<p>Taking the derivative of the goal function w.r.t. <span class="math-container">$w_t^P$</span> you get
<span class="math-container">$$
(1- \vartheta)\frac{1}{V^h_{nPt}}\frac{\partial V^h_{nPt}}{\partial w_t^P} +
\vartheta\frac{1}{V^f_{nPt}}\frac{\partial V^f_{nPt}}{\partial w_t^P} = 0
$$</span>
which, after substituting the definitions is
<span class="math-container">$$
(1- \vartheta)\frac{1}{V^h_{nPt}}\left(\lambda_{ct}(1-\tau_n^t)\right) +
\vartheta\frac{1}{V^f_{nPt}}(-1) = 0.
$$</span>
Rearranging
<span class="math-container">$$
(1- \vartheta)\frac{1}{V^h_{nPt}}\lambda_{ct}(1-\tau_n^t)
= \vartheta\frac{1}{V^f_{nPt}},
$$</span>
rearranging
<span class="math-container">$$
(1- \vartheta)\lambda_{ct}(1-\tau_n^t) V^f_{nPt}
= \vartheta V^h_{nPt},
$$</span>
QED.</p>

CC BY-SA 4.0
2022-02-09T13:56:48.790Z
2022-02-09T13:56:48.790Z
Giskard
<p>Could someone please show how the author derives the the first order condition (14) of this optimization problem using the expressions shown here ?</p>
<p>For context this is taken from a labour market model where the private sector wage, w_p , is the result of Nash bargaining over the respective surpluses of the households V^h and firms V^f</p>
<p><a href="https://i.sstatic.net/CyotC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/CyotC.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-02-09T13:48:58.333Z
Why are people so reluctant to name "the author" or "the paper"? Providing a direct link could also be useful!
Giskard
<mathematical-economics><optimization>
Optimization problem - wage determination
ezekiel91
# Optimization problem - wage determination

(Asked by: username_0 on 2022-02-09)

Could someone please show how the author derives the the first order condition (14) of this optimization problem using the expressions shown here ?

For context this is taken from a labour market model where the private sector wage, w\_p , is the result of Nash bargaining over the respective surpluses of the households V^h and firms V^f

[![enter image description here](https://i.sstatic.net/CyotC.png)](https://i.sstatic.net/CyotC.png)

**Tags:** mathematical-economics, optimization

> **Comments:**
> **username_1**: Why are people so reluctant to name "the author" or "the paper"? Providing a direct link could also be useful!  

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## 1 answers

### Answer by username_1 on 2022-02-09. Score: 2 | Accepted answer

Taking the derivative of the goal function w.r.t. $w\_t^P$ you get
$$
(1- \vartheta)\frac{1}{V^h\_{nPt}}\frac{\partial V^h\_{nPt}}{\partial w\_t^P} +
\vartheta\frac{1}{V^f\_{nPt}}\frac{\partial V^f\_{nPt}}{\partial w\_t^P} = 0
$$
which, after substituting the definitions is
$$
(1- \vartheta)\frac{1}{V^h\_{nPt}}\left(\lambda\_{ct}(1-\tau\_n^t)\right) +
\vartheta\frac{1}{V^f\_{nPt}}(-1) = 0.
$$
Rearranging
$$
(1- \vartheta)\frac{1}{V^h\_{nPt}}\lambda\_{ct}(1-\tau\_n^t)
= \vartheta\frac{1}{V^f\_{nPt}},
$$
rearranging
$$
(1- \vartheta)\lambda\_{ct}(1-\tau\_n^t) V^f\_{nPt}
= \vartheta V^h\_{nPt},
$$
QED.


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