Discussion Paper No. 416
A Two-class Economy from the Multi-sectoral
Perspective: The Controversy between Pasinetti and
Meade—Samuelson—Modigliani Revisited

Kazuhiro Kurose
February 2020
December 2020 (This Version)

TOHOKU ECONOMICS RESEARCH GROUP
Discussion Paper

GRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANA Two-class Economy from the Multi-sectoral
Perspective: The Controversy between Pasinetti and
Meade–Samuelson–Modigliani Revisited
Kazuhiro Kurosey
February 4, 2020
December 23, 2020 (This Version)

Abstract
We examine the Pasinettian two-class multi-sectoral model with a microfoundationof capitalists and workers, speci…cally, two combinations of their behaviour. First, bothact as in…nitely lived agents (ILA) and second, capitalists act as ILA while workers
follow overlapping generations behaviour. We analyse the switch of equilibria simultaneously with the paradox in capital theory. Pasinetti equilibrium is independent oftechnology and the microfoundation. Dual equilibrium depends on technology and differs by microfoundation. Numerical examples of net output/capital ratio and capitalintensity imply we should analyse income and wealth distribution by the models withcapital as a bundle of reproducible commodities.
JEL Classi…cation: B51, D01, D24, E25, D33
Keywords: Pasinetti equilibrium/Dual Equilibrium, Microfoundation of Capitalists’and Workers’Behaviour, Capital Theory Paradox, Multi-sectoral Model

A part of this study was presented at the 9th Summer School on Analytical Political Economy, heldon 26–28 August, 2019 at Doshisya University in Japan. The author is grateful for helpful comments andsuggestions from the participants, especially Hiroyuki Ozaki, Hiroaki Sasaki, and Naoki Yoshihara. He issolely responsible for all remaining errors. Finally, …nancial support from JSPS KAKENHI Grant Number
JP17K03615 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

The distribution of income and capital (or wealth) is one of the major issues in current
economic analysis. Recent research on distribution has been stimulated especially by Piketty(2014). Following him, many economists have studied the degree of concentration of the richer
class’s income and capital (wealth), the kinds of economic models that can …t or predict suchconcentration of income and capital (wealth), and other topics.
The assumptions underlying the behaviour of agents are crucial for analysis of the distribution of income and capital (wealth). Caggetti and De Nardi (2008) and De Nardi and Fella(2017) investigate the various types of general equilibrium models with multiple agents. The…rst type is the general equilibrium model with in…nitely lived agents (ILA). The second typeis overlapping generations (OLG) models, which include some elements of life-cycle structuresand intergenerational links. The third type mixes features of both the ILA and OLG models.
Caggetti and De Nardi (2008) and De Nardi and Fella (2017) argue that the …rst type cannot
generate a level of wealth inequality observed in the data; the second type can explain theobserved inequality much better than the …rst; and the third type simpli…es some aspects ofeither model and thus, makes the analysis more tractable. In addition, they assert that the
introduction of the distinction between entrepreneurs’and workers’decision-making into themodels drastically improves the …t of the models to the data.
When focusing on a speci…c phenomenon, the simple model is very useful to understandthe underlying essential features and mechanism at work. Such a simple model for the analysis
of distribution of income and capital (wealth) would be a two-class model. For the two-classmodel, Pasinetti (1962) is an important point of reference to consider growth and distribution.
This work led to intensive debates with Meade (1963, 1966), Meade and Hahn (1965), andSamuelson and Modigliani (1966a, 1966b), as part of the Cambridge capital controversy, inthe 1960s.1 Interest in the Pasinettian two-class model has been revived recently.
For example, Taylor (2014) criticises Piketty’s (2014) analysis for relying on the neoclassical production function. Taylor (2014) demonstrates that euthanasia, persistence, andtriumph of the rentier are all possible scenarios of the Pasinettian aggregate two-class model,
whereas Piketty (2014) predicts only one scenario, triumph of the rentier. Meanwhile, Taylor’s (2014) model lacks a microfoundation of agents.
Mattauch, Klenert, Stiglitz, and Edenhofer (2017) examine how public investment …nanced by capital tax a¤ects the distribution of wealth when a change in substitutability
between capital and labour is allowed. The basic setting of the model is based on Pasinetti(1962). In other words, the worker earns his income from wage and pro…ts and saves fora life-cycle purpse, which implies that the worker’s behaviour follows the OLG model.2 Bycontrast, the capitalist earns her income solely from pro…t and is assumed to be an ILA. As
a result, it is demonstrated that for any elasticity of substitution greater than a threshold,there exists a capital tax rate at which the capitalist disappears in the steady state, whichcorresponds to the dual equilibrium (DE) in the sense of Samuelson and Modigliani (1966a,
1966b). In addition, for any elasticity of substitution below the threshold there exists a cap1

See also Pasinetti (1964, 1966a, 1966b, 1974). Baranzini and Mirante (2018, Chaps. 6 and 7) provide an
excellent survey of the extensions of the Pasinettian two-class model.
2
Baranzini (1991) is an early example of the models introducing the OLG into the Pasinettian two-classmodel.

2ital tax rate at which the worker disappears in the steady state, which corresponds to theanti-dual equilibrium (ADE) in the sense of Darity (1981). Furthermore, both the capitalist and worker co-exist in the steady state below these capital tax rates, which corresponds
to the Pasinetti equilibrium (PE), shown in Pasinetti (1962) and named in Samuelson and
Modigliani (1966a, 1966b).3
Zamparelli (2016) analyses the Pasinettian two-class economy in the neo-classical framework. He assumes a production function with constant elasticity of substitution (CES)
between capital and labour and no microfoundation of capitalist’s and worker’s behaviour.
The capitalist earns her income solely from pro…t and the worker from wage and pro…t. As
a result, if the capitalist’s saving rate is higher than the worker’s and the elasticity of substitution is high enough to ensure endogenous growth, ADE exists. In addition, he shows that
capital tax can favourably improve the distribution to the worker in the steady state.
Stiglitz (2016) points out new stylised facts on growth and accumulation, and asserts thatthe standard neo-classical models cannot explain the recent movement of the ratio of wealth toincome, even taking technical change into consideration.4 Then, Stiglitz (2016) assumes the
Pasinettian two-class economy in which the capitalist is the ILA and the worker’s behaviour
follows the OLG model. The feature of his two-class model is to introduce land as a factor
of production in the model, as taking land rent and exploitation rent into account better
explains the recent movement of the ratio of wealth to income.
Sasaki (2018) investigates the existence and stability of the steady states obtained by
the Pasinettian two-class economy in which the capitalist is an ILA, the worker’s behaviour
follows the OLG model, and there is a CES production function. It is shown that although
PE and DE exist depending on the combinations of parameters, PE is stable under reasonablevalues of the parameters. Furthermore, Taylor, Foley, and Rezai (2019) consider a Pasinettian
two-class economy in which growth is demand driven and no microfoundation of agents isformulated. Taylor et al. (2019) examine the existence and stability of the PE and DE
obtained as the steady states in the demand-driven growth model. In a context of the richest
1% of US households receiving about 7% of wages, Taylor et al. (2019) interestingly considera case in which capitalists, who have a higher rate of saving than that of workers, receive
some wage income besides pro…t. Then, it is shown that the DE obtained in this case isunstable.
To consider social security, Michl and Foley (2004), Michl (2007, 2009) construct a
Pasinettian two-class growth model in which the capitalist is an ILA and the worker’s be3

Mattauch, Edenhofer, Klenert, and Bénard (2016) investigate the distribution of wealth in relation to
public investment in a model with high-income households that are ILA and middle-income households that
follow the OLG model. Their analysis focuses on the equilibrium at which both types of households survive
in the steady state (i.e. PE).
4
In contrast to Kaldor’s (1961) old stylised facts, the new ones proposed by Stiglitz (2016) are:there is growing inequality in both wages and capital income (wealth), and growing inequality overall;
wealth is more unequally distributed than wages;
average wages have stagnated, even as productivity has increased, so the share of capital has increased;
the wealth ratio has increased signi…cantly;
the return on capital has not declined, even as the wealth–income ratio has increased.

