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An ordered-response income adequacy model.

An Ordered-Response Income Adequacy Model

Individual and social well-being are central to analysis of equity and social progress. It is natural that the measurement of well-being is a point of convergence for economics and other social sciences. Interest currently exists in using subjective data to estimate welfare or utility functions for individuals. There is corresponding interest in the comparison of welfare functions across households. Analyses of the theoretical potential for improvements in welfare, the incidence of poverty, and the formation of preferences can use such estimates and comparisons.

The "Leyden model," so known for its place of origin, dominates work on measuring welfare levels of individuals from subjective data.(1) Analysis using the Leyden model (LM) have worked on specifying poverty lines, measuring economic progress, testing for optimal income distributions, and estimating the value of social service benefits and environmental improvements. This study introduces a new approach to estimating individual welfare levels from subjective data. It also deals with comparing welfare measures across individuals. The types of data used are plentiful in secondary sources. The approach does not require some of the assumptions of the LM that are objectionable to many economists.(2)

The basic tools of the approach developed here, ordinal response models and random utility theory, are not new. The way they are combined is new, as is their use with introspective data on welfare. The neoclassical economists' equivalent and compensating variations follow from the model. In mathematical form, these measures are very similar to measures used in more ad hoc fashion in the LM literature. the approach is a bridge between neoclassical economic welfare measures and LM in the type of data with which it starts and in the form of welfare computations.

This paper assumes familiarity with neoclassical economic welfare measures. It also assumes familiarity with the philosophical underpinnings of ordinal utility and revealed preference and with various approaches to estimating family equivalence scales. The proper provides a brief overview of the LM. There are excellent and inclusive review articles on the LM for the interested reader.


The Leyden model is the most rigorous previous work in economics using subjective assessments of income adequacy. This section gives the model's specification, assumptions, and results, so that they can be compared with those of the ordered-response model. The model starts with a survey question:

Taking into account my (our) condition of living, I would consider a net family income per week/per month/per year as excellant if it were above _____, and good if it were between _____ and _____.

There is a similar question for amply sufficient, sufficient, barely sufficient, insufficient, very insufficient, bad, and very bad (open-ended category). For each household, the verbal descriptions evaluate hypothetical incomes, denoted by [Y.sub.h] can be a representative income for the interval. For instance, [Y.sub.h] may be the midpoint for the income interval associated with "good."

The next step is to associate a numerical utility score with each verbal description. In general, scores depend on the number of questions and assign equal intervals between any two adjacent questions. The result of this step is a set of incomes and corresponding utility numbers for each household.

The LM postulates that utility numbers and incomes for any household follow the log-normal cumulative distribution function,

U([Y.sub.h]) = N[(In Y.sub.h - m)/s].

The utility function for each household has partners m and s. U([Y.sub.h]) maps from income to utility on a zero-one continous scale. Parameter m is the log of income associated U([Y.sub.h]) = 0.5. It is the log of income at which the cumulative normal distribution function that describes utility has its inflection point. At this point, utility switches from being convex in the log of income to being concave. Households with a higher value of m require more money to reach the inflection point.(3) Parameter s describes how utility departs from 0.5 when icome departs from m. The marginal utility of money moves inversely with s.

LM assumes that individuals perceive their welfare cardinally, in a way that tells something about intensity of feeling. Furthermore, LM assumes that the individual has well-defined best possible and worst possible welfare levels (van Praag 1968; 1971, pp. 338-339). LM also requires that the income adequacy question measures individual utility on a cardinal scale.

Whether utility obeys the lognormal transformation is an issue for the validity of measurement in LM. The lognormal implies that utility has the properties of a probability distribution. Another issue is whether, given the lognormal transformation, it is valid to assign equal utility intervals between adjacent questions. In terms of the probability isomorphism, van Praag calls this implication "the same welfare mass" in the intervals (van Praag 1971, p. 334). The implication is the same change in intensity of feeling in the intervals or measurement of utility on a strongly cardinal scale (Morey 1984, p. 166).(4)

The LM implies strong cardinality of its welfare measure. It requires interpersonal comparability of utility functions only in that they all follow the lognormal transformation. The parameters, m and s, can have different values for each household.

Taking the inverse function of the standard normal distribution function and rearranging gives

1n Y.sub.h] = m + [s.sub.[N.sup.-[U(Y.sub.h)].

Appending a stochastic error term to (2) gives an equation for estimating m and s for any individual. The observations consist of ln [Y.sub.h] as dependent variable and corresponding utility scores, [N.sup.-][U(Y.sub.h)] as the independent variable. With the survey question above, there would be eight observations for each individual.(5)

Intercept m in the estimating equation may be related systematically to household characteristics like family income or family size. Empirically, little cross-sectional variation in s is related to household characteristics. For simple exposition and consistency with other studies, this paper deals only with m as a function of household characteristics,

m = [b.sub.0] + [b.sub.1][ln(fs)] +[b.sub.2][ln([ln(y.sub.a)].

In (3), fs is family size and [y.sub.a] is actual income. Different studies under LM use permament income or relative income as [Y.sub.a,] but actual income is most common. The empirical definition of actual income usually is cash income after taxes, excluding the value of government payments and in kind assistance. Previous works have elaborated the psychological and policy implications of (3) (See footnote 1). Shifting of m with actual income is called "preference drift."

Using (2) with the error appended, the analyst can get as estimate of m for each household. [Y.sub.a] and fs are known for each household. Adding a stochastic error term to (3), and substituting the values for m, [Y.sub.A], and fs, gives an estimating equation for [b.sub.0], [b.sub.1], and [b.sub.2]. These estimates describe how m varies among households with household income and family size.

Intuitively, m will increase with actual income and family size. If a family is originally at utility 0.5 and its size increases, per capita income and utility will fall. The larger family will need a higher income to have utility 0.5. Parameter m is the log of income required for utility to be 0.5. A larger income for utility 0.5 implies a larger log of income and a larger value for m. A family that is accustomed to a higher level of consumption will require a higher income to reach any given level of utility. Their expectations will be relatively high. In particular, they will require a higher income to reach utility of 0.5, and parameter m will be relatively large in their utility function.

Formally, (3) is important because it puts income into preferences. Pollak (1977) has analyzed a neoclassical model with prices from the budget constraint in preferences. [Y.sub.h] and [Y.sub.a] are analogous to Pollak's market prices and normal. Pollak concludes that, will price-dependent preferences, welfare measures such as the usual neoclassical measures may be calculated, but their interpretation is more complex (1977, pp. 73-75). As Pollak's work in the neoclassical framework suggests, welfare measures for the LM, or the ordered-response model developed below, require new interpretations.

