<p>Ultimately you can make up any assumption on utility you want so you could in principle say that: <span class="math-container">$U_i^{new} = U_i^{old} - tx_i$</span> however this is very crude way of introducing an effect of taxation.</p>
<p>Here you are basically assuming the tax will not change peoples behavior of how much they are driving. Rather you just assume it affects their welfare and that's it. However, we know that people respond to incentives so it is not good way to model effect of tax this way.</p>
<p>What you should rather do is to explicitly model how distance travelled is affected by tax. Next the utility after tax will be just utility with new distances travelled.</p>
<p>If you want to follow this advice your new utility should look like this:</p>
<p><span class="math-container">$$U_i(x_1(t), x_2(t), \cdots, x_n(t))$$</span></p>
<p>Next add some budget constraint, agents can drive only if they can afford it, so add something like:</p>
<p><span class="math-container">$$w_i=(1+t)px_i$$</span></p>
<p>where <span class="math-container">$w$</span> would be person's budget for travel <span class="math-container">$t$</span> tax <span class="math-container">$p$</span> price of travelling 1 unit of distance by the agent. To make it more interesting you could also add some alternative to travel by car. Maybe agent could take public transport so you would have:</p>
<p><span class="math-container">$$U_i(x_i(t), y_i) \quad s.t. w_i=(1+t)px_i + qy_i$$</span></p>
<p>where <span class="math-container">$y_i$</span> is distance traveled by public transport.</p>

CC BY-SA 4.0
2022-09-23T20:38:56.733Z
2022-09-23T22:27:15.617Z
2022-09-23T22:27:15.617Z
1muflon1
<p>One can model this scenario as a simultaneous-move strategic game, for a given <span class="math-container">$t\geq 0$</span>:</p>
<ul>
<li>Set of Players: <span class="math-container">$N=\{1,2,\ldots,n\}$</span></li>
<li>Action Set <span class="math-container">$A_i$</span> of each player <span class="math-container">$i\in N$</span> is simply <span class="math-container">$\mathbb{R}_+$</span></li>
<li>Preferences are Quasi-linear. For each <span class="math-container">$i\in N$</span>, preferences are represented by some
<span class="math-container">$u_i:A_1\times A_2\times\cdots\times A_n\rightarrow\mathbb{R}$</span> is given by
<span class="math-container">$u_i(x_1,x_2,\ldots,x_n)=v_i(x_1,x_2,\ldots,x_n)-tx_i$</span></li>
</ul>
<p>Now one can solve for the Nash equilibrium of the above game, which may depend on the tax-rate parameter <span class="math-container">$t$</span>.</p>
<p>Valuation functions <span class="math-container">$v_i$</span>, you may consider are</p>
<ul>
<li><span class="math-container">$v_i(x_1,x_2,\ldots,x_n)=a_i\sqrt{x_i}-\sum_{j=1}^{n}x_j$</span>, where <span class="math-container">$\sum_{j=1}^{n}x_j$</span> is total congestion or pollution and affects the valuation of <span class="math-container">$i$</span> negatively, <span class="math-container">$a_i&gt;0$</span> is <span class="math-container">$i$</span>'s parameter</li>
<li><span class="math-container">$v_i(x_1,x_2,\ldots,x_n)=a_i\ln(x_i)-\sum_{j=1}^{n}x_j$</span></li>
<li><span class="math-container">$v_i(x_1,x_2,\ldots,x_n)=f_i(x_i)-\sum_{j=1}^{n}x_j$</span>, where <span class="math-container">$f_i$</span> is increasing and concave.</li>
</ul>

CC BY-SA 4.0
2024-02-16T13:46:14.063Z
2024-02-16T13:57:26.900Z
2024-02-16T13:57:26.900Z
Amit
<p>Suppose there are <span class="math-container">$n$</span> agents in an economy and the utility of person <span class="math-container">$i$</span> is given by <span class="math-container">$U_i(x_1, x_2, \cdots, x_n)$</span> where <span class="math-container">$x_j$</span> denotes the distance driven by each agent in the economy. One's utility depends on how much others drive due to road congestion and pollution. To contain road congestion, the local government decides to impose a unit-tax on the distance covered by each individual. Unit tax here refers to a tax on each kilometre a person drives.</p>
<p>How does the tax interact with a person's utility? I know that less kilometres will be travelled due to the tax, but I can't say anything beyond that. Is <span class="math-container">$U_i^{new} = U_i^{old} - tx_i$</span> a valid way of introducing the tax to the utility function?</p>

<microeconomics><taxation>
How tax relates with utility function?
Bayes
# How tax relates with utility function?

