Adam Szulc Empirical versus policy equivalence scales: matching estimation

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Bank i Kredyt 45(1), 2014, 37–52

Empirical versus policy equivalence scales:

matching estimation

Adam Szulc*

Submitted: 28 January 2013. Accepted: 28 July 2013.

Abstract

Properties of three types of equivalence scales obtained through the matching estimation are compared. They are derived by estimating the effect that demographic variables have on the household expenditures. The households are matched on common values of well-being covariates. Together with the earlier concept by Szulc (2009) two new types of covariates are examined: equivalent income − in this approach the scales minimise the deviation between the estimated and presumed values (solely empirical method), and covariates with imposed economy of scale on selected commodities (policy scales). The empirical part of the study is based on the Polish household budget survey data.

Keywords: equivalence scales, matching estimation JEL: D12, C14

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A. Szulc

38 1. Introduction

Equivalence scales may be generally defined as deflators adjusting the household nominal income or expenditure with respect to its demographic composition. Provided two households of different types, calculation of respective scales allows one to obtain equivalent incomes or expenditures (i.e. ratios of nominal income/expenditures and equivalence scale). Once they are properly estimated, the equivalent income/expenditure may be directly compared between households irrespectively to their demographic composition. Therefore, the equivalence scales are inherent elements of research on income distribution, poverty, inequality or social policy. The question of “true” equivalence scales may be also important in evaluations of household’s credit ability while impact of family composition on cost of living is concerned. Numerous studies confirmed considerable impact of the scales applied on the income distribution (Buchmann et al. 1988; Duclos, Mercader-Prats 1999; Ayala, Martinez, Huerta 2003; Szulc 2006). Relatively large differences between poverty and inequality indices due to scales variation suggest that at least some of the studies on income distribution may be flawed with this respect.

Formally, the equivalence scale is defined as a ratio of expenditures allowing two households under comparison to reach same well-being level. In other words, it answers the question how much more or less a household of given demographic type requires to attain the same well-being level that a household of other type does. The main theoretical problem resulting from abovementioned definition is: how to measure well-being to make it comparable between the households of different type. This issue may be resolved for instance by applying a consumer utility concept or by assuming that the ratio of food expenses in the consumer expenditures is such a measure (Engel approach). In the first method, which gained widest recognition in the scientific literature, the households are assumed to reach the same well-being level if their utility functions reach the same value. On one hand, this assumption is more realistic than that embedded in Engel approach (which assumes that households with the same food ratio reach the same well-being level). On the other hand, it causes identification problems related to non-comparability between individuals’ utility levels (see Pollak, Wales 1979; Blundell, Lewbel 1991). Consequently, they are relatively sensitive to assumptions making those comparisons possible, as well as to functional forms. This sometimes results in rather bizarre empirical results (see Szulc 2003 for a discussion). Moreover, utility based scales require estimation of the complete demand systems with demographic variables and, therefore, seek long time series to ensure sufficient price variation. The comprehensive review of relevant literature as well as estimates for Poland may be found in Dudek (2011). The alternative solution that overcome most of the abovementioned problems is based on subjective income questions (Kapteyn, van Praag 1976; Kot 1997). However, its credibility is restricted due to using verbal people’s declarations rather than observing their behaviour.

In this paper the concept of equivalence scales employing matching estimation (see Szulc 2009) is developed. Contrary to those based on utility theory, this type of scales is attributed by several advantages:

− no assumption on household behaviour nor utility comparability is required; this resolves the identification problem;

− they may be estimated using single period data without price information,

− it is possible, to incorporate multidimensional measures of household’s well-being, also including non-monetary components.

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Empirical versus policy equivalence scales...

39

Comparing to the original study by Szulc (2009), the following concepts are added in the present one.

1. Introducing two other methods of estimation:

− purely empirical that minimises a deviation between resulting equivalence scales and those assumed a priori in the estimation; household income and expenditure are the only data used,

− partly normative, combining empirical scales with some normative assumptions on cost of living (policy equivalence scales).

