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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.

bloopton
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  • 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. – Giskard Feb 15 '16 at 22:17
  • 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. – bloopton Feb 15 '16 at 22:57
  • 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. – bloopton Feb 15 '16 at 23:13
  • 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. – bloopton Feb 15 '16 at 23:17
  • 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. – bloopton Feb 16 '16 at 00:05
  • To be perfectly clear: My comment answered your pre-edit question: Your proposed approach would yield different results. I gave the reason why. – Giskard Feb 16 '16 at 00:48
  • With regard to all your other comments: If you were trying to convince me, linking to one of these "similar" questions would have been helpful. But since in your comment you state "I am not asking for a numerical solution" I am removing that superfluous part from your question. – Giskard Feb 16 '16 at 00:49
  • I didn't think I needed to link considering you were the one who seemed to have been fine with it, but here, pulled one off the front page after half a second of searching: http://economics.stackexchange.com/questions/10691/elasticity-of-demand-equals-1-but-income-decreases – bloopton Feb 16 '16 at 00:55
  • And as for the similar question, here is the link for this one: http://economics.stackexchange.com/questions/5404/calculate-the-substitution-effect I don't think I need to explain just how "numerical" these are even compared to my pre edit question. – bloopton Feb 16 '16 at 00:57
  • And while I am not indeed looking for a numerical solution, I detailed that one would be helpful in demonstrating the conceptual answers given, so your removal of the numerical section of my answer (for no good reason) seems to be more out of pettiness and a draconian pursuit of adherence to flagging rules that, as I have said before, are not consistent with the website's policy. – bloopton Feb 16 '16 at 00:59
  • I really have no idea why you decided to want to close this question, I would like to believe that the concepts I am inquiring about are worth a thread and that you instead are just choosing to be difficult for no good reason. – bloopton Feb 16 '16 at 01:04
  • Let us continue this discussion in chat. – bloopton Feb 16 '16 at 03:27
  • "I would like to believe [...] that you instead are just choosing to be difficult for no good reason." You are entitled to your beliefs. After editing out the numerical part yesterday I withdrew my close vote. I am still unsure how I have not answered your original question. (Perhaps you are too upset with me to think about the answer given in my very first comment.) I hope someone will provide you with a more detailed answer. – Giskard Feb 16 '16 at 09:50

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