Macroeconomic Models and Relationships That May Limit Growth
although international trade widens the market attainable by a successful innovator and thus raises the incentive to do research, it also raises the cost of research by making labor more productive in manufacturing [and therefore raising overall wages], with effects that tend to offset each other.
—Aghion and Howitt, 1998, p. 5,
emphasis added
This chapter differs from the materials in parts one and two of this book in two main ways. First, its analysis is macroeconomic rather than microeconomic. Secondly, and more substantially, it examines some of the forces that may conceivably bring the unprecedented growth performance of the capitalist economies to an end, slowing it to the pace of progress of earlier eras.
The question, then, is, how may such an ending come about? Of course, if the end of spectacular growth is sufficiently far in the future it is almost the same, as a practical matter, as never-ending advance. And such a very remote termination period cannot be ruled out, as will be suggested in this chapter and the next. Still, it is a vast understatement to say that all this is uncertain. Therefore, it is less fruitful to speculate about growth rates of the distant future than to examine some of the forces that work toward termination of rapid growth and those that work in the other direction. That is what this chapter and the next will do.
WHY TURN TO MACROECONOMICS?
I have argued here that the nature of the forces that can be credited with the extraordinary record of economic growth of the free-market economies requires us to study the behavior of individual firms and industries. So far, only microeconomic models have been used to investigate several distinctive features of capitalism that stimulate rapid growth. These models have described, for example, how fierce competition puts managements under constant pressure to avoid falling behind in industries where innovations in products and processes are prime weapons of inter-firm rivalry. But this still leaves unexplored how the innovative activities of all these firms add up. In other words, what is the effect on the economy as a whole of the combined innovative efforts of our rivalrous oligopolists? Here, the microanalyst is in danger of committing the fallacy of composition, that is, of arguing, without evidence, that the behavior of an individual enterprise is simply replicated on a larger scale in the behavior of the economy. We know that is not always so, and the main model of this chapter will provide a direct counterexample, in which the growth-promoting activities of the firm automatically generate forces that inhibit growth of the economy.
To deal with such issues, it is clearly useful to turn to macroeconomic analysis. Aggregating the agents of the economy into a small number of broad sectors facilitates analysis and permits clearer implications to be determined. The result will be a macroeconomic model of endogenous innovation, but one very different from many of the valuable constructs—including the work of Arrow (1962), Lucas (1988), Romer (1986, 1990), and others—that have attracted much deserved attention. The model offered here will attempt to deal explicitly and directly, if in a somewhat primitive manner, with the way in which economic forces direct the economy’s innovation activities. Moreover, the model here is not ahistoric; that is, it contains elements that, it can be argued, are particularly characteristic of a free-market economy. This contrasts with the other macromodels to which I have just referred. There, decisions affecting innovation occur, so to speak, behind the scenes, so that we can see the results but can only indirectly infer the mechanism and the process that are responsible. Moreover, the production functions that arguably are at the heart of those analyses, like the rest of their structures, contain no particular features representing special characteristics of current historical circumstances. To show more specifically what this means, this chapter begins with a brief review of some pertinent features of those macromodels.
THE RELEVANCE OF MACROECONOMIC MODELS WITH ENDOGENOUS INNOVATION
Recent growth analysis had its beginnings in the 1950s with the work of Robert Solow (1956), who deservedly elicited renewed interest in models of growth and who designed approaches compatible with statistical estimation. For our purposes, it is most pertinent to consider the extent to which these models, notably the more recent ones that have been described as “endogenous innovation models,” actually bring out the behavior of the innovation process and its contribution to growth. I will suggest that the endogenous innovation and the growth features of these models are only implicit, while my concerns in this book require me to make the innovation process more explicit, in order to see just how the historical features of the free-market economy affect the innovation and growth processes. This will require construction of a model in which there are at least two sectors: the innovation sector and the rest of the economy.1
The current macromodels, like much of the economic literature, have older roots. Ricardo and other classical economists recognized that innovation does occur.2 It results in a shifting of the production function and postponement of the stationary state, something that can occur repeatedly and can keep the economy expanding indefinitely. What is missing in the Ricardian story is any endogenous explanation of the innovation process and special historical features that affect it. That is why the innovation process in a formalization of the Ricardian model must be represented simply as A(t), a (stochastic) function of time and nothing else, and with no distinguishing features that differentiate the process in a capitalist economy from that in any other form of economic organization.3
The original Solow model—the prototype neoclassical model—contains a representation of innovation not much different from Ricardo’s, with invention also autonomous, and undifferentiated as between free-market economies and other economic forms. The model assumes that there are diminishing returns to capital, an attribute that predicts convergence of productivities and per capita incomes in different economies, because wealthier economies have relatively large capital stocks whose productivities, relative to those of poorer countries, are severely reduced by diminishing returns.
Two observations led Romer to argue that this neoclassical model requires some modification. First, he observed that the prediction of universal convergence—the catch-up of all economies to approximately the same levels of productivity and per capita income—apparently implied by the theory is not confirmed by the facts. The many statistical studies of the convergence hypothesis generally conclude that, although the growth performances of the wealthiest economies have, indeed, been moving closer together, most of the impecunious nations are falling further behind. Secondly, he noted (as students of the subject had long observed) that the innovation process is neither largely autonomous nor largely fortuitous. The amount of activity devoted to innovation, and the output of that activity, are influenced substantially by what else is going on in the economy. This led to a series of constructs referred to as the “endogenous growth models.”
