Thielen, Clemens; Weinschenk, Philipp
A icle — Published Ve sion
Analyzing he e ec s o minimum wages: a mic oeconomic
app oach
Economic Theo y
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Sugges ed Ci a ion: Thielen, Clemens; Weinschenk, Philipp (2024) : Analyzing he e ec s o minimum
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Economic Theo y (2025) 79:945–991
h ps://doi.o g/10.1007/s00199-024-01607-3
RESEARCH ARTICLE
Analyzing he e ec s o minimum wages: a mic oeconomic
app oach
Clemens Thielen1·Philipp Weinschenk2
Recei ed: 18 Oc obe 2022 / Accep ed: 9 Sep embe 2024 / Published online: 22 No embe 2024
© The Au ho (s) 2024
Abs ac
We use a mic oeconomic app oach o analyze he e ec s o minimum wages. Agen s
a e allowed o ha e di e en p oduc i i ies a di e en p incipals as well as di e en
cos s o wo king. We ob ain se e al new and in e es ing e ec s. Minimum wages could
in luence he gene a ed su plus when lea ing employmen una ec ed, and des oy jobs
ha gene a e ela i ely high le els o su plus when a ec ing employmen . Fu he -
mo e, minimum wages could ha m agen s e en i hese s ay employed, while p incipals
could bene i om hem. We p o ide a comple e cha ac e iza ion o he e ec s and
show ha hese hold independen ly o he speci ic ba gaining p ocedu e and in o ma-
ion s uc u e.
Keywo ds Minimum wages ·P incipal-agen model ·Cos s o wo king ·Wel a e
e ec s
JEL Classi ica ion C78 ·D21 ·J31 ·J38 ·K31
1 In oduc ion
The insigh ha wo ke s’ occupa ional decisions depend no only on mone a y com-
pensa ion, bu also on nonmone a y job cha ac e is ics, is widely accep ed.1Employe s
migh ake hese nonmone a y cha ac e is ics in o conside a ion when making wage
1This insigh —al eady concei ed by Smi h (1937)—is, o example, he key assump ion in he li e a u e on
compensa ing wage di e en ials and has ecen ly gained much a en ion, c . Kaplan and Schulho e -Wohl
(2018).
BPhilipp Weinschenk
[email p o ec ed]
Clemens Thielen
[email p o ec ed]
1Campus S aubing o Bio echnology and Sus ainabili y, Technical Uni e si y o Munich,
Am Essigbe g 3, 94315 S aubing, Ge many
2Depa men o Business and Economics, Uni e si y o Kaise slau e n-Landau, Go lieb-Daimle -S .,
67663 Kaise slau e n, Ge many
123
946 C. Thielen, P. Weinschenk
o e s.2S ill, hese aspec s ha e no been ully conside ed when analyzing minimum
wages, which is one o he mos con o e sial, ye impo an opics in economics.3We
con ibu e o closing his gap by conside ing a simple mic oeconomic model whe e an
agen (in e p e ed as a wo ke ) can no only ha e di e en p oduc i i ies a di e en
p incipals (po en ial employe s), bu also di e en cos s o wo king o which p incipals
can adjus hei wage o e s. Agen s a e hus allowed o ha e p e e ences ega ding
hei employmen , and hese p e e ences can be aken in o accoun by p incipals when
o e ing wages. This is impo an since job oppo uni ies usually di e in hei non-
mone a y cha ac e is ics, e.g., how sa is ying, demanding, engaging, haza dous, and
lexible hey a e, and wha commu ing cos s hey cause.4
Ou model eplica es he s anda d esul s we know om he exis ing li e a u e in
case each agen aces he same cos s a all p incipals: Minimum wages edis ibu e
income om p incipals o agen s, such ha p incipals su e , while agen s who s ay
employed bene i , and he e iciency is only a ec ed i employmen is dec eased.
Acco dingly, he e ec s o minimum wages a e a simple ade-o be ween e iciency
(which is only a ec ed i employmen is lowe ed) and edis ibu ion (di ec ed om
employe s o wo ke s).
This is illus a ed in he ollowing example.5Conside a si ua ion wi h wo p in-
cipals and one agen . Le he agen Aha e p oduc i i y θ1=11 when wo king o
p incipal P1, and p oduc i i y θ2=9 when wo king o p incipal P2. The agen ’s
cos s ca e 5 a bo h p incipals. P incipals can make o e s o he agen , who can
P1
P2
A
θ1=11
c=5
θ2=9
c=5
2Despi e he ac ha uck d i e s ea n well abo e he na ional a e age sala y, he U.S. uck indus y
has p oblems inding d i e s (CNN 2015;CNBC2018), which indica es ha ela i ely low-skilled wo ke s
also ca e o nonmone a y job cha ac e is ics. As a eac ion, Walma has inc eased uck d i e sala ies o
$87,500 a yea on a e age (CNBC 2019).
3Acco ding o he (In e na ional Labou O ganiza ion (ILO) 2017), 90% o ILO membe s a es ha e
minimum wages. In he U.S., o example, he e is a con o e sial deba e no only abou he minimum wage
a ede al le el, bu also abou he minimum wage in s a es and ci ies (New Yo k Times 2021a,b,c). The e
is also a discussion in he UK abou he po en ial ise o he minimum wage and li ing wage (Financial
Times 2022).
4Bonhomme and Joli e (2009) empi ically show ha he ype o wo k, he wo king condi ions, he wo king
imes, he job secu i y, and he dis ance o wo k a e indeed impo an job cha ac e is ics o wo ke s. Ca d
e al. (2018) show ha allowing wo ke s o ha e idiosync a ic as es o di e en wo kplaces is help ul o
unde s and many well-documen ed empi ical egula i ies on labo ma ke s.
5In his example as well as in he main pa o he pape , we examine he single-agen case. As we show in
Appendix C, ou esul s ca y o e o he mul i-agen case wi h an a bi a y numbe o agen s i he e a e
no (binding) capaci y cons ain s o he p incipals, and he obse ed e ec s o minimum wages s ill a ise
when capaci y cons ain s a e conside ed.
123
Analyzing he e ec s o minimum wages 947
hen accep one o he o e s o ejec s all o e s. Fo simplici y o exposi ion, we do
no model he agen as a s a egic playe , bu assume he agen o accep he u ili y-
maximizing o e .6In he Nash equilib ium wi hou a minimum wage, he agen wo ks
o P1 o a wage o w=9, which esul s in u ili y w−c=4 o he agen and a su -
plus o θ1−c=6. E e y binding minimum wage w ha lea es he agen employed,
i.e., w∈(9,11], inc eases he agen ’s u ili y ( o w−5) while lea ing he su plus
una ec ed. Highe minimum wages w>11 cause he agen o become unemployed
and lowe he su plus.
The assump ion ha each agen aces he same cos s a all p incipals—which migh
belong o di e en economic sec o s o egions—is s ong and gene ically iola ed.7
We, he e o e, allow agen s o ha e di e en cos s and ob ain new esul s ha , in
pa icula , in alida e he simple ade-o pe spec i e on minimum wages. The main
esul s a e he ollowing.
Fi s , a minimum wage can ad e sely a ec he gene a ed su plus e en i he
employmen le el is unchanged. This e ec a ises in ou model since a minimum wage
could des oy job oppo uni ies wi h ela i ely high le els o su plus while simul ane-
ously main aining oppo uni ies wi h ela i ely low le els o su plus. These wel a e
losses may be ha d o de ec o ou side obse e s (e.g., poli icians o econome i-
cians), since employmen le els s ay cons an and wages inc ease. We can hus speak
o “hidden cos s” o minimum wages.
Second, an agen can also su e om a minimum wage when emaining employed.
The e a e wo di e en easons o his e ec o eme ge: (i) he minimum wage could
o ce an agen o eloca e o ano he p incipal, which we show o dec ease his8u ili y
gene ically, o (ii) he minimum wage could allow an agen o s ay a he same p incipal,
bu cause a educ ion o he wage paymen . This shows ha minimum wages—which
a e usually in ended o help agen s—can ac ually ha m agen s also when hey s ay
employed.
Some o he basic esul s a e illus a ed in he ollowing example (see igu e on he
nex page), which is iden ical o he p e ious example excep ha he agen ’s cos s
a e di e en (e.g., due o di e en commu ing cos s o wo king condi ions). Wi hou
a minimum wage, he agen wo ks o P2 o a wage o 7,9which esul s in he u ili y
w−c2=4 and he su plus θ2−c2=6. Any minimum wage w∈(9,11)causes he
agen o s ay employed (by eloca ing o P1), bu lowe s he agen ’s u ili y o w−7
and he su plus o θ1−c1=4. In e es ingly, no only agen ’s u ili y lowe s, bu also
agg ega e p o i s.
Relaxing he seemingly innocen assump ion ha each agen has iden ical cos s
a all p incipals hus d as ically changes he e ec s o minimum wages. The esul s
a e obus and hold in a a ie y o di e en se ings; see Sec ions 5and 6and he
appendices o nume ous ex ensions and obus ness checks. Ou analysis p o ides
u he obus e ec s.
6In case o indi e ence, we le he agen beha e acco ding o na u al ie-b eaking ules ha , as we show
in Appendix A.1, exac ly model he agen ’s beha io in any subgame-pe ec Nash equilib ium.
7So kin (2016,2018), o ins ance, documen s ha nonpay cha ac e is ics di e subs an ially be ween
economic sec o s.
8We ollow he s anda d con en ion and alk abou emale p incipals and male agen s.
9P1will o e a wage o 11 so ha P2has o o e a leas 7 o a ac he agen .
123
948 C. Thielen, P. Weinschenk
P1
P2
A
θ1=11
c1=7
θ2=9
c2=3
Fi s , wages may o e shoo . Tha is, equilib ium wages may inc ease s ic ly abo e
he imposed minimum wage.
Second, p incipals may bene i om minimum wages. This e ec a ises when he
minimum wage (i) o ces an agen o eloca e o ano he p incipal o (ii) sh inks he
se o compe ing p incipals i no eloca ion occu s. Case (i) a ises in he example
abo e, whe e P1bene i s om any minimum wage w∈(9,11). Case (ii) a ises in he
example i he agen ’s cos a P1is changed o c1=4. He e, he agen wo ks o P1 o
a wage o 10 wi hou a minimum wage, bu any minimum wage w∈(9,10)dec eases
he wage paid by P1 o wsince P2will no longe compe e.
Thi d, when causing unemploymen , minimum wages may des oy jobs ha gen-
e a e ela i ely high le els o su plus. Acco dingly, he jobs ha a e los due o a
minimum wage a e no necessa ily he ones ha gene a e only ma ginal le els o
su plus, and he e ec o minimum wages on e iciency is no only o second o de .
Fou h, minimum wages can cause an inc ease in equilib ium p oduc i i ies by elo-
ca ing agen s om low o high p oduc i i y jobs. The p oduc i i y gains a e, howe e ,
gene ically o e compensa ed by highe cos s. Thus, he p oduc i i y gains caused by
minimum wages a e gene ically accompanied by e iciency losses.
