Online Appendix to the Paper No Claim? Your Gain: Design of Residual Value Extended Warranties under Risk Aversion and Strategic Claim Behavior

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1 Online Appendix to the Pape No Claim? You Gain: Design of Residual Value Extended Waanties unde Risk Avesion and Stategic Claim Behavio Lemma Given any x ě y ě 0, pe x e y q{ is inceasing in Moeove, pe x e y q{ ě x y if ą 0; pe x e y q{ ď x y if ă 0 Poof of Lemma Notice that pe x e y q{ pe pxyq q{ e y Because both pe pxyq q{ and e y ae non-negative fo any and x ě y, and e y is inceasing in fo any given y ě 0, then it is sufficient to show that pe pxyq q{ is inceasing in Conside its fist-ode deivative with espect to B`pe pxyq q{ px yqepxyq pe pxyq q B 2 Let Gpq : e pxyq Then, BGpq{B ě 0 and B 2 Gpq{B 2 ě 0 fo any given x ě y, so Gpq is inceasing convex in Note that Gp0q 0 Appaently, fo any ě 0, Gpq Gp0q ` ż 0 G pτqdτ ď G pq We note that the above inequality also holds fo any ď 0 because Gpq Gp0q 0 G pτqdτ ď p0 qg pq G pq Thus, B`pe pxyq q{ G pq Gpq ě 0, B 2 and given any x ě y ě 0, pe x e y q{ is inceasing in Conside the limit as goes to zeo, lim Ñ0 pe x e y q{ lim Ñ0 pxe x ye y q x y Theefoe, pe x e y q{ ě x y if ą 0; pe x e y q{ ď x y if ă 0 Futhemoe, thee exist tighte bounds fo pe x e y q{, eg, fo any x ě y ě 0, minpe x, e y q px yq ď pe x e y q{ ż x y e τ dτ ď maxpe x, e y q px yq Poof of Theoem Appaently, the optimal claim policy has a theshold stuctue: it is optimal fo a custome with isk attitude to place a claim at time t fo a failue with epai cost C t if and only if C t ě gpt;, q Moeove, it is staightfowad that gpt;, q is deceasing in t and inceasing in, noting that t epesents the time-to-go We next show the monotonic compaative statics of gpt;, q with espect to Suppose gpt;, q ď gpt;, q at time t fo any ą We will next show gpt`δ;, q ď gpt`δ;, q fo a sufficiently small δ ą 0 Accoding to the diffeential equation (), gpt ` δ;, q gpt;, q λtδ minpc `Ee t,gpt;,qq s ` opδq, gpt ` δ;, q gpt;, q λtδ Then, `E e minpc t,gpt;,qq { `E e minpc t,gpt;,qq { minpct,gpt; `Ee,qq s ` opδq ě `E e minpc t,gpt;,qq { `E e minpc t,gpt;,qq { E e minpc t,gpt;,qq e minpc t,gpt;,qq ě E Θ pminpc t, gpt;, qq minpc t, gpt;, qqq ě E Θ pgpt;, q gpt;, qq, whee Θ e minpc t,gpt;,qq if ě 0; Θ e minpc t,gpt;,qq if ď 0 The fist inequality holds because E e minpc t,gpt;,qq { is inceasing in by Lemma ; the second inequality holds by a simila agument in the poof of Lemma ; the thid inequality holds because minpc t, xq minpc t, yq ď px yq fo any x ě y

