De winkel‐Kal , Ma kus; Wey, Ch is ian
A icle — Published Ve sion
Why “Ene gy P ice B akes” Encou age Mo al Haza d, Raise
Ene gy P ices, and Rein o ce Ene gy Sa ings
The RAND Jou nal o Economics
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John Wiley & Sons
Sugges ed Ci a ion: De winkel‐Kal , Ma kus; Wey, Ch is ian (2025) : Why “Ene gy P ice B akes”
Encou age Mo al Haza d, Raise Ene gy P ices, and Rein o ce Ene gy Sa ings, The RAND Jou nal o
Economics, ISSN 1756-2171, Wiley, Hoboken, NJ, Vol. 56, Iss. 2, pp. 129-144,
h ps://doi.o g/10.1111/1756-2171.12489
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The RAND Jou nal o Economics
ARTICLE
Why “Ene gy P ice B akes” Encou age Mo al Haza d, Raise
Ene gy P ices, and Rein o ce Ene gy Sa ings
Ma kus De winkel-Kal 1Ch is ian Wey2
1Max Planck Ins i u e o Resea ch on Collec i e Goods, Uni e si y o Müns e 2Düsseldo Ins i u e o Compe i ion Economics (DICE), Hein ich Heine
Uni e si y Düsseldo
Co espondence: Ma kus De winkel-Kal ([email p o ec ed].de)
Accep ed: 17 Janua y 2025
Funding: This wo k was inancially suppo by he Ge man Resea ch Founda ion (DFG p ojec 490725816, Ma kus De winkel-Kal ; 235577387/GRK 1974,
Ch is ian Wey).
Keywo ds: ene gy c isis | ene gy p ice b ake | ene gy p ice policies | ene gy sa ing
ABSTRACT
To help households and i ms wi h exploding ene gy cos s in he a e ma h o he Uk aine wa , a new policy called he “ene gy
p ice b ake” was implemen ed. A unique ea u e o his elie measu e is ha i p o ides a ans e ha inc eases in he consume ’s
con ac ual pe -uni p ice o ene gy. In a o mal model, we show ha his policy c ea es incen i es o mo al haza d o ene gy
p o ide s o aise pe -uni p ices. Whe eas his mo al haza d p oblem inc eases he policy’s iscal cos s, i also ein o ces ene gy
sa ings. Whe he he policy’s main bene icia ies a e consume s o i ms depends on he ma ke s uc u e.
JEL Classi ica ion: D04, L12, Q48, K33
1 In oduc ion
In he a e ma h o he Russian in asion o Uk aine in Feb ua y
2022, Eu opean go e nmen s ha e implemen ed a ious ene gy
p ice elie p og ams o households and i ms o add ess he
sky ocke ing ene gy cos s. These p og ams, including lump-sum
ans e s, ene gy p ice subsidies, ene gy ax cu s, and p ice caps,
aim o alle ia e he inancial bu den o consume s caused by
he c isis. A oiding a sho age o ene gy—especially, na u al
gas—has become a op p io i y o policymake s, pa icula ly in
Ge many, whe e na u al gas is he majo ene gy sou ce o bo h
la ge i ms and households.1Whe eas con en ional measu es
like p ice caps o ene gy subsidies p o ide immedia e elie o
consume s, such app oaches isk causing a b eakdown in he
ene gy ma ke by c ea ing excess demand.
To add ess hese challenges, a no el policy ins umen called
he “ene gy p ice b ake” was de eloped and implemen ed in
Ge many s a ing om Janua y 1, 2023, wi h a subs an ial
budge o up o 200 billion eu os being alloca ed o i . The
ene gy p ice b ake applies o bo h na u al gas and elec ici y
in Ge many, and could also become a policy ins umen
in he EU in u u e ene gy c ises (see EU 2023). Thus, an
in-dep h analysis o he e ec s o he ene gy p ice b ake is
wa an ed.
The ene gy p ice b ake se es wo p ima y objec i es: (i) incen-
i izing ene gy sa ings among consume s and (ii) p o iding
inancial elie and p o ec ion agains excessi e ene gy p ices.
I ope a es h ough a ans e scheme de ined by he ollowing
equa ion:
“ ans e =(con ac ual pe -uni p ice
−gua an eed pe -uni p ice) ×quo a.” (1)
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion License, which pe mi s use, dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed.
© 2025 The Au ho (s). The RAND Jou nal o Economics published by Wiley Pe iodicals LLC on behal o The RAND Co po a ion.
The RAND Jou nal o Economics, 2025; 56:129–144
h ps://doi.o g/10.1111/1756-2171.12489
129
He e, he “con ac ual pe -uni p ice” e e s o he consume ’s
cu en con ac ual p ice pe uni o ene gy (i.e., pe kWh) ha
he ene gy p o ide de e mines, he “gua an eed pe -uni p ice”
e e s o an ene gy p ice pe kWh ha he go e nmen se s
upon implemen ing he ene gy p ice b ake, and he “quo a”
e e s o a ixed ene gy quan i y ha he consume canno
in luence a e he announcemen o he ene gy p ice b ake
(i could, o ins ance, e e o a pe cen age o he espec i e
consume ’s p e ious consump ion le el o a pe cen age o he
median consume ’s p e ious consump ion le el).2The ans e
scheme behind he ene gy p ice b ake is dis inguishably di e en
om o he ans e schemes as i in ol es a lump-sum ans-
e ha a ies in he consume ’s cu en con ac ual pe -uni
p ice.
In his pape , we p o ide a o mal analysis o he e ec s o
he ene gy p ice b ake on ene gy supplie s and consume s,
which could be households o ene gy-consuming i ms. Fo ha ,
we build on a model o supplie –consume con ac ing. Fi s ,
suppose a monopolis ic ene gy supplie (“he”) o e s a wo-
pa a i con ac con ingen on he p esence o he ene gy
p ice b ake, and a consume (“she”) decides whe he o accep
he con ac and de e mines he ene gy consump ion le el.3
We demons a e ha he ene gy p ice b ake c ea es a mo al
haza d p oblem on he supplie ’s side, d i en by he ac ha
he join su plus o he supplie and he consume inc eases
wi h he con ac ual pe -uni p ice. Because he supplie can
ex ac he whole su plus ia he ixed ee (keeping he consume
indi e en be ween accep ing and ejec ing he con ac o e ),
he has an incen i e o aise he pe -uni p ice and he e o e also
he ans e oppo unis ically. By inc easing he pe -uni p ice,
he ene gy p ice b ake no only sus ains bu also s eng hens
he incen i es o ene gy conse a ion. This, howe e , goes along
wi h wo d awbacks: Fi s , gi en a monopoly supplie , he ene gy
p ice b ake ails o inancially elie e consume s as he supplie
pocke s he en i e addi ional su plus. Second, as he go e nmen-
al ans e depends on he con ac ual p ices, he iscal cos s
o he ene gy p ice b ake will be much highe han expec ed
i policymake s do no ake he mo al haza d p oblem in o
accoun .
We nex show ha compe i ion does no esol e he mo al haza d
p oblem. The in ui ion is ha a consume will choose he con ac
ha gi es he he highes u ili y. The consume bene i s om
a highe pe -uni p ice as his also aises he ans e . Thus,
unlike unde a monopoly, consume s a e he bene icia ies o he
ans e scheme. Also unde compe i ion, he ans e scheme is
exploi ed, bu he e i helps o achie e bo h policy objec i es (i.e.,
incen i izing ene gy sa ings and p o iding consume elie ).4
A e his gene al analysis o he implica ions o an ene gy p ice
b ake, we nex look in o he e ec s o he egula o y cons ain s
ins i u ed in Ge many’s ecen legisla ion o he ene gy p ice
b akes (EWPBG 2022;S omPBG2022). The e o e, we add hese
egula o y cons ain s—which aim o ban he misuse o he
ans e scheme—successi ely o ou model. The i s cons ain
ega ds he ene gy con ac ’s pe -uni p ice, he second he
con ac ’s ixed paymen , and he hi d he o e all ans e o
he ene gy p ice b ake o he consume . In e es ingly, we ind
ha hese egula o y cons ain s do no undamen ally al e ou
co e insigh s.
Fi s , he con ac ual pe -uni p ice ha a supplie can cha ge
is cons ained (bu could well be abo e ma ginal cos s). The
legisla ion on he ene gy p ice b akes s a es ha ene gy p o ide s
mus no inc ease he p ice beyond “an objec i ely jus i ied”
amoun (see §27 in EWPBG 2022 and §39 in S omPBG 2022);
o he wise, he Fede al Ca el O ice could in e ene. As i is
ques ionable whe he he Fede al Ca el O ice has su icien
capaci y o moni o p ice inc eases o all ene gy p o ide s in
he ma ke ,5 i ms’ disc e ion o aise p ices beyond wha is
objec i ely jus i ied is a guably subs an ial, hough no unlimi ed.
We show ha wi h his cons ain in place he main message
o ou p eceding analysis s ays alid: he pe -uni p ice is aised
abo e he equilib ium le el ha would p e ail in he absence o
he ans e scheme.