We s a ou analysis wi h he basic ba gaining p ocedu e whe e he p incipals
make o e s. This simple p ocedu e is s anda d in he p incipal-agen li e a u e bu
qui e speci ic. The e o e, we also conside al e na i e ba gaining p ocedu es as well
as s able ou comes, which abs ac om how pa ies ba gain and wha hey know.
We show ha he se o s able ou comes is gi en by he con ex combina ions o he
ou come ob ained when p incipals make o e s and he ou come when agen s make
o e s. In e es ingly, excep o he bounda y case co esponding o he si ua ion whe e
agen s make o e s, he e ec s o a minimum wage on s able ou comes a e quali a i ely
exac ly he same as in he case whe e he p incipals make o e s.10 The e o e—and his
is impo an —all e ec s o minimum wages we iden i y when p incipals make o e s
a e he consequence o s abili y and no he consequence o he speci ic ba gaining
p ocedu e o in o ma ion s uc u e.
1.1 Rela ed li e a u e
The empi ical li e a u e on minimum wages is ex ensi e; see (Neuma k and Wasche
2008; Manning 2021) o o e iews. Ea ly s udies summa ized by B own e al. (1982)
10 Fo he bounda y case whe e he s able ou come coincides wi h he case whe e agen s make o e s, he
same e ec s a ise as in case whe e he p incipals make o e s, excep ha agen s can ne e bene i om a
minimum wage.
123
Analyzing he e ec s o minimum wages 949
mainly ound nega i e employmen e ec s o minimum wages. This iew was chal-
lenged by he amous s udy o Ca d and K uege (1994), who ind no indica ion o a
educ ion o employmen . The deba e on he employmen e ec o minimum wages
is ongoing and he e is s ill no consensus (Neuma k 2019; Neuma k e al. 2014b,a;
Alleg e o e al. 2011,2017; Dube e al. 2010; Clemens and Wi he 2019; Fang and Lin
2015; Addison e al. 2013; Liu e al. 2016; Thompson 2009; Mu a ye and Oshchep-
ko 2016; Mee and Wes 2016; Wol son and Belman 2019; K eine e al. 2020;
Caliendo e al. 2018; Cengiz e al. 2019; Ha asz osi and Lindne 2019; Aa onson e al.
2018). In he conclusion, we iden i y se e al ac o s ha in luence he employmen
e ec s o minimum wages, which migh help us o unde s and he la ge a ie y o he
empi ical indings.
The e is also a ich heo e ical li e a u e on minimum wages; see Neuma k and
Wasche (2008); Flinn (2010) o o e iews. The adi ional iew is based on he
supply-demand model, also e e ed o as he neoclassical compe i i e model (Mankiw
2017; Ca d and K uege 1995). In i s basic o m, a unique equilib ium wage ob ained
a he in e sec ion o he labo supply and demand cu es is paid o all wo ke s.
A minimum wage is only e ec i e i i is abo e his equilib ium wage. Such an
e ec i e minimum wage lowe s employmen (due o a dec eased labo demand),
causes unemploymen (since labo demand alls sho o labo supply), and lowe s he
gene a ed su plus (by causing a deadweigh loss). Howe e , only he jobs ha gene a e
he lowes le els o su plus a e des oyed. Since only ma ginal wo ke s lose he e jobs,
a minimum wage has only a second-o de e ec on e iciency (Lee and Saez 2012,
page 739). O e all, a minimum wage bene i s wo ke s who s ay employed, bu ha ms
employe s and wo ke s who become unemployed.
Ano he class o models akes in o accoun ha employe s could ha e ma ke powe .
As explained by Robinson (1933) and S igle (1946), a monopsonis op imally chooses
an employmen le el below he compe i i e equilib ium in o de o educe i s o al wage
paymen . A minimum wage emo es he monopsonis ’s incen i e o keep employmen
a i icially low and, hus, inc eases he wage ( o he minimum wage) and he employ-
men . This is bene icial o he wo ke s and he gene a ed su plus (since he deadweigh
loss is educed) bu ha m ul o he employe . In he ela ed se ing o monopsonis ic
compe i ion s udied by Bhaska and To (1999), a minimum wage also inc eases i m-
le el employmen , bu may a he same ime cause an exi o i ms, hus leading o
ambiguous e ec s on he agg ega e employmen le el.
A hi d class o models is based on sea ch heo y. In hese models, he wage o e
dis ibu ion eme ges as he equilib ium o a noncoope a i e wage sea ch and wage
pos ing game be ween wo ke s and employe s, and a minimum wage changes he
game. See Flinn (2010) o a comp ehensi e o e iew. In a seminal pape , Bu de
and Mo ensen (1998) show ha a minimum wage inc eases employmen and shi s he
equilib ium wage o e dis ibu ion o he igh . Hence, he common esul o he exis -
ing li e a u e, acco ding o which employed agen s bene i om a binding minimum
wage, also holds he e. This ype o model is ypically e e ed o as “dynamic monop-
sony” since sea ch- ela ed ic ions induce monopsony-like beha io (see Neuma k
and Wasche (2008, Chap e 3.2.2)).
He e ogeneous p e e ences o e job cha ac e is ics a e conside ed by Bhaska e al.
(2002); Bhaska and To (1999,2003) o analyze minimum wages. These au ho s
123
950 C. Thielen, P. Weinschenk
assume ha i ms a e unable o make indi idual o e s o wo ke s ha depend on he
wo ke s’ speci ic p e e ences. Mo eo e , hei wo k ocuses mo e on he agg ega e
i m and indus y employmen e ec s o minimum wages, while ou pape dis inc ly
emphasizes he po en ial he e ogenei y o wage, employmen , and wel a e ou comes.
In ou model, depending on he speci ic scena io, a (highe ) minimum wage may lead
o agen s con inuing o wo k o he same p incipals and acc uing a highe o lowe
u ili y, eloca ing o di e en p incipals and expe iencing a lowe u ili y, o ending up
ou o wo k en i ely—and hese e ec s migh be he e ogeneous among agen s. Thus,
minimum wages can ha e a ich se o e ec s in ou model, and we p o ide a comple e
cha ac e iza ion o hem.
2 Model desc ip ion
We now in oduce ou model o he case o a single agen , which is ex ended o
mul iple agen s in Sec . 6. Conside an agen (in e p e ed as a wo ke ) who can wo k
o one o np incipals (in e p e ed as po en ial employe s). I he agen wo ks o
p incipal i∈N={1,...,n}, his u ili y is
ui=wi−ci,
whe e wiis he wage paid by p incipal iand ciis he agen ’s cos when wo king
o i.11 The cos depend on he job cha ac e is ics a p incipal i, e.g., commu ing
cos s, ype o wo k, wo king condi ions, wo king imes, and job secu i y. I he agen
does no wo k o any p incipal, we say he agen is unemployed and his u ili y is
u0=0.12 The cos ciis hus he agen ’s oppo uni y cos , i.e., he paymen o which
he is indi e en be ween wo king o p incipal iand no wo king. We analyze bo h
he case o un es ic ed wages and he case o es ic ed wages, whe e a minimum
wage w equi es ha wi≥w. We unde line all a iables in case o es ic ed wages.
The p o i o p incipal ii he agen wo ks o he is
πi=θi−wi,
whe e θiis he agen ’s p oduc i i y13 a p incipal i, while he p o i is no malized o
ze o i he agen does no wo k o he . We conside only p incipals o which he
11 We can also allow o non-linea u ili y unc ions u(wi,ci)=h(wi)−ci(o some mono one ans-
o ma ion o i ). This is equi alen o he case ui=wi−ciwhen ans o ming he p oduc i i ies ia h.
In e es ingly, i he u ili y unc ion is conca e in wi, some o he possible nega i e e ec s o a minimum
wage on he agen ’s u ili y and on he su plus migh become e en s onge .
12 The case whe e he agen has a nonze o ese a ion u ili y is equi alen o he case wi h ze o ese a ion
u ili y and all cos s inc eased by he ini ial ese a ion u ili y.
13 The p oduc i i y θican be in e p e ed as p incipal i’s g oss p o i i he agen wo ks o he . In case
p oduc i i y is s ochas ic, θiis in e p e ed as expec ed g oss p o i . Simila ly, i he agen ’s cos is s ochas ic,
cican be in e p e ed as expec ed cos .
123
Analyzing he e ec s o minimum wages 951
p oduc i i y exceeds he cos (i.e., θi>ci o all i)14 and le θmax :=maxiθideno e
he maximum p oduc i i y.
The su plus gene a ed when he agen wo ks o p incipal iis
si=πi+ui=θi−ci.
I he agen does no wo k o any p incipal, he su plus is s0=0.
We call he p incipal he agen wo ks o he winning p incipal and le i∗deno e
he index. I he agen does no wo k o any p incipal, we se i∗=0. The emaining
p incipals N {i∗}a e e e ed o as he losing p incipals. The maximum su plus is
smax :=maxisiand Nmax :={i∈N:si=smax}is he se o p incipals whe e his
su plus can be gene a ed.
In case o a minimum wage, we deno e he p incipals who can a o d he minimum
wage wi hou making a loss by N:={i∈N:θi≥w}. The maximum su plus among
hese p incipals is smax :=max{0,si:i∈N}. The se o p incipals who can a o d
he minimum wage and yield his su plus is Nmax :={i∈N:si=smax}.
Recognize ha di e en combina ions o p oduc i i y and cos may no only s em
om p incipals who a e possibly loca ed in di e en geog aphic egions o indus-
ial sec o s, bu also om he possibili y o choosing wo king hou s, in es men s in
wo king condi ions, o e o s; see Appendix D.2. In all hese scena ios, a highe p o-
duc i i y is na u ally associa ed wi h a highe cos , bu no necessa ily wi h a highe
su plus.
3 Basic ba gaining p ocedu e
We s a by conside ing he basic ba gaining p ocedu e whe e he p incipals make ake-
i -o -lea e-i o e s. This p ocedu e is p edominan in he agency li e a u e (c . La on
and Ma imo (2002)) and allows o a simple and in ui i e cha ac e iza ion. As we
show la e , he e ec s o minimum wages we ob ain by using his p ocedu e a e obus .
The p ocedu e is as ollows: Fi s , each p incipal io e s a wage wi∈R o he
agen o makes no o e . In case a minimum wage wis imposed, he o e ed wages
mus sa is y wi≥w. A e ecei ing all o e s, he agen ei he accep s one o he
o e s o ejec s all o e s, and he payo s a e ealized.
We assume ha he p incipals know he pa ame e s o he model, which enables
each p incipal o de e mine he bes esponse o he o he p incipals’ o e s. This
assump ion is no needed when in e p e ing he equilib ium o he basic ba gaining
p ocedu e as he esul o ascending o e s made by p incipals (c . Appendix A.5). I
is no needed ei he when examining s able ou comes (c . Appendix B.2), whe e one
abs ac s om how he pa ies ba gain and wha hey know.
Fo simplici y o exposi ion, we do no model he agen as a s a egic playe ha
ac s a e he p incipals ha e made hei o e s. Ins ead, a ionali y o he agen is
modeled by assuming ha he always accep s an o e ha maximizes his u ili y o
no o e i all o e s p o ide nega i e u ili y. In case o indi e ence, we le he agen
14 P incipals i o which θi≤cia e edundan since employing he agen can ne e yield a posi i e
p o i /u ili y o one o he wo pa ies wi hou yielding a nega i e u ili y/p o i o he o he pa y.
