I am given a utility function $U(x,y)$, income $I$ and prices $p_x,p_y$. Then the price of $y$ rises to $p_y' = 3 \cdot p_y$. How are we supposed to calculate the substitution effect and income effect on the demand for x?
I performed a standard Slutsky decomposition, but I mistakenly began with the precondition $x_0 = x_1$. Of course, this meant that the $∆x^s$ and $∆x^i$ were all screwed up (there would be no intermediary step if the beginning and end were the same). I know that substitution/income effects are calculated from relative price changes, so would an alternate approach be to treat the tripling of $p_y$ as equivalent to multiplying $p_x$ by $1/3$? And then simply proceeding from there?
Edit: Due to popular demand, here is a rephrasing of the essence of my question:
How is a Slutsky/Hicksian decomposition calculated under a cross price change ($p_y$ changes, affecting $x$, for example.)? An algebraic walkthrough given a Utility function: $x^{a}y^{1-a}$ and a budget line $P_xX+P_YY = I$, and a given $I_0$ would be helpful.
