4.1 The envelope parameters
There is a unique \(a {\gt} t_0\) such that \(H(a)=B\), and the resulting envelope \(J_G\) lies above the affine function \(u \mapsto B - p_0 u\) on \([0,\infty )\).
Proof
The map \(H(x)=x s_G(x)+\int _x^\infty s_G(t)\, dt\) is strictly decreasing on \((t_0,\infty )\) and decays to \(0\) at infinity, while \(H(t_0)=A=2B\). This gives a unique solution to \(H(a)=B\). The tangent-line lower bound for \(J_G\) follows from the piecewise definition of the envelope: on \([0,a]\) the envelope is exactly the affine line, and on \([a,\infty )\) the tail-integral branch stays above its common value at \(u=a\).