<p>It may depend on what $c$ represents and how the tax is applied</p>

<ul>
<li><p>If $c$ is the quantity actually consumed then it remains $u(c)$</p></li>
<li><p>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 $u((1-\tau)c)$ </p></li>
<li><p>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 $u \left({c}\right)$ </p></li>
<li><p>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 $u \left(\frac{c}{1+\tau}\right)$</p></li>
</ul>

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

CC BY-SA 4.0
2018-08-13T09:36:50.173Z
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
CC BY-SA 4.0
2018-08-13T09:39:39.767Z
@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
CC BY-SA 4.0
2018-08-13T09:49:36.070Z
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
CC BY-SA 4.0
2018-08-12T23:24:50.287Z
2018-08-12T23:24:50.287Z
Henry
<p>U(c). </p>

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

<p>U(c) just measures the utility you get from consuming c. </p>

<p>As an aside, consumption taxes should not be 1-t. </p>

<p>Example: Investment = F - (1+t)C = Net Savings.   </p>

<p>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. </p>

<p><strong>EDIT</strong></p>

<p>The answer by @Henry got me thinking about the following. </p>

<p>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 :</p>

<p>Investment = Output - Price * C  </p>

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

CC BY-SA 4.0
2018-08-10T21:44:58.033Z
2018-08-13T09:48:17.813Z
2018-08-13T09:48:17.813Z
erik
<p>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 </p>

<utility><taxation>
Utility and consumption tax
user19108
# Utility and consumption tax

(Asked by: username_0 on 2018-08-10)

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

**Tags:** utility, taxation


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## 2 answers

### Answer by username_1 on 2018-08-12. Score: 1

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 $u((1-\tau)c)$

> **Comments:**
> **username_2**: 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.  
> **username_1**: @username_2 - No.  I was just trying to think of a case where  the $u((1-\tau)c)$ in the question might actually mean something sensible  
> **username_2**: 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).  

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### Answer by username_2 on 2018-08-10. Score: 1

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 @username_1 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.


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