Sorting and Global Uniqueness in Two-Good HARA Economies with Many Patience Types

We study global uniqueness of competitive equilibrium in two-good pure-exchange economies with heterogeneous impatience types and a common HARA Bernoulli utility. The paper connects the CRRA sorting result of \citet{GeanakoplosWalsh2018} with the line of HARA uniqueness results developed in \citet{LoiMatta2022,LoiMatta2024}. In the CRRA case, ordered endowments provide a sorting mechanism for uniqueness. In the HARA case, uniqueness is known to hold for arbitrary endowments under the curvature bound $\gamma\le I/(I-1)$, where $I$ is the number of impatience types. For two types, the curvature restriction can be removed under a monotone sorting condition linking patience and endowment composition. The present paper shows that this high-curvature HARA sorting mechanism is not specific to the two-type case. Our main result proves global uniqueness for any finite number of impatience types and any $\gamma>1$. If types can be ordered so that more patient agents hold weakly more of the first good and weakly less of the second, then the equilibrium price is globally unique. Thus the paper extends the two-type high-curvature HARA result to a genuinely multi-type setting and complements the arbitrary-endowment low-curvature result by replacing the low-curvature restriction with an economically interpretable sorting restriction. In the CRRA subcase ($b=0$), the ordered-endowment condition coincides with that of \citet{GeanakoplosWalsh2018}, and our corollary recovers their uniqueness result. The contribution of the present paper is therefore not the sorting condition itself but its reach: the same ordered heterogeneity in patience and endowment composition rules out multiplicity throughout the shifted HARA case ($b>0$), for any finite number of types and any $\gamma>1$, through a global coefficient-ratio argument.