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Technical Change in Alternative Theories of Growth*
Luca Zamparelli†
January 25, 2024
Abstract
This paper investigates alternative ways of introducing technological progress in heterodox
theories of economic growth. We model technical change as: i) exogenous and costless; ii)
a positive externality of capital accumulation, the wage share or the employment rate; iii) en-
dogenous and costly. We implement these formalizations in Classical growth theories, where
investments coincide with full capacity savings, and Keynesian theories where capital accumu-
lation is demand constrained. We also distinguish between abundant and inelastic labor market
closures. We discuss the outcomes of these models in terms of long-run growth, functional
income distribution and employment.
Keywords: Technical change, Heterodox growth models, R&D, Factor income shares, Em-
ployment
JEL Classification: D24, E25, D33, O30, O41
1 Introduction
Since the inception of modern political economy, technical change has played a prominent role
in the analysis of economic growth and development. In the introduction to The Wealth of Na-
tions, Adam Smith singled out labor productivity (‘the skill, dexterity, and judgment with which ..
labour is generally applied’) as the ultimate source of economic opulence (Smith, 1776). Classical
economists and Marx also understood that innovation and technology are not simply a fundamental
driver of economic growth, but they are strongly intertwined with the dynamics of income distri-
bution and employment. Both Ricardo and Marx recognized the role of profitability in eliciting
*Chapter prepared for the Handbook of Alternative Theories of Growth, Second Edition, edited by Mark Setterfield.
Sections 2, 3, and 4.1 draw on Tavani and Zamparelli (2017b).
†Department of Social and Economic Sciences, Sapienza University of Rome. Email: Luca.Zamparelli@Uniroma1.it.
1
innovation as new techniques of production would be implemented only if they increased the profit
rate; while Ricardo in the famous chapter "On Machinery" suggested that technical change could
harm employment (‘substitution of machinery for human labour, is often very injurious to the inter-
ests of the class of labourers’ Ricardo , 1821[1951-1973], Works I: 388).
This chapter investigates alternative ways of introducing technological progress in heterodox
theories of economic growth. In particular, the focus will be on how the interaction of different for-
malizations of technical change and alternative closures of the model affects the economy’s growth,
functional income distribution and employment in the long run.
To restrict the scope of our inquiry, we must first clarify what we mean by heterodox growth
theory. We define as heterodox any theory not based on the neoclassical general equilibrium theory
paradigm. In this way, we rule out both the exogenous growth model based on perfect competi-
tion, where factors of production are substitutable, fully employed and paid according to the their
marginal products, and endogenous growth models based on imperfect competition, where factors
of production are still fully employed and substitutable but where the existence of market power
precludes marginal cost pricing. Within this definition, we will look at those theories that allow for
long-run steady states featuring the so-called Kaldor facts (Kaldor, 1961). Specifically, we will re-
strict our analysis to fully adjusted positions where: factor income shares and the output-capital ratio
are constant; technical change is purely labor-saving; and the capital-labor ratio increases at a con-
stant rate. We will distinguish between Classical growth theories where investments coincide with
full capacity savings and Keynesian theories where capital accumulation is demand constrained.
Since the focus is on balanced growth, we will not discuss the prominent role of technical change
in Evolutionary growth theory.
With respect to the formalization of technical change, we will explore three alternatives. The
first one assumes that it is exogenous and costless. A second option models it as a positive ex-
ternality of other economic variables; in particular, capital accumulation, the wage share and the
employment rate may contribute to labor productivity growth. Finally, we will explore the possi-
bility that technical change is endogenous and costly because it requires R&D investment, either
private or public.
The rest of the paper is organized as follows. Section 2 lays out the analytical structure and
the main assumptions of the models we investigate. Section 3 discusses Classical growth under
the two alternative specifications of abundant and scarce labor supply. Section 4 explores three
possible versions of Keynesian growth: a Kaleckian closure with exogenous income distribution and
endogenous capacity utilization; a Harrod-Kaldorian variant with endogenous income distribution
and normal, exogenous, utilization; and, finally, a Sraffian one with exogenous distribution and
normal utilization. Section 5 discusses the possibility that costly R&D is paid by the public sector.
Section 6 concludes summarizing the main results.
2 Analytical Framework
In this section, we outline the analytical structure of the economy and the assumptions on class
2
division, investment, labor supply and technical change.
2.1 Production
Final output Yis produced according to a Leontief production function that combines capital
Kand labor Lin fixed proportions:
Y=min[uBK, AL],(1)
where Bdenotes the ratio of potential output (Yp)to capital, Ais labor productivity, and u≡
Y/Y pis a measure of capacity utilization. The zero elasticity of substitution between factors of
production implies a radical departure from the neoclassical growth model. First, with respect to
income distribution, factors cannot be paid according to their marginal products, which are not
defined at the profit maximizing factor choice uBK =AL. Second, it allows for equilibrium
labor unemployment since there are no endogenous market mechanisms reconciling labor demand
L=uBK/A to labor supply N. When L < N and u= 1,unemployment is purely technological,
or structural. When BK/A < N and u < 1, both technology and lack of aggregate demand
contribute to unemployment.
2.2 Social classes and income distribution
There are two classes of households in the economy: capitalists own the means of production,
receive profit income Πfrom the ownership of the capital stock, and have constant propensity to
save s. Workers supply labor, earn a real wage w, and do not save. If we denote the wage share as
ω≡wL/Y =w/A, capitalists’ profits will be
Π = Y(1 −ω) = uBK(1 −ω).(2)
2.3 Investment
We will distinguish alternative growth theories according to the investment functions they assume.
The first distinction is between supply- and demand-side theories. Supply-side theories assume that
investments (I) coincide with savings (S) at full capacity: I=sY p(1 −ω).We define gK≡I/K
the actual growth rate of capital, which in this case yields:
gK=sY p(1 −ω)/K =sB(1 −ω).(3)
In contrast, investment demand is independent of saving behavior in demand-side theories. We
distinguish between exogenous and induced theories of investment. Exogenous theories posit that
(at least some share) of investment does not depend on income. We can borrow from the post-
Keynesian tradition and assume that, in addition to an autonomous component, investment is a
positive function of measures of aggregate demand and profitability:
gK=γ0+γ1u+γ2(1 −ω).(4)
3
Kaldor (1956, p.94) clarified that the Keynesian principle of aggregate demand can be alterna-
tively used to provide either a theory of the level of economic activity or of income distribution. We
will use the investment function in (4) to discuss two alternative cases. In the first case, we will take
income distribution as given to find the equilibrium level of capacity utilization. In the second one,
we will assume that capacity utilization is at its normal level (u=un= 1) to solve for the equilib-
rium wage share. We will refer to these cases as, respectively, Kaleckian and Harrod-Kaldorian.1
Finally, induced theories of investment assume it is an increasing function of output:
gK=hY/K =huB. (5)
We will explore this function in the context of the so-called ‘Sraffian supermultiplier’.
