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\begin{document}
\title{Romer Problems 3.1, 3.2, 3.4, 3.5, and 3.14}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
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\section*{Problem 3.1}
{\label{240660}}
\begin{quote}
\(Y(t) = A(t) (1 - a_L) L(t)\)
\(\dot A(t) = Ba_L^\gamma L(t) ^\gamma A(t) ^\theta\)
\end{quote}
\textbf{a)} On a balanced growth path,~
\begin{quote}
\(\frac{\dot A(t)}{A(t)} = G_A * = \frac{\gamma n}{1 - \theta}\)
\(\frac{\dot A(t)}{A(t)} = BA_L^\gamma L(t) ^\gamma A(t) ^{\theta - 1}\)
\end{quote}
Setting these equal and simplifying yields:
\begin{quote}
\(A(t) = [\frac{(1 - \theta) BA_L ^\gamma L(t) ^\gamma}{\gamma n}] ^{\frac{1}{1 - \theta}}\)
\end{quote}
\textbf{b)}~Substituting in the equation for knowledge into the first
equation for output gives us:
\begin{quote}
\(Y(t) = [[\frac{(1 - \theta) BA_L ^\gamma L(t) ^\gamma}{\gamma n}] ^{\frac{1}{1 - \theta}}] (1 - a_L) L(t)\)
\end{quote}
Now maximizing the log of output with respect to the fraction of the
labor force used in R\&D:
\begin{quote}
\(\ln Y(t) = \frac{1}{1 - \theta} \ln [\frac{(1 - \theta) B}{\gamma n}] + \frac{\gamma}{1 - \theta} \ln a_L + \ln (1 - a_L) + [(\frac{\gamma}{1 - \theta}) + 1] \ln L(t)\)
\end{quote}
The first order condition is given by:
\begin{quote}
\(\frac{\partial \ln Y(t)}{\partial a_L} = \frac{\gamma}{1 - \theta} \frac{1}{a_L} - \frac{1}{1 - a_L} = 0\)
\end{quote}
Finally, solving for the value of~\(a_L\) that maximizes
output on the balanced growth path:
\begin{quote}
\(a_L * = \frac{\gamma}{(1 - \theta) + \gamma}\)
\end{quote}
This result signifies that the higher theta is, the more important the
stock of knowledge is for the production of new knowledge. Likewise, the
higher gamma, the more important labor is in the production of new
knowledge, thus, the larger fraction of the labor force should be
employed in the R\&D sector.~
\section*{Problem 3.2}
{\label{240660}}
Using the production functions, we substitute into the proportion
equation:
\begin{quote}
\(\frac{Y_1(t)}{Y_2(t)} = [\frac{K_1(t)}{K_2(t)}]^\theta\)
\end{quote}
Taking the time derivative of the log of the above equation gives:
\begin{quote}
\(\frac{\frac{\dot Y_1(t)}{Y_2(t)}}{\frac{Y_1(t)}{Y_2(t)}} = \theta [\frac{\dot K_1(t)}{K_1(t)} - \frac{\dot K_2(t)}{K_2(t)}]\)
\(= \theta [g_{K,1} (t) - g_{K,2}(t)] > 0\)
\end{quote}
Thus, with~\(\theta >1\),~\(g_{K,i}\) will always be
increasing and the gap between the two countries will be positive and
increasing over time.~ To visualize this:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\section*{Problem 3.4}
{\label{240660}}
The relevant equations for the loci and growth rates of K and A are:
\begin{quote}
\(\dot g_K = 0 \Longrightarrow g_K = g_A + n\)~
\(\dot g_A = 0 \Longrightarrow g_A = \frac{(1- \theta)g_A - \gamma n}{\beta}\)
\(g_K(t) = c_K [\frac{A(t) L(t)}{K(t)}] ^{1 - \alpha}\)
\begin{quote}
\(c_K \equiv s(1 - a_K)^\alpha (a - a_L)^{1 - \alpha}\)
\end{quote}
\(g_A(t) = c_A K(t)^\beta L(t) ^\gamma A(t) ^{\theta - 1}\)
\begin{quote}
\(c_A \equiv Ba_K^\beta a_L^\gamma\)
\end{quote}
\end{quote}
\textbf{a)}~Since savings isn't in the first two equations, the locus
won't shift when savings increases. However, it would cause a jump in
the growth rate of capital, as can be seen in the fourth equation. The
figure below demonstrates the occurrence, a leap up to F:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\textbf{b)~}At point F, the growth rate of A is rising. This means that
the increase in savings caused the growth rate of capital to jump and
now the amount of capital devoted to the production of knowledge is
higher, and so, the growth rate of knowledge has risen. Over time, the
economy would drift down, crossing the~\(\dot g_A = 0\) locus, and
fall back to the point of E since there are decreasing returns to
capital.
