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# Utility

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### Utility

Utility, or usefulness, is the (perceived) ability of something to satisfy needs or wants.[1] Utility is an important concept in economics and game theory, because it represents satisfaction experienced by the consumer of a good. Not coincidentally, a good is something that satisfies human wants and provides utility, for example, to a consumer making a purchase. It was recognized that one can not directly measure benefit, satisfaction or happiness from a good or service, so instead economists have devised ways of representing and measuring utility in terms of economic choices that can be counted. Economists have attempted to perfect highly abstract methods of comparing utilities by observing and calculating economic choices. In the simplest sense, economists consider utility to be revealed in people's willingness to pay different amounts for different goods (i.e. in prices; without conflating the two concepts).

## Contents

• Economic definitions 1
• Quantifying utility 2
• Cardinal and ordinal utility 3
• Expected utility 4
• Additive von Neumann–Morgenstern utility 4.1
• Money 5
• Utility as probability of success 5.1
• Discussion and criticism 6
• References 8

## Economic definitions

In economics, utility is a representation of preferences over some set of goods and services. Preferences have a (continuous) utility representation so long as they are transitive, complete, and continuous.

Utility is usually applied by economists in such constructs as the indifference curve, which plot the combination of commodities that an individual or a society would accept to maintain a given level of satisfaction. Individual utility and social utility can be construed as the value of a utility function and a social welfare function respectively. When coupled with production or commodity constraints, under some assumptions, these functions can be used to analyze Pareto efficiency, such as illustrated by Edgeworth boxes in contract curves. Such efficiency is a central concept in welfare economics.

In finance, utility is applied to generate an individual's price for an asset called the indifference price. Utility functions are also related to risk measures, with the most common example being the entropic risk measure.There has been some controversy over the question whether the utility of a commodity can be measured or not. At one time, it was assumed that the consumer was able to say exactly how much utility he got from the commodity. The economists who made this assumption, belong to the 'Cardinalist School' (of Economics).

## Quantifying utility

It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer underlying relative utilities from observed choice. These 'revealed preferences', as they were named by Paul Samuelson, were revealed e.g. in people's willingness to pay:
Utility is taken to be correlative to Desire or Want. It has been already argued that desires cannot be measured directly, but only indirectly, by the outward phenomena to which they give rise: and that in those cases with which economics is chiefly concerned the measure is found in the price which a person is willing to pay for the fulfilment or satisfaction of his desire.[2]:78

## Cardinal and ordinal utility

Economists distinguish between cardinal utility and ordinal utility. When cardinal utility is used, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. On the other hand, ordinal utility captures only ranking and not strength of preferences.

Utility functions of both sorts assign a ranking to members of a choice set. For example, suppose a cup of orange juice has utility of 120 utils, a cup of tea has a utility of 80 utils, and a cup of water has a utility of 40 utils. When speaking of cardinal utility, it could be concluded that the cup of orange juice is better than the cup of tea by exactly the same amount by which the cup of tea is better than the cup of water. One is not entitled to conclude, however, that the cup of tea is two thirds as good as the cup of juice, because this conclusion would depend not only on magnitudes of utility differences, but also on the "zero" of utility.

It is tempting when dealing with cardinal utility to aggregate utilities across persons. The argument against this is that interpersonal comparisons of utility are meaningless because there is no simple way to interpret how different people value consumption bundles.

When ordinal utilities are used, differences in utils are treated as ethically or behaviorally meaningless: the utility index encode a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences. In the above example, it would only be possible to say that juice is preferred to tea to water, but no more.

Neoclassical economics has largely retreated from using cardinal utility functions as the basic objects of economic analysis, in favor of considering agent preferences over choice sets. However, preference relations can often be represented by utility functions satisfying several properties.

Ordinal utility functions are unique up to positive monotone transformations, while cardinal utilities are unique up to positive linear transformations.

Although preferences are the conventional foundation of microeconomics, it is often convenient to represent preferences with a utility function and analyze human behavior indirectly with utility functions. Let X be the consumption set, the set of all mutually-exclusive baskets the consumer could conceivably consume. The consumer's utility function u \colon X \rightarrow \R ranks each package in the consumption set. If the consumer strictly prefers x to y or is indifferent between them, then u(x)\geq u(y).

For example, suppose a consumer's consumption set is X = {nothing, 1 apple,1 orange, 1 apple and 1 orange, 2 apples, 2 oranges}, and its utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 4, u(2 apples) = 2 and u(2 oranges) = 3. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges.

In microeconomic models, there are usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity. This gives a consumption set of \R^L_+, and each package x \in \R^L_+ is a vector containing the amounts of each commodity. In the previous example, we might say there are two commodities: apples and oranges. If we say apples is the first commodity, and oranges the second, then the consumption set X =\R^2_+ and u(0, 0) = 0, u(1, 0) = 1, u(0, 1) = 2, u(1, 1) = 4, u(2, 0) = 2, u(0, 2) = 3 as before. Note that for u to be a utility function on X, it must be defined for every package in X.

