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J Evol Econ (1999) 9: 157±185 Scale eects in Schumpeterian models of economic growth Elias Dinopoulos1 , Peter Thompson2 1 2 Department of Economics, University of Florida, Gainesville, FL 32611, USA Department of Economics, University of Houston, Houston, TX 77204±5882, USA Abstract. Early models of Schumpeterian growth incorporate scale eects predicting that large economies grow faster than small economies, and that population growth causes accelerating per capita income growth. An absence of clear empirical evidence for these scale eects has led some researchers to question the foundations underlying the Schumpeterian approach to growth. This paper reviews empirical evidence on the relationship between scale and growth, and recent attempts to construct Schumpeterian growth models without scale eects. Key words: Economic growth ± R&D ± Scale eects JEL-classi®cation: O2; O3 1 Introduction The idea that scale matters for economic growth is one of the oldest ideas in economics. William Petty (1682), observing the reconstruction of London after the Great Fire, seems to have been the ®rst to identify a possible link: As for the arts of delight and ornament, they are best promoted by the greatest number of emulators. And it is more likely that one ingenious curious man may rather be found amongst 4 millions than among 400 persons. * We would like to thank Steven Klepper for suggesting the topic of this paper, and two referees for some extremely useful suggestions. Correspondence to: E. Dinopoulos 158 E. Dinopoulos, P. Thompson Petty's insight survived through centuries of economic thought1, and disappeared from the study of economic growth only after the development of the neoclassical model, in which scale plays no role. During the last decade, however, the question of scale has received renewed attention from growth theorists. This attention is a direct result of the formalization of Schumpeterian models of R&D-driven growth2, which have made the link between scale and growth precise and explicit. Consider the following simple version of a Schumpeterian model. Output is governed by a Ricardian production function Y t A tLY t 1 in which Y(t) is output, LY(t) is labor devoted to the production of ®nal goods, and A(t) denotes the state of knowledge at time t. Knowledge advances according to _ A t cLA t A t 2 where LA(t) = L(t))LY(t) is labor devoted to knowledge creation, and L is the size of the work force3. Dividing both sides of equations (1) and (2) by population, and noting that the fractions of labor allocated to manufacturing, LY(t)/L(t), and to R&D, LA(t)/L(t), must be constant in any steady state, one obtains _ _ y t A t LA t c L t ; 3 y t A t L t where y(t)=Y(t)/L(t) is output per capita.Equations (2) and (3) together generate testable propositions. Equation (3), relating resource endowments to growth, predicts that a country such as China should have an enormous growth advantage over, 1 The idea can be found in the opening chapter to Wealth of Nations [``Men are much more likely to discover easier and readier methods of attaining any object when the whole attention of their minds is directed towards that single object than when it is dissipated among a great variety of things.'' (Smith, 1776, Book I, ch. 1)], and survived in much the same form to the beginning of the twentieth century [``If we could separate the individuals whose knowledge, correlated and combined, is expressed in the ocean steamship or great modern building, it is doubtful if their separate-knowledge would suce for more than the construction and tools of the savage.'' (George, 1898, Book I, ch. VI)]. Schumpeter's (1942) views on the importance of the size of the resource base for inventive activity are well known. 2 Romer (1983, 1986) initiated this new approach to economic growth, but clear expositions of Schumpererian models can also be found in Aghion and Howitt (1992, 1998), Grossman and Helpman (1991), Romer (1990), and Segerstrom, Anant and Dinopoulos (1990). These papers constitute what we shall refer to as the early Schumpeterian models. Readers not familiar with the microeconomic foundations of Schumpeterian growth models can ®nd a concise overview in Dinopoulos (1994) 3 Romer (1990), along with many other authors, assumes that human capital (or skilled labor) drives the knowledge creation process. The prediction of scale eects survives this relabeling. Scale eects in Schumpeterian models of economic growth 159 say, Hong Kong, and that population growth induces rising income growth. Equation (2), relating input use to growth, predicts that increases in R&D eort should be accompanied by increases in the rate of growth. These predictions are robust to model details and to equilibrium choice. An equation similar to (3) is the outcome of planning problems in all early Schumpeterian models and, even though market imperfections cause competitive equilibria to diverge from optimal plans, fully-speci®ed competitive equilibria also preserve the prediction that growth rates increase with economy size4. Nonetheless, in the decade following Romer's ®rst contributions, theorists have been curiously ambivalent about the importance that should be attached to the eects of scale. Romer (1986, 1994, 1996) views scale eects as a primary motivation for his theory. Grossman and Helpman (1991), and Rivera-Batiz and Romer (1991) have treated scale eects as an important outcome of their models, using them to predict a link between international integration and growth. Other theorists have simply ignored scale eects. Indeed, to Lucas (1993) they were simply ``a nuisance implication that we want to dispose of.'' Two in¯uential papers by Jones (1995a, 1995b) have recently forced economists to take the question of scale eects seriously. Jones' analysis goes to the heart of what we call the ``scale eects problem''. First, he pointed out that scale eects are central to the logic underlying existing Schumpeterian models ± that individuals discover new products and processes ± and they cannot simply be ignored as nuisance parameters. Second, he could ®nd no empirical support for the scale eects predicted so forcefully by these models. Third, he argued that a plausible solution to the scale eects problem leads to a model that fundamentally alters the long-run properties of Schumpeterian models. In particular, Jones' solution removes the property that long-run growth rates are endogenous. In a short period of time, Jones has stimulated several responses. Some of these have followed Jones and constructed models in which the long-run growth rate is exogenous; others have developed alternative models in which there are no scale eects but long-run growth remains endogenous. The development of the new class of models that exhibit long-run growth without scale eects requires a more precise terminology. We will use the term ``Schumpeterian growth'' to refer to a particular type of economic growth that is generated through the introduction of new goods or processes, as opposed to physical or human capital accumulation. We will use the term ``endogenous'' [``exogenous''] to distinguish between Schumpeterian growth models in which long-run growth can [cannot] be aected by permanent policy changes5. For instance, the Romer (1990) model is an 4 Moreover, similar scale eects arise in models of learning by doing. See Krugman (1987), Lucas (1993), Parente (1994), Stokey (1988), and Young (1991, 1993). 5 In our view, the term ``Schumpeterian growth'' is preferable to the term ``R&D-based growth'' because there are models of exogenous long-run growth where the introduction of new products results from deterministic ®xed costs in an environment without knowledge spillovers. Since R&D investment is highly uncertain and is usually associated with knowledge spillovers, the term R&D-based growth is somewhat narrower in scope when applied to this type of models. 160 E. Dinopoulos, P. Thompson endogenous Schumpeterian growth model with scale eects, whereas Jones' (1995b) model could be classi®ed as an exogenous Schumpeterian growth model without scale eects. In this paper we assess the current state of aairs on the scale eects problem. Our analysis focuses on three related questions. First, what can one infer from the empirical evidence on scale eects? Second, can Schumpeterian growth models be constructed that remove the scale eects property? Third, can empirical observations distinguish between competing theoretical solutions to the scale eects problem? 