Ordinal Utility Subjective Poverty Line and Equivalence Scales1
Andrew Grodner
East Carolina University, Greenville, NC 27858-4353 USA
Tel: 252 558-3040. E-mail: [email protected]
Rafael Salas
Universidad Complutense de Madrid, Campus de Somosaguas 28223 Madrid (Spain)
Tel: +34 91 3942512. E-mail: [email protected]
Version 20120314.6:25 EST – edited abstract and introduction
ABSTRACT
Most subjective poverty lines and subjective equivalence scales have been rationalized in the literature by
assuming a cardinal utility framework, an approach which greatly restricts their use within the standard
microeconomic theory. However, we show that an approach based on Minimum Income Needs Question (MINQ) is
consistent with an ordinal utility setting. It requires two basic assumptions: (i) the minimum income perception which
assumes systematic perception errors that can be isolated by the interception method, and (ii) equivalent utility which
assumes that households at the true minimum income have the same utility level across various demographic
characteristics. Neither assumption imposes any additional limitations on the underlying utility function. The
subjective equivalence scales based on MINQ can aide identification of the demand systems without loss of generality
or without putting any additional restrictions on either theoretical model or estimation.
KEY WORDS: Poverty, Equivalence Scales, Subjective Poverty Line, Subjective Method
JEL CODES: D30 and D63.
1
We are grateful for helpful comments from J. Bishop. This paper has benefited from the financial support from the Spanish
Ministry of Science and Technology (ECO2010-17590). The usual disclaimer applies.
1. INTRODUCTION
The proper derivation and use of equivalent scales is a key element in conducting an appropriate
analysis of inequality, poverty, or welfare in a particular society. Well defined equivalence scales can
adjust for economies of scale within the family and thus allow for cross-household and cross-person
comparisons (Coutler at al., 1992). Unfortunately, theoretically sound demand systems equivalence
scales suffer from identification problems (Blundell and Lewbel 1991 IN REF), and empirically
convincing subjective scales are often criticized for their unrealistic assumptions about the underlying
utility function (Seidl Christian 1994 IN REF). In this paper we show that a subclass of subjective
equivalence scales derived from Minimum Income Needs Question (MINQ) are in fact independent of
the underlying utility function, making them an attractive alternative to a researcher looking for
theoretically sound yet empirically robust approach to derive equivalence scales.
Subjective equivalent scale was first proposed by Kapteyn and Van Praag (1976, p. 319). The
approach is a by-product of Leyden Poverty Line (LPL) derivation which uses information from a
particular survey question, Income Evaluation Question (IEQ). The limitation of the approach is due to
the assumption that utility function is described by the Welfare Function of Income (WFI), which
follows log-normal Cumulative Distribution Function (see Hartog 1988 for discussion and critique IN
REF). However, there are other approaches to calculate subjective equivalent scales such as methods
based on Minimum Income Question (MINQ) (Goethard et al 1977), attitude questions (Deleeck et al
1984), measures of income satisfaction (Schwarze 2003, Review of Income and Wealth 49(3)),
consumption adequacy (Pradhan and Ravallion 2000, The Review of Economics and Statistics 82(3),
462-471), subjective evaluation question (Stewart 2009), etc. The limitation of the current literature is
that even though all the methods have the same premise, ie. that households are themselves the best
judges of their own situation, the theoretical foundations have not been studied except for the approach
based on IEQ.
1
The motivation for our study comes from Hartog’s observation that in deriving subjective poverty
line (SPL) using MINQ one only needs to assume that “people (…) associate a certain common,
interpersonally comparable feeling of welfare with verbal description of the minimum level (‘enough to
get along’)” (Hartog 1988, p. 255, citing Hagenaars 1986). Moreover, he adds that in “(SPL)
application of Leyden methodology, the cardinality of the welfare function is not required”. Thus, it
implies that Engel curves can in fact take many analytical forms and demand function for consumer
goods are not confined to any particular shape (Hartog 1988, p. 252). As a result, the subjective
equivalence scales based on MINQ can aide identification of the demand systems without loss of
generality or without putting any additional restrictions on either theoretical model or estimation
(Blundell and Lewbel 1991, p. 57). (footnote: In other words, MINQ-based equivalence scales can
share the same theoretical foundation as the demand based specifications. An illustrative demonstration
using equivalence scales based on consumption adequacy is Ree at al (2010).)
