<p>It's just algebraic rearrangement. Below I suppress the subscripts and the argument of <span class="math-container">$A$</span>:</p>
<p>Take the first term on the RHS of profit,
<span class="math-container">\begin{align}
PA&amp;=\frac{\mathrm dC(A)}{\mathrm dA}\cdot A\\
&amp;=\frac{C(A)}{C(A)}\cdot\frac{\mathrm dC(A)}{\mathrm dA}\cdot A\\
&amp;=C(A)\frac{\mathrm dC(A)}{C(A)}\cdot\frac{A}{\mathrm dA}\\
&amp;=C(A)\cdot \mathrm d\ln C(A)\cdot\frac{1}{\mathrm d\ln A}
\end{align}</span>
The rest should be straightforward.</p>

CC BY-SA 4.0
2022-02-10T21:36:25.493Z
2022-02-10T21:36:25.493Z
Herr K.
<p>I came across the paper &quot;The Microgeography of Housing Supply&quot; (Baum-Snow, 2020)(<a href="https://luhan2.weebly.com/uploads/9/6/0/8/96080580/housing_supply_julyb2020.pdf" rel="nofollow noreferrer">https://luhan2.weebly.com/uploads/9/6/0/8/96080580/housing_supply_julyb2020.pdf</a>) and I must admit that I do not know how to solve the bid-rent function in it as described in the screenshot attached. When I put the optimal living area A=A(P) and the marginal price P=dC(A)/dA, I do not end up with the result implied by the paper. Are they using a particular theorem here?</p>
<p><a href="https://i.sstatic.net/Cb7H0.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Cb7H0.png" alt="enter image description here" /></a></p>

<profit-maximization><profits>
Bid-rent function in The Microgeography of Housing Supply
housing_guy_8192
# Bid-rent function in The Microgeography of Housing Supply

(Asked by: username_0 on 2022-02-10)

I came across the paper "The Microgeography of Housing Supply" (Baum-Snow, 2020)(<https://luhan2.weebly.com/uploads/9/6/0/8/96080580/housing_supply_julyb2020.pdf>) and I must admit that I do not know how to solve the bid-rent function in it as described in the screenshot attached. When I put the optimal living area A=A(P) and the marginal price P=dC(A)/dA, I do not end up with the result implied by the paper. Are they using a particular theorem here?

[![enter image description here](https://i.sstatic.net/Cb7H0.png)](https://i.sstatic.net/Cb7H0.png)

**Tags:** profit-maximization, profits

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## 1 answers

### Answer by username_1 on 2022-02-10. Score: 3

It's just algebraic rearrangement. Below I suppress the subscripts and the argument of $A$:

Take the first term on the RHS of profit,
\begin{align}
PA&=\frac{\mathrm dC(A)}{\mathrm dA}\cdot A\\
&=\frac{C(A)}{C(A)}\cdot\frac{\mathrm dC(A)}{\mathrm dA}\cdot A\\
&=C(A)\frac{\mathrm dC(A)}{C(A)}\cdot\frac{A}{\mathrm dA}\\
&=C(A)\cdot \mathrm d\ln C(A)\cdot\frac{1}{\mathrm d\ln A}
\end{align}
The rest should be straightforward.


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