<p>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.</p>
<p>The production function:
<span class="math-container">$Q_t = \Bigg[(1-\mu)^{\frac{1}{\epsilon}}\bigg(\big(\frac{K_t}{\alpha}\big)^{\alpha}\big(\frac{L_t}{1-\alpha}\big)^{1-\alpha}\bigg)^{\frac{\epsilon-1}{\epsilon}}+(\mu)^{\frac{1}{\epsilon}}(M_t)^{\frac{\epsilon-1}{\epsilon}}\Bigg]^{\frac{\epsilon}{\epsilon-1}}$</span></p>
<p>The log-linearization:
<span class="math-container">$\hat{q}_t = \alpha(1-S_M) \hat{k}_t + (1-\alpha)(1-S_M)\hat{l}_t + S_M \hat{m}_t$</span>
where (<span class="math-container">$S_M = \frac{M}{Q}$</span>)</p>
<p>My approach so far:</p>
<p><span class="math-container">$(Qe^{\hat{q}_t})^{\frac{\epsilon-1}{\epsilon}} = \Bigg[(1-\mu)^{\frac{1}{\epsilon}}\bigg(\big(\frac{Ke^{\hat{k}_t}}{\alpha}\big)^{\alpha}\big(\frac{Le^{\hat{l}_t}}{1-\alpha}\big)^{1-\alpha}\bigg)^{\frac{\epsilon-1}{\epsilon}}+(\mu)^{\frac{1}{\epsilon}}(Me^{\hat{m}_t})^{\frac{\epsilon-1}{\epsilon}}\Bigg]$</span></p>
<p><span class="math-container">$(e^{\hat{q}_t})^{\frac{\epsilon-1}{\epsilon}} = \Bigg[(1-\mu)^{\frac{1}{\epsilon}}\bigg(\big( \frac{K^{\alpha}L^{1-\alpha}}{Q}\big) \big(\frac{e^{\hat{k}_t}}{\alpha}\big)^{\alpha}\big(\frac{e^{\hat{l}_t}}{1-\alpha}\big)^{1-\alpha}\bigg)^{\frac{\epsilon-1}{\epsilon}}+(\mu)^{\frac{1}{\epsilon}} (\frac{M}{Q})^{\frac{\epsilon-1}{\epsilon}} (e^{\hat{m}_t})^{\frac{\epsilon-1}{\epsilon}}\Bigg]$</span></p>
<p><span class="math-container">$\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) $</span></p>
<p><span class="math-container">$(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) $</span></p>
<p>What is irritating me the most are the remaining <span class="math-container">$\epsilon$</span> in the powers of the factor shares. Any hint on how to go further?</p>

CC BY-SA 4.0
2021-02-17T20:04:18.117Z
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
Brennan
CC BY-SA 4.0
2021-02-18T11:32:21.837Z
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.
EconRider
<ces-function><log-linearization>
Log-linearization of CES production function
EconRider
# Log-linearization of CES production function

(Asked by: username_0 on 2021-02-17)

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?

**Tags:** ces-function, log-linearization
> **Comments:**
> **username_1**: 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  
> **username_0**: 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.  

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## 0 answers

