<p>I think Slutsky symmetry also implies <span class="math-container">$\alpha_D\epsilon_{DC} = (1-\alpha_D)\epsilon_{CD}$</span>. Using this, the equality follows.</p>

CC BY-SA 4.0
2024-10-11T14:54:54.093Z
2024-10-11T14:54:54.093Z
VARulle
<p>I am replicating the derivations from the paper <em><strong>Environmental policy, public finance and the labour market in a second-best world</strong></em> (Bovenberg &amp; van der Ploeg, <em>Journal of Public Economics</em>, 1994), but I’m struggling with a key step.</p>
<p>First of all: some context. In this section of the paper, the government is trying to raise a given revenue in the least distortionary way by imposing taxes on a good <span class="math-container">$D$</span> and on labor <span class="math-container">$L$</span>. The economy is completed by a third good <span class="math-container">$C$</span> (Thus, for reference, the budget constraint of individuals is given by <span class="math-container">$P_C C + P_D D = P_L L$</span>). In this section, environmental externalities from the consumption of <span class="math-container">$D$</span> are NOT considered, so this is the classic &quot;Ramsey tax&quot; problem. This is what I'm struggling with:</p>
<p>The authors show that the expression for the optimal tax on <span class="math-container">$D$</span> is:</p>
<p><span class="math-container">$\qquad
\frac{t_D}{1 + t_D} = \left( \frac{\epsilon_{LL} - \epsilon_{DL}}{\epsilon_{LD} - \epsilon_{DD}} \right) \theta_L
$</span></p>
<p>They then rewrite it to:</p>
<p><span class="math-container">$\qquad
\frac{t_D}{1 + t_D} = \left( \frac{\epsilon_{CL} - \epsilon_{DL}}{\epsilon_{CD} - \epsilon_{DD}} \right) \theta_L
$</span></p>
<p>Justifying the passage by saying:</p>
<blockquote>
<p>Since compensated demand functions depend only on relative prices, <span class="math-container">$\epsilon_{iL} = - (\epsilon_{iC} + \epsilon_{iD}), i=D,L$</span> must hold. Hence, using Slutsky symmetry (i.e. <span class="math-container">$\epsilon_{LD} = - \alpha_D\epsilon_{DL}$</span> and <span class="math-container">$\epsilon_{LC} = - (1-\alpha_D)\epsilon_{CL}$</span>), we can write [the first equation as the second].</p>
</blockquote>
<p>To give more info:</p>
<ul>
<li><span class="math-container">$\epsilon_{AB}$</span>: <em>compensated</em> elasticity of commodity <span class="math-container">$A$</span> wrt <span class="math-container">$P_B$</span>, defined as <span class="math-container">$P_B S_{AB}/A$</span>, where S is the Slutsky matrix of Hicksian demand derivatives with respect to price.</li>
<li><span class="math-container">$\alpha_D$</span>: private budget share of good <span class="math-container">$D$</span>, equal to <span class="math-container">$P_DD/P_LL$</span>.</li>
</ul>
<p>I think the first equation in the quote derives from the application of Euler's theorem (Hicksian demand is homogeneous of degree 0, sum of those elasticities must be 0, but why <span class="math-container">$C$</span> is not considered?). I also understood the symmetry of the Slutsky matrix.</p>
<p>At this point, it seems like an algebra issue—I'm struggling with understanding which variables to substitute and how to apply the economic results discussed above. Could anyone help walk me through the algebraic steps between these transformations? Thank you in advance!</p>

CC BY-SA 4.0
2024-10-11T14:14:51.667Z
The second equation in the line about Slutsky symmetry should be $\epsilon_{LC} = - (1-\alpha_D)\epsilon_{CL}$, right?
VARulle
CC BY-SA 4.0
2024-10-11T15:21:22.560Z
@VARulle yes, my bad! I'll edit the question.
lorenzo
<microeconomics><consumer-theory><public-economics><environmental-economics><optimal-taxation>
Difficulty understanding steps in optimal taxation derivations
lorenzo
# Difficulty understanding steps in optimal taxation derivations

(Asked by: username_0 on 2024-10-11)

I am replicating the derivations from the paper ***Environmental policy, public finance and the labour market in a second-best world*** (Bovenberg & van der Ploeg, *Journal of Public Economics*, 1994), but I’m struggling with a key step.

First of all: some context. In this section of the paper, the government is trying to raise a given revenue in the least distortionary way by imposing taxes on a good $D$ and on labor $L$. The economy is completed by a third good $C$ (Thus, for reference, the budget constraint of individuals is given by $P\_C C + P\_D D = P\_L L$). In this section, environmental externalities from the consumption of $D$ are NOT considered, so this is the classic "Ramsey tax" problem. This is what I'm struggling with:

The authors show that the expression for the optimal tax on $D$ is:

$\qquad
\frac{t\_D}{1 + t\_D} = \left( \frac{\epsilon\_{LL} - \epsilon\_{DL}}{\epsilon\_{LD} - \epsilon\_{DD}} \right) \theta\_L
$

They then rewrite it to:

$\qquad
\frac{t\_D}{1 + t\_D} = \left( \frac{\epsilon\_{CL} - \epsilon\_{DL}}{\epsilon\_{CD} - \epsilon\_{DD}} \right) \theta\_L
$

Justifying the passage by saying:

> Since compensated demand functions depend only on relative prices, $\epsilon\_{iL} = - (\epsilon\_{iC} + \epsilon\_{iD}), i=D,L$ must hold. Hence, using Slutsky symmetry (i.e. $\epsilon\_{LD} = - \alpha\_D\epsilon\_{DL}$ and $\epsilon\_{LC} = - (1-\alpha\_D)\epsilon\_{CL}$), we can write [the first equation as the second].

To give more info:

* $\epsilon\_{AB}$: *compensated* elasticity of commodity $A$ wrt $P\_B$, defined as $P\_B S\_{AB}/A$, where S is the Slutsky matrix of Hicksian demand derivatives with respect to price.
* $\alpha\_D$: private budget share of good $D$, equal to $P\_DD/P\_LL$.

I think the first equation in the quote derives from the application of Euler's theorem (Hicksian demand is homogeneous of degree 0, sum of those elasticities must be 0, but why $C$ is not considered?). I also understood the symmetry of the Slutsky matrix.

At this point, it seems like an algebra issue—I'm struggling with understanding which variables to substitute and how to apply the economic results discussed above. Could anyone help walk me through the algebraic steps between these transformations? Thank you in advance!

**Tags:** microeconomics, consumer-theory, public-economics, environmental-economics, optimal-taxation
> **Comments:**
> **username_1**: The second equation in the line about Slutsky symmetry should be $\epsilon_{LC} = - (1-\alpha_D)\epsilon_{CL}$, right?  
> **username_0**: @VARulle yes, my bad! I'll edit the question.  

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## 1 answers

### Answer by username_1 on 2024-10-11. Score: 0 | Accepted answer

I think Slutsky symmetry also implies $\alpha\_D\epsilon\_{DC} = (1-\alpha\_D)\epsilon\_{CD}$. Using this, the equality follows.


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