P ocedia - Social and Beha io al Sciences 213 ( 2015 ) 74 – 79
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1877-0428 © 2015 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license
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Pee - e iew unde esponsibili y o Kaunas Uni e si y o Technology, School o Economics and Business
doi: 10.1016/j.sbsp o.2015.11.406
ScienceDi ec
20 h In e na ional Scien i ic Con e ence Economics and Managemen - 2015 (ICEM-2015)
The use o unc ional di e en ial equa ions in he model o he mea
ma ke wi h supply delay
Bobalo á Ma inaa, No o ná Ve onikab,*
a,b B no Uni e si y o Technology, Kolejni 2906/4, 61200 B no, Czech Republic
Abs ac
In economic applica ions, we ha e o make he assump ion ha ela ions be ween he a iables a y wi h ime. One o he
possible ways o inco po a ing he p ocess dynamics in o he model is o desc ibe he model by unc ional equa ions. The pape is
based on he assump ion ha he balance be ween he demand and supply can be success ully exp essed by a model desc ibed by
di e en ial equa ions, e en i he goods a e supplied wi h a ce ain delay.
The equa ion is sol ed by mode n heo y. Theo e ical esul s a e illus a ed by an example, wi h conc e e esul s p esen ed in
g aphical o m. The solu ion is p esen ed by mode n compu e simula ion and he Maple sys em is used.
The au ho s come o he conclusion ha a delay in he supply o goods can cause an oscilla ion in he p ice. On he o he hand, i
is possible o de ine condi ions unde which he solu ion is mono onous.
© 2015 The Au ho s. Published by Else ie L d.
Pee - e iew unde esponsibili y o Kaunas Uni e si y o Technology, School o Economics and Business.
Keywo ds: Di e en ial equa ions; unc ional; model; Wal as; delay; Maple.
In oduc ion
One o he ypical cha ac e is ics o con empo a y economic de elopmen in he wo ld is he g owing
in e dependence and in e connec ion o he ma ke s o he na ional economies and he mul ina ional g oups, which
is some imes called he globaliza ion p ocess. In o ma ion and he abili y o handle in o ma ion e ec i ely a e he
sou ce o weal h and powe . I is in o ma ion which plays a decisi e ole in he success o e e y indi idual and
* Co esponding au ho . Tel.: +420 54114 3718.
E-mail add ess: NOVO[email p o ec ed].CZ.
© 2015 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY-NC-ND license
(h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
Pee - e iew unde esponsibili y o Kaunas Uni e si y o Technology, School o Economics and Business
75
Bobalo á Ma ina and No o ná Ve onika / P ocedia - Social and Beha io al Sciences 213 ( 2015 ) 74 – 79
o ganiza ion. The changing wo ld needs in o ma ion o su i e and ma ke s and s ock-exchanges a e he main
d i ing o ce o he economy. The en y in o he uni ied ma ke and ha moniza ion o he legal amewo k ha e
adecisi e e ec on he economic posi ion o households and companies. To make his e ec less cos ly and socially
unaccep able, sui able ins umen s o examining he e ec s o he co esponding measu es mus be used in
cons uc ing he s a egies o economic de elopmen and o ecas ing. One o he ques ions mos equen ly
discussed in connec ion wi h u he economic de elopmen and planning is he ques ion o u u e de elopmen o
he ma ke s.
A company owne who wan s o be success ul needs o ha e in o ma ion on he en i onmen and he cu en
ends. Such in o ma ion may p o ec him in he u u e agains sales p oblems, sales decline, o p o i dec eases. The
speed o e alua ing in o ma ion, acquisi ion o new esou ces, e ec i e communica ion, and collabo a ion among
people a e some o he ac o s which a e essen ial o a success ul economic si ua ion. Ag icul u al de elopmen , o
ins ance, in luences o some ex en all sec o s o indus ial p oduc ion which ollow he p oduc ion o ag icul u al
commodi ies, and he e o e i is essen ial o use s o ag icul u al p oduc s o be able o o esee i s u u e
mo emen s. This pape ocuses on demons a ing how unc ional di e en ial equa ions wi h delayed a gumen can
be used o model ma ke ela ions in he poul y mea ma ke in he Czech Republic. The aim o he s udy is o
analyze he beha io o he model om he poin o iew o oscilla o ici y in he case o a change in he inpu
pa ame e . In he applica ions sec ion, we show how he model is cons uc ed and how i may be sol ed in
apa icula case. The model beha io is p esen ed using compu e simula ion and he Maple Sys em is u ilized in
ep esen ing he esul s g aphically.
