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Macroeconomic Policies on Shaky Foundations – Whither Mainstream Economics?
Berlin 31 October – 1 November 2008

Endogenous growth theory twenty years after: a critical assessment

Sergio Cesaratto
University of Siena
Cesaratto@unisi.it
Introduction*
Some years ago I found one ‘key’ of the origins of endogenous growth literature (EGT) in
the dissatisfaction that emerged in the late fifties with one result of the neoclassical growth modelby Robert Solow (1956), that is the independence of the growth rate of the economy from thesaving ratio, the ratio between the (full employment) saving supply and output, a variable subject topolicy influence, e.g. by tax treatment favourable to saving and investment (Cesaratto 1999a,
1999b). I pointed out the difficulties met both by the earlier and the new generations of endogenous
models in trying to relate the growth and saving rates (see also Serrano & Cesaratto, 2002). In the
meanwhile the literature on EGT has been enriched by further models and empirical verifications,that have been endeavoured to face those difficulties. In this paper I will reassess my earlyconclusions about EGT in the light of the most recent contributions. In this paper I shall distinguishamong three generations of EGT models: the old from the sixties; the new from the late eighties;
and the most recent from the second half of the nineties. As we shall see, EGT models can bedivided in two fields: those who defend the Solowian approach (semi-endogenous models) andthose who try to depart from it (full endogenous models).
Quite relevant to corroborate the interpretation of EGT put forward in my earlier
contributions was a paper by Marving Frankel, published in 1962 six months later than the morewell known Arrow’s learning-by-doing model, that I ‘rediscovered’ guided by the key to EGT that
had just come to my mind.1 Frankel observed that in the Harrod-Domar model, where a production
function Y = aK is used (where a is the output-capital ratio), the rate of economic growth depends
on the saving ratio s according to the well known Harrodian formula g = s / v , where v = 1 / a .
*1This is a preliminary and incomplete version. Also the English language has not been revised.

I firstly evoked Frankel’s contribution in Cesaratto (1995). Frankel’s forerunner paper was later‘rediscovered’ by Cannon (2000) who, however, fails to locate Frankel’s original contribution in the context
of the troubles met by the old and new EGT.

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Economists, he argued, “have found such models attractive because of their relatively simple
structure, because of the emphasis they give to capital accumulation as an ‘engine of growth’ – anemphasis with deep routes in economic thought – and because of their pragmatically satisfying
results” (1962, p.996). However, he continued, “the production function Y = aK has nothinginteresting to say about resource allocation or income distribution. Worse than this, as a generalstatement of the resources required in production, it is positively wrong, as any one-factor
production function must be” (ibid). On the opposite, the most agreeable Cobb-Douglas production
function, although satisfactory from the point of view of “resource allocation or income
distribution”, in a growth context does lead to a growth rate that depends on the rate of growth ofthe labour force and where the investment rate – that in this marginalist context coincides with thesaving rate – does not affect neither the aggregate growth rate nor, through productivity growth, thatof output per worker (ibid, pp.996-997). Labour productivity growth could be introduced into themodel, but its sources remained “exogenous” in the distinctive sense that it does not depend upon
the endogenous saving choices of the community (or of a representative agent)2. The same
unhappiness with regard to this specific result of neoclassical Solow’s model was expressed in those
years by other authors (cf. Cesaratto 1999a, 1999b for the references).
In this regard we may start by asking whether the relationship between the investment(saving) ration and the growth rate is just a theoretical supposition – although one ‘with deep routes
in [traditional] economic thought’ -, or is also a proved empirical fact. In this respect in section 1 I
shall first consider the recent controversy on the empirical evidence on this relation. It sounds thatmost of the results confirm the relation, an outcome unfavourable to Solow’s prediction of its
absence in the secular equilibrium. Solow’s model did not deny it, however, in the transition from
one to another secular equilibrium, so that I will briefly look in section 2 to some recent defences ofSolow’s model based on the length of the transition towards the steady state. Given the frailty ofthis defence, we shall then look in section III at proper EGT models recalling Frankel’s andArrow’s seminal and archetypal models, not for the sake of history of thought, but since new andrecent EGT has not moved a single step beyond the stage set by those earlier models, if not byadding further theoretical and empirical considerations to lean in favour of one or the other possible
approaches displayed on that podium. This was one earlier conclusion on EGT, and I shall have notreasons to change it but just to update in the present work (although I respect to the richness of thedebate over the last twenty years). More importantly, the archetypal models posed some problems
that have passed to new and recent EGT. However, these troubles have not passed unnoticed due in2

It is widely recognised that to get the sense of EGT it is irrelevant if the saving rate is taken as given, or if
choices are analysed through a Ramsey model. The latter would just add maths but no substance (see e.g.
Mankiw, 1995, pp.279-280).

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particular to the influential contributions by Charles Jones that in the light of those difficulties, notsurprisingly, argues in favour of sticking to the traditional Solow’s growth framework. The mostrecent generation of EGT models has countered Jones’s criticism by approaches introducing anotherbattery of ad hoc assumption that, however, also remind to the archetypal, as we shall point out insection V. After having discussed in section VI Jones’ neo-Solowian perspective, in the final
remarks, I will deal with a methodological defence of the neoclassical growth modelling according
to which one should not be too critical of the analytical or empirical limitation of single models, butrather pick up the insight provided by each of them. In view of this remark, I will emphasize therelevance of the Sraffian and Keynesian criticism of neoclassical theory.
I. Econometric growth theory
Recent and earlier presentations of growth theory including Jones (2002) and Solow (1970)
have focused over the explanation of the famous Kaldorian six ‘stylized facts’ of economic growththat, as they are well known, it is pointless to recall here. To this list, two additional stylised fact
may be added that growth models may aim to explain that are, however, more controversial: apositive correlation between the investment (saving) rate, that is the ration between investment(saving) and output, and the rate of economic growth, both in aggregate and per-capita terms,
respectively. Not surprisingly, among those who deny these two additional stylised facts we foundthe supporters of the Solowian view and, on the opposite side, those who feel uneasy with thisapproach. The debate has focused on the empirical relation between the investment rate ( I Y ),taken in a close economy as a proxy of the saving rate (s),3 and per capita growth rate ( g y ).4 We
shall see that most contributions find an empirical correlation between I Y and g y . Defendants ofSolow’s model have found various ways to shield their master’s results.
In an earlier study, Hill (1964) examined the mentioned relation in a group of industrialised
countries over the period 1954-62 and found a significant correlation (particularly strong for thelarger economies) between I Y and aggregate growth ( g Y ), and also with g y especially once

3

After Feldstein & Horioka (1980) empirical results, some scepticism has emerged amongst neoclassicaleconomists regarding the influence of foreign capital on domestic investment.

