Timing of investments in oligopoly under uncertainty: A framework for numerical analysis

Transcription

1 Timing of invesmens in oligopoly under uncerainy: A frameork for numerical analysis Pauli Muro, Erkka Näsäkkälä 2, Jussi Keppo 3 Absrac We presen a modeling frameork for he analysis of invesmens in an oligopoly marke for a homogenous commodiy. The demand evolves sochasically and he firms carry ou invesmen projecs in order o adjus heir producion cos funcions or producion capaciies. The model is formulaed as a discree ime sae-space game here he firms use feedback sraegies. The firms are assumed o move sequenially o ensure a unique Markov-perfec Nash equilibrium. Once he equilibrium has been solved, Mone Carlo simulaion is used o form probabiliy disribuions for he firms cash flo paerns and accomplished invesmens. Such informaion can be used o value firms operaing in an oligopoly marke. An example of he model is given in a duopoly marke. The example illusraes he rade-off beeen he value of flexibiliy and economies of scale under compeiive ineracion. Key ords: Invesmen analysis, Game heory, Uncerainy modeling, Real opions Deparmen of Economics, Helsinki School of Economics, P.O. Box 20, 000 Helsinki, Finland. E- mail: pauli.muro@hkkk.fi. Corresponding auhor. 2 Sysems Analysis Laboraory, Helsinki Universiy of Technology, P.O. Box 00, FIN-0205 HUT, Finland. erkka.nasakkala@hu.fi 3 Deparmen of Indusrial and Operaions Engineering, Universiy of Michigan, 205 Beal Avenue, Ann Arbor, MI, , USA. keppo@umich.edu

2 Inroducion In fas developing indusries, he iming of invesmen projecs is an imporan par of firms sraegies. Typically, invesmen decisions are made under high uncerainies and hey are irreversible. If an individual invesmen does no affec oher projecs, real opions heory can be direcly applied in choosing he opimal iming of invesmen. Examples of influenial heoreical papers in his lieraure sream are, e.g., McDonald and Siegel (986) and Pindyck (988), hile he main heory is summarized in Dixi and Pindyck (994) and Trigeorgis (996). Hoever, anoher characerisic feaure of many indusries is he compeiive ineracion beeen he firms. An invesmen of a firm may have a significan effec on marke prices, hich affecs he profiabiliy of he pas and fuure invesmen projecs of all firms. This makes he use of sandard real opions models quesionable in such conexs. Building of elecommunicaions capaciy is a good example of an indusry ih he menioned characerisics. Technologies and demand change consanly and, herefore, neork invesmens mus be carried ou regularly. In a given geographical area here usually exis only fe large elecommunicaions companies capable of carrying ou largescale invesmens. A ne capaciy invesmen usually increases significanly he supply of he marke, hich means ha in he invesmen planning he effecs of ne invesmens on he capaciy price have o be considered. The eviden draback of mos of he exising lieraure of real opions in his conex is ha he effecs of compeiors acions are ignored. There are oher sreams of lieraure, hoever, ha accoun for compeiive ineracion. A relaed lieraure considers invesmen in equilibrium under perfec compeiion and raional expecaions. A seminal paper is Lucas and Presco (97), here he social opimaliy of he equilibrium in a discree-ime Markov chain model is esablished. Leahy (993) discovered ha he equilibrium enry ime under free enry is he same as he opimal enry ime of a myopic firm ho ignores fuure enry by compeiors. Anoher conribuion is by Baldursson and Karazas (997), ho esablish he links beeen social opimum, equilibrium, and 2

3 opimum of a myopic invesor under a general sochasic demand process uilizing singular sochasic conrol heory. Grenadier (999) enriches he analyses by including consrucion delays. Hoever, some assumpions in hese models make heir applicaion resriced. Firs, he assumpion of perfec compeiion is no realisic in many indusries. As menioned earlier, in he elecom indusry here are ypically fe firms ho have he required resources o carry ou ne capaciy invesmens. Second, he papers assume ha invesmens are incremenal, hich means ha hey can be underaken in infiniely divisible unis. In realiy firms ofen face increasing reurns o scale on he projec level, hus ne capaciy is added in lumps. Examples of real opions models ha sudy he iming and size of lumpy invesmens are Dixi (995), Bar-Ilan and Srange (999), and Dangl (999), bu hey ignore compeiion. There is a ne lieraure sream ha combines game heory and real opions heory. Examples of such papers in discree ime are Smi and Ankum (993) and Kulailaka and Peroi (998). Papers ih uncerainy modeled as a diffusion process are Grenadier (996), Lambrech (999), Joaquin and Buler (999), Mason and Weeds (200), Moreo (996), Muro (2002), Muro and Keppo (2002), and Weeds (2002). These models are, hoever, resriced o very simple seings. In all menioned papers he game is played on a single projec, so he full dynamics of he indusry is no analyzed. Excepions in his respec are Baldursson (998) and Williams (993), ho analyse he dynamics of oligopolisic indusries under uncerainy. Hoever, he invesmens are incremenal in hese models. On he oher hand, dynamic oligopoly markes ih muliple lumpy invesmens have been sudied by Gilber and Harris (984) and Mills (990), bu in a deerminisic seing assuming perfec foresigh by he firms. Oher models of dynamic oligopoly in deerminisic seings can be found in, e.g., Fudenberg and Tirole (986). In he presen paper, e are ineresed in he iming of lumpy invesmen projecs under uncerainy and oligopolisic compeiion. The imporan characerisic is ha he oupu price is influenced by o facors: exogenous uncerainy and ne capaciy invesmens. Hoever, modeling he full dynamics of such markes is analyically so difficul ha one has o rely on numerical analysis. For his purpose, e presen a sysemaic modeling frameork o analyze a marke ih he folloing properies. There are several large 3

