Jaspe sen, Johannes G.
A icle — Published Ve sion
When ull insu ance may no be op imal: The case o
es ic ed subs i u ion
Heal h Economics
P o ided in Coope a ion wi h:
John Wiley & Sons
Sugges ed Ci a ion: Jaspe sen, Johannes G. (2022) : When ull insu ance may no be op imal: The case
o es ic ed subs i u ion, Heal h Economics, ISSN 1099-1050, Wiley Pe iodicals, Inc., Hoboken, USA,
Vol. 31, Iss. 6, pp. 1249-1257,
h ps://doi.o g/10.1002/hec.4491
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/264536
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1 | INTRODUCTION
E en a highly subsidized p emiums, a no able sha e o people does no pu chase heal h insu ance co e age (Finkels ein
e al.,2019). Such beha io is in in con as wi h Mossin's(1968) inding ha when insu ance is a ailable a an ac ua i-
ally ai a e, expec ed u ili y decision-make s should pu chase ull co e age. While his esul was de i ed o mone a y
losses, i is commonly applied o heal h insu ance, as well (McGui e,2011). Because ad e se selec ion alone appea s o
be insu icien o econcile empi ically documen ed beha io wi h canonical heo y, explana ions ypically encompass
liquidi y cons ain s (E icson & Sydno ,2018), di e ences in cos s o compensa ed and uncompensa ed ca e (Finkels ein
e al.,2019), weak adminis a i e capaci y (Bane jee e al.,2021), inco ec isk pe cep ion (as documen ed o insu ance
ma ke s o he han heal h by, e.g., Spinnewijn,2015) o choice e o s (Handel & Kols ad,2015).
I de elop an al e na i e explana ion o he unde insu ance phenomenon using a model wi h expec ed u ili y deci-
sion-make s who a e no subjec o any o he abo e lis ed con ounds. They maximize hei u ili y o e heal h and weal h
bu a e unable o make compensa ed adjus men s o hei heal h s a us unless hey a e sick. I decision-make s would
like o ade heal h o weal h in case hey a e heal hy, hey will be unwilling o pay o a ull eco e y in case hey a e
LMU Munich School o Managemen ,
LMU Munich, Ludwig-Maximilians-
Uni e si ä , Munich, Ge many
Co espondence
Johannes G. Jaspe sen, LMU Munich
School o Managemen , Ludwig-
Maximilians-Uni e si ä München,
Geschwis e -Scholl-Pla z 1, Munich
80539, Ge many.
Email: [email p o ec ed]
Abs ac
E en when hea ily subsidized, a subs an ial po ion o people choose o o go
pu chasing heal h insu ance co e age. In his no e, I in oduce an explana ion
o his phenomenon which does no assume choice e o s, inco ec belie s,
di e en ly p iced uncompensa ed ca e, o in o ma ion asymme ies. When
indi iduals a e incapable o eely ading o heal h and weal h and he ini ial
alloca ion o goods is subop imal om hei pe spec i e, he s anda d esul o
demand o ac ua ially ai insu ance in a single good wo ld does no gene alize
o he heal h insu ance con ex . Thus, people migh no pu chase ull heal h
insu ance co e age e en i i is p iced a ac ua ially ai le els. I a gue ha his
si ua ion is pa icula ly likely o occu in he low-income popula ion, and hence
i is ele an o he achie emen o uni e sal heal h co e age.
KEYWORDS
heal h insu ance, unde insu ance
JEL CLASSIFICATION
D14, G22, I13
SHORT RESEARCH ARTICLE
Heal h Economics Le e s
When ull insu ance may no be op imal: The case o
es ic ed subs i u ion
Johannes G. Jaspe sen
DOI: 10.1002/hec.4491
Recei ed: 1 Ap il 2021 Re ised: 19 Oc obe 2021 Accep ed: 18 Feb ua y 2022
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion-NonComme cial 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 and is no used o comme cial pu poses.
© 2022 The Au ho s. Heal h Economics published by John Wiley & Sons L d.
wileyonlinelib a y.com/jou nal/hecHeal h Economics. 2022;31:1249–1257. 1249
sick. I a gue, ha such si ua ions may no be uncommon, pa icula ly o he low-income popula ion. I hey appea ,
decision-make s p e e o pu chase pa ial insu ance, e en i p emiums a e ac ua ially ai .
