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66-469: Consider the universal set , U {\displaystyle U} , to be the set of all products and brands which will satisfy a given need. The awareness or knowledge set , A {\displaystyle A} , is defined as all of the products and brands within the universal set that the consumer is aware of. This awareness can be derived from a product search, brand familiarity, advertisements, word-of-mouth, or any other method which informs

132-528: A real number to each alternative in such a manner that alternative a is assigned a number greater than alternative b if and only if the individual prefers alternative a to alternative b . In this situation someone who selects the most preferred alternative is necessarily also selecting the alternative that maximizes the associated utility function. Suppose James has utility function U = x y {\displaystyle U={\sqrt {xy}}} such that x {\displaystyle x}

198-519: A social welfare function . Instead of giving actual numbers over different bundles, ordinal utilities are only the rankings of utilities received from different bundles of goods or services. For example, ordinal utility could tell that having two ice creams provide a greater utility to individuals in comparison to one ice cream but could not tell exactly how much extra utility received by the individual. Ordinal utility, it does not require individuals to specify how much extra utility he or she received from

264-472: A universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. Russell's paradox concerns

330-409: A certain person has from a certain state of the world. Over time, the term has been used in at least two different meanings. The relationship between these two kinds of utility functions is highly controversial among both economists and ethicists . Consider a set of alternatives among which a person has a preference ordering. A utility function represents that ordering if it is possible to assign

396-416: A consumption set of R + L {\displaystyle \mathbb {R} _{+}^{L}} , and each package x ∈ R + L {\displaystyle x\in \mathbb {R} _{+}^{L}} is a vector containing the amounts of each commodity. For the example, there are two commodities: apples and oranges. If we say apples is the first commodity, and oranges

462-445: A cup of water equal to 1-p. One cannot conclude, however, that the cup of tea is two thirds of the goodness of the cup of juice, because this conclusion would depend not only on magnitudes of utility differences, but also on the "zero" of utility. For example, if the "zero" of utility was located at -40, then a cup of orange juice would be 160 utils more than zero, a cup of tea 120 utils more than zero. Cardinal utility can be considered as

528-415: A cup of water has a utility of 40 utils. With cardinal utility, it can 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. Formally, this means that if a person has a cup of tea, he or she would be willing to take any bet with a probability, p, greater than .5 of getting a cup of juice, with a risk of getting

594-436: A few options (usually 2–5), which reduces the cognitive load and fatigue of an exhaustive search. This screening process is mostly based on heuristics about the product, and is generally considered to be a lower-effort process than the evaluation of the consideration set, once it is formed. Studies on heuristic screening methodologies have shown that consumers are more satisfied with their purchase decision and less stressed by

660-413: A good's marginal utility is positive, additional consumption of it increases utility; if zero, the consumer is satiated and indifferent about consuming more; if negative, the consumer would pay to reduce his consumption. Rational individuals only consume additional units of goods if it increases the marginal utility. However, the law of diminishing marginal utility means an additional unit consumed brings

726-445: A lower marginal utility than that brought by the previous unit consumed. For example, drinking one bottle of water makes a thirsty person satisfied; as the consumption of water increases, he may feel begin to feel bad which causes the marginal utility to decrease to zero or even become negative. Furthermore, this is also used to analyze progressive taxes as the greater taxes can result in the loss of utility. Marginal rate of substitution

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792-501: A manner consistent with Quine's, but this is not possible for Oberschelp's, since in it the singleton function is provably a set, which leads immediately to paradox in New Foundations. Another example is positive set theory , where the axiom of comprehension is restricted to hold only for the positive formulas (formulas that do not contain negations). Such set theories are motivated by notions of closure in topology. The idea of

858-409: A particular consumer's consideration set is not enough to predict their final product choice, and this knowledge is trivial when compared to something like the utility function , which is much more robust. Since neither the utility function or consideration set are directly observable, researchers are still unsure whether either is an accurate or useful model for describing choice. Another criticism of

924-416: A positive linear transformation (multiplying by a positive number, and adding some other number); however, both utility functions represent the same preferences. When cardinal utility is assumed, the magnitude of utility differences is treated as an ethically or behaviorally significant quantity. For example, suppose a cup of orange juice has utility of 120 "utils", a cup of tea has a utility of 80 utils, and

