Utility Functions

11 Key excerpts on "Utility Functions"

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  Intermediate Microeconomics

  An Intuitive Approach with Calculus

  • Thomas Nechyba(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  We will now see that these indifference curves can be interpreted as parts of mathematical functions that summarize tastes more fully. These functions are called Utility Functions , and are mathematical rules that assign numbers to bundles of goods in such a way that more preferred bundles are assigned higher numbers. A mathematical function is a formula that assigns numbers to points. For instance, the function f ( x ) 5 x 2 is a way of assigning numbers to different points in the space R 1 , the real line, the space consisting of points with only a single component. To the point x 5 1/2, the function assigns a value of 1/4; to the point x 5 1, the function assigns a value of 1; and to the point x 5 2, the function assigns the value 4. The full function is depicted in Graph 4.7. 0 1 2 3 4 f ( x ) = x 2 f ( x ) f (1) = 1 2 f (2) = 2 2 f ( x ) = x 2 5 x 1 2 3 4 1 4 1 2 f ( ) = ( ) 2 1 2 1 2 Graph 4.7 An Example of a Function f : R 1 S R 1 In mathematical notation, we would indicate by f : R 1 S R 1 that such a function f is a formula that assigns a real number to each point on the real line. We would read this notation as ‘the function f takes points on the real line R 1 and assigns to them a value from the real line R 1 ’. Such functions are not, how-ever, of particular use to us as we think about representing tastes because we are generally considering bundles that consist of more than one good, bundles such as those consisting of combinations of hoodies and jeans. Thus, we might be more interested in a function f : R 2 1 S R 1 that assigns to each point made up of two real numbers (i.e. points that lie in R 2 1 ) a single real number (i.e. a number in R 1 ). One example of Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

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  Microeconomics

  Equilibrium and Efficiency

  • Thijs ten Raa(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  This shows that national production is not necessary to ensure a population’s high standard of living. Nor is it sufficient. This is, in a sense, the flip-side of the oil example. Surrounding countries may produce and supply the goods and services, but their workers may earn a miserable wage 42 The Demand Side of the Economy and hence enjoy a low standard of living, particularly when their economies are plagued by corruption or other sources of inefficiency. The fulfillment of consumers’ objectives is supreme to the fulfillment of pro-ducers’ objectives. The ultimate goal of an economy is to meet consumers’ needs. This is why we present the theory of the consumer first, even though the non-financial nature of consumers’ objectives makes these objectives more complicated to analyze than those of producers. In the next section we introduce Utility Functions, which are used to model the consumer’s choice problem. In Section 3.3 we test whether the hypothesis of utility maximization is consistent with the data and, if it is, we discuss the con-struction of a utility function. Utility is a slippery concept; utils are not absolute units of measurement. In Section 3.4 we show that utility can be represented by a close cousin, namely the expenditure function, whose values are in convenient monetary units. Section 3.5 introduces a practical, widely used class of Utility Functions, namely those with a constant elasticity of substitution. Section 3.6 shows how the quantities demanded vary with price given a certain standard of living. In Chapter 4 the quantities demanded will be reduced further as a function of price and income. Section 3.7 constructs price indices. 3.2 Utility Functions It is not easy to describe the level of well-being that comes with the consumption of a given quantities of goods and services. If there was only one consumable, say rice, one could simply use a certain quantity of it as the measure of well-being.

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  The Microeconomics of Public Policy Analysis

  • Lee S. Friedman(Author)
  • 2017(Publication Date)
  • Princeton University Press

    (Publisher)

  In the following chapters, we will build more thoroughly upon the general concepts in-troduced here in order to develop skills of application in specific policy contexts. Summary Because of the limits on obtainable data, a central task of the economics profession is to develop tools that allow the analyst to infer important consequences accurately and with a minimum of data. This chapter introduces one of these tools, the utility-maximization model of individual decision-making, which is used in many economic policy analyses. The model consists of these assumptions: Each individual has a preference-ordering of possible consumption choices that is consistent, convex, and nonsatiable, and makes choices in ac-cordance with that preference-ordering. The preference-ordering and choice-making can be represented as an ordinal utility function that the individual acts to maximize. While utility is neither measurable or inter-personally comparable, we can observe and measure an individual’s marginal rate of sub-stitution MRS i,j —the amount he or she can forgo of one good j in order to obtain an additional unit of another good i and remain just indifferent. The MRS i,j varies with the mixture of goods that the individual consumes—it depends not only on the preference-ordering, but also on the relative abundance or scarcity of each item in the consumer’s consumption bundle. We illustrate that utility-maximizing individuals, when subject to a budget constraint, will choose a mix of goods such that the MRS i, j of any two goods i and j that are in the bundle will equal the ratio of their prices P i / P j . The assumptions used in the model are not intended to be literally true; they are intended to yield accurate predictions of many decisions. The analyst always retains discretionary judgment about whether the model works well for the particular decisions a policy might 66 Chapter Three

