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\begin{document}
\title{Macro HW Ch. 9~}
\author[1]{Clara Jace}%
\affil[1]{George Mason University}%
\vspace{-1em}
\date{\today}
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\section*{Question 9.1}
{\label{183123}}
\(Y = K^\alpha L^{1 - \alpha}\)
\(P, Y, W, r_K\)are given
\subsection*{Part A}
{\label{116222}}
Solving for labor with prices, capital, output and rental price of
capital given yields
\begin{quote}
\(L = Y^{\frac{1}{1 - \alpha}} K^{\frac{- \alpha}{1 - \alpha}}\)~
\end{quote}
\subsection*{Part B}
{\label{116222}}
Since profits equal
\begin{quote}
\(\pi = PY - WL - r_K K \)
\end{quote}
Adding in the optimal level of labor will yield
\begin{quote}
\(\pi = PY - W[Y^{\frac{1}{1 - \alpha}} K^{\frac{- \alpha}{1 - \alpha}}] - r_K K\)
\end{quote}
\subsection*{Part C}
{\label{116222}}
The first-order condition for the profit-maximizing choice of K will be
\begin{quote}
\(\frac{\partial \pi}{\partial K} = \frac{\alpha}{1 - \alpha} WY ^{\frac{1}{1 - \alpha}} K^{\frac{- \alpha}{1 - \alpha} - 1} - r_K = 0\)
\end{quote}
The second-order condition should be negative to verify that the
first-order is a maximum. We can see this by
\begin{quote}
\(\frac{\partial \pi ^2}{\partial K ^2} = \frac{1}{1 - \alpha} \frac{\alpha}{1 - \alpha} WY ^{\frac{1}{1 - \alpha}} K^{\frac{ \alpha- 2}{1 - \alpha}}\)
\end{quote}
Since~\(\alpha < 1\), the second-order condition is satisfied.
\subsection*{Part D}
{\label{116222}}
To solve for K, we begin with
\begin{quote}
\(K^{\frac{1}{1 - \alpha}} = (\frac{\alpha}{1 - \alpha}) \frac{WY^{\frac{1}{1 - \alpha}}}{r_K}\)
\(K= Y(\frac{\alpha}{1 - \alpha})^{1 - \alpha} \frac{W}{r_K}^{1 - \alpha}\)
\end{quote}
Given this equation, increases in Y (which could correspond to changes
in P) will raise K. The elasticity of K with respect to W is positive.
However, it's elasticity with respect to the rental price is negative.
We can also see that the elasticity of K with respect to Y is one.~
\section*{Question 9.3}
{\label{183123}}
The after-tax interest rate is
\begin{quote}
\(r - \tau i\)
\end{quote}
We are also given the equation
\begin{quote}
\(r_K(t) = [r(t) + \delta - (\frac{\dot p_K (t)}{p_K(t)})]p_K(t)\)
\end{quote}
in the book. This equation simply shows how a home-owner forgoes the
amount of interest that could be obtained by selling the home and also
pays an amount toward depreciation. By adding in the after-tax real
interest rate, this becomes
\begin{quote}
\(r_K(t) = [r(t) - \tau i (t) + \delta - (\frac{\dot p_K(t)}{p_K(t)})]p_K(t)\)
\end{quote}
Substituting~\(i(t) = r(t) + \pi (t)\) into the above gives
\begin{quote}
\(r_K(t) = [r(t) - \tau r (t) - \tau \pi (t) + \delta - (\frac{\dot p_K t(t)}{p_K(t)})]p_K(t)\)
\end{quote}
To find how an increase in inflation for a given interest rate would
affect the user cost of capital, we can take the first-derivative with
respect to inflation
\begin{quote}
\(\frac{\partial r_K}{\partial \pi (t)} = - \tau p_K (t) < 0\)
\end{quote}
Here we can see than an increase in inflation would reduce the user cost
of housing since it increases the tax-deductible nominal interest rate
payments. Therefore, an increase in inflation increases the desired
stock of owner-occupied housing for the market.~
\section*{Question 9.6}
{\label{183123}}
The equation of motion of the market value of capital:
\begin{quote}
\(\dot q (t) = r q(t) - \pi K(t)\)
\end{quote}
Thus, the condition required for equilibrium is
\begin{quote}
\(q = \frac{\pi K}{r}\)
\end{quote}
The equation of motion for capital:~
\begin{quote}
\(\dot K(t) = f(q(t))\)
\end{quote}
Thus, the condition required for equilibrium is
\begin{quote}
\(q = 1\)
\end{quote}
\subsection*{Part A}
{\label{722234}}
At the time of destruction, capital falls to half its initial level. For
the economy to return to its stable equilibrium, the market value must
adjust to the saddle path. Thus q jumps up immediately, putting the
economy at point A (below). In other words, the capital leftover is now
more valuable due to increased scarcity. Then, the economy moves down
the saddle path as q falls and capital rises. The higher market value
would attract investment, thus raising the stock of capital. Eventually
the economy would return to the original equilibrium point E.
