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, \(\theta < 1\).
\(\frac{\dot g_K(t)}{g_K(t)} = (1 - \alpha)[g_A (t) + n - g_K(t)]\)
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. 
\(g_A * = (\frac{\beta + \gamma}{1 - (\theta + \beta)}) n\)
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