Abstract # Strategic Inventories in a two-period Cournot Duopoly. Vijayendra Viswanathan Jaejin Jang. University of Wisconsin-Milwaukee

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1 Abstrct # Strtegic Inventories in two-period Cournot Duopoly Vijyendr Viswnthn Jejin Jng University of Wisconsin-Milwukee P.O. Box 784 Deprtment of Industril nd Mnufcturing Engineering University of Wisconsin-Milwukee Milwukee, WI 5301 viswn@uwm.edu, jng@uwm.edu +1 (414) , +1 (414) POMS 0th Annul Conference Orlndo, Florid U.S.A. My 1 to My 4, 009

2 1. Introduction nd Literture Review Strtegic Inventories re inventories crried by supply chin entities for purely strtegic resons, even in the bsence of the trditionl resons to hold inventory. Trditionl resons for holding inventory t supply chin entity (e.g., mnufcturer, retiler, nd distributor) hve been economies of scle in production, resulting in cycle inventories; to hedge ginst production or distribution delys nd ensure timely vilbility of goods, resulting in pipeline inventories; inventories held s sfety stock, to hedge ginst demnd nd supply uncertinty; inventories held to hedge ginst price fluctutions, termed specultive inventory (Annd et l. 008); nd lso inventories held to smooth production nd thus lower production costs s in (Holt et l. 1960) Most of the discussions on inventory, in literture on decentrlized supply chin coordintion, hve focused on the development of coordinting contrcts, which develop optiml inventory strtegies in ech or combintion of the bove situtions. Recent literture reviews in the re include Tsy et l. (1998), who review supply chin contrcting literture focusing on stochstic nd deterministic demnd supply chin models. Cchon (003) presents review on just uncertin demnd models. He reviews five different types of contrcts buybck, quntity flexibility, sles rebte, wholesle price nd revenue shring, elucidting the dvntges nd drwbcks of ech kind of contrcting mechnism, using generl newsvendor model exmple, s well s surveying the extnt literture on ech contrct type nd lso generl trends in contrcting literture. He lso ddresses the importnt clsses of supply chin co-ordintion problems such s

3 coordintion with horizontl competition, multi-period ordering, symmetric informtion, informtion shring, two-loction bse-stock models with forecst updting nd such. Annd et l. (008) mke n interesting conjecture on the booming literture in the re of supply chin coordintion tht it lrgely ignores the effect of inventories nd concentrtes on motivting the right ction under symmetric informtion or morl hzrd. This pper mkes n ttempt to bridge this gp, by bringing this literture strem closer to nother prllel strem of literture in economics tht studies verticl control, horizontl competition nd such issues, for exmple, Tirole (1990) nd Deneckere (1996), who both study verticl control issues in non coopertive setting, s well s ppers like Sloner (1996), Rotemberg nd Sloner (1999) nd Mollgrd et l. (000), who ll study inventories in horizontl competition settings, but decoupled from verticl control. Annd et l (008) s well s its predecessor working pper Annd et l. (00) is one of the first ppers in recent times tht studies strtegic inventories nd verticl control together, in multi-period ordering environment. Keskinock et l. (008), extend the bsic one-mnufcturer, two-retiler, two-period model from Annd et l. (00) to study the interply of verticl control nd strtegic inventories in sitution where the mnufcturer s first period cpcity is limited. 33

4 We ttempt here to contribute to expnding this emerging literture strem studying strtegic inventories nd verticl control together with downstrem competition, using one-mnufcturer, two-retiler model, where the two retilers re Cournot Duopoly. A Cournot Duopoly is one of the simplest models of economic competition, owing its existence to its discovery by Antoine Augustin Cournot in the 1800s. It consists of two firms tht compete on the quntity of product ech is selling, which both decide simultneously. The most importnt fetures of Cournot Duopoly re tht both firms produce homogeneous product, the number of firms is fixed through the selling sesons, nd the firms hve mrket power, i.e., ech firm s quntity procurement decision ffects the good s mrket price. Also, there is no collusion llowed between firms. Cournot Duopolies hve been studied extensively in Economics literture. Some recent works in the re tht re relted to our work re the following: Sloner (1987) studies Cournot Duopoly with two ordering periods. Pl (1991) studies two-period Cournot Duopoly with two production periods nd cost differentils. This model is closest to ours, the significnt difference being tht the cost differentil is endogenous to the competing firms, in Pl s model compred with our model, where the procurement cost for retiler will depend on the wholesle price set by the mnufcturer for tht prticulr period (verticl control). Mtsumur (00) studies inventories s strtegic wepon in two-period Cournot Duopoly, but here, too, s in Pl (1991), verticl control is not considered. 44

5 . Model Description We formulte dynmic two-period model of Cournot Duopoly with one upstrem mnufcturer. Demnd is price-dependent, liner nd the model is full-informtion. All informtion required for ech Supply Chin entity to mke decisions is known to them. The retilers re llowed to crry inventory from period 1 to period t finite, but smll, holding cost. Figure 1: Schemtic of the two-period ordering Cournot Duopoly with one upstrem mnufcturer Period 1: Mnufcturer w 1 Retiler 1 Retiler q q 1 I, I 1 Period : Mnufcturer w Retiler 1 Retiler q 1 q The interctions between the retilers nd mnufcturer in ech period re modeled s three-stge gme: 55

