Necessity and sufficiency
In logic and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement "If P Q", we say that "Q is necessary for P" because P cannot be true unless Q is true. We say that "P is sufficient for Q" because P being true always implies that Q is true, but P not being true does not always imply that Q is not true; the assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either true or false. In ordinary English, "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. Being a male sibling is a necessary and sufficient condition for being a brother. In the conditional statement, "if S N", the expression represented by S is called the antecedent and the expression represented by N is called the consequent; this conditional statement may be written in many equivalent ways, for instance, "N if S", "S only if N", "S implies N", "N is implied by S", S → N, S ⇒ N, or "N whenever S".
In the above situation, we say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement the consequent N must be true if S is to be true. Phrased differently, the antecedent S cannot be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named. In the above situation, we can say S is a sufficient condition for N. Again, consider the third column of the truth table below. If the conditional statement is true if S is true, N must be true. In common terms, "S guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that someone has a Name. A necessary and sufficient condition requires that both of the implications S ⇒ N ⇒ S hold. From the first of these we see that S is a sufficient condition for N, from the second that S is a necessary condition for N; this is expressed as "S is necessary and sufficient for N ", "S if and only if N ", or S ⇔ N.
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false P is false". By contraposition, this is the same thing as "whenever P is true, so is Q"; the logical relation between P and Q is expressed as "if P Q" and denoted "P ⇒ Q". It may be expressed as any of "P only if Q", "Q, if P", "Q whenever P", "Q when P". One finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition, as shown in Example 5. Example 1 For it to be true that "John is a bachelor", it is necessary that it be true that he is unmarried, adult, since to state "John is a bachelor" implies John has each of those three additional predicates. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number, both and prime. Example 3 Consider thunder, the sound caused by lightning. We say. Whenever there's lightning, there's thunder; the thunder does not cause the lightning, but because lightning always comes with thunder, we say that thunder is necessary for lightning.
Example 4 Being at least 30 years old is necessary for serving in the U. S. Senate. If you are under 30 years old it is impossible for you to be a senator; that is, if you are a senator, it follows that you are at least 30 years old. Example 5 In algebra, for some set S together with an operation ⋆ to form a group, it is necessary that ⋆ be associative, it is necessary that S include a special element e such that for every x in S it is the case that e ⋆ x and x ⋆ e both equal x. It is necessary that for every x in S there exist a corresponding element x″ such that both x ⋆ x″ and x″ ⋆ x equal the special element e. None of these three necessary conditions by itself is sufficient, but the conjunction of the three is. If P is sufficient for Q knowing P to be true is adequate grounds to conclude that Q is true; the logical relation is, as before, expressed as "if P Q" or "P ⇒ Q". This can be expressed as "P only if Q", "P implies Q" or several other variants, it may be the case that several sufficient conditions, when taken together, constitute a single necessary condition, as illustrated in example 5.
Example 1 "John is a king" implies. So knowing that it is true that John is a king is sufficient to know that he is a male. Example 2 A number's being divisible by 4 is sufficient for its being but being divisible by 2 is both sufficient and necessary. Example 3 An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent. That is, the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can refer to each point on the indifference curve as rendering the same level of utility for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is a device to represent preferences rather than something from which preferences come; the main use of indifference curves is in the representation of observable demand patterns for individual consumers over commodity bundles. There are infinitely many indifference curves: one passes through each combination. A collection of indifference curves, illustrated graphically, is referred to as an indifference map; the theory of indifference curves was developed by Francis Ysidro Edgeworth, who explained in his 1881 book the mathematics needed for their drawing.
The theory can be derived from William Stanley Jevons' ordinal utility theory, which posits that individuals can always rank any consumption bundles by order of preference. A graph of indifference curves for several utility levels of an individual consumer is called an indifference map. Points yielding different utility levels are each associated with distinct indifference curves and these indifference curves on the indifference map are like contour lines on a topographical graph; each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction you are climbing a mound of utility; the higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top," or a "bliss point," a consumption bundle, preferred to all others. Indifference curves are represented to be: Defined only in the non-negative quadrant of commodity quantities. Negatively sloped; that is, as quantity consumed of one good increases, total satisfaction would increase if not offset by a decrease in the quantity consumed of the other good.
Equivalently, such that more of either good is preferred to no increase, is excluded. The negative slope of the indifference curve reflects the assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, the assumption of non-satiation; because of monotonicity of preferences and non-satiation, a bundle with more of both goods must be preferred to one with less of both, thus the first bundle must yield a higher utility, lie on a different indifference curve at a higher utility level. The negative slope of the indifference curve implies that the marginal rate of substitution is always positive. So, with, no two curves can intersect. Transitive with respect to points on distinct indifference curves; that is, if each point on I2 is preferred to each point on I1, each point on I3 is preferred to each point on I2, each point on I3 is preferred to each point on I1. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.
Convex. With, convex preferences imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case as a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged. Preferences are complete; the consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him. Assume that there are two consumption bundles A and B each containing two commodities x and y. A consumer can unambiguously determine that one and only one of the following is the case: A is preferred to B, formally written as A p B B is preferred to A, formally written as B p A A is indifferent to B, formally written as A I B This axiom precludes the possibility that the consumer cannot decide, It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.
