153-471: In economics , the Gini coefficient ( / ˈ dʒ iː n i / JEE -nee ), also known as the Gini index or Gini ratio , is a measure of statistical dispersion intended to represent the income inequality , the wealth inequality , or the consumption inequality within a nation or a social group . It was developed by Italian statistician and sociologist Corrado Gini . The Gini coefficient measures
306-435: A quadratic function across pairs of intervals or building an appropriately smooth approximation to the underlying distribution function that matches the known data. If the population mean and boundary values for each interval are also known, these can also often be used to improve the accuracy of the approximation. The Gini coefficient calculated from a sample is a statistic, and its standard error, or confidence intervals for
459-591: A set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra , as established by the influence and works of Emmy Noether . Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: The study of types of algebraic structures as mathematical objects
612-486: A country's wealth or income distribution deviates from an equal distribution. The Gini coefficient is usually defined mathematically based on the Lorenz curve , which plots the proportion of the total income of the population (y-axis) that is cumulatively earned by the bottom x of the population (see diagram). The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as
765-499: A country. The Gini coefficient on disposable income—sometimes referred to as the after-tax Gini coefficient—is calculated on income after taxes and transfers. It measures inequality in income after considering the effect of taxes and social spending already in place in a country. For OECD countries over the 2008–2009 period, the Gini coefficient (pre-taxes and transfers) for a total population ranged between 0.34 and 0.53, with South Korea
918-702: 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
1071-451: 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
1224-614: A foundation for all mathematics). Mathematics involves the description and manipulation of abstract objects that consist of either abstractions from nature or—in modern mathematics—purely abstract entities that are stipulated to have certain properties, called axioms . Mathematics uses pure reason to prove properties of objects, a proof consisting of a succession of applications of deductive rules to already established results. These results include previously proved theorems , axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of
1377-669: A fruitful interaction between mathematics and science , to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January ;2006 issue of the Bulletin of the American Mathematical Society , "The number of papers and books included in the Mathematical Reviews (MR) database since 1940 (the first year of operation of MR)
1530-420: 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
1683-404: A mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space . Today's subareas of geometry include: Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were
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#17327938097721836-422: A mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture . Through a series of rigorous arguments employing deductive reasoning , a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma . A proven instance that forms part of
1989-483: A measure of inequality of income or wealth . For OECD countries in the late 20th century, considering the effect of taxes and transfer payments , the income Gini coefficient ranged between 0.24 and 0.49, with Slovakia being the lowest and Mexico the highest. African countries had the highest pre-tax Gini coefficients in 2008–2009, with South Africa having the world's highest, estimated to be 0.63 to 0.7. However, this figure drops to 0.52 after social assistance
2142-449: 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
2295-402: A more general finding is termed a corollary . Numerous technical terms used in mathematics are neologisms , such as polynomial and homeomorphism . Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, " or " means "one, the other or both", while, in common language, it
2448-467: 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 ,
2601-535: A population mean with a given level of confidence. Because of its use of optimization , the mathematical theory of statistics overlaps with other decision sciences , such as operations research , control theory , and mathematical economics . Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory ; numerical analysis broadly includes
2754-597: 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
2907-409: 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
3060-411: A separate branch of mathematics until the seventeenth century. At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas. Some of these areas correspond to the older division, as
3213-469: 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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#17327938097723366-424: A single unknown , which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices , modular integers , and geometric transformations ), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of
3519-409: 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
3672-452: 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
3825-435: 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
3978-418: A statistical action, such as using a procedure in, for example, parameter estimation , hypothesis testing , and selecting the best . In these traditional areas of mathematical statistics , a statistical-decision problem is formulated by minimizing an objective function , like expected loss or cost , under specific constraints. For example, designing a survey often involves minimizing the cost of estimating
4131-442: 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