4

The fact that neoclassical economists focus upon only one of the two additional stylised facts should notcome as a surprise. As known, the aggregate growth rate is given by the summation of the growth rate of theworkforce plus labour productivity growth. In a neoclassical full-employment model, the workforce grows at
a natural demographic rate n. That the aggregate growth rate should adjust to the (exogenous) growth rate of
the workforce irrespective of the investment (saving) rate is something that the neoclassical Weltanschauungcannot put into discussion, so that the discussion centred over the rate of technical change. By contrast, for
non orthodox economists it is the aggregate growth rate that is more important since they believe that market
economies are not, on average, in full employment. Productivity growth is also important, of course, tomeasure the competitiveness of a country and as a source of job displacement.

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investment in machinery and equipment was considered. More recently, this kind of investment has
been the object of some much quoted papers by De Long and Summers (1991, 1992). They also
found that productivity growth is positively associated to high investment in equipment in a large
sample of rich and developing countries over the Second World War II period.5 Abel (1992, p.200)
assigns to De Long-Summers’ finding the status of ‘new stylised fact’.
In their influential paper, Mankiw, Romer and Weil (MRW, 1992; see also Mankiw 1995)
show that Solow’s model, once ‘augmented’ to include ‘human capital’, is able to explain a gooddeal of the variations among countries in per capita income levels (not growth rates). They propose
a production function like:
Yt = K tα H tβ ( At Lt )1−α − βwhere H represents ‘human capital’ and A the traditional exogenous technical progress (hereafter
we shall omit the time subscript in the equations), and α + β < 1 . According to MRW this equation
would lead to good predictions of those variations on Solowian lines, that is based on differences inthe saving rates and in population growth, without recurring to exogenous (or endogenous)
technical change.6
These results concern the differentials in the levels of per capital income, but not the growth
rates. In this respect MRW acknowledge the lait motive of EGT, that ‘countries with a higher savingrate grow persistently faster and that they will not converge to a same steady state growth path even
if they have the same technological endowment’ (ibid., p.421-422). Recall that according to
Solow’s model countries with similar s and technology would still show different growth rates
when starting from different initial per capita capital (income) levels, although they will in the longrun converge towards their common growth rate. Taking advantage of this result, Mankiw (1995,
5

These authors attribute big importance to differences in the price of equipment goods across countries. A
high saving ratio associated to a low price of equipment would generate high productivity growth (e.g. De
Long & Summers, 1991, pp.484-485). Yet the causes of the low prices of capital equipment are not clearly
explored by these authors. They reject an accelerator explanation of growth, whereby it is growth that leads
to an high investment shares, because this would be associated, in their view, to a higher equipment price notto the lower one showed by their data (e.g. De Long & Summers, 1991, pp.473-474; 1992, p.176).

6

Without ‘human capital’ the traditional model would only partially explain those inter-country variations(Mankiw 1995, pp.282-284) The reason to include ‘human capital’ is that without it Solow’s model would
still well explain the mentioned variations, but with an income share α accruing to capital over two-third(against a usual value of one third). A high value of α would be justified once the returns to educated labour
are assimilated to capital’s returns. The economics of the model is that a higher ‘human capital’ component
would lead to a higher per capita income and saving supply; the latter does in turn lead to a higher per capita
income (MRW, 1992, p.417; Mankiw, 1995, p.290). Although in their model there is a touch of endogenous
growth, in the sense that the accumulation of ‘human capital’ is an endogenous decisions, the model is notable to generate endogenous growth (that is the endogenous decisions to accumulate physical of ‘human’capital do not affect the rate of growth (the model would become a full endogenous model with α + β = 1 ,see the production function adopted by endogenous models below in sections III and IV). We put ‘humancapital’ in inverted commas given the shaky nature of this magnitude, cf. Steedman 2001).

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p.278) maintains that: “The inability of saving to affect steady state growth (…) might appear
inconsistent with the strong correlation between growth and saving across countries. But thiscorrelation could reflect the transitional dynamics that arise as economies approach their steadystates’.7 In addition, MRW (1992) approvingly quote Barro (1989) that there is no much evidenceof a convergence across countries in the sense that poorer countries (those starting with a lower per
capita capital endowment) do not generally tend to grow faster and catch up richer countries. They
introduce, in this regard, the concept of conditional convergence, thereafter entered in textbook
expositions, that countries do not converge towards a common growth rate, but each one toward her
own one.8 So, they conclude, growth rates can differ both (i) as countries are in ‘transitional
dynamics’, and this may explain the relation between s and g y ; and (ii) because they converge
towards different secular paths, and this is might be explained by the differences in s and population
growth (n) (MRW, 1992, p.423).9
Critical of De Long-Summers results is Jones who argues that whereas in the maineconomies the share on GDP of both total and equipment investment has increased, (per capita)
output growth rates ‘have fallen, if anything, over the post-war era’ (Jones, 1995b, p.508). Jones’s
idea seems to be that although De Long-Summers can be right to envisage a cross-country
correlation between I Y and g y , for each country over a period of time it is not however true that a

7

The empirical results about convergence towards secular growth reported by these authors (e.g. Mankiw,
1995, pp.284-285) suggest a faster convergence towards the steady state than that predicted by Solow’s
model. Assuming a higher value of the capital share α , obtained again by considering part of labour as‘human capital’, would make the predictions of the ‘augmented’ Solow model closer to those data (ibid,
p.291; MRW, 1992, p.428).
8‘[T]he neoclassical model predicts that each economy converges to its own steady state, which in turn isdetermined by its saving and population growth rates’ (Mankiw, 1995, p.284).

