Discussion Paper No. 443
Models of Structural Change and Kaldor’s Facts:
Critical Survey from the Cambridge Keynesian
Perspective
Kazuhiro Kurose
January 2021

TOHOKU ECONOMICS RESEARCH GROUP
Discussion Paper

GRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANModels of Structural Change and Kaldor’s Facts:
Critical Survey from the Cambridge Keynesian
Perspective
Kazuhiro Kurosey
January 12, 2021

Abstract
This study addresses the reconciliation of structural change with Kaldor’s facts,
which is a new research agenda in this area. The mainstream reconciliation strategy isthat the facts are interpreted as a state at which the economy grows along the generalisedbalanced growth path and multi-sectoral models are transformed into the one-sectoralmodel that has the uniquely (saddle-path) stable steady state. We argue that the mainstream strategy is far from Kaldor’s own thoughts and overlooks structural change inphysical capital. The Cambridge Keynesian reconciliation based on Pasinetti’s structural dynamics demonstrates that structural change inevitably accompanies changes insocial institutions to maintain full employment, whereas the mainstream reconciliationis achieved entirely through the market mechanism.
JEL Classi…cation: B24, E12, O14, O41
Keywords: Structural change; Kaldor’s facts; Cambridge Keynesians; Pasinetti’s
structural economic dynamics; Adjustment through market
This paper was presented at the 2015 International Conference on Economic Theory and Policy at Meiji
University, the Japan Association for Evolutionary Economics and the seminar at Kyoto University in 2016,
and the 30th EAEPE (the European Association for Evolutionary Political Economy) annual conference inNice in 2018. The author thanks all the participants, especially Roberto Ciccone and Antonio D’Agata, fortheir valuable comments. All remaining errors are solely the author’s responsibility. Finally, …nancial support
from KAKENHI (26380284, 17K03615) is gratefully acknowledged.
yGraduate School of Economics and Management, Tohoku University, Kawauchi 27-1, Aobaku, Sendai
980-8576, Japan, E-mail: kazuhirokurose@tohoku.ac.jp1

1

Introduction

Since the advent of classical economics, the analysis of economic structures, which refers to
the structures of prices, quantities, expenditure, and employment from the multi-industrial
or multi-sectoral perspective, has been one of the central subjects in the principles of political
economy. Smith (1979) argued for the natural process of economic development from a multiindustrial perspective. Ricardo (1951) constructed a growth model including the corn andgold industries. Marx (1967) constructed a schema of reproduction with two sectors. As iswell known, even Walras (1984), one of the founders of neoclassical economics, constructeda general equilibrium model.
After aggregate models of economic growth such as Solow (1956) became popular following
the Second World War, the attention paid to structural analyses in macroeconomics faded.
Although some multi-sectoral/multi-industrial models à la Leontief (1941) and Neumann(1945) were used even after the Second World War, the focus was on the balanced growthpath, as indicated by the turnpike theorem (Khan and Piazza, 2012; McKenzie, 2008). Only
Goodwin (1949, 1974) and Pasinetti (1965, 1981, 1993) continued to focus on structural
analysis (Kerr and Scazzieri, 2013).
As Arena (2017), Rogerson (2019), and Silva and Teixeira (2008) showed, however, mainstream economics has revived the attention paid to structural change since the 1990s. The
growing attention on structural change is also veri…ed by the fact that the term ‘structuralchange’(Matsuyama, 2008) was added into the 2008 version of The New Palgrave Dictionaryof Economics as well as the term ‘structural economic dynamics’proposed by Pasinetti and
Scazzieri (1987). Moreover, a handbook related to structural change was recently published(Monga and Lin, 2019).
Further, a new research subject related to structural change has emerged, namely, examining whether structural change can be reconciled with Kaldor’s (1961) facts, which can besummarised as follows:
1. Per-capita output grows over time and its growth rate does not tend to diminish;
2. Physical capital per worker grows over time;
3. The rate of return on capital is nearly constant;
4. The ratio of physical capital to output is nearly constant;
5. The shares of labour and physical capital in national income are nearly constant; and6. The growth rate of output per worker di¤ers substantially across countries.
According to Barro and Sala-i-Martin (2004), all these facts except fact 3 seem to …treasonably well with the long-run data for advanced countries, and fact 3 can be replaced withthe fact that the rate of return on capital tends to decline over some range as an economy
grows. Although fact 3 can be corrected slightly, the decline in the return is moderate.
Herrendorf et al. (2019) also con…rmed that Kaldor’s facts continue to hold overall in that
constant trends provide a reasonable …rst-order description of most of the data and that2

sizeable short- and medium-term ‡uctuations around the trends exist. These factors imply
that Kaldor’s facts remain eligible as an analytical point of reference with respect to thelong-run economic growth of advanced countries, at least as a …rst approximation, and thus
they indicate the facts that have empirical regularities— even at present.
Furthermore, some studies examine the possibility of reconciling structural change withnot only Kaldor’s but also Kuznets’ (1973) facts. The latter facts indicate the structuralchanges in employment, consumption, and output in such a way that the shares of employment, consumption expenditure, and output shift from agriculture to manufacturing, andeventually to services, as income grows.1 It has also been pointed out that the evolution of
manufacturing is hump-shaped (e.g. Lin and Wang, 2019).
From a theoretical point of view, structural change occurs for demand-side or supply-side
reasons, or a mixture of both.2 The demand-side reason is represented by non-homothetic
preferences. In other words, it is indicated by non-linear Engel curves. The supply-side reason
implies that industrial or sectoral di¤erences in the growth rates of total factor productivity(TFP) or/and in factor intensities are assumed.
Examples of mainstream studies paying attention to the former reason include AlonsoCarrera and Raurich (2015), Bonatti and Felice (2008), Falkinger (1994), Foellmi (2005),
Foellmi and Zweimüller (2008), Hori et al. (2015), Kongsamut et al. (2001), Laitner (2000),
and Matsuyama (2019).
Mainstream research focusing on the latter reason includes Acemoglu and Guerrieri(2008), Alvarez-Cuadrado et al. (2017), Bonatti and Felice (2008), and Ngai and Pissarides(2007). Furthermore, the following mainstream studies emphasise that structural change iscaused by both reasons: Boppart (2014a, 2014b), Comin et al. (2020), Guilló et al. (2011),
Meckl (2002), Muro (2017), and Świecki¾ (2017).
In addition, we should mention the so-called Baumol (1967) disease. He found structural change in the allocation of labour and consumption in a two-sector pure labour modelwith sectoral di¤erences in the growth rates of labour productivities when demand for thegood produced by the sector with lower productivity growth is su¢ ciently price-inelastic andincome-elastic. He also showed that the real growth rate converges to the growth rate of
the lower labour productivity in a two-sector model. The sector with the lower productivity
growth rate then becomes dominant in the long run.
In conducting the new research agenda, mainstream economists interpret Kaldor’s facts
with reference to the concept of balanced growth path. As shown in Sections 3–5, they
investigate whether the model of structural change is consistent with balanced growth ataggregate levels with the sectoral reallocation of labour/capital by extending the conceptof the balanced growth path (the generalised balanced growth path or GBGP) to deal withstructural change. Moreover, mainstream economists tend to think of a multi-sectoral model
as a natural extension of the one-sector growth model, such as the Ramsey (1928) and
Solow (1956) growth models (Herrendorf et al., 2014). Therefore, such economists somehow
1

Jorgenson and Timmer (2011) pointed out that Maddison (1980) also discovered the same facts as Kuznets(1973). In addition, they closely examined structural change in the service sector in advanced economies.
2
Some studies indicate additional reasons such as the e¤ects of international trade (e.g. Matsuyama, 2009;
Uy et al., 2013; van Neuss, 2019), changes in sectoral interlinkages (e.g. van Neuss, 2019), and intersectoral
labour wedges (e.g. Alonso-Carrera and Raurich, 2018; Świecki,¾
2017).

3attempt to convert the multi-sectoral models of structural change into, at most, a twodimensional di¤erential system of equations, as in the optimal growth model of Ramsey.
Such a conversion is achieved by combining various types of utility and production functions
such as the Cobb–Douglas, constant elasticity of substitution (CES), and constant relativerisk aversion (CRRA) functions.
In this study, we critically review mainstream models reconciling structural change with
Kaldor’s facts from the Cambridge Keynesian point of view.3 The distinctive features ofCambridge Keynesians are given in Section 6. We exclusively focus on the mainstream
theoretical strategy to reconcile structural change with Kaldor’s facts and disregard how well
the results obtained by mainstream models …t the data on structural change and economic
growth.
Such purely theoretical attention to structural change and Kaldor’s facts is relevant toCambridge Keynesian economists. First, as already mentioned, structural change is nothing
but the theoretical …eld of research that Pasinetti, one of the most in‡uential Cambridge
Keynesian economists, pioneered in the early 1960s and has extended since (Baranzini andMirante, 2018). Although some studies such as Rogerson (2019) and Stijepic (2011) review
the relationship between structural change and Kaldor’s facts from mainstream perspectives,
few works are written from the Cambridge Keynesian point of view.4
Second, Kaldor was one of the most in‡uential and prominent …gures among Cambridge
Keynesian economists. Then, whether the mainstream understanding of Kaldor’s facts is
consistent with Kaldor’s own thoughts can be examined. Concretely, whether the mainstream
reduction of Kaldor’s facts to the GBGP is an adequate approach is a theoretically important
issue. Whether the consideration of the multi-sectoral model accompanying structural change
as a natural extension of the one-sector growth model is an adequate treatment also has
theoretical relevance.
Third, Cambridge Keynesian economists criticised the principle of marginal productivity
and asserted the importance of heterogeneous and reproducible capital goods (Harcourt,
1972; Pasinetti, 1977). The factor of production termed capital consists of heterogeneousand reproducible commodities, as in Sra¤a (1960), in the real world. To the best of ourknowledge, capital is assumed to be a homogeneous factor of production in all mainstream
multi-sectoral models reconciling structural change with Kaldor’s facts. Therefore, whether
the treatment of capital can be justi…ed to analyse the relationship between structural change
and Kaldor’s facts is a worthwhile subject to address for Cambridge Keynesian economists.
The rest of this paper is organised as follows. Section 2 summarises the extended concept of the balanced growth path. In addition, we de…ne structural change as the term isused in this study. Section 3 reviews representative mainstream models that reconcile the3See, for example, Marcuzzo and Rosselli (2016) and Pasinetti (2007) with respect to the Cambridge
Keynesian point of view.
4
Arena (2017) referred to the above-described new agenda of the reconciliation of structural change with
Kaldor’s facts to clarify the relationships among the business cycle, economic growth, and structural change as
the mainstream …eld of research and compare it with Pasinetti’s contributions and ‘evolutionary’approaches
to structural change. Arena and Porta (2012) also mentioned it. This study aims to clarify the essential characteristics of the mainstream strategy of the reconciliation and evaluate them from the Cambridge Keynesian
perspective.

