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Taking the ratio of the rearranged second and third equations of The comparative static identities: $$ \begin{array}{r}B-x^{*} P_{x}-y^{*} P_{y} \equiv 0 \\ U_{x}\left(x^{*}, y^{*}\right)-\lambda^{*} P_{x} \equiv 0 \\ U_{y}\left(x^{*}, y^{*}\right)-\lambda^{*} P_{y} \equiv 0\end{array} $$ we get $$ \frac{U_{x}\left(x^{*}, y^{*}\right)}{U_{y}\left(x^{*}, y^{...


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