There is a common element in the approach to endogenization taken by many of these models. Each focuses on some source of enhanced growth, such as education or innovation. Let the value of this variable, call it innovation activity, contributed by agent j be represented by Xj. For the economy as a whole the corresponding value is, then, X = ΣjXj. If each innovating agent’s output of new technology is a function not only of Xj but of X as well, it follows that there are spillovers from this sort of innovative activity. Then all innovators benefit from the activity of all other members of the group, but with the obvious impediment to efficient growth constituted by the resulting difference between private and social benefits of innovation activity. The Lucas model of 1988 is an example of this type of structure. It can be described as taking the production function to be A(H), where H is the investment in human capital of the entire society, as distinguished from Hj, the corresponding investment by agent j. Similarly, the Romer model of 1986 can be taken to use a production function such as A(R), where R is society’s investment in the accumulation of knowledge, as distinguished from Rj, the corresponding variable by agent j.
The central point is that each of these models employs a production function that contains a growth component, A(.), but that none of them has any attribute uniquely related to free-enterprise economies rather than some other economic form. That is what makes these models ahistorical. Moreover, where they do take account of innovation, the mechanism of the activity enters only implicitly. There is no equation or other relationship that attempts to describe, for example, the incentive structure that leads to determination of the magnitude of this activity.
Another significant difference between the approach of recent macrotheory and mine is the focus of the former on the role of such influences as learning-by-doing and accumulation of human capital, while I focus on the powerful pressures exerted by the free competitive market. Economic historians—Moses Abramovitz being a notable example—have concluded that economies cannot attain relatively rapid rates of growth without the human and social capability to absorb technology and employ it effectively, and this surely implies that they cannot create improved technology without these either. Thus, the macrotheorists are undoubtedly right in emphasizing the role of human capital and its use in the research process. There is also good reason to concur with Romer’s judgment that “[t]his … offers one possible way to explain the wide variation in growth rates observed among countries and the fact that in some countries growth in income per capita has been close to zero” (1990, p. S96). On the other hand, it seems hard to explain the poverty in ancient Greece and the apparent absence of rapid growth in per capita incomes in ancient Rome and medieval China as resulting from the absence of human capital. As a matter of fact, in Imperial China human capital was sufficient to produce an astonishing stream of inventions, as we saw in chapter fourteen. But, although invention may have been abundant, innovation, defined as the critical step of effective utilization in the production process, was evidently rare in those economies. It is surely plausible that a critical part of the explanation was the absence of an effective incentive mechanism and, in particular, of a drivingly competitive market that rewarded innovation and mercilessly punished failure to take advantage of innovation opportunities.
I have already suggested one further link in this argument. Investment in human capital, like investment in plant and equipment, no doubt is to some degree itself an endogenous variable, and its magnitude is clearly increased by growth in the wealth of a nation. The free-market mechanism, through its stimulation of innovation, has served to create the national wealth that has permitted increased expenditure on education as well as on physical capital. The evidence is clearly consistent with the conclusion that rapid economic growth since the Industrial Revolution has increased real per capita expenditure on education far beyond anything previously experienced. Moreover, before the surge of innovations that launched the Industrial Revolution, the poverty of the economy meant that national savings were far too low to permit any substantial investment in human capital (or in physical capital).
Thus, whereas an upsurge in innovation was arguably a prerequisite for significantly expanded education, marked educational expansion does not seem to have been required for innovation. It took only a few relatively educated individuals in the eighteenth century to launch the “wave of gadgets” that underlay the Industrial Revolution in England. Indeed, neither James Watt nor John Harrison, the creators of probably the era’s most economically important technological improvements, seems to have had a very advanced education. It can be argued that it was the free market that transformed such technological improvements into rapid economic growth, and that it was the growth, in turn, that underlay the spread of education. If so, although human capital investment must continue to be considered a necessary requirement for substantial technical progress, it can be argued that the capitalist market mechanism is both necessary and (indirectly) sufficient for the task. This is because the capitalist market mechanism serves not only to provide the incentives required for abundant innovative activity but also to stimulate the indispensable investment in human capital.
Earlier chapters sought to deal with the critical role of competition in innovative activity and growth, but did so at a cost. The microeconomic approach employed for the purpose prevented a formal analysis, even at a rudimentary level, of the implications for the economy as a whole. As a step toward elimination of this gap, this chapter provides a macroeconomic model, but one that is very different from the models that have just been discussed. In doing so, I make no claim of superiority of my model. I claim only that it is tied more closely to the microeconomic structures of earlier chapters, and that it seeks to shed light on aspects of the economy’s growth mechanism different from those dealt with in the mainstream macromodels. Specifically, the model focuses on the influence of the cost of the (routine) research and development process, examining how this is affected by the economy’s rate of growth of productivity. My arms-race ratchet model of R&D in chapter four will be taken into account, to show how the demand for R&D can be influenced by this cost. This leads directly to a feedback relationship in which R&D affects the future growth of the economy, and that growth, in turn, influences the volume of R&D. The model that describes this feedback will be used to bring together the disparate elements that have just been mentioned, and to tie them into a coherent whole.