123
952 C. Thielen, P. Weinschenk
beha e acco ding o he ollowing ie-b eaking ules ha , as we show in Appendix A.1,
exac ly model he agen ’s s a egic beha io in any subgame-pe ec Nash equilib ium
o he ex ensi e o m game. Fi s , in case o indi e ence, he agen p e e s o wo k,
i.e., accep an o e ins ead o no o e . Second, i se e al o e s maximize he agen ’s
u ili y, he chooses one ha maximizes he su plus among hese o e s. Thi d, i se e al
o e s maximize he agen ’s u ili y and he su plus, he agen chooses an o e acco ding
o an a bi a y de e minis ic ie-b eaking ule. We concen a e on Nash equilib ia in
undomina ed s a egies.15
Fo some esul s, we equi e he ie-b eaking ule used in case ha se e al o e s
simul aneously maximize he agen ’s u ili y and he su plus o sa is y he p ope y o
independence o i ele an al e na i es:
De ini ion 3.1 A ie-b eaking ule sa is ies independence o i ele an al e na i es i
he ollowing holds o any wo subse s N ⊆N⊆N: I he ie-b eaking ule selec s
p incipal jamong he p incipals in N, i also selec s jamong he p incipals in N
whene e j∈N.
We nex cha ac e ize he pu e-s a egy Nash equilib ia bo h o un es ic ed and
es ic ed wages and hen analyze he e ec s o minimum wages.
3.1 Un es ic ed wages
The ollowing p oposi ion es ablishes he exis ence o a Nash equilib ium. In he
speci ied equilib ium, he p incipals whe e he maximum su plus can be gene a ed
o e wages equal o he second-highes su plus plus he agen ’s cos ,16 while he
emaining p incipals o e wages equal o he p oduc i i y.
P oposi ion 3.1 Fo un es ic ed wages, he s a egy p o ile
wi=max{0,θj−cj:j∈N {i}} + ci o i ∈Nmax
θi o i /∈Nmax
cons i u es a Nash equilib ium yielding he maximum su plus smax.
P oo Obse e ha , o all i∈Nmax and all i/∈Nmax, he o e s in P oposi ion 3.1
yield
ui=wi−ci=max{0,θj−cj:j∈N {i}} ≥ θi−ci=wi−ci=ui.
Hence, ui≥0i∈Nmax and he agen canno do be e han accep ing an o e om
a p incipal i∗∈Nmax such ha he maximum su plus smax is gene a ed.
I emains o show ha no p incipal can imp o e by changing he o e . Each
p incipal i/∈Nmax—i.e., each p incipal whe e he agen canno gene a e he maximum
15 Weakly domina ed s a egies a e discussed in Appendix A.2.
16 In case o n≥2 p incipals, an al e na i e in e p e a ion is ha each p incipal iwhe e he maximum
su plus can be gene a ed o e s a wage equal o he wage o e ed by a bes compe i o j(i.e., a p incipal
wi h he second-highes su plus) plus he cos di e en ial ci−cj.
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Analyzing he e ec s o minimum wages 959
Fig. 1 E ec s on he paid wage
Theo em 4.3 Suppose he e is a minimum wage w≤θmax.
(I) I i∗=i∗and
(1) w>w
i∗, he agen ’s u ili y inc eases, i.e., ui∗>ui∗.
(2) w=wi∗, he agen ’s u ili y emains unchanged, i.e., ui∗=ui∗.
(3) w<w
i∗, he agen ’s u ili y dec eases weakly, i.e., ui∗≤ui∗.
(II) I i∗= i∗, he agen ’s u ili y dec eases weakly, i.e., ui∗≤ui∗.
When he ie-b eaking ule sa is ies independence o i ele an al e na i es, Case (I)
applies i and only i w≤θi∗, and Case (II) applies o he wise.
P oo (I) (1) Follows since w>w
i∗and wi∗≥wsuch ha wi∗>w
i∗as well as
i∗=i∗.
(2) I w=wi∗, hen w−ci∗=wi∗−ci∗. Since i∗=i∗and, by P oposi ion 3.2,
wi∗−ci∗=max{0,θj−cj:j∈N {i∗}}, his yields
w−ci∗=max{0,θj−cj:j∈N {i∗}}.
Because he maximum is no smalle han max{0,θj−cj:j∈N {i∗}},
maxw−ci∗,max{0,θj−cj:j∈N {i∗}}=max{0,θj−cj:j∈N {i∗}}.
By P oposi ions 3.2 and 3.4, his is equi alen o ui∗=ui∗.
(3) I w<w
i∗, he same a gumen a ion as in he p oo o (2) shows ha
w−ci∗<max{0,θj−cj:j∈N {i∗}}.
Since max{0,θj−cj:j∈N {i∗}} ≤ max{0,θj−cj:j∈N {i∗}},
P oposi ions 3.2 and 3.4 imply ha ui∗≤ui∗.
123
960 C. Thielen, P. Weinschenk
(II) P oposi ion 3.2 and i∗= i∗yield
ui∗=max{0,θj−cj:j∈N {i∗}} ≥ θi∗−ci∗≥wi∗−ci∗=ui∗,
whe e he second inequali y ollows since he paid wage wi∗canno exceed θi∗.
The claim ha , when he ie-b eaking ule sa is ies independence o i ele an al e -
na i es, Case (I) applies i and only i w≤θi∗, while Case (II) applies o he wise,
ollows di ec ly om he de ini ion o independence o i ele an al e na i es and he
ac ha i∗∈Nmax i and only i w≤θi∗.
Theo em 4.3 shows ha a minimum wage inc eases he agen ’s u ili y when he
minimum wage exceeds he ini ial wage and he e is no eloca ion. O he wise, he
agen ’s u ili y dec eases a leas weakly. This also implies ha he agen can ne e
bene i om wage o e shoo ing, i.e., he wage inc ease can ne e compensa e he
agen ’s cos inc ease. We show nex ha a minimum wage could cause a s ic dec ease
o he agen ’s u ili y.
Co olla y 4.4 Suppose he e is a minimum wage w≤θmax.
(I) I i∗=i∗, he agen ’s u ili y dec eases (i.e., ui∗<ui∗) i and only i w<w
i∗and
(EBC)holds.
(II) I i∗= i∗, he agen ’s u ili y dec eases (i.e., ui∗<ui∗) i and only i
|Nmax|=1o i∗/∈a gmax{θj−cj:j∈N {i∗}}.(UD)
P oo (I) Since i∗=i∗, he agen ’s cos s ay cons an such ha he agen ’s u ili y
dec eases i and only i he paid wage dec eases. By Co olla y 4.3, his happens i
and only i w<w
i∗and max{0,θj−cj:j∈N {i∗}} + ci∗<max{0,θj−cj:
j∈N {i∗}} + ci∗, whe e he la e condi ion educes o (EBC) since i∗=i∗.
(II) A dec ease in he agen ’s u ili y occu s i and only i a leas one o he wo
inequali ies in Case (II) o he p oo o Theo em 4.3 is s ic , which happens
exac ly i |Nmax|=1o i∗/∈a gmax{θj−cj:j∈N {i∗}}.
A minimum wage can hus educe he agen ’s u ili y o wo easons. Fi s , when
he minimum wage allows he winning p incipal o lowe he wage paymen , which
a ises whene e he minimum wage elimina es he p incipal’s bes compe i o (s). See
Example 4.2 o an illus a ion. Second, when he minimum wage causes he agen
o eloca e o ano he p incipal and condi ion (UD) holds, which is gene ically ue.
This is illus a ed in Example 4.1, whe e he minimum wage lowe s he agen ’s u ili y
om 6 o 4.23 In ui i ely, he agen ’s u ili y gene ically su e s i he eloca es because
he highe cos o e compensa es he highe wage paymen .
Summa izing, e en i he agen s ays employed a e he in oduc ion o inc ease
o a minimum wage, his u ili y may su e —no ma e whe he he swi ches o ano he
p incipal o no . I he swi ches, he agen ’s u ili y gene ically dec eases, and his holds
e en i he minimum wage exceeds he p e iously paid wage. See Figu e 2 o he
o e iew.
23 In his example, o all minimum wages be ween 9 and 12, condi ion (UD) is only iola ed in he
non-gene ic case whe e θ1−c1=θ2−c2.
123
Analyzing he e ec s o minimum wages 961
Fig. 2 E ec s on he agen ’s u ili y
4.4 E ec s on he p incipals’p o i s
By compa ing P oposi ions 3.2 and 3.4, we di ec ly ob ain he e ec s on he p o i s.
Theo em 4.4 Suppose he e is a minimum wage w≤θmax.
(I) P incipal i’s p o i dec eases i and only i ei he (a) i =i∗= i∗and |Nmax|=
1,o (b)i=i∗=i∗and w>w
i∗.
(II) P incipal i’s p o i inc eases i and only i ei he (a) i =i∗= i∗,w= θi∗,
and |Nmax|=1,o (b)i=i∗=i∗,w<w
i∗, and (EBC)holds.
Theo em 4.4 shows ha a p incipal can bene i om a minimum wage o wo
easons. Fi s , i he minimum wage o ces an agen o eloca e, he p incipal who is able
o a ac he agen now gene ically ea ns a posi i e p o i . See Example 4.1. Second, i
he minimum wage sh inks he se o compe ing p incipals when no eloca ion occu s
and he minimum wage alls sho o he ini ial wage. See Example 4.2. In e es ingly, a
minimum wage can simul aneously lowe he agen ’s u ili y and he agg ega e p o i s,
as Co olla y 4.4 and Theo em 4.4 e eal. See he second example in he in oduc ion
o an illus a ion.
5 Al e na i es o he basic ba gaining p ocedu e
In Appendix B, we examine al e na i e ba gaining p ocedu es. Fi s , we le he agen
p opose wages o he p incipals. Second, we abs ac om how he pa ies ba gain
123
962 C. Thielen, P. Weinschenk
and wha hey know by examining s able ou comes. We show ha he se o s able
ou comes is gi en by he con ex combina ions o he ou come ob ained when he
p incipals make o e s and he ou come when he agen makes o e s. Rema kably, o
any ixed ba gaining powe o he agen excep o he bounda y case co esponding
o he si ua ion whe e he agen makes o e s, he e ec s o minimum wages o s able
ou comes a e always quali a i ely iden ical o he e ec s obse ed when he p incipals
make o e s. Tha is, i a minimum wage inc eases [dec eases, does no change] he
su plus, he wage, he agen ’s u ili y, o he p incipals’ p o i s o s able ou comes, he
same holds ue when he p incipals make o e s (and ice e sa). This implies ha
all e ec s o minimum wages we ha e iden i ied be o e o he p ocedu e whe e he
p incipals make o e s a e he consequence o he s abili y equi emen , and no he
consequence o he speci ic ba gaining p ocedu e o in o ma ion s uc u e.