2 The evidence It is worth pointing out immediately that if scale eects are present in the data they are not at all obvious. Figure 1, which plots average annual per capita income growth rates for 1960±88 against initial population for a broad cross-section of countries, reveals no relationship between size and growth. But one should perhaps not expect the evidence to be so obvious. As we have noted, Schumpeterian models have three important observable consequences for long-run economic growth. First, population growth should cause per capita income growth to accelerate. As population has grown signi®cantly everywhere during the twentieth century, time series of income per capita should exhibit accelerating growth. Second, larger economies are predicted to grow faster, a prediction that should be evident in international cross-sectional regressions. Third, changes in the level of inputs used in knowledge creation should be accompanied by changes in the rate of growth. This section brie¯y reviews the evidence that has been brought to bear on each of these three predictions. An assessment of the conclusions that can be drawn is given at the end of the section. Fig. 1. Scale and Per Capita Income Growth, 1960±1988 Scale eects in Schumpeterian models of economic growth 161 Table 1. Romer's and Jones' tests for changes in per capita GDP growth I Romera United Kingdom France Denmark United States Germany Sweden Italy Australia Norway Japan Canada II Jonesb Date of ®rst observation p Change in mean growth rate, 1900±29 vs. 1950±87 (annual percentage rate) 1700 1700 1818 1800 1850 1861 1861 1861 1865 1870 1870 .63* .69** .70** .68** .67* .58 .76** .64 .81** .67** .64 1.64* 1.47* 0.77 0.10c 2.24* 1.19 2.17** 1.83** 1.28 3.99** 0.62 Asterisks denote signi®cance at the (*) ten percent and (**) ®ve percent levels. a p is the probability that, for any two growth rates, the later one is the larger. Asterisks denote the probability of observing a value of p at least as large as the observed value under the null hypothesis that the true probability is 0.5. Romer's last decade is 1970±79. b Jones reports t-tests of the dierence in mean growth rates for the two periods. c First period is 1880±1929. Sources: Romer (1986, Table 3) and Jones (1995a, Table 2). 2.1 Time-series evidence on income growth Romer's evidence for accelerating growth is replicated in column I of Table 1. Comparing average growth rates across decades, Romer calculated ^, that for any two growth rates, the later one is the sample probability, p larger. Rank correlation tests were used to test the null hypothesis that the true probability, p, is equal to 0.5. Of eleven countries tested, all point estimates of p exceeded 0.5, and the null was rejected (at ten percent signi®cance) in eight of them6. Similar evidence is provided by Jones (1995a), who compares the mean growth rates for 1900±1929 with post-war growth over the period 1950±1987. The mean growth rate was higher in the later period for all ®fteen countries examined, although t-tests revealed a signi®cant dierence in only six cases7. While there are problems with these data8 , few economists would argue against the claim that long-run average 6 Romer (1986) also claims additional support for accelerating growth from Reynold's (1983) overview of growth in developing countries: ``growth rates appear to be increasing not only as a function of calendar time but also as a function of the level of development.'' 7 We report in column II of Table 1 Jones' results only for the eleven countries analyzed by Romer. 8 The data are from Maddison (1982). De Long (1988) has shown that selection bias (of ex post industrialized countries) in the data set generates inferences that do not necessarily apply to broader samples. 162 E. Dinopoulos, P. Thompson growth rates have risen over the course of the last few centuries. After all, growth rates during the industrial revolution would today correspond to periods of recession.9 Jones (1995a) has also suggested that one can test his model against earlier Schumpeterian growth models with scale eects, by testing whether growth rates have persistent components. He tested for unit roots in the 1900±87 growth rates of 14 OECD countries. In every case, tests of the null hypothesis that growth rates have a unit root are strongly rejected10. Jones points out that endogenous Schumpeterian growth models with scale eects are consistent with this evidence only if the permanent changes in the underlying variables believed to drive long-run growth have just happened to oset each other. It would, he claims, be little short of miraculous if this had occurred. Interestingly enough, Kocherlakota and Yi (1997) have suggested that the miracle described by Jones may well have happened. They construct and test a simple model in which long-run growth may depend (positively) on public capital and (negatively) on distortionary taxation. Using long timeseries data for the United States and the United Kingdom, they conclude that public capital and taxation both have permanent eects on growth, but that their eects have almost exactly oset each other. Moreover, they point out that one should expect their eects to be osetting: As public capital investment rises and stimulates growth, the government's budget constraint requires that it raises taxes, with osetting growth eects, to ®nance its expenditures. Therefore, absent direct tests on the persistence of shocks to policy, the stationarity of growth rates is at best suggestive of the absence of scale eects, and does not necessarily imply that long-run growth is exogenous. 2.2 International cross-sectional evidence on economy size and growth The simple scatterplot of growth and initial population in Figure 1 revealed no relationship between income growth and population. In cross-sectional 9 Although his data are far less reliable, Kremer (1993) provides evidence that growth rates have been rising since prehistoric times. He combines a scale-dependent production function with a neo-Malthusian population growth function, and argues that accelerating population growth over the last one million years or so is indicative of scale eects. 10 Jones' results are consistent with a large body of evidence on unit roots in per capita GDP levels. Numerous studies since Nelson and Plosser's (1982) pioneering work have concluded that the logarithm of per capita GDP is integrated of order one. If the logarithm of per capita income is integrated of order one, then it necessarily follows that per capita income growth is integrated of order zero, as Jones showed. The case for stationarity in growth rates has been further strengthened by recent evidence (Ben-David and Papell, 1995; Lumsdaine and Papell, 1997) that, once one allows for trend breaks, income per capita levels are themselves stationary between breaks. The trend breaks are, of course, permanent components. But they are most often associated with the Great Depression, World War II, and the ®rst oil-price shock, events that have typically been outside the purview of growth models. Scale eects in Schumpeterian models of economic growth 163 regressions, Backus, Kehoe and Kehoe (1992) similarly found no evidence of scale eects: A regression of per capita GDP growth on the log of initial GDP yielded an insigni®cant slope coecient, and an R2 of just 0.02, half the value one would expect to obtain when the dependent variable is a vector of random numbers11. Controlling for additional determinants of growth did not reverse this result. However, Backus et al. did ®nd evidence of scale eects when attention is restricted to the manufacturing sector. A simple regression of growth in manufacturing output per worker on the log of initial manufacturing output for 67 countries yielded a signi®cant and positive coecient, and initial manufacturing scale explained one third of the sample variation in growth rates. Their results imply that each doubling of the size of the manufacturing sector is on average associated with a 0.6 percent increase in the subsequent growth rate. The scale coecient remained positive through a variety of model speci®cations, and in most cases remained signi®cant. Moreover, the result was robust to variations in sample coverage that exclude particularly high or low incomes per capita. 