  This paper examines the identification power of assumptions that formalize
the notion of complementarity in the context of a nonparametric bounds analysis
of treatment response. I extend the literature on partial identification via
shape restrictions by exploiting cross-dimensional restrictions on treatment
response when treatments are multidimensional; the assumption of
supermodularity can strengthen bounds on average treatment effects in studies
of policy complementarity. This restriction can be combined with a statistical
independence assumption to derive improved bounds on treatment effect
distributions, aiding in the evaluation of complex randomized controlled
trials. Complementarities arising from treatment effect heterogeneity can be
incorporated through supermodular instrumental variables to strengthen
identification in studies with one or multiple treatments. An application
examining the long-run impact of zoning on the evolution of urban spatial
structure illustrates the value of the proposed identification methods.