Preferences are reflexiveThis means that if A and B are identical in all respects the consumer will recognize this fact and be indifferent in comparing A and B A = B ⇒ A I BPreferences are transitiveIf A p B and B p C A p C. If A I B and B I C A I C; this is a consistency assumption. Preferences are continuousIf A is preferred to B and C is sufficiently close to B A is preferred to C. A p B and C → B ⇒ A p C. "Continuous" means infinitely divisible - just like there are infinitely many numbers between 1 and 2 all bundles are infinitely divisible. T
Hal Ronald Varian is an economist specializing in microeconomics and information economics. He is the chief economist at Google and he holds the title of emeritus professor at the University of California, Berkeley where he was founding dean of the School of Information. Hal Varian was born on March 1947 in Wooster, Ohio, he received his B. S. from MIT in economics in 1969 and both his M. A. and Ph. D. from the University of California, Berkeley in 1973. Varian taught at MIT, Stanford University, the University of Oxford, the University of Michigan, the University of Siena and other universities around the world, he has two honorary doctorates, from the University of Oulu, Finland in 2002, a Dr. h. c. from the Karlsruhe Institute of Technology, awarded in 2006. He is emeritus professor at the University of California, where he was founding dean of the School of Information. Varian joined Google in 2002 as a consultant, has worked on the design of advertising auctions, finance, corporate strategy, public policy.
He is the chief economist at Google. Varian is the author of two bestselling textbooks: Intermediate Microeconomics, an undergraduate microeconomics text, Microeconomic Analysis, an advanced text aimed at first-year graduate students in economics. Together with Carl Shapiro, he co-authored Information Rules: A Strategic Guide to the Network Economy and The Economics of Information Technology: An Introduction. Varian has one child, Christopher Max Varian. Varian Rule Varian's theorems Hal Varian's Website Position Auctions Roberts, Russ. "Varian on Technology". EconTalk. Library of Economics and Liberty. Hal Varian publications indexed by Google Scholar "Hal Varian". JSTOR. Appearances on C-SPAN
Economics is the social science that studies the production and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents. Microeconomics analyzes basic elements in the economy, including individual agents and markets, their interactions, the outcomes of interactions. Individual agents may include, for example, firms and sellers. Macroeconomics analyzes the entire economy and issues affecting it, including unemployment of resources, economic growth, the public policies that address these issues. See glossary of economics. Other broad distinctions within economics include those between positive economics, describing "what is", normative economics, advocating "what ought to be". Economic analysis can be applied throughout society, in business, health care, government. Economic analysis is sometimes applied to such diverse subjects as crime, the family, politics, social institutions, war and the environment; the discipline was renamed in the late 19th century due to Alfred Marshall, from "political economy" to "economics" as a shorter term for "economic science".
At that time, it became more open to rigorous thinking and made increased use of mathematics, which helped support efforts to have it accepted as a science and as a separate discipline outside of political science and other social sciences. There are a variety of modern definitions of economics. Scottish philosopher Adam Smith defined what was 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 a plentiful revenue or subsistence for the people... to supply the state or commonwealth with a revenue for the publick services. Jean-Baptiste Say, distinguishing the subject from its public-policy uses, defines it as the science of production and consumption of wealth. On the satirical side, Thomas Carlyle coined "the dismal science" as an epithet for classical economics, in this context linked to the pessimistic analysis of Malthus. John Stuart Mill defines the subject in a social context as: 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 provides a still cited definition in his textbook Principles of Economics that extends analysis beyond wealth and from the societal to the microeconomic level: Economics is a study of man in the ordinary business of life. It enquires 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 developed implications of what has been termed "erhaps the most accepted current definition of the subject": Economics is a science which studies human behaviour as a relationship between ends and scarce means which have alternative uses. Robbins describes the definition as not classificatory in "pick out certain kinds of behaviour" but rather analytical in "focus attention on a particular aspect of behaviour, the form imposed by the influence of scarcity." He affirmed that previous economists have centred their studies on the analysis of wealth: how wealth is created and consumed. 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, generates both cost and benefits. If the war is not winnable or if the expected costs outweigh the benefits, the deciding actors may never go to war but rather explore other alternatives. We cannot define economics as the science that studies wealth, crime and any other field economic analysis can be applied to; some subsequent comments criticized the definition as overly broad in failing to limit its subject matter to analysis of markets. From the 1960s, such comments abated as the economic theory of maximizing behaviour and rational-choice modelling expanded the domain of the subject to areas treated in other fields. There are other criticisms as well, such as in scarcity not accounting for the macroeconomics of high unemployment. Gary Becker, a contributor to the expansion of economics into new areas, describes the approach he favours as "combin assumptions of maximizing behaviour, stable preferences, market equilibrium, used relentlessly and unflinchingly."
One commentary characterizes the remark as making economics an approach rather than a subject matter but with great specificity as to the "choice process and the type of social interaction that 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. A