The Dynamic Approach to Heterogeneous Innovations. (Anton Bondarev)

Transcription

1 The Dynamc Approach to Heterogeneous Innovatons (Anton Bondarev)

2 The author thanks H. Dawd, F. Redel, J.-M. Bonnseau and I. G. Pospelov. Abstract. In ths work the dynamcal framework whch combnes dfferent aspects of nnovatve actvty s analyzed. Frst the basc model wth fnte tme horzon s constructed where the sngle agent (planner) s optmzng hs stream of nvestments nto the process of creaton of new products together wth nvestments nto the mprovement of already nvented products. The range of products whch mght be nvented s gven by the bounded real nterval. Next the role of heterogenety of the nvestment characterstcs of these new products s analyzed and t s demonstrated that ths heterogenety plays the essental role n the dynamcs of the model. Further on the analyss s extended to account for long-run behavor of the planner on the nfnte-tme horzon. Steady-states of the system are derved and ther stablty s analyzed. Two followng chapters of the work deal wth two dfferent extensons of the basc model. Possbltes for further analyss of the gven approach and the dfference n conclusons and polcy mplcatons wth earler approaches to nnovatons analyss are demonstrated. Frst the effect of patentng polcy on the nnovatve actvty s taken nto account. There the dversty of possble outcomes wth respect to the patent s length s demonstrated and t s argued that ths effect may not be captured wthout the precense of heterogenety of nnovatve products under analyss. In the last chapter the extenson whch ntroduces several nnovatng agents s consdered. There the subsequent optmal control problem s transformed nto the dfferental game n nfntedmensonal space. The set of pecewse-constant strateges s derved and t s shown to be the only one stable set n the class of at most lnear feedback strateges. In an effect the specalzaton of nnovatve actvty between agents s observed and t s demonstrated that ths specalzaton has a foundaton n nternal characterstcs of these agents. The suggested work provdes several prospects for further enrchment and development n all the areas beng consdered.

3 Contents Preface 1 Chapter 1. Product and Qualty Innovatons: A Unfed Approach 7 1. Introducton 7 2. Assumptons and Basc Framework Model Theoretcal Results General Soluton Homogeneous Products Lnear Model Qualty Investments Parametrc Analyss Dscusson 26 Chapter 2. Infnte Tme Horzon Problem Introducton Monopolst Problem Formulaton Qualty Growth Problem Varety Expanson Problem Steady States Parameter Influence Dscusson 39 Chapter 3. Patents n Heterogeneous Innovatons Framework Introducton Fnte Tme Patents Qualty Growth n Patent Model Varety Expanson Process n the Patent Model Homogeneous vs Heterogeneous Products n Fnte-Tme Patents Settng Compensaton Effect vs Potental Proft Effect Dstrbutonal Effects Parameters Influence n the Patentng Model Dscusson 66 Chapter 4. Strategc Interactons n Heterogeneous Innovatons Framework Introducton Strategc Interactons n Heterogeneous Innovatons Framework Basc Problem Formulaton Decomposton Method n Dfferental Game Settng Qualty Growth Problem Varety Expanson Problem Dscusson 120 Bblography 125 3

4 4 CONTENTS Appendx 127

5 Preface One of the basc sources of economc growth s technologcal progress, as t s argued by economc growth theory. Technologcal progress emerges as a result of the nnovatve actvty of economc agents. That s why modelng nnovatons s one of the key areas of modern economcs. Startng from 1960 s there have been a lot of attempts of such modelng and ncorporaton of nnovatons nto macroeconomc models. Ths strand of lterature s concentrated on the effects of technologcal progress on the economc growth, rather then on the nature and source of ths progress. That s why nowadays these theores are referred to as theores of exogenous technologcal progress and/or exogenous economc growth. It was soon recognzed, that t s mportant to model explctly the process of nnovatons themselves to endogenze the technologcal progress. Frst such attempts have been made n the era of classcal growth theory n 1960 s. All these theores may be dvded nto three groups: dsemboded technologcal change, emboded technologcal change and nduced technologcal change. In the framework of neoclasscal growth theory the technologcal change has been treated as the only source of economc growth and ths source of growth receved much of the attenton of economsts. Frst explct models of technologcal change were that of dsemboded one. They assumed that technologcal progress s reflected n the growth of productvty of labour and captal n the aggregate producton functon. There was no explct formulaton of mcro-foundatons of such growth of productvty of factors. They have been assumed to grow wth some exogenous rate. Such an approach was not satsfyng to explan technologcal change as t was not compatble wth stylzed facts and moreover, such a technologcal progress had to be Harrod-neutral all the tme (that s, only the productvty of captal may ncrease over tme, not that of labour) whch s not the case. Another approach, that of nduced technologcal change, endogenzed technologcal progress and made t the object of proft maxmzaton on the frms level. However, ths rate of progress does not depend on any resources and s bounded from above by some nventon fronter whch s not allowed to drft n tme. Ths approach s much closer to present-day ones and bears sgns of the mcrofoundaton of technologcal progress. Examples of such models may be found n [47], [48]. It s ths strand of lterature where the research sector of economy was frst modeled as a separate one and the role of human captal has been recognzed, [50], [51]. At that tme the nterest n patent s length has rsen. Ths s closely related to the dscusson of the role of nnovatve enttes n the economy, see [20]. Last neoclasscal concept s that of emboded technologcal change. There the technologcal growth s emboded n factors of producton and s dstrbuted over tme when these factors have been produced. Ths s one of the foundatons for recent vntage models and methods beng used n ths lterature are also employed n the suggested work for other purposes. One of the scarce works on ths s [53]. In summary, some of the neoclasscal models already contaned deas of product varetes, developed further on and that of dstrbuted nature of technologcal progress. However these concepts have been developed n full only later on. 1