3

haviour follows the OLG model. They show the existence of PE and DE in the model, andanalyse various e¤ects of social security. For example, Michl and Foley (2004) show thatan unfunded social security system relying on payroll taxes reduces workers’lifetime welfare
and saving, since the change in their saving a¤ects the level of the share of capital ownedby the workers but not the rate of economic growth in PE. The change in worker’s savings
has no e¤ect on the path of capital owned by capitalists in the model. Therefore, the decrease in the level of workers’share of capital reduces the overall level of capital, output, andemployment without a¤ecting the rate of economic growth. This is called the level e¤ect.
The e¤ect is mitigated by the presence of a reserve fund. Their model identi…es the social
security reserve fund as a potential vehicle for generating capital accumulation and e¤ecting
progressive redistribution of wealth.
The above-mentioned models with a microfoundation of agents assume the neo-classical‘well-behaved’ production function with capital as the primary factor of production or arede-facto one-commodity models. As clari…ed in the Cambridge capital controversy, the assumption of the neo-classical well-behaved production function with capital as the primaryfactor of production by hypothesis excludes some phenomena which may arise if capital isregarded as a bundle of reproducible and heterogeneous commodities.5 Furthermore, thecontroversy reveals that the results obtained by the one-commodity model do not necessarily
hold in a model with multiple commodities (see, e.g., Harcourt, 1972).
As highlighted by Piketty (2014), capital in modern capitalist economies typically consistsof reproducible and heterogeneous commodities. It would be signi…cant to analyse growthand distribution in the Pasinettian two-class economy under the assumption of multiple
commodities and capital as a bundle of reproducible and heterogeneous commodities. In
this study, we examine the Pasinettian two-class economy by a sort of activity analysis (i.e.
Leontief–Sra¤a model). Since, as already mentioned, the kinds of assumptions made about
the micro-behaviour of agents are crucial for our purpose, we consider two combinations of
microfoundation of capitalists’ and workers’ behaviour: …rst, both capitalists and workers
are ILA; and second, capitalists are ILA and workers’ behaviour follows the OLG model.
The scenarios cover all combinations used in the abovementioned models. However, ourexamination is con…ned to the steady states obtained in PE and DE.
The …rst characteristic of our study is that we analyse the switch of the type of equilibria (from PE to DE, or vice versa) simultaneously with paradoxical phenomena in capitaltheory (reswitching of technique and reverse capital deepening). This analysis is certainly
impossible in models which assume the neo-classical well-behaved production function withcapital as the primary factor of production. This is because, as mentioned, the possibilityof such paradoxical phenomena arising is by hypothesis excluded from the models. Although
Pasinetti (1962) and Samuelson and Modigliani (1966a, 1966b) use aggregate macroeconomic
models, Morishima (1969) constructs a multi-sectoral general equilibrium model to analyse
the properties of the steady-state path of PE and DE. Subsequently, Hosoda (1989) extends
Morishima’s model to analyse the switch of the type of equilibria simultaneously with thecapital theory paradoxes. Since capitalists’and workers’saving rates are assumed to be exogenously given in both Morishima’s (1969) and Hosoda’s (1989) models, we further develop
5

Stiglitz (2016) mentions the possibility of such phenomena arising, yet assumes a perfectly neo-classicaltechnology in his model.

4them to consider the abovementioned combinations of the microfoundation of capitalists’andworkers’behaviour.
In analysing the distribution of income/capital (wealth), the movements of the ratio of
output to capital and the pro…t (or the wage) share are important variables to be determinedin a model. Piketty (2015) argues that:the right model to think about rising capital–income ratios and capital shares inrecent decades is a multi-sector model of capital accumulation, with substantial
movements in relative prices, and with important variation in bargaining power.
His claim is based on the re‡ection, which Stiglitz (2016) also asserts, that the recent movements of the ratio of capital to output and the pro…t share cannot be explained by standard
neo-classical models in which a well-behaved production function with capital as the primaryfactor is assumed. Although we do not consider the variation of bargaining power, our multisectoral model based on Morishima (1969) and Hosoda (1989) shown in Section 3 su¢ ciently
satis…es Piketty’s claim. This is because the movements of the ratio of net output to capitaland the income share that are inconsistent with the principle of marginal productivity ofcapital are obtained by the Leontief–Sra¤a model in which capital is composed of a bundle ofreproducible and heterogeneous commodities. This is the second characteristic of our model.
The rest of this paper is organised as follows. Section 2 brie‡y reviews the conceptof the long-run competitive equilibrium in Hosoda (1989), on which our model is based.
Hosoda’s (1989) model has one degree of freedom, as in Sra¤a (1960), and thus, the rateof pro…t is treated as exogenously given throughout his study. We follow his treatmentof the rate of pro…t in our model. Therefore, the rate of economic growth in the steadystates is obtained as the function of the rate of pro…t, unlike in Pasinetti (1962, 1974) andSamuelson and Modigliani (1966a, 1966b), whose rate of pro…t is the endogenous variable
and rate of economic growth is the exogenous variable (natural rate of economic growth).
This is a device to analyse the paradoxes in capital theory, together with economic growth.
Section 3 presents our multi-sectoral two-class model with a microfoundation of capitalists
and workers. We analyse the relationship between the rates of economic growth and pro…tobtained in each combination of the microfoundation of capitalist and worker. We show theexistence of PE and DE and analyse the properties of the steady states. Section 4 presents
numerical examples to examine the working of the model proposed in Section 3; we analyse
the movements of the ratio of net output to capital and the capital intensity in relation to
the change in the rate of pro…t. Section 5 presents concluding remarks.

