Onlin Supplmnt Avanc Slling in a Supply Cain unr Uncrtain Supply an Dman. Proos o Analytical sults Proo o Lmma. Using a = minl 0 ; x g; w can rwrit () as ollows (x ; w ; x ; w ) = a +(m0 w )a +( +" x w )x Tus, i m0 w < l 0 ; tn x = m0 w is t uniqu optimal orr quantity tat maximizs ; otrwis, any quantity x ( l 0 ) incluing m0 w is optimal. Proo o Lmma. W n to consir only t cas wn w m 0 bcaus otrwis x AB = 0 rom Lmma. Lt us rst consir t cass (i) an (ii) in wic m 0 > 0. Obsrv rom () tat M is linar in w wn w m 0 l 0, an quaratic in w wit its maximum at w = m0 wn m 0 l 0 < w m 0 It is asy to sow tat M is unimoal an tat its maximum is attain at w AB = m 0 l 0 i m 0 l 0 > m0 (i.., l0 < m0 ), an at wab = m0 otrwis. In t cas (iii) wn m 0 0, or any w ; x AB = 0 rom Lmma. By substituting w AB in ac cas into x AB in Lmma, w obtain x AB By substituting (w AB ; x AB ) into () an (), w obtain t xprssions or M an ; rspctivly. Proo o Lmma 3. (a) W can rwrit (5) as ollows E ~"; ~ [ (x ; w )] = x + ( w )x g F ( x )g + x 0 () + ( w )()gf () Using t Libniz intgral rul, w obtain @E ~"; ~ [ (x ;w )] @x = x + ( w )g F ( x )g Sinc F ( x ) 0; E ~"; ~ [ (x ; w )] is unimoal an maximiz at x A (w ) = w (b) Sinc M (w ) in (6) is concav in w or any ~ ; E ~ [ M (w )] is also concav in w, nc a n uniqu w A xists. As illustrat in t blow gur, i ~ 05; arg min ~ oi w ; w w = 05; n w otrwis, arg min ~ oi w ; w w = ~ > 05 Tus, i ~ 05 8 ~ ; w A = 05; otrwis, w w A > 05 By substituting wa an xa (wa ) into (5) an (6), w obtain t rsults or EA an E A M mar. W can sow tat " 075 is su cint to guarant positiv pric in quilibrium unr avanc slling as ollows. (Tis conition is also su cint or positiv pric unr ynamic slling bcaus x AB x A ) In quilibrium, xa (wa ) = wa, so p A = + ~" min ~ ; x A (wa )g = + ~" min ~ ; wa g Dpning on t valu o wa ; tr ar tr cass (i) I wa <, tn min ~ ; wa g = wa 8 ~ ; an minp A j~" ["; "]; ~ [; ]g = " + +wa wa, tn min min ~ ; wa gj ~ [; ]g = wa, an minp A j~" ["; "]; ~ (ii) I [; ]g = " + +wa (iii) I wa >, tn min ~ ; wa g = ~ 8 ~, an minp A j~" ["; "]; ~ [; ]g = + " In (i) an (ii), minp A j~" ["; "]; ~ [; ]g > 0 i an only i " +wa +w, wr 075 bcaus w A 05. In (iii), minpa j~" ["; "]; ~ [; ]g > 0 i an only i ", wr < wa = +wa 075 Tror, wn " 075, p A > 0 or all ~" an ~
Figur Manuacturr s Ex-Post Pro t M unr Avanc Slling Proo o Torm. (a) W can rwrit E A M givn in (6) as E A M = max w w min ~ ; w F ( ~ ) = wa min ~ ; wa F ( ~ ) (8) W can also rwrit t manuacturr s x-ant pro t E B M unr rgular slling as E B M = " " B M( ~ ; ~")G(~")F ( ~ ) max w min ~ ; w F ( w ~ ); (9) wr t inquality is u to Jnsn s inquality an t proprty tat B M prsnt in Corollary is incrasing an convx in ~". Finally, obsrv tat t intgran o (9) is gratr tan or qual to tat o (8) or ~, nc E B M EA M EB M = EA M wn on o t ollowing two i conitions ar satis (i) Pr ~ = = Pr +~" ~ i = bcaus B M (~ ; ~") is linar in ~", an w tat maximizs t intgran o (9) is qual to w A = (s t gur abov), or (ii) Pr [~" = 0] = Pr +~" ~ i = bcaus E B M = EA M = 8 rom Lmma 3(i) an Corollary (i). (b) Suppos Pr +~" ~ i = Tis implis tat 05( +") 05, so E A = 6 by Lmma i 3(i). From Corollary (i), E B = E (+~") 6 in tis cas. By Jnsn s inquality, E B EA ; wr t quality ols wn Pr[~" = 0] = Wn Pr +~" ~ i <, Tabl sows tat E B E A can b itr positiv or ngativ. Proo o Torm. (a) T rsult ollows irctly rom Torm (a) an Lmma. (b) Suppos Pr +~" ~ i = Tn, rom Torm (b), E B EA. Sinc EAB E B 8; ~ ; ~", E AB E A, wr t quality ols wn EAB = E B = EA. Wn Pr +~" ~ i < ; Tabl 3 sows tat E AB E A can b itr positiv or ngativ. n Proo o Proposition. From t proo o Lmma 3, M (w ) = min ~ o w ; w w in (6) is concav in ~ or any givn w. By t wll-nown proprty o t scon-orr stocastic ominanc, E ~ u( ~ i ) E ~ u( ~ i ) or any concav unction u. Tus, E ~ [ M (w )] E ~ [ M (w )] or any givn w D n w 0 arg maxe ~ [ M (w )] an w 00 arg maxe ~ [ M (w )] Tn, E ~ [ A M ] = w w E ~ [ M (w 0 )] E ~ [ M (w 00)] E ~ [ M (w 00)] = E ~ [ A M ] T rst inquality ollows rom t
optimality o w 0 an t scon inquality ollows rom t scon-orr stocastic ominanc o ~ ovr ~ Proo o Proposition. (a) From Corollary, B M is incrasing an concav in ~ or ~ < +~" an it is constant or ~ +~". Tus, B M is non-crasing an concav in ~. By ollowing t sam argumnt as in t proo o Proposition, w can sow t sir rsult. Similarly, w can obtain t rsults or w B an xb by sowing tat wb is non-incrasing an convx in ~ an tat x B is non-crasing an concav in ~ (b) From Corollary, B M is incrasing an convx in ~" or ~" ~ an it is linarly incrasing or ~" > ~. Tus, B M is incrasing an an convx in ~". By ollowing t sam argumnt as in t proo o Proposition, w can sow t sir rsult. Similarly, w can sow tat w B is incrasing an convx in ~" an tat x B is non-crasing an concav in ~"; an obtain t sir rsults.. Computation unr Spci c Probability Distributions Tis sction prsnts t clos-orm xprssions o t quilibrium outcoms unr avanc or rgular slling, an xplains t procur to comput t quilibrium outcoms unr ynamic slling wn t spci c istributions o ~ an ~" ar us as scrib in 6. Unr t avanc slling stratgy, a A (w ) pns on t ranom yil ~ in t ollowing mannr i w ~ + r; a A (w ) = w an i r ~ w, aa (w ) = ~ Tus, w can xprss E ~ [ M (w )] in (6) as E ~ [ M (w )] = E ~ [w a A (w )] = +r w a w r + a r w r ; wr a min + r; max r; w. By noting tat a can ta on tr i rnt valus pning 8 on w, E ~ [ M (w )] can b rwrittn as w i 0 w ( + r) >< E ~ [ M (w )] = w 6r [w ( + r + p r)g] [w ( + r p r)g] i ( + r) < w < ( r) > w ( w ) i ( r) w Lmma 3 as sown tat i ( r) 05, w A = 05 I ( r) < 05, w can sow w A = 3 ( r) + 3p ( + r + r ) ( + r) + ( w 0 ) (0) T proo o t abov rsult is as ollows. Lt g (w ) w, g (w ) w 6r [w ( + r + p r)g] [w ( + r p r)g], an g 3 (w ) w ( w ) Not tat g is a cubic unction o w an its omain [ ( + r); ( r)] is contain in [ ( + r + p r); ( + r p r)]. Sinc 3 g w 3 < 0 an (+r p r) > 0; g is unimoal wit its maximum at a largr root o g (w ) w = 0; wic is qual to w 0 givn in (0). (Not tat g (w ) w = 0 is a quaratic quation o w.) Also, g 3 is concav in w wit its maximum at 05. Sinc g w w = (+r) = g w w = (+r) = > 0, 3
