In mathematics , a binary relation R on a set X is transitive if, for all elements a , b , c in X , whenever R relates a to b and b to c , then R also relates a to c .
113-451: In economics , utility is a measure of the satisfaction that 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
226-489: A = b = c = x {\displaystyle a=b=c=x} , and indeed in this case a R c {\displaystyle aRc} , while if the ordered pair is not of the form ( x , x ) {\displaystyle (x,x)} then there are no such elements a , b , c ∈ X {\displaystyle a,b,c\in X} and hence R {\displaystyle R}
339-427: A R c . {\displaystyle aRc.} A term's definition may require additional properties that are not listed in this table. A homogeneous relation R on the set X is a transitive relation if, Or in terms of first-order logic : where a R b is the infix notation for ( a , b ) ∈ R . As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy
452-443: A < b and b < c then a < c ; and if x = y and y = z then x = z . All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle aRb} and b R c {\displaystyle bRc} then
565-518: 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
678-416: 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
791-445: 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
904-415: 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
1017-702: A definition of economics as a study of human behaviour, subject to and constrained by scarcity, which forces people to choose, allocate scarce resources to competing ends, and economise (seeking the greatest welfare while avoiding the wasting of scarce resources). According to Robbins: "Economics is the science which studies human behavior as a relationship between ends and scarce means which have alternative uses". Robbins' definition eventually became widely accepted by mainstream economists, and found its way into current textbooks. Although far from unanimous, most mainstream economists would accept some version of Robbins' definition, even though many have raised serious objections to
1130-451: A distinct field. The book focused on determinants of national income in the short run when prices are relatively inflexible. Keynes attempted to explain in broad theoretical detail why high labour-market unemployment might not be self-correcting due to low " effective demand " and why even price flexibility and monetary policy might be unavailing. The term "revolutionary" has been applied to the book in its impact on economic analysis. During
1243-412: 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
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#17328025368221356-420: A lower relative cost of production, rather relying only on its own production. It has been termed a "fundamental analytical explanation" for gains from trade . Coming at the end of the classical tradition, John Stuart Mill (1848) parted company with the earlier classical economists on the inevitability of the distribution of income produced by the market system. Mill pointed to a distinct difference between
1469-445: 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
1582-449: A more comprehensive theory of costs on the supply side. In the 20th century, neoclassical theorists departed from an earlier idea that suggested measuring total utility for a society, opting instead for ordinal utility , which posits behaviour-based relations across individuals. In microeconomics , neoclassical economics represents incentives and costs as playing a pervasive role in shaping decision making . An immediate example of this
1695-467: A more important role in mainstream economic theory. Also, heterogeneity among the economic agents, e.g. differences in income, plays an increasing role in recent economic research. Other schools or trends of thought referring to a particular style of economics practised at and disseminated from well-defined groups of academicians that have become known worldwide, include the Freiburg School ,
1808-415: 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
1921-597: A proportion of the value their work had created. Marxian economics was further developed by Karl Kautsky (1854–1938)'s The Economic Doctrines of Karl Marx and The Class Struggle (Erfurt Program) , Rudolf Hilferding 's (1877–1941) Finance Capital , Vladimir Lenin (1870–1924)'s The Development of Capitalism in Russia and Imperialism, the Highest Stage of Capitalism , and Rosa Luxemburg (1871–1919)'s The Accumulation of Capital . At its inception as
2034-409: A rapidly growing population against a limited amount of land meant diminishing returns to labour. The result, he claimed, was chronically low wages, which prevented the standard of living for most of the population from rising above the subsistence level. Economist Julian Simon has criticised Malthus's conclusions. While Adam Smith emphasised production and income, David Ricardo (1817) focused on
2147-552: A relation is a transitive relation. The relation "is the birth parent of" on a set of people is not a transitive relation. However, in biology the need often arises to consider birth parenthood over an arbitrary number of generations: the relation "is a birth ancestor of" is a transitive relation and it is the transitive closure of the relation "is the birth parent of". For the example of towns and roads above, ( A , C ) ∈ R * provided you can travel between towns A and C using any number of roads. No general formula that counts
2260-455: A relation is transitive then its transitive extension is itself, that is, if R is a transitive relation then R 1 = R . The transitive extension of R 1 would be denoted by R 2 , and continuing in this way, in general, the transitive extension of R i would be R i + 1 . The transitive closure of R , denoted by R * or R is the set union of R , R 1 , R 2 , ... . The transitive closure of