Pollak's terminology is a convenient bridge between neoclassical works and the LM literature. In Pollak's terms, LM welfare measures are derived for a "steady state" in which [Y.sub.a] = [Y.sub.h], and this common level is denoted by Y. "Steady state" does not describe how Y =[Y.sub.a] = [Y.sub.h] comes about. Alternative scenarios correspond to alternative measures of [Y.sub.a] in (3). The LM derives poverty income levels and family equivalence scales for a steady state.(6) The steady state utility function with (3) substituted into (1) is

U(Y) = N[(1 - b.sub.2)ln Y - b.sub.0 - b.sub.1 (ln fs)]/s).

This is analogous to Polka's steady state conditional demand functions, where normal prices are equal to market prices.(7)

The LM literature defines steady state welfare measures exactly as the neoclassical approach defines the equivalent and compensating variations. For the compensating variation, the approach is to derive that change in income that, in the steady state, leaves welfare unchanged when some other variable changes. For instance, the utility level U(Y) and N.sup.-[U(Y)] may correspond to the poverty level welfare. Then, changes in income required to maintain that utility level define poverty lines for different family sizes. Welfare can be constant at any level for computing such compensating variations.

Equivalent variations are changes in income that produce the same effect on welfare as some other change, such as a change in family size. Equivalent variations or compensating variations and the income levels they imply can define family equivalence scales. Equivalent variations are not appropriate for defining a poverty line, since they produce, by definition, changes in welfare.

Compensating or equivalent variations are computable for a change in any characteristics of the household that affects welfare. The characteristic could be family size, ethnic background, or any other factor that is thought to shift preferences or affect welfare with given preferences. Here, the characteristic is denoted as "A." The characteristic could enter in many functional forms, some of which would not yield closed forms for the welfare measures. Here, A enters linearly. The linear form is consistent with previous work and yields a closed form for the welfare measures.(8)

Studies with the LM have assumed that A enters through preference drift but not directly through felt welfare for given preferences. Characteristics such as family size logically can have both effects. Appending [b.sub.3'A to (3) enters A linearly into preference drift. In (1), appending [b.sub.]A to In Y enters A linearly into welfare for given preferences. With these additions, a steady state utility function like (4) is

U(Y) = N{[(1 - [b.sub.2)(ln Y) + [b.sub.4]A - [b.sub.1](ln fs) - [b.sub.3A]/s}.

This is the key equation for deriving compensating or equivalent variations or family equivalence scales in the LM.

Adding A to the model does not alter the argument that information-maximizing responses provide an estimatable model of a form like (2). However, [b.sub.3] and [b.sub.4] cannot be separately estimated, unless [b.sub.3]A uses a "reference" value for A and [b.sub.4]A uses the actual or some hypothetical value of A. That is, [b.sub.3] and [b.sub.4] are separately recoverable only if they are associated with different As, like the different incomes [Y.sub.h] and [Y.sub.a] (1) and (3). Making the parameters separately recoverable may require changing the welfare questions to include hypothetical and reference values of A. Fortunately, it is not necessary to recover [b.sub.3] and [b.sub.4] separately to derive and compute steady state welfare measures in the LM. This paper deals with those steady state welfare measures. Future data collection should incorporate any changes that are required for the more general model, especially if the analyst wants to know a variable's effects on preferences and felt welfare separately or to compute welfare measures without assuming the steady state.

If A changes from a value of [A.sup.0] to [A.sup.1], the equivalent variation is the difference in income that would have produced the same change in welfare. The level of utility associated with (Y,[A.sub.1]) also would be associated with [A.sub.0] some multiple of income given by Ye, where e[is greater than]0. Factor e is whatever multiple of Y yieles and equivalent level of welfare as [A.sub.1]. the change in income or equivalent variation is EV = Y (1 - e).

The equivalent variation is found by setting the right-hand side of (5) with arguments (Y,[A.sub.1]) equal to the same expression with arguments (Ye,[A.sub.0]). Since the lognormal distribution function is monotonic, this equality will hold when the arguments of N are equal. Dropping terms and factors that are the same on both sides of the equality, the required condition is

(1 - [b.sub.2])ln(Ye) + ([b.sub.4] - [b.sub.3])[A.sub.0] = (l - [b.sub.2])ln(Y) + ([b.sub.4] - [b.sub.3])[A.sub.1].

Algebraic manipulation of this equality gives

e =exp[(b.sub.4] - [b.sub.3])([A.sub.1] - [A.sub.0])/(l - [b.sub.2]), and the equivalent variation is EV = Y(1 - e). Factor e and the EV can be computed without knowing [b.sub.3] and [b.sub.4] separately.

Deriving the compensating variation (CV) proceeds in the same way. The conceptual question is what change in income would leave utility unchanged when A goes from [A.sup.0] to [A.sup.1]. The required change in income is CV. If cY (c[is greater than]0) is the multiple of Y that gives the same utility for (Y,[A.sup.0]) and (cY,[A.sup.1]), then CV = Y (1 -- c). Equating (5) with arguments (Y,[A.sup.0]) to (5) with arguments (cY,[A.sup.1]), manipulation similar to that for EV gives

c = exp [([b.sub.4] - [b.sub.3]) ([A.sup.0] - [A.sup.1]) / (1 - [b.sub.2])]. (8)

EV and CV differ in the sign associated with ([A.sup.0] - [A.sup.1]) in the exponential expressions. Therefore, EV and CV will have different magnitudes. As is conventional in neoclassical economics, e [is greater than] 1 and c [is greater than] 1 corresponds to a negative EV and CV for a gain. A welfare loss gives e [is less than] 1 and c [is less than] 1 and positive EV and CV.

EV or CV can define equivalence scales for households with different characteristics. For instance, a standard household might have values fs and A, and a second household might have some other values fs' and A'. Using CV or EV for the difference between the household characteristics, the second household is equivalent to (Y -- EV)/Y or (Y -- CV)/Y standard households. The equivalence scales will differ, depending on whether CV or EV is used.