w A os not xist in t rst intrval o w. Suppos ( r) 05 Tn, g w w = ( r) = + ( r) 0; nc wa = 05 [ ( r); ]. Nxt, suppos ( r) < 05. Tn, g w w > 0; g w = (+r) w < 0 an g 3 w = ( r) w < 0; nc = ( r) wa = w0 is optimal. In tis cas, w A > 05 bcaus g w w = =05 6r ( r)g i > 0 By substituting w = w A into xa (w ), w can obtain t clos-orm xprssions or x A (wa ) an similarly or EA M an E A. For t rgular slling stratgy, using t x-post quilibrium outcoms givn in Corollary, w comput t x-ant quilibrium outcoms. Using Corollary, w can writ t x-ant xpct wolsal pric as E ~;~" w B = Pr(~" = ) Pr( ~ b ) + + i b r [ + ] r +Pr(~" = ) = r + 3r + ( + ) b + ( ) c g b + c ( r) ; Pr( ~ c ) + c wr b min + r; max r; + an c min + r; max r; Similarly, w can xprss otr x-ant quilibrium outcoms suc as E B M, ExB, an EB. For ac o six possibl cass o aving i rnt pairs o b or c ; w av riv t clos-orm xprssion o t quilibrium outcoms, wic ar availabl upon rqust. For t ynamic slling stratgy, w av prsnt in Lmma t clos-orm xprssions o t quilibrium outcoms in t scon-prio gam. As w sow blow, t trmination o t rtailr s quilibrium pr-boo quantity in t rst prio involvs t analysis o 39 cass. Tus, insta o ning t clos-orm xprssions or ac cas, w vis an cint procur to comput t quilibrium outcoms in t rst prio. First, consir t rtailr s problm in t rst prio. T conitions provi in Lmma can b r-writtn as ollows ~ l 0 ~m0 as ~ +~"+x trsol numbrs min + r; max r; ++x r, ~ l 0 > 0 as ~ > x ; an ~m 0 > 0 as x < +~". D n ; min + r; max r; +x, an min + r; max r; x ; an t inicator unction I(y) = i y is tru an I(y) = 0, otrwis. Tn E ~;~" [ (x ; w )] in (3) can b rwrittn as E ~;~" [ (x ; w )] = +r ( x w ) x r + r ( w ) r + Pr(~" = )I x < + + Pr(~" = )I x < 6 6 ( + x ) r + i ( x ) r +r ( x ) r + i ( x ) r, an I x < ; E~;~" [ (x ; w )] tas on +r Dpning on t valus o ;,, I x < + i rnt unctional orms o x. Not tat ; or is itr a constant or a linar unction o x ; an I x < + or I x < i rnt combination o ;,, I x < + is itr 0 or. Hnc, in ac intrval o x aving a, an I x <, E~;~" [ (x ; w )] is at most a cubic unction o x. T pr-boo orr quantity x AB (w ) tat maximizs E ~;~" [ (x ; w )] or any givn w is itr a bounary point btwn any two intrvals or an intrior point at wic t rst orr conition is satis. Tr ar potntially 39 caniats or x AB (w ) wic consist o 9 bounary points btwn two intrvals o x an 30 intrior optimal points witin any intrval o i [ ] r
