Oxford Review of Economic Policy, Volume 23, Number 1, 2007, pp.3–14

The last 50 years in growth theory and thenext 10

Robert M. Solow∗

Abstract This article offers a personal view of the main achievements of (broadly) neoclassical growththeory, along with a few of the important gaps that remain. It discusses briefly the pluses and minuses oftwo major recent lines of research: endogenous growth theory and the drawing of causal inferences frominternational cross-sections, and criticizes the widespread contemporary tendency to convert the normativeRamsey model into a positive representative-agent macroeconomic model applying at all frequencies. Finally,it comments on the articles appearing in this symposium.

Key words: Solow growth model, growth theory, 1956 anniversary

JEL classification: B22, E13, O41

I am cheered and delighted by the attention paid to this anniversary; but I am also alittle embarrassed. Why embarrassed? Because I really believe that progress in economics(and other similar disciplines) comes more from research communities than from any oneindividual at a time. It is research communities that separate the good stuff from the routine,and see to it that the sillier outcroppings of imagination get sanded down. At least it worksthat way most of the time. We owe more than we acknowledge to our colleagues and graduatestudents.

Here is a partial example that I will come back to in a minute. If you have been interestedin growth theory for a while, you probably know that Trevor Swan—who was a splendidmacroeconomist—also published a paper on growth theory in 1956 (Swan, 1956). In thatarticle you can find the essentials of the basic neoclassical model of economic growth. Whydid the version in my paper become the standard, and attract most of the attention?

I think it was for a collection of reasons of different kinds, none individually of verygreat importance. For instance, Swan worked entirely with the Cobb–Douglas function; butthis was one of those cases where a more general assumption turned out to be simpler andmore transparent. As a result, his way of representing the model diagrammatically was not

∗Massachusetts Institute of Technologydoi: 10.1093/icb/grm004 The Author 2007. Published by Oxford University Press.For permissions please e-mail: journals.permissions@oxfordjournals.org

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4 Robert M. Solow

so clear and user-friendly. A second and more substantial reason was that Swan saw himselfas responding to Joan Robinson’s complaints and strictures about capital and growth, whileI was thinking more about finding a way to avoid the implausibilities of the Harrod–Domarstory (Harrod, 1939; Domar, 1946). (I will tell you a relevant anecdote in a minute.) That is tosay, I happened to be coming at the problem from a more significant direction. A third reasonis that Swan was an Australian writing in the Economic Record, and I was an Americanwriting in the Quarterly Journal of Economics. The community of growth theorists took itfrom there.

When I finished that 1956 paper, I had no idea that it would still be alive and well 50 yearslater, more or less part of the folklore. Nor did I understand that it would be the origin ofan enormous literature and a whole cottage industry of growth-model building that is stillflourishing, as the articles in this issue of the Review demonstrate. So why was it such asuccess? Are there methodological lessons to be learned about How To Make An Impression?

My own favourite how-to-do-it injunctions are: (i) keep it simple; (ii) get it right; and(iii) make it plausible. (By getting it right, I mean finding a clear, intuitive formulation, notmerely avoiding algebraic errors.) I suspect that all three of these maxims were working forthat 1956 paper. It was certainly simple; it didn’t get lost in the complications and blindalleys that beset Trevor Swan’s attempt; and it was plausible in the sense that it fitted thestylized facts, offered opportunities to test and to calibrate, and didn’t require you to believesomething unbelievable.

Here is where the anecdote that I promised comes in. I spent the year 1963–4 in Cambridge,England, engaged in one interminable and pointless hassle with Joan Robinson about some ofthese issues. Interminable is bad enough, pointless is bad enough, and putting them togetheris pretty awful. The details are too lurid to be told to young people. At one point, however,I realized that the discussion had become metaphysical and repetitive, and I decided to try anew tack. So I buttonholed Joan in her office one day and said: ‘Imagine that Mao Tse-Tungcalls you in’—she was in her Chinese period then—‘and asks a meaningful question. ThePeople’s Republic has been investing 20 per cent of its national income for a very long time.There is now a proposal to increase that to 23 per cent. To make a correct decision, we needto know the consequences of such a change. Professor Robinson, how should we calculatewhat will happen if we increase our investment quota and sustain it?’

‘So what will you tell Chairman Mao?’ I asked Joan. She baulked and bridled and dodgedand changed the subject, but for once I was relentless. ‘Come on, Joan, this is Chairman Maoasking a legitimate economic question; the future of the People’s Republic and possibly ofmankind may depend on the answer. What do you tell him?’ Finally, she grumbled: ‘Well, Iguess a constant capital–output ratio will do.’ It made my day; I knew I could do better thanthat, and I knew she had been forced by practicality, even imaginary practicality, to give upthe metaphysical ghost. I was smiling all the way home to tell my wife that Joan had buckled,and violated her own metaphysics.