Many studies have used this method to get equivalence scales (Colasanto, Kapteyn, and van der Gaag 1984; Goedhart, et al. 1977; Hagenaars 1986; Kapteyn and van Praag 1976; van Praag 1971). They use some variation of an income adequacy question like the one above, estimate m and s for each household, and then estimate (3) from the sample of m estimates. The sample mean for s and the estimated relationship (3) provide nonstochastic parameter values for a representative household. Nonstochastic values for CV and EV and family equivalence scales follow from the representative parameter values. The LM introduces stochastic elements in ad hoc fashion for estimation. They disappear, without explanation, in the derivation of welfare measures. (9)


Landmarks for the LM have been van Praag's monograph establishing theoretical foundations (van Praag 1968), subsequent work of Kapteyn elaborating, refining, and reviewing the LM (Kapteyn, Wansbeek, and Buyze 1980; Kapteyn and Wansbeek 1985; Kapteyn, Kooreman, and Willemse 1988), and Hagenaars' book (Hagenaars 1986) reporting in massive and general application of the LM to measuring poverty in European countries. Hartog (1988, pp. 263-265) reviewed Hagenaars' book and, in doing so, summarized the status of the LM. The LM has generated a large number of high-quality papers in leading economics journals. It also has produced massive empirical work that strongly, although sometimes indirectly, supports the validity of the LM. Yet, Hartog notes that

. . . for all its vast output and dissemination of the basic ideas at the most advanced levels, the research project is an isolated one. Neither the research program, nor elements thereof, have been adopted by any economist who was not, at one time or another, a direct associate or staff member of van Praag or Kapteyn. (p. 263)

Hartog suggests the following characteristics of the LM are responsible for its lack of acceptance among economists:

(1) The Cardinality Axiom--The LM, its form, and many of the powerful normative results that flow from it are built on the axiom that utility can be measured on a zero-one scale isomorphic to a probability distribution function. This requires that utility be bounded and that equal intervals on the utility scale represent equally salient changes in welfare for an individual. For economists raised on a philosophy of ordinal utility measurement, this strong cardinality is difficult or impossible to admit.

(2) Theoretical Isolation--The Leyden utility function cannot be interpreted as a neoclassical indirect utility function, despite its resemblance to one. The neoclassical model and the LM have inconsistent assumptions. For instance, van Praag (1968) explicitly dismisses the neoclassical assumption that a consumer can fully solve a utility-maximization problem.

(3) Lack of Behavioral Predictions--While the LM has proven fruitful in normative analysis, it does not yield positive predictions about consumer behavior in markets. Because the Leyden utility function is not derived as the indirect objective function for an optimization problem, applications of the envelope theorem such as Roy's identity do not follow to get market behavior predictions as they do in the neoclassical approach (Varian 1984, pp. 127-128). Another source of this difficulty that Hartog apparently overlooks is the presence of a budget constraint element (income in the LM) in preferences. The work of Pollak (1977), in a neoclassical context, shows that putting budget constraint elements into preferences removes most of the implications of the neoclassical theory for market behavior.

(4) Data Accessibility--Use of income adequacy questions of the LM type is increasing but still rare.

There are obstacles to acceptance of the LM that Hartog does not point out.

(5) Dominance of the Behavioral School--Economists' behavioral upbringing emphasizes objectively (externally) observable behavior and casts suspicion on verbal behavior based on introspection.

(6) Reliance on Hypothetical Questions--Economists are especially mistrustful of asking people about their behavior or feelings in hypothetical circumstances. Applications of the LM are based on assessments of hypothetical incomes ([Y.sub.h] above).

(7) Lack of Rigor in Application--Estimation and application of the LM have sometimes been less than rigorous. In particular, stochastic elements appear in the model when they are convenient for estimation, but they disappear when deriving welfare measures.

(8) Adherence to the Lognormal Functional Form--LM analysts have stuck with the lognormal form for the welfare function because it has performed robustly in applications, it is derived from a very rigorous theoretical foundation, and it lends itself to further normative analysis. The best test to date of the lognormal specification, however, rejects it in favor of a logarithmic specification (transformed to preserve the probability isomorphism). The logarithmic has no theoretical derivation like that of the lognormal (Van Herwaarden and Kapteyn 1981; Hartog 1988, p. 249). Neither the lognormal nor the logarithmic is superior theoretically and empirically.

Derivation of the lognormal as the utility function by van Praag leaves no room for the LM to compromise on functional forms. The derivation also requires strong assumptions of no learning by consumers (Hartog 1988, pp. 247-248). The neoclassical approach may have similar controversy over empirical specifications and strong assumptions. It does not, however, imply a single functional form for indirect utility, so it may escape the rejection of one specification. Both approaches have strong assumptions, but the neoclassical model is entrenched and well known.


Kapsalis (1981) attempted to deal with the evaluation of hypothetical incomes while still using introspective data to develop welfare measures. Van Praag and Van der Sar (1988) attempted to deal with the cardinal utility objection.

Kapsalis suggested scoring answers to an income adequacy question,

How adequate do you consider your family income? (Check one.) 1. Adequate, 2. Fairly Adequate, 3. Barely Adequate, 4. Inadequate by 1.0 if response is Adequate or Fairly Adequate, [Y.sub.adq] 0.5 if response is Barley Adequate, and 0.0 if response is Inadequate,

and estimating (presumably by least-squares)

[Y.sub.adq] = [a.sub.0] + [a.sub.1](fs) + [a.sub.2.y] + z

where y is actual income and z is an error term. (10)

Kapsalis avoids assessments of hypothetical incomes. His approach assume, however, that utility measures are comparable across households, in that different households attach the same verbal descriptions to the same levels of welfare. Least-square estimation of (9) also assumes that [Y.sub.adq] is a "true" underlying strongly cardinal utility scale. Assigning values of 1.0, 0.5, and 0.0 and estimating by least squares requires that "Adequate" and "Fairly Adequate" are equally as much greater than "Barely Adequate" in utility as "Inadequate" is below "Barely Adequate." That is, there is an assumption that equal numerical intervals represent equal differences in intensity of feeling. Collapsing "Adequate" and "Fairly Adequate" into a single value makes the unreasonable assumption that they are equally salient to the individual. (11) Kapsalis does not derive welfare measures.

Kapsalis removes the objectionable assessment of hypothetical incomes, but maintains cardinal utility and makes the further assumption that utility is comparable across households.

Van Praag and Van de Sar (1988) release the assumption that an individual's utility is cardinally measurable. They "come to a new setup of the Leyden methodology, where we strip it of its cardinalistic traits." (p. 194) They start with a question such as:

What income level, in your circumstances, would it take to just make ends meet? What would be the absolute minimum income level for your family?

Alternatively, the question can be about an income level that is excellent, sufficient, or any of the other terms used in the original LM. (See van Praag and Van de Sar 1988, p. 199.) The income responses are the dependent variable in an equation explaining them by household characteristics. Van Praag and Van de Sar call the equation with income as a function of household characteristic a "virtual household cold function." By imposing the steady state condition, they derive a "true household cost" and family equivalence scales. If the cost equations have the same form and parameters for all levels of welfare, a single "general m-scale" prevails (p. 198). Empirical work in the same paper suggests different cost equations and equivalence scales for different levels of welfare (pp. 198-205).