Second, ene gy p ice b ake egula ions equi e ha he ixed
paymen mus e lec cos s, and ha bonus paymen s o con-
sume s o signing a con ac a e i ually elimina ed (see §4
(1) o EWPBG 2022). Gi en hose es ic ions, he supplie can
e ec i ely only se he con ac ual pe -uni p ice. Then, he
mo al haza d p oblem will again a ise unde a monopoly and
unde compe i ion, p o ided ha consume u ili y inc eases in
he con ac ual pe -uni p ice. In ui i ely, a consume bene i s
om a highe pe -uni p ice when i inc eases he ans e by
mo e han he ene gy bill. Wi h linea con ac s, he e is no ixed
ee ia which su plus can be shi ed be ween he consume and
he supplie , and hus a consume does no unequi ocally p e e
high con ac ual p ices. Whe he a consume wan s o sign a
high-p ice con ac now depends on he demand cu e. I he
consume ’s op imal consump ion le el lies abo e he quo a, he
consume p e e s a low-p ice con ac . I i lies below he quo a,
howe e , she is willing o sign a high-p ice con ac and he mo al
haza d p oblem a ises.
Thi d, he ans e o he ene gy p ice b ake i sel is capped in
such a way ha a consume canno pay less han ze o o he
annual ene gy consump ion. Wi hou his cons ain , he ene gy
p ice b ake would allow some ex eme sa e s o lowe hei
annual ene gy bill no only o ze o bu also below ze o. This
ans e cap could inc ease consump ion up o a le el ha he
bill becomes ze o. Thus, i could inc ease an “ex eme sa e ’s”
ene gy consump ion, which is, as we show, he mo e likely he
highe he con ac ual pe -uni p ice. Ne e heless, he mass o
such ex eme sa e s is a guably negligible.
In addi ion, we conside also a scena io ha is concei able, bu
ha is no pa o he Ge man legisla ion on he ene gy p ice
b akes, namely, ha he go e nmen implemen s a cos -based
p ice egula ion ha s ic ly cons ains he con ac ual p ices sup-
plie s can cha ge. Such a egula ion o p ices also does no sol e
he mo al haza d p oblem. A consume could ha e he incen i e
o sign a high-p ice con ac as his ensu es a highe ans e .
Consume s wi h lowe equilib ium ene gy consump ion a e mo e
inclined o a o a high-p ice con ac . Wi h egula ed p ices,
ene gy supplie s could espond o he demand o high-p ice
con ac s wi h he choice o highe -cos wholesale s.
Finally, we discuss o he possible solu ions o he mo al haza d
p oblem a ising om he ene gy p ice b ake. He e we also o e
a emedy which en ails imposing a limi on he ex en o which
he ans e can inc ease in esponse o a highe con ac ual
130 The RAND Jou nal o Economics,2025
pe -uni p ice. Once his maximum le el is eached, any u he
inc ease in he con ac ual p ice does no esul in a highe
ans e . This egula ion allows he policymake o cons ain
he milking incen i es wi hou spoiling sa ing incen i es o
supplie s’ p o i abili y.
1.1 Rela ed Li e a u e
We con ibu e o he li e a u e dealing wi h he ene gy c isis and
he esul ing ene gy policies (Bachmann e al. 2022; Kes e nich,
Von G ae eni z, and Wambach 2022, and K use-Ande sen 2023)
and, mo e gene ally, o he li e a u e ha e alua es ene gy sa ings
policies (e.g., Reiss and Whi e 2008,I o2015,o F ase 2022).
To he bes o ou knowledge, we a e he i s o in es iga e he
ene gy p ice b ake heo e ically. We a e also unawa e o any
wo k in es iga ing a ans e scheme simila o he p oposed
p ice b ake. P ice caps and lump-sum ans e s analyzed in he
li e a u e do no sha e he no el and dis inguishing ea u e o
he p ice b ake ha he join su plus o i ms and consume s
inc eases in p ice (gi en he p ice is no below ma ginal cos ).
Whe eas an al e na i e app oach would be o s a wi h he p in-
cipal’s op imiza ion p oblem and de i e op imal ans e schemes
(see La on and Ti ole 1993; Viscusi, Ha ing on, and Sapping on
2018), ou goal is o assess he e ec i eness o he exis ing ene gy
p ice b ake and compa e i o al e na i e ins umen s.
We con ibu e o he cu en policy deba e on policies in he
ene gy c isis (see Amaglobeli e al. 2023;Fab a2023; Haan and
Schinkel 2023; Si in e al. 2023). No ewo hy in ou con ex is
he (in o mal) policy b ie on he ene gy p ice b ake (A aye and
Hillenb and 2022) ha poin s o he ac ha his measu e educes
consume s’ incen i es o sea ch o be e deals, which mi iga es
p ice compe i ion and migh aise p ices.
Ou pape is o ganized as ollows. In Sec ion 2, we p o ide a
g aphical illus a ion o he incen i es a ising om he ene gy
p ice b ake (in compa ison o al e na i e policies). In Sec ion 3,
we i s p esen he basic se up and he benchma k analysis,
whe e no elie p og am is in place (Sec ion 3.1). He e, we de i e
he ma ke ou comes unde a monopoly and unde compe i ion
when supplie s o e wo-pa a i con ac s. In Sec ion 3.2,
we in oduce he ene gy p ice b ake and show how i could
be exploi ed wi h no cons ain s on he ene gy p ices being
in place, which highligh s he incen i es o mo al haza d. In
Sec ion 3.3, we analyze he (a guably mo e ealis ic) case whe e
he con ac ual pe -uni p ice is es ic ed by legal cons ain s. In
Sec ion 4, we p o ide ex ensions on linea a i s (Sec ion 4.1),
capped ans e s (Sec ion 4.2), egula ed p ices (Sec ion 4.3), and
po en ial solu ions o he mo al haza d p oblem (Sec ion 4.4),
be o e we conclude in Sec ion 5. All p oo s a e elega ed o
he Appendix.
2 A G aphical Illus a ion o he P ice B ake s.
O he Ene gy Policies
The ene gy p ice b ake and i s ela ion o o he inancial elie
p og ams can be illus a ed a he hand o he household’s
budge line. Suppose he household can spend he income 𝑚
FIGURE 1 The hin line gi es he household’s budge line wi hou
he ene gy p ice b ake, and he hick line gi es he budge line wi h he
ene gy p ice b ake.
on ene gy consump ion 𝑥(measu ed in kWh) and on o he
goods 𝐶(measu ed in eu os).6Le 𝑝be he con ac ual ene gy
p ice measu ed in eu os pe kWh, and suppose i exceeds he
gua an eed p ice o he ene gy p ice b ake. In his sec ion,
we abs ac om he ixed paymen , as he e we ocus on he
oppo uni y cos s o ene gy consump ion, which a e independen
o he ixed paymen included in a wo-pa a i . Unde he
ene gy p ice b ake, he household aces he budge line
𝑝𝑥 +𝐶=𝑚+𝑇(𝑝), (2)
whe e 𝑇(𝑝) >0is he ans e speci ied in Equa ion (1).
Figu e 1depic s he budge line o a household wi h and wi hou
an ene gy p ice b ake in place. The ho izon al axis ep esen s he
ene gy consump ion le el 𝑥, and he e ical axis he consump-
ion expendi u es on o he goods 𝐶. In he absence o he ene gy
p ice b ake, he budge line is gi en by he hin line connec ing
he poin s (0, 𝑚) and (𝑚∕𝑝, 0) wi h a slope o 𝑑𝐶∕𝑑𝑥 =−𝑝, which
e lec s he oppo uni y cos o ene gy consump ion in e ms o
o egone expendi u es on o he consump ion goods 𝐶. Wi h he
in oduc ion o he ene gy p ice b ake, he household ecei es a
ans e 𝑇(𝑝), which is independen o i s ene gy consump ion in
he cu en pe iod. Fo a gi en p ice 𝑝, he ans e is, he e o e,
jus like a ixed ans e paymen o he household and does no
a ec he oppo uni y cos s o ene gy consump ion. Thus, he
slope o he budge line is again gi en by −𝑝, exac ly as in he
absence o such a ans e scheme (see he hick line in Figu e 1).
Mo eo e , he o he elemen s o he ene gy p ice b ake—namely,
he gua an eed p ice and he quo a—only a ec he amoun o
he ans e and he e o e do also no a ec he oppo uni y cos s
o ene gy consump ion.
Figu e 2elucida es he implica ions o he ac ha he ans e o
an ene gy p ice b ake depends on he con ac ual ene gy p ice
𝑝. I shows how he household’s budge line is a ec ed when
he con ac ual pe -uni p ice inc eases om 𝑝(solid line) o 𝑝′
(dashed line). Bo h budge lines in e sec a he quo a because
hen he household e ec i ely pays he gua an eed p ice (as
131
FIGURE 2 Budge lines wi h ene gy p ice b ake ans e s o 𝑝
(solid line) and 𝑝′>𝑝(dashed line).
speci ied by he ene gy p ice b ake) o he consumed quo a. Fo
all consump ion le els below he quo a, he budge line o 𝑝′
lies abo e he one o 𝑝. This e eals he incen i e o households
wi h an equilib ium consump ion le el below he quo a o choose
ene gy con ac s wi h high con ac ual p ices.7I consuming less
han he quo a, he household e ec i ely pays he gua an eed
p ice o ene gy consump ion bu also bene i s om he highe
ans e o a highe con ac ual p ice; his elaxes he cons ain
on consuming o he goods.
In he ollowing, we compa e he ene gy p ice b ake o a ious
o he policies and measu es implemen ed in he ene gy c isis.
