Dynamic pricing using flexible heterogeneous sales response models

Asche sleben, Philipp; S eine , Win ied J.
A icle — Published Ve sion
Dynamic p icing using lexible he e ogeneous sales
esponse models
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1 3
ORIGINAL ARTICLE
Dynamic p icing using lexible he e ogeneous sales
esponse models
PhilippAsche sleben1 · Win iedJ.S eine 1
Recei ed: 31 Ma ch 2022 / Accep ed: 12 Feb ua y 2024 / Published online: 29 Ma ch 2024
© The Au ho (s) 2024
Abs ac
We combine nonpa ame ic p ice esponse modeling and dynamic p icing. In pa -
icula , we model sales esponse o as -mo ing consume goods sold by a physi-
cal e aile using a Bayesian semipa ame ic app oach and inco po a e he p ice
o he p e ious pe iod as well as u he ime-dependen co a ia es. All nonlinea
e ec s including he one-pe iod lagged p ice dynamics a e modeled ia P-splines,
and embedding he semipa ame ic model in o a Hie a chical Bayesian amewo k
enables he es ima ion o nonlinea he e ogeneous (i.e., s o e-speci ic) immedia e
and lagged p ice e ec s. The nonlinea he e ogeneous model speci ica ion is used
o p ice op imiza ion and allows he de i a ion o op imal p ice pa hs o b ands o
indi idual s o es o e aile s. In an empi ical s udy, we demons a e ha ou p o-
posed model can p o ide highe expec ed p o i s compa ed o compe ing benchma k
models, while a he same ime no se iously su e ing om bounda y p oblems o
op imized p ices and sales quan i ies. Op imal p ice policies o b ands a e de e -
mined by a disc e e dynamic p og amming algo i hm.
Keywo ds Sales esponse models· Func ional lexibili y· S o e he e ogenei y·
P ice dynamics· P ice op imiza ion· Disc e e dynamic p og amming
1 In oduc ion
Es ima ing p ice esponse unc ions o suppo p icing decisions is a highly ele an
opic in he ma ke ing li e a u e and in ma ke ing esea ch p ac ice. A p ice esponse
unc ion o , mo e gene ally speaking, a sales esponse unc ion ela es he sales o
a b and o own and compe i i e b and p ices, o accompanying ma ke ing ac i i ies
* Philipp Asche sleben
philipp.asche sleben@ u-claus hal.de
Win ied J. S eine
[email p o ec ed]
1 Depa men o Ma ke ing, Claus hal Uni e si y o Technology, Julius-Albe -S . 2,
38678Claus hal-Zelle eld, Ge many
30
P.Asche sleben, W.J.S eine
1 3
(like ea u e o display ad e ising), and/o o u he co a ia es accoun ing o ime-
o s o e-speci ic e ec s. In his pape , we ocus on sales esponse modeling o as -
mo ing consume goods ha a e sold by physical e aile s and o which scanne
da a a e nowadays widely a ailable. Publica ions in his esea ch a ea ha e pa icu-
la ly ocused on he ollowing dimensions:
Fi s , he speci ica ion o he ‘co ec ’ unc ional o m o he ela ionship
be ween sales and p ices was domina ed by s ic ly pa ame ic modeling un il he
2000s, s a ing wi h simple linea eg ession (lin-lin) models and ollowed by non-
linea pa ame ic models, especially mul iplica i e (log-log) and exponen ial (log-
lin) models. To o e come he p oblem ha pa ame ic models can la gely ail o
app oxima e he ue unc ional o m inhe en o eal da a (Hä dle 1990; VanHee de
1999; Lee lang e al. 2000), esea che s s a ed o apply mo e lexible nonpa ame ic
echniques, which a e able o explo e he unc ional shape di ec ly om da a ins ead
o assuming a p ede ined pa ame ic unc ional o m (Hanssens e al. 2002). The e
is e y clea e idence om many s udies ha hese nonpa ame ic me hods can
(g ea ly) imp o e he p edic i e model pe o mance as well as expec ed p o i s o e
pa ame ic modeling.1 Acco ding o VanHee de e al. (2002), manage s should ely
on models which p o ide he mos accu a e p edic ions. Besides, mo e lexible non-
pa ame ic es ima ion me hods ha e also become es ablished in choice modeling,
see, e.g., Abe (1991), Abe (1995), Abe e al. (2004), o Boz uğ e al. (2014).
Second, i is well-known ha he agg ega ion o s o e-le el da a ac oss s o es
o a e ail chain leads o biased es ima es o he e ec s o ma ke ing ac i i ies i
hese e ec s di e o indi idual s o es (as one would expec ) bu e en i ma ke -
ing e ec s a e homogeneous ac oss s o es (e.g., K ishnamu hi e al. 1990; Ch is en
e al. 1997). Fu he , i ma ke ing e ec s we e di e en ac oss indi idual s o es o a
e aile , pooling s o e-le el da a in o a homogeneous model should also cause a bias
in he e ec es ima es. Al hough empi ical indings a e no unequi ocal, he e oge-
neous s o e-le el sales esponse models ha enable he es ima ion o s o e-speci ic
e ec s o en p o ided mo e o a leas no less accu a e sales p edic ions compa ed
o hei homogeneous (pooled) coun e pa s. Mo e impo an ly, only he e ogeneous
sales esponse models allow o s o e-speci ic op imal p icing, also known as mic o-
ma ke ing p icing (Mon gome y 1997). The e is u he empi ical e idence ha
accoun ing o (unobse ed) he e ogenei y can also inc ease expec ed chain p o i s
(e.g., Mon gome y 1997; Lang e al. 2015).
Thi d, i is easonable o assume ha pas p ices o b ands o expec ed p ices in
u u e pe iods migh a ec a b and’s sales o p ice esponse in he cu en pe iod.
The e o e, he accommoda ion o p ice dynamics in sales esponse models has
also been o a long adi ion, be i ia conside ing lagged o lead p ice co a i-
a es (e.g., VanHee de e al. 2000, 2004), allowing o ime- a ying pa ame e s
(e.g., Foekens e  al. 1999; Kopalle e  al. 1999), o inco po a ing ma ke -le el
e e ence p ice a iables (e.g., G eenlea 1995; Fibich e al. 2003; Asche sleben
1 We use he e m nonpa ame ic unc ion o model he ela ionship be ween a speci ic me ic p edic-
o (like p ice) and a me ic c i e ion a iable (like sales) using lexible eg ession echniques, while he
e m semipa ame ic app oach will desc ibe a model whe e one o mo e p edic o e ec s a e cap u ed
pa ame ically, while o he s nonpa ame ically.
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1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
and S eine 2022). Almos all o hese dynamic app oaches ha e been di ec ed a
ei he explaining s uc u al e ec s o sales p omo ions ( o example he lack o
pos p omo ion dips in s o e da a), o showing ha dynamic p icing can inc ease
p o i s o e s a ic models. Impo an ly, dynamic p icing allows he compu a ion
o op imal p ice pa hs, i.e., op imal p ices ha a y o e ime due o he consid-
e a ion o ime-dependen e ec s on p ice o sales esponse.
F om he p e ious explana ions, i is clea ha accommoda ing unc ional
lexibili y in p ice esponse, s o e he e ogenei y in ma ke ing e ec s, and p ice
dynamics can imp o e bo h sales p edic ions and expec ed chain p o i s, which
is why i seems p omising o conside hese ea u es join ly in a s o e-le el sales
esponse model. Howe e , no a single app oach has ye accoun ed o all h ee
dimensions simul aneously. The e o e, ou i s esea ch ques ion is whe he
a sales esponse model ha combines he h ee componen s can p o ide highe
expec ed chain p o i s compa ed o exis ing simple (nes ed) models. In pa icu-
la , ou p oposed app oach will allow o he de i a ion o op imal p ice pa hs by
modeling he e ogeneous nonlinea p ice e ec s ia nonpa ame ic p ice unc ions
(including he one-pe iod lagged p ice e ec ), and we will assess whe he his
highe model complexi y pays o compa ed o simple models.
As indica ed abo e, some esea che s wen beyond he desc ip i e modeling
s age and used an es ima ed sales o p ice esponse model as basis o subsequen ly
de i e op imal p ices o p ice pa hs. Mos o hese op imiza ion models assume
linea p ice e ec s, i.e., p ice e ec s es ima ed by s ic ly pa ame ic esponse
modeling, and almos all o hese pa ame ic app oaches also included p ice
dynamics. The op imiza ion esul s o many o hese pa ame ic app oaches p o-
ide e idence o a leas sugges he exis ence o co ne solu ions, i.e., ha op i-
mized p ices hi he uppe bounda y o obse ed p ices, which would make he
p ice op imiza ion exe cise seem less use ul. A his poin , i is impo an o no e
ha none o he dynamic op imiza ion app oaches has modeled p ice esponse
mo e lexibly, like we p opose in ou model. Ou second esea ch ques ion he e-
o e ela es o his bounda y p icing p oblem, making ou p oposed app oach ec-
ommendable only i op imal p ices do no , o no always o no e y equen ly
hi he bounda ies o obse ed p ices (and as well do no induce bounda y solu-
ions o ela ed sales quan i ies). Mo e explici ly, we wan o analyze how p one
ou p oposed model is o bounda y p ice hi s. I he bounda y p icing p oblem is
no an issue, we a e p obably he i s o enable he calcula ion o op imal p ice
pa hs o each o he s o es o a e ail chain ia nonpa ame ic he e ogeneous
p ice e ec s modeling.
The es o he pape is o ganized as ollows: In Sec .2, we p o ide a e iew o
he ele an li e a u e o ou app oach. In Sec .3, we in oduce ou new dynamic
semipa ame ic Hie a chical Bayesian sales esponse model oge he wi h nes ed
model e sions. We will use he la e o model compa ison. We u he p opose
a disc e e dynamic p og amming app oach o calcula ing op imal p ice pa hs.
Sec ion4 s a s wi h a sho desc ip ion o he s o e-le el scanne da a used in ou
empi ical s udy and p o ides de ails abou model es ima ion and alida ion p oce-
du es. Mo eo e , op imal p ices, op imized p o i s, and expec ed losses when no
32
P.Asche sleben, W.J.S eine
1 3
using he p oposed model a e analyzed. In Sec .5, we conclude wi h a summa y o
indings and discuss limi a ions o ou esea ch.
