This document discusses estimating inter-temporal equivalence scales for Poland from 2005-2010 using microdata on household expenditures. It proposes using stochastic equivalence scales (SES) as a new method that is based on the weaker condition of stochastic indifference rather than exactness or independence of base. Non-parametric and parametric SES are estimated and show diminishing economies of scale over time, with inter-temporal SES estimates being lower than current estimates. The SES approach allows for homogenization of expenditure distributions across time and household groups.

1. INTER-TEMPORAL EQUIVALENCEINTER-TEMPORAL EQUIVALENCE
SCALES BASED ON STOCHASTICSCALES 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 ConferenceThe paper prepared for the 33rd IARIW General Conference
August 24-30, 2014, Rotterdam, the NetherlandsAugust 24-30, 2014, Rotterdam, the Netherlands

2. 2
The PurposeThe Purpose
 EEstimatstimatinging the inter-temporal equivalence scale forthe inter-temporal equivalence scale for
Poland in the years 2005-2010, using micro-data onPoland in the years 2005-2010, using micro-data on
expenditure distributionsexpenditure distributions..
 The idea consists in choosing one reference group ofThe idea consists in choosing one reference group of
households (single adults in 2010) for all years and allhouseholds (single adults in 2010) for all years and all
selected groups of households.selected groups of households.
 Assuming constant prices and ‘benchmark’ referenceAssuming constant prices and ‘benchmark’ reference
household group,household group, thethe equivalence scales allow forequivalence scales allow for
homogenisation of expenditure distributions across timehomogenisation of expenditure distributions across time
and household groups.and household groups.

3. 3
MotivationMotivation
 Heterogeneity of household populations raises seriousHeterogeneity of household populations raises serious
difficulties when addressing inequality, welfare anddifficulties 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 assumptionthe assumption
alternatively calledalternatively 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 theySeveral papers have tested this assumption, but they
ultimately reject itultimately reject it..
 TheThe ESE/IB conditionESE/IB condition seemseemss to beto be too strongtoo strong..
 Stochastic indifference criterion offers a weakerStochastic indifference criterion offers a weaker
condition thancondition than ESE/IBESE/IB

4. 4
MethodsMethods
 The stochastic equivalence scales (SES) areThe stochastic equivalence scales (SES) are
applied for homogenisation of heterogeneousapplied for homogenisation of heterogeneous
populations of householdspopulations of households in different yearsin different years..
 The SES is a new class of equivalence scalesThe SES is a new class of equivalence scales
that base on the concept of stochasticthat base on the concept of stochastic
indifference.indifference.
 We estimate SESsWe estimate SESs using micro-data onusing micro-data on
expenditure distributionsexpenditure distributions for Poland in the yearsfor Poland in the years
2005-20102005-2010 (constant prices, 2010=100).(constant prices, 2010=100).

5. 5
StochasticStochastic dominancedominance
 WW11~R~R11(w)(w) andand WW22(x)~R(x)~R22(w)(w)
 11stst
order stochastic dominance:order stochastic dominance:
WW22 ≥≥FSDFSD WW11 ↔↔ RR11(w )≥R(w )≥R22(w)(w),,
for allfor all w,w, with strict inequality for somewith strict inequality for some ww..
 22ndnd
order stochastic dominance:order stochastic dominance:
WW22 ≥≥SSSDSD WW11 ↔↔
for allfor all w,w, with strict inequality for somewith strict inequality for some ww..
 Corrolary 1.Corrolary 1. 11stst
order stochastic dominance impliesorder stochastic dominance implies
stochastic dominance at 2stochastic dominance at 2ndnd
and higher orders.and higher orders.
 ThisThis implication goes in only one direction.implication goes in only one direction.
∫∫ ≥
ww
dttRdttR
0
2
0
1 )()(

6. 6
Stochastic indifferenceStochastic indifference
 WW11 isis 11stst
orderorder indifferent toindifferent to WW22 ↔↔ RR11(w)=R(w)=R22(w)(w),, for allfor all ww..
 WW11 isis 22ndnd
orderorder indifferent toindifferent to WW22 ↔↔
for allfor all ww..
 CorollaryCorollary 22.. TThe first order indifference implies the indifference of allhe 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.
 CorollaryCorollary 3.3. TThe 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 inSocial welfare in WW11 andand WW22, is the same for all utility functions, is the same for all utility functions
uu ∈∈ UU22..
 Poverty inPoverty in WW11 andand WW22 is the same for all poverty lines and for theis the same for all poverty lines and for the
Atkinson’s (1987) class of poverty indices.Atkinson’s (1987) class of poverty indices.
 Inequalities inInequalities in WW11 andand WW22 are the same.are the same.
∫∫ =
ww
dttRdttR
0
2
0
1 )()(

7. 7
Stochastic indifference and equivalence scaleStochastic 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 variablesLet the positive continuous random variables
XXii~F~Fii(x)(x) andand Y~G(y)Y~G(y) describe expenditure distributions ofdescribe expenditure distributions of
the households withthe households with iith attributes,th attributes, i=1,…,m,i=1,…,m, and theand the
reference households, respectively. Let ESE/IBreference households, respectively. Let ESE/IB
assumption holds which means that there existsassumption holds which means that there exists Δ(p,αΔ(p,αii))
satisfyingsatisfying the following equationthe following equation
 DefineDefine
 ThenThen ZZii is stochastically indifferent tois stochastically indifferent to YY,, i.e.i.e. H(z)=G(z)H(z)=G(z)






∆
= o
i
i
p
x
pvxpv α
α
α ,
),(
,),,(
)(~),(/ zHpXZ iii α∆=

8. 8
The relationThe relationshipship between ESE/IBbetween ESE/IB
and stochastic indifferenceand stochastic indifference
Stochastic
indifference
ESE/IB