x. T bounary points ar 0; +, ; (+r) (+) (at wic + r = ++x (at wic r = ++x ), (+r) ( ) (at wic + r = +x ), ), ( r) (+) ( r) ( ) (at wic r = +x ), ( + r) (at wic + r = x ), an ( r) (at wic r = x ). To n t intrior optimal points, w rst simpliy t xprssion o E ~;~" [ (x ; w )] or ac o t ollowing 30 cass = I x < = 0; so E~;~" [ (x ; w )] can av 3 i rnt xprs- (i) I x +, I x < + sions wn = + r; r or x ; (ii) I x < +, I x < + = an I x < = 0; so E~;~" [ (x ; w )] can av 9 i rnt xprssions wn = + r; r or x, an = + r; r or ++x ; (iii) I x <, I x < + = I x < = ; so E~;~" [ (x ; w )] can av 8 i rnt xprssions wn = + r; r or x, = + r; r or ++x, an = + r; r or +x (not 8 cass xist insta o 7 cass bcaus ). For ac o t abov 30 cass, w can asily obtain an intrior optimal point rom t rst orr conition (wic w omit r). By comparing E ~;~" [ (x ; w )] at ts 39 caniats, w can n t rtailr s bst rspons x AB (w ) or any givn w Nxt, consir t manuacturr s problm in t rst prio. Similar to E ~;~" [ (x ; w )], w can rwrit E ~;~" [ M (x ; w )] in () as E ~;~" [ M (x ; w )] = +r (w x ) r + r (w ) r + Pr(~" = )I x < + +r 8 ( + x ) r + i ( + ) ( x ) r + Pr(~" = )I x < +r 8 ( x ) r + ( ) ( x ) r W can cintly comput t pr-boo wolsal pric w AB tat maximizs E ~;~" M (x AB (w ); w ) as ollows. W rst comput t rtailr s pr-boo quantity x AB (w ) as a unction o w an intiy bounary points btwn any two ajacnt intrvals o w at wic x AB (w ) switcs rom on o t 39 caniat points to anotr. In ac intrval o w, E ~;~" M (x AB (w ); w ) is a continuous unction, nc its local maximum is attain at itr a bounary point or an intrior point at wic t rst orr conition is satis. By comparing local maxima, w can intiy a global optimal point w AB. 3. Invntory Holbac Consir t Staclbrg gam tat tas plac bor t supply an man uncrtaintis ar rsolv. In tis gam, t manuacturr rst sts is pr-boo wolsal pric w an tn t rtailr trmins r pr-boo orr quantity x. In aition, atr obsrving t raliz man an supply, t rtailr as an option o witoling t part o t pr-boo orr quantity tat s as rciv. W us suprscript AH to not quilibrium outcoms in tis gam. For gnral probability istributions o ~ an ~", u to t complxity o t mol, t problm is intractabl. Tus, w assum t sam istributions as in t prvious sction, an compar i 5
quilibrium outcoms in tis gam wit tos unr avanc or rgular slling. Suppos tat ~ is uniormly istribut btwn r an + r, wr r (0; ], an tat ~" = wit probability 0.5 an ~" = wit probability 0.5, wr (0; ]. Lt Q not t quantity tat t rtailr slls to t mart. Atr obsrving ~", t rtailr trmins Q ( x ) to maximiz r x-post pro t ( + ~" Q)Q w x Tus, t optimal Q AH quals +~" Clarly, t rtailr woul not pr-boo mor tan +, wic is QAH wn ~" = ; i.., x +. T x-ant xpct pro t ( o t rtailr in quation () o t bas mol is tn moi into w x + 05( + x )x + 05( x )x i x E ~" [ (x ; w )] = w x + 05( + x )x + 05( ) i x >. Tis can b intrprt as ollows. I t rtailr as rciv x rom r pr-boo orr, it is optimal or r to sll t ntir quantity tat s as rciv rom r pr-boo orr; i.., Q AH = x Otrwis, wn t man turns out to b low (i.., ~" = ), it is optimal or t rtailr to witol x (> 0) an sll only Q AH = ; an wn t man is ig (i.., ~" = ), it is optimal to sll Q AH = x ( ( ; + ]) From t abov quation or E ~" [ (x ; w )], givn t pr-boo ( wolsal pric w ; w obtain t ollowing optimal pr-boo quantity w x AH i x or quivalntly w (w ) = + w i x > or quivalntly w < Not tat + w > w i an only i w <. Compar wit x A (w ) = w or all w in our bas mol, x AH (w ) implis tat, givn t pr-boo wolsal pric w, t rtailr pr-boos mor wit t olbac option tan in t bas mol. In anticipation o t rtailr s bst rspons x AH (w ); t manuacturr trmins is prboo wolsal pric w to maximiz 8 is nxpct pro t < w min ~ o ; w E M (w ) = w a AH i w (w ) = n w min ~ o ; + w i w <. In t ollowing, w will rst analyz (Intrval ) w an (Intrval ) w < sparatly, an tn combin t rsults. W will sow tat E M (w ) is itr unimoal or bimoal in w. (Intrval ) In tis intrval, t rtailr os not witol t invntory. Tus, w can us t rsults o t bas mol wit no olbac option prsnt in t prvious sction. Sinc E M (w ) is unimoal, it is optimal or t manuacturr to st w = max05; g i ( w = maxw 0 ; g otrwis, wr w 0 is givn in (0). r) 05; an (Intrval ) In tis intrval, t rtailr witols t invntory atr obsrving raliz ~ an ~". Dpning on t ranom yil ~, a AH (w ) can b xprss as ollows i + w ~ + r; a AH (w ) = + in (6) as w an i r ~ w, aah (w ) = ~ Tus, w can xprss E ~ [ M (w )] E ~ [ M (w )] = E ~ [w a AH (w )] = +r + w w w r + w r w r ; wr w min + r; max r; + w. By noting tat w can ta on tr i rnt valus pning on w, E ~ [ M (w )] can b rwrittn as 6
8 w i 0 w + ( + r) >< w + r w ( + r + p r) E ~ [ M (w )] = w + ( + r p + i ( + r) < w < + ( r) r) > + (w ) + ( + ) i + ( r) w By ollowing t sam mto as prsnt in Sction o tis onlin supplmnt, w can sow tat E M (w ) is unimoal in tis intrval an tat t optimal wolsal pric in tis intrval is i ( r) + ; w = min + ; g; otrwis, w = minw 00 ; g, wr w 00 = + 3 ( + r) + 6p ( + r + r ) ( + )( + r) + ( + ) (Intrvals & ) Bcaus E M (w ) is unimoal in ac o Intrval an Intrval, E M (w ) is unimoal or bimoal in w across t two intrvals. Tror, a global optimum w AH can b oun by comparing t maximum valu o E M (w ) in ac o t two intrvals. By substituting w AH into x AH (w ), w can obtain x AH (w AH ) an t rsulting xpct pro ts o bot rms. Using t mto scrib abov, w av comput t quilibrium outcoms o 80 scnarios by varying rom 0 to an varying an r rom 0 to 09 wit an incrmnt o 0 W tn compar t quilibrium outcoms wit tos unr avanc or rgular slling. To illustrat, w prsnt t rsults o 7 scnarios in Tabls A an A. From our numrical xprimnts, w raw t ollowing obsrvations (a) In all 80 scnarios, E AH M EA M an xah x A. Howvr, w av obsrv bot wah w A > 0 an wah w A < 0; an bot EAH > EA an EAH < EA. (b) In all 80 scnarios, E B M EAH M. Howvr, w av obsrv bot wb w AH > 0 an w B w AH < 0; bot x B > xah an x B < xah ; an bot E B > EAH an EB < EAH. Obsrvation (a) sows tat t rtailr pr-boos a largr quantity wit t olbac option. Howvr, tis os not ncssarily bn t t rtailr bcaus t manuacturr can anticipat t rtailr s bst rspons an trmin is wolsal pric accoringly. As a rsult, it is only t manuacturr wo will always bn t rom tis aitional option. Obsrvation (b) con rms tat our rsults obtain in t bas mol continu to ol in t olbac mol t manuacturr is always bttr o unr rgular slling tan unr avanc slling, wras t rtailr may