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D. Martimort, J. Pouyet, and &. Ricci, If convergence towards that point occurs in nite time T sb (?) < +?, we demonstrate that output is necessarily null along the whole trajectory. Indeed, by CauchyLipschitz Theorem, there is a unique solution to the system now made of (A.9) and (A.10) with the initial condition (S sb (?, T sb (?)), q sb (?, T sb (?))) = ( ? S(?), 0) and this is

, S(?), 0) itself. But this contradicts the initial condition S sb (?, 0) = ? > ? if ? > ? S(?). Hence, the system cannot converge in nite time towards ( ? S(?), 0) and q sb (?, t) remains positive. Finally, gathering everything the system (A.9

, Suppose instead that ? ? ? S(?), or equivalently (because?Cbecause? because?C(?, S) is decreasing in S) ? C(?, ?) ? ? C(?, ? S(?)) = P (0)