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В1СНИК ХНТУ № 2(85), 2023 р.
ШФОРМАЦШШ ТЕХНОЛОГИ
ШФОРМАЦШШ ТЕХНОЛОГИ
UDC 519.86 https://doi.Org/10.35546/kntu2078-4481.2023.2.17
T. P. BILOUSOVA
Senior Lecturer at the Department of Management and Information Technologies Kherson State Agrarian and Economic University ORCID: 0000-0002-6982-8960
SIMULATION MODELING OF MARKET EQUILIBRIUM
There are many factors in the market that influence its behavior: tastes and preferences of consumers, interests of consumers and sellers, competition, market monopolization, legislation in the country, seasonal changes. Some are random in nature. It is impossible to take everything into account. The market of one product is considered from the point of view of the seller who sells it. At the same time, three cases are possible: a shortage of goods, a surplus of goods and an equilibrium state. A model was built, the purpose of which is to determine the optimal volume of purchases, which provides the seller with the greatest profit. Delivery delays and market inertia are also taken into account. An approach such as simulation modeling is usedfor market research. Application of the simulation model is of great importance for the analysis of economic phenomena. This provides advantages compared to performing experiments on a real system and using other methods. Analyzed Walras-Marshall models and web-like model. In the Walras-Marshall model, market value depends on supply and demand, that is, on the needs and funds of buyers, on the one hand, and on the labor and costs ofproducers, on the other. The dynamic model determines the change of market factors over time. All variables are functions of time. In the spider-like model, the volume of supply reacts to price changes with some delay. Then the analysis of the model is complicated. The amount of demand is determined by the prices of the current period, and the amount of supply is determined by the prices of the previous period, that is, the required amount of goods arrives late. Solving the task of finding optimal purchase volumes, a market model without a supply line is considered. The demand function is assumed to be constant. Delayed deliveries are taken into account. The price is determined by the market, that is, for a fixed volume of goods, the market price is set, it is this price that provides the greatest profit. By changing purchasing strategies and order volumes, you can choose the optimal strategy in such a way as to determine the optimal supply line. Market inertia means that the price is constant over a short period of time. Certain limits limit the trader from significantly increasing or decreasing the price.
Key words: dynamical system, demand function, offer function, optimization, simulation modeling.
Т. П. Б1ЛОУСОВА
старший викладач кафедри менеджменту та шформацшних технологш Херсонський державний аграрно-економiчний ушверситет ORCID: 0000-0002-6982-8960
1М1ТАЦ1ЙНЕ МОДЕЛЮВАННЯ РИНКОВО1 Р1ВНОВАГИ
На ринку е безлЬч чинникЬв , яю впливають на його поведтку: смаки та вподобання споживач1в, iнтереси споживач1в та продавцiв, конкуренцiя, монополЬзацЬя ринку, законодавство у краШ, сезонн змти. Деяк мають випадковий характер. Все врахувати неможливо. Розглянено ринок одного товару i3 погляду продавця, який його реалЬзуе. При цьому можливi три випадки: дефщит товару, надлишок товару та рiвноважний стан. Побудована модель, цЬль яmi: визначення оптимального обсягу закупiвель, що забезпечуе продавцю найбЬльший прибуток. Також враховано запЬзнення постачання та терцттсть ринку. Для дослЬдження ринку застосовано такий пiдхiд, як iмiтацiйне моделювання. Застосування ЬмЬтацШно1' моделi мае велике значення для анализу економiчних явищ. Це дае переваги у порiвняннi з виконанням експериментiв над реальною системою та використанням iнших методiв. Проанальзоваш моделi Вальраса-Маршалла та nавутиноnодiбна модель. У моделi Вальраса-Маршалла ринкова вартЬсть залежить вiд попиту та пропозицп, тобто, вiд потреб i коштiв покупцЬв, з одного боку, та вiд працi i витрат виробниюв, з iншого. Динамiчна модель визначае змту ринкових чинниюв у чаа. УсЬ змшн е функцЬями часу. У павутиноподЬбнт модель обсяг пропозицп реагуе на змти цт Ьз деяким затзненням. ТодЬ аналЬз модель ускладнюеться. РозмЬр попиту визначаеться цтами поточного перюду, а величина пропозицп визначаеться цЬнами попереднього перюду, тобто необхЬдний обсяг товару надходить Ьз затзненням. ВирШуючи завдання пошуку оптимальних обсягЬв закупЬвель, розглядаеться ринкова модель без лтП пропозицп. ФункцЬя попиту вважаеться незмтною. Враховуеться запЬзнення поставок. Цту визначае ринок, тобто за фЬксованого обсягу товарЬв встановлюеться ринкова цта, саме вона забезпечуе найбЬльший прибуток. ЗмЬнюючи стратеги закупЬвель та обсяги замовлень, можна пЬдЬбрати оптимальну стратегт таким чином, щоб визначити оптимальну лЬнт пропозицп. 1нерцШтсть ринку означае, що у невеликому промЬжку часу цта посттна. ПевнЬ рамки обмежують торговця вЬд значного пЬдвищення чи зниження цти.
