Price and Inventory Dynamics in an Oligopoly Industry: A Framework for Commodity Markets

Transcription

1 BGPE Dscusson Paper No. 92 Prce and Inventory Dynamcs n an Olgopoly Industry: A Framework for Commodty Markets Alexander Stenmetz Aprl 2010 ISSN Edtor: Prof. Regna T. Rphahn, Ph.D. Fredrch-Alexander-Unversty Erlangen-Nuremberg Alexander Stenmetz

2 Prce and Inventory Dynamcs n an Olgopoly Industry: A Framework for Commodty Markets Alexander Stenmetz Unversty of Würzburg Aprl 13, 2010 Abstract Ths paper analyzes the nteracton between prce and nventory decsons n an olgopoly ndustry and ts mplcatons for the dynamcs of prces. The work extends exstng lterature and especally the work of Hall and Rust (2007) to endogenous prces and strategc olgopoly competton. We show that the optmal decson rule s an (S, s) order polcy and prces and nventory are strategc substtutes. Fxed orderng costs generate nfrequent orders. Consequently, wth strategc competton n prces, (S, s) nventory behavor together wth demand uncertanty generates endogenous cyclcal patterns n prces wthout any exogenous shocks. Hence, the developed model provdes a promsng framework for explanng dynamcs of commodty markets and especally observed autocorrelaton n prce fluctuatons. Keywords: Inventory dynamcs, prce competton, olgopoly, (S, s) order polcy, commodty markets. JEL classfcaton numbers: D21, D43, E22, L81. 1 Introducton Ths paper analyzes the nteracton between prce and nventory decsons n an olgopoly ndustry and ts mplcatons for the dynamcs of prces such as prce dsperson. Crosssectonal prce dsperson s a common feature n many retal markets. Snce Stgler s (1961) semnal work prce dsperson has usually been explaned by consumer search I am grateful to Norbert Schulz for contnuous encouragement and Víctor Agurregabra, who motvated ths work, for nsghtful gudance and nvaluable advce. I also thank semnar partcpants at the 7th BGPE research workshop for helpful comments and suggestons. Unversty of Würzburg and Bavaran Graduate Program n Economcs, Sanderrng 2, Würzburg, Germany. Phone: , e-mal: alex.stenmetz@un-wuerzburg.de, web: 1

3 costs. In contrast, Agurregabra (1999) shows that retal nventores can generate (S, s) dynamcs of nventores whch n turn can explan tme varablty of prces of supermarket chans. 1 However, as n hs model monopolstc competton s analyzed prce dsperson between dfferent frms can not be observed. Extendng the descrbed work, ths paper addresses the queston how olgopolstc competton affects these dynamcs. 2 Prevous papers have characterzed the optmal decson rules of smlar dynamc models. In addton to Agurregabra (1999) who analyzes prce and nventores wth lump-sum costs under monopolstc competton Hall and Rust (2007) study optmal nventory decsons wth lump-sum costs under perfect competton. Ther paper extends the framework of Agurregabra (1999) n some ways but s otherwse lmted to one decson varable as prces are taken as gven. Hall and Rust (2007) show that n ther perfect competton model the (S,s) polcy s an optmal order strategy. 3 To the best of our knowledge, these two works studyng extreme cases of competton are by far the most elaborated papers nvestgatng these decson problems. 4 The analyss of optmal decson rules under olgopolstc competton forms an obvous gap n the lterature. However, related studes of olgopolstc competton exsts. Dutta and Sundaram (1992) and Dutta and Rustchn (1995) analyze a dscrete choce stochastc duopoly game wth lump-sum costs. In these frameworks the one abstract decson varable affectng both frms payoffs cannot be nterpreted as beng related to nventory. Nevertheless, the optmalty of an (S,s) polcy can also be shown. More recently, Besanko and Doraszelsk (2004) study decsons about prces and capacty. However, the man and mportant dfference between nventory and capacty s that excess capacty s worthless whle keepng nventory affects future competton. Hence, addtonal strategc effects due to kept stock are at place. Ths s especally mportant when nvestgatng olgopolstc competton. Ths paper extends the lterature by characterzng an equlbrum n a model of prce and nventory competton n olgopoly. We allow olgopolstc frms to nteract strategcally. Ths allows for studyng prce dsperson between frms. 1 Under an (S,s) rule nventory moves between the target nventory level, S, and the order threshold, s, wth s < S. Whenever the frm s nventory level falls below the order threshold, a new order s placed such that the target nventory level S s attaned. 2 Addtonally, the man focus of the paper by Agurregabra s an emprcal analyss buldng on a numerc smulaton. The formal theoretcal proof of the optmalty of the consdered nventory decson s therefore not rgourously done and ncomplete. Thus, our paper s the frst to formally prove the optmalty of (S, s) polcy wth endogenous prces. 3 Thereby Hall and Rust (2007) extend earler work lke Seth and Cheng (1997) and Cheng and Seth (1999) to a more general specfcaton of the Markov process. 4 There exst also some papersanalyzng dynamcolgopoly wth nventoreswthout consdernglumpsum orderng cost, lke Krman and Sobel (1974) or more recently Bernsten and Federgruen (2004). However, wthout orderng cost statonary optmal strateges result whch are n essence dentcal to those of the correspondng statc sngle perod game. 2