To analyze the path of the log output per worker, we take the time
derivative of the log of the production function:~
\begin{quote}
\(\frac{\dot Y(t)}{Y(t)} - \alpha g_K (t) + (1 - \alpha) [g_A (t) + n]\)
\end{quote}
This means that total output on the balanced growth path grows at the
rate of the growth of A plus population growth. During the transition,
both growth rates of K and A are higher and so output per worker must be
growing at a rate greater than it's balanced growth path value. Whether
the growth rate in output per worker rises or falls depends on the value
of alpha. The figure below shows the progression:
\par\null\par\null\par\null\par\null\par\null
~
\textbf{c)~}Since there are net decreasing returns to the factors of
production here, the increase in saving only have a level, not growth,
effect on output per worker. This is similar to a rise in saving in the
Solow model, though A rises here as well which makes the quantitative
effect larger.~
\section*{Problem 3.5}
{\label{240660}}
\textbf{a)~}With the assumptions of~\(\beta + \theta = 1 , n = 0\), the growth rate
equations for K and A simplify to:
\begin{quote}
\(g_K(t) = [c_K L^{1 - \alpha}] [\frac{A(t)}{K(t)}]^{1 - \alpha}\)
\(g_A (t) = [c_a L^\gamma] [\frac{K(t)}{A(t)^\beta}]\)
\end{quote}
The ratio of these growth rates determines the growth rates, which are
equal when:
\begin{quote}
~\([c_K L^{1 - \alpha}] [\frac{A(t)}{K(t)}]^{1 - \alpha} = [c_a L^\gamma] [\frac{K(t)}{A(t)^\beta}]\)
\([\frac{A(t)}{K(t)}]^{1 - \alpha + \beta} = [\frac{c_A}{c_K}] L^{\gamma - (1 - \alpha)}\)
\end{quote}
Therefore, the proportion that yields the equal growth rates is:
\begin{quote}
\(\frac{A(t)}{K(t)}= [(\frac{c_A}{c_K}) L^{\gamma - (1 - \alpha)}] ^{\frac{1}{1 - \alpha + \beta}}\)
\end{quote}
\textbf{b)~}To find the growth rate of A and K when they are equal, we
can substitute and simplify the previous equation:
\begin{quote}
\(g* = [c_K L^{1 - \alpha}] [(\frac{c_A}{c_K})L ^{\gamma - (1 - \alpha)}] ^{\frac{1 - \alpha}{1 - \alpha + \beta}}\)
\(g* = [c_K ^\beta c_A ^{1 - \alpha} L^{(1 - \alpha)(\gamma + \alpha)}] ^{\frac{1}{1 - \alpha + \beta}}\)
\end{quote}
\textbf{c)~}To see how an increase in saving effects long-run growth, we
can substitute the definitions of consumption of K and A into the above
equation:
\begin{quote}
\(g* = [s^\beta (1 - a_K)^{\alpha \beta} (1 - a_L) ^{(1 - \alpha) \beta} B^{(1 - \alpha)} a_K ^{\beta (1 - \alpha)} a_L ^{\gamma (1 - \alpha)} L^{(1 - \alpha) - (\gamma + \alpha)}] ^{\frac{1}{1 - \alpha + \beta}}\)
\end{quote}
Taking the natural log and determining the elasticity with respect to
the saving rate:
\begin{quote}
\(\frac{\partial \ln g*}{\partial \ln s} = \frac{\beta}{(1 - \alpha + \beta)} > 0\)
\end{quote}
Therefore, an increase in the saving rate increases the long-run growth
rate of the economy since the resources devoted to physical capital
accumulation have increased.~
\textbf{d)~}Maximizing with respect to~\(a_K\) will
determine the fraction of capital stock that should be employed in the
R\&D sectors if we want to maximize long-run growth. The first-order
condition is:
