Working with the Solow Growth Model

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C h a p t e r 4
Working with the Solow Growth Model
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Key EquationsSolow Growth Model

• ?k/k s (y/k) - sd - n
• k is capital per worker
• y is real gross domestic product (real GDP) per
  worker
• y/k is the average product of capital
• s is the saving rate
• d is the depreciation rate
• n is the population growth rate.

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Solow Growth ModelSteady State
s(y/k)sdn

• We assumed that everything on the right-hand side
  was constant except for y/k.
• In the transition to the steady state, the rise
  in k led to a fall in y/k and, hence, to a fall
  in ?k/k.
• In the steady state, k was constant and,
  therefore, y/k was constant. Hence, ?k/k was
  constant and equal to zero.

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Solow Growth ModelChange in savings rate (s)
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Solow Growth ModelChange in savings rate (s)

• In the short run, an increase in the saving rate
  raises the growth rate of capital per worker.
• This growth rate remains higher during the
  transition to the steady state.

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Solow Growth ModelChange in savings rate (s)

• In the long run, the growth rate of capital per
  worker is the samezerofor any saving rate.
• In this long-run or steady-state situation, a
  higher saving rate leads to higher steady state
  capital per worker, k, not to a change in the
  growth rate (which remains at zero).
• Af(k)/kdn/s

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Solow Growth Modelthe effect of s on consumptions

• In the short run, consumption decreases and k
  arises.
• cy-dk-s(y-dk) y-dk-nk
• ?c ?y-(dn) ?k (MPK-d-n) ?k
• In the long run, whether the consumption in the
  steady state increases depends on MPK.
• Golden Rule

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Solow Growth ModelChange in technology level (A)
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Solow Growth ModelChange in technology level (A)

• In the short run, an increase in the technology
  level, A, raises the growth rates of capital and
  real GDP per worker.
• These growth rates remain higher during the
  transition to the steady state.

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Solow Growth ModelChange in technology level (A)

• In the long run, the growth rates of capital and
  real GDP per worker are the samezerofor any
  technology level.
• In this long-run or steady state situation, a
  higher technology level leads to higher
  steady-state capital and real GDP per worker, k
  and y, not to changes in the growth rates (which
  remain at zero).
• Af(k)/kdn/s

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Solow Growth ModelChange in the labor input
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Solow Growth Model Change in the labor input

• In the short run, an increase in labor input,
  L(0), raises the growth rates of capital and real
  GDP per worker.
• These growth rates remain higher during the
  transition to the steady state.

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Solow Growth Model Change in the labor input

• In the long run, the growth rates of capital and
  real GDP per worker are the samezerofor any
  level of labor input, L(0).
• The steady-state capital and real GDP per worker,
  k and y, are the same for any L.
• In the long run an economy with twice as much
  labor input has twice as much capital and real
  GDP.

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Solow Growth Model Change in population growth
rate
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Solow Growth Model Change in population growth
rate
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Solow Growth Model Change in population growth
rate

• In the short run, a higher n lowers ?k/k and ?
  y/y.
• These growth rates remain lower during the
  transition to the steady state.

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Solow Growth Model Change in population growth
rate

• In the steady state, ?k/k and ?y/y are zero for
  any n.
• A higher n leads to lower steady-state capital
  and real GDP per worker, k and y, not to
  changes in the growth rates, ?k/k and ?y/y (which
  remain at zero).
• A change in n does affect the steady-state
• growth rates of the levels of capital and real
  GDP, ? K/K and ?Y/Y.

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Solow Growth ModelSum up

• k k s, A, n, d, L(0)
• () () (-) (-) (0)

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Solow Growth Model Convergence

• One of the most important questions about
  economic growth is
• whether poor countries tend to converge or
  catch up to rich countries.

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Solow Growth Model Convergence
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Solow Growth Model Convergence

• Economy 1 starts with lower capital per worker
  than economy 2k(0)1 is less than k(0)2.
• Economy 1 grows faster initially because the
  vertical distance between the s(y/k) curve and
  the sdn line is greater at k(0)1 than at k(0)2.