4 firms producing a homogeneous and non-sorable commodiy, he demand for hich evolves sochasically. The firms have a number of invesmen opporuniies available, hich hey can use o adjus heir producion cos funcions or maximum capaciies. In choosing he imings for such invesmens, he firms ake ino accoun he demand uncerainy and he acions of he oher firms. Thus, he marke can be seen as a game ih o opposie facors affecing he opimal invesmen iming. On one hand, uncerainy and irreversibiliy give an incenive o pospone he invesmen. On he oher hand, he firms recognize he preempive effec of he invesmen agains he oher invesors, hich increases he incenive o ac quickly. We assume ha he firms behave non-cooperaively. Their sraegies are assumed o depend on he sae of he marke, hich consiss of he value of he sochasic demand process and he accomplished projecs of all firms. The equilibrium resuling from such sraegies is a Markov-perfec Nash equilibrium. To ensure a unique equilibrium, e make an assumpion ha he firms move sequenially. We presen an algorihm o compue he equilibrium using backard inducion. Our approach has o seps. Firs, he equilibrium sraegies of he firms are solved, and second, he developmen of demand over ime is Mone Carlo simulaed. This allos one o esimae he probabiliy disribuions of he firms discouned cash flos, and hus o derive quaniaive measures on he values of he firms or some of heir operaions under risky marke developmen and compeiive ineracion. The characerisic feaure of our approach is ha i combines he game heoreic equilibrium concep ih he simulaion mehod ha is a sandard ool in analyzing marke values and risk exposures under uncerainy. To illusrae he model frameork, e give an example of a marke, here he demand goes hrough a period of srong groh, bu evenually sabilizes. This kind of dynamics are ypical for emerging markes. The ineresing issue in such a marke is he ay in hich he uncerainy in he demand groh is ransmied hrough he compeiive ineracion o he firms profi disribuions. In he example, e illusrae he disribuions of he firms oal payoffs ha resul from he simulaion. We compare a symmeric case o a case, here one of he firms mus make invesmens in larger lumps han he oher 4

5 firm. The asymmeric case is an illusraion of he rade-off beeen he value of flexibiliy and scale economies. The firm ih he possibiliy o make small invesmens is more flexible, bu he firm ha makes larger invesmens is more cos-efficien. The simulaions demonsrae ho he properies of one firm change he shape of he oher firm s payoff disribuion. The paper is organized as follos. The model frameork and he main noaion are presened in Secion 2. In Secion 3, he sraegies and he equilibrium are defined. The algorihm o compue he equilibrium sraegies is given in Secion 4. Secion 5 conains an illusraive example and Secion 6 concludes. 2 Model frameork In his secion e give a formal descripion of he model. We explore a marke for a homogenous non-sorable commodiy ihin a finie ime horizon. Le = { 0,,...,T} he se of ime periods. By he period direcly afer period. The se of firms is denoed by W {, K, n} se of invesmen opporuniies { I I 2 K} invesmen projec i Γ be Γ e refer o an arbirary ime period and by + Γ o =. For each W, here exiss a counable I =. If firm decides o accomplish he, i I a period, i has o give up he corresponding sunk cos δ () a. The projec ill be finished by he period +. Invesmens canno be reversed. Le [ I ] denoe he collecion of all subses of I including he empy se and he se I iself. By ~ I I, e refer o he se of invesmen projecs firm has accomplished by he ime period. We call he se I ~ firm s capaciy a. The firms produce a given homogenous commodiy. The producion cos is a funcion ~ C : R + [ I ] R +, so ha C = C( q, I ) is he cos a period ih oupu level q and capaciy I ~ ~. If here is an upper limi M ( I ) for oupu ih a given capaciy, e ~ define he producion cos o be infinie for higher oupus, ha is, C ( q, I ) = for 5

6 ~ ~ q > M ( I ). For insance, ih he elecommunicaions capaciy, C ( q, I ) is ~ ypically close o zero if q < M ( ) and oherise i is equal o infiniy. Thus, a firm I can aler is producion cos and/or increase is capaciy by carrying ou cosly invesmen projecs. The price of he commodiy is defined by an inverse demand funcion, hich is subjec o a sochasic demand shock variable X. The value of he demand shock a period is X Θ, here Θ is a finie subse of R. The inverse demand funcion is hen a mapping P : R Θ R, such ha P ( Q, X ) is he marke price a period ih given + indusry oupu Q = p = q and demand shock value X. The shock variable follos a Markov process. Thus, here is a ransiion probabiliy funcion f : Θ Θ Γ [ 0,] such ha he condiional ransiion probabiliy is P{ X x X = x } = f ( x, x, ) Since he sae variable =. X is sochasic, he reamen of he risk faced by he firms is crucial. We assume ha he firms discoun heir cash flos using a fixed discoun facor γ, hich is a sandard assumpion in heoreical papers combining real opions analysis and game heory (e.g. Baldursson, 998; Lambrech, 999; Mason and Weeds, 200; Muro, 2002; Weeds, 2002; Williams, 993). This allos he inerpreaion of he model using he equivalen risk-neural valuaion principle. If here are financial asses raded in a complee and arbirage-free financial marke ha perfecly correlae ih X, all he uncerainies in X can be hedged using hese asses. This corresponds o he frameork of Cox, Ingersoll, and Ross (985), hich exends he basic risk-neural valuaion argumen of Cox and Ross (976) o cover non-raded asses. More precisely, in a complee marke, all derivaive asses can be replicaed ih financial porfolios, and due o he la of one price, he value of a derivaive asse mus be he same as he value of he replicaing porfolio. This means ha here exiss a unique maringale measure such ha he risky cash flos can be valued by discouning hem ih he risk-free ineres rae, and aking he expecaion under he maringale measure. Therefore, he model should be inerpreed so ha f gives he 'risk-neural' ransiion probabiliies, i.e., hose under he maringale 6

