<p>I have an application of the envelope theorem below, and would love some feedback on the steps taken to get from (2) - (3) and (3) - (4) and (4) - (5)</p>
<p>The overall question is: 'derive the marginal value for human capital in the second year' using the second-period value function, with parameters:</p>
<ul>
<li>human capital <span class="math-container">$k_2$</span></li>
<li>savings <span class="math-container">$m_2$</span></li>
<li>consumption good price <span class="math-container">$p_2$</span></li>
<li>labour wage <span class="math-container">$w_2$</span></li>
<li>labour supply <span class="math-container">$l_2$</span></li>
</ul>
<p>The value function is given:</p>
<p><strong>(1)</strong>              <span class="math-container">$$V(k_2,m_2;p_2,w_2) = max_{c_2,l_2} u(c_2, l_2)$$</span>
s.t <span class="math-container">$$ p_2c_2 = w_2k_2l_2+m_2$$</span></p>
<p><strong>(2)</strong> With the budget constraint substituted into the objective we get:</p>
<p><span class="math-container">$$V(k_2,m_2;p_2,w_2)=u(w_2,kl_2(k_2,m_2;p_2,w_2+m)/p_2,l_2(k_2,m_2;p_2,w_2)$$</span></p>
<p><strong>(3)</strong> By the envelope theorem</p>
<p><span class="math-container">$$\frac{dV(k_2;m_2;p_2,w_2)}{dk}= [\frac{d}{dk}u ((w_2kl_2+m_2)/p_2,l_2]_{l_2=l_2(k_2,m_2;p_2,w_2)}$$</span></p>
<p><strong>(4)</strong> <span class="math-container">$$= [u_c((w_2kl_2+m_2/p_2,l_2)\frac{w_2l_2}{p_2}]_{l_2=l_2=(k_2,m_2;p_2,w_2)}$$</span></p>
<p><strong>(5)</strong> <span class="math-container">$$=u_c(k_2,m_2;p_2,w_2),l_2(k_2,m_2;p_2,w_2))\frac {w_2l_2(k_2,m_2;p_2,w_2)}{p_2}$$</span></p>
<p><strong>Screenshot uploaded for clarity</strong></p>
<p><a href="https://i.sstatic.net/AijVD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/AijVD.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2020-12-10T21:07:40.170Z
Is it not something like $\partial V/\partial k = \lambda w l$ where $\lambda = \partial V/\partial m$?
Jesper Hybel
CC BY-SA 4.0
2020-12-10T21:47:25.097Z
Hi Jesper, thanks for your contribution. I've uploaded a screenshot for clarity
EB3112
CC BY-SA 4.0
2020-12-10T21:52:23.950Z
Which is the same because $\lambda = \partial V/\partial m = \frac{\partial u/\partial c} {p}$. I think (fast look could make mistake.. so you should chek)
Jesper Hybel
CC BY-SA 4.0
2020-12-10T21:56:18.493Z
You can measure the value of relaxing the constraint either in terms of getting more $m$ or in terms of getting $\partial u/\partial c$ more utility per dollar/euro therefore divided by $p$ the price of $c$
Jesper Hybel
CC BY-SA 4.0
2020-12-10T21:57:57.120Z
Ok - that makes sense in terms of what you've said. Thank you.
EB3112
<mathematical-economics>
The Envelope Theorem, Some Workings/Substitution
EB3112
# The Envelope Theorem, Some Workings/Substitution

(Asked by: username_0 on 2020-12-09)

I have an application of the envelope theorem below, and would love some feedback on the steps taken to get from (2) - (3) and (3) - (4) and (4) - (5)

The overall question is: 'derive the marginal value for human capital in the second year' using the second-period value function, with parameters:

* human capital $k\_2$
* savings $m\_2$
* consumption good price $p\_2$
* labour wage $w\_2$
* labour supply $l\_2$

The value function is given:

**(1)** $$V(k\_2,m\_2;p\_2,w\_2) = max\_{c\_2,l\_2} u(c\_2, l\_2)$$
s.t $$ p\_2c\_2 = w\_2k\_2l\_2+m\_2$$

**(2)** With the budget constraint substituted into the objective we get:

$$V(k\_2,m\_2;p\_2,w\_2)=u(w\_2,kl\_2(k\_2,m\_2;p\_2,w\_2+m)/p\_2,l\_2(k\_2,m\_2;p\_2,w\_2)$$

**(3)** By the envelope theorem

$$\frac{dV(k\_2;m\_2;p\_2,w\_2)}{dk}= [\frac{d}{dk}u ((w\_2kl\_2+m\_2)/p\_2,l\_2]\_{l\_2=l\_2(k\_2,m\_2;p\_2,w\_2)}$$

**(4)** $$= [u\_c((w\_2kl\_2+m\_2/p\_2,l\_2)\frac{w\_2l\_2}{p\_2}]\_{l\_2=l\_2=(k\_2,m\_2;p\_2,w\_2)}$$

**(5)** $$=u\_c(k\_2,m\_2;p\_2,w\_2),l\_2(k\_2,m\_2;p\_2,w\_2))\frac {w\_2l\_2(k\_2,m\_2;p\_2,w\_2)}{p\_2}$$

**Screenshot uploaded for clarity**

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

**Tags:** mathematical-economics

> **Comments:**
> **username_1**: Is it not something like $\partial V/\partial k = \lambda w l$ where $\lambda = \partial V/\partial m$?  
> **username_0**: Hi username_1, thanks for your contribution. I've uploaded a screenshot for clarity  
> **username_1**: Which is the same because $\lambda = \partial V/\partial m = \frac{\partial u/\partial c} {p}$. I think (fast look could make mistake.. so you should chek)  
> **username_1**: You can measure the value of relaxing the constraint either in terms of getting more $m$ or in terms of getting $\partial u/\partial c$ more utility per dollar/euro therefore divided by $p$ the price of $c$  
> **username_0**: Ok - that makes sense in terms of what you've said. Thank you.  

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## 0 answers

