I'm reading an economic article "Skewing the odds: Taking risks for rank-based rewards"
The author consider the following maximization problem: let $P$ be a continuous increasing function on $[0,\infty$) with $P(0)=\underline{\nu}$ and $P(\infty)=\overline{\nu}$ with $0\leq \underline{\nu}<\overline{\nu}<\infty$. Let $\mu \in (0,\infty)$ be a constant. Suppose in addition $P(\mu)<\overline{\nu}$. Then we consider the following optimization problem:
$$ \begin{array}{r l} \text{maximize} & \int_0^\infty P(x)dF(x) \\ \text{subject to} & \int_0^\infty dF(x) \leq 1\\ & \int_0^\infty xdF(x)dx \le \mu. \end{array} $$
The meaning of $P$ is the contest payoff function, the second constraint is capacity constraint.
Then the associated Lagrangian is $$ L(dF,\alpha,\beta) = \int_0^\infty [P(x) -(\alpha+\beta x)] dF(x) + \alpha+\beta. $$ Then by the Lagrange multiplier theorem, there exist $\alpha^0,\beta^0$ such that $$\begin{aligned} & \sup \left\{\int_0^\infty P(x)dF(x) : \int_0^\infty dF(x) \leq 1, \int_0^\infty xdF(x) \leq \mu \right\}\\ &=\sup_{dF\geq 0} L(dF,\alpha^0,\beta^0). \end{aligned} $$
Here is my question. According to the author, $\beta^0>0$. The reasoning is the following:
Since $P(\mu)<\overline{\nu}$, the contestant does not have sufficient capacity to gurantee the largest possible payoff $\overline{\nu}$. Thus, increased capacity has value. Hence the capacity constraint cannot be slack, i.e., the capacity constraint must be binding at the optimum and hence $\beta^0>0$.
I don't know what theorem does the author used. First, I thought that this is related to complementary slackness, but I'm not sure.
One of my colleague tried to prove this $\beta^0>0$ by using enveloping lemma(He is major in economics), but it seems that there is some mathematical issues on that argument.
How can I prove this?