A agonés, En ique a
A icle
Family es ic ions a wo k
Economies
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MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: A agonés, En ique a (2024) : Family es ic ions a wo k, Economies, ISSN
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Ci a ion: A agones, En ique a. 2024.
Family Res ic ions a Wo k.
Economies 12: 101. h ps://doi.o g/
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economies
A icle
Family Res ic ions a Wo k
En ique a A agones 1,2
1Ins i u d’Anàlisi Econòmica, CSIC, Campus UAB, 08193 Bella e a, Spain; [email p o ec ed]
2Ba celona School o Economics Ramon T ias Fa gas, 25-27, 08005 Ba celona, Spain
Abs ac : This pape analyzes one o he causes o he cu en gende -unbalanced si ua ion in he
labo ma ke : he disc imina ion ha indi iduals ace a wo k due o hei commi men o unpaid ca e
wo k. I aims a inding mechanisms ha may induce a change om he cu en unbalanced si ua ion
o a wo ld in which males and emales a e ound in mo e equal sha es in all p o essions and a all
le els. I cons uc a o mal model ha includes he he e ogenei y o indi iduals ega ding hei amily
commi men s and I in es iga e how i a ec s he indi idual’s op imal labo ma ke pa icipa ion.
The wel a e o indi iduals wi h commi men o amily du ies is educed o wo di e en easons:
o no being able o pa icipa e as much in he labo ma ke and hus ecei e a lowe labo income
and o no being able o con ibu e as much o hei amily commi men s. I compa e he esul s
o he emale and male sec ions o he socie y and I illus a e he obse ed gende gaps in e ms o
labo ma ke pa icipa ion, income le els, and he o e all u ili y ob ained. I ind ha e en hough he
gende wage gap may be alle ia ed wi h educ ions in he cos associa ed o unpaid ca e wo k, he
gende u ili y gap will pe sis .
Keywo ds: disc imina ion; labo ma ke ; unpaid ca e wo k
JEL Classi ica ion: J7; J31
1. In oduc ion
The unde - ep esen a ion o women wi h espec o men in many p o essions a all
le els and in all p o essions a some le els is a ac . Gi en ha women ep esen abou
50 pe cen o he popula ion, i is easonable o desc ibe his si ua ion as gende unbalanced.
The demand o a balanced p opo ion o women in all p o essions and a all le els has
been aised and i s suppo has been inc easing o e ime. This demand can be based on he
claim ha a wo ld in which males and emales a e ound in equal sha es in all p o essions
and a all le els would be op imal. Bu i also can be based on an equi y claim: emales and
males should ha e he same p o essional oppo uni ies.
This pape analyzes one o he causes o he cu en gende -unbalanced si ua ion: he
disc imina ion ha indi iduals ace a wo k due o hei commi men o unpaid ca e wo k.
I aims a inding mechanisms ha may induce a change om he cu en unbalanced
si ua ion o a wo ld in which males and emales a e ound in mo e equal sha es in all
p o essions and a all le els.
The analysis p oposed in his pape con ibu es o he heo e ical li e a u e on disc im-
ina ion in he labo ma ke . The exis ing li e a u e is mos ly de o ed o explaining acial
disc imina ion such us Peski and Szen es (2013) who analyze how acial s e eo ypes may
de e mine he labo ma ke ou comes. The heo e ical models ha ocus on he gende
gaps in he labo ma ke assume speci ic s a egies o he i ms such as incen i e con ac s
Albanesi and Oli e i (2009) o di e en ypes o jobs (F ancois 1998;Dolado e al. 2013)
and hey display mul iple sel - ul illing equilib ia, some o which exhibi gende gaps.
In addi ion, Cella and Manzoni (2023) analyze how disc imina ion in elec o al con es s
p oduce ewe emale candida es and ewe emale elec ed poli icians, e en hough in
a e age hey a e mo e compe en han hei male coun e pa s. This pape p oposes a
Economies 2024,12, 101. h ps://doi.o g/10.3390/economies12050101 h ps://www.mdpi.com/jou nal/economies
Economies 2024,12, 101 2 o 18
heo e ical analysis o he gende disc imina ion ha o igina es in he supply side o he
labo ma ke wi h he aim o p o iding new insigh s ha would complemen he ex ensi e
empi ical li e a u e on he subjec .
Fi s , we b ie ly s a e some ac s abou he gende gaps obse ed in he labo ma ke .
Ac oss he EU, he gende employmen gap ( he di e ence be ween he employmen a es
o men and women o wo king age: 20–64 yea s) was 10.8 pe cen age poin s in 2021,
meaning ha he p opo ion o men o wo king age in employmen exceeded ha o
women by 10.8 pe cen age poin s. Women end o wo k less hou s and hey a e mo e likely
o engage in low paid and in o mal wo k and in pa - ime jobs. Ma e ni y lea es ha e a
long- un nega i e e ec on he pa icipa ion o women in he labo o ce (Be and 2020;
Isen e al. 2017).