4284-477: A wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory , type theory , computability theory and computational complexity theory . Although these aspects of mathematical logic were introduced before the rise of computers , their use in compiler design, formal verification , program analysis , proof assistants and other aspects of computer science , contributed in turn to
4437-415: Is f − u . A more graded distribution with these same values u and f will always have a higher Gini coefficient than f − u . For example, if the wealthiest u = 20% of the population has f = 80% of all income (see Pareto principle ), the income Gini coefficient is at least 60%. In another example, if u = 1% of the world's population owns f = 50% of all wealth, the wealth Gini coefficient
4590-703: Is Fermat's Last Theorem . This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles , who used tools including scheme theory from algebraic geometry , category theory , and homological algebra . Another example is Goldbach's conjecture , which asserts that every even integer greater than 2 is the sum of two prime numbers . Stated in 1742 by Christian Goldbach , it remains unproven despite considerable effort. Number theory includes several subareas, including analytic number theory , algebraic number theory , geometry of numbers (method oriented), diophantine equations , and transcendence theory (problem oriented). Geometry
4743-853: 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 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
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4896-421: 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
5049-438: 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
5202-544: Is at least 49%. In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve . For example, (taking y to indicate the income or wealth of a person or household): The Gini coefficient can also be considered as half the relative mean absolute difference . For a random sample S with values y 1 ≤ y 2 ≤ ⋯ ≤ y n {\displaystyle y_{1}\leq y_{2}\leq \cdots \leq y_{n}} ,
5355-403: Is commonly used for advanced parts. Analysis is further subdivided into real analysis , where variables represent real numbers , and complex analysis , where variables represent complex numbers . Analysis includes many subareas shared by other areas of mathematics which include: Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example
5508-513: Is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic , the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism , as founded by David Hilbert around 1910. The "nature" of
5661-407: Is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called " exclusive or "). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module
5814-460: Is equivalent to the definition based on the Lorenz curve . The mean absolute difference is the average absolute difference of all pairs of items of the population, and the relative mean absolute difference is the mean absolute difference divided by the average , x ¯ {\displaystyle {\bar {x}}} , to normalize for scale. If x i is the wealth or income of person i , and there are n persons, then
5967-429: Is from 0 (total equality) to 1 (absolute inequality). This measure is often rendered as a percentage, spanning 0 to 100. However, if negative values are factored in, as in cases of debt, the Gini index could exceed 1. Typically, we presuppose a positive mean or total, precluding a Gini coefficient below zero. An alternative approach is to define the Gini coefficient as half of the relative mean absolute difference , which
6120-493: Is in Babylonian mathematics that elementary arithmetic ( addition , subtraction , multiplication , and division ) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as
6273-490: Is large, the income distribution may be represented by a continuous probability density function f ( x ) where f ( x ) dx is the fraction of the population with wealth or income in the interval dx about x . If F ( x ) is the cumulative distribution function for f ( x ): and L ( x ) is the Lorenz function: then the Lorenz curve L ( F ) may then be represented as a function parametric in L ( x ) and F ( x ) and
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6426-586: Is mostly used for numerical calculations . Number theory dates back to ancient Babylon and probably China . Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler . The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss . Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example
6579-404: Is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results ( theorems ) and a few basic statements. The basic statements are not subject to proof because they are self-evident ( postulates ), or are part of the definition of the subject of study ( axioms ). This principle, foundational for all mathematics,
6732-1192: Is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations , unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts . Operation and relations are generally represented by specific symbols or glyphs , such as + ( plus ), × ( multiplication ), ∫ {\textstyle \int } ( integral ), = ( equal ), and < ( less than ). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of
6885-547: Is often held to be Archimedes ( c. 287 – c. 212 BC ) of Syracuse . He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections ( Apollonius of Perga , 3rd century BC), trigonometry ( Hipparchus of Nicaea , 2nd century BC), and
7038-433: Is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines , angles and circles , which were developed mainly for the needs of surveying and architecture , but has since blossomed out into many other subfields. A fundamental innovation was the ancient Greeks' introduction of the concept of proofs , which require that every assertion must be proved . For example, it
7191-474: Is perfect income equality when everyone's income x j {\displaystyle x_{j}} equals the mean income x ¯ {\displaystyle {\overline {x}}} , so that r j = 1 {\displaystyle r_{j}=1} for everyone). Measures of inequality, then, are measures of the average deviations of the r j = 1 {\displaystyle r_{j}=1} from 1;
7344-413: 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