2 2 Theefoe, gpt ` δ;, q gpt ` δ;, q ď `gpt;, q gpt;, q with espect to the isk attitude The case with appoaching 8: g pt; 8, q lim λt Ñ8 λ t δθ ď 0 Then, gpt;, q is deceasing E e minpc t,gpt;,qq 0 Then, gpt; 8, q gp0; 8, q ` t 0 g ps; 8, qds The case with appoaching `8: fo any positive time t g pt; 8, q lim λt E e minpc t,gpt;,qq λ t E minpc t, gpt; 8, qq e 8 minpc t,gpt;8,qq Ñ`8 If gpt; 8, q ą 0, then g pt; 8, q 8 and gpt; 8, q gp0; 8, q ` t 0 g ps; 8, qds ă 0 fo any positive t, which is impossible Thus, the only solution to the above diffeential equation is gpt; 8, q 0 fo any positive t Poof of Poposition (a) Note that gpt;, q is deceasing in t and gp0;, q When the time-togo t is vey small, gpt;, q is sufficiently close to, so it is optimal fo the custome not to claim any failue since gpt;, q «ą c Then, the diffeential equation () becomes g pt;, q `ec λt Solving the above diffeential equation and combining with the bounday condition gp0;, q yields gpt;, q pec qλptq Denote the unique solution to the equation { pe c qλptq c with espect to t by t Then, fo any t ě t, it is optimal to claim all the failues so the diffeential equation () becomes g pt;, q `egpt;,q λt, with bounday condition gpt ;, q c Similaly, the unique solution to the above diffeential equation is gpt;, q ln ` eλptq`λpt q p ec q (b) Unde the exponential distibution, the diffeential equation () can be ewitten as follows g pt;, q λt `epµqgpt;,q µ It is staightfowad to veify that function (3) satisfies the diffeential equation () and its bounday condition In paticula, eλptqepµq lim gpt;, q lim ѵ ѵ eλptq ` epµq eλptq Poof of Poposition 2 We fist conside the case ą 0 Conside the fist-ode deivative with espect to, Bw tw pt; q ERptqe Rptq s Ee Rptq s log `Ee s Rptq B 2 Ee Rptq s ě ERptqe Rptq s ERptqsEe Rptq s Ee Rptq s The inequality holds because of the Jensen s inequality: log `Ee s Rptq ď Elogpe Rptq qs ERptqs Fo a simila eason, ERptqe Rptq s ě ERptqse ERptqs We will next show a stonge esult, ie, ERptqe Rptq s ě ERptqsEe Rptq s Fist, suppose that Rptq takes values fom the discete set trptq,, Rptq n u with espective pobabilities α,, α n, whee α ` ` α n Then, nÿ nÿ nÿ ERptqe Rptq s ERptqsEe Rptq s α i Rptq i e Rptqi α i Rptq i α j e Rptqj i nÿ α i Rptq ie nÿ nÿ nÿ Rptqi α j e Rptqj α i Rptq i α j e Rptqi e Rptqj i j i j ÿ α i α j`rptqi Rptq j e Rptqi e Rptqj ě 0, pi,jq whee pi, jq and pj, iq ae consideed the same pai The inequality holds because `Rptq i Rptq j e Rptqi e Rptqj ě 0 fo each pai pi, jq Similaly, we can show that the inequality also holds when Rptq has a continuous suppot set Theefoe, fo ą 0 Bw tw pt; q B ě Ee Rptq s i j ÿ α i α j`rptqi Rptq j e Rptqi e Rptqj ą 0 pi,jq

3 Similaly, we can show that the above inequality also holds fo ă 0 Thus, w tw pt; q is inceasing in We next conside the case with appoaching 0 as follows, lim w twpt; q lim Rptq Ñ0 B ln `Ee s ERptqe Rptq s lim ERptqs Ñ0 lim Ñ0 B{B Ñ0 Ee Rptq s It completes the poof Poof of Poposition 3 Let N t be a andom vaiable following the Poisson distibution with mean Λptq t 0 λ s ds Notice that ř N t C i i is a compound Poisson andom vaiable It is known that the total epai cost Rptq has the same distibution as ř N t C i i assuming that the epai costs ae iid, independent of the failue pocess (see, eg, Ross 995) By equation (5), w tw pt; q log `Ee Rptq s log `Ee ř N t i C i s log `E Ee ř N t i C i N t s log `E Π N t i Ee C i N t s log `E M C pq N t log `e