2.4 The two Harrod’s problems
Modern growth theory conventionally begins with Roy Harrod’s (Harrod, 1939) attempt to ex-
tend the Keynesian analysis to the long-run. He introduced the notions of ‘warranted’ growth rate
(gs
K) as the growth rate that ensures the saving-investment equality, and ‘natural’ growth rate (gp)
as the unemployment stabilizing rate, thus equal to the sum of population and labor productivity
growth. He discussed the issues connected to the difference between actual, warranted and natural
growth rate. We do not analyze the solutions that alternative theories have offered to reconcile the
three growth rates (see Blecker and Settterfield (2019) for a recent account). We will simply as-
sume that the economy is in a long-run state where both the goods market and the labor market are
in equilibrium, so that the three growth rates equalize.
Given our notation and assumptions, the warranted growth rate is
gs
K=S/K =suB(1 −ω),(6)
while the natural rate of growth is
gp=˙
N/N +˙
A/A, (7)
with the additional hypothesis that the labor force is a constant fraction of population. The following
two conditions must hold simultaneously in a long-run steady state equilibrium:
gK=suB(1 −ω) = ˙
N/N +˙
A/A. (8)
We will index by ss the values of the endogenous variables that solve (8).
2.5 Labor supply
We will explore two alternative assumptions regarding labor supply. Under the first assumption,
1The Kaldorian growth model has mostly adopted an open economy framework and has emphasized the role of
export as a source of aggregate demand (see Blecker 2022 for a recent survey). In that context, since labor productivity
growth improves an economy’s competitiveness and access to global markets it is considered a fundamental determinant
of investment. Since we assume a closed economy, that channel is precluded and the investment function ultimately
depends on profitability.
4
labor supply is inelastic as it grows at a constant exogenous rate n, so that ˙
N/N =n. We will
see that, depending on the formalization of technical change, exogenous labor supply may constrain
capital accumulation and growth while affecting income distribution. On the other hand, labor
supply can be infinitely elastic, or abundant, and capable of instantaneously accommodating labor
demand, that is, it is endogenous to economic growth: ˙
N/N =˙
L/L ≡gL.An endogenous, or
abundant, labor force typically applies to emerging economies, where large rural or informal sectors
provide ample labor reserves to be drawn into the formal economy. Still, even mature economies
can be relatively unconstrained by labor supply through loose migration policies.
2.6 Technical change formalizations
Equation (8) shows that in a long-run equilibrium technical change increases labor productivity
while leaving capital productivity constant. In fact, this is a well-established result in growth theory,
known at least since Uzawa (1961). For this reason, we will mostly (but see Section 3.3) analyze
formalizations of technical progress that simply leave the potential-output to capital ratio constant
while producing positive labor productivity growth rates ˙
A/A ≡gA.
We will distinguish three approaches to technical change. The simplest way to model technical
change is by assuming it is exogenous. It is costless and occurs in the economy independently of
economic activity and of the level of any economic variable: gA=a. A first attempt to make techni-
cal change endogenous treats it as a positive externality of some economic variable x:˙
A=H(x)A.
In the heterodox literature, we can find at least three variables that have been associated to positive
effects on productivity growth. The Kaldor-Verdoorn law (Verdoorn, 1949; Kaldor, 1957), or tech-
nical progress function, states that technical change is an increasing function of either (absolute or
per-capita) capital accumulation or output growth. It is based on the notions of economies of scale,
macroeconomic increasing returns, and learning by doing. We restrict our attention to one linear
version of the law that has productivity growth depending on the capital growth rate:
gA=φ0+φ1gK.(9)
A second option makes labor productivity growth a positive function of the wage share. This no-
tion is founded both in the classical-Marxian analysis of the choice of techniques and in the induced
innovation literature discussed below in Section 3.3. The basic intuition relies on the incentive to
introduce labor-saving innovations by firms facing rising labor costs, that is higher wage shares. We
simply assume a reduced form that posits a direct positive relationship from the wage share to the
growth rate of labor productivity:
gA=f(ω), f′>0,(10)
as in Taylor (1991) and Dutt (2013a). However, we will show below that this relation can be
microfounded if we let firms optimally choose the direction or the intensity of technical change. We
will refer to to this formalization as the Classical-Marxian motive or induced innovation theory. The
theory of Marx-biased technical change delivers a similar relation between productivity growth and
the wage share, but we will not explore it as it produces unbalanced growth.
5
A third alternative points to labor market tightness as a source of technical change (Dutt, 2006;
Flaschel and Skott, 2006; Sasaki, 2010; Palley, 2012; Setterfield, 2013), since labor shortages would
provide the incentive to adopt labor-saving innovations. The relevant variable in this case is the
employment rate defined as e≡L/N, so that productivity growth can be represented as
gA=q(e), q′>0.(11)
Note that this specification of technical change is only relevant with exogenous labor supply. When
labor supply is endogenous, labor is never scarce and its limited supply cannot provide any incentive
to innovate. We will treat each type of externality as a single source of technical chance, but multiple
papers have combined two of them in making technical change endogenous (just as examples see
Taylor et al. 2019; Palley 2019; Michl and Tavani 2022; Fazzari et al. 2020; Allain 2021)
Observe that in the three specifications of technical change as an externality (9), (10), and (11)
we have assumed a positive linear spillover from the existing level of technology to the productivity
of the factor responsible for technical progress. This means that, over time, given levels of capital
accumulation, wage share or employment rate produce ever increasing rises in productivity growth.
We are not aware of arguments in the literature that address the plausibility of this issue.