2Long-run Competitive Equilibrium

The main purpose of this section is to review Pasinetti’s (1962) two-class growth model froma multi-sectoral perspective. We focus on the basic structure of the models in Morishima(1969) and Hosoda (1989).
We assume that the primary factor of production is labour alone, joint production isabsent, and the number of reproducible commodities is n; the jth sector has m (j) = 1
processes. Column vectors Aj1 ;; Ajm(j) 2 Rn+ for j = 1;
; n denote processes of thejth sector to produce a unit of the commodity. We de…ne an input matrix as follows:5

A11 ;

A
where m =n
X

; A1m(1) ; A21 ;; A2m(2) ;n m; Anm(n) 2 R+;

; An1 ;

m (j). Letting Ljk be the labour input coe¢ cient corresponding to the process

j=1

Ajk for k 2 f1;; m (j)g, we de…ne such a labour input vector as follows:
L11 ;

L

; L1m(1) ; L21 ;; L2m(2) ;; Lnm(n) 2 Rm+:

; Ln1 ;

Moreover, we de…ne E as
E

;e ;
(e1 ;
;e ;e ;
| {z 1} | 2 {z 2}
m (1)
m (2)

; en ;; e ) 2 Rn m
| {z n}m (n)where ej 2 Rn denote the vector, the jth element of which is unity and the others zero.
We choose only one process from each sector. De…ne a column vector x
x1k(1) ; x2k(2); xnk(n) 2
Rn+ , indicating that the jth sector utilises only one process k (j). Furthermore, we de…ne thefollowing vector in accordance with A:
q

(0;
|

; x1k(1){zm (1); 0; 0;}In addition, we de…ne such a matrix asAA1k(1) ; A2k(2) ;; 0; 0|; xnk(n) ;
{z
m (n)

; Ank(n) 2 Rn+ n , k (j) 2 f1;; 0) 2 Rm+:}; m (j)g, for j = 1;

; n:

The matrix is termed a technical coe¢ cient matrix. Corresponding to A , we construct sucha labour input vector as L
L1k(1) ; L2k(2) ;; Lnk(m) 2 Rn+ , which is termed a technicalnYlabour input vector. There arem (j) such matrices and vectors. We make pairs of the
j=1

technical coe¢ cient matrix and the technical labour input vector (A ; L ), and index them,nYsuch as ; ;. De…ne the set
f ; ; g. Of course, j j =m (j) holds, where j j
j=1

is the number of elements of the set.
Here, we make the following assumptions.

Assumption 1 (A1): Every commodity is basic for any technique in the sense of Sra¤a(1960).
Assumption 2 (A2): For every u 2 Rn+ , there exists g > 0 and q 2 Rm
+ such that[E

(1 + g) A] q = u:6

Assumption 3 (A3): The period of production of all goods is one unit of time and labour
is an indispensable factor of production.
A1 implies that every commodity is required as an input directly or indirectly to produce
a unit of any commodity, which means that every technical matrix is indecomposable. A2
1
1
implies the existence of g 2 (0; G) and q 0, such that [E (1 + g) A] q > 0 (Gandis the minimum Frobenius root of technical coe¢ cient matrices). Certainly, G equals
the maximum rate of pro…t R attainable in the given set . In other words, A2 implies that
A is productive in the sense that positive net output can be produced. A1–A3 are the mild
assumptions usually made in the Leontief–Sra¤a production models.
We assume perfectly competitive markets, so that the rate of pro…t rt and the wage rate
wt established in time t are uniform. Matrix A represents the technology available at eachtime t = 1; 2; .6 The activity level at time t is denoted by q t (or xt ). The amount of input
commodities necessary to produce q t (or xt ) is denoted by the vector of Aq t (or A xt ), andthose input commodities necessary are produced at time t 1, which can be purchased atprices pt 1 2 Rn+ at the beginning of time t. The amount of labour necessary to produce q t(or xt ) is denoted by Lq t (or L xt ). The uniform wage rate wt is ex-post paid to the workers.
Suppose that our economy is composed of agents belonging to the capitalist class andthose belonging to the worker class. Since we assume that all members belonging to eachclass have the same preferences, the behaviour of each class as a whole can be represented
by that of any of its constituents. Therefore, we pay attention to a single capitalist and asingle worker.
Pasinetti (1962) argues that the worker owns a part of the stock of capital when hesaves and thus, receives a part of pro…ts as well as wages. Therefore, Pasinetti, in correcting Kaldor’s (1956) saving functions asserts that the saving function of the capitalist
should be formulated as Stc = sc Ptc and that of the worker as Stw = sw (Wt + Ptw ), where
Stc ; Stw ; Ptc ; Ptw ; Wt ; sc ;, and sw denote the capitalist’s saving, the worker’s saving, the amountof pro…ts distributed to the capitalist, that distributed to the worker, the total amount ofwages, the constant saving rate of the capitalist, and that of the worker, respectively. In
other words, the total saving function is given by St Stc + Stw = sc Ptc + sw (Wt + Ptw ). In
addition, 0 5 sw < sc 5 1 are assumed, and Ptc + Ptw = Pt holds, where Pt denotes the total
pro…ts generated in the economy. Since Ptc is the capitalist’s total income and Wt + Ptw is
the worker’s total income, it is obvious that the capitalist’s consumption demand and theworker’s consumption demand for each good are proportional to their incomes, and the proportionality is given by constant 1 sc in the capitalist’s consumption demand function andconstant 1 sw in the worker’s consumption demand function.
The problem to be addressed here is how to divide Pt into Ptc and Ptw . Pasinetti (1962)
proposes the conditions for the division of pro…ts. The conditions are indicated as follows.
Condition 1 (C1): The stock of capital owned by the capitalist and that by the worker
grow at the same rate:
K_ tc
K_ tw
K_ t
= c = w for 8t;
Kt
Kt
Kt
6

Here, we do not consider technical progress in that the row or the column of A and L changes as timegoes by. In other words, m and n are …xed over time.

7

where Ktc ; Ktw denote the stock of capital owned by the capitalist and that by the worker,
respectively, at time t.
Condition 2 (C2): The rate of pro…t irrespective of its owner is uniform:
Pt
Pc
Pw= tc = tw for 8t;
Kt
Kt
Kt
K_ tw

By combining C1 with C2 and using the de…nitional relationship (i.e. K_ tc
Stw ), we obtain Condition 3.

Stc and

Condition 3 (C3): The following relationship is satis…ed:
Ptw
Ptc
=
for 8t.
Stw
Stc
Following C3 and the above de…ned saving functions of the capitalist and worker, we
obtain
Ptw =

Ptc

sw

Wt =

sw
sc

sw

(1)

wt Lq t ;

sc

sw

Pt

Ptw = rt pt 1 Aq t

sw
sc

sw

wt Lq t .

(2)

As is well known from (2), we can obtain the PE or DE, depending on whether rt pt 1 Aq t S
The capitalist and worker co-exist
in PE, while in DE, the capitalist disappears and the entire stock of capital is owned by theworker.
Based on (2), Hosoda (1989) assumes that the capitalist’s consumption demand ct 2 Rn+
is given as follows:sw
w Lq t hold, which we closely analyse in Section 3.
s c sw t

ct = max rt pt 1 Aq t

sw
sc

sw

wt Lq t ; 0

1 sc(pt ) ;
pt (pt )(3)

where (pt ) 2 Rn+ denotes the capitalist’s consumption basket per unit of her income, whichis assumed to be the function of prices. Based on (1), the worker’s consumption demanddt 2 Rn+ is given as follows:
dt = min

sw
sc

sw

wt Lq t ; rt pt 1 Aq t

1 sw
' (pt ) ;pt ' (pt )(4)where ' (pt ) 2 Rn+ denotes the worker’s consumption basket per unit of his income, which is,
as in the capitalist consumption demand function, assumed to be the function of prices. It is
assumed that both (pt ) and ' (pt ) are homogeneous functions of degree zero with respect
to prices, and pt (pt ) 6= 0 and pt ' (pt ) 6= 0 for 8t. The capitalist’s consumption demand
and the worker’s consumption demand are homothetic functions.
Based on these assumptions, we construct the two-class general equilibrium model asfollows:8

8
pt E 5 (1 + rt ) pt 1 A + wt L;
>
>
>
>
>
>
>
>
pt Eq t 1 = (1 + rt )pt 1 Aq t 1 + wt Lq t 1 ;
>
>
>
>
>
>
>
>
Eq t 1 = Aq t + ct + dt ;
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:

(5)

pt Eq t 1 = pt Aq t + pt ct + pt dt ;pt Eq t 1 > 0;
pt ; q t0; wt0; for all t.

