MATHEMATICAL MODELS OF MONOPOLISTIC COMPETITION: THEORETICAL PRINCIPLES AND APPLICATIONS UDC 51-7:

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1 FACTA UNIVERSITATIS Seres: Economcs and Organzaton Vol. 8, N o, 0, pp. - 9 MATHEMATICAL MODELS OF MONOPOLISTIC COMPETITION: THEORETICAL PRINCIPLES AND APPLICATIONS UDC 5-7: Mlan Božnovć, Vladca Stojanovć Faculty of Economcs, Unversty of Prštna (n K. Mtrovca), Serba mlan_boznovc@yahoo.com Faculty of Scences and Mathematcs, Unversty of Prštna (n K. Mtrovca), Serba vlast70@gmal.com Abstract. In ths paper we wll consder the problem of monopolstc competton of two or more frms that offer the same products. Specal attenton wll be put on the constructon of a general mathematcal model of olgopolstc competton. Ths model was appled n practcal research,.e. t was mplemented n the producton optmzaton of three monopolstc frms. Key Words: monopolstc competton, models of functonal dependence, optmzaton of economcs varables.. INTRODUCTION The contnuously growng antagonsm between sellers and buyers, motvated by maxmalstc goals of achevng the bggest dfference between the cost of producton and sale of goods on the market, represents an everlastng economc overpowerng that we could call the competton. The man generator of competton s certanly the mechansm of settng the market prces, establshed on the bass of economc law the law of supply and demand. Monopolstc competton as a state on the market s completely opposte to the perfect competton. Regardless of how t s created, the man characterstc of monopolstc competton s the presence of monopoly n ether the supply or demand area. Ths means that a manufacturer s confronted wth hgh number of consumers and vce versa, the consumers are able to choose the goods from many producers of smlar products. Therefore, the amount of supply (or demand) s such, that monopoly has the ablty to set the prce of the products regardless of the quantty of the producton. In olgopolstc markets almost whole producton s performed by several companes only. The products Receved May 5, 0

2 M. BOŽINOVIĆ, V. STOJANOVIĆ on ths market may not necessarly be dfferentated. Therefore, olgopolstc company determnes the prce and the amount of producton s based on the strategc assessment of behavor of the competton, and on the other hand, the compettors decsons are hghly dependent on the strategc plannng of the leadng company. Although t mght seem abstract at the frst glance, the monopolstc behavor s completely logcal and t provdes the bass for determnng the equlbrum condtons n the olgopolstc market. In fact, n real stuatons n the market, there s a queston of how to manage the prce and the level of producton so that they could be deally balanced, along wth the queston of ts feasblty. Certanly, n ths case the market prce s not an outcome of the law of supply and demand because there s no possblty of ncreasng the number of bdders, so the products offered on the market have no close substtutes. However, the purchasng power of consumers to pay a certan prce represents a lmtng factor for the prce growth, and an ndvdual problem whch occurs s the formaton of an approprate model n order to formally descrbe the behavor of olgopolstc companes on the market. Guded by ths dea, and begnnng from the standard functonal models of economc varables (see, for example, Henderson and Quandt [3]), we wll present dfferent ways of mathematcal modelng of monopolstc competton and determnaton of the optmal, equlbrum level of prce and demand n whch the monopolstc companes acheve maxmum proft.. THE MODEL OF OLIGOPOLISTIC COMPETITION Suppose that the olgopolstc competton (of order n) occurs when several companes (frms) F,, F n appear on the market offerng the same product X. If q,, q n denotes ther demand and producton volume of each company, respectvely, the total demand for X s equal to the sum of ndvdual demands,.e. q q q, () n whle the law of demand n general, nverse form s gven by p f (q), () where p s the prce of product X. Further, T (q), T (q),,t n (q) denotes the total costs of product X n the companes F, F,, F n, respectvely, and C (q), C (q),, C n (q) ther total ncome. Then, total ncome s generally equal to the product of demands and prce C ( n q) q p q f ( q q ), (3) whle the dfference between total revenue and total costs presents the proft of company, gven by the functons P C T,,, n. (4) We can put equaton (4) n a developed form, takng the functon of the total revenue (3). The proft functons of company F, =,,, n, then as follows