One ob ious measu e is a gene al cu on ene gy axes, which
was obse ed in se e al coun ies du ing he 2022/23 ene gy c isis
(Sga a a i e al. 2023). Such a ax cu educes he ela i e cos s o
ene gy and u ns he hin budge line in Figu e 1ou wa d a ound
he e ical in e cep .
Fu he mo e, i is ins uc i e o compa e he ene gy p ice b ake
wi h a p ice-cap egula ion such as he Du ch ene gy p ice ceiling
sys em ( o de ails see Haan and Schinkel 2023), whe e an ene gy
p ice cap (o , o ins ance, 40 eu o cen s pe kWh o elec ici y
in 2023 in he Ne he lands) applies o a ce ain quo a.8Only o
ene gy consump ion ha exceeds his quo a, he con ac ual pe -
uni p ice applies.9
Figu e 3shows how such a p ice-cap egula ion a ec s he
household’s budge line. The hin solid line ep esen s he budge
line wi hou any in e en ion. The (kinked) dashed line depic s
he budge line unde a p ice-cap egula ion wi h a quo a. Fo
consump ion le els below he quo a, he oppo uni y cos s o
ene gy consump ion a e gi en by he capped p ice 𝑝CAP, which
is smalle han he con ac ual p ice 𝑝. Only o consump ion
le els abo e he quo a (as speci ied in he p ice-cap egula ion),
oppo uni y cos s a e gi en by 𝑝. The hick solid line, on he
con a y, ep esen s he budge line unde he ene gy p ice
b ake, whe e he oppo uni y cos s o ene gy consump ion a e
always gi en by 𝑝. Thus, ce e is pa ibus, he ene gy p ice b ake
should induce highe ene gy sa ing incen i es han unde a
p ice-cap egula ion.
FIGURE 3 Budge line wi hou any in e en ion ( hin solid line),
wi h a p ice cap (dashed line), and wi h a p ice b ake ( hick solid line).
Rela edly, Aus ia has implemen ed an ene gy p ice subsidy,
whe e he go e nmen subsidizes he elec ici y p ice by paying
a ce ain pe cen age o he ene gy p ice up o some consump ion
quo a, bu maximally 30 cen pe kWh. 10 Fo a gi en con ac ual
p ice, he Aus ian elie p og am gi es ise o a budge cons ain
ha esembles he one unde he Du ch ene gy p ice ceiling
sys em, as p ices pe uni a e dampened up o some quo a.
Ene gy ouche s educe sa ing incen i es by e en mo e han he
p eceding policies. Coun ies like C oa ia, F ance, and Po ugal
ha e implemen ed ene gy ouche s o some demanding g oups
(Sga a a i e al. 2023), so ha up o some ce ain quan i y ene gy
is ee. This policy can be ep esen ed by he dashed budge line
in Figu e 3when his is la up o he kink.
Al oge he , he discussed al e na i es o he p ice b ake (namely,
p ice caps, ene gy p ice subsidies, ene gy ouche s, and ax cu s)
educe he oppo uni y cos s o ene gy consump ion (a leas up
o some quo a) and could he e o e agg a a e he ene gy c isis
in he o m o a possible ma ke b eakdown. The ene gy p ice
b ake (like a ixed ans e scheme) does no su e om his
p oblem, so ha his scheme is pa icula ly a ac i e when he
isk o a ma ke b eakdown could pose a eal p oblem. In ac ,
ou analysis below shows ha he ene gy p ice b ake e en ends o
aise ene gy p ices, so ha ene gy sa ing incen i es a e ein o ced
by his policy.
3Model and Analysis
3.1 Benchma k (Wi hou T ans e Scheme)
Suppose a monopolis supplie o e s a wo-pa a i con ac
wi h a con ac ual pe -uni p ice 𝑝≥0anda ixedpaymen 𝐹.
The consume ’s o e all u ili y is
𝐶𝑆 ={𝑈(𝑥) −𝑝𝑥 −𝐹i he con ac is accep ed
𝑅i he con ac is ejec ed, (3)
whe e 𝑈(𝑥) is he u ili y o consuming ene gy quan i y 𝑥≥0,
and 𝑅≥0 ep esen s he consume ’s ese a ion u ili y (i.e, he
maximal u ili y he consume can ob ain when swi ching o he
bes al e na i e). 11
132 The RAND Jou nal o Economics,2025
Le 𝑈(𝑥) be a leas wice con inuously di e en iable o e
ℝ≥0. We impose u he s anda d assump ions; namely, 𝑈(0) =
0,𝑈′∶=𝜕𝑈(𝑥)∕𝜕𝑥 >0,and𝑈′′ ∶=𝜕2𝑈(𝑥)∕𝜕𝑥2<0. Thus, he
u ili y om ze o consump ion is se o ze o, u ili y is s ic ly
inc easing in he amoun o ene gy consumed, and he ma ginal
u ili y dec eases wi h highe ene gy consump ion le els. No e
ha we do no impose es ic ions on any o he highe -o de
de i a i e o 𝑈(𝑥). In pa icula , he hi d de i a i e can be
posi i e o nega i e and may also al e na e i s sign along 𝑥≥0.
We can easily ela e ou se up o he g aphical exposi ion in
Sec ion 2(see Figu e 1) i we assume ha he o e all consume
u ili y depends no only on he u ili y o ene gy consump ion
𝑈(𝑥) bu also linea ly on expendi u es on o he goods 𝐶(i.e., he
o e all consume u ili y is quasi-linea and equal o 𝑈(𝑥) +𝐶).
Fo a wo-pa a i , he budge cons ain is gi en by 𝑝𝑥 +𝐹+
𝐶≤𝑚and hus di e s om (2), whe e we assumed a linea
ene gy p ice. As he consume will exhaus he budge , we
ge 𝐶𝑆 =𝑈(𝑥) −𝑝𝑥 −𝐹+𝑚, which di e s om (3) only in he
cons an 𝑚 ha does no a ec any o ou esul s.12 Fo he sake
o b e i y, we wo k wi h (3), which omi s 𝐶and he associa ed
budge cons ain .
The supplie ’s p o i is gi en by
𝜋∶=(𝑝 −𝑐)𝑥 +𝐹, (4)
whe e 𝑐≥0gi es he ma ginal cos o ene gy supply. The join
su plus o he supplie and he consume is hen gi en by
𝐶𝑆 +𝜋=𝑈(𝑥) −𝑐𝑥. (5)
Le 𝑘∶=lim𝑥→0+𝑈′be he choke p ice, ha is, he lowes p ice
a which demand is ze o. To ob ain a non- i ial solu ion in ou
ollowing analysis, we impose he ollowing assump ion on he
choke p ice and he join su plus.
Assump ion 1. The choke p ice 𝑘sa is ies 𝑐<𝑘<∞.In addi-
ion, he e exis s 𝑥>0so ha he e is scope o Pa e o-imp o ing
ade, ha is,𝑈(𝑥) −𝑐𝑥 >𝑅.
The con ac ing game p oceeds in wo s ages. In he i s s age,
he supplie (“he”) o e s a wo-pa a i con ac ; in he second
s age, he consume (“she”) accep s o ejec s he o e ed con ac .
I she accep s, she de e mines he ene gy consump ion le el 𝑥.
I she ejec s, she ealizes he ese a ion alue 𝑅. We sol e his
game o subgame-pe ec Nash equilib ia.
I he consume accep s he con ac , she sol es max𝑥≥0𝑈(𝑥) −
𝑝𝑥 −𝐹. He ene gy demand 𝑥(𝑝) hen ollows om he i s -o de
condi ion
𝑈′−𝑝≤0, (6)
which holds as an equali y i he solu ion is s ic ly posi i e
(𝑥(𝑝) >0), in which case 𝑑𝑥(𝑝)∕𝑑𝑝 =1∕𝑈′′ <0holds (i.e.,
demand is downwa d sloping). I 𝑈′<𝑝 o all 𝑥>0, hen𝑥(𝑝) =
0. The nex lemma summa izes ou esul s on he consume ’s
demand unc ion.
Lemma 1 (Ene gy demand). Suppose he consume has
accep ed a wo-pa a i con ac wi h a con ac ual pe -uni p ice
𝑝≥0. Then, he demand 𝑥(𝑝) ollows om (6) and depends on 𝑝
as ollows:
i) I 𝑝∈[0,𝑘), hen 𝑥(𝑝) >0and 𝑑𝑥(𝑝)∕𝑑𝑝 =1∕𝑈′′ <0,as
well as lim𝑝→𝑘−𝑥(𝑝) =0.
ii) I 𝑝≥𝑘, hen 𝑥(𝑝) =0.
The consume accep s a con ac o e i he pa icipa ion
cons ain
𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) −𝐹≥𝑅(7)
is sa is ied, wi h 𝑥(𝑝)being cha ac e ized in Lemma 1. I 𝑥(𝑝) =0,
hen an accep able con ac mus sa is y 𝐹≤0wi h |𝐹|≥𝑅; ha
is, a ixed paymen is made om he supplie o he consume . I
(7) is iola ed, he consume ejec s he con ac o e and ealizes
𝑅.
In he i s s age o he game, he supplie an icipa es he
consume ’s demand unc ion 𝑥(𝑝) as well as he pa icipa ion
cons ain (7), and se s a wo-pa a i con ac ha sol es
max
𝐹,𝑝 (𝑝 −𝑐)𝑥(𝑝) +𝐹s. . (7).