2.3 The scale of inputs and economic growth We turn now to the scale of input use. A useful starting place is Jones' (1995b) evidence on R&D eort in the United States. Jones presents evidence similar to that given in Figure 2. Between 1950 and 1988, the number of scientists and engineers engaged in R&D grew by a factor of ®ve. Over the same period, productivity growth was constant, or perhaps even declining. A similar story can be told for most OECD countries. Moreover, the story does not change when one restricts attention to productivity growth in the manufacturing sector. Backus et al. (1992) investigated the eects of input scale in a crosssectional setting. They conducted cross-country regressions on equations of the form A_ LA b0 b1 LA b2 ; 4 A L which allows for discriminating tests between the scale and intensity of inputs used in knowledge creation. Their regressions used the growth rates of GDP per capita and manufacturing per worker, while LA was variously measured by the number of students, the number of scientists, and R&D expenditure. Figure 3 summarizes the relevant point estimates obtained from these speci®cations. The ellipses centered on each pair of point estimates indicate the 95 percent joint con®dence interval. The conclusions that 11 In principle, we prefer some measure of population as a measure of scale. Initial GDP confounds scale eects and variations in initial per capita income that are related to transition dynamics in the neoclassical model caused by initial variations in the capitallabor ratio. As Mankiw, Romer and Weil (1992), among many others, have shown, initial per capita GDP may be negatively related to subsequent growth rates, leading to a downward bias in the scale coecient. 164 E. Dinopoulos, P. Thompson Fig. 2. R&D Eort and GDP Growth in the US, 1950±1990 emerge seem clear. There is no support for scale eects in GDP, while intensity eects matter only when inputs are measured by the number of students. In contrast, scale eects in the manufacturing sector are evident for all three input measures. Fig. 3. Scale and intensity Eects on Growth, 1970±1985 Scale eects in Schumpeterian models of economic growth 165 2.4 Firm growth and scale eects The problem of scale eects arises also in models of industry evolution. After all, the Schumpeterian growth models are little more than industry models to which resource constraints and a large dose of symmetry have been appended. The relationship between ®rm growth and size has been investigated extensively by industrial organization researchers in connection to Gibrat's law, which implies that the growth rate of ®rms is independent of their size (i.e., there are no ®rm-level scale eects)12. The results reported in this literature have been mixed. Several studies have rejected Gibrat's law by reporting a negative statistical correlation between size and ®rm growth (Evans, 1987a,b; Hall, 1987). These studies support the conjecture that Gibrat's law holds for large ®rms. For instance, Hall (1987, p583) states that ``Gibrat's Law is weakly rejected for the smaller ®rms in my sample and accepted for the larger ®rms'', and Evans (1987a, p567) states that ``®rm growth decreases at a diminishing rate with ®rm size even after controlling for the exit of slow-growing ®rms from the sample. Gibrat's Law therefore fails although the severity of the failure decreases with ®rm size''. Despite their mixed results, however, these studies clearly point in the direction of no scale eects at the level of the ®rm. In contrast, several recent studies have investigated the determinants and eects of industry shake outs, and have suggested that scale eects may underlie factors that generate shake outs.13 This literature has postulated scale eects in the form of ``learning'' about a technological breakthrough (Jovanovic and MacDonald, 1994) or R&D scale economies associated with process innovations within the context of multiproduct ®rms (Klepper, 1996). The relatively short duration of shake outs points out to the possibility of transitional scale eects that might be important in understanding historical episodes. However, it would not be risky to conclude that, if anything, ®rm-level evidence does not support the assumption of long-run growth scale eects. 2.5 What can we conclude from the empirical evidence? The empirical evidence on scale eects is, to say the least, unclear. First, growth rates have accelerated over the course of a century or more, but they have failed to accelerate in the face of increasing R&D eort during the last forty years. Second, there is no evidence for scale eects across countries in GDP, but there is in manufacturing output. Third, scale eects do not appear in ®rm-level data, yet they may lie behind the phenomenon of industry shake outs. Not surprisingly, the authors of these various studies 12 Sutton (1997) provides an excellent recent survey of the literature and evidence on Gibrat's hypothesis. 13 Shake outs are associated with a relatively short period of time of about 20±25 years, when the number of producers tends ®rst to rise and then to fall dramatically. Klepper and Simons (1997) have documented the existence of sharp shakeouts for the U.S. automobiles, tires, televisions and penicillin industries. 166 E. Dinopoulos, P. Thompson have reached con¯icting conclusions. Romer (1986) claims that increasing growth rates constitute prima facie evidence for scale eects and, in Romer (1996), he argues that scale eects should no longer be treated the way a growth accountant such as Dennison did, as a kind of afterthought that had something to do with plant size. They should be treated the way Adam Smith did, as one of the fundamental aspects of our economic world. Jones (1995a,b), in contrast, dismisses the evidence on accelerating growth, and points instead to the increases in R&D eort: one might worry about the relevant unit of observation (the world vs. a single country) or the lags associated with R&D, but it should be clear . . . that these concerns cannot overturn the rejection of scale eects. The assumption embedded in the R&D equation that the growth rate of the economy is proportional to the level of resources devoted to R&D is obviously false. Finally, Backus et al. take something of a middle ground: [The] theories do better in general when confronting the data for manufacturing than they do for aggregate output. At least in the case of R&D expenditures this should come as no surprise: Most R&D expenditures that are sector speci®c go to manufacturing. On the face of it, Jones' time-series evidence against scale eects is compelling. However, we think that Jones has overstated to some degree the import of the evidence. First, changes in reporting practices and requirements have increased the fraction of R&D that is now formally documented14. Second, a simple enumeration of scientists and engineers overstates the real increase in R&D intensity15. Third, there is evidence suggesting that diculties in measuring quality growth and service-sector productivity have become more pronounced over time, and consequently that growth rates are understated to a greater extent now than they were, say, thirty years ago16. Despite the importance of these issues, we none14 For example, Hall (1993) has noted that the introduction of tax credits for incremental R&D investment in the United States may have provided new incentives to report R&D expenditures. 