Deep Projections and the Local Nature of the Cass Criterion

This paper defines the deep projections of the indifference and the offer hypersurfaces. These projections are used to measure how much the trade hyperplane must be curved to reach these other two canonical manifolds. In particular, it is shown that the factor $\lambda_{n}(p)\partial e(p,v_{n}(p))/\partial u>0$ measures how much more bent the offer hypersurface is relative to the indifference hypersurface, where $\lambda_{n}(\cdot)$ is the Lagrange multiplier and $v_{n}(\cdot)$ is the indirect utility associated with the normalized Walrasian demand $x_{n}(\cdot)$. These definitions and results are then applied to a consumption-loan overlapping generations economy to provide general statements for the sufficiency and necessity of the Cass criterion based on $\sum^{\infty}_{t=1}1/(\Vert p_{t}\Vert\sum_{h\in G_{t}}\Vert c^{h}_{t}\Vert)=\infty$ (thus allowing unbounded dynamics for both the demography and per capita endowments) under assumptions that reveal its local nature.