Online Supplement for The Value of Bespoke : Demand Learning, Preference Learning, and Customer Behavior

Transcription

1 Online Supplemen for The Value of Bespoke : Demand Learning, Preferene Learning, and Cusomer Behavior Tingliang Huang Carroll Shool of Managemen, Boson College, Chesnu Hill, Massahuses 0467, inglianghuang@bedu Chao Liang Cheung Kong Graduae Shool of Business, Beijing 00738, China, liang@kgsbedun Jingqi Wang Fauly of Business and Eonomis, The Universiy of Hong Kong, Pokfulam, Hong Kong, jingqi@hkuhk This Online Supplemen inludes four seions Online Supplemen provides he ehnial proofs o he proposiions, heorems and orollaries in he paper All he oher seions inlude some supporing maerials and exra analysis for he paper Speifially, Online Appendix inludes supplemenal analysis for 5 of he paper in Online Supplemen 3, e analyze a varian of he basi model in 3 and 4 by alloing he firm o prie afer seeing he demand realizaion Online Supplemen 4 shos ha our model used in 5 of he paper an be operaionalized and suppored by he inerpreaion of produ line design or assormen planning Online Supplemen Proofs Proof of Proposiion Leing uθ, p p θ = 0, e have θ = p Noe 0 θ The demand is p D hen p, is 0 hen p >, and is D hen p < Le p 0, q 0 and π 0 denoe he opimal prie, opimal produion quaniy and opimal profi of he radiional sysem, respeively We mus have p 0 and p 0 d L q0 p 0 d H Noe ha for p and p d L q p hen D = d H, and πp, q = p min p d H, he firm s profi πp, q = p min p d H, q q = p q d L, q q = p p d L q hen D = d L By aking he expeaion over D, e have πp, q = p[q + p d L ] q Noie ha πp,q = p and q πp, q is linear in q For any p here i When q = p p p, he opimal q is eiher d L or p d H d L, e have πp, q = p d L p, hih is maximized a p = p L = + by solving dπ = 0, and he orresponding produion quaniy and profi are q = q dp L = d L and

2 Huang e al: The Value of Bespoke Arile submied o ; manusrip no πp L, q L = d L Noe p L due o + and > ii When q = p d H, e have πp, q = µ D d H µ D pp d H µ D For d H µ D, e have ha µ D pp d H µ D is maximized a p = p H = + by solving dπ = 0, and he orresponding produion dp quaniy and profi are q = q H = d H d H µ D and πp H, q H = µ D d H µ D Noe ha p L p H due o d H µ D and d H µ D For < d H µ D, e kno µ D pp d H µ D 0 hen p Noe ha πp L, q L πp H, q H if and only if d H d L µ D Combining ases i and ii, e d L µ D have he folloing If < d H µ D, hen p 0 = p L = +, q 0 = q L = d L, and π 0 = πp L, q L = d L If d H µ D, hen hen d H d L µ D, e have ha p d L µ 0 = p L = +, q 0 = q L = d L, D µ D and π0 = πp L, q L = d L ; oherise, e have ha p 0 = p H = +, q 0 = q H = d H d H µ D, and π0 = πp H, q H = µ D d H µ D Beause d H µ D < d H d L µ D, Proposiion follos d L µ D Proof of Proposiion Beause he firm deermines is produion quaniy q afer learning he realizaion of D, i alays ses q = p D for any prie p, here D = d L or d H depending on he demand realizaion The profi an be expressed as πp = p d H p + p d L p = µ D p p, hih is maximized a p = p L = +, and he orresponding opimal profi is µ D Proof of Proposiion 3 Beause he firm an ompleely remove he preferene-mismah os, i alays ses he opimal prie p p = and all he usomers ould like o buy a his prie The firm s profi is π, q = E[ mind, q] p q Similar o he analysis in Proposiion, e kno ha he opimal produion quaniy q p saisfies d L q p d H Afer aking he expeaion over µ D µ D D, e have π, q = [q + d L ] p q Noie ha π,q q d H µ D = p and π, q is linear in q We kno ha if p, hen q p = d H and he firm s opimal profi π P = µ D p d H ; oherise, q p = d L and π P = p d L Proof of Proposiion 4 In Bespoke sysem, he firm alays harges he prie sine i an ompleely remove he preferene-mismah, and i alays ses he produion quaniy equal o he demand realizaion d H or d L sine i an learn he demand realizaion before deermining he quaniy Therefore, is opimal profi πap = p d H + p d L = p µ D Proof of Lemma Reall κ = d H d L µ D We an prove ha κ dereases in by d L µ D shoing dκ d µ D < 0 Noe µ D = d H + d L We have µ D d L µ D = d H d L µ D + d L d L µ D d H d L µ D, and he resul κ follos Noie ha d L µ D d L+µ D and κ We have κ d H d L +µ D µ D d L +µ D Proof of Theorem Reall κ = d H = d H d L +d H µ D µ D d L = d H d L + d H d L d H d L µ D d L µ D = d L µ D We have shon ha κ is dereasing in, and κ > There exiss 0 > suh ha κ 0 = We disuss he folloing ases:

3 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 3 i Suppose < min p, κ} Then by Proposiions -4 and simplifying, he suffiien and neessary ondiion π0 + πap πa + πp is equivalen o p Noe ha he ondiion < min p, d H d L µ D } is equivalen o < µ D d p and d H d L µ D > L µ D µ D d Beause κ = L µ D d H d L µ D and is dereasing in, e have ha d H d L µ D > µ D d L µ D µ D d holds if <, and hus L µ } D < min p, d H d L µ D is equivalen o < min d p, L µ 0}, here κ 0 = Hene, in his ase, D µ D he ondiion for omplemens is < min p, 0} and p ii Suppose min p, κ} max p, κ} This ondiion is equivalen o min p, 0} max p, 0} We disuss o subases: iia If 0 < p, hen he supposed ondiion beomes 0 p In his subase, e have p and κ By Proposiions -4 and sim- plifying, he suffiien and neessary ondiion π0 + πap πa + πp [ subase, e have p is equivalen o p ] + d H µ D iib If 0 p, hen he supposed ondiion beomes p 0 In his and κ By Proposiions -4 and simplifying, he suffiien and neessary ondiion π 0 + π AP π A + π p is equivalen o p p+ Nex, e sho p p+ alays holds in his subase Beause 0 in his subase and κ dereases in, i is suffiien o sho κ 0 κ p p+ Noe κ 0 =, κ p p+ p p+ Lemma, and ha p p+ p p+ p p+ is equivalen o + p hih is alays rue due o p > and + Hene, he ondiion in his subase is p 0 Noie ha have p p p p p+ an be rerien as p Sine p in his subase, e, hih is simplified o p Therefore, he ondiions in his subase an also be rien as p 0, and p iii Suppose > max p, κ}, hen by Proposiions -4 and simplifying, he suffiien and neessary ondiion π 0 + π AP π A + π p is equivalen o p + + d H µ D, hih alays holds due o +, p >, and d H µ D Hene, he ondiion in his ase is equivalen o > max p, 0} We are no ready o summarize all hese ases ogeher: If and only if one of he folloing o ondiions holds, demand learning and preferene learning are omplemens: 0 and p ; 0 and p C p, [ ] } here C p = max + d H µ D, Here is from ases i and iib, and is from ases iia and iii Proof of Corollary When 0 p, he suffiien and neessary ondiions an be rien as < p and p in ase i, p 0 in ase ii ie, ase iib, and > 0 in ase iii In order o ge he resul, e jus need o sho ha p is redundan in ase i hen 0 p Beause P 0, i is suffiien o sho 0, equivalenly,