Our objective is to formally expand on Hartog’s argument by presenting two assumptions required
for MINQ and then discussing their implications for the underlying WFI. First, we introduce (i) the
assumption of "Minimum Needs Perception", which says that except for households whose incomes are
exactly equal to the poverty level all other respondents have systematic differences in perception when
answering MINQ. Second, we introduce (ii) "Equivalent Utility" assumption, which states that all those
households whose incomes equal the poverty line will have the same utility level across all
demographic groups. We show that both assumptions are sufficient for consistent derivation of MINQ
subjective equivalence scales. Then, we study implications of assumptions (i) and (ii) on the on the
WFI by explicitly modeling perception function (Kapteyn Arie, Peter Kooreman, and Rob Willemse.
1988, p. 224, IN REF) and deriving the perceived income from MINQ. The perception function
determines where individuals think the true minimum needs income is relative to their individual
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income. The perceived income is used as the argument in the WFI and is equivalent to answers in IEQ
used to derive Leyden Poverty Line.
The result of the paper is that when using MINQ one neither needs to introduce WFI, nor make
any particular assumptions about it. Thus, subjective scales based on MINQ can be regarded as
theoretically sound and robust to most possible utility functions, and yet empirically viable and
identifiable. We recommend using MINQ over other subjective scale approaches for that reason.
2. SUBJECTIVE POVERTY LINE AND THE INTERSECTION METHOD
The derivation of the poverty line using subjective questions was first proposed by Goedhart et al.
(1977). They introduce two approaches: Leyden Poverty Line (LPL) based on based on multi-level
question, and Subjective Poverty Line (SPL) based on a one-level question. Even though Flik and Van
Praag (1991) argue that LPL is theoretically superior to the SPL, we argue that SPL is still preferred
because it is less restrictive and can easily be combined with in any other study which requires
specification of the utility function of income.
The SPL is based on the answer to the minimum needs question (MINQ): "what is the minimum
income that you would have to have to make ends meet? Even though for a given group of individuals
the answers to MINQ vary due to "misperception", Goedhart et al. (1977) argued that there is a
systematic relationship between answers to MINQ and an actual income. We further call it perception
error. In particular, those with actual income above their stated minimum income overestimate the
poverty threshold, whereas those with actual income below their stated minimum income
underestimate the poverty threshold. Therefore, according to their argument only those whose actual
income equals minimum needs income would answer MINQ correctly, which in turn becomes a
definition of the poverty line. The intuition is that at this level of income the household does not have
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any saving nor debt during the survey period, but any shortfall from this level would push them into
poverty.
Unfortunately because most samples do not include individuals whose actual income equals the
answer to MINQ, or their numbers in the population may be very limited, it is not possible to directly
use answers to MINQ in formulation of the subjective poverty line. However, Goedhart et al. (1977)
shows that by modeling the perception error in MINQ one can use data from the entire survey and to
estimate correct answer to MINQ. The objective of their approach is to find such actual income level
where individuals would not make a perception error when reporting MINQ. The approach is called the
intersection method because graphically the poverty line is at the point of intersection between the line
representing actual answers to MINQ conditional on actual income, and the line representing
hypothetical answers to MINQ as if they would always equal to the actual income (45 degree line). A
formal presentation of the intersection method follows.
2.1. Definitions
Let us assume that the population can be partitioned into 𝑛 exclusive groups of households
according to their characteristics 𝑧, where 𝑧 𝜖𝑍 and 𝑍 = {𝑧1 , 𝑧2 , … , 𝑧𝑛 }. For convenience let us order the
characteristics in ascending order, from more less needed to more needed groups (ex. household size).
We denote the total actual income of the household by 𝑦 𝜖 ℝ+ .