One of her major contentions had been that it was illegitimate to think of ‘capital’ as afactor of production with a marginal product. Yes, a single capital good (or its services)was a productive input. But aggregating those goods, whose services are yielded over theirremaining lifetimes, introduces all sorts of complications. It is always a problem in economicsto navigate between pure and abstract conceptions (how would a concept like ‘capital’ fitinto a complete and formal description of an economy) and the needs of practical calculation(Mao’s hypothetical question). It can (almost) never be done perfectly. I thought that JoanRobinson had been unfairly playing on that difficulty in order to undermine the ‘neoclassical’

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attempt to construct a usable model of investment and growth. Faced with the need to bepragmatic, she had no recourse but the kind of statement that she had criticized in others.

That is what I mean by making it plausible: a simple, clear model should tell you howto get from empirical beliefs to practical conclusions. There will always be additions andmodifications to fit the occasion, but the model should provide a road map. I still think thatis how growth theory should be done, though the beliefs and conclusions may, of course,change through trial and error and the passage of time, and reasonable conclusions can’t bemore detailed than the model will bear.

With those general principles as background, I suppose I should say something about thetwo most important innovations to come along in the past 50 years within the framework ofneoclassical growth theory. The two I have in mind are, as I hope you would guess, first,‘endogenous growth theory’ as pioneered by Paul Romer (1986) and Robert Lucas (1988),and then taken up by an army of economists, and, second, the drawing of inferences aboutthe determinants of economic growth from international cross-sections, an activity whosefirst protagonist may have been Robert Barro (1991), also with innumerable followers amongindividuals and institutions. This second line of thought only became thinkable after thepublication of the Penn World Tables by Robert Summers and Alan Heston. No argumentswithout numbers, and they provided the numbers.

The story of endogenous growth theory may (repeat: may) turn out to be a good exampleof the way a research community takes a new thought and moulds it into something useful.One of the earliest products of endogenous growth theory was the so-called AK model, whichI thought from the first to be a distraction. It claimed to endogenize the steady-state growthrate by what amounted to pure surface assumption. It was simple, all right, but neither rightnor plausible. The community eventually made an implicit judgment and sees less of it thesedays.

Then a further thought dawned on me. If you want to endogenize ‘the’ growth rate ofx, you are going to need a linear differential equation of the form dx/dt = G(.)x, wherethe growth-rate G is a function of things you think you know how to determine (but not afunction of x or its growth rate). Exponential curves come from that differential equation. Soburied in every ‘endogenous growth’ model there is going to be an absolutely indispensablelinear equation of that form. And sure enough, if you root around in every such model youfind somewhere the assumption that dx/dt = G(.)x, where x is something related to the levelof output. It may be the production function for human capital, or the production functionfor technological knowledge, or something else, but it will be there. And the plausibility ofthe model depends crucially on the plausibility and robustness of that assumption. I wantto emphasize how special this is: it amounts to the firm assumption that the growth rate ofoutput (or some determinant of output) is independent of the level of output itself.

If you want to endogenize the steady-state growth rate in a model driven by human-capitalinvestment or technological progress, you need precisely this linearity. But then, I think,you owe the community a serious argument that this assumption is either self-evident orrobustly confirmed by observation. My impression is that this demonstration has not beenforthcoming. The literature seems to take it for granted and move on to elaboration.

Is that just all in the game? I think there has been one unfortunate semi-practicalconsequence. Some of the literature gives the impression that it is after all pretty easy toincrease the long-run growth rate. Just reduce a tax on capital here or eliminate an inefficientregulation there, and the reward is fabulous, a higher growth rate forever, which is surelymore valuable than any lingering bleeding-heart reservations about the policy itself. Butin real life it is very hard to move the permanent growth rate; and when it happens, as

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perhaps in the USA in the later 1990s, the source can be a bit mysterious even after the fact.Endogenizing the steady-state growth rate is a serious ambition and deserves serious effort.

An alternative idea may be to focus less on the notion of exponential growth. One caneasily imagine classes of models and exogenous influences that do not even allow for episodesof steady-state exponential growth. That would have to be a more computer-oriented project,but the building-up of simulation experience under varied assumptions may lead to generalunderstanding.

In the meantime, I suspect that the most valuable contribution of endogenous growth theoryhas not been the theory itself, but rather the stimulus it has provided to thinking about theactual ‘production’ of human capital and useful technological knowledge.