2 Li e a u e e iew
Table 1 p o ides a summa y o ele an a icles ela ed o sales esponse mod-
eling in he ield o as -mo ing consume goods ha ha e combined a leas wo
o he dimensions discussed in he in oduc ion (i.e., unc ional lexibili y in p ice
esponse, s o e he e ogenei y in ma ke ing e ec s, p ice dynamics, p ice op imiza-
ion) in hei empi ical applica ions.
As men ioned be o e, nonlinea pa ame ic models we e s a e-o - he-a o a
long ime o cap u e p ice o mo e gene ally sales e ec s o as -mo ing consume
goods (e.g., H uschka 1997; Mon gome y 1997; Foekens e al. 1999; Kopalle e al.
1999; VanHee de e al. 2000, 2002; H uschka 2006b; And ews e al. 2008), and
hey s ill se e as benchma ks o he mo e sophis ica ed nonpa ame ic app oaches
obse ed nowadays in many publica ions. Possible nonpa ame ic ope a ionaliza-
ions ha e anged om ke nel eg ession (e.g., VanHee de e al. 2001) and neu-
al ne s (e.g., H uschka 2006a, 2007) o splines such as s ochas ic cubic splines
(Kalyanam and Shi ely 1998), B-splines (H uschka 2000; Haup and Kage e 2012;
Haup e al. 2014) o Bayesian P-Splines (S eine e al. 2007; B ezge and S eine
2008; Webe and S eine 2012; Lang e al. 2015; Webe e al. 2017). I is clea om
he exis ing s udies in his esea ch a ea ha mo e lexible es ima ion echniques a e
mo e powe ul o unco e complex nonlinea i ies in sales esponse and, i hese a e
a wo k, may lead o much be e sales p edic ions and highe expec ed p o i s.
The conside a ion o (unobse ed) s o e he e ogenei y in sales esponse mod-
eling has become e en mo e es ablished, see he column ‘S o e he e ogenei y’ in
Table1. Some esea che s ha e only included s o e in e cep s in o de o accoun
o he e ogenei y in baseline sales ac oss s o es. The mo e ad anced app oaches
o also cap u e s o e he e ogenei y in ma ke ing e ec s ha e all been embedded
in o Hie a chical Bayesian (HB) es ima ion amewo ks,2 whe e some o hem
ha e accommoda ed bo h he e ogenei y and unc ional lexibili y in p ice e ec s
(H uschka 2006a, 2007; Lang e al. 2015; Webe e al. 2017) and he la e h ee
o hem p ice op imiza ion in addi ion. In e es ingly, some esea che s ound ha
accoun ing o s o e he e ogenei y in a pa ame ic sales esponse model did no
necessa ily imp o e he p edic i e model pe o mance (And ews e al. 2008; Webe
and S eine 2012, 2021), whe eas Webe e  al. (2017) did epo majo ad an-
ages om add essing s o e he e ogenei y as soon as unc ional lexibili y in p ice
esponse was aken in o accoun . Lee lang e al. (2000) al eady e y ea ly called
o mo e in ense esea ch wi h ega d o he compa ison o he e ogeneous e sus
homogeneous s o e-le el sales esponse models, and we pick up hei sugges ion
o model compa ison in ou empi ical s udy la e on.
2 While he single no mal dis ibu ion is ypically used in hese HB models o ep esen s o e he e oge-
nei y on he uppe (popula ion) le el, Wedel and Zhang (2004) used a Di ichle p ocess p io o cap u e
he e ogenei y in p ice e ec s ac oss s o es mo e lexibly.

33
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
Table 1 O e iew o empi ical s udies in he con ex o (s o e-le el) sales esponse modeling conside ing a leas wo o he ollowing dimensions: unc ional lexibili y in
p ice esponse, s o e he e ogenei y in ma ke ing e ec s, p ice dynamics, p ice op imiza ion
S udy P ice dynamics Func ional lexibili y S o e he e ogenei y P ice op imiza ion
G eenlea (1995) Re e ence p ices – – Dynamic p og amming
Mon gome y (1997) – – Hie a chical Bayes (HB) Sequ. quad . p og amming
Foekens e al. (1999) Time- a ying pa am. – S o e in e cep s –
Kopalle e al. (1999) Time- a ying pa am. – S o e in e cep s Dynamic p og amming
VanHee de e al. (2001) – Ke nel eg ession S o e in e cep s –
Fibich e al. (2003) Re e ence p ices – – Analy ical solu ion
VanHee de e al. (2004) Lead and lag p ices Local polynom. eg ession – –
Fok e al. (2006) E o co ec ion model – Hie a chical Bayes (HB) –
H uschka (2006a) – Neu al ne s Hie a chical Bayes (HB) –
H uschka (2006b) – – Hie a chical Bayes (HB) G id sea ch
H uschka (2007) – Neu al ne s S o e clus e s, HB Imp o ing hi -and- un
S eine e al. (2007) – Bayesian P-splines S o e in e cep s –
B ezge and S eine (2008) – Bayesian P-splines S o e in e cep s –
Haup and Kage e (2012) – B-splines S o e in e cep s –
Ho á h and Fok (2013) Lag p ices – Hie a chical Bayes (HB) –
Haup e al. (2014) – B-splines S o e in e cep s –
Lang e al. (2015) – Bayesian P-splines Hie a chical Bayes (HB) G id sea ch
Webe e al. (2017) – Bayesian P-splines Gene al he e ogenei y SUR
model, HB
E olu iona y algo i hm
Asche sleben and S eine (2022) Re e ence p ices Bayesian P-splines – –
This s udy Lag p ices Bayesian P-splines Hie a chical Bayes (HB) Dynamic p og amming
34
P.Asche sleben, W.J.S eine
1 3
P ice dynamics we e only seldom conside ed in models wi h nonlinea p ice
e ec s, VanHee de e al. (2004) and mo e ecen ly Asche sleben and S eine (2022)
a e he excep ions wi h hei empi ical s udies. These wo app oaches, howe e , did
no accoun o s o e-speci ic ma ke ing e ec s. Fok e al. (2006) and Ho á h and
Fok (2013) p oposed Hie a chical Bayesian ec o au o eg ession models o analyze
b and-/ca ego y-speci ic di e ences in dynamic p ice e ec s on sales. Bu he ocus
o hese wo s udies was mo e on he in es iga ion o b and- and ca ego y-speci ic
cha ac e is ics as mode a o s o di e ences in own- o c oss-p ice e ec s be ween
b ands and p oduc ca ego ies a he han on wi hin-b and s o e he e ogenei y in
p ice esponse (especially as Fok e al. 2006 used da a om only one single s o e).
The wo app oaches ha combined p ice dynamics and s o e he e ogenei y (Foekens
e al. 1999; Kopalle e al. 1999) only included s o e in e cep s o conside di e -
ences in baseline sales ac oss s o es. Simila o Lee lang e al. (2000), we he e see
he need o a mo e in ense esea ch o assess he e ec o p ice dynamics in s o e-
le el sales esponse models ha as well accoun o s o e-speci ic e ec s and ideally
also o unc ional lexibili y in p ice esponse. Gene ally, we obse e qui e di e en
s a egies o accommoda e p ice dynamics in sales esponse models: ia lagged and
lead p ices (VanHee de e al. 2004; also, e.g., VanHee de e al. 2000), ia ime- a -
ying pa ame e models (Foekens e al. 1999; Kopalle e al. 1999; also, e.g., A aman
e al. 2010), ia ec o -au o eg essi e speci ica ions (Fok e al. 2006; Ho á h and
Fok 2013; also, e.g., Nijs e al. 2001), as well as ia ma ke -le el e e ence p ices
(G eenlea 1995; Fibich e al. 2003; Asche sleben and S eine 2022).
Only he mino i y o he pape s collec ed in Table1 p o ided no ma i e implica-
ions wi h espec o expec ed p o i s o oexpec ed losses om no using he sales
esponse model wi h he bes p edic i e pe o mance. Like he miles one a icle on
mic o-ma ke ing p icing s a egies by Mon gome y (1997), all mo e ecen op imi-
za ion app oaches used a Hie a chical Bayesian es ima ion amewo k o allow o
s o e-speci ic (o a leas clus e wise) op imal p icing (H uschka 2006a, b, 2007;
Webe e al. 2017; Lang e al. 2015). None o hem, howe e , inco po a ed p ice
dynamics o p ice op imiza ion. On he o he hand, he h ee emaining app oaches
ha conside ed dynamic p ice e ec s on sales esponse (G eenlea 1995; Kopalle
e al. 1999; Fibich e al. 2003) did no allow o p ice op imiza ion by modeling
nonlinea and/o he e ogeneous s o e-le el p icing e ec s. Bu , he indings om he
la e h ee s udies s ill indica ed ha accommoda ing p ice dynamics can inc ease
p o i s o e s a ic p ice op imiza ion.
As ano he impo an issue, se e al o he p ice op imiza ion s udies seem o p o-
ide e idence o co ne solu ions o op imized p ices, i.e., ha op imized p ices
would ha e hi he uppe bounda y o he obse ed p ice ange i his uppe bound-
a y had been imposed as a cons ain in he model.3 In e es ingly, his applies o
all h ee pape s ha inco po a ed p ice dynamics o p ice op imiza ion (G eenlea
1995; Kopalle e al. 1999; Fibich e al. 2003), and hese h ee dynamic op imiza ion
app oaches ha e u he in common ha hey used linea p icing e ec s (i.e., p ice
e ec s es ima ed by a pa ame ic sales esponse model). The co ne solu ion p ob-
lem seems as well e iden in he pape o Mon gome y (1997), who did no conside
3 We hank one e e ee o poin ing us o his bounda y p ice phenomenon.
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1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
p ice dynamics bu also used a pa ame ic sales esponse model (an exponen ial
one) o de e mine op imal p ices. Th ee op imiza ion app oaches we e based on
semipa ame ic sales esponse models, whe e nonlinea p ice e ec s we e es ima ed
using neu al ne s o splines in he desc ip i e model s age (H uschka 2007; Lang
e al. 2015; Webe e al. 2017). Indeed, he indings epo ed in H uschka (2007) and
Webe e al. (2017) sugges ha lexible es ima ion me hods migh be less a ec ed
by he bounda y p icing p oblem.4 No e ha we ound no e idence ha op imized
p ices hi he lowe bound o obse ed p ices. And, nei he pape u he allows con-
clusions abou bounda y e ec s o p edic ed sales quan i ies based on op imized
p ices.