7 measure, and γ corresponds o he risk-free ineres rae (hich is for simpliciy assumed o be consan). These risk-neural probabiliies can be calculaed by using he radable asses ha are perfecly correlaing ih X over he discree ime inervals (see e.g. Jackerh, 999; Rubinsein and Jackerh, 200; Cakici and Foser, 2002). These asses can be, for insance, forards on X and forards on a risk facor ha is perfecly correlaed ih X. Because hey are raded in he marke, heir pricing funcion implicily gives he needed risk-neural probabiliies. In order o see his le us denoe he price of an asse ha depends on X a ime by θ(,x), i.e., for each ime e kno he relaionship beeen he asse price and he sae variable. This means, e.g., ha he cos of carry parameer in he forard pricing formula can be calculaed from he exising marke prices (see e.g. Hull, 2002). On he oher hand, he radable asses saisfy he riskneural pricing equaion (see e.g. Duffie, 996) + + [ X = x ] = γ f ( x, x, ) ( +, x ) + ( x ) = γe θ ( +, X ) θ, θ for all { 0,,..., T } he number of elemens in Θ. On he oher hand, if X is no spanned by he financial markes, he model could be inerpreed so ha f gives he 'rue' ransiion probabiliies, and γ corresponds o some subjecive discouning by he firms, possibly adjused o accoun for he risk preferences of he firms. Theoreically, he discoun facor is hen derived from he uiliy funcions of he firms (see, e.g., Cochrane, 2002). Because uiliy funcions are firm specific, he discoun facors could acually be firm specific, as ell as dependen on ime and he sae variable in his case. Hoever, for simpliciy, e reain a common consan γ in he noaion hroughou he paper (hich is accurae hen spanning condiions hold and he 7 x Θ and x Θ, here he expecaion is ih respec o he riskneural probabiliies. As an example, θ(+,x + ) could be he payoff of a forard conrac and θ(,x ) is value a ime. Thus, according o he above equaion, if here are as many radable asses as here are elemens in Θ, and heir pricing funcions are knon, e can solve for each { 0,,..., T } f ( x,, ) : Θ [ 0,] and x Θ he risk neural probabiliies from Θ linear equaions (one for each radable asse), here Θ is

8 risk-free ineres rae is consan). I should be noed ha if firm-specific uiliy funcions are used, he deerminaion of he discoun facors is in pracice difficul. Moreover, i is knon ha he opimal invesmen policy may be very sensiive o he choice of he discoun facor, hus a proper sensiiviy analysis ould be appropriae in such a case. 4 Given he demand shock value and he oupu levels of all firms, he surplus for firm a period is given by ~ ~ π = π ( X, Q, I ) = q P( Q, X ) C ( q, I ). () The firms an o maximize he discouned surplus minus he invesmen coss over he 0 T ime horizon. Given a realizaion of he demand shock parameer x [ x,..., x ] =, and given some fixed capaciy and oupu levels for each period, he oal payoff for firm is: V ~ ~ ~ i T T ~ T ~ T ( x, Q, I ) δ () γ R ( x, I,..., I ) T ( x, Q, I,..., I n ) = γ π + n,(2) 0 i ~ ~ = i: I ( I + I ) T ~ T ~ T here R ( x I,..., I n ) 0 T capaciies a he end of he ime horizon, Q [ Q,..., Q ] ~ ~ 0 ~ T oupu variables for all firms a all ime periods, and I [ I,..., I ], is he salvage value of ih a given demand shock value and = is a vecor conaining he = conains he evoluion of 's capaciy. Because invesmens are irreversible, i is required ha ~ ~ + I I ~ + for all and all = 0,,..., T. The se I I ~ conains projecs ha has decided o build a period, and hich are hen accomplished a period +. 5 Therefore, he sunk coss for hem have o be paid a period. 4 The book by Dixi and Pindyck (994) is very illusraive concerning he maer. In ha book, he coningen claims analysis (applicable hen spanning holds) and dynamic programming approach (applicable ihou spanning, bu ih no sound heory for he choosing of he discouning) are used in parallel. A consan discoun rae is applied hroughou he book. 5 We have used noaion A B = A C B, here C B is he complemen of B. 8

9 The firms choose heir oupu and invesmen decisions in order o maximize he expecaion of (2). Hoever, as he model is dynamic, and he firms are able o observe he sae of he sochasic variable X during he ime horizon, hey can adap heir decisions accordingly. Therefore, he oupu variables and capaciy ses are sochasic processes adaped o he process X. In (2), he vecors Q and I ~ correspond o one given realizaion of hese processes. Moreover, he oupu decisions of he firms affec each oher s profis and accordingly opimal behavior. Therefore, e have o analyze he equilibrium in he firms sraegies. In he nex secion e define formally he sraegies of he firms and he resuling Markov-perfec Nash equilibrium of he game. 3 Sraegies and equilibrium Le he value of he demand shock variable and he invesmen projecs ha he firms have compleed define he sae of he sysem a period. Denoing he sae space by ~ ~ Ω = Θ [ I ]... [ ] Y = X, I,..., I n, here I ~ is he se of I n, he sae vecor is [ ] Ω projecs compleed by. Laer in his secion e augmen his definiion ih an addiional variable. I should be remarked ha he o decisions variables he firms have, namely he oupu and he invesmen enry imes, are differen in naure. The accomplishmen of an invesmen projec has direc and permanen influences over ime for all firms, hile he oupu choice only affecs he presen period. Therefore, he oupu choice a a given period can be made ihou considering he invesmens. In his model, he emphasis is on he invesmen decisions, and o clarify he formulaion, e make he folloing assumpion. ASSUMPTION 3. The oupus ih given capaciies are defined by a mapping Λ : Ω Q... Q n. This gives he oupu levels q ( Y ),..., qn ( Y ) for all firms given he sae Y. This mapping can be undersood as an implici funcion deermined by he Nash equilibrium for he shor-erm price or quaniy compeiion. The oal oupu o he 9