Λptq pm C pqq Λptq pm Cpq q, whee M C pq is the moment geneating function of the epai cost C, ie, M C pq Ee C s The second last equality holds because Ex N t s M Nt plogpxqq e Λptq pxq Appaently, fo the constant epai cost w tw pt; q Λptq pm C pq q{ Λptqpe c q L Moeove, w tw pt; q Ñ Λptqc as Ñ 0 because lim Ñ0 pe c q L c Fo the exponential distibuted epai cost, w tw pt; q Λptq pm Cpq q Λptq µ The last equality holds because M C pq µ{pµ q fo ă µ Appaently, w tw pt; q Ñ Λptq{µ as Ñ 0 Poof of Poposition 4 The willingness-to-pay w vw pt;, q is the quality such that the utility of buying and not buying an RVW is indiffeent, taking into account the possible out-of-pocket cost and efund fo the RVW, ie, EepvRptqq s epvwvwpt;,q`gpt;,qq Combining with equation (5) yields w vw pt;, q w tw pt; q ` gpt;, q (a) Fist, we conside the isk-neutal case, ie, 0 Note that w tw pt; 0q ERptqs by Poposition 2, so we only need to show hpt; 0, q gpt; 0, q ` ERptqs Recall that B `gpt; 0, q ` ERptqs λt E C t minpc t, gpt; 0, qq λ t E `C t gpt; 0, q ` Bt Plugging hpt; 0, q gpt; 0, q ` ERptqs into equation (4) esults in! ) h pt; 0, q λ t PpC t ě gpt; 0, qq EC t C t ě gpt; 0, qs gpt; 0, q λ t E `C t gpt; 0, q ` Combining it with the bounday condition hp0; 0, q gp0; 0, q ` ERp0qs, we have obtained hpt; 0, q gpt; 0, q ` ERptqs at any t ě 0 fo any given efund Next, we conside the case of ą 0 Suppose hpt;, q ě gpt;, q ` ERptqs at some t ě 0 fo given ą 0 and We will next show that fo any sufficiently small δ ą 0, hpt ` δ;, q ě gpt ` δ;, q ` ERpt ` δqs By the diffeential equations () and (4), ż 8 hpt ` δ;, q gpt ` δ;, q ERpt ` δqs hpt;, q gpt;, q ERptqs ` δλ tˆ c t df t pc t q gpt;,q p F t pgpt;, qqq phpt;, q ERptqsq ` p ` Ee minpct,gpt;,qq sq{ EC t s ` opδq hpt;, q gpt;, q ERptqs ` δλ tˆ ż gpt;,q 0 c t df t pc t q p F t pgpt;, qqq phpt;, q ERptqsq 3

4 4 ` p ` Ee minpct,gpt;,qq sq{ ` opδq `hpt;, q gpt;, q ERptqs ` δλ t p F t pgpt;, qqq ` δλ t EminpC t, gpt;, qqs ` p ` Ee minpct,gpt;,qq sq{ ` opδq ě `hpt;, q gpt;, q ERptqs ` δλ t p F t pgpt;, qqq ě 0 The fist inequality holds because p ` Ee minpc t,gpt;,qq sq{ ě EminpC t, gpt;, qqs fo any ą 0 by Lemma ; the second inequality holds because hpt;, q ě gpt;, q ` ERptqs Theefoe, hpt;, q ě gpt;, q ` ERptqs at any t ě 0 fo given ą 0 and Similaly, we can show hpt;, q ď gpt;, q ` ERptqs at any t ě 0 fo given ă 0 and Fo the compaison between hpt;, q and w vw pt;, q, suppose hpt;, q ě gpt;, q ` w tw pt; q at some t ě 0 fo given ă 0 and We will next show that fo a sufficiently small δ ą 0, hpt ` δ;, q ě gpt ` δ;, q ` w tw pt ` δ; q The willingness-to-pay fo the TW can be expessed as follows w tw pt ` δ; q log `Ee s Rpt`δq log `Ee Rptq e s Rpt,t`δq w tw pt; q ` λtδ C `Ee t s ` opδq, (2) whee Rpt, t ` δq epesents the total epai cost fom time-to-go t ` δ to t The equality (2) holds because log `Ee Rpt,t`δq s log `λ t δee C t s ` p λ t δq ` opδq log ` ` λ t δpee C t s q ` opδq λ t δpee C t s q ` opδq The last equality holds because of the Taylo expansion