<p>From the first order condition for optimality, we know that (equation (4) in the paper):</p>

<p><span class="math-container">$$U_x(x,Z(r)-x,S(r))x'(r) - U_y(x,Z(r)-x,S(r))x'(r) + U_s(x,Z(r)-x,S(r))S'(r)=0.$$</span> </p>

<p>When differentiating <span class="math-container">${\bf U}(r)$</span> with respect to <span class="math-container">$r$</span>, we find your expression:
<span class="math-container">$$U_x(x(r),Z(r)-x(r),S(r))x'(r) + U_y(x(r),Z(r)-x(r),S(r))(Z'(r)-x'(r)) + U_s(x(r),Z(r)-x(r),S(r))S'(r).$$</span> </p>

<p>Replacing the first equation into the last one yields
<span class="math-container">$${\bf U}'(r)=U_y(x(r),Z(r)-x(r),S(r))Z'(r).$$</span> </p>

<p>Now, <span class="math-container">$Z'(r)=1/g(Z(r))$</span> which gives equation (8).</p>

CC BY-SA 4.0
2019-05-13T13:56:48.747Z
2019-05-13T18:48:43.437Z
2019-05-13T18:48:43.437Z
Bertrand
<p>This is from Hopkins and Kornienko (2010). In this model, <span class="math-container">$x$</span> is investments, <span class="math-container">$s$</span> is status, and <span class="math-container">$y=z-x$</span> is leisure, where <span class="math-container">$z$</span> is endowments. <span class="math-container">$x(r)$</span> is the optimal investment, and the relative investments determine the status in the model. Here they characterize the agent with the rank, <span class="math-container">$r$</span>, and derive the reduced form utility. But, I am not quite sure about how the envelope theorem works here. </p>

<p>If I differentiate <span class="math-container">$U(r)$</span> with respect to <span class="math-container">$r$</span> and suppress the arguments, I have that
<span class="math-container">$$U_x x' + U_y(y'-x') + U_s s'.$$</span>
I know that <span class="math-container">$Z'(r) = \frac{1}{g(Z(r))}$</span>, where <span class="math-container">$g$</span> is the density of <span class="math-container">$z$</span>. This implies that the first term and third term and <span class="math-container">$U_yx'$</span> turn out to be zero to make the above equation equal to Equation (8). My understanding is that <span class="math-container">$U_x = U_s = 0$</span> since <span class="math-container">$x(r)$</span> is the optimal investment. But, I don't know how to explain the elimination of <span class="math-container">$U_yx'$</span>. Can anyone give me some help? </p>

<p><a href="https://i.sstatic.net/uPsyM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uPsyM.png" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-05-12T18:34:52.943Z
What does $p$ stand for in $Z'(p)$?
Bertrand
CC BY-SA 4.0
2019-05-13T00:05:47.370Z
@Bertrand It should be $Z'(r)$. Sorry.
shk910
<microeconomics><mathematical-economics><theory>
Envelope Theorem in Hopkins and Kornienko (2010)
shk910
# Envelope Theorem in Hopkins and Kornienko (2010)

(Asked by: username_0 on 2019-05-11)

This is from Hopkins and Kornienko (2010). In this model, $x$ is investments, $s$ is status, and $y=z-x$ is leisure, where $z$ is endowments. $x(r)$ is the optimal investment, and the relative investments determine the status in the model. Here they characterize the agent with the rank, $r$, and derive the reduced form utility. But, I am not quite sure about how the envelope theorem works here.

If I differentiate $U(r)$ with respect to $r$ and suppress the arguments, I have that
$$U\_x x' + U\_y(y'-x') + U\_s s'.$$
I know that $Z'(r) = \frac{1}{g(Z(r))}$, where $g$ is the density of $z$. This implies that the first term and third term and $U\_yx'$ turn out to be zero to make the above equation equal to Equation (8). My understanding is that $U\_x = U\_s = 0$ since $x(r)$ is the optimal investment. But, I don't know how to explain the elimination of $U\_yx'$. Can anyone give me some help?

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

**Tags:** microeconomics, mathematical-economics, theory
> **Comments:**
> **username_1**: What does $p$ stand for in $Z'(p)$?  
> **username_0**: @Bertrand It should be $Z'(r)$. Sorry.  

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## 1 answers

### Answer by username_1 on 2019-05-13. Score: 1 | Accepted answer

From the first order condition for optimality, we know that (equation (4) in the paper):

$$U\_x(x,Z(r)-x,S(r))x'(r) - U\_y(x,Z(r)-x,S(r))x'(r) + U\_s(x,Z(r)-x,S(r))S'(r)=0.$$

When differentiating ${\bf U}(r)$ with respect to $r$, we find your expression:
$$U\_x(x(r),Z(r)-x(r),S(r))x'(r) + U\_y(x(r),Z(r)-x(r),S(r))(Z'(r)-x'(r)) + U\_s(x(r),Z(r)-x(r),S(r))S'(r).$$

Replacing the first equation into the last one yields
$${\bf U}'(r)=U\_y(x(r),Z(r)-x(r),S(r))Z'(r).$$

Now, $Z'(r)=1/g(Z(r))$ which gives equation (8).


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