4 Huang e al: The Value of Bespoke 4 Arile submied o ; manusrip no κ 0 κ Noe κ 0 =, κ Lemma, and ha is equivalen o hih is alays rue due o + Hene, hen 0 p, he suffiien and neessary ondiion in ase i an be simplified o < p, and he desired resul follos Proof of Lemma From equaions -5 in he paper, e an express, in erms of demand learning auray I, he firm s poserior probabiliies of he demand foreas afer observing signal s as P D = d H s = d H = β H = I, P D = d L s = d H = β H = I, and P D = d L s = d L = β L = I, P D = d H s = d L = β L = I d H Suppose ha he firm observes demand signal s = d H, hen is updaed demand Ds = d H = ih probabiliy H I β H = I and Ds = d H = d L = 0 ih probabiliy H I = I Based on he analysis for he radiional sysem, e kno ha e need o disuss he folloing o ases: If H I /, ie, I, hen i is opimal o produe zero quaniy, hih resuls in zero expeed profi, ie, π s=d H = 0 If H I /, ie, I, hen he firm s opimal prie p s=d H = +/ H I, opimal produion quaniy qs=d H = d H H I, and opimal profi π s=d H = H Id H H I Suppose ha he firm observes demand signal s = d L, hen is updaed demand Ds = d L = d H ih probabiliy L I β L = I and Ds = d L = d L = 0 ih probabiliy L I = I Based on he analysis for he radiional sysem, e kno ha e need o disuss he folloing o ases: If L I /, ie, I, hen i is opimal o produe zero quaniy, hih resuls in zero expeed profi, ie, π s=d L = 0 If L I /, ie, I hen he firm s opimal prie p s=d L = +/ LI, opimal produion quaniy q s=d L = d H and opimal profi π s=d L = LId H L I, L I, To rie he firm s expeed profi before observing he demand signal s, e need o ompare I and I I urns ou I I is equivalen o / Noe ha π AI, = π s=d H + π s=d L Then, e an ompue his expeed profi based on differen ombinaions of he ondiions on and I For example, if [0, ], e have I 0 I If e furher assume I [I, ], hen π AI, = I d H I Similarly, e obain he profi funions in oher ases

5 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 5 Proof of Theorem We firs disuss all he possible ases Suppose We disuss he folloing o ases depending on he magniude of I: If I [I, ], hen π0 + πap πa + πp is equivalen o 0 + d H p H Id H H I + 0, hih is simplified o p I I If I [0, I ], hen π 0 = π A = π p = 0 and π AP = d H p I is lear ha i is alays ha demand learning and preferene learning are omplemens in his ase Suppose [, p ] Under his assumpion, e disuss he folloing ases depending on he magniude of I: If I [I, ], hen he ondiion for omplemens is d H +dh p, hih an be simplified o p I H Id H H I If I [0, I, hen he ondiion for omplemens is d H p H Id H H I + L Id H I + d H + L I, hih an be simplified o I p I + I I Suppose p We disuss he folloing ases depending on he magniude of I: If I [I, ], he ondiion for hem being omplemens is d H + dh p H Id H p H I + p d H, hih an be simplified o I I A- Noie ha his inequaliy holds ih he sri inequaliy hen I = as shon in he basi model, sine he RHS an be simplified o [ + ] < = p Hoever, e an sho ha he RHS is a srily inreasing funion a he poin I = To see his, denoe fi I I Then, e have he firs-order derivaive f I = [ ] A I =, e have f = I > 0 Hene, here exiss I 0, suh ha he RHS is equal o, ie, fi < p Hene, for I [maxi, I, ], his inequaliy alays holds If I [0, I ], hen he ondiion for hem being omplemens is d H + dh p HId H H I + p d H, + LId H hih an be simplified o I p I + I L I I A-

6 Huang e al: The Value of Bespoke 6 Arile submied o ; manusrip no Therefore, i beomes sraighforard o hek ha he saemen for a high I holds This omplees he proof Proof of Theorem 3 To pin don he profi funions for he four sysems, e need o disuss heher p + p p holds or no Suppose p + p p We have p p+ p Under his assumpion, e firs onsider he subase, ie, We have π 0 + π AP π A + π p equivalen o 0 + d H p p 0 + d H Simplifiaion of his equaion and subsiuing p = J yields p J J Suppose p + p p We have p p+ p Under his assumpion, e have o subases: i p + p and ii p + p We disuss eah of hese subases i Case ia: If p + p, hen he inequaliy π 0 + π AP π A + π p is equivalen o 0 + d H p p 0 + d H Simplifiaion of his equaion and subsiuing p = J yields p J J Case ib: If p + p, hen inequaliy π 0 + π AP π A + π p is equivalen o 0 + d H p p 0 + d H, simplifying hih yields p J J ii p + p If, e have π0 + πap πa + πp equivalen o 0 + d H p p 0 + d H Simplifiaion of his equaion and subsiuing p = J yields p J J To summarize he resul above, e have obained he resul in par proved: If and p J min J, } J, demand learning and imperfe prefer- ene learning are omplemens If pj pj+ J, e have p + p p If e furher have > folloing subases: a If p, e have π 0 + π AP π A + π p, e need o disuss he + equivalen o d H d H p p 0 + d H Simplifiaion of his equaion and subsiuing p = J [ ] yields p + J Noe ha in his ase, e also have he ondiion p p b If p + p, e have π 0 + π AP π A + π p equivalen o d H + d H p p d H p p + d H Simplifying his inequaliy yields p + We laim ha his inequaliy alays holds for he folloing reasons Sine p, i is suffiien o sho ha, and + + +, hih an be simplified o Bu +, e indeed have Hene, if p p, e alays have π 0 + π AP π A + π p Combining ase a and b, e obain he resul Proof of Proposiion 5 Suppose K AP 0 K A0 + K P 0, I 0 < p and K I 0 A K A Noe and K AP 0 K A0 K P 0 0 From Proposiion, and Proposiions A-3-A-5 in Online Appendix 4, e have he folloing resul: If p J AP 0 and K AP > K AP, p, hen he omplemenary ondiion π 0 +π AP π A +π P is equivalen o d H J AP 0 p K AP 0 K A0 K P 0, hih alays holds If p < J AP 0 and K AP > K AP, hen π 0 + π AP