3 Mathematcal Models of Monopolstc Competton: Theoretcal Prncples and Applcatons 3 P q f ( q q ) T. (5) In a further analyss of the model of olgopolstc competton we wll omt the possblty that n case of changes n producton n -th company other monopolsts are ndfferent to the changes. Specfcally, certanly all the other partcpants wll react n a smlar way,.e. changes n producton volume n the company pulls F change n other busnesses. Based on ths, we conclude that for the demand q there s a correspondng functonal dependency, whch s dsplayed as the followng ( q ), j j j q (6) q j, j. Therefore, a functonal relatonshp between the varables q and q j exsts only for j, whch s consstent wth the fact that ths s the demand of dfferent companes. Now, substtutng (6) nto (5) we obtan the followng system of equatons P q f q q j j n q T ( ), where =,,, n and q s a common demand. Further, by dfferentatng equatons (7) order of varables q, =,,, n, and put that receved partal dervatves equate to zero, we get a system of dfferental equatons f ' T f q ( q ), j q (8) j q that completely descrbes the state of the market wthn a set of monopolstc frms. In economc terms, any soluton of (8),.e. a fnte sequence of numbers (7) n ( q,, q n ) R (9) whch dentcally fulflls ths system of equatons, s the optmal value of demand, or optmal level of producton for whch each of monopolstc frms realze maxmum profts. Specfcally, the equatons of system (8) can be nterpreted as the equlbrum condtons of a monopolstc market. They are realzed when the margnal revenues are equal to margnal costs,.e. f C g (q) = T g (q), (0) where C g (q) and T g (q) are dervates of functon of total ncome C(q) and total costs T(q) expressed through the demand for X. Ths concept n economc theory was gven by John Nash n 95 and t s known as Nash s equlbrum. Its basc dea s the formal, mathematcal descrpton of the prncples that each company "dong busness as best he could", takng nto account "the actons of ts compettors". In ths way, he provdes the bass for the applcaton of game theory to descrbe the nteracton between monopolstc companes, whch wll be dscussed n detals later.

4 4 M. BOŽINOVIĆ, V. STOJANOVIĆ 3. MODEL APPLICATION Let us now consder some specfc possbltes of applyng the model above, especally the system of dfferental equatons (8), n purpose of fndng the optmal level of producton values of olgopolstc companes. Because of ths, we wll frstly pont out some specfc assumptons about functonal dependence between these economc varables. One of typcal features of the organzaton of producton n olgopolstc companes exsts n ther "agreement" on a common market share and n the quantty of goods that each frm wll produce. Ths stuaton can be dsplayed through the rato quanttes of manufactured goods,.e. partcpaton of companes n a gven market segmentaton. Usng these facts, the system of demand equatons (6) can be wrtten as functons of drect proportonalty q k q,, j,,, n, () j j where k j are coeffcents of elastcty of demand q j wth respect to q. Accordng to the ntroduced assumptons, t can easly be seen as vald k j k j, and k. Addtonally, the system of equatons () can be wrtten n the form of matrx equaton where q = (q,,q n ) T R n s, so called, vector of demand, whle k k K kn q Kq, () k k k n kn k n knn s the matrx of correspondng coeffcents of proportonalty demand defned n (). Based on (), then t s clear that the demand vector q s the egenvalue vector of matrx K, whch may be of practcal use n ts calculaton. As the correspondng functonal model of the cost of producton we wll use the parabolc (quadratc) functons (3) T a q b q c,,, n. (4) Then, we wll obtan the cost functons usng standard approxmaton procedures, for whch there s a lot of extensve lterature (see, for example, Božnovć and Stojanovć [] or Render and Star [7]). On the other hand, to the demand functon () we assume the smplest functonal model of dependency the prce p and the demand q of the product X. Ths model can be mathematcally expressed by a lnear functon of demand f mq r, (5) whch s easly determned from emprcal data. Accordngly, the correspondng model s called lnear model of monopolstc competton. Usng the afore mentoned functonal models, we can fnd the optmal levels of producton values, we mark them wth q,,,, n, for whch every company acheves