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  Production, Growth, and the Environment

  An Economic Approach

  • William L. Weber(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  House-holds are demanders in the market for goods and services, but are suppliers of inputs such as land, labor, capital, and entrepreneurship. Likewise, business firms are demanders of inputs, but are suppliers of goods and services. 2.1.1 Utility Theory To study consumer choices between alternative competing wants economists rely on utility theory. Utility measures the amount of satisfaction an individual receives from consuming a given bundle of goods and services and economic theory only requires utility to be ordinal; i.e., consumers can rank various bun-dles according to their preferences, but are not required to determine whether 21 22 Production, Growth, and the Environment or not they like one bundle twice or three times as much as another bundle as would a cardinal utility function. Various assumptions are employed in utility theory: in general, more desirable goods are preferred to less and fewer unde-sirable goods (pollution) are preferred to more, substitution between two or more goods is possible, and preferences are subject to diminishing marginal rates of substitution. Suppose an individual consumes two goods—an environmental good like water (good x ) and other goods and services such as restaurant dinners, blue jeans, and housing (good y ). The two goods have prices p x and p y and the quantities the individual consumes are represented by x and y . An indifference curve represents all the various quantities of the two goods that yield the same level of utility. In general, as an individual consumes more of one good they must consume less of the other good to remain at the same level of utility. For instance, for the utility function u = x × y the equation of the indifference curve is found by solving for the quantity of one of the goods, say good y : y = u x . Table 2.1 gives the alternative quantities of the two goods that yield the same level of utility, say u = 10.

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  Microeconomic Theory

  Basic Principles and Extensions

  • Walter Nicholson, Christopher Snyder(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Good discussion of the foundations of preference theory. Most of the focus of the book is on utility in uncertain situations. Mas-Colell, Andrea, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory . New York: Oxford University Press, 1995. Chapters 2 and 3 provide a detailed development of preference relations and their representation by Utility Functions. Stigler, G. “The Development of Utility Theory.” Journal of Political Economy 59, pts. 1–2 (August/October 1950): 307–27, 373–96. A lucid and complete survey of the history of utility theory. Has many interesting insights and asides. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 111 EXTENSIONS SPECIAL PREFERENCES The utility function concept is a general one that can be adapted to a large number of special circumstances. Discovery of ingenious functional forms that reflect the essential aspects of some problem can provide a number of insights that would not be readily apparent with a more literary approach. Here we look at four aspects of preferences that economists have tried to model: (1) threshold effects, (2) quality, (3) habits and addiction, and (4) second-party preferences. In Chapters 7 and 17, we illustrate a number of additional ways of capturing aspects of preferences. E3.1 Threshold effects The model of utility that we developed in this chapter implies an individual will always prefer commodity bundle A to bundle B , provided U 1 A 2 . U 1 B 2 . There may be events that will cause people to shift quickly from consuming bun-dle A to consuming B . In many cases, however, such a light-ning-quick response seems unlikely. People may in fact be “set in their ways” and may require a rather large change in circumstances to change what they do.

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  Imperfect Knowledge Economics

  Exchange Rates and Risk

  • Roman Frydman, Michael D. Goldberg(Authors)
  • 2023(Publication Date)
  • Princeton University Press

    (Publisher)

  The a priori assumptions concerning individual preferences that are adopted by the contemporary approach are thus often thought of as axioms of rational choice. To represent the choice of a particular option, an economist typically picks the option that yields the greatest utility. However, an economist’s representation of the ranking of the outcomes in terms of his parametric utility function is insufficient to determine which option yields the highest utility. Although the utility numbers generated by u( . ) imply a ranking of the outcomes if they were certain to occur, the consequences of each of the options are uncertain. Option 1, for example, can result in one of the two outcomes, y 11 t +1 or y 12 t +1 . Thus, an economist must construct a specification of preferences that ranks options whose outcomes are uncertain. To this end, conventional economists have relied on the expected utility hypothesis (von Neumann and Morgenstern, 1944): the utility of option i = 1, 2, which we denote by U i , is equal to the expected value of the utilities of the outcomes that are associated with the option: U 1 t +1 = p 11 t +1 u(y 11 t +1 ) + p 12 t +1 u(y 12 t +1 ) (3.1) U 2 t +1 = p 21 t +1 u(y 21 t +1 ) + p 22 t +1 u(y 22 t +1 ). (3.2) In addition to the axioms of rational choice, conventional representa- tions of preferences are often based on the assumption of risk aversion: an individual is risk averse if replacing an uncertain final wealth by its expected value makes her better off. 1 In the appendix to chapter 6, we make use of a typical functional form for u( . ) to represent the well-being of risk-averse rational individuals. That function relates an individual’s utility to the level of her consumption. 3.1.3. Behavioral Representations of Preferences: Prospect Theory Kahneman and Tversky and many others have used experiments to exam- ine the adequacy of the axioms of rational choice and the assumption of risk aversion.

