How do I get to this demand function in the monocentric city model?

Question

I need to get this resulting price and quantity (housing):

It's pretty clear that the denominator of the quantity function is just the price function.

From this utility function:

And this constraint:

Now this is a monocentric city model function, so these methods follow:

or:

However, I do not see the actual process of doing that. I need to make this calculations for other utility functions, but when I follow the steps, I do not get the desired result.

Answer 1

Let $K:=y-(f+fa)-(t+ta)x$, and so $c=K-pq$.

From condition $(\mathrm{A1})$, \begin{align} &&\frac{v_q(\cdot)}{v_c(\cdot)}&=p \\ \quad\Rightarrow\quad && \frac{\alpha[K-pq]^{1-\alpha}q^{\alpha-1}}{(1-\alpha)[K-pq]^{-\alpha}q^\alpha}&=p \\ \quad\Rightarrow\quad && pq&=\alpha K \\ \quad\Rightarrow\quad && q&=\frac{\alpha K}{p}. \tag{*} \end{align} From $(\mathrm{A2})$ and $(*)$, \begin{align} &&[K-pq]^{1-\alpha}q^\alpha &= u \\ \quad\overset{(*)}{\Rightarrow}\quad &&[(1-\alpha)K]^{1-\alpha}\biggl(\frac{\alpha K}{p}\biggr)^\alpha &= u\\ \quad\Rightarrow\quad && (1-\alpha)^{1-\alpha}\alpha^\alpha \frac{K}{u}&=p^\alpha. \end{align} Thus, \begin{align} p(\cdot)&=\left[(1-\alpha)^{1-\alpha}\alpha^\alpha \frac{K}{u}\right]^{1/\alpha}\\ q(\cdot)&=\frac{\alpha K}{\left[(1-\alpha)^{1-\alpha}\alpha^\alpha \frac{K}{u}\right]^{1/\alpha}} \end{align}