Then, conside ż 8 hpt ` δ;, q gpt ` δ;, q w tw pt ` δ; q hpt;, q gpt;, q w tw pt; q ` δλ tˆ c t df t pc t q gpt;,q p F t pgpt;, qqq phpt;, q ERptqsq ` p ` Ee minpct,gpt;,qq sq{ pee C t s q{ ` opδq hpt;, q gpt;, q w tw pt; q ` δλ tˆec t s EminpC t, gpt;, qqs p F t pgpt;, qqq phpt;, q gpt;, q ERptqsq ` p ` Ee minpc t,gpt;,qq sq{ pee C t s q{ ` opδq `hpt;, q gpt;, q w tw pt; q ` δλ t p F t pgpt;, qqq ` δλ t p F t pgpt;, qqqpw tw pt; q ERptqsq EminpC t, gpt;, qqs ` Ee minpc t,gpt;,qq s{ ` EC t s ` Ee C t s{ ` opδq ě `hpt;, q gpt;, q w tw pt; q ` δλ t p F t pgpt;, qqq ě 0 The fist inequality holds because w tw pt; q ď ERptqs fo any t ě 0 and ă 0, and by Lemma, fo any given ă 0, `Ee C t s Ee minpc t,gpt;,qq s { ď EC t s EminpC t, gpt;, qqs; the second inequality holds because hpt;, q ě gpt;, q ` w tw pt; q Theefoe, hpt;, q ě w vw pt;, q at any t ě 0 fo given ă 0 and Similaly, we can pove hpt;, q ď w vw pt;, q at any t ě 0 fo given ą 0 and (b) Fist, conside the case of ă 0 Suppose hpt;, q gpt;, q ď hpt;, q gpt;, q fo some t ě 0 and ą Then, `hpt ` δ;, q gpt ` δ;, q `hpt ` δ;, q gpt ` δ;, q `hpt;, q gpt;, q `hpt;, q gpt;, q ż 8 ` δλ tˆ c t df t pc t q p F t pgpt;, qqq phpt;, q ERptqsq ` p ` Ee minpct,gpt;,qq sq{ gpt;,q ż 8 c t df t pc t q ` p F t pgpt;, qqq phpt;, q ERptqsq p ` Ee minpc t,gpt;, qq sq{ gpt;, q `hpt;, q gpt;, q ` δλ t p F t pgpt;, qqq `hpt;, q gpt;, q ` δλ t p F t pgpt;, qqq ` δλ tˆerptqs EminpC t, gpt;, qqs ` p ` Ee minpc t,gpt;,qq sq{

5 5 `ERptqs EminpC t, gpt;, qqs `F t pgpt;, qq F t pgpt;, qq ERptqs p ` Ee minpc t,gpt;, qq sq{ `hpt;, q gpt;, q `hpt;, q gpt;, q ` δλ t p F t pgpt;, qqq ` δλ tˆ EminpC t, gpt;, qqs ` Ee minpct,gpt;,qq s{ EminpC t, gpt;, qqs ` Ee minpc t,gpt;, qq s{ ` `F t pgpt;, qq F t pgpt;, qq `hpt;, q gpt;, q ERptqs ď `hpt;, q gpt;, q `hpt;, q gpt;, q ` δλ t p F t pgpt;, qqq ď 0 The fist inequality holds because hpt;, q ď gpt;, q ` ERptqs fo ă 0, and by Lemma, `Ee minpc t,gpt;,qq s Ee minpc t,gpt;, qq s { ď EminpC t, gpt;, qqs EminpC t, gpt;, qqs; the second inequality holds because hpt;, qgpt;, q ď hpt;, qgpt;, q Theefoe, hpt;, qgpt;, q is deceasing in fo any given ă 0 Similaly, we can pove that hpt;, q gpt;, q is inceasing in fo any given ą 0 Obviously, gpt;, q Ñ 8 as Ñ 8 We have aleady known that hpt;, q gpt;, q is monotonic in fo any given and is bounded fom below and above, ie, w tw pt; q ą hpt;, q gpt;, q ą ERptqs fo any ą 0 and w tw pt; q ă hpt;, q gpt;, q ă ERptqs fo any ă 0, so it conveges to a limit as appoaches infinity To show its limit, we conside its deivative with espect to t Fom diffeential equations () and (4), B`hpt;, q gpt;, q λ t PpC t ą gpt;, qq!ec t C t ą gpt;, qs `hpt;, q ERptqs ) Bt `λt ` E e minpc t,gpt;,qq Ee C t ż s 8 `e C t λ t λ t gpt;,q C `e gpt;,q t gpt;, q df t pc t q `λ t PpC t ą gpt;, qq `hpt;, q gpt;, q ERptqs Appaently, PpC t ą gpt;, qq `hpt;, q gpt;, q ERptqs Ñ 0 as gpt;, q Ñ 8 because hpt;, q gpt;, q is bounded Assume that the moment geneating function is finite, ie, Ee C t s is finite