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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

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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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