The disp opo iona e ep esen a ion o women in low paid and in o mal wo k also
con ibu es o he obse ed gende ea nings gap. The gende ea nings gap measu es he
impac o he h ee combined ac o s ( he a e age hou ly ea nings, he mon hly a e age
numbe o hou s paid and he employmen a e) on he a e age ea nings o all women o
wo king age compa ed wi h men. In 2018, he gende o e all ea nings gap was 36.2% in
he EU. Ac oss Membe S a es, he gende o e all ea nings gap a ied signi ican ly ( om
20.4% in Li huania and Po ugal o 44.2% in Aus ia).
The cu en gende -unbalanced si ua ion can be explained by causes ha a e ela ed
o speci ic gende condi ions o he supply and demand in he labo ma ke . Rega ding he
causes ha p oduce a low demand o women in some p o essions, i is impo an o e e
o di e en kinds o disc imina ion ha o igina ed in he decisions made by he employe s
ha a e biased in a o o men ela i e o women. Some kinds o disc imina ion a e based
on he ac ha mos employe s a e men and hus i is possible ha he men a e mo e likely
o p e e men as employees o cowo ke s ( as e-based disc imina ion). The as e-based
disc imina ion model was i s p oposed by Becke (1957,1971) and Schelling (1971). Gi en
ha in he pas he emale labo o ce has been disp opo iona ely low wi h espec o
he male one, employe s ha e had mo e expe ience wi h male employees and hus ha e
mo e in o ma ion abou he cha ac e is ics o male labo o ce (s a is ical disc imina ion).
The heo y o s a is ical disc imina ion was pionee ed by A ow (1973), Spence (1973) and
Phelps (1972).
This pape ocuses only on he ac o s ha a ec he emale labo supply ha o igina e
in he indi idual’s commi men o p o iding unpaid ca e wo k, ha is “All unpaid se ices
p o ided by indi iduals wi hin a household o communi y o he bene i o i s membe s, including
ca e o pe sons and domes ic wo k. Common examples include cooking, cleaning, collec ing wa e
and uel and looking a e child en, olde pe sons and pe sons wi h illness o disabili ies. Volun a y
communi y wo k ha suppo s pe sonal o household ca e, such as communi y ki chens o childca e,
a e also o ms o unpaid ca e wo k” as de ined by he Uni ed Na ions (UN Women 2022). The
Uni ed Na ions epo also claims ha unpaid ca e wo k is basically p o ided by women:
“Women and gi ls ha e disp opo iona e esponsibili y o unpaid ca e and domes ic wo k; globally
hey spend h ee imes as much ime on his wo k as do men and boys. Unpaid ca e wo k is one o
he main ba ie s p e en ing women om mo ing in o paid employmen and be e quali y jobs.”.
Unpaid ca e wo k is absolu ely necessa y o a wellbeing o he socie y (Folb e 2001).
The ca e, nu u e and educa ion o child en is he basis o he g ow h o he economy
and he e olu ion o he socie y. The ca e o elde s is becoming mo e impo an o e
ime. Ma e ni y is uncondi ionally needed o he su i al o ou socie ies. Thus, i is
o g ea impo ance o s udy he ole o unpaid ca e wo k in he economy. Unpaid ca e
wo k akes up a g ea amoun o ime and i s esponsibili y alls disp opo iona ely on
women (Samman e al. 2016). The indi iduals, whe he male o emale, ha assume his
esponsibili y will expe ience a es ic ion in he quan i y and quali y o hei labo supply.
This es ic ion can be hough o in e ms o he amoun o ime ha hey can o e o hei
p o essional ac i i ies. The decision o how much ime one should de o e o unpaid ca e
wo k ela i e o p o essional wo k is complica ed o mos people since bo h ac i i ies a e
conside ed as e y ele an o e en necessa y (Folb e 2001). This pape aims a analyzing
Economies 2024,12, 101 3 o 18
he implica ions o such decisions. By making he economic cos s and bene i s o ca e
p o ision mo e isible, we migh be able o ind mechanisms ha alle ia e he gende gaps
obse ed in he labo ma ke .
This pape desc ibes a o mal model o indi idual choice in which he choice a iable
is he amoun o ime ha an indi idual decides o de o e o he labo ma ke . This decision
a ec s he o al amoun o income ha an indi idual may ob ain and also he o al amoun
o cos ha an indi idual has o bea depending on he o his p e ious commi men s o
unpaid ca e wo k. I assume ha di e en indi iduals ha e di e en p opensi ies o engage
in amily ca e; hus, he socie y exhibi s a a ie y o cos s associa ed o hem. Howe e ,
in he eal wo ld i is e iden ha in gene al women a e e y much commi ed o such
ac i i ies while men a e much less commi ed. Thus, women a e expec ed o be mo e
a ec ed by he cos s o igina ing in amily commi men s.
The esul s o e a speci ic explana ion o he lowe labo ma ke pa icipa ion o
women, o he sala y gap ha is obse ed in he labo ma ke due o unpaid ca e wo k
commi men s and i also shows he educ ion in o e all wel a e su e ed by he emale
sec o o he socie y due o bo h he lowe income le els and he highe cos s om amily
commi men s. The esul s ob ained also imply ha he e ec s o unpaid ca e wo k commi -
men s canno be a oided. And since hey a e indispensable o he wellbeing o he socie y,
he only possible way o diminish he disc imina ion agains women and o a ain a mo e
gende -balanced labo ma ke is o induce men o commi o ake amily esponsibili ies.