990-443: A universal set A {\displaystyle A} , with φ ( x ) {\displaystyle \varphi (x)} defined as the predicate x ∉ x {\displaystyle x\notin x} , it would state the existence of Russell's paradoxical set, giving a contradiction. It was this contradiction that led the axiom of comprehension to be stated in its restricted form, where it asserts

1056-442: A universal set can be avoided either by using a variant of set theory in which the axiom of comprehension is restricted in some way, or by using a universal object that is not considered to be a set. There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and V ∈ V {\displaystyle V\in V}

1122-516: A universal set seems intuitively desirable in the Zermelo–Fraenkel set theory , particularly because most versions of this theory do allow the use of quantifiers over all sets (see universal quantifier ). One way of allowing an object that behaves similarly to a universal set, without creating paradoxes, is to describe V and similar large collections as proper classes rather than as sets. Russell's paradox does not apply in these theories because

1188-413: A utility function ranks preferences concerning a set of goods and services. Gérard Debreu derived the conditions required for a preference ordering to be representable by a utility function. For a finite set of alternatives, these require only that the preference ordering is complete (so the individual is able to determine which of any two alternatives is preferred or that they are indifferent), and that

1254-412: Is A {\displaystyle A} , it must be the case that A {\displaystyle A} is disjoint from { A } {\displaystyle \{A\}} , and therefore that A {\displaystyle A} does not contain itself. Because a universal set would necessarily contain itself, it cannot exist under these axioms. Russell's paradox prevents

1320-462: Is u (nothing) = 0, u (1 apple) = 1, u (1 orange) = 2, u (1 apple and 1 orange) = 5, u (2 apples) = 2 and u (2 oranges) = 4. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges. In micro-economic models, there are usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity. This gives

1386-401: Is a function from choices to the real numbers: which assigns a real number to every outcome in a way that represents the agent's preferences over simple lotteries. Using the four assumptions mentioned above, the agent will prefer a lottery L 2 {\displaystyle L_{2}} to a lottery L 1 {\displaystyle L_{1}} if and only if, for

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1452-502: Is a major concept in welfare economics . While preferences are the conventional foundation of choice theory in microeconomics , it is often convenient to represent preferences with a utility function . Let X be the consumption set , the set of all mutually-exclusive baskets the consumer could conceivably consume. The consumer's utility function u : X → R {\displaystyle u\colon X\to \mathbb {R} } ranks each possible outcome in

1518-399: Is indeed proportional to log of income.) The first important use of the expected utility theory was that of John von Neumann and Oskar Morgenstern , who used the assumption of expected utility maximization in their formulation of game theory . In finding the probability-weighted average of the utility from each possible outcome: Von Neumann and Morgenstern addressed situations in which

1584-423: Is possible for rational 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. Economists distinguish between total utility and marginal utility. Total utility is the utility of an alternative, an entire consumption bundle or situation in life. The rate of change of utility from changing

1650-448: Is preferred over B. It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer relative utilities from observed choice. These 'revealed preferences', as termed by Paul Samuelson , were revealed e.g. in people's willingness to pay: Utility is assumed to be correlative to Desire or Want. It has been argued already that desires cannot be measured directly, but only indirectly, by

1716-430: Is the number of apples and y {\displaystyle y} is the number of chocolates. Alternative A has x = 9 {\displaystyle x=9} apples and y = 16 {\displaystyle y=16} chocolates; alternative B has x = 13 {\displaystyle x=13} apples and y = 13 {\displaystyle y=13} chocolates. Putting

1782-417: Is the slope of the indifference curve, which measures how much an individual is willing to switch from one good to another. Using a mathematic equation, M R S = − d x 2 / d x 1 {\displaystyle MRS=-\operatorname {d} \!x_{2}/\operatorname {d} \!x_{1}} keeping U ( x 1 , x 2 ) constant. Thus, MRS is how much an individual

1848-508: Is true). In these theories, Zermelo's axiom of comprehension does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way. A set theory containing a universal set is necessarily a non-well-founded set theory . The most widely studied set theory with a universal set is Willard Van Orman Quine 's New Foundations . Alonzo Church and Arnold Oberschelp also published work on such set theories. Church speculated that his theory might be extended in