DIMINISHING RETURNS TO CAPITAL AND TO INNOVATIVE EFFORT AS LIMITS TO GROWTH
As just noted, a critical feature of the original Solow model is the assumption that the accumulation of capital yields diminishing returns. This premise is necessary to yield the steady state upon which the analysis focuses, along with much of subsequent growth theory that follows its example. Obviously, in a model in which growth is driven by accumulation, diminishing returns to accumulation can be expected to lead to a slowdown.
If, however, innovation is also a primary engine of growth, this premise is not enough to ensure a slowdown. Rather, one must also assume diminishing returns to (accumulated) innovation. The picture must be an intertemporal process, like Ricardo’s parable in which the best and most accessible land is utilized first. The story here is that inventors first undertake the projects that are easiest to carry out and that promise the greatest returns. That is, inventions are explored and carried out in the order of their benefit–cost ranking, either because this is the order in which the ideas occur to inventors, or because this is the trajectory adopted by the innovators’ rational calculations. Then, as time passes, the net yield of inventive activity must grow ever smaller. New ideas become ever harder to obtain and, in any event, the best ones have generally been exploited previously. This can be expected to make such activity ever less attractive to inventors and investors and, even if no slowdown in the activity follows, the gains from a fixed level of activity must decline. The scenario is straightforward enough to require no further explanation. It is certainly a conceivable course of events. Still, one must beware of casually taking too dim a view of the future of innovation. There is always the warning example of Charles H. Duell, commissioner of the U.S. Office of Patents, who in 1899 said, “Everything that can be invented has been invented,” and, in his shortsightedness, recommended that the patent office be shut down (cited in Federal Reserve Bank of Dallas, 1996, p. 5).
I turn next to what seems a less obvious threat to continuation of spectacular innovation and growth in the free-market economies. This involves a mechanism that cumulatively increases the relative cost of the R&D process and consequently threatens to bring a secular decline in the quantity of this activity that is demanded.
COST, DEMAND, AND THE VOLUME OF R&D ACTIVITY
Here, I focus on only two of the endogenous influences that can be expected to affect the volume of innovative activity: the demand for products that use the results of R&D and other innovative activity, and the cost of such innovative effort.
Demand
Schmookler (1966) offers extensive evidence indicating that the amount of innovation is affected by the size of the market for related final products. That is, the flow of patented inventions in a particular industry appears to parallel closely the volume of sales of a product both over the business cycle and in terms of longer-run trends. It follows that growing population and expanding GDP can both speed up the pace of innovation. In terms of our arms-race ratchet model of innovation, this and other evidence on the role of demand suggest an influence in addition to a technical breakthrough that can lead firms in an industry to exceed the current norm in R&D spending. Rising demand can move the industry to a new and higher norm—an enhanced level of expenditure on innovation that firms in the industry feel it necessary to adopt in order to stay abreast of their rivals.
But, while enhanced demand seems to make a clear contribution to the magnitude of R&D activity in an industry, the role of Keynesian aggregate demand in the economy is not so straightforward. A powerful suggestive example is provided by Ester Fano (1987), the distinguished Italian economic historian. She reports that during the Great Depression in the United States the employment of scientists and technicians grew markedly. She tells us, “the evidence shows that industrial research underwent such a sustained boom in the 1930s that it could be expected to produce, in addition to cost-reducing devices, a large number of new products as well” (p. 263). More specifically, she reports:
Between 1921 and 1938 industrial research personnel rose by 300%. In 1927 approximately 25% of its employees reportedly worked on a part-time basis; by 1938 this proportion had fallen to 3%. Laboratories rose from fewer than 300 in 1920 to over 1600 in 1931 and more than 2,200 in 1938; the personnel employed increased from about 6,000 in 1920 to over 30,000 in 1931 and over 40,000 in 1938. The annual expenditure rose, from about $25,000,000 in 1920 to over $120,000,000 in 1931, to about $175,000,000 in 1938. In 1937, industrial research on an organized basis in the United States ranked among the 45 manufacturing industries which provided the largest number of jobs. (1987, pp. 262–63)
There is probably no more dramatic case than that of the Great Depression to test (and reject) the proposition that weakness of aggregate demand invariably handicaps innovative activity.
Though this fascinating history clearly must entail many influences, a plausible major component of the explanation of Fano’s observation, which I believe to be the key to the story, is cost. The Great Depression was a period in which the earnings of scientists, engineers, and technicians were extremely low. Since R&D is a labor-intensive activity, the low remuneration meant a major reduction in its relative cost compared with that of activities that are less labor intensive. This brings us to the influence of cost on innovative activity, the central element of the model of this chapter.
The Cost of Innovative Activity
In routinized innovation, as I have emphasized, expenditure on R&D and other innovative activities is just another of the types of investment the firm can use as an input in its production process, that is, as an instrument for the acquisition of revenues and profits. If the relative prices of such inputs change, we can expect substitution to take place, with some degree of replacement of the input whose relative cost has risen by another input whose price has fallen. This at once suggests that the derived demand for investment in innovation has a nonzero cost elasticity that, as Fano’s evidence indicates, may well be substantial. In particular, sharp increases in the earnings of technical personnel can lead to a significant cutback in real investment in innovation.
