Empirical Applications of Neoclassical Growth Models

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Empirical Applications of Neoclassical Growth
Models

• ECON 401 Growth Theory

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Solow Model with Human Capital

• Extend the model to include Human Capital
• Labor in different economies possess different
  levels of education and different skills
• Suppose that output, Y, is produced by
• Physical capital, K
• and by skilled labor, H
• Production function is of Cobb-Douglas type
• YKa(AH)1-a (3-1)
• where A represent labor-augmenting technology

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Solow Model with Human Capital

• Individuals accumulate human capital by spending
  time to learn new skills instead of working.
• HeyuL , (y is a positive constant) (3-2)
• where u denote the fraction of an individuals
  time spent learning skills, and L denote raw
  labor.
• Note that if P denote the total population, then
  the total amount of labor input is given by
  L(1-u)P
• So skilled labor is generated by unskilled labor
  learning skills for time u
• What if u0?

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Solow Model with Human Capital

• By increasing u, a unit of unskilled labor (L)
  increases the effective units of skilled labor
  (H).
• Apply the trick (take logs and derivatives of eq.
  3-2)

Suppose Du1 and y0.10 ? H rises by 10 percent.
(A large literature in labor economics actually
finds out 10 return through higher wages- to
an additional year of schooling).
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Solow Model with Human Capital

• Physical capital accumulation is given by

Let lower case letters denote, as before,
variables divided by the stock of unskilled
labor, L. Re-write the prod. Function
as (Y/L)y ka(Ah)1-a , (3-5) where heyu and
k(K/L). We will assume u is constant and given
exogeneously (at least until Chp 7) Since
production function is similar to earlier ones,
we can easily say that along a balanced growth
path, y and k will grow at the constant rate g
(rate of technological progress)
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Solow Model with Human Capital

• Denote state variables by dividing with Ah (so
  they are constant along a balanced growth path).
  Re-write production function as

Re-write the capital accumulation equation
following the same logic.
(3-7)
So adding human capital doesnt change the basic
structure of the Solow model.
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Solow Model with Human Capital

• Steady state values are found by setting the
  previous equation to zero.

• Countries are rich because
• Have high investment rates in physical capital
• Spend a large fraction of times accumulating
  skills
• Have low population growth rates
• High levels of technology

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Solow Model with Human Capital

• In addition, as in the original Solow model,
  output per capita grows at the rate of
  technological progress, g, in the steady state.
• How does this model perform empirically in terms
  of explaining why some countries are richer than
  others?
• - analyze by looking at relative incomes
  (incomes are growing over time)
• - define per capita income relative to US

(3-9)
Where the hat () is used to denote a variable
relative to US value, and xndg.
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Solow Model with Human Capital

• Need to assume that countries have the same rate
  of technological progress. Why?
• Is it plausible to make this assumption?
• - if g varies across countries, income gap
  will eventually be infinite.
• - Technology may flow across borders due to
• - international trade, or
• - through scientific journals and newspapers,
  or
• - through immigration of scientists and
  engineers
• Hence, it may be plausible to think that
  technology transfer will keep even the poorest
  nations from falling too far behind.
• Note levels of technology can be different

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Solow Model with Human Capital

• If the countries have the same g as we assumed,
  then can Solow model answer why countries had
  different growth rates over the last 30 years?
  The answer is no.
• However, we can still examine the fit of the
  neoclassical model. Figure 3.1 compares the
  actual levels of GDP per worker in 1997 to the
  levels predicted by equation (3-9)
• - Assumed a(1/3). This choice fits well with
  the observation.
• - Measured u as the average educational
  attainment of labor force (in years)
• - Assumed y0.10. So each year of schooling
  increases workers wage by 10 percent. This is
  consistent with international evidence.
• - Assumed gd0.075.
• - Assumed technology level, A, is the same
  across countries.

