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 $\tau_t$.