As an alternative, we can assume that technical change is the outcome of R&D investment, and
is therefore endogenous and costly. Either physical output or labor inputs must be employed to raise
productivity growth. We will follow the first route and assume that if Ris the amount of final good
invested in R&D, productivity growth follows
gA= (R/K)ϑ,(12)
with ϑ∈(0,1) and the normalization of investment by capital is necessary to avoid explosive
growth. We assume different specifications for Rin supply- and demand- side models. In the
former, Ris a share of full capacity saving so that R=δsBK(1 −ω),where δ∈(0,1) will
be a choice variable for the optimizing firm. In the latter case, we will simply assume that R&D
investment is a positive function of output: R=δuBK. We will not assume that firms choose δ
optimally for two reasons: first, profit-maximizing microfoudations do not belong to the Keynesian
tradition; second, and most importantly, the optimal choice of investment (in physical capital or
R&D) is problematic without knowing future sales, as it is the case in a Keynesian economy (though
see Caminati and Sordi (2019)).
Finally, we will conclude by discussing models where R&D investments are financed through
taxation by the public sector.
3 Classical Growth
Most models belonging to the classical tradition assume Say’s law, that is supply creates its own
demand. This means that investment coincides with full capacity saving, which in our notation
yields gK=gs
K=sB(1 −ω),where we have imposed u=un= 1. The main distinction
6
within the classical tradition regards the labor supply assumptions. We start by discussing the case
of abundant labor to turn next to exogenous labor supply growth.
3.1 Abundant labor
Classical political economy was born during the early stages of the industrial revolution. At the time,
labor in the rural sector was abundant and could always accommodate the manufacturing sector’s
demand for labor. Growth was constrained by capital accumulation rather than the availability of
labor. The abundance of labor was also thought to exert downward pressure on real wages, which
would be constant at the exogenous subsistence level. In fact, real wages stagnated at least until
the second half of the 19th century (Allen , 2009). In the context of labor productivity growth, we
will not assume constant real wages as given, or the wage share would trend toward zero. We will
rather take the wage share as fixed at a conventional value (Foley and Michl, 1999): ω= ¯ω. The
corresponding long-run equilibrium is
gK=sB(1 −¯ω) = gL,ss +gA.(13)
In this framework, aggregate growth is fully determined by full capacity saving while the role of
technical change consists in determining employment growth. Given capital accumulation, there ex-
ists a trade-off between accommodating output growth through employment or productivity growth.
Faster technical change means faster real wage growth that benefits a smaller pool of workers. This
is true both with exogenous technical change gL,ss =sB(1 −¯ω)−a, or if we model productivity
growth according to (9): gL,ss =sB(1 −¯ω)(1 −φ1)−φ0; or (10): gL,ss =sB(1 −¯ω)−f(¯ω).
In contrast, the Kaldor-Verdoorn law and the induced innovation theory imply opposite responses
to distributional shocks. Under the latter assumption a rise in the wage share immediately makes
productivity growth faster (fω>0), while a higher wage share slows down the pace of technical
change by reducing capital accumulation if the Kaldor-Verdoorn law holds.
3.2 Exogenous labor supply
Let us now assume exogenous labor supply. With exogenous technical change, the model pro-
duces the Goodwin’s (1967) growth cycle steady state:
sB(1 −ωss ) = n+a.
Income distribution necessarily becomes endogenous as it is responsible for the adjustment of the
economy’s saving to the exogenous natural growth rate. A rise in productivity growth reduces the
economy’s labor demand and exerts a downward pressure on real wages, which results in a lower
wage share. The saving rate has an opposite effects on the wage share as its rise yields higher saving
growth, which immediately translates into higher capital accumulation and labor demand.
Next, we formalize technical change as an externality. When gAfollows (9), balanced growth
yields
sB(1 −ωss ) = n+φ0
(1 −φ1),(14)
7
with gA,ss = (φ0+nφ1)/(1 −φ1).This result is quite similar to the exogenous technical change
outcome. The labor share is an increasing function of the saving rate, and a decreasing function of
productivity growth. While technical change is not exogenous, its equilibrium value is ultimately
determined by technological and demographic parameters only, despite being affected by capital
accumulation and the saving rate during the transition to the steady state.
When labor productivity growth follows the induced innovation motive as in (10), the wage
share is still the endogenous variable in the long-run equilibrium
sB(1 −ωss ) = n+f(ωss),(15)
and it is still an increasing function of the the saving preferences as total differentiation of (15)
shows: dωss/ds =B(1 −ωss)/(f′(ωss ) + sB)>0.A higher saving rate increases capital ac-
cumulation and labor demand; a higher wage share is then necessary to keep the employment rate
stable as it simultaneously increases labor productivity growth and lowers the growth rate of saving.
The interesting twist is that long-run technical change and growth become positive functions of the
saving rate. A rise in the saving rate increases accumulation, which, however, is constrained by la-
bor supply. This makes the wage share increase; but the higher wage share raises labor productivity
growth and the natural growth rate through its effect on the wage share gA,ss =f[ωss(s)],g′
A,s >0.
In the end, the economy stabilizes at a higher growth rate as the warranted growth rate adjusts to the
higher natural growth rate.2
Finally, if technical progress follows (11) the long-run equilibrium becomes
sB(1 −ωss ) = n+h(ess),(16)
and by itself it cannot determine both income distribution and the employment rate. We have two
options. We can go back to the conventional wage share model and assume ωss = ¯ω. At that
point, the equilibrium condition determine ess and productivity growth as negative function of the
exogenous wage share. As an alternative, we can follow the original Goodwin (1967) model, where
real wages growth is a positive function of the employment rate, as a tighter labor market raises
workers’ bargaining strength, say gw=m(e). Since wages and labor productivity must grow at the
same rate to stabilize the wage share, then m(ess) = h(ess)fixes the equilibrium employment rate.
Labor productivity growth is thus determined in the labor market independently of the saving rate,
which, on the other hand, still has a positive effect on the equilibrium wage share through (16).
We now consider the possibility that technical change is costly, that is it requires resources
R=δsBK(1 −ω)to produce labor productivity growth gA= (δsB(1 −ω))ϑ. The long-run
equilibrium condition must be emended to take into account that the share δof saved profits is spent
in R&D investment:
(1 −δ)sB(1 −ωss ) = n+ (δsB(1 −ωss))ϑ.(17)
2We can see that the net effect of a higher saving rate and a lower profit share on the warranted growth rate is positive:
d
ds gs
K=B(1 −ω)−sBdω/ds =B(1 −ω)(1 −sB/(sB +f′)) >0,since sB/(sB +f′)<1.
