TheInfoList

In
welfare economics Welfare economics is a branch of economics that uses microeconomic Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, tha ...
, a social welfare function is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or for every possible pair of social states. Inputs of the function include any variables considered to affect the
economic welfare The welfare definition of economics is an attempt by Alfred Marshall Alfred Marshall (26 July 1842 – 13 July 1924) was an English economist, who was one of the most influential economists of his time. His book, '' Principles of Economics'' ...
of a society. In using welfare measures of persons in the society as inputs, the social welfare function is
individualistic Individualism is the Ethics, moral stance, political philosophy, ideology and social outlook that emphasizes the intrinsic worth of the individual. Individualists promote the exercise of one's goals and desires and to value independence and self ...
in form. One use of a social welfare function is to
represent Represent may refer to: * Represent (Compton's Most Wanted album), ''Represent'' (Compton's Most Wanted album) or the title song, 2000 * Represent (Fat Joe album), ''Represent'' (Fat Joe album), 1993 * ''Represent'', an album by DJ Magic Mike, 1994 ...
prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income. The social welfare function is analogous to the
consumer theory The theory of consumer choice is the branch of microeconomics Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that ar ...
of
budget constraint In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavi ...

tangency for an individual, except that the social welfare function is a mapping of individual preferences or judgments of everyone in the society as to collective choices, which apply to all, whatever individual preferences are for (variable) constraints on factors of production. One point of a social welfare function is to determine how close the analogy is to an ordinal
utility function As a topic of economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...
for an individual with at least minimal restrictions suggested by welfare economics, including constraints on the number of
factors of production In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavio ...
. There are two major distinct but related types of social welfare functions: * A Bergson–Samuelson social welfare function considers welfare for a ''given set'' of individual preferences or welfare rankings. * An Arrow social welfare function considers welfare across ''different possible sets'' of individual preferences or welfare rankings and seemingly reasonable axioms that constrain the function.Prasanta K. Pattanaik, 2008. "social welfare function," ''
The New Palgrave Dictionary of Economics ''The New Palgrave Dictionary of Economics'' (2018), 3rd ed., is an twenty-volume reference work on economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, ...
'', 2nd Edition
Abstract.
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# Bergson–Samuelson social welfare function

In a 1938 article,
Abram Bergson Abram Bergson (April 21, 1914 in New York City New York, often called New York City to distinguish it from New York State New York is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a mont ...
introduced the ''social welfare function''. The object was "to state in precise form the value judgments required for the derivation of the conditions of maximum economic welfare" set out by earlier writers, including
Marshall Marshall may refer to: Places United States * Marshall, Alaska * Marshall, Arkansas * Marshall, California * Lotus, California, former name Marshall * Marshall Pass, a mountain pass in Colorado * Marshall, Illinois * Marshall, Indiana * Marshall, ...

and , and Barone, and
Lerner Lerner is a German and Jewish family name. Its literal meaning can be either "student" or "scholar". It may refer to: Organizations * Lerner Enterprises, a real estate company * Lerner Newspapers * Lerner Publishing Group, a publisher of childr ...

. The function was real-valued and
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

. It was specified to describe the society as a whole. Arguments of the function included the quantities of different commodities produced and consumed and of
resources Resource refers to all the materials available in our environment which help us to satisfy our needs and wants. Resources can broadly be classified upon their availability — they are classified into renewable A renewable resource, also know ...
used in producing different commodities, including labor. Necessary general conditions are that at the maximum value of the function: * The marginal "dollar's worth" of welfare is equal for each individual and for each commodity * The marginal "diswelfare" of each "dollar's worth" of labor is equal for each commodity produced of each labor supplier * The marginal "dollar" cost of each unit of resources is equal to the marginal value productivity for each commodity. Bergson showed how
welfare economics Welfare economics is a branch of economics that uses microeconomic Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, tha ...
could describe a standard of economic efficiency despite dispensing with ''interpersonally-comparable''
cardinal utility In economics, a cardinal utility function or scale is a utility index that preserves Preference (economics), preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the ...
, the hypothesization of which may merely conceal value judgments, and purely subjective ones at that. Auxiliary specifications enable comparison of different social states by each member of society in preference satisfaction. These help define ''
Pareto efficiency Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after V ...
'', which holds if all alternatives have been exhausted to put at least one person in a more preferred position with no one put in a less preferred position. Bergson described an "economic welfare increase" (later called a ''Pareto improvement'') as at least one individual moving to a more preferred position with everyone else indifferent. The social welfare function could then be specified in a ''substantively'' individualistic sense to derive Pareto efficiency (optimality).
Paul Samuelson Paul may refer to: *Paul (given name) Paul () is a common masculine given name in countries and ethnicities with a Christian heritage (Eastern Orthodoxy The Eastern Orthodox Church, officially the Orthodox Catholic Church, is the List ...

