New Welfare Economics and Arrow’s Contribution to Formal Economic Theory
This section clarifies Arrow's two most famous contributions to formal economic theory and their normative implication.
Welfare economics was established by Arthur Cecil Pigou in 1920.
In this field, new welfare economics gained traction after Lionel Robbins' criticism of how old welfare economics did not satisfy the condition of scientific objectivity (Robbins 1969). In 1938, Abram Bergson successfully formalized the economic welfare function that can deduce a criterion of social welfare from a set of economic variables (Bergson 1938; Backhouse, Chapter 8 in this volume). His study was pivotal because it allowed new welfare economists to formally analyze a criterion of social welfare. Arrow tried to rationalize the construction of a social welfare function solely on the basis of individual preferences. To achieve this, he inquired into the collective decision-making process that deduces a criterion of social welfare from a set of individual orderings as a criterion of individual welfare. He did this in the following two ways.addition, Saito (2018) focuses on the theoretical change in Arrow's view of democracy and his attitude toward war. These studies, however, do not analyze the theoretical change in Arrow's view of the market and his idea of justice.
In his book Social Choice and Individual Values, Arrow proved a theorem that is generally regarded as proof of the logical impossibility of a collective-choice rule. He states that “we ask if it is formally possible to construct a procedure for passing from a set of known individual tastes to a pattern of social decision-making, the procedure in question being required to satisfy certain natural conditions” (Arrow 1963, p. 2). To prove this question, first, he assumes that all the members in a society are rational. This means that each member has an ordering of all possible alternatives.
It is defined that an ordering satisfies two conditions: completeness and transitivity. They then choose their top alternatives in the ordering over a set of feasible alternatives at each step, acting to maximize their welfare in their decision-making. Social ordering, which satisfies the same two conditions, is deduced from individual ordering. Second, it is assumed that an individual’s ordering is the criterion for his or her individual welfare and that social ordering is the criterion for social welfare. The social welfare function is mapped from the individual orderings of the members of a society to social ordering (Arrow 1963, pp. 12-19; Saito 2018, p. 241). His theorem questions whether the collective decision process can meet the following four desirable conditions. The first is the condition of unrestricted domain, which requires the collective decision process to accept any pattern of individual ordering. The second is the Pareto principle, which dictates that society should prefer what its members prefer. The third is the condition of indifference to irrelevant alternatives, which dictates that a decision on one opportunity set should be independent of external alternatives. The fourth is the condition of nondictatorship, which directs that no individual ordering should become a social ordering by disregarding others’ individual orderings (Arrow 1963, pp. 96-103). These four conditions of the collective decisionmaking process are assumed to be axioms. Arrow proved that no social welfare function satisfies these four conditions.Arrow’s theorem also perpetuated the research interests of new welfare economics, especially those of Bergson. Arrow wanted to identify the criterion for social welfare in new welfare economics because he argued for a relationship between his social welfare function and Bergson’s economic welfare function (Arrow 1963, pp. 22-24; Bergson 1938). The theorem, however, encompasses not only new welfare economics but also political science, because it deals with general collective decision-making, such as voting (Samuelson 1967, p. 42).
Arrow and Debreu presented the fundamental theorems of welfare economics in 1954. The first theorem specifies that if a market satisfies three assumptions - that it is a complete market, includes price-taking behavior, and contains local non-satiation of preferences - it will reach a competitive equilibrium that is Pareto optimal (Arrow and Debreu 1954). Arrow summarized his research about general equilibrium in 1971 (Arrow and Hahn 1971).
In short, Arrow proved that there is no desirable process of social decision-making that deduces a criterion of social welfare from individual orderings as a criterion of individual welfare. While he also proved that competitive equilibrium can satisfy Pareto optimality as a criterion of social welfare.
11.2
More on the topic New Welfare Economics and Arrow’s Contribution to Formal Economic Theory:
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- Indeterminacy, Self-Fulfilling Prophecies, and Sunspots