Even though van Praag and Van der Sar remove the strong cardinality assumption of the LM, they keep some objectionable features:

(1) They estimate the cost equations by appending an error term in ad hoc fashion.

(2) The derive nonstochastic welfare measures and equivalence scales disregarding the stochastic elements introduced for estimation. Their model, like the LM, is convenient but internally inconsistent in treating stochastic elements.

(3) They rely on evaluation of hypothetical incomes.

Because van Praag and Van der Sar (1988) abandon the rationale for the LM without an alternative, their choice of a double-log specification in income for the cost equation has "no theoretical justification . . . . , only the strong empirical evidence that it fits rather well" (p. 202). Like Kapsalis, van Praag and Van der Sar estimate with cross-sectional data, assuming ordinal comparability of utility measurement over households.

The approaches of Kapsalis and van Praag and Van der Sar are useful contributions, but neither is fully satisfactory. Kapsalis still assumes strong cardinality of welfare measurement. Van Praag and Van der Sar still use hypothetical incomes. Neither approach treats stochastic elements consistently between estimation and developing welfare measures. Both approaches require comparability of utility across households. Neither approach is anchored in a larger theory, so each is isolated.


This section develops a new model for using subjective, introspective data to estimate welfare measures. It does not require cardinal measurement of any individual's welfare or evaluation of hypothetical incomes (although it does not exclude that possibility). The model is rigorous in treating stochastic elements and yields nonstochastic measures of welfare. The model uses data that are relatively common in secondary sources. It is consistent with neoclassical indirect utility functions. it has thorough grounding in random utility theory and the neoclassical approach.

Consider an indirect utility function that maps from exogenous household characteristics, including elements of the budget constraint, to maximum utility. Random utility theory assumes the analysts samples randomly from households that differ in their indirect utility functions only by an additive error term. That is, suitable order-preserving transformations put the indirect utility functions into the same form with the same parameter values in the systematic component. The error term is stochastic to the analyst, because the factors in it are unmeasured of imperfectly measured, but it is nonstochastic to the household. following convention, and for empirical tractability, the indirect utility function is u = DX* + R.

R is the stochastic error distributed with zero mean and standard deviation S. X is a vector of known exogenous household characteristics, and D is a vector of coeeficients. The assumption is that elemenst of D are comparable across households. For empirical work the model is normalized by dividing by S if R is distributed normally or by 3S/[pi.sup.2] (pi=3.14159...) if R is distributed as a hyperbolic-secant-square ([sech.sup.2]) variable. (12) The normalized indirect utility function is

v = dX* + r,

which preserves the ordering.

An income adequacy question like that used by Kapsalis provides the data for estimating the ordered response model (hereafter, ORIAM). More favorable responses get higher numerical values, providing an ordinal scale of any household.

Estimation of (11) requires threshold values for utility such that

if v[is less than or equal to][V.sub.1] the household reports inadequate income, if [V.sub.1]<v[is less than or equal to][V.sub.2] the household reports barely adequate income, if [V.sub.2]<v[is less than or equal to][V.sub.3] the household reports fairly adequate income, and if [V.sub.3]<v the household reports adequate income.

The same thresholds for all households is an assumption. This yields a model in which the dependent variable takes on numerical values up to order (Maddala 1983, pp. 46-49).

The ordered-response model does not require cardinal measurement of utility for any household. The utility function (11) can be subjected to any order-preserving transformation without losing any information in the original estimates, as long as the thresholds are subjected to the same transformation. (13) The model requires comparability in that the form and parameters of the systematic component of utility and the thresholds associated with the statements can be made consistent across households. Only in this sense does the model require comparability. Since the stochastic elements of utility can differ across households, there is no implication that households have the same utility (ordinally or cardinally) for any values of the independent variables. If the stochastic element for any household can differ with the values of the independent variables, different households do not necessarily have even the same ranking for two sets of values of the independent variables. The model only requires that the systematic components be comparable across households, a very special and weak comparability.

The analyst can use ordered probit or ordered logit to estimate the model, using all four categories of income adequacy. If the response categories are collapsed to two, estimation can proceed with binary probit or binary logit. Any of these techniques produces an estimate of vector d in (11) and the threshold values. Probit and logit use different assumptions about the distribution of r, with different implications for calculating welfare measures.

In principle, individuals could evaluate hypothetical incomes. Then, there would be data to estimate the model for each individual. The function (11) can be lognormal, so long as it has the properties of a neoclassical indirect utility function. (14) Any reference income, including actual income, can be included in X. Thus, the ordered-response appoach could fully incorporate the LM specifications. The complexity of the model would increase substantially, however, without any compelling rationale.


If X=(1,ln fs, ln Y, A), then (11) is comparable to (1) with (3) substituted for m, A appended to the model, and the steady state condition on income imposed. Differences are that utility is not constrained to the zero-one interval and the lognormal function is replaced by the linear function dX*. The elements of d are comparable to the [b.sub.i]/S coefficients in (4). The ORIAM is analogous to the LM in the steady state.

In the work of van Praag and Van der Sar the double-log form of income is used without theoritical basis. In contrast, using ln Y on the right-hand side of (11) can be justified. Work by psychophysical researchers establishes a power function as a general description of psychological response to a stimulus. Breault (1981; 1983) has shown that utility is a power of income. In log form, this function becomes linear in parameters. Breault asserts that his findings support a diminishing cardinal marginal utility of income. All the ORIAM requires is the general form.

The steady state restriction used to derive welfare measures in the LM is automatically in the ORIAM when households evaluate only their actual incomes. EV and CV are defined for the ORIAM by the usual identities. The [d.sub.i] replace the [b.sub.i]/s of the LM, and stochastic components are added to each side of the identities. Adding the stochastic elements complicates welfare calculations. The new identities contain stochastic elements, while EV and CV must be non-stochastic to be useful. Computational formulas for EV and CV depend on the distribution of r and the interpretation of EV and CV. Here, there are derivations for EV. CV derivations follow the same paths.

It is convenient to assume that r varies across households but not with the values of the variables in the indirect utility function (11) for any household. The derivation of EV follows exactly the same steps as in the LM. Utility with (Y, [A.sup.1]) is equated to utility with (Ye, [A.sup.0]). For ORIAM, this means equating (11) with different values in the X vector. The identity is

[d.sub.0] + [d.sub.1] (ln fs) + [d.sub.2] (ln Ye) + [d.sub.3][A.sup.0] + r * [d.sub.0] + [d.sub.1] (ln fs) + [d.sub.2](ln Y) + [d.sub.3][A.sup.1] + r.