Then, ż 8 lim gpt;,qñ8 gpt;,q `e C t C `e gpt;,q t gpt;, q df t pc t q 0 The equality holds because the limit of each integal in the above equation is equal to zeo Thus, it holds that B`hpt;, 8q gpt;, 8q Ee C t s λ t Bt Then, fo any given, we have The second equality holds because Bw tw pt; q Bt hpt;, 8q gpt;, 8q w tw pt ` δ; q w tw pt; q lim δñ0 δ lim δñ0 It completes the poof λ t δpee C t s q ` opδq δ ż t 0 Ee Cs s λ s df s pc s q w tw pt; q log `Ee s Rpt,t`δq log `λ t δee C t s ` p λ t δq ` opδq lim lim δñ0 δ δñ0 δ λ tpee C t s q

6 6 Poof of Theoem 2 The maximum pice of the TW is equal to its willingness-to-pay Then, the pofit of offeing a TW is equal to Fo isk-avese customes with ą 0, we have w tw pt ; q ERpT qs w vw pt ;, q hpt ;, q w tw pt ; q ` gpt ;, q hpt ;, q ă w tw pt ; q ERptqs The inequality holds fo any ą 0 by Poposition 4 Fo the RVW povide with a positive efund eans less pofit than the TW, so the RVW degeneates to a TW in a homogeneous maket with isk-avese customes, ie, 0 Similaly, fo isk-seeking customes, ie, ă 0, w vw pt ;, q hpt ;, q w tw pt ; q ` gpt ;, q hpt ;, q ą w tw pt ; q ERptqs Notice that the TW loses money fo ă 0, so the RVW can balance the evenue and the suppot cost by offeing a sufficiently lage efund because Poposition 4 shows that hpt ;, q gpt ;, q Ñ w tw pt ; q as Ñ 8 Poof of Theoem 3 To investigate the pofitability of the RVW ove the TW, we conside the following compaison max wtw pt ; b q ` gpt ; b, q hpt ; a, q ( ą w tw pt ; b q ERpT qs O equivalently min hpt ; a, q gpt ; b, q ( ă ERpT qs By Theoem 4, hpt ; a, q is fist inceasing in a fo a ď 0 and then is deceasing in it fo a ě 0 Then, max hpt ; a, q gpt ; b, q ( is also fist inceasing and then deceasing in a Denote the two solutions to the following equation with espect to a, min hpt ; a, q gpt ; b, q ( ERpT qs by a and a, and a ď a Theefoe, the RVW is stictly moe pofitable than the TW if and only if a ă a o a ą a Poof of Poposition 5 Let α H`w tw pt ; H q ERptqs w tw pt ; L q ERpT qs We have w tw pt ; H q w twpt ; L q α L ERpT qs α H By Poposition 2, w tw pt ; H q is inceasing in H, so thee exists a unique solution to equation (5) with espect to H, denoted by p H We emak that if H ě p, it is moe pofitable fo the povide to only seve type-h customes by chaging pice w tw pt ; H q Poof of Theoem 4 We will pove this theoem by induction (a) Suppose that fo a given efund, w vw pt;, q ě w vw pt;, q at time-to-go t fo any ą We will next show that the inequality also holds at time-to-go t ` δ fo a sufficiently small δ ą 0, ie, w vw pt ` δ;, q ě w vw pt ` δ;, q Accoding to the diffeential equation (), we have gpt ` δ;, q gpt;, q λtδ E e minpc t,gpt;,qq ` opδq Combing with equation (2), we have gpt ` δ;, q ` w tw pt ` δ; q gpt;, q ` w tw pt; q ` λ t δ EeC t s E e minpc t,gpt;,qq Theefoe, gpt ` δ;, q ` w tw pt ` δ; q gpt ` δ;, q ` w tw pt ` δ; q gpt;, q ` w tw pt; q ` opδq C Ee t s E e minpc t,gpt;,qq gpt;, q ` w tw pt; q ` λ t δ Ee Ct s E e minpc t,gpt;,qq ě gpt;, q ` w tw pt; q gpt;, q ` w tw pt; q ` opδq

