Marginal values of capacity decrease in capacity and increase with the time remaining

Let $(\lambda,\rho)$ be a Markov chain choice model with the standing assumptions and $\sum_{j\in N}\lambda_j\le 1$, let $r\in\mathbb R^n$ be the product revenues and $T$ the number of periods. Write $\Delta V_t(x)=V_t(x)-V_t(x-1)$ for the value functions of the (Single Resource) dynamic program. Then

That is, the marginal value of capacity increases as fewer units remain and as more periods remain.

The paper attributes this to Talluri and van Ryzin (2004), for any choice model, and uses both inequalities in the proof of Theorem 5. The same result for an abstract choice model is published on Prove2Me as RevenueManagement.choice_marginal_values; taking its arrival probabilities equal to $1$ and its choice model equal to $P_{j,S}$ gives this statement.

Formalization Note The paper states the inequalities for $\Delta V_{t+1}$; the index ranges above are the ones the proof of Theorem 5 uses ($t=T+1$ is trivial since $V_{T+1}=0$). One hypothesis is added and disclosed: $\sum_j\lambda_j\le 1$, which the paper's description (at most one customer per period, arriving to purchase $j$ with probability $\lambda_j$) implies and which makes $1-\sum_jP_{j,S}$ a probability. No sign condition is placed on the revenues $r_j$, as on the page (the published Talluri–van Ryzin statement assumes nonnegative prices; this statement does not need them). Capacities are not bounded by $c$; the statement for every $x$ contains the one for $x\le c$.