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Solow Growth Model Convergence

• That is, the distance marked by the red arrows is
  greater than that marked by the blue arrows.
• Therefore, capital per worker in economy 1, k1,
  converges over time toward that in economy 2, k2.

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Solow Growth Model Convergence
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Solow Growth Model Convergence

• Economy 1 starts at capital per worker k(0)1 and
  economy 2 starts at k(0)2, where k(0)1 is less
  than k(0)2.
• The two economies have the same steady-state
  capital per worker, k, shown by the dashed blue
  line.
• In each economy, k rises over time toward k.
  However, k grows faster in economy 1 because
  k(0)1 is less than k(0)2.
• Therefore, k1 converges over time toward k2.

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Solow Growth Model Convergence

• y A f(k) and ?y/y a(?k/k)
• ?k/k was higher initially in economy 1 than in
  economy 2.
• Therefore, ?y/y is also higher initially in
  economy 1. Hence, economy 1s real GDP per
  worker, y, converges over time toward economy 2s
  real GDP per worker.

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Solow Growth Model Convergence

• The Solow model says that a poor economywith low
  capital and real GDP per workergrows faster than
  a rich one. The reason is the diminishing average
  product of capital, y/k.
• The Solow model predicts that poorer economies
  tend to converge over time toward richer ones in
  terms of the levels of capital and real GDP per
  worker.

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Solow Growth Model Convergence
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Solow Growth Model Convergence
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Solow Growth Model Convergence
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Solow Growth Model Convergence
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Solow Growth Model Convergence

• Economy 1 starts with lower capital per worker
  than economy 2
• k(0)1 lt k(0)2.
• Assume that economy 1 also has a lower saving
  rate
• s1 lt s2.
• The two economies have the same technology
  levels, A, and population growth rates, n.
• Therefore, k1 is less than k2 .
• It is uncertain which economy grows faster
  initially. The vertical distance marked with the
  blue arrows may be larger or smaller than the one
  marked with the red arrows.

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Solow Growth Model Convergence
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Solow Growth Model Convergence

• Economy 1 starts with lower capital per worker
  than economy 2
• k(0)1 lt k(0)2.
• The two economies now have the same saving rates,
  s, and technology levels, A, but economy 1 has a
  higher population growth rate, n
• n1 gt n2.
• Therefore, k1 is less than k2 .
• It is again uncertain which economy grows faster
  initially. The vertical distance marked with the
  blue arrows may be larger or smaller than the one
  marked with the red arrows.

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Solow Growth Model Convergence
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Solow Growth Model Convergence

• Economy 1 has a lower starting capital per
  workerk(0)1 lt k(0)2and also has a lower
  steady-state capital per worker k1 (the dashed
  brown line) is less than k2 (the dashed blue
  line).
• Each capital per worker converges over time
  toward its own steady-state value k1 (the red
  curve) toward k1 ,
• And k2 (the green curve) toward k2 . However,
  since k1 is less than k2 , k1 does not converge
  toward k2.

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Solow Growth Model Convergence

• Key Results
• k k s, A, n, d, L(0)
• () () (-) (-) (0)
• ?k/k ? k(0) , k
• (-) ()

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Solow Growth Model Convergence

• Conditional convergence
• a lower k(0) predicts a higher ?k/k, conditional
  on k.
• Absolute convergence
• the prediction that a lower k(0) raises ?k/k
  without any conditioning is called.

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Solow Growth Model the speed of Convergence
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Solow Growth Model the speed of Convergence

• Calibration
• The half-life is roughly 18 years.

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Solow Growth Model Endogenous population growth

• Malthus (1798)
• the increase of y (or k) leads to a higher
  growth rate of population, which reduces the
  level of income per capita.
• Modern growth theory
• the higher income per capita reduces the
  population growth rate.