2. Discussion of two basic assumptions in matching estimation: conditional independence (unconfoundedness) and common support. The first one may be interpreted as a feasibility of all covariates having impact in household consumption. The second assumption means that for any vector of covariates existing for one type of household same (or similar) vector may be found for the second type of compared household. None of those assumptions can be directly checked by formal tests.

The remaining part of this paper is organised as follows. In Section 2 the principles of matching estimation in the equivalence scale context are introduced. Section 3 describes the conditional independence and common support assumptions. In Section 4 selection of household well-being covariates is discussed. The empirical results of estimation by means of the three methods together with comparison of the results are reported in Section 5. Section 6 concludes.

2. Equivalence scale as a matching estimator

The matching technique yields the estimate of an average effect of a binary treatment on a continuous scalar outcome (say X). For each i-th individual (i = 1,..., N) in the sample two potential outcomes may occur: X_i(1) when the individual is exposed to the treatment and X_i(0) otherwise. The population average treatment effect (PATE) is defined as follows:

(i = 1, ... , N) )] 0 ( ) 1 ( [Xi Xi E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

i l j l l M i _M w_wx X i(0) 0 1 0 ~ M i M i X µ µ α β σ σ σ σ µ X i j j j 'Z j 0, 1 1 1 1 ] ) ~ ^ ^ ^ [( 1 N i i M i i X w X N SATT M i X~

[

]

N i i i i i t _W _W _K _Z N V 1 2 2 1 ) ( ) 1 ( 1 ˆ Wi= 1 if Ai= = = = = = A1and Wi= 0 ) ( 2 _Z = =1 : 2 1 2 2 1 ₍ ₎ 2 1 ) ( i W i i l i Z _N _M X X SATT Z W X(0) | 1 ) | 1 (W Z P

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m ₁ ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) (1) In the empirical economic studies a “treatment” may mean receiving a social transfer, participation in training but also belonging to any socio-economic group, for instance in estimation of gender effects. In the equivalence scales context the “treatment” denotes belonging of i-th household to a particular demographic type. Therefore, the PATE may be interpreted as a measure of the effect the demographic attributes have on the expenditure of one type of household (a “treated” one), as compared to the expenditure of a reference household (a “control” household).

It is not possible to estimate the PATE directly as any individual cannot be at the state 0 and 1 simultaneously. This is also true when the demographic effect on a particular household expenditure is considered: when two types of households are compared (say A1 and A0) it may

belong to one class only. The matching estimation resolves this problem by comparing two types of outcome for each household in the sample: (i) the actual expenditure, (ii) the counterfactual expenditure that this household would have, if it was of a supplementary type. The latter is estimated by means of the actual expenditure(s) of the supplementary type of household(s) matched on common or closest values of well-being covariates. This meets the idea of the

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A. Szulc

40

equivalence scales presented in the Introduction: the matched households are assumed to reach the same or similar level of well-being. Therefore, for each element in the sample with attributes Aj the (individual) equivalence scale may be calculated as a ratio of its expenditure to expenditure(s) of a matched household(s) with attributes A1-j.

As an equivalence scale is an expenditure ratio, and not a difference, the average effect of demographic attributes (A0 and A1) on logarithm of expenditure X is estimated. Hence, the

logarithmic equivalence scale (say m) is estimated as a following mean logarithmic demographic effect (hereafter: MLDE):

(i = 1, ... , N) )] 0 ( ) 1 ( [X_i X_i E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) (2)

In such a representation, a logarithm of equivalence scale is an expected value of difference between logarithms of expenditures of both types of households, assuming that both ones are of the same type (A1). Therefore, to estimate the equivalence scales defined above for each household of type