This may appear to conflict with the conclusions of the ratchet model of chapter 4. That model, as we have seen, asserts that innovation expenditure is sticky downward, because in an innovation arms race no firm dares to cut its R&D outlays unilaterally. With no firm willing to take the first step, no such reduction will take place. However, there are two influences that modify this conclusion. First, in a period of extreme economic upheaval, such as a major depression or a serious inflation, clearly the norms guiding business decisions in more settled time are threatened and may well be abandoned.4 Second, and more directly to the point in the discussion that follows, it is my hypothesis that in the short run the R&D investment norms are calibrated in nominal rather than real terms, at least to a considerable degree. That is, when the costs of R&D rise because technical personnel wages or other costs increase, the R&D norms will not rise immediately or by the fully corresponding amount. It is highly implausible, given what we know of business responses to changing cost and price levels, that expenditures of the firm will be adapted to inflation fully and without any lag. It is even conceivable that, for some interval of time, nominal expenditures will not be adjusted at all, so that the price elasticity of demand for innovation investment will be unity. I will discuss this assumption further in chapter sixteen, where some empirical information on the subject is also provided.
It will be argued next that, in the innovation process, such changes in its cost are neither fully exogenous nor exclusively random. On the contrary, the flow of innovation itself has a critical influence on these prices and costs. This follows from the effects of innovation on the rate of productivity growth in the economy as a whole, as well as the persistence of differences in productivity growth rates of different sectors of the economy.
THE COST DISEASE MODEL AND INNOVATION: AN INTUITIVE SUMMARY
A major component of the analysis that follows is based on a model—since called the cost disease model, or the unbalanced growth model—that William Bowen and I introduced some time ago (see, for example, Baumol, 1967) to show how persistent and relatively rapid increases in productivity in some sectors of the economy (the “progressive sectors”) must lead to persistent and cumulative increases in the relative costs of the sectors that constantly grow more slowly (the “stagnant sectors”). I will argue that the cost disease of the technologically stagnant personal services can affect research in somewhat the same way that it does education, health care, legal activity, the performing arts, and a number of other services, and can lead to a persistent rise in the real cost of research, in the manner that characterizes other services whose productivity (at least so far) is not easily increased year after year, so that they do not benefit from as much of a productivity offset to rising wages as other economic sectors do. Of course, there are many differences between the cost structure of R&D and, say, health care. My point is only that they do have a common element: the handicraft component of their production processes and the relative difficulty of achievement of reductions in the overall role of that component.
Some years after the cost disease model was introduced, it was extended to include a hybrid sector, the asymptotically stagnant sector, defined to employ in relatively fixed proportions some inputs supplied by the progressive sector and some by the stagnant sector (see, for example, Baumol, Blackman, and Wolff, 1989, chapter 6). The stagnant sector and its behavior play a critical role in the analysis of this chapter. Two industries that seem to fit the description of asymptotic stagnancy rather closely are television broadcasting (whose primary inputs are electronic equipment and live performance) and computer usage (whose main inputs are sophisticated hardware and human labor devoted to software creation,5 data gathering, and so on). Since the story will be familiar to a number of readers, I will summarize it very briefly and intuitively here, leaving demonstration of its properties to the appendix to this chapter.
The asymptotically stagnant cost is characterized by a distinctive intertemporal cost pattern. Initially, its cost tends to fall rapidly. But after some time this is reversed and the trajectory of the unit cost of the output of the asymptotically stagnant sector approaches closer and closer to that of its stagnant input. The intuitive explanation is simple. The falling cost of the progressive sector input accounts for the initial fall in the real unit cost of the asymptotically stagnant sector. But the very fall in the cost of that input reduces its share in the total costs of the asymptotically stagnant sector, leaving the behavior of those costs to be determined largely by the course of the stagnant sector input. Hence, an initial period of decline in the cost of the asymptotically stagnant sector is followed by a future of rising relative cost.6 It is noteworthy that this sector appears to include some of the economy’s most high-tech activities.
R&D AS ASYMPTOTICALLY STAGNANT ACTIVITY
There is reason to presume that the cost trajectory of R&D activity falls somewhere between those of a sector that is purely stagnant and one that is asymptotically stagnant. R&D may, itself, be thought of as using, preponderantly, two types of input—mental labor (that is, human time) and technological equipment such as computers—making it an activity approximating the characteristics of asymptotic stagnancy, though one with some intertemporal variation in input proportions. Innovation is such an activity. The act of thinking is a crucial input for the research process, but have we become more proficient at this handicraft activity than Isaac Newton, Gottfried Leibniz, or Christiaan Huygens? Probably not. The productivity of labor has risen at an annual rate slightly less than 2 percent compounded since roughly 1830, when the Industrial Revolution really took off, so that the real product of an hour of labor has multiplied by a factor of perhaps twenty since that time. This means that the opportunity cost of an hour devoted to the technologically stagnant process of thinking must have risen by about 1,900 percent! If R&D is interpreted as just another input in the production process, such a rise in its relative price must have cut back its derived demand—inducing some substitution away from this input and toward other inputs whose real cost was reduced by technical change. The cost disease of the stagnant component of research, then, may conceivably be a major impediment to acceleration of innovation.
