Abstract. Abstract Introduction Two Models Classic Optimal Lot Size (OLS) Model Monopolistic Competition 3

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1 Optmal Lot Sze, Inventores, Prces and JIT Under Monopolstc Competton * Mchael C. Lovell Wesleyan Unversty, Mddletown, CT Mlovell@Wesleyan.edu May 17, 2000 [Prelmnary Draft] Abstract Ths paper attempts to demonstrate that the dscplnes of Operatons Resarch/Management Scence and economcs are complementary by drawng on both n analyzng the effects of the ntroducton of Just n Tme on prces and product varety. From the dscplne of OR/MS I draw on the Optmal Lot Sze Model. From the dscplne of economcs I borrow the theory of monopolstc competton. The model of monopolstc competton recognzes the sgnfcance of product dfferentaton: Because no two frms n a monopolstc compettve ndustry produce exactly the same product, the demand curve confrontng each producer s downward slopng, lke monopoly. Because of free entry and ext, economc profts are drven to zero n the long run, lke perfect competton. Ths paper ntroduces nventores nto the model of monopolstc competton by ncorporatng optmzed setup and nventory carryng costs nto the representatve frm s total cost functon. I show that the adopton of Just n Tme (JIT), mplemented by reengneerng the product so as to drastcally reduce or elmnate setup costs, benefts consumers, at least n the long run: compettve forces mean that the adopton of JIT leads to both a reducton n prce and an ncrease n the varety of products offered n the marketplace. But alas, the rewards to the nnovatng busness frm are only transtory. Free entry drves economc profts back to zero for both the nnovatng and the mtatng frms. Abstract Introducton Two Models Classc Optmal Lot Sze (OLS) Model Monopolstc Competton 3 3. Synthess Just n Tme Summary and Concluson... 7 References...7 MS/OR + Economcs * Prelmnary Draft of a paper to be presented at the 11 th Internatonal Symposum on Inventores, ISIR, Budapest, August, 2000.

2 Lovell: Optmal Lot Sze, Inventores, Prces and JIT under Monopolstc Competton page 2 1. Introducton Practtoners of Operatons Research and Management Scence make ther lvng by tellng busness frms how they can operate more proftably, and they pass the market test. Economsts customarly assume that busness enterprses succeed n maxmzng proft. Whle economsts thus assume that the Operatons Researchers have completed ther task, I thnk that much may be ganed by ntegratng these two dscplnes. I thnk t unfortunate that economsts don t try to learn more from our sster dscplnes. 1 Inventory Fgure 1: S-s Inventores S s Days In ths paper, I wll try to demonstrate that these are ndeed complementary dscplnes by mergng two dstnct strands of thought: From the dscplne of Operatons Research I draw on the concept of OLS (Optmal Lot Sze not ordnary least squares). From mcro economc theory, I draw on the theory of Monopolstc Competton. After a bref revew of the relevant features of these two theores, I wll combne them nto a model that can be used to analyze certan economc consequences of adoptng Just-n-Tme. I nvestgate how prces and the number of frms n the ndustry wll adjust as a result of adoptng ths new management style. 2. Two Models 2.1. Classc Optmal Lot Sze (OLS) Model Our frm produces ts product n lots (or batches) of sze D n meetng annual sales q. The goods produced n each batch are placed n nventory and gradually sold off. After a batch s sold off and nventory reduced to zero, another lot s produced and the cycle repeats. The resultng saw-tooth movement of the nventory stock s dsplayed on Fgure 1 Two types of cost must be balanced n decdng how bg a batch D to produce: 1. It costs money to set up the machnery to produce a batch and there may be cleanup or shutdown costs at the end of the run. 2. If we produce larger batches, we wll have fewer setup costs each year and our Fgure 2: Optmal Lot Sze total annual setup cost wll be less. But the bgger the batches the larger our nventory, and holdng nventory costs us money. 1 A major contrbuton of ISIR s to provde a venue for such nteracton, as at the 1994 Lake Balaton Workshop on Mcro Foundatons of Macroeconomc Analyss of Inventores. Costs Setup Costs OLS Lot Sze Total cost Inventory holdng cost