15 As R&D uses multiple inputs, the number of R&D scientists and engineers is not a sucient statistic for R&D eort. Between 1970 and 1988, private sector expenditure on R&D in the United States rose from 1.6 percent to 1.8 percent of GDP, a much more moderate increase. Aghion and Howitt (1998, ch. 12) point out that total expenditure on R&D in the United States has been between 2.2 percent and 2.9 percent of GDP in every year since 1957, with no obvious tendency to rise over time. 16 Following an exhaustive study of measurement errors in many service sectors, Baily and Gordon (1988) conclude that as much as twenty percent of the post-1973 productivity slowdown can be attributed to unobserved productivity growth. Thompson and Waldo (1997) have estimated the consumer's Euler equation obtained from a structural model of process and product innovation in the presence of measurement error, concluding that ®fteen percent of the slowdown can be attributed to unobserved quality growth. For related evidence see Brooke (1992), Griliches (1994), and Hausman (1997). Scale eects in Schumpeterian models of economic growth 167 theless doubt that attempts to deal with them would overturn Jones' empirical puzzle. Accelerating growth rates provide only indirect evidence of scale eects, and the attention we should pay to them depends on whether there exist alternative and more plausible explanations17. There are several competing explanations. First, much of the increase in post-war growth rates can be attributed to standard neoclassical transition dynamics18. Five of the six signi®cant increases in growth rates that Jones found were for countries whose stocks of physical capital had suered signi®cant damage during World War II, and whose rapid post-war growth rates may re¯ect only a return to steady state. Second, the knowledge production function has itself also undergone technical change. It seems self-evident that if technological change aects the ®nal-good production process it can also aect the knowledge creation process. Accelerating growth may consequently be driven by innovations that increase c in equation (2), thereby raising the productivity of resources allocated to knowledge creation. Third, changes in the institutional environment, incidental to the formal model, may have also played a critical role in promoting more rapid technological change. The manufacturing scale eects identi®ed by Backus et al. are dramatic. Should we believe them? Yet again, we think there are good reasons to be cautious about these results. First, scale may be no more than a proxy for omitted variables. Omitted factors that limit growth may be the same as, or at least correlated with, omitted factors that limit the size of the manufacturing sector or the scale of inputs into knowledge creation. Backus et al. do report some supplementary regressions with additional explanatory variables drawn from prior empirical literature. The point estimates of the scale coecient are reduced, although not signi®cantly so19. Second, if growth rates are autoregressive, it is easy to generate spurious scale eects from cross-sectional regressions. We conducted a regression of 1970±1985 growth rates of per capita GDP against their counterparts for the period 1960±70. A sample of 119 countries yielded a slope coecient of 0.27, indicating that countries which grew faster than average during 1970± 85 (the period covered by Backus et al.) were also likely to have grown faster than average during 1960±70. It immediately follows that, even if growth rates in the ®rst period were orthogonal to initial size (i.e. there are no scale eects), growth rates in the second period will be positively correlated with 1970 scale. While we think the empirical question of scale eects requires further work, we have made some tentative conclusions. In our view, the evidence 17 Kremer (1993), for example, acknowledges that his population data are also consistent with numerous models lacking scale eects. 18 Incorporating physical capital into Schumpeterian models is straighforward, and has the eect of combining the transitional dynamics of the Solow model with the endogenous growth properties of the R&D-based growth model. 19 The supplementary regressions are reported for the economy size regressions, but not for the input regressions. The set of supplementary regressors used is also rather small, although Levine and Renelt (1992) and Sala-i-Martin (1997) report that at least 60 variables have been found to be signi®cant in one growth regression or another. 168 E. Dinopoulos, P. Thompson does not provide much support for long-run aggregate scale eects. We have no doubt that economies can be small enough to sti¯e innovation20, but it does not seem plausible that scale is an important determinant of growth in modern economies. As we have argued, evidence to the contrary must be viewed with considerable caution. In short, we are more or less in accord with Romer's (1996) recent position that scale eects may be important for understanding episodes in economic history, but we also think that one should not rely on them to explain dierences in growth rates among modern nations. 3 Removing scale eects Despite considerable ambiguity in the evidence, theorists have begun to construct Schumpeterian growth models that exclude scale eects. This section reviews the theoretical progress that has been made on the scale eects problem. No attempt is made to do justice to the rich detail contained in the models reviewed below. We continue to let equation (1) describe production of the ®nal good, and we restrict attention to steady-state _ growth paths in which A t=A, LY(t)/L(t) and LA(t)/L(t) are constant. Along the steady-state path, per capita income grows at the same rate as technology, and we can focus our attention on the latter variable. 3.1 Technology in Schumpeterian growth models Schumpeterian models provide substance to the technology parameter, A(t), in one of two ways. In the ®rst (e.g. Grossman and Helpman, 1991, ch. 3; Romer, 1990), A(t) is related to the number, v(t), of horizontally dierentiated varieties of goods that have been developed by time t. In these variety-expansion models the level of technology can be written as A t v t1= rÿ1 5 where r is the elasticity of substitution between varieties. Variety matters in these models either because total factor productivity increases when ®rms use a greater variety of intermediate goods, or because consumers have a taste for variety in consumption. In either case, the number of varieties is assumed to depend on eective R&D investment, I(t), according to v_ t I tv t : 6 The second class of models (e.g. Segerstrom, Anant and Dinopoulos, 1990; Grossman and Helpman, 1991, ch. 4; Aghion and Howitt, 1992) focuses on quality improvements, and interpret technology as A t kq t 7 where k > 1 is equal to one plus the quality increment of a good relative to its immediate predecessor in an industry, and q(t) is the number of 20 The isolated communities of feudal Europe provide an obvious example. Scale eects in Schumpeterian models of economic growth 169 innovations that have occurred since time zero21. The arrival of innovations in an industry is random and governed by a homogeneous Poisson process whose intensity (i.e. the expected number of innovations per unit of time) equals the eective R&D investment, I(t). Given a large number of identical independent industries, the law of large numbers implies that aggregate growth is deterministic and satis®es _ I t : q t 8 For both classes of models, one obtains _ A t cI t ; A t 9 where c 1= r ÿ 1 in the variety-expansion class of models, and c ln k in the quality-improvement models. Both classes of models incorporate important knowledge spillovers that ensure that growth persists in the long run. In the variety-expansion setting, the proportional increase in income secured by an additional variety falls as the number of existing varieties rises [see eq. (5)]; to oset this, the cost of developing a new variety, in terms of eective R&D eort, must fall as the number of existing varieties increases [eq. (6)]. In quality-improvement models each innovation secures a constant proportional increase in the level of quality [eq. (7)], so that the expected cost of an innovation can be constant [eq. (8)]. The scale eects property in both classes of models arises from the functional form that relates eective R&D to economy resources. A generalized form of this relationship can be written as I t LA t X t 10 where LA(t) is the amount of labor devoted to R&D, and X(t) captures the degree of R&D diculty in the sense that higher values of X require more labor to achieve the same level of I22. Combining (9) and (10), we have: _ A t LA t L t c ; 11 A t L t X t and, as the fraction of the labor force devoted to R&D, LA(t)/L(t), must be constant in the steady state, the removal of scale eects requires that the steady-state growth rate of X(t) be exactly the same as the population growth rate. In a cross-sectional dimension, scale eects are absent only when X is proportional to L. The challenge for theorists has been to construct plausible models that satisfy this stringent steady-state condition. We review two distinct approaches that have oered possible solutions. 21 Productivity growth in a single-good economy can be modeled in an analogous fashion. In this case, k measures one plus the proportional increase in total factor productivity obtained from an innovation. 22 In multi-factor models LA can be replaced with any constant returns to scale production function that generates R&G services without aecting the essence of the scale of eects problem. 