4

structural change caused by the demand-side reason with Kaldor’s facts. Section 4 reviews
the mainstream models that reconcile the structural change caused by the supply-side reason
with Kaldor’s facts. Section 5 reviews the mainstream models that reconcile the structuralchange caused by both the demand-side and the supply-side reasons with Kaldor’s facts.
Section 6 argues that the mainstream strategy to reconcile structural change with Kaldor’sfacts considers multi-sectoral models to be natural extensions of the one-sector growth model,
which has a uniquely (saddle-path) stable steady state. However, we assert that this is far
from Kaldor’s own thoughts and the Cambridge Keynesian perspective. We show that mainstream models overlook another important structural change: the change in the compositionof physical capital. Furthermore, we argue that the mainstream reconciliation of structuralchange with Kaldor’s facts depends on the perfect adjustment mechanism through markets.
According to the Cambridge Keynesian perspective based on Pasinetti’s (1981) structural
economic dynamics, we assert that structural change inevitably requires social institutions
to change to maintain full employment. The importance of paying attention to changes insocial institutions is increasing. Section 7 concludes.

2

De…nitions of the Extended Concept of the Balanced
Growth Path and Structural Change

As stated in the previous section, mainstream economists consider Kaldor’s facts in reference
to balanced growth path; Kaldor’s facts require the pro…t (or interest) rate and capital–output
ratio to be constant despite growth in aggregate output and labour productivity. These results are obtained under standard neoclassical growth models if Harrod-neutral technicalprogress is assumed (Uzawa, 1961). On the contrary, structural change describes the phenomenon that the structures of prices, quantities, consumption expenditure, and employment
can vary over time. In principle, therefore, it cannot be reconciled with the balanced growth
path in the strict sense. This new research agenda is thus an important and interesting issue
in theories of economic growth.
Mainstream economists extend the concept of the balanced growth path to make it possible for structural change to be reconciled with Kaldor’s facts. This extended concept is the
GBGP, the minimum requirement of which can be speci…ed as follows:
De…nition 1 The GBGP is a path along which one or more variables grow at a constantrate.
The GBGP does not require all the variables of the di¤erential system of equations togrow at the same rate, unlike the balanced growth path; some variables can grow at di¤erent
rates.
Suppose a three-dimensional di¤erential system of equations: x (t) ; y (t) ; and z (t). Then,the GBGP can be exempli…ed by the state at which y grows at a constant rate but not x
and z or y and z grow at an identical constant rate but not x (Stijepic, 2011).
In the context of the presented new research agenda, for example, the GBGP allows
sectoral output shares to grow at di¤erent rates, whereas aggregate consumption and aggregate
capital grow at a constant rate.
5

The mainstream strategy thinks of Kaldor’s facts as the state at which the GBGP exists
and the economy grows along the path. It can be shown that some variables such as therate of interest, the aggregate capital-output ratio, and the share of capital income are kept
constant when the economy grows along the GBGP.
Next, we de…ne structural change in our study, since the term ‘structure’ has broad
meanings in the …elds of economic theory and policy (Monga and Lin, 2013). Too broad ade…nition is not suited to theoretical analyses. Therefore, we de…ne it practically.
De…nition 2 Structural change represents the changes in the sectoral composition of relative
prices, output, consumption (expenditure), and employment.
Since the GBGP does not require all the variables to grow at the same rate, structuralchange (changes in sectoral composition) can be reconciled with the constant growth rate(s)of the aggregate variable(s). De…nition 2 follows the one proposed by Pasinetti (1981),
Pasinetti and Scazzieri (1987), and Scazzieri (2018), although some of the models reviewed
in the forthcoming sections de…ne it more narrowly.

3

Reconciliation of the Structural Change Caused by
the Demand-side Reason with Kaldor’s Facts

In this section, we examine the characteristic of mainstream multi-sectoral models that attempt to reconcile the structural change caused by the demand-side reason with Kaldor’sfacts. As a representative example, we review Kongsamut et al. (2001) and subsequently
examine other examples of models in which the structural change caused by the demand-side
reason is reconciled with Kaldor’s facts.

3.1

Kongsamut et al. (2001)

There are three sectors: agriculture, manufacturing, and services. The output of each sector
in period t is respectively denoted by A (t) 2 A; 1 , M (t) 2 R+ , and S (t) 2 R+ . All thesectors share the standard neoclassical production function, F , which is identical up to theconstant of proportionality. It is assumed that only manufacturing goods can be consumed
and invested and the remaining goods are just consumed. Since structural change is causedby the demand-side reason, the assumptions on technology are standard:
A (t) = BA F A (t) K (t) ; N A (t) X (t) ;M (t) + K_ (t) + K (t) = BM F M (t) K (t) ; N M (t) X (t) ;S (t) = BS F
A
A

(t) +

S

(t) K (t) ; N S (t) X (t) ;

M

(t) +

M

S

(t) = 1;S

N (t) + N (t) + N (t) = 1;
X_ (t) = gX (t) ;6where N i (t) ; i (t) denote the labour employed and share of capital employed in sector iin period t (i = A; M; S), respectively. The total amount of labour is normalised to unity.
X (t) denotes Harrod-neutral technical progress, the rate of which is g > 0. and Bi are thedepreciation rate and parameter denoting the technology level of sector i, respectively.
Since capital and labour are assumed to be freely mobile, the condition for optimal allocation is that the marginal rates of transformation are equal across the three sectors:
M
S
(t)
(t)
(t)
= M
= S :
A
N (t)
N (t)
N (t)
A

Since the proportionality of production functions is assumed, the relative prices of agriculture
and services to manufacturing are given as follows:
pA =

BM
BM
; pS =:BABS

It implies that there is no structural change in relative prices in the equilibrium.
Using the above formulation, the resource constraint for the whole economy is given as
follows:
M (t) + K_ (t) + K (t) + pA A (t) + pS S (t) = BM F (K (t) ; X (t)) :(1)
This transformation crucially depends on the assumption that the production functions are
identical up to the constant of proportionality.
The demand-side factor is characterised by non-homothetic preferences called Stone–
Geary preferences as follows:
Z 1c (t)11 t
U=e dt; where c (t)A (t) A M (t) S (t) + S ;(2)
10
where ; ; ; ; (rate of time preference); A; S are assumed to be strictly positive and ++ = 1. The income elasticity of demand is less than 1 for agricultural goods, equal to 1 formanufacturing goods, and greater than 1 for services. According to Kongsamut et al. (2001),
A and S can be interpreted as the level of subsistence consumption and home production ofservices, respectively.
The problem to solve here is to maximise (2) subject to (1). Thus, the equilibrium real
rate of interest r is given byr (t) = BM f 0 (k (t));(3)
where k (t) K (t) =X (t) ; f (k (t))
sumption across sectors satis…es
pA A (t)A=

F (k (t) ; 1). Moreover, the optimal allocation of conM (t)andpS S (t) + S

=

M (t)

:

(4)

(4) implies that both A (t) A and S (t) + S are proportional to M (t). Using (3) and (4),the optimal path for the consumption of manufacturing goods is given as
M_ (t)
r (t)
=
M (t)
7

:

(5)

Since A; S are positive, there is no balanced growth path in this model; even when the real
rate of interest is constant, (4) and (5) imply that A (t) and S (t) do not grow at a constantrate. However, it is clear from (1) that the balanced growth path requires A (t) ; M (t) ; andS (t) to grow at rate g.
From (3) and (5), the steady-state value of k, if one exists, must satisfy
BM f 0 (k)(6)= g+ :

Now, suppose that ABS = SBA holds. Then, (1) is rewritten as follows:
M (t) + K_ (t) + K (t) + pA A (t)A + pS S (t) + S = BM f (k (t)) X (t) :(7)

If k (t) is kept constant (i.e. if the steady state exists), the right-hand side of (7) grows at
the rate of g and A (t) A and S (t) + S can grow at the same rate, as shown by (4).
Letting m (t) M (t) =X (t), (5) can be rewritten as follows:m_ (t) =

1(r (t)(8)g) m (t) :Similarly, (7) is rewritten as follows:
k_ (t) = BM f (k (t))( + g) k (t)

m (t)

(9)

:

(8) and (9) constitute the two-dimensional di¤erential system of equations, which has thesame properties as the Ramsey growth model. Then, we can prove the unique existence ofthe saddle-path stable equilibrium path converging towards the steady state (k ; m ). In thesteady state, K (t) and M (t) grow at the rate of g. Then, we obtain the following proposition.
Proposition 3 The GBGP exists if ABS = SBA and the transversality condition is satis…ed.
The initial value of k consistent with the GBGP is given by (6).
The properties of economic growth along the GBGP in the model can be summarised
as follows: the rate of real interest is constant, while capital (manufacturing), agriculture,services, and aggregate output grow at the rate of g in the long run. Therefore, the capital–
output ratio remains constant along the GBGP. This implies that Kaldor’s facts can beobtained. Structural changes in production (consumption) and employment occur. In particular, the employment share in agriculture declines and that in services increases, whereasthat in manufacturing remains constant. Although the output of each commodity grows
over time, moreover, its share of each sector follows the same evolutions as the employment
share, since the production functions are identical up to the constant of proportionally. The
impacts of A and S fade over time and the economy converges to the GBGP. Indeed, as A (t)and S (t) become larger, the utility function is getting close to a homothetic utility function(the Cobb–Douglas utility function).
However, the structural changes in the model cannot continue forever and eventually cease
along the GBGP. This is demonstrated by the fact that lim N_ A (t) = lim N_ S (t) = 0 and
t!1

8

t!1

lim A_ (t) =A (t) = lim S_ (t) =S (t) = g (M_ (t) =M (t) = 0 and N_ M (t) = 0 for all t = 0). In
t!1
this sense, we conclude that the model does not show the persistent coexistence of structuralchange and Kaldor’s facts.
ABS = SBA is termed the knife-edge condition. Under this condition, certain parameters or combinations of parameters are constrained to take on speci…c values for a viableequilibrium to exist (e.g. Turnovsky, 2002). According to Kongsamut et al. (2001), theknife-edge condition should be interpreted such that each agent has a positive endowmentof services and a negative endowment of agricultural goods. The endowments in terms of
relative prices are such that pS S = pA A. The knife-edge condition implies a speci…c equality
between technology and the preference parameters, which is obviously restrictive. Indeed,
Herrendorf et al. (2013) argued that the condition is not trivially consistent with …nal consumption expenditure data on the US economy since the relative price of services to goods
has been increasing steadily since the Second World War, whereas A and S are constants.
t!1

3.2

Other examples of the reconciliation of the structural change
caused by the demand-side reason with Kaldor’s facts

