1. Advanced Macroeconomics:
Chapter 6 – second lecture
Introducing
Advanced Macroeconomics:
Growth and business cycles
EDUCATION AND GROWTH: THE SOLOW MODEL WITH HUMAN CAPITAL

2. The Solow model with human capital
Parameters:
Empirical observations indicate that and

3. The law of motion (repetition)
Define:
From we get that
Restatement of the capital accumulation equations:
Dividing on both sides by gives:

4. 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.

5. 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:

6. The Phase diagram for the Solow model with human capital

7. 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!

8. 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 don’t 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 model’s real prediction. The rate of convergence is computed, the convergence equation is tested, and the rate of convergence is estimated.

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

11. Linear approximation
Write the transition equations in generic form: where and
Linearize the first around ’steady state’:
Using that etc. gives:

12. The full linearised system of two equations:

13. 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

14. 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.

15. Testing the convergence equation and estimating
Same difference equation as in the Solow model implies same solution:
Inserting and gives the convergence equation:

16. 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:

17. Unconditional
Conditional on and n
Conditional on and n

18. Conclusions
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.
This is reassuring for policy implications!
However, important parameters are unexplained, e.g and .
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!”.
These critical remarks point towards theories of endogenous growth.