Following Zeckhause (1970), models o heal h insu ance demand wi h wo a ibu es ha e p ima ily been applied
o consump ion and heal h expendi u es – a se -up which does no make he ade-o be ween heal h and consump ion
explici . One excep ion is he model by Jack and Sheine (1997). They, howe e , base hei analysis on he indi ec u ili y
unc ion and hus implici ly assume ha decision-make s can ade-o heal h and weal h in any desi ed quan i y and
di ec ion. Goldman and Philipson(2007) and McGui e(2011) do no make his assump ion. While hey do no analyze
es ic ed subs i u ion be ween heal h and consump ion, hey show ha indi iduals can p e e pa ial insu ance a ac u-
a ially ai p ices i he ma ginal u ili y o consump ion is inc easing in heal h. 1 In a ela ed model, which also equi es a
posi i e c oss de i a i e o pa ial insu ance, Cook and G aham(1977) conside insu ance o an i eplaceable good – a
se ing which can easily be applied o heal h insu ance (Zwei el e al.,2009). In con as o hem, I assume ha heal h
can be eco e ed a e a shock which makes me ocus on a di e en se o illnesses. Mo eo e , hei model assumes a
mone a y compensa ion o a los good while I assume he mo e common se ing in heal h insu ance o expense-based
eimbu semen .
Res ic ed subs i u ion emo es he equi emen o a posi i e c oss de i a i e o he u ili y unc ion. Ra he , my
esul s hold o a la ge se o possible u ili y unc ions wi h only weak condi ions on he c oss de i a i e. This is impo -
an , because empi ical e idence does no sugges a uni e sally posi i e c oss-de i a i e. While Viscusi and E ans(1990),
Finkels ein e al.(2013) and Blundell e al.(2020) es ima e a posi i e e ec o heal h on ma ginal u ili y, E ans and
Viscusi(1991) ind no such e ec o ansi o y changes in heal h. Likewise, Viscusi(2019) a gues o a posi i e c oss
de i a i e, bu only i he heal h shock is se e e.
2 | RESULTS
Decision-make s maximize u ili y o e consump ion and heal h: U(c,h). Using he no a ion ∂U/∂c=Uc, ∂
2U/∂c
2=Ucc,
∂
3U/∂c
3=Uccc, ∂U/∂h=Uh, ∂
2U/∂h
2=Uhh and ∂
2U/∂c∂h=Uch=Uhc, I assume posi i e and dec easing ma ginal u il-
i y o bo h heal h and consump ion as well as p udence o e money (Uc, Uh>0,Ucc, Uhh < 0, and Uccc > 0). 2 Fu he , I
assume ha he c oss de i a i e o he u ili y unc ion does no o e whelm he e ec o ma ginal u ili y on p e e ences
when adjus ing o ela i e p ices. As is de ailed below, decision-make s can eco e om a bad heal h e en a mone-
a y cos s q. Fo mally, he assump ion on he c oss de i a i e implies ha Uch>qUcc and qUch>Uhh. As is explained
in de ail by Chung e al.(2019), his assump ion is wi hou loss o much gene ali y and should co e he as majo i y
o decision-make s in he heal h insu ance ma ke . As mo e o a echnical assump ion, I u he assume ha he c oss
de i a i e is no oo la ge in magni ude such ha insu ance demand and ou -o -pocke paymen s a e de e mined in a
conca e decision p oblem.
Decision-make s ha e ini ial weal h w0 and heal h h0 and ace he isk ha hei heal h dec eases o h0 − L wi h p ob-
abili y p. When sick, decision-make s can pu chase heal hca e o a p ice q. Fo ease o exposi ion, I ocus on he case in
which heal h gains a e linea in heal h ca e expenses. Howe e , he in ui ion o he analysis ca ies o e o non-linea
heal h p oduc ion unc ions as I show in Online Appendix C. In case he insu ed a e heal hy, hey canno change he allo-
ca ion o consump ion and heal h. Tha is, he subs i u ion be ween he wo a gumen s o he u ili y unc ion is es ic ed
o one di ec ion. Also, decision-make s canno inc ease he heal h s a us abo e he ini ial le el o h0. Insu ance agains
he incu sion o medical cos s is a ailable a an ac ua ially ai p ice. Decision-make s can choose he deg ee o insu ance
co e age, deno ed α, on a con inuous scale be ween 0 and 1. The insu ance con ac has an expense-based eimbu se-
men sys em such ha decision-make s ha e o spen he indemni y on heal hca e. The ac ua ially ai p emium, payable
in bo h s a es o he wo ld, is αpqL.
Decision-make s choose he amoun o heal h insu ance o buy. In case hey a e sick and ha e pu chased less han
ull insu ance, hey can also pu chase heal h ca e ou -o -pocke , deno ed (such ha 0≤ ≤1 − α). The objec i e unc ion
eads
max
𝛼≥0,𝑡≥0{𝐸𝑈(𝛼,𝑡) = (1 − 𝑝)𝑈(𝑤
0
−𝛼𝑝𝑞𝐿, ℎ
0
)+𝑝𝑈(𝑤
0
−𝛼𝑝𝑞𝐿 −𝑡𝑞𝐿, ℎ
0
−(1−𝛼−𝑡)𝐿)} .
P oposi ion 1 s a es he main esul .