Assuming that all sectors grows at an exogenously given rate of g 2 [0; G), the economy is on a balanced growth path such that pt 1 = pt ; q t = (1 + g) q t 1 ; ct = ct 1 ; dt =dt 1 ; wt 1 = wt , and rt 1 = rt hold for 8t. As mentioned in Section 1, we treat the rate ofpro…t as the exogenous variable throughout this study. Then, the cost-minimising techniqueand thus, the equilibrium prices are determined independently of the capitalist’s consumption demand and the worker’s consumption demand, owing to the non-substitution theorem.
Let A 2 Rn+ n and L 2 Rn+ be the cost-minimising technical coe¢ cient matrix and thecost-minimising technical labour input vector, which are termed the long-run competitive
equilibrium technical coe¢ cient matrix and the long-run competitive equilibrium labour input
vector, respectively, for the given rate of pro…t r 2 [0; R). In other words, the given rateof pro…t determines the cost-minimising technique (A ; L ), independently of the size andstructure of the …nal demand, for 2 . Given the rate of pro…t, the long-run competitive equilibrium technical coe¢ cient matrix and the long-run competitive equilibrium labour
input vector are represented by A = A and L = L , respectively, for 2 .
Exogenising r 2 [0; R), we can normalise the prices by w = 1. Then, the price p 2nR+ , termed the long-run competitive equilibrium price vector, is determined. The nonnegativeness of the price is guaranteed by A2. The activity vector x 2 Rn+ satisfying (6),which is based on (3), (4), and (5), is termed the long-run competitive equilibrium activity
vector:8p 5 (1 + r) p A + L ;
>
>
>
>
>
>
>
>
p x = (1 + r)p A x + L x;
>
>
>
>
>
>
>
>
x = (1 + g) A x + c + d;
<
(6)
>
>
p x = (1 + g) p A x + p c + p d;
>
>
>
>
>
>
>
>
p x>0
>
>
>
>
>
>
:
x 0;
where

9

8
>
>
<

h
c = max rp A xh>>: d = min L x +isw
L x; 0sc sw

sw
L x; rp A x
sc sw

p

1 sc
(p )

(p ) ;(7)i

1 sw' (p ) :p '(p )

Hosoda (1989) proves the existence of x , together with p , in the set K fx = 0j L x =1g by applying Brouwer’s …xed point theorem. Therefore, the rate of economic growth canbe obtained as the function of the rate of pro…t as follows:8
sc r if rp A x > scswsw (the PE),><(8)g (r) =>sw
: sw r + 1
if
rp
A
x
5
(the
DE).
p A x
sc sw

The …rst function of (8) is substantially equivalent to the Cambridge equation or the Pasinetti
theorem (Samuelson and Modigliani, 1966a, 1966b; Pasinetti, 1974). Note that the Cambridge equation obtains the rate of pro…t given the rate of economic growth (natural rate of
economic growth) in the two-class economy, whereas the …rst function of (8) determines therate of economic growth given the rate of pro…t. However, it has the same macroeconomic
implication as the Cambridge equation: the worker’s saving rate and technology have noe¤ect on the relationship between the rates of economic growth and pro…t in the PE, and therelationship is determined solely by the capitalist’s saving rate. Similarly, the second function of (8) is substantially equivalent to the DE in Samuelson and Modigliani (1966a, 1966b)
and has the same macroeconomic implication: the worker’s saving rate and the technologya¤ect the relationship. We should note that the form of g (r) in DE depends on the choiceof numéraire except when the so-called organic composition of capital is uniform among all
sectors whereas the form in PE is not.
When PE holds, (rp A x > scswsw ), we have:sc r > sw r +

1
pAx

.

Similarly, when DE holds (i.e. rp A x 5 scswsw ), we havesc r 5 sw r +

1
pAx

.

As Hosoda (1989) shows, either rp A x > scswsw or rp A x 5 scswsw holds but not both forx 2 K. We can obtain a higher rate of economic growth for 8r 2 [0; R) in the equilibrium.

3

Multi-sectoral Two-class Models with Microfoundation of AgentsIn this section, we provide the microfoundation of the behaviour of capitalist and worker
with the two-class model, so that we can endogenise the allocation of their incomes between
consumption and saving. As mentioned in Section 1, we consider two combinations of their
behaviour; …rst, both the capitalist and worker act as ILA, and second, the capitalist acts10

as an ILA while the worker’s behaviour follows that of the OLG model. In other words,
two combinations of microfoundation are represented by (capitalist, worker) = (ILA, ILA),(ILA, OLG).
Hereafter, we assume that consumption good consists of a single commodity (commodity1), although there are n kinds of reproducible and heterogeneous commodities. This is a
simpli…cation often made in models in which multiple commodities exist to avoid unnecessary
complications from the existence of multiple consumption goods (see, e.g. Harcourt, 1972).
By so doing, we can concentrate on the analysis of the problems related to the existence ofcapital that consists of a bundle of reproducible and heterogeneous commodities. Following
the two-class models with the microfoundation referred to in Section 1, we use the log utility
function. Moreover, the assumptions made in Section 2 with respect to the technology andthe time structure of the model are kept unchanged.

3.1

Optimal behaviour in the case of (capitalist, worker) = (ILA, ILA)

In this subsection, we consider the case in which the combination of the microfoundation of
the capitalist and worker is represented by (capitalist, worker) = (ILA, ILA) and we analyse
the properties of PE and DE.
3.1.1

Optimal behaviour of the capitalist in (ILA, ILA)

Since the capitalist is assumed to act as an ILA, she maximises her utility subject to theintertemporal budget constraint. The capitalist solves the following problem:

max U

cct;1 ;q t+1

s.t. pt E (1

c

(1

c)

1
X

t 1
ln ct;1ct=1t+1 ) q t+1+ pt ct 5 rt pt 1 A ((1given frt gt=1;2; ; f t gt=1;2; , p0t) qt) ,0,

where U c ; c 2 (0; 1), and t 2 (0; 1] denote the capitalist’s utility function, the discountrate of the capitalist, and the worker’s claim to a share of the pro…t, respectively, at time t.
Furthermore, ct (ct;1 ; 0;; 0) 2 Rn+ is the capitalist’s consumption vector.
We assume that the transversality condition is satis…ed (i.e. pT Eq T +1 = 0 as T ! 1).
We can identify the long-run competitive equilibrium technique A for the given rate of pro…trt 2 [0; R), based on the non-substitution theorem. Therefore, the solution of the problem ispt ct = (1c ) rt pt 1 A((1t ) xt ) ;(1c ) rt pt 1 A((1t ) xt )ct;1 =

pt;1
11

;

i.e.

