UTILITY FUNCTIONS

 

We are now in a position to derive utility functions for our friends, Anna and Bob. Since monetary values are identical for everyone, the expected monetary value criterion (EMV) will recommend identical decisions for everybody. Not good. What we want is a new metric that incorporates the decision maker’s attitude toward risk. A utility function describes a decision maker’s risk profile for a particular decision problem. A useful way of portraying utility functions is by means of a graph.

First of all, we need a utility scale to measure the subjective value of dollars under conditions of risk. A utility scale is an arbitrary interval scale. On an interval scale, each unit measures equal magnitudes of whatever it is the scale measures. The Fahrenheit and Celsius temperature scales, for example, are interval scales: each Fahrenheit degree measures the same magnitude of heat, but a Fahrenheit degree is different from a Celsius degree (the magnitudes they measure are different). On an interval scale, any two points can be chosen arbitrarily, that is, they can be assigned any numerical value. The rest of the points on the scale, however, must conform to these two arbitrary points so that the intervals measure increments correctly. On the SI system, the freezing point of water (at sea level) was called zero degrees Celsius (0° C) while the boiling point was called 100° C. In the Fahrenheit scale, the same two thermal points are 32° F and 212° F. (Daniel Fahrenheit used 0° F for the coldest temperature achievable with early 18th century technology and 100° F for the human body temperature. Actual body temperature in ° F, however, turned out to be 98.6°.)

Customarily, the two arbitrary end points most often used for utility scales are 0 and 1. That is fine. However, to avoid possible confusion with probabilities, we will use a scale that ranges from 0 to 10. Remember, the numbers are purely arbitrary. The 0 does not mean utter worthlessness; it simply designates the lowest monetary amount in the problem. (Neither 0° C nor 0° F denotes an absolute lack of heat.) The 10, in turn, designates the highest dollar amount in the problem.

We add the new utility scale to the game tree alongside the monetary scale:

 [ut2]

The utility of $100 is set to 10 while $0 is given a utility of 0. But why is the CE given a utility of 5? Elementary, my dear Watson. Since the certain equivalent is, by definition, valued as much as the lottery, their utilities must be the same.

 U(CE) = EU(L)

 EU(L) = 0.5(10) + 0.5(0) = 5  ?  U(CE) = 5

Since Anna’s CE is $25, for her $25 has utility of 5. Bob’s CE is $65, so in his scale $65 has utility of 5.

Plotting their utility functions, we obtain:

 [ut3]

Risk aversion is denoted by a curve that is concave downwards. Risk preference shows upward concavity. Risk neutrality is indicated by a straight line. If a person is risk neutral, there is no need to determine his or her utility function: the monetary values suffice for analysis and the EMV criterion is perfectly adequate.