9

It should be observed that these authors do not attribute much importance to international differences intechnological endowments to explain the divergences of per capita income levels growth across countries,
although this sounds strange if we compare the actual techniques in use in poor and rich countries,
respectively. They seem to explain this arguing that although the production function is the roughly same inall countries, since ‘knowledge …travels, around the world fairly quickly’ (Mankiw, 1995, pp.300-301) the
specific technique - say spades or tractors - used in each country depends on their stage of growth. This, inturn, affects factors’ supply: ‘To use the neoclassical model to explain international variations in growthrequires the assumption that different countries use roughly the same production function at a given point intime. …change [of techniques] should be viewed as a movement along the same production function, rather
than as a shift to a completely new production function’ (ibid., p.281). The ‘augmented’ Solow model would
help again. The different investment in ‘human capital’ would explain the divergent ability across countries
to assimilate superior production functions. All neoclassical growth literature never ever mentions that theacquisition of superior foreign technology and education is a costly process in terms of balance of payments.
According to non-orthodox theory, what can be called the ‘foreign liquidity constraint’, the necessity tocollect enough international currencies to finance the acquisition of foreign embodied and disembodies
technology, is one main economic obstacle to economic growth.

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increasing I Y leads to a higher g y .10 Critical of Jones is Li (2002) who conducts time-series
regressions over 24 OECD countries over 1950-1992 and five industrialised economies over 1870
to 1987 finding that total investment is positively related to growth in more than half of the cases.
Jones’s outcome would be different since they rely on durable investment only and because thefocus on the U.S. would be misleading (ibid, p.97).
Comparing their results to those by De Long-Summers, Blomstrom, Lipsey, Zejan (1996)
find that the ‘long term relationship’ between I Y and g y ‘were due more to the effect of growthon capital formation than to the effect of capital formation on growth’, as argued by De LongSummers (ibid, p.269). This sounds of course more in line with less orthodox theory. Vanhoudt(1998) is not surprised by Bolmstrom et al. results since in Solow model a rise in the saving ratio s
is followed by a sudden rise in the growth rate followed by its decline towards its steady state level.
No surprisingly then, statistical tests would suggest a negative effect of saving on growth.11 A
similar result is obtained by Attanasio, Picci e Scorcu (2000, e.g. pp.198-199),12 while a more
ecumenical outcome is attained by Podrecca and Carmeci (2001) according to whom the causality
can go in both directions. A halfway position is also held by Madsen (2002) whose results on 18
countries over the period 1950-1999 would show that investment in equipment does cause growth,whereas non-residential investment in buildings and structures is caused by economic growth (thisappears perplexing to a non orthodox economist who would see investment in equipment as demand
induced, and construction works as an autonomous component).
Bernanke and Gurkaynak (2001) present empirical results strongly in favour of a positive
influence of I Y on g y .13 The two authors also show that the saving rate ( I Y ), but also
population growth, are positively correlated with total factor productivity (TFP), contrarily to the10

Charles Jones was still a student at MIT when, in his Siena doctorate lectures on growth, Solow (1992,p.85) mentioned, not surprisingly approvingly, Jones’ forthcoming results that, he reports, were obtained‘before [Jones] had read De Long-Summers’s paper’.

11‘Since the mentioned Granger causality test control for lagged growth, it is not surprising that positive
Granger causality from saving to growth does not show up’ (ibid, p.78.

12

It is impressive how much Keynesian thought has been lost by these economists that explain the ‘dynamic
link running from growth to investment’ arguing that ‘[h]igher growth might driven saving up, leading inturn to higher investment’ (ibid, p.183). Elsewhere they admit, and their results do not exclude, a ‘Granger
causation running from investment to saving’; however, they continue, ‘the exact mechanisms at work are
hard to spell out in detail, if an increased demand for capital goods stimulates saving – maybe throughinterest rate effects or the endogenous development of the financial instruments that permit the mobilisationof saving – saving might adjust to investment’ (ibidem). No mention by these authors of concepts such as theinvestment accelerator or the Keynesian multiplier (for a Keynesian adjustment mechanism of saving toinvestment in a long run framework cf. Garegnani 1992).
13

As MRW (1992), Bernanke and Gurkaynak use the Penn World Tables, a multicountry data set by Heston
and Summers (MRW over the years 1960-1985, BG 1960-1995).

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Solow model’s prediction that TPF should be exogenous to factor accumulation. This would showthat endogenous mechanism, saving or fertility decisions, have external or scale effects such todetermine endogenous rather than exogenous growth. In their comments David Romer and Mankiw
remark their thesis that economies are not necessarily on their steady state.
While it sounds that most of the empirical outcomes is in favour of a positive association
between the investment (saving) rate and g y (e.g. Mankiw, 1995, p.302), much effort has been paidby both fronts to check this correlation and see whether other variables would play a moresignificant explanatory role. Opinions tend however to converge on the idea that these exercises areof limited value since they are seriously sensitive to the variables included or excluded, the timespan, the country considered and whatever else.14 We may therefore stand with Mankiw (1995,
p.308) when he argues that: ‘Basic theory, shrewd observation, and common sense are surely morereliable guides for policy’. Let us therefore go back to ‘basic theory’ starting from the old EGT
models.
II.

Arrow, Frankel and the AK model.

Which strategies were open to neoclassical growth theory to endogenise growth? As wellknown, according to Solow’s model the secular growth rate is given by: g Y = n + h ,where n is thegrowth rate of the labour force and h is technical change. There is little space to endogenise n,although a number of authors have explored the relation between fertility choices and growth. Moreobvious was the exploration of possible relations between saving choices and technical progress.
Clearly, for the neoclassical economists the saving rate was a main determinant of capital
accumulation, but because of the decreasing marginal productivity of capital, for a given a labour
supply, a higher saving rate could not persistently raise the accumulation rate that, at the end, had toadjust to the exogenous rate of growth of the workforce (in physical or efficiency units). On theother hand, it was this adjustment that permitted to Solow to conclude that full employment was thelong run rule overcoming the opposite conclusions by the Harrod-Domar (HD) model – whohowever, as cleverly pointed out by Frankel, offered a saving-led growth formula. Notably, in the
HD model substitutability amongst ‘production factors’ is ruled out.
Frankel (1962) proposed a new growth model able to preserve the distributive role of the
Cobb-Douglas production function, but in which the growth rate depends on the investment rate as
14

So we hear Solow saying that: ‘the main fact about these empirical studies’ is that ‘they are not robust’(1992, p.78, italics in the original). Similarly, Mankiw (1995, pp.307-308) states that: ‘Using these
regressions to decide how to foster growth is …most likely a hopeless task. Simultaneity, multicollinearity,and limited degree of freedom are important practical problems for anyone trying to draw inferences frominternational data. Policy makers who want to promote growth would not go far wrong ignoring most of thevast literature reporting growth regressions’. Other sceptical views include Rodrik (2005), who also quote
some other critical surveys, and XXX (2008).

