# Utility and consumption tax

If I have a model with taxes on consumption denoted $\tau$ should I write the utility function as $u(c)$ or $u((1-\tau)c)$? Thanks

## 2 Answers

It may depend on what $c$ represents and how the tax is applied

• If $c$ is the quantity actually consumed then it remains $u(c)$

• If $c$ is the quantity purchased and then you have to pass a quantity $\tau c$ over as taxes keeping $(1-\tau)c$ for actual consumption then it is $u((1-\tau)c)$

• If $c$ is the expenditure on consumption which has a unit price of $p$ and you then have to pay an additional $\tau c$ in tax then it is $u \left(\frac{c}{p}\right)$; if the price is $p=1$ then this is $u \left({c}\right)$

• If $c$ is the expenditure on consumption including tax which therefore has a unit price of $p(1+\tau)$ then it is $u \left(\frac{c}{p(1+\tau)}\right)$; if the pre-tax price is $p=1$ then this is $u \left(\frac{c}{1+\tau}\right)$

Note that $u \left(\frac{c}{1+\tau}\right)$ is not the same as $u((1-\tau)c)$

• Nice answer. I think your response is much more general and improved compared to mine. Incidentally, I only assumed the case in your first point, but have you seen any paper modeling along the lines of your second point? Asking out of curiosity. – erik Aug 13 '18 at 9:36
• @erik - No. I was just trying to think of a case where the $u((1-\tau)c)$ in the question might actually mean something sensible – Henry Aug 13 '18 at 9:39
• I see. I made an edit based on an idea I got from your point (2). I have a feeling that it is possible to set up a way in which one can freely move between your points (1) and (2). – erik Aug 13 '18 at 9:49

U(c).

Taxes are an adjustment on your income/resource and show up on your budget constraint.

U(c) just measures the utility you get from consuming c.

As an aside, consumption taxes should not be 1-t.

Example: Investment = F - (1+t)C = Net Savings.

If you wrote (1-t), it would mean, Investment = F -C + tC. Higher is the tax rate, more is the investment or budget left over for investment, ceteris paribus - which does not make sense.

EDIT

The answer by @Henry got me thinking about the following.

If a model is proposed where a consumer purchases C units of consumption goods taxed at rate t, such that the consumer will have to give away tC units in taxes and ultimately will consume (1-t)C, the analysis is based on tangible goods. In the budget constraint of the consumer it will read as :

Investment = Output - Price * C

since the consumer will have paid for C units and then ultimately consumed (1-t)C units due to the tax.