1. Wal as´s Model
The idea o ma ke equilib ium, in ui i ely in oduced by Adam Smi h, was wo ked ou as a sel -con ained heo y
o gene al equilib ium by Leon Wal as. One o Wal as´s con ibu ions consis ed in he inclusion o he household
sec o . As a ounde o he gene al equilib ium heo y, Wal as was ollowed by o he au ho s, who wo ked ou his
heo y in o a o m in which i may be applied o analyzing economic policies.
Leon Wal as´s scien i ic con ibu ion o he de elopmen o economics was no ully app ecia ed un il a e his
dea h, in pa icula a e Wo ld Wa II, owing o J. R. Hicks and J. Fishe . In he 1930s and 40s, ce ain economis s,
such as G. Cassel, F. Hahn, o T. T i in, inspi ed by Wal as´s heo y, connec ed he model wi h impe ec
compe i ion. J. L. Neumann c ea ed a dynamic equilib ium model and a g ea Wal as en husias , J. R. Hicks applied
Wal as´s me hodology a he mac o-le el. A e Wo ld Wa II, his concep was u he expanded, pa icula ly
hanks o G. Deb eu and K. J. A ow. These au ho s “con i med he consis ency o he gene al equilib ium heo y,
including he de ini ion o exis ence condi ions”. Unde ce ain s ic condi ions, hey ga e a heo e ical
ma hema ical p oo o Wal as´s conjec u e which says ha adap a ion p ocesses will inally lead o a s able
equilib ium.
Recen ly, a numbe o pape s has appea ed which s udy Wal asian dynamics in expe imen al ma ke s (Plo ,
2000; Ande son e al., 2003; C ocke e al., 2011; Hi o a e al., 2005). O he s udies deal wi h he economic
equilib ium (Anello e al., 2010; Dona o e al., 2014; Jo e e al., 2005).
Wal as´s model ep esen s a mo e complex app oach o mic o-economic analysis as compa ed wi h he pa ial
equilib ium me hodology ep esen ed by he Camb idge School. Howe e , due o i s appa en complexi y, he model
is no o en p esen ed in con empo a y medium-le el ex books o mic oeconomics. Va ious ways o sol ing he
model a e s udied, o ins ance, in Danielle (2006) and Anello e al. (2010) books. In he la e , we can ind a
numbe o p oblems which ha e a backg ound in economics o inance and a e o mula ed in e ms o Lebesgue
spaces.
An analysis o he dynamics o p ices, p oduc ion and consump ion, especially in he case o commodi ies, can be
based on he assump ion o he Wal asian heo y which says ha ela i e change in he p ice ݌(ݐ)in ime ݐis
go e ned by he equa ion o equilib ium be ween he demand ܦ(݌(ݐ))andܵ(݌(ݐ)). The dynamic o maliza ion o
his ela ionship is as ollows:݌´=݂(ܦ(݌)െܵ(݌)) (1)
Fu he , we assume ha ݔ݂(ݔ)>0p o ݔ്0.
76 Bobalo á Ma ina and No o ná Ve onika / P ocedia - Social and Beha io al Sciences 213 ( 2015 ) 74 – 79
In he sequel, he p ope ies o his model will be analyzed using he linea app oxima ion
݂(ݔ)=݇ଵݔ,݇ଵ>0. (2)
Also, we assume ha ܦ(݌(ݐ)) =݀ଵ+݀ଶ݌(ݐ), ݀ଶ<0, (3)
ܵ(݌(ݐ)) =ݏଵ+ݏଶ(݌(ݐ)), ݏଶ>0,
whe e ݀ଵ,݀ଶ,ݏଵ,ݏଶ,݇ଵa e pa ame e s.
1.1. The model in he case o he delay assump ion
We will now accep he idea ha he supply side esponds o changes in he p ice wi h ce ain delay. This
si ua ion can be ma hema ically o malized as ollows:
ܦ(݌(ݐ)) =݀ଵ+݀ଶ݌(ݐ), ݀ଶ<0, (4)
ܵ(݌(ݐ)) =ݏଵ+ݏଶ݌(ݐെο), ݏଶ>0,
whe e ο ep esen s he ime equi ed o p oduce a change in he supply, depending on he p ice de elopmen . In
wha ollows, his will be e e ed o as he delay.
The o iginal equa ion (1) can hen be w i en as:
݌´(ݐ)=݇ଵ(݀ଵെݏ
ଵ+݀ଶ݌(ݐ)െݏ
ଶ݌(ݐെο)). (5)
2. Cons uc ing he solu ion
Phenomena which a e based on economic eali y and desc ibed by s a is ical da a can be con enien ly modeled
by me hods based on such b anches o ma hema ics as s a is ics, nume ical me hods, ope a ional esea ch, linea and
dynamic p og amming, op imiza ion e c. (Da id DQG.ĜiSHN, 2013; Fumi e al., 2013).