A1 the counterfactual ( (i = 1, ... , N) )] 0 ( ) 1 ( [Xi Xi E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A₁ A₀ E Xi Xi Ai A₁ A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

i l j l l M i _M w_wx X i(0) 0 1 0 ~ M i M i X µ µ α β σ σ σ σ µ X i j j j 'Z j 0, 1 1 1 1 ] ) ~ ^ ^ ^ [( 1 N i i M i i X w X N SATT M i X~ [ ] N i i i i i t _W _W _K _Z N V 1 2 2 1 ) ( ) 1 ( 1 ˆ Wi= 1 if Ai= = = = = = A1and Wi= 0 ) ( 2 _Z = =1 : 2 1 2 2 1 ₍ ₎ 2 1 ) ( i W i i l i Z _N _M X X SATT Z W X(0) | 1 ) | 1 (W Z P } {_Y _Sa _Y _X _Sa Z / , ( )/

}

{

Y/(1 b CH),(Y X)/(1 b CH) Z SATT Sa a min CH b m ₁ ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0)

) expenditures1 _{must be obtained. They are calculated as actual expenditures}

of household of type A0 for which the metric

(i = 1, ... , N) )] 0 ( ) 1 ( [Xi Xi E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A₁ A₀ E Xi Xi Ai A₁ A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

i l j l l M i _M w_wx X i(0) 0 1 0 ~ M i M i X µ µ α β σ σ σ σ µ X i j j j 'Z j 0, 1 1 1 1 ] ) ~ ^ ^ ^ [( 1 N i i M i i X w X N SATT M i X~ [ ] N i i i i i t _W _W _K _Z N V 1 2 2 1 ) ( ) 1 ( 1 ˆ Wi= 1 if Ai= = = = = = A1and Wi= 0 ) ( 2 _Z = =1 : 2 1 2 2 1 ₍ ₎ 2 1 ) ( i W i i l i Z _N _M X X SATT Z W X(0) | 1 ) | 1 (W Z P } {_Y _Sa _Y _X _Sa Z / , ( )/

}

{

Y/(1 b CH),(Y X)/(1 b CH) Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0)

takes the smallest values across the entire subsample, where Z stands for a vector of the household well-being covariates and V is a weighting matrix. In the present study V is the diagonal matrix with inverses of the variances of each element of Z. For instance, using an Euclidean norm results in matching on mean squared differences between Zj and Zi.

PATE defined by equation (1) is an average treatment effect for the whole population while in estimation of equivalence scales the focus is on the subpopulation of the “treated” (as in the present study), i.e. households attributed by A1 (the effect on treated is hereafter denoted as PATT).

In these cases the counterfactual expenditures are constructed among households of reference type only. Formally, they take the form:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

where Γ_i(0) denotes the set of M closest matches for the i-th household. It is also possible to apply population weights w_l which may be useful while using, like in the present case, household survey data (for instance, to handle non-response). Each household may be used as a match more than once, which is equivalent to matching with replacement. Abadie and Imbens (2006) pointed out that this reduces the estimator bias, though increases its variance.

If the matches are not exact (i.e.

∑

Γ ∈ Γ

∑

i l j l l M i _M w_wx X i(0) 0 1 0 ~ M i M i X µ µ _α β σ σ σ σ µ X i j j j 'Z j 0, 1 1 1 1 ] ) ~ ^ ^ ^ [( 1 N i i M i i X w X N SATT M i X~

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0)

for any observation, which is very likely when continuous covariates are applied) the estimator based on (3) is biased. Imbens (2004) proposed the unbiased method of estimation of PATE or PATT using the following formula for calculation of counterfactual expenditures: (i = 1, ... , N) )] 0 ( ) 1 ( [Xi Xi E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

1_{They may be interpreted as hypothetical household of type A}

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41

where _μ_j ( j = 1, 0) is a regression estimate of actual expenditure X on covariates Zj for a respective subsample. (i = 1, ... , N) )] 0 ( ) 1 ( [X_i X_i E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

If any household of type A0 is used more than once as a match, then number of uses should

be applied as a weight in estimation _μ₀ (which is equivalent to replication of respective observation in the sample).