A his poin , i is e y impo an o men ion ha he indings om ou explo a-
i e esea ch on bounda y p ice e ec s should be ea ed wi h cau ion since he co -
ne solu ion p oblem is no explici ly add essed in any o he pape s and in o ma ion
on bounda y e ec s is spa se. In gene al, op imal p ices depend on p ice elas ici ies
and on a iable cos s (in ou case he wholesale p ices o a e aile ). S ill, he p ice
elas ici y in pa ame ic models is much mo e “ igid”, as i depends on one o only
ew p ice pa ame e es ima es which globally de e mine he shape o he esponse
unc ion o e he en i e obse ed p ice ange. Nonpa ame ic es ima ion echniques
like splines, on he o he hand, i he da a locally, en ailing g ea e lexibili y o
unco e ing complex p ice esponse pa e ns and allowing locally a ying p ice elas-
ici ies. This locally i ing p ope y can make lexible models less p one o co ne
solu ions o op imal p ices. No e ha none o he dynamic op imiza ion app oaches
has modeled p ice esponse mo e lexibly, like we p opose in ou dynamic model.
And di e en om all p e ious s udies in his ield, we will pu a special ocus on
he bounda y p icing p oblem in ou empi ical s udy.
Based on he li e a u e e iew, he cen al con ibu ion o his pape is ha ou
p oposed model will allow o p ice op imiza ion by modeling he e ogeneous non-
linea p icing e ec s h ough nonpa ame ic unc ions o own-, c oss-, and lagged
p ice esponse. In o he wo ds, we a e p obably he i s o enable he compu a ion
o uly dynamic p ice pa hs5 o each s o e o a e ail chain due o he e ogeneous
nonlinea p icing e ec s embedded in a semipa ame ic sales esponse model. None
o he p e iously p oposed sales esponse models has combined he h ee ea u es
unc ional lexibili y, s o e he e ogenei y, and p ice dynamics o de i e op imal p ic-
ing s a egies o as -mo ing consume goods.
4 H uschka (2007) de e mined an eigh -clus e solu ion o he s o es o a e ail chain wi h equal op imal
b and p ices o all s o es wi hin a clus e , and op imized p ices la gely a ied ac oss clus e s o mos
b ands conside ed. On he one hand, he la ge a ia ion o op imal p ices migh indica e less bounda y
p oblems. I emained unclea , howe e , how o en he op imal clus e p ices hi he uppe bounda y o
obse ed p ices a he indi idual s o e le el. Webe e al. (2017) p o ided a plo o op imal s o e-speci ic
b and p ices o one b and as example, sugges ing a ai ly low numbe o bounda y e ec s o hei lex-
ible model.
5 No e ha ime- a ying op imal p ices can also esul om s a ic p ice op imiza ion i a iable cos s a e
no cons an o e ime.
36
P.Asche sleben, W.J.S eine
1 3
3 Model speci ica ion andop imiza ion app oach
In he ollowing, we in oduce a semipa ame ic, he e ogeneous, and dynamic s o e-
le el sales esponse model. We use Bayesian P-splines as nonpa ame ic me hod o
es ima e immedia e and lagged p ice e ec s lexibly, allowing us o unco e possi-
bly excep ional p icing e ec s (like dis inc h eshold o sa u a ion e ec s) di ec ly
om he da a. An ad an age o P-splines is ha hey can easily be cons ained o
p o ide mono onic shapes o p ice esponse (B ezge and S eine 2008), which is
easonable om an economic poin o iew o as -mo ing consume goods. Fol-
lowing Lang e al. (2015), he e ogenei y in p ice esponse ac oss s o es is cap u ed
ia s o e-speci ic scaling ac o s, which can be as well easily embedded as addi ional
pa ame e s in o he Gibbs sampling p ocedu e o ou Hie a chial Bayesian es ima-
ion amewo k. Since he shape o he p ice esponse is pooled o e s o es by his
he e ogenei y speci ica ion (i.e., unc ional lexibili y and he e ogenei y in p ice
esponse a e no decoupled), we u he p o ide obus ness checks by a ying he
numbe o kno s and he deg ee o he unde lying B-spline basis unc ions. Dynam-
ics a e accommoda ed ia he one-pe iod lagged own-i em p ice since mo e ime
lags we e no suppo ed by he da a o almos all b ands conside ed. We will also
check he assump ions equi ed o a p ope es ima ion o he models (mul icollin-
ea i y, he e oskedas ici y, au oco ela ion). We apply a disc e e dynamic p og am-
ming algo i hm o de i e op imal p ice pa hs.
3.1 Semipa ame ic he e ogeneous dynamic model
We use he ollowing addi i e sales esponse model wi h smoo h nonpa ame ic and
mul iplica i e andom p ice e ec s.6 Acco dingly, he (log) uni sales o a b and
in a speci ic s o e and week a e assumed o depend on own- and compe i i e p ice
e ec s (whe e he la e a e cap u ed a he quali y ie le el), as well as a dynamic,
one-week lagged own-p ice e ec . We u he include p omo ional ac i i ies o
he b and (use o a display, odd p icing) and accommoda e seasonali y e ec s ia a
smoo h mon hly end and unobse ed s o e-speci ic e ec s. Models a e es ima ed
o each b and sepa a ely, hus b and indices a e omi ed.
whe e
0
is an unknown smoo h nonlinea ime end o he calenda mon h (
m
);
1
is an unknown smoo h nonlinea dec easing unc ion o he b and’s own p ice
(
ps
);
2
is an unknown smoo h nonlinea inc easing unc ion o he b and’s one-
week lagged own p ice (
ps; −1
);
ci
a e unknown smoo h nonlinea inc easing unc-
ions o c oss-p ice e ec s (
pci
s
) cap u ed a he le el o he p ice-quali y ie s
ci
,
(1)
log(q
s
)=𝜂
s
+𝜀
s
= 0(m )+(1+𝛼s1) 1(ps )+(1+𝛼s2) 2(ps; −1
)
+
∑i
(1+𝛼s,c
i
) c
i
(pci
s )+ �
s
𝜸s+𝛽s+𝜀s ,
6 E ec s o ma ke ing ins umen s o he han p ices as well as s o e in e cep s a e cap u ed pa ame i-
cally, which is why he model is called a semipa ame ic model, also see oo no e 1.
43
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
e sion o compa e p obabilis ic and de e minis ic ou comes. Then, i can be used o
measu e he squa ed e o be ween a p obabilis ic o ecas and a poin measu e, he
o me ela ing o he d aw-based p edic ions o a b and’s uni sales and he la e
ela ing o he co esponding obse ed sales o a b and’s uni sales in a ce ain s o e
and week in ou con ex . In his a ian , he
CRPS
can be seen as a con inuous e -
sion o he well-known B ie sco e ha is de ined as he mean squa ed e o be ween
a disc e e ou come
y
and a co esponding p obabilis ic o ecas based on he cd
F
(see, e.g., Gnei ing and Ra e y 2007; Jo dan 2016,pp.37–38):
In empi ical applica ions, he unde lying ue dis ibu ion
F
is commonly unknown.
Using an empi ical cd

Fecd
n
, based on
n
obse a ions, as an app oxima ion o
F
, he
CRPS
sco e can be compu ed by he ollowing consis en modi ica ion o Eq. (11)
(Jo dan 2016, Sec .6; K üge e al. 2021):
Figu e1 illus a es he concep behind he
CRPS
o a be e unde s anding: as he
sco e is calcula ed as he in eg al o e squa ed di e ences be ween wo dis ibu-
ions, wi h one o hem being degene a ed o a poin measu e, he
CRPS
ep esen s
he size o he g ay-shaded a ea be ween he cumula i e dis ibu ion unc ions. In
o he wo ds, he dis ibu ion unc ion o he p edic i e dis ibu ion is compa ed o
he ideal dis ibu ion unc ion o a poin mass in a newly obse ed alue by o ming
he squa ed dis ance and in eg a ing o e i . The mo e he wo dis ibu ion unc ions
o e lap, i.e., he mo e hey coincide, he lowe is he dis ance be ween hem.
In ou case, he d aws sa ed om he Ma ko chain a e con e gence a e samples
om an (unknown) pos e io dis ibu ion
Fpos
, and using his sco ing ule we again
accoun o pa ame e unce ain y. No e ha we ob ain one
CRPS
alue as “coun e -
pa ” o e e y obse a ion
qs
, and a e aging he
CRPS
alues again ac oss all weeks
and s o es p o ides us wi h a mean
CRPS
alue (
MCRPS
) o each holdou :
(11)
CRPS
(F,y)=
�ℝ
(F(z)−1z≥y)2dz
.
(12)
CRPS
(
Fecd
n,y)= 2
n2
n
∑
i=1
(X(i)−y)
(
n1y<X(i)−i+1
2
).
Fig. 1 Example igu e o he
CRPS
sco ing ule: he g ay-shaded a ea ep esen s he in eg a ed squa ed
di e ence be ween he empi ical cumula i e dis ibu ion unc ion

Fecd
n
o a sample o F and he single-
poin empi ical cd (ecd ) o obse a ion y

44
P.Asche sleben, W.J.S eine
1 3
Finally, a e aging he indi idual
MCRPS
alues o e he
C
olds esul s in he
AMCRPS
alue as ou second p edic i e accu acy measu e co esponding o he
ARMSE
. We use he R package sco ingRules (Jo dan e al. 2019) o compu e
he
CRPS
alues acco ding o Eq. (12).
4.3 Es ima ion esul s andp edic i e model pe o mance
4.3.1 Es ima ion esul s
Es ima ion esul s o he mos complex
DynFlexHe
model as desc ibed in Eq. (1)
a e illus a ed o he b and “Minu e Maid” as example in Fig.2. Remembe ha his
model accoun s o he e ogeneous e ec s ac oss s o es, unc ional lexibili y in p ice
e ec s, and p ice dynamics ep esen ed by he one-pe iod lagged own-p ice e ec .
Depic ed a e he he e ogeneous spline es ima es oge he wi h he pa ial esiduals
as well as iolin plo s o he display and 9- o 99-ending p ice e ec s.
