THE CONCEPT OF THE STOCHASTIC EQUIVALENCE

SCALES. THEORY AND APPLICATIONS

By

Stanislaw Maciej Kot

Discussant: Michael Ward

Context and Issues

• Rich data in a raw or derived state are not necessarily representative data.

• The World Development Report, World Development Indicators and UNDP Human Development Report all contain a variety of per capita measures based on total population counts. But these are dominantly ‘mean’ population indicators that tend to confound rather than illuminate proper international comparative analysis relating to input, output and impact because of significant differences in national demographic characteristics.

• They do not serve the role of proxy indicators nor of metaphors that can aid the understanding of one type of issue or event in terms of another.

Representative Equivalence Scales

• While politicians are keen on talking about ‘every man, woman and child’ as the appropriate public reference base, the subject matter of the aggregate variable – be it consumption, production, income, wealth, energy use, calorie intake, health expenditure, education outlays, teachers, etc demands a different ‘base’ or denominator for the measure to be truly meaningful for policy purposes and analysis

• Even in the case of consumption or household expenditure, the non-homogeneity of household groups creates problems both of inter-HH comparisons and of marginal versus average per capita consumption outlays

Theory versus Practice

• All current equivalence scales – of which there are many with each type specifically depending on the characteristic under investigation – are arbitrary

• Only in the case of expenditures is there a genuine possibility of applying economic [consumption] theory to define equivalent virtual per capita values

• Such a set of scales needs to mimic, for each household size, the total utility values reflected in individual welfare functions, personal consumer preference and behaviour

• The author sets out to determine, theoretically, such a set of individual scales

Author’s Objectives

• to define a concept of stochastic equivalence scales (SES) constituted by an holistic paradigm of welfare

• illustrate with empirical examples from household surveys [using Poland 2000 as a case study]

• use non-parametric and parametric applications to demonstrate relevance of approach by reference to the one person household distribution function

Methodology

• Starting point is individual welfare reflected in a given person’s income utility function

• This is assumed to be a convenient and valid representation of consumer preference. This implies adpting a disposable income concept to reflect consumption, acceptable at lower levels of income but not so for higher overall levels associated with larger families

• Familiar PROBLEM: To aggregate individual welfare to derive a total social welfare function

• As Arrow convincingly demonstarted, this is not possible in rigorous practical terms [although this has not stopped many economists from attempting the social aggregation exercise]

The Impossibility of aggregating individual welfare functions

• The problem of inter-personal comparability is found to be unacceptable to most economic theorists

• In practice, analysts cannot assume individual preferences are independent and isolated from the preferences, and revealed spending, of others.

• This problem spills over into the related issue of deriving, conceptually, relevant equivalence scales for converting consolidated data of one form or another into an hypothetical individual function

• This has to be in conformity with the conventionally recognised theory of consumer behaviour.

• consumer theory may be even less appropriate than other premisses about individual spending drawn from market research and observed personal behaviour

Author’s solution

• Determine an holistic concept of welfare or social preference relations based on a benefit function for the population (the welfare of society) as a whole

• Find a benefit function [BF] consisting of an income distribution, f(x) that can be transformed into a welfare distribution f(w) via the transformation of a random variable x into a new random variable w. The BF takes the form of b.R*->R where b is estimated from the distribution data

• [See author’s paper for the applied algebra!]

Logic of Equivalence

• Need to compare the welfare of households with various needs

• Differentiation of needs is usually associated with the differences in household demographic structure – size of HH, age and sex profile.

• Compare populations of various decomposable homogeneous groups with standard reference group, in this case, single-person household

• Main aim to obtain equivalent welfare level of comparable households of different sizes and composition.

EXAMPLES(Using 2000 HBS for Poland)

1] Abbreviated Social Welfare Function, non-parametric form

u= u´(1-G) where G is Gini coefficient and u is an approximation for u´ and the deflator value is defined accordingly from estimates of the mean income of the sub-group concerned and its specific gini coefficient:

Example: Nine groups, ranked by hh sizeWith a mean and gini for each size groupNote: closeness of fit of each derived distribution for

different HH size with the single household reference group across every income level

2] Example of 25 groups

• Non-parametric application with adults and children less than 18 years old (more complex divisions)

• Also parametric forms represented by a power equivalence scale with constant elasticity in relation to hh size (a condition that can be relaxed)

• Compare methods of Buhmann et al (Rainwater and Smeeding), OECD, Coutler and Katz

Continuous Power Functions

• Parametric approaches also demonstrate a trajectory for power functions that flatten off very quickly, revealing very little difference in the adjustment factors to be applied for 4, 5, 6 [and above] household sizes; can this be real [especially as some equivalent scales for larger families have lower values] and be attributable to household economies of scale? Or is this a statistical artefact akin to curve fitting?

Outcomes, problems and questions

• The practical application of the SES methodology appears simple, seems to work but is it conceptually valid? In particular, is the transformation factor/scalar used clearly related to economies of scale?

• Is Poland 2000 with an even income distribution a special case? Can we really assume that the gini is so similar in each income sub-group as HH size and incomes rise?

• Is the income distribution ’neutral’ in the sense that the single group may be based in an urban location and larger families live on the land?