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  Prices and Quantities

  Fundamentals of Microeconomics

  • Rakesh V. Vohra(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  122 5 Preferences and Utility In the remainder of this chapter we restrict ourselves to the case of just two goods. The insights obtained carry over to three or more goods. With two goods, the generic consumer choice problem will be max U(x 1 , x 2 ) s.t. p 1 x 1 + p 2 x 2 ≤ I x 1 , x 2 ≥ 0. Denote an optimal solution/bundle to this problem by (x 1 (p 1 , p 2 , I ), x 2 (p 1 , p 2 , I )). This highlights the dependence of the optimal bundle on prices and income. The utility an agent enjoys at her utility-maximizing bundle is called her indirect utility function. It depends on p 1 , p 2 , and I and is denoted V (p 1 , p 2 , I ). Therefore: V (p 1 , p 2 , I ) = U(x 1 (p 1 , p 2 , I ), x 2 (p 1 , p 2 , I )). Example 41 Recall Example 38. In that example: V (p 1 , p 2 , I ) = 16x 1 (p 1 , p 2 , I ) 4 x 2 (p 1 , p 2 , I ) 8 = 16  I 3p 1  4  2I 3p 2  8 . 5.4 Utility and Consumption Knowing an agent’s utility function and their income allows one to predict how they will allocate their income amongst different goods. To illustrate, suppose we have a rational agent and two goods: sugar and salt. The agent’s income is $12. So as to establish a connection between income and the two commodities, let $2 be the price per unit of sugar and $1 the price per unit of salt. 8 How does the agent choose between the two commodities? From among all combinations of salt and sugar with a total cost of no more than $12 she chooses the combination with highest utility. If x 1 and x 2 represent the amount of sugar and salt she buys, respectively, then the amount she spends is 2x 1 + x 2 . If her total wealth is $12, then 2x 1 + x 2 ≤ 12. In order to decide which combination of salt and sugar she likes best, we need her utility function. For illustrative purposes suppose her utility function is U(x 1 , x 2 ) = x 1 x 2 . For example, her utility from consuming 2 pounds of sugar (x 1 = 2) and 5 pounds of salt (x 2 = 5) will be 2 × 5 = 10.

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  The Economics of Resource Allocation in Health Care

  Cost-utility, social value, and fairness

  • Andrea Klonschinski(Author)
  • 2016(Publication Date)
  • Taylor & Francis

    (Publisher)

  In summary, the basic ingredients of Jevons’ economics are Bentham’s utilitarianism, psychophysiology, and the use of the differential calculus. Jevons strictly rejected the labor theory of value and made the individuals’ feelings of pleasures and pains the building blocks of his theory. Psychophysiology’s mechanistic conception of the human mind allowed him to apply the methods of the natural sciences and, in particular, mathematics to the investigation of individual decision making as a calculus of pleasure and pain. The application of the calculus, in turn, permitted Jevons to focus on marginal changes in the amounts of utility or, what comes to the same thing in a psychophysiological framework, on the marginal units of pleasure the individual economic agent derives from commodities. These changes are not proportional to the increase in the amount of a commodity but are in fact decreasing. This principle of diminishing marginal utility enabled Jevons to derive the equimarginal principle according to which the individual allocates his resources so as to maximize his own pleasure or, as it were, utility. Methodologically, the economic theory of utility-maximizing behavior can hence be considered the “child of the marriage of utility with the technique of marginal increments and decrements, which itself led directly to the consideration of extremal problems” (Dobb 1973: 172).