Finally, the corresponding estimator of the sample average treatment effect (SATT) for the sub-sample of the “treated” takes the form:

(i = 1, ... , N) )] 0 ( ) 1 ( [X_i X_i E PATE = = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

_Y _Sa _Y _X _Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) (6) where (i = 1, ... , N) )] 0 ( ) 1 ( [X_i X_i E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) is defined by (4) and (5).

Estimates of SATT are equal to PATT estimates, however their variance is lower. Replacing expenditures by their logarithms allows estimation of the right-hand side of (2), i.e. the sample average equivalence scale comparing the households attributed by A1 to those attributed by A0.

The variance of the sample average treatment effect defined by (6) is estimated as:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

where W_i = 1 if A_i = A1 and Wi = 0 otherwise and Ki stands for the number of times the i-th house-hold is used as a match.

Under assumption of constant treatment effect and constant variance2_{of X the conditional}

error variance (i = 1, ... , N) )] 0 ( ) 1 ( [Xi Xi E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

may be calculated as:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Γ_i(0) is described after equation (3).

3. Two basic assumptions in matching estimation

Matching estimation is based on two assumptions that are sufficient for identification of the average treatment effect. The first one, usually termed as unconfoundedness or conditional independence, implies that systematic differences between two sub-samples under comparison are due to treatment.

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A. Szulc

42

The second one, termed as common support or overlap, implies that individuals with the same covariates’ Z values have positive probabilities of being both in treated and control group.

If the mean effect for the treated (SATT) is the only objective of the interest (the present case) both abovementioned assumptions may be weakened, as only the values for the controls are to be estimated. The unconfoundedness assumption is reduced to assumption that under given set of covariates (Z) potential outcome for the controls is independent of the treatment assignment:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m ₁ ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) (10)

where ⊥ denotes independence and W is defined after (8). Weak overlap assumption may be formally written as:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Assumption (10) means that all variables that influence simultaneously potential outcome

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) and treatment assignment are observable and included into the set of covariates Z. Informally speaking, the better set of well-being covariates, the closer unconfoundedness assumption is satisfied. The common support assumption may be interpreted that for both types of household the range of covariates Z is similar and for each treated individual it is possible to find its counterpart in the control group. This assumption may be problematic for some types of covariates (for instance, for equivalent income) when completely different households, like childless couple and couple with three children, are compared. To relax impact of potential violation of the common support assumption the scales were estimated also using a type of “chaining”: a two person household was compared to a single person, three persons to two persons etc. The results were quite close to those obtained with a fixed reference household which suggests that the common support assumption may be ignorable.

4. Selection of the covariates

Three types of the covariate sets are examined in the present study. Each one represents a separate theoretical approach to the estimation and yields a different type of information on relative cost of living. This issue is investigated in succeeding section, especially in Section 5.4.

4.1. Generalised Engel scales

This type of scale utilises a set of variables correlated with household well-being that are comparable between their various types. A food ratio (being used in the Engel method of estimation as a single well--being indicator) is an example of such a variable. The full set of covariates is provided in Appendix. This method, following generally original concept by Szulc (2009), allows expanding the set of well-being covariates by non-monetary indicators. The equivalence scale utilising such set may be considered

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43

a generalisation of the idea underlying the Engel scale. Two households are assumed to achieve the same level of well-being provided that certain observable covariates are equal. Due to employing matching estimation instead of regression model it is possible to use more than one variable as a comparable between households, irrespectively to their type, measure of household well-being. This idea is in line with attempts to include into comparisons also non-monetary variables. Lelli (2005) estimated equivalence scales expanding welfare covariates using “functioning variables” like housing or education. Ebert and Moyes (2003), investigating formal properties of equivalence scales generalised single measures by means of multidimensional welfare (or utility) proxies. Both approaches can be captured by employing an appropriate set of covariates Z.