2373-469: A set of stable preferences, a definite overall guiding objective, and the capability of making a choice. There exists an economic problem, subject to study by economic science, when a decision (choice) is made by one or more players to attain the best possible outcome. Keynesian economics derives from John Maynard Keynes , in particular his book The General Theory of Employment, Interest and Money (1936), which ushered in contemporary macroeconomics as
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#17328025368222486-409: A single tax on income of land owners. In reaction against copious mercantilist trade regulations, the physiocrats advocated a policy of laissez-faire , which called for minimal government intervention in the economy. Adam Smith (1723–1790) was an early economic theorist. Smith was harshly critical of the mercantilists but described the physiocratic system "with all its imperfections" as "perhaps
2599-452: A social science, economics was defined and discussed at length as the study of production, distribution, and consumption of wealth by Jean-Baptiste Say in his Treatise on Political Economy or, The Production, Distribution, and Consumption of Wealth (1803). These three items were considered only in relation to the increase or diminution of wealth, and not in reference to their processes of execution. Say's definition has survived in part up to
2712-435: A sought after end). Some subsequent comments criticised the definition as overly broad in failing to limit its subject matter to analysis of markets. From the 1960s, however, such comments abated as the economic theory of maximizing behaviour and rational-choice modelling expanded the domain of the subject to areas previously treated in other fields. There are other criticisms as well, such as in scarcity not accounting for
2825-442: A synthesis emerged by the 2000s, often given the name the new neoclassical synthesis . It integrated the rational expectations and optimizing framework of the new classical theory with a new Keynesian role for nominal rigidities and other market imperfections like imperfect information in goods, labour and credit markets. The monetarist importance of monetary policy in stabilizing the economy and in particular controlling inflation
2938-412: 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
3051-462: 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
3164-401: 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
3277-502: 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
3390-421: Is a term for the "way (nomos) to run a household (oikos)", or in other words the know-how of an οἰκονομικός ( oikonomikos ), or "household or homestead manager". Derived terms such as "economy" can therefore often mean "frugal" or "thrifty". By extension then, "political economy" was the way to manage a polis or state. There are a variety of modern definitions of economics ; some reflect evolving views of
3503-438: Is also applied to such diverse subjects as crime , education , the family , feminism , law , philosophy , politics , religion , social institutions , war , science , and the environment . The earlier term for the discipline was "political economy", but since the late 19th century, it has commonly been called "economics". The term is ultimately derived from Ancient Greek οἰκονομία ( oikonomia ) which
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3616-682: Is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie. On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does not follow that Alice is the birth mother of Claire. In fact, this relation is antitransitive : Alice can never be the birth mother of Claire. Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'. The examples "is greater than", "is at least as great as", and "is equal to" ( equality ) are transitive relations on various sets. As are
3729-503: Is difficult. See also Brinkmann and McKay (2005). Since the reflexivization of any transitive relation is a preorder , the number of transitive relations an on n -element set is at most 2 time more than the number of preorders, thus it is asymptotically 2 ( 1 / 4 + o ( 1 ) ) n 2 {\displaystyle 2^{(1/4+o(1))n^{2}}} by results of Kleitman and Rothschild. Note that S ( n , k ) refers to Stirling numbers of
3842-495: Is even and y is odd is both transitive and antitransitive. The relation defined by xRy if x is the successor number of y is both intransitive and antitransitive. Unexpected examples of intransitivity arise in situations such as political questions or group preferences. Generalized to stochastic versions ( stochastic transitivity ), the study of transitivity finds applications of in decision theory , psychometrics and utility models . A quasitransitive relation
3955-399: 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
4068-422: 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
4181-550: Is possible to assign 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}
4294-448: 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
4407-413: Is promoting it. By preferring the support of domestic to that of foreign industry, he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it
4520-598: Is the consumer theory of individual demand, which isolates how prices (as costs) and income affect quantity demanded. In macroeconomics it is reflected in an early and lasting neoclassical synthesis with Keynesian macroeconomics. Neoclassical economics is occasionally referred as orthodox economics whether by its critics or sympathisers. Modern mainstream economics builds on neoclassical economics but with many refinements that either supplement or generalise earlier analysis, such as econometrics , game theory , analysis of market failure and imperfect competition , and