8
Capitalists face a trade-off regarding the allocation of their savings. They can either increase the
capital stock or labor productivity growth. If we assume like in Tavani and Zamparelli (2021)
that they choose δto maximize the rate of growth of profits gK+gAω/(1 −ω), we find δ=
1
sB(1−ω)ϑω
1−ω1
1−ϑand gA=ϑω
1−ω
ϑ
1−ϑ.The maximizing choice of the size of R&D investment
makes labor productivity growth become a positive function of the wage share, and in fact provides
a microfoundation for (15). The intuition is straightforward, the higher the unit labor cost the higher
the incentive to save labor inputs relative to increase the capital stock. If we then plug the optimal
values of δand gAinto the long-run equilibrium condition we find
sB(1 −ωss ) = n+ωss
1−ωss
ϑ
1−ϑ1−ωss(1 −ϑ)
1−ωss ,(18)
which is just a specific case of (15) and confirms that the saving rate has a positive effect on the
equilibrium wage share, which, in turn, raises productivity growth.
3.3 The induced innovation hypothesis
In the context of the Classical model with exogenous labor supply we need to discuss the induced
innovation hypothesis. It departs from our general assumption that technical change only improves
labor productivity growth while leaving capital productivity constant. The output-capital ratio will
still be a constant in the long run, but as the result of the economy’s dynamics rather than an as-
sumption on the innovation technology.
Originated in the 1960s by Kennedy (1964) and von Weizsacker (1962), this theory has expe-
rienced a recent revival within the Classical-Marxian tradition (Foley, 2003; Julius, 2005; Rada,
2012; Tavani, 2012; Zamparelli , 2015; Rada et al., 2023). Its central element is the innovation
possibility frontier (IPF), which inversely relates the attainable growth rate of labor productivity to
the growth rate of capital productivity: gB=ϵ(gA), ϵ′<0, ϵ′′ <0. Firms choose the direction
of technical change, that is a point (gA, gB)on the IPF, to maximize the growth rate of the profit
rate: ωgA+ (1 −ω)gB. As a result, labor productivity growth becomes an increasing function of the
wage share: gA=f(ω), f′>0. The induced innovation theory thus provides a microfoundation for
equation (10). However, in contrast with the microfoundation based on costly endogenous technical
change, it does not assume constant capital productivity. This has dramatic implications on the na-
ture of the long-run equilibrium. In fact, the long-run stability of the output-capital ratio determines
the steady state wage share as solution to gB=ϵ[f(ωss)] ≡gB(ωss) = 0,so that ωss =g−1
B(0).
At that point, the equality between warranted and natural growth rates is only responsible for the
determination of the equilibrium capital productivity
sBss(1 −g−1
B(0)) = n+f(g−1
B(0)).(19)
Accordingly, the saving rate only affects the output-capital ratio with no influence on long-run
wage share. Still, a recent generalization of this framework to include costly innovation (Zamparelli
, 2024) has clarified under what technological conditions the saving rate may affect the steady state
income distribution.
9
4 Keynesian Growth
In contrast to Classical growth, Keynesian models rejects Say’s law so that output is demand-
determined. Within the Keynesian framework, we discuss three specifications of the investment
function that corresponds to three different closures: a Kaleckian closure, where the rate of capac-
ity utilization adjusts to ensure the equilibrium in the goods market; a Harrod-Kaldorian closure,
where the saving-investment equilibrium occurs through changes in income distribution; and, fi-
nally, a Sraffian closure, where the investment to output ratio adjusts to ensures normal utilization
rates despite exogenous income distribution. Multiple contributions (Cassetti, 2003; Palley, 1996;
Sasaki, 2010; Allain , 2015; Lavoie, 2016; Taylor et al., 2019; Palley , 2019) may simultaneously
belong to more than one of these categories, but this strict classification is a useful expositional
device.
4.1 Kaleckian closure
In a basic Kaleckian model, firms operate with slack productive capacity and set prices (p) by
charging a constant mark-up µover unit labor costs: p≡1 = (1 + µ)ω, so that µfully determines
the labor share ω= 1/(1+µ) = ¯ω. Therefore the Kaleckian distributive assumption is analogous to
the classical closure with endogenous labor supply, but it is founded on firms’ market power rather
than on the tendency of real wages towards a socially determined subsistence level. Using (4) and
ω= ¯ωin the saving-investment equality we find the short-run equilibrium capacity utilization and
growth rate as
u(¯ω, s) = γ0+γ2(1 −¯ω)
sB(1 −¯ω)−γ1
,
gK(¯ω, s) = sB(1 −¯ω)
sB(1 −¯ω)−γ1
(γ0+γ2(1 −¯ω)) .
Capacity utilization, or aggregate demand, is wage-led and decreasing in the saving rate. Growth,
however, is still a negative function of the saving rate, but can be either wage-led or profit-led de-
pending on how strong the profitability effect on investment γ2is relative to the aggregate demand
effect γ1.
With endogenous labor supply the long-run equilibrium requires:
gK(¯ω, s) = gL,ss +gA.
When growth is unconstrained by labor supply the short-run effects of income distribution and
the saving rate fully translate to the long run, with employment growth operating as a buffer. Sim-
ilarly to the Classical model with abundant labor, technical change determines employment growth
given capital accumulation. When technical change is exogenous gL,ss =gK(¯ω, s)−a, which
shows the trade-off between employment growth on one hand and productivity and real wage
growth on the other hand. When we model productivity growth along Kaldor-Verdoorn lines, it
retains the wage- or profit-led nature of capital accumulation gA=φ0+φ1g(¯ω, s), and so does
employment growth gL,ss =gK(¯ω, s)(1 −φ1)−φ0.Finally, with the Marxian motive we find
10
gL,ss =g(¯ω, s)−f(¯ω).In this case, a higher wage share necessarily decreases employment growth
when growth is profit-led, whereas when growth is wage-led the effect on employment will depend
on whether the wage share positive shock raises accumulation more or less than productivity growth.