An important example of progress in this direction is the body of work on ‘Schumpeterian’models, much of it focused on the idea of ‘creative destruction’. There have been severalcontributions, dominated by the impressive collection of results by Philippe Aghion and PeterHowitt, together and separately (see, for example, Aghion and Howitt, 1992). I cannot dojustice here to their translation of Schumpeter’s imprecise notion into explicit models thatcan be and have been pursued to a very detailed level. But it illustrates how progress can bemade. Their 1998 book is a monumental compilation (Aghion and Howitt, 1998).

I have no idea whether it will be possible to reduce these motives and processes to thesimple formulas that can go into a growth model. It is not so important; any understandingthat is gained can probably be patched into growth theory, formally or informally. Preciselyfor that reason, one wonders why there has been so little contact with those scholars whostudy the organization and functioning of industrial laboratories and other research groups.

Now, what about international cross-section regressions, or what is sometimes called‘empirical growth theory’? There are two distinct varieties. The first, which is primarily aimedat using cross-section observations to learn something about the aggregative technology, isa serious matter. I think one has to be precise about what the countries in the sample areassumed to have in common and what is allowed to differ among them. The literature hasnot always been careful about this. The paper in this issue by Erich Gundlach is an excellentexample of the genre. I think I will save my handful of comments for later.

The second variety proceeds by regressing the country-specific growth rates during somemedium-long period on a potentially long list of country characteristics. Many of theright-hand-side variables are socio-political, some are intended as indicators of regulatoryinefficiency, some are ‘cultural’. Here I think a little modesty is in order. At a minimum, thoseregressions provide interesting descriptive statistics. It can only be useful to have a good ideaof which national characteristics are associated with faster growth during a fairly long periodacross a large sample of countries. It is when the regressions are interpreted causally that Ibegin to look for an exit.

Reverse causation is only the most elementary of the difficulties. Maybe democracy andsocial peace lead to growth; I certainly hope so. But growth may also lead to democracyand social peace; and since both sides of that relation are likely to change slowly, the usualeconometric dodge of lagging a variable cannot convincingly settle the issue. There is alsoreason to wonder about the robustness of the regression coefficients against variations insample period, functional form, choice of regressors, and so on.

But there are other, deeper, problems. The proper left-hand-side variable is growth of totalfactor productivity (TFP) rather than of output itself, because that is what the right-hand-sidevariables are likely to be able to affect. The array of non-economic influences on TFP iscertainly large and interrelated. Anyone who wants to interpret a cross-country regressioncausally has to believe that a particular coefficient really tells you what will happen to the

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growth rate of a country that experiences an increase of the variable in question by �, andalso tells you what will happen if, a few years later, the same variable decreases by thesame �. Maybe so; but it is surely troubling that we have not observed such manoeuvres.I will continue to think of these things as shorthand descriptions rather than as recipes foreconomic change. You should, of course, ask yourself whether one of those regressionsplausibly represents a surface along which countries can actually ‘decide’ to move (back andforth, remember). That is the acid test.

There is very little space left for stray thoughts about the future of growth theory, which isprobably a good thing; no one can know what the next advance (or fad) is likely to be. I willjust mention a couple of issues that seem to me to have been under-researched until now. Thefirst is open-economy growth theory—the incorporation of trade, capital movements, andtechnology transfer into a multi-country model of growth. Grossman and Helpman (1991)was the pioneering text; but it attracted attention more for its quality-ladder models than forits analysis of trading economies. There have been a few further contributions, but nothingdefinitive.

Laundry-list regressions have sometimes found an association between an economy’sopenness to trade and its growth rate. Classical gains-from-trade theory would suggest anassociation between openness and the level of output. If there is a connection between tradeand growth, one ought to be able to model it convincingly. I am not aware that there is anygenerally accepted story about this. Perhaps there is and I have missed it. Otherwise onewonders why more growth theorists aren’t trying. Foreign direct investment plays a veryimportant role in practice. Why not in theory?

That brings me naturally to a second analytical gap that could perhaps be filled in the nextfew years. We have watched the major European economies almost reach US productivitylevels, and then fall back slightly; we remember the years of extremely fast growth in Japan,once the source of much hand-wringing in the all-too-scrutable West; we now see China,or at least part of China, growing faster than we can imagine. Inevitably we see all these asinstances of ‘catch-up’ to a technological leader, the USA. In the background is always theneed to evolve a skilled labour force.

This seems to be another modelling opportunity. How does, or how should, an economydeploy its resources when it has the opportunity, via foreign investment, to attract bothcapital and already-known technology from abroad? Among the resources I have in mind areintellectual resources. Imitation of known technology is not always effortless. How shouldresearch capacity be divided between imitation–adaptation on one side, and the search forbrand-new technology on the other? I am not sure that theory has much to say about aquestion like this, at least partly because the ‘ripeness’ of a particular technological area hasto matter, and this is something that theories of endogenous technological change seem toignore. It may even have a significant exogenous element.