4 Besdes, such a model that s ncorporatng nventory and olgopoly n dynamc competton provdes the most plausble framework for retal ndustres. Retal ndustres have become hghly concentrated,.e., n most categores lke grocery, supermarkets, and offce supples just a handful of rvals compete locally. In the supermarket ndustry for example a small number of frms capture the majorty of sales as supermarkets compete n tght regonal olgopoles. Thus, ths ndustry s a prme example of olgopoly. Besdes, nventory costs are of major mportance. Supermarkets nvest n state of the art dstrbuton systems to mnmze storage and transportaton costs (see e.g. Beresteanu & Ellckson, 2006; Ellckson, 2007). Hence, decdng the optmal nventory and store offer forms an mportant optmzaton problem for supermarket chans. In ths work we study the decson problem of a central store,.e., ts decson about retal prces and orders to supplers, facng olgopolstc competton and takng nto account the exstence of lump-sum orderng cost. We develop a model of retal competton n whch the mpact of nventores on competton and prces can be evaluated. We analyze the characterstcs of the optmal decson rule. The man fndngs of our theoretcal model of olgopoly support the smulaton results of Agurregabra (1999) studyng monopoly. Key factors for prce fluctuatons are lumpsum orderng costs and demand uncertanty. Lump-sum orderng cost generate (S, s) nventory behavor. Demand uncertanty creates a postve probablty of excess demand,.e., stockouts. The postve stockout probablty has a negatve effect on expected sales whch n turn creates substtutablty between prces and nventores n the proft functon such that n equlbrum prces depend negatvely and very sgnfcantly on the level of nventores. Ths results n a cyclcal pattern of nventores and prces where prces declne sgnfcantly when an order s placed and consequently nventory reducton generates prce ncrease. The prcng behavor n ths model can generate cross-sectonal prce dsperson wth cyclcal patterns even wthout menu costs. The rest of the paper s organzed as follows. Secton 2 ntroduces the model and shows mportant characterstcs of frms expected sales. Secton 3 characterzes the optmal decson rules. Secton 4 concludes whle the Appendx contans the proofs of the results stated n the text. 2 The Model Consder an olgopoly market where rsk neutral frms, ndexed by {1, 2,..., N}, sell dfferentated storable products. Each frm sells a varety of the product. Frms compete n prces and they have uncertanty about temporary demand shocks. In the short run, frms cannot respond to these temporary shocks nether by changng prces nor by ncreasng supply, n case of excess demand. Frms do not face any delvery lags and cannot backlog unflled orders. Thus, whenever demand exceeds quantty on hand, 3

5 the resdual unflled demand s lost. Therefore, the quantty sold by frm at perod t s the mnmum of supply and demand: y t = mn{s t +q t,d t }, (1) where y t s the quantty sold; s t s the level of nventores at the begnnng of perod t; q t represents new orders to wholesalers durng perod t; and d t s consumers demand. Every perod t a frm knows the levels of nventores of all the frms n the market,.e., the vector s t {s 1t,s 2t,...,s Nt }. 5 Gven ths nformaton, the frm decdes on prces and new orders (p t,q t ) to maxmze ts expected value E t ( r=1 βr Π,t+r ), where β (0,1) s the dscount factor and Π t s the current proft of frm at perod t. cost: A frm s current proft s equal to revenue mnus orderng cost and nventory holdng Π t = p t y t c q t k I{q t > 0} h s t, (2) where c s the unt orderng cost; k s the fxed or lump-sum orderng cost; and h s the nventory holdng cost. The transton rule of nventores,.e., state varables, s: s t+1 = s t +q t y t = max{0,s t +q t d t }. (3) s Frms have uncertanty about current demand. The demand of product at perod t d t = exp{ε t }d e t. Here, ε t s a temporary and dosyncratc demand shock that s ndependently and dentcally dstrbuted over tme wth cumulatve dstrbuton functon F( ) that s contnuously dfferentable on the Lebesgue measure. These shocks are unknown to frms when they decde prces and orders. Furthermore, d e t s the expected demand that depends on the endogenous prces and the exogenous qualtes of all products. The expected demand d e t s a functon of the prces of all frms such that t s strctly ncreasng n the own prce, strctly decreasng n the prces of compettors, and the revenue functon p d e s strctly concave n p. By defnton of expected demand, we have that E(exp{ε t }) = 1. For techncal reasons t s useful to assume that F( ) s such that the respectve hazard rate h( ) = f( ) 1 F( ) s smaller than one.6 For examples and numercal exercses t may be useful to consder a logt demand model for the expected demand: d e t = exp{w αp t } 1+ N j=1 exp{w j αp jt }, (4) where {w : = 1,2,...,N} are exogenous parameters that represent product qualtes, and α s a parameter that represents the margnal utlty of ncome. The logt demand 5 Ths s a very reasonable assumpton as frms can observe prces and are therefore able to learn and deduce stock levels. 6 Ths assumpton s especally helpful for provng Lemma 2, although t s only a suffcent but not necessary condton. 4

