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.

  • $\begingroup$ No. If you are only looking at ratios you would also have to change the income, so that the $\frac{I}{p_x}$ ratios are the same. I am voting to close this question because its first half is a numerical problem. $\endgroup$
    – Giskard
    Feb 15, 2016 at 22:17
  • $\begingroup$ I looked up related questions and found a similar question (not involving cross price changes) that had an algebraic component. That question was not "shut down" and received several good answers that were much less conceptual than the answer I have asked for. I am not asking for a numerical solution, I want to know how cross price changes effect Slutsky decomp and if you cannot offer a reasonable explanation for how the variation in this problem changes the decomposition process I would ask that you not comment at all, or alternatively edit it to omit the parts you find so disturbing. $\endgroup$
    – bloopton
    Feb 15, 2016 at 22:57
  • $\begingroup$ And from your comment, I hope you can see why a numerical/analytical walk through is so needed, as your own comment really had nothing to do with my question and would perhaps be better expressed in relation to my own hypothesized approach of changing the prices in a way that maintained the price ratio. $\endgroup$
    – bloopton
    Feb 15, 2016 at 23:13
  • $\begingroup$ And in addition, not two posts down there is a problem that has a very explicitly numerical component specifically labeled as an "exercise" that you yourself edited so I have flagged your comment as unconstructive and would request that you are at least consistent and fair in your unconstructive comments in the future. $\endgroup$
    – bloopton
    Feb 15, 2016 at 23:17
  • $\begingroup$ As a closing statement, the flag/report conditions are as follows: This question is completely unclear, incomplete, overly-broad, primarily opinion-based or is not about economics as described in the help center, and it is unlikely to be fixed via editing. $\endgroup$
    – bloopton
    Feb 16, 2016 at 0:05


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