7 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 7 π A + π P if and only if d H [ J AP 0 p ] K AP 0 K A0 K P 0, hih alays holds due o p < J AP 0 3 If p J AP 0 and K AP K AP, p, hen π 0 + π AP π A + π P if and only if d H p K AP J AP 0 K AP 0 K A0 K P 0, hih alays holds due o K AP K AP, p and K AP, p = d H p J AP 0 [ p J AP 0 ] 4 If p < J AP 0 and K AP K AP, hen π 0 +π AP π A +π P if and only if d H p K AP J AP 0 K AP 0 K A0 K P 0, hih alays holds This is beause ih p < J AP 0, e have p > J AP 0 Wih K AP K AP, K AP = d H, e furher have d H p > K AP J AP 0 and hus d H p > K AP J AP 0 From -4, e kno ha demand learning and preferene learning are omplemens ih K AP 0 K A0 + K P 0, and K I 0 A > K A Beause K A = d H I 0, e kno ha K A > K A is equivalen o K A I 0 d H Noe ha I 0 0 is equivalen o I 0 Le = min, K A I 0 I 0 d H } and he desired resul follos Proof of Lemma 3 We solve for sub-game perfe equilibria by analyzing he firm s deision in he seond period firs Beause r s, he firm ill never hoose an iniial invenory level higher han he firsperiod sales in high demand ase Therefore, he firm ill have remaining invenory q in he seond period only if he demand is lo Given p, a usomer loaed a θ buys if r p θ 0 and θ θ Solving r p θ = 0, e have θ p = r p Then he period demand is d L θ p θ + = d L r p θ + And he firm s period profi maximizaion problem is max π p = min p r θ } r p d L θ, q p + s [ ] + r p q d L θ Solving his opimizaion problem, e have: if r θ > s, ie, θ < r s/, hen r θ +s if q r s θ dl, p q, θ = r θ q d L if q < r s θ dl, A-3 hereas if r θ s, ie, θ r s/, he firm does no an o sell in he seond period and all q unis are salvaged a s Subsiuing he opimal p bak, e have he opimal period- profi sq if θ r s π d q, θ = L r θ s + sq if r s q d L θ r s, q r θ q d L if θ < r s q d L A-4 We no onsider he firs period An individual onsumer does no kno he demand realizaion The uiliy of purhasing in period is p θ If θ r s/, hen he firm does no sell

8 Huang e al: The Value of Bespoke 8 Arile submied o ; manusrip no in he seond period, and θ p, p = p If θ < r s/, he expeed uiliy of purhasing in period is r p θ here p is onsumers raional expeaion of he period prie hen he demand is lo Though he opimal p depends on θ, for any individual onsumer, p does no depend on her on deision Solving p θ = r p θ, e have θ p, p = p r p Therefore, usomers opimal purhasing deision in he firs period is θ p, p = p r p p The firs-period demand is Dθ p, p if θ < r s/, if θ r s/ A-5 We nex analyze he firm s soking/invenory deision q in period Noe ha θ p, p does no hange ih respe o he firm s iniial invenory deision Hene, aking p and θ as given, he firm should hoose q o maximize is expeed profi Beause p is alays higher han, given θ, he opimal invenory level should be higher han θ d L In addiion, as r s, he opimal invenory level should never exeed θ d H When he demand is lo, q = q θ d L So q r s follos: θ dl is equivalen o q r s + θ dl The firm s iniial invenory deision is as max p q + p θ d L + π q θ d L, θ q, θ d L q θ d H here π q, θ is given in equaion A-4 If θ r s, he firm s opimal invenory problem is max p q + p θ d L q + sq θ d L θ d L q θ d H The firs order derivaive is π q = p + s Hene, he opimal iniial quaniy is q = θ d L if p < s, and q = θ d H if p s In his ase, he firm does no sell in he seond period, and θ = p Subsiuing θ = p In his equilibrium, if p < s, and if p s, π p = p p π p = p p d H + p p bak o θ r s, e have p r + s p d L + p d L, If θ < r s, he firm s opimal invenory/soking problem is d L + s p d H d L max π q; p, θ, θ d L q θ d H A-6

9 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 9 here p q + if θ d L q < r s + θ dl, π q; p, θ = p q + p θ d L + d Lr θ s if r s + θ dl q θ d H p θ d L + q θ d L r θ q θ d L d L q q + sq θ d L When θ d L q < r s + θ dl, e have π q = p + A-7 r + θ q d L, π q; p, θ = /d L < 0, and he soluion o he firs-order ondiion π q; p, θ = 0 is q = r + θ dl + d L p When r s hen q r s + θ dl q θ d H, π q = p + s If p < θ r + + θ dl and π q 0 for r s < s, + θ dl q θ d H In addiion, e an verify ha q < d L θ Therefore, he opimal iniial invenory is q = d L θ and he firm does no sell in period Then, θ = p Hene, θ < r s is equivalen o p > r + s The firm s profi is π p Subsiuing θ bak o p < θ r +, e have p < + r If θ r + p < s, hen q r s + θ dl and π q 0 for r s + θ dl q θ d H In addiion, e an verify ha q > d L θ In his ase, he opimal invenory is q = q Then q = r θ dl + d L p and p = r θ q d L = r θ p Subsiuing p = r θ p ino equaion A-5 and solving for θ, e have θ = r +p + Then + θ < r s is equivalen o p > + r+s+ +r+, and q = d + p + L Subsiuing + θ and q bak o equaion A-7, e have he firm s profi as π 3 p Subsiuing θ bak o p θ r +, e have p + r If p + s > 0, ie, p > s, hen q > r s + θ dl and π q > 0 for r s + θ dl q θ d H In his ase, he opimal invenory is q = θ d H Then q = θ d H d L and p = r θ +s Subsiuing p = r θ +s ino equaion A-5 and solve for θ e have θ = r p + s + r p Then q = d + s H Subsiuing θ + = r p + s + bak o θ < r s, e have p > r + s Subsiuing θ and q bak ino equaion A-7, e have he firm s profi as π 4 p Comparing he hresholds of p, e have o ases depending on he primiive parameers: Case : When s r + s r + s, hen r+++s s + If p < s, hen beause p < s r ++s equilibrium θ r s Therefore, θ = p and q = p d L If s p < r + s, hen beause s Therefore, θ = p and q = p d H, and + r and p < s r + s, in r +s+, in equilibrium θ r s

10 Huang e al: The Value of Bespoke 0 Arile submied o ; manusrip no If p r + s, hen beause p > s, in equilibrium θ r s Therefore, θ = r p + s + and q = d H r p + s + To summarize, in his ase, he firm s profi as a funion of p is π p if p < s, π p = π p if s p < r + s, π 4 p if p r + s A-8 The profi π 4 p an be obained by subsiuing q = d H r p + s + and θ = r p + s + bak ino equaion A-7 he par ih r s + θ dl q θ d H Case : When s > r+s, hen + r > r+s, + r > r+++s + and + r < s If p < r + s, hen beause p < r + s < s θ = p and q = p d L, in equilibrium θ r s Therefore, If r + s < p < + r, hen in equilibrium θ < r s Therefore, θ = p q = p d L If + r p < s, hen beause p + r > r++s+, in + equilibrium θ < r s Therefore, θ = r +p + + and and q = d L +r+ + p + If p s, hen p > r + s, and hus in equilibrium θ r s Therefore, θ = r p + s + and q = d H r p + s + To summarize, in his ase, he firm s profi as a funion of p is π p if p < + r, π p = π 3 p if + r p < s π 4 p if p s, A-9 The profi π 3 p an be obained by subsiuing q = d L +r+ + p + and θ = r +p + + bak ino equaion A-7 he par ih θ d L q < r s + θ dl Proof of Lemma 4 In he preferene learning sysem, all usomers obain heir ideal produs sine here is no preferene mismah Thus, in he seond period, he firm s opimal prie is p = r and usomers buying in he seond period ill ge zero uiliy Therefore, he firm s opimal prie in he firs period is p = and all usomers an o buy in period The only remaining deision is he iniial invenory level I is lear ha he opimal invenory level is beeen d L and d H The firm s expeed profi is max dl q d H π q = q + d L + s q d L p q Then, π q = + s p Hene, he resuls follo Proof of Proposiion 6 Wih d L = 0, e an use Proposiion for he radiional sysem as: i if, he π 0 = 0; ii if >, he π 0 = d H Wih d L = 0, e an use Proposiion 4