JASPERSEN1250
P oposi ion 1 I qUc (w0, h0) > Uh (w0, h0) and insu ance is ac ua ially ai ,
(i) decision-make s do no pu chase ull insu ance, and
(ii) i qUc (w0, h0 − L) < Uh (w0, h0 − L), >0 i and only i Uch > 0.
All p oo s a e p o ided in he appendix. qUc (w0, h0) > Uh (w0, h0) implies ha in hei ini ial alloca ion (w0, h0), deci-
sion-make s would be willing o dec ease hei heal h s a us i hey could gain he equi alen ea men cos s in addi ional
consump ion. 3 The in ui ion o i em (i) is hus ha decision-make s do no wan o pu chase ull insu ance, because i
o ces hem o alloca e mo e money in o he ea men cos han hey wan o spend on heal h. I em (ii) ollows, because
i consump ion is wo h less in bad heal h, decision-make s alloca e mo e heal hca e cos s o ha s a e o he wo ld. The
in ui ion behind his esul is equal o ha behind he esul o Cook and G aham(1977) and Zwei el e al.(2009). The
esul e lec s he ac ha ou -o -pocke paymen s a e no uncommon, pa icula ly in he low-income popula ion.qUc
(w0, h0 − L) < Uh (w0, h0 − L) simply es ablishes ha some amoun o ea men is demanded in case o sickness.
Unde which ci cums ances will he condi ion qUc (w0, h0) > Uh (w0, h0) mos likely hold? Heal hca e is gene ally
conside ed a no mal good (see, e.g., E ans & Viscusi,1993; Al onso e al.,2016). Thus, he condi ion is mos likely o be
ue in he low-income popula ion. Fo he gi en model, I de i e his esul o mally in he nex p oposi ion which shows
ha all o he hings emaining equal, qUc (w0, h0) > Uh (w0, h0) is mo e likely o be ue as w0 dec eases.
P oposi ion 2 Ce e is pa ibus, he exp ession qUc (w0, h0) − Uh (w0, h0) is dec easing in w0.
As indi iduals ob ain mo e weal h, he ma ginal u ili y o non-medical consump ion dec eases. The cos o imp o -
ing one's heal h is una ec ed by weal h. Because he ma ginal u ili y o heal h a he e y leas dec eases less han he
ma ginal u ili y o consump ion, spending money on heal hca e becomes mo e a ac i e. Wi h dec easing weal h, qUc
(w0, h0) − Uh (w0, h0) is inc easing and qUc (w0, h0) > Uh (w0, h0) is mo e likely o hold. Hence, unde insu ance due o
es ic ed subs i u ion be ween heal h and weal h is mo e likely in he low-income popula ion.
Figu e1 gi es a nume ical illus a ion o he esul s. I se h0=q=1, L=0.8, p=0.1 and U(c, h)=c
γ + h
γ + kc
γh
γ,
such ha Uch has he same sign as k. To ensu e isk a e sion o bo h heal h and weal h, I se γ=0.7. 4 The igu e shows
he op imal insu ance co e age and ou -o -pocke paymen s o weal h anging be ween 0 and 1.5qh0. The su icien
condi ion o P oposi ion 1 is ul illed o any w0 smalle 1qh0.
I espec i e o Uch, decision-make s pu chase ull insu ance i ini ial weal h is somewha la ge han 1qh0. 5 Below
his poin , co e age d ops below 1 and is dec easing in weal h. This demons a es ha he e ec desc ibed in P oposi ion
1 is no mino and may e en lead decision-make s o comple ely o go pu chasing heal h insu ance. Also, he p oposed
mechanism inc eases in ele ance as he decision-make s’ weal h dec eases.
3 | DISCUSSION
Bo h P oposi ion 2 and Figu e1 imply ha es ic ed subs i u ion will mos likely in luence how he low-income popu-
la ion pu chases heal h insu ance. This co esponds o empi ical indings, which documen lacking heal h insu ance
demand o he low-income popula ion (Finkels ein e al.,2019), while inding no such esul s o highe income s a a
JASPERSEN 1251
FIGURE 1 Op imal le els o
insu ance demand (α) and ou -o pocke
paymen s ( ) in a se ing whe e U(c, h)=c
γ
+ h
γ + kc
γh
γ, h0=q=1,p=0.1,L=0.8
and γ=0.7
(Hackmann e al.,2015). Take-up o subsidized heal h insu ance migh hus be inc eased i he subsidies we e accompa-
nied by a social sa e y ne which p o ec s indi iduals om si ua ions wi h e y low le els o weal h.
My esul s depend on he p e e ence condi ion qUc (w0, h0) < Uh (w0, h0), di ec empi ical e idence o which is
a e. Saba and Gallaghe (2019) show ha he low-income popula ion o en o egoes pu chasing heal hca e e en in he
p esence o liquidi y ese es. In hei da a, o egoing heal hca e also is, on a e age, he i s d as ic sa ings measu e in
inancial ha dship. 6 G oss e al.(2021) ind ha indi iduals may no ill d ug p esc ip ions when hey a e low on liquidi y
e en i he copaymen s o hese p esc ip ions a e so low ha liquidi y cons ains a e unlikely. Income is also a decisi e
ac o in he mone a y e alua ion o quali y-adjus ed li e yea s (Bobinac e al.,2010) and he ake-up o den al ca e (Allin
e al.,2009). Bo h esul s gi e c edibili y o he p e e ence condi ion being mo e likely o he low-income popula ion.