6 model s convenent for the dervaton and llustraton of some future results, but t can be relaxed for all our results Implcatons of Demand Uncertanty for Expected Sales As a frm does not know the temporary demand shock ε t, t does not know actual sales y t. Expected profts are Π e t = p t y e t c q t k I{q t > 0} h s t, where y e t represents expected sales,.e., y e t = E[mn{d t,s t +q t }]. Demand uncertanty has mportant mplcatons for the relatonshp between prces and nventores. Lemma ) 1. Expected sales yt e are equal to expected demand d e t tmes a functon,.e., λ( st +q t d e t ( yt e = d e st +q t tλ d e t ). (5) The functon λ(x) s defned as mn{x,exp(ε)}df(ε) and t has the followng propertes: () It s contnuously dfferentable; () t s strctly ncreasng; () λ(0) = 0; (v) λ( ) = E(exp(ε)) = 1; and (v) for x > 0, λ (x) = ln(x) df(ε) = 1 F(ln(x)) (0,1). Proof: See Appendx A.1. In case of a very small (close to zero) supply-to-expected-demand-rato s t+q t d e stockout probablty s very large such that expected sales are much lower than expected t demand (approachng zero). On the other hand, a hgh rato (approachng nfnty) yelds low probablty for stockouts such that expected sales are almost equal to expected demand. The hgher the supply-to-expected-demand-rato the lower gets the probablty of stockout and the more do expected sales converge to expected demand. Ths s formalzed n propertes () - (v). From property (v) yeldng λ (x) < 0 t s now clear that the gan of a hgher supply-to-expected-demand-rato for expected sales s hgher the lower the rato. For low ratos the gan s almost equal to the ncrease of stock as one unt more n stock n essence s a unt more sold. For hgh ratos the probablty of sellng an addtonal unt n stock decreases to zero. Therefore, varablty over tme n the supply-to-expected-demand-rato can generate sgnfcant fluctuatons n expected sales and thus n optmal prces. 7 See Agurregabra (2007) for a dervaton of ths demand model from a model of consumer behavor under possble excess demand. 5

7 2.2 Markov Perfect Equlbrum The model has a Markov structure and we assume that frms play Markov strateges. That s, a frm s strategy depends only on payoff relevant state varables, whch n ths model s the vector of nventores s t. Therefore, a strategy for frm s a functon σ (s t ) from the space of the vector of nventores,r N +, nto the space of the decson varables (p t,q t ),R 2 +,.e., σ (s t ) s a functon fromr N + ntor 2 +. Let σ {σ : = 1,2,...,N} be a set of strategy functons, one for each frm. Suppose that frm consders the rest of the frms to behave accordng to ther respectve strateges n σ. Under ths condton, other frms nventores, s t, follow a Markov transton probablty functon F σ s (s t+1 s t ). Note that ths transton probablty functon depends on the other frms strateges n σ. Takng F σ s as gven, frm s decson problem can be represented usng the Bellman equaton: { V σ (s t ) = max Π σ (p,s t +q )+β {p,q } } V σ (s,t+1,s t+1 )df(ε t )dfs σ (s t+1 s t ). The (expected) proft functon s contnuously dfferentable and the standard regularty condtons apply such that the value functon s unquely determned as the fxed pont of a contracton mappng. Note that ths value functon s condtonal to the other frms strateges. A Markov perfect equlbrum (MPE) s a set of equlbrum strateges σ such that for every frm and for every vector s t R N + we have that { (6) σ (s t ) = arg max {p,q } Π σ (p,s t +q ) +β (s,t+1,s t+1 )df(ε t )df σ s (s t+1 s t ) }. (7) 3 Optmal Decson Rule Let us now characterze the optmal decson rule for a frm n ths game of olgopolstc competton. In ths secton we wll show that the (S,s) rule s ndeed the best response not only to an (S,s) rule but to any gven strategy of the opponents. Ths, of course, mples that the equlbrum resultng from (S, s) strateges by all players s a MPE. In order to represent the optmal decson rule of the olgopolsts, t s convenent to representthedecsonproblemntermsofthevarablesp t andz t s t +q t.thevarable z t represents the total supply of the product durng perod t. It s also useful to defne the followng value functon whch s ndependent of the frm s own current nventory,.e., the only state varable the frm can nfluence (however, t s not ndependent of the 6

