Tail Elasticities and Threshold-Dependent Aggregation

This paper studies aggregation in economic models in which agents’ decisions are governed by endogenous thresholds. The location of a threshold and the distribution of characteristics relative to it jointly determine aggregate outcomes. Examples include heterogeneousfirm models such as Melitz (2003) and financial accelerator models following Bernanke, Gertler, and Gilchrist (1999). We identify the generalized failure rate, which measures the elasticity of the upper-tail probability with respect to the cutoff, as the key selection statistic. We show that an increasing generalized failure rate is equivalent to higher cutoffs making the relative tail distribution smaller in the first-order stochastic order. We then define log-log concavity of the density, a condition equivalent to decreasing density elasticity, and show that it implies an increasing generalized failure rate. Building on this structure, we derive general monotonicity results for ratios of weighted upper-tail integrals, a class of objects that includes many aggregate distortions and equilibrium wedges. The framework yields tractable comparative statics for average productivity, aggregate markups, market concentration, trade elasticities, equilibrium default, and conditional tail risk. It also clarifies the special role of Pareto tails, for which relative tail distributions are invariant to the selection threshold and aggregate elasticities are constant.