6 1. Mnufcturer nnounces the wholesle price for tht period. Retilers quote their respective order quntities 3. Mnufcturer delivers the ordered quntity, before the strt of the selling seson In the first period, from the received quntity, retilers sell the quntity they hve ermrked for sle in the first period nd crry the rest forwrd to the second period. In the second period, the totl selling quntity of ech retiler equls the quntity ordered in the first period, plus whtever is crried forwrd from the first period. Both retilers sell the sme product nd it is ssumed tht ech retiler cn ctully sell whtever quntity he wnts to, in both periods. The price relized per unit of product sold, by either retiler in either period, is given by p = bq, where nd b re constnt prmeters nd Q is the totl quntity being sold in the mrket, by both retilers combined, in tht period. 3. Anlysis Nomenclture: q ji = order quntity of retiler i in period j (i = 1,; j = 1,) I ji = inventory crried by retiler i from period j to period j+1 w j = wholesle price set by the mnufcturer in period j h i = holding cost of retiler i, to crry one unit of inventory from period 1 to period We solve for sub-gme perfect Nsh equilibrium for this two-period gme, with three stges in ech period, to get closed form expressions for the order quntities of the 66

7 retiler in both periods, the wholesle price set by the mnufcturer in ech period, s well s the inventory crried by the retilers from period 1 to period, in equilibrium. We first strt with the second period decisions of the retilers nd the mnufcturer nd use those equilibrium quntities, while solving out the first period gme. Second period decisions of retiler 1: The second period profit function of retiler 1 is given by, П 1 = b( q1 + I + q + I1 )]( q1 + I) wq1 For mximizing Π 1, we hve, first order condition, the optiml q 1 s w q + 1 q 1 = I I.. (1) b This is the best-response q 1 for ny q set by retiler, nd is function of the second period wholesle-price nd the inventories crried by both retilers from period 1. By Symmetry, we hve, q = w q1 + I 1 b I.... () From eqution (1) nd eqution (), we obtin w q 1 = I... (3) nd w q = I 1... (4) Mnufcturer s second period wholesle price decision. The mnufcturer s second period profit function cn be written s: 77

8 Π m = w ( q1 + q ).. (5) nd is set s solution to mx Π m w. Substituting equilibrium q 1 nd q from (3) nd (4) into (5) nd tking first order condition, with respect to w, we hve the optiml w for the mnufcturer in period s w = ( I + I 1 ) (6) 4 Substituting (6) into (3) nd (4), we get the equilibrium profit function of retiler 1 in period s 36b 1 16 Π r = + ( )( I + I1 ) ( )( I + I1 ). (7) 1 Similrly from symmetry of retilers, Π 36b 1 16 r = + ( )( I + I1 ) ( )( I + I1 ) Retilers first period decisions: Retiler 1 s 1 st period profit function is Пr = [ b( q + q1 )]( q + q1 ) w1 ( q + I) h1 I Where q nd I re set by retiler 1, in period 1, to mximize П + second period equilibrium profit. Writing this out, we hve the first period problem s q > 0, I 0 mx> [ b( q h I + + ( )( I + I1 ) ( )( I + I1 ) (8) + q1 )]( q + q1 ) w1 ( q + I) 1 36b 1 16 Grouping terms, we hve 88

9 q mx> > 0, I 0 bq I + ( bq1 w1 ) q + ( w1 h1 I1 ) I + ( q1 bq1 + I1 + ( ) I 1 36b Now, setting the prtil derivtive of the bove function with respect to q to zero nd solving for q, we hve: q ( w1 ) q1 =.. (9) b Similrly, tking prtil derivtive of the bove function with respect to I nd solving for I I 8 = I1... (10) 9b ( w1 + h1 ) Similrly, from symmetry of the retilers q 1 ( w1 ) q =. () b I 8 = I. (1) 9b 1 ( w1 + h ) Also, since the retilers re complementry, I = I 1 in equilibrium. Using the bove in (10) nd (1), nd substituting (9) into (), we get q = q 1 = w 1 4 I = ( w 1 + h 1 ) 9b 4 I 1 = ( w 1 + h ) 9b Becuse, I = I 1, for this to hold, we should hve h 1 = h Then we cn write 99

10 4 I = I 1 = ( w 1 + h), where h 1 = h = h (13) 9b The first period wholesle price is decided by the mnufcturer s the solution to the problem mx 1 w ( q + q1 + I + I1 ) w1 Tking first order condition with respect to w 1 nd solving, we get the optiml w 1 s w 1 = ( h). (14) 5 3 Summrizing the results we hve obtined so fr, we hve First Period Decisions: w 1 = ( h) (15) 5 3 w q = q 1 = = ( + h) 15b 3 1 I = I 1 = (8h ) ; is positive for only 8h 15b Second Period Decisions: 3 4h w = 5 q 1 = w 1 I = (3 4h) = q ; is positive only when 4h/3 15b 4. Discussion: We observe two insights from the nlysis section. One is tht the strtegic inventory quntity is positive only if the reservtion price of the product is less thn or equl to 8h. This implies tht strtegic inventory is recommended in two-period ordering 10

