Class Lecture on Romer
Taking Stock
Main problem: growth of A is exogenous
Solution: endogenize A by including R&D sector?
Thus... fraction of each of the factors of production will be devoted to producing knowledge
\(Y(t) = [(1 - a_K) K(t)]^\alpha [A(t) (1 - a_L) L(t)]^{1 - \alpha}\)
To answer the question, why is A going up or down? :
\(\dot A(t) = B[a_K K(t)]^\beta [a_L L(t)]^\gamma A(t)^\theta\)
Understanding the Model
Now there are two state (accumulated) variables, K and A.
For simplicity's sake, we shut down the capital sector, focusing only on the knowledge sector
\(\frac{\dot A}{A} \equiv g_A(t) = Ba_L ^\gamma L(t) ^\gamma A(t) ^{\theta - 1}\)
In words, the fraction of the labor force and current stock of knowledge impact the rate of growth of A. To make this equation linear (remove the exponents) we take logs:
\(\frac{\dot g_A(t)}{g_A(t)}= \gamma n + (\theta - 1) g_A(t)\)
In words, the exponents have gone in front, and we have substituted in the growth rate of A for the other constants. So long as we have population growing, we will have more g. Most likely, theta is less than one (based upon empiric evidence). Recall that theta is the measure in which existing technology is useful in producing new knowledge. Focus on this case, \(\theta < 1\).
- Dynamics of K (growth rate of the growth rate):
\(\frac{\dot g_K(t)}{g_K(t)} = (1 - \alpha)[g_A (t) + n - g_K(t)]\)
In words, since we took logs, the exponents moved down and now the growth rates of the growth rate of capital depends on the growth rate of each input.
- Steady state values when \(\beta + \theta > 1\)
\(g_A * = (\frac{\beta + \gamma}{1 - (\theta + \beta)}) n\)
Or, \(g_K * = g_A * + n\)
In words, more people means more ideas.
Some Issues
We didn't define knowledge, thus cannot expect all to be created in the same way.
Why would a market-based approach be incomplete? Knowledge is non-rival -- marginal cost after discovery is zero-- and excludable.
Implications of these observations: scientific research could be subsidized, private incentives could be useful in excludable knowledge, institutions matter a lot here, as well as learning by doing.
Empirical Implications
More population growth, more knowledge (applicable worldwide)
Population Growth and Technological Change: One Million B.C. to 1990
- Main prediction of endogenous growth models: high population spurs technological change
- Kremer begins by reminding us of how much our population has grown (6.5 billion) and also the rate of growth has been growing. This is inconsistent with other animals.
- The Malthusian assumption of constant output per worker but growing number of workers, thus the growth rate of labor is proportional to the growth rate of knowledge
- What can we learn from Kremer's paper? Simple endogenous growth models can explain the spread of knowledge around the world and is useful for many fields