Mo e indi ec ly, he ac ha some people a e willing o ade-o hei (expec ed) heal h o weal h is also e iden by
he willingness o accep po en ially heal h- h ea ening jobs. Viscusi and Aldy(2003) show ha such beha io is mo e
common in he low-income popula ion. Las ly, Di Ma eo(2003) shows ha he income elas ici y o heal h expendi u es
in he US low-income popula ion is abo e 1. This sugges s ha willingness o pay o heal h ca e d ops as e han income
a low income b acke s, making qUc (w0, h0) > Uh (w0, h0) mo e likely a low le els o w0.
The esul s o his no e ely on he assump ions ha Uccc > 0 and ha he magni ude o Uch is no oo la ge. These wo
assump ions a e su icien condi ions o a ul illed second o de condi ion and hus a conca e decision p oblem, bu hey
a e no necessa y. Mo eo e , he goal o his no e is no o p o ide a uni e sal esul on heal h insu ance beha io , bu
a he o o e a po en ial explana ion o an obse ed esul unde a easonable se o assump ions. I is unnecessa y ha
he assump ions hold o all decision-make s – hey me ely need o hold o a subse o he popula ion.
While he abo e model uses a simple se -up o ease o exposi ion, wo ex ensions a e possible. The i s co e s he
case in which heal h can only be es o ed up o some le el
𝐴𝐴
h
≤ h0. This equi es adjus men o he p e e ence condi ion
such ha i holds in he bes possible heal h s a e which can s ill be achie ed when sick: qUc(w0,
𝐴𝐴
h
) > Uh(w0,
𝐴𝐴
h
). The
second ex ension, de ailed in Online Appendix C, co e s a non-linea heal h p oduc ion unc ion.
ACKNOWLEDGEMENT
I would like o hank he edi o Eleono a Fiche a as well as wo anonymous e iewe s o hei eedback and Benjamin
Collie , The esa Eden, Richa d Pe e , Ma c Ragin and Casey Ro hschild o aluable commen s. Joëlle Näge p o ided
aluable esea ch assis ance. All emaining e o s a e mine.
CONFLICT OF INTEREST
The au ho epo s no con lic s o in e es ( inancial o o he wise) and no compe ing in e es s.
DATA AVAILABILITY STATEMENT
The code o c ea e he nume ical da a ha suppo he indings o his s udy a e openly a ailable on he au ho 's home-
page a h ps://si es.google.com/jaspe sen.com/jgj/home.
ORCID
Johannes G. Jaspe sen h ps://o cid.o g/0000-0002-3599-8988
ENDNOTES
1 I is wo h men ioning ha his idea is no explo ed in de ail in ei he Goldman and Philipson(2007) o McGui e(2011). The o me ocus
on he implica ions o mo al haza d, while he la e s udy ocuses on he case in which ma ginal u ili y o consump ion is dec easing in
weal h.
2 Risk a e sion o e heal h and consump ion a e s anda d assump ions and p udence has been well es ablished o mone a y s akes (e.g.,
Ebe & Wiesen,2011).
3 This can also be exp essed as Uc (w0, h0)/Uh (w0, h0) > 1/q o cla i y ha i is a condi ion on he ma ginal a e o subs i u ion be ween heal h
and consump ion.
JASPERSEN1252

4 Resul s o al e na i e alues o γ,p and L can be ound in Online Appendix B. None o he pa ame e choices ma e ially a ec he esul s.
5 Op imal co e age is pa ial o some weal h le els abo e qh0 because qUc (w0, h0) > Uh (w0, h0) is a su icien bu no necessa y condi ion. The
necessa y and su icien condi ion is qUc (w0−pqL, h0) > Uh (w0−pqL, h0).
6 O he analyzed measu es a e skipping meals, skipping en paymen s and no paying u ili y bills.
REFERENCES
Al onso, Y. N., Ding, G., & Bishai, D. (2016). Income elas ici y o accines spending e sus gene al heal hca e spending. Heal h Economics,
25(7), 860–872.
Allin, S., Masse ia, C., & Mossialos, E. (2009). Measu ing socioeconomic di e ences in use o heal h ca e se ices by weal h e sus by income.
Ame ican Jou nal o Public Heal h, 99(10), 1849–1855.
Bane jee, A., Finkels ein, A., Hanna, R., Olken, B. A., O naghi, A., & Suma o, S. (2021). The challenges o uni e sal heal h insu ance in
de eloping coun ies: Expe imen al e idence om Indonesia's na ional heal h insu ance. Ame ican Economic Re iew, 111(9), 3035–3063.
