Depth Projections and the Local Nature of the Cass Criterion

Leandro Lyra Braga Dognini

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Abstract: This paper defines the depth projections of the indifference and offer hypersurfaces, and of the transformation frontier. These projections are functions conceived to measure curvature (thus are not numbers, as the Gaussian curvature is) and their second-order behavior is governed by the Hessians of the expenditure and profit functions. In particular, up to second order, the depth projection of the offer hypersurface is always exactly twice that of the indifference hypersurface. In an overlapping generations economy with production, these projections lead to the derivation of general statements for the necessity and sufficiency of the Cass criterion in the form $\sum^{\infty}_{t=1}1/(\Vert p_{t}\Vert\sum_{h\in G_{t}}\Vert c^{h}\Vert)=\infty$, which allows unbounded dynamics for the demography and for per capita endowments, and reconciles the Cass criterion with the Balasko--Cass--Shell relabeling algorithm. Furthermore, the assumptions behind these statements reveal the local nature of the Cass criterion: only the behavior of the indifference hypersurfaces and transformation frontiers in a neighborhood of the equilibrium allocation matters.

Date: 2026-06, Revised 2026-09
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