The nex sec ion in oduces he o mal model. Sec ion 3desc ibes he op imal indi-
idual choices o labo pa icipa ion. Sec ion 4analyzes he e ec s o disc imina ion in
he socie y and compa es he e ec s o disc imina ion be ween wo sec ions o he socie y:
emale and male. Sec ion 5o e s a discussion o he implica ions o he esul s which
includes some policy implica ions de i ed om he analysis o he o mal model. Finally,
Sec ion 6con ains some concluding ema ks and possible ex ensions.
2. The Model
This model conside s he choice o an indi idual abou he o his pa icipa ion in
he labo ma ke . Le
s∈[0, 1]
deno e he co esponding choice a iable and i is o be
in e p e ed as ollows: la ge alues o
s
deno e highe le els o labo ma ke pa icipa ion,
which could be hough o as ime de o ed o wo k bu may include o he conside a ions
such as he quali y o he ime de o ed o wo k, o he quali y o he jobs ha may be
a ained. And o cou se, highe alues o he choice a iable co espond o highe alues o
he indi idual’s labo income.
Indi iduals a e cha ac e ized by hei ype. The ype o an indi idual is a measu e
o he o his le el o commi men o ac i i ies ha a e ela ed o he labo ma ke ela i e
o he indi idual’s commi men o unpaid ca e wo k. Le
∈[0, 1]
deno e he indi idual
ype and i is o be in e p e ed as he amoun o ime ha an indi idual can de o e o
p o essional ac i i ies ha is ee o he cos de i ed om amily commi men s. This implies
ha lowe alues o
e e o indi iduals wi h s ong amily commi men s, and highe
alues o
co espond o indi iduals wi h ew amily commi men s. Fo an indi idual o
ype
, de o ing an amoun o ime la ge han
o he labo ma ke implies a educ ion
in he o his commi men o amily ac i i ies, and i ep esen s a educ ion in he o his
wel a e. In pa icula , i an indi idual o ype
chooses o dedica e an amoun o ime
s=
o he labo ma ke , his indi idual does no su e any addi ional cos . Howe e , i an
indi idual o ype
chooses o dedica e an amoun o ime
s>
o he labo ma ke , his
indi idual will su e a cos de i ed om he o his amily es ic ions.
This cos can be hough o as he amoun o income ha has o be de o ed o pay
someone else o he p o ision o ca e, o i can be hough as he loss p oduced due o he
diminished ca e on o he amily membe s. I is easonable o hink ha his cos is small
o small de ia ions om he ype because i p oduces a small dis o ion in he household
o ganiza ion. Howe e , i is also easonable o hink ha i becomes inc easingly la ge o
la ge de ia ions om he ype, because la ge dis o ions in he household o ganiza ion
Economies 2024,12, 101 4 o 18
may equi e d as ic solu ions. In o de o o malize his a gumen I assume ha his cos
is ep esen ed by a con ex unc ion o he dis ance be ween he indi idual’s ype and he
chosen le el o dedica ion o labo ma ke ac i i ies. Le
C(s) = γ
2( −s)2
and
γ>
0 deno e
he cos o labo ma ke pa icipa ion o an indi idual o ype .
Indi iduals ob ain hei income om hei pa icipa ion in he labo ma ke . We
assume ha his income inc eases wi h he indi idual’s le el o pa icipa ion in he labo
ma ke . The indi idual income is ep esen ed by
W(s) = ωs
whe e
ω>
0 deno es he
income pe uni o ime de o ed o he labo ma ke ha an indi idual may ob ain. In
pa icula ,
ω
also e e s o he o al income ha he indi idual ob ains when he o she
decides on ull- ime pa icipa ion in he labo ma ke , ha is s=1.
The o e all indi idual’s wel a e is measu ed by a u ili y unc ion ha combines he
indi idual’s income and he cos ha he o she has o bea and i is ep esen ed by he
ollowing unc ion:
U(s) = W(s)−C(s) = ωs−γ
2( −s)2
La ge alues o he cos pa ame e ela i e o he income pa ame e imply la ge
educ ions in wel a e. Thus,
γ
ω
can be in e p e ed as a measu e o he le el o disc imina ion
ha an indi idual su e s due o he o his amily commi men s. Since he main ade-o is
de e mined by he ela ionship be ween he pa ame e s
ω
and
γ
, wi hou loss o gene ali y
I no malize he wage o be
ω=
1. This implies ha he in e p e a ion o he pa ame e
γ
is
he weigh o he cos ela i e o he maximal wage and he e o e we can conside
γ
as he
disc imina ion index de i ed om unpaid ca e wo k commi men s.
I analyze he indi idual’s op imal choice o labo ma ke pa icipa ion as a unc ion o
i s ype and o he disc imina ion index
γ
. In pa icula , I am in e es ed in he e ec s ha
γ
may ha e on he op imal indi idual choice and how i a ec s he o his labo income and
o al wel a e. I also analyze he e ec s o he disc imina ion index on he o e all socie y by
conside ing he agg ega e le els o labo ma ke pa icipa ion, income and wel a e ha he
socie y may ob ain, and also he agg ega e le el o cos ha he socie y has o bea . Finally
and mos impo an ly, I analyze hese e ec s on wo seg ega ed sec ions o he socie y:
emale and male. This dis inc ion is impo an acco ding o he empi ical da a since i is he
emale sec ion o he socie y ha bea s mos o cos s de i ed om he amily commi men s
o mo e gene ally om he unpaid ca e wo k. Thus, I compa e he e ec s o disc imina ion
be ween he emale and male sec ions o he socie y.