Blundell, R., Bo ella, M., Commaul , J., & De Na di, M. (2020). Why does consump ion luc ua e in old age and how should he go e nmen insu e
i ? Wo king Pape No. 27348. Na ional Bu eau o Economic Resea ch.
Bobinac, A., Van Exel, N. J. A., Ru en, F. F., & B ouwe , W. B. (2010). Willingness o pay o a quali y-adjus ed li e-yea : The indi idual pe spec-
i e. Value in Heal h, 13(8), 1046–1055.
Chung, H.-K., Glimche , P., & Agniezka, T. (2019). An expe imen al compa ison o isky and iskless choice – limi a ions o p ospec heo y
and expec ed u ili y heo y. Ame ican Economic Jou nal: Mic oeconomics, 11(3), 34–67.
Cook, P. J., & G aham, D. A. (1977). The demand o insu ance and p o ec ion: The case o i eplaceable commodi ies. Qua e ly Jou nal o
Economics, 91(1), 143–156.
Di Ma eo, L. (2003). The income elas ici y o heal h ca e spending. The Eu opean Jou nal o Heal h Economics, 4(1), 20–29.
Ebe , S., & Wiesen, D. (2011). Tes ing o p udence and skewness seeking. Managemen Science, 57(7), 1334–1349.
E icson, K. M., & Sydno , J. R. (2018). Liquidi y cons ain s and he alue o insu ance. Wo king pape . Na ional Bu eau o Economic Resea ch,
No. 24993.
E ans, W. N., & Viscusi, W. K. (1991). Es ima ion o s a e-dependen u ili y unc ions using su ey da a. The Re iew o Economics and S a is ics,
73(1), 94–104.
E ans, W. N., & Viscusi, W. K. (1993). Income e ec s and he alue o heal h. Jou nal o Human Resou ces, 28(3), 497–518.
Finkels ein, A., Hend en, N., & Shepa d, M. (2019). Subsidizing heal h insu ance o low-income adul s: E idence om Massachuse s. Ame -
ican Economic Re iew, 109(4), 1530–1567.
Finkels ein, A., Lu me , E. F., & No owidigdo, M. J. (2013). Wha good is weal h wi hou heal h? The e ec o heal h on he ma ginal u ili y
o consump ion. Jou nal o he Eu opean Economic Associa ion, 11(1), 221–258.
Goldman, D., & Philipson, T. J. (2007). In eg a ed insu ance design in he p esence o mul iple medical echnologies. Ame ican Economic
Re iew, 97(2), 427–432.
G oss, T., Lay on, T. J., & P inz, D. (2021). The liquidi y sensi i i y o heal hca e consump ion: E idence om social secu i y paymen s. Ame ican
Economic Re iew: Insigh s, in p ess.
Hackmann, M. B., Kols ad, J. T., & Kowalski, A. E. (2015). Ad e se selec ion and an indi idual manda e: When heo y mee s p ac ice. Ame i-
can Economic Re iew, 105(3), 1030–1066.
Handel, B. R., & Kols ad, J. T. (2015). Heal h insu ance o " humans": In o ma ion ic ions, plan choice, and consume wel a e. Ame ican
Economic Re iew, 105(8), 2449–2500.
Jack, W., & Sheine , L. (1997). Wel a e-imp o ing heal h expendi u e subsidies. Ame ican Economic Re iew, 87(1), 206–221.
McGui e, T. G. (2011). Demand o heal h insu ance. In Handbook o Heal h Economics (Vol. 2, pp. 317–396). Else ie .
Mossin, J. (1968). Aspec s o a ional insu ance pu chasing. Jou nal o Poli ical Economy, 76(4, Pa 1), 553–568.
Saba , J., & Gallaghe , E. (2019). Rules o humb in household sa ings decisions: Es ima ion using h eshold eg ession. Uni e sidad Diego
Po ales.
Spinnewijn, J. (2015). Unemployed bu op imis ic: Op imal insu ance design wi h biased belie s. Jou nal o he Eu opean Economic Associa-
ion, 13(1), 130–167.
Viscusi, W. K. (2019). U ili y unc ions o mild and se e e heal h isks. Jou nal o Risk and Unce ain y, 58(2), 143–166.
Viscusi, W. K., & Aldy, J. E. (2003). The alue o a s a is ical li e: A c i ical e iew o ma ke es ima es h oughou he wo ld. Jou nal o Risk
and Unce ain y, 27(1), 5–76.
Viscusi, W. K., & E ans, W. N. (1990). U ili y unc ions ha depend on heal h s a us: Es ima es and economic implica ions. The Ame ican
Economic Re iew, 80(9), 353–374.
Zeckhause , R. (1970). Medical insu ance: A case s udy o he adeo be ween isk sp eading and app op ia e incen i es. Jou nal o Economic
Theo y, 2(1), 10–26.