4633-430: 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
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4746-417: 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
4859-551: Is the smallest binary relation on X such that R 1 contains R , and if ( a , b ) ∈ R and ( b , c ) ∈ R then ( a , c ) ∈ R 1 . For example, suppose X is a set of towns, some of which are connected by roads. Let R be the relation on towns where ( A , B ) ∈ R if there is a road directly linking town A and town B . This relation need not be transitive. The transitive extension of this relation can be defined by ( A , C ) ∈ R 1 if you can travel between towns A and C by using at most two roads. If
4972-465: Is vacuously transitive. A transitive relation is asymmetric if and only if it is irreflexive . A transitive relation need not be reflexive . When it is, it is called a preorder . For example, on set X = {1,2,3}: As a counter example, the relation < {\displaystyle <} on the real numbers is transitive, but not reflexive. Let R be a binary relation on set X . The transitive extension of R , denoted R 1 ,
5085-607: Is viewed as basic elements within economies , including individual agents and markets , their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyses economies as systems where production, distribution, consumption, savings , and investment expenditure interact, and factors affecting it: factors of production , such as labour , capital , land , and enterprise , inflation , economic growth , and public policies that have impact on these elements . It also seeks to analyse and describe
5198-423: 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
5311-780: The School of Lausanne , the Stockholm school and the Chicago school of economics . During the 1970s and 1980s mainstream economics was sometimes separated into the Saltwater approach of those universities along the Eastern and Western coasts of the US, and the Freshwater, or Chicago school approach. Within macroeconomics there is, in general order of their historical appearance in
5424-433: The macroeconomics of high unemployment. Gary Becker , a contributor to the expansion of economics into new areas, described the approach he favoured as "combin[ing the] assumptions of maximizing behaviour, stable preferences , and market equilibrium , used relentlessly and unflinchingly." One commentary characterises the remark as making economics an approach rather than a subject matter but with great specificity as to
5537-467: The neoclassical model of economic growth for analysing long-run variables affecting national income . Neoclassical economics studies the behaviour of individuals , households , and organisations (called economic actors, players, or agents), when they manage or use scarce resources, which have alternative uses, to achieve desired ends. Agents are assumed to act rationally, have multiple desirable ends in sight, limited resources to obtain these ends,
5650-432: 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
5763-415: The societal to the microeconomic level: Economics is a study of man in the ordinary business of life. It enquires how he gets his income and how he uses it. Thus, it is on the one side, the study of wealth and on the other and more important side, a part of the study of man. Lionel Robbins (1932) developed implications of what has been termed "[p]erhaps the most commonly accepted current definition of
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#17328025368225876-400: The "choice process and the type of social interaction that [such] analysis involves." The same source reviews a range of definitions included in principles of economics textbooks and concludes that the lack of agreement need not affect the subject-matter that the texts treat. Among economists more generally, it argues that a particular definition presented may reflect the direction toward which
5989-486: The 1970s and 1980s, when several major central banks followed a monetarist-inspired policy, but was later abandoned because the results were unsatisfactory. A more fundamental challenge to the prevailing Keynesian paradigm came in the 1970s from new classical economists like Robert Lucas , Thomas Sargent and Edward Prescott . They introduced the notion of rational expectations in economics, which had profound implications for many economic discussions, among which were
6102-505: 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
6215-428: 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
6328-407: The analysis of wealth: how wealth is created (production), distributed, and consumed; and how wealth can grow. But he said that economics can be used to study other things, such as war, that are outside its usual focus. This is because war has as the goal winning it (as a sought after end ), generates both cost and benefits; and, resources (human life and other costs) are used to attain the goal. If
6441-479: The area of inquiry or object of inquiry rather than the methodology. In the biology department, it is not said that all biology should be studied with DNA analysis. People study living organisms in many different ways, so some people will perform DNA analysis, others might analyse anatomy, and still others might build game theoretic models of animal behaviour. But they are all called biology because they all study living organisms. According to Ha Joon Chang, this view that
6554-423: 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
