<p>Suppose we have the following utility function defined over two commodites,
<span class="math-container">$c_{1}$</span> and <span class="math-container">$c_{2}$</span>. The function is:
<span class="math-container">$$
U\left(c_{1},c_{2}\right)=\ln\left(c_{1}\right)+\ln\left(c_{2}\right)
$$</span>
subject to
<span class="math-container">$$
Y=p_{1}c_{1}+p_{2}c_{2}
$$</span>
.We can back out demand functions for <span class="math-container">$c_{1}$</span> and <span class="math-container">$c_{2}$</span>based on
income and relative prices. In particular, the following tangency
condition should hold after formulating the Lagrangean:
<span class="math-container">$$
\frac{c_{1}}{c_{2}}=\frac{p_{2}}{p_{1}}
$$</span>
Now, imagine that there is a per-unit tax levied on~<span class="math-container">$c_{2}$</span>, such that the
new price is <span class="math-container">$p_{2}\left(1+\tau\right)c_{2}$</span>.</p>
<p>The tangency condition
is affected:
<span class="math-container">$$
\frac{c_{1}}{c_{2}}=\frac{p_{2}(1+\tau)}{p_{1}}
$$</span>
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:
<span class="math-container">$$
Y+p_{2}\tau c_{2}=p_{1}c_{1}+p_{2}\left(1+\tau\right)c_{2}
$$</span>
Notice that the income of the consumer has not changed, as we can
subtract <span class="math-container">$p_{2}\tau c_{2}$</span> on both sides and get back the original
budget constraint:
<span class="math-container">$$
Y=p_{1}c_{1}+p_{2}c_{2}
$$</span>
My question is: why does the optimal value of <span class="math-container">$c_{1}$</span> and <span class="math-container">$c_{2}$</span>
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?</p>

CC BY-SA 4.0
2024-03-15T18:28:08.267Z
Hi! I think your description is a little imprecise. Can you give us the source of your claim "*why does the optimal value of $c_1$ and $c_2$ changed*" so that we can have more context on how exactly this tax works?
Giskard
CC BY-SA 4.0
2024-03-16T13:17:58.377Z
@Giskard Im not fully sure I follow. Essentially, the tangency condition is affected from the F.O.C, while the budget constraint is left unchanged, due to the revenue neutrality of the tax (the government returns it all). Consequently, the optimal values of c1 and c2 following the tax will change, with an unambiguous decrease in the level of c2. Why should this happen when consumers know they will get all of their taxes returned to them?
ChinG
CC BY-SA 4.0
2024-03-16T13:54:08.867Z
Okay, I will try to write it down differently. Why do you think this should happen? Did you read this somewhere? Do you have a source?
Giskard
CC BY-SA 4.0
2024-03-16T15:21:12.497Z
@Giskard Thanks for your response. I was simply trying to write down how a revenue neutral tax is levied in a standard economic model. The taxed amount is returned to households. Intuitively to me, the government is compensating the household for the income effect, but there is a sub effect still at play (relative price channel). While it makes sense mathematically, it does not make sense to me intuitively. If I am to be returned the full taxed amount of a taxed commodity, why should the tax matter in my calculus?
ChinG
CC BY-SA 4.0
2024-03-17T00:07:35.760Z
Where have you heard about this "standard model" and "revenue neutral tax"? Once you post your source, I will try to point out the half sentence I think you missed that changes a variable. Please post your source!
Giskard
<consumer-theory>
Why do revenue neutral taxes result in a change in behaviour
ChinG
# Why do revenue neutral taxes result in a change in behaviour

(Asked by: username_0 on 2024-03-15)

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?

**Tags:** consumer-theory

> **Comments:**
> **username_1**: Hi! I think your description is a little imprecise. Can you give us the source of your claim "*why does the optimal value of $c_1$ and $c_2$ changed*" so that we can have more context on how exactly this tax works?  
> **username_0**: @username_1 Im not fully sure I follow. Essentially, the tangency condition is affected from the F.O.C, while the budget constraint is left unchanged, due to the revenue neutrality of the tax (the government returns it all). Consequently, the optimal values of c1 and c2 following the tax will change, with an unambiguous decrease in the level of c2. Why should this happen when consumers know they will get all of their taxes returned to them?  
> **username_1**: Okay, I will try to write it down differently. Why do you think this should happen? Did you read this somewhere? Do you have a source?  
> **username_0**: @username_1 Thanks for your response. I was simply trying to write down how a revenue neutral tax is levied in a standard economic model. The taxed amount is returned to households. Intuitively to me, the government is compensating the household for the income effect, but there is a sub effect still at play (relative price channel). While it makes sense mathematically, it does not make sense to me intuitively. If I am to be returned the full taxed amount of a taxed commodity, why should the tax matter in my calculus?  
> **username_1**: Where have you heard about this "standard model" and "revenue neutral tax"? Once you post your source, I will try to point out the half sentence I think you missed that changes a variable. Please post your source!  

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## 0 answers