7 ě C Ee t s E e minpc t,gpt;,qq ` λ t δ gpt;, q ` w tw pt; q gpt;, q ` w tw pt; q Ee Ct s E e minpc t,gpt;,qq ě 0 ` opδq 7 The fist inequality holds because gpt;, q ď gpt;, q and E e minpc t,gpt;,qq { ě E e minpc t,gpt;,qq { ; the second inequality holds because by Lemma, Ee C t s E e minpc t,gpt;,qq ě Ee Ct s E e minpc t,gpt;,qq ; the last inequality holds because gpt;, q ` w tw pt; q ě gpt;, q ` w tw pt; q Thus, fo any given time-to-go t and efund, w vw pt;, q is deceasing with espect to no matte whethe is positive o negative Notice that w vw pt;, q w tw pt; q ` gpt;, q and gpt;, q is deceasing in, so w vw pt;, q is inceasing at a lowe ate than w tw pt; q (b) We fist conside the isk-avese customes, ie, ą 0 Suppose that fo a given efund, hpt;, q ď hpt;, q fo any ą ą 0 Then, we compae hpt ` δ;, q and hpt ` δ;, q Fom the diffeential equation (4), hpt ` δ;, q hpt;, q ` δλ t PpC t ą gpt;, qq!ec t C t ą gpt;, qs `hpt;, q ERptqs ) ` opδq Then, we have hpt;, q ` δλ t ˆż 8 gpt;,q c t df t pc t q p F t pgpt;, qq`hpt;, q ERptqs ` opδq ż gpt;,q hpt ` δ;, q hpt ` δ;, q hpt;, q hpt;, q ` δλ tˆ c t df t pc t q ` `F t pgpt;, qq F t pgpt;, qq gpt;,q ERptqs `hpt;, q hpt;, q ` `F t pgpt;, qqhpt;, q F t pgpt;, qqhpt;, q ` opδq ď `hpt;, q hpt;, q p δλ t q ` δλ tˆ`ft pgpt;, qq F t pgpt;, qq `gpt;, q ` ERptqs ` `F t pgpt;, qqhpt;, q F t pgpt;, qqhpt;, q ď `hpt;, q hpt;, q p δλ t q ` δλ tˆ`ft pgpt;, qq F t pgpt;, qq `gpt;, q ` ERptqs hpt;, q ď `hpt;, q hpt;, q p δλ t q ď 0 The fist inequality holds because gpt;, q ď gpt;, q; the second inequality holds because hpt;, q ď hpt;, q; the thid inequality holds because gpt;, q ` ERptqs hpt;, q ď 0 fo ą 0 by Poposition 4 Theefoe, fo any given time-to-go t and efund, hpt;, q is deceasing in fo isk-avese customes Next, conside the isk-seeking customes, ie, ă 0 Suppose that fo a given efund, hpt;, q ě hpt;, q at some t ě 0 fo any ă ă 0 Similaly, we have ż gpt;,q hpt ` δ;, q hpt ` δ;, q hpt;, q hpt;, q ` δλ tˆ c t df t pc t q ` `F t pgpt;, qq gpt;,q F t pgpt;, qq ERptqs ě `hpt;, q hpt;, q p δλ t q ` δλ t`ft pgpt;, qq F t pgpt;, qq `gpt;, q ` ERptqs ` `F t pgpt;, qqhpt;, q F t pgpt;, qqhpt;, q ě `hpt;, q hpt;, q p δλ t q ` δλ t`ft pgpt;, qq F t pgpt;, qq `gpt;, q ` ERptqs hpt;, q ě `hpt;, q hpt;, q p δλ t q ě 0 The fist inequality holds because gpt;, q ď gpt;, q; the second inequality holds because hpt;, q ě hpt;, q; the thid inequality holds because gpt;, q ` ERptqs hpt;, q ě 0 fo ă 0 by Poposition 4 Thus, fo given time-to-go t and efund, hpt;, q is inceasing in fo isk-seeking customes

