1. Empirical Applications of Neoclassical Growth Models
ECON 401: Growth Theory

2. 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
Y=Ka(AH)1-a (3-1)
where A represent labor-augmenting technology

3. Solow Model with Human Capital
Individuals accumulate human capital by spending time to learn new skills instead of working.
H=eyuL , (y is a positive constant) (3-2)
where u denote the fraction of an individual’s 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 u=0?

4. 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 Du=1 and y=0.10  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).

5. 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 h=eyu 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)

6. 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 doesn’t change the basic structure of the Solow model.

7. 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

8. 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 x=n+d+g.

9. 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

10. 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 y=0.10. So each year of schooling increases worker’s wage by 10 percent. This is consistent with international evidence.
- Assumed g+d=0.075.
- Assumed technology level, A, is the same across countries.

11. Fig. 3.1

12. 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 country’s output and capital.
We can use this equation to estimate actual levels of A, using GDP per worker, capital per worker, and educational attaınment for each country as inputs. Figure 3-2 report these estimates.

13. Fig. 3.2

14. 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 ın TFP across countries are large

15. 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 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?

16. 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

17. 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.

18. 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 country’s initial per capita GDP (in 1885) against the country’s growth rate from 1885 to 1994.
Reveals a strong negative relationship between the growth rate and initial per capita GDP

19. Fig. 3.3

20. Fig. 3.4

21. 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
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?

22. Fig. 3.5

23. Fig. 3.6

24. 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

25. Fig. 3.7

26. 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.

27. 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?

28. Fig. 3.8

29. 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.

30. 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 country’s capital stock
Other shocks like large changes in oil prices, hyperinflations (e.g., observed in Latin America), mismanagement of the macroeconomy

31. 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

32. Fig. 3.9

33. 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.

34. Fig. 3.10