Suppose that for all households in the population with given characteristics 𝑧, there exists unique
∗
(𝑧). (footnote:
true minimum needs income (poverty line) which is unobservable, defined as 𝑦𝑚𝑖𝑛
∗
(𝑧) is the focus of the intersection method). What is observable are the
Deriving correct level of 𝑦𝑚𝑖𝑛
answers to MINQ which are “distorted by the fact that (respondent’s) actual income is not equal to his
minimum income” (Goedhard et al 1977, p. 514). Thus, to represent actual answers to MINQ we define
minimum needs income perception function (IPF), 𝑓𝑝 (𝑦; 𝑧), 𝑓𝑝 : ℝ+ x 𝑍 → ℝ, which depends on actual
4
income 𝑦 and characteristics 𝑧, where
𝜕𝑓𝑝 (𝑦;𝑧)
𝜕𝑦
∗
(𝑧) is the
> 0. The difference between IPF and 𝑦𝑚𝑖𝑛
∗
(𝑧) < 0, and nonperception error which is systematic, such that poor individuals have 𝑓𝑝 (𝑦; 𝑧) − 𝑦𝑚𝑖𝑛
∗
(𝑧) > 0. Therefore, we can define perceived income, which is
poor individuals have 𝑓𝑝 (𝑦; 𝑧) − 𝑦𝑚𝑖𝑛
conceptually equivalent to the answers of IEQ when deriving LPL, as the difference between actual
income and the perception error:
∗
(𝑧))
𝑦 𝑝 = 𝑦 − (𝑓𝑝 (𝑦; 𝑧) − 𝑦𝑚𝑖𝑛
It has to be noted that the perception error is different than random error, which is present when
estimating IPF using survey data. If we denote 𝑦𝑚𝑖𝑛 as the reported answer to MINQ in survey data, the
random error is the difference: 𝑦𝑚𝑖𝑛 − 𝑓𝑝 (𝑦; 𝑧). Therefore, the full decomposition of the errors
becomes (see Figure 1):
∗
∗
(𝑧) = [𝑓𝑝 (𝑦; 𝑧) − 𝑦𝑚𝑖𝑛
(𝑧)] + [𝑦𝑚𝑖𝑛 − 𝑓𝑝 (𝑦; 𝑧)]
𝑦𝑚𝑖𝑛 − 𝑦𝑚𝑖𝑛
total reported error = perception (systematic) error + random error
(7)
Throughout the paper we ignore random error (assume it is equal to zero) because it is dataspecific and the derivation of SPL does not depend on the properties of the random error. In other
words, modeling random error is the subject of econometric specification which is not the subject of
the present paper.
Finally, let us define household utility, 𝑈: ℝ+ x 𝑍 → ℝ, whose typical image 𝑈(𝑦 𝑝 ; 𝑧) indicates the
utility associated with a household with perceived income 𝑦 𝑝 and characteristics 𝑧 with the property
that ∆𝑈⁄∆𝑦 𝑝 >0 and ∆𝑈⁄∆𝑧 <0, where z is ordered in a decreasing manner according to needs. We
can alternatively define utility in terms of actual income V(𝑦; 𝑧), where 𝑉: ℝ+ x 𝑍 → ℝ, with
∆𝑉⁄∆𝑦 >0 and ∆𝑉⁄∆𝑧 <0.
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2.2. Intersection Method
The objective of the intersection method is to use answers to MINQ in order to find the unobserved
∗
(𝑧). The ingenuity of the approach is that we can find such income even
minimum needs income, 𝑦𝑚𝑖𝑛
∗
(𝑧) (otherwise the solution if obvious).
