i Online Aendix for When Conumer Antiiate Liquidation Sale: Managing Oeration under Finanial Ditre JR Birge, RP Parker, MX Wu, SA Yang Aendix D: Proof of Lemma C1 Proof of Sulemental Reult in Aendix C We onider two enario: when utomer urhae in the firt eriod and when utomer wait in the firt eriod The enario where utomer urhae in the firt eriod i exatly the ame a in the inventory-learane ae Therefore, we fou on the ae where trategi utomer wait Under the objetive of revenue maximization, the eond-eriod rie i when the firm doe not go bankrut In bankruty, with waiting trategi utomer, if the firm et the rie at, it revenue i minq 1 αd, αd, and if it et the rie at b, it revenue i b[q 1 αd] + If q 1 αd < αd, etting the rie at i learly better Otherwie, etting the rie at i better if and only if αd > bq 1 αd, ie D < Proof of Lemma C2 51 and i omitted here bq α+1 αb The rofit funtion in the buy-region, ie when, q Ω B, i diued in Setion Within the wait-region, ie q > Q b, the firm effetively fae firt-eriod demand 1 αd a only myoi utomer make urhae We onider two further enario 1 When [1 α ]q, that i, q d W, the firm rofit deend on the realized demand D aording to the following three ae a for D eriod; thu, R 1 = q and R 2 = [ b for D d W, q, the firm doe not go bankrut and ha no leftover inventory to alvage in the eond 1 α q 1 α, the firm doe not go bankrut However, it doe have leftover inventory q 1 αd and thi inventory i liquidated at rie Therefore, R 1 = 1 αd and R 2 = [q 1 αd] for D < d W, the firm goe bankrut and the leftover inventory q 1 αd i liquidated at rie b, leading to R 1 = 1 αd and R 2 = b[q 1 αd] Integrating D over the above three enario lead to 24 2 When > [1 α ]q, ie q < d W, we an diu the firm rofit aording to three imilar enario, leading to 25 Proof of Lemma C3 Firt, we firt how that W, = v by ontradition Aume, q Ω W with < v i globally otimal Now onider olution, q Ω with, v] Conider two enario 1 When, q Ω B, for any demand realization D [, d h ], we have minq, D minq, 1 αd Therefore, the firm ell more unit under, q at a higher rie than under, q Hene, the firm firteriod revenue R 1 i higher under, q, leading to a lower robability of liquidation ale, and the average eond-eriod rie i therefore alo higher under, q Therefore, π, q > π, q Eletroni oy available at: htt://rnom/abtrat=2896581
ii 2 When, q Ω W A myoi utomer urhae a long a v and trategi utomer do not urhae, the firm ell the ame unit minq, 1 αd under, q a under, q and, hene, ha a higher R 1 due to the higher rie In the eond eriod, it ha the ame amount of leftover inventory [q 1 αd] + under, q a under, q and a higher exeted eond-eriod rie due to the lower robability of bankruty Therefore, π, q > π, q Combining the two enario, we have that the rofit under, q i greater than, q, ontraditing the aumtion that, q i globally otimal Therefore, = v i neeary for, q Ω W to be globally otimal In other word, W, = v Given the above reult, when identifying q W,, we an imlify the rofit funtion to π W L v, q and π W H v, q In the following, we firt examine the eudo-onavity of π W L v, q and π W H v, q, whih allow u to ue the firt-order ondition only to identify the global otima For eae of exoition, we ure v in the rofit funtion for the ret of thi roof We firt how that π W H q i eudo-onave Taking derivative of π W H with reet to q, we have: dπ W H q = v F + b dq 1 α F q [1 αqhq] b 27 At dπw H dq =, we have: v q = f b 1 α 1 α F q[2 αqhqhq + qh q] 28 v q f b 1 α 1 α F b v F q 1 α q 1 + b F hq q [ ] v q q = F h 1 αhq [ b 1 α 1 α 1 α F q + b]hq < d 2 π W H dq 2 Therefore, π W H q i uedo-onave in q Setting dπw H dq = lead to q W H Seond, we how that π W L q i eudo-onave for [1 αv q, ie the region where π W L q i the relevant rofit funtion Define kq a follow: kq = 1 d W v 1 αv = [v q ] 29 1 αv2 It i eay to hek that kq > for v q Taking derivative, we have dπ W L q = v F + b dq 1 α F [ d W 1 kqhd W ] b 3 Similarly to the ae of π W H, we an alo how that d2 π W L v,q dπ W H dq = lead to q W L a in 4 dq 2 < at dπw L dq = when v q Setting In addition, note that π W L i relevant when [1 αv ]q and π W H i relevant otherwie; at q = 1 αv auming thi value i in the region of, 1 αq NV ], dπ W H dq dπw L dq [1 αv ]q = bfq[kq αq] = bfq v < 31 1 αv Therefore, q = annot be a loal otima, and hene, within the wait-region, only 1 αv qw L or q W H a defined above an be the global otima Eletroni oy available at: htt://rnom/abtrat=2896581
iii Proof of Corollary C1 Firt, note that by definition, q W L and q W H are maller than 1 αq NV For 1 αv q NV, q 1 αq NV uh that v, q Ω B Therefore, we have Π B L v, q > Π B L v, q W i > Π W i v, q W i for i = H, L Therefore, neither v, q W L nor v, q W H an be the global otima Combining thi with Corollary 2, we an immediately get the firt reult To ee the eond reult, note that when > 1 αq NV, the olution with, q = v,, whih i in the buy-region, dominate v, q W L and v, q W H Proof of Corollary C2 Note that q W L maximize π W L v, q, ubjet to [1 αv ]q A the feaible et dereae a inreae, π W L v, q W l dereae in On the other hand, π W H v, q W h i indeendent of, while the feaible et exand a inreae Therefore, πv, q W L π v, q W H dereae in In addition, note that q W L beome irrelevant when i uffiiently large and q W H beome irrelevant when i uffiiently mall Therefore, there exit a threhold T W α uh that πv, q W L π v, q W H if and only if T W α Proof of Lemma C4 Taking the derivative of ˆπ B H with reet to q, [ dˆπ B H dq = πb H q + πb H = bf q bfq q q A = v bf q Therefore, q dˆπ B H dq = F q ] q xdf x 32 = bfq, and hene, [ q ] b bhq F xdx b 33 q Aording to IFR, hq inreae in q and, hene, dˆπb H dq Therefore, ˆπ B H i eudoonave dereae in q when dˆπb H dq Proof of Lemma C5 Similarly to the roof of Lemma C3, we an how that for 1, q and 2, q that are both in Ω B and with 1 < 2, the rofit of 