In economics, utility provides a measure of the satisfaction that a consumer derives from a good or service that they have purchased. In finance, the concept is usually used to measure how satisfaction changes with differing levels of (terminal, i.e., end of period) wealth or with risk and return. Utilities constitute a useful illustration of how the concepts of functions and differentiation can be applied in finance.

Let us start by considering utility as a function of wealth. We would write the utility function as U = f (W). Many such utility functions would be possible, e.g.

1.

U = 5 + 8W 2.

U = 30 − e 0.5W 3.

U = 100W + 0.5W2 4.

U = ln(W)

But would they all make sense as utility functions and what are the properties that we would want a utility function to have?