# How do I derive optimal tax on pollution causing intermediate products?

I am reading "The Environment and Directed Technical Change" 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}

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 $$\tau_t$$.