Algebraic manipulation gives an expression exactly equivalent to (6) above. Following the same steps as for (6) gives

e = exp[[d.sub.3]([A.sup.1] - [A.sup.0])/[d.sub.2]],

and EV = Y(1 - e). Factor [d.sub.3] in (13) replaces ([b.sub.4] - [b.sub.3] in (7), and [d.sub.2] replaces (1 - [b.sub.2]). EV is the amount of money that, if taken away from (or given to) the household, would have the same impact on welfare as the change [A.sup.0] to [A.sup.1]. The difference ([A.sup.1] - [A.sup.0]) could be in family size, household composition, housing, or any other exogenous characteristic. Replacing d's with their estimated values and Y and the As with their sample values will give a nonstochastic estimate of EV for each household. This derivation of EV is called M(EV) below.

While the aassumption that r varies across households but not with X yields a convenient formula for EV, this assumption may not be plausible. It also is at odds with the discrete choice literature. Discrete choice analysis represents the most rigorous application of random utility theory and a logical grounding for empirical work with ORIAM.

In discrete choice analysis it is conventional to assume that the random component of utility may differ with a given household's characteristics and/or the choice made. That is, r may differ with X for a given household. The values of the stochastic component across households reflect imperfectly measured differences in tastes and aspects of the budget constraint. There is no reason for the stochastic components to be equal on the two sides of (12). The conventional assumption is that the stochastic components represent different draws from the same distribution.

It is consistent with discrete choice analysis and random utility theory to assume that different values of Y, fs, and A may produce different realizaations of r. Component r is known to the household for each (Y,fs,A), but stochastic to the analyst. A draw of the stochastic component is taken from identical and independent distributions for each combination (Y,fs,A). The stochastic element for the ith combination is [r.sub.i].

The new specification is still amenable to estimation by probit or logit. Factor e, however, becomes

e = exp[[d.sub.3]/[d.sub.2]([A.sup.1] - [A.sup.0] + r*]

where r* = [r.sub.1] - [r.sub.0]. Draw [r.sub.1] corresponds to (Y,[A.sup.1]), and [r.sub.0] corresponds to (Ye,[A.sup.0]). EV is still given by Y(1 - e). This formulation of EV has all of the assumptions of discrete choice analysis and is well grounded in random utility theory. A problem with this formulation is that e is stochastic to the analyst. The analyst can only describe EV and family equivalence sacles in terms of a probability distribution. Such a description is cumbersome and uninformative.

A natural way to get a representative nonstochastic value for EV is to take its expectation across households with the same (Y,fs,A). That is, the stochastic component is averaged out. The expected value of EV is

E(EV) = Y{1 - exp[([d.sub.3]/[d.sub.2])([A.sup.1] - [A.sup.0]]}E


The last factor in (15) is the moment generating function E[exp(wq)] with q = r* and w = 1/[d.sub.2]. If the distribution of r is standard normal, r* will be distributed normally with zero mean variance two. For this case, E[exp(1[d.sub.2])r*] = exp(1/[d.sub.2]). This formula for EV corresponds to probit estimation of (11). If the distribution of r is [sech.sup.2], r* is distributed as the difference between two independent and identically distributed [sech.sup.2] variables. In this case, the last factor in (15) is [[pi cosecant (pi/[d.sub.2])/[d.sub.2]].sub.2].

As the mean of the EV distribution over r*, (15) is sensitive to the tails of the distribution. The median is an alternative measure of central tendency that is not sensitive to extreme values. EV is a strictly decreasing function of r*, so that, for givem Y and [A.sup.1] - [A.sup.0], the median of EV occurs at [r.sup.*] where prob(r* < [r.sup.*]) = .5. That is, the median of EV will occur at the median of r*. Probit estimation implies that r* has a normal distribution with zero median. For probit, the median of EV across households with Y and [A.sup.1] - [A.sup.0] is given by the formula already labeled M(EV). the median of EV implied by logit estimation is not apparent. (15)

There are two other ways of conceptualizing an equivalent variation that give M(EV) as the measure of welfare change, regardless of whether estimation is to be by probit ot logit. First, E(EV) and M(EV) arise from supposing that each hosehold in the distribution for r* for given Y and [A.sup.1] - [A.sup.0] is given a change in money income that confers the same welfare change as the change from [A.sup.0] to [A.sup.1] in the systematic and stochastic components of utility. (16) If the household is given income only for the systematic component, M(EV) gives the amount of compensation for any household. This is the "best guess" of compensation, because the systematic component of utility is the only part that can be computed by the analyst.

The argument for M(EV) above is informal. M(EV) can also be derived as the proper measure of EV from a formal argument that applies equally to the probit and logit formulations. There are three draws of r implied by the ORIAM and computation of EV. Draw [r.sub.0] corresponds to (Ye,[A.sup.0]), and [r.sub.1] corresponds to (Y,[A.sup.1]). The third draw is [r.sub.2], and it corresponds to (Y,[A.sub.0]). This third draw is relevant to two hypothetical discrete choices. The household chooses between (Y,[A.sup.0]) and (Y,[A.sup.1]) or between (Y,[A.sup.0]) and (Ye,[A.sup.0]). M(EV) answers the question, "What dollar amount (Ye) would give equal probabilities of a household drawn at random from the population choosing (Ye,[A.sup.0]) over (Y,[A.sup.0]) and choosing (Y,[A.sup.1]) over (Y,[A.sup.0])?" This rationale for M(EV) subtly shifts the meaning of "equivalent" in "equivalent variation." EV becomes that amount of money that, when substituted for a change in A, gives the same probability of being chosen over a common alternative (Y,A). (17)

It is necessary to deal with complications that arise in the discrete choice literature if random utility theory is to provide a rigorous basis for the ORIAM. One of these complications is that the exact welfare measure for a household is stochastic to the analyst. This follows naturally from the assumption that part of utility is stochastic to the analyst. Policy analysis requires a nonstochastic welfare measure. E(EV) and M(EV) are formulas for nonstochastic measures. E(EV) represents the intuitively appealing notion of of averaging out the stochastic components. This is true whether underlying model is probit or logit, although the exact E(EV) formulas differ. E(EV) is subject to extremes of the distribution giving implausible figures. M(EV) is analogous to the LM formula, which facilitates comparisons between the LM results and ORIAM results. M(EV) is the median of the distribution of exact EVs if the true underlying model is probit. M(EV) also follows from other appealing interpretations of a "representative" welfare measure. The meaning of "equivalent" in equivalent variation differs slightly, depending on the rationale for M(EV). The weight of advantages is with M(EV), and the following exploratory applications use this formula.