8 8 Poof of Theoem 5 Fo L ă H ď p, the TW captues both type-l and type-h customes and its pofit is equal to w tw pt ; L q ERpT qs An RVW with efund and pice w vw pt ; L, q captues both maket segments Conside the following inequality max o equivalently,! w tw pt ; L q ` gpt ; L, q α L hpt ; L, q α H hpt ; H, q ) ą w tw pt ; L q ERpT qs,! ) min gpt ; L, q ` α L hpt ; L, q ` α H hpt ; H, q ă ERpT qs (22) Since hpt ; H, q is deceasing in H fo any given by Theoem 4, then min gpt ; L, q`α L hpt ; L, q` α H hpt ; H, q ( is also deceasing in H Theefoe, thee exists a theshold H such that inequality (22) holds fo any H ą H Appaently, H ă p H We have aleady shown that the RVW is stictly moe pofitable than the TW fo H P ` H, p H We emak that the RVW may be moe pofitable fo H vaying in an even lage ange Poof of Poposition 6 (a) We will use sample path agument to show the monotonic popety Fo a Poisson pocess with a stationay failue ate λ H, the optimal claim policy has a theshold gpt; λ H, q fo each failue at time t Assume that the custome with a stationay failue ate λ L adopts the same policy with theshold gpt; λ H, q Appaently, the expected efund net of out-of-pocket epai cost taking into account the isk attitude is geate than that unde the failue pocess with a ate λ H because each failue occus with a lowe pobability Theefoe, gpt; λ H, q, that is the net value coesponding to ate λ L unde the optimal claim policy is even highe (b) Fo a given efund, suppose w vw pt; λ H, q ě w vw pt; λ L, q at some t ě 0 Next, we will show that the inequality also holds at time t ` δ fo a sufficiently small δ ą 0, ie, w vw pt ` δ; λ H, q ě w vw pt ` δ; λ L, q Simila to the poof of Theoem 4, we have the following equations Compaing them, we have w vw pt ` δ; λ H, q w vw pt; λ H, q ` λ H δ EeC t s E e minpc t,gpt;λ w vw pt ` δ; λ L, q w vw pt; λ L, q ` λ L δ EeC t s E e minpc t,gpt;λ w vw pt ` δ; λ H, q ě w vw pt ` δ; λ L, q H,qq L,qq ` opδq, ` opδq The inequality holds because w vw pt; λ H, q ě w vw pt; λ L, q, λ H ě λ L and gpt; λ H, q ď gpt; λ L, q Notice that the net value gpt; λ, q of the RVW is deceasing in failue ate λ, so the willingness-to-pay fo the RVW is inceasing at a lowe ate than that fo the TW Poof of Theoem 6 (a) Notice that the willingness-to-pay can be expessed as follows w vw pt ; λ, q w tw pt ; λq ` gpt ; λ, q ΛpT q pm Cpq q Conside its fist-ode deivative with espect to λ B w vw pt ; λ, q T ˆM eλpt q eλpt q C pq Bλ p eλpt q q ` eλpt q ln `p eλpt q q ` eλpt q ą T ˆ M C pq ě 0 p eλpt q q The last inequality holds because of the condition in pat (a) of Theoem 6 (b) Dividing the both sides of equation (5) by pe gpt;λ,q q{, taking integals with espect to t and combining the bounday condition yields the closed-fom solution fo gpt; λ, q gpt; λ, q ln `p eλptq q ` eλptq Fo notational convenience, let Π vw pq w vw pt ; λ L, q `α L h vw pt ; λ L, q ` α H h vw pt ; λ H, q It can be futhe expessed by Π vw pq w tw pt ; λ L q `α L λ L ` α H λ H T c ln p eλlt q ` eλl T `α L eλlt ` α H eλh T

