Question 9.1

\(Y = K^\alpha L^{1 - \alpha}\)
\(P, Y, W, r_K\)are given

Part A

Solving for labor with prices, capital, output and rental price of capital given yields
\(L = Y^{\frac{1}{1 - \alpha}} K^{\frac{- \alpha}{1 - \alpha}}\) 

Part B

Since profits equal
\(\pi = PY - WL - r_K K \)
Adding in the optimal level of labor will yield
\(\pi = PY - W[Y^{\frac{1}{1 - \alpha}} K^{\frac{- \alpha}{1 - \alpha}}] - r_K K\)

Part C

The first-order condition for the profit-maximizing choice of K will be
\(\frac{\partial \pi}{\partial K} = \frac{\alpha}{1 - \alpha} WY ^{\frac{1}{1 - \alpha}} K^{\frac{- \alpha}{1 - \alpha} - 1} - r_K = 0\)
The second-order condition should be negative to verify that the first-order is a maximum. We can see this by
\(\frac{\partial \pi ^2}{\partial K ^2} = \frac{1}{1 - \alpha} \frac{\alpha}{1 - \alpha} WY ^{\frac{1}{1 - \alpha}} K^{\frac{ \alpha- 2}{1 - \alpha}}\)
Since \(\alpha < 1\), the second-order condition is satisfied.

Part D

To solve for K, we begin with
\(K^{\frac{1}{1 - \alpha}} = (\frac{\alpha}{1 - \alpha}) \frac{WY^{\frac{1}{1 - \alpha}}}{r_K}\)
\(K= Y(\frac{\alpha}{1 - \alpha})^{1 - \alpha} \frac{W}{r_K}^{1 - \alpha}\)
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. 

Question 9.3

The after-tax interest rate is
\(r - \tau i\)
We are also given the equation
\(r_K(t) = [r(t) + \delta - (\frac{\dot p_K (t)}{p_K(t)})]p_K(t)\)
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
\(r_K(t) = [r(t) - \tau i (t) + \delta - (\frac{\dot p_K(t)}{p_K(t)})]p_K(t)\)
Substituting \(i(t) = r(t) + \pi (t)\) into the above gives
\(r_K(t) = [r(t) - \tau r (t) - \tau \pi (t) + \delta - (\frac{\dot p_K t(t)}{p_K(t)})]p_K(t)\)
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
\(\frac{\partial r_K}{\partial \pi (t)} = - \tau p_K (t) < 0\)
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. 

Question 9.6

The equation of motion of the market value of capital:
\(\dot q (t) = r q(t) - \pi K(t)\)
Thus, the condition required for equilibrium is
\(q = \frac{\pi K}{r}\)
The equation of motion for capital: 
\(\dot K(t) = f(q(t))\)
Thus, the condition required for equilibrium is
\(q = 1\)

Part A

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.

Part B

This changes the condition required for equilibrium to 
\(q = \frac{(1 - \tau) \pi K}{r}\)
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:
 

Part C

This sort of tax impacts the way a firm invests, which is given by
\(q(t) = 1 + \gamma + C'(I(t)) \)
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.

Question 9.7

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. 

Question 9.8

\(I = I(p_H), I'(*) > 0\)
\(\dot H = I - \delta H\)
\(R = R(H), R'(*) < 0\)
\(r = \frac{(R + \dot p_H)}{p_H}\)

Part A

Given the evolution of the stock of housing function, the equilibrium condition will be
\(I(p_H) = \delta H\)
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
\(\frac{I'(p_H)p_H}{d H} = \delta\)
Solving the rental income equation yields 
\(\dot p_H = rp_H - R(H)\)
Thus the condition for equilibrium will be
\(\dot p_H = 0 = rp_H - R(H)\)
Differentiating will give the slope of that locus
\(\frac{d p_H}{dH} = \frac{R'(H)}{r}\)
The set of points would look as such:

Part B

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:

Part C

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:
 

Part D

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:

Part E

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. 

Part F 

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.