Alonso-Carrera and Raurich (2015) also constructed a multi-sectoral model of structuralchange in which the GBGP exists by assuming the following utility function:#
Z 1 " Qm
)
i (1
(c
c
e
)
i
i
i=1
U=
e t dt;10where ci and cei respectively denote the consumption and minimum consumption requirementof good i for i = 1;
; m, among which only good m can be both consumed and invested
and the remaining goods for i = 1;
; m 1 are just consumed.
> 0 is the inverse of
the intertemporal elasticity of substitution when cei = 0 for all i. The utility function is nonhomothetic when cei > 0 for some i. i 2 (0; 1) denotes the weights of the consumption goodsmP
in the utility function such thati = 1. The assumptions on technology are characterised
i=1

by the Cobb–Douglas production functions Yi (t) = [si (t) K (t)] [Ai (t) ui (t) L (t)]1 , where2 (0; 1), si (t) ; ui (t) denote the shares of capital and labour employed in sector i in periodt and Ai (t) represents the TFP of sector i, the growth rate of which is assumed to be
identical across sectors: A_ i (t) =Ai (t) = for i = 1;
; m. There is no supply-side reason
for structural change here.
Unlike in the standard Ramsey model, Alonso-Carrera and Raurich (2015) derived thethree-dimensional di¤erential system of equations
respect to zP(t) K (t) =Am L (t)Pwith
Pm; e (t)mm
E (t) =Y (t) ; q (t) Q (t) =Y (t), where Y (t)ci .
i=1 pi Yi ; E (t)i=1 pi ci ; Q (t)i=1 pi e
Q (t) and q (t) denote the aggregate value of the minimum consumption requirements and its
intensity in period t, respectively. Using the conditions for pro…t maximisation, Y (t) can betransformed into Y (t) = Am (t) L (t) z (t) .
Assume that the transversality condition is satis…ed. The three-dimensional di¤erential
system of equations of the model can be summarised as follows:

9

z_
q_
e_
= f1 (e; z) ; = f2 (e; z) ; and = f3 (e; z; q) :zqeHere, the initial values of both z and q are given and they are chosen independently of Am (0).
Unlike the standard Ramsey growth model, the steady state, if it exists, is de…ned using one
control variable e and two state variables z; q.
The above three-dimensional di¤erential system of equations can be considered to be
the generalisation of Kongsamut et al. (2001). This is because the knife-edge condition of
Kongsamut et al. (2001) is equivalent to assuming Q = 0 (or q = 0). By assuming q = 0, we
can reduce the dimensionality of the steady state from the model. In other words, the knifeedge condition selects a particular equilibrium path of the two-dimensional manifold. Then,
we can prove the unique existence of the GBGP, which is saddle-path stable, whenever q = 0.
This means that along the GBGP, aggregate capital, output, and consumption expendituregrow at the rate of , and the rate of interest is also kept constant since z is kept constant.
Then, Kaldor’s facts are obtained. It is also shown that Kuznets’facts are obtained in themodel with three sectors.
Since Alonso-Carrera and Raurich (2015) can be considered to be the generalisation of
Kongsamut et al. (2001), they have similar implications. First, structural change eventuallyceases along the GBGP. Moreover, the implausibility of the knife-edge condition, as pointed
out by Herrendorf et al. (2013) with respect to Kongsamut et al. (2001), also applies to
Alonso-Carrera and Raurich (2015).
Furthermore, Li et al. (2019) extended Kongsamut et al. (2001) by introducing Romer’s(1990) endogenous technological change into the three-sector model. The structure of themodel is equivalent to Kongsamut et al. (2001), but an intermediate sector and a researchsector are introduced in addition to the …nal goods sector. The …nal goods (agriculture,
manufacturing, and services) are produced using labour, human capital, and all the typesof intermediate goods designed by the knowledge. The types of intermediate goods are
determined by the knowledge stock created by the research sector. The number of typesof intermediate goods in period t is expressed by the knowledge stock t . The researchsector creates new knowledge using human capital and the existing knowledge t , and thelinear production function of the knowledge is assumed, as is usual in endogenous growthmodels. The utility function assumed by Li et al. (2019) is slightly di¤erent from that ofKongsamut et al. (2001); c (t) = A (t) A
M (t) + M
S (t) + S , where A,M ; S > 0(the …rst denotes the subsistence consumption of agriculture and the second and third thehome production of manufacturing and services, respectively). A term denoting humancapital is included in the consumer’s budget constraint in Li et al. (2019).
Since it is assumed that manufacturing can be both consumed and invested, we cantransform the multi-sectoral model into a type of one-sector optimal growth model using
a similar procedure to that of Kongsamut et al. (2001). Then, we can prove the uniqueexistence of the GBGP whenever the knife-edge condition BAA = BMM + BSS , where Bi denotesthat the technology parameter of sector i = A; M; S, is satis…ed. The growth rate of aggregate
variables along the GBGP is endogenously determined by the total stock of human capital,
rate of time preference, and technology parameters. Moreover, Li et al. (2019) investigated10the e¤ect of human capital on structural change but did not consider the plausibility of theknife-edge condition.
Foellmi (2005) and Foellmi and Zweimüller (2008) presented multi-sectoral models thatreconcile the structural change caused by the demand-side reason with Kaldor’s facts. Therefore, the technology assumptions are neoclassical and identical across all the sectors in themodels; capital and consumption goods are produced by the same neoclassical production
function as consumption goods: F [K (i; t) ; A (t) L (i; t)], where K (i; t) and A (t) L (i; t) denote the amount of capital and e¢ ciency unit of labour employed in sector i in period t,
respectively. A (t) is the stock of labour-augmenting technical knowledge, which increases
at an exogenous rate of g > 0. Since all the sectors have the same production functions,each sector produces at the same capital per-e¢ ciency unit of labour in the equilibrium
K(i;t)= k for all i). Thus, the marginal costs and hence the prices can be(k (i; t)
A(t)L(i;t)
normalised to unity without loss of generality. This means that structural change in prices
does not occur in the models. However, the model allows for the emergence of new goods.
The characteristic of the model is to introduce a ‘hierarchy’of wants using the following
utility function:
Z 1( )1e ( t) d ;u (t) =
1tZ
1 1iwhere ( ) =s2 (s c (i; ))2 di:
2 0
i and i are the index of consumption goods and hierarchy function, respectively, and s > 0,
2 (0; 1). Goods with lower i have higher weights than goods with higher i. It is shown that
consumption demand for a particular good, derived by the above utility function, depends
on the relative position in the hierarchy of wants; goods at a lower position in the hierarchy,
which are given relatively higher priority, are consumed in higher quantity. The income(t)), where N (t) denotes theelasticity of consumption demand for good i is given by 1 (i=N
(i=N (t))
number of consumption goods in period t. It su¢ ces for our purpose to assume that N (t)exogenously increases, although Foellmi and Zweimüller (2008) considered R&D activity as
well. Since the number of goods increases over time, the relative position of good i in thehierarchy of wants, shown by N i(t) , declines over time. This means that the income elasticity ofdemand for good i monotonically declines as new goods emerge over time and consumption
demand for the good …nally reaches saturation. As a result, non-linear Engel curves are
obtained.
RN
Aggregate consumption expenditure can be de…ned as E (t) = 0 pj cj dj, where pj ; cj
denote the price and consumption of good j. Remember that the prices are normalised. Let
aggregate expenditure in the e¢ ciency unit be e (t) = E(t). Then, the model is summarisedA(t)by the two-dimensional di¤erential system of equations with respect to e (t) and k (t). The
method used to solve this model is basically the same as the Ramsey optimal growth model.
Then, the unique existence of the saddle-path steady state (i.e. the GBGP) can be proven.
Along the GBGP, aggregate consumption expenditure and capital in the e¢ ciency units areconstant at the steady state. This implies that aggregate output, consumption, and capitalgrow at a constant and identical rate and that the rate of interest is kept constant. Thus,
11

Kaldor’s facts are obtained. Reducing the generally multi-sectoral model into a model withthree sectors (agriculture, manufacturing, and services), furthermore, it is shown that theevolution of the employment share of manufacturing is hump-shaped and thus Kuznets’facts
are also obtained.
Particular attention should be paid to the fact that the model shows the persistent coexistence of structural change and Kaldor’s facts. This is because the composition of consumption
demand changes along the GBGP. In other words, structural change in sectoral consumption
occurs along the GBGP due to the emergence of new goods, although aggregate consumption grows at a constant rate. The success of proving the persistent coexistence of structuralchange and Kaldor’s facts crucially depends on the form of hierarchy function i . Thanks
to this, the income elasticity of demand for various goods changes over time, as already mentioned, whereas the indirect utility function is equivalent to CRRA in the one-sector modelfrom the viewpoint of a single individual (Foellmi, 2005, p. 21).

4

Reconciliation of the Structural Change Caused by
the Supply-side Reason with Kaldor’s Facts

In this section, we …rst take Ngai and Pissarides (2007) as a representative example of multisectoral models that reconcile the structural change caused by the supply-side reason with
Kaldor’s facts. Subsequently, we review other multi-sectoral models that address the reconciliation of the structural change caused by the supply-side reason with Kaldor’s facts.

4.1

Ngai and Pissarides (2007)

There are m sectors, among which m 1 sectors (i = 1;
; m 1) produce pure consumption
goods and the last one (i = m) produces a good that can be both consumed and invested.
Moreover, it is assumed that the labour force grows at the exogenous rate of n > 0.
The household’s preferences are represented by the following utility function:
Z 1
U=
e t [c1 (t) ;; cm (t)] dt; where
(10)
0

1

[c1 (t) ;; cm (t)]

()
1

1

; ()

m
X
i=1

! i ci (t)(" 1)="!"=(" 1);and ci (t) = 0 denote the per-capita consumption of good i in period t. Moreover, ; "; ! i > 0,mPand
! i = 1 are satis…ed. If = 1, then [c1 (t) ;; cm (t)] = ln ( ), and if " = 1,i=1then ln ( ) =mP! i ln ci (t). These are standard assumptions on preferences; the demandi=1functions have constant price elasticity " and unit income elasticity.
On the contrary, the production function of each sector is distinctively formulated as

12

follows:
ci (t) = Ai (t) F (ni (t) ki (t) ; ni (t)) ; for i = 1;
k_ (t) = Am (t) F (nm (t) km (t) ; nm (t))

cm (t);m1;( + n) k (t) ;where ni (t) ; ki (t) ; k (t) = 0 denote the employment share, capital–labour ratio in sector i,and aggregate capital–labour ratio in period t, respectively. F is the standard neoclassicalproduction function, which is common to all sectors, and Ai (t) (i = 1;
; m) denotes Hicks_
neutral technical progress such that Ai (t) =Ai (t) = i is assumed: Ai (t) is TFP. i 6= j if
i 6= j is the supply-side reason for structural change. The free mobility of both factors is
assumed. Moreover, the following constraints are satis…ed:mXni (t) = 1;

i=1

m
X

(11)

ni (t) ki (t) = k (t) :i=1

The optimal allocation condition requires that the marginal rates of substitution are equal
to the marginal rates of transformation, which implies the following:
i (t)
m (t)

=

Am (t) Fm1
Am (t) Fm2=; for i = 1;
Ai (t) Fi1
Ai (t) Fi2

;m

(12)