As a last comment, I would like to drag my feet about a methodological fashion—butone with real substantive implications—that seems to have taken root in growth theory, andappears likely to persist. Fifty years ago, the research community would have made a sharpdistinction between descriptive models of economic growth and normative models of optimalgrowth. In that view, the Ramsey model was important precisely because it would define agrowth trajectory quite different from the paths actually followed by observed economies.Indeed, the first calibrations of the Ramsey model suggested optimal saving-investment ratesfar higher than anything to be found in modern capitalist economies. The excess was largeenough to constitute a serious puzzle (to which Olivier de La Grandville’s article in this issueproposes a resolution).

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More recently, it has become almost universally the custom to use the Ramsey constructionas if it described macroeconomic fact, rather than a hypothetical social-consensus target.The omniscient social planner has morphed into an immortal representative household; andother economic institutions are assumed to have just those characteristics that will inducethem straightforwardly to carry out the owner–worker–consumer household’s desires. Forexample, the identical perfectly competitive firms have to share the household’s version ofperfect foresight or rational expectations. Some minor imperfections may be allowed, but notsuch as to get in the way of the basic formulation. Groups of agents are not allowed to havedifferent beliefs about the way the economy works, or conflicting objectives. All this is toowell known to require elaboration.

The neatness-freak in me can see why this conversion of normative into positive mighthave some initial intellectual appeal. But the pages of a Review of Economic Policy are anappropriate place to say that the model lacks plausibility as a basis for practical proposalsabout growth policy. The only sort of empirical argument in its favour that has been offered byprotagonists is surprisingly weak. The idea is to ‘calibrate’ the model by choosing parametervalues that have respectability in the literature of economics generally. (Understandably,some tweaking is permitted.) When the model is simulated with those parameter values, it canmatch some very general properties of observed time series, usually the absolute or relativemagnitudes of significant variances and co-variances. This is a very lax sort of criterion, andcannot hope to earn much in the way of credibility. There must be scores of quite differentmodels that could pass the same test, but would have different implications for policy. Noone could claim that this sort of model has won its popularity by empirical success.

Instead, the main argument for this modelling strategy has been a more aesthetic one: itsvirtue is said to be that it is compatible with general equilibrium theory, and thus it is superiorto ad hoc descriptive models that are not related to ‘deep’ structural parameters. The preferrednickname for this class of models is ‘DSGE’ (dynamic stochastic general equilibrium). Ithink that this argument is fundamentally misconceived.

We know from the Sonnenschein–Mantel–Debreu theorems that the sole empiricalimplication of a classical general-equilibrium genealogy is that excess-demand functions arecontinuous and homogeneous of degree zero in prices, and satisfy Walras’s Law. Thoseconditions can be imposed directly on a large class of macroeconomic models. I havemade this point in another context, the example being the monetary macro-models of JamesTobin (see Solow, 2004). It applies just as forcefully here. The cover story about ‘micro-foundations’ can in no way justify recourse to the narrow representative-agent construct.Many other versions of the neoclassical growth model can meet the required conditions; it isonly necessary to impose them directly on the relevant building blocks.

The nature of the sleight-of-hand involved here can be made plain by an analogy. I tellyou that I eat nothing but cabbage. You ask me why, and I reply portentously: I am avegetarian! But vegetarianism is reason for a meatless diet; it cannot justify my extremeand unappetizing choice. Even in growth theory (let alone in short-run macroeconomics),reasonable ‘microfoundations’ do not demand implausibility; indeed, they should excludeimplausibility.

Maybe it would be helpful (to myself, at least) if I said in a couple of sentences whatI think the function of growth theory is. It would go something like this. The long-runbehaviour of a (fully employed) modern economy is the outcome of the interplay of someidentifiable forces. The main ones seem to be the volume of investment in tangible and humancapital, the strength of diminishing returns, the extent of economies of scale, the pace anddirection of technological and organizational innovation. Even this sample list leaves out some

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significant and interesting factors: natural-resource availability, environmental constraints,and the complex question of the relation between short-run fluctuations and medium-run andlonger-run growth. The function of an aggregative growth model is to provide a handy wayof describing how these factors interact, partly to help clear thinking, and partly to guideempirical research.

The articles in this issue of the Oxford Review all fit within this framework. Two preliminarycomments may be in order. First, my picture of the convergence issue is that the key questionturns on which of the main forces are held in common by various national economies, andhow they differ on the others. Presumably that is how one would define ‘convergence clubs’.It might take a more subtle specification of factors than I suggested here; so much the better.

