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I am trying to recover the Log-linearisation of a CES production function in a paper. Although I am fairly confident with Log-linearisations, I simply do not find the supposed result.

The production function: $Q_t = \Bigg[(1-\mu)^{\frac{1}{\epsilon}}\bigg(\big(\frac{K_t}{\alpha}\big)^{\alpha}\big(\frac{L_t}{1-\alpha}\big)^{1-\alpha}\bigg)^{\frac{\epsilon-1}{\epsilon}}+(\mu)^{\frac{1}{\epsilon}}(M_t)^{\frac{\epsilon-1}{\epsilon}}\Bigg]^{\frac{\epsilon}{\epsilon-1}}$

The log-linearization: $\hat{q}_t = \alpha(1-S_M) \hat{k}_t + (1-\alpha)(1-S_M)\hat{l}_t + S_M \hat{m}_t$ where ($S_M = \frac{M}{Q}$)

My approach so far:

$(Qe^{\hat{q}_t})^{\frac{\epsilon-1}{\epsilon}} = \Bigg[(1-\mu)^{\frac{1}{\epsilon}}\bigg(\big(\frac{Ke^{\hat{k}_t}}{\alpha}\big)^{\alpha}\big(\frac{Le^{\hat{l}_t}}{1-\alpha}\big)^{1-\alpha}\bigg)^{\frac{\epsilon-1}{\epsilon}}+(\mu)^{\frac{1}{\epsilon}}(Me^{\hat{m}_t})^{\frac{\epsilon-1}{\epsilon}}\Bigg]$

$(e^{\hat{q}_t})^{\frac{\epsilon-1}{\epsilon}} = \Bigg[(1-\mu)^{\frac{1}{\epsilon}}\bigg(\big( \frac{K^{\alpha}L^{1-\alpha}}{Q}\big) \big(\frac{e^{\hat{k}_t}}{\alpha}\big)^{\alpha}\big(\frac{e^{\hat{l}_t}}{1-\alpha}\big)^{1-\alpha}\bigg)^{\frac{\epsilon-1}{\epsilon}}+(\mu)^{\frac{1}{\epsilon}} (\frac{M}{Q})^{\frac{\epsilon-1}{\epsilon}} (e^{\hat{m}_t})^{\frac{\epsilon-1}{\epsilon}}\Bigg]$

$\frac{\epsilon_Q-1}{\epsilon} (1+\hat{q}_t) = (1-\mu)^{\frac{1}{\epsilon}} (1-S_M)^{\frac{\epsilon-1}{\epsilon}}\frac{\epsilon-1}{\epsilon}(\hat{k}_t+\hat{l}_t)+(\mu)^{\frac{1}{\epsilon}}S_M^{\frac{\epsilon-1}{\epsilon}} \frac{\epsilon-1}{\epsilon}(1+\hat{m}_t) $

$(1+\hat{q}_t) = (1-\mu)^{\frac{1}{\epsilon}} (1-S_M)^{\frac{\epsilon-1}{\epsilon}}(\hat{k}_t+\hat{l}_t)+(\mu)^{\frac{1}{\epsilon}}S_M^{\frac{\epsilon-1}{\epsilon}} (1+\hat{m}_t) $

What is irritating me the most are the remaining $\epsilon$ in the powers of the factor shares. Any hint on how to go further?

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    $\begingroup$ Are they supposed to be epsilon subscript Q? It is really unclear as it stands, you should consider dropping it to just epsilon, much clearer that way and no need to differentiate it from other epsilons as there are none $\endgroup$
    – Brennan
    Feb 17, 2021 at 20:04
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    $\begingroup$ I have different elasticities in my model, so that's where the Q subscripts come from - of course I can and should drop them here for clarity. $\endgroup$
    – EconRider
    Feb 18, 2021 at 11:32

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