TECHNOLOGICAL PROGRESS AND GROWTH: THE GENERAL SOLOW MODEL

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Chapter 5 second lecture
Introducing Advanced Macroeconomics Growth and
business cycles
TECHNOLOGICAL PROGRESS AND GROWTH THE GENERAL
SOLOW MODEL
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The complete Solow model
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Analyzing the Solow model (see previous lecture)

1. Define and
2. From we get
3. From and we get
4. Dividing by on
   both sides gives

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1. Inserting gives the transition
   equation
2. Subtracting from both sides gives the Solow
   equation
3. Dividing by on both sides to get the modified
   Solow equation

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Previous lecture

• Using the transition equation and the transition
  diagram we showed convergence to steady state
  .
• We derived some relevant steady state growth
  paths e.g.
• We showed that there is balanced growth in steady
  state with and growing at the same
  positive growth rate, , and with a constant
  real interest rate, .
• We discussed structural policy aimed at affecting
  steady state.
• We showed and discussed empirics for steady
  state The model substantially underestimates
  the impact of the structural parameters on GDP
  per worker!

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This lecture

• Comparative analysis in the Solow diagrams.
• The convergence process.
• Growth accounting.

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The Solow diagram
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The modified Solow diagram
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Comparative analysis in the Solow diagrams

• Initially the economy is in steady state at
  parameters and . The savings rate
  increases permanently from to .

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• Old steady state and
  , and and both grow at rate ,
  the lower line below
• New steady state and
  , and and grow at rate , but along a
  higher growth path, the upper line above.

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• Transition grows from up to
  . The growth rate of jumps up and then
  gradually falls back to zero. From
  follows that . Hence, during the
  transition, grows at a larger rate than ,
  and jumps up and then falls gradually back to
  . The growth rate of jumps too, since
  .

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Convergence in the Solow model

• The modified Solow diagram again

There is accordance with conditional
convergence Two countries with the same

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The process of convergence empirically

• What does the model tell us more precisely about
  the process of convergence , i.e., how does
  growth in each country depend on structural
  parameters and initial position?
• Using the transition equation,
• we derive a formula for the growth rate of .
  To do this we linearize around steady state etc.
• Differentiating wrt. and evaluating in
  steady state gives
• Its easy to see that .

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• Mathematical note consider a differentiable
  function, going through the point
  , so . Then If furthermore,
  then
• Use (1) on in to
  find that
• This is an approximation of the dynamics of the
  Solow model, and its a linear one!
• The linear difference equation implies stability
  of , since .

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• Use again ,
  this time on
• Use this to rewrite
  
• Use that from one has
  
• The convergence property in each period, a
  constant fraction of the remaining gap is
  closed. The rate of convergence is

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• We now derive the solution to the difference
  equation
• The characteristic polynomial is
  and the associated root is . The complete
  solution to the homogeneous equation is .
• A specific solution to the non-homogeneous is
  , implying that the complete solution
  is
• For the constant to fit with the initial
  situation .
• Hence the solution is

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• Using the solution for etc. gives
• Inserting for and our
  expresssion for gives the convergence
  equation
• Rewrite slightly

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• Assuming that and are the same in all
  countries, this suggests a regression
  equation
• OLS estimation across 90 countries over the
  period 1960-2000 gives
• Plotting against gives

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• Positive aspects significant parameters,
  relatively high R2 etc. This does not crucially
  conflict with the assumption of the same and
  in all countries. But
• We have two estimates of the rate of convergence
• From theory
• From empirics
• Together with an estimated of and
  this gives
• The model substantially overestimates the rate
  of convergence!
• Confronted with empirics, the Solow model does
  quite well, both with respect to its steady state
  and its convergence prediction. However, in both
  cases there is an empirical problem of
  magnitudes. It could have been better, but its
  still a very well-performing model.

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Growth accounting

• With data for and (which
  often is availible) and with we can
  compute as a residual. We call this the
  Solow residual.
• Why not growth accounting in levels?

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Growth accounting per capita (worker)

• With data for and and with we
  can once more compute the Solow residual, .
• We can use this residual to check the underlying
  technological growth

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Conclusions, the general Solow model

• Implications for economic policies are more or
  less the same as those derived from the basic
  Solow model.
• The model implies convergence to a steady state
  with balanced growth and with a constant,
  positive growth rate of GDP per worker. Thus, the
  steady state prediction of the model is in
  accordance with a fundamental stylized fact.
  However, the underlying source of growth,
  technological progress, is not explained.
• The steady state prediction performs quite well
  empirically, but the model underestimates the
  effect of the savings rate and the growth rate of
  the labour force on income per worker.
• The convergence prediction also performs well
  empirically, but the model overestimates the rate
  of convergence.