6 Mul iple agen s
In Appendix C, we show ha he model is eadily gene alized o he case whe e
mul iple agen s in e ac wi h mul iple p incipals. An impo an insigh we ob ain is
ha , in he si ua ion whe e he p incipals ha e no (o no binding) capaci y cons ain s
ha limi he numbe s o agen s hey can employ, he p incipals compe e o each
o he magen s independen ly o he p esence o he o he agen s. Acco dingly, he
si ua ion decomposes in o msingle-agen p oblems, o which he analysis om he
p e ious sec ions applies. In he absence o binding capaci y cons ain s, all esul s
shown o he single-agen case hus ully ca y o e o case wi h mul iple agen s. The
same holds ue i each p incipal has a capaci y o one and he mul i-agen model is
gene a ed by a duplica ion o a single-agen model. Fo he gene al case o capaci y
cons ain s, we p o e he exis ence o s able ou comes ia an algo i hmic app oach
and show ha he p e iously ob ained e ec s o minimum wages s ill a ise.
7 Conclusion
This pape uses a mic oeconomic app oach o s udy he e ec s o minimum wages.
We allow ha agen s can ha e di e en p oduc i i ies a di e en p incipals as well
as di e en cos s o wo king. We iden i y a ich se o e ec s. We in e alia show ha
minimum wages may also lowe e iciency when lea ing employmen una ec ed, and
des oy jobs ha gene a e high le els o su plus when a ec ing employmen . Minimum
wages could ha m agen s e en i hese s ay employed, while p incipals could bene i
om minimum wages. Ou analysis u he e eals ha he p oduc i i y gains caused
by minimum wages a e gene ically accompanied by e iciency losses.
We p o ide a comple e cha ac e iza ion o he e ec s o minimum wages and show
ha all e ec s a e obus . The e ec s a e he consequence o s abili y, and no he
consequence o speci ic ba gaining p ocedu es o in o ma ion s uc u es.
The esul s a e policy- ele an . Suppose, o ins ance, ha a go e nmen execu es
a minimum wage e o m, and he unemploymen a es s ay unchanged. Ou insigh s
imply ha we hen canno conclude ha he e o m necessa ily helps wo ke s and
123
Analyzing he e ec s o minimum wages 963
lea es e iciency unchanged. Ano he impo an issue is ha we ha e o dis inguish
be ween he p oduc i i y e ec s and he e iciency e ec s caused by minimum wages.
I we obse e ha a minimum wage e o m inc eases wo ke s’ p oduc i i ies, hese
gains should no be aken as e idence o a posi i e e iciency e ec .
The model also p o ides es able implica ions. Fi s , an impo an mechanism in he
model is ha minimum wages may cause he eloca ion o wo ke s o o he employe s
whe e hey su e om highe cos s. One could es whe he minimum wages a ec
key componen so wo ke s’cos s,e.g., inc easecommu ingcos s/dis ances.24 Second,
he model p edic s ha minimum wages educe employmen o a lesse ex en (a) in
en i onmen s whe e pa ies ha e mo e in o ma ion,25 (b) in mo e di e se economic
en i onmen s,26 and (c) when be e anspo a ion echnologies a e a ailable.27 Since
he in o ma ion and anspo a ion echnologies ha e imp o ed subs an ially o e he
las decades, he model p edic s ha he employmen e ec o minimum wages should
be weake nowadays han in he pas . This is p ecisely he pa e n me a s udies ind.28
Finally, he model migh also help us o unde s and he a ie y o he empi ical indings
in gi en ime pe iods, which is a no able puzzle o he empi ical li e a u e on minimum
wages (c . he b ie o e iew in he in oduc ion). Some o he a ie y o he empi ical
indings could be due o di e ences in he a o e-desc ibed ac o s (a)–(c) among he
di e en s udies.
Appendices
A Discussion o he ba gaining p ocedu e and equilib ium concep
In his sec ion, we discuss and subs an ia e he basic ba gaining p ocedu e and equi-
lib ium concep used in Sec s. 3and 4. In Sec . A.1, we jus i y he ie-b eaking ules
used in his ba gaining p ocedu e. Sec . A.2 shows he e ec s o allowing weakly
domina ed s a egies o be played by he p incipals, while Sec . A.3 conside s he gen-
e aliza ion o he equilib ium concep o mixed-s a egy Nash equilib ia. Sec ion A.4
shows ha he e ec s obse ed when in oducing a minimum wage ca y o e o he
case whe e an exis ing minimum wage is inc eased. Finally, Sec . A.5 analyzes an
24 Dus mann e al. (2022) documen ha he in oduc ion o a minimum wage in Ge many has inc eased he
commu ing dis ances o low-wage wo ke s by 1.5 km (o 8%) ela i e o high-wage wo ke s. Mo eo e , as
p edic ed by ou model, he minimum wage has led o a eloca ion o low-wage wo ke s o mo e p oduc i e
i ms. These obse a ions can also be explained by he model o Bhaska and To (1999), who also no e ha
some wo ke s migh ha e o accep less p e e ed jobs a e he in oduc ion o a minimum wage.
25 Fo mally, mo e in o ma ion leads o mo e employmen oppo uni ies, which expands he se Nand,
hus, causes an agen o emain employed o a la ge se o minimum wages.
26 A mo e di e se economic en i onmen can be cap u ed by a mean-p ese ing sp ead o he p oduc i i ies,
which causes an agen o emain employed o a la ge se o minimum wages.
27 Lowe anspo a ion cos s lead o an expansion o he se o p incipals o whom an agen can p o i ably
wo k and, hus, o a la ge se o minimum wages o which he agen s ays employed.
28 In hei e iew, B own e al. (1982) es ablish a much-ci ed consensus ha he employmen elas ici y o
minimum wages was be ween −0.3and−0.1. Wol son and Belman’s me a-analysis (Wol son and Belman
2019) uses mo e ecen da a and es ablishes ha he ange has shi ed o −0.13 and −0.07.
123
964 C. Thielen, P. Weinschenk
al e na i e bidding p ocedu e in which he p incipals make ascending o e s o he
agen .
A.1 Jus i ica ion o he ie-b eaking ules
We now discuss he ie-b eaking ules used in he basic ba gaining p ocedu e by
showing ha , in he na u al wo-s age game whe e he p incipals i s make hei
o e s and he agen hen ei he chooses one o hese o e s o ejec s all o e s, he
always beha es acco ding o hese ules in any subgame-pe ec Nash equilib ium.
While we concen a e on he case o un es ic ed wages in he ollowing discussion,
he a gumen s eadily ca y o e o he case o es ic ed wages.
The i s p oposi ion shows ha , in any subgame-pe ec Nash equilib ium o he
wo-s age game, he agen always chooses an o e maximizing he su plus i se e al
o e s maximize his u ili y:
P oposi ion A.1 Suppose ha he p incipals’ o e s a e such ha i,i ∈a gmaxi∈Nwi
−ciwi h si>si . Then he agen choosing p incipal i does no cons i u e a subgame
-pe ec Nash equilib ium.
P oo I wi >θ
i , hen πi =θi −wi <0, so p incipal i can inc ease he p o i
o ze o by educing he o e o θi .I wi ≤θi , hen ui =wi −ci ≤θi −ci =
si <si=θi−ci. Consequen ly, p incipal ican inc ease he p o i by o e ing
θi− o some 0 <<θ
i−ci−si (which makes he agen accep he o e
o i).
The nex p oposi ion shows ha , in any subgame-pe ec Nash equilib ium, he
agen always accep s an o e ins ead o no o e in case o indi e ence:
P oposi ion A.2 Suppose ha he p incipals’ o e s sa is y maxi∈Nwi−ci=0. Then
he agen choosing o ejec all o e s does no cons i u e a subgame-pe ec Nash
equilib ium.
P oo I he agen ejec s all o e s, he assump ion ha θi>ci o all iimplies
ha any p incipal i∈Ncan inc ease he p o i by o e ing wi=ci+ o some
0<<θ
i−ci(which makes he agen accep i’s o e ).
Ou las assump ion on he agen ’s beha io s a es ha he always chooses an o e
acco ding o an a bi a y de e minis ic ie-b eaking ule (e.g., choosing he o e o he
p incipal wi h he lowes index) in case ha se e al o e s simul aneously maximize
bo h his u ili y and he su plus. While using a s ochas ic ie-b eaking ule in his case
would make he winning p incipal and (possibly) he paid wage s ochas ic, i would
no in luence he agen ’s u ili y, he gene a ed su plus, o he p incipals’ p o i s.
A.2 Allowing o weakly domina ed s a egies
In he analysis o he basic ba gaining p ocedu e, we concen a e on equilib ium s a e-
gies ha a e no weakly domina ed. I weakly domina ed s a egies a e allowed, Pa (I)
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Analyzing he e ec s o minimum wages 965
o P oposi ion 3.2 and Pa s (I) and (II), (1) o P oposi ion 3.4 s ill hold by he same
a gumen a ion as in he o iginal p oo s. Consequen ly, he winning p incipal and he
equilib ium su plus emain unchanged and a e s ill uniquely de e mined. Howe e ,
he wage and, hence, he dis ibu ion o he su plus be ween he winning p incipal
and he agen a e, in gene al, no unique anymo e e en o a ixed winning p inci-
pal i∗∈Nmax. We now demons a e his o he case o un es ic ed wages – he case
o es ic ed wages in simila . Speci ically, we now show ha , o un es ic ed wages,
any wage wi∗∈[max{0,θj−cj:j∈N {i∗}} + ci∗,θ
i∗]can be ob ained in a
Nash equilib ium i weakly domina ed s a egies a e allowed. This holds since he
ollowing s a egy p o ile cons i u es a Nash equilib ium: p incipal i∗∈Nmax o e s
wi∗∈[max{0,θj−cj:j∈N {i∗}} + ci∗,θ
i∗], a leas one o he p incipal
j∈N {i∗}o e s wj=wi∗+cj−ci∗, and each o he emaining p inci-
pals k∈N {i∗,j}o e s a wage wkweakly below wi∗+ck−ci∗. No e ha agen j’s
wage o e wjexceeds he p oduc i i y θj—and is, hus, weakly domina ed—unless
j∈a gmax{θj−cj:j∈N {i∗}} and wi∗=max{0,θj−cj:j∈N {i∗}} + ci∗.
Two ema ks a e in o de . Fi s , an equilib ium in which weakly domina ed s a e-
gies a e played equi es an implausibly high deg ee o coo dina ion among playe s.
A leas one p incipal jwho plays a weakly domina ed s a egy mus exac ly ma ch
he u ili y o e ed o he agen by he winning p incipal i∗∈Nmax, despi e he ac
ha he e is a con inuum o possible u ili ies he agen could be o e ed by i∗.Simul-
aneously, p incipal j akes he chance o a loss i coo dina ion ails, i.e., i she o e s
a sligh ly highe u ili y. This is why equilib ia in weakly domina ed s a egies a e
usually uled ou . Second, i is in e es ing o ecognize ha , al hough Nash equilib-
ia in weakly domina ed s a egies a e implausible in he basic ba gaining p ocedu e
whe e he p incipals design he o e s, he se o possible equilib ium wages ha can
esul om hese equilib ia is iden ical o he se o wages in s able ou comes; c . he
cha ac e iza ion o s able ou comes in Sec . B.2.