8 current state per se), and takng the other frms strateges n σ and so F σ s as gven: Q σ (z t,p t ;s t ) cz t +p t +β such that mn{z t ;e ε t t (p t )}df(ε t ) (max{0;z t e ε t t (p t)};s t+1 )df(ε t )df σ s (s t+1 s t ) (8) V σ { } (s t ) = max Q σ (s t +q,p ;s t ) (h c )s t k I {q >0}. {p,q } Gven the functon Q σ, t s clear that an olgopolst chooses (z t,p t ) as a best response to the other frms strateges n σ,.e., other frms order and prcng decsons, to maxmze Q σ (z t,p t ;s t ) ki{z t > s t }. Makng use of ths value functon Q σ we can derve mportant characterstcs of competton n prces and nventores: Lemma 2. The functon Q σ s such that: () Q σ s strctly concave n prces,.e., 2 Q σ (z,p )/ < 0. () Prces and total supply are strategc substtutes,.e., 2 Q σ (z,p )/ z 0. Proof: See Appendx A.2. The postve stockout probablty has a negatve effect on expected sales whch n turn creates substtutablty between prces and nventores n the proft functon. Ths s the case as wth low nventory optmal expected demand (under gven demand uncertanty) s low and thus optmal prce s hgh. Usng σ σ p(s) and σ σ z(s) to represent the optmal response rules for p and z, respectvely, we have {σz σ (s),σσ p (s)} = arg max {z s,p 0} {Qσ (z,p ;s t ) ki{z > s }}. We defne the optmal prce as a functon of current supply: p σ (z ;s ) argmax {p } Qσ (z,p ;s ). (9) Snce Q σ s contnuously dfferentable and strctly concave n prces, p σ (z;s ) s mplctly defned by the frst order condton Qσ (z, p ;s ) p = 0. It s now possble to show that the best response to any strategy s an (S,s) rule: Proposton 1. Frm consders the rest of the frms to behave accordng to ther respectve strateges n σ. Takng F σ s as gven, let frm s best response rule for total supply and prces be σz σ(s) and σσ p (s), respectvely. These functons are such that: 1. σ σ p (s) = pσ (σ z(s);s ), where p σ (z ;s ) s contnuous and strctly decreasng n z ; and 7

9 2. σz σ (s) has the followng form: σz(s) σ s σ (s ) f s t s σ = (s ) f s t > s σ (s ), s t (10) where s σ and s σ are scalars, wth s σ > s σ s, and the followng defntons: Proof: See Appendx A.3. s σ (s ) argmax {z } Qσ (z, p (z );s ), (11) s σ (s ) nf{s Q σ (s σ, p (s σ );s ) k Q σ (s, p(s );s )}. (12) The proposton shows that consderaton of olgopolstc competton does not affect the optmalty of (S,s) nventory rules. 8 Fxed orderng costs generate nfrequent orders. The upper band s σ s defned as the optmal order quantty when the frm has no nventory on hand,.e., the optmal nventory level. The lower band s σ s the smallest value of nventory such that the desred order quantty s zero. Ths order polcy mght appear to be a very natural and ntutve strategy. However, as shown n the appendx the value functon s not concave such that a much more complex decson rule could n prncple be optmal. Addtonally, olgopolstc competton assures that no addtonal assumpton on prces lke the no expected loss condton of Hall and Rust (2007) s necessary for the optmal tradng strategy to be of the (S,s) form. 9 Ths (S, s) nventory behavor together wth demand uncertanty generates cyclcal patterns n prces. The optmal prce s a strctly decreasng functon of a frm s nventory on hand z as the postve probablty of stockouts creates strategc substtutablty between prces and nventores. Thus, the prce ncreases between two orders when the stock level decreases and t drops down when new orders are placed. Ths s the case as wth low nventores the optmal expected demand s lower and hence the optmal prce s hgher. When the level of nventores decreases between two orders, the probablty of stockout ncreases and so expected sales decrease and become more nelastc wth respect to the prce. Thus, the optmal prce ncreases between two orders, and decreases when the elastcty of sales goes up as the result of postve orders. The largest prce ncrease occurs just after a postve order and the ncrements tend to be smaller when we approach to the next postve order. The reason for ths behavor s that the cyclcal path of prces generates a cyclcal behavor n sales. The largest sales and, consequently, the largest stock reductons and prce ncreases, occur just after a postve order. 8 However, as thresholds depend on the compettors nventores, we have an (S(s ),s(s )) decson rule. 9 The no expected loss condton requres that the exogenous nonconstant retal prce exceeds a certan (endogenous) nonconstant threshold any tme. Wth endogenous prces, we do not need to mpose such a condton. 8