3 Lovell: Optmal Lot Sze, Inventores, Prces and JIT under Monopolstc Competton page 3 Fgure 1 The optmal lot sze mnmzes the sum of these two types of cost. Let q equal annual sales and D the sze of a batch. Then there wll be q/d setups a year. If c s s the cost of a setup, then total setup costs for the year wll be c s q/d. Assumng sales take place at a roughly unform rate throughout the year, nventory wll range from 0 to D wth an average level of D/2. If the cost of carryng a unt of output n nventory for a year s c, then yearly nventory cost wll be c D/2. If we let k 0 denote fxed costs and k 1 the cost of the labor and materals and so forth that are requred to produce a unt of output, then our total costs wll be C = k 0 + k 1 q + c D/2 + c s q/d. (1) The optmal lot sze s that D whch mnmzes the annual costs of meetng sales demand q. The standard textbook graph s reproduced as Fgure 2. Dfferentatng (1), we have as a necessary condton for a maxmum C/ D = c /2 - c s q/d 2 = 0, or D = 2 cq/ c. (2) s Ths s the famous square-root rule for the optmal lot sze. 2 Whle the rule s famous, whch s why I use t n ths paper, t may not be the best choce; Lnda G. Sprague and John G. Wacker[1996] explan why t s not used that much n practce.fgure 2: Optmal Lot Sze 2.2. Monopolstc Competton The theory of monopolstc competton was advanced ndependently seven decades ago on dfferent sdes of the Atlantc by Edward Chamberln [1933] and Joan Robnson [1933]. Ther key contrbuton was to recognze that even when there are a large number of frms n an ndustry, the demand curve facng each frm wll be downward slopng because of product dfferentaton, just lke monopoly. But at the same tme, the theory encompasses the case n whch there s free entry and ext nto the ndustry, whch means that economc proft (net of the costs of owner suppled captal and labor) wll be drven n the lmt to zero, just lke pure competton. The standard textbook graph of a representatve frm n monopolstc compettve equlbrum s presented as Fgure 3. The demand curve facng the ndvdual frm, labeled dd, s downward slopng because of product dfferentaton. If our frm rases ts prce, whle other frms keep thers fxed, our frm wll loose some sales; but unlke perfect competton, t wll retan some loyal customers wth strong preferences for ther product. Perfect competton s Law of One Prce does not hold true under monopolstc competton, at least n the short run. The steeper DD curve shows how sales would adjust f all frms n the ndustry were to charge the Fgure 3: Monopolstc Competton dd' ($13.5) dd' ($6) DD'(n=46) Average Total Cost e quantty 2 The equvalent concept, effcent order quantty, arses n procurement. If a retaler orders tems n larger quanttes t wll ncur lower orderng costs each year, but ts nventores wll on average be larger. Precsely the same square root relatonshp arses from ths argument; equaton (2) holds wth c s now representng orderng costs.

4 Lovell: Optmal Lot Sze, Inventores, Prces and JIT under Monopolstc Competton page 4 dentcal prce. But t s assumed that there are enough other frms n the ndustry to make t reasonable for each frm to assume that the others wll not respond when t changes ts own prce. 3 The graph s drawn for the case of long run equlbrum the number of frms has adjusted through free entry and ext untl economc proft s zero at pont e. Ths s the best the frm can do, but t just breaks even. Fgure 3: Monopolstc Competton 3. Synthess As a frst step toward a synthess we must determne the frm s total cost functon, gven that whatever q turns out to be our frm wll use the optmal lot sze specfed by (2). It s easly shown that annual nventory carryng costs wll be c D/2 = [c c s q/2] 1/2, and that ths also equals the costs of q/d 0 setups durng the year. Substtutng back nto (1) yelds the total cost functon, gven that the optmal order quantty specfed by (2) s used: 4 C = k 0 + k 1 q + (2c c s q) 1/2. (3) Lettng α = (2c c s ) 1/2 we may wrte our total cost functon as C = k 0 + k 1 q + αq 1/2. (4) We have average costs C/q = k o /q + k 1 + αq -1/2. (5) The average cost functon on Fgure 2 was drawn for k 0 = 64, k 1 = 4, c = 10 and c s = 75; margnal cost s dc/dq = k 1 + αq -1/2 /2. (6) Our task s to work out the market mplcatons of ths type of cost functon. We shall suppose that the product s sold n a monopolstcally compettve market wth nverse demand functon p (q, p,n) = d 1 + d 2 p - q n 1/2 /d 0, (7) where n s the number of frms n the ndustry. 5 Ths functon mples that an ncrease n the number of compettve frms (.e., a larger n) wll reduce the quantty that our frm can sell at any gven prce. Also, a decrease n the average prce charged by the other frms n the ndustry would mean that our frm wll ether have to lower ts prce or sell less. The propertes of ths demand functon are dscussed at length n Lovell [2000]. The demand functon on Fgure 2 was drawn wth hypothetcal parameter values d 0 = 10, d 1 = 11, d 2 = 0.75 and n = 46. Demand functon (7) mples that revenue wll be Therefore, proft wll be 2 R(q ) = d 1 q + d 2 p q q n 1/2 /d 0. (8) π(q) = R(q ) C(q) = 3 Chamberln[1950] called ths the large group case; he also consdered the more complcated small group case n whch each frm consders the possble reacton of other frms n the ndustry n decdng what prce t should set. 4 It s customary n modelng monopolstc competton to assume constant margnal cost;.e., to use a cost functon nvolvng the frst two terms of (3). See, for example, Lovell [1970] and Dxt-Stgltz[1977]. 5 For smplcty we nvoke Chamberln s symmetry assumpton that all frms have the same cost and demand functons.

