<p>In Chapter 1. General Framework, the authors introduce their model there and explain it step by step.</p>
<blockquote>
<p>[...] The two inputs, <span class="math-container">$Y_c$</span> (clean) and <span class="math-container">$Y_d$</span> (dirty), are produced
using labour and sector-specific machines (intermediates), and the
production of <span class="math-container">$Y_d$</span> may also be a natural exhaustible resource:</p>
<p><span class="math-container">$$Y_{ct} = L_{ct}^{1-\alpha} \int_0^1 A_{cit}^{1-\alpha}
x_{cit}^\alpha di \;\;\; \text{and} \;\;\; Y_{dt} =
R_t^{\alpha_2}L_{dt}^{1-\alpha} \int_0^1 A_{dit}^{1-\alpha_1}
x_{dit}^{\alpha_1} di,  \tag{5} \label{eq5} $$</span></p>
<p>where <span class="math-container">$\alpha, \alpha_1, \alpha_2 \in \left(0,1 \right),
\alpha_1+\alpha_2 = \alpha, A_{jit}$</span> is the quality of machine is the
quality of machine of type <span class="math-container">$i$</span> used in sector <span class="math-container">$j \in \left\{c, d
\right\}$</span> at time <span class="math-container">$t$</span>, <span class="math-container">$x_{jit}$</span> is the quantity of this machine, and
<span class="math-container">$R_t$</span> is the flow consumption from an exhaustible resource at time
<span class="math-container">$t$</span>. [...]</p>
</blockquote>
<p>Acemoglu et al. (2012) is defining <span class="math-container">$\lambda_{jt}$</span> as the Lagrange multiplier for \eqref{eq5}, then the ratio <span class="math-container">$\frac{\lambda_{jt}}{\lambda_{t}}$</span> can be interpreted as the shadow price of input <span class="math-container">$j$</span> at time <span class="math-container">$t$</span> (relative to the price of the final good). Next, to emphasize this interpretation, the author denotes this ratio by <span class="math-container">$\hat{p}_{jt}$</span>.</p>
<p>The first-order conditions with respect to <span class="math-container">$Y_{ct}$</span>
and <span class="math-container">$Y_{dt}$</span> then give:</p>
<p><span class="math-container">$$ \begin{equation}\begin{aligned}
Y_{ct}^{\frac{-1}{\epsilon}}\left(Y_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} &amp;= \hat{p}_{ct} \\
Y_{dt}^{\frac{-1}{\epsilon}}\left(Y_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} - \frac{\omega_{t+1} \xi}{\lambda_t} &amp;= \hat{p}_{dt} \\
 \end{aligned} \end{equation} \tag{A.9} $$</span></p>
<p>And so, the carbon tax is equal to:</p>
<p><span class="math-container">$$ \tau_t = \frac{\omega_{t+1} \xi}{\lambda_{dt}} = \frac{\omega_{t+1} \xi}{\lambda_t \frac{\lambda_{dt}}{\lambda_t}} = \frac{\omega_{t+1} \xi}{\lambda_t \hat{p}_{dt}}, \tag{A.10} $$</span></p>
<p>where <span class="math-container">$\omega_{t+1}$</span> is the shadow value of one unit of environmental quality at time <span class="math-container">$t + 1$</span>, <span class="math-container">$\xi$</span> denotes the rate of environmental degradation resulting from the production of dirty inputs. And so the environmental externality is corrected by introducing a wedge between the marginal product of dirty input in the production of the final good and its shadow value which corresponds to the carbon tax <span class="math-container">$\tau_t$</span>.</p>

CC BY-SA 4.0
2023-06-27T19:25:31.413Z
2023-06-27T19:25:31.413Z
bajun65537
<p>I am reading &quot;<a href="https://www.aeaweb.org/articles?id=10.1257/aer.102.1.131" rel="nofollow noreferrer">The Environment and Directed Technical Change</a>&quot; by Acemoglu et al. (2012).</p>
<p>I cannot understand how the optimal tax in \eqref{eqA10} is derived.
<span class="math-container">$$\tau_t = \frac{\omega_{t+1} \xi}{\lambda_t \hat{p}_{dt}} \tag{A.10}\label{eqA10}$$</span></p>
<p>I see why <span class="math-container">$w_{t+1} \xi $</span> is there and that the denominator is equal to <span class="math-container">$\lambda_t \hat{p}_{dt}$</span> but not sure where this is coming from.</p>
<p>Why wouldn't the Pigouvian tax just be the second term in the price of the dirty good in \eqref{eqA9}?</p>
<p><span class="math-container">$$\begin{equation}\begin{aligned}
Y_{ct}^{\frac{-1}{\epsilon}}\left(Y_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} &amp;= \hat{p}_{ct} \\
Y_{dt}^{\frac{-1}{\epsilon}}\left(Y_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} - \frac{\omega_{t+1} \xi}{\lambda_t} &amp;= \hat{p}_{dt}
 \end{aligned} \end{equation} \tag{A.9}\label{eqA9}$$</span></p>

<production-function><environmental-economics><externalities><endogenous-growth><pigouvian-taxes>
How do I derive optimal tax on pollution causing intermediate products?
EngineerinEcon
# How do I derive optimal tax on pollution causing intermediate products?

