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In the example on Voluntary Real Time Pricing in "Chapter 8 Slutsky Equation", anyone can help to explain why the mentioned pricing scheme is a "Slutsky pivot"?

In the textbook, it says that "they receive a rebate based on the high real time price for every kilowatt of reduced usage". How can this rebate compensate the user to achieve the original purchasing power?

The following is my understanding: Suppose in normal times, the user choose $(x_1^*, x_2^*)$ with the budget constraint $$ p x_1 + x_2 = m. $$ When the temperature rises, the electricity prices rises but with a rebate based on the high real time price for every kilowatt of reduced usage. Then, the new budget constraint is $$ p' x_1 + x_2 = m + p'(x_1^* - x_1). $$ We plug $(x_1^*, x_2^*)$ into the new budget constraint and the leftside is not equal to the rightside. Hence, $(x_1^*, x_2^*)$ is not on the new budget line and we cannot compensate the user. This indicates that it is not like a slutsky pivot. Where am I wrong here?

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The book does not claim that $$ p' x_1 + x_2 = m + p'(x_1^* - x_1) $$ is the equation describing the new budget line. The budget line with slope $-p'$ that passes through $(x_1^*,x_2^*)$ is $$ p' x_1 + x_2 = p' x_1^* + x_2^*. $$ If you wish you can add zero to this $$ p' x_1 + x_2 = p' x_1^* + x_2^* + (m - p x_1^* - x_2^*) $$ and reorganize it to $$ p' x_1 + x_2 = (p'-p) x_1^* + m, $$ which shows that the consumer is compensated by $(p'-p) x_1^*$ dollars. I agree that this is not best described as

they receive a rebate based on the high real time price for every kilowatt of reduced usage

It is fair to say that the rebate incentivizes reduced electricity ($x_1$) usage, as you can again reorganize the above to $$ x_2 = m -p x_1^* + p' (x_1^* - x_1), $$ which shows how much money you have left for other goods, and this indeed depends on the high real time price $p'$ and the marginal reduction $x_1^* - x_1$.

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