11 Huang e al: The Value of Bespoke Arile submied o ; manusrip no for he Bespoke sysem as π AP = d H p Noe + J is equivalen o J We disuss he omplemenary ondiion in elve differen parameer ranges: Suppose J, I [0, I A, p ] and < Then, by Proposiions A-6 and A-7 in Online Appendix 4 and he profi expressions for he radiional and he Bespoke sysems lised above, he suffiien and neessary ondiion π 0 + π AP π A + π p an be rien as 0 + d H p d H J + 0, hih an be simplified as p Suppose J, I [0, I J A, p ] and < p Then, π 0 + π AP π A + π p if and only if p [ J I [0, I B, p ] and p Then, π 0 + π AP π A + π p if and only if p ] 3 Suppose J, [ J ] 4 Suppose J, I [I A, p, ] and < Then, π 0 + πap πa + πp if and only if J π A + π p I 5 Suppose J if and only if J and p Then, π 0 + π AP π A + π p + I, I [I A, p, ] and < p Then, π 0 + π AP 6 Suppose J, I [I B, p, ] if and only if J 7 Suppose + I J >, I [0, I A, p ] and < Then, π 0 + πap πa + πp if and only if p + J 8 Suppose J >, I [0, I A, p ] and < p Then, π 0 + πap πa + πp if and only if p + π 0 +π AP π A +π p if and only if p J + 9 Suppose J >, I [0, I B, p ] and p Then, [ J ] 0 Suppose J >, I [I A, p, ] and < Then, π 0 + π AP π A + π p if and only if J I Suppose J >, I [I A, p, ] and < p Then, π 0 + π AP π A + π p if and only if J I Suppose J >, I [I B, p, ] and p Then, π0 + πap πa + πp if and only if J I No e invesigae he pariular ase saed in his proposiion Suppose I [0, I B, p ] and J [0, ] From laim 3 analyzed above, he ondiions are [ J ] p < Nex, e prove ha here exiss Ĵ > 0 suh ha hen J Ĵ, p ] alays holds Noe ha [ J ] p > [ ] is equivalen o < p p >, < and +, e kno ha p > J=0 [ [ J = [ ] Beause + + hih is alays rue due o J ] alays holds Beause [ J ] is oninuous and inreasing a J, e an alays find suh an Furhermore, by solving p [ J=0 J ], e have Ĵ = p + Ĵ > 0 Suppose I [0, mini A, p, I B, p }] and J [0, min, p }] Then + from laims -3 of above, demand learning and preferene learning are omplemens if and only if one of he hree ondiions holds: and p J, < p p [ J ], and 3 p < Proposiion 6 follos by ombining ondiions and 3 and

12 Huang e al: The Value of Bespoke Arile submied o ; manusrip no Online Supplemen Supplemenal Maerials for 5 In his seion, e provide supplemenal maerials for 5: Exensions o he Basi Model Online Supplemen Analysis for 5 Imperfe Preferene Learning We firs analyze he preferene learning sysem For analyial onveniene, e also normalize d L o zero in his seion We ill abuse noaion by riing p = p J given ha J is fixed The proposiion belo provides he firm s opimal expeed profi for an imperfe preferene learning sysem J: Proposiion A- In an imperfe preferene learning sysem ih he level of usomizaion i If [ p + J, +, he firm opimal expeed profi π pj, = [ J p]d H ii If [ p, p + J], he firm opimal expeed profi π pj, = iii If [0, p ], he firm opimal expeed profi π pj, = 0 d H J p Proof of Proposiion A- To find he onsumers ho are indifferen o purhasing he produ versus no purhasing loaed a θp for a given prie p, e le uθ, p p θ p = 0 We have θp = p p If p [ p, ], e have θp [0, ] so ha he number of onsumers ho purhase he produ is D p p = pd p If p [0, p], all he onsumers ill purhase he produ, ie, D p p = D If p >, D p p = 0 The firm hooses is prie p and produion quaniy q o maximize he expeed profi πp, q = pe mind p p, q p q p mind H, q p q, if p [0, p] πp, q = p min pd H p, q p q, if p [ p, ] p q, if p [, + Le us disuss differen ases depending on he range of prie p For p [0, p], he opimal prie has o be p sine πp, q srily inreases in p Then, he opimal quaniy q = d H if p p and q = 0 if p < p Hene, in his region, he maximum expeed profi πp, q = [ p p ]d H p < p if p p and πp, q = 0 if For p [ p, ], e have πp, q = p min pd H p, q p q p p q and q pd H p Therefore, πp, q = p p pd H p Firs-order ondiion ih respe o prie p yields p = + p Then, e obain he orresponding quaniy q = d H p p Finally, e need o make sure ha he prie is ihin his region, ie, p = + p p, hih is simplified o p p + p Under his ondiion, he maximized expeed profi πp, q = d H p p If his ondiion does no hold, e disuss o subases: If > p + p, hen he opimal prie should

13 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 3 be on he boundary, p = p and he opimal quaniy q = d H The opimal expeed profi is πp, q = [ p p ]d H ; If < p, hen he opimal prie p =, opimal quaniy q = 0, and expeed profi is zero sine no onsumers purhase a his prie To sum, he opimal expeed profi is [ p p ]d H, if [ p + p, + π d p = H p p, if [ p, p + p] 0, if [0, p ] Sine p = J, e an express he profi funion in erms of J Nex, e analyze he Bespoke sysem here preferene learning is imperfe ih he level of usomizaion J and demand learning is perfe Proposiion A- In a Bespoke sysem of imperfe preferene learning ih he level of usomizaion J and perfe demand learning: i If p + J, he firm opimal expeed profi π AP J, = d H J p ii If p + J, he firm opimal expeed profi π AP J, = d H J p Proof of Proposiion A- In his sysem, he firm an observe he demand realizaion D before making is quaniy deision Hene, he firm an perfely mah supply ih demand Suppose D = d H, hen he number of onsumers ho ill purhase he produ is d H, if p [0, p] pd D AP d H = H p, if p [ p, ] 0, if p [, + We hen disuss under eah of he hree prie ranges, ha he opimal prie should be If p [0, p], he firm profi is π dh p, q = p p d H The opimal prie p d H = p and opimal quaniy q d H = d H if p p Hene, π d H = p p d H if p + p If p [ p, ], hen q d H = pd H p The firm profi π dh p, q = p p pd H p Firs-order ondiion ih respe o prie p yields p p p = 0 Hene, p = +p For i o be opimal, e require p = +p [ p, ], hih is simplified o [ p, p + p ] Hene, under his ondiion, e have p d H = +p, q d H = d H p, and he opimal profi πd H = d H p p If his ondiion p does no hold, p + p, e have p d H = p, q d H = d H, and he opimal profi π d H = p p d H The opimal expeed profi is πap = πd H Hene, e obain dh π AP = p p, if p + p d H p p, if p + p Sine p = J, e an express he profi funion in erms of J

