This paper develops a model based on Schumpeter's process of creative destruction. It departs from existing models of endogeneous growth in emphasizing obsolescence of old technologies induced by the accumulation of knowledge and the resulting process or industrial innovations. This has both positive and normative implications for growth. In positive terms, the prospect of a high level of research in the future can deter research today by threatening the fruits of that research with rapid obsolescence. In normative terms, obsolescence creates a negative externality from innovations, and hence a tendency for laissez-faire economies to generate too many innovations, i.e too much growth. This "business-stealing" effect is partly compensated by the fact that innovations tend to be too small under laissez-faire. The model possesses a unique balanced growth equilibrium in which the log of GNP follows a random walk with drift. The size of the drift is the average growth rate of the economy and it is endogeneous to the model ; in particular it depends on the size and likilihood of innovations resulting from research and also on the degree of market power available to an innovator.  This paper presents a model of endogenous stochastic growth, based on Schuxnpecers idea of creative desiruction. We begin from the belief, supported by many empirical studies starting with Solow (1957) , that a large proportion of economic growth in developed counthes is atthbutable to improvement in technology rather than the accumulation of capital. The paper models technological progress as occurring in the form of innovations, which in turn result from the activities of research firms. We depart from existing models of endogenous growth (Rorner, 1986(Rorner, , 1988 and Lucas, 1988) in two fundamental respects. First, we emphasize the fact that technological progress creates losses as well as gains, by rendering obsolete old sidils, goods, markets, and manufacturing processes. This has both positive and normative implications for growth. In positive terms, the prospect of a high level of research in the future can deter research today by threatening the fruits of that research with rapid obsolescence. In normative terms, obsolescence creates a negative externality from innovations, and hence a tendency for laissez-faire to generate too much growth. Obsolescence does not fit well into existing models of endogenous growth. Those models have only positive externalities, in the form of technology spillovers, and thus tend to generate too little growth. The closest in spirit to our model is that of Romer (1988) allowing old goods to be displaced by the inuoduction of new goods, may eliminate growth in this nd of model (see Deneckere and Judd, 19861). Our second departure is that we view the growth process as discontinuous. Existing models of endogenous growth do not produce a random end, unless exogenous technology shocks are added to the model, as is done by King and Rebelo (1988) . We take the view that technology shocks should not be regarded as exogenous in an analysis that seeks to explain the economic decisions underlying the accumulation of knowledge. Instead, we suppose that they are entirely the result of such decisions. Statistical innovations in GNP are produced by economic innovations, the distribution of which is determined by the equilibrium amount of research. This makes the disthbution of technology shocks endogenous to the model.3 In maldng both of these departures from recent models of endogenous growth we take our inspiration from Schumpeter (1942, p.83, his emphasis): The fundamental impulse that sets and keeps the capitalist engine in motion comes from the new consumers' goods, the new methods of production or nansportarion, the new marlcets,...(This processj incessantly revolutionizes the economic smicture within, incessantly destroying the old one, incessantly creating a new one. This process of Creative Destruction is the essential fact about capitalism. In Schumpeter's view, capitalist growth is inherently uncertain. Fundamental breakthroughs are the essence of the process, and they affect the entire economy. However, the uncertainty is endogenous to the system, because the probability of a breakthrough depends upon the level of research, which in turn depends upon the the monopoly rents that constitute "the prizes offered by capitalist society to the successful innovator." (1942, p.102) Thus, "economic life itself changes its own data, by fits and starts." (1934, p.62). The present paper develops a simple model that articulates these basic Schumpeterian elements. Although it is quite special in some dimensions we believe it is also simple and flexible enough to serve as a prototype model of growth through creative destruction.4 2. Tnere are four classes of uadeable objects: land, labor, a consumption good, and a continuum of intermediate goods i e fO,1]. There is also a continuum of identical infinitely-lived individuals, with mass N, each endowed with a one-unit flow of labor, and