Question

How can two different benefit functions reflect exactly the same preferences? Explain.

Answer 1

Suppose I strictly prefer Apple (AA) to Banana (BB) and that these are the only two goods in existence.

Consider the following functions:†

• The function u with u(A)=1u(A)=1, u(B)=0u(B)=0.
• The function v with v(A)=−99v(A)=−99, v(B)=−100v(B)=−100.
• The function w with w(A)=0.1w(A)=0.1, w(B)=0.01w(B)=0.01.

Each of the above functions correctly represents my preference because in each case, the function's value at AA exceeds that at BB.

A utility function (representation) is merely a (very) convenient way to numerically encode my preference. Here there are only two goods and I prefer AA to BB. So, any function whose value at AA is greater than that at BB would correctly represent my preference.

In this example, we have only two goods, so a utility function isn't particularly useful or convenient.

But with more goods (and sometimes also more assumptions), utility functions can help to produce useful results that tell us something about the real world.