14 Huang e al: The Value of Bespoke 4 Arile submied o ; manusrip no Online Supplemen Analysis for 53: Invesmen Coss and Endogenous Learning Levels In his seion, e provide he deailed analysis for he hree sysems in he linear-os seing The resul for radiional sysem remains he same as in he basi model Proposiions A-3-A-5 belo summarize he resuls for demand learning, preferene learning and Bespoke sysems, respeively We firs inrodue some noaions for exposiional onveniene Le K A K A d H I 0 [ I 0 ][ [ I 0 ][ I 0 ] I 0 [ I 0 ] }, and K A3 I 0 ] } Proposiion A-3 In a demand learning sysem ih linear oss, e have: d H I 0, d H I 0 i The opimal learning level I = I 0, if and only if one of he folloing hree ondiions is saisfied: and K I 0 A > K A, < I 0 and K I 0 A > K A, 3 < and K I 0 A > K A3 The orresponding opimal profis are πa = K A0, π A = [ I 0 ]d H [ I 0 ] K A0 I 0 d H [ I 0 ] K A0 and π A = [ I 0 ]d H [ I 0 ] + ii Oherise, he opimal learning level I =, and π A = d H K A I 0 K A0 Proof of Proposiion A-3 We prove he resul in o ases and > Case : When, from Proposiion 5 and CAI = K A0 + K A I I 0, e have: π π A I = A I K A0 K A I I 0 if I [I 0, I ] πai [ I ]d H [ I ] K A0 K A I I 0 if I [maxi 0, I }, ], here I = Beause dπ A I = K di A, d πa I = 0, dπ di A I = d H [ ] di [ I ] K A, and d πa I > 0, e kno ha he opimal learning level I is eiher I di 0 or Noe ha I 0 I is equivalen o For For For I 0 and ha I 0 < πai = I 0 πai if I [I 0, I ] πai if I [I, ] <, π I 0 AI = πai for I [I 0, ] alays holds We have he folloing: I 0, by omparing π AI 0 = K A0 and π A = d H K A0 K A I 0, e kno ha if K A > K A, hen I = I 0 and πa = K A0 ; oherise, I =, and πa = d H K A I 0 K A0 For I 0 <, by omparing π AI 0 = [ I 0 ]d H [ I 0 ] K A0 and π A = d H K A0 K A I 0, e kno ha if K A > K A, hen I = I 0 and π A = [ I 0 ]d H [ I 0 ] K A0 ; oherise, I =, and πa = d H K A0 K A I 0

15 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 5 Case : When >, from Proposiion 5 and CAI = K A0 + K A I I 0, e have: π 3 A I [ I ]d H [ I π AI = ] + I d H [ I ] K A0 K AI I 0 if I [I 0, I ] πa 4 I [ I ]d H [ I ] K A0 K AI I 0 if I [maxi 0, I }, ], here I = Noe ha π4 AI = πai in ase Beause dπ3 A I = d H [ di [ I] ] K [ I ] A, d πa 3 I > 0, dπ4 di A I = d H [ ] K di [ I ] A, and d πa 4 I > 0, e di kno ha he opimal learning level I is eiher I 0 or Noe ha I 0 I is equivalen o I 0 and ha For < For I 0 > alays holds We have:, π I 0 AI = πai 4 for I [I 0, ] πai = I 0 πai 3 if I [I 0, I ] πai 4 if I [I, ] For < I 0, by omparing π AI 0 = [ I 0 ]d H [ I 0 ] K A0 π A = d H K A0 K A I 0, e kno ha if K A > K A, hen I = I 0 and π A = [ I 0 ]d H [ I 0 ] K A0 ; oherise, I =, and πa = d H K A0 For and K A I 0 I 0 <, by omparing π AI 0 = [ I 0 ]d H [ I 0 ] + I 0 d H [ I 0 ] K A0 and π A = d H K A0 K A I 0, e kno ha if K A > K A3, hen I = I 0 and π A = [ I 0 ]d H [ I 0 ] + I 0 d H [ I 0 ] K A0 ; oherise, I =, and πa = d H K A I 0 K A0 Proposiion A-3 shos ha for any given, he opimal demand learning level is high hen he invesmen os K A is small, hih is onsisen ih our inuiion ha he opimal learning auray level is higher hen learning is less expensive We also observe ha demand learning level is high hen is inermediae When is small, he average demand d H is small, hih implies a high learning os per uni When is large, he demand unerainy is lo he oeffiien of variaion is lose o zero and hus he firm has less inenive o learn he demand Le K P, p d H J P 0 [ p p J P 0 ] and K P d H In preferene learning and Bespoke sysems, e may abuse noaion by riing J p = J and J AP = J henever no onfusion arises Proposiion A-4 In a preferene learning sysem ih linear oss, e have: i The opimal learning level J = J P 0, if and only if one of he folloing hree ondiions is saisfied: p, p < p and K J P 0 P > K P, p, 3 > p and K J P 0 P > K P The orresponding opimal profis are π P = K P 0, π P = π P = [ J P 0] p }d H K P 0 d H J P 0 p K P 0 ii Oherise, he opimal learning level J =, and π P = p d H K P 0 K P J P 0 and

16 Huang e al: The Value of Bespoke 6 Arile submied o ; manusrip no Proof of Proposiion A-4 From Proposiion A-, e have πp J K P 0 K P J J P 0 if < p π P J = πp J d H J p K P 0 K P J J P 0 if p p + J πp 3 J [ J] p}d H K P 0 K P J J P 0 if > p + J Beause dπ P J dj = K P, d π P J dj = 0, dπ P J dj = d H p J K P, d π P J dj = d H p J 3 > 0, dπ3 P J dj = d H K P, and d πp 3 J = 0, he opimal learning level J is eiher J dj P 0 or due o d πp i J 0 dj i =,, 3 Noe ha p o + J is equivalen o J p p J P 0 We have he folloing: If p, hen π P J = π P J for J P 0 J If p < If > p J P 0, hen π P J = πp J πp 3 J p J P 0, hen π P J = π 3 P J for J P 0 J By omparing π P J P 0 and π P in p, p < desired resuls if J P 0 J p if p < J and ha p J P 0 is equivalen p and > p J P 0 J P 0, e have he Nex, e presen he resul for a Bespoke sysem ih linear oss Le K AP, p d H p J AP 0 [ p J AP 0 ] and K AP d H Proposiion A-5 In a Bespoke sysem ih linear oss, e have: i The opimal learning level J = J AP 0, if and only if one of he folloing o ondiions is saisfied: p J AP 0 and K AP > K AP, p, p < J AP 0 and K AP > K AP The orresponding opimal profis are π AP = J AP 0 p ] K AP 0 d H J AP 0 p K AP 0 and π AP = d H [ ii Oherise, he opimal learning level J =, and π AP = d H p K AP 0 K AP J AP 0 Proof of Proposiion A-5 From Proposiion A-, e have π π AP J = AP J d H J p K AP 0 K AP J J AP 0 if p + J πap J d H [ J p] K AP 0 K AP J J AP 0 if > p + J Sine dπ AP J dj d π AP J dj = d H p J K AP, d π AP J dj = d H p J 3 > 0, dπ AP J dj = d H K AP, and = 0, he opimal learning level J is eiher J AP 0 or due o d π i AP J dj 0 i =, Noe ha p + J is equivalen o J p o p J AP 0 We have he folloing: and ha J AP 0 p is equivalen