2, q i higher than that of 1, q Furthermore, a Ω B i a loed et, for any, q Ω B but < v and q < Q b, ϵ > uh that +ϵ, q Ω B Therefore,, q Ω B an be otimal only if = v or q = Q b Furthermore, for, q with < v and q = Q b, aume dq b ϵ > uh that + ϵ, q Ω B and d the orreonding rofit i higher than, q Therefore,, q i otimal only if dq b < d Proof of Lemma C6 Aording to Lemma C5, for T D α, B,, q B, need to atify q B, = Q b B, Deending on the relative magnitude of q B,, d B, and d W, we an diu three further enario 1 If q B, d W, the relative rofit funtion i π B L and q B, = 1 αb, Q B, Let the otimal olution be B LL, q B LL and the orreonding rofit under thi olution be Π B LL, α for, α It i lear that dπb LL d < and dπb LL dα < 2 If d W > q B, > d B, the relative rofit funtion i π B L and q B, = Q B, Let the otimal olution be B LH, q B LH and the orreonding rofit under thi olution be Π B LH, α for, α It i lear that dπb LH d < and dπb LH dα = 3 If d B q B, : In thi enario, the relative rofit funtion i π B H, and q B, = Q B, Aording to Lemma C4, the global otima i determined by dˆπb H dq =, ie q B, = q B H, and B, = v bf q B H, and the orreonding rofit under thi olution i Π B H, α for, α It i lear that dπb H d = dπb H dα =
iv Comaring the otimal rofit under the three enario, a Π B H, α i indeendent of and Π B LL, α dereae in, there exit threhold T B,L α > T D α uh that Π B H, α Π B LL, α if and only if T B,L α Similarly, there exit threhold T B,H α uh that Π B H, α Π B LH, α if and only if T B,H α To omare Π B LL, α and Π B LH, α, we onider q a a funtion of Aording to the enveloe theorem, = πb L q 1 < In addition, note that Π B LH < max[v bf q] q F xdx q < F xdx q NV Therefore, there exit a threhold T B,LL α uh that Π B LL Π B LH if and dπ B LL dπb LH d v q NV only if T B,LL α Finally, given the above reult, we an ontrut T B,1 α = min T B,LL α, T B,L α and T B,2 α = max T B,L α, T B,H α to obtain the reult a tated Proof of Lemma C7 Firt, note that the funtion bf x + tf, we have Qdd Therefore, Q dd,a +Qdd ; therefore, bf Q dd + Q dd + tf +x inreae in x When Q dd t + bf Q dd = v + t 34 Q dd All of the above te an be revered, and hene, the firt and eond ondition are equivalent Similarly, we an how that the firt and third tatement are equivalent Proof of Lemma C8 By the definition of Q dd, we have: Therefore, Q dd Qdd,b + 1 α Proof of Lemma C9 ]Q dd,a lead to Q dd,b bf Therefore, Q dd,b [ Q dd,a bf Q dd Q dd,a + 1 α That i, Qdd,b + tf [1 αq dd ] < b + tf Q dd = v + t 35 1 αqdd Similarly, we an how that Q dd,b Qdd Thi follow diretly from the definition For examle, to how that [1 α, note that a Q dd,a ] Q dd,a + + tf Q dd,a, a deired > 1 α < bf Q dd,a, we have Qdd,a > Qdd,a + 1 α Q dd,a + tf + Therefore, = v + t 36 Proof of Lemma C1 In thi ae, q < d B, utomer exeted urlu in the buy-equilibrium i v In the wait-equilibrium, it i bf Therefore, the buy-equilibrium i more aealing if and only q+ 1 α if v bf q or, equivalently, q Q Thi i exatly the ondition under whih the wait-equilibrium exit Therefore, utomer refer the wait-equilibrium [ ] When q d B q+, utomer exeted urlu in the buy-equilibrium i v + t 1 F In the wait-equilibrium, it i bf q Therefore, the buy-equilibrium i more aealing if and only if [ ] q + v + t 1 F bf q 37 or, equivalently, q Q dd,a [ ] Finally, when q d B q+, utomer exeted urlu in the buy-equilibrium i v + t 1 F In the wait-equilibrium, it i bf Therefore, the buy-equilibrium i more aealing to utomer if and only if or, equivalently, q Q dd,b q+ 1 α [ v + t 1 F q + ] q + bf 1 α 38
v Proof of Lemma C11 In rearation, we rearrange the exitene ondition for the buy-equilibrium in Statement 1 of Prooition 4 by onidering the following three enario ] 1 When Q,, Qdd, and min Q, = Therefore, in thi region, the buy- equilibrium exit if and only if q Qdd 2 When [ Q, Q dd ],, Qdd the buy-equilibrium exit if and only if q Q or q ] 3 When > Q dd,, Qdd region, the buy-equilibrium exit if and only if q Q ] And min ], Qdd Q, = Note that a Q dd Q, min = Q Therefore, in thi region, Q, = Q Therefore, in thi Symmetrially, we rearrange the exitene ondition for the wait-equilibrium in Statement 2 of Prooition 4 by onidering the following three enario 1 When > [1 α ]Q dd, Q, Therefore, the waitequilibrium exit if and only if q > Q 1 α 2 When [[1 α ]Q, [1 α ]Q dd ],, and Q, Therefore, the wait-equilibrium exit if and only if q > 1 αqdd 3 When < [1 α ]Q, Q, 1 α wait-equilibrium exit if and only if q > 1 αqdd > 1 αqdd 1 α 1 α or q, and Q, =, and a Q Q dd, 1 αqdd 1 α 1 α 1 αqdd 1 α Therefore, the Aording to the above ondition, note that when > Q dd, the buy-equilibrium exit if and only if q Q and the wait-equilibrium exit if any only if q > Q Thi i onitent with the no-deferred-diount ae, in whih multile equilibria do not exit when > Q Therefore, we only need to onider the region with Q dd We further divide thi region into two ae Cae I: [1 α ]Q dd,a In thi region, we have Q dd,a Q dd Seond, aording to Lemma C9, a [1 α ]Q dd,a, Q dd,a roertie, we further divide [1 α ]Q dd,a for the buy- and wait-equilibria to exit a eified above and, aording to Lemma C7, Q dd,b With thee into the following region deending on the ondition 1 [1 α ]Q, the buy- and wait-equilibria both exit if and only if q 1 αq dd, Qdd whih i not emty for α > For utomer to hooe the buy-equilibrium, note that for q > 1 αqdd have q > Q dd, and hene, [1 α ]q > Therefore, q q+ = 1 α dw more aealing to utomer if and only if q Q dd,b, whih fall in thi region 1 αq dd ],, we > d B Hene, the buy-equilibrium i 2 [1 α ]Q, [1 α ]Q dd ]: In thi region, the wait-equilibrium exit if and only if q > or q Q, However, the region for the buy-equilibrium to exit deend on the relative 1 α magnitude of Q and [1 α ]Q dd We onider three further ub-enario Q, a Q < [1 α ]Q dd and [1 α ]Q, Q : Both equilibria exit when q ] [ ] 1 αq dd or q, Qdd In the firt ub-region, q [d B, d W ] Therefore, utomer refer 1 α the buy-equilibrium if and only if q Q dd,a Note that a Q dd,a, utomer alway refer the 1 α buy-equilibrium in thi ub-region In the eond ub-region, imilarly to the ae with < [1 α ]Q, utomer refer the buy-equilibrium if and only if q Q dd,b Combining the two ub-region, utomer refer the buy-equilibrium if and only if q Q dd,b