Things are particularly interesting when productivity growth is costly. If R=δuBK the equi-
librium utilization and growth rates turn into
u(¯ω, s, δ) = γ0+γ2(1 −¯ω)
B(s(1 −¯ω)−δ)−γ1
gK(¯ω, s, δ) = 1 + γ1
B(s(1 −¯ω)−δ)−γ1(γ0+γ2(1 −¯ω)).
Capacity and growth are increasing in δsince R&D investment is a component of aggregate de-
mand. Long-run productivity growth is gA= (δu( ¯ω, s, δ)B)ϑand it has the short-run properties of
capacity utilization. Thus, it is wage-led and decreasing in the saving rate. Additionally, short-run
demand shocks, say changes in γ0, have permanent effect on technical change. It is important to
emphasize that, as long as demand is wage-led, technical change will be wage-led even when capital
accumulation is profit-led because it is a mere function of output. Of course, things would change
if we made R&D investment a function of both demand and profitability or, as we did in Section
3.2, if we assumed that it is a function of profits rather than output. If labor is abundant, in balanced
growth we find
gK(¯ω, s, δ)) = gL,ss + (δu(¯ω, s, δ)B)ϑ.
The effect of changes in δand son employment growth are ambiguous since both accumulation
and productivity growth move in the same direction. Positive shocks to ωnecessarily reduce em-
ployment growth if capital accumulation is profit-led, while their effect is uncertain when growth is
wage-led.
When labor supply is exogenous, the properties of the Kaleckian model change dramatically
and we can no longer safely assume that income distribution is exogenous. If technical change is
exogenous the long-run equilibrium condition becomes
gK(ωss, s) = n+a. (20)
The right hand side is fully exogenous, so that the equality is achieved by adjustment in the wage
share, which becomes the steady state endogenous variable. The effect of changes in aon in-
come distribution will depend on the wage- or profit- led nature of the short-run growth rate as
a higher or lower wage share will be needed to accommodate the rise in the natural growth rate
dωss/da = 1/g′
ω. On the other hand, a higher saving rate raises (lowers) the wage share if the
economy is wage- (profit-) led: dωss /ds =−g′
s/g′
ω≷0⇐⇒ g′
ω≷0.Since higher savings depress
accumulation, income distribution must raise growth to preserve the equilibrium; hence the wage
share will increase (decrease) when the economy is wage- (profit-) led.
When technical change evolves according to the Kaldor-Verdoorn law, income distribution is
11
still the adjusting variable in the long-run, while the natural growth rate and productivity growth
only depends on technological and demographic parameters:
gK(ωss, s) = n+φ0
(1 −φ1), gA,ss =nφ1+φ0
(1 −φ1).
If we let technical progress follow the classical-Marxian motive the balanced growth condition
is:
gK(ωss, s) = n+f(ωss ).(21)
Income distribution is still endogenous, and now the wage share directly influences the natural
growth rate. Any factor affecting income distribution will have permanent effects on capital accu-
mulation and labor productivity growth. For example, higher autonomous investments (γ0) raise
the wage share and growth when the economy is profit-led in the short-run since dωss/dγ0=
gγ0/(fω−gω)>0. When the economy is wage-led, faster accumulation will raise the wage share
only if fω> gω, that is when the response of labor productivity growth to income distribution is
stronger than the response of short-run accumulation.
When productivity growth react to the labor market tightness, we can acquire the employment
rate as the adjusting variable in the long-run and return the wage share to its exogenous role
gK(¯ω, s) = n+q(ess).(22)
The employment rate and labor productivity growth will improve in line with capital accumulation.
Positive shocks to the wage share will raise (lower) employment and technical change when growth
is wage (profit-) led. They will also rise following a positive demand shock or a drop in the saving
rate. Similar analyses can be found in Flaschel and Skott (2006); Lavoie (2006); Sasaki (2010).
Finally, income distribution becomes again endogenous when technical change is costly since
the wage share is the only adjusting variable in the long-run equilibrium
gK(ωss, s, δ) = n+ (δu(ωss , s, δ)B)ϑ.
Productivity growth is endogenous and it responds to changes in the propensity to invest in R&D
δdirectly and indirectly through its effect on the wage share and capacity utilization. Productivity
growth is a positive function of both δand ω, but we cannot establish the overall effect of δon the
natural growth rate since the sign of dωss/dδ is ambiguous.
4.2 Harrod-Kaldorian closure
If we assume that capacity utilization adjusts to its long-run value, u=un= 1, the investment
function (4) simplifies to gK=γ+γ2(1 −ω),with γ≡γ0+γ1.The saving-investment equality
is sB(1 −ωss ) = γ+γ2(1 −ωss), and determines the long-run profit share
(1 −ωss) = γ
sB −γ2
.
The profit share is a positive function of autonomous investment as prices rise faster than wages
in response to higher aggregate demand: investment ‘forces saving’ through distributional changes.
12
The corresponding growth rate is gK=sBγ
sB−γ2.
If labor supply is abundant, technical change has no effect on growth and distribution, but it de-
termines how much capital accumulation is accommodated by employment or productivity growth.
With exogenous technical change find gL,ss =sBγ
sB−γ2
−a. Whereas when technical change is
modeled an externality we have gL,ss =sBγ
sB−γ2(1 −φ1)−φ0in the Kaldor-Verdoon case, and
gL,ss =sBγ
sB−γ2
−f(sB−γ2−γ
sB−γ2)when we assume the Marxian motive. The only relevant differ-
ence between the last two cases is that productivity growth reacts in opposite way to changes in
autonomous investment. It rises in the former, while it declines in the latter due to a fall in the wage
share.
We now consider costly technical change. R&D investment is R=δBK and affects the saving-
investment equilibrium which becomes sB(1 −ωss) = γ+γ2(1 −ωss ) + δB. Since R&D invest-
ments contribute to aggregate demand the profit-share is a positive function of δ
(1 −ωss) = γ+δB
sB −γ2
.
The corresponding growth rate is also increasing in δthrough its positive effect on profitability:
gK=B(sγ+γ2δ)
sB−γ2. Labor productivity growth is gA= (δB)ϑso that both accumulation and technical
progress increase with δ, while the effect on employment will depend on which effect is stronger
gL,ss =B(sγ+γ2δ)
sB−γ2
−(δB)ϑ.