Ключовi слова: динамЬчна система, функцЬя попиту, функцЬя пропозицп, оптимЬзацЬя, ЬмтацШнемоделювання.
BICHHK XHTy № 2(85), 2023 p.
IHOOPMA^Hm TEXHOnOUi
Analysis of recent research and publications
The problem of building a market model, modeling and forecasting its development is one of the most important problems of the economy in connection with Ukraine's transition to market relations. Most of the market models were built according to the principle of establishing a competitive equilibrium, the existence of which was declared in the work of Walras [1]. Mathematical substantiation of the Walras hypothesis was carried out in the 1950s in the works of Arrow-Debre [2], McKenzie, Gale, Nikaido. Further work was carried out on the improvement of models and their generalization. These studies are considered quite fully in the monographs of Morishima, Nikaido, Lancaster and other modern authors. Most of these works analyzed the balance of aggregate supply and demand (Market equilibrium) [3, 4]. These market models established a balance between supply and demand, but could not be a market model, since, firstly, there was no competition between both producers and consumers in them, and secondly, the purposefulness of the actions of market participants was not reflected ( producers and consumers), which is the basis of competition. The market model should reflect not only the balance between supply and demand, but also the purposefulness of each market participant, taking into account their overall relationship. A vector (multi-criteria) problem of mathematical programming [5] is such a mathematical model that, along with the balance sheet, can reflect the purposefulness of each market participant. To solve this problem, the methods of solving the vector problem, based on the normalization of criteria and the principle of guaranteed result, have been developed [6, 7].
Formulation of the goals of the article
Study of the effect on the dynamics ofthe productpriceof randomfluchiationsindemand, the positionofthedemand line, the purchase price of the product, the strategy of ordering the product. Statistical assessment of the seller's profit at a fixed price of the product and the strate^ of orderinn ths ^odnet. DetemtineSron of the optimcl price ond tttder sSruCegy taking into account the delay. P racllcal impSemeniotion op c moPdemnnopO modef tliai simulntat the meckeo of sno peodnct, taking into account randomfluctuonont and Ue tees, teen h cnoots yott te ecttmoip Su ntSt oS tht oslloc oc well tit Sd fincf the price of the product and the m^urne c° pucrtoves toat provirle the teller w^ toe prsctesS prpfo.