A 1978-1979 survey of persons 60 years and older in seven counties of northern California provides data for one application of the ORIAM techniques. Data from a four-state survey of households eligible for food stamps in 1979-1980 are material for another application. The surveys are described elsewhere (Kushman and Lane 1980; Kushman and Freeman 1986; Lane, Kushman, and Ranney 1983), so this paper omits detailed descriptions. The questionnaires and survey techniques were similar to those used in many other surveys from which data would be available for ORIAM. In particular, the survey of older Californians used a standard needs assessment instrument developed by the Administration on Aging of the former Department of Health, Education, and Welfare. Many agencies, nationwide, used the instrument.

Six hundred and thirty-seven older Californians were asked, "How well does the amount of money you have take care of your needs: very well, fairly well, not very well, or not at all?" For binary probit and binary logit, the dependent variable in ORIAM was defined to equal one if the response was "very well" and zero otherwise. The dependent variable for ordered probit was equal three, two, one, and zero in descending order of income adequacy. Ordered logit also could be used for estimation, but the required software was not available.

For the survey of food stamp eligible households, the income adequacy question was:

To what extent do you think your income is enough to live on?

a. can afford about everything and still save money.

b. can afford about everything you want but not some things.

c. can afford what you need and some wants, but not all.

d. can meet necessities only.

e. not at all adequate.

A binary dependent variable was zero for "can meet necessities only" or "not at all adequate" and one if the response was more favorable. For ordered probit, the dependent variable was integers zero through four, with zero corresponding to "not at all adequate." There were 770 responses for the food stamp eligible sample. For both samples, the income adequacy question imposed the steady state by asking respondents to evaluate their actual incomes.

Table 1 contains variable definitions and descriptive statistics. Ordered-response estimates for binary logit, binary probit, and ordered probit are in Table 2 for the older Californians and Table 3 for the food stamp eligible households. All of the estimated equations were statistically significant at less than one percent.

Unless samples are very large with strong influence from observations with extreme values, probit and logit will give similar results. Different coefficients arise from their different normalizations (Maddala 1983, pp. 10-11). The converted logit coefficients in Table 2 and Table 3 show this consistency. Binary probit and ordered probit are based on the same distribution for the stochastic component of utility, so they should yield similar estimates. For the older Californians, the binary and ordered probit results are similar except for the HOMEOWNER coefficient's significance level. For the food stamp eligible households, there is less agreement between the techniques, and the t-statistics suggest more precision in estimating the parameters with binary probit. (18)

The LM and the ORIAM both use introspective data to estimate consumer welfare measures, but they have very different foundations. It is interesting to speculate whether they yield similar equivalence scales. Lacking a single data set on which to estimate both models, it is feasible only to compare results from this study with results of roughly similar studies. The only comparison is equivalence scales for number of household members, a variable consistently defined across data sets. To increase comparability, equivalence scales from the ORIAM results used M(EV). Using the entire samples for estimation in the ORIAM assumes a general m-scale, in the terminology of van Praag and Van de Sar (1988). With larger samples estimation could be on subsamples to obtain different equivalence scales over different family size ranges.

Colasanto, Kapteyn, and van der Gaag (1984) estimated the LM and associated poverty lines for Wisconsin households who generally were low-income. Their sample is crudely comparable to the sample of food stamp eligible households. Colasanto, Kapteyn, and van der Gaag estimated a family equivalence scale of 1.39 for a household twice as large as the standard household. The food stamp sample yielded an equivalence factor of 1.41 (binary probit or binary logit) or 1.44 (ordered probit). The scales agree closely. (19)

Danziger, et al. (1984) estimated equivalence scales for couples with heads 65 years or older versus one-person elderly households. They used the Income Survey Development Program Research Panel of the Social Security Administration from 1979. The older Californians' data used here are from the same period. Danziger, et al., estimated equivalence factors separately for male and female singles. A relatively small sample prohibits dividing the older Californians' data by sex or using only respondents 65 and older.

Based on the older Californians' data, equivalence between an older couple and an older single requires a ratio of incomes of 1.60 (binary logit), 1.57 (binary probit), or 1.41 (ordered probit). The mean of the male-versus-couple and female-versus-couple equivalence factors from Danzinger, et al., is 1.51.

Consistency between the LM equivalence scales and the ORIAM scales for roughly comparable populations is striking. More extensive comparisons and tests for robustness would require a large data set with questions for estimating various versions of each. The present results, however, are encouraging.


From its genesis, the Leyden model emphasized the potential of introspective data on consumer welfare. Such data has powerful appeal. If one wants to know how people are affected by income, family size, environmental changes, or other circumstances, it makes sense to ask them. Their responses have obvious relevance to public policy questions. The Leyden approach was applied by a few analysts to putting values on family sizes, welfare program benefits, and various public works. Economists, however, are trained and conditioned to raise elaborate defenses against such straightforward data. The Leyden approach put the data in a context wholly foreign to mainstream economists. Economists rejected, or at least neglected, the data and the Leyden approach. Continuing work with the original form of the Leyden model pressured the economists' defenses, but it did not breach them.

Recent work by Kapsalis (1981) and van Praag and Van der Sar (1988) compromises the technical context of the Leyden model to take up using introspective data on ground closer to the neoclassical economist's base. The ordered response income adequacy model presented here is an approach from the other side. It begins with tools that are familiar to the neoclassical economist and incorporates introspective data.

Linking the ordered response model to random utility theory provides rigor and consistency in developing the model. Stochastic elements enter estimation and welfare calculations naturally and consistently. Two formulas for welfare measures emerge. Each formula has one or more precise, rigorous interpretations. Each formula is linked precisely to an estimation technique. One formula is simpler, has several appealing interpretations, and is closely analogous to the Leyden model formula.

The ordered-response model is adaptable to alternative (for instance, nonlinear) indirect utility functions and to alternative assumptions about the stochastic element of utility. These alternative specifications probably would make the model more complex in estimation and computation of welfare measures. The ordered response model for a single individual may be estimated. Then, the model could yield estimates of preference drift. This flexibility would require using questions about hypothetical incomes.

Developing the ordered-response model does not require reference to the Leyden model. Making the connections unifies work on subjective evaluation of income and allows for accumulation of empirical findings from different approaches. The crude comparisons of empirical results presented here suggest that various approaches will yield consistent estimates of welfare measures. This consistency would be powerful in establishing a place for introspective data in policy analysis.