A.3 Mixed-s a egy nash equilib ia
We nex show ha , o bo h un es ic ed and es ic ed wages, he gene a ed su plus, he
agen ’s u ili y, and he p incipals’ p o i s in any (mixed-s a egy) Nash equilib ium a e
as in e e y pu e-s a egy Nash equilib ium wi h p obabili y 1. Consequen ly, he e ec s
o a minimum wage on he su plus, u ili y, and p o i s do no depend on whe he pu e- o
mixed-s a egy equilib ia a e conside ed. Wi h espec o he paid wage, his also holds
i ei he we a e in he gene ic case whe e |Nmax|=|Nmax|=1, o |Nmax|,|Nmax|≤2
and a de e minis ic ie-b eaking ule is used in case ha se e al o e s simul aneously
maximize bo h he agen ’s u ili y and he su plus. I |Nmax|≥3o |Nmax|≥3, he
equilib ium wage can be s ochas ic in some mixed-s a egy equilib ia.
Theo em A.1 Suppose ha he wages a e un es ic ed and |Nmax|=1, i.e., he e is a
unique p incipal imax ∈Nmax. Then e e y (mixed-s a egy) Nash equilib ium sa is ies:
(I) Wi h p obabili y 1, p incipal imax o e s a wage o ˇwimax :=max{0,θj−cj:j∈
N {imax}} + cimax .
(II) Wi h p obabili y 1, he winning p incipal is i∗=imax and wi∗=ˇwimax .
123
966 C. Thielen, P. Weinschenk
Consequen ly, wi h p obabili y 1, he agen ’s u ili y, he p incipals’ p o i s, and he
gene a ed su plus a e as in e e y pu e-s a egy Nash equilib ium (c . P oposi ion 3.2).
P oo (I) Since no p incipal ie e o e s a wage wi>θ
i, p incipal imax wins wi h
p obabili y 1 when o e ing wimax =ˇwimax , so o e ing a highe wage could only
dec ease he expec ed p o i . Consequen ly, we mus ha e wimax ≤ˇwimax wi h
p obabili y 1 and i only emains o show ha also wimax ≥ˇwimax wi h p obabili y 1.
Suppose o he sake o a con adic ion ha p incipal imax o e s a wage lowe han
ˇwimax wi h posi i e p obabili y. I |N|=1, i.e., i imax is he only p incipal, we
ha e ˇwimax =cimax , meaning ha imax o e s a wage lowe han he agen ’s cos
wi h posi i e p obabili y. This is a con adic ion since o e ing a wage below he
agen ’s cos yields p o i ze o o p incipal imax whe eas she can ob ain posi i e
p o i by o e ing wimax :=ˇwimax =cimax <θ
imax .
Fo he case whe e |N|≥2, we le suppideno e he suppo o i’s mixed s a egy
o i∈N. Then, since imax o e s a wage below ˇwimax wi h posi i e p obabili y,
we ha e in (suppimax )< ˇwimax , and we claim ha he o e s o he o he p incipals
sa is y
max{wi−ci:i∈N {imax}} >in (suppimax )−cimax (A.1)
wi h p obabili y 1. No e ha i his was no he case, we would in pa icula ha e
in (suppi)−ci≤in (suppimax )−cimax o any p incipal i∈a gmaxi=imax si,so
shehas o e s inhe suppo ha makehe lose wi h p obabili y1and, consequen ly,
yield expec ed p o i ze o. Fo any 0 << ˇwimax −in (suppimax ), howe e , an
o e o wi:=θi−would make p incipal iwin wi h posi i e p obabili y and,
hus, yield a posi i e expec ed p o i , which is a con adic ion.
Consequen ly, (A.1) holds wi h p obabili y 1, which means ha he e exis o e s
in suppimax o which p incipal imax loses (and ob ains p o i ze o) o su e. This
yields a con adic ion since p incipal imax can ob ain posi i e p o i by o e ing
wimax :=ˇwimax .
(II) Since no p incipal ie e o e s a wage wi>θ
iand p incipal imax o e s ˇwimax wi h
p obabili y 1, i ollows di ec ly ha i∗=imax and wi∗=ˇwimax wi h p obabili y 1.
Theo em A.2 Suppose ha he wages a e un es ic ed and |Nmax|≥2. Then e e y
(mixed-s a egy) Nash equilib ium sa is ies:
(I) A leas wo p incipals i ∈Nmax o e wages o wi=θiwi h p obabili y 1.
(II) Wi h p obabili y 1, he winning p incipal i∗belongs o Nmax and wi∗=θi∗.
Consequen ly, wi h p obabili y 1, he agen ’s u ili y, he p incipals’ p o i s, and he
gene a ed su plus a e as in e e y pu e-s a egy Nash equilib ium (c . P oposi ion 3.2).
P oo (I) We i s show ha a leas one p incipal i∈Nmax o e s wi=θiwi h
p obabili y 1. Suppose o he sake o a con adic ion ha no p incipal i∈Nmax
o e s a wage o wi=θiwi h p obabili y 1. Then, since no p incipal i∈Nmax
e e o e s a wage wi>θ
iin equilib ium, we mus ha e in (suppi)<θ
i o all
i∈Nmax,somin{θi−in (suppi):i∈Nmax}=
:>0.
123
Analyzing he e ec s o minimum wages 967
We now dis inguish wo cases:
Case 1: a gmax{θi−in (suppi):i∈Nmax}Nmax.
Conside any p incipal i0∈a gmax{θi−in (suppi):i∈Nmax}. Because
a gmax{θi−in (suppi):i∈Nmax}Nmax, he e exis s a subse o suppi0
wi h posi i e p obabili y mass such ha all o e s in his se make p incipal i0lose
wi h p obabili y 1 and, hus, yield expec ed p o i ze o o i0. Howe e , deno ing
he di e ence be ween he maximum su plus and he second-highes su plus by
:=smax −max{0,si:i∈N Nmax}>0, p incipal i0can win wi h posi i e
p obabili y and ob ain a posi i e expec ed p o i by o e ing wi0=θi0−min{,}
2,
which is a con adic ion.
Case 2: a gmax{θi−in (suppi):i∈Nmax}=Nmax. We dis inguish wo sub-
cases:
Case 2.1: Fo some p incipal i1∈Nmax, he o e wi1=in (suppi1)has p obabil-
i y 0 (i.e., he e is no mass poin a in (suppi1)).
Le i0∈Nmax {i1}be a bi a y. Then, as in Case 1, o e ing wi0=θi0−min{,}
2
yields a posi i e expec ed p o i o p incipal i0, which we deno e by ˜πi0. Since
he p obabili y o p incipal i1o e ing wi1=in (suppi1)is ze o, howe e , p inci-
pal i0’s p obabili y o winning ends o ze o as wi0app oaches in (suppi0). Thus,
since p incipal i0’s p o i ob ained om any o e is uppe bounded by si0,also
p incipal i0’s expec ed p o i ends o ze o as he o e app oaches in (suppi0).
Consequen ly, he e exis s a subse o suppi0wi h posi i e p obabili y mass such
ha all o e s in his subse yield an expec ed p o i lowe han ˜πi0 o p incipal i0,
which is a con adic ion.
Case 2.2: E e y p incipal i∈Nmax o e s wi=in (suppi)wi h posi i e p oba-
bili y (i.e., he e is mass poin a in (suppi) o e e y i∈Nmax).
Then, wi h posi i e p obabili y, all p incipals in Nmax o e wi=in (suppi)
a he same ime. I some p incipal i0∈Nmax had expec ed p o i o ze o in
his si ua ion (and hence, also uncondi ional expec ed p o i ze o om he o e
wi0=in (suppi0)), his p incipal could again ob ain posi i e expec ed p o i by
o e ing wi0=θi0−min{,}
2(whe e is as in Case 1), which is a con adic ion.
Consequen ly, all p incipals in Nmax mus ha e posi i e expec ed p o i in his
si ua ion and, in pa icula , each one mus ha e a posi i e p obabili y o winning.
Thus, each p incipal in Nmax mus also ha e a posi i e p obabili y o losing when
all p incipals i∈Nmax o e wi=in (suppi). This yields a con adic ion since
each p incipal in Nmax could hen imp o e he (uncondi ional) expec ed p o i by
sligh ly aising he o e .
Hence, in all cases, a leas one p incipal i∈Nmax o e s a wage o wi=θi
wi h p obabili y 1. Now suppose ha only one p incipal i0∈Nmax o e s a wage
o wi0=θi0wi h p obabili y 1. Then, in (suppi)<θ
i o all i∈Nmax {i0}
and, hus, ¯:=min{θi−in (suppi):i∈Nmax {i0}} >0. Then, o e ing
wi0=θi0−min{¯,}
2 o as in Case 1 abo e yields posi i e expec ed p o i o
p incipal i0whe eas i0’s cu en o e o wi0=θi0yields expec ed p o i ze o,
which is a con adic ion. Consequen ly, a leas wo p incipals i∈Nmax mus
o e wages o wi=θiwi h p obabili y 1 as claimed.
123
968 C. Thielen, P. Weinschenk
(II) Since no p incipal ie e o e s a wage wi>θ
iand a leas wo p incipals i∈Nmax
o e wages o wi=θiwi h p obabili y 1, i ollows di ec ly ha i∗∈Nmax and
wi∗=θi∗wi h p obabili y 1.
Theo em A.3 When a minimum wage wis imposed, e e y (mixed-s a egy) Nash equi-
lib ium sa is ies:
(I) I N =∅, hen, wi h p obabili y 1, no p incipal makes an o e .
(II) I N =∅:
(1) I |Nmax|=1, i.e., he e is a unique p incipal imax ∈Nmax, hen, wi h p ob-
abili y 1, p incipal imax o e s a wage o ˇwimax :=maxw,max{0,θj−cj:
j∈N {imax}} + cimax and i∗=imax.
(2) I |Nmax|≥2, hen, wi h p obabili y 1, a leas wo p incipals i ∈Nmax o e
wages o wi=ˇwi=θiand he winning p incipal i∗belongs o Nmax.
Consequen ly, wi h p obabili y 1, he agen ’s u ili y, he p incipals’ p o i s, and he
gene a ed su plus a e as in e e y pu e-s a egy Nash equilib ium (c . P oposi ion 3.4).
P oo (I) I N=∅and some p incipals o e wages wi h posi i e p obabili y, hen
a leas one p incipal’s expec ed p o i will be nega i e, so she could imp o e by
ne e making an o e .
(II) Fo any p incipal i∈N N, all mixed s a egies in which imakes an o e wi h
posi i e p obabili y a e weakly domina ed by he s a egy in which ine e makes an
o e . Hence, he p incipals in N Nne e make o e s wi h posi i e p obabili y
and he game educes o a game be ween he p incipals in N. Then, Claim (1)
ollows since ei he |N|=1, in which case he only emaining p incipal imax
o e s a wage o max{w, cimax }= ˇwimax ,o |N|≥2, in which case one can a gue
as in he p oo o Theo em A.1 excep ha he o e ed wages canno all sho
o w. Simila ly, o Claim (2), one can a gue as in he p oo o Theo em A.2 (again
aking in o accoun ha he o e ed wages canno all sho o w).
