Suppose we have the following utility function defined over two commodites, $c_{1}$ and $c_{2}$. The function is: $$ U\left(c_{1},c_{2}\right)=\ln\left(c_{1}\right)+\ln\left(c_{2}\right) $$ subject to $$ Y=p_{1}c_{1}+p_{2}c_{2} $$ .We can back out demand functions for $c_{1}$ and $c_{2}$based on income and relative prices. In particular, the following tangency condition should hold after formulating the Lagrangean: $$ \frac{c_{1}}{c_{2}}=\frac{p_{2}}{p_{1}} $$ Now, imagine that there is a per-unit tax levied on~$c_{2}$, such that the new price is $p_{2}\left(1+\tau\right)c_{2}$.
The tangency condition is affected: $$ \frac{c_{1}}{c_{2}}=\frac{p_{2}(1+\tau)}{p_{1}} $$ However, the government returns all the taxable income to the consumer (such that it is a revenue neutral tax) such that the budget constraint is now: $$ Y+p_{2}\tau c_{2}=p_{1}c_{1}+p_{2}\left(1+\tau\right)c_{2} $$ Notice that the income of the consumer has not changed, as we can subtract $p_{2}\tau c_{2}$ on both sides and get back the original budget constraint: $$ Y=p_{1}c_{1}+p_{2}c_{2} $$ My question is: why does the optimal value of $c_{1}$ and $c_{2}$ change relative to the case when there was no tax? It obviously will because of the relative price change, and it can be seen from the new tangency condition. Put differently, if I know that all of the taxed income will be returned to me, then why does the model predict that \emph{any }change in behaviour will take place?