each with identical intertemporally additive preferences defined over lifetime consumption, with the constant rate of time preference r> 0. Except in section 5 below, we assume a constant marginal utility of consumption ar each date; thus r is also the unique rate of interest in the economy. There is no disutility from supplying labor. There is also a fixed supply H of land. The consumption good is produced using land and the intermediate goods, subject to constant returns. Since H is fixed we can express the production function as: The innovator offers each patent for sale at a price equal to the expected present value of the monopoly rents accruing to the patent. The buyer pays that price as a lump sum in exchange for unconsuained use of the patent. Thus we implicitly rule out royalties and other conungent connacts between the innovator and the local monopolist. The force of this resniction will be discussed in (d) below where it is argued that without the resthction the two parties would want a connact with negative royalties. The resuiction might therefore be rationalized either by noting that negative royalties are not in fact observed, or by referring to costs of monitoring the output of the intermediate good. No patent covers the use of a new intermediate good in the consumption-good sector. The idea behind this assumption is that potential uses of the new good are too obvious, even before its invention, to be patented. The assumption might also be rationalized by the observation that important innovations like the ones mentioned above have had widespread uses too numerous and diffuse for anyone to monopolize them. The local monopolist's objective is to maximize the expected present value of the flow of profits over the cutrent intervaL When the interval ends so do the rents. The only uncertainty in the decision problem arises because the length of the interval is random. Because no one operates in a positive measure of sectors the monopolist takes as given the amount of research at each time, and hence also takes as given the length of the interval. Because of this and because the cost of the patent is sunk, the monopolist's sategy will be simply to maximize the flow of operating profits it at each instant. In equilibrium all prices and quantities are constant throughout the interval. Also, by symme'y, the same quantity xt will be chosen by each monopolist. This quantity will also be the total output of intermediate goods in period t, which by (2.2) equals the total employment of labor in manufacmring i.e.: We can then re-express (2.1) as: The inverse demand curve facing a monopolist charging the price Pt is the marginal product schedule: Thus the decision problem is to choose x so as to maximize fF'(x)Ict -w]x, tang c and w as given. The necessary first-order condition is Assume that the monopolist's marginal-revenue schedule is downward-sloping: This condition holds automatically when F" < 0; we show in Appendix I that it also holds when F comes from a CES production function. It follows from (2.5) and A.1 that there is a unique solution to the decision problem: where is strictly decreasing. Thus the demand for labor in the intermediate industry is a decreasing function of the cost (in terms of the consumption good) of producing one efficiency unit of an intermediate good. Likewise we can express the monopolist's price and flow of profits as: where is sthctly increasing and is sthctly decreasing. For future use, we also assume: Assumption A.2: (0) = , (o.) = 0. We have not allowed the local monopolist's decision problem to be consttuined by potential competition from the holder of the previous patent. This is because if the consaint were potentially binding; i.e. if the innovation were non-drastic in the usual sense (see, for example Tirole, 1988, ch.1O) then the current patent would be of greatest value to the holder of the previous patent, who would face no such competition. Thus our model predicts that non-drastic innovations would be implemented by incumbent producers, whereas drastic innovations would generally be implemented by new firms. This is in accordance with several empirical studies to the effect that incumbent firms implement less fundamental innovations than do new ennants. (See, for example, Scherer, 1980.) Whether or not the innovation is drastic is determined by whether or not a competitive producer of the consumption good would incur a loss buying from the previous monopolists, at a price equal to the unit cost w. Thus the condition for a drastic innovation is C(p,w) > where CC) is the unit-cost function uniquely associated with the production function F, and p is the equilibrium price of land. The latter is detexmined by the conditions for competitive equilibrium in the markets for land and the consumption good. where Pt is the equilibrium market price of all other intermediate goods, given by (2.7). The special case of a Cobb-Douglas production is defined by: It is straightforward to verify that in this case: There are no contemporaneous