vi, b Q < [1 α ]Q dd and Q, [1 α ]Q dd : Both equilibria exit when q [ ] 1 αq dd or q, Qdd In the firt ub-region, q [d B, d W ] Therefore, utomer refer 1 α the buy-equilibrium if and only if q Q dd,a Similarly to the reviou ae, we again have Q dd,a >, 1 α and hene, utomer alway refer the buy-equilibrium in the firt ub-region In the eond ub-region, imilarly to the ae with [1 α ]Q, utomer refer the buy-equilibrium if and only if q Q dd,b Combining the two ub-region, utomer refer the buy-equilibrium if and only if q Q dd,b Q [1 α ]Q dd and [1 α ]Q, [1 α ]Q dd : In thi region, both equilibria o-exit if and only if q Q, or q, Qdd In the firt ub-region, q < d W ] 1 αq dd 1 α A q > Q >, we have q > db Therefore, utomer refer the buy-equilibrium if and only if q Q dd,a Therefore, in the firt ub-region, utomer alway refer the buy-equilibrium In the eond ub-region, ] 1 αq dd that i, q, Qdd, imilarly to the firt ae, we have q > d W Therefore, utomer refer the buy-equilibrium if and only if q Q dd,b 3 [1 α ]Q dd, [1 α ]Q dd,a ]: In thi region, the wait-equilibrium exit if and only if q > Q However, the region for the buy-equilibrium to exit deend on the relative magnitude of Q [1 α ]Q dd and [1 α ]Q dd,a Therefore, we onider four further ub-enario a Q < [1 α ]Q dd and [1 α ]Q dd, [1 α ]Q dd,a : A Q <, both equilibria exit if and only if q In thi region, q > d B However, a, the relative magnitude of q and d W two ub-region: q, 1 α, Qdd < Qdd 1 α i undetermined in thi region Therefore, we further divide thi region into [ and q In the firt ub-region, q [d B, d W ], and, Qdd 1 α hene, utomer refer the buy-equilibrium if and only if q Q dd,a A Q dd,a, utomer alway 1 α refer the buy-equilibrium in thi ub-region In the eond ub-region, q > d W, and hene, utomer refer the buy-equilibrium if and only if q Q dd,b Note that a Q dd,b >, and Qdd,b 1 α refer the buy-equilibrium if and only if q Q dd,b in the eond ub-region b Q [1 α ]Q dd,a if and only if q Q, Qdd ] A Qdd However, note that Q dd i undetermined in thi region Therefore, we further divide thi region into two ub- [ ] and q In the firt ub-region, q < d W, and hene, utomer magnitude of q and d W region: q Q, 1 α 1 α and [1 α ]Q dd < Qdd and, utomer, [1 α ]Q dd,a : Both equilibria exit > Q dd Q, thi interval i not emty A q > Q >, q > db > Q dd > ; therefore, < Qdd 1 α, and hene, the relative, Qdd 1 α refer the buy-equilibrium when q Q dd,a, whih i greater than In the eond ub-region, q 1 α dw, and hene, utomer refer the buy-equilibrium when q Q dd,b Qdd, whih fall between and 1 α Therefore, in thi region, utomer refer the buy-equilibrium if and only if q Q dd,b Q [[1 α ]Q dd, [1 α ]Q dd,a, and [1 α ]Q dd, Q : Thi enario ha exatly the ame reult a in the enario with Q [1 α ]Q dd,a [1 α ]Q dd Q dd,b and, [1 α ]Q dd,a, and hene, utomer refer the buy-equilibrium if and only if q d Q [[1 α ]Q dd, [1 α ]Q dd,a, and Q, [1 α ]Q dd,a : A Q < ], both equilibria exit if and only if q, Qdd, whih obviouly i not emty In thi interval, imilarly to the reviou ae, q > d B Again, we divide the region into two ub-region: q, 1 α
[ and q, Qdd 1 α ] A in the reviou ae, utomer alway refer the buy-equilibrium in the firt ub-region In the eond ub-region, utomer refer the buy-equilibrium if and only if q Q dd,b Summarizing the above enario, we an onlude that when [1 α ]Q dd,a, the buy-equilibrium i more aealing to utomer if and only if q Q dd,b Cae II: [1 α ]Q dd,a, Q dd ] In thi region, aording to Lemma C9, > 1 α Qdd,a > Q dd,b Furthermore, in thi region, the wait-equilibrium exit if and only if q > Q However, imilarly to the reviou ae, the exitene region for the buy-equilibrium deend on the relative magnitude of Q and [1 α ]Q dd,a Thi require u to onider three ub-enario 1 Q [1 α ]Q dd,a if q, Qdd and [1 α ]Q dd and [1 α ]Q dd,a vii, Q dd : Both equilibria exit if and only The only differene between thi enario and that with Q [1 α ]Q dd,a, [1 α ]Q dd,a i that Q dd,b the buy-equilibrium if and only if q Q dd,a 2 Q > [1 α ]Q dd,a ] if q Q, Qdd < Q dd,a <, utomer refer the buy- 1 α equilibrium if and only if q Q dd,a < Q dd,a < Therefore, utomer refer 1 α and [1 α ]Q dd,a, Q : Both equilibria exit if and only Similarly to the reviou ae, a Q dd,b 3 Q > [1 α ]Q dd,a and [ Q, Q dd : Again, the only differene between thi enario and that with Q Q, [1 α ]Q dd,a i that Q dd,b < Q dd,a < Therefore, utomer refer the buy- 1 α equilibrium if and only if q Q dd,a [[1 α ]Q dd Therefore, we an onlude that when [1 α ]Q dd,a aealing to utomer if and only if q Q dd,a, [1 α ]Q dd,a, and, Q dd ], the buy-equilibrium i more
viii Aendix E: Detail on numerial reult In the aer, we have onduted extenive numerial exeriment to verify that the numerial reult reented in the main body of aer are robut under different arameter eifiation Table 3 ummarize the arameter we have ued to ondut our numerial reult Table 3 Parameter ued in numerial exeriment Parameter Demand Ditribution D v 1 Value Ued Uniform[, 3], Uniform[1,2], Uniform[8, 22], Uniform[2, 28] Triangular[, 1, 5] reented in the aer, Triangular[, 1, 25], Triangular[, 1,75], Triangular[, 3, 5], Triangular[2, 28, 5] Trunated normal with µ = 15, σ = 5, and trunated at and 3 Binary with different arameter both analytial and numerial reult obtained 55, 6 reented in the aer, 65, 7, 75 4, 45, 5 reented in the aer b 2, 25, 3 reented in the aer, 35, 4 α [, 5] with an inrement of 1 [-6, 6] with an inrement of 1
