The diffidence theorem, together with complementary tools, can aid in illuminating a broad set of questions about how to mathematically characterize the set of utility functions with specified economic properties. This article establishes the technique and illustrates its application to many questions, old and new. For example, among many other older and other technically more difficult results, it is shown that (1) several implications of globally greater risk aversion depend on distinct mathematical properties when the initial wealth level is known, (2) whether opening up a new asset market increases or decreases saving depends on whether the reciprocal of marginal utility is concave or convex, and (3) whether opening up a new asset market raises or lowers risk aversion toward small independent risks depends on whether absolute risk aversion is convex or concave.