The way in which this works out is best shown with the aid of a model of the feedback relationship between the production and dissemination of information and the rate of growth of productivity in industry. Here R&D is considered to be a sector of the economy that is engaged in the production of information. The magnitude of its information production clearly influences the rate of productivity growth. However, as we will see, that growth rate in turn will affect the output of information, thereby closing the feedback loop, with effects on the trajectory of innovation that may not be obvious without a formal model.
In brief, the analysis has three elements (three sequential steps):
1. Production of new information through R&D activity stimulates productivity growth in industry.
2. As a result, the price (real cost) of information production and dissemination rises. This is because these activities are what my colleagues and I have called “asymptotically stagnant,” characterized by productivity growth that is initially high but that, with the passage of time, for reasons to be explained, tends to lag further and further behind that of industry.
3. As information grows relatively more costly, other inputs tend to be substituted for information in the production process. For example, when R&D costs have risen, a firm that wants to increase its output may decide not to invest more in R&D designed to raise the productivity of its machines but, instead, to buy additional machines of the current type. Thus, the rising cost of the innovation process can cut the derived demand for innovative activity. That, in turn, impedes productivity growth, thus reversing the first of the three steps of the intertemporal process in its next iteration.
All of this indicates why R&D activity can be expected to behave very much like an asymptotically stagnant sector. However, this conclusion also suggests others. One of the immediate implications is that, depending upon the price (and income) elasticity of the (derived) demand for R&D, it is possible that its rising relative cost will reduce its use relative to other inputs with the passage of time. Moreover, it can be shown that, with time, in my model (a) output of R&D may decline, (b) total expenditure on R&D may rise, both absolutely and as a share of GDP, and (c) the amount of R&D labor time may fall relative to GDP.
FEEDBACK MODEL: PRODUCTIVITY GROWTH AND ENDOGENOUS INNOVATION
In the basic cost disease model it is assumed that in the progressive sector the productivity of labor grows at a constant percentage rate, g. However, this premise ignores a crucial relationship—that between R&D (information production activity) and the technical change in the other sectors that thereby obtain their productivity growth. Thus, instead of g being a constant, it must be a function of the quantity of information produced by R&D that is used as an input by other sectors.
To combine all of the resulting relationships and determine their effects, we must modify the formal model further.7 In the notation here, the subscripts indicating the sector of the economy in question are not needed. Let
gt = |
the rate of growth of productivity outside R&D (information-producing) industries in period t; |
yt = |
the output (level of activity) of R&D activity; |
pt = |
the price of the product of R&D; |
h = |
a parameter determining price-insensitive R&D activity. |
Here I assume that R&D comes from two sources and will refer, correspondingly, to price-sensitive R&D, ypt, which is generated preponderantly by routine business activity, and price-insensitive R&D, yit, so that
Then one can formulate the following illustrative relationships:
(R&D contributes to productivity growth),
(the price of R&D output grows proportionately to gt);
and, for the case where price-insensitive R&D is zero, so that ypt = yt,
(the R&D demand function),
where E > 0 is a bastard intertemporal demand elasticity which, for simplicity of illustration, is assumed to be constant.8 For simplicity, I will also adopt as the price-insensitive R&D output function
This implies that R&D activity encourages and facilitates independent R&D activity, and that the relationship is proportionate. Later, I will briefly consider some alternative forms of the relationship (15.5).
First, let us assume that there is no price-insensitive R&D, so that h = 0 in (15.5). Then, substituting (15.2) and (15.3) into (15.4), we have at once
or
which has the two equilibrium points, at which ye = yt = yt + 1,
We obtain from (15.7)
Next, dealing with the case in which there is price-insensitive R&D, as given by (15.5), I assume that its amount is set using some average or target share, a, of total R&D as a base, so that the price-sensitive portion of the demand function satisfies, instead of (15.4),
Consequently,
which is now our feedback relationship, yielding a trajectory whose properties are studied next.
Setting ye = yt + 1 = yt, we now obtain the two equilibrium values
To test for their stability we note that by (15.12)
which equals
and
To determine the implications for the intertemporal trajectory of R&D activity, we must first discuss the parameter values. As in (15.6)–(15.11), I begin with the case h = 0, meaning that all innovation is price sensitive. Then we expect a = 1 because a is the (target) share of price-sensitive R&D in total R&D. With h = 0 and a = 1, it is easy to verify that the relationships for the general case (15.12)–(15.16) each reduce to the corresponding relationships (15.6)–(15.11). We can assume that a, h, s, and b are all less than unity except where h = 0. This is obvious for h and a, because they can be interpreted, respectively, as the target shares of price-insensitive and price-sensitive R&D as a proportion of total R&D. Parameter s is the rate of growth of productivity in the absence of all R&D activity, and consequently should be very small. Similarly, it is implausible that b, which is dgt + 1/dyt by (15.2), should exceed unity. We have, finally, k = vE, where v is the effect on the rate of growth of R&D cost of a change in productivity growth rate in the remainder of the economy, and is presumably small. E, the (absolute value of the) price elasticity of demand for R&D, is likely to be considerably smaller than 2 (indeed, my ratchet model for nominal expenditure implies E is near unity). This suggests that k can be assumed not to be far above unity, and presumably below that figure. All of these parameters are presumably nonnegative.