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8in the HD model. The trick, that has became a sort of cliquet far all the endogenous growth theory
up to now, was to craft a relation between labour augmenting technical change and the saving rate.
In the recent literature this has been done, following Arrow, Phelps, Uzawa and other authors fromthe sixties, by linking labour productivity growth to externalities from (the endogenous) capital
accumulation, or by relating it to surrogate saving decisions as those associated to the resources
devoted to R&D or education that also imply diversion of resources from present to future
consumption. In the new EGT literature a short cut has instead been taken by those economists thathave just come back to HD through the so-called AK model, not surprisingly in view of Frankel’s
just recalled account of the neoclassical growth theory dilemma. Frankel’s model is particularly
intriguing in this regard.
As recalled, given a production function Y = AK 1−α ( HL) α , where H represents labouraugmenting technical progress,15 the growth rate of the economy is given by the summation of thegrowth rates of the labour force n plus that of the labour augmenting technical progress h . Let us
then assume with Frankel that H depends on per-worker capital endowment, that is:
H = ( K / L) γ

[1]what Frankel names ‘modifier’, where (rather miraculously) Frankel set γ = 1 . In order to preserve
competition, technical change takes the form of an externality due to (per capita) capital
accumulation.16 Suppose then that the saving rate s rises. According to Solow’s model K / L
would thus rise and so H . Technical progress is thus clearly ‘endogenous’. Substituting H = K / L
in the production function yields Y = AK .17 Logarithmic differentiation gives Yˆ = Aˆ + Kˆ (wherethe ‘hat’ stays for growth rate). Assuming no exogenous technical progress, that is Aˆ = 0 , and sinceKˆ = sA ,18 then Yˆ = sA . Interpreting A as the inverse of the capital-output ratio v , we get Yˆ = s ,vthat is the well-know Harrod’s warranted growth equation. Per capita growth is: g y = s − n .19v

15

As known, the stability of the neoclassical growth model implies that technical change assumes this form.
This is of course a further limitation of this model (cf. e.g. Serrano & Cesaratto, 2002, p.9)..
16“If one enterprise alone were to add to its capital, it would encounter diminishing returns to that factor. But
when all do so, all are beneficiaries of compensatory shifts in the modifier” (ibid, 1004).

17Frankel used ‘a’ instead of ‘A’.

18

As K& = sY = sAK .

19

Frankel maintains that is reasonable to assume that sv> n (1962, p.1003 e 1005 footnote).

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9Frankel succeeds therefore to retain both the neoclassical production function, with its ‘nice’distribution properties, and Harrod’s growth equation with its ‘deep routes in economic thought’. Of
course Harrod’s and Frankel’s models are only superficially similar.20
Six months before Frankel’s paper, Arrow (1962) had published another, more famous,article on endogenous growth that, however, failed to deliver endogeneity.21 The technical progress
function (hereafter TPF) selected by Arrow was
H = Kγ[2]where Arrow sets γ < 1. The idea, not dissimilar from Frankel’s own, was that the experience from
capital accumulation brought about an externality on the design of new machinery and, as a result,on the efficiency of the labour force. Substituting in the production function and taking the log
derivatives obtains an aggregate growth rate equal to:
g=

n
1−γ

[3]

This is clearly an exogenous rate that depends on the labour force growth rate. It is not an
encouraging result, although Arrow defended it by arguing that it “seems to be that under full
employment, the increasing labor force permits a more rapid introduction of the newer machinery”(1962, p.166). Note that in this model the production function shows increasing returns to scale but,
nonetheless growth is still exogenous. This is not surprising given that Hˆ = γKˆ : that is capital
accumulation is not such to generate a parallel increase in the labour efficiency, so that growth will
run out of steam without an exogenous growth force.22

20

In particular, the source of endogenous growth in Harrod derives from the fact that growth is constrained
from the capital side only – at least assuming that labour is abundant and that there is no possibility offactors’ substitution in production. That is Yˆ = Kˆ , where Kˆ = sY K . The source of endogenous growth inFrankel is in technical change function. Look at the production function Y = AK 1−α ( HL) α . Suppose for astart that L is constant and normalised to 1. A raise of s is such that K and H raise by the same proportionand so Y. There are not decreasing returns to the variation of K (for a given L) because the variation of K
implies a corresponding variation of H, that is of the amount of labour in efficiency units, so that K and HL
can proceed, so to speak, in parallel. Note that there are not decreasing marginal returns to capital
accumulation because labour in efficiency units accumulate pari passu with the capital stock so that no
relative factors’ scarcity, the source of marginal decreasing returns, does arise. Suppose then that n > 0 . On
the one side this positively affect output Y, but it also correspondingly negatively affect H, the net effect isnil, and capital accumulation and the amount of labour in efficiency units HL can till proceed in parallel. Not
surprisingly labour growth does not affect the aggregate growth rate as much as in Harrod. Of course, it
affects g y with regressive effects: a raise of n reduces the rate of technical change (see Serrano & Cesaratto,
2002, p.19).
21
22

We refer here to a simplified version attributed to Sheshinsky (1967).