PurtcnOiinh mie muluriut
1. Description of the modeling alherithm. A orogtnm wos mod hmo slmdlofed pcc-pcscc lu fif mtnkut and Ptpployinh the results in the form of graphrn. T°p papamolerc od ihe model can be Po^^pec: tiie pre seme op abrencn of delcy und random fluctuations. Using ouch a modd, ycu eiso tert urricnp hnrcnhrmp slOtaiteg^o;^ wd pet zdcuai remlts. ecc^hloi"^ PCs influence of deterministic rsld tandcm e^c^niis (tn SSo rt^occis^ of ce^i_lnii c^o^ds. Moneien nuapIgier: tsSae numbec od geod^ in stock, the volume of purchas es,demand. ic^pty, ^ofir^he amaucfceresources.Voriadiepused mlhe program: penalty function coefficient, delay, quoiHf? e° qoo-) 0t yehlmp cf nurch:lseg] sunpin yhuZt dimand aske ^eiice, stust chase price, profit, order pnymeul, storage p^ymrnri^l^ timu Sd^sceetfoil^riai^e m1zgvh]] hournne m^xrn^utn pro:ELSl] ac\cer^i^e profil, optimal purchase search boundanes,constanS ^chaee vahie, optimuc purchase. Sak nf guhas al the current mmert at the market price, in accoManne idpn H1:^ demand ^ccn^ia^^can:
ec(o)=qo00)OC0) (i)
and formulas:
qqo=Q2)()+QqO~tC oQC>QC)+Qz0->) (2) 1 Q0)t(, Q0(t) <Q(t)+-Qz(t oqt)
Q( lQ J <0 Q0(t)> Q(t)+Qz(tcQ
, mt+Qzh+sC-Qooc qO(0CQ0)+Qz0-Q) 1)
QQ)> Q0(l )i Qe(l) n 0 Q)
When finding the value of demand, a random variable is added — £(t). This value is generated by matlab. The program calculates the values of variables Qccm 2<etУ)tm)ysmtulcSed spI-us c(( demzuCiCupp]lu,profiSoun other smelted values.
J(t) nQs(f). P(t)-Qs.OoqSp oQQ).PS oh0))) (5)
Profit as a function of price, with a fixed supply volume, taking into account the penalty function with the coefficient alf. The result of the function is taken with a "minus" sign, and when searching for an extremum, the function is examined for a minimum. This is done so that when investigating a function, it is possible to use MATLAB computational functions that allow you to search only for minimum points. Qz(t) - profit as a function of purchase volume. Determining the market price at each step, we sell goods, estimate the profit, thus determining what profit we will receive in the future, depending on the volume of purchases at the moment. We establish this dependence in order to search for the optimal purchase volume. Due to the action of random factors, the actual profit value will differ from the value estimated by this function. Due to fluctuations in demand, unsold goods may remain, or vice versa, there may not be enough of them, which means lost profits. The result of the function for the reason described above is taken with a negative sign. Modeling the processes
В1СНИК ХНТУ № 2(85), 2023 p.
1НФ ОРМАЦ1ЙН1 ТЕХНОЛОГИ
of purchasing and selling goods with a delay is implemented as follows. We set the initial quantity of goods Qsk{ 1), time interval t. During the entire time interval, we first receive supplies of goods, then we sell goods, then we set the volume of the next purchase of goods. Delivery of goods: to the number of goods in the warehouse, we add the volume of goods ordered on x earlier. While / < x the seller does not receive any deliveries, he only sells the initial quantity of goods. The greater the delay in deliveries, the more goods must be installed at the initial moment in order to make a profit. However, it is assumed that the initial amount has little effect on further profit or average profit over a long time interval. When t > x, the seller receives shipments and the quantity of goods offered is the sum of the balance in the warehouse and the volume of shipments received. The sale of goods is carried out in accordance with the demand model with random fluctuations, the values are calculated by formulas (2), (3). If the volume of demand is greater than the volume of supply, then all goods are sold out, the number of goods sold is equal to the volume of supply. If the volume of demand is less than the volume of supply, then the quantity of goods sold is equal to the difference between the supply and demand. Profit is calculated according to the formula (4). Then we set the volume of purchase of goods. The purchase price is considered constant. The volume of purchases can be set based on various considerations, thus realizing various strategies.
2 Simulation results. Search for the optimal supply strategy. The created model was applied to the problem of finding the optimal supply strategy. For the study and analysis of the market model, the line of demand for oranges was taken.