170 E. Dinopoulos, P. Thompson 3.2 Exogenous Schumpeterian growth models Jones (1995b), and Segerstrom (1998) have removed scale eects by assuming that R&D becomes more dicult over time. In Scherer's (1965) terminology technological opportunities are assumed to diminish systematically over time. Jones solves a variety expansion model identical to Romer (1990) except for an assumption that X(t)=A(t)1)/. Segerstrom analyzes a quality improvement model that incorporates the assumption that X_ t=X t 1 ÿ /I t. Except for inconsequential dierences in X(0) and A(0), the two assumptions are equivalent and yield a knowledge production function of the form _ A t LA t c ; 12 A t A t 1ÿ/ which states that the proportional advances in the level of technology (whether measured by the number of varieties or the level of quality) obtained from a given amount of R&D eort declines monotonically as technology advances. Jones and Segerstrom justify this assumption with an identical assertion: R&D becomes more dicult over time because ``the most obvious ideas are discovered ®rst.'' But, in an innovative paper, Kortum (1997) has provided some interesting theoretical microfoundations for the assumption of increasing R&D diculty, without requiring that more obvious ideas be discovered ®rst. Kortum models research as a search process that consists of drawing random eciency levels from a probability distribution that represents technological opportunities. Given a current productivity level A(t), the probability that a draw is more productive than the current state of the art is given by S th 1 ÿR F A t, where h > 0. F(A(t)) is the stationary t search distribution, S t ÿ1 LA sds is the research stock, and S(t)h is the spillover function. Kortum's central result is that, irrespective of the value of h and the particular distribution F, the productivity of R&D declines over time23. In the special case that F is a Pareto distribution, Kortum derives a simple expression for productivity growth _ LA t A t j 1 h R t ; 13 A t ÿ1 LA s ds where j is the parameter of the Pareto distribution denoting the mean magnitude of quality improvements (i.e., the size of innovations). Equations (12) and (13) are equivalent and each can be derived by differentiating A(t)=S(t)b with respect to time. Equation (12) corresponds to b 1= 1 ÿ /, while equation (13) is obtained by setting b j 1 h. Thus Kortum's model provides a novel interpretation to the diminishing-opportunities parameter / in the exogenous Schumpeterian growth models 23 While this result at ®rst seems quite remarkable, the fact that it does not depend on the value of h should not be taken to imply that knowledge spillovers are unrestricted. As Kortum readily acknowledges, the assumed form of the search distribution imposes a limit on the potency of spillovers even for large values of h. Scale eects in Schumpeterian models of economic growth 171 developed by Jones and Segerstrom. Because 1 ÿ / 1=j 1 h, diminishing opportunities are stronger if R&D spillovers are weak (i.e., h is small) or if the stationary search Pareto distribution has a low mean (i.e., j is small). _ Let g t A t=A t denote the growth rate of technology, and let gL denote the constant growth rate of population. Dierentiating (12), using the fact that LA(t) grows at the rate gL in steady state, yields: _ g t gL / ÿ 1g t : g t 14 If g(t) is greater than [less than] gL = 1 ÿ /, the growth rate is falling [rising]. Thus, the steady-state path is stable, satisfying g gL = 1 ÿ /), and the level of technology is related to R&D eort by 1 c 1 ÿ /LA t 1ÿ/ : 15 A t gL That is, the growth rate does not depend on any measure of scale. Increases in the steady-state level of R&D raise technology and income per capita at any point in time, but they do not raise the growth rate. Consequently, policies that increase R&D have level eects but not growth eects. 3.3 Endogenous Schumpeterian growth models without scale eects A second approach to the scale eects problem, suggested by Young (1998) and subsequently analyzed independently by Aghion and Howitt (1998, ch. 12), Dinopoulos and Thompson (1998), Howitt (1999), Peretto (1998), and Peretto and Smulders (1998), introduces the concept of localized intertemporal R&D spillovers to generate long-run endogenous Schumpeterian growth without scale eects. These studies remove the scale eects property by essentially the same mechanism as the one employed by exogenous Schumpeterian growth models, namely by assuming that aggregate R&D is becoming more dicult over time as it is spread over more ®rms. However, their novel contribution is that they append other growth channels that maintain the endogeneity of long-run Schumpeterian growth. We will illustrate the concept of localized intertemporal knowledge with a taste structure that, following Dinopoulos and Thompson (1998), incorporates horizontal and vertical product dierentiation. We will assume knowledge spillovers in the vertical (quality) dimension but none in the horizontal (variety) dimension. To this taste structure, we will append the simplest variety creation process introduced in Aghion and Howitt (1998), namely that variety creation is the result of costless imitation. Consider a closed economy consisting of a continuum of in®nitely-lived identical households. The size of each dynastic household grows at an exogenous constant rate, gL>0, which equals the rate of population growth. The representative household maximizes the following intertemporal utility function: 172 E. Dinopoulos, P. Thompson Z1 U e gL ÿqt ln D t dt : 16 0 where q > 0 is the constant subjective discount rate and D(t) is a subutility de®ned by 2 3r= rÿ1 Zv t 6 7 : 17 D t 4 a i; t1=r d i; t rÿ1=r di5 0 D(t) is the quality-augmented Dixit-Stiglitz consumption index; v(t) denotes the number of varieties that have been developed at time t; a(i,t) and d(i,t) are the level of quality and per capita consumption of variety i at time t, and r > 1 is both the constant elasticity of substitution between varieties and the elasticity of demand for any single variety v(i) Let p(i,t) denote the price of good i, and let E(t) denote per capita consumption expenditure. At each instant of time, the R v tconsumer maximizes D(t) subject to the budget constraint E t 0 p i; t d i; tdi. This standard maximization problem yields the instantaneous per capita demand d i; t a i; t p i; tÿr E t : v t R 1ÿr a i; tp i; t 18 0 The instantaneous aggregate demand for each variety is c(i,t)=d(i,t)L(t), where L(t) is the level of population at time t. Symmetry of tastes across varieties implies that ®rms charge an identical and constant price, p i; t r= r ÿ 1, for every product, and (18) can be written as r ÿ 1 a i; t E t ; 19 d i; t r A t v t R v t where A t 0 a i; t di =v t is the economy's average quality level at time t. Per capita consumption of good i depends on its quality level relative to the economy's average and on expenditure per variety. Since all ®rms charge the same price, varieties with higher quality levels command higher market shares. The evolution of per capita expenditure, E(t), is governed by the familiar dierential equation _ E t r t ÿ q ; E t 20 where r(t) is the instantaneous market interest rate. Assume that labor is the only factor of production, and that it can be allocated between two activities: Manufacturing of ®nal products, where one unit of labor produces one unit of output, and quality-enhancing R&D. The level of quality associated with variety i at time t is given by a i; t kq i;t , where k > 1 denotes the quality improvement between two Scale eects in Schumpeterian models of economic growth 173 adjacent innovations within variety i, and q(i,t) is the number of quality innovations that variety i has experienced since time zero. The arrival of innovations is governed by a Poisson process with intensity I(i,t). Assuming symmetry and applying the law of large numbers, it can be shown that the evolution of average quality A t kq t is given by _ A t I t ln k ; A t 21 where I(t) is the R&D investment per variety. It is immediately obvious from this formulation that there are knowledge spillovers related to quality improvements, that any change in I(t) alters long-run growth, and that the removal of scale eects requires a steady-state equilibrium with constant I(t). Following Aghion and Howitt (1998, Chapter 12) we assume that the creation of new varieties is the result of costless imitation and, for simplicity, we suppose that the quality level associated with a variety js created at time s equals the average quality at time t [i.e., a js ; t A s]. Each individual has the same exogenous probability of imitating and the arrival of imitation is governed by an exogenously given intensity h. Therefore, the aggregate rate of variety accumulation is simply v_ t hL t, where _ gL L t. The ratio k(t)=L(t)/v(t) is governed by the equation L t _k t gL k t ÿ hk t2 , and so in steady state the number of varieties is proportional to the level of population: v t h L t : gL 22 In the long run, the number of varieties grows at the rate of population growth, gL, and, in the absence of international technological linkages, larger economies will produce a larger number of varieties. The model is closed with the full employment of labor condition, which can be written as Zv t c i; t L t v tI 0 I r ÿ 1E ; 23 L t k r where R v t v(t)I is the amount of labor devoted to quality-enhancing R&D and c i; tdi is the amount of labor devoted to manufacturing. 