1;

where i (t)@ =@ci , and Fij denotes the partial derivatives of the F of sector i withrespect to the jth variable (j = 1; 2). From the properties of the production functions that
we assume, conditions (11) and (12) imply
ki (t) = k (t) for 8i; andpi (t)=pm (t)

i (t)
m (t)

=

Am (t)
for i = 1;
Ai (t)

;m

1:

(13)

The optimal condition for the representative consumer yields_ m (t)= Am (t) Fk
m (t)

(14)

( + n + );where Fk @F.
@kGiven utility function (10), (13) yields
pi (t) ci (t)=pm (t) cm (t)

!i
!m

"

Am (t)Ai (t)1 "xi (t) ; for i = 1;

;m

1:

(15)

xi (t) is a variable denoting the ratio of consumption expenditure on good i to that on
manufacturing good in period t.
Let us de…ne aggregate consumption expenditure and output per-capita in terms of manmmPPpi (t)pi (t)
c
(t)
;
y
(t)
A (t) F (ni (t) ki (t) ; ni (t)), which
ufacturing as follows: c (t)ipm (t)pm (t) i
i=1

i=1

can be rewritten using (15) and the production functions:c (t) = cm (t) X (t) ; y (t) = Am (t) F (k (t) ; 1) ;13

where X (t)mP

xi (t).

i=1(13) and (15) imply that structural changes in relative prices and consumption expenditure occur since i 6= j if i 6= j. Furthermore, we obtainn_ i (t)d (c=y) =dt=+ (1 ") ( (t)ni (t)
c=yn_ m (t)d (c=y) =dt=+ (1 ") ( (t)nm (t)c=y1 c=yd (c=y) =dt+
1 c=y
nm (t)
where

(t)

m
P

i=1

xi (t)X(t)i) ;for i = 1;;m1;(c=y) (xm =X)
nm (t)

m);i , which is a weighted average of the sectoral TFP growth rates,with the weight given by each good’s consumption share. These show that structural change
in the employment share occurs if " 6= 1 and c=y is kept constant since i 6= j if i 6= j isassumed. Note that d dt(t) 7 0 if and only if " 7 1 (see Lemma A3 in Ngai and Pissarides,2007). Furthermore, if c=y is kept constant and " < 1 (i.e. consumption demand is priceinelastic), employment shifts from the sector with the highest TFP growth rate to that withthe lowest TFP growth rate, and the converse is true if " > 1.
Here, let us specify the production functions in a Cobb–Douglas form: F (ni (t) ki (t) ; ni (t))(ni (t) ki (t)) ni (t)1 = ki (t) ni (t) for 2 (0; 1). Note that is a common parameter toall sectors; this implies that the factor intensities are equal in all the sectors. Then, we canobtaink_ (t) = Am (t) k (t) nm (t) cm (t) ( + n) k (t)= Am (t) k (t)c (t) ( + n) k (t) :(16)

From (14), we havec_ (t)=(c (t)1) ( m(t)) + Am (t) k (t)1( + n + ):(17)(16) and (17) constitute the two-dimensional di¤erential system of equations determining the
motions of the variables in the model. Then, we obtain the following proposition:
Proposition 4 Given any initial k (0) > 0, the necessary and su¢ cient condition for the
existence of the GBGP is given by= 1; " 6= 1; and 9i 2 fi = 1;;m

1 j i 6=mg :In Proposition 4, the multi-sectoral model is transformed into the one-sector model, andthen the existence of the GBGP (i.e. the saddle-path stable steady state ek ;ec , where
14

1

1

e
k = kAm (t) 1 and ec = c (t) Am (t) 1 ) is proven. Along the GBGP, therefore, bothk (t) and c (t) grow at the rate of 1 m , and thus aggregate output y (t) also grows at thesame rate. As in standard models where balanced growth path exists, this means that
aggregate consumption must be a constant proportion of aggregate output along the GBGP.
Given (10), this can hold either when consumption is independent of the rate of interest
or when the rate of interest is constant. Since the rate of interest is determined by themarginal productivity of capital in this model, the constant rate of interest is inconsistent
with structural change. Therefore, consumption must be independent of the rate of interest,
which implies the logarithmic utility function. Then, = 1 is required to prove the existenceof the GBGP in the model.
Moreover, the employment shares of sector m and the sector whose TFP growth rate is
the lowest (highest) in the case of " < 1 (" > 1) converge to positive values and those of
the remaining sectors converge to zero in the long run along the GBGP.5 It is intuitive thatthe employment share of sector m converges to a positive value since capital is produced
at a constant rate along the GBGP. The positive convergence in the employment share of
the sector with the lowest TFP growth rate in the case of " < 1, which is more empirically
relevant than " > 1, depends on the same mechanism as that causing the Baumol (1967)
disease. In other words, capital except that allocated to sector m tends to be absorbed by
the sector with the lowest TFP growth rate along the GBGP when " < 1.
From the production functions, however, the output growth rate of sector i = 1;;m 1
d(c=y)=dt
n_ i
k_ ialong the GBGP is given by " i + ki + ni = " i + gm + (1 ") (t) (note c=y = 0 along
the GBGP), where gm denotes the growth rate of capital (sector m). Therefore, structuralchanges in relative prices, output, and consumption persistently occur along the GBGP,
even though sectors vanish in the employment share in the long run. Therefore, Ngai and
Pissarides (2007) found the persistent coexistence of structural change and Kaldor’s facts.

4.2

Other examples of the reconciliation of the structural change
caused by the supply-side reason with Kaldor’s facts

Acemoglu and Guerrieri (2008) presented a model that reconciles the structural change causedby the supply-side reason with Kaldor’s facts. Their model pays particular attention to theresource reallocation during the growth process and cannot analyse the structural change in
consumption expenditure caused by income growth, since it has only one consumption good.
Suppose an economy with three sectors, one of which produces a consumption good andtwo of which produce di¤erent intermediate goods. The consumption good is produced using
the intermediate goods following the CES production function, without employing the direct
labour, and can be both consumed and invested. The budget constraint is thus given by
K_ (t) + K (t) + c (t) L (t) 5 Y (t), where c (t) ; L (t) ; Y (t) denote per-capita consumption,
population, and the output of the …nal good. L_ (t) =L (t) = n is assumed. The intermediate
goods are produced using capital and labour and the production functions take the Cobb–
Douglas form with sectoral di¤erences in factor intensities and TFP growth rates.
5

Ngai and Pissarides (2007) indicated that the employment shares of the remaining sectors are either
hump-shaped or monotonically declining. Therefore, the model can account for Kuznets’facts as well.

15

The technology assumptions are summarised as follows:
Y (t) =h(" 1)="Y1 (t)+ (1(" 1)=") Y2 (t)i"=(" 1);(18)

where 2 (0; 1), and Y; Y1 ; Y2 respectively denote the amount of consumption good andthose of the two intermediate goods produced by such Cobb–Douglas functions as Yi (t) =Mi (t) Li (t) i Ki (t)1 i for i = 1; 2 with M_ i (t) =Mi (t) = i and 1 > 2 (sector 1 is morelabour-intensive). The factor intensities di¤er in contrast to Ngai and Pissarides (2007). The
resource constraints are given as follows: K1 (t) + K2 (t) 5 K (t) and L1 (t) + L2 (t) 5 L (t).
Concerning the preferences, the CRRA utility function is assumed. This is maximised subject
to the budget constraint.
Here, let us concentrate on the case of " < 1, which has more empirical relevance. Since
it is assumed that capital and labour are homogeneous and the production functions are
speci…ed in the Cobb–Douglas form, the multi-sectoral model can be transformed into a
type of one-sector Ramsey optimal growth model, but it consists of the three-dimensionalc(t)di¤erential system of equations that has one control variable (ec (t) M (t)1= 1 ) and two state
1
K(t)
K
(t)
1
variables (ek (t)and (t)). This is because an equation determining the1=
L(t)M1 (t)

1

K(t)

sectoral allocation of capital must be added. Assuming the knife-edge condition, however,
the three-dimensional di¤erential system of equations can be reduced to the two-dimensional
system, as in Alonso-Carrera and Raurich (2015). Then, the unique existence of the (locally)saddle-path stable steady state ek ;ec , given ek (0) and (0) > 0, can be proven if thetransversality condition is satis…ed.
Along the GBGP, therefore, per-capita consumption c (t) grows at the rate of 1 = 1 (theaugmented rate of technical progress), output Y (t) and capital K (t) grow at the rate ofn + 1 = 1 , and the rate of interest is kept constant. Thanks to the knife-edge condition,
sector 1, which is more labour-intensive, is the asymptotically dominant sector along the
GBGP, meaning that it determines the long-run growth rate of the economy. In other words,
the shares of both capital and labour allocated to sector 1 converge to unity in the long
1 (t)
1 (t)
= limt!1 LL(t)= 1), since the augmented rate ofrun along the GBGP (i.e. limt!1 KK(t)technical progress of sector 2 is assumed to be higher than that of sector 1 (i.e. the relative
output of sector 1 is zero in the long run along the GBGP: limt!1 YY12 (t)= 0). The asymptotic(t)
6
dominance of sector 1 is nothing but the result of Baumol (1967). Moreover, the growth
rates of sectors 1 and 2 di¤er along the GBGP and this di¤erence persists even in the longrun. We can conclude that structural change can be reconciled with Kaldor’s facts in themodel.
Despite the emergence of the dominant sector, structural change in sectoral output persistently occurs along the GBGP; sector 2 always grows faster than sector 1. When there iscapital deepening and both capital and labour are allocated to the two sectors in a constant
proportion, sector 2 can grow faster than sector 1 because it is the more capital-intensive. As
a result, the price of good 2 declines, which leads to a reallocation; however, this reallocation
6

According to Acemoglu and Guerrieri (2008), the sectors growing faster tend to have higher capitalintensity— at least in the United States. Therefore, the assumed knife-edge condition seems plausible.

