<p>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.</p>
<p>Suppose the original price of fuel is <span class="math-container">$p$</span>, tax is <span class="math-container">$t$</span>, <span class="math-container">$x$</span> is the amount of fuel purchased, <span class="math-container">$y$</span> the expenditure on other goods, and <span class="math-container">$m$</span> is income.</p>
<p>Now, the text says that if the original bundle consumed satisfied <span class="math-container">$px+y=m$</span>, the new bundle will satisfy <span class="math-container">$(p+t)x'+y'=m+tx'$</span>(the rebate) , the final consumption of fuel <span class="math-container">$x'$</span> will be less than <span class="math-container">$x$</span>, and it satisfies the equation <span class="math-container">$px'+y'=m$</span> which means the consumer will end up worse off.</p>
<p>Ok, so my question is basically, why doesn't the consumer consume the original bundle? How is the optimal bundle chosen?</p>
<p>Thanks</p>
<p>EDIT: Ok, so does this work?</p>
<p>We start with the equation <span class="math-container">$(p+t)x+y=m$</span> , i.e. before we receive the rebate. Then the increase in the price of the fuel means we consume (assuming the original bundle is <span class="math-container">$(x_0,y_0)$</span>) <span class="math-container">$(x_1,y_1)$</span> with <span class="math-container">$x_1&lt;x_0$</span>. Then we receive a rebate of <span class="math-container">$tx_1$</span>, so we can consume more of fuel and other goods. So we want to solve for <span class="math-container">$(p+t)x+y=m+tx_1$</span>, which gives <span class="math-container">$(x_2,y_2)$</span> , with <span class="math-container">$x_1&lt;x_2&lt;x_0$</span>, then <span class="math-container">$(x_3,y_3),\ldots$</span> finally converging to an <span class="math-container">$(x^*,y^*)$</span> bundle with <span class="math-container">$x^*\le x_0$</span>.</p>
<p>Finally, <span class="math-container">$(x^*,y^*)$</span> will satisfy <span class="math-container">$(p+t)x^*+y^*=m+tx^*$</span>, and so it lies on the original budget line, and the consumer has ended up worse off.</p>

CC BY-SA 4.0
2021-07-13T20:10:14.037Z
"*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?
Giskard
CC BY-SA 4.0
2021-07-13T20:38:42.567Z
I didn't understand why the original bundle could not be chosen. In my edit I tried to give an explanation for why not..
AnotherJohnDoe
CC BY-SA 4.0
2021-07-14T03:41:01.773Z
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.
Giskard
CC BY-SA 4.0
2021-07-14T13:50:50.250Z
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 $\Delta m' = tx'$?
BrsG
CC BY-SA 4.0
2021-07-14T14:58:23.597Z
Then in case the consumer knows the rebate, they will get back to the original bundle?
AnotherJohnDoe
<taxation><slutsky-equation>
Effect of tax plus rebate on fuel consumption
AnotherJohnDoe
# Effect of tax plus rebate on fuel consumption

(Asked by: username_0 on 2021-07-13)

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.

**Tags:** taxation, slutsky-equation

> **Comments:**
> **username_1**: "*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?  
> **username_0**: I didn't understand why the original bundle could not be chosen. In my edit I tried to give an explanation for why not..  
> **username_1**: 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.  
> **username_2**: 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 $\Delta m' = tx'$?  
> **username_0**: Then in case the consumer knows the rebate, they will get back to the original bundle?  

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