A ound he middle o he 20 h cen u y, new ma hema ical models began o appea which can be used o desc ibe
a ious p ac ical-li e si ua ions (Alu , 2014; Balasub amaniam e all, 2014). This de elopmen is closely ela ed o
he de elopmen o he „Ca a heodo ian“ heo y o di e en ial equa ions, pa icula ly he inc easingly sophis ica ed
heo y o unc ional di e en ial equa ions. A gene al heo y conce ned wi h he solu ion o he abo e-men ioned
SUREOHP DQGUHODWHGSUREOHPVFDQEHVWXGLHGIRULQVWDQFHLQWKH PRQRJUDSK .LJXUDG]H DQG 3ĤåD (2003), which
deals wi h he linea p oblem. Applica ion p oblems based on he solu ion o sys ems o di e en ial equa ions wi h a
delayLQFOXGLQJDGHVFULSWLRQRIKRZWKH VROXWLRQLVFRQVWUXFWHGFDQEHIRXQGIRU LQVWDQFHLQ.XFK ĖNRYiDQG
0DĖiVHN (2006) and in he li e a u e ci ed he ein.
Ou cons uc ion o he solu ion uses he Maple sys em. One o he ad an ages o his ma hema ical so wa e is
ha i allows o ind he solu ion symbolically. To ind he solu ion o ou p oblem we use ma hema ical p ocedu es
which a e buil in o he Maple Sys em and which a e designed o sol ing o dina y di e en ial equa ions and he
abo e-men ioned mode n heo y o di e en ial equa ions wi h delayed a gumen .
Le us now conside he ma hema ical model o he economic p ocess desc ibed in he in oduc ion, on he
in e al [െο,ܶ], in he o m:
݌´(ݐ)=݇ଵ(݀ଵെݏ
ଵ+݀ଶ݌(ݐ)െݏ
ଶ(߯(ݐെο)݌(ݐെο)+(1െ߯(ݐെο))݌௛(ݐെο))), (6)
݌(ݐ)=݌௛(ݐ), ݐא[െο,0], (7)
whe e ߯(ݐ)=ቄ 1 ݐ൒0
0 ݐ<0
77
Bobalo á Ma ina and No o ná Ve onika / P ocedia - Social and Beha io al Sciences 213 ( 2015 ) 74 – 79
A na u al consequence o he ac ha he solu ion ݌(ݐ) on he in e al [0, ܶ]is a con inuous ex ension o he
“his o ical“ unc ion ݌௛(ݐ) (ݐ߳[െ߂,0
])is he ac ha he ini ial condi ion o he solu ion ݌(ݐ)o equa ion (7) has
he o m
݌(0) = ݌௛(0). (8)
I ollows om .LJXUDG]HDQG3ĤåD(2003) and om he li e a u e, ci ed he ein, on ully gene al linea bounda y
alue p oblems o unc ional di e en ial equa ions ha :
x he p oblem (6), (8) has a unique solu ion
x he solu ion can be cons uc ed by he me hod o successi e app oxima ions.
Unde hese assump ions, ou p oblem (6), (8) has a unique solu ion and his solu ion is con inuously
di e en iable on he in e al [0, ܶ].
To u he in es iga e he p ope ies o he solu ion o he p oblem (6), (8) we will use well-known esul s on he
oscilla o ici y o he solu ion o a i s -o de linea di e en ial equa ion wi h a cons an delay. F om he heo y
gi en, o ins ance in Aga wal (2012), Tinbe gen (1931), we can easily deduce ha unde he assump ion
ଵ
௘<݇ଵݏଶ߬݁ି௞భௗమఛ<గ
ଶ (9)
e e y solu ion ݌(ݐ)o he equa ion (6) oscilla es a ound he equilib ium p ice ݌௘and o ݐ՜λ he solu ion ends o
he equilib ium p ice ݌௘.The equilib ium poin ݌௘is asymp o ically s able.
3. Poul y mea ma ke
Fo he pas se e al decades, in pa icula om he 1960s, mea consump ion in he Czech Republic has been
inc easing. The eason o his inc ease is ha in he pas , p oduc ion o ag icul u al commodi ies and p oduc s was
subsidized by he s a e. Mea consump ion in he Czech Republic eached i s peak in 1989 and 1990 –97.4 and 96.5
kg pe pe son and yea , espec i ely. A e wa d he e was a decline in mea sales which was due o an inc ease in
p ices and p ice libe aliza ion. A new inc ease in he consump ion o his mea was caused by a o able p ice
ela ions in compa ison wi h o he kinds o mea , he sp ead o highe inaliza ion p oduc s on he domes ic ma ke ,
easy and as ki chen p epa a ion, and in he 1990s also indings on die ological p ope ies o di e en kinds o
mea . In he las ew yea s, poul y mea has been he cheapes mea on he domes ic ma ke in compa ison wi h
o he kinds o mea .
