Innovation, specialization and growth in a model of structural change

Andergassen, Rainer; Nardini, Franco; Ricottilli, Massimo Working Paper Innovation, specialization and growth in a model of structural change Quaderni - Working Paper DSE, No. 718 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Andergassen, Rainer; Nardini, Franco; Ricottilli, Massimo (2010) : Innovation, specialization and growth in a model of structural change, Quaderni - Working Paper DSE, No. 718, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4523 This Version is available at: https://hdl.handle.net/10419/159559 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Innovation, specialization and growth in a model of structural change Rainer Andregassen Franco Nardini MassimoRicottilli Quaderni - Working Paper DSE N° 718 Innovation, specialization and growth in a model of structural change Rainer Andergassen1,2,4, Franco Nardini3, Massimo Ricottilli1,2 April 1, 2011 1Department of Economics, University of Bologna, P.zza Scaravilli 2, 40126 Bologna (Italy) 2CIG, Centro Interdipartimentale Luigi Galvani 3Department of Mathematics for the Economic and Social Sciences, University of Bologna, Viale Filopanti 5, 40126 Bologna (Italy) 4RCEA, Rimini Centre for Economic Analysis (Italy) Abstract The aim of this paper is to investigate the nexus between demand patterns and innovation as it stems from research efforts and the extent of specialization. In the proposed model an innovation race conducted by entrants investing in research and development against established incumbents raises productivity at the industry level and leads to a shift in the aggregate demand pattern and consequently to a redistribution of the profit fund among industries and a restructuring of the production process in each industry. The paper argues that the degree of development as reflected in a demand share distribution is characterized by a corresponding distribution of specialized sectors that becomes more even across industries as the development process proceeds and investigates the consequences in terms of economic growth. Keywords: Innovation; development; structural change JEL: O10; O30; O40 1 1 Introduction The history of the last two centuries and a half of economic development cogently suggests that the increase in productivity due to technical advancement has played a major role in fostering a growth process that has then become largely self-sustaining. It is a well documented fact that the progress in technology and science applied to industry has brought about an extraordinary expansion of implements assisting labour in producing final commodities. This expansion has, of course, been made possible by the deepening of specialization, itself the outcome of ever-increasing division of labour. Yet, it is also equally well recorded that, concomitant to this trend, entire industrial sectors have been subject to sometimes radical restructuring: production processes have become leaner, the input structure simplified and real costs streamlined. The growth of the size of the market, in a Smithian sense, lies at the heart of this movement but evidence also suggests that there is a narrow relationship between increasing productivity and the pattern of final demand. This observation vouchsafes the view that economic development has progressed through stages hallmarked by patterns of final demand matching productivity as well as income per head levels. Amongst other possible criteria, a development stage taxonomy can, indeed, be fashioned by identifying typical consumption standards. Broadly speaking, low productivity and low income economies are by necessity constrained to afford only staple consumption goods required to support a basic livelihood: food, shelter and the bare means to entertain social intercourse. As productivity rises and income per head is augmented, the economy is reshaped to accommodate a demand pattern made up of goods of lesser priority, possibly allowing greater comfort and affluence. In this sense, a whole sequence of steps each characterizing a specific stage can be envisaged, from mere subsistence to mass consumption of durable goods (cars, home appliances, holidays) to luxury goods. The link between demand patterns and income levels has been stylized by the well-known Engel’s curves showing that as income rises the weight of some commodities increases whilst that of others declines as demand is driven to saturation for the latter and as acceleration for the former occurs. A simple and stylized way to formalize this approach, however, is to assume a hierarchy of preferences that ranks goods from very high to very low priority ones, the share of the latter rising as productivity and income per head rise. These demand share shifts subject the economic system to structural changes that produce breaks in the pattern of long term growth and reshape the production structure.1 1These type of events have been studied in two strains of relevant literature; for the demand driven approach centered on hierarchical preferences see Matsuyama (2002), Foellmi and Zewilmueller (2006), (2008);for the one that considers different income elasticities see Kongsamut, Rebelo and Xie (2001) and Laitner (2000);for the supply driven approach 2 The aim of this paper is to investigate the nexus between demand patterns, hallmarks of development stages, and innovation as it stems from research efforts and the extent of specialization. Accordingly, the model we consider strives to put the latter at the centrestage of our analysis to determine the productivity growth allowing for income per head increases generating demand shifts and hence structural change. This is an implication-laden event that leads to both further specialization and to rationalization, the former where demand expands while the latter where it contracts. Thus, they are processes that are endogenously determined, see also Romer (1987) and Ciccone (2002). On the demand side, we consider the consumer’s problem to be a quantity and a variety choice problem, and describe how changes in prices eventually lead to changes in aggregate demand pattern. On the production side, the economy is structured by vertically integrated industries the final sectors of which manufacture consumption goods whilst a number of specialized sectors provide the intermediate goods that are required by the former as inputs. Given this framework, the main source of growth is, of course, innovation the burden of which is mainly laid upon these manufacturers. Innovations, however, are idiosyncratic events that require investment in specific