INTER-TEMPORAL EQUIVALENCE INTER-TEMPORAL EQUIVALENCE SCALES BASED ON STOCHASTIC SCALES BASED ON STOCHASTIC

INDIFFERENCE CRITERION: INDIFFERENCE CRITERION: THE CASE OF POLANDTHE CASE OF POLAND

Stanislaw Maciej KotStanislaw Maciej Kot

Gdansk University of TechnologyGdansk University of Technology, Poland, Poland..

The paper prepared for the 33rd IARIW General Conference The paper prepared for the 33rd IARIW General Conference August 24-30, 2014, Rotterdam, the NetherlandsAugust 24-30, 2014, Rotterdam, the Netherlands

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The PurposeThe Purpose

EEstimatstimatinging the inter-temporal equivalence scale for the inter-temporal equivalence scale for Poland in the years 2005-2010, using micro-data on Poland in the years 2005-2010, using micro-data on expenditure distributionsexpenditure distributions..

The idea consists in choosing one reference group of The idea consists in choosing one reference group of households (single adults in 2010) for all years and all households (single adults in 2010) for all years and all selected groups of households. selected groups of households.

Assuming constant prices and ‘benchmark’ reference Assuming constant prices and ‘benchmark’ reference household group, household group, thethe equivalence scales allow for equivalence scales allow for homogenisation of expenditure distributions across time homogenisation of expenditure distributions across time and household groups. and household groups.

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MotivationMotivation

Heterogeneity of household populations raises serious Heterogeneity of household populations raises serious difficulties when addressing inequality, welfare and difficulties when addressing inequality, welfare and poverty.poverty.

Conventional equivalence scales are not identifiable, Conventional equivalence scales are not identifiable, given consumer demand datagiven consumer demand data, unless , unless the assumption the assumption alternatively called alternatively called independence of baseindependence of base (IB) or (IB) or equivalence-scale exactnessequivalence-scale exactness (ESE) (ESE) holds. holds.

Several papers have tested this assumption, but they Several papers have tested this assumption, but they ultimately reject itultimately reject it..

The The ESE/IB condition ESE/IB condition seemseems s to be to be too strongtoo strong.. Stochastic indifference criterion offers a weaker Stochastic indifference criterion offers a weaker

condition than condition than ESE/IBESE/IB

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MethodsMethods

The stochastic equivalence scales (SES) are The stochastic equivalence scales (SES) are applied for homogenisation of heterogeneous applied for homogenisation of heterogeneous populations of householdspopulations of households in different years in different years..

The SES is a new class of equivalence scales The SES is a new class of equivalence scales that base on the concept of stochastic that base on the concept of stochastic indifference.indifference.

We estimate SESs We estimate SESs using micro-data on using micro-data on expenditure distributionsexpenditure distributions for Poland in the years for Poland in the years 2005-20102005-2010 (constant prices, 2010=100). (constant prices, 2010=100).

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Stochastic Stochastic dominancedominance

WW11~R~R11(w) (w) and and WW22(x)~R(x)~R22(w)(w)

11stst order stochastic dominance: order stochastic dominance:

WW2 2 ≥≥FSDFSD WW11 ↔ ↔ RR11(w )≥R(w )≥R22(w)(w), ,

for all for all w, w, with strict inequality for some with strict inequality for some ww.. 22ndnd order stochastic dominance: order stochastic dominance:

WW2 2 ≥≥SSSDSD WW11 ↔↔

for all for all w, w, with strict inequality for some with strict inequality for some ww.. Corrolary 1.Corrolary 1. 11stst order stochastic dominance implies order stochastic dominance implies

stochastic dominance at 2stochastic dominance at 2ndnd and higher orders. and higher orders. This This implication goes in only one direction.implication goes in only one direction.

∫∫ ≥ww

dttRdttR0 20 1 )()(

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Stochastic indifferenceStochastic indifference

WW11 is is 11stst order order indifferent to indifferent to WW22 ↔↔ RR11(w)=R(w)=R22(w)(w),, for all for all ww..

WW11 is is 22ndnd order order indifferent to indifferent to WW22 ↔ ↔

for all for all ww.. Corollary Corollary 22.. TThe first order indifference implies the indifference of all he first order indifference implies the indifference of all

higher orders and this implication goes in both directions.higher orders and this implication goes in both directions. Corollary Corollary 3.3. T The following statements are equivalent:he following statements are equivalent: RR11(w)=R(w)=R22(w), (w), for all w≥ for all w≥0.0.

