proposition. A renewal bound valid for every admissible allocation [ftip-00O6]
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    Every allocation in Definition [ftip-00O5] satisfies

    \[\int _0^T v(t)\,dt\leq \min \left \{\frac {Ah_0(e^{gT}-1)}{g}, \frac {h_0e^{gT}-h_{\min }}{\eta }\right \}.\]

    Proof. 

      The stock equation gives \((e^{-gt}h(t))'=-\eta e^{-gt}v(t)\leq 0\) almost everywhere. Hence \(h(t)\leq h_0e^{gt}\). Integrating \(v\leq Ah\) gives the first bound. Integrating the discounted stock equation gives

      \[\eta \int _0^T e^{-gt}v(t)\,dt =h_0-e^{-gT}h(T)\leq h_0-e^{-gT}h_{\min }.\]

      Since \(v\geq 0\) and \(e^{-gt}\geq e^{-gT}\), multiplication by \(e^{gT}\) gives the second bound. The argument is pathwise, so it also covers randomized causal allocations satisfying the hypotheses.

      The bound is necessary and need not be attained. With a positive service-price lower bound \(p_*\), division by \(p_*\) supplies a compute envelope and can be used in proposition [ftip-00O3]. The relevant autonomous lower bound is still independent of this stock calculation. The bound increases with \(T\); it makes no finite lifetime claim.