Signalling Model (Rearranging the FOCs of UMP)

Question

I'm currently reading "SAVING RATES AND POVERTY: THE ROLE OF CONSPICUOUS CONSUMPTION AND HUMAN CAPITAL" 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!!

Answer 1

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

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