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Chapter 6
Economic
Growth:
Solow Model
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Chapter 6 Topics
• Economic growth facts
• Solow growth model
• Growth accounting
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Growth Facts
• Before the Industrial Revolution (about 1800),
standards of living differed little over time and across
countries.
• Since the Industrial Revolution, per-capita income
growth has been sustained in the riches countries.
• Ex: in the United States, average annual growth in percapita income has been about 2% (excepting the Great
Depression and World War II) since 1900.
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Figure 6.1 Natural Log of Real PerCapita Income in the United States,
1869–2005
A straight line is a good fit
to the natural log of per-capita
income over most of the period
1869-2005.
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Facts – cont’
• Across countries, real per-capita income and the
investment rate are positively correlated.
• Across countries, real per-capita income and the
population growth rate are negatively correlated.
• Difference in per-capita incomes increased dramatically
among countries of the world between 1800 and 1950,
with the gap widening between rich (like OECD)
countries and the rest of world.
– Divergence between richest and poorest countries.
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Real per capita income and per
capita income growth
• There is essentially no correlation across countries
between the level of output per capita in 1960 and the
average rate of growth in output per capita for the years
1960-2000.
– No correlation between initial wealth and average growth in
per-capita income.
– Ex: there is no tendency for rich countries to grow faster than
poor countries, and vice-versa.
• Rich countries are more alike in terms of rates of
growth than are poor countries.
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Figure 6.4 Growth Rate in Per Capita
Income vs. Real Income Per Capita for
the Countries of the World
1. Over the period of 1960-2000,
one did not see that poor countries
grow faster than rich ones.
(No negative relationship)
2. There is much wider vertical scatter
in the points on the left-hand side than
on the right-hand side.
(Much more disparity in growth rates
among poor countries than rich ones)
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Solow Growth Model
(Exogenous Growth)
• This is a key model which is the basis for the modern
theory of economic growth.
• It provides interesting answers to following questions:
– Source of economic growth
– Why living standards rise over time
– What happens to the level and growth rate of income when
saving rate or population growth rate changes
– What happens to the living standards across countries over
time?
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Equation 6.9: Population growth
• In the Solow growth model, population is
assumed to grow at a constant rate n.
Two periods: current and future periods.
Exogenous population growth.
n>0, positive growth; n<0, negative growth.
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Labor Supply
• Every consumer is endowed with one unit of time in
each period.
• Consumers do not value leisure. So only consumption
goods are valued in utility function.
• Consumers supply labor inelastically. That is, in each
period, they allocate all time endowment in work.
• Hence, the population equals labor force. N is the
number of labor force and n is the growth rate of labor
force.
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Equation 6.10: ConsumptionSavings Behavior
• Consumers are assumed to save a constant
fraction s of their income, consuming the rest.
Y: all income available to consumers
No government, so no taxes.
Consumption-saving problem.
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Equation 6.11: Representative
firm’s production function
Constant returns to scale in K, N
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Equation 6.12
Constant returns to scale implies:
Define y=Y/N, and k=K/N.
y= z f ( k ), where f ( k ) = F( k , 1 )
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Figure 6.12 The Per-Worker
Production Function
1. Slope of prod. Function is MPk.
2. Since MPk is diminishing in k,
Production function is concave in k.
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Equation 6.13: Evolution of the
capital stock
Future capital equals the capital remaining after
depreciation, plus current investment.
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Competitive Equilibrium
• Two markets:
– Labor market: trade consumption goods for labor
– Capital market: trade consumption goods for capital
• Markets clear in CE
– Wage is adjusted so that firms are willing to hire N
workers (fixed labor supply).
– S = I (capital supply = capital demand).
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Equation 6.14: IncomeExpenditure Identity
The income expenditure identity holds as an
equilibrium condition.
S = I (capital market clears), and
S = Y – C (consumer’s budget)
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Equation 6.15
In equilibrium, future capital equals total
savings (= I ) plus what remains of current K.
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Equation 6.16
Substitute for output from the production function.
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Equation 6.17
Rewrite in per-worker form.
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Equation 6.18
Re-arrange, to get:
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Figure 6.13 Determination of the
Steady State Quantity of Capital per
Worker
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Equation 6.19
Equation determining the steady state quantity
of capital per worker, k*:
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Figure 6.14 Determination of the
Steady State Quantity of Capital per
Worker – Solow Diagram
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An increase in the savings rate, s
• In the steady state, this increases capital per
worker and real output per capita.
• In the steady state, there is no effect on the
growth rates of aggregate variables.
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Figure 6.15 Effect of an Increase in the
Savings Rate on the Steady State
Quantity of Capital per Worker
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Figure 6.2 Real Income Per
Capita vs. Investment Rate
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Transitional Dynamics
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Figure 6.16 Effect of an Increase
in the Savings Rate at Time T
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Figure 6.17 Steady State
Consumption per Worker
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Figure 6.18 The Golden Rule
Quantity of Capital per Worker
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An increase in the population
growth rate, n
• Capital per worker and output per worker
decrease.
• There is no effect on the growth rates of
aggregate variables.
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Figure 6.19 Steady State Effects of an
Increase in the Labor Force Growth Rate
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Figure 6.3 Real Income Per Capita vs.
the Population Growth Rate
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Increases in Total Factor
Productivity, z
Sustained increases in z cause sustained
increases in per capita income.
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Figure 6.20 Increases in Total Factor
Productivity in the Solow Growth Model
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Convergence
For countries that have identical total factor
productivity z, labor force growth rate n, and
saving rate s, poor country (with low k) grows
faster than rich country (with high k). In the long
run, both countries will converge to the same
level of capital per worker and output per worker.
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Transitional Dynamics
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Empirical Evidence
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Empirical Evidence
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Conditional Convergence
The further an economy is “below” its steady
state, the faster the economy should grow. The
further an economy is “above” its steady state,
the slower the economy grows.
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Empirical Evidence
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Growth Accounting
An approach that uses the production function
and measurements of aggregate inputs and
outputs to attribute economic growth to: (i)
growth in factor inputs; (ii) total factor
productivity growth.
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Equation 6.20: Cobb-Douglas
Production Function
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Equation 6.21
A labor share in national income of 64% gives:
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Equation 6.22
The Solow residual is calculated as:
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Figure 6.21 Natural Log of the
Solow Residual, 1948–2005
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Table 6.1 Average Annual Growth
Rates in the Solow Residual
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Table 6.2 Measured GDP, Capital Stock,
Employment, and Solow Residual
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Table 6.3 Average Annual
Growth Rates
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Table 6.4 East Asian Growth Miracles
(Average Annual Growth Rates)
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