10 marke is n Q( Y ) = q ( Y ). According o Assumpion 3. he oupu variables are = funcions of he sae, and hus, he surplus for firm a sae ( Y ) = q( Y ) P Q Y ~ ( ( ), X ) C ( q ( Y ), I ) 0 Y can be rien as: π = π. (3) Nex e sar he formulaion of he firms sraegies. Since he oupu of a firm is defined by Λ, he sraegy only defines he invesmen decisions. As a firs sep, consider defining he sraegy of firm as a mapping S Ω Γ [ ] : such ha S ( Y, ) I conains he invesmen projecs ha firm carries ou a ime given sae value This definiion for he sraegies does no guaranee a unique Nash equilibrium, because he opimal acion of a firm ihin a given period is condiional on he acions of is compeiors acions a he same period. Thus, here can be several equilibrium acions for he firms ihin he period. To ensure a unique equilibrium, e make an assumpion ha only one of he firms can make an invesmen a each ime period. Thus e have a sequenial game here he decision makers move one a a ime. I is sraighforard o sho ha such a game has a unique sub-game perfec equilibrium and i can be solved using backard inducion (see e.g. Fudenberg and Tirole, 99). ASSUMPTION 3.2 A each ime period, only one of he firms ges he opporuniy o decide heher i ill underake one or more of is invesmen opporuniies. This assumpion seems resricive from he poin of vie of acual firm managemen, because i ses an exogenous consrain on hen i is possible o ac. Hoever, i can be argued ha such a sequenial decision srucure approximaes he acual sraegic behavior, here delay and shor-erm commimen effecs characerize firms' acions. Besides, i should be noed ha he assumpion allos differen ays o deermine hich firm ges o inves a a specific period. To naural choices are possible. Firs, e can assume ha here is a deerminisic fixed order in hich he firms ac. Maskin and Tirole (988) use a similar assumpion in a heoreical duopoly model, and hey jusify i by he noion of shor-erm commimen. I means ha once a firm has decided o ac, i is commied o he acion for some ime and he oher firm ill have a chance o reac. I Y.

11 seems ha as he number of periods is increased, he alernaing moves of he firms resul in a reasonable approximaion of he acual decision srucure. In he second possible formulaion he firm ho ges he invesmen opporuniy a a given period is dran randomly. This can be inerpreed as a shor-erm commimen of an uncerain duraion. Thus, he sraegy of every firm ha may ge he invesmen opporuniy a period mus deermine ha he firm ould do in such a case. Also, he firms sraegies have o ake ino accoun he probabiliies in hich hey and heir compeiors ge he chance o inves in he folloing periods. The symmeric siuaion beeen he compeiors can be modeled easily ih his kind of a formulaion. Namely, he probabiliies are se so ha a each period all he firms have equal probabiliies o ac before he ohers. The inerpreaion of he seing is ha he firms do no kno ho ill be he fases o inves if he condiions urn favorable. Therefore, hen a firm has a chance o inves, i has o ake ino accoun he fac ha before i ges anoher chance, anoher firm may already have invesed. Nex, e augmen he model so ha i allos he formulaion of he sraegies so ha Assumpion 3.2 is aken ino accoun. We do ha by adding one variable in he sae vecor. This ne variable, denoed by U, ges values in W, and period, firm has he opporuniy o make an invesmen. The variable U = means ha a U, defined for all, is eiher a deerminisic or a sochasic process. To reain as much generaliy as possible, e define he evoluion of he process in he same ay as in he case of he demand shock variable. Thus, here is a ransiion probabiliy funcion : W W Γ [ 0,] g such ha he condiional probabiliy { u U = u } = g( u, u, ) P U =. If every firm has he same probabiliy a each period o ge he chance o inves, hen e have g ( ',, ) = / n for all, ', and. Le Ψ = Θ [ I ]... [ I n ] W vecor as be he augmened sae space and define he ne sae ~ ~ [ X I,..., I U ] Ψ Z =,,. (4) n No e can formally define he sraegies of he firms.

12 DEFINITION 3. The sraegy of he firm W is a mapping S : Ψ Γ [ ] such I ha he se S ( Z, ) conains he invesmen projecs ha firm carries ou a ime given sae value Z. S [ S,..., S n ] = is called a sraegy profile. Le Φ denoe he se of admissible sraegies of firm. The sraegy space S [ ] : Ψ Γ saisfying he o condiions: I Φ conains all mappings ) ( ) C S ~ ( Z, ) I for all and all 2) S ( Z, ) = for all such Z ha Z, U. Condiion ) means ha can only inves in projecs ha i has no ye accomplished. Condiion 2) simply formalizes he assumpion of sequenial invesmen: can no inves if some oher firm ges he opporuniy o inves. As he projecs ha firm decides o carry ou a period are accomplished a period +, he se of projecs I ~ compleed by evolves herefore according o equaion ~ ~ I + I S ( Z =, ). Since he sraegies are funcions of he sochasic variables X and U, he ses of compleed projecs are sochasic processes adaped o X and U. The sae ransiion rule of he model is summarized in he folloing hree equaions: ~ ~ I + I S ( Z =, ), (5) P { X x X = x } = f ( x, x, ) { u U = u } = g( u, u, ) P U =, (6) =. (7) Equaions (6) and (7) describe he evoluion of he processes X and U, and (5) describes ho he firms capaciies evolve according o heir sraegies given a realizaion of he processes X and U. The firm chooses he opimal sraegy in erms of maximal expeced profis. Le V ( Z, S, ) be he expeced ne presen value of all cash flos from period o he las period for firm given sraegies S [ S,..., ] =. This saisfies he recursive equaions S n 2

