Von Neumann-Morgenstern Utility Function

John von Neumann and Oskar Morgenstern extended the theory of consumer preferences by incorporating a theory of behavior toward risk variance. The utility function that bears their names arises from the expected utility hypothesis. Basically, when a consumer is faced with a choice of items or outcomes subject to various levels of chance, the optimal decision will be the one that maximizes the expected value of the utility (i.e., satisfaction) derived from the choice made. Expected value is the sum of the products of the various utilities and their associated probabilities. The consumer is expected to be able to rank the items or outcomes in terms of preference; but the expected value will be conditioned by their probability of occurrence.

The von Neumann-Morgenstern utility function can be used to explain risk-averse, risk-neutral, and riskloving behavior. For example, consider a project a firm undertook last year having particular probabilities for three possible payoffs. These payoffs were $10, $20, or $30. The respective probabilities of receiving them were 20%, 50%, and 30%. Thus, expected payoff from the project was $10(0.2) + $20(0.5) + $30(0.3) = $21. This year, the firm will again undertake the project, but this time, the respective probabilities for the payoffs have changed to 25%, 40%, and 35%. It is easy to verify that the expected payoff is still $21. In other words, mathematically speaking, nothing has changed. It is also true that the probabilities of the lowest and highest payoffs rose at the expense of the middle one, which means there is more variance (or risk) associated with the possible payoffs. The question to pose to the firm is whether or not it will adjust its utility derived from the project despite the project's having the same expected value from one year to the next. If the firm values both years equally, it is said to be risk neutral. The implication is that it equally values a guaranteed payoff of $21 with any set of probabilistic payoffs whose expected value is also $21.

If the firm preferred last year's project environment to this year's, it places higher value on less variability in payoffs. In that regard, by preferring more certainty, the firm is risk averse. Finally, if the increase in variability is actually preferred by the firm, it is said to be risk loving. In a gambling context, a risk averter puts higher utility on the expected value of the gamble than on taking the gamble itself. Conversely, a risk lover prefers to take the gamble rather than settle for a payoff equal to the expected value of that gamble. The implication of the expected utility hypothesis, therefore, is that consumers and firms seek to maximize the expectation of utility rather than monetary values alone. Since utility functions are subjective, different firms and people can approach any given risky event with quite different valuations. For example, the agency problem recognizes that a corporation's board of directors may be more risk loving than its shareholders and, therefore, would evaluate the choice of corporate transactions and investments quite differently even when all monetary values are known by all parties.

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