Beyond correlation: Measuring interdependence through complementarities

Given two sets of random variables, how can one determine whether the former variables are more interdependent than the latter? This question is of major importance to economists, for example, in comparing how various policies affect systemic risk or income inequality. Moreover, correlation is ill-suited to this task as it is typically not justified by any economic objective. Economists' interest in interdependence often stems from complementarities (or substitutabilities) in the environment they analyze. This paper studies interdependence using supermodular objective functions: these functions treat their variables as complements, and their expectation increases as the realizations of the variables become more aligned. The supermodular ordering has a linear structure, which we exploit to obtain tractable characterizations and methods for comparing multivariate distributions, and extend when objective functions are also monotonic or symmetric. We also provide sufficient conditions for comparing random variables generated by common and idiosyncratic shocks or by heterogeneous lotteries, and illustrate our methods with several applications.