ix Aendix F: The imat of trategi fration α: An alternative illutration In thi aendix, we reent the imat of trategi fration α on the firm oerational and finanial metri urrently illutrated in Figure 5 in the main body of the aer in an alternative form, a in Figure 8 Figure 8 The imat of fration of trategi utomer α anevel of finanial ditre on oerational -15% -2% -25% deiion and erformane a Inventory % -1% -2% -3% b Prie -3% -4% -35% 3 1-1 -4% 5 1 15 2 25 3-5% -6% 3 1-1 5 1 15 2 25 3 1% Probability of bankruty d Strategi hare of ditre ot 8% 5% 4% 6% 3% 4% 2% 3 1-1 5 1 15 2 25 3 2% 3 1% 1-1 % 5 1 15 2 25 3 Note All D T riangular, 1, 5, v = 1, = 6, = 5, b = 3 Different line rereent different level of finanial ditre, a marked in the legend Figure 8a 8b rereent the inventory rie hange in erentage relative to the the newvendor benhmark v, q NV Figure 8d rereent the trategi hare of ditre ot, defined Π,α Π, a the roortion of total ditre ot aued by trategi onumer, ie, where Π, α i the firm Π,α Π, otimal rofit under, α In Figure 8d, the trategi hare of ditre ot i not defined when i low a the total ditre ot i zero A hown in Figure 8a, regardle of, the firm hould alway lower it inventory level when faing more trategi onumer, while the magnitude of the inventory redution inreae a the firm beome more finanially ditreed
x The trend on riing, a reented in Figure 8b, however, i le traightforward Indeed, we find that when the firm finanial ditre i in the medium level a rereented by = 1 in the figure, the firm tart to lower rie even when there only exit a mall fration of trategi utomer Thi mirror the boundary between Region BL and NV in Figure 4 However, a the firm beome more ditreed, it only offer a rie diount when the trategi fration i high, yet when the rie diount i very dee Thi ehoe the bifuration between ISC and BH, and alo the finding Figure 5b Regarding bankruty robability, a hown in Figure 8, when the firm robability of bankruty may dro a hown in the two line with = 1 and 1 a α inreae Thi orreond to the hange in the firm riing trategy Finally, Figure 8d ha onfirmed that a α inreae, the art of ot of finanial ditre that i due to trategi utomer behavior beome more and more imortant On the imat of α on the magnitude and effetivene of deferred diount, by rotating Figure 7 in a imilar way, we onfirm the inight we highlight deferred diount i mot valuable to the firm when the firm finanial ditre i at a medium level, and the fration of trategi onumer i high
xi Aendix G: Comarative tati on,, and b To better undertand how different arameter influene the interation between finanial ditre and trategi utomer behavior, we onduted extenive numerial exeriment to etablih omarative tati on the dee liquidation rie b, whih ature the inentive for waiting, the regular alvage rie, and the rourement ot, whih ature the regular rofit margin A rereentative et of reult are illutrated in Figure 9 11 In all figure, we ue the bae arameter D T rianglar, 1, 5, α = 3, v = 1, = 6, = 5 and b = 3 We then vary b, and in eah figure Figure 9 The imat of b on oerational deiion and erformane a Inventory b Prie -5-1 -1-15 -2-2 -25 2 3 4-3 -4 2 3 4-3 -35-5 -4-6 -45 1 9 8 7-5 -4-3 -2-1 1 2 3 4 Probability of bankruty 65% 6% -5-4 -3-2 -1 1 2 3 4 d Strategi hare of ditre ot 2 3 4 6 5 4 3 2 2 3 4 55% 5% 45% 1-5 -4-3 -2-1 1 2 3 4-5 -4-3 -2-1 1 2 3 4 Note: D T riangular, 1, 5, α = 3, v = 1, = 6, = 5 Figure 9a and 9b ature the relative differene of inventory and rie from the newvendor benhmark v, q NV We tart with the imat of b on the firm oerational deiion and finanial erformane, a illutrated in Figure 9 A a arameter unique to our model, b ature bargain hunter valuation in a liquidation ale In ome ene, a high b orreond to a liquidation ale that i well-run, o that utomer deferred
xii Figure 1 The imat of on oerational deiion and erformane a Inventory b Prie -5-1 -1-2 -15-2 -25 4 45 5-3 -4 4 45 5-3 -5-35 -6 1 9 8-5 -4-3 -2-1 1 2 3 4 Probability of bankruty 7% 65% -5-4 -3-2 -1 1 2 3 4 d Strategi hare of ditre ot 4 45 5 7 6 5 4 3 2 4 45 5 6% 55% 5% 1-5 -4-3 -2-1 1 2 3 4-5 -4-3 -2-1 1 2 3 4 Note: D T riangular, 1, 5, α = 3, v = 1, = 6, b = 3 Figure 1a and 1b ature the relative differene of inventory and rie from the newvendor benhmark v, q NV diount in a liquidation ale doe not differ too muh from a regular alvage ale A uh, a higher b alo lead to a lower inentive for trategi utomer to wait Thi i exatly atured in Figure 9 A hown in Figure 9a, a higher b lead to a maller inventory redution Similarly, regarding rie diount, when the level of finanial ditre i not too high, a higher b lead to a maller rie diount However, when i uffiiently high, if b i uffiiently low a atured by b = 2 in Figure 9b, then it i too otly for the firm to indue utomer to urhae early Intead, the otimal deiion i to revert bak to full rie and ignore trategi utomer In Figure 9, we an ee that again, for low, beaue of the more aggreive inventory redution and rie diount, a lower b orreond to a lower robability of bankruty However, a inreae, uh inventory redution and rie diount beome too otly to the firm, who then atually give u anet the robability of bankruty for lower b to jum u Thi ehoe Figure 5, whih how that a higher fration of trategi utomer alo lead to imilar behavior Finally, on trategi hare of ditre ot, thi i not traightforward a a dereae in b inreae both the diret