6 2 PREFACE At the begnnng of 1990 s two new approaches to nnovatons n growth theory emerged, namely, Romer s (1990) model of expandng varety of products and Aghon&Howtt s (1993) model of qualty ladders. Each of them addressed dfferent aspects of nnovatve actvty, but they both have been bult on the dea of endogenzng technologcal change through means of modelng nnovatve actvty. Frst of them explaned technologcal progress as the process of nventon of new goods. Ths dea orgnated from works of Dxt&Stglz, Ether, Spence [45], [44], [46]. The fnal wdely accepted form of ths approach s represented by works of Romer (1990) and Grossman&Helpman (1991). In these works the technologcal progress s modeled as the process that expands the varety of products avalable on the market, thus stmulatng growth through ncrease n the consumer demand (Dxt-Stglz theory) or through ncrease n the productvty (Romer s model). However, qualty (or productvty n the case of nvestment goods) was assumed to be constant. Ths dea lacked the precense of captal or other durable goods whch would grant rse n productvty and ths approach was senstve to scale effects. To overcome these dffcultes a number of extensons to ths approach has been consdered n last two decades. Namely the assumpton of rsng costs of R&D has been employed to weaken the scalng effect and ths s also one of the reasons of adopton of the smlar assumpton n the gven work. Some compettve effects has been consdered also, as well as ntroducton of knowledge as one of the factors of producton [54], [55], [26]. Current work benefts from some of the deas used n these extensons, namely the ntal knowledge about varetes s one of the key factors of the dynamcs as well as rsng costs of nnovatons at the qualty sde (decreasng effcency of nvestments). Second approach explaned technologcal change as the process of creatve destructon of products, based on the dea of Schumpeter, [7]. Every product s assumed to have varyng qualty, whch may be ncreased through nvestments, whle the outcome of these nvestments s assumed to be uncertan. However, quantty or varety of goods avalable on the market s assumed to be constant and every new better product destroys the precedng one upon ts nventon. Ths approach gave brth to a vast strand of lterature startng from early 1990 s. Here man subject of study s competton between nnovatve frms and thus t s related to IO lterature too. Among further extensons of ths strand of lterature are those consderng the possblty of mtaton of the leadng technology as well as some more dense structure of qualty ladders allowng for ntermedate levels of qualty to be acheved, [27], [28]. One of the extensons allowed for subsequent competton on the products market between frms. The basc dea of the current work also follows ths lne. The suggested research uses the man dea of qualty ladders lterature of allowng for qualty growth of gven products as one of the sources of nnovatve actvty. It s argued, that both these approaches are complementary n nature, descrbng two aspects of the same sngle process, whch are gong on smultaneously. At the same tme, there s no unfed model, whch would take nto account both these aspects n the dynamcal framework and allow for heterogenety of nnovatons. Current work uses these deas as gudelnes. Namely the process of creatve destructon has ts place n the growth of qualty of products, as t s n the Aghon s approach and every ncrease n qualty of a gven product does not ncrease the overall varety of products but rather ths margnally mproved product replaces ts predecessor. However, n the Aghon s approach there are explct dscrete generatons of products whle n the suggested work the whole process of qualty mprovement s the smooth one wthout jumps. On the other hand the whole range of products does not need to reman constant but rather s expandng governed by the dynamcal law

7 PREFACE 3 n the nature of Romer s work. Unlke the latter, qualty of all these newly nvented products s subject to change and ths change s descrbed for each product by the qualty mprovng process. Innovatve actvty has also been consdered n IO lterature. The process of nnovatons on ndustral level got attenton of economsts rather early, snce the end of 1960 s. Frst works n ths area were concerned wth patentng problems and patent races. The patent s vewed as the necessary mechansm to protect the nnovator and to create necessary level of ncentves to nnovate for the economc agents. Too long patents would create less stmul for new nventons and hence the noton of optmal length of the patent was born. Frst t was ntroduced by Nordhaus, [20]. Ths semnal paper gave brth to a wde strand of lterature on patents n IO lterature where the man emphass was on sngle-agent (stand alone) models of nnovatons. At the same tme the lterature on patent races assumed the precense of several competng frms n the nnovatve sector of the economy and ths gave brth to the noton of competton n nnovatons. Then the queston whether the compettve envronment should boost nnovatve actvty or not has been rased. One of the frst formal models wth patent races s of Loury, [19]. In all ths early lterature the process of nnovatons themselves remaned somethng lke a black box, that s, t was assumed that the process of nnovatons, although dependng on some exogenous factors s just emergng. Later on t has been notced, that the market structure, such as the number of competng nnovatve frms present, may substantally nfluence the speed of nnovatons also. See [18] as an example of such nfluence of the number of compettors on the nnovatve actvty. Even more later on the nature of nteractons between competng nnovatve agents has been taken nto account and t has been modeled explctly by means of varous statc games as well as of dfferental games as n [22], [21], [17], where the noton of mtaton (costly or costless) as well as R&D cooperatons have been used to descrbe the nature of strategc nteractons between dfferent nnovatve agents. Later on some uncertanty has been consdered as one of the fundamental features of the nnovatve process and formal models of patent races under uncertanty have been constructed. However, tll not that long ago the process of nnovatons tself even f assumed to be the dynamc one, has been vewed upon as a monotonc sngle-shot process of nvestng somethng lke t s n the papers on patent races, where the race s gong on for recevng one sngle patent for a gven nnovaton. In the mddle of 1990 s ths has been replaced by the wdely acknowledged noton of sequental or cumulatve nnovatons framework. In ths approach there s a sequence of dfferent nnovatons gong on n the same market (economy) one after another and whch are based on precedng nnovatons. See [16] as one of the frst examples of such a framework. Stll, all these nnovatons have been of the same nature and have been bult up one on the base of another. At the same tme t was wdely acknowledged that from the IO pont of vew there are at least two types of nnovatve actvty, namely, cost-reducng nnovatons and product-ntroducng ones. It has been noted, that the gven R&D frm may have several dfferent research projects at a tme and they may have dfferent nature and/or complexty. Hence not long ago the noton of heterogeneous nnovatons has been born. It s ths framework n whch the suggested work belongs. One of the frst models n such a framework may be consdered the work of Hopenhayn, [15] whch s manly devoted to the problem of patentng n the precense of multple research projects for a sngle agent. He consders as examples both prevously referred basc models of qualty ladders and varety expanson, but stll he does not unfy them. Moreover, hs framework s more or less statc n nature as are later works n the feld [12], [13], [14]. Current work suggests the unfed

8 4 PREFACE model of heterogeneous nnovatons n dynamc context. Proceedngs n ths work are related to these strands of lterature n several ways. It benefts from deas of nnovatve actvty as t s represented n New Growth theores. At the same tme t s based on the dea of heterogeneous nnovatons whch belongs to the IO lterature on Economcs of Innovatons. Ideas of patents as lmted lfe-cycles of products, as well as deas of mtaton and cooperaton are taken from lterature on nnovatons. Dynamc framework beng constructed follows manly the gudelnes of patent lterature of the 1980 s wth extenson on dstrbuted systems. The basc formulaton does not nclude any uncertanty, competton or notons of patents, there are no notons of consumer, economy, socal planner as well. It s manly concentrated on the analyss of the technologcal sde of nnovatve process. However, t s demonstrated that mere technologcal constrants may govern much of the behavor of nnovatng agents on the ndustry level. The framework does not nclude proftablty, prces, or supply-demand nteracton. Nevertheless, notons of patent and strategc nteracton of agents may be ncluded rather naturally n t. At the same tme snce the suggested framework s free from market-specfc mechansms, t also may be consdered as a prototype for the extenson of lterature on technologcal change n the sense of generalzng results of Aghon and Romer. All these defne the area of current research as n between standard lterature on nnovatons at IO level and New Growth theores. After constructng such a unfed dynamcal framework the role of heterogenety n the characterstcs of nnovatve products s studed n the frst chapter of the work. Then the basc analyss s extended to nfnte-tme horzon to obtan nformaton on the steady-states and ther stablty. Ths part of the work s nspred manly by the New Growth Theory rather then by the recent fndngs n Economcs of Innovatons. However due to the restrctons beng made n the basc model t belongs to the lterature on heterogeneous nnovatons. The mportance of dfferences between homogeneous and heterogeneous nnovatons s explored and the role of dynamc framework s also llustrated there. In two subsequent parts ths basc model s modfed to consder effects of patents and ther length as well as of competton n the space of nnovatons. For that the Hamlton-Jacob-Bellman dynamc programmng approach together wth the Maxmum Prncple are used and then a dfferental game n the space of nnovatons s formulated. To our knowledge t s one of the few examples of dfferental game wth explct soluton n nfnte-dmensonal space. The role of products dversty and dynamcs n the results obtaned and ther dfference from prevous works n the feld s dscussed at the end of each chapter. In the chapter wth patents the man pont of nterest s how the lmted lfecycle for all nvented products may affect the behavor of the nnovator. Notce that n the framework wth only one agent and n the absence of any regulatng mechansm and consumers ths s equvalent to the lmted length of patents for newly nvented products. It s ponted out, that n the developed framework the queston of lmted optmal patent s length n the precense of sequental and heterogeneous nnovatons s also resolved postvely and yeld the fnte length of a patent and ths s n agreement wth the lterature. However the full rgorous proof of that has stll to be obtaned under the framework dscussed, as t turns out to be much more dfcult then n homogeneous case. At the same tme, there s no need to employ notons of socal welfare functons or to model the consumer sde of the market to make ths concluson and ths dstngushes the approach of the chapter from beng prevously used. It s shown that t s the heterogenety of nnovatons whch stmulates nnovatve actvty wth lmted patent s length even wthout any