In other words, there are still marginal decreasing returns to capital. What it is necessary to generateendogenous growth is that there are not decreasing returns to capital or to any other factor that can be14/10/2008

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While Arrow did not assume γ ≥ 1 then? Suppose that γ = 1 . The production function
would then look Y = AKLα . Supposing no exogenous technical change ( Aˆ = 0) , the growthequation would look like Yˆ = Kˆ + αn . A positive labour growth rate ( n > 0) would clearly lead theeconomy outside a uniform secular growth rate (e.g. Ramanathan, 1982, pp. 95-96, Serrano &Cesaratto, 2002, p.17). As rather harshly commented by Cesaratto & Serrano (2002, p.18):‘Therefore if the labour force grows we see here that, rather than accumulate capital more quickly
to seize the externality, what rational agents should do is to save very little and generate ademographic explosion, which in any case need not be too big because any positive rate of growthof the population quickly leads the economy to growth rates that tend to infinity! The result is evenmore disputable than that of the learning by doing model in which, due to the increasing returns of
the economy, a constant rate of growth of the labour force generates a positive per capita growthrate. The reason for this even less reasonable result is that the learning by doing model still retains
the decreasing returns to capital, which guaranteed that a constant positive growth rate of the
population failed to accelerate the growth rate continually, since there was a counteracting tendency
for the capital-output ratio to increase.’
One can now appreciate why the particular shape of Frankel’s modifier permitted topreserve the gravitation towards a secular uniform growth even with γ = 1 and n > 1 : while with

γ = 1 the externality from capital accumulation is sufficient to determine a parallel raise of labour
in efficiency units, a positive growth in physical labour has no net effect on growth for the reasons
explained above.23
As known, the famous AK model of new EGT can be regarded as an Arrow model with thestrong assumption γ = 1 and n = 0 .24 In this case we are back again to the HD model since Yˆ = Kˆand Kˆ = sA , where A = Y

K, that is Yˆ = sA = s . The trenchant comments by Solow were that:v‘The essence here is that there is no primary factor, labor has disappeared’, and ‘It is rather
amazing. …modern literature is in part just a very complicated way of disguising the fact that it isgoing back to Domar, and, as with Domar, the rate of growth becomes endogenous’ (1992, p.18 and
32 respectively).

III. Transitional dynamics

accumulated. This can obtained by assuming very strong IRS, or by specific functions that govern theaccumulation of the ‘accumulable factor’: capital, ‘knowledge’, or ‘human capital’.
23

An akin ‘modifier’ was advanced by Conlisk (1967): see Cesaratto (1995), fn. 24 and (1999b), pp.251252.
24

Romer (1987) and Rebelo (1991) are usually quoted in this regard (cf. e.g. Jones, 2002, 162-163).

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As seen, MRW and other economists accommodate Solow’s model to the empirical results,
generally favourable to a positive association of I Y and g y , by relying on a sluggish ‘transitional
dynamics’. In the earlier days of neoclassical growth theory a fast time-convergence towards thesteady state was regarded as validating the description of secular growth provided by Solow, while
presently - virtue out of necessity - a slow convergence rate is seen as supporting that model insofar
as during the transition the saving (investment) rate, and the policies that affect it, do influence g y .
As well known, the seminal contribution of the speed of convergence was Ryuzo Sato (1963) who
showed that it was a slow process. A relevant assumption that affects the sped of adjustment
concerns the capital share ( α ). Having in mind the standard graphical representation of Solow’s
model and using a Cobb-Douglas,25 the curvature of the sy = sk α function hinges on the value ofthe capital share α : given an, say, upsurge in s that shift upward the sy curve, the higher α thefarer away is the new stationary level of k from an initial k 0 .26 As seen above, MRW rely on‘human capital’ to raise the capital share to two-third of output so as to obtain a slower convergencerate. An analogous suggestion was advanced by Conlisk (1966, p.553 and passim) who notes:‘Human capital is accumulated by diverting resources from other uses; and the amount of humancapital accumulated depends on the amount of resources diverted. Hence, logically, human capitalshould be included in the factor K, and not in the factor L …If human capital is included in K, thena substantially larger value of α is called for than if human capital is not included in K ’, as thenumerical examples confirm (see also Ramanathan 1982, pp.245-248, to which we refer for a wider
discussion). More recently, King & Rebelo (1993) have shown that convergence can be faster, so as
to rule out a role for policies in Solow’s model. The intuition is that if an economy is hit by a shockthat reduces her capital stock, the marginal product of capital would be very high. Rational savers
would react by increasing their saving supply thus accelerating accumulation and the convergence
towards the steady state. According to King & Rebelo this result is not favourable to Solow model
in so far as ‘transitional dynamics …cannot account for important part of sustained cross-country

25

As known, given the neoclassical growth equation k& = sf (k ) − (n + δ )k (standard notation), the secularequilibrium is were k& = 0 , that is were sy = (n + δ )k .
The intuition provided by Ramanathan (1982, p.246) is that when α is high and capital relatively more
important in production, if capital (saving) becomes more abundant, the scarce factor, labour, takes moretime to limit growth (the marginal product of capital falls more slowly): ‘the exogenous labor input…bottlenecks the growth rate. …if the share of capital is larger, then firms can substitute capital for laborand thus evade the bottleneck for longer period of time’. As noted by Jones (2002, p.159), in the limitingcase of α = 1 and constant labour, the case of the AK model, the production function is a straight line andthe marginal product of capital never falls, so that we have an endless transition, a ‘perfect’ endogenous
growth situation.
26

14/10/2008

12

differences in rates of economic development (ibid., p.929). Slower convergence speeds could beobtained, but ‘even if one makes agents very unwilling to substitute over time’, at the price of very
high initial marginal product of capital of the order of 500% or more (ibidem).27
To sum up, as often, the conclusions of different authors do strongly depends upon the valueof the chosen analytical functions and on the value attributed to the parameters, while the empiricaltests do not provide unequivocal results. While this would be a cheap criticism, since this happens
in most fields of the discipline, including in non orthodox fields, it is worth noticing the switch of
position in the Solowian camp, from seeing a fast convergence as a confirmation of the practicalrelevance of the secular neoclassical path to defending a lethargic gravitation to show the relevanceof policy in a Solowian context. Given the host of special assumptions that EGT models have to
make to bring endogenise growth home, neo-Solowian economists have a point in sticking to theirwell established framework stressing the length of the transitional dynamics. There are twoquestions, however. A long transition process presupposes a high capital share by including theaccumulation of ‘human capital’ along that of physical capital, what is analytically a shaky
operation if one think of the insurmountable troubles that neoclassical theory has to overcome inaggregating and measuring in value traditional capital goods. Secondly, as we shortly see, neoSolowian models present explanations of the growth sources that may also be considered little
promising.