  As to the adoption of the Benthamite utility concept it deserves emphasis again that while Bentham’s Principles first and foremost addressed the legislator who should build institutions to the advantage of all, Jevons’ individualistic account of utility maximization was totally detached from any societal concerns. Here, the utility concept serves an explanatory function within demand theory. To accomplish that task, it referred to “subjective scales of valuation which were supposed to reside in the consumer’s mind” (Endres 1999: 602). Thereby, pleasure or utility are considered as being quantities that provide “agents with a monotonic criterion by which to carry out the ordering of the alternative outcomes they face” (Warke 2000a: 20). Put differently, utility provided for an ordering principle, explaining how subjects generate their preference rankings (see Mandler 2001: 374). The maximization of pleasure, then, was regarded as the subjects’ aim and motive for action. Henceforth, utility maximization in economics became more and more associated with the idea of individual rationality (see Cudd 1993: 106) and the problem an economic agent faces became framed as the problem of allocating his resources “in such a way that his well-being is enhanced to the greatest degree possible” (Colvin 1985: 9). The publication of TPE can thus be conceived as the hour of birth of the economic man, i.e., of the “discrete, self-contained, self-interested” individual of modern microeconomics (Colvin 1985: 5), aiming at the maximization of pleasure (see Little 1957: 10). Put differently, the TPE gave rise to the fundamental principle of modern economics that “economic behaviour is maximising behaviour subject to constraints” (Blaug 1997: 280).42

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  Game Theory

  • Michael Maschler, Eilon Solan, Shmuel Zamir(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Utility theory Chapter summary The objective of this chapter is to provide a quantitative representation of players’ preference relations over the possible outcomes of the game, by what is called a utility function. This is a fundamental element of game theory, economic theory, and decision theory in general, since it facilitates the application of mathematical tools in analyzing game situations whose outcomes may vary in their nature, and often be uncertain. The utility function representation of preference relations over uncertain outcomes was developed and named after John von Neumann and Oskar Morgenstern. The main feature of the von Neumann–Morgenstern utility is that it is linear in the probabilities of the outcomes. This implies that a player evaluates an uncertain outcome by its expected utility. We present some properties (also known as axioms) that players’ preference relations can satisfy. We then prove that any preference relation having these properties can be represented by a von Neumann–Morgenstern utility and that this representation is determined up to a positive affine transformation. Finally we note how a player’s attitude toward risk is expressed in his von Neumann–Morgenstern utility function. 2.1 Preference relations and their representation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • A game is a mathematical model of a situation of interactive decision making, in which every decision maker (or player) strives to attain his “best possible” outcome, knowing that each of the other players is striving to do the same thing.

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  Game Theory

  • Michael Maschler, Eilon Solan, Shmuel Zamir(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Utility theory Chapter summary The objective of this chapter is to provide a quantitative representation of players’ preference relations over the possible outcomes of the game, by what is called a utility function. This is a fundamental element of game theory, economic theory, and decision theory in general, since it facilitates the application of mathematical tools in analyzing game situations whose outcomes may vary in their nature, and often be uncertain. The utility function representation of preference relations over uncertain outcomes was developed and named after John von Neumann and Oskar Morgenstern. The main feature of the von Neumann–Morgenstern utility is that it is linear in the probabilities of the outcomes. This implies that a player evaluates an uncertain outcome by its expected utility. We present some properties (also known as axioms) that players’ preference relations can satisfy. We then prove that any preference relation having these properties can be represented by a von Neumann–Morgenstern utility and that this representation is determined up to a positive affine transformation. Finally we note how a player’s attitude towards risk is expressed in his von Neumann–Morgenstern utility function. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2.1 Preference relations and their representation A game is a mathematical model of a situation of interactive decision making, in which every decision maker (or player) strives to attain his “best possible” outcome, knowing that each of the other players is striving to do the same thing.

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  A Short Course in Intermediate Microeconomics with Calculus

  • Roberto Serrano, Allan M. Feldman(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  We will describe and discuss the consumer’s rate of tradeoff of one good against another (called her marginal rate of substitution ). After discussing the consumer’s preferences, we will turn to her utility function . A utility function is a numerical representation of how a consumer feels about alter-native consumption bundles: if she likes the first bundle better than the second, then the utility function assigns a higher number to the first than to the second, and if she likes them equally well, then the utility function assigns the same number to both. We will analyze Utility Functions and describe marginal utility , which, loosely speaking, is the extra utility provided by one additional unit of a good. We will derive the relationship between the marginal utilities of two goods and the marginal rate of substitution of one of the goods for the other. We will provide various algebraic examples of Utility Functions, and, in the appendix, we will briefly review the calculus of derivatives and partial derivatives. 8 2 Preferences and Utility In this chapter and others to follow, we will often assume there are only two goods available, with x 1 and x 2 representing quantities of goods 1 and 2, respectively. Why only two goods? For two reasons: first, for simplicity (two goods gives a much simpler model than three goods or five thousand, often with no loss of generality); and, second, because we are often interested in one particular good, and we can easily focus on that good and call the second good “all other goods,” or “everything else,” or “other stuff.” When there are two goods, any consumption bundle can easily be shown in a standard two-dimensional graph, with the quantity of the first good on the horizontal axis and the quantity of the second good on the vertical axis.

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