if there are no households in the data for whom 𝑦 = 𝑦𝑚𝑖𝑛
Following the specification in Kapteyn, Kooreman, and Willemse (1988), the method is based on the
∗
(𝑧) is the solution to:
existence of IPF such that 𝑦𝑚𝑖𝑛
∗
∗
(𝑧) = 𝑓𝑝 (𝑦𝑚𝑖𝑛
(𝑧); 𝑧)
𝑦𝑚𝑖𝑛
∗
(𝑧)
where for 𝑦 < 𝑦𝑚𝑖𝑛
(3)
∗
(𝑧)
and for 𝑦 > 𝑦𝑚𝑖𝑛
we have that 𝑦 >
∗
(𝑧)
𝑦𝑚𝑖𝑛 (𝑧) . Because there is no random error it means that for 𝑦 < 𝑦𝑚𝑖𝑛
we have that 𝑦 <
we have that 𝑦 < 𝑦𝑚𝑖𝑛 (𝑧)
∗
(𝑧) we have that 𝑦 > 𝑓𝑝 (𝑦; 𝑧) . Solution to (3) can be presented in figure
𝑓𝑝 (𝑦; 𝑧) and for 𝑦 > 𝑦𝑚𝑖𝑛
1, where vertical axis represents values of 𝑓𝑝 (𝑦; 𝑧) (assumed linear for demonstration purposes)
conditional on actual income 𝑦 (horizontal axis). The 45 degree line includes all the points where the
∗
(𝑧) is satisfied. Therefore intersection of 45 degree line with the function 𝑓𝑝 (𝑦; 𝑧)
condition 𝑦 = 𝑦𝑚𝑖𝑛
is the solution to the problem in (3).
In practice, because of existence of random error, one needs econometric model to estimate IPF.
However, the unique feature of SPL is that the functional form of IPF is irrelevant as long as it is
monotonically increasing in both y, and its distance from actual income is always smaller than the
∗
(𝑧)|. We
distance between actual income and true minimum needs income, |𝑦 − 𝑓𝑝 (𝑦; 𝑧)| < |𝑦 − 𝑦𝑚𝑖𝑛
want to note that it is a very important distinction between SPL and LPL because the function form of
IPF in LPL is the double-log specification which is derived from the assumed functional form of WFI.
2.4. Equivalence scales
The subjective equivalence scales based on SPL and LPL compare households with different
characteristics z whose incomes equal the subjective poverty line. Thus, we try to find such level of
6
compensation so that they would feel equally poor regardless of their different characteristics z. The
approach is unique in that there is no flexibility as to what level of utility is picked for comparison (see
equivalence scale derived from demand system). In our framework, the subjective equivalence scale is
described by a function 𝑔(𝑧𝑖 , 𝑧𝑟 ) of household type 𝑧𝑖 with respect to household type 𝑧𝑟 :
∗
(𝑧𝑖 )
𝑦𝑚𝑖𝑛
𝑔(𝑧𝑖 , 𝑧𝑟 ) = ∗
𝑦𝑚𝑖𝑛 (𝑧𝑟 )
where 𝑦∗𝑚𝑖𝑛 (𝑧𝑟 ) is the benchmark level of income for single person household whose income equals the
poverty line. Notice that function 𝑔(𝑧𝑖 , 𝑧𝑟 ) depends on 𝑓𝑝 (𝑦; 𝑧𝑖 ) and 𝑓𝑝 (𝑦; 𝑧𝑟 ) and therefore in LPL it
has a predetermined functional form, whereas in SPL there is a range of possible functional forms.
3. ASSUMPTIONS UNDERLYING MINQ AND IMPLICATIONS FOR UTILITY FUNCTION
In the following we present underlying assumptions for the SPL which are implied by the methods
originally presented in Goedhart et al. (1977). The focus is on demonstrating that those assumptions do
not restrict the underlying utility function to any particular functional form and thus ordinal utility
specification is acceptable.
The first assumption deals with the properties of the IPF:
Assumption 1: Minimum Income Perception. For a given 𝑦𝜖ℝ+ and 𝑧𝜖𝑍:
∗
∗
(𝑧) ⇔ 𝑦 ≥ 𝑓𝑝 (𝑦; 𝑧) > 𝑦𝑚𝑖𝑛
(𝑧)
𝑦 > 𝑦𝑚𝑖𝑛
and
(4a)
∗
∗
(𝑧) ⇔ 𝑦 ≤ 𝑓𝑝 (𝑦; 𝑧) < 𝑦𝑚𝑖𝑛
(𝑧)
𝑦 < 𝑦𝑚𝑖𝑛
(4b)
(The ⇐ is obvious and can be omitted). The interpretation of assumption 1 is that even though
income perception function can take very flexible functional forms, it is has to be restricted by
condition which will guarantee unique solution to (3). Graphically it means that there cannot be
∗
multiple intersection points. The intuition is that households whose 𝑦 is lower than 𝑦𝑚𝑖𝑛
cannot
7
∗
perceive MINQ lower than their 𝑦, and households whose 𝑦 is higher than 𝑦𝑚𝑖𝑛
cannot perceive MINQ
higher than their 𝑦 (and the reverse). In other words, poor households at the very least must be aware
that they are poor even though they may not be able to evaluate the degree of their poverty.