13 V + ~ + ~ + + ( ) + γ { E [ V ( X, I,..., I, U, S,..., S, + ) ]} i ( Z, S, ) = π ( Z ) δ X, U n i i: I S ( Z, ) here he sae evolves according o (5), (6), and (7). A all possible saes, he firms choose sraegies ha give hem he highes possible expeced payoffs. The equilibrium of he game is defined as: DEFINITION 3.2 The Nash equilibrium is a sraegy profile n, (8) S = [ S S n ] Φ... Φ n,..., such ha V ( Z, S,..., S n, ) V ( Z, S,..., S,..., S n, ) W, Γ, Z Ψ, S Φ (9) In a sochasic game, sae dependen sraegies as given in Definiion 3. are called Markov sraegies. A profile of sraegies giving Nash equilibrium in every proper subgame, as he Definiion 3.2 requires, is called a Markov-perfec equilibrium (MPE). Since only one firm can ac a each period, here is cerainly one unique MPE (see, e.g., Fudenberg and Tirole, 99). 4 Compuing he equilibrium * Le V ( Z, ) denoe he expeced discouned value of he cash flos for a Z hen all he firms use heir MPE sraegies from here on. These sraegies can be compued using backard inducion as follos: T. Sar from period T and calculae he presen value of surplus π ( Z ) a all possible saes for all using equaion (3). The value funcion a he las period is given value * T T ~ T ~ T T T V Z, T ) = π ( Z ) + R( X, I,..., I n ). Se S ( Z, T ) = for all and Z. Se he ( ime index a = T. 2. Using recursion (8), calculae V ( Z, ) for all a all saes a period. In compuing V ( Z, ) for he firm, hich has he opporuniy o inves, assign he se S ( Z, ) ha * + gives he larges V ( Z, + ) hen Z evolves according o (5), (6) and (7). For he oher firms, se S ( Z, ) =. 3

14 3. If > 0, se := - and go back o sep 2. Oherise he equilibrium sraegies S ( Z, ) and values V ( Z, ) have been calculaed for all and Z. Afer he equilibrium sraegies have been calculaed, he marke can be simulaed. The sae is iniiaed a = 0 by giving iniial values for ~ I 0, 0 X, and 0 U. The sochasic variable is given a value for he nex period, and he ses using he previously compued sraegies. This is coninued unil I ~ are updaed according o (5) = T. X and U can be chosen randomly using equaions (6) and (7), or differen scenarios can be invesigaed by deerminisically assigning cerain pahs for he variables. 5 Example case In his secion e presen an example case, here a number of specificaions are inroduced in he general model developed in secions 2 and 3. The purpose of his example is o illusrae he modeling frameork raher han o derive conclusions concerning he economic heory of dynamic oligopoly. To carry ou he compuaions, e have rien a compuer program for his specific purpose in C programming language. This program implemens he algorihm described in secion 4. The example considers a marke, here he demand shock process goes hrough hree phases. In he firs phase he demand is sable. In he second phase he demand gros srongly, unil i again sabilizes in he hird phase. Such dynamics ake ofen place in an emerging marke. The firms kno in advance ha in he fuure he demand ill gro rapidly and evenually sabilize. In any case, here are high uncerainies on he exac pah ha he demand developmen ill follo. The differen phases of he demand process are modeled ih differen parameerizaions of he sochasic process. We make he folloing assumpions o specify he example: ) There are o firms W = {,2 } ih he folloing feaures: a) Each firm has invesmen opporuniies of one single ype b) The number of invesmen opporuniies is unlimied 4

15 c) Each invesmen coss δ unis of money d) Each invesmen increases he maximum capaciy by an amoun e) The iniial capaciy is 0 for boh firms f) The variable producion cos is zero up o he capaciy limi. In he noaion of secion 2, e have: C ~ ( q I ) + K + 0, hen q mk, =, here m is he number of elemens in +, hen q > mk 2) The number of periods is 0, i.e. Γ = {0,,...,00} 3) The demand shock process is specified by he ransiion rule: I ~ X + u X ih probabiliy p = ( u) X ih probabiliy p, here p 0.5 = for for for { 0,,...,32} { 33,...,66} { 67,...,00} 4) The inverse demand funcion is ime-invarian and has consan elasiciy: ( Q ) ε P ( Q, X, ) = X a, here price elasiciy saisfies: ε >. 5) Each firm has an equal probabiliy o ge an opporuniy o inves a each period. Formally: g (,, ) = g(,2, ) = g(2,, ) = g(2,2, ) = 2 Γ ~, ~, T T T 6) The salvage value is ( ) R X I I 2 = = + T We nex discuss hese assumpions. Assumpions a e specify ha he firms carry ou invesmens in order o increase heir producion capaciies in lumps of consan size. Each firm has an unlimied se of such invesmen opporuniies 6, bu due o echnological resricions, each firm can underake projecs of one single ype only. We allo firm specific invesmen coss and projec sizes, hich reflecs possibly differen γ T π T 6 In he acual compuaions e have used fixed maximum capaciies, bu se hese maximums so high ha i is never relevan for a firm o use all of is invesmen opporuniies. 5

