Yau [Math. Ann. 207 (1974), pp. 269–270] proved that: There is no non-trivial continuous concave function on a complete manifold with finite volume. We prove analogue theorems for several metric spaces, including Alexandrov spaces curvature bounded below/above, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript alpha...

Zeldes (1989), Carroll (1992; 1993), and others have shown that optimal consumption behavior for consumers facing income uncertainty can be remarkably di erent from the certainty-equivalent case. Carroll (1992; 1993) observes that many of the di erences can be attributed to the concavity of the consumption function under uncertainty, but he does not describe the conditions under which the consu...