(2004, p. 26) notes that Bergson's function "could derive Pareto optimality conditions as ''necessary'' but not sufficient for defining interpersonal normative equity." Still, Pareto efficiency could also characterize ''one'' dimension of a particular social welfare function with distribution of commodities among individuals characterizing ''another'' dimension. As Bergson noted, a welfare improvement from the social welfare function could come from the "position of some individuals" improving at the expense of others. That social welfare function could then be described as characterizing an equity dimension. Samuelson (
1947 It was the first year of the Cold War The Cold War was a period of geopolitical tension between the United States and the Soviet Union and their respective allies, the Western Bloc and the Eastern Bloc, which began following World W ...
, p. 221) himself stressed the flexibility of the social welfare function to characterize ''any'' one ethical belief, Pareto-bound or not, consistent with: * a complete and transitive ranking (an ethically "better", "worse", or "indifferent" ranking) of all social alternatives and * one set out of an infinity of welfare indices and cardinal indicators to characterize the belief. He also presented a lucid verbal and mathematical exposition of the social welfare function (1947, pp. 219–49) with minimal use of Lagrangean multipliers and without the difficult notation of differentials used by Bergson throughout. As Samuelson (1983, p. xxii) notes, Bergson clarified how production and consumption efficiency conditions are distinct from the interpersonal ethical values of the social welfare function. Samuelson further sharpened that distinction by specifying the ''Welfare function'' and the ''Possibility function'' (1947, pp. 243–49). Each has as
arguments In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stat ...
the set of utility functions for everyone in the society. Each can (and commonly does) incorporate Pareto efficiency. The Possibility function also depends on technology and resource restraints. It is written in implicit form, reflecting the ''feasible'' locus of utility combinations imposed by the restraints and allowed by Pareto efficiency. At a given point on the Possibility function, if the utility of all but one person is determined, the remaining person's utility is determined. The Welfare function ranks different hypothetical ''sets'' of utility for everyone in the society from ethically lowest on up (with ties permitted), that is, it makes interpersonal comparisons of utility. Welfare maximization then consists of maximizing the Welfare function subject to the Possibility function as a constraint. The same welfare maximization conditions emerge as in Bergson's analysis.

# Arrow social welfare function (constitution)

Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist {{unreferenced, date=June 2015 A political theorist is someone who engages in constructing or evaluating politica ...
(
1963 Events January * January 1 January 1 or 1 January is the first day of the year in the . There are 364 days remaining until the end of the year (365 in s). This day is known as since the day marks the beginning of the year. __TOC__ ...
) generalizes the analysis. Along earlier lines, his version of a social welfare function, also called a 'constitution', maps a set of individual orderings (
ordinal utility function In economics, an ordinal utility function is a function representing the Preference (economics), preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but ...
s) for everyone in the society to a social ordering, a rule for ranking alternative social states (say passing an enforceable law or not,
ceteris paribus ' or ' () is a Latin phrase meaning "other things equal"; English translations of the phrase include "all other things being equal" or "other things held constant" or "all else unchanged". A prediction or a statement about a ontic, causal, epist ...
). Arrow finds that nothing of behavioral significance is lost by dropping the requirement of social orderings that are ''real-valued'' (and thus
cardinal Cardinal or The Cardinal may refer to: Christianity * Cardinal (Catholic Church), a senior official of the Catholic Church * Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral Navigation * Cardin ...
) in favor of orderings, which are merely ''complete'' and ''transitive'', such as a standard
indifference curve In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behavi ...