10 The nterestng result here s that the prcng behavor n ths model can generate cross-sectonal prce dsperson wth cyclcal patterns even wthout menu cost. The magntude of ths prce dsperson wll depend on the magntude of lump-sum orderng costs, the senstvty of the prce elastcty of sales to changes n the probablty of stockout, and the degree of correlaton between the demand shocks at ndvdual frms. 4 Concluson We have shown that the best response not only to (S,s) strateges but to any strategy s an (S, s) rule. Ths result extends earler fndngs of models wthout prce competton (Hall & Rust, 2007) and models wthout strategc competton (Agurregabra, 1999) where fxed orderng costs generate nfrequent orders. Thus, the (S,s) polcy mght appear to be a very robust strategy. However, t s not hard to change assumptons n ways that destroy ts optmalty. Addtonally, wth strategc competton n prces (S, s) nventory behavor together wth demand uncertanty generates cyclcal pattern n prces. The model developed n ths paper provdes a very promsng alternatve for studyng commodty markets. Commodty prces are extremely volatle and papers of the respectve lterature strand are concerned whether theory s capable of explanng the actual behavor of prces. The more recent lterature on ths topc (see for example Deaton & Laroque, 1992, 1996, and Pndyck, 1994) bulds on the supply and demand tradton (see e.g. Ghosh, Glbert, & Hughes Hallett, 1987, for a revew), but wth explct modelng of the behavor of compettve speculators who hold nventores of commodtes n the expectaton of makng profts. 10 However, perfect competton and the absence of lump-sum orderng cost s always assumed n these papers. The studes are tryng to explan extremely volatle prces as a result of exogenous shocks by modelng the behavor of compettve speculators holdng nventores. Results are rather unsatsfyng: In contrast to the models predctons, real prce fluctuatons are not randomly dstrbuted over tme and ths autocorrelaton cannot be explaned by these types of models. In addton, some probably mportant characterstcs of commodty markets are not captured n ths lterature. Studes of these characterstcs (e.g. Carter & MacLaren, 1997, and Slade & Thlle, 2006) fnd that commodty markets are best descrbed by olgopoly nstead of perfect competton. Besdes, lump-sum orderng cost are realstc n some markets (e.g. at London Metal Exchange where orders can result n physcal delvery and all contracts assume delvery). Incorporatng olgopoly competton and lump sum orderng costs could be mportant to study the dynamcs of some commodty prces. In a model lke ours we are able to generate some knd of tme dependent pattern whch s apparently n lne wth emprcal evdence. Ths s n contrast 10 As even estmatng the models s computatonal demandng authors mostly use smulatons. 9

11 to the usual hypothess that prce fluctuatons are the result of exogenous shocks and therefore randomly dstrbuted over tme. Makng use of the developed model t should now be possble to relate fndngs to commodty prce dynamcs and show that lump-sum orderng cost and olgopoly competton can be mportant to explan extremely volatle prces and especally tme dependences n prce fluctuatons. However, due to the relatvely hgh complexty of the framework further research requres numercal experments. By ths means, other topcs lke precse reactons of frms on compettors orders provde scope for nterestng studes. Ths mportant work s left for future research. 10

12 A Appendx A.1 Expected Sales: Proof of Lemma 1 Proof. For notatonal smplcty, we omt here the frm and tme subndexes. By defnton, expected sales y e are: ( ) s+q y e = mn{s+q,d e exp(ε)}df(ε) = d e λ where λ(x) s defned as mn{x,exp(ε)}df(ε). The functon λ(x) has the followng propertes: Also, Fnally, lm λ(x) = x 0 lm λ(x) = lm x x λ (x) = mn{0, exp(ε)})df(ε) = 0. mn{x, exp(ε)})df(ε) = d e I{x < exp(ε)}df(ε) = 1 F (lnx). exp(ε)df(ε) = 1. A.2 The Value Functon: Proof of Lemma 2 Proof. We use backwards nducton and frst show that the propertes of Lemma 2 hold for the fnte horzon problem wth tme horzon equal to T. and Let us consder Q σ T ( ) to represent the proft functon n the last perod,.e., Therefore, Q σ T(z,p ;s ) = cz +p y eσ (z,p ) Q σ T ( ) 2 Q T ( ) p 2 = cz +p λ = cz +p ( z (p ) ) mn{z ;e ε }df(ε ). = y eσ y eσ (z,p ) (z,p )+p, = 2 yeσ (z,p ) ( Gven that y eσ (z,p ) = λ z and 2 y eσ (z,p ) p 2 y eσ (z,p ) = 2 p 2 = deσ 2 y eσ +p (z,p ). (13) ), we have that p 2 F(lnz ln ), ( d F(lnz ln eσ ) ) 2 f(lnz ln ). (p ) 11