In equilib ium, he consume ’s pa icipa ion cons ain mus
bind. Subs i u ing his in o he supplie ’s p o i unc ion, he
supplie sol es
max
𝑝≥0𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) −𝑅, (8)
which gi es he i s -o de condi ion 𝑈′−𝑐=0. By Assump ion
1, he e exis s a unique p o i -maximizing wo-pa a i con ac
(𝐹∗,𝑝∗)wi h 𝑝∗=𝑐,so ha 𝑥∗∶=𝑥(𝑐) >0. This solu ion maxi-
mizes he join su plus (5), because any p ice 𝑝la ge o smalle
han 𝑐dec eases he join su plus ( o a p oo see Lemma 2 in he
Appendix). Wi h he ixed paymen 𝐹∗, he monopoly supplie
ex ac s he en i e join su plus ne o he consume ’s ese a ion
u ili y, so ha 𝐹∗=𝑈(𝑥∗)−𝑝∗𝑥∗−𝑅>0holds.
We p oceed wi h he analysis o compe i ion, whe e a leas wo
supplie s wi h ma ginal cos s 𝑐compe e o he consume . The
con ac ing game unde compe i ion is as ollows. In he i s
s age, supplie s simul aneously o e wo-pa a i con ac s o
he consume ; in he second s age, he consume accep s one o
he con ac s o ejec s all o e s. I he consume accep s one o
he con ac s, she de e mines he ene gy consump ion le el 𝑥.I
he consume ejec s all con ac s, she ealizes 𝑅. Again, we sol e
his game o subgame-pe ec Nash equilib ia.
I he consume accep s one o he con ac s, he demand is gi en
by Lemma 1. When acing mo e han one accep able con ac , he
consume selec s he con ac wi h he highes o e all u ili y; in
case o indi e ence, he consume selec s each o he con ac s
wi h a s ic ly posi i e p obabili y.
Unde compe i ion, all i ms make ze o p o i , so ha 𝐹∗∗ =0and
𝑝∗∗ =𝑐hold. In his case, he consume pocke s he en i e join
133
su plus, which is maximal as in he monopoly case. P oposi ion 1
summa izes he benchma k esul s.
P oposi ion 1 (Benchma k esul ). Assume ha a sup-
plie o e s a wo-pa a i con ac . Then, he equilib ium bo h
unde monopoly and compe i ion implemen s he join su plus
maximizing solu ion:
i) The con ac ual ene gy p ice pe uni is equal o ma ginal cos s:
𝑝∗=𝑝∗∗ =𝑐.
ii) The consume ’s ene gy consump ion is 𝑥∗=𝑥(𝑐) >0.
Mo eo e , unde a monopoly he ixed paymen is 𝐹∗=𝑈(𝑥∗)−
𝑐𝑥∗−𝑅>0and he consume ealizes o e all u ili y o 𝐶𝑆 =𝑅,
whe eas unde compe i ion he ixed paymen is 𝐹∗∗ =0and he
consume ealizes 𝐶𝑆 =𝑈(𝑥∗)−𝑐𝑥∗>𝑅.
We can now analyze he mos basic ene gy elie measu e,
namely, an uncondi ional ixed ans e o consume s, 𝑇>0. This
ans e is no pa o he join su plus o he supplie and he
consume ne o he ese a ion u ili y, so ha 𝑇does no a ec
he pa icipa ion cons ain (7), bu only inc eases he consume ’s
o e all u ili y by 𝑇. This holds ob iously bo h unde monopoly
and unde compe i ion.
Co olla y 1 (Uncondi ional ixed ans e ). An uncondi-
ional ixed ans e 𝑇>0 om he go e nmen o he consume
only inc eases he consume ’s o e all u ili y by 𝑇, and does no a ec
he ma ke ou come unde monopoly o compe i ion.
3.2 Ene gy P ice B ake: Uncons ained
Con ac ual Pe -Uni P ice
Suppose ha p io o he con ac ing game, he go e nmen
implemen s an ene gy p ice b ake wi h a ans e 𝑇(𝑝) de ined
by
𝑇(𝑝) ∶=max{(𝑝 −𝑠)𝛼𝑥,0}, (9)
whe e 𝑝is he con ac ual pe -uni p ice ha applies when he
ans e scheme is in place, 𝑠>0is he gua an eed pe -uni
p ice, 𝑥>0gi es a e e ence ene gy consump ion le el, and 𝛼∈
(0, 1) gi es some sha e o he e e ence consump ion le el; we
call 𝛼𝑥 he “quo a.” No e ha 𝑝is se by he supplie , whe eas
he go e nmen se s 𝑠and 𝛼.13 The ans e 𝑇(𝑝) is he e o e a
lump-sum paymen ha inc eases linea ly in he p ice se by
he supplie , 𝜕𝑇(𝑝)∕𝜕𝑝 =𝛼𝑥>0,aslongas𝑝>𝑠. By his, he
consume ’s oppo uni y cos o any uni o gas consump ion is
le unchanged and gi en by he cu en con ac ual ene gy p ice
𝑝.
We assume ha 𝑠∈(0,𝑐), so ha he ans e is s ic ly posi i e
o all 𝑝≥𝑐. Thus, gi en he benchma k equilib ium ou come
(see P oposi ion 1), he ans e scheme, ce e is pa ibus, o e s
inancial elie o consume s in he o m o a ans e paymen
𝑇(𝑝∗), which is independen o he ma ke s uc u e (monopoly
o compe i ion).
Wi h a ans e scheme 𝑇(𝑝) in place, o e all consume u ili y is
gi en by
𝐶𝑆 ={𝑈(𝑥) −𝑝𝑥 −𝐹+𝑇(𝑝), i he con ac is accep ed
𝑅, i he con ac is ejec ed.
(10)
I ollows ha he in oduc ion o 𝑇(𝑝) does no a ec ene gy
demand 𝑥(𝑝) (as gi en by Lemma 1) because he ans e does
no depend on he ene gy consump ion le el 𝑥.
C i ically, he ans e scheme a ec s he consume ’s u ili y om
accep ing a ce ain con ac , because he ans e depends di ec ly
on he con ac ual pe -uni p ice. I ollows ha he consume
canno ealize he ans e wi hou accep ing he espec i e
con ac . Consequen ly, he in oduc ion o he ans e scheme
a ec s he consume ’s pa icipa ion cons ain , which is now
gi en by
𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) −𝐹+𝑇(𝑝) ≥𝑅, (11)
wi h demand 𝑥(𝑝) ollowing om Lemma 1. Nex , we analyze he
i s s age o he con ac ing game, bo h o he monopoly case
and he compe i ion case.
3.2.1 Monopoly
An icipa ing he consume ’s decisions in he second s age o he
game, he supplie sol es
max
𝐹,𝑝 𝜋=𝐹+(𝑝 −𝑐)𝑥(𝑝) s. . (11).
In he op imal solu ion, he pa icipa ion cons ain (11)mus
bind. This can be achie ed by se ing 𝐹=𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) +
𝑇(𝑝) −𝑅, which gi es he educed p oblem
max
𝑝≥0ˆ
𝜋(𝑝) ∶=𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝) −𝑅. (12)
The supplie ’s maximiza ion p oblem (12) depends on he sum o
he join su plus, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝), and he ans e , 𝑇(𝑝), which
gi es he “new” join su plus o he supplie and he consume
unde he ene gy p ice b ake. Taking he de i a i e o ˆ
𝜋(𝑝) wi h
espec o 𝑝gi es
𝜕ˆ
𝜋(𝑝)
𝜕𝑝 =𝑈′−𝑐
𝑈′′ +𝜕𝑇(𝑝)
𝜕𝑝 o 𝑥(𝑝) >0and
𝜕ˆ
𝜋(𝑝)
𝜕𝑝 =𝜕𝑇(𝑝)
𝜕𝑝 o 𝑥(𝑝) =0. (13)
Wi hou a ans e 𝑇(𝑝), he op imal p ice would be he join
su plus maximizing p ice 𝑝∗=𝑐wi h 𝑥∗>0(see P oposi ion
1). In oducing he ans e scheme c ea es an incen i e o aise
he con ac ual pe -uni p ice abo e 𝑐, because now 𝜕ˆ
𝜋(𝑝)∕𝜕𝑝 =
𝜕𝑇(𝑝)∕𝜕𝑝 >0holds a 𝑝=𝑐. Thus, he supplie aises he
con ac ual pe -uni p ice abo e he join su plus maximizing
p ice 𝑝∗=𝑐(see P oposi ion 1), so ha ene gy consump ion is
educed (acco ding o Lemma 1) below he socially op imal le el
𝑥∗.
134 The RAND Jou nal o Economics,2025
Equa ion (13) unco e s he undamen al d awback o an ene gy
p ice b ake ha is no p o ec ed by supplemen a y egula ions
agains misuse. Inspec ing he de i a i es in (13) e eals he
incen i e o a bi a ily in la e he con ac ual pe -uni p ice. Fo
p ices 𝑝∈(𝑐,𝑘), he supplie ’s p o i unc ion could, in gene al,
ake many o ms depending on he highe -o de de i a i es o
𝑈(𝑥) (see Lemma 3 in he Appendix). Howe e , in his egion,
i is clea ly bounded om abo e, because he join su plus
𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) is s ic ly dec easing in 𝑝and becomes ze o o
𝑝→𝑘
−(see Lemma 2 in he Appendix), whe eas 𝑇(𝑝) is linea ly
inc easing in 𝑝. As he ans e paymen o he ene gy p ice b ake
inc eases e en o con ac ual p ices abo e he choke p ice, he
supplie ’s p o i ˆ
𝜋(𝑝) is unbounded in 𝑝. Thus, by aising 𝑝abo e
𝑘, he supplie can ealize a highe p o i han o any p ice below
𝑘. The supplie can he e o e ealize an a bi a ily la ge p o i by
ex ac ing he ans e om he ene gy p ice b ake wi h he ixed
paymen , while allowing he consume an o e all u ili y o a
leas 𝑅.