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Fig. 3.1
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Solow Model with Human Capital

• Without accounting for differences in
  technology, the model still describes the per
  capita income distribution across countries
  pretty well.
• Main failure is the departure from the 45 degree
  line.
• Poorest countries are predicted to be richer.
• How can we incorporate actual technology levels?
• Can use production function to solve for A,
  consistent with each countrys output and
  capital.

We can use this equation to estimate actual
levels of A, using GDP per worker, capital per
worker, and educational attainment for each
country as inputs. Figure 3-2 report these
estimates.
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Fig. 3.2
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Solow Model with Human Capital

• Levels of A (calculated) are strongly correlated
  with the levels of output per worker
• Rich countries have high levels of A so they do
  not only have high levels of physical and human
  capital, they also use these inputs very
  productively
• This correlation is far from perfect.
• Countries like Singapore and Italy have much
  higher levels of A than expected (even higher
  than US level)
• Remember level of A is calculated as a residual
  so it incorporates any differences in production
  not factored in through these inputs.
• Qulality of educational systems, on the job
  training, general health of the labor force are
  some examples of other factors
• Hence, it might be more appropriate to refer to
  these estimates as total factor productivity
  (TFP) rather than technology levels.
• Differences in TFP across countries are large

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Solow Model with Human Capital

• Remember equation 3-9

• Based on actual data (in the appendix of the
  book), richest countries of the world have an
  output per worker that is roughly 32 times that
  of the poorest countries. This difference can be
  broken down into
• Investment rates in physical capital Richest
  countries have investment rates around 25
  percent, while poorest countries have rates
  around 5 percent.
• Investment rates in human capital Workers in
  rich countries have about 10-11 years of
  education on average. In poor countries, it is
  less than 3 years. Assuming return to schooling
  is about 10

That is, differences in educational attainment
also contribute a factor of just over 2 to
differences in output per worker
What accounts for the remainder?
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Solow Model with Human Capital

• By construction, it is the differences in total
  factor productivity (TFP).
• This difference should contribute the remaining
  factor of 8 to the differences in output per
  worker
• In summary, Solow model is successful to
  understand the variation in the wealth of
  nations.
• Countries who invest a large fraction of their
  resources in physical and human capital are rich
• Countries who use these inputs productively are
  rich.
• However, Solow model does not help to understand,
• Why some countries invest more than others
• Why some countries attain higher levels of
  technology or productivity

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Convergence and Explaining Differences in Growth
Rates

• How well does it explain the differences in
  growth rates across countries?
• An hypothesis under certain circumstances,
  backward countries would tend to grow faster than
  rich countries in order to close the gap between
  the two groups. This is known as convergence.
• So the question of convergence is whether the
  enormous differences among rich and poor
  countries are getting smaller over time.

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Convergence and Explaining Differences in Growth
Rates

• Baumol (1986) was one of the first to provide
  empirical evidence documenting convergence among
  some countries and absence of convergence among
  others.
• This evidence is displayed at Figure 3.3 which
  plots per capita GDP for several industrialized
  countries from 1870 to 1994.
• Gaps between countries are getting narrower.
• Figure 3.4 explains why some countries grew fast
  and others grew slowly over time. It plots
  countrys initial per capita GDP (in 1885)
  against the countrys growth rate from 1885 to
  1994.
• Reveals a strong negative relationship between
  the growth rate and initial per capita GDP

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Fig. 3.3
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Fig. 3.4
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Convergence and Explaining Differences in Growth
Rates

• Figures 3.5 and 3.6 plot growth rates versus
  initial GDP per worker for the countries that are
  members of the OECD and for the world for the
  period 1960-1997.
• Figure 3.5 shows that convergence hypothesis
  works quite well.
• However, Figure 3.6 shows that the convergence
  hypothesis fails to explain differences in growth
  rates across the world as a whole.
• It does not appear that poor countries grow
  faster than rich countries.
• Why do we see convergence among some countries
  but lack of convergence among the countries of
  the world as a whole?

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Fig. 3.5
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Fig. 3.6
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Convergence and Explaining Differences in Growth
Rates

• Consider the key differential equation (3-7).