13 Insertng these expressons n equaton (13), we get: 2 Q σ T ( ) p 2 ( 2 +p =2 deσ F(lnz ln ) p 2 F(lnz ln ) (2 deσ ( ) ) d eσ 2 f(lnz ln ) (p ) ( =F(lnz ln ) (2 deσ 2 )) +p p 2 ( ) d eσ 2 f(lnz ln ). (p ) ( )) The frst term s negatve because +p 2 s just the second dervatve of the functon p, that s strctly concave by assumpton. It s clear that the second term s also negatve. Therefore, 2 Q T ( ) p 2 < 0. p 2 Furthermore, snce Qσ T ( ) = y eσ y (z,p )+p eσ (z,p ), we have that 2 Q σ T ( ) = yeσ (z,p ) z As we have shown above, yeσ (z,p ) z also shown that yeσ (z,p ) = deσ 2 y eσ (z,p ) z = λ ( z +p 2 y eσ (z,p ) ) z. (14) = 1 F(lnz ln ). We have F(lnz ln ), and therefore = deσ f(lnz ln ) z. Insertng these expressons nto the equaton (14), we get: 2 Q σ T ( ) = 1 F(lnz ln )+ p f(lnz ln ). z z Wth η d (p ) deσ p ) ) ( ( η z λ λ z expresson can be wrtten as wth p > 0 as the elastcty of expected demand, and < 0 as the elastcty of the λ( )-functon the above z λ( ) 2 Q σ T ( ) = λ ( )(1 η d ( )(1 η λ ( )))+λ( )η d ( )η λ ( ) (15) z z η λ ( ) = λ ( ) 2 +λ( )(λ ( )+ z λ ( )) λ( ) 2. The term η λ ( ) s negatve as λ ( )+ z λ ( ) = 1 F( ) f( ) s postve for 1 F( ) > f( ) whch s fulflled by assumpton. Thus, the second term of equaton (15) s negatve. Now, let s partcularze expresson (15) at (z, p T (z)). We can wrte Q σ T ( ) = y eσ (z,p )(1 η d ( )(1 η λ ( ))) 12

14 such that 1 η d ( )(1 η λ ( )) can never be postve at the optmal decson and therefore 2 Q σ T ( ) z < 0 holds. We wll now show that f Q σ t+1 ( ) s strctly concave n prces and prces and supply are strategc substtutes n t+1, then Q σ t ( ) s strctly concave n prces and prces and supply are strategc substtutes n t as well. We make use of the fact that the proft functon s bounded from above. More specfcally, max s 0 max {z s,p 0} { ( ) } p z λ c z k I (p ) {z >s } s smaller than some constant τ <. Ths property guarantees that for any values of z and p Q σ (z,p ) = lm T Qσ T (z,p ). Thus, as n t+1 the value functon gven as Q σ (z t+1,p t+1 ; ) cz t+1 +p t+1 mn { z t+1 ;e ε t+1 t+1 (p t+1) } df(ε t+1 ) +β ( max { 0;zt+1 e ε t+1 t+1 (p t+1) } ; )df(ε t+1 )df σ s (s t+2 s t+1 ) s strctly concave n prces and prces and supply are strategc substtutes, so s the functon n t. Ths completes the proof. A.3 Optmal Decson Rule: Proof of Proposton 1. Followng Scarf (1960), the key to provng that the optmal strategy s of the (S,s) form s to show that the value functon V s k-concave. Our proof explots several propertes of k-concave functons. A real-valued functon f(s) s a k-concave functon f and only f for every s 0 and s 1 such that s 0 s 1 and every scalar δ (0,1): δf(s 0 )+(1 δ)f(s 1 ) (1 δ)k +f(δs 0 +(1 δ)s 1 ). (16) Consder the followng propertes of k concave functons: () If f s strctly k-concave t has a unque global maxmum. () If f s strctly k-concave, and s s the global maxmum, then the equaton f(z) = f(s ) k has two solutons, s L and s H wth s L < s H. Furthermore, f(s) > f(s ) k f and only f s (s L,s H ). () If f(x,y) s k-concave n x for any value of y, and k-concave n y for any value of x, and y (x) argmax y f(x,y), then g(x) f(x,y (x)) s k-concave. (v) If f 1 ( ) s k 1 -concave, f 2 ( ) s k 2 -concave, and α 1, α 2 are two postve scalars, then α 1 f 1 +α 2 f 2 s (α 1 k 1 +α 2 k 2 )-concave. 13

15 Before startng wth the formal proof, we wll brefly llustrate the man dea of why k-concavty s mportant. Consder the k-concave functon V(s) to be a frm s value functon. If V s a contnuous dfferentable functon from k-concavty V(s 1 ) k V(s 0 ) (s 1 s 0 )V (s 0 ) 0 drectly follows. Thus for each local extremum s wth V (s ) = 0, t s the case that V(s ) V(s) k s s. Ths means that each local extremum (mnmum) s s at most k unts below a functon s maxmum rght of ths local mnmum. Ths property s llustrated n Fgure 1. The functon on the left hand sde s an arbtrary value functon that s not k-concave, whle the functon on the rght graph fulflls the condton above. V V k k k k Order No Order Order No Order Order No Order * s s 1 s s 2 s Order No Order s 1 * s s s 2 s Fgure 1: Non-concave value functon and respectve order decsons when the value functon s not k-concave (left) and when t s k-concave (rght). Wth lump-sum orderng cost of k and a frm s value functon lke the one depcted on the left hand sde a complex optmal order polcy results where the frm orders when nventory s below s or around s 1 such that nventory level s s attaned. Addtonally, the frm orders such that an even hgher target level s reached when nventory s around s 2 (whch s even above s ). Wth the value functon beng k-concave lke the one depcted on the rght hand sde, t s easy to see that the optmal strategy s of (S,s) type. In that case frms never order wth nventory above s and frms never order around a local mnmum n between the nventory threshold s and the optmal nventory level s. In the followng we wll make use of ths dea wth regard to the decson problem of our model. Proof. Suppose that Q σ s strctly k-concave n z for any value of p and strctly k-concave n p for any value of z for all values of s t. The optmal prce decson can be wrtten as σ σ p (s) pσ (z ;s ). 14