Second, I interpret the recent surge of interest in the aggregative elasticity of substitution asa very useful attempt to probe more deeply into the sources and consequences of diminishingreturns. At the aggregative level, this has to be more than a merely technological fact.Substitution on the demand side must play an equal role, and also any institutional factors thataffect the geographical, occupational, and industrial mobility of labour and capital. Sortingall this out, theoretically and empirically, is an exciting and important task. We now know,for example, that the elasticity of substitution has, in a well-defined sense, implications forthe level of output: if two economies are in other respects identical, and start from the sameinitial conditions, the one with a larger elasticity of substitution will have a higher growthtrajectory, and the benefit is comparable in size to what would be achieved from a somewhatfaster rate of technological progress,

It is interesting that most (not all) recent attempts to estimate the economy-wide elasticityof substitution have come up with values far smaller than one. That suggests a sharperrole for diminishing returns than we are used to imagining. This finding, if it holds up,could liberate growth theory from the grip of the Cobb–Douglas function, whose specialproperties get embedded in many model-building exercises for no better reason than itssoothing convenience. It is worth noting, on the other side, that large, but not extreme, valuesof the elasticity of substitution allow sustained growth without technological progress.

These matters are at the very heart of neoclassical growth theory and what it has to sayabout the constraints on growth (other than those connected with natural resources). Thereis, therefore, every reason to welcome continued empirical research on these topics, like thatcontained in the paper by Rainer Klump, Peter McAdam, and Alpo Willman in this issue.

I would like to conclude with a few, necessarily brief and sketchy, comments on theresearch papers that follow. They are very diverse in content and method. I found every oneof them interesting and provocative. It says something about the neoclassical growth modelthat it can provide the framework for such a varied collection of investigations.

It is convenient to start with the paper by Erich Gundlach. I am entirely in sympathy withhis basic insistence: if you want to use the neoclassical growth model to understand thedifferences between countries, you have to be clear from the beginning about what parametersthey have in common, and in what ways they are allowed to differ. For simplicity, supposethere are just two countries. One very common, probably too common, assumption is thatthey have the same depreciation rate, the same population growth rate, and the same rate oftechnological progress. But they have different saving-investment rates and different currentlevels of labour-augmenting technology. In effect, we assume they are in or near their steadystates when we observe them, and we want to know how much of the observed differencein output per head results from the difference between s1 and s2, and how much from thedifference between A1 and A2.

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Gundlach observes that empirical capital–output ratios vary very little across countries, andpoints out that this is exactly what you would expect, according to the model, if differences inproductivity reflect primarily differences in technological level. I would put this in a slightlydifferent way, without the possibly misleading Cobb–Douglas assumption. One can solvethe model to give the steady-state productivity (y) and capital intensity (k) as a function ofthe parameters that are allowed to differ, A and s. If only A varied from country to country, yand k for each country would lie on a ray from the origin. They don’t quite do that, accordingto Gundlach’s scatter diagram—though nearly. What kind of and how much variation ins would account for the deviation from a ray, and does that correspond in any way to theobserved cross-country differences in s? My guess is that Gundlach’s evaluation is aboutright, but I would like to see how the fuller treatment works out. In any case, accountingfor the cross-country relation between y and k—which covers both very rich and very poorcountries—is an important matter.

Suppose that the data in Gundlach’s diagram do not quite fit with theory, when accountis taken of differences in both A and s? There would be at least two implications worthconsidering. One is that the model is simply inadequate. Another is that the assumptionthat all the observations describe steady states is seriously misleading. Remember that allcross-country growth regressions of this kind find a negative and statistically significantcoefficient on the country’s initial level of income. That by itself contradicts the steady-stateassumption; the question is how much this matters.

One minor detail: Gundlach remarks, correctly, that we lack a good index of technologicallevel (A) for each country. One device that he tries is to use a conventional measure ofinstitutional quality, on the hypothesis that this is likely to be correlated with technologicallevel. My inclination would be to try for something more direct, if possible, such as industrialelectricity consumption per unit of output, or the number of computers.

The very valuable paper by Kieran McQuinn and Karl Whelan carries this general lineof thought in a different direction, with some exciting results. Nearly everyone takes it forgranted that the rate of growth of TFP is the same everywhere. The only thing that justifies thisremarkable presumption is the fairly mechanical thought that knowledge of new technologydiffuses rapidly around the world. Maybe so, but productivity performance depends on manyother influences besides the content of the latest engineering textbook. (The paper by DavidAudretsch, to which I will come in a moment, is precisely about one set of forces that drivesa wedge between mere ‘knowledge’ and TFP.) Even if TFP is likely to increase more or lessuniformly across regions on the time-scale of centuries, common observation suggests (andmore than suggests, according to McQuinn and Whelan) that rates of TFP growth can differsubstantially even among advanced national economies on the time-scale of decades. Thisseems correct and theoretically and empirically important to me.

