Utility Function

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Utility Function

“Economics is the science which studies human behavior as a relationship between given ends and scarce means which have alternative uses.” This definition of economics, stated by Lionel Robbins in his landmark work An Essay on the Nature and Significance of Economic Science (1932), established the fundamental nature of economics. Given a population with seemingly unlimited wants and needs, the questions of how goods and services are consumed, produced, and distributed are the primary ones to be answered. As a consequence, the amount of satisfaction or happiness derived from consuming resources is of particular interest. Are goods and services being distributed and consumed so as to provide the greatest amount of happiness to individuals and society as a whole? To address this problem, economists require some way of measuring the happiness that is attained from consumption. Utility can be defined as the amount of usefulness or satisfaction or happiness gained by consuming goods and services. Utility is dependent on the bundles of goods consumed by an individual, with greater quantities of goods representing greater levels of happiness or utility. Thus, we can define a utility function as a positive relationship between the consumption of goods and services and the amount of utility received from that consumption. If a utility function can be clearly defined, then it is possible to address the question of maximizing an individual’s utility, based on the size of their consumption bundle. Comparisons can also be made between different bundles, and we can make conclusions regarding individual’s preferences, based on their ranking of various bundles. A well-defined utility function is also relevant when considering the welfare of an entire society. It enables social planners to determine how to distribute goods to individuals so as to maximize the total utility of a society.

The concept of utility and the doctrine of utilitarianism are rooted in Jeremy Bentham’s Principles of Morals and Legislation (1789). In that work’s introduction, Bentham declares that humans are placed “under the governance of two sovereign masters, pain and pleasure” and that the “principle of utility recognizes this subjection, and assumes it for the foundation of that system, the object of which is to rear the fabric of felicity by the hands of reason and of law” (p. 1). Bentham’s “felicific calculus” was an attempt to construct a measure of utility, based on the following dimensions of pleasure and pain: intensity, duration, certainty or uncertainty, and propinquity or remoteness. The difficulties associated with such measurements were enormous, however. There was no clear way to compare feelings that were qualitatively unlike each other, nor any way to measure the intensity of feelings, much less make any comparison between the feelings of different individuals. Nonetheless, by the 1870s utility theory had come to be widely accepted, due to the independent contributions of William Jevons, Carl Menger, and Léon Walras, the leaders of the so-called Marginalist Revolution. Their works initiated a movement away from the classical theory of value set forth by Adam Smith and David Ricardo, among others, which held that the value of a product was based on its production costs. Following Jevons, Menger, and Walras, the emphasis came to be placed on the perceived value of a good and the utility a consumer would receive from the consumption of the good. The existence of utility was accepted by all three; however, the problem of the measurability of utility and the issue of defining the exact form of the utility function remained. None of the three explicitly addressed the problem of measurability; they either merely assumed the existence of such measurements, or stated that utility was measurable but not at the present time.

There were, however, attempts to develop the utility function, which is based on the quantities of the goods consumed. For the goods x ₁, x ₂, x ₃, … an individual’s utility was written as a function of those goods that represented ordinal utility: φ (x ₁, x ₂, x ₃, …). This generalized form was proposed by Vilfredo Pareto, who called it an index function, and then also later on by Francis Edgeworth. Greater levels of consumption were believed to provide greater levels of utility. Consumers were assumed to have rationally ordered preferences and to choose a consumption bundle so as to maximize utility. Utility was also assumed to diminish with each additional unit consumed of each good. Additional units consumed of a good will increase an individual’s total utility, but at a diminishing rate. In the works of several economists, including William Lloyd, N. W. Senior, and Richard Jennings, clear statements of diminishing marginal utility were given. By using the concept of diminishing marginal utility, Adam Smith’s water/diamonds paradox—water is necessary for life and diamonds are not, but the price of diamonds is many times higher—could now be resolved. The relative abundance of water versus the relative scarcity of diamonds is the key here, along with the marginal utility of the last unit consumed of each good. Water is abundant enough that the marginal utility obtained from the last gallon consumed is rather low, compared to the marginal utility of the last diamond consumed.