xiii Figure 11 The imat of on oerational deiion and erformane a Inventory b Prie -5-1 -1-2 -15-2 -25 6 65 7-3 -4 6 65 7-3 -5-35 -6-5 -4-3 -2-1 1 2 3 4-5 -4-3 -2-1 1 2 3 4 Probability of bankruty d Strategi hare of ditre ot 1 9 8 7 64% 6 65 62% 7 6% 58% 6 5 4 6 65 7 56% 54% 52% 3 5% 2 48% 1-5 -4-3 -2-1 1 2 3 4 46% -5-4 -3-2 -1 1 2 3 4 Note: D T riangular, 1, 5, α = 3, v = 1, = 5, b = 3 Figure 11a and 11b ature the relative differene of inventory and rie from the newvendor benhmark v, q NV and trategi ortion of ditre ot Indeed, a hown in Figure 9d, a b inreae, the trategi hare of ditre ot atually inreae, uggeting that the exitene of trategi utomer may atually be more harmful relatively to thoe firm that an run liquidation ale more effiiently The imat of on the firm oerational deiion and erformane i illutrated in Figure 1 On inventory redution Figure 1a, firt note that unlike the above ae with b, whih doe not influene the newvendor benhmark q NV, a larger houlead to a greater q NV Similarly, a greater alo inreae the average alvage rie the firm fae, and hene, even in the reene of finanial ditre, the above effet houlead to a higher inventory level However, a the differene between and b alo determine the trategi utomer waiting inentive, an inreae in alo ugget that the firm need to ut inventory more aggreively in order to indue trategi utomer to urhae Balaning the above two fore, Figure 1a how that the inventory redution relative to the orreonding newvendor quantity q NV inreae a
xiv inreae, uggeting that the trategi waiting hannel dominate For the ame reaon, a hown in Figure 1b, a higher alo lead to a larger rie diount The imat of on the robability of bankruty, a illutrated in Figure 1, alo how a imilar attern: A larger lead to more aggreive inventory redution and rie diount, and hene reult in a maller robability of bankruty for low, but it alo lead to an earlier jum to a high robability of bankruty a inreae On trategi hare of ditre ot, a hown in Figure 1d, imilar to Figure Figure 9d, finanial ditre aued by trategi utomer are relatively more imortant when i mall, or equivalently, when the overall ditre i le evere Finally, Figure 11 illutrate the imat of on the firm oerational deiion and erformane For it imat on inventory, note that imilar to, alo ha a diret imat on the newvendor benhmark q NV Seifially, a greater lower q NV In addition, alo influene the firm finanial ditre through two ometing fore: Firt, a larger ugget a lower rofit margin, leading to more evere finanial ditre Seond, a larger reult in a maller inventory level, alleviating finanial ditre The aggregated effet, however, i le lear The above imat alo interat with the waiting inentive of trategi utomer: while a low inventory level weaken utomer inentive to wait, more evere finanial ditre trengthen it Combining all the above effet, Figure 11a how that a inreae, the relative inventory redution dereae, ie the firm beome le aggreive in utting inventory Thi ugget that the effet of inventory redution aoiated with the inreae in dominate the finanial ditre For the ame reaon, the rie diount offered by the firm alo dereae in Baed on the imat of on inventory and rie, the firm robability of bankruty, a hown in Figure 11 follow the ame attern a in the ae with b and Finally, Figure 11d ugget that the trategi hare of ditre ot i lightly higher when i lower, alo onitent with the attern with and b
xv Aendix H: Rational exetation equilibrium with unobervable q In the main body of the aer, we aume that utomer oberve the inventory level q Thi i ued in Liu and van Ryzin 28, and i examined a the quantity ommitment in Su and Zhang 28, 29 We adot thi aroah a the bai model mainly beaue of our fou on how trategi utomer behavior an aggravate finanial ditre A we already know from Su and Zhang 28, 29, ommitting on quantity imrove the firm rofit Therefore, fouing on uh a enario, we highlight that even when the firm adot valuable oerational trategie to mitigate trategi utomer behavior, uh behavior till an aount for a ignifiant ortion of the firm ditre ot, and other mitigating mehanim uh a deferred diount an till imrove the firm rofitability To how the robutne of our main reult, in thi Aendix, we examine the ae when utomer oberve only rie, but not inventory q Thi i onitent with the rational exetation equilibria framework ued in the bae model in Su and Zhang 28 and Cahon and Swinney 29 In the following, we reent the bai te to etablih the rational exetation equilibrium, and then ue a et of numerial reult to illutrate the robutne of the inight we highlighted in the aer Under thi framework, after oberving the firm rie, trategiutomer form a rational belief on it inventory level, whih we denote a ˆq, and make their urhaing deiion baed on the belief ˆq Cutomer urhaing behavior i ummarized in the following reult Lemma H1 Under the belief that the inventory i ˆq, trategi utomer urhae if and only if 1 αq ˆq Q b := max, Q, 39 where Q = F 1 v b A hown under a ertain belief on the firm inventory level ˆq, the utomer urhaing behavior i exatly the ame a we identify in the main body of the aer Therefore, the elf-fulfilling roerty of bankruty, a highlighted in the aer, i alo reerved On the firm ide, there are two oibilitie deending on the firm belief of trategi utomer urhaing deiion 1 If the firm believe that trategi utomer wait, the analyi i the ame a what we have in the main body of the aer, ie the rie i v, and the otimal inventory q i the ame a in the otimal wait-region inventory Let the orreonding firm rofit to be Π W 2 If the firm believe that trategi utomer buy, under any, it olve q to maximize: πq; = q q q xdf x b minq,d B q xdf x 4 For any, let Q m = arg max q πq; Combining the firm ide and the utomer ide, a, q form a Rational Exetation Equilibrium if and only if, q atify: q = Q m = Q b In the reene of multile REE, the firm will obviouly hooe the one that lead to the highet rofit Thi an be ahieved a there i a one-to-one maing between and q Let the rofit be Π B Intuitively, thi hould be lower than the otimal buy-region olution in the main body of the aer
