Dynamic Equilibria with Nonsmooth Utilities and Stocks: An L∞ Differential GQVI Approach

We develop a comprehensive dynamic Walrasian framework entirely in $L^{\infty}$ so that prices and allocations are essentially bounded, and market clearing holds \emph{pointwise almost everywhere}. Utilities are allowed to be \emph{locally Lipschitz and quasi-concave}; we employ Clarke subgradients to derive generalized quasi-variational inequalities (GQVIs). We endogenize inventories through a capital-accumulation constraint, leading to a \emph{differential} QVI (dQVI). Existence is proved under either strong monotonicity or pseudo-monotonicity and coercivity. We establish \emph{Walras’ law, complementarity, stability and sensitivity} of the equilibrium correspondence in $L^2$-metrics, incorporate \emph{time-discounting} and \emph{uncertainty} on $\Omega\times[0,T]$, and present convergent \emph{numerical schemes} (Rockafellar–Wets penalties and extragradient). Our results close the “in mean vs pointwise” gap noted in dynamic models and connect to modern decomposition approaches for QVIs.