0 The model exhibits a steady-state equilibrium in which the intensity of quality-enhancing R&D per variety and per capita consumption expenditure are constant over time. The scale of the economy, measured by L(t), does not aect the full employment condition in the long-run, and equation (18) implies that r t q Substituting (19) into (17) and performing the integration yields D t r ÿ 1 EA t1= rÿ1 v t1= rÿ1 ; r 24 174 E. Dinopoulos, P. Thompson dierentiation of which provides an expression for the endogenous rate of Schumpeterian growth: _ _ D t 1 A t 1 v_ t D t r ÿ 1 A t r ÿ 1 v t 1 ln k1A gL : r ÿ 1 25 Because the number of varieties is proportional to the scale of the economy, (24) implies that the level of instantaneous utility depends positively on scale. However, the rate of growth of per capita instantaneous utility is unrelated to scale, as (25) demonstrates. The long-run growth rate of the economy depends positively on the rate of population growth as in models of exogenous Schumpeterian growth [compare equations (25) and (14)], but it also depends positively on the magnitude of quality innovations, and on the intensity of quality-enhancing R&D. Any policy that aects the magnitude or intensity of innovations has long-run growth eects. In the absence of quality growth (i.e., when k 1), long-run growth becomes exogenous and depends only on the rate of population growth. In general, however, the absence of population growth does not reduce the long-run growth rate to zero. The role of variety-creation in removing the scale eects property is also apparent from equations (22) and (23). If the evolution of varieties is characterized by intertemporal knowledge spillovers [i.e. v_ t=v t hL t], then the term gL in (25) should be replaced by hL t, and the long-run rate of growth depends on scale and eventually becomes unbounded as L(t) increases exponentially over time. In addition, if the long-run number of varieties v(t) remains constant, the full employment condition (23) implies that I and long-run growth increase exponentially over time. Therefore the endogenous Schumpeterian growth models without scale eects oer a generalization both of earlier Schumpeterian growth models with scale effects and of exogenous Schumpeterian growth models without scale eects. One way to interpret the channel through which localized knowledge spillovers remove scale eects is to rewrite the aggregate (economy wide) R&D-based knowledge production function, (21), as _ A t ln kv tI k ln kLA t LA t ln kI c ; A t v t L t X t 26 where c ln k is a positive constant, and X(t)=v(t)=L(t)/k can be interpreted as the degree of aggregate R&D diculty. In other words (26) is identical to (11), where the degree of R&D diculty at time t, equal to the number of varieties, grows at the rate gL 24. Finally, it is worth mentioning that, in the presence of localized intertemporal knowledge spillovers the aggregate knowledge production function can be written as a function of the share of labor devoted to R&D, 24 This speci®cation of the aggregate knowledge production function has been used in several recent studies (see Dinopoulos and Segerstrom 1998; Sener 1997; among others) under the name of the permanent eects on growth (PEG) model. Scale eects in Schumpeterian models of economic growth 175 LA(t)/L(t), [see (26)] contrary to theoretical objections raised by Jones (1995b) regarding the microfoundations for this speci®cation25. Costless imitation might be the simplest assumption that generates a linear equation relating the number of varieties to the level of population, but it is not the only one. Young (1998) and Dinopoulos and Thompson (1998) assume that the establishment of a new variety entails a ®xed cost, and derive an equation that is equivalent to (22). Peretto (1998) and Peretto and Smulders (1998) employ a similar assumption in a model of growth through variety accumulation where, instead of quality improvements, ®rms produce a ¯ow of varieties and the establishment of a new (multiproduct) ®rm requires a ®xed cost. Free entry in the process of variety accumulation results in an equation similar to (22). Howitt (1999) obtains the same result by assuming that the variety-creation function depends on labor and ®nal output through a constant returns to scale production function, and that there is free entry in varieties. 4 On the empirics of Schumpeterian growth models without scale eects All the models reviewed in Section 3 are consistent with the post-war time series evidence presented by Jones (1995a,b) and the cross-country evidence reported by Backus et al. (1992): Resources devoted to R&D increase exponentially, whereas the rate of long-run growth is constant over time. Initially, several studies (e.g., Jones, 1995a,b; Feenstra et al., 1997; Eaton and Kortum, 1999a) focused on testing the dierences between endogenous Schumpeterian growth models with scale eects and exogenous Schumpeterian growth models without scale eects. These studies lend empirical support to the latter class of models and have been interpreted, somewhat prematurely, as providing evidence for the exogeneity of long-run Schumpeterian growth. The recent development of endogenous Schumpeterian growth models without scale eects has now provided a more demanding alternative against which to test exogenous Schumpeterian growth models. While much remains to be done in this area (indeed, this section will read much like a list of ideas for future research), we summarize here some observations on the empirics underlying dierences between the models. 4.1 Patent statistics One testable proposition that dierentiates models incorporating variety creation from those that are based on only quality improvements is the 25 Jones (1995b) states: ``First, equation (26) is inconsistent with the microfoundations of the R&D models developed by Romer/Grossman-Helpman/Aghion-Howitt. These microfoundations imply that new ideas are discovered by individuals so that the number of innovations is inherently tied to the number of persons engaged in R&D. A speci®cation devoid of scale such as (26) has the counterfactual implication that an economy with only one unit of labor can produce as many innovations (or at least can generate equivalent TPF growth) as an economy with 1 million units of labor''. 