🤖 AI Summary
This study investigates the necessary and sufficient conditions for dynamic efficiency, as characterized by the Cass criterion, in an overlapping generations model with consumption and borrowing where both population and per capita endowments grow without bound. Employing methods from differential geometry, the paper projects indifference hypersurfaces and aggregate supply hypersurfaces onto the trading hyperplane to quantify the curvature required for their tangency. By integrating Walrasian demand normalization with indirect utility analysis, the work reveals the local geometric nature of the Cass criterion. The main contribution is a general efficiency test: the economy is dynamically efficient if and only if the series ∑₁^∞ 1/(‖pₜ‖∑_{h∈Gₜ}‖cₜʰ‖) diverges, a condition that applies broadly to unbounded dynamic settings.
📝 Abstract
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 $λ_{n}(p)\partial e(p,v_{n}(p))/\partial u>0$ measures how much more bent the offer hypersurface is relative to the indifference hypersurface, where $λ_{n}(\cdot)$ 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.