Substitute Valuations with Divisible Goods∗

In a companion paper, we showed that weak and strong notions of substitutes in economies with discrete goods have different implications for auction theory and equilibrium theory. In contrast, for the divisible goods case with concave valuations, natural extensions of these concepts coincide. Concave substitute valuations are characterized by submodularity of the dual profit function over nonlinear prices and are robust with respect to additive concave perturbations, which extends a related notion of robustness established for complements, as in Milgrom (1994). When all bidders have concave substitute valuations, the Vickrey outcome is in the core but the law of aggregate demand, which holds for strong substitutes in discrete economies, can fail.