16 echnological resources possessed by differen firms (for example paens or echnological compeence). In some conexs, e.g. in elecriciy producion, his migh also reflec specific naural resources or local condiions available o he firms. The assumpion of zero variable cos (assumpion f) is a plausible assumpion for elecommunicaions capaciy, here he cos of ransmiing a uni of daa is negligible given ha he sufficien ransmission infrasrucure is in place (for more on his, see e.g. Clark, 997; MacKie-Mason and Varian, 996). Thus, all coss associaed ih producion arise hrough he fixed invesmen coss. Hoever, his kind of a cos srucure ould no be realisic in some oher indusries of homogenous commodiies. For example, in elecriciy poer generaion consideraions of variable producion coss are very imporan ih some echnologies, hereas ih some oher echnologies, e.g., hydro poer, ind poer, or geohermal poer, he zero variable cos assumpion can be ell jusified. Noe ha in our model during he long opimizaion period he fixed invesmen coss can be vieed as variable coss because he companies choose heir invesmen amouns. The number of periods (Assumpion 2) is chosen large enough o have a reasonable approximaion of he decision srucure, here he firms decide abou he iming of invesmens in coninuously evolving marke. The demand shock process in Assumpion 3 reflecs he hree phases of marke dynamics, he sable beginning and end, and he high groh in he middle. 7 As discussed in Secion 2, if here are enough financial insrumens ha perfecly correlae ih X hen hese ransiion probabiliies can be calculaed from hese insrumens (for more on his kind of esimaion see e.g. Jackerh, 999; Rubinsein and Jackerh, 200; Cakici and Foser, 2002). For insance, in he elecommunicaions marke hese ransiion probabiliies can be calculaed from 7 Varying he probabiliies of going up acually also changes he volailiy of he demand process. Wih he parameer value u =.07, hich e use in he compuaions (see Table ), he sandard deviaion of he percenage change of X ihin one period is abou 6.8% a he beginning and end of he ime horizon and abou 4% during he high groh phase. 6

17 bandidh leasing conracs (see e.g. Keppo, 2002). The expeced demand shock value ih 68% confidence inerval is illusraed in Figure Demand Shock Period Figure : Expeced value of he demand shock process The inverse demand funcion ih consan elasiciy (assumpion 4) has been chosen for o reasons: firs because i has he realisic propery ha he price ges alays a posiive value, and second because i is convenien for specifying he opimal oupus of he firms given fixed capaciies. Namely, given zero marginal coss and given ha he elasiciy is high enough (more precisely hen ε > ), i is easy o sho ha he marginal revenue is posiive for any firm, no maer ha he oupus of he firms are. This implies ha in equilibrium he firms an o uilize fully heir capaciies, ha is, e have + ( Y ) = mk Y Ω q. Thus, e have an example of a realisic siuaion, here he sraegic behavior concerns only he iming of invesmens, bu no he oupu decisions. The assumpion of posiive marginal revenues has also been used in many influenial heoreical models of dynamic oligopolies, e.g., Gilber and Harris (984), Ghemaa and Nalebuff (985), and Mills (990). Alernaively, in case of a loer elasiciy, e could simply specify he mapping Λ so ha he oupus are deermined as he Courno- Nash equilibrium of he shor-erm quaniy game ih fixed capaciies. 7

18 Assumpion 5 simply means ha he urns in hich he firms are alloed o move are randomized symmerically. For simpliciy, e assume ha he agens are risk-neural ih respec o his uncerainy. Assumpion 6 means ha he salvage value for each firm a he las period is equal o he discouned sream of fuure cash flos assuming ha he las period demand shock value and capaciies ill prevail forever. The equilibrium sraegies are calculaed according o he algorihm given in secion 4. As given in Definiion 3., he sraegy deermines hich projecs he firm underakes a a given sae. In he conex of his example, here each firm can only underake one ype of projecs, i is possible o express he sraegies graphically. Namely, given some fixed capaciies for boh of he firms, i is easy o undersand ha he higher he demand shock value, he more aracive i mus be for a firm o ake he nex invesmen projec. Thus, he sraegies can be expressed as "hreshold curves" such ha a hreshold curve value ih given capaciies represens he loes demand shock value a hich he firm is illing o ake he nex projec. This ype of characerizaion of he opimal invesmen rules is ypical in he real opions lieraure (see, e.g., Dixi and Pindyck, 994). Figure 2 shos he loes hreshold curves for firm in a symmeric siuaion here δ = δ 2 and K (in his symmeric siuaion he sraegies for firm 2 are idenical). + + = K 2 The exac parameer values are given in Table. The o numbers associaed ih each hreshold curve refer o he capaciies of firms and 2, respecively. For example, he curve 0- represens he loes demand shock value a hich firm is illing o build one more capaciy uni given ha i has 0 unis of capaciy hile firm 2 already has uni of capaciy. The gray borders represen he highes and loes possible sae values ha can ever be reached. One can see ha he hreshold curves are no monoonic, hich is due o he differen phases of he demand groh. In he beginning, here he high groh phase is approaching, i becomes more and more aracive o inves, hus he hresholds are decreasing. Hoever, hen he high groh period has sared, and he evenual sabilizaion of he marke is approaching, he invesing becomes less aracive. Finally, he end ransiion ih he high salvage value associaed ih he exising capaciies and he absence of any furher invesmen opporuniies makes he invesmen increasingly aracive oards he end of he ime horizon. 8