IV. The new EGT and its neo-Solowian critics
Simplifying a bit, new EGT has taken two roads: one was to follow Arrow’s contribution tolink increasing returns to capital accumulation in a way summarised by the AK model alreadymentioned at the bottom of section II. The other was to see the source of EG in the R&D (oreducation) effort. The second route was anticipated in the sixties by Phelps, Shell, Uzawa andothers.28 The troubles with both these directions are similar, so it is not really necessary to deal withthem separately. In order to grasp this similarity let us look at the exposition and criticism byCharles Jones. Jones’ theoretical work on neo-solowian lines has perhaps been the most influential
in the last ten years or so, paralleling the earlier empirical contribution by MRW. Indeed, the firstinfluential contribution by Jones has been also empirical.

27

In spite of this, Solow (2000, pp.164-165) seems to applaud to King-Rebelo’s point about a fast
convergence rate, likely because he regards it as a confirmation of the practical relevance of the secular
growth rate predicted by his model. Rather inconsistently, in his Siena doctoral lectures (1992, p.82) Solowargues that De Long-Summers’ results about the influence of the investment rate on growth might be due toa slow convergence path, so as not to disconfirm his model.
28

Cf. Cesaratto (1999a, 1999b), Cesaratto & Serrano (2002).

14/10/2008

13

In a number of papers Jones (1995a; 1995b; 2003) summarized a number of ‘R&D basedmodels’ (defined ‘R/GH/AH models’) 29 through the following equations:
Y = K 1−α ( HLY ) α[4]

H& = δH φ LλH[5]

L = LY + LH and LH = s H L
where the symbols are by now obvious to the reader. R/GH/AH models typically assume

φ = 1 and λ = 1 . As it is clear applying these assumptions to the TPF [5],H&= δs H L
H[6]

The growth rate is endogenous as it hinges upon the term s H that indicates the choices of thecommunity between using labour for direct production or for the production of ‘ideas’. The trouble
with equation [6] is that a positive growth in the scale of the R&D labour force, due cet.par. to a
positive growth rate of the labour force, will correspondingly raise the productivity growth rate,
what the good sense would lead to reject. Jones has named this as ‘strong scale effect’ (or ‘growthscale effect’).30 In a particularly influential paper, Jones (1995b) showed that while the U.S. per
capital growth rate had been approx constant over the period 1880-1987 (1.81 % annually), thenumber of R&D scientists had increased five-fold. Similar results apply to other OECD countries.
Therefore, Jones concluded: ‘[t]hese models predict that growth rates should be proportional to thelevel of R&D, which is clearly falsified by the tremendous rise in R&D over the last 40 tears’(1995b, p.513). The parallel with Arrow’s (1962) model is clear (Jones 1995a, fn.10; Cesaratto1995): as much as the assumption of γ = 1 in Arrow’s TPF led that model into troubles in that it
could not accommodate a positive growth rate of the labour force, in the present ‘class’ of R&D
models the assumption φ = 1 and λ = 1 leads to similar difficulties.

V. Modified modifiers
In an exemplary clear paper Jones (1999) summarises his criticism to the R/GH/AH models
and extends it to another ‘class of models’, the Y/P/AH/DT models that has tried to save

29
30

After the contributions by Romer (1990), Grossman and Helpman (1991), Aghion and Howitt (1992).

A definition of strong scale effects is the following: ‘In models that exhibit “strong” scale effects, thegrowth rate of the economy is an increasing function of scale (which typically means overall population or
the population of educated workers)’ (Jones 2004, p.38). We have ‘weak scale effects’ when ‘the level of percapita income long run is increasing function of the size of the economy’ (ibid). The problems with the
R/GH/AH models and with some of the earlier generation of EGT models were pointed out in various placesof my 1995 paper. Of course these problems were well known before Jones (and me).

14/10/2008
endogenous growth

14
without incurring in the ‘scale growth effects’.31 These models are thus

synthesised. Suppose that output Q is composed by a variety B of consumption goods Y, allproduced in the same amount, that is Q = BY .32 Simplifying equation [4], each product is produced
using labour LY and a stock of ‘ideas’ H: Y = H σ LY . In turn, the variety B depends on the
population level according to a function like B = Lβ . Finally, and this is the key assumption, thestock of ideas evolves according to the TPF:
H& LH
δ
H B

[7]

The rationale of having the term B in the TPF is that population growth has two effects. On
the one hand it has a positive effect on the production of ideas through a larger amount of LH
labour, but, on the other hand, it also leads to a greater variety of products, so that the amount of

LH per product line does not increase. As above, the amount of labour devoted to the generation ofideas depends on the endogenous preferences of the community about the shares of LY and LH
respectively: LH = sL and LY = (1 − s) L . The TPF can be therefore written as:g H = δsL1− β .
Given these assumptions, the aggregate output growth is given by: g q = g B + g y . The firstterm on the left hand side is g B = βn . With regard to the second term, since each product grows inaggregate at a rate: g Y = σg H + n , per capita growth will be: g y = σg H + n − n = σg H , or:g y = δsL1− β . We get therefore an aggregate output growth rate equal to:g q = βn + σδsL1− β .
If we assume with the Y/P/AH/DT models that β = 1 to obtaing q = n + σδs[8]

we see that growth does depend on the endogenous preferences of the community expressed by theterm s, and growth is positive even with zero population growth. There are no growth scale effects
since the term L has disappeared from the growth equation:

31

After the models by Young (1998), Peretto (1998), Aghion & Howitt (1998), Dinopolous & Thompson(1998)
32

We leave aside aggregation problems and focus only on the assumptions concerning TPFs that assure
stable growth. In the specific example, the various goods are produced using labour only, so an aggregation
problem does not seem to arise. Otherwise one can maintain that the Ys represents varieties of a same good,say corn.