\begin{quote}
\(\frac{\partial \ln g*}{\partial a_K} = \frac{\beta}{(1 - \alpha + \beta)} [\frac{- \alpha}{1 - a_K} + \frac{1 - \alpha}{a_K}] = 0\)
\end{quote}
Solving for the optimal level and simplifying yields:
\begin{quote}
\(\frac{\alpha}{1 - a_K} = \frac {1 - \alpha}{a_K} \Longrightarrow a_K * = 1 - \alpha\)
\end{quote}
Therefore, the optimal proportion of the capital stock to employ in R\&D
is equal to effective labor's share in the production of output. In the
case we're considering, the effects of a rise in beta (making capital
more important in R\&D but also making the production of new capital
more valuable) would cancel each other out.~
\section*{Problem 3.14}
{\label{240660}}
\textbf{a)~}This will mean finding the value of tau such that the ratio
of output per worker in the north to the south is equal to 10. Taking
the time derivative of the natural log of the north's production
function:~
\begin{quote}
\(\frac{\frac{\dot Y_N(t)}{L_N}}{\frac{Y_N(t)}{L_N}} = \frac{\dot A_N (t)}{A_N(t)} = 0.03\)
\end{quote}
So the growth rate of output per worker and knowledge is 3\% per year,
thus:
\begin{quote}
\(A_N (t) = e^{0.03 \tau} A_N (t - \tau)\)
\end{quote}
Dividing this by the souther production function will yield:
\begin{quote}
\(\frac{\frac{Y_N(t)}{L_N}}{\frac{Y_S(t)}{L_S}} = \frac{A_N (t) (1 - a_L)}{A_S(t)} \approx \frac{A_N(t)}{A_N (t - \tau)} = e^{0.03 \tau}\)
\end{quote}
Therefore we can solve for tau:
\begin{quote}
\(e^{0.03 \tau} = 10 \Longrightarrow \tau \approx 76.8\)
\end{quote}
Thus, poor countries would be using technology developed many years
ago-- to explain a 10-fold difference in income per person would point
to technology developed in the late 1920s.~
\textbf{b) i)~}For the north, the value of k on the balanced growth path
would be:
\begin{quote}
\(sf(k_N *) = (n + g + \delta) k_N *\)
\end{quote}
Here, the growth rate of knowledge in the north is g. We know that only
the growth rates of technology are different for the north and south,
specifically:
\begin{quote}
\(A_S (t) - \dot A_N ( t - \tau)\)
\end{quote}
Because the growth rate of northern knowledge is constant and equal to
g, we have:
\begin{quote}
\(\frac{\dot A_S(t)}{A_S(t)} = g\)
\end{quote}
And so for the south, balanced growth capital is:
\begin{quote}
\(sf(k_S *) = (n + g + \delta) k_S *\)
\end{quote}
We see that both northern and southern balanced growth path capital must
be equal.~
\textbf{ii)~}Introducing capital will not impact the previous result.
Since the balanced growth path capital is equal in both regions, output
per unit of effective labor is equal as well, meaning that the addition
of capital would come across evenly. This is shown by:
\begin{quote}
\(\frac{Y_N(t)}{L_N(t)} \equiv A_N(t) y_N*\)
\end{quote}
Dividing this by the southern equation yields:
\begin{quote}
\(\frac{\frac{Y_N(t)}{L_N(t)}}{\frac{Y_S(t)}{L_S(t)}} = \frac{A_N(t)}{A_N(t - \tau)} = e^{0.03 \tau}\)
\end{quote}
Again, the gap must be about 76.8 years between the north and south's
growth in technology, which capital has not changed.~~
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