A.4 Inc easing an exis ing minimum wage
Ano he na u al ques ion is whe he hee ec sobse edin ou model when in oducing
a minimum wage ca y o e o he case whe e an exis ing minimum wage is inc eased.
To his end, conside an ins ance in which an ini ial minimum wage wini ial al eady
exis s and is inc eased o wnew >w
ini ial.Le Nini ial :={i∈N:θi≥wini ial}and
Nnew :={i∈N:θi≥wnew}deno e he se o p incipals o which he p oduc i i y
weakly exceeds he ini ial/new minimum wage, espec i ely. Mo eo e , le Nini ial
max :=
a gmaxi∈Nini ial si. We claim ha he ins ance can be ans o med in o an equi alen
ins ance in which no minimum wage exis s ini ially.
Fo he cons uc ion o he ans o med ins ance, we dis inguish wo cases:
Case 1: |Nini ial
max |≥2, o |Nini ial
max |=1 and cimax ≥wini ial o imax ∈Nini ial
max .
123
Analyzing he e ec s o minimum wages 975
I |Nmax|=1, he winning p incipal is again unique and wNE
i∗<ˆwNE
i∗. Conse-
quen ly, he unique alue αcan again be in e p e ed as he agen ’s ba gaining powe ,
and we can exp ess he agen ’s u ili y and he winning p incipal’s p o i as
ui∗=α·ˆuNE
i∗+(1−α) ·uNE
i∗=: uα
i∗and (B.5)
πi∗=α·ˆπNE
i∗+(1−α) ·πNE
i∗=: πα
i∗,(B.6)
which implies ha ui∗∈[uNE
i∗,ˆuNE
i∗]and πi∗∈[ˆπNE
i∗,πNE
i∗] o any s able ou -
come. Simila o he case o un es ic ed wages, he agen ’s ba gaining powe α hus
de e mines how he pa ies sha e he excess su plus c ea ed by hei ela ionship. Fo -
mally, ui∗∈[uNE
i∗,ˆuNE
i∗]=[σ, smax]and πi∗∈[ˆπNE
i∗,πNE
i∗]=[0,smax −σ], whe e
σ:=max{w−ci∗,ssecond}wi h ssecond :=max{0,θj−cj:j∈N {i∗}} deno ing
he second-highes su plus among he p incipals in N.
B.2.3 E ec s o minimum wages
In o de o explo e he e ec s o a minimum wage, we conside he case whe e he
agen does no ge unemployed, i.e., whe e w≤θmax. Mo eo e , in o de o a oid
edious case dis inc ions, we concen a e on he gene ic case whe e |Nmax|=1 and
|Nmax|=1, in which case he e exis s a unique winning p incipal i∗∈Nmax wi hou
a minimum wage and a unique winning p incipal i∗∈Nmax wi h a minimum wage.
We addi ionally assume ha he ba gaining powe o he agen does no change when
in oducing he minimum wage, i.e., α=α.31
As we can see om Equa ions (B.1) o(B.6), he e ec s o a minimum wage w
on he wage, u ili y, and winning p incipal’s p o i o any ixed alue o α∈(0,1)
a e gi en by he weigh ed e ec s a ising in he ex eme cases α=0 and α=1.
The ollowing heo em shows ha , ac ually, he quali a i e e ec s on each o hese
a iables o any α∈(0,1)a e he same as o α=0.
Theo em B.2 Suppose ha |Nmax|=|Nmax|=1. The ollowing holds o each o
he a iables gene a ed su plus, paid wage, agen ’s u ili y, and winning p incipal’s
p o i : I he minimum wage wdec eases he a iable / lea es he a iable unchanged /
inc eases he a iable o α=0, hen he same applies o all α∈[0,1).
P oo We conside each o he a iables sepa a ely.
Fo he gene a ed su plus, we no e ha , by P oposi ions B.3 and B.4, wi hou a
minimum wage, e e y s able ou come gene a es a su plus o smax and, wi h a minimum
wage w, e e y s able ou come gene a es a su plus o smax. Thus, he change in he
gene a ed su plus is iden ical o all α∈[0,1].
31 No e ha , as desc ibed be o e, he agen ’s ba gaining powe de e mines how he pa ies sha e he excess
su plus smax −ssecond (o smax −σin case o es ic ed wages) c ea ed by hei ela ionship. Hence, while
bo h he gene a ed su plus and he excess su plus—and, hus, he amoun o su plus o e which he pa ies
ba gain—may change when in oducing a minimum wage, i is na u al o assume ha he agen ’s ba gaining
powe emains unchanged.
123
976 C. Thielen, P. Weinschenk
Fo he paid wage, we no e ha , wi h α=αand he unique winning p incipals i∗
and i∗wi hou and wi h a minimum wage, espec i ely, Eqs. (B.1) and (B.4)show
ha
wα
i∗−wα
i∗=α·wα=1
i∗−wα=1
i∗+(1−α) ·wα=0
i∗−wα=0
i∗.(B.7)
I w≤wα=1
i∗, hen wα=1
i∗−wα=1
i∗=0 by Theo em B.1,sowα
i∗−wα
i∗=(1−α) ·
(wα=0
i∗−wα=0
i∗), which shows ha he change in he paid wage o any α∈(0,1)
is quali a i ely he same as o α=0. I w>w
α=1
i∗, hen also w>w
α=0
i∗since
wα=1
i∗=θi∗≥wα=0
i∗by P oposi ion B.1. Consequen ly, he minimum wage causes
all wages o inc ease since wα=1
i∗−wα=1
i∗>0 and wα=0
i∗−wα=0
i∗>0in(B.7), so
also wα
i∗−wα
i∗>0. Hence, he change in he paid wage o any α∈(0,1)is again
quali a i ely he same as o α=0.
Fo he agen ’s u ili y, we simila ly ha e
uα
i∗−uα
i∗=α·uα=1
i∗−uα=1
i∗+(1−α) ·uα=0
i∗−uα=0
i∗(B.8)
by Equa ions (B.2) and (B.5). I w≤maxj∈Nmax θj=θi∗, hen uα=1
i∗−uα=1
i∗=0by
Theo em B.1,so
uα
i∗−uα
i∗=(1−α) ·uα=0
i∗−uα=0
i∗,
which shows ha he change in he agen ’s u ili y o any α∈(0,1)is quali a i ely
he same as o α=0. I w>maxj∈Nmax θj=θi∗, hen i∗= i∗and he agen ’s u ili y
dec eases in all cases since uα=1
i∗−uα=1
i∗<0 by Theo em B.1 and uα=0
i∗−uα=0
i∗<0
by Co olla y 4.4. Thus, he change o any α∈(0,1)is again quali a i ely he same
as o α=0.
Fo he winning p incipal’s p o i , we ha e
πα
i∗−πα
i∗=α·πα=1
i∗−πα=1
i∗+(1−α) ·πα=0
i∗−πα=0
i∗(B.9)
by Equa ions (B.3) and (B.6). Exploi ing ha πα=1
i∗=πα=1
i∗=0 by Theo em B.1,
his shows he desi ed esul .
Theo em B.2 e eals ha he quali a i e e ec s o minimum wages on each single
a iable o he model ob ained o he basic ba gaining p ocedu e (whe e he p incipals
ha e all ba gaining powe , i.e., α=0) ex end o all cases whe e he p incipals ha e
nonze o ba gaining powe (i.e., α<1).32
32 E en hough he heo em ocuses on he gene ic case whe e |Nmax|=|Nmax|=1, an e en s onge
esul holds in he nongene ic case whe e |Nmax|,|Nmax|≥2. Then, we ha e si∗=ui∗=smax,πi∗=0,
123
Analyzing he e ec s o minimum wages 977
Conce ning wage o e shoo ing (which is o mally de ined o an a bi a y alue o
αin he ollowing de ini ion), howe e , he quan i a i e e ec o he minimum wage
on he paid wage is also impo an , as we now demons a e.
De ini ion B.2 Fo a gi en ba gaining powe α∈[0,1]o he agen , a minimum
wage w>w
α
i∗causes wage o e shoo ing i he agen s ays employed and he paid
wage inc eases o wα
i∗>w.
The ollowing heo em shows ha , i wage o e shoo ing occu s o a minimum
wage win one o he ex eme cases α=0o α=1, hen i also occu s o all
in e media e cases.
Theo em B.3 Suppose ha |Nmax|=|Nmax|=1. I a minimum wage wcauses wage
o e shoo ing o ei he α=0o o α=1, hen he minimum wage walso causes
wage o e shoo ing o all α∈(0,1).
P oo No e ha |Nmax|=1 implies ha he agen s ays employed in all cases, so his
p e equisi e o wage o e shoo ing does no ha e o be conside ed in he es o he
p oo .
Fi s assume ha wcauses wage o e shoo ing o α=0. Fix some α∈(0,1).
Then by De ini ion 4.1 and Co olla y 4.2:
wNE
i∗>w>w
NE
i∗,(B.10)
i∗= i∗.(B.11)
Toge he wi h (B.4) and ˆwNE
i∗≥w, his yields
wα
i∗=α·ˆwNE
i∗
≥w
+(1−α) ·wNE
i∗
>w
>w.
I emains o show ha wα
i∗<w. To his end, i s no e ha i∗∈Nwould imply
i∗∈Nmax,soi∗=i∗, which con adic s (B.11). Thus, we mus ha e i∗/∈N, which
implies ha θi∗<wand also ˆwNE
i∗<w. Hence, by (B.1) and (B.10)
wα
i∗=α·ˆwNE
i∗
<w
+(1−α) ·wNE
i∗
<w
<w.(B.12)
Now assume ha wcauses wage o e shoo ing o α=1. Again ix some α∈(0,1).
Then by he de ini ion o wage o e shoo ing gi en in Co olla y B.1:
ˆwNE
i∗>w>ˆwNE
i∗.(B.13)
and wi∗=θi∗in any s able ou come wi hou a minimum wage and, simila ly, si∗=ui∗=smax,
πi∗=0, and wi∗=θi∗in any s able ou come wi h a minimum wage. Consequen ly, he quali a i e
and quan i a i e e ec s on he gene a ed su plus, he agen ’s u ili y, and he winning p incipal’s p o i a e
comple ely independen o which s able ou comes a e conside ed, while he e ec on he paid wage only
depends on he choice o he winning p incipal wi hou and wi h he minimum wage om he se s Nmax
and Nmax, espec i ely.
123
978 C. Thielen, P. Weinschenk
Toge he wi h (B.4), his yields
wα
i∗=α·ˆwNE
i∗
>w
+(1−α) ·wNE
i∗
≥w
>w.
In addi ion, since ˆwNE
i∗=θi∗by P oposi ion B.1 and wNE
i∗≤θi∗, he second inequali y
in (B.13) yields ha also wNE
i∗<w. Hence, by (B.1), we ob ain
wα
i∗=α·ˆwNE
i∗
<w
+(1−α) ·wNE
i∗
<w
<w.