1914-424: Is willing to pay for consuming a greater amount of x 1 . MRS is related to marginal utility. The relationship between marginal utility and MRS is: Expected utility theory deals with the analysis of choices among risky projects with multiple (possibly multidimensional) outcomes. The St. Petersburg paradox was first proposed by Nicholas Bernoulli in 1713 and solved by Daniel Bernoulli in 1738, although

1980-476: The axiom of regularity and axiom of pairing . In Zermelo–Fraenkel set theory , the axiom of regularity and axiom of pairing prevent any set from containing itself. For any set A {\displaystyle A} , the set { A } {\displaystyle \{A\}} (constructed using pairing) necessarily contains an element disjoint from { A } {\displaystyle \{A\}} , by regularity. Because its only element

2046-438: The decision-making process when the consideration set is smaller. This is particularly true when information overload is high; that is, the quantity of available information exceeds the processing power a consumer is willing to dedicate to it. Researchers have several theories as to why the consideration set is formed. The formation of the consideration set is considered the first step in common decision-making frameworks, with

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2112-432: The optimal attainable value of a given utility function, which depends on the prices of the goods and the income or wealth level that the individual possesses. One use of the indirect utility concept is the notion of the utility of money. The (indirect) utility function for money is a nonlinear function that is bounded and asymmetric about the origin. The utility function is concave in the positive region, representing

2178-501: The Swiss mathematician Gabriel Cramer proposed taking the expectation of a square-root utility function of money in an 1728 letter to N. Bernoulli. D. Bernoulli argued that the paradox could be resolved if decision-makers displayed risk aversion and argued for a logarithmic cardinal utility function. (Analysis of international survey data during the 21st century has shown that insofar as utility represents happiness, as for utilitarianism , it

2244-428: The above example, it would only be possible to say that juice is preferred to tea to water. Thus, ordinal utility utilizes comparisons, such as "preferred to", "no more", "less than", etc. If a function u ( x ) {\displaystyle u(x)} is ordinal and non-negative, it is equivalent to the function u ( x ) 2 {\displaystyle u(x)^{2}} , because taking

2310-423: The assumption that utility can be measured by quantifiable characteristics, such as height, weight, temperature, etc. Neoclassical economics has largely retreated from using cardinal utility functions as the basis of economic behavior. A notable exception is in the context of analyzing choice with conditions of risk (see below ). Sometimes cardinal utility is used to aggregate utilities across persons, to create

2376-418: The axiom of comprehension operates on sets, not on classes. The category of sets can also be considered to be a universal object that is, again, not itself a set. It has all sets as elements, and also includes arrows for all functions from one set to another. Again, it does not contain itself, because it is not itself a set. Utility In economics , utility is a measure of the satisfaction that

2442-567: The combinations of commodity X and Y along the same indifference curve are regarded indifferently by individuals, which means all the combinations along an indifference curve result in the same value of utility. 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, by some assumptions these functions can be used to analyze Pareto efficiency , such as illustrated by Edgeworth boxes in contract curves . Such efficiency

2508-435: The consideration set is formed, the next step is to select an alternative from the evaluated options. There are several models that describe how consumers make these selections, however many researchers believe that this process is concurrent with the formation process; that is, product selection is often being contemplated while the consideration set is forming. Nonetheless, the following methodologies are widely considered to be

2574-561: The consideration set is how it is applied. Marketers often assume that all consumers have the same consideration set; that is, they assume all consumers are selecting between the same set of options. The consideration set is actually theorized to be highly individualistic — and the products within it reflect a variety of factors such as the consumer's socioeconomic status , attitudes, and perceptions. This implies that marketers should treat consideration set formation as probabilistic , rather than objective. Universal set In set theory ,

2640-434: The consumer of a viable option. While the awareness set is largely composed of products that reside in the consumer's long-term memory , the awareness set can also be expanded by products found during the search process – such as recommended products on an e-commerce site or shelves in a supermarket. Thus A ⊆ U {\displaystyle A\subseteq U} . The awareness set can be further divided into

2706-402: The consumption set. If the consumer strictly prefers x to y or is indifferent between them, then u ( x ) ≥ u ( y ) {\displaystyle 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 his utility function

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2772-404: The existence of a subset of a given set rather than the existence of a set of all sets that satisfy a given formula. When the axiom of restricted comprehension is applied to an arbitrary set A {\displaystyle A} , with the predicate φ ( x ) ≡ x ∉ x {\displaystyle \varphi (x)\equiv x\notin x} , it produces