Then (15.10) tells us the equilibrium point ye = 0 will be monotonic and stable when h = 0. That is, the slope of the phase graph of (15.9) will be positive but less than unity at the origin, yt + 1 = yt = 0 (figure 15.1 on page 279). Similarly, by (15.11), equilibrium point ye = −s/b will be negative and unstable so the time path of R&D will go steadily toward the origin, moving monotonically toward zero R&D and productivity growth rate. That is, we are led in this case to the grim conclusion that successful innovation can handicap and slow further innovation effort, driving it toward a stationary state with zero further outlay on this activity. It can do so by increasingly raising the relative cost of the handicraft portion of research and development activity, leading investment to search for alternatives whose relative cost does not grow so persistently.
As is to be expected, this pattern continues if h becomes positive, though remaining very small, and a declines slightly below unity, as is shown by (15.15) and (15.16). However, when h becomes sufficiently large, that is, if future independent innovation is stimulated sufficiently by the amount of current (and successful) innovation, then, if a does not decline, so that a + h > 1 by a sufficient amount, the nonzero equilibrium point will take a positive value and become stable (figure 15.2 on page 280). This is the case that is to be hoped for, with both innovation and output growth expanding “forever,” though they both must gradually slow down and approach a fixed upper limit. Then the cost disease will no longer be able to engender stagnation, and outlays on innovative activity, though approaching a positive equilibrium level, which they will have no tendency to surpass, will also have no tendency to decline, with real outlays approaching the equilibrium level for the indefinite future.9
With h sufficiently large, so that dyt + 1/dyt becomes negative, the trajectory will grow oscillatory, finally lapsing into explosive oscillation with cycles of increasing amplitude. Ultimately, it may well lead to a limit cycle, to the left of the maximum of the parabola that is the graph of our feedback relationship (15.9). It is even possible that h will reach a value at which a chaotic regime results.
Intuitively, such cycles are generated by the sequence generated by an initial leap in R&D activity. This, thereafter, substantially increases the productivity growth of the economy, causing a sharp rise in the cost of R&D, which in turn slows down productivity growth, and so on and on.
But much more pertinent to our central subject is the implication of the case h = 0, a = 1, or, more generally, the case a + h = 1. Then the only stable equilibrium is ye = 0. As we have just noted, this means that, in an economy that behaves as this model does, rising productivity and the consequently rising cost of R&D will in the long run drive R&D expenditure steadily toward zero. To inquire a bit further into the intuitive reason, roughly, a is the share of past R&D expenditure that the business sector and other price-responsive innovators plan to undertake in the next period. Similarly, h can be viewed, roughly, as the share of past R&D expenditure that other innovators plan to undertake in the following period. Then, a + h < 1 implies that the two innovator groups together plan to undertake less innovation in the next period than they did in the past. But this effect is exacerbated by the negative slope of the demand curve for R&D of the first of the two innovator groups. When a + h = 1, although the two groups together would have spent the same amount as they did in the previous period if the price of the R&D input had not risen, the previous period’s R&D will have increased the economy’s productivity and so it will have increased that price. The net result is that, even when the sum of the two parameters is unity, in this model R&D spending will decline steadily. In the next chapter we will see why the scenario of reality is not quite so dismal.
GRAPHIC ANALYSIS OF THE FEEDBACK MODEL
It is perhaps possible to get a clearer picture of the behavior of the model with the aid of the standard pseudo-phase diagram for a difference equation. Figure 15.1 depicts the case in which no price-insensitive R&D is generated, that is, in which h = 0 in (15.5). Then, the basic feedback equation (15.7) can be represented in (yt, yt + 1) space as the parabolic locus LL in figure 15.1. The equilibrium points are, of course, the points 0 and E where LL intersects the 45° line, that is, where yt = yt + 1. Since, as has been shown, the nonzero equilibrium point in this case is negative, E lies to the left of the vertical axis. This equilibrium represents a (nonsensical) negative amount of R&D activity, and is really irrelevant. What is much more to the point is what happens after beginning with an initial position y0 > 0. We see that with the relationships as shown in figure 15.1, with the 45° line above LL, there will be a time path y0ab … that leads monotonically (that is, without any deviation), toward y = 0, period after period.
FIGURE 15.1
Determining the Time Path of Innovation Output: Decreasing Output Case
FIGURE 15.2
Determining the Time Path of Innovation Output: Stationary Equilibrium Case
Figure 15.2 shows how matters change when the economy also provides some price-insensitive R&D whose quantity follows the expression yit + 1 = hyit, h > 0, as assumed in my expanded illustrative model. Since this last relation obviously is represented by a positively sloping straight line through the origin, its addition to (15.7), to obtain (15.12), simply results in a counterclockwise rotation of LL in figure 15.1 around the origin, moving it to the position of LL in figure 15.2. We see that this tends to move the position of equilibrium point E to the right. For h sufficiently large, it transfers E to the right of the origin, as shown in figure 15.2. The new equilibrium point, with its positive level and growth rate of R&D, can be stable since at that point LL can cut the 45° line from above.