The concept of diminishing marginal utility led in turn to the concept of the downward sloping demand curve. Walras successfully established the link between utility and demand, by using equations expressing maximum satisfaction for an individual. For some given number of m commodities, he derived the demand function as a relationship between the quantity and price of a commodity and the prices of all the other commodities, ceteris paribus —that is, with all other variables (such as money income and tastes) held constant. In 1892 Pareto rigorously showed that diminishing marginal utility directly implied negatively sloped demand curves. Alfred Marshall’s Principles of Economics, first published in 1890, also formally constructed a demand curve based on utility and marginal utility. Marshall stated that “[t]here is then one law and only one law which is common to all demand schedules[,] … that the greater the amount to be sold, the smaller will be the price at which it will find purchasers” (pp. 159–160)—a conclusion he drew from the concept of diminishing marginal utility. Eugen Slutsky and John Hicks, both working with the assumptions that the utility function was additive and that consumers faced the diminishing marginal utility of consumption, also derived the downward sloping demand curve. The total change in demand for a good could be decomposed into two elements: the substitution effect and the income effect. For instance, if the price of a good increases the substitution effect dictates that a consumer will substitute consumption for a cheaper good in place of the now more expensive good. The income effect dictates that due to the price increase, a consumer effectively has less real income to spend, and will therefore consume less of both goods. For a normal good, this means that a price increase will mean a decrease in quantity demanded, and thus the demand curve will be downward sloping.

Another important extension of the utility function is the indifference curve, devised initially by Edgeworth. Consider a case involving two goods, and the levels of utility gained from consuming various combinations of the goods. A three-dimensional graph of this function, with two coordinates representing quantities of the two goods, and the third coordinate representing utility level, will give a utility surface, which rises with increasing quantities of the two goods consumed. For varying levels of utility, an indifference curve can be represented by “slices” of the utility surface that are parallel to the plane of the two commodities. An indifference curve is a two-dimensional graph of all the possible combinations of two goods that will give a consumer a given amount of utility. The downward slope of an indifference curve indicates the rate of tradeoff between the two goods, known as the marginal rate of substitution. Because a consumer is assumed to experience diminishing marginal utility for each additional unit consumed, decreasing consumption of one good must be offset by increasingly greater consumption of the other good, if the same level of utility is to be maintained. This fact gives rise to the shape of the indifference curve, which is convex to the origin. A graph of indifference curves for varying levels of utility is known as an indifference map. Consumers prefer to be on higher indifference curves, because these represent greater levels of consumption bundles and therefore greater levels of utility. Paul Samuelson’s revealed preference theory was another area of work that described consumer behavior. Instead of assuming the existence of a utility function and establishing its properties, Samuelson stated that consumers revealed their preferences by what goods they purchased. From observing consumer behavior, preference relationships between bundles of goods could be established, and downward sloping indifference curves could be constructed, based on diminishing marginal utility associated with increasing consumption.

The advent of modern utility theory came with the publication of John von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior in 1947. In this work, it was established that individual agents make decisions so as to maximize the expected amount of utility they received from their choice of a consumption bundle. Because these choices are assigned probabilistic outcomes, utility theory could be considered to have a game theoretic framework that includes risk and an individual’s attitudes toward risk-taking. Decisions regarding utility-maximization are influenced by the probabilities of outcomes. If a consumer’s utility is assumed to be based on consumption as a whole and not just on one good, measures of risk aversion can be written in the form developed by economists Kenneth Arrow and John Pratt. Measurements of risk aversion are of particular importance in analyzing the behavior of consumers who make investment decisions under conditions of uncertainty.

In addition to examining economic activity as it is, without value judgment, the discipline of economics also attempts to establish normative guidelines, based on how things ought to be. Welfare economics considers the normative aspects of economic outcomes and analyzes whether these outcomes can be improved through benevolent social planning. Thus, the question of maximizing social utility becomes relevant. Edgeworth and Marshall, among others, assumed that utility was cardinal —that is, measurable in absolute terms—and that it was possible to make interpersonal comparisons of individual utility functions. This led to their conclusion that a social welfare function could be constructed based on a summing of all individual utility functions. Pareto, Hicks, and Nicholas Kaldor took a different approach, using an ordinal utility, which considers only the rankings of different bundles of goods. Social welfare was analyzed on the basis of Pareto efficiency, the principle that a society’s condition was optimal if no further improvements in the allocation of goods could be made without making someone worse off. Kaldor-Hicks efficiency was an extension of Pareto efficiency. In this model, an outcome could be improved if the individuals who would be made better off by a reallocation of goods compensated those who would be made worse off.

SEE ALSO Bentham, Jeremy; Expected Utility Theory; Marginalism; Objective Function; Pareto, Vilfredo; Preferences; Preferences, Interdependent; Principal-Agent Models; Representative Agent; Ricardo, David; Risk; Smith, Adam; Social Welfare Functions; Utilitarianism; Utility, Objective; Utility, Subjective; Utility, Von Neumann-Morgenstern; Von Neumann, John; Walras, Léon