176 E. Dinopoulos, P. Thompson long-run prediction on the ¯ow of new products. The former class of models (e.g., Jones, 1995b; Dinopoulos and Thompson, 1998; and Young, 1998) predict that the ¯ow of innovations increases at the rate of population growth, and therefore that the ¯ow of innovations per R&D worker is constant over time. In contrast, in models of exogenous Schumpeterian growth that abstract from variety accumulation (e.g., Kortum, 1997; and Segerstrom, 1998), the ¯ow of innovations is constant and the ¯ow of innovations per researcher declines over time. Assuming that a constant fraction of all innovations are patented, patent statistics can be used, in principle, as an approximate measure of the ¯ow of innovations to distinguish between these two classes of models26. Consider ®rst the standard time-series evidence on patents. Since the turn of the century, patents granted annually by the U.S. patent oce to U.S. residents have ¯uctuated between 25,000 and 60,000, with no consistent trend (see series B and C in Fig. 4). This evidence clearly lends prima facie support to the models of Kortum (1997) and Segerstrom (1998). However, the evidence is muddied by several features of the data. First, the aggregate data confound two interesting phenomena: The constant rate of patent awards comprises the sum of a strong positive trend in patents granted to U.S. corporations (which rose from less than 5,000 at the beginning of the century to almost 40,000 by 1970; see series A in Fig. 4) and a countervailing decline in patents granted to individuals (which fell from 30,000 early in the century to less than 14,000 by 1970; see series B in Fig. 4). As the models we have reviewed are clearly intended to represent ®rm behavior, there is some doubt about which time series we should consider. What did it mean to be an individual inventor in, say, 1920? Is the shift from individual to corporate patenting a re¯ection of changes in who is inventing, or merely changes in who appropriates the rents from invention? And are there dierences in the propensity of individuals and corporations to patent inventions? We do not know the answers to these questions, yet obtaining them is clearly a precursor to carrying out reliable tests between competing models. A second complication in the data is that the behaviors of patent grants and patent applications have been somewhat dierent. In particular, applications for U.S. patents by U.S. residents have doubled since 1985, reaching over 120,000 in 1995 (Kortum and Lerner, 1997). The surge in patents since 1985 has occurred in a period of constant and even declining R&D intensity which suggests a rising patents per researcher ratio. One possible explanation is that the recent surge in patenting re¯ects institutional changes, possibly the establishment of the Court of Appeals of the Federal Circuit in 1982. However, Kortum and Lerner (1997), simulating a version of Kortum's (1997) model, cast doubt on the institutional explanation and conclude instead that the surge in patent applications re¯ects an 26 However, it must be emphasized that patent statistics should be used with caution for inferences regarding the empirical performance of growth models that do not explicitly model the microfoundations of endogenous patenting behavior of ®rms, and that abstract from other mechanisms that provide protection to intellectual property rights. (see Griliches, 1990). Scale eects in Schumpeterian models of economic growth 177 Fig. 4. U.S. Patent Grants and Applications, 1990±1993 increase in U.S. innovation27. The surge in applications has not yet been matched by an increase in patent grants, however, and it remains to be seen whether the incremental applications actually do constitute inventions. A third complication is that patent data do not readily capture changes in the value of patents. Schankerman and Pakes (1986) have attempted to estimate the value of patents in three European countries from patent renewal rates, concluding that patent values had risen over time to oset the decline in patents per researcher28. Thompson (1996) produces some contradictory evidence. Using stock market data from U.S. publicly traded corporations, he concluded that the stock market valuation of an innovation declined between 1973 and 1991, and that this decline was due in part to a rise in technological opportunity. Finally, Kortum (1997) points to a rapid increase in the fraction of domestic innovations which are also patented internationally. This trend to broader patent protection, he contends, 27 Note, however, that Kortum's model predicts not only a temporary surge in patenting caused by a permanent increase in research productivity, but also a temporary increase in R&D intensity. The latter prediction is not consistent with the U.S. evidence as Kortum and Lerner (1997) acknowledge. 28 We are somewhat skeptical of the reliability of evidence from patent renewal data. Patent renewal costs are extremely low and, at least for corporations, it may be cheaper to renew a patent than to calculate the bene®ts of renewal. Currently, the basic ®ling fee in the U.S. is $790 ($395 for small entities). Patent attorney oces advertising on the internet are oering patent ®ling services for as low as $1,600, including all application fees. 178 E. Dinopoulos, P. Thompson is consistent with the claim that patents have become more valuable over time. It is fair to conclude that the time series evidence on the ¯ow of innovations is mixed and does not yet provide decisive support to any class of Schumpeterian growth models without scale eects. 4.2 Scale eects in income levels All the models reviewed in section 3 predict scale eects in income levels. For instance, in models with variety accumulation the level of instantaneous utility is an increasing function of the number of varieties, and the level of varieties is proportional to the level of population [see eqs. (15) and (23)]. Thus, these models predict that larger economies have higher income per capita in the long run. The models do not predict that (abstracting from international linkages) China will grow more rapidly that Hong Kong, but they do predict that it will be enormously wealthier. However, one needs to distinguish between models where varieties take the form of intermediate goods (e.g., Aghion and Howitt, 1998; Howitt, 1999; Jones, 1995b; Peretto, 1996; Young, 1998) and models where varieties are ®nal consumption goods (e.g., Dinopoulos and Thompson, 1998). The former predict that measured GDP per capita exhibits scale eects, since the accumulation of varieties is captured by increases in total factor productivity, which are re¯ected in ®nal output. The latter predict that the level of instantaneous utility exhibits scale eects and that these level eects refer to dierences in the cost of living. However, it is now well-known that GDP is not a measure of the cost of living29. In fact, GDP de¯ators do not even attempt to capture the welfare gains from increased variety of ®nal consumption goods. In practice, a small number of products within a narrowly de®ned category (e.g., 30 inch color televisions) are sampled to obtain an average price, and the consumer price index is not enhanced by increased number of varieties within each category. That is, GDP statistics treat all goods within a product category as perfect substitutes. As Hausman (1997) has shown, failure to account for new ready-to-eat cereal brands has resulted in an overestimation of the consumer price index for cereals by about 20 to 25 percent. The prediction of scale level (as opposed to growth) eects is an issue that awaits further empirical investigation regarding the relevance of both endogenous and exogenous long-run Schumpeterian growth models. 4.3 Closed-economy models in an open world It might also be useful to remind the reader that all these growth models have abstracted from the role international market linkages and the international diusion of knowledge issues30. These linkages create diculties in 29 Armknecht et al. (1997) have an excellent discussion on the limitations of the U.S. CPI in accounting for new varieties. 30 Notable exceptions are the theoretical models of Aghion and Howitt (1998, ch. 12), Dinopoulos and Segerstrom (1998), Eaton and Kortum (1999a, b), and Howitt (1997), who develop multi-country models of Schumpeterian growth without scale eects. Scale eects in Schumpeterian models of economic growth 179 interpreting and using international cross-sectional data to test some of the implications of growth models without scale eects. Consider, for example, Kortum's (1997) model and assume that there is perfect international diffusion of knowledge which renders the P location of research irrelevant. Then the world ¯ow of research is LA t ni1 LiA (t), where LiA t is research employment in country i. Equation (13) then determines the world growth rate, which will be common for all countries. Although all countries share the same level of productivity, the inventive output of each country (and hence the ¯ow of innovations per unit of time) becomes proportional to its research eort, but there are no scale eects on growth because all countries grow at the same rate. Eaton and Kortum (1998b) analyze the more realistic case of imperfect international technology diusion which allows for differences in productivity levels across countries, but equal growth still prevails in steady state31. Notwithstanding their possible sensitivity to open-economy extensions, the models do oer a novel insight concerning the determinants of crosscountry income per capita dierences through level scale eects. For