P oposi ion 2 (Uncons ained con ac ual ene gy p ice
unde monopoly). Suppose a ans e scheme 𝑇(𝑝) gi en by
(9), and suppose ha a monopoly supplie o e s a wo-pa a i
con ac o he consume such ha he consume ’s pa icipa ion
cons ain (11) holds. Then, he supplie can inc ease his p o i by
any amoun by aising he con ac ual p ice pe uni by su icien ly
much. Thus, we ha e 𝑝→∞,𝑇(𝑝) → ∞,and𝜋→∞, while 𝑥=0.
3.2.2 Compe i ion
Unde compe i ion, he consume selec s he con ac which
o e s he highes o e all u ili y. Because he ene gy p ice b ake,
as de ined abo e, allows o milking he ans e scheme by any
amoun , i ollows ha 𝑝→∞and 𝑇(𝑝) → ∞,so ha 𝑥=0
mus hold. The main di e ence o he monopoly case is ha
compe i ion o ces i ms o o e he mos a ac i e con ac o
he consume as, o he wise, she would no buy. Compe i ion o
he consume ’s con ac accep ance, he e o e, ine i ably induces
i ms o in la e he con ac ual ene gy p ice as his makes he
con ac o e mos a ac i e. We summa ize hose esul s as
ollows.
P oposi ion 3 (Uncons ained con ac ual ene gy p ice
unde compe i ion). Suppose a ans e scheme 𝑇(𝑝) gi en
by (9), and suppose ha a leas wo supplie s o e wo-pa
a i con ac s o he consume , such ha i ms make non-
nega i e p o i s. Then, 𝑝→∞,𝑇(𝑝) → ∞,and𝐶𝑆 → ∞, while
𝑥=0.
P oposi ion 3 shows ha compe i ion leads essen ially o he
same ou come as a monopoly. Ye , in he monopoly case,
i is he supplie who bene i s om aising he con ac ual
pe -uni p ice abo e ma ginal cos s because his inc eases his
p o i di ec ly (gi en some consume u ili y such ha he
consume ’s pa icipa ion cons ain is sa is ied); unde compe-
i ion, i is he consume ’s decision ule o selec he mos
a ac i e con ac which induces i ms o aise he con ac-
ual ene gy p ice (gi en some non-nega i e p o i le el i hey
sell).
3.3 Ene gy P ice B ake: Cons ained Con ac ual
Pe -Uni P ice
Ou esul s on uncons ained milking o he ene gy p ice b ake
illus a e he incen i es a ising om his policy ins umen ,
bu uncons ained milking ep esen s an ob ious misuse o he
ene gy elie scheme.14 Hence, assuming ha he con ac ual
p ice is cons ained by some p ice 𝑝, which ensu es a s ic ly
posi i e ene gy consump ion le el, is easonable. Thus, in he
ollowing, we impose a maximum con ac ual ene gy p ice pe
uni , 𝑝,wi h𝑝∈(𝑐,𝑘),so ha 𝑥(𝑝) >0.15 No e ha his p ice
cons ain would ne e be binding in he benchma k case (see
Sec ion 3.1), whe e 𝑝∗=𝑝∗∗ =𝑐holds; ha is, he pu pose o 𝑝
is o cons ain po en ial misuse o he ans e scheme bu no o
lowe egula ene gy p ices.
3.3.1 Monopoly
The ene gy p ice cons ain 𝑝≤𝑝e ec i ely cons ains he
monopolis ’s abili y o ake ad an age o he ans e scheme.
Gi en 𝑝≤𝑝, he ollowing p oposi ion speci ies he equilib ium
con ac and i s p ope ies.
P oposi ion 4 (Cons ained con ac ual pe -uni p ice
unde monopoly). Suppose a ans e scheme 𝑇(𝑝) gi en by (9),
and suppose he addi ional cons ain 𝑝≤𝑝∈(𝑐,𝑘)holds. Then,
he monopoly supplie ’s equilib ium ( wo-pa a i ) con ac o e
ul ills ei he (i) o (ii):
i) In e io solu ion:𝑝 ul ills (𝑈′−𝑐)∕𝑈′′ +𝜕𝑇(𝑝)∕𝜕𝑝 =0,so
ha 𝑝∗<𝑝≤𝑝.
ii) Co ne solu ion:𝑝 ul ills 𝑝∗<𝑝=𝑝.
Mo eo e , he supplie ’s p o i is 𝜋=𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝) −𝑅
and he consume ge s 𝑅, while 𝑇(𝑝) >𝑇(𝑝∗)and 0<𝑥(𝑝) <
𝑥(𝑝∗)always hold.
P oposi ion 4 shows ha he in oduc ion o he ans e scheme
aises he pe -uni p ice also when i is e ec i ely cons ained.
I aises bo h he equilib ium p ice and he ans e beyond he
le els ha would p e ail wi hou he ene gy p ice b ake. The
op imal p ice ei he ul ills he condi ion o a local maximum, o
i is ob ained a 𝑝. In any case, ene gy sa ing incen i es a e always
ein o ced by he in oduc ion o he ene gy p ice b ake ( ela i e
o he e icien ene gy consump ion le el 𝑥∗) bu consume s
a e no elie ed; hey ecei e hei ese a ion u ili y 𝑅and a e
indi e en o he si ua ion wi hou he ans e scheme.
3.3.2 Compe i ion
Compe i ion does no a ec he con ac ual pe -uni p ice and he
ene gy consump ion le el as de i ed o he monopoly case. No e
i s ha a consume who aces mo e han one accep able con ac
o e chooses he con ac ha yields he highes o e all u ili y
(10). Unde compe i ion, i ms o e wo-pa a i con ac s ha ,
again, maximize he join su plus, bu hey make ze o p o i s.
Hence, he ixed paymen mus be nega i e wi h 𝐹=−(𝑝 −
𝑐)𝑥(𝑝).Thus,on opo 𝑈(𝑥(𝑝)), he consume also ully pocke s
135
he ans e 𝑇(𝑝) as well as he supplie ’s p o i ma gin (𝑝 −
𝑐)𝑥(𝑝), and he expenses a e 𝑝𝑥(𝑝).
P oposi ion 5 (Cons ained con ac ual pe -uni p ice
unde compe i ion). Suppose a ans e scheme 𝑇(𝑝) gi en by
(9), and suppose he addi ional cons ain 𝑝≤𝑝∈(𝑐,𝑘)holds.
Then, unde compe i ion, 𝑝and 𝑥(𝑝) a e he same as unde a
monopoly (i.e., hey a e as speci ied in P oposi ion 4), supplie s
ealize 𝜋=0and consume s ge 𝐶𝑆 =𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝) >
𝑅.
P oposi ion 5 again shows ha compe i ion leads o he same
ma ke ou come as a monopoly. In bo h cases, he con ac ual
pe -uni p ice is inc eased abo e he socially e icien le el (i.e.,
𝑐). As in he monopoly case, he supplie s maximize he join
su plus (5) plus he ans e o he ene gy p ice b ake 𝑇(𝑝).
Howe e , compe i ion be ween supplie s d i es p o i s o ze o,
which implies a nega i e ixed paymen .
P oposi ions 4 and 5 show ha in oducing an ene gy p ice
b ake educes ene gy consump ion compa ed o he benchma k
si ua ion wi h no ans e scheme (o when compa ed wi h a
ixed ans e paymen ; see Co olla y 1). This esul is a di ec
esul o he ac s ha (i) he con ac ual ene gy p ice inc eases
abo e 𝑐in he p esence o he ene gy p ice b ake, and (ii)
ene gy demand is downwa d sloping (see Lemma 1). Thus, he
ene gy p ice b ake achie es he objec i e o educing ene gy
demand.
Mo eo e , P oposi ions 4 and 5 show ha whe he he objec i e o
he ene gy p ice b ake o elie e consume s inancially is eached
depends on he ma ke s uc u e. This objec i e can be achie ed
wi h compe i ion among supplie s because consume s hen
ully pocke he ene gy p ice b ake ans e s. On he con a y,
consume s a e no elie ed in he p esence o a monopolis ic
supplie ha ully pocke s he ans e s om he ene gy p ice
b ake.
4Ex ensions
In he ollowing, we analyze how addi ional es ic ions on
ene gy supply con ac s a ec he equilib ium ou come unde an
ene gy p ice b ake. Fi s , we suppose ha a supplie can only
eely choose he con ac ual pe -uni p ice (i.e., we ha e a egime
o “linea ene gy con ac s”); second, we examine he case o
“capped ans e s” (as speci ied in Ge many’s ene gy p ice b ake
legisla ion whe eby a consume ’s ene gy bill canno be nega i e);
and hi d, we conside (cos -based) “ egula ed ene gy p ices.”
Those es ic ions could limi he mo al haza d p oblem induced
by he ene gy p ice b ake, bu hey canno elimina e i en i ely.
Finally, in a ou h ex ension, we discuss po en ial solu ions o
he mo al haza d p oblem.