xvi Next, the firm omare Π W and Π B and ik the higher one Intuitively, a Π W i the ame a in the urrent verion and Π B i lower, the ISC region in Figure 4 hould exand To ee if our main inight in the aer remain valid, we ondut a et of numerial exeriment under the ame arameter ued in Figure 5 The orreonding reult are reented in Figure 12 Figure 12 Oerational deiion and erformane when inventory q i unobervable a Inventory b Prie 5 1 1 2 15 2 25 15 3 3 4 15 3 3 5 35 6 1 9 8 5 4 3 2 1 1 2 3 4 Probability of bankruty 6 5 5 4 3 2 1 1 2 3 4 d Strategi hare of ditre ot 7 6 5 4 15 3 4 3 15 3 3 2 2 1 1 5 4 3 2 1 1 2 3 4 5 4 3 2 1 1 2 3 4 Note: D T riangular, 1, 5, v = 1, = 6, = 5, b = 3 Panel 12a 12b rereent the relative differene between the q and the orreonding newvendor benhmark q NV v By omaring Figure 5 and 12, we note that the firm oerational deiion and erformane follow very imilar attern whether inventory i obervable or non-obervable to utomer, uggeting that our main inight remain unhanged That aid, we an alo oberve that when inventory q i unobervable, the firm need to lower inventory and rie for a maller threhold Thi ehoe the finding etablihed in Su and Zhang 28, 29 that quantity ommitment an mitigate trategi waiting and imrove the firm rofit
xvii Aendix I: Single la of utomer with a diount fator In thi aendix, we modify the model in the main body of the aer by altering the following two aumtion 1 All firt-eriod utomer are forward-looking 2 Firt-eriod utomer eond-eriod valuation i v 2 [b, ] For thi aumtion, note that while the aer tudie the interation between finanial ditre and trategi onumer behavior, it emhaize more on finanial ditre by ointing out that trategi onumer behavior an be another oure of finanial ditre Therefore, we et u the model o that the value of trategi waiting only ome from the oible liquidation ale For thi uroe, we onfine v 2 [b, ] a for v 2 >, utomer may alo wait even in the abene of finanial ditre v 2 an be viewed a utomer atient level A higher v 2 ugget that utomer are very atient, and are hene more likely to wait, kee everything ele the ame, while a lower v 2 weaken the inentive to wait To an extreme, when v 2 = b, utomer have no inentive to wait We an how that the firm oerational deiion and rofit under v 2 = b i exatly the ame a in Lemma 2, ie α = in the model of the main body of the aer I1 Under thi model, we firt etablih utomer behavior and then tudy the firm oerational deiion Strategi utomer urhae deiion under a rational belief Similar to Setion 4 in the main body of the aer, utomer behavior an be etablihed a follow Prooition I1 Let Q u = F 1 v, and Q ub = max, Q u v 2 b 1 A buy-equilibrium exit if and only if q 2 A wait-equilibrium exit if and only if q > Q ub or q Q u 3 The wait-equilibrium i more aealing to utomer when both equilibria o-exit Figure 13 Strategi utomer behavior in equilibrium q q = q = B;W W Q u B Q u Q u Prooition I1 i illutrated in Figure 13 Comare thi with equilibrium ondition a ummarized in Prooition 1 and orreondingly, Figure 3 in the main body of the aer, a hown, the exitene ondition for the wait-equilibrium ha a imilar iee-wie linear truture a the in Figure 3: when the inventory level i uffiiently high, antiiating other utomer wait for the liquidation ale, an individual
xviii utomer alo find it more benefiial to wait Indeed, by etting v =, the exitene ondition for the wait-equilibrium in thi model i the ame a the one in the main body of the aer when α = 1, ie all utomer are trategi The exitene ondition for the buy-equilibrium differ from the orreonding reult in the main body of the aer However, the reult remain unhanged that when both the buy- and wait-equilibrium o-exit, the wait-equilibrium i more aealing to trategi utomer Therefore, when induing utomer to urhae, the firm ha to enure that, q atifie q Q ub, imilar to the reult in the main body of the aer I2 Otimal rie and inventory deiion With an undertanding of utomer urhaing behavior, we haraterize the firm otimal rie and inventory deiion Clearly, a all firt-eriod utomer are homogenou, the only enible trategy the firm an adot i to indue all of them to urhae in the firt eriod, imilar to Su and Zhang 28 Under, q uh that all firt-eriod utomer urhae, the firm rofit deend on the firt-eriod realized demand D in the following way 1 For D < q, the firm firt-eriod revenue i D In the eond eriod, if D q, or equivalently, D d B := +q, the firm goe bankruty, and the revenue from the liquidation ale i bq D On the other hand, for D > d B, the firm urvive and the revenue from the alvage ale i q D 2 For D q, the inventory i old out, and hene the firm rofit i q Combining the above enario, the firm total exeted rofit i: π u = q + q xdf x + b = q q d B q xdf x + q xdf x b q d B d B q xdf x + q[1 F q] 41 q xdf x 42 Regarding the otimal olution, imilar to the main body of the aer,, q hould either be the unontrained otima, or lay on the buy-wait boundary a illutrated in Figure 13 Therefore, there are three oible andidate 1 The unontrained otimal, whih i v, q NV Clearly, thi olution atifie q Q ub if and only if q NV 2 If, q lay on the downward loing art of the buy-wait boundary, the otimal olution i learly = v, and q = Clearly The orreonding rofit funtion i: v / π L u = 1 + v x df x 43 Clearly, π L u dereae in, but it i indeendent of v 2, the utomer atiene level Alo, note that d B = in thi etting In other word, under v, q, to avoid trategi waiting, the firm omletely eliminate the robability of bankruty 3 If, q lay on the flat art of the buy-wait boundary, q, ie = v v 2 bf q, q an be determined by olving the following rofit funtion π H u = q q q xdf x b d B q xdf x 44