7497-567: Is sometimes mistranslated as a condemnation of mathematicians. The apparent plural form in English goes back to the Latin neuter plural mathematica ( Cicero ), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ) and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after
7650-419: Is taken into account, and drops again to 0.47 after taxation. The country with the lowest Gini coefficient is Slovakia, with a Gini coefficient of 0.232. The Gini coefficient of the global income in 2005 has been estimated to be between 0.61 and 0.68 by various sources. There are some issues in interpreting a Gini coefficient, as the same value may result from many different distribution curves. To mitigate this,
7803-598: 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
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#17327938097727956-574: Is the error function ( since G = 2 Φ ( σ 2 ) − 1 {\displaystyle G=2\Phi \left({\frac {\sigma }{\sqrt {2}}}\right)-1} , where Φ {\displaystyle \Phi } is the cumulative distribution function of a standard normal distribution). In the table below, some examples for probability density functions with support on [ 0 , ∞ ) {\displaystyle [0,\infty )} are shown. The Dirac delta distribution represents
8109-405: Is the fraction of the population with income or wealth y i > 0 {\displaystyle y_{i}>0} , the Gini coefficient is: where If the points with non-zero probabilities are indexed in increasing order ( y i < y i + 1 ) {\displaystyle (y_{i}<y_{i+1})} , then: where When the population
8262-461: Is the mean of the distribution, and the lower limits of integration may be replaced by zero when all incomes are positive. While the income distribution of any particular country will not correspond perfectly to the theoretical models , these models can provide a qualitative explanation of the income distribution in a nation given the Gini coefficient. The extreme cases are represented by the most equal possible society in which every person receives
8415-418: Is the purpose of universal algebra and category theory . The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces ; this particular area of application is called algebraic topology . Calculus, formerly called infinitesimal calculus,
8568-405: Is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms —especially their implementation and computational complexity —play a major role in discrete mathematics. The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of
8721-508: Is true regarding number theory (the modern name for higher arithmetic ) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations . Number theory began with
8874-586: The Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It
9027-768: The Golden Age of Islam , especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra . Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarizmi , Omar Khayyam and Sharaf al-Dīn al-Ṭūsī . The Greek and Arabic mathematical texts were in turn translated to Latin during
9180-511: The Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements , is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity
9333-536: The Renaissance , mathematics was divided into two main areas: arithmetic , regarding the manipulation of numbers, and geometry , regarding the study of shapes. Some types of pseudoscience , such as numerology and astrology , were not then clearly distinguished from mathematics. During the Renaissance, two more areas appeared. Mathematical notation led to algebra which, roughly speaking, consists of
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#17327938097729486-780: 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
9639-408: The X k indexed in increasing order ( X k – 1 < X k ), so that: If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and: is the resulting approximation for G. More accurate results can be obtained using other methods to approximate the area B, such as approximating the Lorenz curve with
9792-446: The controversy over Cantor's set theory . In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour . This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory . Roughly speaking, each mathematical object
9945-413: The inequality among the values of a frequency distribution , such as levels of income . A Gini coefficient of 0 reflects perfect equality, where all income or wealth values are the same, while a Gini coefficient of 1 (or 100%) reflects maximal inequality among values, a situation where a single individual has all the income while all others have none. The Gini coefficient was proposed by Corrado Gini as
10098-433: 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
10251-467: 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,
10404-415: 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
10557-400: 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
10710-400: The 17th century, when René Descartes introduced what is now called Cartesian coordinates . This constituted a major change of paradigm : Instead of defining real numbers as lengths of line segments (see number line ), it allowed the representation of points using their coordinates , which are numbers. Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry
10863-486: 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
11016-405: The 19th century, mathematicians discovered non-Euclidean geometries , which do not follow the parallel postulate . By questioning that postulate's truth, this discovery has been viewed as joining Russell's paradox in revealing the foundational crisis of mathematics . This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not
11169-532: The 20th century. The P versus NP problem , which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. Discrete mathematics includes: The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic , although used for mathematical proofs, belonged to philosophy and
11322-437: The Gini coefficient G is given by: When the income (or wealth) distribution is given as a continuous probability density function p ( x ), the Gini coefficient is again half of the relative mean absolute difference: where μ = ∫ − ∞ ∞ x p ( x ) d x {\displaystyle \textstyle \mu =\int _{-\infty }^{\infty }xp(x)\,dx}
