Given are the following two functions: $ g(x,\theta)=1-\frac{\left( 1-\theta\right) }{\theta}\left( \frac{(x-1)R} {(1-(1-\pi)i)R +\left( 1-\pi\right) i x}\right) \tag{NAG} $ and $f(x)=\frac{\pi xR}{\left[ \left( 1-i\right) R+ix\right] } \cdot\frac{(1-i)}{\left( 1-(1-\pi)(1-i)\right) } \tag{CMP}$
What I want to show is that the intersection point of $g(x,\theta)$ and $f(x)$ increases with $\theta$. I mean that an increase in $\theta$ leads to a new intersection which is characterized by both higher x and higher function value.
This can be shown graphically but what I need is a formal proof. The problem is that solving for the equilibrium is a solution of a complex (but just) quadratic equation.
Note that the CMP is no function of $\theta$ but an increasing function of x $\forall x \in \mathbb{R}_{\leq0}$, thus a change in $\theta$ moves the intersection along CMP. Additionally, NAC is a decreasing function of $x$ $(\frac{\partial g(x,\theta)}{\partial p_{n}}<0$) but the slope of the tangent increases with $\theta$ (\frac{\partial^2 g(x,\theta)}{\partial p_{n} \partial \theta}>0). Since the NAC passes independently of the parameter setting through $(1,1)$, an increase in $\theta$, from $\theta^{**}$ to $\theta^{*} $leads to a raise of the angel $\alpha$ moving a low intersection point $(x^{**},y^{**})$ to a higher $(x^{*},y^{*})$. To summarize, what I need is a formal proof that shows that the intersection point increases with an increase in $\theta$. I hope someone can help me to show this with a without the need of solving for the intersection point. (I actually do not know how to start). Thx for your help!!!