16 That means, gvng the optmal prcng functon p σ (z;s ) the frm chooses nventory level σz σ(s) whch results n prcng σσ p (s) as a functon of the pre-order nventory. As Q σ ( ) s strctly k-concave, s σ (s ) and p σ (s σ (s ),s ) are unque and p σ (, ) s a real functon. Furthermore, Q σ (z, p σ (z );s ) s also strctly k-concave. By defnton of σz σ (s), s σ (s ), and p σ (s σ ( ), ), t s clear that s σ (s ) f Q σ (s σ, p σ (s σ ); ) k > Q σ (s, p σ (s ); ) σz σ (s) = s f Q σ (s σ, p σ (s σ ); ) k Q σ (s, p σ (s ); ). Due to the k-concavty of Q σ (z, p σ (z ); ) the equaton Q σ (s σ, p σ (s σ ); ) k = Q σ (s, p σ (s ); ) has only two solutons. Let these two solutons be s L ( ) and sh ( ), where sl k-concavty mples ( ) s σ ( ) s H Q σ (s σ, p σ (s σ ); ) k Q σ (s, p σ (s ); ) s L ( ) s ( ) s H ( ). ( ). Then, It s clear that the condtons s > s H and s s H do not play any role because the stock level s always lower or equal to s σ. Wth s σ as the smaller of the two solutons by defnton we can wrte the optmal decson as σz σ s σ f s s σ =, s f s > s σ. The accordng optmal prcng decson for the nventory before orderng s σp σ (s) = p σ (s σ pσ ) f s s σ (σ z (s)) =, p σ (s ) otherwse. It further remans to show that Q σ s ndeed k-concave. We proceed n three steps: (a) If (s) s strctly k-concave n s, then Q σ ( ) s strctly k-concave n z for any value of p. (b) If (s) s strctly k-concave n s, then Q σ ( ) s strctly k-concave n p for any value of z. (c) (s) s strctly k-concave n s. (a) We wll now show that f (s) s strctly k-concave n s, then Q σ (z,p ;s ) s strctly k-concave n z for any value of p. 15

17 By the frst part of the proof, there exst s σ and s σ satsfyng 0 sσ s σ for whch V σ can be represented as V σ (s) = V(Q) σ (s,σp(s)) σ Q σ = (s σ, p (s σ );s )+cs h(s ) k f s [0,s σ ), Q σ (s,σ p (s );s )+cs h(s ) f s s. V σ (s) can be extended to be a functon defned onr Rn 1 + : (s) = (0,s )+cs f s 0, (s) else, whch s needed as the proof of (c) mples that V s k-concave n s overr. We can wrte Q σ as Q σ ( ) = QσR ( )+βq σv ( ), (17) where and Q σv ( ) Q σr ( ) cz +p mn{z ;e ε }df(ε ) ( ) = cz +p z λ (p ) (max{0;z t e ε t t (p t )};s t+1 )df(ε t )df σ s (s t+1 s t ). Let usnow consder thefuncton (s e ε ( ); )df(ε )df σ s. Snceeach ( ) s k-concave n s over R, and snce postve lnear combnatons of pontwse lmts of k-concave functons are k-concave, t follows that (s e ε ( ); )df(ε )df σ s s k-concave n s onr. Wth ε ( ) as the value of ε for whch demand s equal to supply z,.e. z = exp( ε (z )) ( ), we have V σ (s e ε ( ); )df(ε )dfs σ = = ε (s ) + ε (s ) ε (s ) +c (s e ε ( ); )df(ε )df σ s (s e ε ( ); )df(ε )df σ s (s e ε ( ); )df(ε )df σ s + (0; ) ε (s ) =Q σv (s,p, )+c (s e ε ( ))df(ε ) ε (s ) (s e ε ( ))df(ε ). ε (s ) df(ε )df σ s 16