17 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 7 If p + J AP 0, hen π AP J = πap J πap J if J AP 0 J p if p < J If > p + J AP 0, hen π AP J = π AP J for J AP 0 J By omparing π AP J AP 0 and π AP in p + J AP 0 and > p + J AP 0, e have he desired resul Proposiions A-4-A-5 sho ha he opimal preferene learning auray levels in boh preferene and Bespoke sysems end o be high hen he invesmen oss K P and K AP are small We also observe ha he firm ends o hoose a high preferene learning auray level hen he expeed demand is high ie, a large and/or d H or hen p lo average invesmen os per-uni produ A small p is small This is beause a high expeed demand implies a relaively o learn he preferene Our numerial sudy also shos ha hen K P = K AP means a high margin and hus more inenive and J P 0 = J AP 0, he firm alays hooses a higher preferene learning auray level in he Bespoke sysem ompared o he preferene learning sysem A higher J means a higher invesmen os The firm s learning os may no pay off in he preferene learning sysem hen he realized demand is less han is produion quaniy, hile he firm never faes his senario in he Bespoke sysem beause i an exaly mah supply and demand Based on he resuls for all he four sysems, e are ready o invesigae he inerrelaionship beeen demand learning and preferene learning We are able o provide all he neessary and suffiien ondiions for demand learning and preferene learning being omplemens For breviy, hey are no presened here bu available from he auhors Online Supplemen 3 Analysis for 54: Impa of Salvage Value In his seion, e provide a represenaive numerial sudy o demonsrae he impa of salvage value s in he onex of sraegi usomers and bargain-huning onsumers From our numerial analysis, e observe ha as he salvage value s inreases, demand learning and preferene learning are more likely o be omplemenary regardless ho sraegi usomers are; see Figure A- for represenaive examples Sine here is no lefover invenory in boh demand learning and Bespoke sysems ie, heir profis are invarian ih respe o s, his observaion implies ha he radiional sysem benefis more from high salvage value ompared o he preferene learning sysem The explanaion is as follos Wih a higher salvage value, he firm ends o order a higher quaniy in boh radiional and preferene learning sysems In he preferene learning sysem, he firm has reahed he highes quaniy d H hen p is small and he highes prie i an harge even ih zero salvage value, hile i does no in he radiional sysem This implies ha he firm has more flexibiliy o adjus he prie and order quaniy o beer leverage he benefi from a higher salvage value in he radiional sysem Furhermore, ih he presene of sraegi behavior in he radiional sysem, a higher prie in period resuling from a higher salvage value redues usomers aiing inenives and hus inreases he firm s profi

18 Huang e al: The Value of Bespoke 8 Arile submied o ; manusrip no Figure A- The Impa of Salvage Value d H = 60, d L = 0, = 36, = 8, = 8 Online Supplemen 4 Analysis for 55: Correlaion Beeen Demand and Preferene Learnings Proposiion A-6 belo summarizes he resul for he demand learning sysem in hih onsumer preferene is also imperfely learned We follo he frameork in Seion 5 o model he imperfe preferene learning Wihou losing managerial insighs, e normalize d L o zero for analyial raabiliy in his seion J: Proposiion A-6 In a perfe demand learning sysem ih imperfe preferene learning of auray i If + J, he firm s opimal expeed profi π A = d H [ J ] ii If < + J, he firm s opimal expeed profi π A = d H J Proof of Proposiion A-6 In his sysem, he firm an observe he demand realizaion D before making is quaniy deision Hene, he firm an perfely mah supply ih demand Suppose D = d H, hen he number of onsumers ho ill purhase he produ is d H, if p [0, p] pd D A d H = H p, if p [ p, ] 0, if p [, + We hen disuss under eah of he hree prie ranges, ha he opimal prie should be If p [0, p], he firm profi is π dh p, q = p d H The opimal prie p d H = p and opimal quaniy q d H = d H p Hene, π d H = p d H if p + If p [ p, ], hen q d H = pd H p The firm profi π dh p, q = p pd H p Firs-order ondiion ih respe o prie p yields p p = 0 Hene, p = + For i being opimal, e require p = + [ p, ], hih is simplified o [, + p ] Hene, under his ondiion, e have p d H = +, q d H = d H p, and he opimal profi π d H = d H p If his ondiion does no hold, ie, > + p, e if

19 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 9 have p d H = p, q d H = d H, and he opimal profi π d H = p d H The opimal expeed profi is π A = π d H Hene, e obain π A = dh p, if + p d H p, if + p Sine p = J, e an express he profi funion in erms of J I is expeed ha he firm s profi inreases as he preferene learning auray J inreases Proposiion A-7 belo summarizes he resul for he preferene learning sysem in hih demand is also imperfely learned We follo he frameork in Seion 5 o model he imperfe demand learning For exposiional onveniene, e inrodue he noaions: I A, p p and I B, p p Proposiion A-7 In a perfe preferene learning sysem ih imperfe demand learning of auray I: i If [0, p ] and I [I A, p, ], he firm s opimal expeed profi π p = d H [ I ] p } ii If [0, p ] and I [0, I A, p ], he firm s opimal expeed profi π p = 0 i If [ p, ] and I [I B, p, ], he firm s opimal expeed profi π p = d H [ I ] p } i If [ p, ] and I [0, I B, p ], he firm s opimal expeed profi π p = p d H Proof of Proposiion A-7 Wih perfe preferene learning, he firm also ses p = and all usomers ould like o buy a his prie The firm s expeed profi an be expressed as π p q = E[minD, q] p q Suppose ha he firm observes demand signal s = d H, hen is updaed demand Ds = d H = d H probabiliy H I I and Ds = d H = d L = 0 ih probabiliy H I = I So he profi π s=dh = H I p q When I I A, p ie, H I p 0, q s=d H = d H and π s=d H = H I p d H ; hen I < I A, p, q s=d H = 0 and π s=d H = 0 Suppose ha he firm observes demand signal s = d L, hen is updaed demand Ds = d L = d H probabiliy L I I and Ds = d L = d L = 0 ih probabiliy L I = I So he profi π s=dl = L I p q When I I B, p ie, L I p 0, q s=d L = d H and π s=d H = L I p d H ; hen I > I B, p, q s=d L = 0 and π s=d L = 0 To rie he firm s expeed profi before observing he demand signal s, e need o ompare I A, p p and I B, p p I urns ou I A, p I B, p is equivalen o p / Noe ha π pi, a = π s=d H + π s=d L Then, e an ompue his expeed profi based on differen ombinaions of he ondiions on and I For example, if [0, p ], e have I A, p 0 I B, p If e furher assume I [I A, p, ], hen π pi, = d H [ I ] p } Similarly, e obain he profi funions in oher ases ih ih One may argue ha alhough he firm imperfely learns he onsumer preferene ihou spending speifi effors in he demand learning sysem, he produion os migh sill inrease above We keep he produion os unhanged as in his seion in order o isolae he impa of he learning-mehanism orrelaion