An interesting sufficient condition for conditions (4a) and (4b) to be satisfied is that 0 <
Critical consequence of assumption 1 is that under
𝜕𝑓𝑝 (𝑦;𝑧)
𝜕𝑦
𝜕𝑓𝑝 (𝑦;𝑧)
𝜕𝑦
≤ 1.
≤ 1, 𝑦1𝑝 is a strictly increasing function
of 𝑦 for the same z. Therefore 𝑈(𝑦 𝑝 ; 𝑧) and V(𝑦; 𝑧) are ordinal equivalent given any 𝑦1 and 𝑦2 :
𝑈(𝑦1𝑝 ; 𝑧) ≥ 𝑈(𝑦2𝑝 ; 𝑧) ⇔ 𝑉(𝑦1 ; 𝑧) ≥ 𝑉(𝑦2 ; 𝑧). In other words, the existence of perception function does
not impose additional restrictions on individual utility whether the argument is perceived income or
actual income.
The second assumption allows results of the intersection method to be used in derivation of
subjective equivalence scales because it guarantees equal utility levels for different levels of z:
Assumption 2: Equivalent utility.2
Given any 𝑦1 𝑦2 𝜖ℝ+ and 𝑧1 , 𝑧2 𝜖𝑍
∗
∗
(𝑧1 ) ∧ 𝑦2𝑝 (𝑦2 ; 𝑧2 ) = 𝑦𝑚𝑖𝑛
(𝑧2 ) ⇒ 𝑈(𝑦1𝑝 (𝑦1 ; 𝑧1 ); 𝑧1 ) = 𝑈(𝑦2𝑝 (𝑦2 ; 𝑧2 ); 𝑧2 )
𝑦1𝑝 (𝑦1 ; 𝑧1 ) = 𝑦𝑚𝑖𝑛
We can interpret assumption 2 as requiring households whose incomes are equal to poverty line to
have the same utility level. In fact, it is the premise of the generally defined equivalence scale.
However, in SPL we need to be able to make that claim for only one utility level.
∗
∗
(𝑧1 ) ∧ 𝑦2 = 𝑦𝑚𝑖𝑛
(𝑧2 ) ⇒ 𝑈(𝑦1 ; 𝑧1 ) = 𝑈(𝑦2 ; 𝑧2 ), since at
Formally, it can be stated as 𝑦1 = 𝑦𝑚𝑖𝑛
∗
∗
𝑝
𝑦𝑚𝑖𝑛 (𝑧), 𝑦 (𝑦; 𝑧) = 𝑦𝑚𝑖𝑛 (𝑧) = 𝑦, ∀𝑦, 𝑧. Alternatively and in connection with the V function,
∗
∗
(𝑧1 ) ∧ 𝑦2 = 𝑦𝑚𝑖𝑛
(𝑧2 ) ⇒ 𝑉(𝑦1 ; 𝑧1 ) = 𝑉(𝑦2 ; 𝑧2 ).
assumption 2 can be restated as 𝑦1 = 𝑦𝑚𝑖𝑛
2
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4. IMPLICATIONS TO SUBJECTIVE POVERTY LINES AND EQUIVALENCE SCALES
A first implication of Assumptions 1 and 2 is that utility is also equalized in terms of actual
income, at the minimum value:
∗
∗
(𝑧1 ) ∧ 𝑦2 = 𝑦𝑚𝑖𝑛
(𝑧2 ) ⇒ 𝑈(𝑦1 ; 𝑧1 ) = 𝑈(𝑦2 ; 𝑧2 ).