map. The earlier analysis mapped any set of individual orderings to ''one'' social ordering, whatever it was. This social ordering selected the top-ranked ''feasible'' alternative from the economic environment as to resource constraints. Arrow proposed to examine mapping different sets of individual orderings to possibly different social orderings. Here the social ordering would depend on the set of individual orderings, rather than being ''imposed'' (invariant to them). Stunningly (relative to a course of theory from
Adam Smith Adam Smith ( 1723 – 17 July 1790) was a Scottish economist, philosopher as well as a moral philosopher Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and ...

and
Jeremy Bentham Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 4 February 1747">Old_Style_and_New_Style_dates.htm ...

on), Arrow proved the '' general impossibility theorem'' which says that it is impossible to have a social welfare function that satisfies a certain set of "apparently reasonable" conditions.

# Cardinal social welfare functions

A cardinal social welfare function is a function that takes as input numeric representations of individual utilities (also known as
cardinal utility In economics, a cardinal utility function or scale is a utility index that preserves Preference (economics), preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the ...
), and returns as output a numeric representation of the collective welfare. The underlying assumption is that individuals utilities can be put on a common scale and compared. Examples of such measures can be: *
life expectancy Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, its current age, and other demographic Demography (from prefix ''demo-'' from Ancient Greek Ancien ...

, * per capita income. For the purposes of this section, income is adopted as the measurement of utility. The form of the social welfare function is intended to express a statement of objectives of a society. The
utilitarian Utilitarianism is a family of normative Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
or
Benthamite Jeremy Bentham (; 15 February 1748
ld Style and New Style dates, O.S. 4 February 1747 LD may refer to: Arts and entertainment Film and television *Lorraine "L.D." Delacorte, a character on the TV series ''Degrassi ''Degrassi'' is a Canadian teen drama In film A film, also called a movie, motion picture or moving pi ...
– 6 June 1832) was an English philosopher, jurist, and social reformer regarded as the founder of modern utilitarianism. Bentham defined as the "fundamen ...
social welfare function measures social welfare as the total or sum of individual incomes: :$W = \sum_^n Y_i$ where $W$ is social welfare and $Y_i$ is the income of individual $i$ among $n$ individuals in society. In this case, maximizing the social welfare means maximizing the total income of the people in the society, without regard to how incomes are distributed in society. It does not distinguish between an income transfer from rich to poor and vice versa. If an income transfer from the poor to the rich results in a bigger increase in the utility of the rich than the decrease in the utility of the poor, the society is expected to accept such a transfer, because the total utility of the society has increased as a whole. Alternatively, society's welfare can also be measured under this function by taking the average of individual incomes: :$W = \frac\sum_^n Y_i = \overline$ In contrast, the max-min or Rawlsian social welfare function (based on the philosophical work of
John Rawls John Bordley Rawls (; February 21, 1921 – November 24, 2002) was an American moral A moral (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was ori ...
) measures the social welfare of society on the basis of the welfare of the least well-off individual member of society: :$W = \min\left(Y_1, Y_2, \cdots, Y_n\right)$ Here maximizing societal welfare would mean maximizing the income of the poorest person in society without regard for the income of other individuals. These two social welfare functions express very different views about how a society would need to be organised in order to maximize welfare, with the first emphasizing total incomes and the second emphasizing the needs of the worst-off. The max-min welfare function can be seen as reflecting an extreme form of uncertainty aversion on the part of society as a whole, since it is concerned only with the worst conditions that a member of society could face.
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom ( ...

proposed a welfare function in 1973: :$W_\mathrm = \overline \left(1-G\right)$ The average per capita income of a measured group (e.g. nation) is multiplied with $\left(1-G\right)$ where $G$ is the
Gini index In economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those so ...
, a relative inequality measure. James E. Foster (1996) proposed to use one of Atkinson's Indexes, which is an entropy measure. Due to the relation between Atkinsons entropy measure and the
Theil index #REDIRECT Theil index The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation. The first presentation of this method of measuring inequal ...
, Foster's welfare function also can be computed directly using the Theil-L Index. :$W_\mathrm = \overline \mathrm^$ The value yielded by this function has a concrete meaning. There are several possible incomes which could be earned by a ''person'', who randomly is selected from a population with an unequal distribution of incomes. This welfare function marks the income, which a randomly selected person is most likely to have. Similar to the
median In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a m ...