(13)
MCRPS
=1
S
S
∑
s=1
1
Ts
T
s
∑
=1
CRPS(Fpos
s ,qs )
,
(14)
⇒
AMCRPS =1
C
C
∑
c=1
1
S
S
∑
s=1
1
T
s
T
(c)
s
∑
=1
CRPS(Fpos ;(c)
s ,q(c)
s )
.
Fig. 2 Es ima ion esul s o he
DynFlexHe
model using he b and "Minu e Maid" as example: es i-
ma ed e ec s and pa ial esiduals o own, lagged, and compe i i e p ice a iables (accoun ing o he -
e ogenei y ia scaled spline slopes) as well as es ima ed e ec s o display and odd p ice endings (he e o-
geneous e ec s displayed by iolin plo s)
45
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
Fi s , es ima es and he co esponding e ec sizes sugges ace alidi y. The s ong-
es e ec is obse ed o he own-p ice e ec , and i is also he one o he p ice e ec s
showing a la ge amoun o he e ogenei y ac oss s o es as is ep esen ed by he ela-
i ely la ge bandwid h o he scalings o he spline unc ions. Mo e speci ically, 21%
o (a o al o 3240) pai wise 80% c edible in e als o he es ima ed scaling ac o s
do no o e lap be ween s o es, con i ming ha own-p ice e ec s signi ican ly di e
be ween many s o es. We will u he discuss below in mo e de ail ha , as a ule, own-
p ice e ec s o all eigh b ands a e much mo e he e ogeneous ac oss s o es han bo h
c oss-p ice e ec s and lagged p ice e ec s (see Fig.4). The o e all slope o he own-
p ice e ec e eals a h eshold e ec nea 1.75$ beyond which sales o “Minu e Maid”
s ongly inc ease. The lagged p ice e ec shows a s eepe slope o small p ice le els
han o medium and high p ice le els (whe e he lagged p ice e ec is a he la ), and
in addi ion a less dis inc h eshold e ec a 2.50$. This sugges s ha cus ome s espond
less o high p ices o “Minu e Maid” in he p e ious pe iod, in e ms o buying mo e
o he b and in he cu en pe iod, han o a e y low p e ious p ice in e ms o buying
less o “Minu e Maid” in he cu en pe iod. In o he wo ds, less pu chases seem o be
pos poned i he p e ious p ice is e y high while a ce ain le el o s ockpiling appea s
o occu in case o a e y low one, albei hese conclusions a e di icul o alida e wi h
agg ega e da a. Some pu chase decele a ion howe e ob iously exis s o p ices la ge
han 2.50$. Compa ed o he own-p ice e ec s (including he lagged p ice e ec ),
c oss-p ice e ec s u n ou a he la wi h sales o “Minu e Maid” being leas (mos )
a ec ed by he p i a e label b and (p emium b ands). In e es ingly, he sales e ec o a
9-ending p ice (excluding a p ice ending in 99) is equen ly (much) la ge han he one
o a 99-ending p ice (excluding o he p ice endings in 9) and u ns ou signi ican ly
posi i e o
93.8
% o he s o es (
D99
:
59.3
%). The display e ec is no signi ican o all
s o es bu one.
Figu e 3 displays he lagged p ice e ec s in he ou dynamic models o he
b and “Minu e Maid”, see he homogeneous and he e ogeneous pa ame ic models
( op le and igh panels) and he co esponding lexible model e sions (bo om
panels). The ad an age o using a lexible app oach ins ead o modeling he e ec
in a pa ame ic way becomes ob ious again, as was al eady isible in Fig.2: non-
linea i ies wi h piecewise s eepe o la e slopes can be modeled wi h Bayesian
P-splines7 bu no wi h pa ame ic unc ions. Ne e heless, a close look a he bo -
om le panel e eals wide con idence bands a he lowes p ice le els, whe e only a
ew obse a ions a e a ailable.8
No e ha he e is i ually no di e ence in he es ima ed own-p ice e ec s
be ween he espec i e s a ic and dynamic model a ian s, as illus a ed in Fig.9
in he Appendix. This implies ha he immedia e own-p ice e ec emains highly
obus e en i he models a e ex ended o cap u e p ice dynamics. Plo s o he es i-
ma ed lagged p ice e ec s o all eigh b ands ob ained by he
DynFlexHe
models
7 No e ha we use he cen e ed sampling me hod o spline es ima ion p o ided in BayesX such ha he
sum o he spline unc ion o e all obse a ions equals ze o.
8 We did no include he con idence bands in he lowe igh panel o he lexible he e ogeneous
dynamic model o p e en clu e . The co esponding con idence bands a he lowe bound o he p ice
ange look highly simila .
46
P.Asche sleben, W.J.S eine
1 3
a e p o ided in Fig.10 in he Appendix, showing e y di e en shapes ac oss b ands
and hence con i ming he bene i s o nonpa ame ic es ima ion as well o he
dynamic p ice e ec s. The comple e es ima ion esul s o all b ands a e a ailable
om he au ho s upon eques .
Figu e4 shows densi y plo s o he es ima ed s o e-speci ic scaling o andom
e ec s pa ame e s o he
DynPa He
and
DynFlexHe
models in o de o ge a
deepe unde s anding abou how much he e ogenei y is inhe en o ou da a and how
Fig. 3 Es ima ed e ec o he
lagged p ice on he sales o
he b and “Minu e Maid” o
he di e en dynamic model
a ian s
Fig. 4 Densi y plo s o s o e-speci ic scaling o andom e ec pa ame e es ima es (cen e ed a ound hei
mean) o he
DynPa He
and
DynFlexHe
models
47
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
his he e ogenei y is handled by hese wo model a ian s9. Some poin s om Fig.4
a e in pa icula no ewo hy. Fi s , as al eady illus a ed in Fig.1 o he b and “Min-
u e Maid”, he e ogenei y ac oss s o es is especially dis inc o he own-p ice e ec ;
his applies mo e o less o all eigh b ands independen whe he he own-p ice e ec
is cap u ed lexibly o pa ame ically. Second, he e ogenei y be ween s o es does no
seem o be an issue o bo h he lagged p ice e ec no o c oss-p ice e ec s i p ice
e ec s a e modeled pa ame ically (
DynPa He
). Howe e , once p ice esponse is
modeled lexibly bo h lagged p ice e ec s and c oss-p ice e ec s do e eal a mod-
e a e amoun o s o e he e ogenei y o all b ands. Lang e al. (2015) ha e p e i-
ously p o ided a possible explana ion o his phenomenon, albei no in he con ex
o p ice dynamics. And hi d, he e is also some he e ogenei y ac oss s o es in he
e ec s o using odd p ices and a display o all b ands and, like o he own-p ice,
he e ec s a e a he s able ac oss he wo models (
DynPa He
,
DynFlexHe
).
4.3.2 P edic i e pe o mance
We e alua e he p edic i e pe o mance o he compe ing models in e ms o he
A e age Roo Mean Squa ed Sales P edic ion E o (
ARMSE
) and he A e age
Mean Con inuous Ranked P obabili y Sco e (
AMCRPS
), which we e in oduced in
Sec .4.2. We use a 9- old subsampling, whe e 8/9 o he da a ( andomly d awn wi h-
ou eplacemen ) a e used each ime o model es ima ion and he emaining 1/9 o
he da a se es as holdou sample. The p edic i e alidi y esul s o he wo pe -
o mance measu es a e epo ed in Table3 o he
ARMSE
measu e and in Table4
o he
AMCRPS
measu e, oge he wi h ela i e imp o emen s o de e io a ions o
he di e en models in p edic i e accu acy compa ed o he p oposed
DynFlexHe
model ( o each b and and agg ega ed a he median) and equencies in how many
cases each model p o ided he bes pe o mance ( ow ‘# bes ’).
F om Table3, we a i s obse e ha di e en models pe o m bes o di e -
en b ands in e ms o he
ARMSE
measu e, i.e., he e seems o be no clea a o i e
model unde his e o measu e a i s glance. Fo se en ou o eigh b ands, a lex-
ible model e sion p o ides he bes p edic i e accu acy, as does a dynamic model
e sion o six o he b ands, while o h ee b ands he bes model is a he e ogene-
ous one. On he o he hand, when agg ega ed ac oss b ands, he p oposed
DynFlex
-
He
model is only sligh ly ou pe o med by he
DynFlexHom
model and p o ides
he second-bes p edic ions, see he median in ela i e changes in he
ARMSE
meas-
u e a he bo om o he able.
In Table4, he pic u e is comple ely di e en and e y clea . Using he
CRPS
measu e o assess he p edic i e model pe o mance, he mos complex model
(
DynFlexHe
) always p o ides he mos accu a e p edic ions, and conce ning he
h ee model dimensions unc ional lexibili y, s o e he e ogenei y, and p ice dynam-
ics he mo e complex model a ian ou pe o ms i s simple coun e pa (i.e.,
9 We he e abs ain om displaying he co esponding densi y plo s o he s a ic he e ogeneous coun e -
pa models (
S a Pa He
and
S a FlexHe
) because he e is i ually no di e ence be ween he s a ic
and dynamic model a ian s excep ha he lagged p ice is no included in he s a ic models.
48
P.Asche sleben, W.J.S eine
1 3
he e ogeneous models pe o m be e han homogeneous ones, lexible models be e
han pa ame ic ones, and dynamic models be e han s a ic ones).