4.2. Recursive equivalence scales

This approach is based on correlates for which equivalence scales’ values must be assumed a priori. Equivalent income is such a variable. It is assumed that the “true” equivalence scale minimises the absolute difference between presumed scale and that obtained by matching estimation. The scales obtained in this way may be considered purely empirical, i.e. based on observation of household behaviour only. The households are being matched on equivalent incomes (Y). It would be ideal to make matching on equivalent expenditures (X), however the scalar transformation of the outcome variable cannot be used as a covariate (under exact matches the resulting treatment effect would be always equal to presumed value). Supplementary, equivalent income surplus (a difference between income and expenditure divided by the equivalence scale) is examined as an additional covariate, however adding this variable does not change the final results on equivalence scales. The scales used for calculation of equivalent income are assumed to be an exponential or linear function (to provide better exposition, depending on the type of comparisons3_{), defined uniquely}

by one unknown parameter a or b and by the demographic attributes. Hence, two considered sets of covariates Z appear in the form:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) or (i = 1, ... , N) )] 0 ( ) 1 ( [X_i X_i E PATE= = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m 1 ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) (12) where S stands for a household size and CH stands for a number of children in the household.

Once parameters a or b are set, a standard matching estimation is applied to obtain equivalence scales in the way described in section 4.1 and defined by equation (6). Naturally, the resulting scales depend on a or b and are different from those included into (12). To obtain “true” values, a or b are generated within a reasonable range4_{and then checked for reaching a minimum}

absolute difference between both scales. More formally, the scales depending on the household size are calculated by the formula m = Sa' where a' is a solution to the minimisation problem:

3_{As comparisons are being made between two types of the household only, the specification of the function does not affect}

the final result.

4_{For instance, to compare two persons with one a values are generated within interval [0.59; 079] which is equivalent to}

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A. Szulc

44

(i = 1, ... , N) )] 0 ( ) 1 ( [Xi Xi E PATE = = = = = = = = = = > < = = – – – -] | ) 0 ( (ln ) 1 ( [ln )] , ( ln[m A1 A0 E Xi Xi Ai A1 A1 j M i X V i j Z Z – _ _ _ _ _ _ _ _ + + + + _ _ + + j i Z Z 1 i ) 0 ( A A if 1

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

Z SATT Sa a min CH b m ₁ ' SATT CH b b (1 ) min ) 0 ( Γ ∈ ⊥ i l Γi(0) (13)

The scales for estimation of the cost of children are calculated in an analogous way:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

where b′ is a solution to the minimisation problem:

∑

Γ ∈ Γ

∑

[

]

}

{

Y Sa Y X Sa Z / , ( )/

}

{

Y

/(

1 b CH

),

(

Y X

)

/(

1 b CH

)

4.3. Partly normative equivalence scales

In the previous methods the ratio of actual household expenditure is the only information utilised in calculation of the scales. Those methods do not distinct between “rational” and “non--rational” expenses. In policy applications some normative presumptions may be introduced, for example to ensure adequate diet (see Nelson 1993, pp. 472−474). Technically, this may be implemented by imposing commodity specific equivalence scales in vector Z and then employing matching estimation described in Section 4.1. In the present study housing is such a commodity. The modified generalised Engel equivalence scales are estimated under assumption that adding one person to the household increases housing costs by certain proportion. In other words, housing equivalence scales are assumed rather than estimated. As this type of expenditure is characterised by very high economy of scale, the respective equivalence scales are set well below those estimated for the whole consumption by means of the previous methods.

5. Estimates of three types of equivalence scales

The scales in the present study are estimated using the data coming from the 2005 household budget survey collected by the Central Statistical Office. The estimations of demographic effect are performed with respect to the household size (irrespectively to the members’ age) and, alternatively, with respect to the number of children below 16 years of age. As comparison between those two estimations is not the goal of this study, they are performed on different subsamples. The estimation intended to calculate the cost of children is run on households with the head below 56 years of age to reduce the impact of age while in the previous one no restriction on head’s age is put. Only urban households were used in both estimations. The algorithm employed for calculations has been developed by Abadie et al. (2004) as STATA command nnmatch. It allows unbiased estimation of SATT using the formula described by equations (4) and (5).