3. The Op imal Indi idual Choice
In o de o ind he op imal le el o labo ma ke pa icipa ion o an indi idual o
ype
∈[0, 1]
, I sol e he maximiza ion p oblem o he o his u ili y unc ion wi h espec
o he choice a iable s∈[0, 1]. This esul is s a ed in he nex p oposi ion.
P oposi ion 1. The op imal le el o labo ma ke pa icipa ion o an indi idual o ype is
s∗( ,γ)=(1
γ+ i ≤γ−1
γ
1i ≥γ−1
γ
All p oo s a e elega ed o Appendix A.
No ice ha he op imal le el o labo ma ke pa icipa ion o all ypes is posi i e; hus,
e e yone chooses o de o e some ime o wo k. Indi iduals wi h la ge ypes decide o
de o e mo e ime o he labo ma ke and only he highes ypes decide o ully pa icipa e
in he labo ma ke , ha is s∗( ,γ)=1.
No ice ha
γ−1
γ
inc eases wi h
γ
. Thus, la ge alues o
γ
imply smalle p opo ions
o indi iduals ha choose a ull labo ma ke pa icipa ion. Fo 0
<γ<
1, we ha e ha all
indi iduals decide o ully pa icipa e in he labo ma ke and ob ain he maximal wage,
and hus hey decide o bea all he cos ha is equi ed o i . E en i in his case all ypes
ob ain he same income, hey do no ob ain he same u ili y, because each indi idual has
Economies 2024,12, 101 5 o 18
o bea a di e en cos . Thus, we also ha e some disc imina ion: lowe ypes bea highe
cos s and ob ain lowe u ili y le els.
Howe e , ull emale pa icipa ion in he labo ma ke is no wha we obse e happen-
ing in he eal wo ld. Since we aim a explaining he exis ing gende -unbalanced ea u es
o he eal labo ma ke o mos o he pape we conside ha 1
<γ
. In his case, we ha e
ha only some indi iduals, hose wi h highe ypes, decide o ully pa icipa e in he labo
ma ke and ob ain he maximal wage. Ins ead indi iduals wi h lowe ypes op o pa ial
labo ma ke pa icipa ion, which is exac ly wha he empi ical esul s show. The e o e, o
mos o he pape we assume ha 1
<γ
, which implies ha he commi men o unpaid
wo k is conside ed as e y impo an o all indi iduals ela i e o he labo income. This
assump ion is elaxed owa ds he end o he pape , when we conside possible policies
ha may alle ia e he exis ing gende gaps in he eal labo ma ke .
Assump ion 1. γ>1.
P oposi ion 1 highligh s he wo old ele ance o he disc imina ion in he indi idual’s
op imal decision abou labo ma ke pa icipa ion: i de e mines which pa o he socie y
decides o pa icipa e ull ime in he labo ma ke and i also explains he dis ibu ion o
he pa - ime jobs de i ed om he indi iduals’ op imal choice.
On he one hand, he a io
γ−1
γ
cha ac e izes all he indi idual ypes whose decision
on how much o pa icipa e in he labo ma ke is nega i ely a ec ed by he cos bea ing
o amily commi men s. All ypes wi h alues below
γ−1
γ
op o a pa ial labo ma ke
pa icipa ion and hus su e om he e ec s o he disc imina ion because hey a e no able
o ob ain he ull income. While all ypes wi h alues abo e
γ−1
γ
op o a ull labo ma ke
pa icipa ion and ob ain he ull income om i . No ice ha la ge le els o disc imina ion
imply la ge se s o ypes ha decide on a pa ial labo ma ke pa icipa ion.
On he o he hand, he disc imina ion index also a ec s he le el o pa icipa ion in
he labo ma ke o hose indi iduals ha op o pa ial labo ma ke pa icipa ion: hei
op imal choice o labo ma ke pa icipa ion dec eases wi h
γ
. Thus, la ge alues o
γ
imply ha mos indi iduals decide o wo k less.
The gains and losses p oduced on he indi iduals’ o al wel a e a e ep esen ed by
he indi idual’s income, cos and u ili y compu ed a he op imal le el o labo ma ke
pa icipa ion gi en by p oposi ion 1 and hey a e s a ed in he nex p oposi ion.
P oposi ion 2. The op imal income ob ained by an indi idual o ype is
W∗( ,γ) = (1
γ+ i ≤γ−1
γ
1i ≥γ−1
γ
The op imal cos suppo ed by an indi idual o ype is
C∗( ,γ) = (1
2γi ≤γ−1
γ
γ
2(1− )2i ≥γ−1
γ
The op imal u ili y ob ained by an indi idual o ype is
U∗( ,γ) = (1
2γ+ i ≤γ−1
γ
1−γ
2(1− )2i ≥γ−1
γ
Obse e ha he indi idual income and he indi idual u ili y ecei ed by all ypes a e
posi i e and hey bo h inc ease wi h he ype o hose indi iduals ha op o a pa ial
labo ma ke pa icipa ion, who in u n ob ain only a ac ion o he ull income. Indi iduals
wi h a la ge ype
≥γ−1
γ
decide o ully pa icipa e in he labo ma ke ; hey ob ain
Economies 2024,12, 101 6 o 18
he maximal wage
ω
, and hei indi idual u ili y inc eases wi h hei ype. Howe e , his
inc ease becomes smalle o la ge alues o .