resources. For simplicity’s sake but, we believe, without loss of generality, we restrict the resources to be employed for the purpose to highly specialized manpower. Successful innovation leads to the acquisition of exclusive rights to the production and sale of these goods to final users but once gained a monopoly position, innovators enjoy quasi rents and cease the research effort leaving this task to followers who, by taking up the challenge, attempt to oust them (see, for example, Aghion and Howitt, 1992). Successful efforts by innovators do not remain confined within the sector in which they occur: it has often and persuasively been observed that as a consequence of their innovation technological imbalances arise involving all the complementary inputs that make up the concerned technology. These imbalances, however, although involving only the industry to which the relevant sectors belong to, open the way to seek the solutions that are necessary to heal them by acting as focusing devises (Rosenberg, 1990; David, 1976; Mokyr, 1990) thus spreading the productivity increase to all sectors within the industry with which they are technologically bound. These considerations lead us to envisage productivity growth as an industry-wide phenomenon and not just as a sector-confined event. It is the promise of future productivity-linked profits that motivate the protagonists of technological advance but it must also be recognized that the size of profitability is set by the size of the market for each final good which, in turn, depends on the demand pattern prevailing in the economy in each point in time. The extent see Ngai and Pissarides (2007) and Acemoglou and Guerrieri (2008).See Matsuyama (2008) and Buera and Kaboski (2009) for a survey. 3 of the profit fund, we hold, is relatively larger where demand and in equilibrium supply are also relatively larger defining an opportunity domain that can be exploited by contributing to the technical profile of each final good. This domain of opportunity acts by signalling new, would-be producers of intermediate goods that they can appropriate a share of the total profits made available by demand by introducing new specialized inputs that will then increase productivity throughout the industry. This has historically been another major source of innovations. New capital goods, in this paper new intermediates, are not however conjured up in a technological vacuum. On the contrary, they are the actual result of a process of learning and searching construing the awareness that it is technologically not only feasible but also profitable to further specialize a production process. The protagonists are those agents that at various levels are directly involved in the current manufacturing of already existing inputs: they are those who possess the knowledge, the know-how and the skills to realize how the production process can be further perfected and complemented. This is basically why new specialized inputs proceed from within the industry as spin-offs of already operating firms. They may be the very entrepreneurs but also workers, engineers and whoever is therein employed. If this leads to lengthening the chain, or string, of specialized inputs, it may also lead to its rationalization, meaning by this term the trimming and shortening of this chain, if the need and opportunity arise, by easing the industry cost structure. These further observations lead us to argue that as the relative size and the distribution of demand shares identifies a stage of development this is also identified by a corresponding extent of specialization in each industry. We deem this to be an important point since it allows to associate to the evolution of innovation-led productivity and income per head an evolution of demand shares and, therefore, of a pattern of specialization. The economy we analyze is normalized by a constant total employment that accordingly defines the extent of the market. Furthermore, the time scale that the model applies is two-faceted. Demand shocks that engender structural change are deemed to be relatively rare events hence occurring on a slow time scale in contrast to innovations improving existing intermediate goods, theirs being accordingly a faster one. Thus, the economy is assumed to have the time required to settle onto stationary states therefore marking as many stages of development. It will be shown that the expected economy’s growth rate in the stationary state, that is for any given size of total employment and demand share distribution, is eventually determined by the rate at which innovations arrive as a direct outcome of followers’ investment in R&D employment. Yet, a crucial ingredient is the distribution of demand shares. It will be shown that as this distribution becomes less lopsided and more even, the expected 4 growth rate decreases. This is due to the fact that the contribution to growth is higher where arrivals are higher as a consequence of each sector innovation effort. Economies in which there is a greater number of such sectors grow faster. Thus, the more intermediate-producing sectors concentrate on fewer industries the higher is the average growth rate and as they get more evenly distributed the lower it becomes. Specialization in our model is demand driven: as the growth process continues, consumers’ demand is distributed on a wider range of goods stimulating an innovative process that is diluted on a larger number of industries thus reducing the growth rate. This carries an important implication: as an economy progresses towards higher stages of development keeping the same extent of the market and the same number of intermediate-producing sectors its growth rate slows down. By the same effect and conditions, a less developed country manages to have a higher growth rate than a developed one, exhibiting a process of convergence. The same effect applies if, thanks to international trade, it is able to concentrate its innovative effort on a fewer number of industries. An important question arises at this stage of our investigation since what remains to be seen is what happens when demand shares actually change. This event forebodes momentous adjustments as some industries are witness to demand increase whilst others are likely to be involved in the opposite event. Would-be investors attempting to innovate have to take due notice of these occurrence and modify their expectations, differently according to whether the change over is likely to involve them in an increasing demand environment or in a decreasing one. We formally analyze a simplified version of the model and