(9)

where pt;1 denotes the price of commodity 1 determined by rt . (9) implies that the capitalist
consumes commodity 1 proportionally to her income rt pt 1 A ((1t ) xt ). Therefore, thecapitalist’s saving rate is c .
3.1.2

Optimal behaviour of the worker in (ILA, ILA)

The worker, like the capitalist, maximises his utility subject to the intertemporal budget
constraint. Then, he solves the following problem:

max U

w

cw
t;1 ;q t+1

s.t. pt E

(1

w)

1
X

t 1
w ln dt;1t=1t+1 q t+1

+ pt dt 5 Lq t + rt pt 1 A ( t q t ) ;given frt gt=1;2; ; f t gt=1;2; ; p00,where U w and w 2 (0; 1) denote the worker’s utility function and the discount rate of theworker, respectively.7 Similarly, in the case of the capitalist, dt (dt;1 ; 0;; 0) 2 Rn+ is theworker’s consumption vector. Note that the wage rate is normalised as wt = 1. In addition,
the following assumption is made:
w <

c:

(10)

(10) means that the capitalist’s saving rate must be higher than the worker’s. As is mentioned
in Section 2, it is nothing but the assumption made by Pasinetti (1962).
If the transversality condition is satis…ed, as in the case of the capitalist’s problem, thesolution of the problem ispt dt = (1w)

L xt + rt pt 1 A ( t xt ) , i.e.

(1w)

L xt + rt pt 1 A ( t xt ):pt;1

dt;1 =(11)(11) demonstrates that the worker consumes commodity 1 proportionally to his income
L xt + rt pt 1 A ( t xt ), and the worker’s saving rate is w .
7

Although it is assumed that both the capitalist and worker control q t+1 , we do not need to apply thedi¤erential game to our problem. This is because the non-substitution theorem holds in the economies
assumed in this study. Thanks to the theorem, any q t+1 0 is e¢ cient and, as we show in Subsection 3.1.4,
the solution of the capitalist’s problem is always consistent with that of the worker’s problem in an economic
system. Regarding the application of the di¤erential game to the Pasinettian two-class model, see Chappelland Latham (1983).

12

3.1.3Distribution of pro…ts in (ILA, ILA)

In this subsection, we consider the distribution of pro…ts between the capitalist and worker.
Doing so implies specifying the condition for which t must be satis…ed to obtain the longrun competitive equilibrium de…ned in Section 2. Then, we obtain the following relationship,
following C3:(1c (1t ) rt pt 1 A((1t ) rt pt 1 A ((1t ) xt )t ) xt )t rt pt 1 A xt=
wt rt pt 1 A xt =L xt + rt pt 1 A ( t xt )w L xtc; i.e.

;w

In the steady state (pt 1 = pt ; xt = (1 + g) xt 1 ; ct = ct 1 ; dt = dt 1 ; rt 1 = rt , andt ), we haverp A x =w L xt
c

t 1 =

(12)

:

w

(12) implies that (10) is an indispensable condition for the existence of PE in the case of(ILA, ILA), as Pasinetti (1962, 1966a, 1974) asserts.
3.1.4

The long-run competitive equilibrium in (ILA, ILA)

Based on (6), (9), (11), and (12), the long-run competitive equilibrium in this case is obtainedby the following system for the given rate of pro…t r 2 [0; R):8
p 5 (1 + r) p A + L ;
>
>
>
>
>
>
>
>
p x = (1 + r)p A x + L x;
>
>
>
>
>
>
>
>
x = (1 + g) A x + c + d;
<
(13)
>
>
p
x
=
(1
+
g)
p
A
x
+
p
c
+
p
d;
>
>
>
>
>
>
>
>
p x>0
>
>
>
>
>
>
:
x 0;
where

8
>
>
<

h
c1 = 1 p1 c max rp A x>>: d1 = 1w

p1

h

L x + min

w
c

w
c

w

L x; 0wi

L x; rp A x

;
i

(14)
:

From the fourth equation of (13) and (14), the following relation holds when PE is obtained:13

p x = (1 + g) p A x + (1c)

w

max rp A x
c

+ (1

w)

w

L x + min

L x; rp A xc

L x; 0w:

w

For the rate of pro…t such that rp A x > c w w L x, the capitalist and worker co-exist, since
the capitalist’s income is kept positive.Therefore, we obtainp x = (1 + g) p A x + (1c)

w

rp A xL x + (1c

= f1 + g + (1

w)

w

L x+

w

c

Lx
w

c ) rg p A x + L x.

By comparing it with the second equation of (13), we obtain
g=

c r:

Similarly, for the rate of pro…t such that rp A x 5 c w w L x, as pointed out in Section
2, the capitalist disappears, since her income is zero. Thus, all incomes are distributed to
the worker. Therefore, we obtainp x = (1 + g) p A x + (1= 1 + g + (1w ) [L x + rp A x]
w

w) r

p Ax

Lx

p A x + L x.

Therefore, we obtain
g=

r+

w

Lx
p Ax

=

w

p (I A ) x:
p Ax

By the same method as that of Hosoda (1989), we prove the existence of the long-run
competitive activity vector 0 x 2 K. Therefore, we obtain the rate of economic growth,satisfying (13) and (14), as a function of the rate of pro…t, as follows:g (r) =

8
>
<
>
:

cr

if rp A x >w
cw(the PE),(15)w

r + p A1 x=

p (I A )x
w p A xif rp A x 5w
cw(the DE).

Note that the activity level is normalised as L x = 1.
(15) is the exactly same result as that of Pasinetti (1962, 1974), Samuelson and Modigliani(1966a, 1966b), Hosoda (1989), and Morishima (1969). Then, it shows that the PE is characterised by the Cambridge equation, regardless of the long-run competitive equilibrium technique (i.e. irrespective of the elements of A and L ; note that the elements vary, dependingon the level of the rate of pro…t). The rate of economic growth is determined by the worker’s
14

saving rate multiplied by the ratio of net output to capital in DE, which implies that therelationship between the rates of economic growth and pro…t depends on the technology andthe worker’s saving rate.
Furthermore, just like in Hosoda (1989), we have

cr >

w

r+

1
pAx

in the PE, and

cr 5

w

r+

1
pAx

in the DE.

The result implies that we obtain an equilibrium with a higher rate of economic growth for
8r 2 [0; R).

3.2

Optimal behaviour in the case of (capitalist, worker) = (ILA, OLG)

In this subsection, we examine the case in which the combination of the microfoundation of
the capitalist and worker is represented by (capitalist, worker) = (ILA, OLG), and analyse
the properties of the equilibria obtained in the case.
3.2.1

Optimal behaviour of the capitalist in (ILA, OLG)

The capitalist’s behaviour is essentially the same as in the previous case. Following thenotation used in the previous subsection, the problem addressed by the capitalist is given as
follows:

max U

ct;1 ;xt+1

c

(1

c)

1
X

t 1
ln ct;1ct=1s.t. pt Eq t+1 + pt ct 5 rt pt 1 A ((1t) qt) ;given frt gt=1;2; ; f t gt=1;2; ; p00.