4.1 Linea Ene gy Con ac s
I he supplie can se a wo-pa a i con ac , he can maximize
he join su plus wi h he pe -uni p ice 𝑝, and sha e i e icien ly
wi h he ixed paymen 𝐹. Ene gy ma ke egula ions, howe e ,
could cons ain he p o ide s’ abili y o se o al e he ixed
paymen .16 How do ou esul s change when he supplie can
only se linea ene gy con ac s, ha is, he pe -uni p ice? As in
he p e ious sec ion, we assume some maximal p ice 𝑝 ha he
con ac ual pe -uni p ice 𝑝canno su pass; ha is, 𝑝≤𝑝,wi h
𝑝∈(𝑐,𝑘).
4.1.1 Monopoly
Assume a monopoly supplie and he same wo-s age con ac ing
game as be o e, wi h he only di e ence ha he supplie can now
only se a “linea ” ene gy p ice 𝑝. Consume u ili y is gi en by (3),
wi h 𝐹=0. I a consume accep s he o e , he demand 𝑥(𝑝) is
gi en by Lemma 1.
Fo a gi en ans e scheme 𝑇(𝑝) and ene gy demand 𝑥(𝑝), he
pa icipa ion cons ain o he consume is gi en by
𝐶𝑆(𝑝) ∶=𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) +𝑇(𝑝) ≥𝑅, (14)
whe e he le -hand side o he inequali y is he o e all u ili y
om accep ing a con ac wi h p ice 𝑝<𝑘. An icipa ing ene gy
demand 𝑥(𝑝), he supplie sol es
max
𝑝𝜋(𝑝) ∶=(𝑝 −𝑐)𝑥(𝑝) s. . (14). (15)
Unlike he wo-pa a i case, whe e he ans e scheme 𝑇(𝑝)
can ende an ou come wi h 𝑥(𝑝) =0p o i able (see P oposi ion
1), wi h linea p ices, he p o i is always ze o when he pe -uni
p ice su passes he choke p ice.
The p o i unc ion 𝜋(𝑝) has a leas one local maximum.17 To
p oceed pa simoniously, we impose he s anda d assump ion ha
he ma ginal p o i unc ion changes i s sign only once so ha
he e is only one local maximum. In addi ion, we assume ha 𝑝
does no es ic he a ainabili y o he unique local maximum
(in ac , we can hink o 𝑝being close o 𝑘). Hence, as be o e, he
maximal p ice 𝑝is no used as some o m o p ice-cap egula ion
o es ic he monopoly supplie ’s p ice-se ing beha io in he
absence o he ene gy p ice b ake; howe e , i could es ic he
exploi a ion o he ans e paymen when an ene gy p ice b ake
is in place.18
Assump ion 2. The supplie ’s ma ginal p o i 𝜕𝜋(𝑝)∕𝜕𝑝 has a
mos one ze o o e (𝑐, 𝑘),which we deno e 𝑝𝐼.Mo eo e ,𝑝𝐼<𝑝.
The analysis o he e ec s o he ans e scheme 𝑇(𝑝) depends
on he con ac ing ou come in he absence o i . Gi en ene gy
demand (Lemma 1), we ha e o dis inguish wo cases depending
on whe he o no he pa icipa ion cons ain
𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝) ≥𝑅(16)
is binding.
Case I (Pa icipa ion cons ain (16)no binding).The
unique local maximum 𝑝𝐼gi es he monopoly solu ion, and
ollows om he i s -o de condi ion
𝜕𝜋(𝑝)
𝜕𝑝 =𝑥(𝑝) +𝑝−𝑐
𝑈′′ =0. (17)
136 The RAND Jou nal o Economics,2025
By (6), 𝑈′−𝑐
𝑈′′
>
=
<0⇔𝑝<
=
>𝑐. No e also ha lim𝑝→𝑘−[𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝)]=0
ollows om lim𝑝→𝑘−𝑥(𝑝) =0and 𝑈(0) =0, and pa ii ollows om
𝑥(𝑝) =0 o all 𝑝≥𝑘(Lemma 1). Thus, he join su plus has a unique
maximum a 𝑝∗=𝑐.□
Pa i o he p oposi ion o he monopoly case ollows immedia ely om
Lemma 2 because he solu ion o (8)mus be hesameas hesolu ion
o he maximiza ion o he join su plus. Pa ii o he monopoly case
ollows om Assump ion 1, so ha he ene gy consump ion le el is s ic ly
posi i e and he socially op imally one. In equilib ium, he consume ’s
pa icipa ion cons ain (7) mus bind, so ha he ixed ee is gi en by he
maximal join su plus ne o he consume ’s ou side op ion u ili y; ha is,
𝐹=𝑈(𝑥∗)−𝑐𝑥∗−𝑅, which is also he supplie ’s equilib ium p o i . The
consume hen ob ains 𝐶𝑆 =𝑅.
Unde compe i ion, supplie s a e pe ec ly subs i u able om he con-
sume ’s pe spec i e (as he e is no p oduc di e en ia ion; mo eo e ,
ma ginal cos s a e cons an and he same o all supplie s), so ha he
consume always chooses he con ac o e wi h he highes o e all u ili y
𝐶𝑆. I is hen s aigh o wa d o see ha he equilib ium con ac o e
mus maximize he consume ’s o e all u ili y (i.e., he con ac ual pe -
uni p ice is se o ma ginal cos s o maximize he join su plus acco ding
o Lemma 1), whe eas no supplie can ealize s ic ly posi i e p o i s wi h
𝐹>0(as he supplie could be unde cu by 𝐹−𝜀,wi h𝜀>0). I ollows
ha 𝑝∗=𝑐and 𝐹=0mus hold unde compe i ion, so ha any supplie ’s
p o i is ze o and he consume ’s o e all u ili y is equal o he maximal
join su plus; ha is, 𝐶𝑆 =𝑈(𝑥∗)−𝑐𝑥∗.
P oo o Co olla y 1. An uncondi ional ixed ans e 𝑇>0nei he a ec s
he consume ’s pa icipa ion cons ain (7) no he consume ’s ene gy
demand acco ding o Lemma 1. I , he e o e, does also no a ec he
supplie ’s maximiza ion p oblem, so ha he ma ke equilib ium as
desc ibed in P oposi ion 1 emains he same. □
P oo o P oposi ion 2. The supplie aces maximiza ion p oblem (12).
To unde s and i s solu ion, i is help ul o examine how 𝑝a ec s he
sum o he join su plus (see Lemma 2) and he ans e 𝑇(𝑝); his hen
de e mines he ma ginal p o i (see Equa ion (13)). Gi en ene gy demand
𝑥(𝑝) acco ding o Lemma 1, he ollowing lemma speci ies he p ope ies
o he join su plus, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) (see Lemma 2), augmen ed by he
ene gy p ice b ake 𝑇(𝑝), which we de ine by
Π(𝑝) ∶=𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) +𝑇(𝑝). (A2)
Lemma 3 (Join su plus wi h ene gy p ice b ake). Assume he go e nmen
o e s an uncons ained ene gy p ice b ake (9) o he consume . Then Π(𝑝),
ul ills he ollowing p ope ies:
i) I is con inuous e e ywhe e and i is di e en iable o all 𝑝≥0excep
a 𝑝=𝑠and 𝑝=𝑘, whe e i has wo kinks.
ii) I is s ic ly inc easing o all 𝑝∈[0,𝑐], ob ains he alue 𝑇(𝑘) a 𝑝=
𝑘, and i has he cons an slope 𝜕Π(𝑝)∕𝜕𝑝 =𝜕𝑇(𝑝)∕𝜕𝑝 =𝛼𝑥>0 o
all 𝑝>𝑘.
iii) I is bounded om abo e and om below on [𝑐, 𝑘].
i ) On [𝑐, 𝑘],Π(𝑝) has a maximum ei he a some 𝑝∈(𝑐,𝑘)(in e io
solu ion) o a 𝑝=𝑘(co ne solu ion).
P oo o Lemma 3. No e i s ha Π(𝑝) is he sum o 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝)
(which p ope ies a e gi en in Lemma 2) and 𝑇(𝑝). No e ha 𝑇(𝑝) =0
o 𝑝≤𝑠and 𝑇(𝑝) linea ly inc easing wi h slope 0<𝜕𝑇(𝑝)∕𝜕𝑝 <∞ o
all 𝑝>𝑠.
Pa i). By Lemma 2, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) is con inuous in 𝑝and has a
kink a 𝑝=𝑘, and 𝑇(𝑝) is linea in 𝑝 o 𝑝>𝑠.Mo eo e ,𝑇(𝑝)
is ze o o 𝑝≤𝑠and linea ly inc easing o all 𝑝>𝑠, so ha
𝑇(𝑝) is also con inuous and has a kink a 𝑝=𝑠. I ollows ha
Π(𝑝) is con inuous in 𝑝wi h wo kinks a 𝑝=𝑠and 𝑝=𝑘.
Likewise, Π(𝑝) is di e en iable e e ywhe e excep a poin s
𝑝=𝑠and 𝑝=𝑘.