5.1. Generalised Engel scales

The generalised Engel scales are estimated using one and four matches (M in equation 3). The latter estimation should ensure lower standard errors than the previous, however at the cost of higher bias. In Table 1 the estimates of MLDE (mean logarithmic demographic effects, see equation 2)

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with standard errors as well as equivalence scales are displayed. To facilitate the comparisons the estimates are supplemented by OECD 70/50 equivalence scales.5_{It should be mentioned, however,}

that in original OECD formula children are defined as persons below 15 years of age, i.e. younger by one year than those in the present research.

The general conclusion is that OECD 70/50 scales seem to underrate, with one exception, economy of scale, especially for larger households (of three or more persons). Economy of scale varies with respect to the number of persons and number of children in the household. It is higher for households of three or more persons. For a single adult with one child the OECD and the empirical scales are virtually same. The results of estimation are sensitive to the choice of some covariates. This is especially true for the dwelling size being a variable presuming some sort of commodity specific equivalence scale. If it is defined as number of squared meters per person rather than per square root of the number of persons, the equivalence scales increase by 4−6%.

The standard errors (see equation 8) for the mean demographic effect are reasonably low yielding all t-statistics well above 10. This hardly can surprise, as it is only the proof of significant differences between total expenditures for households of different types. As suggested by the theory, the results obtained with the use of four matches yields lower standard errors than those obtained by means of one match. No regularities are observable when comparing the values of both estimates.

5.2. Recursive equivalence scales

When the household size is the variable of interest the estimates of the household demographic effect reveal usually lower economy of scale than in the case of the previous method. This is especially true for households of three and four persons and for households with children. In other words, the recursive method suggests that the cost of children is higher than that indicated by the previous method. The numerical results are quite close to those obtained by means of the OECD 70/50 equivalence scales and even higher for the households of three children. The differences between the method are discussed in more details in Section 4.4.

5.3. Partly normative equivalence scales

The presumed housing costs of adding one person, children or adult, is equal to 10% of a cost for single adult household6_{. The assumed values for all types of households included are displayed in}

Table 3 together with the estimates of the scales for the whole consumption.

In this estimation only a subset of covariates employed in the estimation reported in Section 4.1 is utilised. The whole set includes some non-monetary correlates of well-being, like education. In this estimation only the variables that may be used as proxy variables for housing equivalent expenditures are employed: dwelling size, food ratio, presence of car and satelite/cable TV. These

5_{Normative indicators giving weight 1 to the first adult, 0.7 to any other person aged over 14 and 0.5 to every child below 15.} 6_{As the scales are estimated only to illustrate the method, this assumption is not based on any actual policy scales nor}

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covariates are supplemented by total housing expenditures divided by presumed equivalence scale. For comparison purposes, the “unrestricted” estimates, i.e. obtained without the latter variable are also displayed in Table 3.

When the household size is the only household demographic attribute explored, the scales with presumed housing economy of scale are considerably below those obtained by the “unrestricted” estimation. This is not true, however, when households with children are compared with a childless couple with same housing equivalence scales assumed. Both estimations yield very similar results. This means that assumed economy of scale for housing is close to that actual. In other words, children below 16 years of age cause much lower housing costs than older persons.

The present method allows also estimation of commodity specific equivalence scale for any group of expenditures. It requires simulations of presumed values to reach approximate equality between the scales estimated with and without presumptions. Such equity may be observed for a household of two adults with one child as well as for two adults with three children, both compared to a childless couple. For housing equivalence scales equal to 1.09 (1.2/(1+0.1)), and 1.26 (1.3 /(1+0.1)), respectively, the scales resulting from the “unrestricted” and restricted estimations are similar, therefore 1.09 and 1.26 may be considered close approximations to the actual housing equivalence scales.