The indi idual cos suppo ed by all ypes is also posi i e. The cos suppo ed by he
indi iduals ha op o a ull labo ma ke pa icipa ion dec eases wi h he o his ype, and
his educ ion becomes smalle o la ge ypes. And all hose ypes ha op o a pa ial
labo ma ke pa icipa ion end up suppo ing he exac same o al amoun o indi idual
cos (see Figu e 1).
Figu e 1: Op imal indi idual decisions.
Figu e 2: Compa ing o al income and o al u ili y o emale and male ypes.
Op imal labo ma ke pa icipa ion
01
1
"
1
" − 1
"
$∗%
Op imal income dis ibu ion
01
1
1
"
"∗#
$ − 1
$
Op imal cos dis ibu ion
01
1
2#
!∗"
# − 1
#
Op imal wel a e dis ibu ion
01
1
2#
1
1 − 1
2$
$∗%
# − 1
#
TWF(!)
!
1 2
TUF(!)
!
1 2
1/2
3
3/8
1/8
TWM(!)TUM(!)
Figu e 1. Op imal indi idual decisions.
Inc eases in he disc imina ion index
γ
p oduce dec eases in he indi idual income,
in he indi idual cos and in he indi idual u ili y o hose ypes ha choose a pa ial
labo ma ke pa icipa ion. No ice ha on he one hand, hey decide o wo k less when
he disc imina ion index inc eases, and ha is why hey ob ain a lowe income. Howe e ,
hei indi idual cos also inc eases and o e all i p oduces a la ge educ ion in indi idual
u ili y. The indi idual income o he la ge ypes, hose ha choose a ull labo ma ke
pa icipa ion, is no a ec ed by inc eases in he cos pa ame e . Howe e , hei indi idual
cos inc eases and hus hei indi idual u ili y also dec eases. The e o e, inc eases in he
alue o
γ
imply lowe u ili y o all ypes. The o mal desc ip ion o he compa a i e
s a ics is included in he p oo o he p oposi ion con ained in Appendix A.
4. Disc imina ion in Socie y
In his sec ion, we p esen and discuss he main esul s. Fi s , we ocus on he agg e-
ga ed economic a iables o he socie y and hen we compa e hese a iables o he wo
sec ions o he socie y: emale and male ypes. Suppose ha he indi idual ypes in he
socie y a e dis ibu ed acco ding o a uni o m p obabili y dis ibu ion unc ion o e he
suppo
[0, 1]
. We compu e he labo ma ke pa icipa ion o he socie y
T∗(γ)
, o al income
o he socie y
TW∗(γ)
, he o al cos de i ed om labo pa icipa ion
TC∗(γ)
and he o al
u ili y ob ained TU∗(γ)as s a ed in he nex p oposi ion.
Economies 2024,12, 101 7 o 18
P oposi ion 3. The labo ma ke pa icipa ion, o al income, o al cos and o al u ili y o he
socie y a e:
T∗(γ)=TW∗(γ)=γ2+2γ−1
2γ2
TC∗(γ)=3γ−2
6γ2
TU∗(γ)=3γ2+3γ−1
6γ2
No ice ha since we ha e no malized he wage o be 1 we now ha e ha he measu e
o he labo ma ke pa icipa ion is equal o he measu e o he o al income. We ind ha
he labo ma ke pa icipa ion and o al income dec ease wi h
γ
because lowe ypes choose
o dedica e less ime o wo k when i s associa ed cos becomes mo e expensi e. To al u ili y
dec eases wi h
γ
because an inc ease in he cos pa ame e educes he indi idual u ili y
o all ypes: lowe ypes wo k less and highe ypes pay a highe cos .
The o al cos may inc ease o dec ease wi h
γ
depending on he alue o he disc imi-
na ion. Recall ha he indi idual cos o lowe ypes dec eases wi h he cos pa ame e
while he indi idual cos o highe ypes inc eases wi h he cos pa ame e . O e all, we
ha e ha he o al cos inc eases wi h
γ
i he disc imina ion is e y low
1<γ<4
3
because in his case mos indi iduals decide o ully pa icipa e in he labo ma ke and
hus hei indi idual cos inc eases wi h
γ
. Howe e , he o al cos dec eases wi h
γ
i
he disc imina ion is mo e se e e
γ>4
3
because in his case mo e indi iduals op o a
pa - ime pa icipa ion in he labo ma ke and hey educe hei pa icipa ion when he
cos pa ame e inc eases. This implies ha he educ ion in cos due o he educ ion in
pa - ime labo ma ke pa icipa ion compensa es o he inc ease in cos ha he la ge
ypes su e . The eason is wo old: he cos ha highe ypes ha e o pay is ela i ely
small compa ed o ha o lowe ypes, and o la ge alues o he disc imina ion index he
p opo ion o ypes ha decide o wo k ull ime is smalle .