then simulate a more complex version to show that results that have been obtained do hold more generally. By studying a traverse path over a fixed time period, we are able to show that the number of specialized sectors does increase where demand shares increase and decrease where the latter fall: thus, industries where demand expansion occurs are subject to deepening specialization while industries where it diminishes are subject to rationalization by lessening the number of intermediate sectors. An important side effect is that owing to uncertainty connected with the eventual disappearance, concomitantly coupled with the appearance, of some sectors generated by demand shifts, the level of innovation-oriented employment contracts during the traverse period to be restored when it is expected to be finally over. The paper is structured as follows. Section 2 describes the demand side of the economy, Section 3 sets out the production structure while Section 4 describes the innovative process that takes place within the sphere of intermediate goods producers as well as specialization and rationalization that there occur. In Section 5 we calculate the stationary state expected growth rate when demand shares 5 are fixed and then analyze the traverse path when demand shares change as a consequence of the innovation process. Section 6 draws the paper to a close. Proofs are placed in the appendix. 2 Consumption pattern 2.1 The individual demand function Consider an economy with 1, ..., j, ..., J differentiated goods with prices at time t p1,t, ..., pj,t, ..., pJ,t and populated by Lconsumers. We conceive the consumer’s problem as a problem where, for given prices and income Ri,i= 1,2...L, an optimal decision is to be taken concerning how many goods, and how much of each, are to be consumed: a variety and quantity choice problem. Moreover, as prices change, a revision is to be made on whether to change the variety and quantity proportions of the consumption bundle. To solve this quantity and variety choice problem, it is stipulated that consuming yi,1,t,yi,2,t, ..., yi,j,t of jgoods at time tyields to the individual ian instantaneous utility rendered by the following logarithmic function ui(j, t) = Log (Ci,j) + αj j X h=1 Log (yi,h,t) where Ci,j is a weight that accounts for individual i’s impatience to consume good j. Formally, for given prices p1,t, ..., pJ,t, and a constant income Ri, and assuming that the individual’s intertemporal discount factor coincides with the constant interest rate r, the individual i’s problem can be written as a sequence of static problems Ui(j, t) = max yi,1,t...,yi,j,t ui(j, t) s.t. j P h=1 ph,tyi,h,t ≤Ri (1) and maxj∈{1,...,J}Ui(j, t)(2) In (1) individual isolves the quantity choice problem while in (2) the variety choice problem is solved. We postulate the following Assumption. Assumption 1 1. The difference αj+1 −αjis positive and non-decreasing in j; 2. constants Ci,j for i= 1, ..., Land j= 1, ..., J satisfy the condition that χi(j)≡Ci,j+1 Ci,j (Ri j+1 )(j+1)αj+1 (Ri j)jαj 6 is non-increasing in j. 3. prices are such that pj+1,t > pj,t >1for each t≥0. Condition 1. requires that the degree of concavity of the individual’s utility function does not increase as variety jis increased. Condition 2. constrains the rate of change of the indirect utility obtainable by equally distributing income on the number of goods in a given basket, hence independently of prices, not to increase in j.2This property generates a hierarchy such that high priority goods weigh more in determining indirect utility than those of lesser priority. Condition 3. makes this hierarchy explicit by stating that prices of lower priority goods, the luxury ones, are higher than those that apply to higher priority goods, the basic ones. The solution to the consumer’s problem is as follows. Let j∗ i,t be the optimal variety at time t, that is Uij∗ i,t, t> Ui(j, t)for each jother than j∗ i,t, then yi,j,t =       1 j∗ i,t Ri pj,t 0 for each j≤j∗ i,t for each j > j∗ i,t (3) from which the indirect utility can be obtained. Assumption 1 states sufficient conditions for the existence of j∗ i,t. More particularly, it guarantees that the utility is monotonically increasing in jfor the consumption bundles with variety lower than j∗ tand monotonically decreasing for baskets with variety larger than j∗ t, that is, Ui(j, t)< Ui(j+ 1, t)for each j= 1, ..., j∗ i,t −1(4) and Ui(j+ 1, t)< Ui(j, t)for each j=j∗ i,t, ..., J −1(5) To see this, taking into account (3), write U(j, t)≶Ui(j+ 1, t), i.e. the inequality of indirect utilities, as Ω (j, t)≶χi(j, t), where Ω (j, t)≡pαj+1 j+1,t Πj h=1pαj+1−αj h,t . Assumption 1 (i) and (iii) guarantee that Ω (j, t)is increasing in j. Conditions (4) and (5) can be written as follows Ω (j, t)< χi(j, t)for each j= 1, ..., j∗ i,t −1 2To better grasp this point, consider that should a decrease of indirect utility occur by adding one more good in the equally distributed basket, a further increase of such goods would yield an even larger drop in indirect utility. 7 When specialization is involved, because of the complex processes of learning-by-doing and byusing that constantly take place within the entire industry and more specifically in capital goods production, opportunities to further specialize are caught through spin-offs that lead to the setting up of new sectors, deepening the industry’s capital structure. Thus, the extant length of the input string catches the extent to which the process of specialization has gone in a specific industry j. As later discussed, owing to the possibility of earning monopoly profits, there is an outstanding incentive to devise yet more specific implements in the industry which appears to insure larger profits. As shown by (18), the existence of a profit fund that can further accommodate more intermediate producers provides incentives for spin-offs. It is quite clearly the case that this opportunity is greater where total profits are higher. An event such as this is not without consequences. By the very fact that specialization is indeed furthered, tasks that were previously carried out by the joint contribution of existing inputs are now performed in a different and innovative way by new intermediates. The upshot is that all the technical norms that are incorporated in the already-in-use kj,t−intermediates are changed: it is an adjustment process that is driven by the focusing device brought about by the new element of production (Assumption 4 (ia)); for the sake of simplicity we are assuming that the productivity increase brought by specialization is the same as that brought by the