, this income will be smaller than the average per capita income. :$W_\mathrm = \overline \mathrm^$ Here the Theil-T index is applied. The inverse value yielded by this function has a concrete meaning as well. There are several possible incomes to which a ''Euro'' may belong, which is randomly picked from the sum of all unequally distributed incomes. This welfare function marks the income, which a randomly selected Euro most likely belongs to. The inverse value of that function will be larger than the average per capita income. The article on the Theil index provides further information about how this index is used in order to compute welfare functions.

## Axioms of cardinal welfarism

Suppose we are given a
preference relation The term preference relation is used to refer to orderings that describe human preferences for one thing over an other. * In mathematics, preferences may be modeled as a weak ordering or a semiorder, two different types of binary relation. One speci ...
''R'' on utility profiles. ''R'' is a weak
total order In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
on utility profiles—it can tell us, given any two utility profiles, if they are indifferent or one of them is better than the other. A reasonable preference ordering should satisfy several axioms: 1. Monotonicity, i.e., if the utility of an individual increases while all other utilities remain equal, ''R'' should strictly prefer the second profile. E.g., it should prefer the profile (1,4,4,5) to (1,2,4,5). This is related to
Pareto optimality Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be better off without making at least one individual or preference criterion worse off or without any loss thereof. The concept is named after V ...

. 2. Symmetry, i.e., ''R'' should be indifferent to permutation of the numbers in the utility profile. E.g., it should be indifferent between (1,4,4,5) and (5,4,1,4). 3. Continuity: for every profile ''v'', the set of profiles weakly better than ''v'' and the set of profiles weakly worse than ''v'' are
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
s. 4. Independence of unconcerned agents, i.e., ''R'' should be independent of individuals whose utilities have not changed. E.g., if ''R'' prefers (2,2,4) to (1,3,4), then it also prefers (2,2,9) to (1,3,9); the utility of agent 3 should not affect the comparison between two utility profiles of agents 1 and 2. This property can also be called locality or separability. It allows us to treat allocation problems in a local way, and separate them from the allocation in the rest of society. Every preference relation with properties 1–4 can be represented as by a function ''W'' which is a sum of the form: :$W\left(u_1,\dots,u_n\right) = \sum_^n w\left(u_i\right)$ where ''w'' is a continuous increasing function. It is also reasonable to require: 5. Independence of common scale, i.e., the relation between two utility profiles does not change if both of them are multiplied by the same scalar (e.g., the relation does not depend on whether we measure the income in cents, dollars or thousands). If the preference relation has properties 1–5, then the function ''w'' belongs to the following one-parameter family: * $w_p\left(x\right) = x^p$ for $p>0$, * $w_0\left(x\right) = \ln\left(x\right)$ for $p=0$, * $w_p\left(x\right) = -x^p$ for $p<0$. This family has some familiar members: * The limit when $p\to -\infty$ is the ''leximin'' ordering; * For $p=0$ we get the Nash bargaining solution—maximizing the product of utilities; * For $p=1$ we get the
utilitarian Utilitarianism is a family of normative Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
welfare function—maximizing the sum of utilities; * The limit when $p\to \infty$ is the ''leximax'' ordering. If, in addition, we require: 6. the Pigou–Dalton principle, then the parameter ''p'', in the above family, must be at most 1.

*
Aggregation problem An ''aggregate'' in economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuse ...
*
Gorman polar form Gorman polar form is a functional form for indirect utility function__NOTOC__ In economics, a consumer's indirect utility function v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of in ...
*
Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
* Community indifference curve *
Distribution (economics) In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant ...
*
Economic welfare The welfare definition of economics is an attempt by Alfred Marshall Alfred Marshall (26 July 1842 – 13 July 1924) was an English economist, who was one of the most influential economists of his time. His book, '' Principles of Economics'' ...
*
Extended sympathyExtended sympathy in welfare economics Welfare economics is a branch of economics that uses microeconomics, microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level. Attempting to apply the principles of wel ...
*
Justice (economics) Justice Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" being impacted upon by numerous fields, with many differing viewpoints and persp ...
*
Liberal paradox The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoni ...
*
Social choice theory Social choice theory or social choice is a theoretical A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense o ...
*
Welfare economics Welfare economics is a branch of economics that uses microeconomic Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, tha ...
* Production-possibility frontier

