We decompose the generalized Lorenz order into a size and a distribution component. The former is represented by stochastic dominance, the latter
by the standard Lorenz order. We show that it is always possible, given generalized Lorenz dominance between two distributions F and G, to find distributions H1 and H2 such that F stochastically dominates H1 and H1 Lorenz-dominates G, and such that F Lorenz-dominates H2 and H2 stochastically dominates G. We also show that generalized Lorenz dominance is characterized by this property and discuss the implications of these results for choice under risk.