<p>could anyone please help to explain how equation (4) in <a href="https://onlinelibrary.wiley.com/doi/full/10.1111/j.1542-4774.2011.01049.x" rel="nofollow noreferrer">this</a> paper is derived? I understand that it's an application of first order conditions using the chain rule from equations (1)-(3), but the notation is very confusing and I can't follow the math at all. Thankyou in advance to anyone that can help</p>
<p>For reference, equation (1) is:</p>
<p><span class="math-container">$ Y_t = A_t[\theta_t L_{1t}^\rho + L_{2t}^\rho]^{1/\rho} $</span></p>
<p>Equation (2) is:</p>
<p><span class="math-container">$ L_{et} = (\sum_{a}\alpha_{ea}L_{ea}^\eta)^{1/\eta},  e = (1,2) $</span></p>
<p>Equation (3) is:</p>
<p><span class="math-container">$ L_{eat} = (N_{eat}^\delta + \beta_{eat}M_{eat}^\delta)^{1/\delta} $</span></p>
<p>And from this, equation 4 is derived as:</p>
<p><span class="math-container">$ \ln W_{eat}^S = \ln A_t + 1/\sigma_E \ln Y_t + \ln\theta_{et} + \ln\alpha_{e\alpha} + \ln\beta_{eat}^S + (1/\sigma_A - 1/\sigma_E)\ln L_{et} + (1/\sigma_I - 1/\sigma_A)\ln L_{eat} - (1/\sigma_I) \ln S_{eat},  S = {M,N} $</span></p>
<p>Where</p>
<p><span class="math-container">$ \sigma_I = (1/{1-\delta}), \sigma_A = (1/{1-\eta}), \sigma_E = (1/{1-\rho}) $</span></p>

CC BY-SA 4.0
2021-02-08T12:23:59.950Z
please consider making the question self contained by including the equation in the question - you can do it here the same way as you would do in Latex (if you are not familiar you do it by typing out the equation and enclosing it in dollar signs e.g. `$ x+5=y$`will be rendered as $x+5=y$, for Greek letters you can use \letter e.g. `$y = \beta x + \gamma$` renders as $y = \beta x + \gamma$
1muflon1
CC BY-SA 4.0
2021-02-08T13:00:11.897Z
I have done so now, thankyou
wui
<labor-economics><wages><ces>
Derivation of wage rate from nested CES production function
wui
# Derivation of wage rate from nested CES production function

(Asked by: username_0 on 2021-02-08)

could anyone please help to explain how equation (4) in [this](https://onlinelibrary.wiley.com/doi/full/10.1111/j.1542-4774.2011.01049.x) paper is derived? I understand that it's an application of first order conditions using the chain rule from equations (1)-(3), but the notation is very confusing and I can't follow the math at all. Thankyou in advance to anyone that can help

For reference, equation (1) is:

$ Y\_t = A\_t[\theta\_t L\_{1t}^\rho + L\_{2t}^\rho]^{1/\rho} $

Equation (2) is:

$ L\_{et} = (\sum\_{a}\alpha\_{ea}L\_{ea}^\eta)^{1/\eta}, e = (1,2) $

Equation (3) is:

$ L\_{eat} = (N\_{eat}^\delta + \beta\_{eat}M\_{eat}^\delta)^{1/\delta} $

And from this, equation 4 is derived as:

$ \ln W\_{eat}^S = \ln A\_t + 1/\sigma\_E \ln Y\_t + \ln\theta\_{et} + \ln\alpha\_{e\alpha} + \ln\beta\_{eat}^S + (1/\sigma\_A - 1/\sigma\_E)\ln L\_{et} + (1/\sigma\_I - 1/\sigma\_A)\ln L\_{eat} - (1/\sigma\_I) \ln S\_{eat}, S = {M,N} $

Where

$ \sigma\_I = (1/{1-\delta}), \sigma\_A = (1/{1-\eta}), \sigma\_E = (1/{1-\rho}) $

**Tags:** labor-economics, wages, ces

> **Comments:**
> **username_1**: please consider making the question self contained by including the equation in the question - you can do it here the same way as you would do in Latex (if you are not familiar you do it by typing out the equation and enclosing it in dollar signs e.g. `$ x+5=y$`will be rendered as $x+5=y$, for Greek letters you can use \letter e.g. `$y = \beta x + \gamma$` renders as $y = \beta x + \gamma$  
> **username_0**: I have done so now, thankyou  

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## 0 answers