Zwei el, P., B eye , F., & Ki mann, M. (2009). Heal h economics (2nd ed.). Sp inge .
How o ci e his a icle: Jaspe sen, J. G. (2022). When ull insu ance may no be op imal: The case o es ic ed
subs i u ion. Heal h Economics, 31(6), 1249–1257. h ps://doi.o g/10.1002/hec.4491
JASPERSEN 1253
APPENDIX A: PROOFS OF PROPOSITIONS
A.1 | P oo o P oposi ion 1
I will abb e ia e w0 − αpqL, h0 as yI and w0 − αpqL − qL, h0 − (1 − α − )L as yII such ha yI and yII deno e weal h and
heal h in he s a e wi hou and wi h a heal h shock, espec i ely.
I em (i): The decision-make maximizes expec ed u ili y o e α≥0 and ≥0 wi h he es ic ions α + ≤1 such ha
he se o possible choices is closed, con ex and a la ice. Fu he , ∂
2EU/∂α∂− =pL
2[q(1+p)Uch (yII)−Uhh ( yII)−pq
2Ucc
( yII)]=pL
2[pq(Uch ( yII)−qUcc ( yII))+(qUch ( yII)−Uhh ( yII))] > 0. Such ha expec ed u ili y has inc easing di e ences
in (α, − ).
I we se =0, we can sol e o an op imal deg ee o coinsu ance wi hou ou -o -pocke paymen s, deno ed
𝐴𝐴 𝐴𝐴𝐴
. I ul ills
∂EU/∂α| =0=Uh(w0−
𝐴𝐴 𝐴𝐴𝐴
pqL,h0−(1−
𝐴𝐴 𝐴𝐴𝐴
)L)−q[(1−p)Uc(w0−
𝐴𝐴 𝐴𝐴𝐴
pqL,h0)+pUc(w0 −
𝐴𝐴 𝐴𝐴𝐴
pqL,h0−(1−
𝐴𝐴 𝐴𝐴𝐴
)L)] = 0. We es ablish
ha ∂
2EU/∂α
2| =0 < 0, by seeing ha q
2p[(1−p)Ucc( yI)+pUcc( yII)]−2pqUch( yII)+Uhh( yII)<q
2pUcc( yII)−2pqUch( yII)+
Uhh( yII) = qp[qUcc( yII)−Uch( yII)]+[Uhh−qpUch] < 0, whe e he i s inequali y ollows om Uccc > 0 and he second one
ollows om ou assump ions on Uch. ∂
2EU/∂α
2| =0<0 shows ha he second o de condi ion is ul illed and hus
𝐴𝐴 𝐴𝐴𝐴
< 1
i ∂EU/∂α| =0,α=1<0. ∂EU/∂α| =0,α=1=Uh(w0−pqL, h0)−qUc(w0−pqL, h0). F om qUc(w0, h0) > Uh(w0, h0), we know
ha Uh(w0, h0)−qUc(w0, h0) < 0. Fu he , due o Uch>qUcc, Uh(w0, h0)−Uh(w0−qpL, h0)> qUc(w0, h0)−qUc(w0−pqL,
h0). Rea anging ende s ∂EU/∂α|α=1<Uh(w0, h0)−qUc(w0, h0) < 0 and hus
𝐴𝐴 𝐴𝐴𝐴
< 1.
Because EU has inc easing di e ences in (α,− ), α will be lowe wi h inc easing . We hus know ha he op imal
le el o α when is no es ic ed o 0 is weakly smalle han he op imal le el o α when is es ic ed o 0. I ollows ha
α
∗ ≤
𝐴𝐴 𝐴𝐴𝐴
< 1.
I em (ii): We now conside he decision-p oblem using a Lag angian app oach wi h wo inequali y cons ain s. We i s
no e ha EU is conca e in (α, ) by seeing ha due o Uccc > 0 i holds ha ∂
2EU/∂α
2=pL
2[q
2pEUcc−2pqUch(yII)+Uhh(yII)] <
pL
2[qp(qUcc(yII)−Uch(yII))+(Uhh(yII)−pqUch(yII))] < 0, ha ∂
2EU/∂
2=pL
2[q(qUcc(yII)−Uch(yII)+(Uhh(yII)−qUch(yII))]
< 0 and ha he de e minan o he Jacobian is posi i e. The la e can be seen by D=∂
2EU/∂α
2 × ∂
2EU/∂ −(∂
2EU/∂α∂ )
2
=p
2L
4[pq
2EUcc−2pqUch(yII)+Uhh(yII)][q
2Ucc(yII)−2qUch(yII)+Uhh(yII)]−p
2L
4[pq
2Ucc(yII)−q(1+p)Uch(yII)+Uhh(yII)]
2.