The significant role of price-insensitive innovation activity makes it important to emphasize that in our model it can be taken to refer to more than the work of the lone, dedicated inventor who works fanatically in basement or garage. It also can include, for example, the unexpected breakthrough originating in a bureaucratically run corporate R&D facility that induces the firm to undertake a substantial increase in the resources it devotes to innovation. Although this will obviously differ from industry to industry, and can be expected to occur only sporadically, if at all, in any particular firm, in the economy as a whole its expected value may conceivably be fairly predictable.
The discussion of the graphs also indicates how one can analyze cases in which price-insensitive R&D activity is described by possibly more complicated relationships than yt + 1 = hyt. For example, one may want to explore nonlinear relationships with diminishing returns as enhanced current R&D diminishes the stock of possible innovations that it is easy to explore and develop in the future. Or, alternatively, one can consider the possibility of increasing returns as successful innovation makes further innovation activity less costly and more powerful. Since, in analyzing such relationships, they are added to the part of (15.7) that describes routine business-provided R&D, one need merely see what happens when the graph representing such a relationship is added to LL in figure 15.1.
We can readily confirm that the behavior of the model does not depend on the particular functional forms employed in (15.1)–(15.5), and that its qualitative properties remain robust under a considerable broadening of my premises. The analysis indicates that, if there really is a feedback relationship between information activity and productivity growth in industry, this raises the possibility of a non-self-terminating trajectory, with monotonically declining productivity growth and information production. This is certainly a disturbing prospect for productivity policy.
CONCLUDING COMMENTS
A primary purpose of my model has been to show how it is possible to analyze more explicitly the ways in which outlays on R&D and the rate of growth of productivity can be determined endogenously and simultaneously. The model has sought to introduce the role of the price mechanism and significant productivity growth explicitly, taking a first step toward elimination of the ahistorical character of the mainstream macromodels of growth. This at least suggests one of the sources of strength and some of the vulnerability of the market economy as a mechanism that produces innovation and growth.
Specifically, the discussion has taken account of the likelihood that an increase in R&D expenditure will stimulate productivity growth, albeit with some lag. However, it has also recognized that the very rate of productivity growth in other sectors contributed by R&D affects the relative cost of R&D and, arguably, tends to increase that relative cost. Thus, the very success of the work of the R&D sector may sow the seeds of a future price impediment to demand for its output.
This rather primitive macroeconomic model appears to paint a rather pessimistic picture for the future. There are, however, built-in offsets that raise the possibility of more promising prospects. This is the subject of the next chapter.
APPENDIX
DERIVATION OF THE PROPOSITIONS ON THE COST DISEASE AND ASYMPTOTIC STAGNANCY
This appendix reviews the relevant derivations of the properties of the cost disease model, which were described earlier in the chapter. In the deliberately oversimplified cost disease model, the economy is taken to be composed of two sectors: a progressive sector, 1, in which productivity grows exponentially, and a stagnant sector, 2, in which productivity remains constant. There is a single input, labor, with quantities L1 and L2 used by the respective sectors, and whose outputs are given by
Letting w represent the common wage rate, we have the unit-cost figures
This yields for the stagnant sector a relative cost that grows exponentially with time:
Thus we have
PROPOSITION 15.A1. In the cost disease model, the per unit cost of the output of the stagnant sector will rise without limit relative to that of the progressive sector.
Next, I introduce the third prototype sector, the “asymptotically stagnant” sector, a simplified representation of sectors of the economy that include R&D activity. This sector is composed of activities that use in (more or less) fixed proportions two different types of input, one produced by progressive sector 1, and the other of which either is obtained from stagnant sector 2 or is composed of pure labor (or some combination of the two).10 Assume for simplicity that input–output proportions are absolutely fixed. Let y3 represent the output of the sector and y13 and y23 be the inputs of the other two sectors used in the production of y3. We can then write
where by choice of units we can set k2 = 1. We then obtain for the average cost of sector 3
Measured in terms of labor units, that is, holding w constant, the first term of (15.A5) must approach zero asymptotically. Consequently, the behavior of AC3 over time will approach that of the stagnant sector, 2.
Thus, we have
PROPOSITION 15.A2. The behavior of the average cost of an asymptotically stagnant sector will approach, asymptotically, that of the stagnant sector from which the former obtains some of its inputs.
What is surprising about the phenomenon we are discussing is that the sectors of the economy suffering from the asymptotic stagnation problem in its most extreme form include, in reality, some of those providing the most “high-tech” activities—those in the vanguard of innovation and change. That this is predicted by the theory should be clear from (15.A5), which shows that the more rapid the rate of productivity growth of the sector 1 input—that is, the greater the value of r—the more rapidly will the intertemporal behavior of AC3 approach that of the nonprogressive sector.