spiliovers in the research sector, that is, a research firm employing z will experience innovations with a Poisson arrival rate z, and these arrivals will be independent of other firms research employment i = -z.9 Let W be the value of a research firm after the 1th innovation, and let Vt be the value of patents from the 1th innovation. If the firm makes an innovation it will receive V÷1. The density of this event at the current instant is z. Thus the expected payoff per unit time from successful innovation is zV11. The firm will realize acapital gainon W+i_W1 when any flrmmakesan innovation. Thus its expected rate of capital gain is Xn 't+1 -W1). Its flow of labor cost is wZ. It takes w and as given. Thus we have the Bellman equation:10 (2.14) Assume free enny into research. Then W =0, so (2.14) can be expressed as: The Kuhn-Tucker condition is: (2.16) w z 0, with at least one equality (waloe). The value V1 is the expected present value of the constant flow over an interval whose length is exponentially disthbuted with parameter Xn1: Note that there is an important intertemporal spillover in this model. An innovation reduces costs forever. It allows each subsequent innovation to reduce the unit-cost parameter c by the same fraction 7, and with the same probability but from a starting value c_1 that will be lower by the fraction '(than t would otherwise have been. The producer of an innovation will capture (some of) the rents from that cost reduction, but only during the next interval. After that the rents will be captured by other innovators, building upon the basis of the present innovation, but without compensating the present innovator.11 This interremporal spillover will play a role in the welfare analysis of section 4 below. x It is clear now why a successful innovator would want to offer a conttact with negative royalties. Such a connact would induce each local monopolist to produce more than This would have two effects on V, as given by (2.17). First, it would reduce the numerator below the maximal value given by (2.8). Second, it would also reduce the denominator by reducing n, through (2.19). In the neighborhood of the zero-royalty solution analyzed above the former effect would be a second-order small by the envelope theorem, but the latter would not. Thus the seller of the th line of patents would want negative royalties so as to discourage creative desmiction duiing interval r. The local monopolists would not attempt to discourage creative desttuction on their own by producing more than (cw) in the absence of negative royalties, because each one is too small to affect n. Using (2.6) and the fact that is an invertible function, we can write w = Multiplying both sides of (2.20) by c and using the identity: c1 = we obtain: which we can re-express by the following first-order difference equation: where: We define a perfect foresight equilibrium (PFE) as a sequence (xe) in (0, N] satisfying (3.2) for all t 0. In PFE everyone can predict the future evolution of the endogenous variables (we, x, y ire, n, V) with certainty.'4 However, the length of each interval (in real time ic), as well as the identity of each innovator, is random. The characterization of PFE is simplified by the following: Lemma 1: The mapping x i-G(x) is decreasing in x. Lemma 1 follows from the fact that both and are decreasing functions. The economic interpretation is as follows: A foreseen increase in x1 will raise the reward . to the next innovator on the one hand it raises the flow of monopoly rents and on the other hand it reduces the amount of research and hence the amount of creative desuuction next interval (the effect in the denominator). The increase in V1 will raise the equilibrium wage w this period (if x <N), which in turn will induce the intermediate monopolists to reduce their demand for labor this period, x. 12 Because (3.2) is forward-looking, history does not determine a unique value of x0 in PFE. Typically, there will be a continuum of PFE indexed by the initial value x0. However there will only be a finite number of periodic uajectories to which any PFE could converge asymptotically. More precisely: Proposition 1: All PFE are. or converge asymptotically to one of the fol1owin: -a stationary equilibrium (SE) with positive growth. -SE with zero growth. -a "real" 2-cycle, or -a "no-growth teap" (NGT'). Furthermore there always exists a unique SE. We define as a steady state = 5(i). In SE the economy experiences balanced growth in the sense that the allocation of employment between manufacturing () and research (fi=N-)remainsconstanL Gmwthispositiveifi<Nandzeroif=N,because there will be innovations if and only if labor is allocated to research. We define real 2-cycle as a pair (x0, x1) such than (3.3) x°= (x1), x1 = 5(x°), x1 s x° and N x°, x1). Then a real 2-cycle corresponds to PFE in which manufacturing employment oscillates between two different values with each succeeding innovation. High manufacturing employment in odd intervals raises the reward to research during even