Rewrite it as
(3-10)
Remember average product of capital declines as
(k/Ah) increases diminishing returns to capital
accumulation (why?) Figure 3-7
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Fig. 3.7
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Convergence and Explaining Differences in Growth
Rates

• Remember the difference between the two curves in
  Figure 3.7 is the growth rate of (k/Ah).
• Note also that growth rate of (y/Ah) is simply
  proportional to this difference.
• Since growth rate of A is constant, any changes
  in the growth rates of (y/Ah) and (k/Ah) must be
  due to changes in the growth rates of k and y.
• Suppose the economy of InitiallyBehind is at
  (k/Ah)IB , while InitiallyAhead is at (k/Ah)IA.
  If they have the same A, same sK, and same n,
  then InitiallyBehind should grow faster initially
  than InitiallyAhead . Both approach same
  steady-state.
• ? Among countries that have the same
  steady-state, the convergence hypothesis should
  hold.
• For industrialized countries this might not be a
  bad assumption.

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Convergence and Explaining Differences in Growth
Rates

• However, all countries of the world do not have
  the same steady-state, which explains the lack of
  convergence across the world.
• In fact, the differences in income levels around
  the world (remember Figure 3.2) reflect the
  differences in steady-states.
• ? Hence, the countries are not expected to grow
  toward the same steady-state target.
• Remember the principle of transition dynamics
  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 should grow.
• This prediction/principle can explain differences
  in growth rates.
• Figure 3.8 plots growth rate of GDP per worker
  against the deviation of GDP per worker (relative
  to that of US) from its steady-state value.
• How do you know steady-state value?

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Fig. 3.8
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Convergence and Explaining Differences in Growth
Rates

• According to this figure, poorer countries do not
  grow faster, but countries that are poor relative
  to their own steady-states (ratio closer to 1
  closer to steady-state) tend to grow more
  rapidly.
• Examples are Japan. Korea, Singapore and Hong
  Kong in 1960.
• This is sometimes called conditional convergence
  because it reflects the convergence of countries
  after we control for differences in
  steady-states.
• Extensions of this analysis of convergence
• US states
• Regions of France
• Prefectures in Japan
• all exhibit unconditional convergence.

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Convergence and Explaining Differences in Growth
Rates

• Why did we see wide differences in growth rates
  across countries in chapter 1?
• Countries that do not at their steady-states are
  not expected to grow at the same rate. There are
  many reasons why they might not be in their
  steady-states.
• An increase in the investment rate
• A change in the population growth rate
• A change in the level of technology
• Or a War that destroys a countrys capital stock
• Other shocks like large changes in oil prices,
  hyperinflations (e.g., observed in Latin
  America), mismanagement of the macroeconomy

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The Evolution of Income Distribution

• Are the rich countries getting richer and poor
  ones are getting poorer? Are poorest countries
  falling behind while the countries with
  intermediate incomes converging toward the rich?
  These are questions about the evolution of
  distribution of per capita incomes around the
  world.
• Figure 3.9 shows that, for the world as a whole,
  enormous gaps in incomes across countries have
  now narrowed over time.
• Pritchett (1997) in a paper titled Divergence,
  Big Time calculates the ratio of per capita GDP
  between the richest and poorest countries in the
  world. This ratio was
• 8.7 in 1870, and
• 45.2 in 1990

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Fig. 3.9
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The Evolution of Income Distribution

• Figure 3.10 examines changes in each point of the
  income distribution.
• According to this figure,
• In 1960, 50 of the countries had relative
  incomes that were less than 15 of US GDP per
  worker.
• By 1997, this number improved slightly to about
  20.
• In poorest economies, those below the 30th
  percentile, had relative incomes in 1997 lower
  than in 1960.
• There seems to be a convergence at the middle and
  top of this distribution, while we observe a
  divergence at the lower end.
• Quah (1996) suggest that this tendency will
  result in an income distribution with twin
  peaks, a mass of countries at both ends of the
  income distribution.

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Fig. 3.10