So far, it has been assumed for simplicity that there is a fixed ratio between the input quantities sector 3 derives from the progressive and the stagnant sectors. But this premise was used only for expository convenience. It is easy to show that, over a broad range of patterns of behavior of input proportions, the same sort of cost problems arise for the asymptotically stagnant sector.
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1. Though my discussion focuses on endogenous innovation, it is, of course, not meant to imply that exogenous or purely fortuitous developments play no role.
2. And they were far less pessimistic about the future than is sometimes suggested. For the evidence, see Hollander (1998, p. 232) and Ricardo’s parliamentary speeches, as reported in Gordon (1976).
3. The following discussion draws heavily on the symposium on New Growth Theory in the Journal of Economic Perspectives 8 (Winter 1994), pp. 3–72, and on Paul Romer’s contribution in particular (1994b).
4. This is related to an important observation that follows from the valuable work in evolutionary (or out-of-equilibrium) economics of Nelson and Winter (1982) and Amendola and Gaffard (1998). They note that in an intertemporal process analysis, such as the one that follows, it may not be legitimate to assume that the parameter values or the structure of the model itself remain unchanged with the passage of time. Developments that are particularly extreme, including internally generated disturbances, can force the pertinent model to change sharply. Thus, the appropriate model itself can be subject to endogenous modifying forces.
5. However, as has been shown in an unpublished article by François Horn of Université Charles de Gaulle in Lille (2000), productivity in software creation has been rising rapidly, even if not as quickly as in hardware. Thus, it is surely incorrect to characterize all non-hardware inputs into the computation process as stagnant in terms of productivity growth.
6. For some empirical evidence on the cost behavior of the sectors used here as illustrations of asymptotic stagnancy, see Baumol, Blackman, and Wolff (1989, pp. 131–40).
7. This model is based on earlier writings by Baumol and Wolff (1983).
8. This negatively sloping demand curve for R&D activity seems an obvious assumption. It is supported by some evidence (see, for example, Schmookler, 1966) and, as described above, in the work of Fano. The assumption is also employed elsewhere in the literature, for example in Grossman and Helpman (1991). It should also be noted that the equation is a generalization of the premise, mentioned earlier in this chapter, that oligopolistic competition forces the firms discussed in chapter 4 to fix their nominal R&D expenditure at the level, K, the current industry standard. For then (omitting subscripts) y = K/p so that y′p = −p′y, and hence y′/y = −p′/p. The subject is discussed further in the next chapter.
9. It may be thought, as I once did, that, even in this more fortunate case, there must arise another impediment to growth: the slowing of average productivity growth in the economy as increasing innovative activity leads to a transfer of labor from the (productivity) progressive final-product sector to the asymptotically stagnant innovation sector. For then the average productivity of the two sectors, weighted by the sizes of their labor inputs, must ultimately decline.
However, Nicholas Oulton of the Bank of England (2000) has contributed an important new theorem that entails very different productivity behavior in the economy when, as is true of innovation, this (ultimately) more stagnant activity supplies an intermediate product that is used as an input elsewhere in the economy rather than being used directly by consumers. Oulton shows that, when the allocation of inputs is efficient and entails a shift of primary input from the progressive final-product sector to the stagnant or asymptotically stagnant intermediate-product sector, then the economy’s overall productivity can, in rather general circumstances, actually be expected to grow faster. That is, the shift of labor to the stagnant sector can in this case be expected to increase overall productivity growth rather than depressing it. It does so because in this case productivity growth in the two sectors is additive, with growth of labor productivity in innovation, however limited, indirectly adding to the given total factor productivity (TFP) growth in the final-product sector, thereby enhancing productivity growth in the final product of the given labor force of the economy. As Oulton explains the seemingly “very paradoxical” result (assuming, for simplicity, that innovation is produced by labor alone, whereas final product uses both labor and innovation as inputs, as in my model here):
There are two ways in which the economy can obtain more final product, given that total employment is fixed. One is if TFP rises in final product supply, the other is if TFP rises in innovation. … TFP growth in innovation raises the productivity of labor employed there. … Hence TFP growth in innovation causes higher final-product output, since the final-product sector buys in the output of the innovation sector. The higher the proportion of the labor force employed in innovation, the bigger the impact on TFP growth in the final-product sector [given the rate of TFP growth in the final-product sector’s utilization of its two inputs]. … The reason is that such a shift will raise the contribution to the aggregate coming from innovation without reducing the contribution coming from the final-product sector. (2000, pp. 14–15; throughout the quotation, for more direct applicability to the current discussion, I have substituted “final-product output” for “car output” and “innovation” for “business services”)
10. It makes no difference to the analysis whether sector 3’s second input is y2 or labor itself, since both costs behave similarly. That is, AC2 = w/b, by (15.A2), and the average cost of labor to industry is simply w. Indeed, we can aggregate the supply of y2 and labor to industry 3 into a single broader sector that offers them both and that is clearly stagnant. If sector 3 uses L3 hours of labor and y23 units of y2 per unit of y3 produced, then writing y23 = aL23 it is clear that a unit of output 3 requires L3 + L23 hours of labor altogether. Thus, we may invent a fictitious output y2* satisfying y2*/a = L3 + L23 as the product of the aggregated stagnant sector that supplies both labor and y2 to sector 3.