prr itr stratgy. T comparison btwn avanc slling wit t olbac option an ynamic slling is similar to t comparison btwn avanc slling witout tis option an ynamic slling. Tabl A3 summarizs t comparativ statics o tis gam. It sows tat t cts o capacity an supply uncrtainty r on t quilibrium outcoms ar t sam as tos unr avanc slling witout t olbac option. Wil man uncrtainty os not a ct t quilibrium outcoms witout t olbac option, it os a ct t quilibrium outcoms o tis gam. T cts o on t xpct pro ts o bot rms ar t sam as tos unr rgular slling man uncrtainty bn ts t manuacturr but os not always bn t t rtailr. In aition, w av obsrv tat t rtailr s pr-booing quantity is incrasing in man uncrtainty, but t 7
manuacturr s wolsal pric is not monotonic in. Tabl A. Equilibrium Outcoms unr Avanc Slling E cts o t Holbac Option. Paramtrs Equilibrium Outcom Comparisons r E AH M EA M EAH E A wah w A x AH x A 0. 0. 0. 0.000 0.000 0.000 0.000 0. 0. 0.5 0.000 0.000 0.000 0.000 0. 0. 0.9 0.000 0.000 0.000 0.000 0. 0.5 0. 0.000 0.000 0.000 0.000 0. 0.5 0.5 0.000 0.000 0.000 0.000 0. 0.5 0.9 0.000 0.000 0.000 0.000 0. 0.9 0. 0.006-0.00 0.05 0.003 0. 0.9 0.5 0.008-0.003 0.06 0.03 0. 0.9 0.9 0.00-0.00 0.068 0.05 0. 0. 0. 0.000 0.000 0.000 0.000 0. 0. 0.5 0.000 0.000 0.000 0.000 0. 0. 0.9 0.000 0.000 0.000 0.000 0. 0.5 0. 0.05 0.035-0. 0. 0. 0.5 0.5 0.006 0.06-0.093 0.08 0. 0.5 0.9 0.003 0.03-0.5 0.09 0. 0.9 0. 0.090 0.03 0.06 0.36 0. 0.9 0.5 0.070 0.03 0.057 0.3 0. 0.9 0.9 0.060 0.05 0.03 0.5 0.7 0. 0. 0.000 0.000 0.000 0.000 0.7 0. 0.5 0.000 0.000 0.000 0.000 0.7 0. 0.9 0.000 0.000 0.000 0.000 0.7 0.5 0. 0.06 0.039-0.5 0.5 0.7 0.5 0.5 0.05 0.037-0.0 0.0 0.7 0.5 0.9 0.009 0.036-0.0 0.08 0.7 0.9 0. 0.0 0.05-0.05 0.5 0.7 0.9 0.5 0.096 0.037 0.006 0.9 0.7 0.9 0.9 0.078 0.037 0.000 0.88 Tabl A. Equilibrium Outcoms gular Slling vs. Avanc Slling wit t Holbac Option Paramtrs Equilibrium Outcom Comparisons r E B M EAH M E B EAH Ew B w AH Ex B x AH 0. 0. 0. 0.00-0.00 0.00-0.005 0. 0. 0.5 0.006-0.003 0.09-0.0 0. 0. 0.9 0.008-0.003 0.085-0.03 0. 0.5 0. 0.00-0.00 0.00-0.005 0. 0.5 0.5 0.006-0.00 0.05-0.06 0. 0.5 0.9 0.008-0.005 0.097-0.09 0. 0.9 0. 0.00-0.00 0.033-0.05 0. 0.9 0.5 0.00-0.005 0.063-0.075 0. 0.9 0.9 0.006-0.009 0.093-0.05 0. 0. 0. 0.00 0.00 0.000 0.000 0. 0. 0.5 0.00 0.00-0.05 0.008 0. 0. 0.9 0.007 0.003 0.05-0.008 0. 0.5 0. 0.06-0.00 0.6-0.5 0. 0.5 0.5 0.0-0.06 0.08-0.089 0. 0.5 0.9 0.0-0.05 0.5-0.0 0. 0.9 0. 0.005 0.005 0.0-0.73 0. 0.9 0.5 0.05 0.007 0.0-0.67 0. 0.9 0.9 0.07-0.00 0.06-0.96 0.7 0. 0. 0.00 0.00 0.000 0.000 0.7 0. 0.5 0.00 0.00 0.000 0.000 0.7 0. 0.9 0.00 0.003 0.00-0.00 0.7 0.5 0. 0.06-0.03 0.5-0.5 0.7 0.5 0.5 0.06-0.0 0.0-0.0 0.7 0.5 0.9 0.0-0.03 0.3-0.5 0.7 0.9 0. 0.00-0.00 0.05-0.5 0.7 0.9 0.5 0.00 0.008 0.005-0.00 0.7 0.9 0.9 0.0-0.00 0.0-0.09 Tabl A3. Comparativ Statics o t Gam unr Avanc Slling wit Holbac " # " r # " # x AH 56 0 3 88 35 07 0 309 w AH 0 56 3 35 88 07 39 8 309 E AH 56 0 3 59 35 07 5 96 309 E AH M 56 0 3 0 509 07 0 309 8