14/10/2008

15

At this point, watchful readers, having registered that what distinguishes the Y/P/AH/DT’s
TPF from R/GH/AH’s one is the term B, that with β = 1 is equal to L, will have noticed that this issimilar to the distinction between Arrow’s and Frankel’s TPFs.33
Jones’s (1999, pp. 142-143) criticism points to the ad hoc nature of the assumption β = 1 .
Indeed, with β < 1 the number of sectors would grow less than proportionally with population, sothat the model would still exhibit growth scale effects. With β > 1 the number of sectors wouldgrow more than proportionally with population, and this has a negative effect on productivity
growth as a rising LH workforce would be spread over an even more rapidly rising number ofsectors. Asymptotically growth would then still depend upon population growth. This is indeed thestandard criticism raised against new EGT since its very beginning (Cesaratto 1999b, p.788).
Besides the analogy between Frankel’s and Y/P/AH/DT’s TPF, what Jones fails to notice in
his otherwise illuminating paper is the similarity of Y/P/AH/DT’s equation [8] to the TPF proposedby Lucas’ (1988) in another seminal paper of the new EGT, in turn similar to Frankel’s modifier as
showed by Serrano & Cesaratto 2002, pp. 22-24).34 In Lucas we find a TPF like:H&

H

where z =

= ϑz

LH

L

is the share of labour devoted to education, R&D or the like. Notably, thedifference with equation [6] is that L appears at the dominator, as much as in Frankel’s equation [1]and Y/P/AH/DT’s equation [7]. As the latter TPFs, Lucas’ TPF does indeed fit well in the Solow’s
model without perturbing its main features (Lucas 1988, p.19-20).

35

As argued in Serrano & Cesaratto(2002, p…): in these models ‘the increase of efficiency does not depend on knowledge

33

Respectively: K = H(Arrow with γ = 1 ) and H =H&K(Frankel);= δLH (R/GH/AH) and
H
L

L
H&
= δ H (Y/P/AH/DT).
H
B
34
35

Jones quotes in passing Frankel in 2002, p.162 ,fn. 5.

Jones (1995a, p.762-63; 2004, pp. 42-43) does not regard Lucas’ TPF an appropriate solution since theidea that the absolute number of R&D employees matters for technical progress would be lost in the Lucas’
TPF, only the share on total workforce do indeed matter (we shall come back on this, that Jones names‘Mozart effect’, soon). In addition, the empirical evidence is that in the U.S. a threefold raise in the R&D
effort (z) would have not been followed by a parallel increase in the secular growth rate (the same would
apply to other countries).

14/10/2008

16

accumulation itself, but on knowledge accumulation per worker (in close analogy with the Frankel
modifier that enabled efficiency to grow with the quantity of physical capital per worker)’.36

VI. Neo-Solowian (semi-endogenous) models
Having thus pointed out both the troubles with the R/GH/AH models (growth scale effects)
and with Y/P/AH/DT models (ad hoc assumptions in order to avoid the growth scale effects), Jones
proposes a model that avoids the growth scale effects although at the price of sacrificing
endogenous growth. Let us reconsider equation [5]: H& = δH φ LλH . Jones assumes φ < 1 and λ < 1 .
His explanation of these assumptions are as follows (e.g. Jones 1995a, pp.764-766; 2002, pp. 99100). The simplest TPF inspired by P.Romer would be: H& = δ LH , that is the number of new ‘ideas’is proportional to the number of individuals engaged in R&D activities. The term δ measures therate at which new ideas are caught. δ can in turn be seen as a function of the ideas alreadydiscovered: δ = δH φ . Now, if φ > 0 this means that the generation of new ideas is a positive
function of the stock of earlier ideas – that is past discoveries open the way to new ones, the‘standing on shoulder effect’-, whereas φ < 0 would correspond to the ‘fishing out’ case in which
new discoveries are arduous to be done over time - the most obvious but breakthrough discoveries
are done earlier. If φ = 0 , the two facts precisely compensate each other. Finally, the exponent

λ < 1 in the equation [5] suggest that there might be duplications in R&D. Jones (1995a, p.766)
argues that φ > 0 is a plausible assumption, whereas the value φ = 1 assumed by R/GH/AH would
be an arbitrary condition.
Given equation [5] and the assumptions φ < 1 and λ < 1 , the growth rate of the stock ofideas is given by:

H&

λ

H

=δ L

H 1−φ

[9]

&
Along a balanced growth path the ideas growth rate, g H = H

H, should be constant. Thisimplies that, taking the logarithmic derivatives of both sides of equation [9] it must be:&
0 = λ LH

LH&
− (1 − φ ) H

H. It is reasonable to assume that the number of ideas hunters does grow at&
the same rate of population, that is LH

36

LH= n , so that we have:

Alternatively the similarity between Lucas and Frankel can be seen by writing H& = jzLe , wheree=H

measures the ‘per capita’ amount of knowledge. The endogenous rate of technical change would be

L&
H / H = jz .

14/10/2008
gH =

17

λn
1−φ

[10]

In Jones’ parlance (2002, p.106), this model would show ‘weak scale effects’ or ‘level
effects’ in so far as it shows ‘the somewhat surprising implication that this eliminates the long-run
growth effects of policy’. With φ = 1 , as note above, policy is effective, but ‘this assumption
generates the counterfactual prediction that growth rates should accelerate over time with a growing
population’. The similarity of equation [10] with Arrow’s growth equation [3], and of theconclusions drawn on their basis (see above section II), is striking as Jones fully acknowledge (e.g.
1995a, fn.10, p.768), so that the origin of his ‘surprise’ is mysterious.

