# Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition

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?

• Hicksian demand is not a function of income. – Giskard Aug 31 '19 at 7:18
• Thanks--edited for clarity. My math is done for $h(\mathbf{p}, \bar{u}) = \frac{p_y}{p_x} y \implies \frac{\partial h}{\partial p_x} = - \frac{p_y}{p_x^2} y$ – aintgeorge Aug 31 '19 at 19:05
• 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? – Giskard Sep 1 '19 at 6:12