇒ c∗ = 0: no investment, means no capital, so GDP falls to
zero
◦ s = 100% ⇒ c∗ = 0: if you investment everything, you never
consume!
⇒ answer is somewhere in between.
• Simple math gives (if Y = Kα L1−α )
s∗ = α = 1/3
Intuition: α is capital’s share of income. If you are saving more than your
capital earns, you are saving too much (you never eat the reward!)
A country should never have an investment rate permanently greater
than 1/3. So we know investment rates will decline!
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What we learn from the Solow Framework
• In the long run, a country is rich or poor in part based on what fraction of
its output it invests for the future.
◦ Similar reasoning suggests that the fraction of the population
employed, the number of hours worked per person, and
investment in human capital play a similar role.
• Through the principle of transition dynamics, the model helps us
understand why some countries grow rapidly and others slowly.
• Capital accumulation — the accumulation of more machine tools,
computers, and factories per worker — is not a mechanism that leads to
sustained long-run growth.
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Shortcomings of the Solow Framework
• The question of why countries have different investment rates remains
on the table
◦ Similarly, TFP differences actually become more important. Why?
Think about what happens in the Solow diagram if there is an
increase in TFP...
• How do we understand long run economic growth?
◦ The Solow model does not answer...
◦ But see the next several classes!
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Reading: No Need to Dig (The Economist)
• Questions:
◦ What is the main point of the reading?
◦ How does it relate to the Solow model?
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Questions for Review
• What is the mechanism in the Solow model that generates growth?
• Why does it fail to sustain growth in the long run?
• What determines the real interest rate in an economy?
• How does the Solow framework explain differences in growth rates
across countries?
• Why do poor countries in the OECD grow faster than rich countries in
the OECD, but poor countries in the world do not grow faster than rich
countries in the world?
• What is the principle of transition dynamics?
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