Suppose that the regulator relies on the lobbying efforts to decide whether the product can be marketed. We look for an equilibrium of the contest game, where lobby \(I\) with type \(\delta\) plays \(x^{*}\left(\delta\right)\) and lobby \(E\) plays \(y^{*}\). We show that a unique equilibrium exists, where all players play interior strategies.
Consider first the equilibrium behavior of lobby \(I\). Anticipating lobby \(E\)’s effort \(y\), lobby \(I\) of type \(\delta\) chooses \(x\) to maximize his expected utility
\begin{equation} \pi\left(x,y\right)\left(b-\min\left(\delta,k\right)\right)-x\mbox{.}\nonumber \\ \end{equation}If \(y=0\), as \(\pi\left(x,y\right)=1\) for all \(x\), then \(x=0\) is optimal. Otherwise, we can differentiate it with respect to \(x\) to obtain the first-order condition describing lobby \(I\)’s optimal effort
\begin{equation} \left(\frac{y}{\left(x+y\right)^{2}}\right)\left(b-\min\left(\delta,k\right)\right)-1\leq 0\text{,}\\ \end{equation}which holds with equality for an interior solution. It follows that the lobby \(I\)’s reaction function is