Sufficient statistics for economic mobility

Empirical studies of intergenerational mobility measure it with different statistics-intergenerational elasticities, rank-rank slopes, transition matrices, axiomatic indices-and sometimes reach different conclusions about which places or periods experienced more mobility. I show that a single property of the joint parent-child distribution determines when all measures agree: concordance, which captures how rank-preserving the distribution is. When one economy is everywhere more rank-preserving than another, all standard relative-mobility measures produce identical orderings. Under this condition, the measures become substitutable-a researcher observing any one can infer what all others would show. I prove this holds for regression-based, transition-matrix, and axiomatic measures; remains valid under proxy outcomes and measurement error; and arises naturally in a Becker-Tomes-Loury model under an expansion in parental investments. When concordance fails to rank distributions, different mobility measures disagree because each apply different weights to regions of the joint distribution. These weights matter typically because inequality changes across contexts. Twentieth-century U.S. data validate the predictions: rank-based measures move together while elasticity-based measures diverge as inequality rises.