innovation (14). Assumption 4 (ib) indicates that the productivity of the new input improves the average productivity in the relevant industry6. Until the appearance of a spin-off, researching followers remain the owners of an outdated but still effective technology, the productivity gap of which, relatively to monopolists in charge, is given by (14). Because of a specializing spin-off, this gap can potentially double (see (16)). Again for the sake of simplicity, we assume (see Assumption 4 (ic)) that followers can, by imitation and possibly reverse engineering, seize the opportunity provided by the new pervasive change to leapfrog on a technology with kj,t intermediates and bridge one of the two productivity gaps. As soon as a specialization conducive spin-off appears, followers are able to adopt it and adapt their outdated technology thus keeping their gap from widening (see Assumption 4 (ic)). On the contrary, history has recorded rationalization processes through which the introduction of innovations has actually simplified and cut short the string of intermediate sectors. These events have generally been the consequence of adverse effective demand shifts that have jeopardized their profitability although creating an incentive to introduce cost-reducing innovations. The pruning of intermediate sectors has therefore had the effect of forcing those that have managed to remain to 6In the Appendix we show that by this assumption the mark-up does not increase after the occurrence of specialization. 14 restructure and become more productive. This restructuring process is the result of a profit decline that happens to hit some industries as a consequence of a relative demand shortfall that leads to a shrinking of the available profit fund. In this case, the length of the intermediate good string that characterizes the indirect cost structure becomes a burden that requires some leaning if profitability is to be restored. Yet, as in the case of specialization, this is a process that involves the whole industry leading to a change of the overall production technique. In both cases, the consequence is that an entirely new technique becomes available (Assumption 4 (iia)). Likewise, followers are able to replicate this process keeping the gap from widening (see Assumption 4 (iib)). 4.3 Profits and competitive threats Given the assumptions mentioned above the profit to be earned in the k-th sector of the j-th industry is πk j,t =1 ηck j,txk j,tw=1 ηck j,t bj,0 ak j,0 ly j,tw(19) Assumption 5 The absence of arbitrage opportunities among different sectors of the same industry insures that the employment of the same quantity of labour affords an equal profit flow πk j,t =πk0 j,t k, k0= 1, ..., kj,t ,j= 1, .., J . Assumption 5 and (19) allow us to distinguish two different mark-up components: an industry wide term cj,t and a sector specific correcting factor ak j,0which is linked to the k-good productivity. It is, indeed, cj,t that enables producers to earn monopoly profits. Yet, in order that this be actually the case it must be tuned as to forbid previous incumbents to remain in the industry as competitors. Thus, the size of cj,t is to be such that the productivity increase be entirely appropriated by the entrant and the old incumbent’s profits be driven to zero to oust him or her out of the market. The critical size of cj,t can be derived by considering that the price of the final good is the same no matter who produces the intermediate good to be employed. Taking Assumptions 3 and 5 into account, we obtain the following. Lemma 1 The mark-up of intermediate good producer kof industry jis ck j,t =cj,tak j,0,(20) 15 while the industry wide mark-up is cj,t =η bj,0kj,t (1 + kj,tδj,t)eλ−1.(21) Proof. In the Appendix. By (10) and (21), the price of a final good jbecomes simply pj,t =w bj,t (1 + δj,tkj,t)eλ(22) Remark 1 On account of the non-arbitrage Assumption 5, the profit of an intermediate good producer in any industry jdepends only on the specific industry, but not on the specific intermediate good k πj,t =πk j,t =1 kj,t (1 + δj,tkj,t)eλ−1wly j,t .(23) The appearance of new sectors carries with it the burden of a new production process, no matter how simple, encumbering the economy with more employment, a new technique and yet another monopolist enjoying exclusive ownership rights upon it. It follows that some conditions must be satisfied for the lengthening of the process to be feasible and further specialization take place. As mentioned, because of specialization the whole process becomes more productive. On account of (16), a productivity increase, say at time t, must translate into lower prices; it follows that furthering specialization is feasible if and only if pj,t ≤pj,t−(24) The price pj,t that rules when such an event occurs is determined by a new mark-up that successful incumbents charge on their production cost and which depends on the competitive threats on their monopoly position. The adoption of a new specialized technology in which a new incumbent joins the existing ones lengthening the input string to kj,t intermediate goods must reckon with a twofold competitive threat. On the one hand, erstwhile monopolists and now followers, although availing themselves of an outdated technology, are in a position, see Assumption 4 (ic), to incorporate the new specializing input and narrow the gap separating them from the more productive one. On the other hand, the kj,t−=kj,t −1incumbents in charge 7are likely to resist the adoption of the new technology 7Since specialization lowers the mark-up (see Assumption 4 (ib)), previous incumbents will not freely adopt the new technology. 16 implied by specialization that would, in fact, decrease their share of the industry total profits. The first threat may be averted by limiting8the mark-up to cj,t since the gap between the two technologies is still eλ. As to the second one, the new technology with kj,t intermediate inputs will be adopted only if this new mark-up cj,t is larger or equal to the bottom-line one, denote it by c0 j,t, that the previous kj,t −1incumbents can charge without allowing their followers to enter the market (again, see Assumption 4 (ic)). Lemma 2 (24) holds if and only if eλ−1≥ 1 akj,t j,0 η bj,0+ kj,t− P k0=1 1 ak0 j,0 (25) Moreover, if (25) holds, then c0 j,t ≤cj,t (26) Proof. In the Appendix. This lemma simply states that the increase in productivity λmust be sufficiently high as to more than offset the increase in real direct and indirect labour costs implied by the lengthening of the specialization string. If this is the case, then the productivity of the new technology is sufficiently high to force the old incumbents to fine tune their own technology, allow a specializing spin-off to introduce a new intermediate input and share9their monopoly position with this new producer. Remark 2 We explicitly remark that the right hand side of inequality (25) is a decreasing function of kj,t. Thus, if kj,t is above a certain threshold, then (25) holds and this is all the more true at every successive specialization event. If this is not the case, then specialization may not be viable. 