6667-443: The author believes economics is evolving, or should evolve. Many economists including nobel prize winners James M. Buchanan and Ronald Coase reject the method-based definition of Robbins and continue to prefer definitions like those of Say, in terms of its subject matter. Ha-Joon Chang has for example argued that the definition of Robbins would make economics very peculiar because all other sciences define themselves in terms of
6780-517: The colonies. Physiocrats , a group of 18th-century French thinkers and writers, developed the idea of the economy as a circular flow of income and output. Physiocrats believed that only agricultural production generated a clear surplus over cost, so that agriculture was the basis of all wealth. Thus, they opposed the mercantilist policy of promoting manufacturing and trade at the expense of agriculture, including import tariffs. Physiocrats advocated replacing administratively costly tax collections with
6893-567: 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
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#17328025368227006-402: 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
7119-488: The design of modern monetary policy and are now standard workhorses in most central banks. After the 2007–2008 financial crisis , macroeconomic research has put greater emphasis on understanding and integrating the financial system into models of the general economy and shedding light on the ways in which problems in the financial sector can turn into major macroeconomic recessions. In this and other research branches, inspiration from behavioural economics has started playing
7232-506: The distribution of income among landowners, workers, and capitalists. Ricardo saw an inherent conflict between landowners on the one hand and labour and capital on the other. He posited that the growth of population and capital, pressing against a fixed supply of land, pushes up rents and holds down wages and profits. Ricardo was also the first to state and prove the principle of comparative advantage , according to which each country should specialise in producing and exporting goods in that it has
7345-469: The economy can and should be studied in only one way (for example by studying only rational choices), and going even one step further and basically redefining economics as a theory of everything, is peculiar. Questions regarding distribution of resources are found throughout the writings of the Boeotian poet Hesiod and several economic historians have described Hesiod as the "first economist". However,
7458-751: The first large-scale macroeconometric model , applying the Keynesian thinking systematically to the US economy . Immediately after World War II, Keynesian was the dominant economic view of the United States establishment and its allies, Marxian economics was the dominant economic view of the Soviet Union nomenklatura and its allies. Monetarism appeared in the 1950s and 1960s, its intellectual leader being Milton Friedman . Monetarists contended that monetary policy and other monetary shocks, as represented by
7571-540: The following decades, many economists followed Keynes' ideas and expanded on his works. John Hicks and Alvin Hansen developed the IS–LM model which was a simple formalisation of some of Keynes' insights on the economy's short-run equilibrium. Franco Modigliani and James Tobin developed important theories of private consumption and investment , respectively, two major components of aggregate demand . Lawrence Klein built
7684-438: 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
7797-566: 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
7910-509: The global economy . Other broad distinctions within economics include those between positive economics , describing "what is", and normative economics , advocating "what ought to be"; between economic theory and applied economics ; between rational and behavioural economics ; and between mainstream economics and heterodox economics . Economic analysis can be applied throughout society, including business , finance , cybersecurity , health care , engineering and government . It
8023-424: The growth in the money stock, was an important cause of economic fluctuations, and consequently that monetary policy was more important than fiscal policy for purposes of stabilisation . Friedman was also skeptical about the ability of central banks to conduct a sensible active monetary policy in practice, advocating instead using simple rules such as a steady rate of money growth. Monetarism rose to prominence in
8136-463: The importance of various market failures for the functioning of the economy, as had Keynes. Not least, they proposed various reasons that potentially explained the empirically observed features of price and wage rigidity , usually made to be endogenous features of the models, rather than simply assumed as in older Keynesian-style ones. After decades of often heated discussions between Keynesians, monetarists, new classical and new Keynesian economists,
8249-400: 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
8362-410: The literature; classical economics , neoclassical economics , Keynesian economics , the neoclassical synthesis , monetarism , new classical economics , New Keynesian economics and the new neoclassical synthesis . Transitive relation Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If
8475-449: The market's two roles: allocation of resources and distribution of income. The market might be efficient in allocating resources but not in distributing income, he wrote, making it necessary for society to intervene. Value theory was important in classical theory. Smith wrote that the "real price of every thing ... is the toil and trouble of acquiring it". Smith maintained that, with rent and profit, other costs besides wages also enter