Social welfare in Social welfare in WW11 and and WW22, is the same for all utility functions , is the same for all utility functions

u u ∈∈ U U22..

Poverty in Poverty in WW11 and and WW22 is the same for all poverty lines and for the is the same for all poverty lines and for the

Atkinson’s (1987) class of poverty indices. Atkinson’s (1987) class of poverty indices. Inequalities in Inequalities in WW11 and and WW22 are the same. are the same.

∫∫ =ww

dttRdttR0 20 1 )()(

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Stochastic indifference and equivalence scale Stochastic indifference and equivalence scale exactness (ESE)/ indepedence of base (IB)exactness (ESE)/ indepedence of base (IB)

Theorem 1.Theorem 1. Let the positive continuous random variables Let the positive continuous random variables XXii~F~F ii(x)(x) and and Y~G(y)Y~G(y) describe expenditure distributions of describe expenditure distributions of

the households with the households with iith attributes, th attributes, i=1,…,m,i=1,…,m, and the and the reference households, respectively. Let ESE/IB reference households, respectively. Let ESE/IB assumption holds which means that there exists assumption holds which means that there exists Δ(p,αΔ(p,α ii))

satisfyingsatisfying the following equation the following equation

Define Define

Then Then ZZ ii is stochastically indifferent to is stochastically indifferent to YY, , i.e. i.e. H(z)=G(z) H(z)=G(z)

∆

= oi

i p

xpvxpv α

αα ,

),(,),,(

)(~),(/ zHpXZ iii α∆=

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The relationThe relationshipship between ESE/IB between ESE/IB and stochastic indifferenceand stochastic indifference

Stochastic indifference

ESE/IB

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The relationship between the strict stationarity The relationship between the strict stationarity of time series and the weak stationarityof time series and the weak stationarity

XXtt is sis stationary tationary ((in the strict in the strict

sensesense) ↔ ) ↔ F(xF(x11,...,x,...,xnn)=F(x)=F(x1+k1+k,...,x,...,xn+kn+k) ) (1)(1)

TheoremTheorem. . If If XX tt is stationary is stationary

(in the stric sense) then:(in the stric sense) then: E[E[XX tt]= ]= m = m = const. (2)const. (2)

DD22[[XX tt]=]=ss22 = const. (3) = const. (3)

cov(tcov(t11,t,t22) = cov(t) = cov(t22-t-t11) ) (4)(4)

XXt t is s is stationary in the tationary in the weakweak

sensesense ↔ (2), (3), and (4) hold. ↔ (2), (3), and (4) hold.

Weak stationarity

Strict stationarit

y

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Stochastic Equivalence Scales (SES)Stochastic Equivalence Scales (SES)

LLet et Y~G(y)Y~G(y) describe the reference expenditure distribution and describe the reference expenditure distribution and XX ii~F~F ii(x)(x) be the evaluated expenditure distribution of the ith household be the evaluated expenditure distribution of the ith household

group, group, i=1,…,m.i=1,…,m. LetLet ss((⋅⋅)) = [ = [ss11((⋅⋅),…,s),…,smm(∙) (∙) ] ] bebe a continuous and strictly monotonic real- a continuous and strictly monotonic real-

valued vector functionvalued vector function.. Let Let ZZ ii = s = s ii(X(X ii)~H)~H ii(z)(z) be the transformation of the evaluated expenditure be the transformation of the evaluated expenditure

distribution distribution XX ii. .

Define the random variable Define the random variable Z~H(z)Z~H(z) as a mixture of the as a mixture of the mm transformed transformed distributions distributions ZZ ii::

DefinitionDefinition 1 1. . TThe function he function ss((⋅⋅) will be called the ) will be called the stochastic stochastic equivalence scaleequivalence scale (SES) if and only if the following equality holds: (SES) if and only if the following equality holds:

∀∀z ≥0, H(z) = G(z)z ≥0, H(z) = G(z) (20)(20)

∑ == m

i ii zHzH1

)()( π ∑ ==∧>=∀ m

i iimi1

10,,..,1 ππ

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Stochastic Equivalence Scales (cont...)Stochastic Equivalence Scales (cont...)

The definition of an The definition of an SESSES is ‘axiomatic’ in the sense that it is ‘axiomatic’ in the sense that it only defines the only defines the propertiesproperties of a functionof a function that can be that can be recognised as an recognised as an SESSES. This definition does not describe . This definition does not describe how an how an SESSES should be constructed. should be constructed.