16

never o¤sets the greater increase in sector 2 because of 1 > 2 . Then, this model accounts
for the persistent coexistence of structural change and Kaldor’s facts.
Alvarez-Cuadrado et al. (2017) extended Acemoglu and Guerrieri (2008) by introducing a
new supply-side reason (sectoral di¤erences in the elasticities of substitution between capitaland labour). If the aggregate capital–labour ratio and wage–interest ratio increase, thesector with the higher elasticity of substitution is in a better position to substitute capitalfor labour. The sectoral di¤erences cause structural change. According to Alvarez-Cuadrado
et al. (2017), the signi…cance of the introduction of sectoral di¤erences in the elasticities
of substitution is, …rst, that they have been con…rmed by many empirical studies. Second,
empirical evidence con…rms the sectoral di¤erences in the growth rates of the capital–labour
ratios and in the evolution of the factor income shares.
Then, the following production functions of the two intermediate goods are assumed:hYi (t) = (1

( i 1)= i+i ) (Ai (t) Li (t))( i 1)= i
i Ki (t)i i =( i 1);

where i 2 (0; 1), i 2 [0; 1) ; and A_ i (t) =Ai (t) = i > 0 for i = 1; 2. Without loss ofgenerality, 2 > 1 can be assumed. Sector 2 is the more ‡exible sector in that it is easier
to substitute in sector 2 than in sector 1. The production function of the consumption
good is assumed to be the same as (18). The model can be reduced to Ngai and Pissarides(2007), where structural change is caused by the sectoral di¤erences in the TFP growth rates
if " 6= 1; 1 = 2 ; 1 = 2 ; 1 6= 2 , while it can be reduced to Acemoglu and Guerrieri(2008), where structural change is caused by the sectoral di¤erences in the factor intensities
if " 6= 1; 1 = 2 ; 1 = 2 ; 1 6= 2 .
For tractability, Alvarez-Cuadrado et al. (2017) paid attention only to the case of " =1; 1 = 2 = ; 2 > 1 = 1; 1 = 2 = with respect to the dynamics of the model. In otherwords, the consumption good and intermediate good 1 are produced by the Cobb–Douglas
production function and intermediate good 2 is produced by the CES production function;
moreover, no sectoral di¤erences in the factor intensities and TFP growth rates exist. Hence,
the Baumol disease does not emerge. In this case, any result derived from the model is
attributable to the sectoral di¤erences in substitution.
To avoid unnecessary complication and concentrate on the analysis of the supply-side
reasons, the model does not formulate the consumer’s optimisation problem (the introductionof the problem into the model is straightforward). Since the constant saving rate is assumed
to be exogenously given, as in the Solow model, the motion of capital accumulation follows
K_ (t) = Y (t)K (t).
Combining the motion of capital accumulation with the static optimisation conditions, we
K1 (t)
can thus obtain the non-linear di¤erential equation with respect to: _ (t) = h ( (t))K(t)
dhwith the property of d < 0. Therefore, the unique existence and local stability of the steadystate can be proven, just as in the Solow model. This means the unique existence of thelocally stable GBGP, along which Y (t) ; Y1 (t) ; Y2 (t) ; K (t) ; K1 (t) ; K2 (t) grow at the rateof n + and L1 (t) ; L2 (t) grow at the rate of n, while the rate of interest is kept constant.
Therefore, Kaldor’s facts are obtained.
Structural change ends once the economy reaches the GBGP. It occurs only during thetransition towards the GBGP. When the economy starts from K lower than its steady-state17

value, K grows faster than AL, and thus the sector with the higher elasticity of substitution
absorbs more capital and releases labour. This is because that sector tends to substitute thecheaper factor (capital) for the more expensive one (labour) as capital is accumulated andthe wage–interest ratio rises. Therefore, the sectoral capital–labour ratios and factor income
shares evolve di¤erently in the two sectors. Hence, structural change occurs. However, thecoexistence of structural change and Kaldor’s facts is not shown in the model. If structuralchange occurs, then Kaldor’s facts cannot be obtained; by contrast, if structural changeceases, then Kaldor’s facts are obtained.

5

Reconciliation of the Structural Change Caused by
Both Reasons with Kaldor’s Facts

In this section, we review mainstream multi-sectoral models which reconcile structural change
caused by both the demand-side and the supply-side reasons with Kaldor’s facts. Boppart(2014a), Herrendorf et al. (2020), Comin et al. (2020), Guilló et al. (2011), and Meckl (2002)
are recent examples of such models.

5.1

Boppart (2014a)

This model has two consumption goods: good (G) and service (S). It is assumed that thehousehold is indexed by i 2 [0; 1], each of which consists of N (t) identical members, where
N (t) = exp [nt], n > 0. Each member of household i is endowed with li 2 l; 1 ; l > 0 unitsof labour and labour is supplied inelasticallyin every period. Therefore, the aggregate labour
R1
supply is de…ned by L (t) N (t) 0 li di, the growth rate of which is given by n. Householdi, which is indexed by i 2 [0; 1], has the following intertemporal preferences:
Z 1
Ui =exp [ (n) t] (pG (t) ; pS (t) ; ei (t)) dt, where0(pG (t) ; pS (t) ; ei (t)) =1"ei (t)pS (t)"pG (t)pS (t)

1
+ ;
"

(19)

where is the rate of time preference, and > n > 0 is assumed. (19) is the indirect instantaneous utility function, where 0 5 " 5 < 1 and > 0 are assumed. " is the parametera¤ecting the degree of non-homotheticity of the preferences; if " = 0, the preferences are
homothetic. is the parameter a¤ecting the degree of substitutability between G and S. If" = = 0, the preferences are reduced to the Cobb–Douglas form. As shown later, is theparameter a¤ecting the expenditure share on G; if = 0, the model is reduced to a one-sector
model and the preferences are reduced to the CRRA form. pG (t) ; pS (t) ; ei (t) are the priceof goods, services, and nominal per-capita expenditure of household i, respectively.
(19) is a preference termed price-independent generalised linearity. This makes the aggregation of households’expenditure trivial since the aggregate expenditure share coincides
with that of a representative household whose expenditure is the same share as that of theaggregate economy. Moreover, it ensures that the representative expenditure is independentof prices (Muellbauer, 1976).
18

The technology assumptions are similar to those of Ngai and Pissarides (2007):7
Yj (t) = Aj (t) F [Kj (t) ; exp (gt) Lj (t)] ; for j = G; S;
YI (t) = F [KI (t) ; exp (gt) LI (t)] ;where index I denotes an investment good and F is the identical neoclassical production
A_ (t)
function. The TFP growth rates di¤er: Ajj (t)j = 0 for j = G; S. The price of theinvestment good is adopted as the numéraire in each period. As in (13), pp_GG pp_SS = S
Gholds, which means that structural change in prices occurs. In addition, the sectoral factor
Kj (t)) are uniform in the equilibrium:
intensities in terms of the e¢ ciency unit (kj (t) exp(gt)L
j (t)
K(t)
S (t)+KI (t)kG (t) = kS (t) = kI (t) = k (t), where k (t)= KG (t)+K
and L (t) =exp(gt)L(t)exp(gt)L(t)LG (t) + LS (t) + LI (t).
In each period, household i maximises (19) subject to the budget constraint: ei (t) =pG (t) xiG (t) + pS (t) xiS (t), where xiG (t) ; xiS (t) denote the per-capita consumption of goods
and services, respectively.8Z Then, their aggregate consumption can be respectively de…ned1

N (t)xij (t)di for j = G; S. Aggregate consumption expenditure can
Z 1 0
ei (t)di = pG XG (t) + pS (t) XS (t). To analyse structural change,
be de…ned as E (t)as follows: Xj (t)0pG XG (t).
E(t)

de…ne the aggregate expenditure share of good as 'G (t)E(t)
Letting e (t) exp(gt)L(t), capital accumulation is determined by the di¤erential equation,
as in Ngai and Pissarides (2007):

k_ (t) = f [k (t)]( + n + g) k (t)(20)

e (t) ;

where f [k (t)] = F (k (t) ; 1). In addition, we can obtain the di¤erential equation with respect
to e (t): corresponding to the Euler equation:r (t)e_ (t)
=
e (t)
1

+" S
"

(21)

g;

where r (t) = f 0 [k (t)]is satis…ed by the …rms’optimisation.
(20) and (21) constitute the two-dimensional di¤erential system of equations. The method
used to solve this system is the same as for the Ramsey optimal growth model. Therefore,
we can prove the following proposition:
7

Boppart (2014a) used the AK production function in the investment good sector to avoid the analysis oftransition dynamics. However, we use the standard neoclassical production function in the sector, as shown
by Boppart (2014b), to understand the mainstream strategy to reconcile structural change with Kaldor’sfacts.
8
As a result, the following consumption demand functions are obtained:xiG (t) =ei (t)pG (t)pS (t)ei (t)"pG (t)pS (t)and xiS (t) =ei (t)1
pS (t)pS (t)ei (t)"pG (t)pS (t):The Engel curves are non-linear if " < 1. Moreover, the elasticity of substitution between goods andservices is less than unity for all the households in each period thanks to the assumption of 0 5 " 5 < 1.

19

Proposition 5 The two-dimensional di¤erential systems of equations (20) and (21) havethe uniquely saddle-path stable steady state (k ; e ), given k (0) > 0, if the transversality
conditions,n > "eg , and f 0 (k ) >n ge are satis…ed.

Proposition 5 indicates the existence of the GBGP, along which the aggregate capital,
consumption expenditure, and output grow at the constant rate of g + n. Moreover, theK (t)capital–labour ratio ( Ljj(t) ) grows at the rate of g in all the sectors, whereas the rates ofinterest and saving are kept constant along the GBGP. Then, Kaldor’s facts are obtained.
Since the steady state value of k is constant along the GBGP, each sector grows at the'_ (t)
constantly speci…c rate. It is also proven that 'G (t) 5 0 holds along the GBGP, implyingG
that the expenditure share of the good is decreasing. This implies that a structural change
in expenditure occurs along the GBGP because the relative price of the good changes along'_ (t)
the GBGP and " 5 < 1 is assumed. 'G (t) 5 0 immediately implies that lim 'G (t) = 0
G
t!1
holds along the GBGP.
By the same logic as in Ngai and Pissarides (2007), however, structural changes in relative
prices, output, and consumption persistently occur along the GBGP; the structural changes
in output and consumption are veri…ed by the consumption demand functions in footnote 8.
Therefore, the persistent coexistence of structural change and Kaldor’s facts is ensured.
The novelty of the model is that preference (19) enables us to address the heterogeneityof households. Although the evolution of the macroeconomic variables can be indicated by
the representative household, each household’s behaviour is also understood. It is shown that
poorer households, represented by those with lower ei (t), tend to spend a larger proportionof their income on the good than richer ones.