5.4. Comparison of the estimates obtained by three methods

The results for all three methods are summarised in Figures 1 and 2. Each type of the estimation yields different information on cost of living comparisons. The generalised Engel scales use non--monetary well-being covariates like dwelling size, proxy variables like education or main source of income as well as consumption patterns. Therefore, those comparisons introduce a type of normative judgements referring to the household needs. The recursive equivalence scales are based solely on observed household consumption and income which not necessary are equivalent to well-being measures in a broad sense. There are at least two reasons for such a reservation. First, some expenses, for instance on alcohol or tobacco, hardly improve the household well- -being. Second, some households, especially those poor, may be unable to satisfy their increased needs resulting from changing their demographic composition. Therefore, applying actual (low) expenditures observed for a large group of households of particular type (e.g. those with three children) may result in underestimation of their needs. Comparison of the empirical results for both methods does not necessary suggest that this may be the case for the Polish households. “Purely empirical” (recursive) equivalence scales for three or four person households are still well above those obtained with the use of some normative non-monetary covariates of well-being. This is also true for households with two or three children.

The third method, employing a priori housing equivalence scales hardly can be compared with two previous ones. The numerical results strongly depend on assumed commodity specific equivalence scales. The restrictions imposed on the housing economy of scale lower the estimates of the households equivalence scales, as compared to those obtained without restrictions. This results directly from the definition of equivalence scale, which is a mean ratio of total expenditures for two

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types of households under comparison provided their equal well-being level. Imposing lower than actual housing equivalence scales results in comparing non-reference households with a reference type households whose housing expenditures pushes its well-being up. Thus, the average ratio is underestimated which is a consequence of a priori policy suppositions on relative cost of living.

6. Concluding remarks

In this paper the matching estimation approach to equivalence scales proposed by Szulc (2009) is developed. Comparing to the original study, in which households are matched on a set of well-being correlates, two new methods of estimation are added: (i) recursive estimation in which deviation between presumed and resulting from estimation scales is minimised, (ii) estimation incorporating a priori assumptions on commodity specific equivalence scales, combining in this way empirical and normative approach. Three abovementioned methods of estimation are operationalised by defining a set of well-being indicator(s) on which household are being matched. They capture, respectively:

1. Set of monetary and non-monetary well-being correlates (generalised Engel method). 2. Equivalent income in which initial equivalence scales are presumed (recursive method).

3. Set of indicators including housing equivalent expenditure in which equivalence scale may be presumed on a normative basis (policy equivalence scales).

The main findings allows the following conclusions. OECD 70/50 equivalence scales generally overestimate large households’ cost of living due to ignoring variability of economy of scale, which is captured by matching estimation. Comparing generalised Engel and recursive scales does not suggests that households with large number of children are forced to reduce their increased needs resulting from changing their demographic composition. Policy commodity specific equivalence scales based on normative assumptions may be incorporated into formal algorithm of estimation. The numerical values do not contradict expectations based on household behaviour observation.

The matching estimation is characterised by the following advantages. The estimation and the identification do not require assumptions on the household behaviour. It is possible to compare the household types of any composition without assuming particular functional form of interaction between expenditures and demographic variables. The assumption on separability is also redundant. It is possible to decide, due to elasticity in well-being covariates selection, what type of comparisons is performed. They may capture material needs only but also functionings. With respect to social policy needs, it is possible to impose economy of scale for some types of expenditures and to estimate the scales for the remaining consumption empirically. The matching estimation is not data consuming. It may be performed without price information, for a single period sample and only on households of interest. The results obtained may be easily interpreted, as the equivalence scales are defined in the form of mean ratio of empirical expenditures.

The disadvantages of the proposed method of estimation include necessity to compare each two types of the households separately. As a consequence, the scales for the types of households underrepresented in the sample may be difficult to estimate precisely. The aforementioned disadvantages also include impossibility to test formally the unconfoundedness and common support assumptions. On the other hand, informal test suggests that the latter one is not violated by the empirical data.