McQuinn and Whelan then go on to make a neat analytical point: the model says thatthe law of motion of the capital–output ratio is independent of the TFP growth rate, unlikethe dynamics of output per unit of labour. One important consequence of this insight is thatinferences from cross-country observations can be made without assuming a common TFPgrowth rate if the analysis is carried on in terms of the capital–output ratio. I don’t supposeI had noticed this in 1956, although I certainly messed around with the capital–output ratio,because the idea of cross-country inferences was not in my head. I was thinking only aboutsingle closed-economy time series. (I have never had any sympathy for the uniform-TFP-growth-rate assumption.) The reader of this paper will see that doing the analysis their wayleads to some substantial revisions of conventional estimates.

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I think this is a real advance. Two further questions occur to me. One is minor. They findthat their empirical estimate of the speed of adjustment to steady state is always close tobut a little larger than the value suggested by the model, using standard stylized facts. Whatmight account for this directional difference? The second question is much broader. Oncegrowth theory abandons the implausible limitation to uniform TFP growth rates, it is naturalto wonder about the actual pattern of national growth rates, and about the likely determinantsof this pattern.

David Audretsch observes that it is useful—and correct—to say that an important functionof what we call entrepreneurship is precisely to bridge the gap between specific pieces oftechnological knowledge and innovations in actual production, often through the creationof new firms. He remarks in passing that the efficiency of this nexus is a major source ofregional differences in the growth of TFP. This is easy to believe, especially for anyone whohas eavesdropped on discussions of growth policy. If this idea can be embodied in empiricalgrowth accounting, it would add a lot to the explanatory power of growth theory.

There is also a connection to another long-standing worry of mine. We estimate time seriesof TFP in the conventional way, more or less completely detached from the narrative ofidentifiable technological changes that a historian would produce for the same stretch of time.There are reasons for this disjunction. TFP is estimated for aggregates, for a whole industryat a minimum, whereas the historical narrative is usually about single firms or even singleindividuals. Both temporal aggregation and cross-sectional aggregation will mask individualevents. Besides, a lot of productive innovation has nothing to do with research or withresearch workers; it is created in the act of production through ‘learning by doing’ or somesimilar process. And then there are what I have already vaguely called ‘other influences’.Nevertheless, it would be interesting to see if any connection can be made, perhaps in aspecific industry, between the time series of TFP and an informed narrative of significantinnovations and their diffusion. (One can see in principle how TFP should be related tonew-product innovations, but it is not clear what would happen in practice.)

This train of thought leads naturally to the very attractive paper by Klump et al. Theconnection is that one of their goals is a flexible estimate of the character of factor-augmenting technical progress from US and euro-area time series. Their preferred findingis a combination of exponential labour-augmenting and sub-exponential capital-augmentingtechnical progress. In the long run, then, Harrod-neutrality dominates. (They do not tell ushow long a run that is.) So steady-state growth is possible eventually, but perhaps not now.(When, exactly?)

Among the other nice aspects of this paper is the effort to put together a consistent dataset, and the use of a three-equation model for estimating the elasticity of substitution. Thethree equations are the production function itself and the two first-order conditions on labourand capital. (The original 1961 paper by Arrow, Chenery, Minhas, and me used only thecondition for labour; we were doing cross-sections and did not have data on capital stocks.)Their main finding for both economies is an elasticity of substitution significantly less thanone, in fact about 0.6.

I have one major reservation about this. The empirical basis consists of annual time seriesfor the USA from 1953 to 1998 and quarterly for the euro area from 1970 to 2003. The tacitassumption is that the business cycle can be ignored, which means in effect that the capitalstock is assumed to be fully utilized all the time. Think what this does. Recessions tend to befairly short, but they must leave traces in annual data. In a recession, the capital–labour ratiowill appear to rise, because recorded employment will catch the diminished use of labour,but the recorded capital stock does not catch idle capacity. Also, in recessions the income

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share of capital tends to be depressed, for the usual reasons. This is the standard recipefor an elasticity of substitution less than one: the increasingly abundant factor loses relativeshare. So I wonder if the Klump et al. finding contains some downward bias. I worriedabout this under-utilization problem in 1957; the best I could do then was to assume thatthe unemployment rate of capital was the same as that of labour in the same year. It shouldbe possible to do better now. I don’t know how serious this bias could be, but I want toassure Rainer Klump and his co-authors that I don’t mention it just because Olivier de LaGrandville and I have found interesting implications of a large elasticity of substitution. I ammore worried about the tendency of modern (American) macroeconomists to forget about thepathology of business cycles.