5 Innovation, growth and structural change In this section we specify the innovation processes and determine the economy’s stationary state as well as its expected growth rate. Furthermore, we characterize a traverse process as demand undergoes a structural shift. Three different innovation events are actually dealt with, two of them being concomitant. The first is a vertical innovation,a firm-specific occurrence, that results from 8See Lemma 1 and (21). 9We stress again that previous incumbents will not freely adopt the new technology (see (48) in the proof of Lemma 2 in the appendix). 17 followers’ researching efforts and that comes to pass with a Poisson arrival rate. Since this rate is increasing with the number of employees (h) hired to carry out this process, it is expedient to normalize it as hsuch that the probability of an innovation event in a period dt of time is hdt. These efforts in the same time period imply a cost that for simplicity’s sake is rendered by C(h) = a 2h2+F 2. The second and third are a specialization spin-off and a rationalization process occurring in consequence of a demand shift that is assumed to happen according to an arrival rate µd: the former where the demand share becomes larger, the latter where it contracts. As discussed in the Section 2, demand shifts depend on relative price changes.Reference is made to a consumption pattern which arranges goods according to their priority in terms of the quality of life that they can afford and thus in accordance to income effective purchasing power. The price decline is made possible by technology-driven productivity increases being the result of firms’ innovative efforts. In this context, it is expedient to assume that the Poisson arrival rate µddepends on the research efforts of firms engaged in high priority good production 10. The value of an innovation to those who attempt it depends on the profit flow and, crucially, on the occurrence of these events according to whether they happen to be in expanding or contracting industries. To an innovator in an expanding industry, the value of an innovation depends on the flow of profits and on the likelihood of being ousted by the next generation of vertical innovators who will be employing hf,t workers for this purpose defining an arrival rate of equal magnitude. To one in an industry that stands a chance of contracting, the value, besides on the profit flow, depends on the likewise probability of being ousted and on the probability of being involved in a rationalization process that makes his sector redundant. Let Vj,t designate the value of an innovation in the j-th industry. If equilibrium prevails, innovators in industry jwill reckon that their expected flow of profits based on the likely value of their innovation is, given a discount rate r, rVj,t =πj,tw−hf,tVj,t −µdVj,t max {∆kj,t,0} kj,t (27) where ∆kj,t =kj,t −k0 j,t is the difference between the number kj,t of intermediate producers in industry jat time tand their expected number k0 j,t after the demand shock. Thus, max{∆kj,t,0} kj,t indicates the probability of an intermediate producer in that industry of being caught in the rationalization process, given the probability of demand shock µddt : this probability is positive if the demand shock is negative and zero otherwise. Solving (27): 10A more precise formulation of this assumption will be given below (see Assumption 7). 18 Vj,t =πj,t r+hf,t +µd max{∆kj,t,0} kj,t (28) Maximization of the net expected value of a vertical innovation for a follower in the intermediate sector of industry jreads11 Wj,t = max hhVj,t −a 2h2−F 2for j= 1..., J ; (29) here we assume that firms do not internalize the effect of their own research efforts on the arrival rate µd. The Jfirst order conditions resulting from the maximization problems (29) yield the followers’ innovative efforts hj,t as functions of the number of monopolists in each industry and of the expected efforts of future and present competitors. Since we are assuming that there is no entry barrier, new spin-offs occur as long as the net expected value of a vertical innovation remains positive; this means that, after a demand shock, the number of monopolists kj,t in industry jwill increase [decrease] if the share βj,t increases [decreases], as long as Wj,t >0. Since in equilibrium expected values equal current ones, the free entry condition determines the size of kj,t, for each jand t. In the following subsections we are going to characterize the stationary state levels of both employment and output as well as stationary state growth rates. In subsection 5.1 the structure of demand shares is assumed as given and coinciding with a level of per capita income denoting a corresponding development level. In subsection 5.2 shares evolve due to price changes and the stationary states thereof implied are considered. Furthermore, a traverse from a stationary state to another is investigated as a consequence of a demand share shift. For the following we introduce the simplifying assumption that r= 0 and consider kj,t to be real valued. 5.1 Stationary state We first characterize the static product and labor market equilibrium equilibrium. Given aggregate demand for good jat time t(6), equilibrium requires that at any point of time t, real demand and supply match: ys j,t =yd j,t =βj,t Yt pj,t . (30) 11We recall that, thanks to Assumption 5 of no arbitrage, the expected value depends only on the industry jand not on sector k. Therefore all followers in the same industry jwill choose the same optimal effort hj,t. 19 From this the supply of intermediate good kand the employment in the final sector of industry j follow at once: xk j,t =βj,t 1 ak j,t Yt pj,t , (31) ly j,t =βj,t 1 bj,t Yt pj,t . (32) The k-th sector realized profits (23) in this industry are, in consequence, a mere proportion of aggregate output: πj,t =1−e−λ1 kj,t βj,tYt, (33) showing that the sector-wise flow of profits, given the productivity rate of increase and the demand share, depends only on the size of aggregate output and the extent of specialization. As mentioned above, the total industry