The solution is obtained as follows:pt ct = (1ct;1 =(1c ) rt pt 1 Ac ) (1((1t ) xt ) , i.e.

t ) rt pt 1 A xt

pt;1

(16)

:

As in Subsection 3.1.1, (16) implies that the capitalist consumes commodity 1 proportionally to her income rt pt 1 A ((1t ) xt ), and thus, the capitalist’s saving rate is c .
15

3.2.2

Optimal behaviour of the worker in (ILA, OLG)

Since the worker’s behaviour is characterised by the OLG model, he lives for two periods.
The worker, who is born at time t, supplies the labour force and obtains wage Lq t (notethat the wage rate is normalised as wt = 1) at time t. This is his young age. At this age, hemust solve the problem of allocating the wage between consumption and saving, and gains
his livelihood at time t + 1 by this saving. This time represents his old age. Therefore, thebudget constraint for the young age of the worker born at time t is written as pt dbt +St 5 Lq t ,dbt;1 ; 0;
; 0 2 Rn+ and St denote the consumption vector for the young age
where dbtof the worker born at time t and the saving, respectively. Moreover, the budget constraint
for the old age of the worker born at time t is given by pt+1 dat 5 (1 + rt+1 )St , where; 0 2 Rn+ denotes the consumption vector for the old age of the worker
dat;1 ; 0;datborn at time t. By combining these budget constraints, we obtain the worker’s intertemporalbudget constraint.
Therefore, the worker born at time t solves the following problem:max (1

dbt;1 ;dat;1bw ) ln dt;1 +aw ln dt;1

1p da 5 Lq t ;1 + rt+1 t+1 t
given frt gt=1;2; , fq t gt=1;2 , p0 0.
s.t. pt dbt +The solution is given as follows:

pt dbt = (1pt+1 dat =w ) L xt ;i.e. dbt;1 =w (1 + rt+1 ) L xt ;(1

i.e. dat;1 =w ) L xt

pt;1

(17)

;

w (1 + rt+1 ) L xt

pt+1;1:(18)(17) implies that the worker in young age consumes consumption 1 proportionally to his
income L xt . Therefore, the worker’s saving rate is w . (18) means that the worker in old
age maintains his livelihood by dissaving (i.e. using his saving).
3.2.3

Distribution of pro…ts in (ILA, OLG)

The property of the OLG model means that the pro…ts distributed to the worker at time
t must equal the income that the worker, born at time t 1, receives in old age (i.e.
w (1 + rt ) L xt 1 = t rt pt 1 A xt ). Following C3, we obtain the distribution of pro…ts
between the capitalist and worker as follows:

16

(1
c (1

t ) rt pt 1 A xt

t r t pt 1 A x t=t ) rt pt 1 A xt

w L xtw

t r t pt 1 A x t =, i.e.

L xt :c

As argued in detail in Subsection 3.4, (18) implies that we need not assume w < c to ensure
the existence of PE in the case of (ILA, OLG), in contrast to (12). We obtain (19) in thesteady staterp A x =

w

(19)

L x.

c

3.3

The long-run competitive equilibrium in (ILA, OLG)

The property of the OLG model means that workers in young age and old age co-exist withina period of time. Therefore, based on (16)–(19), the steady-state consumption vectors of thecapitalist and workers are
8
i
h
1 c
w
>
c1 = p max rp A x
L x; 0
>
<
c
1
(20)
h
i
>
>
1
: d1 = da + db =min w L x; rp A x + (1w) L x11p1crespectively. Then, the long-run competitive equilibrium can be de…ned by (13) for the given
rate of pro…t r 2 [0; R). Similarly, in Subsection 3.1.4, we obtain the following from thefourth equation of (13) and (20):p x = (1 + g) p A x + (1c ) maxw

rp A x

L x; 0

c

+ min

w

L xt ; rp A x + (1w ) L x.

c

For the rate of pro…t such that rp A x >p x = (1 + g) p A x + (1

c)

w
c

L x, it can be transformed as follows:w

rp A xL xt +c= f1 + g + (1c ) rg p A x + L x.

By comparing it with the second equation of (13), we obtaing=
For the rate of pro…t such that rp A x 5
17

c r.
w
c

L x, we obtainwc

L xt + (1w) L x

p x = (1 + g) p A x + rp A x + (1w) L x
wL x
p A x + L x.
= 1+g+r
p Ax
By comparing it with the second equation of (13), we obtain
g=

w

Lx
.
p Ax

Using the same method as that of Hosoda (1989), we can prove the existence of the longrun competitive activity vector x
0, together with p
0, in set K. Therefore, we obtain
the rate of economic growth, satisfying (13) and (20), as a function with respect to the rateof pro…t, as follows:8
if rp A x > wc (the PE),cr<g (r) =
(21):w
w
if rp A x 5 c (the DE).
p A x

Note that the activity level is normalised as L x = 1.
(21) demonstrates that the PE is substantially characterised by the Cambridge equation,
whichever technique is cost minimising, and the rate of economic growth in the DE is determined by the worker’s saving rate and the ratio of labour to capital in the case of (ILA, OLG).
As is the case of (ILA, ILA), the relationship between the rates of economic growth and pro…tdepends on the technology and the worker’s saving rate.
Moreover, when the PE holds in this case (rp A x > wc ), we have
cr >w

Similarly, when the DE holds (rp A x 5

w

cr 5

c

1
.
p Ax

), we have:
w

p Ax

.

The result implies that we obtain an equilibrium with a higher rate of economic growth for
8r 2 [0; R), as in Subsection 3.1.4.

3.4

Short remarks of Section 3

As shown by (15) and (21), the relationship between the rates of economic growth and pro…tin PE, in which the capitalist and worker co-exist in the steady state, is given by g = c rirrespective of the microfoundation of the capitalist and worker. Thus, the relationship is
independent of the long-run competitive equilibrium technique and has substantially thesame implication as the Cambridge equation. In other words, the relationship is independentof the technology and the worker’s saving rate, and is determined solely by the capitalist’s
saving rate. Since the choice of the assumptions on individuals’behaviour can be regarded
as the institution in the sense of Pasinetti (2007), our results con…rm Pasinetti’s (2012, p.
18

1440) statement that ‘the Cambridge equation is independent of the institutional set-up of
the society that is considered’.
On the contrary, the relationship between the rate of economic growth and that of pro…tin DE, in which the capitalist disappears and all the stock of capital is owned by the worker,
depends entirely on the assumptions of the microfoundation. Whereas (15) demonstrates
that the rate of economic growth is determined by the worker’s saving rate multiplied by theratio of net output to capital in the case of (ILA, ILA), (21) means that it is determined by
the worker’s saving rate multiplied by the ratio of labour to capital in the case of (ILA, OLG).
In both cases, the relationship is dependent on the technology, which is in contrast to the PE.
Based on Pasinetti (1966a, 1974), the dependence of the relationship between the rates of
economic growth and pro…t on the technology in the DE provides Samuelson and Modigliani(1966a, 1966b) with the opportunity to resurrect the neo-classical well-behaved production
function.
The di¤erence in the relationship in the DE is solely attributed to that of the worker’s
behaviour, which di¤erentiates the worker’s saving functions in the case of (ILA, ILA) and(ILA, OLG). Pasinetti (1962, p. 270) criticises the Kaldorian saving function (St = sc Pt +sw Wt , where sc > sw ), as ‘a logical slip’and argues that the Cambridge equation is obtainedin Kaldor (1956) only under the restrictive assumption of sw = 0. If the worker saves a partof his income, he must be allowed to own it and thus, receives the income from not only
wage but also capital. Then, Pasinetti (1962) proposes a new saving function alternative tothe Kaldorian, as shown in Section 2: St = sc Ptc + sw (Wt + Pcw ), where sc > sw must be
assumed. As implied by (9) and (11), Pasinetti’s total saving function corresponds to thecase of (ILA, ILA). As implied by (11), the worker in this case saves from not only his wage
but also the pro…t distributed to him.
It is obvious that if a worker saves, then he receives the pro…t. However, it does notnecessarily mean that the pro…t is saved. This is demonstrated by the case of (ILA, OLG).
Although the worker certainly receives not only wage but also pro…t in this case, he saves
only the wage part and consumes the pro…t entirely. Following the notation used in Section
1, the total saving function in this case can be written as St = sc Ptc + sw Wt . As suggestedby (19), we should pay attention to the result that (10) – whether the capitalist’s savingrate is greater than the worker’s – is dispensable for the existence of the PE in the case of(ILA, OLG). We …nd that the PE exists even when the worker’s saving rate is higher thanthe capitalist’s in this case, as indicated by the numerical example in Section 4. Caggetti and
De Nardi (2008) and De Nardi (2015) imply that people whose behaviour follows the OLG
model cannot be thought of as an unrealistic or extreme case, as mentioned in Section 1,
since the introduction of their behaviour into the models leads to a better …t with the data.
Concerning the Kaldorian saving function, Samuelson and Modigliani (1966a) suggest
reinterpreting the case in which the worker saves a part of earned pro…t as if he were acapitalist, and thus, they argue that there is no ‘logical slip’. Pasinetti (1983) criticises
Samuelson and Modigliani’s (1966a) re-interpretation for being ‘incompatible with steady
growth’. Here, ‘steady growth’means PE (i.e. the growth path on which both the capitalist
and worker co-exist). The case of (ILA, OLG) is the combination of the microfoundation
under which there is no ‘logical slip’and the compatibility with the ‘steady growth’is ensured,
irrespective of whether the capitalist’saving rate is greater than the worker’s, if C3 holds.