Pa ii). By Lemma 2, 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) is inc easing in 𝑝 o 0≤𝑝≤𝑐
(wi h a ze o a 𝑝=𝑐), and 𝑇(𝑝) is s ic ly inc easing in 𝑝 o all
𝑝>𝑠,wi h𝑠<𝑐. Thus, Π(𝑝) is s ic ly inc easing in 𝑝 o all
0≤𝑝≤𝑐; and in pa icula , a 𝑝=𝑐. By Lemma 2, 𝑈(𝑥(𝑝)) −
𝑐𝑥(𝑝) is ze o o 𝑝≥𝑘, and 𝑇(𝑝) is linea o all 𝑝>𝑠.
Pa iii). An uppe bound is gi en by Π(𝑝) <Π(𝑐) +Π(𝑘) <∞, which
ollows om 𝑈(𝑥(𝑝)) −𝑐𝑥(𝑝) s ic ly dec easing in 𝑝(see pa
ii o Lemma 2) and 𝑇(𝑝)linea ly inc easing in 𝑝.Alowe bound
is gi en by Π(𝑝) >0, which ollows om lim𝑝→𝑘−𝑈(𝑥(𝑝)) −
𝑐𝑥(𝑝) =0(Lemma 2) and 𝑇(𝑝) >0 o all 𝑝∈[𝑐,𝑘].
Pa i ). Because o pa ii, he e canno be a maximum a 𝑝≤𝑐.
Then, he e a e wo possible cases: ei he 𝜕Π(𝑝)∕𝜕𝑝 >0 o all
𝑐≤𝑝<𝑘,wi hlim𝑝→𝑘−Π(𝑝) =𝑇(𝑘), o he e exis s a leas
one p ice 𝑝∈(𝑐,𝑘), whe e he condi ion o a local maximum
𝜕Π(𝑝)∕𝜕𝑝 =0holds. In he o me case, he unique maximum
is eached a 𝑝=𝑘, and in he la e case, he e a e wo possible
candida es o a maximum: ei he a a p ice 𝑝∈(𝑐,𝑘),whe e
𝜕Π(𝑝)∕𝜕𝑝 =0holds, o a 𝑝=𝑘. The o me solu ion gi es he
in e io solu ion and he la e one he co ne solu ion.
□
No e ha a solu ion o he supplie ’s maximiza ion p oblem (12)mus
also maximize Π(𝑝) as hey di e only in he cons an 𝑅. Thus, Lemma
3 allows us o cha ac e ize also he ma ginal p o i (see Equa ion (13))
and he e o e p o e he p oposi ion. Fi s , 𝑝∗=𝑐canno be a solu ion
because a his poin he i m’s ma ginal p o i s ic ly inc eases (by pa
ii o Lemma 3). Fo 𝑐<𝑝≤𝑘 he e may exis an in e io maximum (by
pa s iii and i o Lemma 3) and/o a maximum a 𝑝=𝑘whe e he p o i
is gi en by 𝑇(𝑘). As he i m’s p o i can be aised by any amoun due o
𝑇(𝑝) o 𝑝>𝑘wi h 𝑥(𝑝) =0, he i s pa o he p oposi ion ollows. I
is hen ob ious ha he e is also a la ge enough con ac ual ene gy p ice
wi h 𝑇(𝑝) >𝑅, which allows he supplie o ex ac he a bi a ily la ge
ans e which gi es he las pa o he p oposi ion. □
P oo o P oposi ion 3. Suppose a i m o e s a con ac (𝑝, 𝐹) ha is
accep ed by he consume . Gi en he consume ’s ene gy demand, he
i m’s p o i is 𝜋=(𝑝 −𝑐)𝑥(𝑝) +𝐹, which gi es 𝐹=𝜋−(𝑝 −𝑐)𝑥(𝑝).
Subs i u ing his in o he consume ’s o e all u ili y (3), we ge 𝐶𝑆(𝑝,𝐹) ∶=
Π(𝑝) −𝜋. Thus, when he consume aces di e en con ac s ha sa is y
he pa icipa ion cons ain (11), he consume selec s he con ac wi h
he highes o e all u ili y. Fi ms hus compe e in wo-pa a i s (𝑝, 𝐹)
o maximize he consume ’s o e all u ili y which mus lead o 𝑝→∞,
because hen he ans e om he ene gy p ice b ake also becomes
a bi a ily la ge. Sub ac ing a ixed p o i le el does no comp omise he
a ac i eness o he con ac . □
P oo o P oposi ion 4. The supplie sol es (12) unde he addi ional
cons ain 𝑝≤𝑝. The s a emen s o he p oposi ion hen ollow om
Lemma 3. In pa icula , pa i o Lemma 3 also applies o he subin e al
𝑝∈[𝑐,𝑝]. Thus, Π(𝑝) ei he has maximum a some 𝑝∈(𝑐,𝑝], whe e he
condi ion o a local maximum
𝜕𝜋
𝜕𝑝 =𝜕Π(𝑝)
𝜕𝑝 =𝑈′−𝑐
𝑈′′ +𝜕𝑇
𝜕𝑝 =0
holds (“in e io solu ion”), o i ob ains a global maximum a 𝑝=𝑝
(“co ne solu ion”). Whe eas Π(𝑝) is no di e en iable a 𝑝=𝑘,i is
di e en iable a 𝑝, which implies ha he condi ion o a local maximum
could be sa is ied a 𝑝=𝑝.
By Assump ion 1, bo h he in e io and he co ne solu ion can be
implemen ed by he supplie wi h a ixed paymen , which lea es an
o e all u ili y o 𝑅 o he consume . The supplie hen ealizes he p o i
as s a ed in he p oposi ion. Bo h he in e io and he co ne solu ion
inc ease he con ac ual ene gy p ice abo e 𝑝∗=𝑐, so ha ene gy demand
dec eases wi h 𝑥(𝑝) <𝑥(𝑝∗). Clea ly, he ans e o he ene gy p ice
b ake is also la ge when compa ed wi h he ans e ha would esul
om he p ice in he benchma k case whe e 𝑝∗=𝑐. This p o es he
p oposi ion. □
P oo o P oposi ion 5. To p o e his p oposi ion, we can use P oposi ion
3. Again, i ms compe e in o e ing wo-pa a i con ac s, and he
consume selec s he con ac ha gi es he he maximal o e all u ili y
143
𝐶𝑆(𝑝,𝐹) =Π(𝑝) −𝜋. I is hen ob ious ha a i m canno make a posi i e
p o i , and 𝑝is chosen o maximize Π(𝑝) (i.e., he sum o he join
su plus plus he ans e om he ene gy p ice b ake). Only i a leas wo
supplie s o e such a con ac , he e is no p o i able unila e al de ia ion
incen i e; ha is, we ha e eached a subgame-pe ec equilib ium. Thus,
he equilib ium p ice is ei he ob ained as an in e io solu ion o a co ne
solu ion. I hen mus also hold ha ene gy consump ion is s ic ly lowe
han in he benchma k case wi hou an ene gy p ice b ake. Finally, and
in con as o he monopoly ou come as desc ibed in P oposi ion 5, i ms
now make ze o p o i s whe eas he consume ully pocke s he join
su plus including he ans e o he ene gy p ice b ake. No ably, he e
he ixed paymen is s ic ly nega i e (i.e., he consume ge s a bonus
paymen ), so ha 𝐹=−(𝑝 −𝑐)𝑥(𝑝) holds. □
P oo o P oposi ion 6. We i s analyze he equilib ium wi hou he ene gy
p ice b ake 𝑇(𝑝).Gi en(16) holds, he monopoly supplie se s he p ice
𝑝𝐼acco ding o (17), whe e Assump ion 2 gua an ees ha 𝑝𝐼is unique
and easible (i.e., 𝑝𝐼<𝑝). Thus, he s anda d monopoly solu ion, 𝑝𝐼,is
he equilib ium ou come whene e (16) is no binding a his p ice. I , o
he con a y, (16)is iola eda 𝑝𝐼, hen he monopolis se s he highes
possible p ice 𝑝𝐼𝐼 whe e (16) holds as an equali y. This ollows om
no icing ha he le -hand side o (16)—namely, consume u ili y—is
s ic ly dec easing in 𝑝,wi h
𝜕
𝜕𝑝[𝑈(𝑥(𝑝)) −𝑝𝑥(𝑝)]=−𝑥(𝑝) <0, (A3)
whe e we used (6). Thus, he e is a unique p ice 𝑝𝐼𝐼 <𝑝𝐼whe e he
consume ’s pa icipa ion cons ain (16) holds as an equali y (exis ence
ollows om Assump ion 1). By Assump ion 2, he supplie ’s p o i is
s ic ly inc easing in 𝑝 o all 𝑝<𝑝𝐼, so ha i is indeed op imal o he
supplie o se he p ice 𝑝𝐼, whe e (16) holds as an equali y.
Nex assume ha an ene gy p ice b ake 𝑇(𝑝) >0is in place. The in o-
duc ion o 𝑇(𝑝) has he e ec ha i elaxes he consume ’s pa icipa ion
cons ain (16), which is now gi en by (14). Consequen ly, when 𝑝𝐼
(acco ding o Equa ion (17)) is he solu ion o (15) o 𝑇(𝑝) =0(i.e., wi h
no ene gy p ice b ake in place), hen his mus also be he solu ion when
𝑇(𝑝) >0holds. Thus, he in oduc ion o he ene gy p ice b ake has no
e ec on he con ac ual ene gy p ice 𝑝𝐼, ene gy demand 𝑥(𝑝𝐼)>0, he
monopolis ’s p o i 𝜋(𝑝𝐼), and i only inc eases he consume ’s u ili y
by 𝑇(𝑝𝐼) om 𝑈(𝑥(𝑝𝐼)) −𝑝𝐼𝑥(𝑝𝐼) o 𝑈(𝑥(𝑝𝐼)) −𝑝𝐼𝑥(𝑝𝐼)+𝑇(𝑝𝐼). This
p o es pa i o he p oposi ion.