We show ha D/p
2L
4>0 and hus D>0. Due o Uccc > 0, D/p
2L
4 > [pq
2Ucc(yII)−2pqUch(yII)+Uhh(yII)][q
2Ucc(yII)−2qUch
(yII)+Uhh(yII)]−[pq
2Ucc(yII)−q(1+p)Uch(yII)+Uhh(yII)]
2=pq
2Ucc[(1−p)q
2Ucc−2pqUch]+Uhh[(1−p)q
2Ucc−2pqUch]
−q
2(1−p)
2(Uch)
2 which is posi i e i Uch is no oo la ge in absolu e magni ude, as is assumed.
Conca i y o EU in (α, ) and maximiza ion o e a closed and con ex se ensu e ha he Ka ush-Kuhn-Tucke (KKT)
condi ions o he Lag angian a e bo h necessa y and su icien o a maximiza ion. These condi ions a e gi en by
(1): ∂ ℒ/∂α = pL[Uh(yII)−q(1 − p)Uc(yI)−qpUc(yII)]+λ1 = 0,
(2): ∂ ℒ/∂ = pL[Uh(yII)−qUc(yII)]+λ2 = 0,
(3) –(6): λ1, λ2 ≥ 0, λ1α = 0,λ2
∗ = 0.
We i s es ablish ha unde he condi ions o he p oposi ion, α
∗ and
∗ canno be 0 a he same ime. This would
imply ha Uh(yII)−qUc( yII)=Uh(w0, h0−L) − qUc(w0, h0 − L)>0, such ha KKT condi ions (2) and (4) con adic one
ano he . Thus, ei he α
∗ o
∗ o bo h need o be posi i e.
We now show ha Uch ≤ 0 and >0 canno lead o ul illed KKT condi ions. F om >0, λ2=0 ollows. This implies
pLUh(yII)=pLqUc(yII) om (2) which we subs i u e in o (1) o ob ain pLq(1−p)[Uc(yII)−Uc(yI)]+λ1=0. (3) hus implies
ha Uc(yII)−Uc(yI) ≤ 0. Howe e , we know om i em (i) ha α<1, making consump ion in yII smalle han in yI. We also
know ha because heal h in yII canno be la ge han in yI. Combining bo h obse a ions, Ucc < 0, and Uch > 0 shows ha
Uc(yII) > Uc(yI) and hus ende s a con adic ion.
Las ly, we show ha Uch > 0 and
∗=0 canno ul ill he KKT condi ions, bu ha hey can be ul illed o
∗ > 0. S a -
ing a
∗=0, we know α
∗ > 0 and hus λ1=0. F om (1), we can hen de i e Uh(yII)=q(1 − p)Uc(yI)−qpUc(yII). Subs i u ing
in o (2) ende s pLq(1 − p)[Uc(yI)−Uc(yII)] + λ2=0. Howe e , a
∗=0 his implies Uc(w0−α
∗pqL, h0)−Uc(w0−α
∗pqL,
h0−(1−α
∗)L) ≤ 0. This is impossible o Uch > 0 as long as α
∗ < 1, which we know is ue om i em (i). Thus Uch > 0
con adic s
∗=0. Fo
∗ > 0, λ2=0 and hus Uh(yII)=qUc(yII). Subs i u ing in o (1) implies Uc(yII) − Uc(yI)=Uc(w0
− α
∗pqL −
∗qL, h0 − (1 − α
∗ −
∗q)L) − Uc(w0−α
∗pqL, h0) ≤ 0. A
∗=0, Uch implies Uc(yII)−Uc(yI)<0 and because
Uc(yII)−Uc(yI) is dec easing in , his will also be ue o
∗ > 0.
JASPERSEN1254
A.2 | P oo o P oposi ion 2
The de i a i e o he exp ession owa d w0 is qUcc(w0, h0)−Uch(w0, h0). This is nega i e unde he assump ions made on
he p e e ences.
APPENDIX B: NUMERICAL RESULTS WITH ALTERNATIVE PARAMETRIZATIONS
In his online appendix, I show he esul s o he nume ical illus a ion o di e en alues o γ, p and L. Figu eB1 shows
he esul s o γ=0.2 and γ=0.5. Lowe alues o γ imply highe le els o u ili y cu a u e o consump ion and heal h.
Resul s a e e y simila o hose in he main analysis. Dec easing γ leads o less dis inguished esul s o he di e en
alues o k. Addi ionally, smalle le els o γ dec ease he sha e o ou -o -pocke paymen s ela i e o ha o expenses
co e ed by insu ance.
Figu esB2–B4 demons a e he esul s o di e en le els o p and L. In Figu eB2, I keep he le el o he loss cons an
a 0.8, while inc easing p o 0.2 and dec easing i o 0.05. In Figu eB3, he p obabili y is kep cons an , while he loss is
inc eased o 1.0 ( he maximal possible le el) and dec eased o 0.4. Las ly, Figu eB4 keeps he expec ed alue o he heal h
loss cons an a 0.08 as in he main analysis, bu inc eases he p obabili y o loss, while simul aneously dec easing he
loss amoun . Ac oss all h ee igu es, he p obabili y o loss seems o ha e e y li le in luence on he esul s. In con as ,
lowe ing he loss amoun makes he esul s mo e ex eme in he sense ha insu ance demand d ops mo e quickly as
weal h dec eases.