Od it) = 360 - 3 • P(t)+c (/) .Various strategies for delivering goods to the market are considered. First, the supply strategy was considered, which determines the optimal value at each point in time. Finding this value is done as a search for the extremum of the function Qz{t), which estimates the future profit. Modeling this strategy showed the following results: Model parameters: Od (/) = 360-3-P(t), Qsk{l) = 300 - initial quantity of goods; x = 20 -lag; «// = 2 - penalty function coefficient; T=100 - time interval.
A large delay x = 20 led to a shortage of goods in the initial period. Then the ordered goods arrived, but there was no overstocking. On the purchase schedule, you can see that the amount of purchases has remained the same, but purchases have become more rare, more than zero purchases. The shortage of goods in the initial period caused the price to rise rapidly and remain at that value until supplies were delivered. At a high price, demand decreased, and increased only after its decrease. Then the price began to fluctuate around a constant value. Purchases either increased or decreased. The number of unsold goods was insignificant and arose only due to random fluctuations in demand. After the deficit was eliminated, supply and demand were about the same, except for zero intervals. Due to condition (4), some deliveries were zero, but lagged behind the appearance of unsold balances. This led to the fact that the number of goods offered and the profit were equal to 0.
Thus, the initial quantity of goods was insufficient. But then the strategy of determining purchases led to finding optimal purchases that provide more profit.
Another purchasing strategy implemented with the model is constant supply. Let us determine the optimal constant supply empirically. Before determining it, you need to evaluate the upper and lower bounds of the search. Consider the following situations: 1) Market overstocking: Qz(t) = 300, but subject to condition (4).
Rice 1. Schedule for changing the offer
Rice 2. Graph of change in demand
Rice 3. Schedule of purchases
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80
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аАл Ы to .а л ; A J. А, ЛЛ
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Rice 4. Schedule of unsold balances of goods
BICHHK XHTY№ 2(85), 2023 p.
MGOPMA^HHI TEXHOiïOnï
Rice. 5. Graph of price changes
Rice 6. Graph of profit change
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.............¡/A
V
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Hce7. Scheduleofchanges insupply when overstocked
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etice h.GraphoS chaece iehemanddf ring overstockhng
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Riee9. Graph of profit when overstocked
The price changes similarly to the previous situation, so the chart is not shown. Purchases: either Qsk(tt) = 300 or Qs() = 0. In this case, the supply of goods significantly exceeds the demand, which leads to the accumulation of unnecessary volumes of goods. Obviously, this leads to unnecessary storage costs. With a surplus of goods, profit is reduced by 2times.
2)Shortageof goodson the market: Qsk(t) =100.
In conditions of scarcity, the price rises, because of this, demand falls. Profit falls from the initial point in time and then keeps at a lower level. Thus, with a constant value of supply equal to 300, there was an excess of goods, with a constant value of supply equal to 100, there was a shortage of goods. Next, we determine the empirically optimal amount of supplies. It turned out to be 175. By setting the supply equal to this value, taking into account condition (4), we obtain thefollowingresults.
Rice10. Supply schedulewith a shortage ofgoods
Rice 11. Demand schedule with a shortage of goods
BICHHK XHTY № 2(85), 2023p.
IH&OPMAIJIHHI TEXHOMOn 'i
20
40
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1CX
Rice 12. Price chart with a shortage of goods
10000 3000
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o a.
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nr
0
,'Vi
0
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Rice 13. Graph of profit with a shortage of goods
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0
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Rice 14. Graph of purchases at a constant value of purchases
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r'l
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Rice. 15. Graphs of supply and demand for a constant supply of 175
100 BO BO 40 20 0
. . A
a N\ ^ A
Rice 16. Price chart at a constant supply value of 175
Rice 17. Graph of product balances at a constant value of purchases equal to 175
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8000
S 6000 a.
4000 2000 0
> i
....
40 60
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Rice 18. Grap h of profit at a c ons taint value of purchases equal to 175
В1СНИК ХНТУ № 2(85), 2023 р.