19

4Type of Equilibria and Choice of TechniqueIn the previous sections, we concentrate on the relationship between the rates of economic
growth and pro…t obtained in PE and DE. As pointed out in Section 1, the advantage ofour model is that the switch of the type of equilibria can be examined simultaneously withthe choice of technique (the occurrence of paradoxes in capital theory). We analyse them byusing the numerical examples, and then derive the propositions from our model.
First, we con…rm the concepts used in this section. Let b 2 Rn+ be a bundle of commodities
adopted as the numéraire. For the rate of pro…t such that r 2 [0; R ), where R denotes the
maximum rate of pro…t attainable by technique 2 , the wage curve of technique measured
by b can be de…ned as follows:
1.
L [I (1 + r) A ] 1 b
w (r) is a function of r that corresponds to (A ; L ) for 2 . Furthermore, we de…ne thefactor price frontier ! (r) as follows:
w (r) =! (r)max w (r) :2

In other words, the factor price frontier is the outermost envelope of the wage curves, which
implies that the cost-minimising technique for the given rate of pro…t is that by which thehighest wage rate can be o¤ered (Kurz and Salvadori, 1995, p. 142).

4.1

Numerical Example

We consider the following numerical example:
Example 1: We suppose a two-class economy with two commodities. Commodity 1 has only
2=5
one process: [a1 ; L1 ] =; 1 and commodity 2 has two alternative processes and :2
1=40
0:0001[a2 ; L2 ] =; 1 and [a2 ; L2 ] =; 275 .8 The technology available to1=10
113=232 464
the whole economy are given as follows:A=

2=5 1=40 0:0001;L = 1 1
2 1=10 113=232

275
464

.

Therefore, we have
[A ; L ] =2=5 1=40; (1; 1) ;2 1=10[A ; L ] =Here, the set

275
2=5 0:0001; 1;
2 113=232
464.

= f ; g is de…ned. We consider the wage curves of technique for 2
1
obtained by adopting a bundle of commodities b =as the numéraire. Figure 1 depicts0
the wage curves and thus, the factor price frontier as their outermost envelope.
8

The numerical example is from Vienneau (2005).

20

Insert Figure 1 here.
The intersection of the wage curves, called the switch points, are given by r t 0:197;
0:807. As the long-run competitive equilibrium technique (i.e. cost-minimising technique for
given rate of pro…t), is chosen at r 2 [0; 0:197), is chosen at r 2 (0:197; 0:807), and ischosen again at r 2 (0:807; 1:04), where R = 1:04.9 Figure 1 shows that both reswitchingof technique and reverse capital deepening occur.10 Since technique , which is chosen inr 2 [0; 0:197), is chosen again in r 2 (0:807; 1:04), the reswitching of technique occurs. Since
the switch of technique from to at r t 0:8007 implies that technique with higher capitalintensity is chosen despite the rise of the rate of pro…t, reverse capital deepening occurs. In
the following part, we assume c = 0:7 and w = 0:3, unless otherwise stated.
4.1.1

Relationship between the rate of economic growth and that of pro…t in
the case of (ILA, ILA)

Under the above assumptions, we obtain the relationship between the rates of economic
growth and pro…t, given by (15). Figure 2 shows the relationship under the technology given
in Example 1.
Insert Figure 2 here.
In Figure 2, g and g denote the rate of economic growth as a function of the rateof pro…t in the DE when and , respectively, are activated as the long-run competitive
equilibrium technique. The rate of economic growth as a function of the rate of pro…t in the
PE is indicated by g = 0:7r, whichever is the long-run competitive equilibrium technique.
As con…rmed in Figure 1, techniqueis the long-run competitive equilibrium technique for r 2 [0; 0:197) and thus, is chosen in the interval of the rate of pro…t. Note thatg (r), where A = A , holds for r 2 [0; 0:197). Since we have
0:7r < 0:3 r + p A1 x
a higher rate of economic growth in the steady state, as argued in Section 3.1.4, the DE is
obtained under techniquein the interval of the rate of pro…t. For r 2 (0:197; 0:807),
as shown in Figure 1, the long-run competitive equilibrium technique is . Because ofg (r) 0:3 r + p A1 x , where A = A , the intersection between g = 0:7r and g = g (r)is given by r t 0:325. Then, 0:7r 5 g (r) holds for r 2 (0:197; 0:325]. This result implies
9Note that each commodity has the same price at the switch points irrespective of whether it is produced
by any technique or any convex combination of techniques. See Kurz and Salvadori (1995, p. 142) andPasinetti (1977, pp. 158–159).
10
These phenomena are inconsistent with the principle of marginal productivity of capital. According
to the principle, there is a one-to-one correspondence between the rate of pro…t and the cost-minimisingtechnique. The reswitching of technique means that one technique, which is chosen as the cost-minimisingtechnique at the rate of pro…t, is also chosen at other rate of pro…t. Moreover, according to the principle,
the capital intensity (value of capital per worker) is a monotonically decreasing function of the rate of pro…t.
Reverse capital deepening (or capital reversing) means that the increase in the rate of pro…t raises the capitalintensity. See, for example, Kurz and Salvadori (1995, pp. 147–149) and Pasinetti (1977, pp. 169–173) for
details.

