Cubo A Mathematicai Joumal Vol.05/~0S - OCTOBER 2003 Control of Dynamic Oligopsonies with Production Factors Laszlo Kapolyi Systems Consulting Rt, 1535 Budapest, 114, Pf. 709, Hv.ngary and Ferenc Szidarovszky Systems and Industrial Engineering Department, University of A rízona, Tucson, Arízona 85721-0020,U.S.A. ABST RACT. Oynamic o!igopsony is examined with discrete time scales. The control- lability of t he dynamic system is investigated where the competitive price is select ed as the control variable. For the single product factor case complete characterization is given. A simple example illustrates that for the more general case no general solution can be given. We al so elaborated the case when the fixed price of the production factors is the control variable. We show that the system is uncontrol!able in this case. Introduction Oligopoly and related models have been examined intensively during the last 4-5 decades . This research area goes back to Cournot (1838), who is considered the founder of this field. A comprehensive summary of results up to the mid seventies is presented in Okuguchi (1976) . Variants of the classical Cournot model with applications to natural resources rnanagement are discussed in Okuguchi and Szidarovszky ( 1999). Most studies considered the cornpeti- tion among the firm in the product market only. However there is a competition among th.e firms in the factor market as well. Oligopsonies include this kind of competition in the oligopo\y models, as it is shown in Szidarovszky and Okuguchi (2001). Recently a d y namic <132 Control of Dynamic O/igopsonies with Production Fac tors ollgopsony model was introduced by Kapolyi and Szidarovszky {2001) , in which the cxis- t ence , uniqueness and the global asymptotical stability of the Nash-Cournot equilibrium wcre prove O a nd c1r > O for ali k, and d1 and d2 are both positive, the system is control\able if and only if d1 :f:. d2, or equivalently c 1 '# c2. If N ~ 3, then we will prove that the system is not 434 Control of Dynamic Oligopsonies with Production Factors controllab le. Notice first that H~ B(l-L) where 1 is the N x N identity matrix , and ali e lements of l are equal to l. Then !i' ~ 1!_2 ((N - 2)1 + L) whi ch is a linear combination of 1 and Ji, and similarly, ll.3 ,li.4 , .• are ali linear combina- tion s of L and lf.... Therefo re lf..2r;., H.3f , .. are linear comb inations off and fu, so the rank of K is nt most two. Hence the system is not controllabl e. Next we show that for M > 1 we cannot give a general answer. To illustrate the problem assume that N = 2, and matrix 11.. is diagonal. So let Then further more and fu~(~ ff)(~)~(¡~;) . !i'o ~ ( ~ ~ ) ( ~~'. ) ~ ( ~:~; ) ' Therefore the Kalman-matrix has the special form Subtract t he a 2 -multiple of th e first column from the third column, and the a: 2 -multiple of t he seco nd co lumn from t he last column t o have matrix ( d, ad, dz {Jd4 dJ adi d4 fJ d2 (13' - a 2 )d2 P(IJ' ~ a 2 )d, ) o o ((32 - a2)d4 /3(/32 - a2) d2 1 Laszlo Kapolyi & Ferenc Szidarovszky with determinant (expanded with respect to its last column) -/3({fl - a 2)ch{,B2 - a 2)d2(adi - ad~) +/3(/32 - a 2)d4(/32 - a 2)d4 (adi - ad~) = a/3(.82 - a2)2(di - d~)(d~ - 4 ) 435 which is zero if o = /3, or d1 = d3 , o r d2 = d4 , otherwise nonzero. ln t he first case the system is not contro\lable, and in the second case it is. For nond iagonal f1 and N > 2 matrix K is more comp\icated , a nd therefore no general couditions can be given. 4 Control by Fixed Labor Cost Consider again t he d ynamic system (3) and assume that t he fixed price g of t he production factors is controlled. lf al! factors are cont ro lled in the same way, then g has to be replaced by a· u(t ), where 11(t ) shows the control. If the prod uction factors are controlled differently, t hen !! is replaced by d iag (a1 , . . , UM ):!!(t), where !!(t) is a n M -dimensiona l control vector . In these cases the coefficient mat rix 11. is the same as in the previous section, however d,i; (k = l , 2, ... , N ) has t o be replaced by either Notice t hat for N 2: 2 matrix K has rank at most M , since its first M rows are identica l to the second M rows, which are the same as the third M rows, and so on. Therefore the system is a\ways uncontrol\able. References (1] C OURNOT, A. , Recherchessur les Principies Mathém a.tiq ues de la. Théorie des Richcsses, Hachett e, Paris . (English Trans lation ( 1960) Resear ches into t he Mat hema tical Prin- cipies of the Theory of Wealth, Kelley, New Yor k), 1838. [2J ÜKUGUCHI , K. , Expectations a.nd Stability in Oligopoly Models, Springer-Verlag , Berlin/ Heidelberg/New York, 1976. [3] Ü KUGUCHI , K. ANO SZJOAROVSZKY , F ., The Tlieory of Oligopoly witli Mu/ti-Product f'irms (2"d edition), Springer-Verlag, Berlin/ He idelberg/ New York, 1999. 436 C0ntr0l ef Dynamic Oligopsonies with Production Fa.ctors [4] KAPOLYT, L. ANO Sz.rn>AROVSZ.KY, F., Dynamic Qligopolies with Producbien Factors, Southwest J. of Pur.e and AppJ.ied Mat:h., Vol. 2001, Issue 2 (20011), pp. 73-76. [5] SZIDAROVSZKY, F. ANO BAHILL, A.T., Linear Systems Theory. CRC Press, Boca Raton/Londen (2"d edition), 1998. [6] SZIDAROVSZKY, F. ANI!l ÜKUGUCHI, K., Dynamic Analysis ofOJigepsony under Adap- tive Expeetat;ions, Seuthwest J. of Pure and Applied Math., Vel. 20011, Issue 2 (2001), pp. 53-60.