Question:
Let $V(x, y) = (1-\overline{p}) U(x) + \overline{p} U(y) - \overline{p} U(F)$ where $U$ is a strictly concave function ($U'>0$ and $U''<0$) with $U(0)=0$ and $0<\overline{p}<1$ is some constant. Also, we have the restriction $0 \le x \le y \le F$ and if $x=0$ then $y=F$. Consider the following system of equations: \begin{align} \tag 1 V(x,y) = 0 \\ \tag 2 -\frac{1-p_i}{p_i} = -\left(\frac{1-\overline{p}}{\overline{p}}\right) \frac{U'(x)}{U'(y)} \end{align} where $0<p_i<1$, $i=c, d$, and $p_d>p_c$.
Denote the solution to the above system of equations as $(x_c^*, y_c^*)$ when $i=c$ and $(x_d^*, y_d^*)$ when $i=d$. Show that $x_d^* > x_c^*$ and $y_d^* < y_c^*$.
My attempt:
Note that $(0, F)$ satisfies Equation $(1)$ since $$V(0, F) = (1-\overline{p})\underbrace{U(0)}_{=0} + \overline{p}U(F) - \overline{p}U(F) = 0$$
Implicitly differentiating Equation $(1)$ yields: $$\tag 3 \frac{\partial y}{\partial x} = -\left(\frac{1-\overline{p}}{\overline{p}}\right) \frac{U'(x)}{U'(y)} <0 $$ which is exactly the RHS of Equation $(2)$. Taking derivatives again with respect to $x$ of the above yields: $$\tag 4 \frac{\partial^2 y}{\partial x^2} = -\left(\frac{1-\overline{p}}{\overline{p}}\right) \left[\frac{U''(x)U'(y) - U'(x)U''(y) \frac{\partial y}{\partial x}}{[U'(y)]^2} \right] >0 $$
Thus, if we sketch the level curve of $V(x,y)=0$, it must be a concave up function that passes through $(0, F)$, which looks something like the red curve in the following picture:
The graphical "interpretation" of Equation $(2)$ is then to equate the tangent of the level curve of $V(x,y)=0$ to $-(1-p_i)/p_i$. Note that since $p_d>p_c$ then $$-\frac{1-p_c}{p_c} < -\frac{1-p_d}{p_d} <0 $$
To illustrate this in the above picture, the solution $(x_c^*, y_c^*)$ corresponds to the intersection of the orange tangent line with the red curve while $(x_d^*, y_d^*)$ corresponds to the intersection of the green tangent line with the red curve. From the picture, it is clear that $x_d^* > x_c^*$ and $y_d^* < y_c^*$ because of the curvature of the red curve.
Queries:
How can I "convert" my pictorial proof above into a more rigorous mathematical written proof? That is, can I prove the required result without a graphical approach?
