For a fixed number R, define () as a random variable that takes value with probability 1. Let us say that preference relation is risk-averse if for every random variable X it is true that (EX) X. Consider rational decision maker with strictly increasing utility function. Prove that the following statements are equivalent. (a) u is concave. (b) The preference relationship induced by utility function u is risk-averse (in the sense defined above). (c) If c(X) is the certainty equivalent of X, then c(X) EX.