The ideal next step is experimentation with the Leyden and ordered-response models on a data set that would support both. It would be possible to test for robustness of family equivalence scales and other empirical results across models, estimation techniques, and socioeconomic groups. Investigating robustness to different measures of reference income ([Y.sub.a]) and to the use of questions about hypothetical incomes might reveal more about the nature and significance of preference drift.

Evidence from the Leyden model to date suggests that a preference drift exists. This poses a problem for testing the neoclassical economic theory of consumer behavior. When preferences shift with changes in the budget constraint, the comparative static implications of neoclassical economic theory do not hold. New testable implications do not emerge. It is generally impossible to discover preference drift from the market transactions data on which neoclassical analysis typically relies (Basmann, Molina, and Slottje 1983a, 1983b; Pollak 1977). Using subjective, introspective data and the ordered-response model allows testing for preference drift within the neoclassical framework. The ordered-response model permits a test of the fixed-preference foundation for neoclassical economics of consumer behavior. Implications for positive neoclassical consumer economics of rejecting the fixed-preference axiom go far beyond what this paper can explore.

The steady state equivalent and compensating variations derived here are applicable in all the public policy analyses that currently use welfare tests like those of Hicks, Kaldor, and Scitovsky and combinations and refinements of them (for example, Ng 1984). Any analysis that currently uses a variation of consumer surplus could use the steady state welfare measures instead. Allowing rigorous analysis of introspective data opens up new sources of data. This type of data is especially important when public policy involves goods that are not traded in competitive markets (environmental externalities, illegal goods). In these cases, market prices and quantities that are the usual basis of empirical work are not available. Hypothetical questions can generate the relevant data, but it will be subjective and introspective.

This paper does not address the philosophical issue of how much economic progress is possible when there is preference drift or how overall progress can be measured. Neoclassical economists have long recognized that, if preferences shift, the choice of any base preferences for welfare calculations is arbitrary. That choice affects any welfare measures (Samuelson and Swamy 1974, pp. 585-587). If tests with the ordered-response model confirm the Leyden findings of preference drift, all public policy analysts will have to give more attention to the few thinkers who have addressed welfare in relative rather than absolutist terms (for instance, Sen 1983). This paper does not pursue these highly abstract and value-laden questions. It does foreshadow their ultimate importance.

(1) For detailed reviews and citations to empirical experiments, elaborations, and applications of the Leyden Model, see Hagenaars and van Praag (1985); Hartog (1988); Kapteyn, Kooreman, and Willemse (1988); Kapteyn and Wansbeek (1985); van Praag and Van der Sar (1988), and sources cited therein.

(2) The objectionable assumptions include cardionality of the Leyden utility numbers, a lognormal utility function, and boundedness of the utility numbers.

(3) The cumulative normal distribution function as used here does not describe the distribution of a random variable. Therefore, the parameter m and the inflection point do not represent a mean or median.

(4) The Leyden individual utility function is arguably purely cardinal, in Morey's terminology (1984, p. 166, footnote 7). the ratios of the utility numbers may indicate relative proportions of the difference between the worst possible and best possible states attained by the individual. This implies that the worst possible state is a meaningful zero.

(5) The lower income bound of the highest category is reduntant with the upper bound of the next highest of the nine categories.

(6) Estimation of the model parameters does not invoke the steady state, but computaion of welfare measure usually does.

(7) Using Pollak's terminology emphasizes analogies welfare measures derived in the neoclassical approach and the LM. His terminology also is not associated with any particular interpretation of preference drift. The LM literature uses the steady state to derive "true" or "short term" family equivalence scales (Kapteyn and van Praag 1976, pp. 325-326). The short-term LM equivalent scales concern welfare actually experienced when compensation is given. This seems a close parallel to neoclassical equivalent and compensating variations in which (conceptually) the household receives compensation. In the LM "Long run" no compensation is given, and the family adjsuts its expectations. Long run welfare measures for the ordered response model developed below do not impose the steady condition on incomes. Obvious changes in the derivations will give the long-run measures.

(8) The steps below can be used to derive welfare measures for any functional form. The derivation will reveal whether a close form is possible.

(9) For instance, in Colasanto, Kapteyn, an van der Gaag (1984, pp. 131-132), error terms are appended where they were needed for estimation and suppressed to derive welfare measures.

(10) Factory apparently was inadvertently omitted in the printed article. Kapsalis noted that the logs of fs and y are used in the LM literature, but he used the nonlog forms for simplicity.

(11) Goedhart, et al. (1977) is the immediate predecessor to Kapsalis. the model proposed there is a bridge between the LM and Kapsalis' approach, but discussing it her is not necessary.

(12) It is possible to derive the qualitative response model from random utility theory without assuming linearity in parameters. The normalization is different for nonlinear models, and the literature on discrete choice and qualitative response emphasize the linear case. Maddala (1983, pp. 9-11) describe alternatives to the normal and sec[h.sup.2] distributions.

(13) Interpersonal comparability of utility functions and utility measurement, which are required for cross section estimation of (11), may seem to imply that utility is measured cardinally. this is not the case. Individuals are required to associate the same statement about income adequacy with the same number on each person's utility scale. It does not follow that all individuals associate the same intensity of feeling with the statements not that the intervals between utility numbers imply anything more than order about the intensity of feeling. Utility must be measured comparably but not necessary cardinally.

(14) Estimatting the model for an individual from responses to various hypothetical levels of income requires r to differ with income for the same household. This is fully consistent with random utility theory. It would rule out some derivations of welfare measures below that assume r varies across household but not with the circumstances of any one household. Applications f theLM usually suppress prices. The applications reported below take prices into account in keeping with the neoclassical indirect utility function. A lognormal utility function would require nonlinearity in the parameters and a suitable estimation technique and would change the welfare measure formulas. These variations are beyond the scope of this paper.

(15) If r' has the sec[h.sup.2] distribution, the median would be where r' = 0. For the ORIAM, however, r' is the different between two random variables, each of which has a sec[h.sup.2] distribution. The distribution of a variable that is the difference between two sec[h.sup.2] variables is not known.

(16) As before, "systematic" and "stochastic" refer to the components as known or observered by the Analyst. Utility is entirely nonstochastic to the household.