5 Mathematcal Models of Monopolstc Competton: Theoretcal Prncples and Applcatons 5 maxmum proft by sellng products X. By smply replacng the functon () (5) nto the system of equatons (8) and ts resolved by q, we fnd where,,, n. q n q ( b r), mk a (6) j j To llustrate the prevous procedure, we wll descrbe a typcal example of monopolstc producton optmzaton of three frms A, B, C, dealng wth the chemcal ndustry. These companes produce a certan knd of santary products ntended for hotel chans and larger companes, as prmary consumers. Usng data of Wens,.e. Egwald Web Servces [8], we take the partcpaton of these frms n a gven market segmentaton s ::3, respectvely. In ths way, f we mark wth q, q, q 3 the approprate volume of daly producton of the product (000 unts), the matrx coeffcents of proportonalty crossed demand forms (3) s 3 K 3. (7) 3 3 In the next step, as we have already ponted out above, we can determne the approprate functonal dependence of market prces p, ncome C and costs T compared to the volume of producton q and the total demand q. If the prce of the product expressed n euros, usng the prevously descrbed approxmatons we obtaned Demand functon: p = q Income functon: C (x) = 0.87q q Cost functons: Frm A: T (q) = q q Frm B: T (q) = q q Frm C: T 3 (q) = q q. As we apply the optmzaton procedure descrbed above, based on solvng systems of equatons (8), or drect applcaton of equalty (6), we fnd the optmal level of producton value (n 0 3 unts of product): q 5.09; q 8.067; q (8) Note that the obtaned values correspond to a somewhat "deal" proporton of partcpaton of these companes n the market. Based on them, by smply replacng n the prevous functon, we get the correspondng values of market prces and total revenues, producton costs and profts, as shown n Table.

6 6 M. BOŽINOVIĆ, V. STOJANOVIĆ Table. Estmated values of producton parameters of monopoly companes Optmal estmated values Frms A B C Σ Demand /000 unts/ Prce / / Income /000 / Costs /000 / Proft /000 / Fg.. Charts representng ncome, costs and profts of the companes together wth optmal levels of producton Say, for example, that the market prces of that product for all three companes share a lot of balanced values and they are n a range from 6.6 to 7.3 euros. On the other hand, total proft of all frms, whch amounts to euros, s the sum of ndvdual maxmum profts whch are realzed as the dfference of total revenue and producton costs of the product. Functonal dependence of ncome, costs and profts, along wth optmal levels of producton of all three companes, can be dsplayed graphcally as n Fg..

7 Mathematcal Models of Monopolstc Competton: Theoretcal Prncples and Applcatons 7 4. THE MODEL OF "BIPOLISTIC" COMPETITION In the smplest case, when only two compettors perform n olgopolstc market,.e. when t comes to so-called bpolstc competton, every company has only one "compettor" whch must be taken nto account when makng ts decsons. Then, both companes have a common goal,.e. ther nterest s to "set up" ther producton of product X so that they acheve maxmum profts n the market. In ths secton we descrbe the mathematcal algorthm of determnng the maxmum proft of two opposng monopoly enterprses F and F. It s clear that all dscusson descrbed above s vald f n=, and we begn from that pont of vew. Therefore, demand functon for product X n the market be gven n the form (), where the total demand s q q q. (9) The system of dfferental equatons (8) then becomes f q f q f ' T ( q) q q f ' T ( q ). q q (0) Note that functonal dependency between demand q and q generally has the form q ), () ( q whereas the "other" demand functon ξ s the nverse of ξ,.e. t s vald q q ) ( ). () ( q In ths way, each equaton of system (0) can be expressed as a functon of sngle producton volume q or q, because q q q ) ( q ). (3) ( q In the prevous revew, one of the basc assumptons was that any company reacts to change,.e. ncreases the volume of producton n another company. Ths stuaton n practce s certanly real, because t s natural to assume that each company wll react to the ncrease n producton volume of ts compettor. However, n some stuatons we can conclude that by determnng the maxmum proft of one of the companes F, =, other company F j, j wll not change ts output and vce versa. In ths case jont demand q, smlarly to before, can be showen as a varable whch depends only on one of the varables q, q. Moreover, then s ' ( q ) ' ( q ) 0, so the system of equatons (0) becomes T f q f ', (4) q where =,. Fnally, f we ntroduce the labels