19 Table : Parameer values ih symmeric firms Parameer δ 2 + δ K + K 2 γ u a ε 0 X Value Demand shock Period Figure 2: Threshold curves From he hreshold curves, one can conclude he mos likely order in hich he firms carry ou invesmens. Since he firms have idenical sraegies, i is equally likely for boh of he firms o be he firs o inves (under Assumpion 5 ha assigns equal probabiliies for he invesmen urns). Once one of he firms has invesed in one capaciy uni, he quesion is ho ill underake he nex invesmen. Since he curve 0- is belo ha of -0 (hich acually lies ouside he figure area), i is more aracive for he firm ih no capaciy o build is firs capaciy uni han for he firm ih uni o build is second uni. Therefore, he second invesmen ill be underaken by he firm, hich does no have any capaciy ye. This same paerns ill coninue furher, ha is, one can see ha he firms make invesmens sequenially so ha he difference in capaciies is a mos one uni. This resembles he resul in Baldursson (998), here asymmeries in iniial capaciies are gradually evened ou in a dynamic oligopoly, here firms increase capaciy coninuously. 9

20 Even if one can ge some insigh on ho he marke is likely o develop by looking a he hreshold curves, he acual developmen of he marke depends of course on he realizaions of he sochasic processes. Figure 3 illusraes he equilibrium invesmens for one randomly chosen realizaion of X and U. Also he price is presened in he figure. The parameers are given in Table. The firs invesmen occurs a period 4 here firm 2 increases is capaciy by one uni. Firm follos a period 47. A period 65 firm 2 underakes is second invesmen, and firm follos a period 68. Therefore, he oal capaciy a he final period is 4 unis. Each capaciy invesmen has a decreasing effec on he marke price, as seen in he figure Demand shock Capaciy Capaciy 2 Price Period Figure 3: Example ih symmeric firms Nex, e change he parameers so ha he firms are differen ih respec o he invesmen opporuniies available. Firm 2 can no only make invesmens ice as large as firm, bu on he oher hand, invesmen cos per uni of capaciy is considerably loer. Figure 4 illusraes he equilibrium invesmens for one randomly chosen realizaion. The parameers used are summarized in Table 2. This ime, firm increases is capaciy a period 3 by one uni, hile firm 2 ais unil he period 44 before increasing is capaciy by o unis. Afer ha, firm sill makes o invesmens, a periods 63 and 82. This ime, he oal capaciy a he final period is five unis, here firm has hree unis and firm 2 has o unis. 20

21 Table 2: Parameer values ih asymmeric firms Parameer δ 2 + δ K + K 2 γ u a ε 0 X Value Demand shock Capaciy Capaciy 2 Price Period Figure 4: Example ih asymmeric firms I should be emphasized ha he sraegies as given in Definiion 3. are deerminisic decision rules ha adap o all marke saes. The opimal sraegies are deermined by ~ ~ Z = X I, I, U and heir probabiliies., 2 aking ino accoun all possible saes of [ ] Hoever, he acual invesmens and profis are sochasic, because hey depend on he realizaion of he processes X and U. Therefore, Mone Carlo simulaion can be used o analyze he probabiliy disribuions of he firms cash flo paerns. This is done by repeaedly simulaing X and U, and using he equilibrium sraegy profile a each realizaion of he simulaion o calculae he resuling cash flos. Figure 5 shos he scaer plo of he firms oal discouned profis for 5000 realizaions ih he symmeric parameerizaion of Table. Figure 6 presens he oal profis for 5000 realizaions ih he asymmeric parameerizaion of Table 2. 2

22 Figure 5: Scaer plo of payoffs in symmeric case Figure 6: Scaer plo of payoffs in asymmeric case As could be expeced, he firms payoff disribuions in Figure 5 are similar due o he idenical cos srucures. The oucomes are evenly spread around he dashed line, along hich boh of he firms have exacly he same payoffs. In Figure 6 he firms have asymmeric cos srucures, so he oal profi disribuions are no idenical eiher. The scaer plo is curved and he oucomes are more disincly spli ino differen areas. 22

23 When boh firms ge relaively lo payoffs, i is ypically firm 2 ha profis more, hich can be seen as an accumulaion of poins above he dashed line here he curvaure of he disribuion is highes. On he oher hand, hen boh firms ge high payoffs, hich happens hen he demand realizaion is high, i is ypically firm ha profis more, hich can be seen as an accumulaion of poins a he upper righ hand corner belo he dashed line. There are also some poins along he line y = 0, hich reflecs he fac ha in some realizaions he demand groh is so eak ha firm 2 does no make even a single invesmen. This example is an ineresing illusraion of he rade-off beeen he value of flexibiliy and scale economies under compeiion. Firm is more flexible ih he possibiliy o make smaller invesmens, bu firm 2 is more cos-efficien since he cos per uni of capaciy is loer. I is ineresing o noe ha ih some realizaions of he shock process flexibiliy is a more advanageous propery, hile for oher realizaions he converse is rue. The fac ha firm is able o make he larges payoffs may seem counerinuiive. I could be expeced ha hen demand groh is very high, firm 2 ould be in an advanageous posiion ih he possibiliy o make large invesmens. Hoever, a closer examinaion of individual simulaion resuls reveals ha along hose realizaions here demand gros srongly, firm gradually moves far ahead of firm 2 in capaciy. Typically, in high demand realizaions he oal capaciies a he las period are around 6 9 for firm, bu only 4 6 for firm 2. Thus, i seems ha i is easier for firm o keep up ih he groing demand, since he small invesmens allo gradual capaciy expansion as he demand groh goes along, ihou having so much donside risk due o large individual invesmens. On he oher hand, in he cases here demand groh is no so high, even one large invesmen made by firm 2 prevens firm o fully benefi from he advanage of being able o make small invesmens. Figure 7 presens a hisogram of he profis in 5000 simulaions for boh firms in he symmeric seing ih he parameerizaion of Table. Figure 8 conains he profi hisograms for he firms in he asymmeric seing ih he parameerizaion of Table 2. 23