(Asked by: username_0 on 2023-06-22)

I am reading "[The Environment and Directed Technical Change](https://www.aeaweb.org/articles?id=10.1257/aer.102.1.131)" by Acemoglu et al. (2012).

I cannot understand how the optimal tax in \eqref{eqA10} is derived.
$$\tau\_t = \frac{\omega\_{t+1} \xi}{\lambda\_t \hat{p}\_{dt}} \tag{A.10}\label{eqA10}$$

I see why $w\_{t+1} \xi $ is there and that the denominator is equal to $\lambda\_t \hat{p}\_{dt}$ but not sure where this is coming from.

Why wouldn't the Pigouvian tax just be the second term in the price of the dirty good in \eqref{eqA9}?

$$\begin{equation}\begin{aligned}
Y\_{ct}^{\frac{-1}{\epsilon}}\left(Y\_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y\_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} &= \hat{p}\_{ct} \\
Y\_{dt}^{\frac{-1}{\epsilon}}\left(Y\_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y\_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} - \frac{\omega\_{t+1} \xi}{\lambda\_t} &= \hat{p}\_{dt}
\end{aligned} \end{equation} \tag{A.9}\label{eqA9}$$

**Tags:** production-function, environmental-economics, externalities, endogenous-growth, pigouvian-taxes


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## 1 answers

### Answer by username_1 on 2023-06-27. Score: 1

In Chapter 1. General Framework, the authors introduce their model there and explain it step by step.

> [...] The two inputs, $Y\_c$ (clean) and $Y\_d$ (dirty), are produced
> using labour and sector-specific machines (intermediates), and the
> production of $Y\_d$ may also be a natural exhaustible resource:
>
> $$Y\_{ct} = L\_{ct}^{1-\alpha} \int\_0^1 A\_{cit}^{1-\alpha}
> x\_{cit}^\alpha di \;\;\; \text{and} \;\;\; Y\_{dt} =
> R\_t^{\alpha\_2}L\_{dt}^{1-\alpha} \int\_0^1 A\_{dit}^{1-\alpha\_1}
> x\_{dit}^{\alpha\_1} di, \tag{5} \label{eq5} $$
>
> where $\alpha, \alpha\_1, \alpha\_2 \in \left(0,1 \right),
> \alpha\_1+\alpha\_2 = \alpha, A\_{jit}$ is the quality of machine is the
> quality of machine of type $i$ used in sector $j \in \left\{c, d
> \right\}$ at time $t$, $x\_{jit}$ is the quantity of this machine, and
> $R\_t$ is the flow consumption from an exhaustible resource at time
> $t$. [...]

Acemoglu et al. (2012) is defining $\lambda\_{jt}$ as the Lagrange multiplier for \eqref{eq5}, then the ratio $\frac{\lambda\_{jt}}{\lambda\_{t}}$ can be interpreted as the shadow price of input $j$ at time $t$ (relative to the price of the final good). Next, to emphasize this interpretation, the author denotes this ratio by $\hat{p}\_{jt}$.

The first-order conditions with respect to $Y\_{ct}$
and $Y\_{dt}$ then give:

$$ \begin{equation}\begin{aligned}
Y\_{ct}^{\frac{-1}{\epsilon}}\left(Y\_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y\_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} &= \hat{p}\_{ct} \\
Y\_{dt}^{\frac{-1}{\epsilon}}\left(Y\_{ct}^{\frac{\epsilon-1}{\epsilon}} + Y\_{dt}^{\frac{\epsilon-1}{\epsilon}} \right)^{\frac{1}{\epsilon-1}} - \frac{\omega\_{t+1} \xi}{\lambda\_t} &= \hat{p}\_{dt} \\
\end{aligned} \end{equation} \tag{A.9} $$

And so, the carbon tax is equal to:

$$ \tau\_t = \frac{\omega\_{t+1} \xi}{\lambda\_{dt}} = \frac{\omega\_{t+1} \xi}{\lambda\_t \frac{\lambda\_{dt}}{\lambda\_t}} = \frac{\omega\_{t+1} \xi}{\lambda\_t \hat{p}\_{dt}}, \tag{A.10} $$

where $\omega\_{t+1}$ is the shadow value of one unit of environmental quality at time $t + 1$, $\xi$ denotes the rate of environmental degradation resulting from the production of dirty inputs. And so the environmental externality is corrected by introducing a wedge between the marginal product of dirty input in the production of the final good and its shadow value which corresponds to the carbon tax $\tau\_t$.


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