Complementarity and aggregate implications of assortative matching: A nonparametric analysis

This paper presents econometric methods for measuring the average output effect of reallocating an indivisible input across production units. A distinctive feature of reallocations is that, by definition, they involve no augmentation of resources and, as such, leave the marginal distribution of the reallocated input unchanged. Nevertheless, if the production technology is nonseparable, they may alter average output. An example is the reallocation of teachers across classrooms composed of students of varying mean ability. We focus on the effects of reallocating one input, while holding the assignment of another, potentially complementary, input fixed. We introduce a class of such reallocations—correlated matching rules—that includes the status quo allocation, a random allocation, and both the perfect positive and negative assortative matching allocations as special cases. We also characterize the effects of small changes in the status quo allocation. Our analysis leaves the production technology nonparametric. Identification therefore requires conditional exogeneity of the input to be reallocated given the potentially complementary (and possibly other) input(s). We relate this exogeneity assumption to the pairwise stability concept used in the game theoretic literature on two‐sided matching models with transfers. For estimation, we use a two‐step approach. In the first step, we nonparametrically estimate the production function. In the second step, we average the estimated production function over the distribution of inputs induced by the new assignment rule. Our methods build upon the partial mean literature, but require extensions involving boundary issues and the fact that the weight function used in averaging is itself estimated. We derive the large‐sample properties of our proposed estimators and assess their small‐sample properties via a limited set of Monte Carlo experiments. Our characterization of the large‐sample properties of estimated correlated matching rules uses a new result on kernel estimated “double averages,” which may be of independent interest.