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 patience types and a common HARA Bernoulli utility. The HARA utility is posed on its natural domain; after the affine shift Xᵢ=xᵢ+h X i = x i + h , Yᵢ=yᵢ+h Y i = y i + h , the consumer problem becomes a CRRA problem with shifted endowments, so the first-order demand used in the proof is genuinely interior. The main result is a global coefficient-ratio criterion: after a change of variables, market clearing can be written as r^θ =A(r)/B(r) r θ = A ( r ) / B ( r ) , where A A and B B have positive coefficients built from elementary symmetric polynomials in the patience parameters. If the associated coefficient ratios form a monotone chain, then $$A(r)/B(r)$$ A ( r ) / B ( r ) is nonincreasing while r^θ r θ is strictly increasing, and the equilibrium price is unique. Sorting of patience and endowment composition—more patient agents holding weakly more of the first good and weakly less of the second—is an economically interpretable sufficient condition for this chain. In the sorted case, the result recovers the ordered-endowment CRRA benchmark of Geanakoplos and Walsh (2018) for every b≥ 0 b ≥ 0 . The autonomous contribution is the coefficient-ratio criterion itself: it is strictly more general than sorting, as shown by a non-sorted, non-collinear example satisfying the chain.