Nex , suppose ha he emale ypes a e ep esen ed by hose
such ha 0
≤ ≤1
2
and he male ypes a e ep esen ed by hose
such ha
1
2≤ ≤
1. The eason is ha i is a
ac ha women a e much mo e commi ed o unpaid ca e wo k han men. I compu e he
labo ma ke pa icipa ion, he o al income, he o al cos and he o al u ili y e alua ed
a he op imal indi idual choice o emale ypes (
TF∗(γ)
,
TWF∗(γ)
,
TCF∗(γ)
,
TUF∗(γ))
and o male ypes (
TM∗(γ)
,
TWM∗(γ)
,
TCM∗(γ)
,
TUM∗(γ))
sepa a ely. I analyze how
hese a iables a e a ec ed by changes in he disc imina ion index γ. Then, I compa e he
esul s ob ained o each sec ion o socie y in o de o e alua e he ex en o he e ec o
he gende disc imina ion in he labo ma ke o e pa icipa ion, income, cos and u ili y.
The nex p oposi ion shows he esul s ob ained o he agg ega ed economic a iables
co esponding o he emale ypes. No ice ha o his analysis I ha e o conside wo cases
depending on he alue o he pa ame e
γ
. Fo
γ>
2, I ha e ha all emale ypes decide
o wo k pa ime while o
γ<
2 I ha e ha some emale ypes decide o wo k pa ime
and some o hem decide o wo k ull ime.
P oposi ion 4. The labo pa icipa ion, o al income, o al cos and o al u ili y o emale ypes a e:
TF∗(γ)=TWF∗(γ)=(2γ−1
2γ2i γ≤2
4+γ
8γi γ≥2
TCF∗(γ)=(24γ−16−γ3
48γ2i γ≤2
1
4γi γ≥2
Economies 2024,12, 101 8 o 18
TUF∗(γ)=(γ3+24γ−8
48γ2i γ≤2
2+γ
8γi γ≥2
Recall ha o
γ≥
2 all emale ypes wo k pa ime, while o
γ≤
2 some o hem
decide o wo k ull ime. In all cases, he o al income and he o al u ili y o emale ypes
dec ease wi h he disc imina ion index, as we ound in he o e all socie y. Fo
γ≥
2,
we ha e ha he o al cos o emale ypes dec eases wi h
γ
, because as we ha e seen
be o e pa - ime wo ke s dec ease hei labo ma ke pa icipa ion when he cos pa ame e
inc eases; hus, in his case he o al cos is educed. Fo
γ≤
2, we ha e ha some emale
ypes wo k ull ime. In his case, he, compa a i e s a ics o he emale ypes esembles
e y much ha o he socie y o e all: he o al cos o emale ypes dec eases wi h
γ
o
ela i ely small alues o he disc imina ion index (
γ<γ<4
3)
and i inc eases wi h
γ
o
la ge alues o he disc imina ion index. The only di e ence is ha o alues o
γ
such
ha
γ<γ<4
3
he o al cos o he emale ypes inc eases wi h
γ
while he o al cos o
he socie y dec eases wi h
γ
. The eason is ha o alues o
γ
such ha
γ<γ<4
3
he
p opo ion o emale ypes ha decide o wo k ull ime ela i e o he emale popula ion
is no la ge enough and hus he o al cos o he emale ypes dec eases wi h he cos o
he majo i y because o pa - ime wo ke s ha decide o wo k less. While o he same
pa ame e alues, he p opo ion o ull- ime wo ke s in he socie y is la ge enough ela i e
o he o al popula ion and hus he o al cos o he socie y inc eases.
Now I eplica e he p e ious analysis o he male ypes. Again, o his analysis I
ha e o conside wo cases depending on he alue o he pa ame e
γ
. Fo
γ<
2, I ha e all
male ypes deciding o wo k ull ime while o
γ>
2 I ha e ha some male ypes decide
o wo k pa ime.
P oposi ion 5. The labo pa icipa ion, o al income, o al cos and o al u ili y o male ypes a e:
TM∗(γ)=TWM∗(γ)=(1
2i γ≤2
3γ2+4γ−4
8γ2i γ≥2
TCM∗(γ)=(γ
48 i γ≤2
3γ−4
12γ2i γ≥2
TUM∗(γ)=(24−γ
48 i γ≤2
9γ2+6γ−4
24γ2i γ≥2
In his case, we also ha e o conside wo di e en si ua ions. When disc imina ion is
low (
γ≤
2
)
, all male ypes decide o wo k ull ime and he o al income is no a ec ed by
changes in
γ
. To al cos inc eases wi h
γ
because all male ypes a e wo king ull ime, and
he e o e hei o al u ili y dec eases wi h
γ
. O cou se, inc eases o
γ
beyond 2 imply ha
some male ypes decide o educe hei labo ma ke pa icipa ion.