8588-437: The most famous passages in all economics," Smith represents every individual as trying to employ any capital they might command for their own advantage, not that of the society, and for the sake of profit, which is necessary at some level for employing capital in domestic industry, and positively related to the value of produce. In this: He generally, indeed, neither intends to promote the public interest, nor knows how much he
8701-499: The norm for most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period. Economics Economics ( / ˌ ɛ k ə ˈ n ɒ m ɪ k s , ˌ iː k ə -/ ) is a social science that studies the production , distribution , and consumption of goods and services . Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyses what
8814-675: The number of transitive relations on a finite set (sequence A006905 in the OEIS ) is known. However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in the OEIS ), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and antisymmetric. Pfeiffer has made some progress in this direction, expressing relations with combinations of these properties in terms of each other, but still calculating any one
8927-484: 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
9040-465: 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
9153-503: The pessimistic analysis of Malthus (1798). John Stuart Mill (1844) delimited the subject matter further: The science which traces the laws of such of the phenomena of society as arise from the combined operations of mankind for the production of wealth, in so far as those phenomena are not modified by the pursuit of any other object. Alfred Marshall provided a still widely cited definition in his textbook Principles of Economics (1890) that extended analysis beyond wealth and from
9266-575: 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
9379-458: 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
9492-437: 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
9605-487: The present, modified by substituting the word "wealth" for "goods and services" meaning that wealth may include non-material objects as well. One hundred and thirty years later, Lionel Robbins noticed that this definition no longer sufficed, because many economists were making theoretical and philosophical inroads in other areas of human activity. In his Essay on the Nature and Significance of Economic Science , he proposed
9718-409: The price of a commodity. Other classical economists presented variations on Smith, termed the ' labour theory of value '. Classical economics focused on the tendency of any market economy to settle in a final stationary state made up of a constant stock of physical wealth (capital) and a constant population size . Marxist (later, Marxian) economics descends from classical economics and it derives from
9831-459: 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
9944-429: The purest approximation to the truth that has yet been published" on the subject. The publication of Adam Smith 's The Wealth of Nations in 1776, has been described as "the effective birth of economics as a separate discipline." The book identified land, labour, and capital as the three factors of production and the major contributors to a nation's wealth, as distinct from the physiocratic idea that only agriculture
10057-519: 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
10170-567: 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
10283-412: The scope and method of economics, emanating from that definition. A body of theory later termed "neoclassical economics" formed from about 1870 to 1910. The term "economics" was popularised by such neoclassical economists as Alfred Marshall and Mary Paley Marshall as a concise synonym for "economic science" and a substitute for the earlier " political economy ". This corresponded to the influence on
10396-421: The second kind . A relation R is called intransitive if it is not transitive, that is, if xRy and yRz , but not xRz , for some x , y , z . In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive, but not antitransitive. The relation defined by xRy if x
10509-449: 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
10622-494: The set of real numbers or the set of natural numbers: More examples of transitive relations: Examples of non-transitive relations: The empty relation on any set X {\displaystyle X} is transitive because there are no elements a , b , c ∈ X {\displaystyle a,b,c\in X} such that a R b {\displaystyle aRb} and b R c {\displaystyle bRc} , and hence
10735-486: The so-called Lucas critique and the presentation of real business cycle models . During the 1980s, a group of researchers appeared being called New Keynesian economists , including among others George Akerlof , Janet Yellen , Gregory Mankiw and Olivier Blanchard . They adopted the principle of rational expectations and other monetarist or new classical ideas such as building upon models employing micro foundations and optimizing behaviour, but simultaneously emphasised
10848-444: The source of the word economy. Joseph Schumpeter described 16th and 17th century scholastic writers, including Tomás de Mercado , Luis de Molina , and Juan de Lugo , as "coming nearer than any other group to being the 'founders' of scientific economics" as to monetary , interest , and value theory within a natural-law perspective. Two groups, who later were called "mercantilists" and "physiocrats", more directly influenced
10961-710: 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
11074-406: The state or commonwealth with a revenue for the publick services. Jean-Baptiste Say (1803), distinguishing the subject matter from its public-policy uses, defined it as the science of production, distribution, and consumption of wealth . On the satirical side, Thomas Carlyle (1849) coined " the dismal science " as an epithet for classical economics , in this context, commonly linked to
