Proposition 4 — with perfectly reliable resources, $\Delta\ge 0$ for every nondecreasing utility, the loss-averse and the CVaR objective

Consider $N$ products with common margin $p>0$, served either by $N$ dedicated resources of marginal cost $c>0$ (network SD) or by one flexible resource of marginal cost $c_{N+1}$ (network SF), all perfectly reliable. The firm has initial wealth $w_0$, and the demand vector $\tilde X$ is nonnegative with an arbitrary joint distribution. Let $\beta\ge 1$ and $\eta\in(0,1]$. Then for every flexible cost $c_{N+1}\le c$:

1. for every nondecreasing utility $u$ (the class $U_1$), every nonnegative SD investment $K$ is matched by a nonnegative SF investment $K_{N+1}$ with

2. the same holds for the loss-averse objective (6);
3. for the CVaR objective (8), every pair $(K,v)$ with $K\ge0$ is matched by a pair $(K_{N+1},v')$ with $K_{N+1}\ge 0$ and a CVaR bracket at least as large.

In the paper's terms: the flexibility premium satisfies $\Delta\ge 0$ for all $u_1\in U_1$, and in particular for the loss-averse and the CVaR objective functions. Together with Proposition 1 (risk neutrality), this identifies when the intuition "flexibility is worth paying for" is valid: whenever supply is perfectly reliable, whatever the demand distribution and whatever the (monotone) risk attitude.

Formalization Note The flexibility premium $\Delta=(c^I_{N+1}-c)/c$ of Definitions 1–2 is not defined as a number (the indifference cost need not be unique); "$\Delta\ge 0$" is encoded as "SF is weakly preferred for every $c_{N+1}\le c$", which is how p. 41 reads $\Delta$. "Weakly preferred" ($V^{SF,*}\ge V^{SD,*}$) is encoded without suprema, as the matching statements above. Perfect reliability ($\theta=1$) is built into the terminal wealths, which then no longer involve $\lambda$. Nonnegative, measurable demand and $p,c>0$ are stated explicitly; no density, independence or integrability of demand is assumed ("Let $\tilde X$ have any joint distribution").