11 Cournot Duopoly, only when the reservtion price () of the product being sold is resonbly smll. Another observtion is tht the second period ordering quntity of the retilers is positive only for 4h/3. This lso is significnt in the sense tht, when < 4h/3, no second period ordering is needed nd ll second period demnd is met from inventory crried from the first period. So, when <4h/3, the inventory crried from the first period is enough to fully meet the second period demnd. Combining both the results, we cn sy tht for 4h/3 8h, it is optiml for the retilers in Cournot Duopoly to crry strtegic inventory to the second period, nd this inventory is ctully fully sufficient to meet second period demnd. This lso intuitively seems like good strtegy in tht price rnge, since we observe tht the second period equilibrium wholesle price set by the mnufcturer is much greter thn the first (compring eqn. (6) to eqn. (14)), but this informtion is not known to the retiler in period 1. However this is only nrrow bnd of reservtion prices, nd generlly low rnge, since h is smll positive quntity. As such we cn sy tht strtegic inventory is optiml in two-period ordering Cournot Duopoly, only for low-priced products nd too only in nrrow rnge. Annd et l. (008) show tht the strtegic inventory quntity is positive, when > 4h, in one-retiler, one-mnufcturer, two-period ordering problem, without ny competition. So, we cn conclude tht, when we expnd the sid model to consider two-retiler duopoly, most of this incentive to hold strtegic inventory disppers. Also, we observe

12 tht the equilibrium wholesle prices set by the mnufcturer in the two periods is higher in our cse (Cournot Competition), in both periods, thn the corresponding prices considered by Annd et l. (008), which hs no competition, (ll other conditions similr), implying tht retiler competition forces regime of higher wholesle prices. We lso observe n interesting trend with the equilibrium wholesle prices under Cournot Competition. If we compre the prices we hve obtined to those obtined by Annd et. l. (008), under two-period ordering dynmic contrct, but without competition, (Compre Equtions (14) nd (15) to Tble 1 of Annd et. l. (008)) we see tht the prices we obtin re lower in both periods, possibly implying tht retiler competition forces the mnufcturer to lower his wholesle price. The second period equilibrium wholesle price is higher thn the first, in Cournot Competition, which is similr to the result under dynmic contrct without competition obtined in Annd et. l. (008) Competition lso seems to hve interesting effects on the totl system output. We define the totl system output s the totl quntity of product sold in the two periods, which is the sum of the quntities ordered in both periods (becuse, we re ssuming tht the retilers cn sell everything they wnt to sell in ech period, it is only the price-per-unit tht vries, depending on the totl quntity in the mrket). Compring the totl system output cross the two periods in competition (q + q 1 + q 1 + q from bove, to q 1 +q from Tble 1 of Annd et l (008), we observe tht the system output is higher in Cournot Duopoly, thn for the cse with single monopolistic retiler, implying tht 1

13 competition promotes ech retiler to order more nd sell more, thereby incresing the system output. 5. References: 1. Annd, Krishnn, Anupindi, Rvi, Bssok, Yehud. Strtegic Inventories in Verticl Contrcts. Mngement Science, 54: , Published online before print October 1, 008, DOI: /mnsc Cchon, G Supply chin coordintion with contrcts. Hndbooks in Opertions Reserch nd Mngement Science: Supply Chin Mngement, edited by Steve Grves nd Ton de Kok. North-Hollnd. 3. Holt, C., F. Modiglini, J. Muth, H. Simon Plnning Production, Inventories, nd Work Force. Prentice Hll, Inc. 4. Mtsumur, Toshihiro, Cournot Duopoly with Multi-period Competition: Inventory s Coordintion Device, Austrlin Economic Ppers, Blckwell Publishing, vol. 38(3), pges 189-0, September. 5. Mollgrd, H. Peter, Sougt Poddr, Dn Sski Strtegic inventories in two-period oligopoly. Deprtment of Economics, University of Exeter, Exeter, UK. 6. P. Keskinock, K. Chrnsiriskskul, nd P. Griffin (008), Strtegic Inventory in Cpcitted Supply Chin Procurement, Mngeril nd Decision Economics, Vol. 9, No. 1,

14 7. Pl, Debshis, "Cournot duopoly with two production periods nd cost differentils," Journl of Economic Theory, Elsevier, vol. 55(), pges , December. 8. Rotemberg, Julio J., Grth Sloner Cyclicl behvior of strtegic inventories. Qurterly Journl of Economics 104(1) Sloner, Grth The role of obsolescence nd inventory costs in providing commitment. Interntionl Journl of Industril Orgniztion Tsy, A.A., S. Nhmis, nd N. Agrwl, "Modeling Supply Chin Contrcts: A Review," Quntittive Models for Supply Chin Mngement (Volume 17 of Interntionl Series in Opertions Reserch nd Mngement Science), S. Tyur, R. Gneshn, nd M. Mgzine (Eds.), Kluwer Acdemic Publishers, Boston, MA, 1998, pp