9 PREFACE 5 competton beng present. Note also that ths framework may be easly extended to the varable length of patents by choosng some specfcaton of ths patent s length as a functon of the product s ndex. In the last part of the gven work effects of strategc nteractons n the dynamc context of two multproduct nnovatve agents are consdered. Ths work may be consdered as an extenson of results of Ln and Lambertn, [14], [13] nto the dynamc envronment and uncountable number of products. From the other hand t complements the recent model of Lambertn, [56] by allowng for the nfntedmensonal space of products and not only for the dynamc nteracton. However such an extenson proofs to be essentally dfferent from prevous fndngs. In partcular, t supports and extends the lterature on jont R&D ventures as the nature of strategc nteracton but combnes t wth mtatve behavor n qualty growth. It has been noted n the begnnng of 1990 s that n real world economes nnovatve frms do not compete wth each other n the nventon of new products, f ths nventon requres substantal amount of efforts. They nstead cooperate n the creaton of new products and compete only n product markets further on. See [23], [24]. However due to the more general framework adopted n the suggested work t s possble to extend ths dea and allow for mtaton n qualty (cost-reducng) nnovatons on the second phase of nnovatve actvty. It s demonstrated, that olgopolstc market wth such a structure may be at least as effcent n terms of the rate of nnovatons as t s n the case of the monopoly. It s analyzed, what condtons make olgopolstc envronment more productve and what condtons make monopoly to produce more nventons per unt of tme. In ths way the gven work contrbutes to the dscusson on the optmal market structure for nnovatons as n late [25] where the dfferental-game approach s also used for ths. Unlke ths work, no ambguty s found n the ncentves and behavor of the agents due to the precense of dfferent knds of nnovatons n the model. Instead, t s shown that the equlbrum wth mtaton and cooperaton may exst only f both knds of nnovatons (qualty mprovng and varety enhancng) are consdered wthng the unfed framework. In such a stuaton ncentves to nvest less because of the mtaton n qualty and the ncentve to nvest more n varety enhancng whle beng the mtator are mutually balanced and ths gves a possblty for the desred outcome for both agents. Ths framework may be also extended to the arbtrary number of agents. In the followng chapters frst the basc model s constructed and then both of these extensons are consdered.

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11 CHAPTER 1 Product and Qualty Innovatons: A Unfed Approach 1. Introducton In ths chapter the constructon of basc fnte-tme dynamcal model of qualty&product nnovatons s carred out. The man purpose s to demonstrate the mportance of such an approach when comparng to separate analyss of these two processes. It has to be noted, that to our knowledge there are no models whch permt the smultaneous dynamc optmzaton of product and qualty nnovatons of heterogeneous type. However some attempts has been made to brng together heterogeneous process and product nnovatons n a statc context. Frequently ths purpose s acheved through the constructon of the 2-stage statc game, where on the frst stage the decson upon the ntroducton of new good s beng made and on the second - how much nvestment to put nto the development of qualty of ths newly ntroduced product (condtonal upon the successful ntroducton of t on the frst stage). One example of such papers s [10] whch s manly devoted not to the nteracton between both types of nnovatons themselves, but to the relaton between organzatonal structure of the frm and ts nnovatve decsons. It s shown, that the complementarty between process and product nnovatons s the drect consequence of the complementarty between frm s manufacturng capabltes and ts research capabltes. Current work correlates wth ths knd of lterature n the dea of smultaneous decson makng upon nnovatons of both types. However, here the stuaton wth multple products to be ntroduced on the market wth some speed, whch s controlled by the nnovatng frm s consdered. One other paper whch corresponds to some extent to the suggested analyss s of Boone, [11]. Ther the process of nnovatons s also formulated as the 2-stage game, but the author tackles manly wth ncentves to nnovate and ther relaton to the partcular characterstcs of the proft functon. The current work abstracts from the market characterstcs of the frm (partcularly from proft functon characterstcs). It s suffcent to assume some lnear and constant return from the ncrease n qualty and range of products to yeld the results of the model. Both these examples are statc n nature and they do not handle multproduct stuatons. Later on t has been noted, that real nnovatve companes are often multproduct monopoles. Papers by Lambertn, [12], [13] study the equlbrum characterstcs of nvestments of such a monopoly. He allows for multproduct nvestments, and the number of exstng products may also ncrease n the result of product nnovatons. However the whole model s statc because t handles only the equlbrum ponts of nnovatve polcy of a monopolst. Author does not study any dynamcal characterstcs of product and process nnovatons but only the equlbrum dstrbuton of nvestments. In the second paper Lambertn clams that the equlbrum level of qualty nvestments s hgher for the monopolst then the socal optmum. However, more recent paper by Ln [14] suggests that ths heavly depends on the level of economes of scope for the monopolst. In general to be able to answer 7