Now suppose ha (16) is binding in he p o i maximizing solu ion o (15)
o 𝑇(𝑝) =0, so ha he monopolis se s 𝑝𝐼𝐼 <𝑝𝐼, whe e (16)holdsas
an equali y. Again, he only e ec o he in oduc ion o he ene gy p ice
b ake, wi h 𝑇(𝑝) >0, is o elax he consume ’s pa icipa ion cons ain
(16), which is now gi en by (14). Clea ly, he consume ’s pa icipa ion
cons ain (14)mus beslacka 𝑝𝐼𝐼. As he supplie ’s p o i is s ic ly
inc easing in 𝑝a 𝑝𝐼𝐼 <𝑝𝐼(Assump ion 2), he will always inc ease he
p ice abo e 𝑝𝐼𝐼. This p o es pa ii.a o he p oposi ion.
As shown in he main ex , he le -hand side o (14)(i.e.,𝐶𝑆(𝑝)) ei he
inc eases o dec eases a any 𝑝≤𝑝(acco ding o Equa ion (20)), and i is
s ic ly con ex (see Equa ion (19)). Thus, i 𝜕𝐶𝑆(𝑝)∕𝜕𝑝 >0a 𝑝𝐼𝐼 (which
holds i 𝛼𝑥>𝑥(𝑝𝐼𝐼)acco ding o Equa ion (20)), hen he supplie can
p o i ably inc ease he p ice abo e 𝑝𝐼𝐼 up o 𝑝𝐼( he s anda d monopoly
solu ion Equa ion (17)), because any such p ice inc ease mus u he
elax he consume ’s pa icipa ion cons ain (14); ha is, any inc ease
in 𝑝also inc eases he consume ’s o e all u ili y 𝐶𝑆(𝑝). Thus bo h he
supplie and he consume s ic ly bene i om he in oduc ion o he
ene gy p ice b ake. This p o es pa ii.b o he p oposi ion.
The in oduc ion o he ene gy p ice b ake, he e o e, always inc eases
he con ac ual ene gy p ice, so ha pa ii.c o he p oposi ion ollows
di ec ly om Lemma 1. □
P oo o P oposi ion 7. Unde compe i ion, he consume selec s he
con ac which gi es he highes u ili y (3), gi en ha he u ili y is
no smalle han 𝑅. I he e is mo e han one such con ac , hen he
consume selec s each o he con ac s wi h a s ic ly posi i e p obabili y.
As ene gy is homogeneous, supplie s compe e in Be and ashion.
The e o e, wi hou loss o gene ali y, we conside he duopoly case wi h
wo supplie s, using he indices 𝑖and 𝑖′ o ep esen a supplie ’s iden i y.
Fi s , conside he case wi hou a ans e scheme. Take i m 𝑖’s con ac
o e 𝑝𝑖. Suppose he consume ’s pa icipa ion cons ain (16) holds, hen
i m 𝑖’s p o i unc ion 𝜋𝑖is gi en by
𝜋𝑖(𝑝𝑖,𝑝
𝑖′)=⎧
⎪
⎨
⎪
⎩
(𝑝𝑖−𝑐)𝑥(𝑝𝑖), i 𝑝𝑖<𝑝𝑖′
𝛽𝑖(𝑝𝑖−𝑐)𝑥(𝑝𝑖), i 𝑝𝑖=𝑝𝑖′
0, i 𝑝𝑖>𝑝𝑖′,
o 𝑖≠𝑖′
whe e 𝑥(𝑝𝑖)is he consume ’s ene gy demand (acco ding o Lemma 1)
and 𝛽𝑖∈ (0, 1) is he p obabili y ha he consume selec s i m 𝑖’s o e
when 𝑝𝑖=𝑝𝑖′,wi h𝛽𝑖+𝛽𝑖′=1.
He e, consume u ili y is s ic ly dec easing in 𝑝(see (A3)). Thus, i
𝑝𝑖′=𝑐, hen i m 𝑖canno do be e han also se ing he p ice 𝑝𝑖=𝑐,in
which case p o i s a e ze o. Clea ly, all o he p ices canno cons i u e an
equilib ium, so ha 𝑝∗=𝑐is he unique equilib ium p ice.
Now, conside he in oduc ion o a ans e scheme 𝑇(𝑝) >0so ha he
consume ’s pa icipa ion cons ain o a selec ed con ac is gi en by
(14). Facing wo con ac o e s 𝑝𝑖and 𝑝𝑖′(bo h mee ing he consume ’s
pa icipa ion cons ain ), he consume selec s he con ac wi h a highe
o e all u ili y 𝐶𝑆(𝑝).I 𝑝𝑖=𝑝𝑖′, so ha 𝐶𝑆(𝑝𝑖)=𝐶𝑆(𝑝𝑖′),wi hou losso
gene ali y, 𝛽𝑖∈ (0, 1) gi es he p obabili y ha he consume selec s i m
𝑖’s o e .
Assume 𝜕𝐶𝑆(𝑝)∕𝜕𝑝 ≥0a 𝑝=𝑐(which holds i 𝛼𝑥≥𝑥(𝑐) acco ding o
Equa ion (20)). Fi m 𝑖’s p o i unc ion o 𝑝𝑖,𝑝
𝑖′≤𝑝is hen gi en by
𝜋𝑖(𝑝𝑖,𝑝
𝑖′)=⎧
⎪
⎨
⎪
⎩
(𝑝𝑖−𝑐)𝑥(𝑝𝑖), i 𝑝𝑖>𝑝𝑖′
𝛽𝑖(𝑝𝑖−𝑐)𝑥(𝑝𝑖), i 𝑝𝑖=𝑝𝑖′
0, i 𝑝𝑖<𝑝𝑖′,
o 𝑖≠𝑖′
whe e 𝑥(𝑝𝑖)is he consume ’s ene gy demand (acco ding o Lemma 1)
and 𝛽𝑖∈ (0, 1) is he p obabili y ha he consume selec s i m 𝑖’s o e
when 𝑝𝑖=𝑝𝑖′,wi h𝛽𝑖+𝛽𝑖′=1. He e, supplie 𝑖’s con ac is selec ed o
su e by he consume whene e 𝑝𝑖>𝑝𝑖′holds, whe eas i is selec ed wi h
some posi i e p obabili y 𝛽𝑖whene e 𝑝𝑖=𝑝𝑖′holds. I is hen immedia e
o see ha 𝑝is he unique equilib ium ene gy p ice. Clea ly, any pai o
p ices wi h 𝑝𝑖<𝑝𝑖′≤𝑝, can be uled ou because hen i m 𝑖has a s ic
incen i e o inc ease i s p ice o 𝑝𝑖′. Mo eo e , any pai o p ices wi h
𝑝𝑖=𝑝𝑖′<𝑝can also be uled ou , because hen i m 𝑖could inc ease i s
p ice by 𝜀>0 o gain he en i e ma ke and hus o inc ease i s p o i . This
p o es pa i o he p oposi ion.
In all o he cases, ha is, when 𝜕𝐶𝑆(𝑝)∕𝜕𝑝 <0a 𝑝=𝑐holds, hen 𝐶𝑆(𝑝)
ei he has a global maximum a 𝑝=𝑐o a 𝑝=𝑝, which ollows om
he s ic con exi y o 𝐶𝑆(𝑝) (see Equa ion (19)). I 𝐶𝑆(𝑝) ≥𝐶𝑆(𝑐), hen
𝑝=𝑝is he unique equilib ium. I , o he con a y, 𝐶𝑆(𝑐) >𝐶𝑆(𝑝) ≥𝑅,
hen bo h 𝑝=𝑐and 𝑝=𝑝a e bo h possible equilib ia, and he la e one
is s ic ly p e e ed om he i ms’ pe spec i e. By he same easoning as
abo e, any o he p ice pai canno be an equilib ium ou come. □
P oo o Co olla y 2. Suppose he cons ain 𝐹≥0. Then, he equilib ium
as speci ied in P oposi ion 5 is no easible, because he e 𝐹<0.We
show ha unde compe i ion, 𝐹>0can be uled ou , so ha 𝐹=0mus
hold in equilib ium. Suppose 𝐹>0. Bo h i ms mus make ze o p o i s
(o he wise, one i m could inc ease i s p o i by educing 𝐹sligh ly o gain
he en i e ma ke ). Then, 𝐹=|(𝑝 −𝑐)𝑥(𝑝)|wi h 𝑝<𝑐mus hold by he
ze o-p o i condi ion. Fo 𝑝<𝑐, he join su plus including he ans e ,
Π(𝑝), is inc easing in 𝑝by pa ii o Lemma 3. Thus, inc easing 𝑝s ic ly
inc eases he join su plus, and by educing he ixed paymen in he igh
way, he addi ional join su plus can be di ided in a such way ha he
consume is s ic ly be e o and he i m can win he consume , and
also he i m is s ic ly be e o . Thus, he e canno be an equilib ium
wi h 𝐹>0. I hen ollows ha 𝐹=0mus hold, so ha he equilib ium
mus be he same as in he linea case, which is gi en in P oposi ion 7. □
144 The RAND Jou nal o Economics,2025
