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I'm studying the Slutsky equation and an example in the text discusses the effect of a tax plus rebate on the consumption of fuel.

Suppose the original price of fuel is $p$, tax is $t$, $x$ is the amount of fuel purchased, $y$ the expenditure on other goods, and $m$ is income.

Now, the text says that if the original bundle consumed satisfied $px+y=m$, the new bundle will satisfy $(p+t)x'+y'=m+tx'$(the rebate) , the final consumption of fuel $x'$ will be less than $x$, and it satisfies the equation $px'+y'=m$ which means the consumer will end up worse off.

Ok, so my question is basically, why doesn't the consumer consume the original bundle? How is the optimal bundle chosen?

Thanks

EDIT: Ok, so does this work?

We start with the equation $(p+t)x+y=m$ , i.e. before we receive the rebate. Then the increase in the price of the fuel means we consume (assuming the original bundle is $(x_0,y_0)$) $(x_1,y_1)$ with $x_1<x_0$. Then we receive a rebate of $tx_1$, so we can consume more of fuel and other goods. So we want to solve for $(p+t)x+y=m+tx_1$, which gives $(x_2,y_2)$ , with $x_1<x_2<x_0$, then $(x_3,y_3),\ldots$ finally converging to an $(x^*,y^*)$ bundle with $x^*\le x_0$.

Finally, $(x^*,y^*)$ will satisfy $(p+t)x^*+y^*=m+tx^*$, and so it lies on the original budget line, and the consumer has ended up worse off.

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  • $\begingroup$ "How is the optimal bundle chosen?" I am confused. Did you start your textbook with the chapter on the Slutsky equation? Wasn't there a chapter on Choice earlier? What exactly are you asking here? $\endgroup$
    – Giskard
    Jul 13 at 20:10
  • $\begingroup$ I didn't understand why the original bundle could not be chosen. In my edit I tried to give an explanation for why not.. $\endgroup$ Jul 13 at 20:38
  • $\begingroup$ For the original bundle $px+y=m$ holds. Adding $tx$ to both sides we get $(p+t)x+y=m+tx$, hence the original bundle can be chosen in the tax + rebate situation as well. So I still do not understand your question, I think it is unclear. Hopefully one of the upvoters of your question understands it better and will return with an answer. $\endgroup$
    – Giskard
    Jul 14 at 3:41
  • $\begingroup$ For a worse-off outcome, do you not need to assume that the choice maker does not know that the rebate is directly related to the taxes and amount bought. In other words, is the income in the tax case $m'=m+\Delta m'$, where only the authorities know that $\Delta m' = tx'$? $\endgroup$
    – BrsG
    Jul 14 at 13:50
  • $\begingroup$ Then in case the consumer knows the rebate, they will get back to the original bundle? $\endgroup$ Jul 14 at 14:58

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