<p>They are simply equating <span class="math-container">$\frac{\partial u(x,y)}{\partial x}$</span> to 0 and rearranging:</p>
<p><span class="math-container">$$
\frac{\partial u(x,y)}{\partial x}:= 0 \iff \lambda(y-x)^{1-\lambda}[\tilde y(h,x)]^{\lambda - 1}\frac{\partial \tilde y(h,x)}{\partial x} -(1-\lambda)(y-x)^{-\lambda}[\tilde y(h,x)]^{\lambda} = 0
$$</span></p>
<p>Rearranging this, you get indeed</p>
<p><span class="math-container">$$
\frac{\lambda}{1-\lambda}\frac{y-x}{\tilde y (h,x)} = \frac{1}{\partial \tilde y (h,x) /\partial x}
$$</span></p>

CC BY-SA 4.0
2024-01-17T10:13:23.500Z
2024-01-17T10:13:23.500Z
Matteo Bulgarelli
<p>I'm currently reading "<a href="https://academic.oup.com/ej/article-abstract/122/563/933/5079744" rel="nofollow noreferrer">SAVING RATES AND POVERTY: THE ROLE OF CONSPICUOUS CONSUMPTION AND HUMAN CAPITAL</a>" by Omer Moav and Zvika Neeman, and I encounter some problem deriving one of the equations.</p>

<p>Utility function, where <span class="math-container">$y$</span> is income, <span class="math-container">$\widetilde y$</span> is the belief of <span class="math-container">$y$</span> based on <span class="math-container">$x$</span> and human capital <span class="math-container">$h$</span>, <span class="math-container">$x$</span> is the product of interest:</p>

<p><span class="math-container">$$u(y,x) = (y-x)^{1-\lambda} \widetilde y(h,x)^\lambda$$</span></p>

<p>Differentiate with respect to <span class="math-container">$x$</span>:</p>

<p><span class="math-container">$$\frac{\lambda}{1-\lambda} \frac{y-x}{\widetilde y(h,x)} = 
\frac{1}{d \widetilde y (h,x) / d x}$$</span></p>

<p>This is the confusing part, how do I get this from above knowing that <span class="math-container">$y=\widetilde y(h,x)$</span>?</p>

<p>Given this <span class="math-container">$\underline y$</span>, the lowest possible income an individual of <span class="math-container">$h$</span> can have (<span class="math-container">$\pi$</span> is an uncertain component of income that is not observed and has mean zero):</p>

<p><span class="math-container">$$\underline y (h) \equiv h + \underline \pi (h)$$</span></p>

<p><span class="math-container">$$\widetilde y (h,x)^{1/(1-\lambda)} - \frac{x}{\lambda} \widetilde y (h,x)^{\lambda / (1-\lambda)} = \underline y (h)^{1/(1-\lambda)}$$</span></p>

<p>I think it's something to do with utility being indifferent for the marginal person who consume <span class="math-container">$x&gt;0$</span> and <span class="math-container">$x=0$</span>, but I don't know how to work it out to get the equation shown above.</p>

<p>Thanks!!</p>

CC BY-SA 4.0
2019-07-11T09:16:02.820Z
This is not so much a question about signaling, but on how to solve a differential equation. In separating equilibrium, you have $y=\widetilde y$ and then you use $\underline y$ as the boundary.
Bayesian
CC BY-SA 4.0
2019-07-11T18:38:42.743Z
Thanks for the edit on the question. Yup, I'm not sure how to solve this :(
Estee L
<microeconomics><difference-equations><signaling>
Signalling Model (Rearranging the FOCs of UMP)
Estee L
# Signalling Model (Rearranging the FOCs of UMP)

(Asked by: username_0 on 2019-07-10)

I'm currently reading "[SAVING RATES AND POVERTY: THE ROLE OF CONSPICUOUS CONSUMPTION AND HUMAN CAPITAL](https://academic.oup.com/ej/article-abstract/122/563/933/5079744)" by Omer Moav and Zvika Neeman, and I encounter some problem deriving one of the equations.

Utility function, where $y$ is income, $\widetilde y$ is the belief of $y$ based on $x$ and human capital $h$, $x$ is the product of interest:

$$u(y,x) = (y-x)^{1-\lambda} \widetilde y(h,x)^\lambda$$

Differentiate with respect to $x$:

$$\frac{\lambda}{1-\lambda} \frac{y-x}{\widetilde y(h,x)} =
\frac{1}{d \widetilde y (h,x) / d x}$$

This is the confusing part, how do I get this from above knowing that $y=\widetilde y(h,x)$?

Given this $\underline y$, the lowest possible income an individual of $h$ can have ($\pi$ is an uncertain component of income that is not observed and has mean zero):

$$\underline y (h) \equiv h + \underline \pi (h)$$

$$\widetilde y (h,x)^{1/(1-\lambda)} - \frac{x}{\lambda} \widetilde y (h,x)^{\lambda / (1-\lambda)} = \underline y (h)^{1/(1-\lambda)}$$

I think it's something to do with utility being indifferent for the marginal person who consume $x>0$ and $x=0$, but I don't know how to work it out to get the equation shown above.

Thanks!!

**Tags:** microeconomics, difference-equations, signaling
> **Comments:**
> **username_1**: This is not so much a question about signaling, but on how to solve a differential equation. In separating equilibrium, you have $y=\widetilde y$ and then you use $\underline y$ as the boundary.  
> **username_0**: Thanks for the edit on the question. Yup, I'm not sure how to solve this :(  

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## 1 answers

### Answer by username_2 on 2024-01-17. Score: 0

They are simply equating $\frac{\partial u(x,y)}{\partial x}$ to 0 and rearranging:

$$
\frac{\partial u(x,y)}{\partial x}:= 0 \iff \lambda(y-x)^{1-\lambda}[\tilde y(h,x)]^{\lambda - 1}\frac{\partial \tilde y(h,x)}{\partial x} -(1-\lambda)(y-x)^{-\lambda}[\tilde y(h,x)]^{\lambda} = 0
$$

Rearranging this, you get indeed

$$
\frac{\lambda}{1-\lambda}\frac{y-x}{\tilde y (h,x)} = \frac{1}{\partial \tilde y (h,x) /\partial x}
$$


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