The paper by Davide Fiaschi and Andrea Mario Lavezzi is hard to discuss in just afew minutes, because its relation to the others is indirect. It also deals with internationalcross-sections, but in a very different way; in other respects it stands by itself, but still broadlywithin the neoclassical growth framework. All I can do is to make one or two casual remarks.

The descriptive basis of the analysis is a plot of dlog GDP/dt against log GDP for a largegroup of countries and times. Then, in a key step, the observations are translated into thetransition matrix of a Markov chain, where the states are defined by the same two dimensions.This is an interesting approach. Notice that the name of the country has disappeared fromview. The Markov hypothesis says that the probability of moving from one class to eachother class depends only on the starting state; no longer is history relevant. Is it easy tobelieve that? In the language I have been using, a particular country at a particular time ischaracterized by values of A and s, technology level and rate of investment. Do two countriesin the same state, but with different values of A and s necessarily share the same transitionprobabilities? I suppose this could be tested; in fact, when the authors sort observations byrate of investment, they are testing it.

The only other observation I have time for relates to a very interesting analytic step in thepaper. Fiaschi and Lavezzi try to interpret their data in light of a standard neoclassical modelwith a novel twist. They introduce a level-of-technology parameter and then suppose thereare technological spillovers from each country to other countries. The strength of the spilloverbetween any pair depends on the ‘distance’ between them; and distance means economic, notgeographical, distance, measured by the disparity between their levels of income per head.

What is more, a country receives positive spillovers from more advanced countries,and negative spillovers from less advanced countries. I find that a little hard to believe;it seems to treat having a low technological level like a sort of contagious disease. Butthe focus on international technological spillovers strikes me as important and relevant.Casual observation says that the catch-up process is a vital part of the evolution of nationaleconomies. Its relation to trade and to foreign direct investment needs to be incorporated intoa theory of open-economy growth.

Here I come to the highly interesting paper by my friend and collaborator Olivier de LaGrandville. I leave aside the kind things he said about me; he was not under oath (and neitheram I). He goes on to raise a question that has bothered me, and many others, for a verylong time. Richard Goodwin was one of my teachers; I probably read his 1961 paper onoptimal growth in manuscript. Goodwin found, and de La Grandville verifies on a broader andmore detailed scale, that straightforward application of the Ramsey principle to a reasonablycalibrated growth model leads to absurdly high estimates of the socially optimal ratio ofsaving-investment to income (and may sometimes lead to mathematical pathologies). Whatshould we think?

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The last 50 years in growth theory and the next 10 13

I spoke of a reasonably calibrated model. But we really don’t know how to calibrate anessential part of the model: the function that gives the per capita social utility of per capitaconsumption. De La Grandville shows that to get the optimal saving rate down to a reasonablesize would require so much concavity in the social utility function as to cast doubt on theverisimilitude of the whole procedure. Then he takes an altogether different tack that requiresmuch more discussion than I can give it here.

He gives up on fiddling with the utility function, and simply maximizes the time-discountedsum of consumption per head. That seems to be moving in the wrong direction: a linear utilityfunction calls for a bang-bang solution to the problem. (I wonder if the fact that the Fisherequation appears as the formal Euler equation for this problem is a reflection of this.) Buthe also confines the applicability of the model to situations in which the marginal product ofcapital is already very near the social rate of discount. That gets the optimal saving rate backin the common sense range, as he shows.

It also forces us to rethink the original question in a different way: suppose that acomfortable situation of the de La Grandville type were suddenly to be disturbed by thedestruction of a substantial part of the capital stock, by a natural disaster, for instance. Arewe not right back in the Goodwin–Ramsey problem?

Now comes a further radical suggestion: if the Ramsey formulation is really intuitivelysatisfying only near the steady state, why should we use it in the recovery-from-catastrophecontext? So he proposes a different sort of rule of thumb, and restores common sense onceagain. Does that way of thinking truly resolve the Ramsey paradox? That is a question thatshould be discussed at leisure. But it is a useful question not only for its own sake, but becauseit reminds us that every abstract model needs this kind of plausibility–reasonableness smelltest before one starts just applying it. We need the reminder because that sort of considerationis too often omitted.