8 8 M. BOŽINOVIĆ, V. STOJANOVIĆ T ( q, q ) f q f ', (5) q then the system of equatons (4) can be wrtten n mplct form as ( q, q ) 0,,. (6) Implct functons gven by equatons (6) graphcally represent the two curves n the coordnate plane q Oq, whch ntersect at a pont M (Fg., left) whose coordnates (q, q ) represent the optmal level of demand,.e. the solutons that companes F and F provde maxmum proft. Fnally, f the functons of total cost T (q) and T (q) are equals, then the functons (6) are nverse to each other. As s known, the graphcs of functons φ and φ are symmetrcal n relaton to the lne q = q. Hence the ntersecton of ther graphcs le exactly on that lne, whch s why we conclude that the optmal level of demand from both companes, s gven by equason q q q ( q0 ). (7) Now, a common demand of both companes we can get from equaton (4),.e. q q f f ' T ', (8) u where T u (q) = T (q) + T (q) s the functon of the total cost of the product X, whle the correspondng prce p 0 s found from the equaton p0 f ( q / ). Fg. provdes a graphcal vew of the above-descrbed process of fndng the optmal level of q 0 and p 0. Fg.. Determnng the optmal level of demand and prce n the case of two "bpolstc" competng frms

9 Mathematcal Models of Monopolstc Competton: Theoretcal Prncples and Applcatons 9 5. CONCLUSION The form of olgopoly as a model of the market structure s qute prevalent, and there are plenty of examples: automotve, steel ndustry, precous metals ndustry, ol ndustry, computer equpment, etc. Lookng at ndvdual sectors of the economy where a small numbers of compettors appear, t s clear that each, whle makng ther decsons, must take nto account the way n whch management actons can affect competton,.e. how each of them wll respond. Regardng ths, the above descrbed procedure to determne the equlbrum level of producton may be the bass of the formal descrpton of the jont acton of monopolstc frms n the market. As the consequences of monopolstc power, there arse the economc neffcency and loss of consumers ncome n favor of monopoles. Therefore, precse analyss of the behavor of monopolstc enterprses may be the bass for strengthenng the free market competton, prce controls and the applcaton of other ant-monopolstc measures. REFERENCES. Behrens, K. and Murata, Y., General equlbrum models of monopolstc competton: A new approach. Journal of Economc Theory, Vol. 36, Issue, pp , Božnovć, M. and Stojanovć, V., Mathematcal methods and models n economy of enterprses (n Serban). Leposavć: Hgh School of Economy, Henderson, J.M. and Quandt, R.E., Mcroeconomc Theory: A Mathematcal Approach. New York: McGraw-Hll, Hummels, D. and Levnsohn, J., Monopolstc Competton and Internatonal Trade: Reconsderng the Evdence. Oxford Journals: Quarterly Journal of Economcs, Vol. 0, Issue3, pp , La, H. and Trefler, D., The Gans from Trade wth Monopolstc Competton: Specfcaton, Estmaton, and Ms-Specfcaton. NBER Workng Paper No. 969, Petkovć, D., The mpact of market structure on economc effcency - an emprcal analyss of Markets mlk n Serba (n Serban). Serban economc forum - Workng document, Render, B. and Star, R., Quanttatve analyss for management. London: Alln and Bacon, Wens, E.G. Monopolstc competton. MATEMATIČKI MODELI MONOPOLISTIČKE KONKURENCIJE: TEORIJSKI PRINCIPI I PRIMENE Mlan Božnovć, Vladca Stojanovć U ovom radu razmatramo problem monopolstčke konkurencje dva l vše preduzeća sa stm prozvodom. Posebna pažnja posvećena je konstrukcj opšteg matematčkog modela olgopolne konkurencje koj je zatm prmenjen u optmzacj prozvodnje trju monopolstčkh frm. Ključne reč: monopolstčka konkurencja, model funkconalnh zavsnost, optmzacja ekonomskh velčna.