FIGURE B2 The g aph shows he esul s o Figu e1 o di e en alues o p. Lines show he nume ically de e mined op imal le els
o insu ance demand (α) and ou -o pocke paymen s ( ) in a se ing whe e U(c,h)=c
γ+h
γ+kc
γh
γ, h0=q=1, γ=0.7 and L=0.8. (a) Resul s
o p=0.2 and L=0.8. (b) Resul s o p=0.05 and L=0.8
JASPERSEN
FIGURE B1 The g aph shows he esul s o Figu e1 o di e en alues o γ. Lines show he nume ically de e mined op imal le els
o insu ance demand (α) and ou -o pocke paymen s ( ) in a se ing whe e U(c,h)=c
γ+h
γ+kc
γh
γ, h0=q=1, p=0.1 and L=0.8. (a) Resul s
o γ=0.2. (b) Resul s o γ=0.5
1255
FIGURE B3 The g aph shows he esul s o Figu e1 o di e en alues o L. Lines show he nume ically de e mined op imal le els
o insu ance demand (α) and ou -o pocke paymen s ( ) in a se ing whe e U(c,h)=c
γ+h
γ+kc
γh
γ, h0=q=1, γ=0.7 and p=0.1. (a) Resul s
o p=0.1 and L=1.0 (b) Resul s o p=0.1 and L=0.4
APPENDIX C: EXTENSION TO NON-LINEAR HEALTH PRODUCTION FUNCTION
A po en ial c i icism o he model analyzed in he main ex is ha a linea bene i unc ion o mone a y paymen s o
heal h ca e is un ealis ic. I is indeed he case ha cheape p ocedu es o medica ions can o en achie e some amoun
o heal h gains bu all sho o achie ing heal h le els close o he op imum, which equi e mo e expensi e ea men s.
An a h i ic join in an elde ly pa ien , o example, can be ea ed wi h pain medica ion o wi h su gical inse ion o a
p o hesis. While bo h es o e he mobili y o he pa ien , he cheape ea men wi h pu e medica ion has side e ec s on
li e and kidneys, while he ope a ion is disp opo ionally mo e expensi e, bu has ewe side e ec s. Fo medica ion,
diabe es ea men s pose a good example o a conca e bene i unc ion. Diabe es can be ea ed wi h human insulin o
human-analog insulin. The cheape human insulin can be used o medica ion, bu wo ks in a delayed ashion, allow-
ing less p ecise s ee ing o he insulin le el, which equi es a lowe a ge le el o insulin han is used when medica ing
wi h he as e ac ing and mo e easily s ee able human-analog insulin. Thus, some inc eases om a bad heal h s a e
can be achie ed by he cheap medica ion, bu imp o emen s beyond hese ini ial inc eases a e disp opo ionally mo e
expensi e.
I model such a heal h p oduc ion unc ion o mally, by assuming ha mone a y paymen s ans e in o he heal h
s a us conca ely wi h unc ion : [0,1] → [0,1] wi h (0)=0, (1)=1, ′ > 0and ″ ≤ 0. This also changes he o mal o m
o he condi ion ha he exchange a e adjus ed c oss de i a i e o he u ili y unc ion does no o e whelm he e ec o
ma ginal u ili y on p e e ences. Speci ically, his condi ion now dic a es ha q/ ′(x)Ucc(c, h0 − (1 − (x))L) < Uch(c, h0 − (1
− (x))L) and Uhh(c, h0 − (1 − (x))L) < q/ ′(x)Uch(c, h0 − (1 − (x))L) o all le els o c and x.
The objec i e unc ion o his ex ended model is
max
𝛼≥0,𝑡≥0{𝐸𝑈(𝛼)=(1−𝑝)𝑈(𝑤
0
−𝛼𝑝𝑞𝐿, ℎ
0
)+𝑝𝑈(𝑤
0
−𝛼𝑝𝑞𝐿 −𝑡𝑞𝐿, ℎ
0
−(1−𝑟(𝛼+𝑡))𝐿)} .
FIGURE B4 The g aph shows he esul s o Figu e1 o di e en alues o p and L bu he same expec ed alue o he loss as in he
main analysis. Lines show he nume ically de e mined op imal le els o insu ance demand (α) and ou -o pocke paymen s ( ) in a se ing
whe e U(c,h)=c
γ+h
γ+kc
γh
γ, h0=q=1, γ=0.7 and pL=0.08. (a) Resul s o p=0.2 and L=0.4. (b) Resul s o p=0.4 and L=0.2
JASPERSEN1256