profit fund Πj,t =πj,tkj,t =1−e−λβj,tYtis a function of share βj,t and aggregate output; a fact that indicates that deepening specialization implies dividing up in smaller slices the same volume of profits, for a constant productivity rate of increase. As discussed below in greater detail, Πj,t has the further implication that, ceteris paribus, profits are higher where the demand share is higher. As it is to be expected, incentives clearly lie where demand for final goods is relatively higher and it pays to further specialize in that process of final production the demand for which is proportionately higher than the relatively longer string length. We assume that the available labour force does not change over time. Assumption 6 The size of overall employment Lis kept constant L=Ly t+Lx t+Ht. Here Ly t= J P j=1 ly j,t is the total final-good employment while Lx t=PJ j=1 Pkj,t k=1 lk j,t is the total employment of intermediate goods producing sectors across all industries, finally Ht= J P j=1 hk j,t is the employment of manpower that followers use to conjure up the next round of innovations where hk j,t measures the innovative effort of firm kin industry jat time t. Lemma 3 Profits in each intermediate sector of any industry jare proportional to the final goods 20 industries’ total employment: πj,t = βj,t kj,t J P j0=1 βj0,t 1+δj0,tkj0,t eλ−1wtLy t(34) Proof. In the appendix. We next define the stationary state. Definition 1 A stationary state is the state in which demand shares do not and are not expected to change over time. The following proposition characterizes the economy’s stationary state. Proposition 1 In the stationary state the number of intermediate goods sectors in industry jamounts to: kj=βj1−e−λY F,(35) from which the overall total number of sectors: ¯ k= J X j=1 kj=1−e−λY F(36) and aggregate output is Y=L 1 weλ+q1 F a (1 −e−λ) .(37) Proof. In the Appendix. Note that Yremains constant over time and that it is an increasing function of L. Furthermore kj=βjf(L)while ¯ k=f(L)remains constant over time, f(L)increasing in Land decreasing in F. Remark 3 Inserting (33) into (37) we find a simple expression for the aggregate nominal profit Πt= PJ j=1 Πj,t Πt=L 1 w(eλ−1) +q1 F a Πtis also constant in time. It is interesting to note that, in the stationary state, while the extent of specialization in each sector depends on the extant demand share and hence on the current stage of development, aggregate profits 21 depend only on the extent of the market as measured by total employment. Having characterized stationary state levels, the ensuing subsection will address the issue of the economy’s long-run growth rate highlighting some of its properties concerning the role of specialization and the size of the market. 5.2 The stationary state growth rate Although the view we hold of this economy is one in which a development process continuously reshapes the pattern of final demand, we nevertheless assess these long-run properties in the stationary state, namely when demand shares remain constant and no drive to both specialization and rationalization occurs. Thus, the focus will lie on a comparison of growth rates for different but constant demand shares. Because of the assumptions that have been made (see Assumption 3 and 6 and (48)), the economy growth is essentially due to productivity growth which is, in turn, explained by the innovations that followers conjure up in their strive to oust reigning monopolists, a feat that is achieved thanks to investment in research and development. Proposition 2 The stationary state growth rate is gYR=rF a J X j=1 βjeλβjf(L)−1.(38) Proof. In the Appendix. This result lends itself to some interesting interpretations. The first observation is that gYRdepends on total employment L, effectively a proxy of the extent of the market. In an economy in which it is held constant, the sum of the intermediate sectors, k, also remains constant, specialization having gone further where demand shares had increased and restructuring eased sectorial costs where they had declined. Thus, since the growth rate in each point of a stationary state sequence depends on a specific distribution of demand shares, a stage of development as denoted by the corresponding demand shares is accordingly identified. Furthermore, it follows that a larger Limplies a greater number of intermediate sectors hence a higher aggregate growth rate: a larger extent of the market in a Smithian sense as measured by Ldeepens overall specialization and enhances the economy’s long-run growth rate. The second observation is, as noted above and given L, that the growth rate depends on the distribution of the intermediate sectors amongst the various industries. It is indeed straightforward to see that the aggregate growth factor is an average of the various industries’ own factor weighted 22 by aggregate demand shares, the latter having been shaped by the very development process. As this process unfolds assigning greater weights to goods that are less essential with real income growth, the impact of innovations owing to monopolists’ followers’ research and development efforts is spread more evenly over a larger number of industries. The implication is that the overall growth factor is lessened on account of a more balanced distribution of kover the entire number Jof industries. Notice that the comparatively lowest growth factor occurs when βj=β=1 J.This point has an interesting implication. If the development process gets under way, overall specialization being furthered and the size of the market increased, the growth rate of the less developed economies rises whilst that of the more developed ones slows down generating a process of convergence. A third observation follows immediately from the previous two. An economy that manages to concentrate its aggregate output on fewer industries, other things being equal, achieves higher aggregate growth for the simple reason that specialization is also more concentrated: the same ¯ kdistributed on fewer j0s. This configuration may, for instance, occur in economies that, in spite of possessing a high real income per capita, through foreign trade have specialized in the production of a relatively small number of goods that it exports while importing many more allowing comparative advantage and higher growth. It is, furthermore, a well established result of the relevant theory that international trade by increasing the demand for the goods subject to relative specialization enlarges the size of the market, L, leading to an yet higher long-run growth rate. It must, however, be stressed that graduating from a stage of development to the next depends crucially on the research and development process that finally yields innovations, productivity growth and ultimately the increase of income per head that reshapes the distribution of demand shares. It is on this logical sequence of events that the development process hinges upon, the sooner the innovation-led virtuous circle of productivity growth is ignited, the faster will growth be and the more effective the catching-up path. What remains to be seen are the implications of passing from a demand pattern to another as a consequence of the above stated process. This is the topic of the following subsection. 