9 9 Conside the deivatives of Π vw pq with espect to BΠ vw pq eλl T L LT B p eλl T `α eλ ` α, H eλh T q ` eλl T B 2 L Π vw pq p T eλ q eλl T B `p 2 eλ L T 2 ă 0 q ` eλl T So, the pofit function Π vw pq is stictly concave in Solving the fist-ode condition BΠ vw pq{b 0 yields the optimal efund eλ LT ln ` pα L eλlt ` α HeλH T q p eλl T q pα L eλlt ` α HeλH T q It can be veified that ą 0 because fo any λ H ą λ L, eλlt ` pα L eλlt ` α H eλh T q ą p eλlt q pα L eλlt ` α H eλh T q (c) Solving the equation w tw pt ; λ L q `α L λ L ` α H λ H T c α H`w tw pt ; λ H q λ H T c fo λ H yields pλ H λ L pm Cpq q{ α L c α H pm C pq q{ To show that the RVW is always stictly moe pofitable than the TW, we compae Πp q to the pofit of the TW, which is equal to w tw pt ; λ L q `α L λ L ` α H λ H T c, ie, Πp q ą w tw pt ; λ L q `α L λ L ` α H λ H T c Notice that the RVW degeneates to the TW when the efund is equal to zeo, ie, Πp0q w tw pt ; λ L q `αl λ L ` α H λ H T c Because Πpq is stictly inceasing in and ą 0, the above stict inequality holds Poof of Poposition 7 (a) We can use the simila aguments as in the poof of Poposition 6 to show the monotonic popety with espect to the epai cost (b) Fo a given efund, suppose that w vw pt; C H, q ą w vw pt; C L, q at some t ě 0 Next, we will show w vw pt ` δ; C H, q ě w vw pt ` δ; C L, q Similaly, w vw pt ` δ; C, q w vw pt; C, q ` λ t δ EeC t s E e minpc t,gpt;c,qq ` opδq We next compae Moeove, we have H w vw pt ` δ; C H, q w vw pt; C H, q ` λ t δ EeC s E e minpch,gpt;c H,qq ` opδq, Ls w vw pt ` δ; C L, q w vw pt; C L, q ` λ t δ EeC E e minpcl,gpt;c L,qq ` opδq Ee CH s E e minpch,gpt;c H,qq ě Ee CH s E e minpch,gpt;c L,qq ě Ee CL s E e minpcl,gpt;c L,qq The fist inequality holds because gpt; C H, q ď gpt; C L, q; the second inequality holds because function `ex e minpx,gq is inceasing in x and C H is stochastically lage than C L Theefoe, fo any given efund w vw pt ` δ; C H, q ě w vw pt ` δ; C L, q, which completes the poof Poof of Poposition 8 Fo any given efund, poblem (7) is a linea pogam Fom the constaints, we have p vw ă p tw ` gpt ; L, q ď w tw pt ; H q ` gpt ; L, q Recall that p vw ď w vw pt ; L, q w tw pt ; L q ` gpt ; L, q Because w tw pt ; L q ď w tw pt ; H q, then the maximum pice of the RVW is p vw w vw pt ; L, q Because p tw ď p vw gpt ; H, q, then the maximum pice of the TW is p tw w vw pt ; L, q gpt ; H, q

10 0 Poof of Theoem 7 Compaison between poblems (8) and (3) yields that the waanty menu eans stictly moe pofit than the RVW alone fo any given efund, ie, max!gpt ; L, q α L hpt ; L, q α H`gpT! ; H, q ` ERpT qs ) ą max gpt ; L, q α L hpt ; L, q ) α H hpt ; H, q The inequality holds because hpt ; H, q ą gpt ; H, q ` ERpT qs fo any isk-avese custome and any positive efund by Poposition 4 To show the pofit advantage of the menu ove the TW alone, conside the following inequality max o equivalently,!w tw pt ; L q ` gpt ; L, q α L hpt ; L, q α H`gpT ; H, q ` ERpT qs ) ą w tw pt ; L q ERpT qs,! ) min gpt ; L q ` α L hpt ; L, q ` α H gpt ; H, q ă α L ERpT qs (23) Since gpt ; H, q is deceasing in H fo any given by Theoem, then min gpt ; L, q`α L hpt ; L, q` α H gpt ; H, q ( is also deceasing in H Theefoe, thee exists a theshold H such that inequality (23) holds fo any H ą H Compaing inequalities (22) and (23) and ecalling hpt ; H, q ą gpt ; H, q ` ERpT qs fo H ą 0 by Poposition 4, we have H ă H It completes the poof