EDUCATION%20AND%20GROWTH:%20THE%20SOLOW%20MODEL%20WITH%20HUMAN%20CAPITAL

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Chapter 6 second lecture
Introducing Advanced Macroeconomics Growth and
business cycles
EDUCATION AND GROWTH THE SOLOW MODEL WITH HUMAN
CAPITAL
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The Solow model with human capital

• Parameters
• Empirical observations indicate that
  and .

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The law of motion (repetition)

1. Define
2. From we get that
3. Restatement of the capital accumulation
   equations
4. Dividing on both sides by gives

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• Inserting gives the transition
  equations
• The law of motion two coupled first-order
  difference equations in and . Given
  and they decide and .
• follows from , and then
  follows from and from
  .
• We have shown/will show
• There is a well-defined steady state.
• Numerical simulations suggest convergence towards
  a steady state for reasonable parameter values.
• Convergence holds for a linear approximation
  around a steady state.

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• Subtracting and , respectively, on both
  sides of the transition equations gives the Solow
  equations
• From these we found in the previous lecture a
  well-defined steady state , , implying
  also a steady state growth path, , for income
  per worker.
• From the Solow equations also follow
• We use these to construct

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The Phase diagram for the Solow model with human
capital
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Comparative analysis in the phase diagram

• An increase in shifts upwards. At
  first, only increases, but then begins to
  increase as well.
• What happens to and to ? The growth
  rates jump!

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Stability of steady state

• In the previous lecture we saw that the steady
  state prediction does well empirically and has
  interesting policy implications. It also exhibits
  balanced growth
• and are constant and
  grow at a constant rate, .
• grows at the
  constant rate, , too.
• constant.
• We dont know yet if the model implies
  convergence to steady state. We will
• Show convergence in a numerical simulation.
• Show convergence analytically for a linear
  approximation around steady state. This also
  gives us the convergence equation, the models
  real prediction. The rate of convergence is
  computed, the convergence equation is tested, and
  the rate of convergence is estimated.

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Simulation

• Let
  and .
• In steady state and .
• Start in and , and simulate
  using the transition equations
• The result is shown in the phase diagram

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Linear approximation

• Write the transition equations in generic
  formwhere and
  .
• Linearize the first around steady state
• Using that etc.
  gives

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• The full linearised system of two equations

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• From we have that
  . Multiplying the first of our two equations
  by and the second by , and adding the two
  resulting equations give

• The figure shows convergence as
  .

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The rate of convergence

• Rewriting the above difference equation in
  gives
• In the Solow model, the rate of convergence was
  .
• From empirical observations (
  etc.), should now be around 2.5 (in the Solow
  model it was 5). This was one of the motives for
  including human capital to lower the
  theoretically predicted rate of convergence.

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Testing the convergence equation and estimating

• Same difference equation as in the Solow model
  implies same solution
• Inserting and gives
  the convergence equation

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• The convergence equation suggests the regression
• OLS estimation across 80 countries gives
• . Inserting
  and
  gives . Uncertainty
  95 confidence interval for is from 0.8 to
  2.3.
• Furthermore, and implied are
  and .
• Illustration of the convergence process

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• Unconditional
• Conditional on and n
• Conditional on and n

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Conclusions

1. The Solow model with human capital performs very
   well empirically, both wrt. its steady state
   prediction and wrt. its convergence prediction,
   even under the assumption of the same level of
   and the same growth rate in technology in all
   countries.
2. This is reassuring for policy implications!
3. However, important parameters are unexplained,
   e.g. and .
4. In particular, the rate of technological growth,
   , that determines the long run rate of growth
   in GDP per worker remains unexplained we have
   understood almost everything, and yet almost
   nothing!.
5. These critical remarks point towards theories of
   endogenous growth.