A utility function u \colon X \rightarrow \R represents a preference relation \preceq on X iff for every x, y \in X, u(x)\leq u(y) implies x\preceq y. If u represents \preceq, then this implies \preceq is complete and transitive, and hence rational.

In order to simplify calculations, various assumptions have been made of utility functions.

Most utility functions used in modeling or theory are well-behaved. They are usually monotonic and quasi-concave. However, it is possible for preferences not to be representable by a utility function. An example is lexicographic preferences which are not continuous and cannot be represented by a continuous utility function.[3]

## Expected utility

The expected utility theory deals with the analysis of choices among risky projects with (possibly multidimensional) outcomes.

The expected utility model was first proposed by Nicholas Bernoulli in 1713 and solved by Daniel Bernoulli in 1738 as the St. Petersburg paradox. Bernoulli argued that the paradox could be resolved if decisionmakers displayed risk aversion and argued for a logarithmic cardinal utility function.

The first important use of the expected utility theory was that of game theory.

When comparing objects it makes sense to rank utilities, but older conceptions of utility allowed no way to compare the sizes of utilities — a person may say that a new shirt is preferable to a baloney sandwich, but not that it is twenty times preferable to the sandwich.

The reason is that the utility of twenty sandwiches is not twenty times the utility of one sandwich, by the law of diminishing returns. So it is hard to compare the utility of the shirt with 'twenty times the utility of the sandwich'. But Von Neumann and Morgenstern suggested an unambiguous way of making a comparison like this.

Their method of comparison involves considering probabilities. If a person can choose between various randomized events (lotteries), then it is possible to additively compare the shirt and the sandwich. It is possible to compare a sandwich with probability 1, to a shirt with probability p or nothing with probability 1 − p. By adjusting p, the point at which the sandwich becomes preferable defines the ratio of the utilities of the two options.

A notation for a lottery is as follows: if options A and B have probability p and 1 − p in the lottery, write it as a linear combination:

L = p A + (1-p) B

More generally, for a lottery with many possible options:

L = \sum_i p_i A_i,

where \textstyle\sum_i p_i =1.

By making some reasonable assumptions about the way choices behave, von Neumann and Morgenstern showed that if an agent can choose between the lotteries, then this agent has a utility function which can be added and multiplied by real numbers, which means the utility of an arbitrary lottery can be calculated as a linear combination of the utility of its parts.

This is called the expected utility theorem. The required assumptions are four axioms about the properties of the agent's preference relation over 'simple lotteries', which are lotteries with just two options. Writing B\preceq A to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are:

1. completeness: For any two simple lotteries L and M, either L\preceq M or M\preceq L (or both, that is: M\ equally \ preferable\ to\ L).
2. transitivity: for any three lotteries L,M,N, if L\preceq M and M\preceq N, then L\preceq N.
3. convexity/continuity (Archimedean property): If L \preceq M\preceq N, then there is a p between 0 and 1 such that the lottery pL + (1-p)N is equally preferable to M.
4. independence: for any three lotteries L,M,N, L \preceq M if and only if pL+(1-p)N \preceq pM+(1-p)N. Intuitively: If the lottery formed by the combination of L and N has less or equal utility (is no more preferable) than the lottery formed by the combination of M and N then and only then L \preceq M. And this formulation goes with the idea that L,M,N are probabilistically independent because then and only then we can combine them with complementary to 1 possibilities.

Axioms 3 and 4 enable us to decide about the relative utilities of two assets or lotteries.

In more formal language: A von Neumann–Morgenstern utility function is a function from choices to the real numbers:

u \colon X \rightarrow \R

which assigns a real number to every outcome in a way that captures the agent's preferences over simple lotteries. Under the four assumptions mentioned above, the agent will prefer a lottery L_2 to a lottery L_1 if and only if the expected utility of L_2 is greater than the expected utility of L_1:

L_1\preceq L_2 \text{ iff } u(L_1)\leq u(L_2).

Repeating in category language: u is a morphism between the category of preferences with uncertainty and the category of reals as an additive group.

Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom.

## Money

One of the most common uses of a utility function, especially in economics, is the utility of money. The utility function for money is a nonlinear function that is bounded and asymmetric about the origin. These properties can be derived from reasonable assumptions that are generally accepted by economists and decision theorists, especially proponents of rational choice theory. The utility function is concave in the positive region, reflecting the phenomenon of diminishing marginal utility. The boundedness reflects the fact that beyond a certain point money ceases being useful at all, as the size of any economy at any point in time is itself bounded. The asymmetry about the origin reflects the fact that gaining and losing money can have radically different implications both for individuals and businesses. The nonlinearity of the utility function for money has profound implications in decision making processes: in situations where outcomes of choices influence utility through gains or losses of money, which are the norm in most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period.[4]

### Utility as probability of success

Castagnoli and LiCalzi and Bordley and LiCalzi (2000) provided another interpretation for Von Neumann and Morgenstern's theory. Specifically for any utility function, there exists a hypothetical reference lottery with the utility of a lottery being its probability of performing no worse than the reference lottery. Suppose success is defined as getting an outcome no worse than the outcome of the reference lottery. Then this mathematical equivalence means that maximizing expected utility is equivalent to maximizing the probability of success. In many contexts, this makes the concept of utility easier to justify and to apply. For example, a firm's utility might be the probability of meeting uncertain future customer expectations. [5][6][7][8]