5.2

Other examples of the reconciliation of the structural change
caused by both reasons with Kaldor’s facts

In addition to that of Boppart (2014a, 2014b), other multi-sectoral models reconcile structuralchange with Kaldor’s facts.99Here, let us refer to Echevarria (1995, 1997, 2000). Although her model did not focus on Kaldor’sfacts, it presented an innovative model to address structural change and economic growth, which promoted
the emergence of the new research agenda described above. The economy she supposed has three goods;
agriculture (indexed by sector 1), manufacturing (sector 2), and services (sector 3). Manufacturing can beboth consumed and invested and the remaining goods are pure consumption goods. The preferences are
1
3
3
P
P
P
t
j
assumed as follows: U =; where2 (0; 1) ; j >j ln Cj (t)j = 1; j > 0;
j Cj (t)
t=0

j=1

j=1

0; j = 0. If at least one of j is strictly positive,term j Cj (t) j indicates the demand-side reason forstructural change. The evolution of consumption of each good is similar to that obtained by the Stone–Geary
type of preferences used in Kongsamut et al. (2001). The advantage of the preferences by Echevarria is thatit can avoid some unpleasant features of the Stone–Geary preferences: it cannot be de…ned for A (t) < A in(2). Hence, the interior solution to the static optimisation problem always exists in Echevarria’s speci…cation.
Moreover, she assumed Cobb–Douglas technologies with sectoral di¤erences in both TFP growth rates andfactor intensities. These are the supply-side reasons for structural change. She proved the existence of the
GBGP if j = 0 for j = 1; 2; 3. In other words, capital in the three sectors, aggregate capital, investment,
and consumption of manufacturing grow at the same constant rate. Moreover, consumptions of agriculture

20Herrendorf et al. (2020) also constructed a multi-sectoral model of the structural change
caused by both reasons. Their model has the same structure as that of Boppart (2014a), except the assumption of the production of an investment good. In other words, the preferences
are assumed to take the price-independent generalised linearity form. The characteristic of
the model lies on the supply side. In the model, a representative household consumes agood and a service, which are produced by inputting capital and labour using Cobb–Douglas
production functions with identical factor intensities but di¤erent TFP growth rates. The
production functions are de…ned in terms of the value-added. Although it is often assumed
that only the good can be both consumed and invested, the model assumes that the investment good is produced by combining the good and service, the aggregator of which is givenby the CES form with exogenously investment-speci…c technical change. Thanks to this assumption, aggregate output (i.e. the sum of investment and consumption) in the equilibrium
can be expressed in the Cobb–Douglas form, with the investment-speci…c TFP growth rate
endogenously determined by the composition of investment input, which in turn depends on
the TFP growth rates of the production of the good and service.
Making a set of assumptions on the parameters, the unique existence of the GBGP can beproven, along which the aggregate capital, output, consumption expenditure, and investment
grow at the same constant rate determined by the investment-speci…c TFP growth rate andfactor intensity, Moreover, the rate of interest and capital–output ratio are kept constant.
Therefore, Kaldor’s facts are obtained. Furthermore, structural change in relative prices andconsumption expenditure persistently occurs along the GBGP, as in Boppart (2014a). Then,
service is the dominant sector in the long run.
Comin et al. (2020) assumed pure reproducible capital (not consumable) and pure consumption goods, all of which are produced by capital and labour under Cobb–Douglas production functions. The TFP growth rates and factor intensities are assumed to di¤er bysector, which are the supply-side reasons driving structural change. Since a pure capital good is assumed, capital accumulation is determined by a single equation: Ym (t) =Am (t) Km (t) m Lm (t)1 m = K (t + 1) (1) K (t), where the subscript m denotes thesector producing the pure capital good.
The preferences in Comin et al. (2020) generalise those in Comin et al. (2018). Here, we
review the preferences in the latter version, in which some functional forms are speci…ed. The
preferences are given over a bundle of consumption goods C (t) (C1 (t) ;; CN (t)) suchthat C, an index of real income measuring consumer utility, is implicitly de…ned through theconstraint:
NX

1( i C "i ) Ci

1

= 1;

(22)

< 1, and "i > 0 for i = 1;

; N . The sectoral di¤erence "i

i=1

where

i > 0; "i > 0;and services grow at the di¤erent constant rates with each other (see Appendix in Echevarria, 1997). Then,
Kaldor’s facts are obtained if j = 0 for j = 1; 2; 3. Furthermore, she showed that the economy with j > 0 for
j = 1; 2; 3 asymptotically converges to the GBGP. This is because income (and thus consumption) increases,jbecomes negligible. The utility function is getting close to the Cobb–Douglas form for high levels
j Cj (t)of consumptions. However, the model indicates the persistent coexistence of structural change and Kaldor’sfacts, due to the same logic of Ngai and Pissarides (2007).

21

controls for the relative income elasticity of demand and leads the demand functions to beNP
non-homothetic. Let E =pi Ci be consumption expenditure. Solving the expenditurei=1

C "i . The properties of the nonminimisation problem subject to (22) yields Ci = i pEi
homothetic CES in (22) are that non-homothetic features do not systematically vanish as@ ln(Ci =Cj )income (and thus utility) rises (i.e.
= "i "j ; for 8i; j = 1;
; N ) and that@ ln C@ ln(Ci =Cj )the elasticity of substitution between di¤erent goods is constant (i.e. @ ln(pi =pj ) = ; for8i; j = 1;
; N ).
Each household inelastically supplies one unit of labour and is endowed with a homogeneous asset A (0) in the initial period. The utility function is de…ned by the index C andassumed to have the CRRA form:
!
1
1
X
C
(t)
1
t
U=
:
1
t=0

Maximising the utility function subject to the budget constraint yields the Euler condition. Capital accumulation follows the equation shown above. Therefore, we obtain thetwo-dimensional di¤erential system of equations with respect to C (t) and K (t).
Further imposing the assumptions and conditions, the unique existence of the GBGP for
any initial value of capital stock is proven. It is shown that along the GBGP, i) the real rateof interest is constant; ii) aggregate consumption expenditure, total nominal output, andcapital stock grow at the same constant rate; and iii) the dominant sector in the long run
emerges. Then, Kaldor’s facts are obtained.
Although the dominant sector surviving in the long run is determined solely by the TFP
growth rate in Ngai and Pissarides (2007), it is also determined by the income elasticity ofdemand "i and sectoral factor intensity i in this model.

6

Structural Change and Kaldor’s Facts from Cambridge Keynesian Perspectives

So far, we have extensively reviewed the mainstream multi-sectoral models that attempt toreconcile structural change with Kaldor’s facts. Although there are some exceptions,10 we can10

For example, Muro (2017) presented a growth model with three factors and three goods and attempted
to reconcile structural change with Kaldor’s facts. The author proved the unique existence of the GBGP
and (su¢ cient) conditions for saddle-path stability by transforming the three-good model into a type of twosector optimal growth model à la Uzawa (1964). The structural change was caused by both non-homothetic
preferences and sectoral di¤erences in factor intensities. Second, Hori et al. (2015) constructed an endogenous
growth model with two sectors. Structural change is driven by the demand-side reason in the model, whichassumes that utility depends on not only the level of consumption but also the reference levels of consumption
represented by the stock of external habits. External habits cause endogenous growth, and these make theEngel curves non-linear. We obtain the three-dimensional di¤erential system of equations. Although themodel obtains the various types of equilibria, depending on the values of the parameters, it shows thepossibility of the existence of the local saddle-path stable steady state. Third, Alonso-Carrera and Raurich(2018) built a two-sector model reconciling structural change with Kaldor’s facts. In their model, both22

con…rm the mainstream strategy of the reconciliation: holding Kaldor’s facts means the stateat which the aggregate variables such as capital, output, and consumption expenditure growat constant rates, and the multi-sectoral models are transformed into the one-sector growthmodel that has a unique (saddle-path) stable steady state. This transformation guarantees
the existence of the GBGP.
Herrendorf et al. (2014) characterised mainstream multi-sectoral models of structuralchange as ‘a natural extension of the one-sector growth model that incorporates structuraltransformation’. The dynamics of the one-sector growth model transformed from multisectoral models can be described by, at most, a two-dimensional di¤erential system of equations.
As con…rmed in Section 1, Kaldor’s facts can be considered to be empirical regularities
of long-run economic growth, at least in advanced economies. To the best of our knowledge,
there is no research arguing that the facts have already lost its relevance to the analysis
of economic growth. The problem to address here is whether the mainstream strategy is
consistent with Kaldor’s own thoughts. In this section, we evaluate the mainstream strategy from the Cambridge Keynesian perspective (Pasinetti, 2007), in which Kaldor’s own
thoughts are included. We …rst argue that the mainstream strategy is entirely inconsistent
with Kaldor’s own thoughts. Second, we indicate that the mainstream strategy overlooks
another important structural change seen from the Cambridge Keynesian perspective. To
allow the alternative strategy to reconcile structural change with Kaldor’s facts, moreover,
we criticise such a mainstream characteristic of the reconciliation that crucially relies on aperfect adjustment mechanism through markets.
The Cambridge Keynesian group was formed after the Second World War and its founders
were the pupils of Keynes, such as Kahn, J. Robinson, Sra¤a, Kaldor, and others. According
to Pasinetti (2007, pp. 219–237), the features of Cambridge Keynesians can be summarised
as follows:
1. Reality (not simply abstract rationality) as the starting point of economic theory;
2. Economic logic with internal consistency (not only formal rigour);
3. Malthus and the Classics (not Walras and the Marginalists);
4. Non-ergodic (in place of stationary, timeless) economic systems;
5. Causality vs. interdependence;
6. Macroeconomics before microeconomics;
7. Disequilibrium and instability (not equilibrium) as the normal state of industrial economies;
reasons for structural change are included: the minimum consumption requirement for agriculture and sectoraldi¤erences in both the TFP growth rates and the factor intensities. It takes into account the additional reason
to a¤ect structural change: sectoral di¤erences in wages. It is argued that the additional reason is required toaccount for the relationship between the sectoral evolution of output and that of employment in the UnitedStates. They obtain the four-dimensional di¤erential system of equations with three state variables. The
unique existence of the saddle-path stable steady state is proved.

23

8. Need of …nding an appropriate analytical framework for dealing with technical change
and economic growth;
9. A strong, deeply felt social concern.
According to him, these features are the most important of Cambridge Keynesians, as
they are not always shared by all members and not exactly found in their works.

6.1

Mainstream interpretation of Kaldor’s facts

First, we must con…rm Kaldor’s statement: ‘none of these “facts” can be plausibly “explained”by the theoretical constructions of neo-classical theory’(Kaldor, 1961, p. 179). However, mainstream economists assert that Uzawa (1961) proved that the neoclassical growthmodel with Harrod-neutral technical progress can account for Kaldor’s facts.11
Although the basic idea of Kaldor (1961) had already been presented at the conference
on the theory of capital held on Corfu in 1958, it is inconceivable that Kaldor did not know
about Uzawa’s paper when completing his paper because Kaldor was the chair of the editorialcommittee of the Review of Economic Studies when the paper was published in that journal.
In fact, Kaldor was one of the most distinguished …gures of the journal, veri…ed by the factthat he served as chair from Vol. 9, No. 1 (1941) to Vol. 28, No. 3 (1961). Therefore,
we conjecture that Kaldor must have known Uzawa’s paper, but considered that it failed to
explain the facts.
Kaldor (1961) noted the inherent logical di¢ culties of de…ning capital using the neoclassical production function and criticised the smooth substitutability between capital and labour
as an unrealistic assumption. Instead, he assumed strict complementarity between capitaland labour that, according to him, has more a¢ nity with the classical economics of Ricardo
and Marx as well as Neumann (1945) model (Lutz and Hague, 1961, pp. 289–403; Kaldor,
1975). Moreover, he asserted that marginal productivity has no relevance in determining theshare of factor income.
In addition, Kaldor (1957) had already pointed out that the ‘constancies’of the capital–
output ratio, pro…t rate, and pro…t share are observed in many advanced economies, statingthat ‘existing theories are unable to account for such constancies except in terms of particularhypotheses (unsupported by any independent evidence), such as the unitary-elasticity of substitution between Capital and Labour, or more recently, constancy of the degree of monopoly
or the “neutrality”of technical progress’. The multi-sectoral models reviewed in the preceding sections are transformed using the CES and Cobb–Douglas functions. Moreover, they
assume Harrod-neutral technical progress somewhere. Otherwise, the steady state does notexist in the neoclassical growth model, as Uzawa (1961) showed. However, Kaldor explicitly
stated that he was trying to get away from the rigid idea that if the capital–output ratio
remained constant, this was caused by a peculiarity of technical progress— by its ‘neutrality’11

Jones and Romer (2010) argued that there is no longer any interesting debate about the properties of
Kaldor’s …rst …ve facts that a model must contain to explain them; only fact 6 continues to have analytical relevance today. Moreover, they asserted that ideas, institutions, populations, and human capital are
important to the growth of modern advanced countries.

