By Satya R. Chakravarty
In this booklet we're all in favour of source of revenue profile established moral social index numbers. a moral index is designed from an particular social overview functionality with a selected goal in brain. for instance, a moral relative inequality index determines the fraction of overall source of revenue which may be kept with none welfare loss if society distri buted earning both. moral indices distinction with descriptive indices that are de rived with no utilizing any suggestion of social welfare. understand that, moral indices aren't intended to supplant descriptive indices, really they're built with diversified goals. we start bankruptcy 1 with a proper dialogue at the idea of a social overview functionality. ordinarily physique of this bankruptcy we examine the matter of rating source of revenue profiles utilizing a social assessment functionality. In bankruptcy 2 we set approximately interpreting adjust local methods to the size of inequality. In bankruptcy three we concentration our consciousness at the Gini index, the main often used index of inequality, and its extensions. In bankruptcy four we formulate the idea of a moral distance functionality that measures welfare of 1 inhabitants relative to a different. bankruptcy five is dedicated to quantifications and discussions of different definitions of relative deprivation recommend by means of Runci man(1966).
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13) is continously differentiable. 16) A + B . x) be same as 1~(x) for all c > 0 and x E Dn. 17) 1 we must have Atkinson(1970) stated that this theorem can be proved by applying results of Arrow (1971) and Pratt(1964) in the theory of choice under uncertainty. Here we give a simple proof using functional equations. ) 1 d(c'Xi) d U(CX e) d(cx e) OXe --=-=n· . __ . 21) is independent of i. This shows that U' (xj)/U' (xi) is homogeneous of degree zero in (x j, xi )6, where U' stands for the derivative of U.
That is, 'the whole set of indifference curves can be generated by radial expansion or contraction (by means of rays emanating from the origin) of a single curve' (Graaff(1977». Thus, the SEF should be an ordinal transform of a linearly homogeneous function. All such SEFs are called homothetic. 28) 47 where Wn is linearly homogeneous and ¢ is increasing in its argument. An SEF W: D ---l1R 1 is homothetic if for all n E IN, W n is homothetic. 7 (Blackorby and Donaldson 1978) The Atkinson-Kolm-Sen inequality index I3 is a relative index if and only if W is homothetic.
In Sections 3 and 4 of this chapter we discuss alternative approaches to the construction of cardinally significant inequality indices. Simple proofs of the theorems of Atkinson, and Blackorby and Donaldson are provided in these sections. However, according to the Blackorby-Donaldson procedure an ordinal transform of a given inequality index may lead to an SEF which need not be ordinally related to the SEF consistent with the given index. That is, if the concept of inequality is assumed to be a purely ordinal one, then the Blackorby-Donaldson method is an inappropriate procedure.