24 Number of observaions Number of observaions Payoff of firm Payoff of firm 2 Figure 7: Payoff hisograms in symmeric case Number of observaions Number of observaions Payoff of firm Payoff of firm 2 Figure 8: Payoff hisograms in asymmeric case I is ineresing o noe ha he shapes of he payoff disribuions change for boh of he firms hen moving from he symmeric case o he asymmeric one. For firm he payoff disribuion ges ider ih more oucomes giving high payoffs. An explanaion is ha hen resriced o large lumpy invesmens, firm 2 is no able o adap as ell o demand changes, and hus, facing less compeiive response, firm can beer uilize he realizaions ih high demand groh o is advanage. Indeed, in mos of he realizaions 24

25 firm 2 makes 3 invesmens, hereas he number of invesmens underaken by firm ranges much more, ypically from o 9. As here is lile variaion in he invesmen behavior of firm 2, and on he oher hand he invesmen behavior of firm adaps more effecively o demand groh hus having a sabilizing effec on he price, here is lile payoff variaion for firm 2. This can be seen as a more concenraed payoff disribuion. Overall, an ineresing poin o noe is ha he change in he parameers of one of he firms no only has an effec on is on payoff, bu due o he sraegic ineracion i changes he shape of he payoff disribuion for he oher firm as ell. I is also eviden from figures 7 and 8 ha he profi disribuions are no normally disribued. There is a considerable ail in he disribuions a high oal profis, hile here is much less variaion a he lef hand side of he mean value. This is no surprising, because he disribuion of he incremen in he shock variable ihin one of he hree separae groh phases approaches log-normal disribuion as he number of periods is increased. Even hough here is a non-linear relaionship beeen he payoff per period and he shock variable, i can be expeced ha he oal payoff is skeed o lef. Finally, i should be noed from figures 7 and 8 ha he oal payoffs for boh of he firms are raher small hen compared o he invesmen coss. This reflecs he ren dissipaion effec due o he preempive compeiion. This can be furher illusraed by comparing he duopoly profis o he monopoly case. Figure 9 shos he hisogram of he oal indusry payoffs (sum of he firms' payoffs) in he symmeric duopoly ih parameers of Table. As a comparison, he figure shos also he hisogram of a single firm ih he same parameer values, bu in he absence of compeiion. One can see ha he compeiive pressure drives he indusry profis o relaively lo levels even in he case of only o firms. 25

26 Number of observaions Number of observaions Duopoly payoff Monopoly payoff Figure 9: Indusry payoffs in duopoly and monopoly This example considers a duopoly, bu an exension o more han o firms ould be sraighforard in principle. Hoever, as he increase in he number of firms increases he dimension of he sae space, such an exension increases significanly he required compuing capaciy. To exend our example o an oligopoly ih hree or more asymmeric firms ould in pracice require a reducion in he number of periods. Alhough i ould be ineresing o look a he effec of varying he number of firms, such an exension is no expeced o add any concepually imporan ne feaure in he model. I can be expeced ha he increased number of firms ould drive he indusry profis o even loer levels han in duopoly. 6 Conclusions This paper considers he dynamics of a marke, here here are several large producers of a homogenous commodiy, and here each producer faces a se of discree invesmen opporuniies. By accomplishing invesmen projecs, he firms loer heir producion coss or increase heir maximum capaciies (or, in general, aler heir producion cos funcions). The se of invesmen opporuniies can be seen as a porfolio of real opions, 26

27 and he firms have o decide he opimal iming of invesmens in an uncerain environmen. Wha makes he seing difficul o analyze is he compeiive ineracion beeen he firms. The profis of all firms are affeced by he oupu choice of any individual firm hrough he inverse demand funcion. Therefore, in deciding he opimal invesmen iming he firms have o ake ino accoun he effecs of heir acions on he behavior of he oher firms. We have no aemped o conduc an analyical reamen of he equilibrium in such a seing. Insead, e have proposed a sysemaic o-sage approach for analyzing numerically such a marke. In he firs sage, he equilibrium sraegies of he firms are compued. The indusry equilibrium can be undersood as a se of such sraegies, or decision rules, here no firm has an incenive o change her sraegy a any possible sae of he game. In oher ords, e refer o he Markov-perfec Nash equilibrium as he soluion concep. In he second sage he equilibrium sraegies are used o simulae he marke. This makes i possible o analyze he saisical properies of he cash flos resuling from such a compeiive marke. In oher ords, he model allos one o make conclusions on ho he primary uncerainy in he exogenous demand shock parameer is ransmied hrough he compeiive ineracion o he resuling profi flos. We have illusraed he model ih an example of o firms ha compee agains each oher in building, e.g., elecommunicaions capaciy. We have assumed ha he marke for such capaciy gros sloly in he beginning, hen goes hrough a period of srong groh, and evenually sabilizes. These hree sages of groh are modeled by using differen parameers in he sochasic process characerizing he demand groh. We have illusraed he oal payoff disribuions of he firms resuling from he Mone Carlo simulaions. The payoff disribuions are skeed o lef. The asymmeric case, here one of he firms can only make invesmens in larger lumps han he oher firm, illusraes he rade-off beeen flexibiliy and scale-economies. In our example, flexibiliy is a more advanageous propery ih some realizaions of he sochasic demand groh, hile he converse is rue for ohers. We also demonsrae ha he change in he parameers of one 27

28 firm no only has an effec on is on payoff, bu changes he saisical properies of he oher firm's payoff disribuion as ell. Acknoledgemens We ould like o hank Fridrik Baldursson, Pierre-Olivier Pineau, Rune Senbacka, Jon Thor Surluson, and o anonymous referees for helpful commens on earlier versions of he paper. Parial financing by he Nordic Energy Research is graefully acknoledged. 28

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