11187-408: The subject of mathematical methods used in the natural sciences . Neoclassical economics systematically integrated supply and demand as joint determinants of both price and quantity in market equilibrium, influencing the allocation of output and income distribution. It rejected the classical economics' labour theory of value in favour of a marginal utility theory of value on the demand side and
11300-402: The subject or different views among economists. Scottish philosopher Adam Smith (1776) defined what was then called political economy as "an inquiry into the nature and causes of the wealth of nations", in particular as: a branch of the science of a statesman or legislator [with the twofold objectives of providing] a plentiful revenue or subsistence for the people ... [and] to supply
11413-471: The subject": Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses. Robbins described the definition as not classificatory in "pick[ing] out certain kinds of behaviour" but rather analytical in "focus[ing] attention on a particular aspect of behaviour, the form imposed by the influence of scarcity ." He affirmed that previous economists have usually centred their studies on
11526-826: The subsequent development of the subject. Both groups were associated with the rise of economic nationalism and modern capitalism in Europe. Mercantilism was an economic doctrine that flourished from the 16th to 18th century in a prolific pamphlet literature, whether of merchants or statesmen. It held that a nation's wealth depended on its accumulation of gold and silver. Nations without access to mines could obtain gold and silver from trade only by selling goods abroad and restricting imports other than of gold and silver. The doctrine called for importing inexpensive raw materials to be used in manufacturing goods, which could be exported, and for state regulation to impose protective tariffs on foreign manufactured goods and prohibit manufacturing in
11639-422: The transitivity condition is vacuously true . A relation R containing only one ordered pair is also transitive: if the ordered pair is of the form ( x , x ) {\displaystyle (x,x)} for some x ∈ X {\displaystyle x\in X} the only such elements a , b , c ∈ X {\displaystyle a,b,c\in X} are
11752-458: 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
11865-456: 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
11978-403: 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,
12091-467: The war is not winnable or if the expected costs outweigh the benefits, the deciding actors (assuming they are rational) may never go to war (a decision ) but rather explore other alternatives. Economics cannot be defined as the science that studies wealth, war, crime, education, and any other field economic analysis can be applied to; but, as the science that studies a particular common aspect of each of those subjects (they all use scarce resources to attain
12204-423: 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
12317-515: The word Oikos , the Greek word from which the word economy derives, was used for issues regarding how to manage a household (which was understood to be the landowner, his family, and his slaves ) rather than to refer to some normative societal system of distribution of resources, which is a more recent phenomenon. Xenophon , the author of the Oeconomicus , is credited by philologues for being
12430-460: The work of Karl Marx . The first volume of Marx's major work, Das Kapital , was published in 1867. Marx focused on the labour theory of value and theory of surplus value . Marx wrote that they were mechanisms used by capital to exploit labour. The labour theory of value held that the value of an exchanged commodity was determined by the labour that went into its production, and the theory of surplus value demonstrated how workers were only paid
12543-421: Was no part of it. By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it. The Reverend Thomas Robert Malthus (1798) used the concept of diminishing returns to explain low living standards. Human population , he argued, tended to increase geometrically, outstripping the production of food, which increased arithmetically. The force of
12656-892: Was productive. Smith discusses potential benefits of specialisation by division of labour , including increased labour productivity and gains from trade , whether between town and country or across countries. His "theorem" that "the division of labor is limited by the extent of the market" has been described as the "core of a theory of the functions of firm and industry " and a "fundamental principle of economic organization." To Smith has also been ascribed "the most important substantive proposition in all of economics" and foundation of resource-allocation theory—that, under competition , resource owners (of labour, land, and capital) seek their most profitable uses, resulting in an equal rate of return for all uses in equilibrium (adjusted for apparent differences arising from such factors as training and unemployment). In an argument that includes "one of
12769-437: Was recognised as well as the traditional Keynesian insistence that fiscal policy could also play an influential role in affecting aggregate demand . Methodologically, the synthesis led to a new class of applied models, known as dynamic stochastic general equilibrium or DSGE models, descending from real business cycles models, but extended with several new Keynesian and other features. These models proved useful and influential in
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