Sales and Markup Dispersion: Theory and Empirics

We characterize the relationship between the distributions of two variables linked by a structural model. We then show that, in models of heterogeneous firms in monopolistic competition, this relationship implies a new demand function that we call "CREMR" (Constant Revenue Elasticity of Marginal Revenue). This demand function is the only one that is consistent with productivity and sales distributions having the same form (whether Pareto, lognormal, or Frechet) in the cross section, and it is necessary and sufficient for Gibrat's Law to hold over time. Among the applications we consider, we use our methodology to characterize misallocation across firms; we derive the distribution of markups implied by any assumptions on demand and productivity; and we show empirically that CREMR-based markup distributions provide an excellent parsimonious fit to Indian firm-level data, which in turn allows us to calculate the proportion of firms that are of suboptimal size in the market equilibrium.