12 8 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH ths queston one has to account for dynamcal perspectve of multple products development and the evoluton of the product space. Methodologcally the current model s closer to the recent lterature on vntage captal models although t concentrates on another type of questons. It s ths strand of lterature where the dstrbuted parameter optmal control models are extensvely used to descrbe the nvestment polcy of an agent whch has captal wth dfferent dates of appearance at hand. Then hs polcy should depend on the dstrbuton of the mass of hs captal n past tme and hence the dynamc problem the agent has to solve s of dstrbuted parameter optmal control type. Examples of such models are [29], [31] and others. Ths strand of lterature uses vntage captal dea to descrbe polcy of nvestments on ndustral level, lke n [31], [43] and also to contrbute to the growth theores wth emboded technologcal progress of the neoclasscal type. Somehow dfferent approach of Hrtonenko, [32] uses ntegral equatons of Lotka-Volterra type to descrbe the dynamcs of the overall mass of captal nstead of partal dfferental equatons but also n the optmal control framework. To our purpose the dstrbuted optmal control method s more revelant. The dfference of the suggested work methods s that t does not use delayed structure of captal as n vntage captal models. For purposes of ths work t s suffcent to assume the dstrbuton over the products space. Ths sgnfcantly smplfes the analyss whle the delayed tme structure of the mass of nnovatons have lttle of nterest at ths stage. One of the few dynamc approaches to modelng heterogeneous nnovatons s the work of Hopenhayn&Mtchell, [15] whch handles the nnovatve process n a rather general way usng operators n a Banach space. However ther work s manly concentrated on the patent polcy and handles nnovatve process n a sense of prevous theores, namely of Shapro, [34]. There s no underlyng process of generaton of new products, nstead there s a statc space of deas from whch an nnovator s makng a random draw of an dea to develop. At the same tme all these deas may dffer from each other n there value and thus the heterogenety of nnovatons s observed. But ths heterogenety s not that mportant for the dynamcs of the overall model, snce only one nnovaton s developed at every gven tme so the nnovatve agent does not have a choce between dfferent nvestment/nnovatve strateges. The suggested approach combnes deas of Hopenhayn&Mtchell and of Lambertn and Ln n a way that nnovatons are assumed to dffer n ther characterstcs from each other as n [15] and n the same tme the appearance of the new products on the market as n [12], [14] s allowed n the dynamc context. So t s argued, that to handle the nnovatve actvty n multproduct stuatons one have to consder the dynamc context. At the same tme, there s no such a model, whch would take nto account these dynamc nteractons between dfferent types of nvestments nto multple products smultaneously n the lterature on economcs of nnovatons. Ths chapter tres to fll ths gap. It concentrates on two questons, namely: How expanson of products varety nfluences qualty nnovatons of all dfferent already exstng and new products? What s the role of structural characterstcs of these products by themselves n nnovatve process? To answer these and related questons the model s bult usng optmal control theory methods. The basc framework dscussed below allows for a very general formulaton of nnovatve actvtes. However n the current chapter the smplfed approach s chosen as the man goal s just to demonstrate the mportance of such a unfed approach to nnovatons. For that uncertanty s neglected n the model

13 1. INTRODUCTION 9 despte the mportance of uncertanty n nnovatons whch s wdely acknowledged now. More then ths, only one agent s modeled and there are no possble strategc nteractons here. It s also assumed that all products are present snce ther nventon tll the termnal tme of the overall model. Two last smplfyng assumptons are relaxed n subsequent chapters of the work. The analyss of the ntroduced model reveals the mportance of the nteracton between varety expanson process and the qualty nnovatons. It turns out, that the most essental characterstc of products to be nvented ( product space ) s the heterogenety of some parameters across products. If all products have smlar propertes, then the overall nnovatve process may be descrbed by a smple optmal control model wth only two state varables as n [56]. However, f these potental products are not smlar to each other n ther nvestments effcences or some other technologcal parameters, explct formulaton of the nfnte-dmensonal model s the natural way to capture the lnk between nvestment strateges and choces between varety expanson ntensty and speed of qualty growth. Ths lnk has essental nfluence both on varety expanson and qualty growth processes. It s demonstrated that overall dynamcs of nnovatons s dfferent under the assumpton of the absence of such a lnk and wth the explct ntroducton of t nto the system. Such more complete system has much more complcated dynamcs. Despte of the number of smplfyng assumptons (lnearty, sngle agent, compactness, etc.) the model ntroduced below s capable of reproducng the number of well known facts concernng nnovatons and reacts on the change of parameters n a very ntutve way. Due to the smplcty of the structure, the model has plenty of useful future extensons. The structure of the chapter s as followng. Frst the basc framework suggested and mportance of the assumptons beng made n the process are dscussed. Second the general model n the form of dstrbuted parameter optmal control one s constructed. After that some necessary theoretcal propertes of such a constructon, namely the structure of state-space and control space are analyzed as well as the result on the exstence of non-zero adjont varables s proved. In the subsequent secton general soluton and ts propertes are descrbed. It turns out that not much may be stated on ths stage concernng the model s dynamcs, except of the general nature of nteractons between dfferent types of nnovatons. Specfcally, t turns out that only the characterstcs of the boundary product s qualty dynamcs are revelant for the varety expanson dynamcs, not those of all the mass of products nvented. Ths justfes the choce of dstrbuted parameter optmal control method n the form of dfferental equatons rather then ntegral ones. However, the explct soluton may not be acheved at ths stage. That s why n the followng secton the model s smplfed by allowng all products to be dentcal, thus turnng the model nto the homogeneous one. Ths smple case helps to analyze more general and complcated propertes of the model. It turns out that even n homogeneous case the lnk between dfferent types of nnovatons nfluences the dynamcs substantally although due to the smlar nature of all products ths lnk s rather statc. Rest of the chapter s devoted to the heterogeneous case. However, only some specal type of heterogenety whch yelds lnear dfferental system as the result s allowed. Namely decreasng effcency of nvestments n qualty growth across products s assumed. The overall dynamcs of qualty nvestments s dscussed and the dynamcs of the whole model s reconstructed, combnng results of prevous sectons. Then the analyss of nfluence of varous parameters changes on the dynamcs of the system s carred out. In the last secton results and understandng acheved upon the unfed process of heterogeneous nnovatons are descrbed and some possble future extensons and refnements of the model suggested here are mentoned.

14 10 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH 2. Assumptons and Basc Framework To model the process of expanson of products varety and qualty growth smultaneously the noton of the products space s ntroduced. Ths space contans as elements all products whch are already nvented as well as potental products that may be nvented n the future. Every product has ts own characterstc - ts qualty. Ths characterstc s not constant but s the functon of tme and nvestments. From ths pont of vew, space of products s the functonal space and ts elements are qualty functons for every product. Then products themselves (both already nvented and potental ones) are dmensons of such a space. Assume that number of products (that s, dmensons of the product space) s nfnte. More than ths, assume that ths number s the real number, so dmensonalty of the product space s uncountable. In such a framework t s no longer correct to speak about number of products but rather about the range of them. Ths range s assumed to be bounded from above by some maxmal range of products whch can be nvented n a gven system (economy, market). The space of products thus conssts of nfnte-dmensonal vector-valued functons Q(t), whch descrbe evoluton of all products qualtes over tme. Such a space s hard to analyze n general and addtonal structure s put onto t to make n manageable. For ths assume that qualty growth of every product does not explctly depend on other products. Then every functon Q(t) may be represented by the nfnte dmensonal system of real-valued functons q (t) or, equvalently, by the functon of two arguments q(, t), where s the ndex of a product and t s tme. In the last representaton t may be shown usng standard arguments from functonal analyss, [1], that the space of functons q(, t) s somorphc to the L 2 space, provded t has a compact support. So assume that the range of product as well as tme are compact subsets of R: (1.1) t [0,.., T ] = T R + ; [0,.., N] = I R +. Boundedness of Lebesgue ntegrals over that space s assumed but ths requrement s not essental for the analyss, as the problem may be casted n a Banach space. Denote the space of such functons by L 2 (T I; Q), where Q R + as well. Process of the expanson of varety of actual products s descrbed by a onedmensonal functon of tme, n(t), whch takes values n the space I. So range of exstng products s constantly changng over tme. It s then natural to requre that qualty for products whch are not yet nvented, cannot change from ts ntal level. Then n(t) dynamcs represents the moton n the space of potental products along the subspace represented by q(, t) functons. It descrbes maxmal ndex of ths functon wth non-zero value. Analytcally the last requrement can be expressed as a constrant upon the functon q(, t): (1.2) q(, t) = 0 n(t) ; I. The whole process of nnovatons s then descrbed by contnuous expanson of the range of products avalable and by smultaneous growth of qualty of all products whch are already nvented n some nfnte dmensonal product space Q, whle qualty growth process for each product s ndependent from other processes and s launched at the tme when ths product s actually nvented, (1.2). To mpose control over these processes denote nvestments beng made n the varety expanson and n qualty growth of every product separately, u(t) and g (t) respectvely. To ensure exstence of optmal behavor assume these two process to be postve real valued and bounded from above. Dynamcs of varety expanson