While wage o e shoo ing in one o he ex eme cases α=0o α=1 is su icien
o wage o e shoo ing in all in e media e cases α∈(0,1), i is no necessa y. Indeed,
as he ollowing example demons a es, wage o e shoo ing can occu o in e media e
cases when i does no occu a he ex emes:
Example B.1 Conside a si ua ion wi h wo p incipals, whe e θ1=12, θ2=10 and
c1=2, c2=2. Wi hou a minimum wage, we ha e i∗=1, wi∗=10, and ˆwi∗=12.
Now ix some α∈(0,1)and conside a minimum wage wwi h 10 +2α<w<12.
Then Nmax =Nmax ={1}and he Nash equilib ium wages a he winning p inci-
pal i∗=1a ewNE
i∗=wand ˆwNE
i∗=12. In pa icula , no wage o e shoo ing occu s i
he agen has no o all ba gaining powe . Howe e , when he agen ’s ba gaining powe
is α, wage o e shoo ing does occu since we ha e
wα
i∗=α·12 +(1−α) ·10 =10 +2α<w and wα
i∗=α·12 +(1−α) ·w>w.
C Analysis o he mul i-agen model
We p e iously examined how a minimum wage a ec s he in e ac ion be ween an
agen and a se o p incipals. In his sec ion, we gene alize he model o he case o
m≥2 agen s. We deno e he se o agen s by M={1,...,m}, and le ci,jand θi,j
deno e agen j’s cos and p oduc i i y, espec i ely, when wo king o p incipal i∈N.
Simila ly, ui,j:=wi,j−ci,jdeno es he u ili y o agen ji he wo ks o p incipal i
a wage wi,j, and πi,j:=θi,j−wi,jdeno es he p o i esul ing o p incipal i om
his employmen . Each agen can wo k o a mos one p incipal, bu a p incipal may
employ se e al agen s. The o al p o i πio p incipal iamoun s o he sum o he
p o i s esul ing om all hese employmen s, whe e πi:=0 i she does no employ
any agen . Simila ly, we le ujdeno e agen j’s u ili y a he p incipal he wo ks o ,
whe e uj:=0 i agen jdoes no wo k o any p incipal. The su plus gene a ed when
agen jwo ks o p incipal iis deno ed by si,j:=θi,j−ci,j.
No e ha , while we assumed (wi hou loss o gene ali y) ha θi>ci o all i∈N
in he single-agen case, we now allow ha θi,j≤ci,j o some i,j(which means ha
123
Analyzing he e ec s o minimum wages 979
agen jcan ne e be p o i ably employed by p incipal i). This allows us o conside
si ua ions whe e each agen is only p oduc i e (i.e., has p oduc i i y exceeding cos )
a a subse o he p incipals.
In he ollowing, i will be use ul o summa ize he cos s and p oduc i i ies in wo
(n×m) ma ices C=(ci,j)i,jand =(θi,j)i,j, espec i ely. He e, each ow co e-
sponds o a p incipal i∈Nand each column o an agen j∈Mand he co esponding
en ies ci,jo Cand θi,jo a e he cos and p oduc i i y, espec i ely, esul ing when
p incipal iemploys agen j. Simila ly, he ma ices W=(wi,j)i,j,U=(ui,j)i,j, and
=(πi,j)i,jsumma ize he (o e ed) wages, u ili ies, and p o i s, espec i ely. We
se wi,j:=NO i p incipal imakes no o e o agen j. The assignmen s a ing which
p incipals employ which agen s is summa ized in he bina y ma ix A=(ai,j)i,j,
whe e ai,j=1 i p incipal iemploys agen jand ai,j=0, o he wise. A pai (A,W)
consis ing o he assignmen Aand he wages Wwill be e e ed o as an ou come.
In he si ua ion in which he p incipals ha e no capaci y cons ain s limi ing he
numbe s o agen s hey can employ, he p incipals compe e o each agen indepen-
den ly o he p esence o he o he agen s. Consequen ly, he si ua ion decomposes
in o msingle-agen p oblems, o which he analysis om he p e ious sec ions applies.
Hence, in he absence o capaci y cons ain s, all esul s shown o he single-agen
case ca y o e o case o mul iple agen s.
In he si ua ion whe e each p incipal has a capaci y o one, i.e., a p incipal can only
employ a single agen , i is easy o see ha he esul s om he single-agen case ca y
o e o he mul i-agen case i he economy is simply duplica ed a gi en numbe o
imes, i.e., he e a e kiden ical agen s and kcopies o each p incipal o some k≥2.
We now seek o explo e he gene al mul i-agen case wi h capaci y cons ain s.
He e, each p incipal i∈Nhas a posi i e in ege capaci y κi∈N>0 ha speci ies he
maximum numbe o agen s she can employ and we summa ize he capaci ies in a
ec o κ=(κ1,...,κ
n). We s a by conside ing he canonical ex ension o ou basic
ba gaining p ocedu e in which each p incipal can o e wages o a mos as many agen s
as he capaci y allows he o employ. The ollowing example, howe e , demons a es
ha , in his case, pu e-s a egy Nash equilib ia may ail o exis and mixed-s a egy
Nash equilib ia may no be s able – e en when no minimum wage is imposed:
Example C.1 Conside he si ua ion wi h wo p incipals wi h uni capaci ies κ1=
κ2=1 and h ee agen s shown in Figu e 3, whe e no minimum wage is imposed
and all cos s a e ze o. Ob iously, he e canno be any pu e-s a egy Nash equilib ium
in which ei he no p incipal makes an o e o agen 2 o bo h p incipals make hei
o e s o agen 2. Hence, in a pu e-s a egy equilib ium, exac ly one p incipal—say,
p incipal 1—would ha e o make an o e o agen 2. Gi en ha p incipal 2 does
no make an o e o agen 2, p incipal 1’s op imal o e o agen 2 is w1,2=0. Bu
p incipal 2 could hen inc ease he p o i by o e ing w2,2=1 o agen 2.
The e does, howe e , exis a (unique) symme ic, mixed-s a egy Nash equilib ium
in which each p incipal makes an o e o agen 2 wi h p obabili y 1/6, in which
case she dis ibu es he o e ed wage on [0,2], and o e s wage ze o o he agen who
yields p oduc i i y 10 o he , o he wise. Then, howe e , agen 2 ecei es no o e wi h
p obabili y (5/6)2=25/36, in which case each p incipal could imp o e he p o i ex
pos by employing agen 2.
123
980 C. Thielen, P. Weinschenk
Fig. 3 Example C.1.
Connec ions ha a e omi ed
yield nega i e su plus
We now show ha s able ou comes do always exis in he se ing wi h mul iple agen s
and capaci y cons ain s bo h wi h and wi hou a minimum wage. He e, s abili y is
o mally de ined as ollows:
De ini ion C.1 I a minimum wage wis imposed, an ou come (A,W)is called s able
i i espec s he capaci ies and he minimum wage, i.e., j∈Mai,j≤κi o all i∈N
and wi,j≥w o all i,jwi h ai,j=1, is indi idually a ional, i.e., πi,j,ui,j≥0 o
all i,jwi h ai,j=1, and, o all i,jwi h ai,j=0, we ha e:
(a) I j∈Mai,j<κ
i, hen uj≥si,jo θi,j≤w.
(b) I j∈Mai,j=κi, hen uj+min{πi,j:ai,j=1}≥si,jo θi,j≤w+
min{πi,j:ai,j=1}.
Fo un es ic ed wages, s abili y is de ined analogously, excep ha he condi ions
in ol ing he minimum wage wa e omi ed.
No e ha , e en hough an ou come (A,W)con ains wage o e s wi,j o all pai s
(i,j), only hose o e s whe e ai,j=1 a e ele an o he s abili y o he ou come.
An ou come espec ing bo h he capaci ies and he minimum wage is s able i and
only i all employmen s in he cu en assignmen yield nonnega i e p o i and u ili y
o he co esponding p incipals and agen s, espec i ely, and no p incipal ican o e
a wage o an agen jcu en ly no wo king o he in a way ha bo h iand jwould
imp o e and he capaci ies as well as he minimum wage a e s ill espec ed.
In Case (a), whe e p incipal i’s capaci y κiis no exhaus ed in he cu en assign-
men , he condi ion ha p incipal icanno o e such a wage o agen jmeans
ha ei he j’s cu en u ili y ujal eady weakly exceeds he su plus si,jhe would
gene a e when wo king o i, o he minimum wage weakly exceeds j’s p oduc-
i i y θi,ja p incipal i. I bo h o hese condi ions a e iola ed, he e exis wage
o e s imp o ing j’s u ili y, while a he same ime espec ing i’s capaci y and
he minimum wage and yielding a posi i e p o i πi,j o p incipal i, o ins ance
w
i,j:=θi,j−1
2min{si,j−uj,θ
i,j−w}.
In Case (b), whe e p incipal i’s capaci y κiis al eady exhaus ed in he cu en
assignmen , he condi ions a e simila excep ha p incipal i hen has o ge id o one
123
Analyzing he e ec s o minimum wages 981
(leas p o i able) agen she cu en ly employs, which yields a p o i loss o min{πi,j:
ai,j=1}.33
In he ollowing, we e e o a wage o e o a p incipal i o an agen j ha would
inc ease bo h p incipal i’s p o i and agen j’s u ili y by a leas a gi en amoun >0
as an -imp o ing o e :
De ini ion C.2 Gi en an ou come (A,W), a minimum wage w, and >0, a wage
o e w
i,j≥wo p incipal i o agen jis called an -imp o ing o e i w
i,j−ci,j≥
uj+and ei he j∈Mai,j<κ
iand θi,j−w
i,j≥,o j∈Mai,j=κiand
θi,j−w
i,j≥min{πi,j:ai,j=1}+. Fo un es ic ed wages, he de ini ion is he
same excep ha he condi ion w
i,j≥wis omi ed. The ou come (A,W)is called
-s able i i espec s he capaci ies and he minimum wage i one exis s, is indi idually
a ional, and no p incipal ican make an -imp o ing o e o any agen j.
The ollowing obse a ion is ob ained di ec ly om he de ini ion:
Obse a ion C.1 Fo any ou come (A,W), he ollowing holds:
(I) I p incipal i can make an -imp o ing o e o agen j, hen ai,j=0, i.e., j does
no wo k o i in he cu en assignmen .
(II) I (A,W)is -s able o some >0, hen i is also -s able o all >.
The nex p oposi ion p o ides he connec ion be ween s abili y and -s abili y:
P oposi ion C.1 An ou come (A,W)is s able i and only i i is -s able o all >0,
i.e., i and only i he e does no exis an -imp o ing o e o any >0.
P oo I he e exis s an -imp o ing o e o some p incipal i o some agen j o
some >0, hen ai,j=0 and he condi ions in a) and b) o De ini ion C.1 a e clea ly
iola ed o i,j.
Con e sely, i he ou come (A,W) espec s he capaci ies and he minimum wage
i one exis s bu is no s able, hen some p incipal ican make an o e o some agen j
such ha bo h i’s p o i and j’s u ili y imp o e. This o e is hen an -imp o ing o e
when choosing as he minimum o he wo imp o emen s.