Fo highe alues o he disc imina ion index (
γ>
2
)
, some male ypes decide o
wo k pa ime and in his case we ha e ha o al income and o al u ili y o male ypes
dec ease wi h
γ
while o al cos only dec eases wi h
γ
o la ge alues o he disc imina ion
index (
γ>8
3)
. The only di e ence wi h espec o he compa a i e s a ics o he socie y
is ound o alues o
γ
such ha
4
3<γ<8
3
. Fo hese alues, he o al cos o he male
ypes inc eases wi h
γ
while he o al cos o he socie y dec eases wi h
γ
. The eason is
ha o alues o
γ
such ha
4
3<γ<8
3
he p opo ion o male ypes ha decide o wo k
ull ime ela i e o he male popula ion is la ge enough and hus he o al cos o he
male ypes inc eases wi h he cos o he majo i y because ull- ime wo ke s ha e o pay a
la ge cos . While o he same pa ame e alues, he p opo ion o ull- ime wo ke s in
he socie y is no la ge enough ela i e o he o al popula ion and hus he o al cos o he
socie y dec eases.
Economies 2024,12, 101 15 o 18
TCF∗(γ)=R1
2
0CF∗( ;γ)=R1
2
01
2γd =h1
2γ i1
2
0=1
4γ
TUF∗(γ)=1
21
γ+1
4−1
4γ=1
41
2+1
γ=2+γ
8γ
Compa a i e s a ics:
∂TWF∗(γ)
∂γ =−1
2γ2<0
∂2TWF∗(γ)
∂γ2=1
γ3>0
limγ→∞TWF∗(γ)=1
8
∂TCF∗(γ)
∂γ =−1
4γ2<0
∂2TCF∗(γ)
∂γ2=1
2γ3>0
limγ→∞TCF∗(γ)=0
∂TUF∗(γ)
∂γ =−1
4γ2<0
∂TUF∗(γ)
∂γ =1
2γ3>0
limγ→∞TUF∗(γ)=1
8.
O e all, we ha e ha :
TF∗(γ)=TWF∗(γ)=(2γ−1
2γ2i γ≤2
4+γ
8γi γ≥2
TCF∗(γ)=(24γ−16−γ3
48γ2i γ≤2
1
4γi γ≥2
TUF∗(γ)=(γ3+24γ−8
48γ2i γ≤2
2+γ
8γi γ≥2
∂TWF∗(γ)
∂γ <0
∂TCF∗(γ)
∂γ <0 i γ>γ o some 1 <γ<4
3
∂TUF∗(γ)
∂γ <0 .
P oo o P oposi ion 5. We conside wo sepa a e cases depending on whe he γ≷2.
Case 1: Fo γ≤2 we ha e γ−1
γ≤1
2and:
TM∗(γ)=TWM∗(γ)=R1
1
2WM∗(γ)=R1
1
2d =[ ]1
1
2=1−1
2=1
2
TCM∗(γ)=R1
1
2CM∗(γ)=R1
1
2
γ
2(1− )2d =h−γ
6(1− )3i1
1
2
=γ
48
TUM∗(γ)=1
2−γ
48 =24−γ
48
Compa a i e s a ics:
∂TWM∗(ω,γ)
∂γ =0
∂TCM∗(ω,γ)
∂γ =1
48 >0
∂TUM∗(ω,γ)
∂γ =−1
48 <0.
Case 2: Fo γ≥2 we ha e γ−1
γ≥1
2and:
TWM∗(γ)=R1
1
2WM∗(γ)=R
γ−1
γ
1
21
γ+ d +R1
γ−1
γd =
h1
γ + 2
2iγ−1
γ
1
2
+[ ]1
γ−1
γ
=3γ2+4γ−4
8γ2
TCM∗(γ)=R1
1
2CM∗(γ)=R
γ−1
γ
1
2
1
2γd +R1
γ−1
γ
γ
2(1− )2d =
h1
2γ iγ−1
γ
1
2
+h−γ
6(1− )3i1
γ−1
γ
=3γ−4
12γ2
TUM∗(γ)=3
8+γ−1
2γ2−3γ−4
12γ2=9γ2+6γ−4
24γ2
Economies 2024,12, 101 16 o 18
Compa a i e s a ics:
∂TWM∗(γ)
∂γ =2γ2−4γ(γ−1)
2γ2=2−γ
2γ3<0 i γ>2 which always holds in his case.
∂2TWM∗(γ)
∂γ2=γ−3
γ4>0 i γ>3
limγ→∞TWM∗(γ)=3
8
∂TCM∗(γ)
∂γ =3γ−2(3γ−4)
12γ3=8−3γ
12γ3<0 i γ>8
3
∂2TCM∗(γ)
∂γ2=γ−4
2γ4>0 i γ>4
limγ→∞TCM∗(γ)=0
∂TUM∗(γ)
∂γ =4−3γ
12γ3<0 i γ>4
3which always holds in his case.
∂2TUM∗(γ)
∂γ2=γ−2
2γ4>0 i γ>2
limγ→∞TUM∗(γ)=3
8
O e all, we ha e ha :
TM∗(γ)=TWM∗(γ)=(1
2i γ≤2
3γ2+4γ−4
8γ2i γ≥2
TCM∗(γ)=(γ
48 i γ≤2
3γ−4
12γ2i γ≥2
TUM∗(γ)=(24−γ
48 i γ≤2
9γ2+6γ−4
24γ2i γ≥2
and
∂TWM∗(γ)
∂γ ≦0 i γ≷2
∂TCM∗(γ)
∂γ <0 i γ>8
3
∂TUM∗(γ)
∂γ <0 o all γ.