15 3. MODEL 11 and qualty growth are determned through nvestment polcy, that s the actual choce of nvestments at each pont n tme n both drectons. No uncertanty s present n the model for smplfcaton purposes. 3. Model Under the basc framework descrbed n prevous secton to cast the model nto the optmal control framework, the scheme of so-called planned nnovatons s adopted: there s only one agent (socal planner or, alternatvely, the monopolst) who maxmzes the output of nnovatons n any gven perod of tme over the fxed tme horzon accordng to some objectve functonal. It s defned as: (1.3) J def = T 0 e rt ( n(t) 0 [ q(, t) 1 2 g(, t)2] d 1 2 u(t)2 )dt max Planner s maxmzng ntegral sum of qualtes of all products nvented untl each tme t mnus nvestments beng made to every nvented product s qualty and to the overall expanson process over the plannng horzon. There s no sgn of prces or proft n ths formulaton. Market clearng mechansm and all the mechancs behnd the market structure are neglected. One way to motvate such a formulaton of the objectve functonal s to assume lnear n every product proft functon and untary prce of each product. However, the man pont of nterest here s not the proft maxmzaton, but the maxmzaton of the output of nnovatons n every gven moment of tme. It s equvalent to the lnearty of proft functon whch s a standard assumpton n nnovaton lterature (consder Lambertn, [12] as one of later examples). One may treat the planner here ether as the central authorty n some centralzed planned economy or as the monopolst on some market. Dynamcs of qualty growth and varety expanson processes are governed by subsequent dynamc equatons: n(t) = αu(t); (1.4) and statc constrants: (1.5) q(, t) = γ()g(, t) β()q(, t); [0,.., N] = I R + ; t [0,.., T ] = T R +. u(t) 0; g(, t) 0; 0 n(t) N; q(, t) =n(t) = 0; q(, 0) = 0, I; n(0) = n 0 0. Ths assumes zero ntal qualty for all products and some fxed ntal range of products avalable. Observe, that the fourth constrant n (1.5) s equvalent to (1.2), provded nvestments to qualty growth are nonnegatve. Next observaton concerns γ() and β() functons. These are functons of effcency of nvestments to every product s qualty and rate of qualty decay n the absence of nvestments dependng on the product s ndex respectvely. These two functons represent structural characterstcs of the products space beng consdered as a whole, as they defne relatve dfferences n products as functons. Expressons (1.3), (1.4), (1.5) together consttute a dstrbuted parameter control system. In general such systems may be hard to solve, but makng use of the

16 12 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH assumpton of the ndependence of qualty growth processes for every t can be equvalently wrtten n the form of nfnte-dmensonal optmal control system wth respect to q (t), g (t) functons. Infnte dmensonal representaton of the system (1.3), (1.4), (1.5) allows for the applcaton of Maxmum Prncple, whle each q (t) functon represents a separate state varable. As a result, one have system of N + 1 frst order condtons as well as adjont equatons. Some detals of the theoretcal treatment of the ntroduced model are presented below. 4. Theoretcal Results In ths secton some theoretcal results concernng the model (1.3), (1.4), (1.5) are presented. One of the reasons to transform the model from dstrbuted parameter form to nfnte dmensonal one s that maxmum prncple s easer to prove for nfnte dmensonal problem then for the dstrbuted system. Note, that any dstrbuted parameter system may be transformed nto the nfnte dmensonal ODE system. However, preservaton of optmalty results s not granted. That s, soluton to nfnte dmensonal model may be not an optmal soluton to the subsequent dstrbuted parameter problem due to the form of objectve functonal (1.3). In formal terms ths means that convergence wth respect to L 2 (T I; Q) norm mples but s not mpled by convergence n components (projecton spaces for each I) n general. However n the case of the model (1.3), (1.4), (1.5) one may show such an equvalence. For ths t s suffcent to show the contnuty of all projectons of the functonal (1.3) along the I ndex space. Proposton 1. Objectve functonal (1.3) s contnuous wth respect to I and T spaces. Proof. One need only to show the contnuty of n(t) q(, t) g(, t)2 d term n (1.3). For every gven n(t) t s a functon of the nterval n the subsequent L 2 space, generated by q(, t) and g(, t) functons. Ths functon of an nterval s contnuous f the generatng functon of a pont has countable number of dscontnutes. Ths s true for the functon gven, as q(, t) s a contnuous functon (snce ts a state varable) and g(, t) s assumed to have fnte number of dscontnutes n t for every as any control functon. These mples contnuty wth respect to T space. The only problem s possble dscontnuty along I space, snce ths ndex space s a subset of real numbers. Consder projectons of the form g t (), q t (). These are functons of the ndex for every t. Note that functons q t () are contnuous as the last condton n (1.5) s exactly the contnuty requrement. Regularty of g t () functons depends on the regularty of parameter functons γ(), β() only whch are arbtrary at the moment. It s suffcent to assume some regularty condtons on functons γ(), β() to grant objectve functonal s contnuty. For purposes of the rest of the work assume them to be contnuous functons of. Then functons g t () can have at most countable number of dscontnutes and ths s suffcent for functons of nterval n(t) q(, t) g(, t)2 d to be contnuous. Wth contnuty of the objectve functonal one may freely transform the problem (1.3), (1.4), (1.5) to the nfnte dmensonal framework. Frst note that (1.5) mply compactness of state space for every q (t) and for n(t) both. Second observaton s that the whole system of dfferental equatons (1.4) may be decoupled nto equatons for n(t) and for q(, t). The only lnk between the two components of the system s through the (1.5). The system for

