Abstract
A utility function is a tool used to assign numerical values to a set of goods or services in order to allow for a numerical representation of the preferences of a rational individual. Though its first use dates back to the 18th century, the term “utility” does not appear until the late 19th century in Welfare Economics, and it is fully axiomatised only in the mid-20th century with the birth of Decision Theory, where the utility function is the central concept. Indeed, in several studies focused, e.g., on analysing the roles of the rationality axioms and their weakening, or in completing partial preferences, a utility function is always considered to be the proper tool to fully depict the attitude of an investor toward risk.
TopBackground
The so-called “St. Petersburg paradox” was first raised by N. Bernoulli (1713), and can be shortly exposed as follows. Suppose that a game is proposed, based on repeatedly throwing a coin and paying, on the first occurrence of “heads,” a prize that doubles at each previous occurrence of “tails.” If, for instance, the base prize is “one bean,” then one bean is paid if the first toss yields “tails,” otherwise the prize becomes two beans, and so forth. In such a way, if the coin has to be tossed, say, five times before it yields the first occurrence of “tails,” the prize paid at the fifth toss is 25 – 1 = 16 beans. It is straightforward that the expected value of the win is infinite; yet, and hence the term “paradox,” it seems unrealistic that a rational investor would accept to trade his entire wealth with the right of playing such a game a single time.
A few years later, Cramer (1728) and D. Bernoulli (1738) proposed two solutions to such a problem. Although apparently rival to each other, both their approaches were based on the idea that a player would focus, rather than on (the expected value of) the “material satisfaction” measured by the monetary value of a win, on the “moral satisfaction” which is supposed to be a mathematical function of that win. Such function was the square root for Cramer and the logarithm for D. Bernoulli: notably, both of them were increasing and concave, or marginally decreasing (that is to say, with a decreasing derivative function).
The actual development of the concept of marginal utility had to wait until the second half of the 19th century, when it was almost simultaneously proposed by Jevons (1871), Menger (1871). and Walras (1874). Their works, implying the need to abandon the concept of “objective” value of goods in favor to a “subjective” value essentially determined by the equilibrium of supply and demand, can be considered to mark the transition from Classical to Neoclassical economics.
In the spirit of Bentham (1776) and Mill (1863), the utility they had in mind was a cardinal, “objective” unit of measure for people’s satisfaction. For instance, if an individual’s utility for a cup of tea and a glass of milk were, respectively, 10 and 5, it would be acceptable to conclude that such an individual values tea “twice as” milk, and sometimes even that (s)he would consider “equivalent” a cup of tea and two glasses of milk. Moreover, utilities could even be added across individuals in order to obtain the “total amount” of satisfaction of a community, whose maximisation had to be the main concern of political choices.
Key Terms in this Chapter
Pareto Optimality: A consumption allocation is Pareto optimal if it is not possible to increase the utility of some individual without decreasing the utility of someone else.
Utility Function: A function associating a real number to every item of consumption goods or items in such a way to mirror the desirability values that an individual attaches to them.
Preferences: An individual’s conviction on the relative rankings in a set of alternatives allowing her or him to choose his favourites from that set. If preferences satisfy some sensibleness requirement, usually called rationality axioms , then they can be represented by means of a utility function.
Decision Theory: Field of study crossing, among others, Economics, Mathematics, and Psychology, which aims at identifying the issues that are relevant in making choices.
Rational Individual: One whose preferences satisfy the “rationalty” axioms required by the model under consideration. Stronger rationality axioms lead to more significant and compelling representations of the individual choices by means of a utility function.
Risk Aversion: The reluctance of an individual to trade a certain payoff for an uncertain one, unless the latter has a greater expected value than the traded amount.
Lottery: The typical object of choices in von Neumann and Morgenstern framework. A lottery is a random variable whose possible determination and whose probability law are known to the decision maker.