## Discussion and criticism

Cambridge economist Joan Robinson famously criticized utility for being a circular concept: "Utility is the quality in commodities that makes individuals want to buy them, and the fact that individuals want to buy commodities shows that they have utility"[9]:48 Robinson also pointed out that because the theory assumes that preferences are fixed this means that utility is not a testable assumption. This is because if we take changes in peoples' behavior in relation to a change in prices or a change in the underlying budget constraint we can never be sure to what extent the change in behavior was due to the change in price or budget constraint and how much was due to a change in preferences.[10] This criticism is similar to that of the philosopher Hans Albert who argued that the ceteris paribus conditions on which the marginalist theory of demand rested on rendered the theory itself an empty tautology and completely closed to experimental testing.[11] In essence, demand and supply curve (theoretical line of quantity of a product which would have been offered or requested for given price) is purely ontological and could never been demonstrated empirically.

Another criticism comes from the assertion that neither cardinal nor ordinary utility is empirically observable in the real world. In the case of cardinal utility it is impossible to measure the level of satisfaction "quantitatively" when someone consumes or purchases an apple. In case of ordinal utility, it is impossible to determine what choices were made when someone purchases, for example, an orange. Any act would involve preference over a vast set of choices (such as apple, orange juice, other vegetable, vitamin C tablets, exercise, not purchasing, etc.).[12][13]

Other questions of what arguments ought to enter into a utility function are difficult to answer, yet seem necessary to understanding utility. Whether people gain utility from coherence of wants, beliefs or a sense of duty is key to understanding their behavior in the utility organon. [14] Likewise, choosing between alternatives is itself a a process of determining what to consider as alternatives, a question of choice within uncertainty. [15]

An evolutionary psychology perspective is that utility may be better viewed as due to preferences that maximized evolutionary fitness in the ancestral environment but not necessarily in the current one.[16]

## References

2. ^ Marshall, Alfred (1920). Principles of Economics. An introductory volume (8th ed.). London: Macmillan.
3. ^ Ingersoll, Jonathan E., Jr. (1987). Theory of Financial Decision Making. Totowa: Rowman and Littlefield. p. 21.
4. ^ Berger, J. O. (1985). "Utility and Loss". Statistical Decision Theory and Bayesian Analysis (2nd ed.). Berlin: Springer-Verlag.
5. ^ Castagnoli, E. and M. LiCalzi. "Expected Utility Theory without Utility." Theory and Decision, 1996
6. ^ Bordley, R. and M. LiCalzi. "Decision Analysis with Targets instead of Utilities," Decisions in Economics and Finance. 2000.
7. ^ Bordley,R. And C.Kirkwood. Multiattribute preference analysis with Performance Targets. Operations Research. 2004.
8. ^ Bordley, R.; Pollock, S. (2009). "A Decision-Analytic Approach to Reliability-Based Design Optimization". Operations Research 57 (5): 1262–1270.
9. ^ Robinson, Joan (1962). Economic Philosophy. Harmondsworth, Middlesex, UK: Penguin Books.
10. ^ Pilkington, Philip. 'Joan Robinson’s Critique of Marginal Utility Theory'. Fixing the Economists. 17 February 2014. Available at: http://fixingtheeconomists.wordpress.com/2014/02/17/joan-robinsons-critique-of-marginal-utility-theory/
11. ^ Pilkington, Philip. 'utility Hans Albert Expands Robinson’s Critique of Marginal Utility Theory to the Law of Demand'. Fixing the Economists. 27 February 2014. Available at: http://fixingtheeconomists.wordpress.com/2014/02/27/hans-albert-expands-robinsons-critique-of-marginal-utility-theory-to-the-law-of-demand/
12. ^ Archived July 16, 2011 at the Wayback Machine
13. ^ http://elsa.berkeley.edu/~botond/mistakeschicago.pdf
14. ^ Klein, Daniel (May 2014). "Professor". Econ Journal Watch 11 (2): 97-105. Retrieved November 15, 2014.
15. ^ Burke, Kenneth (1932). Towards a Better Life. Berkeley, Calif.:: University of California Press.
16. ^ Paul H. Rubin and C. Monica Capra. The evolutionary psychology of economics. In Roberts, S. C. (2011). Roberts, S. Craig, ed. "Applied Evolutionary Psychology". Oxford University Press.

• Anand, Paul (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press.
• Georgescu-Roegen, Nicholas (Aug 1936). "The Pure Theory of Consumer's Behavior". Quarterly Journal of Economics 50 (4): 545–593.
• Nicholson, Walter (1978). Microeconomic Theory (Second ed.). Hinsdale: Dryden Press. pp. 53–87.
• Plous, S. (1993). The Psychology of Judgement and Decision Making. New York: McGraw-Hill.