20 Huang e al: The Value of Bespoke 0 Arile submied o ; manusrip no Online Supplemen 3 Responsive Priing In he basi model, e assumed ha he firm deermines he prie before unerain demand is realized or learned This assumpion is onsisen ih he exensive operaions lieraure see, eg, Peruzzi and Dada 999 and referenes herein This fis he senarios here he firm observes he demand afer selling he produ, or he firm needs o use he prie for adverisemen before observing he demand, as i is ypially he ase for many produs Hoever, in some seings, he firm may be able o se he prie afer he demand is realized or learned In his seion, e invesigae ho responsive/oningen priing may affe our main resul in he basi model We denoe q 0 and π 0 as he firm s opimal produion quaniy and opimal expeed profi respeively For he radiional sysem, he folloing proposiion haraerizes he equilibrium ouome Proposiion A-8 In a radiional sysem ih responsive priing: i When, e have: ia For, d L dh q 0 = and d + π 0 = ; H d + L d H d L ib For >, d L dh q 0 = dh and π 0 = d H + d L ; i For any given demand realizaion D and produion quaniy q, he opimal prie pq, D = and pq, D = q D oherise ii When >, le + d L iia For min,, q 0 = d H + d L d + H, + d L d H, and e have: and π 0 = d L iib If <, hen for < <, q0 = d L and π0 = d L d L d H + d L ; ii If >, hen for < <, π 0 = max orresponding q 0 = d + H d L iid For > max,, q 0 = d H or q 0 = d H ; ; + d H d L d + H d L D if q, d H + d L, and he and π 0 = d H + d L; iie For any given demand realizaion D and produion quaniy q, he opimal prie pq, D = if q D, and pq, D = q D oherise Proof of Proposiion A-8 We ork bakards: firs solve for he opimal prie pq, D given produion quaniy q and demand realizaion D, and hen solve for q 0 Clearly, for he opimal prie p e have p, and he demand is p D The firm profi an be expressed as πp; q, D = p minq, p D} q When p q p, e have q D and πp; q, D = p q In his ase, πp; q, D is maximized a D pq, D = q D q q and he resuling profi is πq, D = q When p > D D and πp; q, D = D p p q In his ase, πp; q, D is maximized a pq, D = is πq, D = D q D p, e have q > D and he resuling profi q Noe ha q is equivalen o D q, is equivalen o, D is equivalen o D q, q > q is equivalen o >, q D < and < Nex, e disuss pq, D in differen regions When >, e have q > q a For D < q, e have q < < q < Beause <, e kno ha for q p, e have p > and D D D hus πp; q, D = D p p q The opimal prie pq, D = and he resuling profi πq, D = D b For q D q, e have < q D < < Combining he opimal prie pq, D for p q

21 Huang e al: The Value of Bespoke Arile submied o ; manusrip no q q and for p, e have ha for he hole range p, he opimal prie pq, D = D D and he resuling profi πq, D = D he opimal prie pq, D for p q D for q p, he opimal prie pq, D = D q For D > q, e have < < q < Combining D q and for p, e have ha for he hole range D q and he resuling profi πq, D = q D Combining resuls a-, e have he laim i in Proposiion A-8 as q q D ih πq, D = q if q D D pq, D =, ih πq, D = D q if q > D A-0 Finally, e solve for he opimal quaniy q 0 Noe D is eiher d H ih probabiliy, or d L ih probabiliy We disuss hree possible regions: q d L, d L < q d H and q > d H Based on equaion A-0, e ompue π q = E[πq, D] by aking he expeaion of πq, D over D: π q = q q d H + d L if q d L, π q = π q = q d H q + d L q if d L < q d H, π 3 q = µ D q if q > d H Noie ha π q is onave and maximized a q = d H, and π 3 q dereases in q Beause equivalen o >, and dh d L dh d + H d L d + H d L, π q is onave and maximized a q = d L is equivalen o d L, dh dh > d L d H is alays rue, e have he folloing resul If d L dh, hen q0 = By subsiuing d + q 0 = ino π H d + q, e have π0 = If >, L d H d + L d H d d L L dh hen q0 = d H By subsiuing q 0 = d H ino π q, e have π0 = d H + d L We have finished he proof for laim i in Proposiion A-8 In a similar ay, e an derive he resul for laim ii > Wih responsive priing, he profi in radiional sysem should be higher han ha in he basi model ihou responsive priing The prie influenes he number of usomers ho an o buy he produ Wih responsive priing, he firm an se he prie based on he realized demand o beer mah he number of usomers ho an o buy he produ ih he produion quaniy and hus improves is profi Oher han he radiional sysem, he opimal sraegy and profi do no hange in he oher hree sysems: Regardless of he demand realizaion, he firm alays ses p = in preferene learning and Bespoke sysems, and alays ses p = + in demand learning sysem The equilibrium resul for he demand learning sysem as saed in Proposiion A-9 belo is he same as ha in he basi model bu he proof is slighly differen due o responsive priing is + Proposiion A-9 In a demand learning sysem ih responsive priing: The firm s opimal prie p A =, opimal produion quaniy q AD = D for a demand realizaion D, and opimal profi π A = µ D Proof of Proposiion A-9 The firm deermines is prie p and produion quaniy q afer learning he realizaion of D Obviously, p, and in his ase he number of usomers ho an o buy he