2838-739: The existence of a universal set in set theories that include Zermelo 's axiom of restricted comprehension . This axiom states that, for any formula φ ( x ) {\displaystyle \varphi (x)} and any set A {\displaystyle A} , there exists a set { x ∈ A ∣ φ ( x ) } {\displaystyle \{x\in A\mid \varphi (x)\}} that contains exactly those elements x {\displaystyle x} of A {\displaystyle A} that satisfy φ {\displaystyle \varphi } . If this axiom could be applied to

2904-510: The following sets: Thus, C ∪ R ∪ P = A ⊆ U {\displaystyle C\cup R\cup P=A\subseteq U} The consideration set models how humans behave when faced with many choices. While the consideration set is not directly observable, researchers believe that its existence is evident by a logical conjunction of prominent economic and psychological theories. In this model, consumers screen many product options before fully evaluating just

2970-438: The function itself, and which plot the combination of commodities that an individual would accept to maintain a given level of satisfaction. Combining indifference curves with budget constraints allows for derivation of individual demand curves . A diagram of a general indifference curve is shown below (Figure 1). The vertical axes and the horizontal axes represent an individual's consumption of commodity Y and X respectively. All

3036-566: The function needs to be defined for fractional apples and oranges too. One function that would fit these numbers is u ( x apples , x oranges ) = x apples + 2 x oranges + 2 x apples x oranges . {\displaystyle u(x_{\text{apples}},x_{\text{oranges}})=x_{\text{apples}}+2x_{\text{oranges}}+2x_{\text{apples}}x_{\text{oranges}}.} Preferences have three main properties : Assume an individual has two choices, A and B. By ranking

3102-400: The idea of a universal set concerns the power set of the set of all sets. Because this power set is a set of sets, it would necessarily be a subset of the set of all sets, provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself. The difficulties associated with

3168-443: The impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither contain itself (because its members all do not contain themselves) nor avoid containing itself (because if it did, it should be included as one of its members). This paradox prevents the existence of a universal set in set theories that include either Zermelo 's axiom of restricted comprehension , or

3234-401: The individual prefers bundle A to bundle C. (If a ≥ b and b ≥ c , then a ≥ c for all ( a , b , c )). If a bundle A contains all the goods that a bundle B contains, but A also contains more of at least one good than B, then the individual prefers A over B. If, for example, bundle A = {1 apple,2 oranges}, and bundle B = {1 apple,1 orange}, then A

3300-484: The outcomes of choices are not known with certainty, but have probabilities associated with them. A notation for a lottery is as follows: if options A and B have probability p and 1 −  p in the lottery, we write it as a linear combination: More generally, for a lottery with many possible options: where ∑ i p i = 1 {\displaystyle \sum _{i}p_{i}=1} . By making some reasonable assumptions about

3366-467: The outward phenomena which they cause: and that in those cases with which economics is mainly concerned the measure is found by the price which a person is willing to pay for the fulfillment or satisfaction of his desire. Utility functions , expressing utility as a function of the amounts of the various goods consumed, are treated as either cardinal or ordinal , depending on whether they are or are not interpreted as providing more information than simply

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3432-575: The phenomenon of diminishing marginal utility . The boundedness represents the fact that beyond a certain amount money ceases being useful at all, as the size of any economy at that time is itself bounded. The asymmetry about the origin represents the fact that gaining and losing money can have radically different implications both for individuals and businesses. The non-linearity of the utility function for money has profound implications in decision-making processes: in situations where outcomes of choices influence utility by gains or losses of money, which are

3498-459: The preference order is transitive . If the set of alternatives is not finite (for example because even if the number of goods is finite, the quantity chosen can be any real number on an interval) there exists a continuous utility function representing a consumer's preferences if and only if the consumer's preferences are complete, transitive and continuous. Utility can be represented through sets of indifference curve , which are level curves of

3564-437: The preferred bundle of goods or services in comparison to other bundles. They are only required to tell which bundles they prefer. When ordinal utilities are used, differences in utils (values assumed by the utility function) are treated as ethically or behaviorally meaningless: the utility index encodes a full behavioral ordering between members of a choice set, but tells nothing about the related strength of preferences . For