18 Usng the defnton of Q σ, we have Q σ ( ) =Q σr ( )+βq σv ( ) =p mn{z ;e ε }df(ε ) cz +β +β βc ε (z ) ε (z ) ε (z ) (z e ε ( ); )df(ε )df σ s (z e ε ( ); )df(ε )df σ s (z e ε ( ))df(ε ). The sum of the thrd and fourth terms n the last equaton s k-concave snce V σ (s e ε ( ); )df(ε )dfs σ s k-concave. Snce cz s a lnear and hence convex functon of z, a suffcent condton for the k-concavty of Q σ ( ) s that the functon ( ) p z λ βc (z e ε ( ))df(ε ) (p ) ε (z ) s concave n z. The functon s contnuously dfferentable n z wth second dervatves (p βc)(1 F(lnz ln ( ))). AsF( ) < 1,thsexpressonsnon-postveandhenceQ σ sk-concave aslongasp βc. (Obvously, a weaker condton for that result exsts.) Recall and For provng that Q σ s ndeed k-concave we need to show that σp σ (s) βc 0 holds. ( ) Q σ (z,p ;s ) cz +p z λ (p ) +β V σ (max{0;z e ε };s t+1 )df(ε )dfs σ (s t+1 s ) { } V σ (s t) = max Q σ (s t +q,p ;s t ) (h c )s t k I {q >0}. {p,q } ( ) where the expected sales λ z are always smaller than or equal to total supply z. Let s suppose to the contrary that there s an optmal prce σp σ < βc < c. ) In that case cz +p λ would be negatve. Thus, wthout a new order ( z the current value V σ(s t) would be smaller than the expected value V σ(s t+1) after sellng the goods at prce σ p (s t ) although the nventory s larger,.e., s t > s t+1. Ths cannot be the case n equlbrum. The same s true n the case wth orderng. Orderng goods and smultaneously sellng them for a prce lower than the purchase prce cannot be an optmal strategy. Thus, the optmal prce σp σ s always greater c. 17

19 (b) We wll show that f V σ(s) s strctly k-concave n s, then Q σ ( ) s strctly k- concave n p for any value of z. We can represent the functon Q σr ( ) as cz +p y σe (z,p ;s ), where y σe ( ) s the expected sales functon. The functon Q σr ( ) s the same as the functon Q σ at the last perod Q σ T. We have shown n the proof of Lemma 2 that ths functon s convex. Therefore, 2 Q σr ( ) < 0. p 2 An argumentaton analogous to part (a) yelds a smlar suffcent condton for the k-concavty of Q σ ( ) n p, namely that the functon p λ ( z ) βc (p ) ε (z ) (z e ε ( ))df(ε ) s concave n p. The functon s contnuously dfferentable n p wth a second dervatve that s negatve. Therefore, Q σ ( ) s k-concave n p. (c) Fnally, we show that (s) s strctly k-concave n s. Lke n proof of Lemma 2 we make use of the fact that the proft functon s bounded from above. Ths property guarantees that for any value of s V σ (s ; ) = lm V T(s σ ; ) T wth T (s ) as the value functon for the fnte horzon problem wth tme horzon equal to T. We prove k-concavty by nducton. For T = 1 we have Q σ 1 ( ) s strctly concave n z and p due to (a) and (b). Usng the result of the frst part of the proof, the optmal decson for ths one-perod problem has the form of equatons (9) and (10). Hence, the value functon of ths one perod problem s 1(s, ) = I(s < s σ 1)(Q σ 1(s σ 1, p σ 1(s σ 1 )) k)+i(s s σ 1)Q σ 1(s, p σ 1(s, )) (h c )s. Wth Q σ 1 ( ) beng concave, t s smple to verfy that V σ 1 (s, ) fulflls the defnton of strct k-concavty. Assume now that for any t 1, Vt σ(s, ) s strctly k-concave. Then, ( Q σ t+1(z, p σ t+1( );s t+1 ) = cz + p σ t+1( ) ( p σ z t+1( ))λ +β ( p σ t+1 ( )) ( max { 0;z e ε ( p σ t+1 ( ))} ; )df(ε )df σ s (s t+2 s t+1 ). ( ) As p σ t+1 ( )deσ ( p σ t+1 ( ))λ z ( p σ t+1 ( )) cz s agan strctly concave and Vt σ(s, ) s strctly k-concave, dueto property (v) of k-concave functons, Q σ t+1 (z, p σ t+1 ( ); ) s also strctly k-concave. Hence, the optmal decson has agan the form of equatons (9) and (10) and the value functon of ths fnte-horzon problem s t+1(s, ) =I(s < s σ t+1) ( Q σ t+1(s σ t+1, p σ t+1(s σ t+1)) k ) +I(s s σ t+1)q σ t+1(s, p σ t+1(s, )) (h c )s. ) 18

20 Smlar to V1 σ(s, ), ths value functon s strctly k-concave whch completes the proof by nducton. Therefore, V σ(s ; ) = lm T VT σ(s ; ) s strctly k-concave. Ths completes the proof of the optmalty of the descrbed orderng strategy. Propertes of the optmal prce. We complete the proof of Proposton 1 by showng that p( ) s a contnuous and strctly decreasng functon. The functon p σ s the value of p that maxmzes Q σ n p for a gven z. Snce Q σ s contnuously dfferentable and strctly concave n prces, p σ (z;s ) s mplctly defned by the frst order condton Qσ (z, p;s ) = 0. By the mplct functon theorem, we have that d p (z ) σ dz = 2 Q σ (z, p )/ z 2 Q σ (z, p )/, that by Lemma 2 s negatve. Ths completes the proof. 19

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