ШФОРМАЦШШ ТЕХНОЛОГИ
Conclusions
The optimal strategy involves not only finding the supply that leads to the greatest profit, but also the fulfillment of other important conditions. The quantity of offered goods, which is the sum of the balance of goods and the supply of goods, depends on the prices of the previous period and should correspond as much as possible to the demand in currently. The price dynamics is not profitable for the trader, because this reduces his profit, so the price of the goods must be stable.
Bibliography
1. Walras L. Elements d'Economie Politique Pure. Revue de Théologie et de Philosophie et Compte-rendu des Principales Publications Scientifiques. 1874. Vol. 7. P. 628-632. URL: https://www.jstor.org/stable/44346456?seq = 1#meta-data_info_tab_contents
2. Arrow K. J., Debreu G. Existence of an Equilibrium for a Competitive Economy. Econometrica. 1954. Vol. 22. Issue 3. P. 265-290.
3. Козак Ю. Г., Мацкул В. М. Математичш методи та моделi для MaricrpiB з eROHOMÎKH. Практичш застосу-вання: навч. noci6. Кив: Центр учбово1 лггератури, 2017. 254 с.
4. Бшоусова Т. П., Лi В. Е. Математичне моделювання рiвноваги функцш попиту та пропозицп. Сучасна молодь в ceimi iнформацшних технологш: матер1али IIВсеукр. наук.-практ. ттернет-конф. молодих вчених та здобувачгв вищо'1 оcвimи, присвячено'1 Дню науки (м. Херсон, 14 травня 2021 р.). Херсон: Книжкове видавництво ФОП Вишемирський В. С., 2021. С. 152-155.
5. Дебела I. М. Економжо-математичне моделювання: навч. поаб. Херсон: 2011. 348 с.
6. Бшоусова, Т. П. Математична модель оптимального ринку. Тавршський науковий вюник. Сергя: Економта. Херсон, 2021. № 8. С. 70-75.
7. Бшоусова, Т. П. Математична модель оптимального ринку одного товару. Тавршський науковий вюник. Серiя: Економка. Херсон, 2021. № 9. С. 101-108.
References
1. Walras L. (1874) Elements d'Economie Politique Pure. Revue de Théologie et de Philosophie et Compte-rendu des Principales Publications Scientifiques. 7, 628-632. Retrieved from https://www.jstor.org/stable/44346456?seq=1#meta-data_info_tab_contents
2. Arrow K. J., Debreu G. (1954) Existence of an equilibrium for a competitive economy. Econometrica. 22, 3, 265-290.
3. Kozak Yu. H. Matskul V. M. (2017) Matematychni metody ta modeli dlia mahistriv z ekonomiky. Praktychni zastosuvannia: Navch. posib. [Mathematical Methods and Models for Masters in Economics. Practical Applications: a textbook]. K.: Tsentr uchbovoi literatury.
4. Bilousova T. P., Li V. E. (2021) Matematychne modeliuvannia rivnovahy funktsii popytu ta propozytsii. [Mathematical Modeling of the Balance of Supply and Demand Functions]. Suchasna molodv sviti informatsiinykh tekhnolohii: materialy II Vseukr. nauk.-prakt. internet-konf. molodykh vchenykh ta zdobuvachiv vyshchoi osvity, prysviachenoi Dniu nauky (Kherson, 14 May, 2021). Kherson: Knyzhkove vydavnytstvo FOP Vyshemyrskyi V. S. pp. 152-155.
5. Debela I. M. (2011) Ekonomiko-matematychne modeliuvannia: navchalnyi posibnyk. [Economic and Mathematical Modeling: a textbook]. Kherson: Khersonska miska drukarnia. (in Ukrainian)
6. Bilousova T. P. (2021) Matematychna model optymalnoho rynku. [Mathematical model of the optimal market]. Taurian Scientific Bulletin. Series: Economics, vol. 8, pp. 70-75.
7. Bilousova T. P. (2021) Matematychna model optymalnoho rynku odnoho tovaru. [Mathematical model of the optimal market ofjne goods]. Taurian Scientific Bulletin. Series: Economics, vol. 9, pp. 101-108.