5 Lovell: Optmal Lot Sze, Inventores, Prces and JIT under Monopolstc Competton page 5 = k 0 + (d 1 + d 2 p - k 1 )q q 2 n 1/2 /d 0 - αq 1/2. (9) As a necessary condton for proft maxmzaton we must have dπ/dq = d 1 + d 2 p - k 1 2 n 1/2 d 0 q - αq -1/2 /2= 0 (10) Long-run monopolstc compettve equlbrum also requres that p = p and that n adjusts so that π = 0. The detals of the OLS- Monopolstc Compettve equlbrum are plotted on Fgure 4 for the specfed values of the parameters. At equlbrum pont e we have p = p = $13.5, q = 11.3, n = 46 and π = 0. Ths s a Nash equlbrum because t does not pay for any frm to change ts prce, gven what the other frms are chargng. It s also a long-run equlbrum because the entry of another frm would lead to negatve profts. Fgure 4: OLS-Monopolstc Compettve Equlbrum 4. Just n Tme Fgure 4: OLS - MC Equlbrum Just n Tme (JIT), a manageral concept orgnatng n Japan, s an alternatve to batch processng. 6 The JIT strategy s to re-engneer the producton process, and perhaps also modfy the desgn of the product, so as to elmnate or to reduce drastcally the set-up-cost. At an ISIR conference at Wesleyan n 1987, Professor Hajme Yamashna of Kyoto Unversty llustrated what s nvolved wth a dramatc example. Toyota desgned an automated factory for makng the wde varety of gauges requred for ts autos. Wth batch processng substantal setup costs were ncurred because of the dled labor and the loss of producton durng down tme whle the machnes were adjusted to swtch to the producton of a dfferent type or model of gauge. In order to mplement JIT, the robots were reengneered Fgure 5: JIT to - MC automatcally Equlbrum swtch from producng one type of gauge to another. The swtch was accomplshed by sendng a dummy 25 sgnal gauge down the assembly lne, whch was sensed by each robot. Instead of dong ts thng DD' (n=100) to the dummy gauge as t progressed to ts staton on the assembly lne, each robot n turn would 20 dd' read the dummy gauge s nstructons to self-adjust ts settngs to reflect the next type of gauge to 15 = $12 be produced. Thus, the setup cost s reduced to the opportunty cost e of Average not Total producng Cost one part. Wth neglgble setup cost the optmal nventory s reduced 10 to essentally zero, as can be seen from equaton (2). A varety of benefts accrue to the frm that swtches 0 to JIT. Advocates of JIT argue that n addton to reducng nventores and ther carryng cost t also contrbutes to tghter qualty control. Inventores hde defects! Wth batch processng, a maladjustment quantty of a machne may lead to the producton of an entre lot of defectve parts before the problem s recognzed. The problem s lkely to be compounded because an assembly-lne worker encounterng a defectve part wll be tempted to throw t back nto the nventory ple and take another part rather than addressng the problem. Not only does ths adverse selecton mean that the next shft may nhert an nventory /q+4 Margnal cost DD'(n=46) e quantty Average Total Cost 6 See also the onlne OR notes of J.E. Beasley,