24

(Lutz and Hague, 1961, p. 370). Thus, it is impossible to consider that Kaldor was satis…edwith the neoclassical explanation of the facts based on Uzawa (1961).
Furthermore, the Cambridge Keynesian feature 1, namely Reality (not merely abstract
rationality) as the starting point of any economic theory, means that ‘any theory needs to be
based on factual evidence, to be evaluated right from the start and not only to be empiricaltested at the end’(Pasinetti, 2007, p. 220). According to him, Keynes expressed the featureby stressing that his analysis refers to a ‘monetary theory of production’(Keynes, 1973, pp.
253–255) and Kaldor did so by proposing ‘stylised facts’. Kaldor used the term to mean
‘empirical regularities that are su¢ ciently general and persistent as to be able to capture thecorresponding objective features of reality’(Pasinetti, 2007, p. 220) but never intended thefacts to be used to justify the balanced growth achieved by forms of production functions
with a smooth substitution between the factors of production and Harrod-neutral technicalprogress (or the coincidence that capital-saving and labour-saving technical progress happen
to precisely o¤set one another, as Kaldor (1957, p. 593) said). Therefore, the mainstream
strategy is also inconsistent with Cambridge Keynesian thought.
Additionally, it is unclear how Kaldor considered the relationship between structuralchange and the facts. However, he recognised that economic growth entails structural change.
He observed that aggregate capital–output ratios have great stability in the long run in many
developed countries, stating that ‘this stability in over-all ratios concealed much variation
over time in the capital-output ratios of individual industries and particular sectors of theeconomy’. The coexistence of a stable aggregate ratio and variable sectoral capital–output
ratios was considered to be ‘very puzzling’(Lutz and Hague, 1961, pp. 339–340).
Furthermore, Kaldor (1984) constructed a two-sector growth model featuring agriculture(corn) and manufacturing (steel). The interrelation between these two sectors is the same as
in Marx’s (1967) schema of reproduction in that corn is the wage good and steel is a pure
capital good used in both sectors. He showed that the equilibrium growth rate of the two
sectors is equal. However, Kaldor (1984, p. 45) added, ‘In practice, the growth of industry is
likely to be greater than the growth of agriculture because the income elasticity of demand for
manufactured goods is higher than the income elasticity of demand for agricultural goods’.
The demand-side reason for structural change is indicated here.

6.2

Mainstream treatment of structural change

Feature 1 implies a further inconsistency with Cambridge Keynesians. Mainstream economics
only cares about how well the data …t empirical tests or model predictions when the models
assume the Cobb–Douglas or/and CES production function. On the basis of the Cambridge
Keynesian perspective, however, theoretical research that only cares about the empirical test
or model predictions is insu¢ cientFirst, the mainstream strategy of the reconciliation overlooks another important structural change: changes in the sectoral composition of physical capital. The questions at thestarting point of model building include ‘what is capital’and ‘of what does capital consist’.
The realistic answer is that a large part of capital consists of a bundle of heterogeneous andreproducible commodities.
Remember that all the mainstream multi-sectoral models reviewed so far assume that25

capital is reproducible but consists of a single commodity. In other words, the mainstream
models consider capital to be homogeneous.12 Homogeneous capital is an important assumption for mainstream economics. If capital consists of a bundle of heterogeneous andreproducible commodities, one cannot aggregate such commodities without them being multiplied by prices. If heterogeneous capital goods exist, the aggregate value is de…ned as
n
P
K (r)
pj (r) Kj (r), where Kj (r) denotes the amount of commodity j used as the capj=1

ital input, which is produced by the neoclassical production function Fj (K1j ; ;; Knj ; Lj )with Fj 6= Fi if j 6= i. As is well known (see Burmeister, 1980, pp. 122–123), dK< 0 doesdr
not necessarily hold. In other words, the general monotonically decreasing relation between
capital intensity and the interest rate no longer holds unlike in the one-sector model in which
single homogeneous capital exists.
This gives rise to di¢ culties proving the unique existence of the steady-state rate ofinterest (pro…t) and convergence of the economic system to the steady state. We can obtainthe condition for the existence of the index of aggregate capital K and aggregate production
function F (K) satisfying all the properties of a well-behaved neoclassical production function
around the steady state, as Burmeister (1980, pp. 131–134) showed, although it is restrictive.
More importantly, the aggregate production function is only useful for comparing alternative
steady-state equilibria and cannot be used to study more general dynamic paths (Burmeister,2000).
Mainstream economics tends to focus only on the heterogeneity of consumption goods
and exclude that of capital goods in the dynamic analysis because the more capital goods areincluded in the model, the higher the dimension of the di¤erential system of equations thatmust be analysed. In general, the number of consumption (capital) goods corresponds to thecontrol (state) variable. As the number of state variables increases, the stability analysis (inparticular, the proof of global stability) becomes harder and more complicated, as shown by
Alonso-Carrera and Raurich (2015) and Acemoglu and Guerrieri (2008) than when only one
state variable exists (Stijepic, 2011, p. 104).
In other words, mainstream economists avoid assuming heterogeneous capital goods forpurely technical reasons. Moreover, the Ramsey optimal growth model, into which many
mainstream multi-sectoral models of structural change are transformed, had been regarded
as a normative one but has recently been used to describe actual phenomena observed in themarket. This seems to re‡ect the mainstream con…dence of adjustment mechanism throughmarkets, reinforced from the 1980s, as Pasinetti (2018) argued. The optimal growth pathobtained by the Ramsey model, to which optimal control theory is applied, coincides with thedecentralised market equilibrium (Barro and Sala-i-Martin, 2004). How stringent conditions
are required for the existence of the market equilibrium is seldom examined seriously. The
mainstream strategy is chosen from the viewpoint of purely abstract rationality, which isclearly against Cambridge Keynesian feature 1.
12

Herrendorf et al. (2015) argued that the structural change in the agriculture, manufacturing, and services
sectors in the United States after the Second World War can be su¢ ciently captured by Cobb–Douglas
production functions with sector-speci…c TFP growth rates. To estimate the production functions, however,exogenous exponent capital is assumed. That is, the existence of heterogeneous and reproducible capital is
not taken into consideration.

26

The Cambridge Keynesian features re‡ect that industrial production, not exchange, isthe dominant activity. The emergence of classical economics corresponds to this, and thusclassical economists adopted the production paradigm (Pasinetti, 1977, 1981, 1986, 2007).
The industry, which does not need to be con…ned to only manufacturing, is a dynamic
concept, meaning that every component of the economic system continues to change forever(Pasinetti, 1981, 1993, 2007). Solow (2014) stated that multi-sectoral and one-sector growthmodels are complements, not rivals. Pasinetti objected to his opinion and said:
It [structural dynamic approach to growth] goes much beyond complementarity,
to such an extent that, when we consider the passage of time, one-sector approach
becomes incompatible with structural analysis. Essentially, the two approaches
embody two di¤erent visions of the industrial world. The vision behind structuraldynamics originates from the consideration of permanently evolving economic
system. The vision behind the aggregate model of traditional growth model
embodies a static, or at most a stationary, view of the economic system,(Pasinetti, 2014, p. 284)
The statement implies that Pasinetti denies the mainstream perception that multi-sectoralmodels are a natural extension of the one-sector model.
Are the properties of the one-sector model desirable? Burmeister and Turnovsky (1972)
stated so because this model exhibits capital deepening when the steady state having a higher
rate of interest (pro…t) is compared with that having a lower rate. Here, capital deepening
can be de…ned as the increase in the physical capital–labour ratio. In the regular economy(Burmeister and Turnovsky, 1972; Burmeister, 1980), lower steady-state rates of interest(pro…t) are always associated with higher steady-state consumption. This neoclassical parable
has ‘considerable heuristic values in giving insights into the fundamentals of interest theory’(Samuelson, 1962, p. 193). This parable must not be violated in mainstream economics.
Furthermore, as Burmeister (1980) and Hahn (1966) clari…ed, there is the inconvenience formainstream economics that the stability of equilibria is di¢ cult to obtain in models withheterogeneous capital goods.
If capital consists of a bundle of heterogeneous and reproducible commodities, capital
deepening cannot be de…ned physically, as already pointed out. Therefore, the parable holding in the one-sector growth model cannot be necessarily generalised to multi-sectoral models with heterogeneous capital goods. Since phenomena inconsistent with the parable isundesirable, mainstream economists call them ‘paradoxical’, ‘perverse’, ‘exceptional’, and‘anomalous’(Pasinetti, 1966, 2000).
In the production paradigm, capital is not acceptable as the primary and homogeneous
factor of production. There is no intrinsic reason to believe that the parable holding in the
one-sector growth model is desirable, given that capital in reality consists of a bundle ofheterogeneous and reproducible commodities such as equipment, machinery, buildings, facilities, and vehicles (see also Hagemann and Scazzieri, 2009, with respect to capital structureand economic dynamics). The importance of capital as a bundle of heterogeneous and reproducible commodities was recognised by classical economists and Marx. Economic growthin capitalist economies, as Marx (1965, p. 622) clearly understood, is typically characterised27

by the reduction in the mass of labour in proportion to the mass of the means of production
moved by labour. The fact that some means of production are composed of heterogeneousand reproducible commodities (i.e. past labour) was also clearly understood by Marx, sothat labour productivity rises in the growth process.
Indeed, much research insists on the importance of the heterogeneity of capital goods ineconomic growth. First, the composition of heterogeneous capital changes as the economy
grows. For example, Nomura (2004, p. 155) showed that the proportion of construction tototal capital stock declined by about 13% in real terms in Japan from 1960 to 2000 and thatthe average growth rate of construction (5.8%) is lower than that of total capital stock (6.8%)
during the same period. On the contrary, the proportions of general instrumentation andelectric machinery tend to increase and their average growth rates are much higher than thatof total capital stock.13 These results imply that the composition of physical capital changes
as income grows.
Mutreja (2014) asserted that the relation between the composition of physical capitaland income di¤erences has lacked scholarly attention while the relation between the capital–
output ratio and income di¤erences has been closely analysed. She demonstrated that thecomposition of physical capital is systematically related to income, which, according to her,is an important factor in explaining income di¤erences across countries.14 She also showedthat cross-country di¤erences in equipment capital are much larger than those in structure
capital; the equipment capital–output ratio is a factor of approximately 7 between rich andpoor, while the structure capital–output ratio is a factor of only 3. These results should
be considered carefully when we pay attention to fact 6, although Jones and Romer (2010)
asserted the e¤ects of ideas, institutions, and human capital.
Moreover, the cross-country dispersion in the equipment capital–output ratio has also
increased over time, while the dispersion in the structure capital–output ratio has declined.
Moreover, standard growth accounting has attributed a larger proportion of income differences to the TFP di¤erences in the models that excluded heterogeneous capital goods.
Mutreja’s (2014) results imply that when a country’s income is rising, the composition ofphysical capital changes in such a way that the proportion of equipment capital to aggregate capital increases. These results strongly support the importance of the existence ofheterogeneous capital goods in the analysis of structural change.15
The importance of structural change in physical capital is reinforced by the …nding that13