If we turn to exogenous labor supply, accumulation may be constrained by the natural growth
rate, depending on the formalizations of technical progress. When gA=a, long-run income distri-
bution is anchored by technical change and population growth rather than accumulation:
sB(1 −ωss ) = n+a.
Faster technical change reduces the wage share as it lowers labor demand. The interesting
result is that the equality of actual and natural growth rate requires capital accumulation to lose its
independent nature. In the long-run, not only income distribution but also autonomous investment
must become endogenous to ensure equilibrium
sBγss
sB −γ2
=sB(1 −ωss ) = n+a.
In fact, aggregate demand plays no role and the model becomes indistinguishable from the
Classical model with exogenous labor supply. The same is true when technical change follows the
Kaldor-Verdoorn law: sBγss
sB −γ2
=sB(1 −ωss ) = n+φ0
(1 −φ1).
In both cases, a higher saving rate increases autonomous investment and the wage share with
no effect on growth. On the one hand, an increase in sreduces accumulation so γss must rise to
offset its effect; on the other hand, ωss increases as redistribution in favor of the class with the
lower (zero) propensity to save is necessary to keep the warranted growth rate constant. The model
can regain its Keynesian demand-led flavor if the parameters of the technical progress function
13
become endogenous. Just as an example, if we assumed ϕ1=ϕ1(e),ϕ′
1>0(as in Setterfield,
2013), a demand shock would increase capital accumulation, labor demand, the employment rate
and, finally, the natural growth rate.
When we follow the Marxian assumption, income distribution is still fixed by the equality be-
tween warranted and natural growth rates: sB(1−ωss ) = n+f(ωss)and investment must adjust to
satisfy the saving-investment equality: sB γss
sB−γ2=sB(1−ωss).However, since the equilibrium wage
share is a positive function of the saving rate, productivity growth is endogenous and increasing in s:
gA,ss =f[ωss(s)], g′
A>0.We also have a reversal of the paradox of thrift as capital accumulation
must rise in line with the higher natural growth rate following the increase in s.
In contrast, the Keynesian character of the model can be fully restored when technical change
responds to labor market tightness. Accumulation is exogenous once again and determines income
distribution and productivity growth
sBγ
sB −γ2
=sB(1 −ωss ) = n+q(ess).
Faster accumulation creates larger saving by reducing the wage share and faster productivity growth
by increasing the employment rate through higher labor demand.
If we turn to costly innovation, long-run growth depends on the propensity to invest in R&D
as the natural growth rate is n+ (δB)ϑ.However, the Keynesian flavor of this closure is only
partial since we cannot simultaneously retain the autonomous character of both R&D and physical
capital investment. In fact, once δpins down income distribution, sB(1 −ωss) = n+ (δB)ϑ, the
saving-investment equality requires γto adjust endogenously sB(1 −ωss) = γss +γ2(1 −ωss ).
4.3 Sraffian closure
A relatively recent and rapidly growing literature has used the notion of ‘supermultiplier’ to rec-
oncile exogenous income distribution, normal capacity utilization, and demand-led growth (see
Serrano 1995 for the seminal version of the model). The starting point is that investment is induced
according to equation (5), but there is at least one component of aggregate demand, Z, which is
autonomous, or independent of output, does not generate additional productive capacity and grows
at the exogenous rate gZ.Different types of expenditures, such as autonomous consumption, gov-
ernment expenditure, export, or residential investment have been considered plausible examples of
autonomous demand. The crucial point is that the investment to output ratio in (5) is not a constant
but adjusts to its steady state value hss to ensure that capacity utilization reaches its normal level
u=un= 1. If income distribution is exogenous (ω= ¯ω), the steady state saving-investment
equation yields
BK (s(1 −¯ω)−hss) = Z, (23)
which implies that in the long-run gK=gZ.By definition gK=I/K =hssB, so that the
exogenous rate of autonomous expenditure gZfixes hss.Going back to (23) and dividing both sides
by K, we find
14
sB(1 −¯ω)−gZ=zss,
where we defined z≡Z/K. Notice that higher autonomous demand growth reduces zss as in
equilibrium it requires higher investment thus reducing the autonomous component of demand.
We now turn to the labor market and assume that labor supply is abundant. Once again, technical
change has a passive role, it does not affect growth and distribution but residually determines em-
ployment growth given capital accumulation. When productivity growth is exogenous, we simply
have gL,ss =gz−a. Similarly, when we assume the Kaldor-Verdoorn law gL,ss =gZ(1−φ1)−φ0,
while gL,ss =gZ−f(¯ω)when technical change is induced by income distribution.
If productivity growth is costly, the saving-investment equilibrium is affected by R&D invest-
ment: BK (s(1 −¯ω)−hss −δ) = Z. Hence gZ=sB(1 −¯ω)−δB −zss = (δB)ϑ+gL,ss .
Growth is fully determined by the exogenous rate of growth of autonomous demand, while changes
in R&D expenditure simply affects (in opposite way) the level of autonomous demand and em-
ployment growth. We could also think of R&D investment as the autonomous component of ag-
gregate demand, that is R=Z. In this case, however, the level of R&D expenditure would not
be an exogenous variable anymore as it would adjust to ensure the saving-investment equilibrium:
gZ=sB(1 −¯ω)−zss =zssϑ+gL,ss.
When labor supply is exogenous, the demand-led properties of the model are compromised
under some specifications of technical change. If productivity growth is exogenous, the natural
growth rate is fully exogenous and the growth rate of autonomous demand thus loses its independent
nature: gZ,ss =sB(1 −¯ω)−zss =n+a. The Kaldor-Verdoorn case is qualitatively identical
as we have gZ,ss =sB(1 −¯ω)−zss =n+φ0
(1−φ1).Things are more interesting if we assume the
Marxian motive. If we keep distribution exogenous, gZstill loses its independent character gZ,ss =
sB(1 −¯ω)−zss =n+f( ¯ω). Growth will be supply-side and wage-led. Positive shocks to the wage
share raise the natural growth rate while demand and capital growth follow. We can re-establish
the exogenous nature of gzif we let income distribution be endogenous. Changes in gZwill be
accommodated by the wage share and the level of autonomous expenditure gZ=sB(1 −ωss)−
zss =n+f(ωss); this solution is clearly closer in spirit to the Kaldorian closure than to the Sraffian
one, but produces the unusual result of simultaneous positive correlation between aggregate demand
growth and the labor share. When technical change reacts to the employment rate, gZregains its
independent status even with exogenous distribution: gZ=sB(1 −¯ω)−zss =n+q(ess). Faster
autonomous demand growth is enabled by a higher employment rate that raises the natural growth
rate; at the same time, zss falls to make room for higher investment. Similar analyses are found in
Fazzari et al. (2020); Palley (2019).