xix q = [v v 2 bf q]q [v v 2 bf q] q xdf x b q+ v v 2 bf q q xdf x In thi ae, the firm mitigate trategi waiting by diretly ontrolling inventory It i obviou that at the otimal, q for thi enario, we have < v, ie the firm ull both the inventory and rie lever to eliminate trategi waiting In addition, by the Enveloe Theorem, it an alo be hown that under the otimal, q, the rofit π H u hould dereae in through d B Note that the above three andidate otimal olution orreond aroximately to Region NV, BL, and BH in Figure 4, uggeting that the ingle utomer la model hare ome imilaritie with the bai model in the aer on the mehanim to that indue utomer to urhae 45 Figure 14 Otimal deiion under a ingle utomer la with a diount fator a Inventory b Prie -5-5 -1-15 -1-2 -25-15 -3-2 1 8 6 4 3 35 4-5 -4-3 -2-1 1 2 3 4 Probability of bankruty -35 3-4 35 4-45 -5-4 -3-2 -1 1 2 3 4 7% 6% 5% 4% 3% d Strategi hare of ditre ot 2 3 35 4-5 -4-3 -2-1 1 2 3 4-5 -4-3 -2-1 1 2 3 4 Note: D T riangular, 1, 5, v = 1, = 6, = 5, b = 3 Different line rereent different v 2, with the orreonding number in the legend Figure 14a and 14b lot the relative differene between the orreonding quantitie and the newvendor benhmark v, q NV 2% 1% % 3 35 4 The otimal deiion and the orreonding erformane i illutrated in Figure 14 A hown in Figure
xx 14a, with a ingle la of trategi utomer, q in general dereae in and v 2, whih ature how atient or equivalently, trategi utomer are Thi attern i imilar to that under the urrent model in the main body of the aer Figure 5a On rie Figure 14b, we oberve that when utomer are more trategi, in general the firm ha to offer a deeer rie diount, imilar to the model in the aer However, a we argue above, with a ingle la of trategi utomer, the firm alway indue utomer to urhae A a reult, the otimal rie never revert to v a in the model in the aer Combining inventory and rie, we oberve the three tage a diued above: when i low, the firm imly offer the newvendor benhmark v, q NV A inreae, the firm foue on reduing inventory, while keeing rie at v Finally, for uffiiently high, the firm ue both rie and inventory to indue utomer to urhae Thi attern i alo onitent with the eking order reult we highlight in the main body of the aer Regarding the robability of bankruty, a hown in Figure 14, imilar to Figure 5, for low, under the otimal deiion, the robability of bankruty i lower when utomer are more trategi Finally, Figure 14d how that the trategi hare of ditre ot i higher when utomer are more trategi while the level of finanial ditre i at a medium level, alo imilar to the attern under the urrent model in the aer Figure 5d In ummary, the only major differene between the otimal deiion and erformane under the ingle utomer la model and thoe under the urrent model in the aer i that the firm doe not revert it otimal rie to v for high I3 Proof Proof of Prooition I1 We firt roof the exitene ondition of the two equilibria earately, and then omare the two Firt, on the exitene ondition of the buy-equilibrium, if q and all other utomer buy, waiting ha no value, and hene the buy-equilibrium exit If > q and all other utomer buy, the firm annot urvive, and the exeted value of waiting i v 2 bf q, ie the demand i lower than inventory Therefore, all utomer urhae i an equilibrium if v v 2 bf q Seond, on the exitene ondition of the wait-equilibrium, if q, even auming all other utomer wait, waiting ha no value for a ingle utomer So wait-equilibrium doe not exit For > q, if all other utomer wait, the firm annot urvive Therefore, the exeted value of waiting i v 2 bf q Therefore, all utomer wait i an equilibrium if and only if > v v 2 bf q Finally, when both equilibria o-exit, the wait-equilibria are more aealing to utomer for the ame reaon in Prooition 1 and the detail are omitted
xxi Aendix J: Maximizing the robability of urvival In thi aendix, we examine the firm otimal deiion with the objetive to maximize it urvival robability To guarantee the uniquene of the olution, when there are multile olution that lead to the ame robability of urvival, we aume that the firm refer the one that maximize rofit We fou on the ae without deferred diount, orreonding to Setion 5 in the aer Clearly, given rie and inventory, q, trategi utomer behavior remain the ame a haraterized in Setion 4 in the aer Under uh behavior, the firm otimal deiion, q and the orreonding robability of bankruty i ummarized in the following rooition Prooition J1 Let T m = max v F 1 v, and T = max Q When the firm rimary objetive i to minimize the robability of bankruty, 1 for T D α, the firm otimal deiion are, q = v, q NV b 2 for T D α, v, the firm otimal inventory q < q NV Under the otimal trategy, the firm robability of bankruty i zero 3 for v, max [1 αv ]d h, T m, the otimal deiion are: v, if α 1 T, q 1 αv v = T, otherwie T and the orreonding robability of bankruty i F q 4 for max [1 αv ]d h, T m, the otimal deiion, q are the ame a the rofit maximization ae, and the firm robability of bankruty i 1% 46 Figure 15 Illutration of the firm otimal trategie when maximizing the robability of urvival α α = 1 T v PB-B NB BC = 1 αv q NV = [1 αv ]d h NV v PB-W T m Note NV rereent the region where the newvendor olution i otimal; NB for the region with no bankruty yet the newvendor olution i not otimal; PB-B rereent the region where the otimal trategy indue the trategi onumer to buy and the reulting robability of bankruty i between %, 1%; PB-W indue the trategi onumer to wait and the reulting robability of bankruty i between %, 1%; BC rereent the region where the otimal trategy lead to bankruty with ertainty The rooition i illutrated in Figure 15 Clearly, when T D α a defined in the main body of the aer Region NV, the firm an avoid bankruty even if he tok the newvendor quantity and doe