11475-503: The Gini coefficient can be misleading when used to make political comparisons between large and small countries or those with different immigration policies (see limitations section). The Gini coefficient for the entire world has been estimated by various parties to be between 0.61 and 0.68. The graph shows the values expressed as a percentage in their historical development for a number of countries. Economics Economics ( / ˌ ɛ k ə ˈ n ɒ m ɪ k s , ˌ iː k ə -/ )
11628-686: The Gini coefficient will be a function only of a, b, c... . For example, for the exponential distribution , which is a function of only x and a scale parameter, the Gini coefficient is a constant, equal to 1/2. For some functional forms, the Gini index can be calculated explicitly. For example, if y follows a log-normal distribution with the standard deviation of logs equal to σ {\displaystyle \sigma } , then G = erf ( σ 2 ) {\displaystyle G=\operatorname {erf} \left({\frac {\sigma }{2}}\right)} where erf {\displaystyle \operatorname {erf} }
11781-637: The Middle Ages and made available in Europe. During the early modern period , mathematics began to develop at an accelerating pace in Western Europe , with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of logarithms by John Napier in 1614, which greatly simplified numerical calculations, especially for astronomy and marine navigation ,
11934-407: 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
12087-479: 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
12240-443: 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
12393-583: The beginnings of algebra (Diophantus, 3rd century AD). The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics . Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine , and an early form of infinite series . During
12546-411: The case where everyone has the same wealth (or income); it implies no variations between incomes. Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated using various techniques for interpolating the missing values of the Lorenz curve. If ( X k , Y k ) are the known points on the Lorenz curve, with
12699-517: 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
12852-511: The concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new fields. Since then,
13005-399: The current language, where expressions play the role of noun phrases and formulas play the role of clauses . Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is
13158-571: The demographic structure should be taken into account. Countries with an aging population, or those with an increased birth rate, experience an increasing pre-tax Gini coefficient even if real income distribution for working adults remains constant. Many scholars have devised over a dozen variants of the Gini coefficient. The Gini coefficient was developed by the Italian statistician Corrado Gini and published in his 1912 paper Variabilità e mutabilità (English: variability and mutability ). Building on
13311-569: The derived expression mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη ), meaning ' mathematical science ' . It entered the English language during the Late Middle English period through French and Latin. Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely
13464-488: 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
13617-412: The distance of the inequality ratios (the r j ) from 1. Gini coefficients of income are calculated on a market income and a disposable income basis. The Gini coefficient on market income—sometimes referred to as a pre-tax Gini coefficient—is calculated on income before taxes and transfers. It measures inequality in income without considering the effect of taxes and social spending already in place in
13770-506: 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
13923-469: 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,
14076-449: The error distributions and the independence of error terms. These assumptions are often not valid for real data sets. There is still ongoing debate surrounding this topic. Guillermina Jasso and Angus Deaton independently proposed the following formula for the Gini coefficient: where μ {\displaystyle \mu } is mean income of the population, P i is the income rank P of person i, with income X, such that
14229-400: The estimate of A can be used to derive the estimate of G directly without using a jackknife. This method only requires using ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improving with increasing sample size. However, it has been argued that this depends on the model's assumptions about
14382-428: The expansion of these logical theories. The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory . Statisticians generate data with random sampling or randomized experiments . Statistical theory studies decision problems such as minimizing the risk ( expected loss ) of
14535-751: 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
14688-567: The first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. In Latin and English, until around 1700, the term mathematics more commonly meant " astrology " (or sometimes " astronomy ") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This change has resulted in several mistranslations: For example, Saint Augustine 's warning that Christians should beware of mathematici , meaning "astrologers",
14841-540: 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
14994-509: 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
15147-434: The greater the average deviation, the greater the inequality. Based on these observations the inequality indices have this common form: where p j weights the units by their population share, and f ( r j ) is a function of the deviation of each unit's r j from 1, the point of equality. The insight of this generalized inequality index is that inequality indices differ because they employ different functions of
15300-424: 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
15453-463: 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,
