Proposition A.4 (ii) — the optimal payment rate $z_{SB}$ lies in $[v_x,\frac p{r+p}v_x]$ for $v_x\le0$ and equals $\frac p{r+p}v_x$ for $v_x\ge0$

Fix real numbers $y$ (standing for $v_x$) and $k$ (standing for $v_{xx}$), and consider the objective of the producer's HJB equation (A.11) in the limit $A\nearrow\infty$:

where $F_0$ is the function of Lemma A.1. Then:

1. if $y\ge0$, the point $z=\frac p{r+p}y$ minimises $\Phi$ over $\mathbb R$;
2. if $y\le0$, $\Phi$ has a minimiser over $\mathbb R$ in the interval $\big[y,\frac p{r+p}y\big]$.

This locates the second-best payment rate for consumption reduction: in off-peak periods it is explicit, and in peak periods it lies between the producer's marginal value $v_x$ and its fraction $\frac p{r+p}v_x$.

Formalization Note. The page states the claim "for large $A$", with the cap term $\eta_A(v_x,z)=(v_x+(z^--A)^+)^2-v_x^2\to0$ as $A\nearrow\infty$ in (A.11); it is formalized with $\eta_A\equiv0$, the limit the page names (with $\eta_A$ at finite $A$ the claim can fail). The page's open interval $(v_x,\frac p{r+p}v_x)$ is empty at $v_x=0$, where the minimiser is $0$; the closed interval is used. "The minimiser" is read as "a minimiser": when $\bar\mu=0$ and $d=0$ the minimiser need not be unique.