The compact paper by Philippe Aghion and Peter Howitt teaches a useful lesson aboutinterpreting growth theory, even if it is more directly relevant for growth accounting. Thegeneral admonition is that the choice of what is exogenous and what is endogenous is anintrinsic part of any theory. The particular application to growth theory is not necessarilynew, but is certainly still worth stating. One striking conclusion from the original neoclassicalmodel was that the long-run growth rate is independent of the saving-investment quota. Butthat was under the assumption that technological change entered exogenously. Aghion andHowitt exhibit a model which is like the original one in every respect but one: technologicalchange is endogenized in a particular way. In their modified model, the saving-investmentrate does influence the steady-state growth rate. No mechanism in the original model iscontradicted; but the size of the capital stock has an effect on the rate of technologicalinnovation, and that relationship opens a channel from the saving-investment rate to thegrowth rate.

This is a worthwhile and interesting reminder. It also gives me a chance to ride a few paceson an old hobby-horse that made a brief appearance earlier in these notes. The Cobb–Douglasproduction function is a wonderful vehicle for generating instructive examples. But it hasspecial Santa Claus properties, and one must not be misled about the generality of thoseexamples. The Aghion–Howitt machinery resembles something I proposed in my own firstpaper on ‘embodiment’. To get clean results I had to assume that technological changewas purely capital-augmenting, just the opposite of the conventional assumption of Harrod-neutrality. (See the paper by Klump et al. in this issue for empirical indications.) In the caseof the Cobb–Douglas function (and only then) the distinction between labour-augmenting,capital-augmenting, and output-augmenting (Hicks-neutral) technological change evaporates.

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14 Robert M. Solow

I wonder how much of the Aghion–Howitt story will hold up outside the Cobb–Douglascase, without some other restrictive assumption.

However that turns out, the basic reminder is valid. Exogeneity assumptions matterbeyond themselves. I wrote that 1960 paper trying to find a path by which the saving-investment quota could after all affect the asymptotic growth rate. The particular mechanismI explored—embodiment itself—did not have that effect. We all believe that the determinantsof long-run growth are somehow endogenous, but the ‘somehow’ is not obvious, nor is iteasy to test hypotheses. Aghion and Howitt have found one hypothesis that does implicatethe saving-investment quota, but the range of its field of application should be investigated.They go about it in the right way, and they may be on the right track. The more the merrier.

References

Aghion, P., and Howitt, P. (1992), ‘A Model of Growth through Creative Destruction’, Econometrica,60(March), 323–51.

— — (1998), Endogenous Growth Theory, Cambridge, MA, MIT Press.— — (2007), ‘Capital, Innovation, and Growth Accounting’, Oxford Review of Economic Policy, 23(1),

79–93.Audretsch, D. B. (2007), ‘Entrepreneurship Capital and Economic Growth’, Oxford Review of Economic

Policy, 23(1), 63–78.Barro, R. J. (1991), ‘Economic Growth in a Cross Section of Countries’, Quarterly Journal of Economics,

106(May), 407–43.Domar, E. (1946), ‘Capital Expansion, Rate of Growth, and Employment’, Econometrica, 14(April), 137–47.Fiaschi, D., and Lavezzi, A. M. (2007), ‘Appropriate Technology in a Solovian Nonlinear Growth Model’,

Oxford Review of Economic Policy, 23(1), 115–133.Grossman, G., and Helpman, E. (1991), Innovation and Growth in the Global Economy. Cambridge, MA,

MIT Press.Gundlach, E. (2007), ‘The Solow Model in the Empirics of Growth and Trade’, Oxford Review of Economic

Policy, 23(1), 25–44.Harrod, R. (1939), ‘An Essay in Dynamic Theory’, Economic Journal, March, 13–33.Klump, R., McAdam, P., and Willman, A. (2007), ‘The Long-term SucCESs of the Neoclassical Growth

Model, Oxford Review of Economic Policy, 23(1), 94–114.La Grandville, O. de (2007), ‘The 1956 Contribution to Economic Growth Theory by Robert Solow: A Major

Landmark and Some of its Undiscovered Riches’, Oxford Review of Economic Policy, 23(1), 15–24.Lucas, R. E., Jr (1988), ‘On the Mechanics of Economic Development’, Journal of Monetary Economics,

22(July), 3–42.McQuinn, K., and Whelan, K. (2007), ‘Solow (1956) as a Model of Cross-country Growth Dynamics’, Oxford

Review of Economic Policy, 23(1), 45–62.Romer, P. (1986), ‘Increasing Returns and Economic Growth’, Journal of Political Economy, 94(October),

1002–37.Solow, R. (2004), ‘The Tobin Approach to Monetary Economics’, Journal of Money, Credit and Banking,

36(4), 657–63.Swan, T. (1956), ‘Economic Growth and Capital Accumulation’, Economic Record, 32(November), 334–61.

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