17 4. THEORETICAL RESULTS 13 qualty growth s then wrtten n the form of nfnte dmensonal system of ODEs: q (t) = γ g (t) β q (t), (1.6) [0,.., N] = I R +. Whch may be wrtten n operator form: (1.7) q(t) = Aq(t) + Bg(t). Ths system of controlled equatons s lnear and as such s object of Hlle-Yosda theory. To grant mplementaton of Maxmum Prncple one have to be sure that the uncontrolled part of the system (1.6) s a well-posed Cauchy problem. Ths can be done through Hlle-Yosda theorem [3]. For the system (1.7) t s partcularly smple, snce operator A has dagonal form and does not depend on t explctly. The formal proof s not gven here as t follows standard textbooks [3]. It s suffcent to note that ths reles on the proof that operator A has a full rank and ts nvert s bounded. Ths obvously depends on the choce of γ(), β() functons, so assume these to be not zero everywhere and wth bounded nverses. Note, that ths s the second regularty assumpton mposed on parameter functons. Next observe that optmal controls exst. Frst observe that control space possesses product topology and may be decoupled nto controls over varety expanson process and qualty growth. Frst of these s one dmensonal and bounded, second s nfnte dmensonal and bounded n each coordnate. Defne control space as (1.8) J = U G. Takng nto account constrants (1.5) one may defne admssble control set as (1.9) J ad = U ad G ad. where both subspaces are compacts snce boundedness of nvestments. Then the whole admssble control space s compact also. Ths yelds the exstence result. Proposton 2. Optmal controls u(t), g(t) exst. Proof follows from compactness of admssble control space. And one obtans a useful corollary: Corollary 1. Admssble control space s spke-complete. Proof follows drectly from compactness of the admssble control space. Spkecompleteness of the control space s the necessary condton for the formulaton of Maxmum Prncple and does not concde n general wth compactness for nfntedmensonal spaces. In plan words spke-completeness means that control space s closed under the operaton of spke-perturbatons of any control trajectory whch belongs to the admssble space. As spke-completeness s necessary for the exstence of optmal controls, n ths case ths mportant property s just a drect consequence of the exstence result above. For more detals see [3]. Now one may make use of Maxmum Prncple approach to obtan optmal controls. For that some standard propertes of control space and control system have to be fulflled, ncludng regularty of the B operator n (1.7) and completeness of the control space wth respect to perturbatons. These two are straghtforward to show and rgorous proofs are not gven here. It s suffcent to note that completeness of the control space wth respect to spke perturbatons s the consequence of compactness of admssble control space and regularty of operator B follows from the fact that t s dagonal and does not depend on tme. Then the only regularty requrement s on the functons γ(), whch s already assumed above. The dffculty of nfnte dmensonal problem s that the exstence of non zero optmal set of adjont varables s not granted. It has to be proved separately. For ths make use of the followng lemma:

18 14 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH Lemma 1. [3] Let {t n }, { j n }, {ỹ n } be the sequences of tme, controls and states convergng to optmal soluton of the control problem. Assume that there exsts such ρ > 0 and a precompact sequence {Q n }, Q n E, such that: (1.10) n=n 0 {t 1 n R(0, t n ; J, j n ) K Y (ỹ n ) B(0, ρ) + Q n } contans an nteror pont for n 0 large enough. Then the multpler z s not zero. It can be shown that requrements of ths lemma hold for the gven problem due to the specal structure of the control and state spaces (they are compact and possess product topology). Full proof may be found n Appendx. In concluson note that all exstence and regularty results presented here drectly depend on the regularty assumptons on γ(), β() functons. Up to now they are not specfed explctly but t s assumed that: These are contnuous functons of ; Inverse functons γ() 1, β() 1 exst. No monotoncty or dfferentablty requrements are necessary. To ensure economcally meanngful values, α, β(), γ() are restrcted to be postve: (1.11) α > 0; β() > 0, [0, N] ; γ() > 0, [0, N). Note that γ() s allowed to be zero for the last product to be nvented. Ths s done to allow for those nvestment effcency functons whch yeld decreasng down to zero effcency of nvestments. Observe, that these functons represent measure of heterogenety of the product space Q: the more regular these functons are, the more close dfferent products are to each other n ther structural characterstcs. 5. General Soluton General soluton to the problem (1.3), (1.4), (1.5) s obtaned through applcaton of the Maxmum Prncple to the subsequent nfnte dmensonal system. Hamltonan functon s gven by (1.12) H = λ(t) αu(t) + n(t) 0 N 0 {q (t) 1 2 g (t) 2 }d 1 2 u(t)2 + {ψ (t) (γ g (t) β q (t))}d. For each one may derve frst order condtons for optmal control and state trajectores for qualty of every product: (1.13) g (t) = γ ψ (t); ψ (t) = (r + β )ψ (t) 1; q (t) = γ g (t) β q (t).

19 5. GENERAL SOLUTION 15 Ths system yeld optmal control and state trajectores as functons of γ and β : ( ) 1 e (r+β)(t T ) g (t) = γ ; r + β ( (e rt β (t+t ) e (r+β)(t t) )β + (1 e βt )(2β + r) ) γ 2 (1.14) q (t) =. (r + β )(2β + r)β However, takng nto account (1.5) these solutons are effectve only for > n(t) as nvestments cannot be postve untl that tme and qualty level remans zero. Ths means qualty dynamcs has a pecewse form: ( ) 1 e γ (r+β )(t T ) r+β g (t) =, n(t) ; 0, n(t) < ; { rt β ((e (t+t ) e (r+β )(T t) )β +(1 e β t )(2β +r))γ 2 q (t) = (r+β )(2β +r)β, n(t) ; (1.15) 0, n(t) <. Note, that all product s qualtes has some maxmal attanable level, whch s never reached n the fnte tme. Ths level s computed as a fxed pont of a gven dynamcal system (1.13): (1.16) q = γ 2 (r + β )β. Frst order condtons for n(t) yeld the system of two dfferental equatons on n(t) and ts costate: (1.17) wth boundary condtons (1.18) λ(t) = rλ(t) g n(t)(t) 2 ψ n(t) (t)γ n(t) g n(t) (t); n(0) = n 0 ; λ(t ) = 0. n(t) = αλ(t). Varety expanson process however, s the process across all the states q. Hence, to obtan dynamcs of n(t) one should aggregate qualty dynamcs across states to return to the dstrbuted parameter form of the problem. In that way system (1.17) depends on functons γ() and β(). After substtuton for g n(t) (t) from (1.15) t can be seen that system (1.17) s not tme nvarant for any specfcaton of γ() and β() functons. Dynamcs of ths system depends at each pont n tme from the value of nvestments effcency to the growth of qualty of the next product to be nvented and do not depend on any other ones. Ths system represents a recurrence relaton n the space of products Q, as γ() functon depends on, whch s also the value of n(t) functon. That means that at = n(t) γ() functon s the functon of n(t). Actual shape of dynamcs of the process of varety expanson heavly depends on the shape of ths functon then. Through drect ntegraton one may obtan general equaton of moton for n(t), as an ntegro-dfferental equaton: (1.19) n(t) = t s 0 T γ(n(τ)) e β(n(τ)) f(τ,t ) dτds. Ths s hard to analyze and exstence of the soluton s not granted for arbtrary γ( ), β( ) functons. Some general propertes of the system may be captured from the form of the