In other words, In other words, anyany function function ss((⋅⋅)) that fulfils the condition that fulfils the condition (20) has to be recognised as an (20) has to be recognised as an SESSES. .

Let Let dd = = [[dd ii], ], i = 1,…,m,i = 1,…,m, be the vector of positive numbers be the vector of positive numbers called ‘deflators’ that transform the evaluated called ‘deflators’ that transform the evaluated expenditure distributions expenditure distributions XX11,…,X,…,Xmm thusly: thusly:

ZZ ii = X = X ii/d/d ii ∼∼ HH ii(z),(z), i = 1,…,m, i = 1,…,m, (21)(21) Definition Definition 22.. Under the above notations, the vector Under the above notations, the vector d d will will

be called the be called the relativerelative SES SES if and only if the deflators if and only if the deflators dd11,,…,d…,dmm are such that equality (20) holds. are such that equality (20) holds.

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Statistical issues concerning SESStatistical issues concerning SES

HH00:: H(z) = G(z) H(z) = G(z)) against ) against HHaa:: H(z) ≠ G(z) H(z) ≠ G(z), for all , for all z.z.

Kolmogorov-Smirnov (K-S) test:Kolmogorov-Smirnov (K-S) test:

EstimationEstimation

or equivalentlyor equivalently

The estimator The estimator s*s* can be found using the grid-search can be found using the grid-search

method.method.

nl

nlzGzHU

z +⋅−= |)()(ˆ|max

ln

lnzGzHU

zs +⋅−= |)()|(|maxmin*)(

ss

α>∧≥= *)()]()([max)( spuUPp calc sss*s

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Estimating the Estimating the dd deflator of a non-parametric deflator of a non-parametric SESSES

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Estimating the Estimating the θθ    parameter of the power parameter of the power SESSES ,,d=hd=hθθ, ,  hh -- household size household size

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TABLE 1. Estimates of non-parametric inter-temporal SESs

Size 2005 2006 2007 2008 2009 2010

1 .883p=.335

.919p=.196

.914p=.692

.942p=.946

.985p=.961

1

2 1.416p= .135

1.475p=.316

1.530p=.344

1.603p=.619

1.656p=.481

1.687p=618

3 1.642p=.374

1.776p=.144

1.856p=.469

1.985p=.411

2.037p=.199

2.069p=.091

4 1.789p= .127

1.903p=.278

2.028p=.157

2.162p=.141

2.220p=.116

2.217p=.228

≥5 1.665p= .214

2.002p=.000

2.126p=.001

2.261p=.009

2.326p=.001

2.372p=.000

p (K-S) .069 .061 .069 .137 .077 .040

p(K-W) .599 .766 .487 .853 .946 .910

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Estimates of parametric inter-temporal Estimates of parametric inter-temporal SESs SESs and current and current SESsSESs

Year Inter-temporal SES.Reference households:

single adults, 2010

Current SES.Reference households:

single adults, current year

θ p-value θ p-value

2005 0.401(0.400;0.402)

0.063 0.519(0.513; 0.526)

0.382

2006 0.459(0.457; 464)

0.076 0.539(0.537;0.541)

0.113

2007 0.501(0.499; 0.506)

0.099 0.589(0.582; 0.597)

0.477

2008 0.555(0.550; 0.571)

0.238 0.610(0.606; 0.622

0.241

2009 0.581(576; 0.595)

0.168 0.597(0.592; 0.609)

0.250

2010 0.594(0.591; 0.603)

0.153 0.594(0.591; 0.603)

0.153

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ConclusionsConclusions

The application of SES to inter-temporal The application of SES to inter-temporal comparisons of expenditure distributions has comparisons of expenditure distributions has turned out very useful.turned out very useful.

Polish households exhibited large Polish households exhibited large butbut diminishing economies of scale in the years diminishing economies of scale in the years 2005-2010.2005-2010.

TThe inter-temporal estimates of economies of he inter-temporal estimates of economies of scale are lower than the current estimates. scale are lower than the current estimates.

Thank youThank you

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Parametric SES based on disposable incomeParametric SES based on disposable income

0 . 5 8 8 0 . 5 9 0 0 . 5 9 2 0 . 5 9 4 0 . 5 9 6 0 . 5 9 8 0 . 6 0 0 0 . 6 0 2

θ 1

5 E - 1 0

5 E - 0 9

5 E - 0 8

5 E - 0 7

5 E - 0 6

5 E - 0 5

5 E - 0 4

5 E - 0 3

5 E - 0 2

5 E - 0 1

5 E + 0 0

log

[p(θ1)

]