When productivity growth is costly and R&D investments are induced, only one between gZand
the propensity to invest in R&D can be independent. If δis an exogenous variable, growth is fully
determined from the supply side and demand growth adjusts: gZ,ss =sB(1−¯ω)−zss =n+(δB)ϑ.
Nomaler et al. (2021) find an analogous result, though with a more complex productivity growth
function that depends on the accumulated stock of R&D capital rather than on the flow of R&D
investment. Otherwise, we need to assume that R&D investment would adapt to exogenous demand
15
growth: gZ=sB(1 −¯ω)−zss =n+ (δssB)ϑ.Finally, we face an analogous dilemma if we take
R&D investment as autonomous expenditure. When we are free to choose R&D investment, demand
growth is not independent anymore as it adjusts to the natural growth rate: gZ,ss =sB(1−¯ω)−z=
n+zϑ.If, on the other hand, we choose to maintain the exogenous nature of demand growth, R&D
investment is constrained in the long-run by demand: gZ=sB(1 −¯ω)−zss =n+zϑ
ss.
5 Public R&D
We now turn to the public sector and consider the possibility that government expenditure affects the
evolution of technology. While most heterodox research has focused on public physical investment
and its complementarity with private capital (see Dutt 2013b; Tavani and Zamparelli 2016, 2017a),
here we will study the impact of public R&D investment on labor productivity growth. Government
collects taxes on profit income by charging the tax rate t. Taxes are fully spent in R&D investment
so that R=tuB(1 −ω)K, and gA= (tuB(1 −ω))ϑ.
In the Classical model with endogenous labor supply, the steady growth condition becomes
(1 −t)sB(1 −¯ω) = gL,ss + (tB(1 −¯ω))ϑ.
The tax rate plays a role analogous to the R&D investment share of saving δ, but it is now a policy
instrument. Therefore, resolving the conflict between increasing labor productivity or employment
growth becomes a government’s prerogative. Given the wage share, raising tlowers capital accu-
mulation and increases labor productivity growth, thus reducing employment growth. As t→1,
growth stops while productivity growth reaches its maximum and employment falls (asymptotically
to zero): gL,ss =−(tB(1 −¯ω))ϑ<0.
When labor supply grows at a constant rate, the wage share becomes endogenous:
(1 −t)sB(1 −ωss ) = n+ (tB(1 −ωss ))ϑ.
The tax rate affects both income distribution and growth, so there will be a wage share- (tω) and
growth rate- (tg) maximizing tax rate. The relation between the two is not obvious in principle.
In a more complex framework, Tavani and Zamparelli (2020) show that tω< tgso that, on the
one hand, policy-makers face a trade-off if they try to simultaneously maximize growth and reduce
inequality, but, at the same time, for all t < tωraising the tax rate increases both growth and the
wage share.
Turning to the Kaleckian model, public R&D investment is R=tuB(1 −¯ω)so that the saving-
investment equilibrium yields (1 −t)suB(1 −¯ω) = γ0+γ1u+γ2(1 −¯ω).Hence,
u(¯ω, s, t) = γ0+γ2(1 −¯ω)
(1 −t)sB(1 −¯ω)−γ1
gK(¯ω, s, t) = 1 + γ1
(1 −t)sB(1 −¯ω)−γ1(γ0+γ2(1 −¯ω)),
16
with du/dt, dg/dt > 0.Both capacity utilization and capital accumulation increase in the tax rate
because taxes are fully spent in R&D, which is a component of aggregate demand. With abundant
labor supply, the balance growth condition now becomes
gK(¯ω, s, t) = gL,ss + (tu(¯ω, s, t)B(1 −¯ω))ϑ.
Productivity growth is also increasing in tboth directly and through its positive effect on taxable
income, while the effect on employment growth depends on the relative strength of the two positive
effects on accumulation and technical change. On the other hand, if labor supply is exogenous the
wage share becomes endogenous in steady state
gK(ωss, s, t) = n+ (tu(ωss , s, t)B(1 −ωss))ϑ.
The policy-maker can manipulate the tax rate to affect income distribution and growth similarly to
the Classical case; but the relation between tωand tgis further complicate by the possibility that the
economy may be wage or profit led.
In the Harrod-Kaldorian case, the equilibrium in the goods market determine the profit share as
a positive function of the tax rate because higher taxes increase demand:
(1 −ωss) = γ
(1 −t)sB −γ2
;
and since accumulation depends on profitability growth is also increasing in t
g=(1 −t)sBγ
(1 −t)sB −γ2
.
When we move to balanced growth with endogenous labor supply we find
(1 −t)sBγ
(1 −t)sB −γ2
=gL,ss +tBγ
(1 −t)sB −γ2ϑ
.
Productivity growth, gA= (tB(1−ωss))ϑ,increases in the tax rate both directly and through its pos-
itive effect on taxable income since traises the profit share. Once again, the effect on employment
is ambiguous. With exogenous labor supply we cannot simultaneously implement a discretionary
fiscal policy while maintaining the exogenous nature of autonomous investment. Either one will
have to go. For example, public investment in R&D determine private capital accumulation
(1 −t)sBγss
(1 −t)sB −γ2
=n+tBγss
(1 −t)sB −γ2ϑ
.
In the Sraffian framework the saving-investment condition in steady state becomes
BK (s(1 −t)(1 −¯ω)−hss) = Z.
17
Given the exogenous growth rate of autonomous expenditure, zss is a negative function of the tax
rate:
s(1 −t)B(1 −¯ω)−gZ=zss.
Higher taxes means higher R&D, which means a lower level of autonomous expenditure if invest-
ment growth is given. If labor supply is endogenous, the tax rate determines productivity growth
and employment growth with no effect on capital accumulation
gZ=gK=gL,ss + (tB(1 −¯ω))ϑ.