3630-459: The properties of the agent's preference relation over 'simple lotteries', which are lotteries with just two options. Writing B ⪯ A {\displaystyle B\preceq A} to mean 'A is weakly preferred to B' ('A is preferred at least as much as B'), the axioms are: 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

3696-520: The quantity of one good consumed is termed the marginal utility of that good. Marginal utility therefore measures the slope of the utility function with respect to the changes of one good. Marginal utility usually decreases with consumption of the good, the idea of "diminishing marginal utility". In calculus notation, the marginal utility of good X is M U x = ∂ U ∂ X {\displaystyle MU_{x}={\frac {\partial U}{\partial X}}} . When

3762-569: The rank ordering of preferences among bundles of goods, such as information concerning the strength of preferences. Cardinal utility states that the utilities obtained from consumption can be measured and ranked objectively and are representable by numbers. There are fundamental assumptions of cardinal utility. Economic agents should be able to rank different bundles of goods based on their own preferences or utilities, and also sort different transitions of two bundles of goods. A cardinal utility function can be transformed to another utility function by

3828-452: The second step being choosing an alternative from the set. Consumer behavior researchers have identified many frameworks, methodologies, and heuristics consumers often use to form a consideration set. It is important to note that this is not an exhaustive list, and that the formation of the consideration set varies significantly depending on the consumer and context of the decision. Nonetheless, some commonly observed methodologies are: Once

3894-454: The second step of consumer decision making: The decision-making process is still not well enough understood to clarify the distinction between the models used to represent the process and the process of decision-making itself. Many researchers reject the idea of a two-step decision-making process using a consideration set, and instead insist on viewing the consideration set as simply an indicator of preferences. Many researchers claim that knowing

3960-449: The second, then the consumption set is X = R + 2 {\displaystyle X=\mathbb {R} _{+}^{2}} and u (0, 0) = 0, u (1, 0) = 1, u (0, 1) = 2, u (1, 1) = 5, u (2, 0) = 2, u (0, 2) = 4 as before. For u to be a utility function on  X , however, it must be defined for every package in  X , so now

4026-710: The square is an increasing monotone (or monotonic) transformation . This means that the ordinal preference induced by these functions is the same (although they are two different functions). In contrast, if u ( x ) {\displaystyle u(x)} is cardinal, it is not equivalent to u ( x ) 2 {\displaystyle u(x)^{2}} . In order to simplify calculations, various alternative assumptions have been made concerning details of human preferences, and these imply various alternative utility functions such as: Most utility functions used for modeling or theory are well-behaved. They are usually monotonic and quasi-concave. However, it

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4092-621: The subset of elements of A {\displaystyle A} that do not contain themselves. It cannot be a member of A {\displaystyle A} , because if it were it would be included as a member of itself, by its definition, contradicting the fact that it cannot contain itself. In this way, it is possible to construct a witness to the non-universality of A {\displaystyle A} , even in versions of set theory that allow sets to contain themselves. This indeed holds even with predicative comprehension and over intuitionistic logic . Another difficulty with

4158-458: The two choices, one and only one of the following relationships is true: an individual strictly prefers A (A > B); an individual strictly prefers B (B>A); an individual is indifferent between A and B (A = B). Either a ≥ b OR b ≥ a (OR both) for all ( a , b ) Individuals' preferences are consistent over bundles. If an individual prefers bundle A to bundle B, and prefers bundle B to bundle C, then it can be assumed that

4224-456: The utility function characterizing that agent, the expected utility of L 2 {\displaystyle L_{2}} is greater than the expected utility of L 1 {\displaystyle L_{1}} : Of all the axioms, independence is the most often discarded. A variety of generalized expected utility theories have arisen, most of which omit or relax the independence axiom. An indirect utility function gives

4290-403: The values x , y {\displaystyle x,y} into the utility function yields 9 × 16 = 12 {\displaystyle {\sqrt {9\times 16}}=12} for alternative A and 13 × 13 = 13 {\displaystyle {\sqrt {13\times 13}}=13} for B, so James prefers alternative B. In general economic terms,

4356-423: 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 such that the desirability of an arbitrary lottery can be computed as a linear combination of the utilities of its parts, with the weights being their probabilities of occurring. This is termed the expected utility theorem . The required assumptions are four axioms about

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