To alida e ou conjec u e ha he
ARMSE
measu e is mo e p one o ex eme
obse a ions (e.g., e y high sales a e y low p ices) han he
AMCRPS
meas-
u e, we compu ed he A e age Roo Median Squa ed E o (
ARMedSE
) as u he
pe o mance measu e, which is known o be much mo e obus agains ex eme
obse a ions compa ed o he
ARMSE
 (F anses and Ghijsels 1999). Acco ding
o he
ARMedSE
measu e, he
DynFlexHom
model pe o ms bes o all b ands,
and dynamic and lexible models almos always pe o m be e han hei s a ic
and pa ame ic coun e pa s – jus as when using he
AMCRPS
. Di e en om
he
AMCRPS
esul s, homogeneous models a e p e e ed o he e ogeneous ones in
Table 3 Ou -o -sample p edic i e pe o mance o he compe ing models e alua ed by he A e age Roo
Mean Squa ed Sales P edic ion E o (
ARMSE
) in holdou samples compa ed o he p oposed
DynFlex
-
He
model. Bes models pe b and a e ma ked in bold
The bo om lines show he median ela i e imp o emen / de e io a ion o each model (i.e., ac oss
b ands) compa ed o he
DynFlexHe
model and he numbe o imes a conside ed model ou pe o med
he o he models
S a Dyn
Pa Flex Pa Flex
Hom He Hom He Hom He Hom He
Flo . Na l. 26.5 26.8 25.4 28.5 23.8 23.824.6 26.6
(−0.2%)
(+1.1%)
(−4.5%)
(+7.5%)
(−10%)
(−10%)
(−7.2%)
–
T opic. Pu e 57.4 57.1 53.0 52.1 55.9 55.1 52.2 51.1
(+12%)
(+12%)
(+3.8%)
(+1.9%)
(+9.4%)
(+7.8%)
(+2.1%)
–
Ci us Hill 120.7 122.1 101.1 135.0 104.7 107.1 86.0107.4
(+12%)
(+14%)
(−5.8%)
(+26%)
(−2.5%)
(−0.3%)
(−20%)
–
Flo . Gold 57.5 57.7 55.6 55.9 55.5 55.8 53.453.8
(+
6.8
%)
(+
7.3
%)
(+
3.4
%)
(+
3.9
%)
(+
3.1
%)
(+
3.7
%)
(−
0.8
%)
–
Min. Maid 53.8 53.3 52.5 52.054.2 53.6 53.2 52.8
(+1.8%)
(+0.9%)
(−0.6%)
(−1.6%)
(+2.6%)
(+1.4%)
(+0.7%)
–
T ee F esh 83.6 83.1 64.483.7 81.1 80.6 64.5 79.7
(+4.8%)
(+4.2%)
(−19%)
(+5.0%)
(+1.7%)
(+1.1%)
(−19%)
–
T opicana 110.4 112.3 105.4 108.1 100.9 101.9 93.794.6
(+17%)
(+19%)
(+11%)
(+14%)
(+6.7%)
(+7.7%)
(−1.0%)
–
Dominick’s 311.7 316.1 307.5 313.8 300.9 306.6 278.5281.1
(+11%)
(+12%)
(+9.4%)
(+12%)
(+7.0%)
(+9.1%)
(−0.9%)
–
median
+8.9%
+9.5%
+1.4%
+6.2%
+2.8%
+2.5%
−1.0%
–
# bes – – 1 1 – 1 4 1

49
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
nea ly all cases.10 Based on he o e all conside a ion o he esul s o all h ee p e-
dic i e alidi y measu es (
ARMSE
,
AMCRPS
,
ARMedSE
) we can ecommend he
use o ei he he
DynFlexHe
o
DynFlexHom
o sales p edic ions.
4.3.3 Fu he es s and obus ness checks
We also conduc ed es s o all models (by b and) o assess he assump ions
equi ed o a p ope es ima ion o hem (mul icollinea i y, he e oskedas ici y,
Table 4 Ou -o -sample p edic i e pe o mance o he compe ing models e alua ed by he A e age Mean
Con inuous Ranked P obabili y Sco e (
AMCRPS
) in holdou samples compa ed o he p oposed
Dyn
-
FlexHe
model. Bes models pe b and a e ma ked in bold
The bo om lines show he median ela i e imp o emen / de e io a ion o each model (i.e., ac oss
b ands) compa ed o he
DynFlexHe
model and he numbe o imes a conside ed model ou pe o med
he o he models
S a Dyn
Pa Flex Pa Flex
Hom He Hom He Hom He Hom He
Flo . Na l. 10.1 9.8 9.5 8.6 9.2 8.8 9.0 8.1
(+24%)
(+20%)
(+17%)
(+6.0%)
(+14%)
(+8.1%)
(+11%)
–
T opic. Pu e 24.4 23.2 22.8 20.8 23.3 21.8 22.1 19.9
(+23%)
(+17%)
(+15%)
(+4.5%)
(+17%)
(+9.6%)
(+11%)
–
Ci us Hill 28.9 27.8 24.7 21.9 24.8 23.6 21.5 18.8
(+53%)
(+47%)
(+31%)
(+16%)
(+32%)
(+25%)
(+14%)
–
Flo . Gold 20.4 19.9 19.5 18.8 19.6 19.2 18.6 18.0
(+
14
%)
(+
11
%)
(+
8.5
%)
(+
4.4
%)
(+
8.8
%)
(+
6.4
%)
(+
3.5
%)
–
Min. Maid 21.8 20.6 20.9 19.3 20.8 19.5 20.1 18.6
(+17%)
(+11%)
(+12%)
(+3.9%)
(+12%)
(+4.8%)
(+8.1%)
–
T ee F esh 19.2 18.5 15.9 13.7 18.8 18.1 15.8 13.5
(+42%)
(+37%)
(+17%)
(+0.8%)
(+39%)
(+33%)
(+17%)
–
T opicana 55.0 53.3 52.0 49.8 50.9 49.4 46.7 44.2
(+25%)
(+21%)
(+18%)
(+13%)
(+15%)
(+12%)
(+5.8%)
–
Dominick’s 150.9 148.3 145.0 140.2 144.4 141.4 134.6 128.2
(+18%)
(+16%)
(+13%)
(+9.4%)
(+13%)
(+10%)
(+5.0%)
–
median
+24%
+19%
+16%
+5.2%
+14%
+9.9%
+9.5%
–
# bes – – – – – – – 8
10 The
ARMedSE
esul s can be ob ained om he au ho s upon eques . We u he conduc ed a small
simula ion s udy o es he sensi i i y o he
CRPS
in compa ison o he
RMSE
agains ou lie s. Fo his,
we i s gene a ed 100 obse a ions om he s anda d no mal and added in a second s ep one o wo
ou lie s o he obse a ions. The
RMSE
inc eased conside ably, while he
CRPS
emained highly obus
e e y ime. This means ha he
CRPS
weighs cen al obse a ions mo e s ongly han mo e ex eme
obse a ions compa ed o he
RMSE
.
50
P.Asche sleben, W.J.S eine
1 3
au oco ela ion), and b ie ly summa ize esul s o he p oposed
DynFlexHe
model
in he ollowing.11 Mul icollinea i y was no a p oblem in any case: a iance in la-
ion ac o s (VIF) we e ne e highe han 2.5 ac oss b ands and hus a om being
c i ical, whe e as a ule he highes VIFs could be obse ed o he own-i em p ice
co a ia es (immedia e and lagged p ices). He e oskedas ici y was e alua ed using
he B own-Fo sy he es , which is obus agains de ia ions om no mali y (B own
and Fo sy he 1974). We applied he es o compa e he a iance o he esiduals
agains he i ed alues, and esul s we e mo e mixed he e. The B own-Fo sy he es
was no signi ican o ou b ands, indica ing ha he e oskedas ici y is no an issue
he e (e.g., wi h
p=0.82
o “Minu e Maid”), while i u ned ou signi ican o he
o he ou b ands (
p<0.01
). We u he used a modi ied e sion o he Du bin-Wa -
son es o panel da a o assess au oco ela ion o he esiduals, which is as well
obus agains de ia ions om he no mal dis ibu ion (Bha ga a e al. 1982; Ali and
Sha ma 1993). Gi en he numbe o co a ia es and sample size pe b and (be ween
6624 and 6827 obse a ions), he lowe c i ical alue a ound 1.85 is sligh ly unde -
cu o wo b ands (
𝛼=0.05
), indica ing posi i e au oco ela ion only o hese wo
b ands. We ound no e idence o nega i e au oco ela ion ac oss b ands. He e -
oskedas ici y and au oco ela ion lead o biased es ima es o s anda d e o s and can
a ec he alidi y o s a is ical es s, wi h he esul o a possibly misleading s a is i-
cal in e ence. Howe e , as ou ocus lies on he p edic i e model pe o mance and
ela ed p o i implica ions, iola ions o hese assump ions seem less se e e.
We u he pe o med obus ness checks s a ing wi h modi ica ions o he speci-
ica ion o he P-splines. By de aul , we used 20 kno s and B-spline basis unc ions
o deg ee 3 o es ima e p ice e ec s lexibly. In o de o assess he sensi i i y o
he p edic i e powe (
AMCRPS
,
ARMSE
) o he
DynFlexHe
and
DynFlexHom
models depending on he numbe o kno s and/o he deg ee o he B-spline basis
unc ions, we u he es ima ed hem wi h a lowe deg ee o he B-splines (deg ee 1)
and/o mo e kno s (40 kno s, co esponding o he sugges ed uppe bound by Eile s
and Ma x 1996) o he immedia e and lagged own-p ice e ec s. The
CRPS
sco es
u ned ou highly obus o all b ands and, excep o one b and, he
RMSE
measu e
as well, indica ing ha 20 kno s a e su icien and B-splines o deg ee 1 would pe -
o m compa able. De ails a e p o ided in Tables7 and 8 in he Appendix.
Finally, we conside ed only one ime lag in he dynamic model e sions o con-
side p ice dynamics. By de ini ion, his implies ha only sho - e m p ice-change
o sho -li ed pos -p omo ion e ec s can be analyzed, while longe pe sis ing ma -
ke ing e ec s canno be accommoda ed. To es i he da a suppo highe o de
au o eg essi e s uc u es, we added wo mo e lags o he own p ice o he
Dyn
-
FlexHe
and
DynFlexHom
models (wi h he de aul se ings o 20 kno s and deg ee
3 o he P-splines), cap u ing he highe -o de lag dynamics (lag 2, lag 3) pa ame -
ically o keep he model complexi y manageable. Again, he
CRPS
sco es emained
ex emely obus in bo h models excep o he s o e b and, whe e adding a second
p ice lag somewha imp o ed he p edic i e pe o mance. In e ms o
ARMSE
,
11 The comple e esul s o all es s and models can be ob ained om he au ho s upon eques . No e ha
mul icollinea i y measu es only di e ac oss b ands and no ac oss models, because all models sha e he
same co a ia es pe b and.
51
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
adding mo e lags did no pay o o all b ands in he
DynFlexHom
model and o
six b ands (including he s o e b and) in he
DynFlexHe
model. De ails abou hese
dynamic model ex ensions a e p o ided in Tables9 and 10 in he Appendix.
