Concave Function

A function is concave on an interval, if in a certain environment of any point within this interval the graph of the function lies below the tangent line constructed at that point.

A type of concave utility function that is sometimes used in decision analysis is the exponential function of the form

u(x)=a – be^{-x/R },

where

*u(x)* = the utility value for payoff *x*

*a,b* = parameters that can be set to scale the function (to define the 0 and 1.0 points, for example)

*R* = the risk aversion parameter

The mentioned function has the special property that it exhibits constant risk aversion over the whole range of money values (the risk premium does not change in different parts of the utility curve).