# References

*
Kenneth J. Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and political theorist {{unreferenced, date=June 2015 A political theorist is someone who engages in constructing or evaluating politica ...
, 1951, 2nd ed., 1963, ''
Social Choice and Individual Values #REDIRECT Social Choice and Individual Values #REDIRECT Social Choice and Individual Values#REDIRECT Social Choice and Individual Values Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951, 2nd ed., 1963) and a theorem within ...
'' *
Abram Bergson Abram Bergson (April 21, 1914 in New York City New York, often called New York City to distinguish it from New York State New York is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a mont ...
(Burk),"A Reformulation of Certain Aspects of Welfare Economics," ''Quarterly Journal of Economics'', 52(2), February 1938, 310–34
Bergson–Samuelson social welfare functions
in Paretian welfare economics from the New School. * James E. Foster and
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom ( ...

, 1996, ''On Economic Inequality'', expanded edition with annexe, . * John C. Harsanyi, 1987, “interpersonal utility comparisons," ''
The New Palgrave: A Dictionary of Economics ''The New Palgrave Dictionary of Economics'' (2018), 3rd ed., is an twenty-volume reference work on economics published by Palgrave Macmillan. It contains around 3,000 entries, including many classic essays from the original Inglis Palgrave Dictiona ...
'', v. 2, 955–58 * ::Also available as
a journal article.
* Jan de Van Graaff, 1957, "Theoretical Welfare Economics", 1957, Cambridge, UK: Cambridge University Press. *
Lionel Robbins Lionel Charles Robbins, Baron Robbins, (22 November 1898 – 15 May 1984) was a British economist An economist is a professional and practitioner in the social science Social science is the Branches of science, branch of science devot ...

, 1935, 2nd ed.. ''
An Essay on the Nature and Significance of Economic Science Lionel Robbins Lionel Charles Robbins, Baron Robbins, (22 November 1898 – 15 May 1984) was a British economist, and prominent member of the economics department at the London School of Economics (LSE). He is known for his leadership at LSE, hi ...
'', ch. VI * ____, 1938, "Interpersonal Comparisons of Utility: A Comment," ''Economic Journal'', 43(4), 635–41 * Paul A. Samuelson, 1947, Enlarged ed. 1983, ''
Foundations of Economic Analysis ''Foundations of Economic Analysis'' is a book by Paul A. Samuelson published in 1947 (Enlarged ed., 1983) by Harvard University Press. It is based on Samuelson's 1941 doctoral dissertation at Harvard University. The book sought to demonstrate a c ...
'', pp. xxi–xxiv & ch. VIII, "Welfare Economics," * _____, 1977. "Reaffirming the Existence of 'Reasonable' Bergson–Samuelson Social Welfare Functions," ''Economica'', N.S., 44(173),
pp. 81
88. Reprinted in (1986) ''The Collected Scientific Papers of Paul A. Samuelson'', pp
47–54.
* _____, 1981. "Bergsonian Welfare Economics", in S. Rosefielde (ed.), ''Economic Welfare and the Economics of Soviet Socialism: Essays in Honor of Abram Bergson'',
Cambridge University Press Cambridge University Press (CUP) is the publishing business of the University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowled ...
, Cambridge, pp. 223–66. Reprinted in (1986) ''The Collected Scientific Papers of Paul A. Samuelson'', pp.
–46.
* (1963). "Distribution, Transitivity and Little's Welfare Criteria," ''Economic Journal'', 73(292)
pp. 771
78. * _____, 1970
984 Year 984 ( CMLXXXIV) was a leap year starting on Tuesday A leap year starting on Tuesday is any year with 366 days (i.e. it includes 29 February) that begins on Tuesday, 1 January, and ends on Wednesday, 31 December. Its dominical letters hence ...
''Collective Choice and Social Welfare'
(description)
ch. 3, "Collective Rationality." * _____ (1982). ''Choice, Welfare and Measurement'', MIT Press
Description
and scroll to chapter-previe