6 Lovell: Optmal Lot Sze, Inventores, Prces and JIT under Monopolstc Competton page 6 wth a hgh number of defects. It also means that a substantal number of defectve parts wll have been produced before the defect s properly recognzed and correctve actons taken. That s why reducng nventory wth JIT can make a key contrbuton to mproved qualty control. 7 How wll a monopolstcally compettve ndustry respond to the adopton of just-n tme? Clearly the nnovatng frm wll reap hgher profts, but how wll prce and the number of frms be affected f other frms also make the swtch? Wll consumers beneft from the swtch to JIT? The predctons that the theory of monopolstc competton makes about the effects of shftng to JIT management are llustrated on Fgure 5. 8 The elmnaton of set-up and nventory carryng costs reduces the average cost functon for the nnovatng frm to the new Average Total Cost curve. Wth ts reduced cost structure, the best prce for the JIT nnovatng frm to charge, gven that the other frms n the ndustry contnue to charge p = $13.50, s p = (d 1 + d 2 p + k 1 )/2 = $ (11) The correspondng quantty demanded, gven that there are n = 46 frms n the ndustry, s q = (d 1 + d 2 p k 1 )d 0 /2n 1/2 = (12) The nnovatng frm wll enjoy profts of ( ) = $43.8, less the annual amortzed cost of the nvestment requred to nsttute JIT. When other frms n the ndustry adopt JIT, ther lower cost structure wll ntensfy market pressures. The new Nash equlbrum, after all the frms have adopted JIT, gven that n = 46, s wth quantty p N = (d 1 + k 1 )/(2-d 2 ) = $12 for all, (13) q N = d 0 [d 1 (1-d 2 )k 1 )]/(c(n)(2-d 2 ) = (14) Profts per frm are now π N = q N (p N -k 1 ) k 0 = $ (15) Thanks to the copycats, the nnovatng frm now enjoys lower profts than antcpated, but customers beneft from consumng more of the product at lower prces. Ths s unlkely to be the end of the story. Postve profts wll attract more frms nto the ndustry. Entry wll contnue untl economc proft s drven down to zero. For ths numercal example, the long run equlbrum predcted by the theory of monopolstc competton nvolves a prce of $12 but the number of frms ncreases to Output for each frm s n = {d 0 [d 1 (1-d 2 ) p ]( p -k 1 )/ k 0 } 1/(1-γ) = 100. (16) q( p,n) = k 0 /( p -k 1 ) = d 0 [d 1 (1-d 2 ) p ]/c(n) = 8. (17) 7 In Lovell [1993] I show that the economy-wde adopton of JIT mght contrbute to economc stablty wthn the context of a mult-sector model, but the role of prce complcatons was excluded from the analyss. 8 The graph and followng equaton are from Lovell [2000].

7 Lovell: Optmal Lot Sze, Inventores, Prces and JIT under Monopolstc Competton page 7 5. Summary and Concluson Ths paper draws upon the dscplnes of Operatons Research/Management Scence and economcs n demonstratng that the ntroducton of JIT may result n both lower prces and ncreased product varety. There may be added complcatons to the story. On the one hand, the analyss presented here neglected the possble effects of the adopton of just n tme on fxed costs k 0. Ths coeffcent may go up f t has to absorb the captal costs ncurred n adoptng the JIT technology. On the other hand, the adjustment may nvolve the adopton of multple product lnes by exstng frms as well as the entry of new frms. If so, the fxed cost k 0 may be spread over several products and addtonal economes to scale may be realzed. Never the less, at the end of the day both the nnovatng frm and ts mtators are lkely to fnd that the gans from mproved manageral practce are far from permanent, but customers wll beneft both from lower prces and from an ncrease n product varety. Fgure 5: JIT Monopolstc-Compettve Equlbrum References Chamberln, E. H., The Theory of Monopolstc Competton, 6 th edton, Cambrdge Dxt, Avnash K and Joseph E. Stgltz, Monopolstc Competton and Optmum Product Dversty, Amercan Economc Revew, June, 1977, Hotellng, Harold, Stablty n Competton, Economc Journal, March, 1929, Lovell, Mchael C., Product Dfferentaton and Market Structure, Western Economc Journal, June 1970, Lovell, Mchael C., Smulatng the Inventory Cycle, Journal of Economc Behavor and Organzaton, June, 1993, Lovell, Mchael C., Monopolstc Competton Theory and Applcatons (workng paper), May 7, Robnson, Joan, The Economcs of Imperfect Competton, Macmllan, Spence, Mchael, Product Dfferentaton and Welfare, Amercan Economc Revew, May, Sprague, Lnda G. and John G. Wacker, Mcroeconomc Underpnnngs of Macroeconomc Analyss of Inventores, Reflectons on Frm and Natonal Inventores, eds: A.Chkán, A. Mlne, and L. G. Sprague, Internatonal Socety for Inventory Research, Yamashna, Hajme, A New Formulaton of Flow Shop and Job Shop Schedulng for Just-In- Tme Producton, unpublshed paper delvered at the Conference on the Economcs of Inventory Management, Wesleyan Unversty, August, 1987.