15606-436: The integration is taken from minus infinity to plus infinity.) The Gini coefficient may be expressed in terms of the quantile function Q ( F ) (inverse of the cumulative distribution function: Q(F(x)) = x) Since the Gini coefficient is independent of scale , if the distribution function can be expressed in the form f(x,φ,a,b,c...) where φ is a scale factor and a, b, c... are dimensionless parameters, then
15759-491: The interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method , which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than sixty first-level areas of mathematics. Before
15912-400: The introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716). Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and
16065-733: The literature; classical economics , neoclassical economics , Keynesian economics , the neoclassical synthesis , monetarism , new classical economics , New Keynesian economics and the new neoclassical synthesis . Mathematics Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), analysis (the study of continuous changes), and set theory (presently used as
16218-461: The lowest and Italy the highest. The Gini coefficient (after-taxes and transfers) for a total population ranged between 0.25 and 0.48, with Denmark the lowest and Mexico the highest. For the United States, the country with the largest population among OECD countries, the pre-tax Gini index was 0.49, and the after-tax Gini index was 0.38 in 2008–2009. The OECD average for total populations in OECD countries
16371-409: The manipulation of numbers , that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term
16524-449: 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
16677-424: The measurement of concentration and variability of characters in 1914. Here, he presented the concentration ratio , which further developed in the Gini coefficient used today. Secondly, Gini observed that his proposed ratio can be also achieved by improving methods already introduced by Lorenz, Chatelain, or Séailles. In 1915, Gaetano Pietra introduced a geometrical interpretation between Gini's proposed ratio and
16830-437: 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
16983-400: The natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer , who promoted intuitionistic logic , which explicitly lacks the law of excluded middle . These problems and debates led to
17136-467: The need to divide by N² instead. FAO explains another version of the formula. The Gini coefficient and other standard inequality indices reduce to a common form. Perfect equality—the absence of inequality—exists when and only when the inequality ratio, r j = x j / x ¯ {\displaystyle r_{j}=x_{j}/{\overline {x}}} , equals 1 for all j units in some population (for example, there
17289-536: The objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains
17442-521: The pattern of physics and metaphysics , inherited from Greek. In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math . In addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. Evidence for more complex mathematics does not appear until around 3000 BC , when
17595-503: 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
17748-431: The population Gini coefficient, should be reported. These can be calculated using bootstrap techniques, mathematically complicated and computationally demanding even in an era of fast computers. Economist Tomson Ogwang made the process more efficient by setting up a "trick regression model" in which respective income variables in the sample are ranked, with the lowest income being allocated rank 1. The model then expresses
17901-487: 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
18054-409: 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
18207-658: The proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss , who made numerous contributions to fields such as algebra, analysis, differential geometry , matrix theory , number theory, and statistics . In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems , which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. Mathematics has since been greatly extended, and there has been
18360-429: 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
18513-408: The rank (dependent variable) as the sum of a constant A and a normal error term whose variance is inversely proportional to y k : Thus, G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculation of the jackknife estimate for the standard error. Economist David Giles argued that the standard error of
18666-498: The ratio between the area of observed concentration and maximum concentration. This altered version of Gini coefficient became the most commonly used inequality index in upcoming years. According to data from OECD , Gini coefficient was first officially used country-wide in Canada in the 1970s. Canadian index of income inequality ranged from 0.303 to 0.284 from 1976 to the end of 1980s. OECD started to publish more countries’ data since
18819-427: The ratio of the area that lies between the line of equality and the Lorenz curve (marked A in the diagram) over the total area under the line of equality (marked A and B in the diagram); i.e., G = A /( A + B ) . If there are no negative incomes, it is also equal to 2 A and 1 − 2 B due to the fact that A + B = 0.5 . Assuming non-negative income or wealth for all, the Gini coefficient's theoretical range
18972-549: The richest person receives a rank of 1 and the poorest a rank of N . This effectively gives higher weight to poorer people in the income distribution, which allows the Gini to meet the Transfer Principle . Note that the Jasso-Deaton formula rescales the coefficient so that its value is one if all the X i {\displaystyle X_{i}} are zero except one. Note however Allison's reply on
19125-401: The same income ( G = 0 ), and the most unequal society (with N individuals) where a single person receives 100% of the total income and the remaining N − 1 people receive none ( G = 1 − 1/ N ). A simple case assumes just two levels of income, low and high. If the high income group is a proportion u of the population and earns a proportion f of all income, then the Gini coefficient