A signi ican change in he poul y mea consump ion occu ed in 2003, when his mea expe ienced a mode a e
decline. The nex yea , howe e , he consump ion inc eased again o 25.3 kg pe pe son and yea . A eco d inc ease
in poul y mea consump ion occu ed in 2005 –26.1 kg pe pe son. Fo a compa ison, consump ion in EU coun ies
is a ound 23 kg. In 2007, he e was a decline in poul y mea consump ion, which was mos p obably caused by
consume s ha ing conce ns abou bi d lu. A he momen , poul y mea consump ion in he Czech Republic anges
be ween 24 and 25 kg pe pe son and yea .
In compiling ou model, da a om si ua ion and ou look epo s o he Minis y o Heal h o he CR was used.
We u ilized annual alues o he yea s 2002 –2014. The co esponding coe icien s we e ob ained eg ession
analysis. The leng h o he delay is conside ed cons an , i co esponds o he poul y mea p oduce s´ abili y o
adap o ma ke changes. The coe icien ݇ଵ, which ep esen s sensi i i y o he ma ke o changing condi ions, is le
as an unknown alue.
Equa ion o mea poul y ma ke :
݌´(ݐ)=݇ଵ(375,2 െ166 െ0,78݌(ݐ)െ1,78݌(ݐെ0,5)).
Equa ion o his o ical unc ion:
78 Bobalo á Ma ina and No o ná Ve onika / P ocedia - Social and Beha io al Sciences 213 ( 2015 ) 74 – 79
݌௛(ݐ)=0,181ݐଶ+3,4ݐ+67,ݐא[െ0,5,0].
F om he condi ion (8) abo e we a e able o conclude ha in ou case, he solu ion oscilla es a he momen when
he coe icien ݇ଵ–ma ke sensi i i y –exceeds he alue 1.0779. I he coe icien exceeds he alue 3.402, hen
he solu ion ceases o be asymp o ically s able.
The solu ions o di e en alues o ݇ଵa e shown in Fig. 1. The alue ݇ଵ=0.1co esponds o he si ua ion
when he ma ke sensi i i y is e y low and he solu ion ends o an equilib ium s a e wi hou oscilla ions. The i s
g aph shows ha he solu ion is mono onous and ends o a cons an alue. I ݇ଵ=1, he condi ion unde which he
solu ion oscilla es is ul illed and, simul aneously, we a e able o obse e ha he solu ion slowly ends o an
equilib ium s a e wi h a dec easing ampli ude o he oscilla ions. The las g aph is o ݇ଵ=7. In his g aph we a e
able o see sine wa es wi h an ampli ude inc easing in ime. This means ha no equilib ium s a e can be eached.
Fig. 1. The solu ions o di e en alues o ݇ଵ
Conclusions
The ools p o ided by unc ional analysis can be success ully used o p ocessing ce ain ype o in o ma ion in
o de o main ain o e en inc ease a company´s compe i i eness and educe i s isks in he compe i ion s uggle. The
modeling o economic p ocesses gi es us a clea idea o he ac i i ies and ela ed aspec s o he modeled a ea. A
o mal desc ip ion o he model en o ces a clea , b ie and g aphic ca ching o eali y and e y o en e eals
inconsis en beha io o he modeled p ocess.
This pape p esen s one o he possible ways o sol ing he Wal as model o ma ke equilib ium in he case
whe e he model is desc ibed by an o dina y di e en ial equa ion wi h delayed a gumen . Mode n heo y is used in
sol ing he equa ion and condi ions unde which he solu ion is oscilla o y a e analyzed.
Theo e ical esul s a e illus a ed by an example and conc e e esul s a e desc ibed g aphically. The ma ke unde
in es iga ion is he poul y mea ma ke in he Czech Republic. As we can see om he esul s, he beha io o he
ma ke in a ious si ua ions can be simula ed by sui able so wa e which pe mi s o imp o e he quali y o he
decision-making p ocess.
The model p esen ed in he pape can be success ully expanded o ob ain mo e accu a e in o ma ion on he
beha io o he model unde in es iga ion. One o he a ac i e aspec s o he mode n me hod o solu ion used in
he pape is ha p o ided sui able so wa e, conside ably wide possibili ies o analyzing he p oblem can be
ob ained. This may p o e help ul in u he esea ch. Also, a gene aliza ion o he model which would admi non-
cons an delay can be conside ed.

79
Bobalo á Ma ina and No o ná Ve onika / P ocedia - Social and Beha io al Sciences 213 ( 2015 ) 74 – 79
Acknowledgemen s
This pape was suppo ed by g an FP-S-15-2787 ”E ec i e Use o ICT and Quan i a i e Me hods o Business
P ocesses Op imiza ion’’ om he In e nal G an Agency a B no Uni e si y o Technology.
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