𝑦1 = 𝑦𝑚𝑖𝑛
A second implication of Assumptions 1 and 2 is the following. By assumption 1 we obtain:
∗
∗
(𝑧1 ) = 𝑦1𝑝 (𝑦1 ; 𝑧1 ) ⇔ 𝑈(𝑦1𝑝 (𝑦1 ; 𝑧1 ); 𝑧1 ) = 𝑈(𝑦𝑚𝑖𝑛
(𝑧1 ); 𝑧1 )
𝑦1 = 𝑦𝑚𝑖𝑛
𝑝
𝑝
∗
∗
𝑦2 = 𝑦𝑚𝑖𝑛 (𝑧2 ) = 𝑦2 (𝑦2 ; 𝑧2 ) ⇔ 𝑈(𝑦2 (𝑦2 ; 𝑧2 ); 𝑧2 ) = 𝑈(𝑦𝑚𝑖𝑛 (𝑧2 ); 𝑧2 )
And by assumption 2 we conclude that:
𝑝
𝑝
∗
∗
(𝑧1 ); 𝑧1 ) = 𝑈(𝑦𝑚𝑖𝑛
(𝑧2 ); 𝑧2 ),
𝑈(𝑦1 (𝑦1 ; 𝑧1 ); 𝑧1 ) = 𝑈(𝑦2 (𝑦2 ; 𝑧2 ); 𝑧2 ) = 𝑈(𝑦𝑚𝑖𝑛
which obviously implies 𝑦∗𝑚𝑖𝑛 (𝑧1 ) < 𝑦∗𝑚𝑖𝑛 (𝑧2 ) due to ∆𝑈⁄∆𝑦 𝑝 >0 and ∆𝑈⁄∆𝑧 <0. We get higher
subjective poverty lines for more needed households. Implicit equivalent scale for family 𝑧2 with
respect to 𝑧1 at the poverty threshold is just 𝑦∗𝑚𝑖𝑛 (𝑧2 )/𝑦∗𝑚𝑖𝑛 (𝑧1 ).3
Same conclusion follows from starting from the function V(𝑦; 𝑧) under alternative assumption
proposed in footnote 2.
∗
∗
(𝑧1 ) ⇔ 𝑉(𝑦1 ; 𝑧1 ) = 𝑉(𝑦𝑚𝑖𝑛
(𝑧1 ); 𝑧1 )
𝑦1 = 𝑦𝑚𝑖𝑛
∗
∗
𝑦2 = 𝑦𝑚𝑖𝑛 (𝑧2 ) ⇔ 𝑉(𝑦2 ; 𝑧2 ) = 𝑉(𝑦𝑚𝑖𝑛 (𝑧2 ); 𝑧2 )
∗
∗
(𝑧1 ); 𝑧1 ) = 𝑉(𝑦𝑚𝑖𝑛
(𝑧2 ); 𝑧2 )
𝑉(𝑦1 ; 𝑧1 ) = 𝑉(𝑦2 ; 𝑧2 ) = 𝑉(𝑦𝑚𝑖𝑛
3
∗
∗
(𝑧1 ) < 𝑦𝑚𝑖𝑛
(𝑧2 ) due to ∆𝑉⁄∆𝑦 >0and ∆𝑉⁄∆𝑧 <0.
which obviously implies 𝑦𝑚𝑖𝑛
9
[Figure 2 about here]
5. CONCLUSION
We have rationalized the use of subjective poverty line and implicit subjective equivalent scales
using an ordinal utility framework. Two weak assumptions are needed. The first one concerns the
existence of systematic perception error that households commit when answering minimum income
questions. This error can be isolated, by using the standard intersection method proposed in the
literature. The second assumption equalizes utilities across households at the minimum income. Our
framework allows us to define the utility of the households in terms of their perceived income.
Subjective equivalent scales are equally rationalized from both settings: where utility is defined in
terms of either perceiver or actual income.
Further extensions: to different utility or income levels (not just the minimum). We have to extend
the intersection method and Assumption 1 to multiple income levels, that are the answers to multiquestions questionnaires, and we will get different equivalent scales at different levels. This will be the
purpose of future research.
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