4.4 Op imiza ion
4.4.1 Se ings andcons ain s
In o de o ind op imal (i.e., p o i maximizing) s o e-speci ic p ices o each b and
m
, we use he dynamic p og am
(P)
in oduced in Sec .3.3. As he ac ion space o
de e mining a b and’s op imal p ice in pe iod
is independen o he cu en s a e
(i.e., he b and’s p ice in he p e ious pe iod), he se o gene ally possible p ices
o he b and is cons an o e ime, hence in each pe iod he same wi h
X (s )=X
.
And since
s +1=g(x ,s )=x
, he s a e space equals he ac ion space (
S=X
). Fo
each b and and s o e, we se indi idual s a e and ac ion spaces as o con ain a g id o
possible p ices in 0.01$ s eps be ween he minimum and maximum obse ed p ice
le els, deno ed as [ p
obs
m,i,min
; p
obs
m,i,max
], o he conside ed b and
m
in s o e
i
:
The es ic ion o op imized p ices o he ange o obse ed p ice le els is chosen o
p ese e he e aile ’s p ice image pe cei ed by cus ome s. I u he helps o ob ain
ealis ic p edic ions, especially when using lexible unc ions ha i he unde lying
da a locally such ha ex apola ions could be s ill mo e p oblema ic compa ed o
pa ame ic unc ions.
To s a he ecu si e p ocess and sol e he dynamic op imiza ion p oblem, he
ini ial s a e
s1=p0
has o be ixed. As s a ing alue, we decided o use he p ice in
he middle o he obse ed p ice ange o a conside ed b and and s o e. By conduc -
ing some sensi i i y analyses, we checked ha op imal p ice pa hs we e ai ly obus
agains a ia ions o his s a ing alue. We u he se he discoun a e
o 0 wi h-
ou loss o gene ali y.
As a second cons ain , we limi he p edic ed sales
q
o a b and in a ce ain s o e
o he la ges numbe o sales
qobs
m,i,max
obse ed o his b and in his s o e in ou
da a. Using his second cons ain le s us s ay conse a i e and p e en s us om an
o e es ima ion o a b and’s uni sales beyond he uppe bound o obse ed uni sales
pe week and s o e o ( e y) low p ices. This gua an ees ealis ic sales scena ios
as obse ed in he da a on he one hand, bu may im he g ea e lexibili y o he
spline es ima es on he o he hand. In o he wo ds, i he sales o ecas o a b and
in a s o e om one o ou es ima ed models is highe compa ed o he maximum
obse ed sales, we unca e he p edic ion o he maximum o obse ed sales:
(15)
S
=Smi =
{
pobs
m,i,min,pobs
m,i,min +0.01, …,pobs
m,i,max
}
=Xmi =X
.
(16)

q
=min
{
q,qobs
m,i,max
}.
52
P.Asche sleben, W.J.S eine
1 3
No e ha p edic ions
q
a e again made a he d awle el as al eady desc ibed in Eq.
(8), i.e., we in eg a e he op imiza ion c i e ion o e he pos e io dis ibu ion o he
pa ame e s by a e aging he op imiza ion c i e ion ac oss d aws, be o e op imizing
i . To u he speed up op imiza ion, we use e e y en h d aw om he sa ed d aws
o he Ma ko chain a e con e gence.
4.4.2 Op imal p ice pa hs
We use o wa d ecu sion o sol e he op imiza ion p oblems and gene a e 5184
op imal p ice pa hs in o al (81 s o es imes 8 b ands imes 8 sales esponse models).
I seems easonable o assume ha op imal p ice pa hs migh a y depending on he
s o e-speci ic scaling ac o s
(1+𝛼s1)
. Fo illus a ion, we con inue wi h he b and
“Minu e Maid” as example and choose o his b and h ee ou o he 81 s o es wi h
a low s.medium s.high scaling ac o o he own-p ice e ec es ima ed by he
DynFlexHe
model.
Figu e5 shows op imized p ice pa hs o he b and “Minu e Maid” o he h ee
selec ed s o es and all he e ogeneous model a ian s. Fu he depic ed a e a iable
cos s (also p o ided in he da a) and obse ed p ices as benchma ks o op imized
p ices (panels in he op ow). Obse ed p ices show a e y high a ia ion du ing
he i s 25 weeks, swi ching back and o h mos ly be ween only wo di e en p ice
le els wi h no mo e han h ee consecu i e weeks a he high p ice le el and no mo e
han wo consecu i e weeks a he lowe p ice le el (no e ha he high p ice le el is
highes in he s o e wi h he high scaling pa ame e , 3.17$, and lowes o he s o e
wi h he low scaling pa ame e , 2.62$.) This clea hi-lo p ice pa e n dilu es a e he
i s 25 weeks, bu some p ice le els s ill show up mo e o en han o he s (e.g., 1.99$
in all h ee selec ed s o es), and p ice cu s o a ying dep hs con inue o occu . Also
no e ha he gene al p ice le el o “Minu e Maid” dec eases wi h dec easing a i-
able cos s (wholesale p ices), as expec ed.
The op imized p ices esul ing om he
S a Pa He
model ( ed lines in he
middle ow) “ ollow” he cos s e y closely, i.e., inc eased cos s lead o inc eased
p ices and ice e sa, and he co ela ion be ween obse ed cos s and op imized
p ices (0.92) is conside ably highe han he co ela ion be ween obse ed cos s
and obse ed p ices (0.60). I is u he easonable ha op imized p ices end in 9
o 99 since odd p icing was shown o ha e posi i e e ec s on expec ed sales (see
Sec .4.3.1). This op imized p ice pa e n in pa s also holds o he
S a FlexHe
model (middle ow; blue lines), bu wi h a mo e clea hi-lo p icing scheme and a
smalle numbe o di e en p ice le els. Comple ely di e en op imal p ice pa hs
a e ob ained o he dynamic models. Fo he s o e wi h he low scaling pa ame-
e o he own-p ice e ec (bo om le panel), op imized p ices esul ing om he
DynPa He
model a y in a much smalle bandwid h. Small p ice cu s a e obse ed
p ima ily in he second hal o he da a ime window, and he high p ice le el is
cons an o e ime. Op imized p ices ob ained om he
DynFlexHe
model ha dly
show any a ia ion (bo om ow; blue lines). Deep p ice cu s a e obse ed in only
h ee weeks, in all o he weeks he op imal p ice is se consis en ly high a he same
unique p ice le el as sugges ed by he
DynPa He
model (bo om ow; ed lines).
59
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
di e en p edic i e alidi y measu es used. Ne e heless, mo e complexi y always
p o ided a be e o ecas ing accu acy acco ding o he
AMCRPS
measu e, sugges -
ing ha he
DynFlexHe
model was he model o choice he e, while he
ARMedSE
measu e unequi ocally ended o he
DynFlexHom
model ins ead. Rega ding he
ARMSE
measu e, a leas one o hese wo models (
DynFlexHe
,
DynFlexHom
)
also pe o med bes o i e b ands as well as no d ama ically wo se o he o he
h ee b ands. The e o e, wi hou loss o gene ali y, we choose he
DynFlexHe
model o compu e expec ed losses nex .
In pa icula , we ake he op imal p ices o a b and de e mined by he
DynFlex
-
He
model and calcula e he o al (s o e) p o i o his b and (i.e., summing up o e
all weeks) unde he
DynFlexHe
model. Expec ed losses in o al (s o e) p o i can
hen be compu ed when using op imal p ices de e mined by a di e en model bu
inse ed in he
DynFlexHe
model o expec ed p o i calcula ions (c ., e.g., Lang
e al. 2015). No e ha expec ed s o e p o i s om he
DynFlexHe
model based on
op imal p ices o a di e en model can u n ou highe han expec ed p o i s om he
DynFlexHe
model based on i s ue (i.e., own) op imal p ices in pa icula weeks
bu no when summed up o e all weeks. The e o e, expec ed losses in o al (s o e)
p o i s a e nega i e by de ini ion compa ed o he assumed “ ue” model.
Figu e7 shows box plo s o expec ed losses by s o es o he di e en he e o-
geneous model al e na i es (homogeneous models a e omi ed in he igu e as hey
p o ided pa e ns highly simila o hose o hei he e ogeneous coun e pa s). One
impo an inding he e is ha expec ed losses ela i e o he
DynFlexHe
model a e
gene ally no la ge han 10% o he
DynPa He
model ac oss all b ands (al hough
e en a loss o “only” 10% can ansla e in o a huge loss exp essed in mone a y
uni s). The leas complex he e ogeneous model (
S a Pa He
) ei he leads o he
highes expec ed losses o six b ands (wi h a maximum median expec ed loss o
26
% and la ges indi idual s o e losses o up o
−40
% ac oss b ands) o expec ed
losses simila ly high as o i s lexible coun e pa (
S a FlexHe
). The
DynFlex
-
Hom
model on he o he hand p o ides he mos inconsis en esul s he e: i shows
(much) highe expec ed losses han he
DynPa He
model o i e b ands bu
pe o ms excellen ly o he o he h ee b ands, whe e i comes up wi h only e y
small o almos negligible expec ed losses. O e all, he
DynPa He
seems o be he
mos obus model a ian o educe expec ed losses ela i e o he p oposed
Dyn
-
FlexHe
model. The e o e, a leas o he da a a hand, he use o a he e ogeneous
dynamic model o p icing decisions is s ongly sugges ed. The main (p elimina y)
conclusion esul ing om ou op imiza ion exe cise is ha nonlinea he e ogeneous
dynamic p icing p o ides be e p icing decisions and highe expec ed chain p o i s,
and can p e en losses due o subop imal p icing. Tha ’s why we ecommend o p e-
e he p oposed
DynFlexHe
model.
Finally, i is impo an o no e ha op imizing p o i s o e alua ing expec ed
losses based on an econome ic model like we do is p one o he Lucas c i ique, a
i s glance. T ans e ed o ou con ex , Lucas (1976) s a es ha since he s uc u e o
an econome ic model in ol es op imal decision ules o b and manage s and since
hese ules change sys ema ically wi h he s uc u e o he ime se ies da a ele an
o he e aile ’s policy, any change in he e aile ’s policy will al e he s uc u e o
he econome ic model (Lucas 1976,p.41). Howe e , he ocus o his pape does

60
P.Asche sleben, W.J.S eine
1 3
no lie on he e alua ion o he e aile ’s p icing policy bu on he compa ison o
he s a is ical and no ma i e capabili ies o di e en a ian s o econome ic models
and in pa icula on assessing he bene i s o including unc ional lexibili y in p ice
esponse, s o e he e ogenei y, and/o p ice dynamics in a sales esponse model.