(Asked by: username_0 on 2022-09-23)

Suppose there are $n$ agents in an economy and the utility of person $i$ is given by $U\_i(x\_1, x\_2, \cdots, x\_n)$ where $x\_j$ denotes the distance driven by each agent in the economy. One's utility depends on how much others drive due to road congestion and pollution. To contain road congestion, the local government decides to impose a unit-tax on the distance covered by each individual. Unit tax here refers to a tax on each kilometre a person drives.

How does the tax interact with a person's utility? I know that less kilometres will be travelled due to the tax, but I can't say anything beyond that. Is $U\_i^{new} = U\_i^{old} - tx\_i$ a valid way of introducing the tax to the utility function?

**Tags:** microeconomics, taxation

---

## 2 answers

### Answer by username_1 on 2022-09-23. Score: 1

Ultimately you can make up any assumption on utility you want so you could in principle say that: $U\_i^{new} = U\_i^{old} - tx\_i$ however this is very crude way of introducing an effect of taxation.

Here you are basically assuming the tax will not change peoples behavior of how much they are driving. Rather you just assume it affects their welfare and that's it. However, we know that people respond to incentives so it is not good way to model effect of tax this way.

What you should rather do is to explicitly model how distance travelled is affected by tax. Next the utility after tax will be just utility with new distances travelled.

If you want to follow this advice your new utility should look like this:

$$U\_i(x\_1(t), x\_2(t), \cdots, x\_n(t))$$

Next add some budget constraint, agents can drive only if they can afford it, so add something like:

$$w\_i=(1+t)px\_i$$

where $w$ would be person's budget for travel $t$ tax $p$ price of travelling 1 unit of distance by the agent. To make it more interesting you could also add some alternative to travel by car. Maybe agent could take public transport so you would have:

$$U\_i(x\_i(t), y\_i) \quad s.t. w\_i=(1+t)px\_i + qy\_i$$

where $y\_i$ is distance traveled by public transport.


---

### Answer by username_2 on 2024-02-16. Score: 0

One can model this scenario as a simultaneous-move strategic game, for a given $t\geq 0$:

* Set of Players: $N=\{1,2,\ldots,n\}$
* Action Set $A\_i$ of each player $i\in N$ is simply $\mathbb{R}\_+$
* Preferences are Quasi-linear. For each $i\in N$, preferences are represented by some
  $u\_i:A\_1\times A\_2\times\cdots\times A\_n\rightarrow\mathbb{R}$ is given by
  $u\_i(x\_1,x\_2,\ldots,x\_n)=v\_i(x\_1,x\_2,\ldots,x\_n)-tx\_i$

Now one can solve for the Nash equilibrium of the above game, which may depend on the tax-rate parameter $t$.

Valuation functions $v\_i$, you may consider are

* $v\_i(x\_1,x\_2,\ldots,x\_n)=a\_i\sqrt{x\_i}-\sum\_{j=1}^{n}x\_j$, where $\sum\_{j=1}^{n}x\_j$ is total congestion or pollution and affects the valuation of $i$ negatively, $a\_i>0$ is $i$'s parameter
* $v\_i(x\_1,x\_2,\ldots,x\_n)=a\_i\ln(x\_i)-\sum\_{j=1}^{n}x\_j$
* $v\_i(x\_1,x\_2,\ldots,x\_n)=f\_i(x\_i)-\sum\_{j=1}^{n}x\_j$, where $f\_i$ is increasing and concave.


---