intervals, and hence depresses manufacturing employment during even intervals, through the mechanism discussed above. Likewise, low manufacturing employment in even intervals raises manufacturing employment in odd intervals. A "no-growth' n'ap is a pair (x°, x1) such that the first three conditions of (3.3) hold, but N E (x0, x1). As shown in Figure 2 below, NGT exists when (3(N) <G'(N) 3 X. Even though iNGT defines an infinitesequence (x}0, the oscillation of the economy will stop afterperiod 1. Fromthenon the economywiflpezformas ifin SE with zerogrowth. The interpieraflon of NGT is that the prospect of low manufacturing employment in even periods so depresses the incentive to research in odd periods that research stops. As we shall see, this can happen even in an economy that possesses a positive balanced-growth equilibrium. Proposition 1 rules out complicated periodic ajectories such as k-cycles with k 3, or chaotic PFE. It follows immediately from Lemma 1, which implies that x in odd periods follows a monotonic path in [0, N). Therefore the sequence converges. By continuity, the limit point is a fixed point of the second-iterate map 2, which corresponds to either SE or a In particular, for a fixed value of y, ?, r, N, a necessary and sufficient condition for positive growth is that the parameter a be sufficiently small: a < a* /( + 'er) < 1. In particular NGT will always exist when the interest xte r is saaicientiy small.16 Note that the smaller r (or rA), the easier it is to satisfy (3.5): in other words, "no-.growth tap" equilibria are more likely to exist in an economy that possesses a positive balanced-growth equilibrium! Then, from Proposition 2, we know that there exists a real 2-cycle whenever a is small and G(N) <xC (rA small). We can surrmiarize the conclusions obtained in the Cobb-Douglas case 17 by the following picture. Again can be interpreted as an inverse measure of the degree of monopoly power in the intermediate indusny. 2-cycle (either real or NGT). Since Y(x) is non-increasing in The reason for cycles in this model is sirnilr, but not identical, to the reason for cycles in the model of Shleifer (1986) .'9 In the Shicifer model innovations are exogenous, and there are multiple equilibrium sategies for implementing them, because of an aggregate-demand externality. That externality is present in this model in an intertemporal rather than intersectoral form. As (3.1) makes clear, higher manufacturing output next period raises the flow of monopoly rents next period. As we have seen, it is this effect, together with the effect of creative destruction, that makes x depend inversely upon xt+l, and which therefore makes high manufacturing employment in odd intervals together with low manufacturing employment in even intervals a possible equilibrium configuration. There are several questions that would need to be addressed before judging the likely empirical relevance of these 2-cycles. One is their observability under realistic expectational assumptions; equilibria that are stable under perfect foresight (as in Figure 4above ) are often unstable under learning (Grandmont and Laroque, 1986). Another is the size of the resulting output fluctuations; modern economies typically allocate no more than 2 or 3 percent of their what appear in the model as technology shocks depends upon the same economic factors that determine AGR In the extreme case where the incentive to do research is so small as to result in no growth, VGR will also be zero. We shall now proceed under the assumption that the economy is always in a situation of positive balanced growth. Thus:20 where is defined by: The following Lemma is an immediate consequence of Lemma 1 and the fact that and are decreasing: Lernma2: d=G-1<O; >O; J<O; >O; We then obtain: Proposition 3: The amount of research performed in a positive balanced growth equilibrium (a) decrease with the rate of interest r. (b) increase with the arrival parameter . (a) an increase in the interest rare r means an increase in the required rate of rennn on research, whose effect will be to reduce the total investment in R & D. (b) an increase in the arrival rate ). will have both the positive effect of raising the speed at which research pays (the effect on the numerator of 4.1), and the negative effect of increasing creative desmiction (effect on the denominator). The first effect dominates.21 (c) an increase in the size of each innovation (decrease in y) also increases research by increasing the size of next interval's monopoly rents relative to today's research costs. (d) an increase in total labor endowment N increases research by increasing, for a given fi, the size of the market that can be "monopolized" by a successful innovator. We now complete our cornparadve stacs analysis by studying the effect of the degree of monopoly power in the intermediate indusny. In the Cobb-Douglas case, we saw that a good (inverse) measure of monopoly power was the parameter a. In the general case we can proceed in a similar way and define: where 0 <a < 