Fig. 7 Box plo s o expec ed losses by s o e (agg ega ed o e weeks) o he he e ogeneous model a i-
an s using he
DynFlexHe
model as “ ue” model
Fig. 8 Dis ibu ions o s o e-speci ic sha es o op imized p ices hi ing he uppe bound o he obse ed
p ice ange o he dynamic model a ian s
DynPa He
,
DynFlexHom
, and
DynFlexHe
61
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
4.4.5 Bounda y p icing e ec s
All op imiza ion esul s discussed abo e would be o less alue i op imized p ices
always o equen ly hi he bounda ies o he obse ed s o e-speci ic p ice anges. In
he ollowing, we illus a e he bounda y p icing issue o he ollowing h ee mod-
els: he
DynFlexHe
, which p o ided he highes expec ed p o i s o mos b ands
(and as well on a e age ac oss b ands) and he mos accu a e sales p edic ions based
on he
CRPS
sco e; he
DynFlexHom
, which as well showed an excellen o ecas ing
accu acy based on all h ee p edic i e alidi y s a is ics (
CRPS
,
RMSE
,
RMedSE
);
and he
DynPa He
, which consis en ly (i.e., ac oss b ands) led o he leas expec ed
losses compa ed o he
DynFlexHe
. We included he la e model also because
he e seemed o be e idence om ou li e a u e sea ch ha dynamic pa ame ic
models we e especially p one o uppe bound co ne solu ions, compa e Sec .2. Fig-
u e8 displays he dis ibu ions o s o e-speci ic sha es o op imized p ices hi ing
he uppe bound o he obse ed p ice ange, bo h by b and and ac oss all b ands.
The plo s indica e ha bounda y p icing is no a gene al p oblem by b and o model
(in he sense ha he bounda y is always o mos ly hi o ce ain b ands o models)
and ha he occu ence o co ne solu ions is no c i ical o mos b ands. Fo ou
example b and “Minu e Maid”, op imized p ices almos ne e hi he uppe bound
in mos s o es and in he wo s case in 12% o he weeks in a s o e. Fo wo b ands
(“Ci us Hill”, “T ee F esh”), howe e , he uppe p ice bound is hi (much) mo e
o en in a la ge numbe o s o es. On he o he hand, we obse e highly di e en
sha es o bounda y hi s ac oss he s o es o hese wo b ands (especially o he
he e ogeneous models), including s o es wi h ze o sha e (no uppe bounda y hi s) as
well. Ac oss b ands and he h ee dynamic models shown, 50% (75%) o he s o es
show uppe bounda y p ice hi s in less han 10% (35%) o he weeks (no displayed
in he igu e). As i is u he ob ious, he uppe p ice bound pa e ns di e mo e by
b ands han by models, which consis en ly applies o he o he models no shown,
oo.
Ou indings con adic he assump ion ha pa ame ic models o models wi h
linea p ice e ec s migh be gene ally p one o bounda y p icing e ec s. Howe e ,
we consis en ly ind mo e s o es wi h highe sha es o uppe bounda y p ice hi s o
he dynamic models compa ed o hei s a ic coun e pa s. As a ule, we also see a
(much) g ea e bandwid h o s o es wi h di e en sha es o uppe bounda y p ice
hi s o he e ogeneous models compa ed o hei homogeneous coun e pa s ac oss
b ands, as one migh ha e expec ed. No such sys ema ic can be de i ed o lexible
e sus pa ame ic models.
In con as o uppe bounda y p ice hi s, lowe bounda y p ice hi s as well as
uppe bounda y sales hi s we e ha dly obse ed. The sha e o lowe bound p ice
hi s a e aged ac oss s o es is no la ge han 1% by bo h b and and ype o model.
The numbe o uppe bounda y sales hi s ne e exceeds 5 imes in a s o e ac oss
b ands and models, co esponding o a maximum sha e o 6%. The de ailed esul s
on bounda y e ec s by b and o all models can be ob ained om he au ho s upon
eques .
62
P.Asche sleben, W.J.S eine
1 3
5 Conclusions andlimi a ions
In his pape , we p oposed a Hie a chical Bayesian semipa ame ic s o e sales model
wi h p ice dynamics o op imal p icing o as -mo ing consume goods o e ed in
physical s o es o a e aile and maximizing ela ed expec ed b and p o i s. Using
Bayesian P-splines as a nonpa ame ic echnique o es ima e p ice e ec s allowed us
o dispense wi h he speci ica ion o a speci ic unc ional o m o own- and c oss-
p ice esponse a p io i and o iden i y possible “i egula ” p icing e ec s (e.g., kinks
and s eps in p ice esponse) ha a e di icul o cap u e wi h pa ame ic unc ions.
He e ogenei y o p ice e ec s ac oss s o es was accommoda ed ia scaling ac o s
o he P-splines, which se e as andom e ec pa ame e s o scale he p ice unc-
ions up- o downwa ds o indi idual s o es while p ese ing hei o e all shape.
Accommoda ing he e ogenei y has become s a e-o - he-a i no a mus in econo-
me ic ma ke ing models, a beyond he p esen con ex o s o e-le el sales esponse
models. P ice dynamics we e accoun ed o ia one-week lagged own p ices, which
allowed us o implici ly add ess s ockpiling o cus ome -holdo e e ec s, al hough
such dynamic e ec s a e mo e di icul o explo e based on agg ega e sales da a.
Fo his eason, we u he p o ided obus ness checks o he dynamic a ian s o
he lexible nonlinea models by including mo e ime lags o he own-i em p ice,
indica ing ha highe o de au o eg essi e s uc u es we e (mos ly) no suppo ed by
he da a. Op imal p ice pa hs o b ands we e de e mined by a disc e e dynamic p o-
g amming algo i hm. We u he imposed addi ional cons ain s o he p ice op i-
miza ion s ep on p ice anges and uppe bounds o b and sales o p ese e a ealis ic
scena io o p o i implica ions. To he bes o ou knowledge, add essing he h ee
dimensions unc ional lexibili y, s o e he e ogenei y, and p ice dynamics simul ane-
ously in one app oach (bo h o sales esponse modeling and subsequen p ice op i-
miza ion) has no been p oposed p e iously. The e o e, his app oach allows us o
he i s ime o disen angle he e ec s o hese h ee dimensions on sales, p icing
and p o i s. In addi ion, we in oduced he Con inuous Ranked P obabili y Sco e as
a new measu e o assess he p edic i e model pe o mance, a measu e ha has no
been used be o e in he sales esponse modeling ield.
We applied ou p oposed app oach in an empi ical s udy o da a o e ige a ed
o ange juice b ands o e ed in physical s o es o a la ge e ail chain and compa ed
i o se e al benchma k models (which igno e he e ogenei y, unc ional lexibili y,
and/o p ice dynamics) in e ms o o ecas ing accu acy, op imal p icing o p ice
pa hs, op imized p o i s, and expec ed losses. Based on h ee di e en p edic i e
pe o mance measu es, we ound ha accommoda ing bo h unc ional lexibili y in
p ice esponse and p ice dynamics p o ided he bes sales p edic ions. Mo eo e ,
sales p edic ions om all models we e almos ai ly well dimensioned wi h ega d
o obse ed b and sales, i.e., he cons ain s imposed on he p ice op imiza ion
s ep wo ked well and p o ec ed ou models om o e - o unde es ima ion biases
in sales p edic ions and as a esul in expec ed p o i calcula ions, oo. In addi ion,
op imized expec ed p o i s we e highes unde he p oposed lexible, he e ogeneous,
and dynamic sales esponse model o i e b ands and by ei he again a lexible o a
63
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
dynamic model o he o he h ee b ands. Impo an ly, op imized expec ed p o i s
we e highes o six b ands based on models including p ice dynamics, sugges ing
ha i is e y impo an o manage s o accommoda e p ice dynamics o p icing
s a egies as well, a leas o ou da a a hand. No leas , he bes models by b and
we e all he e ogeneous ones, and he p oposed lexible, he e ogeneous, and dynamic
sales esponse model p o ided he highes expec ed p o i s on a e age ac oss b ands.
As ou i s esea ch ques ion was whe he a sales esponse model ha combines
nonlinea p icing, s o e he e ogenei y in p ice (and o he ma ke ing) e ec s, and
p ice dynamics can p o ide highe expec ed chain p o i s, he answe o his ques-
ion is yes. The bene i s om accommoda ing p ice dynamics o op imal p icing
decisions we e also clea ly isible in ou analyses on expec ed losses. Fu he mo e,
and his is a e y impo an poin , we could show ha uppe bounda y p icing
e ec s we e usually mode a e ac oss b ands and ne e c i ical. In addi ion, he s o e-
speci ic sha es o p ice hi s a he uppe bound u ned ou highly he e ogeneous o
some b ands. And lowe bounda y p ice hi s as well as uppe bounda y sales hi s
occu ed only in e y ew cases. As such, ou second esea ch ques ion can also be
posi i ely e alua ed in he sense ha bounda y p icing e ec s a e no a c i ical issue
o he p oposed model, a leas no o he da a a hand. This inding oge he wi h
he o he ones summa ized abo e le us ecommend he use o he new model o
p ice op imiza ion.
Some limi a ions o ou s udy should also be add essed. Fi s , we ook he pe -
spec i e o a b and manage , who should be in e es ed in analyzing he consequences
o subop imal p icing s a egies om using a “w ong” sales esponse model o he /
his b and, assuming ixed p icing pa e ns o subs i u e b ands and he possibili y
o adjus he p ice o he own b and in he e aile ’s asso men . This b and pe spec-
i e is also an op ion o p oduc ca ego y manage s o e aile s o analyze how p ice
changes o one b and in hei asso men would a ec ca ego y sales and p o i s.
Ne e heless, a nex s ep could hen be an ex ension o ou model o op imal ca -
ego y p icing, using a seemingly un ela ed eg ession (SUR) app oach o model
es ima ion as p oposed, e.g., by Webe e al. (2017). Second, o he op ions o accom-
moda ing p ice dynamics could be conside ed, o example by allowing o ime- a -
ying p ice e ec s ins ead o using simple lagged p ices (leaning on epa ame iza-
ion app oaches such as p oposed by Foekens e al. 1999 o Kopalle e al. 1999).