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\subsection*{Part B}
{\label{722234}}
This changes the condition required for equilibrium to~
\begin{quote}
\(q = \frac{(1 - \tau) \pi K}{r}\)
\end{quote}
Thus the slope of the~\(\dot q = 0\) line is flatter and the new
locus is lower. At the time of the tax, q can jump down immediately but
it takes longer for the stock of capital to respond. The economy then
moves up the new saddle path as the stock of capital falls as q rises.
Finally, the market value of capital returns to the long-run equilibrium
value of 1 though the capital stock is at a permanently lower level.
This can be seen below:
\par\null\par\null\par\null\par\null\par\null\par\null
~
\subsection*{Part C}
{\label{722234}}
This sort of tax impacts the way a firm invests, which is given by
\begin{quote}
\(q(t) = 1 + \gamma + C'(I(t)) \)
\end{quote}
Here, the~\(\dot K = 0\) when~\(q(t) = 1 + \gamma\), meaning the
locus has shifted up. In words, this means that the q jumps up at the
time of the tax, while K shrinks as the economy moves along the new
saddle path until it reaches a new equilibrium at point E'. This is
because the tax has distorted the market so the lower level of existing
capital stock is worth more than before.
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\section*{Question 9.7}
{\label{183123}}
If the capital levy is levied, q will jump discontinuously. This point
is in a region where both q and K are falling. Since owners of the
capital will maximize and not expect avoidable capital losses, the value
of the capital before and after the levy must be equal as firms try to
get rid of capital in anticipation of the levy. At the time of the levy,
q will jump back up so that the economy is right back on the saddle
path. Therefore, over time, the capital stock accumulates again and the
economy is back at equilibrium.~
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\section*{Question 9.8}
{\label{183123}}
\begin{quote}
\(I = I(p_H), I'(*) > 0\)
\(\dot H = I - \delta H\)
\(R = R(H), R'(*) < 0\)
\(r = \frac{(R + \dot p_H)}{p_H}\)
\end{quote}
\subsection*{Part A}
{\label{196509}}
Given the evolution of the stock of housing function, the equilibrium
condition will be
\begin{quote}
\(I(p_H) = \delta H\)
\end{quote}
In words, that is the new investment in housing must offset the
depreciation of the existing housing exactly for the stock to remain
constant. The slope is given by the differentiation with respect to H
\begin{quote}
\(\frac{I'(p_H)p_H}{d H} = \delta\)
\end{quote}
Solving the rental income equation yields~
\begin{quote}
\(\dot p_H = rp_H - R(H)\)
\end{quote}
Thus the condition for equilibrium will be
\begin{quote}
\(\dot p_H = 0 = rp_H - R(H)\)
\end{quote}
Differentiating will give the slope of that locus
\begin{quote}
\(\frac{d p_H}{dH} = \frac{R'(H)}{r}\)
\end{quote}
The set of points would look as such:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\subsection*{Part B}
{\label{196509}}
Since~\(I'(p_H) > 0\), we know that the growth of housing is
increasing in price. Thus above the equilibrium locus, the growth rate
as well as stock of housing is also rising. Below the locus, it will be
falling as investment is too low to offset depreciation.
Since~\(R(H)' < 0\), the price of housing is increasing with the
stock of housing. This means that to the right of the locus the growth
of the price is rising and to the left it will be falling.~ Intuitively,
this means that the rent must be offset by capital gains.~
The dynamics along with the saddle path would look at such:
\par\null\par\null\par\null\par\null\par\null\par\null
\subsection*{Part C}
{\label{196509}}
Since the equilibrium price level is defined by~\(p_H = \frac{R(H)}{r}\), a
rise in the rate of return means that the equilibrium level will be
lower than before. Additionally, the slope will be less negative and
flatter than the one before. The housing level equilibrium locus is
unaffected by a permanent increase in the rate of return. At the time of
the increase, the stock of housing will slowly fall after a
discontinuous jump down in the price of housing onto a new saddle path,
caused by decreased investment. Shown below:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
~
\subsection*{Part D}
{\label{196509}}
If there is a permanent increase at a given time in the future, when
this news is known, there will be no change until the actual increase in
rate of return. At the time of the news, the price of housing had
shifted down, thus lowering investment, and eventually, the stock of
housing. Thus between the time of the news and increase in rate of
return, the stock is falling and rent is increasing.~
At the time of the increase, the equilibrium price locus shifts to the
left and becomes flatter, having the economy at a new saddle path.
Still, the stock of housing falls and rent rises. As the economy moves
along the new saddle path,~ the price of housing and investment rise
until it reaches a new long run equilibrium. Shown below:
\par\null\par\null\par\null\par\null\par\null\par\null\par\null
\subsection*{Part E}
{\label{196509}}
In this model, adjustment costs are external in this model. No internal
costs can be seen by the fact that there are no direct costs for
building new capital (houses). However, since the costs are external,~
this means that the real price of housing adjusts in the market as firms
make their investment decisions.~
\subsection*{Part F~}
{\label{196509}}
The~\(\dot H = 0\) locus is not horizontal because the investment
depends on the real price of housing. Here, depreciation is shown to be
proportional to the stock, thus higher levels of housing require more
investment to maintain it. We can see through~\(I'(p_H) > 0\) that
higher investment means the price of housing will be higher, and thus
the locus is upward sloping.~
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