20 16 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH general soluton though. Qualtes growths for all products have smlar form and dffer from each other only by values of γ() and β() functons. Startng from the tme when the product s nvented, ts growth process s ndependent of any other varables of the system. However, tme when ths process starts s defned through the expanson process, n(t). Ths last one depends heavly on the parameters functons of the product space. Ths means that no sngle product nor ts qualty process affect varety expanson. Instead ths last one depends on some aggregate characterzaton of the product space, whch s gven by γ() and β() functons. Mere exstence of soluton to the problem tself depends on the propertes of these functons whch may be vewed as fundamental characterstcs of the product space tself. That s, the more regularty requrements one puts on them, the more homogeneous space of products n terms of ther dversty one s consderng. Note that ths dversty s dfferent from the range of products N, whch smply denotes some dstance between dfferent products. Parameter functons are measures of effcency of nvestments across products. They may or may not depend on the maxmal range of potental products. Here the smplest case when they do depend on N s consdered. That means the products space has only one effcent characterstc of ts dversty - that s range of potental products. However the general framework s not lmted to these partcular spaces and more general structure may be consdered. To obtan some partcular soluton to the problem one have to specfy the form of these parameter functons. Consder as two examples the smplest case wth constant functons (transformng the problem to the homogeneous one) and the case whch lnearzes the system (1.17) n the rest of the chapter. 6. Homogeneous Products To observe the role of dversty of the product space, frst account for the homogeneous verson of t. For that assume both γ(), β() functons to be constant across dfferent products: (1.20) γ() = γ; β() = β; I. Note, that ths would not mean that products are dentcal n ther consumpton characterstcs, but only that effcency of nvestments nto the qualtes of dfferent products do not change wth the expanson of varety of products. Ths s the reason to refer to such a product s space as homogeneous. In such a homogeneous case every product s qualty has essentally the same shape of dynamcs. The only dfference s n the startng date of nvestments (beng defned from varety expanson). Then the only source for dentfyng dfferent products and separatng them from each other les n the n(t) space, snce the pecewse form of qualty functons (1.15). Solutons for qualty growth n homogeneous case then are of the form (for any product ): ) (1.21) q homo (t) = ( 1 e g homo γ (r+β)(t T ) r+β, n(t) ; (t) = 0, n(t) < ; { rt β(t+t ((e ) e (r+β)(t t) )β+(1 e βt )(2β+r))γ 2 (r+β)(2β+r)β, n(t) ; 0, n(t) <.

21 6. HOMOGENEOUS PRODUCTS 17 All product s qualtes have the same maxmal attanable level, whch s never reached n the fnte tme. Ths level s: (1.22) q = γ 2 (r + β)β ; I. It may be obtaned by equatng to zero lefthandsde of the system (1.13) wth homogeneous effcency functons. Wth constant β( ), γ( ) functons ths level s unque and the same for all products beng nvented. Ths level s also the saddlepont of the system, snce the characterstc equaton has two dstnct real roots of dfferent sgns. Ths means that asymptotcally all products qualtes would reach that pont and stay there n nfnte-tme horzon wth gven ntal condtons. Wth the assumpton of homogenety between products nvestment characterstcs the system (1.17) s reduced to (1.23) λ(t) 1 = rλ(t) 2(r + β) 2 γ2 (1 e (r+β)(t T ) ) 2 ; n(t) = αλ(t). Ths s stll a non-autonomous system but may be solved n elementary functons. Homogeneous soluton for varety expanson process s then: = n homo (t) = 1 (β + r) 3 (2β + r)βr 2 ( C 1 C 2 e 2(r+β)(t T ) + C 3 e (r+β)(t T ) C 3 e r(t T ) + C 4 t ) ; (1.24) λ homo (t) = γ 2 (β + r) 2 (2β + r)βr ( C 5 C 6 e 2(r+β)(t T ) + C 7 e (r+β)(t T ) C 8 e r(t T )). where C 1,.., C 8 denote some combnatons of parameters of the model. Ths soluton s an ncreasng functon of tme n n(t) and decreasng n λ(t) parts. Wth soluton at hand, one can easly analyze the nfluence of dfferent parameters on the system s behavor. Note frst, that the qualtatve dynamcs of every product s qualty does not change rrespectvely of the chosen specfcaton of the parameter functons. Ths happens because for any gven qualty functon, q (t) wth fxed, these paramaters are constants. The only requrement s ther postvty. Thus, t s exactly the same as n the general model wth excepton that γ(), β() functons are now constants across ndces also. Actual projectons of qualty functons n q t plane are plotted below for some arbtrary underlyng process of varety expanson (t defnes the q = 0 pont at t axs for all soluton curves).

22 18 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH.. Ths set of qualty functons s evaluated for the set of parameters: (1.25) SET H := [β = 0.2, γ = 0.4, r = 0.05, T = 10]. Below several soluton curves correspondng to (1.24) wth the same parameter set plus α = 0.5 and varyng ntal range values are plotted... Ths soluton does not have any fxed ponts, as t can be readly seen from system (1.23) and t s characterstc equaton, whch has one real postve root and one zero root. One has the unbounded growth wth decreasng speed of n(t) functon. Next observe the parameter nfluence: Range of products N does not nfluence dynamcs of the system n the case of nteror soluton as long as parameter values do not depend on t. However f γ = γ(n), β = β(n), as t was assumed earler, ts nfluence s equvalent to the nfluence of these parameters; Length of the plannng horzon, T, postvely nfluences the range of products nvented untl ths horzon. Ths s the drect consequence of the monotoncty of soluton wth respect to tme. Shadow prce of nvestments, λ(t) has lesser ntal values for shorter horzons;

23 7. LINEAR MODEL 19 Rate of decay of the qualty, β whch s here constant across products, sgnfcantly reduces the speed of varety expanson. However, even for β > γ varety expanson has a postve dynamcs; Effcency of nvestments nto products qualtes, γ postvely nfluences the dynamcs of varety expanson. The same s true for effcency of nvestments nto varety expanson tself. The last one has more sgnfcant nfluence. All the parameters nfluence dynamcs n a quet ntutve way. Note however, that as long as γ, β are constants, N, the maxmal range, dsappears from the system completely. Now observe the character of nterdependence between varety expanson and qualty growth processes. In ths homogeneous case the only lnk s one noted above, that s, qualty nvestments to every product starts only when t s nvented. At the same tme all nvestments to qualty after they start, are dentcal n ther speed. Varety expanson process does depend on the characterstcs of the product space. However ths dependence s also quet fragle: the only dfference between the ndependent varety expanson process and the one accounted here s n constants γ, β. Obvously, ths does not change dynamcal characterstcs of the process, but only the mass of nventons, acqured at each pont n tme. So one may conclude, that qualtatve behavor of the system (1.23) does not dffer substantally from ndependent development of qualty and product nnovatons. Observe that the varety expanson process wll not evolve f all products qualtes would be zero all the tme, that s f γ = 0. However varety expanson process does not depend on fronter product nvestment characterstcs as t s n the general case, (1.17). So there s some but very weak dependence of varety expanson from nvestments to qualty growth. Ths observaton leads to the concluson that some suffcent degree of heterogenety of the product space s essental for non-trval nterdependence of qualty growth and varety expanson processes. On the other hand, the smple verson of the model studed n ths secton demonstrates, that standard models of nnovatons whch treat both processes ndependently may be casted nto the suggested framework as specal cases for homogeneous (n the sense defned above) space of products. If one wants to descrbe evoluton of range and qualty of products whch are not that dentcal n ther nvestment characterstcs, one has to model both processes smultaneously. 7. Lnear Model From now on some specfc form of parameter functons s assumed. Assume that γ( ) functon does depend on N and s monotonc and decreasng n, whle β( ) functon s constant for smplcty: (1.26) β() = β; γ() = N γ. Wth such a form of parameter functons system (1.17) s a lnear non-autonomous system: (1.27) λ(t) = rλ(t) (N n(t))(1 e(β+r)(t T ) ) 2 (r + β) 2 ; n(t) = αλ(t). Ths system does not have solutons n elementary functons (see [4]), but t does have real-valued solutons accordng to the general Sturm-Luvlle theory, [5]. So one can analyze dynamcs of ths system. It can be also noted that ths system