(17) It may seem that the choice between (Y,[A.sup.0] and (Ye,[A.sup.0] would always be resolved in favor of the larger amount of money and the choice between (Y,[A.sup.1] would always be resolved in favor of the larger amount of A (assuming A is a "good"). This is not true in general. The stochastic component need not have the same realization for a given household, even with regard to sign, for different (Y,A) combinations. The analyst can only describe any choice probabilistically. If the stochastic component does have the same realization for all (Y,A) combinations, it cancels out, and M(EV) arises as discussed above.

(18) The results might be made more alike by collapsing the ordered responses to three or four categories rather than five. There was no further experimentation in this study.

(19) For example, in the binayr logit estimates, [d.sub.1] = .385, and [d.sub.2] = .779. The equivalence scale for family size is derived with [d.sub.1] replacing [d.sub.3] in the formula for facotr e. That is, fs is playing the part of the more general variable A. If the standard family has size fs, one twice as large has size 2fs. Then, for a doubling of family size, e = exp{[d.sub.1[ln(2fs) - ln(fs)]/[S.sub.2}. because ln(2fs) = ln(2) + ln(fs), = 1n(2) + 1n(fs), the binary logit factor is e = exp[(- .385) (1n 2)/.779] = .71. The equivalence scale is given by Y/Ye = Y/ .71Y = 1.14.


Basmann, R. L., D. J. Molina, and D. J. Slottje (1983a), "budget Constraint Prices as Preference Changing Parameters of Generalized Fechner-Thurstone Direct Utility Functions," American Economic Review, 73 (3) (June): 411-413.

Basmann, R. L., D. J. Molina, and D. J. Slottje (1983b), "Variable Consumer Preferences, Economic Inequality, and Cost-of-Living: Part One," Paper presented to the 96th Annual Meeting of the American Economic Association. San Francisco, CA (December 18-20).

Breault, K. D. (1981), "The Modern Psychophysical Measurement of Cardinal Utility: A Return to Introspective Cardinality?" Social Science Quarterly, 62 (4) (December): 672-684.

Breault, K. D. (1983), "Psychophysical Measurement and the Validity of the Modern Economic Approach: A Presentation of Methods and Preliminary Experiments," Social Science Research, 12 (2) (June): 187-203.

Colasanto, Diane, Arie Kapteyn, and Jacques van der Gaag (1984), "Two Subjecive Definitions of Poverty: Results From the Wisconsin Basic Needs Study," Journal of Human Resources, 19 (1) (Winter): 127-138.

Danzieger, Sheldon, et al. (1984), "The Direct Measurement of Welfare Levels: How much Does It Take to Make Ends Meet?" Review of Economics and Statistics, 66(3) (Augugst): 500-505.

Goedhart, Theo, et al. (1977), "The Poverty Line: Concept and Measurement," Journal of Human Resources, 12 (4) (Fall): 482-502.

Haenaars, Aldi J. M. (1986), The Perception of Poverty, Amsterdam: North Holland.

HAgernaars, Aldi J. and Bernard M.S. van Praag (1985), "A Synthesis of Poverty Line Definitions," Review of Income and Wealth, 31(2)(June): 139-154.

Harto, Joop (1988), "Poverty and the Measurement of Individual Welfare: A Review of A. J. M. Hagenaar's The Perception of Poverty," Journal of Human Resources, 23 (2) (Spring): 243-266.

Kapsalis, Constantine (1981), "Poverty Lines: An Alternative Method of Estimation," Journal of Human Resources, 16 (3) (Summer): 477-480.

Kapteyn, Arie, Tom Wansbeek, and Jeannine Buyze (1980). "The Dynamics of Preferences Formation," Journal of Economic Behavior and Organization, 1 (2) (June): 123-157.

Kapteyn, Arie and Barnard van Praag (1976), "A New Approach to the Construction of Family Equivalent Scales," European Economic Review, 7 (4) (May): 313-335.

Kapteyn, Arie, peter Kooreman, and Rob Willemse (1988), "Some Methodological Issues in the Implementation of Subjective Poverty Definitions," Journal of Human Resources, 23 (2) (Spring): 222-242.

Kapteyn, Arie and Tom Wansbeek (1985), "The Individual Welfare Function," Journal of Economic Psychology, 6 (4) (December): 333-363.

Kushman, John E. and Beth Freeman (1986), "Service Consciousness and Service Knowledge Among Older Americans," International Journal of Aging and Human Development, 23 (3): 217-237.

Kushman, John E. and Sylvia Lane (1980), "A Multivariate Analysis of Factors Affecting Perceived Life Satisfaction and Psychological Well-Being the Elderly," Social Science Quarterly, 61 (2) (September): 264-277.

Lane, Sylvia, John E. Kusman, and Christine K. Ranney (1983), "Food Stamp Program Participation: An Exploratory Analysis," Western Journal of Agricultural Economics, 8 (1)(July): 3-26.

Maddala, G. S. (1983), Limited-Dependent and Qualitative Variables in Econometrics, Cambridge MA: Cambridge University Press.

Morey, Edward R. (1984), "Confuser Surplus," American Economic Review, 74 (1) (March): 163-173.

Ng, Yew-Kwang (1984), "Quasi-Pareto Social Improvements," American Economic Review, 74 (5) (Decemnber): 1035-1050.

Pollak, Robert A. (1977), "Price Dependent Preferences." American Economic Review, 67 (2) (March): 64-75.

Samuelson, P. A. and S. Swamy (1974), "Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis," American Economic Review, 64 (4) (September): 566-593.

Sen, Amartya (1983), "Poor, Relatively Speaking," Oxford Economic Papers, 35 (2) (July): 153-169.

Van Herwaarden, Floor G. and Arie Kapteyn (1981), "Empirical Comparison of the Shape of Welfare Functions," European Economic Review, 15 (2) (March): 261-286.

van Praag, Bernard M. S. (1971)), "The Welfare Function of Income in Belgium: An Empirical Investigation," European Economic Review, 2 (3) (Spring): 337-369.

van Praag, Bernard M. S. and Nico L. Van der Sar (1988), "Household Cost Functions and Equivalence Scales," Journal of Human Resources, 23 (2) (Spring): 193-210.

Varian, Hal. R. (1984), Microeconomic Analysis, 2nd Edition, New York, NY: W. W. Norton.

John E. Kushman is a Professor and the Chair of the Department of Textiles, Design, and Consumer Economics at the University of Delaware, Newark, DE, and Christine K. Ranney is an Association Professor in the Department of Agricultural Economics at Cornell University, Center Associate, Western Rural Development Center, Oregon State Univesity, Corvallis, OR.
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Author:Kushman, John E.; Ranney, Christine K.
Publication:Journal of Consumer Affairs
Date:Dec 22, 1990
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