9. 9
The relationship between the strict stationarityThe relationship between the strict stationarity
of time series and the weak stationarityof time series and the weak stationarity
 XXtt is sis stationarytationary ((in the strictin the strict
sensesense) ↔) ↔
F(xF(x11,...,x,...,xnn)=F(x)=F(x1+k1+k,...,x,...,xn+kn+k)) (1)(1)
 TheoremTheorem.. IfIf XXtt is stationaryis stationary
(in the stric sense) then:(in the stric sense) then:
 E[E[XXtt]=]= m =m = const. (2)const. (2)
 DD22
[[XXtt]=]=ss22
= const. (3)= const. (3)
 cov(tcov(t11,t,t22) = cov(t) = cov(t22-t-t11)) (4)(4)
 XXtt is sis stationary in thetationary in the weakweak
sensesense ↔ (2), (3), and (4) hold.↔ (2), (3), and (4) hold.
Weak
stationarity
Strict
stationarit
y

10. 10
Stochastic Equivalence Scales (SES)Stochastic Equivalence Scales (SES)
 LLetet Y~G(y)Y~G(y) describe the reference expenditure distribution anddescribe the reference expenditure distribution and
XXii~F~Fii(x)(x) be the evaluated expenditure distribution of the ith householdbe 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..
 LetLet ZZii = s= sii(X(Xii)~H)~Hii(z)(z) be the transformation of the evaluated expenditurebe the transformation of the evaluated expenditure
distributiondistribution XXii..
 Define the random variableDefine the random variable Z~H(z)Z~H(z) as a mixture of theas a mixture of the mm transformedtransformed
distributionsdistributions ZZii::
 DefinitionDefinition 11.. TThe functionhe function ss((⋅⋅) will be called the) will be called the stochasticstochastic
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 zHzH 1
)()( π ∑=
=∧>=∀
m
i iimi 1
10,,..,1 ππ

11. 11
Stochastic Equivalence Scales (cont...)Stochastic Equivalence Scales (cont...)
 The definition of anThe definition of an SESSES is ‘axiomatic’ in the sense that itis ‘axiomatic’ in the sense that it
only defines theonly defines the propertiesproperties of a functionof a function that can bethat can be
recognised as anrecognised as an SESSES. This definition does not describe. This definition does not describe
how anhow an SESSES should be constructed.should be constructed.
 In other words,In other words, anyany functionfunction ss((⋅⋅)) that fulfils the conditionthat fulfils the condition
(20) has to be recognised as an(20) has to be recognised as an SESSES..
 LetLet dd == [[ddii],], i = 1,…,m,i = 1,…,m, be the vector of positive numbersbe the vector of positive numbers
called ‘deflators’ that transform the evaluatedcalled ‘deflators’ that transform the evaluated
expenditure distributionsexpenditure distributions XX11,…,X,…,Xmm thusly:thusly:
 ZZii = X= Xii/d/dii ∼∼ HHii(z),(z), i = 1,…,m,i = 1,…,m, (21)(21)
 DefinitionDefinition 22.. Under the above notations, the vectorUnder the above notations, the vector d d willwill
be called thebe called the relativerelative SESSES if and only if the deflatorsif and only if the deflators dd11,,
…,d…,dmm are such that equality (20) holds.are such that equality (20) holds.

12. 12
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 estimatorThe estimator s*s* can be found using the grid-searchcan be found using the grid-search
method.method.
nl
nl
zGzHU
z +
⋅
−= |)()(ˆ|max

ln
ln
zGzHU
zs +
⋅
−= |)()|(|maxmin*)(

ss
α>∧≥= *)()]()([max)( spuUPp calc sss*
s

13. 13
Estimating theEstimating the dd deflator of a non-parametricdeflator of a non-parametric SESSES

14. 14
Estimating theEstimating the θθ parameter of the powerparameter of the power SESSES ,,
d=hd=hθθ
, , hh -- household sizehousehold size

15. 15
TABLE 1. Estimates of non-parametric inter-temporal SESs
Size 2005 2006 2007 2008 2009 2010
1 .883
p=.335
.919
p=.196
.914
p=.692
.942
p=.946
.985
p=.961
1
2 1.416
p= .135
1.475
p=.316
1.530
p=.344
1.603
p=.619
1.656
p=.481
1.687
p=618
3 1.642
p=.374
1.776
p=.144
1.856
p=.469
1.985
p=.411
2.037
p=.199
2.069
p=.091
4 1.789
p= .127
1.903
p=.278
2.028
p=.157
2.162
p=.141
2.220
p=.116
2.217
p=.228
≥5 1.665
p= .214
2.002
p=.000
2.126
p=.001
2.261
p=.009
2.326
p=.001
2.372
p=.000
p (K-S) .069 .061 .069 .137 .077 .040
p(K-W) .599 .766 .487 .853 .946 .910

16. 16
Estimates of parametric inter-temporalEstimates of parametric inter-temporal SESsSESs
and currentand 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

17. 17
ConclusionsConclusions
 The application of SES to inter-temporalThe application of SES to inter-temporal
comparisons of expenditure distributions hascomparisons of expenditure distributions has
turned out very useful.turned out very useful.
 Polish households exhibited largePolish households exhibited large butbut
diminishing economies of scale in the yearsdiminishing economies of scale in the years
2005-2010.2005-2010.
 TThe inter-temporal estimates of economies ofhe inter-temporal estimates of economies of
scale are lower than the current estimates.scale are lower than the current estimates.

18. Thank youThank you

19. 19
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)]