1. The elasticity r again depends positively on a.22 We can then show: This expression decreases with a. The lemma then follows from (4.1). n We then obtain the following intuitive result: Proposition 4: An increase in monopoly power (reduction in a) increases the amount of research in a balanced growth eguiiibrium We now derive an explicit expression for both the average growth rate (AGR) and the variability of the growth rate (VOR) in a balanced growth equilibrium. The volume of real output (i.e. the flow of consumption goods) during interval t is:  Thus the dme path of the log of real output âi y(r)-where r is real time-will be a random The positive effect of N on AGR has the unfortunate implication, which Romer (1988) has noted in a similar context, that larger economies should grow faster. In fact, (4.1') in foomote 20 above implies that, in the Cobb-Douglas case, a doubling of population should more than double the growth rate. (These comparisons must be intertemporal rather than international, since we have not dealt with the question of transboundary technology flows.) We accept the obvious implication that this class of models has little to say, without considerable modification, about the relationship between population size and growth rate. The variability of output in this model should be distinguished from its variability in existing real business cycle models, for the following reasons. First, the variability of shocks has been endogenized here. An innovation in the statistical sense in real output is the effect of an innovation in the economic sense, the distribution of which depends as we have seen upon economic decisions that respond to parameter changes. We now compare the laissez-faire AGR derived above with the AGR that would be chosen by a social planner whose objective was to maximize the expected present value of utility of consumption y(c). The maximized value of the planner's objective function, Z, when c is historically given, is determined by the following Bellman equation: The first-order condition an interior solution for this problem is: It is easily checked that the solution to (4.9) is: There are three differences between (4.12) and (4.14): The first difference is the presence of the term in the denominator of (4.12). This corresponds to the intertemporal spillover effect discussed in section 2(c). A private innovator will capture rent from his innovation for one interval only, whereas the social rent continues forever. Formally it comes from the presence of the term Z÷1 in the definition (4.9) of the social planner's value function Z, whereas no such term appears in the definition (2.17) of the private value of an innovation, V. This effect leads the laissez-faire economy to underinvest in research: i.e. The second difference is the presence of the term . y in the numerator of (4.14). This corresponds to a "business-stealin' effect. The private innovator does not internalize the loss of surplus to the monopolist whose rents he destroys. It comes formally from the presence of -X(N -x)Z in the social planner's maximand in (4.9), whereas no corresponding subtraction appears in the problem (2.15) solved by the private research firm. This effect leads pnvate firms to ovennvest in research: i.e. x <x The third difference is the presence of the term ( -1) instead of the numerator of (4.14). This corresponds to the monopolistic distortion effect. It arises because the flow of returns whose capitalized value the social planner attempts to maximize is total output, whereas the private value V is the capitalized value of profits, itt. This distortiofl can work in either direction. In the Cobb-Douglas case it is non-existent, since in that case = (1-_ct)y, so the effect amounts to nothing more than a muldplicative constant in the social-planner's decision-problem.) The intertemporal spiilover effect will dominate when the size of innovations is large (i.e. y is small)-in this case the private reward to research will be small compared to the big social reward-and laissez-faire will generate an AGR less than optimaL2' On the other hand when there is much monopoly power (a close to zero in the Cobb-Douglas case) and innovations are not too large (i.e. y is not too small), the business-stealing effect will dominate, leading to an AGR under laissez-faire which exceeds the optimal level of average Let denote the value to a reseaxth firm of m2ldrlg the th innovation at time t. Define xCt), w(t) and xt) analogously. We confine attention to balanced growth equilibria, in which x(t) = (constant). Then % = (c('r)w(t)) as before, so c(r)w(t) = cw (constant). Likewise ,tt) = (cw)/c(t). As before, V('t) is the expected present value of it evaluated over interval t. Since icc) grows at the constant exponential raze over this interval, it foUows that it (jv) (4.16) () = t t r+Xâ-g Therefore: (ii))/c(v)
The endogenous firm growth theories by REF on the other hand view innovation as part of a process of creative-destruction.
A model of growth through creative destruction