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\begin{document}
\title{Endogenous Growth Lecture}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
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\date{\today}
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\section*{Class Lecture on Romer}
{\label{337808}}
\subsection*{Taking Stock}
{\label{891888}}
Main problem: growth of A is exogenous
Solution: endogenize A by including R\&D sector?~
Thus\ldots{} 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? :
\(\dot A(t) = B[a_K K(t)]^\beta [a_L L(t)]^\gamma A(t)^\theta\)
\subsection*{Understanding the Model}
{\label{129547}}
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
\begin{itemize}
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Growth of A:
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\begin{quote}
\(\frac{\dot A}{A} \equiv g_A(t) = Ba_L ^\gamma L(t) ^\gamma A(t) ^{\theta - 1}\)
\end{quote}
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:
\begin{quote}
\(\frac{\dot g_A(t)}{g_A(t)}= \gamma n + (\theta - 1) g_A(t)\)
\end{quote}
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.~
\textbf{Focus} on this case,~\(\theta < 1\).
\begin{itemize}
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\item
Dynamics of K (growth rate of the growth rate):
\end{itemize}
\begin{quote}
\(\frac{\dot g_K(t)}{g_K(t)} = (1 - \alpha)[g_A (t) + n - g_K(t)]\)
\end{quote}
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.~
\begin{itemize}
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Steady state values when~\(\beta + \theta > 1\)
\end{itemize}
\begin{quote}
\(g_A * = (\frac{\beta + \gamma}{1 - (\theta + \beta)}) n\)
Or,~\(g_K * = g_A * + n\)
\end{quote}
In words, more people means more ideas.~
\subsection*{Some Issues}
{\label{146662}}
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.~
\subsection*{Empirical Implications}
{\label{178859}}
More population growth, more knowledge (applicable worldwide)~
\section*{Population Growth and Technological Change: One Million B.C. to
1990}
{\label{846965}}
\begin{itemize}
\tightlist
\item
Main prediction of endogenous growth models: high population spurs
technological change
\item
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.
\item
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
\item
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
\end{itemize}
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