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I've read (pg 10) in Gourinchas' notes on consumption that the income and substitution effects cancel for log preferences, and I tried to prove this to myself doing the Slutsky decomposition for the simple case of $U(x,y) = \ln(x) + \ln(y)$ s.t. $p_x x + p_y y = I$. I get that the Hicksian and Marshallian demand functions are $x^* = \frac{p_y}{p_x} y = \frac{I}{2 p_x} $ respectively. Plugging these in gives me this for the decomposition:

$$ \frac{\partial x^*}{\partial p_x} = - \frac{p_y}{p_x^2} y - \frac{1}{2 p_x} \frac{p_y}{p_x} y $$

which is definitely not zero. Am I misunderstanding?

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    $\begingroup$ Hicksian demand is not a function of income. $\endgroup$ – Giskard Aug 31 at 7:18
  • $\begingroup$ Thanks--edited for clarity. My math is done for $h(\mathbf{p}, \bar{u}) = \frac{p_y}{p_x} y \implies \frac{\partial h}{\partial p_x} = - \frac{p_y}{p_x^2} y$ $\endgroup$ – aintgeorge Aug 31 at 19:05
  • $\begingroup$ When you have $h(\textbf{p},\bar{u})$, shouldn't the stuff on the other side be only a function of the prices and $\bar{u}$? Yet you also have $y$ in $\frac{p_y}{p_x}y$. Perhaps it is this and the missing $\bar{u}$ that is causing the problem? $\endgroup$ – Giskard Sep 1 at 6:12
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You interpreted the passage you read as essentially saying that "with log preferences, the Marshallian demand for the good does not depend on its price." That doesn't sound very plausible, don't you think?

But the passage clearly refers to an intertemporal choice problem, it is about log-preferences over a single good and over time, it does not relate to the Slutsky decomposition of the static Marshallian demand function.

I would suggest to continue reading the notes up to page 14.

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