Also, lead p ice e ec s could be added o accoun o an icipa o y esponses o con-
sume s as a esul o p ice p omo ion announcemen s ( ollowing, e.g., VanHee de
e al. 2000, 2004). Thi d, ou app oach could be u he ex ended o allow o syn-
e gy e ec s be ween ma ke ing ins umen s. Fo example, A aman e al. (2010) p o-
posed a dynamic model ha ela es b and sales o he long- e m ma ke ing s a egy
ep esen ed by ad e ising, p oduc , and place in addi ion o p ice. Finally, we did
no ea he issue o p ice endogenei y, which, i igno ed, could lead o biased e ec
es ima es. A good ins umen o accoun o endogenei y in sales esponse models is
he lagged p ice. Howe e lagged p ices a e no exogenous in ou dynamic model(s)
and he e o e couldno be aken as ins umen he e. Wholesale p ices, in ou case
64
P.Asche sleben, W.J.S eine
1 3
used as cos s in ou p o i unc ions, ep esen ano he candida e o an ins umen-
al a iable. As objec ed by Rossi (2014), howe e , e ail p ices o en show a much
highe a ia ion han wholesale p ices. This also holds o many s o es in ou da a, as
was isible om Fig.2 ( op ow) o he b and “Minu e Maid” as example. Fu he -
mo e, wo-s ep es ima ion as bes known ins umen al a iable echnique is es ic ed
o linea models and canno be applied o ou nonlinea models (see, e.g., H uschka
2017). Mo eo e , Ebbes e al. (2011) showed ha models which accoun o endoge-
nei y ia ins umen al a iables do no necessa ily p o ide be e p edic ions. In he
absence o adequa e ins umen s one could use ins umen - ee echniques o accom-
moda e endogenei y. H uschka (2017) has ecen ly p o ided an o e iew o he a i-
ous op ions he e, emphasizing, howe e , ha cu en ly only he copula-based me hod
by Pa k and Gup a (2012) could be ex ended o nonlinea models like he ones p o-
posed in his a icle (c . H uschka 2017,p.28). To he bes o ou knowledge, such
an ex ension o he copula-based me hod has no ye been de eloped o he lexible
scaling models we conside . We he e o e lea e he issue o endogenei y as well as
he o he limi a ions men ioned abo e, o u u e esea ch.
Appendix
Addi ional igu es
See Figs.9, 10 and 11.
Fig. 9 Compa ison o es ima ed own-p ice e ec s be ween s a ic and dynamic model a ian s o he
b and “Minu e Maid”

65
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
Fig. 10 Es ima ed lagged p ice e ec s esul ing om he
DynFlexHe
model o all eigh b ands in he
e ige a ed o ange juice ca ego y
66
P.Asche sleben, W.J.S eine
1 3
Fig. 11 Fo each b and, he numbe s in he uppe iangula pa o he panels e e o he ela i e e-
quencies wi h which each wo di e en models sha e he same op imal p ice le els in a pa icula s o e
and week. Addi ionally, he diagonal elemen s indica e he ela i e equencies whe e he op imized p ice
and he obse ed p ice pe s o e and week coincide o a pa icula model. Highligh ed by colo ed bo -
de s a e he pai wise compa isons be ween models ha a e mo e simila o each o he , i.e., ha di e in
only one o he h ee model dimensions ( unc ional lexibili y, s o e he e ogenei y, p ice dynamics). Fo
example, he compa ison be ween he
DynFlexHom
model and he
S a FlexHom
model is amed in ed
as bo h models a e lexible and homogeneous and jus di e in he dynamic componen
67
1 3
Dynamic p icing using lexible he e ogeneous sales esponse…
Addi ional ables
See Tables6, 7, 8, 9 and 10.
Table 6 Desc ip i e s a is ics o weekly p ices, ma ke sha es, and uni sales pe s o e
*The sha e o p ice changes ep esen s he sha e o weeks wi h a di e en s o e-speci ic p ice compa ed
o he p e ious week (
p ≠p −1
)
B and Re ail p ice ($)/P ice changes (%)* M. sha e (%) Uni sales
Range Mean SD
p ≠
p
−1
Mean SD Mean SD
P emium b ands
Flo ida Na u al [1.54, 3.35] 2.85 0.33 38.6 4.7 6.5 27.2 46.2
T opicana Pu e [1.29, 3.87] 2.96 0.57 46.7 12.3 13.6 74.8 98.2
Na ional b ands
Ci us Hill [0.99, 3.07] 2.31 0.35 42.8 8.0 12.7 53.5 157.4
Flo ida Gold [0.99, 3.08] 2.18 0.39 43.3 5.2 7.9 33.4 64.0
Minu e Maid [0.99, 3.17] 2.22 0.42 55.2 10.2 13.8 52.0 76.8
T ee F esh [0.99, 2.69] 2.15 0.31 42.3 7.7 8.6 48.8 92.5
T opicana [1.41, 2.99] 2.20 0.38 56.1 18.4 21.1 112.8 159.0
P i a e b and
Dominick’s [0.99, 2.69] 1.76 0.42 47.3 34.0 25.5 308.4 538.8
Table 7 Ou -o -sample p edic i e pe o mance o he
DynFlexHom
and
DynFlexHe
models o di -
e en numbe s o kno s and di e en deg ees o he B-spline basis unc ions, e alua ed by he A e -
age Roo Mean Squa ed E o (
ARMSE
) in holdou samples ( ela i e imp o emen s/de e io a ions
inARMSE alues e e o he de aul speci ica ion o he P-splines wi h 20 kno s and deg ee 3)
B and
DynFlexHom DynFlexHe
20/1 20/3 40/1 40/3 20/1 20/3 40/1 40/3
Flo . Na l. 24.9 25.0 24.9 25.0 27.0 27.2 27.0 26.9
(
−
0.2%) – (
−
0.3%) (
−
0.1%) (
−
0.7%) – (
−
0.4%) (
−
0.8%)
T opic. Pu e 52.8 52.9 52.9 53.0 51.5 51.6 51.7 51.7
(
−
0.1%) – (
+
0.1%) (
+
0.2%) (
−
0.3%) – (
+
0.1%) (
+
0.2%)
Ci us Hill 84.6 87.3 84.0 86.3 107.2 112.0 107.2 110.4
(
−
3.1%) – (
−
3.8%) (
−
1.1%) (
−
4.2%) – (
−
4.3%) (
−
1.4%)
Flo . Gold 54.0 54.1 54.0 54.1 54.5 54.6 54.5 54.5
(
−
0.2%) – (
−
0.1%) (±0.0%) (
−
0.1%) – (
−
0.1%) (
−
0.1%)
Min. Maid 53.8 54.1 53.7 54.1 53.7 53.8 53.5 54.2
(
−
0.5%) – (
−
0.7%) (
−
0.1%) (
−
0.2%) – (
−
0.5%) (
+
0.8%)
T ee F esh 65.8 65.7 65.8 65.8 76.5 78.4 76.9 78.5
(
+
0.1%) – (±0.0%) (
+
0.1%) (
−
2.4%) – (
−
1.9%) (
+
0.1%)
T opicana 91.5 91.8 91.7 91.8 92.7 93.0 93.0 93.1
(
−
0.3%) – (
−
0.1%) (±0.0%) (
−
0.3%) – (
−
0.1%) (±0.0%)
Dominick’s 282.9 283.6 283.5 284.5 285.1 286.2 286.3 286.9
(
−
0.3%) – (±0.0%) (
+
0.3%) (
−
0.4%) – (±0.0%) (
+
0.3%)
68
P.Asche sleben, W.J.S eine
1 3
Table 8 Ou -o -sample p edic i e pe o mance o he
DynFlexHom
and
DynFlexHe
models o di e en numbe s o kno s and di e en deg ees o he B-spline basis
unc ions, e alua ed by he A e age Mean Con inuous Ranked P obabili y Sco e (
AMCRPS
) in holdou samples( ela i e imp o emen s/de e io a ions in AMCRPS alues
e e o he de aul speci ica ion o he P-splines wi h 20 kno s and deg ee 3)
B and
DynFlexHom DynFlexHe
20/1 20/3 40/1 40/3 20/1 20/3 40/1 40/3
Flo . Na l. 8.9 9.0 8.9 8.9 7.9 8.0 7.9 8.0
(
−
0.7%) – (
−
0.7%) (
−
0.4%) (
−
0.9%) – (
−
0.8%) (
−
0.5%)
T opic. Pu e 22.3 22.4 22.3 22.3 20.1 20.1 20.1 20.1
(
−
0.1%) – (
−
0.2%) (
−
0.2%) (
−
0.2%) – (
−
0.2%) (
−
0.1%)
Ci us Hill 21.5 21.8 21.5 21.7 18.8 19.1 18.9 19.0
(
−
1.4%) – (
−
1.4%) (
−
0.5%) (
−
1.7%) – (
−
1.2%) (
−
0.5%)
Flo . Gold 18.9 19.0 18.9 18.9 18.3 18.4 18.3 18.3
(
−
0.2%) – (
−
0.1%) (
−
0.1%) (
−
0.3%) – (
−
0.3%) (
−
0.2%)
Min. Maid 20.3 20.4 20.3 20.4 18.9 19.0 18.8 18.8
(
−
0.6%) – (
−
0.7%) (
−
0.4%) (
−
0.4%) – (
−
1.0%) (
−
0.7%)
T ee F esh 16.1 16.0 16.0 16.0 13.6 13.7 13.7 13.8
(
+
0.1%) – (
−
0.2%) (±0.0%) (
−
0.3%) – (
+
0.2%) (
+
0.5%)
T opicana 45.3 45.5 45.3 45.4 43.3 43.4 43.4 43.5
(
−
0.5%) – (
−
0.4%) (
−
0.2%) (
−
0.4%) – (±0.0%) (
+
0.1%)
Dominick’s 137.7 138.4 138.0 138.5 131.6 132.2 132.0 132.3
(
−
0.6%) – (
−
0.3%) (±0.0%) (
−
0.5%) – (
−
0.1%) (
+
0.1%)