We now algo i hmically p o e he exis ence o -s able ou comes o e e y >0,
and hen use P oposi ion C.1 in o de o de i e also he exis ence o a s able
ou come. While Algo i hm 1is o mula ed o he case wi h a minimum wage,
i s aigh o wa dly applies also o he case wi hou a minimum wage by se ing
˘wi,j:=ci,j+uj+in line 3.
As long as he e exis s a p incipal who can make an -imp o ing o e o some
agen , Algo i hm 1chooses such a p incipal iwho hen makes an -imp o ing o e
o an agen jin a way ha maximizes he p o i . To do so, p incipal icould ha e o
eplace a leas p o i able agen lshe has p e iously employed by agen jin case ha he
capaci y κiis al eady exhaus ed. Such a eplacemen , howe e , ne e ac ually occu s
in any i e a ion o he algo i hm, as he p oo o he ollowing p oposi ion shows.
33 I bo h condi ions in b) a e iola ed, a possible wage o e imp o ing j’s u ili y and i’s p o i and
espec ing he minimum wage is gi en by w
i,j:=max w,θ
i,j−min{πi,j:ai,j=1}− δ
2,whe e
δ:=si,j−uj−min{πi,j:ai,j=1}.
123
982 C. Thielen, P. Weinschenk
Algo i hm 1 -s able ou come
Inpu : Se No p incipals, se Mo agen s, minimum wage w, p oduc i i ies ,cos sC,
capaci ies κ,and>0.
Ou pu : An -s able ou come (A,W).
1 Ini ialize ai,j:=0, wi,j:=NO, πi,j:=0, uj:=0 o all i∈N,j∈M.
2while he e exis s a p incipal iwho can make an -imp o ing o e o some agen do
3 Choose j∈a gmax
j∈M
{θi,j−˘wi,j},whe e ˘wi,j:=max{w,ci,j+uj+}.
4i
j∈M
ai,j=κi hen
5 Choose l∈a gmin
j∈M:ai,j=1
{πi,j}.
6Se ai,l:=0, πi,l:=0, and ul:=0.
7end i
8Se ai,j:=1andai,j:=0 o all i∈N {i}.
9Se wi,j:=˘wi,jand wi,j:=NO o all i∈N {i}.
10 Se πi,j:=θi,j−˘wi,jand πi,j:=0 o all i∈N {i}.
11 Se uj:=˘wi,j−ci,j.
12 end while
13 e u n (A,W)
P oposi ion C.2 Algo i hm 1 e mina es a e a ini e numbe o i e a ions wi h an
-s able ou come (A,W).
P oo Since he ou come e u ned by he algo i hm a e mina ion is clea ly -s able,
we only ha e o show ha he algo i hm e mina es a e a ini e numbe o i e a ions
o he while loop.
To his end, we now show ha he u ili y ujo each agen jis mono onously
inc easing du ing he while loop. Since he u ili y o some agen jinc eases by a
leas >0 in each i e a ion o he while loop and he sum o all u ili ies o he agen s
(which is ze o be o e he i s i e a ion) is bounded om abo e (e.g., by he sum o all
si,j), his will di ec ly imply ha he numbe o i e a ions o he while loop is ini e.
In o de o show he desi ed mono onici y o he u ili ies, we i s obse e ha an
agen k’s u ili y ukcan ne e dec ease as long as he agen s ays employed (i.e., as
long as i∈Nai,k=1 du ing he while loop). This holds since he agen hen s ays
employed a he same p incipal o he same wage un il he ecei es an -imp o ing
o e , which inc eases his u ili y by a leas . Mo eo e , obse e ha he only poin in
he algo i hm whe e a p e iously employed agen can become unemployed is in lines
4–6, whe e a p incipal i eplaces an agen lby ano he agen jbecause he capaci y κi
is al eady exhaus ed when making an -imp o ing o e o agen j. Consequen ly,
he claim ollows i we show ha he condi ion o he i s a emen in line 4 is ne e
sa is ied, i.e., ha no p incipal e e eplaces an agen by ano he agen because he
capaci y is al eady exhaus ed.
Suppose ha i e a ion is he i s o he while loop in which some p incipal i
eplaces an agen lby ano he agen j.Le < deno e he la es p e ious i e a ion in
which p incipal io e ed a new wage ˘wi,l( ) o agen l. Then, since no agen was e e
eplaced by ano he agen du ing he i e a ions 1,..., −1, agen j’s u ili ies uj( )
and uj( )a he s a o i e a ions and mus sa is y uj( )≤uj( ). Thus, he
123
Analyzing he e ec s o minimum wages 983
alues ˘wi,j( )and ˘wi,j( )in i e a ions and also sa is y ˘wi,j( )≤˘wi,j( ). Hence,
deno ing he alues πi,ka he beginning o i e a ion by πi,k( ),weha e
θi,j−˘wi,j( )≥θi,j−˘wi,j( )≥πi,l( )+=θi,l−˘wi,l( )+>θ
i,l−˘wi,l( ),
whe e he second inequali y ollows since l∈a gmin j∈M:ai,j=1{πi,j( )}and ˘wi,j( )
is an -imp o ing o e in i e a ion . This yields he desi ed con adic ion since agen i
should ha e made an -imp o ing o e o agen jins ead o agen lin i e a ion
acco ding o line 3.
In pa icula , P oposi ion C.2 yields he ollowing esul :
Co olla y C.1 Fo e e y >0, he e exis s an -s able ou come.
We now use P oposi ion C.1 and Co olla y C.1 o es ablish he exis ence o a s able
ou come. To his end, o any ixed assignmen A espec ing he capaci ies, conside
he ollowing mixed in ege linea p og am (MILP), which compu es he smalles
alue ≥0 such ha he assignmen A oge he wi h sui able wages W o ms an
ou come (A,W) ha is -s able o all >
:
min s. . (C.1)
uj=
i:ai,j=1
(wi,j−ci,j)∀j∈M(a)
πi,j=θi,j−wi,j∀(i,j)wi h ai,j=1(b)
max{w, ci,j}≤wi,j≤θi,j∀(i,j)wi h ai,j=1(c)
si,j·xi,j≤uj+2∀(i,j)wi h ai,j=0,
j∈M
ai,j<κ
i(d)
si,j·xi,j≤uj+πi,˜
j+2∀(i,j)wi h ai,j=0,
j∈M
ai,j=κi,∀˜
jwi h ai,˜
j=1(e)
(θi,j−w) ·yi,j≤∀(i,j)wi h ai,j=0,
j∈M
ai,j<κ
i( )
(θi,j−w) ·yi,j≤θi,˜
j−wi,˜
j+∀(i,j)wi h ai,j=0,
j∈M
ai,j=κi,∀˜
jwi h ai,˜
j=1
(g)
xi,j+yi,j≥1∀(i,j)wi h ai,j=0(h)
xi,j,yi,j∈{0,1}∀(i,j)(i)
≥0
(j)
No e ha he a iable wi,j ep esen ing he wage paid by p incipal i o agen j
only exis s in case ha ai,j=1, i.e., i agen jwo ks o p incipal i.In heMILP,(a)
and (b) link he u ili ies ujand he p o i s πi,j, espec i ely, o he paid wages wi,j.
The cons ain s (c) ensu e indi idual a ionali y and ha all paid wages adhe e o
123
984 C. Thielen, P. Weinschenk
he minimum wage. The cons ain s (d)–(i) ensu e ha no p incipal ican make an
-imp o ing o e o any agen j o any >
: Ei he xi,j=1 and he su plus si,j
allows a join imp o emen o a mos 2(by (d), o (e)) o yi,j=1 and p incipal i
can imp o e by a mos due o he minimum wage (by ( )o (g)).
Fo each assignmen A espec ing he capaci ies, we deno e he op imum objec i e
alue o MILP (C.1)by˜(A). No e ha an op imal solu ion always exis s as long
as (C.1) is easible (c . Sch ij e (1998)). I no easible solu ion exis s o a gi en
assignmen A,wele ˜(A):=+∞. We hen ha e ˜(A)≥0 o e e y assignmen A
due o cons ain (j). Mo eo e , we le ˜:=minA˜(A)≥0 deno e he minimum o he
alues ˜(A)o e all he ( ini ely many) assignmen s A. No e ha ˜is ini e since (C.1)
has a easible solu ion o a leas one assignmen A espec ing he capaci ies: I
ai,j=0 o all i,j, hen se ing xi,j:=1 and yi,j,π
i,j,uj:=0 o all i,j, and
choosing :=maxi,j
si,j
2is clea ly easible.
We now use MILP (C.1) o es ablish he exis ence o s able ou comes:
Theo em C.1 The e exis s a s able ou come (A,W) o any p oduc i i ies , cos s C,
capaci ies κ, and any minimum wage w.
P oo We show ha ˜=0 o any p oduc i i ies , cos s C, capaci ies κ, and any
minimum wage w. This will p o e he claim since a co esponding assignmen A
o which ˜(A)=0 oge he wi h he wages wi,j om an op imal solu ion o (C.1)
o his Acons i u e an ou come (A,W) ha is -s able o all >0. Thus, his
ou come (A,W)is s able by P oposi ion C.1.
Suppose o he sake o a con adic ion ha ˜>0. Then, by Co olla y C.1, he e
exis s an ou come (A,W) ha is ˜
2-s able, and by Obse a ion C.1, his ou come is
also -s able o all > ˜
2. Thus, he wages gi en by Winduce a easible solu ion
o (C.1) wi h objec i e alue a mos ˜
2,so˜(A)≤˜
2, which yields a con adic ion
since ˜(A)≥˜by de ini ion o ˜.
Theo em C.1 also holds in case no minimum wage is imposed, since a minimum
wage w≤mini,jci,jhas no e ec .34 Fu he , ecognize ha he esul in Theo em C.1
is cons uc i e in he sense ha , o any assignmen A espec ing he capaci ies, we can
use MILP (C.1) in o de o compu e wages Wsuch ha he ou come (A,W)is s able
( hese wages a e gi en by he a iables wi,jin an op imal solu ion i he op imum
objec i e alue ˜(A)is ze o) o decide ha no such wages exis o assignmen A
(which is he case i ˜(A)>0).
The ollowing example demons a es ha he e ec s obse ed in he single-agen
case s ill a ise in he case o mul iple agen s:
Example C.2 Conside he si ua ion wi h six p incipals wi h uni capaci ies κi=1 o
all iand ou agen s shown in Figu e 4. When no minimum wage is imposed, he
only possible assignmen in a s able ou come is a1,1=a3,2=a4,3=a6,4=1 and
34 Fo he case whe e no minimum wage is imposed, he exis ence o a s able ou come can also be shown
by using linea p og amming duali y in o de o calcula e he u ili ies and p o i s in a s able ou come;
see Shapley and Shubik (1971). This echnique, howe e , does no easily gene alize o he case in which a
(non i ial) minimum wage exis s, since he minimum wage may p e en ce ain combina ions o u ili ies
and p o i s o each p incipal-agen pai .
123
Analyzing he e ec s o minimum wages 991
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