P oo o P oposi ion 6. We conside wo sepa a e cases depending on whe he γ≷2.
Case 1: Fo γ≤2 we ha e:
TF∗(γ)
TM∗(γ)=TWF∗(γ)
TWM∗(γ)=
2γ−1
2γ2
1
2
=2γ−1
γ2∈3
4, 1<1
TCF∗(γ)
TCM∗(γ)=
24γ−16−γ3
48γ2
γ
48
=24γ−16−γ3
γ3∈[3, 7]
TUF∗(γ)
TUM∗(γ)=
24γ−8+γ3
48γ2
24−γ
48
=24γ−8+γ3
24γ2−γ3∈h6
11 ,17
23 i
and
∂TWF∗(γ)
TWM∗(γ)
∂γ =2(1−γ)
γ3<0; TWF∗(1)
TWM∗(1)=1; TWF∗(2)
TWM∗(2)=3
4
∂TCF∗(γ)
TCM∗(γ)
∂γ =48(1−γ)
γ4<0; TCF∗(1)
TCM∗(1)=7; TCF∗(2)
TCM∗(2)=3
TUF∗(1)
TUM∗(1)=17
23 ;TCF∗(2)
TCM∗(2)=6
11
∂TUF∗(γ)
TUM∗(γ)
∂γ =24γγ3+2γ2−25γ+16
(24γ2−γ3)2<0 i γ3+2γ2−25γ+16 <0
No ice ha o γ=1 we ha e ha γ3+2γ2−25γ+16 =−6<0;
o γ=2 we ha e ha γ3+2γ2−25γ+16 =−18 <0
and in addi ion we ha e ha γ3+2γ2−25γ+16 dec eases wi h γ o 1 <γ<2.
Case 2: Fo γ≥2 we ha e:
TWF∗(γ)
TWM∗(γ)=
4+γ
8γ
3γ2+4γ−4
8γ2
=4γ+γ2
3γ2+4γ−4∈h1
3,3
4i
TCF∗(γ)
TCM∗(γ)=
1
4γ
3γ−4
12γ2
=3γ
3γ−4∈[1, 3]
Economies 2024,12, 101 17 o 18
TUF∗(γ)
TUM∗(γ)=
2+γ
8γ
9γ2+6γ−4
24γ2
=6γ+3γ2
9γ2+6γ−4∈h1
3,6
11 i
and
∂TWF∗(γ)
TWM∗(γ)
∂γ =−42+γ+γ2
(3γ2+4γ−4)2<0<0; TWF∗(2)
TWM∗(2)=3
4; limγ→∞
TWF∗(γ)
TWM∗(γ)=1
3
∂TCF∗(γ)
TCM∗(γ)
∂γ =−12
(3γ−4)2<0; TCF∗(2)
TCM∗(2)=3; limγ→∞
TCF∗(γ)
TCM∗(γ)=1
∂TUF∗(γ)
TUM∗(γ)
∂γ =−22+2γ+3γ2
(9γ2+6γ−4)2<0; TUF∗(2)
TUM∗(2)=6
11 ; limγ→∞
TUF∗(γ)
TUM∗(γ)=1
3
O e all, we ha e:
TF∗(γ)
TM∗(γ)=TWF∗(γ)
TWM∗(γ)∈h1
3, 1i
TCF∗(γ)
TCM∗(γ)∈[1, 7]
TUF∗(γ)
TUM∗(γ)∈h1
3,17
23 i
and
∂TWF∗(γ)
TWM∗(γ)
∂γ <0;
∂TCF∗(γ)
TCM∗(γ)
∂γ <0;
∂TUF∗(γ)
TUM∗(γ)
∂γ <0.
P oo o P oposi ion 7. Fo γ≤1 we ha e ha s∗( ,γ)=1 and
W∗( ,γ) = 1
C∗( ,γ) = γ
2(1− )2
U∗( ,γ) = 1−γ
2(1− )2.
Fo emales ypes we ha e:
TF∗(γ)=TWF∗(γ)=R1
2
0WF∗( ;γ)=R1
2
0d =[ ]1
2
0=1
2
TCF∗(γ)=R1
2
0CF∗( ;γ)=R1
2
0
γ
2(1− )2d =h−γ
6(1− )3i1
2
0=7γ
48
TUF∗(γ)=1
2−7γ
48 =24−7γ
48
Fo male ypes we ha e:
TM∗(γ)=TWM∗(γ)=R1
1
2WM∗(γ)=R1
1
2d =[ ]1
1
2=1
2
TCM∗(γ)=R1
1
2CM∗(γ)=R1
1
2
γ
2(1− )2d =h−γ
6(1− )3i1
1
2
=γ
48
TUM∗(γ)=1
2−γ
48 =24−γ
48
Compa ing emale and male ypes we ob ain:
TWF∗(γ)
TWM∗(γ)=1
2
1
2
=1
TCF∗(γ)
TCM∗(γ)=
7γ
48
γ
48
=7
TUF∗(γ)
TUM∗(γ)=
24−7γ
48
24−γ
48
=24−7γ
24−γ
wi h
TUF∗(γ)
TUM∗(γ)
γ=−144
(24−γ)2<0, TUF∗(1)
TUM∗(1)=17
23 , and TUF∗(0)
TUM∗(0)=1.
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