VII. Millenarian explanations of Mozart production
Jones fervently defends the idea that productivity growth depends on population growth(e.g. 2002, pp.103-104). After all, this is the idea, humans are the ultimate fuel of the search
process, and it should not be surprising that a larger population growth has a positive effect on thegeneration of new ideas. Jones’ favourite quotation is from Phelps (1968, pp.511-512) according towhich: ‘One can hardly imagine …how poor we would be today were it not for the rapid population
growth of the past to which we owe the enormous number of technological advances enjoyed today. …If I could re-do the history of the world, halving population size each year from the
beginning of time on some random basis, I would not do it for fear of losing Mozart in the process’.
One might certainly argue that by cutting by half the German speaking population in the eighteen
and nineteen centuries would risk to loose many of the greatest musicians ever, but that could bedone to other populations of comparable size, at that or other times, without much fear of losing any
sort of talent. Ruling out genetic factors, something seems missing.37 Jones (2004, pp. 48-56)
discusses these possible objections at some length. Looking at different world regions over the very
long run, 12,000 years or so, it appears some relation between population size at the beginning of
the period and their technological rank measured at the year 1000/1500 or so (before the European
exploration broke the isolation of different areas). The rationale of this correlation (ibid, p. 56)
would lay in the following virtuous circle: at the beginning a small population could generate ideas
only over long periods of time. Low productivity and subsistence levels kept population constant.
However, once one idea was produced subsistence levels and fertility rise, causing a larger
population. This in turn facilitates the production of new ideas over shorter lapses of time and so on
and so forth. 38
37

May be what is relevant is the diffusion of musical learning amongst the population, so that the number ofMozarts could be increased by fostering musical education (for a similar remark see Temple, 2003, p.506).

38

Limitating himself to his ‘weak scale effects’, Jones (2004, p.45) seems to arrive to the point ofsuggesting higher fertility as a way to raise per-capital income. He defends this stance arguing that this

14/10/2008

18

This kind of reasoning certainly facilitate the construction of nice mathematical models,
however it is difficult to think that this kind of explanation can be no more than one aspect of somemore complex historical process.39 In addition, Jones recognises that the most recent experience
seems to disconfirm this explanation since ‘countries with the most rapid rates of population growth– many in Africa – are among the countries with the slowest rates of per capita income growth’(Jones, 2004, p.50). This would be due, according to Jones, to the ‘different levels of human capitaland different policies, institutions, and property rights’ and we should rely on ‘econometric
evidence that seeks to neutralize these differences’ (ibidem). Some ‘econometric evidence’ ispresented in order to confirm the idea that one factor to be ‘neutralised’ is openness to international
trade. This in turn would be associated to openness to international ‘flow of ideas’. Once controlled
for both trade and institutional quality, the evidence would suggest a significant elasticity of GDP
per worker with respect to the size of the workforce. In my opinion this is not enough to dissipate
some scepticism with regard to Jones’ thesis. Indeed, one thing is to argue that economic growth isaccompanied by population growth, on the other hand it is difficult to attribute to regard the later asa main cause of recent episodes of economic development. Over a millennium perspective
population configurations might certainly be a significant factor, amongst others structural factors,
to explain secular backwardness (dispersed population in the vast territory of Africa is quoted inthis respect as an explanation of its backwardness). Whether this is a useful perspective to explain
the secular economic delay of China and her recent rocketing economic growth is doubtful.

Concluding remarks
Modern mainstream growth theory seems therefore split into two camps: those who favour
endogenous models with their policy effectiveness over long-run growth, although at the price ofspecial assumptions such as the knife-edge value in the exponents of the TPFs, or zero population
growth - this is avoided by the use of the various ad hoc modifiers a là Frankel -; and the neoSolowian economists (semi-endogenous models) which limit the policy impact over the transition
periods and attribute a questionable role to population growth in explaining not only aggregate
growth (which sounds quite odd to a Keynesian economist), but even productivity growth. Senioreconomists such as Frank Hahn (1994, p.1) have been quite critical of the ‘backward reasoning’
employed by the new EGT in selecting their TPFs: the description of reality is bended in order that
it fits a steady state growth model, and not the other way round as it would appear natural. Indeed,
too much attention is paid to long run ‘balanced’ growth, notes Temple (2003, p.501 and passim) inwould be a way to endogenise growth by making it dependent on the policy choices of the community overfertility.
39

An author often quoted for having pointed out a number of material geographical and biological factorsthat affected the millenarian process of growth differentiation among world regions is Diamond (1997).

14/10/2008

19

an stimulating article, whereas empirically we do not really know whether the economy is moving
along a steady state or, may be, converging to it.40 Balanced growth requires the very peculiar
knife-edge assumptions recalled above even in the exogenous Solowian framework (ibid, pp.499500). Temple quotes here a point made by Jones (2004, pp.60-64) that what is required for steadystate growth is the linearity in the TPFs, whether of the exogenous Solowian variety or of theendogenous species.41 Once we are not obsessed with balanced growth, Temple concludes, theseknife edge assumption my be taken at their face value, as simplifying hypothesis to make theoretical
explorations that provide different perspectives of reality: ‘growth models are often best seen aslaboratories for thought experiments, and apparently competing frameworks can form a usefulcomplement to one another’ (Temple, 2003, p. 508). This is a sensible position, that may also
explain the sense of accomplishment that, after all, mainstream growth economists display withregard to the past twenty years of revival of growth theory. Of course, if we pass from the academic
results to the capacity to provide practical policy suggestions the bottle would appear half full, to begenerous, as Temple (ibid, p.501) or Mankiw (1995, p.309) are next to admit.
We may finally ask ourselves if a different approach, alternative to neoclassical theory,
would provide a more satisfactory explanation of economic growth. In this regard should be easily
appreciated by non orthodox economists that neoclassical growth theory – in any of its
ramifications - is the natural victim of the capital theory critique. Alternative theoretical accounts ofcapital accumulation appear more amenable to the stylised facts that conventional growth analysis
is at pain to allow for. This is the case of the relation between the investment rate and the growthrate (cf. Serrano & Cesaratto, pp.25-27). The decisive role of aggregate demand, both in the shortand in the long run periods, as the key of economic growth, is of course the differentia specifica, of
non orthodox schools (Garegnani, 1992). On the supply side, no doubt that the introduction oftechnical progress in growth models, including those of non orthodox orientation, is a very artificial
operation that may be useful for some purposes, but that cannot be the basis for a satisfactory
account of a phenomenon whose understanding does require, in my opinion, historical researchrather than economic modelling. But, after all, this true as well with regard to the historical andgeopolitical sources of aggregate demand in the real episodes of economic growth.

40

The only ‘piece of evidence we have for the existence of a balanced growth path is that the real interest
rate does not display any clear trend’ (ibid, p.507). Note that it is only within neoclassical theory that a
constant real interest rate requires balanced growth, in particular the constancy of the relative proportions oflabour (in efficiency units) and capital.
41&

Also exogenous technical progress requires a linear TPF, e.g. m = H

H.

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20

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