As Herrendorf et al. (2014) pointed out, the change in capital structure considerably di¤ers depending onwhether it is measured in real or nominal terms. In general, the proportions are more stable when measured
in nominal terms.
14
Caselli (2005) also showed the e¤ect of the capital composition on income di¤erences.
15
Mutreja et al. (2018) also constructed a multi-sectoral and multi-country Ricardian model with economic
growth to investigate the e¤ect of the international trade of capital goods through two channels (capitalformation and TFP) on economic development. There are consumption goods, intermediate goods, capital
goods (machinery and equipment), and structure (other manufacturing) in the economies. Intermediate
and capital goods are tradable. It was shown that importing capital goods from developed countries allows
developing ones to access more e¢ cient technologies for capital goods. This reduces the relative price of capital
goods in developing countries and increases their investment rates as well as the steady-state capital–output
ratio. Combining this with the comparative advantage raising the TFP relatively, the income di¤erences
between developed and developing countries are reduced.

28

skilled labour is more complementary to equipment capital than unskilled labour (e.g. Krusell
et al., 2000). This implies that unskilled labour is easily substituted as the growth in thestock of equipment capital increases. Therefore, as the share of equipment capital increases,
as Mutreja (2014) pointed out, the inequality of labour income between skilled and unskilled
labour grows. This phenomenon is termed capital–skill complementarity (Hornstein et al.
2005; Griliches, 1969). Although Krusell et al. (2000) did not treat the heterogeneity ofphysical capital in their model, they found that the stock of equipment capital measured ine¢ ciency units grew at about twice the rate of both capital structures and consumption in the
post-Second World War period, with its growth rate accelerating since the late 1970s. Hence,
the importance of analysing the change in the composition of physical capital is increasing.
In addition, the importance of the structure of physical capital in economic growth andincome distribution is supported by Piketty (2014, p. 22), although some of the conceptualand methodological problems with his de…nition of capital were pointed out by Garbellini(2020), who showed the evolution of the prices and quantities of various capital goods (e.g.
construction, housing, equipment, and metal products and machinery). Models omitting
heterogeneous capital goods might worsen the analysis of capitalist economies. Moreover,
Herrendorf et al. (2014, p. 901) confessed that the mainstream multi-sectoral models thathave the GBGP are overly stringent. The validity of some assumptions and conditions
made on the parameters are seldom examined fully. Hence, it is a pity that the mainstream
avoidance to include the existence of heterogeneous capital goods into their models comesfrom the above-mentioned technical (mathematical) reasons.
Second, we pay attention to the mainstream characteristic that the reconciliation isachieved entirely by the adjustment mechanism through markets. Mainstream multi-sectoralmodels assume perfect competition and no rigidities in markets. The adjustment mechanism
through markets means that changes in (factor) prices guarantee that demand and supply
are equal in the markets. Mainstream economics believes that Kaldor’s facts are obtained asa result of the adjustment mechanism working perfectly. The reliance on the market mechanism has become stronger over time, especially since the collapse of socialist economies.
Now, such a reliance is the common belief of mainstream economics. Indeed, the mainstream
strategy is consistent solely with the institution of perfect competition, which does not exist
in any economies in the real world.
Pasinetti’s structural economic dynamics include both the demand-side and the supplyside reasons behind structural change, and heterogeneous and reproducible capital goods arealso taken into consideration. The novelty of Pasinetti’s structural economic dynamics is thecapability of addressing the ‘institutional problem’as well as structural change, thanks to theseparation theorem (Pasinetti, 2007). Here, the institution means a set of procedures, rulesof individual and social behaviour, regulations, and administrative bodies. The institutional
problem of a society is to ‘construct(ing) its institutions, adopting and modifying them astime goes on, perfecting them and ... even discarding some of them, while inventing newones’ (Pasinetti, 2007, p. 306). Furthermore, the institutional problem is solved to search
for and lead to the natural positions within relevant and reasonably acceptable span oftime (Pasinetti, 1993, p. 117). Put another way, it is solved to obtain a set of the natural
magnitudes indicated below and ful…ll the e¤ective demand condition (Pasinetti, 1981, 1993).
Obviously, ful…lling the condition is one of the policy objectives of almost all governments.
29

The …rst stage of investigation in the theorem is to construct the natural economic system,
which is an abstract economic system de…ned independently of any institution as well as thecultural, geographical, and historical circumstances. The natural economic system represents
a logically ideal state of economies. Hence, it comprises a set of the natural magnitudes,
namely natural prices, natural quantities, natural rates of pro…t, natural wage rate, andnatural rate of interest. If we mention the ideal level of employment, which might be called
the natural level of employment, it must be the level of full employment. Then, the natural
economic system possesses the normative property.
To analyse economies under the speci…c institution, we must depart from the natural
economic system. As Pasinetti (1981, pp. 151–152) argued, the natural rates of pro…t areinconsistent with the capitalist economic system, since the competition leads them to equalise
and consumption of capital owners must be positive in the system. Thus, he argued that suchrates should be replaced with the equilibrium pro…t rate given by the Cambridge equation(Pasinetti, 1974, Chaps. 5 and 6): e = s1c (g + r ), where sc ; r denote capitalists’ savingrate and the weighted average growth rate of per-capita demand, respectively.16
According to our understanding, the essential features of the separation theorem isthe theoretical schemes erected at the …rst stage of investigations cannot be closed.
They must contain a su¢ cient number of degrees of freedom to allow the insertionof whatever type of rules of behaviour that may then emerge from carrying out thesecond (the more practically oriented, more down-to-earth) stage of investigation(Pasinetti, 2007, pp. 276–277).
In other words, the solution of the natural economic system must be indeterminate,
meaning that the natural magnitudes above are obtained as the functions with respect tocertain parameters. In the second stage of investigation, the model is closed by determining
these parameters. In other words, solving the institutional problem is nothing to determine
the parameters (Kurose, 2018).17 Such determination is open to all disciplines, not only
economics. Therefore, we no longer need to cling to rigid formulations such as maximisationand minimisation.
The important implication derived from Pasinetti’s structural economic dynamics and theseparation theorem is that the institutions must be changed to maintain full employment.
The institutional problem is not a once-and-for-all condition, and must be solved to maintainthe e¤ective demand condition over time since the ful…lment does not entail an automatic
self-adjusting process in structural economic dynamics.
The equilibrium rate of pro…t is obtained under the assumption of PScc = PSww , where P; S denote theamount of pro…t and saving, respectively, and subscripts c; w denote the capitalist and worker. Under what
conditions it holds in structural economic dynamics must be scrutinised.
17
In this sense, the separation theorem positively attaches a signi…cance to the indeterminacy of equilibria,
although it is in general interpreted as indicator of model misspeci…cation and thus detested in mainstream
economic theories. The indeterminacy of equilibria has a signi…cant implication for neo-Ricardian economists. This is because it enables them to assert that income distribution cannot be determined solely by thedemand-supply mechanism. Furthermore, they argue that some outside and non-economic factors, such as
the collective bargaining by labour union, are necessary to determine the income distribution and relative
prices. See Kurose (2018) and Kurose and Yoshihara (2019) concerning this point.
16

30

Since per-capita demand growth eventually reaches saturation and productivity growth
rates are generally increase, the e¤ective demand condition is unlikely to be satis…ed. Thisis represented by Cambridge Keynesian feature 7: Disequilibrium and instability. Hence, thecoordination problem necessarily occurs. One of the solution to the problem is given by theinstitutional changes.
It would be relevant to consider the possibility of the e¤ects of the institutional changes onreconciling structural change with Kaldor’s facts. The income distribution, which is related
to fact 5, is not entirely determined by the market forces. It is obviously a¤ected by theinstitutional factors in Pasinetti’s sense. In particular, the labour market regulations greatly
have an e¤ect on the income distributed to workers, and the labour market institutions havebeen changed in many countries (e.g. Boeri, 2011; MacLeod, 2011).18
As Arena (2017, pp. 116–117) argued, it is di¢ cult for Pasinetti’s structural dynamic
model to analyse the standard mathematical stability of equilibrium, because no adjustment mechanism is built into the model. However, this is not necessarily a disadvantage of
Pasinetti’s structural economic dynamics. Instead of the analysis of the stability, the separation theorem has the potential to conduct institutional analysis, which is impossible formainstream economists. The relationship between the reconciliation and the institutionalchanges remains an open question.

7

Concluding Remarks

Kaldor’s facts are considered to be ‘empirical regularities that are su¢ ciently general andpersistent’. In this study, we review mainstream multi-sectoral models in which structuralchange is reconciled with Kaldor’s facts. The mainstream strategy of reconciliation is thatholding the facts is regarded as the state at which the economy grows along the GBGP and
the multi-sectoral model is transformed into the one-sector growth model (e.g. the Ramsey
and Solow models) that has the uniquely (saddle-path) stable state. In this sense, mainstream
economists consider multi-sectoral models to be a natural extension of the one-sector growthmodel.
We argue that the mainstream strategy is far from Kaldor’s own thoughts and the Cambridge Keynesian perspective. Moreover, the mainstream analysis of structural change overlooks another important structural change, namely, changes in the sectoral composition ofphysical capital, because all mainstream multi-sectoral models assume homogeneous capital.
As is shown by Mutreja (2014), structural change in physical capital is systematically related
to income. Therefore, structural change in physical capital is an important characteristic ofeconomic growth that should not be omitted from theoretical studies.
Moreover, the attempt to reconcile structural change with Kaldor’s facts from the Cambridge Keynesian perspective is expected to pave the way for institutional analysis to berequired for maintaining full employment. Such analysis can be feasible only by adopting
18
In this context, parameters and appearing in the e¤ective demand condition in Pasinetti (1981, 1993),
which respectively denote the proportion of the active to the total population and the ratio of working hours
to the total number of hours forming the unit of time considered, are of great signi…cance. The ful…llment of
the condition over time requires for the parameters to change in Pasinetti’s structural economic dynamics.

31

the Cambridge Keynesian perspective since the mainstream strategy assumes a perfect adjustment mechanism. Adopting the Cambridge Keynesian perspective to reconcile structuralchange with Kaldor’s facts is also an attempt to connect economic theory with institutions.
As Pasinetti (2020) argued, this attempt ought to be paid more attention in economics.

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