5.3 Traverse dynamics We analyze the traverse dynamics between two different demand patterns in two ways. Firstly, by introducing some simplifying assumptions we describe the main properties in an analytical way. Secondly, we confirm the main result through numerical simulations. The level of generality that has so far been found expedient to illustrate the model does not 23 [5] David, P., 1976: Technical Choice, Innovation and Economic Growth. London, Cambridge University Press. [6] Foellmi, R. and J. Zweimüller, 2006. Income Distribution and Demand-Induced Innovations. Review of Economic Studies, 73, 941-960. [7] Foellmi, R. and J. Zweimüller, 2008. Structural change, Engel’s consumption cycles and Kaldor’s facts of economic growth. Journal of Monetary Economics, 55, 1317 - 1328. [8] Kongsamut, P., S. Rebelo and D. Xie, 2001. Beyond Balanced Growth. Review of Economics Studies, 68, 869 - 882. [9] Laitner, J., 2000. Structural Change and Economic Growth. Review of Economic Studies, 67, 545 - 561. [10] Matsuyama, K., 2002. The Rise of Mass Consumption Societies. Journal of Political Economy, 110, 1035 - 1070. [11] Matsuyama, K., 2008. Structural Change. In: Durlauf, S.N. and L.E. Blume (Eds.), The New Palgrave Dictionary of Economics, 2nd edition. Palgrave McMillan. [12] Mokyr J.,1990: The Lever of Riches. New York, Oxford University Press. [13] Ngai, L.R. and C.A. Pissarides, 2007. Structural Change in a Multisector Model of Growth. American Economic Review, 97, 429 - 443. [14] Romer, P., 1987. Growth Based on Increasing Returns Due to Specialization. American Economic Review, 77, 56 - 63. [15] Rosenberg, N.,1990. Inside the Black Box. Cambridge, Cambridge University Press. Appendix Proof of Lemma 1. Since the new technique is more productive than the preceding one, the innovator can push the price of the final good down to a level p0 j,t, driving the mark-up c0 j,t of the previous incumbents to zero, thus ousting them from the market. By (10) and (13) with c0 j,t = 0 we 30 obtain p0 j,t =w bj,te−λ 1 + kj,t X k=1 bj,0 ηak j,0  pj,t =w bj,t  1 + kj,t X k=1 1 + cj,tak j,tbj,0 ηak j,0  The equation p0 j,t =pj,t together with (15) gives (21). Proof of Lemma 2. First we show that necessary and sufficient condition for (26) is eλ−1≥ ω η bj,0 akj,t j,0 1 + ωδj,tkj,t . (46) To see this, note that the maximum mark-up, which prevents followers to enter the market with an outdated but specialized technology, can easily be calculated by equating the latter unitary cost of production to the cost of production plus the mark-up that incumbents can charge by still using the shorter, pre-specialization one 13   1 bj,t− +ω kj,t X k=1 1 ηak j,t− wt=  1 bj,t− +ω kj,t−1 X k=1 1 ηak j,t−1 + c0 j,tak j,0 wt, (47) whence c0 j,t =1 akj,t j,0kj,t .Substituting this (21) and (15) in (26), we obtain (46). It is clear that if (25) holds, then also (46) does. We next show that under Assumption 2, condition cj,t−≥cj,t (48) holds. Note that (48) is equivalent to 1 kj,t−≥ω η bj,0 akj,t j,0−1 k kj,t− P k0=1 bj,0 ak0 j,0!; this may be written as 1 akj,t−+1 j,0≤ 1 kj,t− η ωbj,0+ kj,t− P k0=1 1 ak0 j,0!, which in turn holds iff bj,0 ηak j,0≤δj,t−for k= 1,2, ... and j= 1, ..., J (see 17), whence (18). Proof of Lemma 3. Employment in both final and intermediate sectors can be characterized in 13We stress explicitly that in (47) the productivity bj,t−is the same for the incumbents’ old technology and for that available to followers. This holds since we have assumed that specialization may also be exploited by followers and the productivity gain is the same in the case of pure innovations and of specialization Assumption (4(ic)), (14) and (16)). 31 terms of aggregate output. From (22), (31) , (32) workers engaged in producing a final good jnumber: ly j,t =βj,t Yt weλ(1 + δj,tkj,t)(49) Bearing in mind (49), Ly tis seen to be a function, other things being equal, of the extent of specialization: Ly t=Yt weλv J X j=1 βj,t 1 + kj,tδj,t (50) From (50) Yt=eλ J P j=1 βj,t 1+δj,tkj,t wLy t(51) The expression within the sum in (50) is the value of effective demand reaching industry jin terms of its real labour cost; as such it is a measure of demand in real terms. It follows that its sum is an index of real aggregate demand that divided by the productivity augmented wage rate yields the volume of employment it can afford. Given (49) and (51) the industry jlabour force can simply be viewed as a proportion of total employment where the proportional factor is the ratio of an index of real effective demand of industry jto the index of real aggregate demand: ly j,t = βj,t 1+δj,tkj,t J P j0=1 βj0,t 1+δj0,tkj0,t Ly t(52) Notice, finally, that (34) follows from (23), (52) and (11), (9). Proof of Proposition 1. Given the stationary state definition, kj,t remains invariant, hence (28) becomes Vj,t =πj,t h; consequently the first order conditions14 for problem (29) yield hk j,t =hj,t =Vj,t a, πj,t =F, (53) and hj,t =rF a. While from (11): 14Since πj,t is decreasing in kj,t and increasing in ki,t for i6=j, see (34), the solution is unique. 