19278-604: The sample Gini coefficient is a consistent estimator of the population Gini coefficient, but is not in general unbiased . In simplified form: There does not exist a sample statistic that is always an unbiased estimator of the population Gini coefficient. For a discrete probability distribution with probability mass function f ( y i ) , {\displaystyle f(y_{i}),} i = 1 , … , n {\displaystyle i=1,\ldots ,n} , where f ( y i ) {\displaystyle f(y_{i})}
19431-412: 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
19584-486: 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
19737-444: 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
19890-413: The start of the 21st century. Central European countries Slovenia , Czechia , and Slovakia have had the lowest inequality index out of all OECD countries ever since the 2000s. Scandinavian countries also frequently appeared at the top of the list of equality in the last decades. The Gini coefficient is an index for the degree of inequality in the distribution of income/wealth, used to estimate how far
20043-406: 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
20196-657: The study and the manipulation of formulas . Calculus , consisting of the two subfields differential calculus and integral calculus , is the study of continuous functions , which model the typically nonlinear relationships between varying quantities, as represented by variables . This division into four main areas—arithmetic, geometry, algebra, and calculus —endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet did not become
20349-568: The study of approximation and discretization with special focus on rounding errors . Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic- matrix -and- graph theory . Other areas of computational mathematics include computer algebra and symbolic computation . The word mathematics comes from the Ancient Greek word máthēma ( μάθημα ), meaning ' something learned, knowledge, mathematics ' , and
20502-408: 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
20655-402: 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
20808-471: 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
20961-773: 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
21114-672: The theory under consideration. Mathematics is essential in the natural sciences , engineering , medicine , finance , computer science , and the social sciences . Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent of any scientific experimentation. Some areas of mathematics, such as statistics and game theory , are developed in close correlation with their applications and are often grouped under applied mathematics . Other areas are developed independently from any application (and are therefore called pure mathematics ) but often later find practical applications. Historically,
21267-487: The title of his main treatise . Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas . Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra ), and polynomial equations in
21420-508: The two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' that he used for naming one of these methods in
21573-445: The value of B can be found by integration : The Gini coefficient can also be calculated directly from the cumulative distribution function of the distribution F ( y ). Defining μ as the mean of the distribution, and specifying that F ( y ) is zero for all negative values, the Gini coefficient is given by: The latter result comes from integration by parts . (Note that this formula can be applied when there are negative values if
21726-467: 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
21879-463: 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
22032-460: 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
22185-440: The work of American economist Max Lorenz , Gini proposed that the difference between the hypothetical straight line depicting perfect equality, and the actual line depicting people's incomes, be used as a measure of inequality. In this paper, he introduced the concept of simple mean difference as a measure of variability. He then applied the simple mean difference of observed variables to income and wealth inequality in his work On
22338-521: Was 0.46 for the pre-tax income Gini index and 0.31 for the after-tax income Gini index. Taxes and social spending that were in place in 2008–2009 period in OECD countries significantly lowered effective income inequality, and in general, "European countries—especially Nordic and Continental welfare states —achieve lower levels of income inequality than other countries." Using the Gini can help quantify differences in welfare and compensation policies and philosophies. However, it should be borne in mind that
22491-462: Was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book Elements . The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane ( plane geometry ) and the three-dimensional Euclidean space . Euclidean geometry was developed without change of methods or scope until
22644-414: Was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz . It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis"
22797-421: 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
22950-437: Was not specifically studied by mathematicians. Before Cantor 's study of infinite sets , mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration . Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument . This led to
23103-892: 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
23256-437: 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
23409-571: Was split into two new subfields: synthetic geometry , which uses purely geometrical methods, and analytic geometry , which uses coordinates systemically. Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions , the study of which led to differential geometry . They can also be defined as implicit equations , often polynomial equations (which spawned algebraic geometry ). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. In
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