Before revision:  We show that under non-parametric pooling (where \gamma-heterogeneity must typically be estimated numerically ), the multiplicative decomposition holds such that \gamma \geq \alpha for all values of the elasticity parameter q > 0, and \beta-heterogeneity amounts to the discrete number of distinct mixture components in the system.
After revision:  Under non-parametric pooling (where \gamma-heterogeneity must typically be estimated numerically ), the multiplicative decomposition holds such that \gamma \geq \alpha for all values of the elasticity parameter q > 0, and \beta-heterogeneity amounts to the discrete number of distinct mixture components in the system.