When labor supply is exogenous, on the other hand, we cannot simultaneously have exogenous
distribution, exogenous growth of autonomous expenditure and independent choice of the tax rate.
If growth is anchored by gZ, changes in the tax rate necessarily affect income distribution:
gZ=gK=n+ (tB(1 −ωss ))ϑ.
6 Conclusion
This paper has reviewed alternative ways of modeling technical change within heterodox growth
theories. We have formalized technological progress in three ways. First, we have simply taken it as
exogenous and costless. Second, we have assumed that productivity growth occurs in the economy
as an externality of other economic variables, namely capital accumulation, the wage share and the
employment rate. Finally, we have posited that technical change is endogenous and costly because
it requires private or public R&D investment.
We have coupled these formalizations of technical change with growth models classified by
their different investment functions. We have started with a supply-side Classical model where in-
vestments coincide with full-capacity savings. We have then explored three models featuring Key-
nesian, demand-led, investment functions: a Kaleckian model with endogenous capacity utilization
and exogenous income distribution; a Harrod-Kaldorian model with normal capacity utilization and
endogenous income distribution; and a Sraffian model with both normal capacity utilization and ex-
ogenous distribution. All cases are studied under the alternative assumptions of inelastic or abundant
labor supply.
We shall now try to summarize a few general lessons of our analysis. Overall, we have shown
that changes in capital accumulation tend to have distributional effects with inelastic labor supply
while they produce growth effects when the labor force is abundant; but the specific formalizations
of technical change matter, particularly in labor constrained economies. When labor supply is
abundant, and productivity growth is either exogenous or of the externality variety, technical change
plays virtually no role in long-run growth and distribution outcomes. Whether capital accumulation
be supply or demand determined, it is unconstrained by the labor market and fully determines long-
run growth; while income distribution is either exogenous or decided in the goods market. Technical
change merely determines the level of employment growth compatible with capital accumulation.
18
Productivity growth becomes more relevant when it is costly. Since R&D investments are part
of aggregate demand, they will influence capacity utilization in the Kaleckian model and income
distribution in the Harrod-Kaldorian model; changes in utilization and the wage share will, in turn,
have permanent effects on growth.
The picture changes drastically and becomes more diverse when the economy is labor con-
strained. If technical change is exogenous or evolves along the Kaldor-Verdoorn lines, the long-run
natural growth rate becomes a function of purely technological and demographic parameters. Both
in the Classical and in Keynesian models, long-run growth is supply side and independent of saving
or investment propensities. In the Classical, Kaleckian and Harrod-Kaldorian models capital accu-
mulation adjusts to the natural growth rate through changes in income distribution; but while with
the Kaleckian closure investment retains its independent exogenous nature, it becomes constrained
by the natural growth rate in the Harrod-Kaldorian model. In fact, under these two formalizations
of technical change, the Harrod-Kaldorian model is virtually identical to the Classical model in the
long run, unless the parameters of the technical progress function are further assumed to be en-
dogenous. With the Sraffian closure, autonomous demand growth rather than the wage share is the
adjusting variable.
Things are more interesting when technical change follows the Marxian motive. Income distri-
bution is still the adjusting variable in the Classical, Kaleckian and and Harrod-Kaldorian models,
but long-run productivity and capital growth become sensitive to saving and investment preferences.
The Classical and Harrod-Kaldorian closures retain their supply side nature but an increase in the
saving rate raises the natural growth rate through its positive effect on the wage share. In the Kaleck-
ian model, investment affects long-run productivity growth through its influence on the wage share,
with the sign of the effect depending on whether the economy is wage- or profit- led and on the
strength of the induced innovation effect. In the Sraffian model we obtain a wage-led supply side
result if we keep income distribution exogenous, whereas long-run growth regain its demand-led
nature if we let the wage share be the adjusting variable.
When productivity growth reacts to labor market tightness, the drivers of capital accumulation
fully regain their ability to determine long-run growth. In the Classical model with exogenous wage
shares, a positive shock to the saving rate increases capital accumulation, which is accommodated by
higher employment and natural growth rates. The same occurs for demand shocks in the Kaleckian
model, where, in addition, the employment productivity effect allows long-run growth to retain
its wage- or profit- led short-run nature. In similar fashion, exogenous investment or autonomous
demand growth are the ultimate sources of steady state growth when we assume, respectively, the
Harrod-Kaldorian or Sraffian closures.
Next, we have turned to costly innovation with exogenous labor supply. We have obtained R&D
investment as a profit maximizing choice only in the Classical model. The result is a microfoun-
dation of the Marxian productivity function and, as a consequence, the wage share and the natural
growth rate are increasing in the saving rate. In the Keynesian models, we have taken the propensity
to invest in R&D as a given parameter. In the Kaleckian variant, both income distribution and long-
19
run productivity growth are endogenous and depend on R&D investment, but we could not establish
the sign of the two relations. In the Harrod-Kaldorian version of the model, R&D investments raise
long-run growth but reduce the wage share, while investment in capital accumulation loses its ex-
ogenous nature. Similarly, growth of autonomous demand becomes endogenous to adjust to the
R&D investment-determined natural growth rate under the Sraffian closure. The analysis of public
R&D mostly resembles the discussion of private costly technical change, with the difference that
the amount of R&D investment in the economy becomes a policy variable and can be used to target
specific growth and distribution targets.
As parting thoughts, we should mention three subjects we did not address in our analysis. First,
both in Classical (Michl and Tavani , 2022) and in Keynesian (Dutt, 2006, 2010) models, some of the
technical change formalizations we discussed have been used to produce indeterminacy and path-
dependence. The appeal of such possibilities for heterodox economics is apparent, but we have con-
fined our discussion to unique steady state equilibria. Second, by assuming that workers do not save
and accumulate wealth we have avoided to investigate the impact of technical change on class wealth
inequality. This literature appears is still in a nascent stage (see Taylor et al. 2019; Cruz and Tavani
2023) but appears destined to become ever more relevant in the context of increasing automation
of production. Finally, heterodox economists have recently begun to introduce technical change in
growth models concerned with climate mitigation (Taylor et al., 2016; Naqvi and Stockhammer ,
2018; Rezai et al., 2018; de Oliveira and Lima , 2022).
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