22 Huang e al: The Value of Bespoke Arile submied o ; manusrip no Figure A- The Inerrelaionship of Demand Learning and Preferene Learning ih Responsive Priing d H = 60, d L = 0, = 30, = 9, = produ is p D The profi an be expressed as πp, q; D = p minq, p D} q I is easy o sho ha he firm alays ses q = p D for any p and D Then, he profi an be expressed as πp; D = p min p D, p D} p D = D pp hih is maximized a p = +, and he orresponding opimal quaniy is q AD = opimal profi given D is D and he expeed opimal profi is µ D + D = D The We expe ha demand learning and preferene learning are more likely o be omplemenary Numerial resuls onfirm our inuiion, as shon in Figure A- here e use he same se of parameers as in Figure for he basi model Furhermore, onsisen ih our basi model, demand learning and preferene learning are omplemenary hen he os of usomizaion is lo and he probabiliy of having high demand is large Online Supplemen 4 Conneion o Produ Line Design The model seup follos he basi model in he main body of he paper Suppose usomers/onsumers are uniformly disribued on a Hoelling line [ /, /] There is a monopolis firm selling n horizonally differeniaed produs, hih are posiioned a loaion θ i, i,,, n} on he Hoelling line, ih orresponding pries p i, i,,, n} Wihou loss of generaliy, e assume ha θ θ θ n Consumers ge uiliy from he produ, and inur he ravelling os and pay he prie The per uni ravelling os is denoed as If a onsumer a loaion θ buys produ i, her ne uiliy is θ θ i p i A onsumer buys he produ ha gives her he highes non-negaive ne uiliy Consumers may subsiue for anoher produ only if he produ also gives her a non-negaive ne uiliy if her favorable produ is ou of sok The firm s per uni produion os is For any given number of produs n, he firm needs o deermine boh pries and soking levels for all he produs

23 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 3 Lemma A- In any opimal sraegy, all produs are sold a he same prie, no onsumer ges srily posiive uiliy from muliple produs, and onsumers on he boundaries -/ and / ge non-posiive uiliy from purhasing Proof of Lemma A- Define he marke overage of a produ as he onsumers ho ge non-negaive uiliies from purhasing his produ Clearly, he marke overage of any produ is oniguous in ha if usomers a boh loaions buy produ i, hen any usomer loaed beeen hese o loaions buys produ i Firs, here is no produ overage overlapping in any opimal sraegy Oherise, here are some usomers ho ge posiive uiliies from muliple produs Then he firm an inrease he produ prie or shif he produ loaion o inrease profi To see his, suppose produ i and j have overlapping overage, hen if one produ ihou loss of generaliy, assuming i is i only overlaps on one side ih he oher one, hen e an inrease p i and shif he posiion of produ i aay from produ j o inrease he profi If boh produs has overage overlapping on boh sides, hen e an repea he argumen and sho ha all produs have overlapping on boh sides, and usomers a boh ends ge srily posiive uiliy from purhasing Then, e an inrease all pries by he same amoun ihou hanging demands for all produs, hih ill inrease he oal profi Therefore, in any opimal sraegy, here should be no produ overage overlapping Seond, a he opimal sraegy, usomers a boh ends -/ and / ge non-posiive uiliy from purhasing Suppose no, hen a usomer a one end ihou loss of generaliy, assuming loaion 0 a srily posiive uiliy from purhasing produ produ is loses o loaion 0, hen he firm an inrease p o p + ɛ here ɛ is an suffiienly small amoun and inrease θ o θ + ɛ Then, he demands are he same and he profi is higher Given here is no overage overlapping and usomers a boh ends ge non-posiive uiliy from purhasing, L i define as L i = p i / is he overage of produ i and DL i is he demand for produ i Then, he invenory problem for produ i is max q i min d H L i, q i } p i + min d L L i, q i } p i q i Solving his nesvendor problem e have he opimal invenory d H L i if p i q i = d L L i if p i < Subsiuing q i bak, e have he opimal profi d H + d L p i p p i d i H if p i π i p i = p d i L p i if p i < Third, all produs mus have he same prie Suppose no, hen here exis o adjaen produs ih differen pries Tha is, here exiss a produ i, suh ha p i p i+ Wihou loss of generaliy, assume p i > p i+ Then L i < L i+ and he oal overage is L i + L i+ = p i p i+ / As long as e keep he

24 Huang e al: The Value of Bespoke 4 Arile submied o ; manusrip no oal prie p i + p i+ he same, e an keep he same oal overage by he o produs L i + L i+ e may need o shif he loaions and do no affe oher produs We an verify ha π i p i is onave in he o domains [, and [/,, so if p i > p i+ / or p i+ < p i < /, e an replae p i and p i+ ih heir average p i + p i+ / and reloae he o produs so ha he oal overage remains he same and e ge a higher profi If p i > / and p i+ < /, hen π p i+ = d L + p i+, and π p i = µ D sho ha π p i+ = π p i 0 an never be he ase if p i > / and p i+ < / Solving π p i+ = π p i gives us p i+ = d H d L d L + µ D d L p i p i + d H We Simplifying π p i 0, e have p i µ D+d H µ D Then p i+ = d H d L d L + µ D d L p i d H d L d L + µ D +d H d L = Therefore, π p i+ = d L + p i+ d L + = 4d L > 0, hih onra- dis ih π p i+ 0 Hene, π p i+ = π p i 0 anno be he ase given p i > / and p i+ < / Then, here remain o possibiliies: π p i+ π p i or π p i+ = π p i > 0 We nex onsider hese o ases one by one Case : π p i+ π p i If π p i+ > π p i, e an inrease p i+ by ɛ and redue p i by ɛ o inrease he profis from he o produs ihou affeing oher produs Similarly, if π p i+ < π p i, e an redue p i+ by ɛ and inrease p i by ɛ o inrease he profis from he o produs ihou affeing oher produs Case : π p i+ = π p i > 0 We an hen inrease boh p i and p i+ o inrease he profis from he o produs Wih he higher pries, he overage of he o produs ill be smaller, and hus on affe oher produs Therefore, in any opimal sraegy, i mus be ha all pries are equal Then all produs should have equal overage ih no overlapping and hus equal demand, and heir opimal invenory levels are he same Then e only need o deide one prie for all produs, and he oal invenory level for all produs Beause here should be no overlapping, eah produ overs a mos /n on he Hoelling line Therefore, solving p i / /n, he opimal prie mus saisfy p i /n The nex proposiion shos ha a he opimal sraegy, offering n produs is equivalen o reduing he ravelling os o /n Proposiion A-0 Given any fixed and, denoe he oal profi fp, n, as a funion of prie p, number of produs n, per-uni raveling os under he opimal produ posiioning and invenory sraegy Then, e have fp, n, = fp,, n Proof of Proposiion A-0 In he opimal sraegy, all n produs have he same prie and he same soking level Denoe he opimal oal soking level given prie p as q p Sine all produs should have he same soking level, he opimal soking level for eah produ is q p /n The demand for eah produ is D p / Then, he oal expeed profi from all produs is [ D p f p, n, = ne min, q p }] p q p n

25 Huang e al: The Value of Bespoke Arile submied o ; manusrip no 5 [ }] D p = E min, q p p q p /n Subsiuing n = and = /n e have fp,, }] D p [min n = E, q p p q p /n Clearly, fp, n, = fp,, n Mapping he number of produs n o he level of usomizaion J Proposiion A-0 allos us o esablish a one-o-one mapping of he number of produs n in his seion o he level of usomizaion/personalizaion J in he main body of he paper When he firm offers n produs, he redued ravelling os is p = /n Then he level of usomizaion is J = p / = /n In he exreme, hen n =, he level of usomizaion is zero; hereas as n goes o infiniy, he level of usomizaion approahes one Due o his equivalene, hoosing he number of produs n is he same as deermining he level of usomizaion J Therefore, he produ line design inerpreaion jusifies our sylized model of using J for preferene learning and personalizaion