24 20 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH provdes monotonc motons n n(t) drecton gven ntal and termnal condtons: change of n(t) cannot be negatve and s decreasng to zero at the termnal tme. These observatons gve one the possblty to analyze the propertes of the system (1.27). For that one may employ the method of analyss, whch s vald for lnear dynamcal systems (non-autonomous) only, [4]. That s, one may ntroduce some artfcal varable, x(t), whch would make the system autonomous: (1.28) λ(t) (N n(t))x(t) = rλ(t) (r + β) 2 ; n(t) = αλ(t); x(t) = (1 e (β+r)(t T ) ) 2. In ths way by fxng certan levels of x(t) varable one obtans autonomous lnear system n λ(t), n(t) whch has usual soluton and may be analyzed by standard technques. The x(t) varable s decreasng functon of tme, varyng from zero to one. Wth r = 0.05, β = 0.2, T = 10 t looks lke:.. It may be treated as a contracton operator of the system (1.28), whch transforms the phase space of the system. For a gven system ths operator spans the phase space along the λ(t) drecton due to the negatve sgn of t. To observe ths, consder gradent felds for dfferent levels of x (0.2, 0.5, 1) for the same parameter values:... The only locus of the phase space whch s consstent wth boundary value of the costate varable (λ(t ) = 0) s the locus constraned by the sold black lne from above. Durng the evoluton of a system, ths locus s spanned along the lambda axs. Then one may move ths system and ts resultng solutons along the x(t) axs wth the gven speed to reconstruct the dynamcs of ntal non-autonomous system. Observe also, that n termnal tme, t = T, x(t) operator vanshes makng the

25 7. LINEAR MODEL 21 system (1.28) autonomous. Gven arguments reveal, that the acton of the mappng x(t) on the system preserves ts fxed ponts and changes the dynamcs wth respect to shadow prce movements. So one may obtan stablty results through nvestgaton of the (1.28). Note, that by smple change of varables, lettng (N n(t)) = y(t), the (1.28) can be made homogeneous. Subsequent autonomous system has the fundamental system of solutons of the followng form: (1.29) n(t) = C 1 e 1 r 2 +rβ+ 2 +C 2 e 1 r 2 +rβ 2 r 4 +2r 3 β+r 2 β 2 +2α 2 γ 2 (r+β) + r 4 +2r 3 β+r 2 β 2 +2α 2 γ 2 (r+β) + N; λ(t) = n(t). Where C 1, C 2 are constants of ntegraton. Ths system of lnearly ndependent solutons has rather smple dynamcs n the λ(t) n(t) phase space. Gven boundary condtons (1.18) one has monotonc moton of the system. The n(t) varable s growng steadly wth decreasng speed, whle λ(t) - shadow prce of nvestments - s decreasng untl zero. Ths s the standard dynamcs of the captal accumulaton problem. Observe, that the gven autonomous system has a saddle type dynamcs wth the fxed pont at n(t) = N. Ths can be checked through computng egenvalues of the homogeneous autonomous problem, resultng from (1.28) for fxed x(t): α J δi = δ(δ r) (r + β) 2 x = α λ2 rλ (r + β) 2 x; δ 1,2 = r r 2 α ± (r+β) x (1.30) 2. 2 It s clear, that both egenvalues are real and of dfferent sgns, snce x 0. It s also clear, that ths autonomous system has the unque fxed pont at λ(t) = 0; (1.31) n(t) = N. whch may be reached only at the termnal tme. Observe, that the orgnal non-autonomous system (1.27) wll reach ths level at t = T also, because the x(t) functon goes to zero wth t T and at tme close to the termnal one the orgnal system s behavor s the same as of the autonomous one. Snce the x(t) s the decreasng functon of tme from one to zero, t would push n(t) level constantly away from ts steady-state level, as shadow prce of nvestments wll decrease much slower, then n the (1.28), due to the acton of x(t). However, at T ths x(t) term would eventually go close to 1 for all t and hence asymptotcally at long tme horzons n(t) dynamcs may be descrbed by the means of autonomous system (1.28) wth x(t) = 1 for all t, not only at the end perod. It also has to be noted, that the type of dynamcs s dfferent f all products are dentcal n terms of nvestments effcency (that s, when γ( ) functon s also constant across products). It has been shown, that the homogeneous system (1.23) do not have any fxed ponts and n(t) growth s unbounded. However our man nterest s to analyze the dfferences n the behavor of the system n the presence of heterogenety of products beng developed. Ths s what γ( ) functon accounts for as well as x(t) term n the (1.28). In the smplest case beng studed here wth parameters specfcaton lke n (1.26) there s no very much structural dfference n the system (1.28) behavor n comparson to autonomous system. Movement along the x(t) axs wth some exponental speed brngs possblty for temporary shadow prce ncreases, whle expanson process s speedng up n comparson wth

26 22 1. PRODUCT AND QUALITY INNOVATIONS: A UNIFIED APPROACH homogeneous case. Shadow prce of nvestments to n(t) may have temporary growth perod n the begnnng of the plannng horzon. At tmes close to t = 12 T shadow prce reaches ts maxmum and begns to decrease steadly untl zero. Wth longer plannng horzons there mght be longer fluctuatons of shadow prce of nvestments. Below s the schematcal reconstructon of the 3-dmensonal system movement... It s more mportant that the heterogenety of products brngs heterogenety of qualty nvestments nto the model and through that some sgnfcant changes to the overall system s behavor. 8. Qualty Investments Now take a closer look on the 2nd part of the dynamcal system - nvestments to the qualty growth. For any gven dynamcs of qualty s of the saddle-type. Ths can be demonstrated both graphcally and analytcally. Observe that the system (1.13) takes the form: (1.32) ψ (t) = (r + β )ψ (t) 1; = γ 2 ψ (t) β q (t). q (t) after substtuton of optmal controls g (t) for each nto the system. Ths s the usual system of two 1-st order ODE s, whch may be analyzed through conventonal methods, [4]. For that change varables n such a way as to make the system homogeneous: ψˆ(t) = (r + β )ψ (t) 1; (1.33) qˆ(t) = β q (t) γ2. (r + β ) The homogeneous system defned n such a way has two egenvalues whch are real, dstnct and have dfferent sgns: J λ I = λ2 rλ β (β + r); p r2 + β (β + r) r (1.34) λ = ± 2 2 Obvously, these roots are of dfferent sgns, snce expresson under square root s bgger then r. Ths means exactly the saddle - type dynamcs of the system. One can easly compute sngular ponts of ths system for each. For that just equate