example, consider the thought experiment of opening trade between two identical economies with the structure described in Section 3.3. There will be intra-industry trade in varieties which will generate income per capita level eects, but which will not have long-run growth eects32. Since intraindustry trade is more prevalent among advanced countries with similar factor endowments, the level eects of trade on income per capita should be greater among advanced countries. This prediction has profound implications for the performance of cross-country income per capita regressions and indicates that multisectoral and multicountry models that incorporate both trade and technology transfer might enhance our understanding of cross-country productivity dierences and of the wealth of nations33. Schumpeterian growth models without scale eects, but incorporating variety expansion, have provided a new link between the static trade literature on patterns of trade and gains from trade, and the new growth literature. Incorporating open economy features into the models might also moderate the strong degree of level scale eects implied by the closed economy models of growth without scale eects. This is certainly desirable, as the scale eects in a closed economy may be staggering. Consider, for example, the U.S. and France, two countries with similar R&D intensities. As the U.S. employs four times as many researchers as does France, the 31 The same result is also obtained in Dinopoulos and Segerstrom (1999). Indeed, the income per capita level eects are identical to the ones generated by static models of intra-industry trade with monopolistic competition and manufacturing scale economies (e.g., Krugman, 1979). 33 For example, it may explain why Mankiw, Romer and Weil (1992) found that the income per capita predictions of the closed-economy neoclassical model do not perform well on a cross-section restricted to OECD countries. However, Dinopoulos and Thompson (1999) and Klenow and RodrõÂ guez-Clare (1997) have provided evidence that the neoclassical model does not perform well on a cross-section of any sample of countries. 32 180 E. Dinopoulos, P. Thompson steady-state equation (15) implies that the U.S. should be at least four times richer. For / 0:5, the U.S. should be 16 times richer than France, while for / 0:9 it should be one million times richer! The extent to which multicountry models will moderate this implication of the closed-economy models awaits further research. 4.4 Does variety creation require R&D? Finally, the variety expansion models of Section 3 raise an interesting measurement issue associated with resources devoted to variety creation. On the one hand, the process of variety accumulation does not entail any uncertainty and, what is more important, there are no knowledge spillovers. These features suggest that resources devoted to variety development might not be associated with R&D investment, but rather should be interpreted as ®xed capital start-up costs. This view is reinforced by the obvious kinship between growth models with variety accumulation and no scale eects, and the literature on variety-based static monopolistic competition introduced by Dixit and Stiglitz (1977), in which the number of varieties depends on ®xed manufacturing costs. On the other hand, unlike the static models where, by necessity, the creation of varieties occurs instantly (as opposed to sequentially) and manufacturing scale economies limit the number of available products, this class of models assumes that the ®xed costs are paid before any manufacturing is taken place. This assumption is standard is R&D-based models of economic growth. The treatment of variety creation in Section 3.3 therefore raises the following question: Does variety creation require R&D, or physical capital investment? The answer matters, of course, for how one should test the models with R&D and investment data. But it also matters for a rather dierent reason. In Young's (1998) model, in which vertical and horizontal innovations occur simultaneously as a result of the same R&D investment, proportional R&D subsidies do not aect the steady-state intensity of R&D per variety. Put another way, the standard policy tool to raise R&D eort has no eect, even though increases in R&D eort do have permanent eects on growth. Dinopoulos and Thompson (1998) have argued that this awkward result turns on the question of whether or not the costs of variety creation are, in fact, R&D. This issue might be resolved in the future, but these considerations provide support for our use of the term ``Schumpeterian'' instead of ``R&D-based'' growth to characterize this strand of literature. 5 Conclusions This paper addressed three questions. First, what evidence is there for the scale eects predicted by early Schumpeterian growth models? Second, how can one modify the early models to remove the scale eects? Third, what empirical evidence might be brought to bear on the modi®ed models? Scale eects in Schumpeterian models of economic growth 181 Our answer to the ®rst question is brief: The evidence for the existence of scale eects is weak, although they might have been present in history. There is no clear evidence to suggest that larger economies grow faster. We described two main theoretical approaches that have generated Schumpeterian growth models without scale eects. In the ®rst, aggregate R&D eort aimed at proportional technology increments is assumed to become more dicult over time, perhaps because the obvious ideas are discovered ®rst. In the second, ®rm-level R&D aimed at proportional technology increments is not becoming more dicult over time, but economywide R&D is, because it is diused over a greater number of varieties in larger economies. Both approaches generate steady states in which scale has level, but not growth, eects. Exogenous Schumpeterian growth models predict that long-run per capita income growth is proportional to the rate of population growth. Endogenous Schumpeterian growth models remove the scale eects property in much the same way as exogenous Schumpeterian growth models, but they maintain the endogeneity of long-run growth by introducing the notion of localized intertemporal knowledge spillovers. The latter models generate an expression for long-run growth that consists of two additive terms: A term that is proportional to the rate of population growth and a term that can be aected by a variety of permanent policy changes. The identi®cation of the scale eects property of earlier Schumpeterian growth models, and the response to Jones' criticism of them, have profound implications for the theory of economic growth. At the theoretical level, the existence of endogenous Schumpeterian growth models without scale eects serves as a strong reminder that the removal of scale eects does not necessarily imply the removal of policy endogeneity of long-run Schumpeterian growth. However, it must be emphasized that the removal of scale eects reduces considerably the set of policies that aect long-run Schumpeterian growth. In addition, the construction of Schumpeterian growth models without scale eects provides the ®rst step towards a generalization of the neoclassical growth model. The generalization, we anticipate, will eventually result in a uni®ed theory of economic growth in which endogenous technology levels and growth rates are appended to the standard neoclassical model of economic growth. We are not fully satis®ed with the empirical testing of the Schumpeterian growth models summarized in this paper. Theoretical solutions to the scale eects problem have been naturally couched in terms of closed economy models, yet the empirical evidence that can be brought to bear on the models is muddied by measurement problems and by the potentially important eects of trade and international technology diusion. Jones (1995b) has suggested that because of international technology ¯ows one should not even attempt to test the models against modern international cross-sectional data. We do not share this view, although we do acknowledge that one must be very cautious when interpreting results. There are of course numerous empirical studies on technology diusion34, and there is a 34 Griliches (1992) and Nadiri (1993) review the empirical literature. For subsequent research see Coe and Helpman (1995), Houser (1996), Keller (1997), Lichtenberg (1996) and Rogers (1996). 182 E. Dinopoulos, P. 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