xxii not offer rie diount A inreae Region NB, the firm an till omletely eliminate the robability of bankruty, but may need to lower rie and/or inventory In artiular, when = v, the firm otimal deiion i = v and q =, ie the firm et her inventory level to the lowet oible demand realization It reflet that in thi region, to maximize the robability of bankruty, the firm behave very onervatively Aordingly, the value of the firm uffer from evere under-invetment A ontinue to inreae, it i imoible to omletely eliminate bankruty Aordingly, the otimal trategy an be further laified into two ategorie deending whether trategi utomer are indued to buy Region PB-B or to wait Region PB-W Both region hare two ommon feature Firt, the firm an only avoid bankruty if and only if all inventory i old in the firt eriod For examle, when α i mall, the otimal wait-region olution i otimal Under, the firm tok at demand realization that allow the firm to avoid bankruty, whih i exatly the mallet 1 αv Seond, in both Region PB-B and PB-W, the amount of inventory q inreae in the level of finanial ditre For intane, while q = at = v, when = [1 αv ]d h for mall α, the otimal inventory level i atually d h The two feature ugget that when bankruty annot be omletely eliminated, the firm atually adot more aggreive oerational trategie to reah for urvival Therefore, in thi region, the value of the firm may uffer from evere over-invetment Finally, when i uffiiently large Region BC, bankruty i unavoidable even at the highet oible demand realization Therefore, the firm imly adot the trategy that maximize her rofit a in the main body of the aer Comaring the above reult with the otimal trategy under the objetive of rofit maximization, we note that there are ome imilaritie between the two trategie Firt, when i very mall or very large, the two trategie are idential Seond, in general, both trategie indue utomer to urhae early when α i uffiiently high However, the two trategie are alo different, mot ignifiantly when i in the middle range Region PB-B, PB-W In thi region, when the firm minimize the robability of bankruty, he atually order more when her finanial ditre deeen, while the otimal order quantity dereae in under the objetive of rofit maximization a in the aer Although we do not have diret evidene in whih objetive manager adot in ratie, the reult under rofit maximization eem to be more onitent with emirial evidene that firm often holower level of inventory under finanial ditre Chevalier 1995, Mata 211 The above reult are further illutrated in Figure 16 baed on the ame et of arameter ued in Figure 5 The non-monotoniity of inventory level q, a hown in Figure 16a, i artiularly notable Oberve that at =, to omletely eliminate the robability of bankruty, the firm imly doe not order anything However, for uffiiently large, the firm may order above the newvendor level Suh behavior i refleted in Figure 16d, whih how that the firm arifie mot value around thee extreme inventory level J1 Proof Proof of Prooition J1 We rove thi rooition by three te
xxiii Figure 16 The imat of uing urvival maximization a the objetive a Inventory b Prie 4 2-1 -2-3 -2-4 -4-5 -6-6 -8 15 3-7 -8 15 3-5 -4-3 -2-1 1 2 3 4-5 -4-3 -2-1 1 2 3 4 Probability of bankruty d Profit 1 9-1 8-2 7-3 6-4 5-5 4-6 3-7 2 1 15 3-5 -4-3 -2-1 1 2 3 4-8 -9 15 3-5 -4-3 -2-1 1 2 3 4 Note: D T riangular, 1, 5, v = 1, = 6, = 5, b = 3 Figure 16a how the relative differene between the otimal inventory and the newvendor benhmark q NV Figure 16d how the relative differene between the otimal rofit under the urvival maximization objetive and that under the value/rofit maximization objetive 1 We how that the firm an guarantee urvival if and only if v For the uffiient art, note that for any in thi range, it i eay to how that under = v and q =, all utomer urhae, and the firm never goe bankrut For the neeary art, note that the maximal guaranteed revenue for the firm i v Alo, if the firm order le than, the firm minimal rofit an be imroved if the firm order more Therefore, if > v, the firm will alway go bankrut with a oitive robability In thi region, a the realized bankruty robability i zero, utomer have no inentive to wait, and hene we only need to fou on the buy-region When 1 αv q NV, v, q NV lead to zero bankruty robability, and hene the olution i otimal under the objetive of the urvival maximization 2 We how that for v, max[1 αv ]d h, T m, the otimal trategy i a tated in Statement 2 in the rooition
xxiv In thi region, to identify the olution that minimize robability of bankruty, we onider two enario: the otimal olution in the buy-region, and the otimal olution in the wait-region a Conider the wait-region olution firt Clearly, the rie i till v, and the robability of bankruty q+ q+ under q [, d h ] i F if q, and 1% otherwie Therefore, the robability of bankruty i 1 αv 1 αv minimized at the boundary, ie q = q+, or equivalently, 1 αv q =, and the robability of bankruty 1 αv i F For > [1 αv ]d h, there exit no olution in the wait-region that an avoid bankruty F 1 αv q+ if q+ b Conider the buy-region olution Similarly, under, q Ω B, the robability of bankruty i q, and 1% otherwie Therefore, the otimal trategy olve for the following otimization roblem: q + 47 t q 48 q max Q, 1 αq 49 min, q Note that given fixed, the objetive funtion dereae in q, and therefore the eond ontraint i irrelevant unle there exit no feaible olution Therefore, the otimal q =, and hene the above otimization roblem i imlified to maximize ubjet to the ontraint max Q, 1 αq Examining the ontraint, we note that Q 1 αq if and only if [1 α ]Q Under thi ondition, the ontraint, 1 αq, i automatially atified On the other hand, for [1 α ]Q <, the ontraint an be imlified to Q Combining the above two enario, the ontraint an be imlified to Q Clearly, for > T m, the ontraint i infeaible For T m, the otimal rie in the buy-region i T a defined Furthermore, it i lear that T dereae in Combining the otimal olution in the buy-region and that in the wait-region, we find that for v, max[1 αv ]d h, T m, under the otimal olution, the robability of bankruty i F min, whih i between % and 1% not inluive 1 αv, T 3 when max [1 αv ]d h, T m, it i imoible to avoid bankruty, hene the otimal olution i the ame a the rofit maximization ae, and the firm robability of bankruty i 1%