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?
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.