32 Lx t=Yt weλv J X j=1 kj,tδj,t 1 + kj,tδj,t βj,t(54) Given Assumption 6 , (49) and (54) it follows that sectorial employment in each industry is lx j,t = kj,t X k=1 lk j,t =βj,t kj,t X k=1 bj,t ηak j,t Yt weλ(1 + δj,tkj,t)=βj,tYt δj,tkj,t weλ(1 + δj,tkj,t) and total manufacturing employment : Lt= J X j=1 ly j,t +lx j,t=Yt weλ. (55) From (33) and the free entry condition in the stationary state (35) and (36) are obtained. The third of (53) and (36) allow us to calculate the employment Htthat followers hire to conjure up the next round of innovations. Considering (50) and (54) , Htis, then, equal to L− Yt weλv=Ht(56) Finally from (56), (36) and the third of (53), L=Y(1 weλ+r1 Fa 1−e−λ). (57) aggregate final good output (37) follows. Proof of Proposition 2. The accounting definition of final goods aggregate growth, given (33), demand and supply being in equilibrium15, is: · Yt Yt = J X j=1 βj,t · pj,t pj,t +· yj,t yj,t !. In this model, however, the nominal wage rate is kept constant and productivity gains translate into proportionally lower prices such that the real wage rate and likewise real monopolists’ profits grow in step with productivity. Because of the stationary state assumption in which the economy is taken to be in a notional state in which neither demand shares nor specialization strings vary, we can account 15By (30) this means that pj,tyj,t Yt =βj,t. 33 for real growth by assuming them to be βjand kjrespectively, both remaining invariant at a base point in time. In this case, long-run real growth YR,t turns out to be: · YR,t YR,t = J X j=1 βj · yj,t yj,t , from which we can characterize the expected long-run growth rate, recalling that the Poisson arrival rate is hj,t =qF a, as: gYR=E  · YR YR =rF a J X j=1 βjekjλ−1, since kj=βjf(L), the expression in (38) follows. Proof of Proposition 3. Thanks to the role of expectations in our model, the proofs are obtained beginning from the final stage τ=Tand then proving backward recursion formulae for kj,τ and Lτ. In the final state ∆k1,τ =k1,τ −k0 1,τ = 0 because no further demand shock is expected. From (53) (15) and (28), (34) hI j,τ =Vj,τ a=πj,τ ahI j,τ = βj,τ kj,τ eλ−1wLτ ahI j,τ . (58) Substituting from (58) and (53) the total employment Lbecomes a function of Lτ L=Lτ+k1hI 1,τ +k1hI 1,τ = 1 + eλ−1w √aF !Lτ. (59) Thus the labour force employed in manufacturing diminishes when the productivity gain λof an innovation increases or the wage rate wincreases, whereas it diminishes when either the fixed cost of research For the variable one a, or both, increase Lτ=L 1 + (eλ−1)w √aF . (60) The reverse is true for employment hired to search for innovations HI τ=k1hI 1,τ +k2hI 2,τ = (eλ−1)w √aF L 1 + (eλ−1)w √aF . (61) Consider the next to the last state, where ∆k1,τ =k1,τ −k0 1,τ >0. From the first of (53), (28), 34 (15) and (34) with r= 0 and j= 2 hI 2,τ =V2,τ a=πk 2,τ ahI 2,τ = β2,τ k2,τ eλ−1wLτ ahI 2,τ ; (62) again from (53), (28), (15) and (34) with r= 0 and j= 1 hI 1,τ =V1,τ a=πk 1,τ ahI 1,τ = β1,τ k1,τ eλ−1wLτ a1 + µd ∆k1,τ k1,τ hI 1,τ . (63) We conclude with total employment; from the last of (53)gain w L=Lτ+k1hI 1,τ +k2hI 2,τ =Lτ+ (k1,τ +k2,τ )rF a; (64) substituting from the last of (53), (44), (45), rearranging terms and solving for Lτ Lτ=L−qF a µd 1+µdk0 1,τ 1 + (eλ−1)w √F a 1−µd (1+µd)β1,τ . (65) Consider next the necessary and sufficient condition for the monotonicity of the sequence {k1,τ }T t=1. Since we are assuming that, at every period τ, agents correctly forecast the future value of k1, putting k0 1,τ =k1,τ+1, (44), (45), (65) become k2,τ =β2,tra FΦLτ(66) k1,τ =β1,τ ra FΦ1 1 + µd Lτ+µd 1 + µd k1,τ+1 (67) Lτ=L−qF a µd 1+µdk1,τ+1 1+Φ1−µd 1+µdβ1,τ (68) where Φ = eλ−1 √aF w. Now, k1,τ > k1,τ+1 if k1,τ+1 < β1,τ ra FΦ1 1 + µd Lτ+µd 1 + µd k1,τ+1 , (69) 35 substituting (68) into (69) we obtain k1,τ+1 < β1,τ ra FΦ1 1 + µd L−qF a µd 1+µdk1,τ+1 1+Φ1−µd 1+µdβ1,τ +µd 1 + µd k1,τ+1 . (70) Elementary simplifications give the following result. The necessary and sufficient condition for the sequence {k1,τ }T t=1 to be decreasing is k1,τ+1 <LΦ (1 + Φ)β1,τ ra F. (71) Consider the recurrence formula for the sequence {k1,τ }T τ=1. Substituting (68) into (67) we obtain k1,τ =β1,τ ra FΦ1 1 + µd L−qF a µd 1+µdk1,τ+1 1+Φ1−µd 1+µdβ1,τ +µd 1 + µd k1,τ+1 , rearranging terms k1,τ =Lβ1,τ pa FΦ1 1+µd 1+Φ1−µd 1+µdβ1,τ +µd 1 + µd k1,τ+1 1+Φ−Φβ1,τ 1+Φ1−µd 1+µdβ1,τ . (72) Using the recurrence formula (72), it is possible to prove that, if (71) holds, then it holds also for k1,τ and β1,τ−1; k1,τ <Lβ1,τ pa FΦ1 1+µd 1+Φ1−µd 1+µdβ1,τ +(73) µd 1 + µdLΦ (1 + Φ)β1,τ ra F 1+Φ−Φβ1,τ 1+Φ1−µd 1+µdβ1,τ ; eliminating the common factor LΦ 1+Φ β1,τ pa F, rearranging terms and remembering that β1,τ is a decreasing function, we obtain k1,τ <LΦ 1+Φβ1,τ−1ra F. (74) We are ready to prove part (a) of the Proposition. Since (71) implies (74) for every τ, it is sufficient 36 to prove that (71) holds for τ=T−1. Since k1,T =k1,T +1 (67) yields k1,T =β1,τ ra FΦLT(75) which, together with (68), gives k1,T =LΦ 1+Φβ1,T ra F; (76) finally, since β1,T > β1,T −1 k1,T <LΦ 1+Φβ1,T −1ra F. Remark 4 From (75) and (76) we immediately get LT=L1 1+Φ (77) and k1,T =LΦ 1+Φβ2,T ra F.(78) Substituting (76) into (68) it is easy to calculate LT−1 LT−1=L1 1+Φ 1+Φ1−µd 1+µdβ1,T  1+Φ1−µd 1+µdβ1,T −1;(79) clearly LT−1> LT. Consider next part (b) of the Proposition. From (66) and (67) we get k2,τ =β2,τ β1,τ (1 + µd)k1,τ +β2,τ β1,τ µdk1,τ+1 . (80) Inserting the recurrence formula (72) into (80) and simplifying we obtain k2,τ =β2,τ Φ 1+Φ1−µd 1+µd1−β2,τ Lra F−µd 1 + µd k1,τ+1. (81) Since τ→Lpa F−µd 1+µdk1,τ+1is positive and increasing by Proposition 3 and τ→β2,τ 1+Φ“1−µd 1+µd(1−β2,τ )” is positive and increasing as well, the same holds for τ→k2,τ . 37 Part (c) of the Proposition. First we show that the following recurrence formula holds for Lτ Lτ=1 + β2,τ+1ΦLτ+1 +L 1+µd µd(1 + Φ) −β1,τ Φ. (82) From (64) k2,τ +k1,τ =ra F(L−Lτ). (83) From (81) we get µd 1 + µd k1,τ+1 =Lra F−k2,τ β2,τ Φ 1+Φ“1−µd 1+µd(1−β2,τ )” , (84) substituting k2,τ from (66) in (84) and adding to both sides the term µd 1+µdk2,τ+1 we obtain µd 1 + µd k2,τ+1 +µd 1 + µd k1,τ+1 =Lra F−(85) 1+Φ1−µd 1 + µd1−β2,τ ra FLτ+µd 1 + µd k2,τ+1 substituting k2,τ+1 again from (66) and k1,τ +1 +k2,τ+1 from (83) and arranging terms we obtain the recurrence equation Lτ+1 =1+µd µd(1 + Φ) −β1,τ ΦLτ−1 µdL 1 + β2,τ+1Φ. (86) Inverting (86) we obtain the backward recurrence equation (82). Next we show that the necessary and sufficient condition for Lτ+1 < Lτis Lτ+1 <L 1+Φ−µdβ2,τ+1 −β2,τ . (87) From (82) Lτ+1 < Lτmeans Lτ+1 <1 + β2,τ+1ΦLτ+1 +L 1+µd µd(1 + Φ) −β1,τ Φ. (88) Simplifying and rearranging terms we obtain (87). By (82) and (77) Lτ> LTbecomes 1 + β2,τ+1ΦLτ+1 +L 1+µd µd(1 + Φ) −β1,τ Φ>L 1+Φ . (89) 38 Solving for Lτ+1 (89) gives Lτ+1 >L 1+Φ 1 + β2,τ Φ 1 + β2,τ+1Φ(90) which implies Lτ> LTsince the sequence β2,τ is increasing and LT−1> LTby (79). Figure 1 0. 2 0. 3 0. 4 0. 5 E 2 5 10 15 k 1 and k 2 Equilibrium Values for k 1 and k 2 as Functions of E 2 1 2 3 4 5 W 0.20 0.21 0.22 0.23 0.2 4 0.2 5 0.26 J Real GDP Growth Rate 0 .2 0.30.40.5 E 2 9.99981  10 6 9.99981  10 6 9.99982  10 6 9.99982  10 6 L t x  L t y Equilibrium Values for L t y  L t x as a Function of E 2 Figure 2 0. 2 0. 3 0. 4 0. 5 E 2 5 10 15 k 1 and k 2 Equilibrium Values for k 1 and k 2 as Functions of E 2 1 2 3 4 5 W 0.20 0.21 0.22 0.23 0.2 4 0.2 5 0.26 J Real GDP Growth Rate 0 .2 0.30.40.5 E 2 9.99981  10 6 9.99981  10 6 9.99982  10 6 9.99982  10 6 L t x  L t y Equilibrium Values for L t y  L t x as a Function of E 2 39