21

that the DE is obtained under technique for r 2 (0:197; 0:325]. For r 2 (0:325; 0:807),
on the contrary, 0:7r > g (r) holds, implying that the PE is obtained under techniquein the interval of the rate of pro…t. Furthermore, for r 2 (0:807; 1:04), the long-run competitive equilibrium technique is again. Recall that reswitching of technique and reverse
capital deepening take place in the switching technique fromtoat r
0:807. For
r 2 (0:807; 1:04), 0:7r > g (r) holds. This result means that the PE is obtained for in theinterval of the rate of pro…t. Note that the relationship between the rate of economic growth
and that of pro…t in the PE, represented by g = 0:7r, is unchanged even though the long-run
competitive equilibrium technique changes from to .
Since the cost of production is always minimised at the switch points regardless of which
technique or convex combination of existing techniques is used, as mentioned in footnote 9,
the long-run competitive equilibrium is consistent with the multiple rates of economic growthat the switch points. However, it would be reasonable to suppose that the highest rate of
economic growth is preferred. Then, the DE is obtained under technique at r 0:197 and
the PE at r 0:807.
g (r)for 2 . From the de…nition of g = g (r), (r) denotes the ratio of
Let (r)w
net output to capital when 2is the long-run competitive equilibrium technique at r 2[0; R ). Under the technology given in Example 1, 0:268 g (0:807) < g (0:807) 0:294.11
This result implies that
(0:807) <(0:807). In other words, the ratio of net output tocapital rises when the long-run competitive equilibrium technique switches from to at
the switch point r 0:807. The movement of (r) around r 0:807 is compatible with theprinciple of marginal productivity of capital. However, the movement of the capital intensity
when the long-run competitive equilibrium technique switches from to at r
0:807 isinconsistent with the principle, since reverse capital deepening occurs. Figure 2 demonstrates
that among two indexes characterising the technique in the neo-classical production function,
one movement (ratio of net output to capital) is compatible with the principle but the other(capital intensity) is not. The result cannot be obtained by the model in which capitalis treated as the primary factor of production. When capital is composed of a bundle ofreproducible and heterogeneous commodities, the phenomena incompatible with the principle
are not necessarily considered as paradoxical.
4.1.2

Relationship between the rates of economic growth and pro…t in the caseof (ILA, OLG)

Next, we examine the case of (ILA, OLG) under the technology given in Example 1. Figure
3 shows the relationship between the rates of economic growth and pro…t, given by (21):Insert Figure 3 here.
In Figure 3, similar to Subsection 4.1.1, g and g denote the the rate of economic growth
as a function of the rate of pro…t in the DE whenand , respectively, are utilised as
the long-run competitive equilibrium technique. Note that g (r) is the worker’s saving rate
11

Note that g = g (r) and g = g (r) in this case have the intersection at r

22

0:796.

multiplied by the ratio of labour to capital (reciprocal of capital intensity) in this case. The
rate of economic growth as a function of the rate of pro…t in the PE is given by g = 0:7r,
which is the same function as that obtained in Subsection 4.1.1. This veri…es the result
obtained in (15) and (21).
The intersection between g = g (r) and g = 0:7r is given by r 0:191. Therefore, the DE
is obtained for r 2 [0; 0:191] and the PE for r 2 (0:191; 0:197) under technique . The DE is
obtained for r 2 [0:197; 0:223] and the PE is obtained for r 2 (0:223; 0:807) under technique, where r 0:223 is the intersection between g = g (r) and g = 0:7r. For r 2 (0:807; 1:04),
moreover, the PE is obtained under technique .
4.1.3

The case ofw >c in (ILA, OLG)

As already pointed out in Subsection 3.4, (10) is an indispensable assumption to ensure theexistence of the PE in the case of (ILA, ILA) but not in the case of (ILA, OLG). In thissubsection, we examine the case of w > c in (ILA, OLG) by assuming the technology given
in Example 1 and w = 0:6; c = 0:35. This case is unrealistic but is a logical exercise to
test the working of the Pasinettian two-class model with a microfoundation of capitalist andworker.
Figure 4 shows the relationship between the rates of economic growth and pro…t in thiscase:
Insert Figure 4 here.
The …gure indicates that the DE is obtained under technique for r 2 [0; 0:197), and theDE is obtained under technique for r 2 [0:197; 0:735], where r 0:735 is the intersection
between g = g (r) and g = 0:35r. Furthermore, the PE is certainly obtained under techniquefor r 2 (0:735; 0:807) and under technique for r 2 [0:807; 1:04).

4.2

Short remarks of Section 4

In Section 4, we closely examine the results obtained in Section 3 by using the numericalexamples. Although these examples may be speci…c, they provide useful insight of the analysisof the distribution of income and capital (wealth).
From Figures 2, 3, and 4, we observe that even when techniques switch from one toanother, the relationship between the rates of economic growth and pro…t obtained in thePE, which is given by g = c r, is independent of the technology and the microfoundationof the capitalist’s and worker’s behaviour. In addition, we observe from the …gures thatthe relationship obtained in the DE depends on the technology and the microfoundation of
agents for the given rate of pro…t.
Although the movement of the capital intensity around the switch point exhibits paradoxical behaviour in Figure 2, that of the ratio of net output to capital is compatible with theprinciple of marginal productivity of capital. This interesting result is obtained only in themodel in which capital is treated as a bundle of reproducible and heterogeneous commodities.

23

Moreover, the phenomena inconsistent with the principle of marginal productivity of capitalare not necessarily considered as a paradox.
The fundamental cause of such paradoxical phenomena arising is our assumption that
capital consists of a bundle of reproducible and heterogeneous commodities and not that ourmodel lacks a continuous and di¤erentiable production function. Obviously, our assumption
on capital is realistic. Burmeister (1981) shows that even when neo-classical well-behaved
production functions are assumed, the possibility of paradoxical phenomena arising cannot
be excluded if capital consists of reproducible and heterogeneous commodities. To exclude
this possibility, a peculiar assumption which is unnecessary in the one-commodity modelneeds to be made.12
The rate of economic growth obtained in the PE is necessarily an increasing function ofthat of pro…t, irrespective of the microfoundation of agents and the technology. Although
the relationships characterising the DE in the cases of both (ILA, ILA) and (ILA, OLG) arealso an increasing function of the rate of pro…t in Example 1, we can easily …nd numericalexamples in which the rate of economic growth is a decreasing function of that of pro…t in
the DE, depending on the assumption of the microfoundation of the agents.13 Complicatedrelationships between the rate of economic growth and income distribution can be obtained
only by employing multi-sectoral models with capital as a bundle of reproducible and heterogeneous commodities. In fact, we know little about the movement of g (r) for 2
unless the neo-classical production function is assumed, since it depends on the property ofthe technology and there are complicated e¤ects on the aggregation of income and capitalwhen capital consists of a bundle of reproducible and heterogeneous commodities.

5

Concluding Remarks

In this study, we examine the distribution of income and capital (wealth) by a multi-sectoral
two-class model with a microfoundation of capitalists and workers. The characteristic of
our model is to treat capital as a bundle of reproducible and heterogeneous commodities.
Clearly, capital in that sense, rather than capital as the primary factor of production, is atypical factor of production in modern capitalist economies. Our model can be considered
as a summary of the Cambridge capital controversy in that it enables us to comprehensively
analyse the major issues of the controversy (i.e. controversy about the Cambridge equationand the choice of technique) by using a single model. The aggregate macroeconomic modelis used to address the former issue and the multi-sectoral model à la the Leontief–Sra¤a isused to address the latter.
12
According to Burmeister (1981), this peculiar assumption is that the so-called real Wicksell e¤ect isnegative for all feasible r > 0:nX
i=1

pi (r)dki (r)< 0;
drwhere pi (r) and ki (r) denote the relative price of commodity i and the per capita amount of the commodity
necessary as the capital input, respectively. As our analysis shows, it is not necessary to assume that theinequality is likely to be satis…ed.
13
See Appendix A for a numerical example in which the rate of economic growth is a decreasing functionof that of pro…t.

24

The movements of the ratio of net output to capital, the capital intensity, and the rateof pro…t (or wage rate) are important variables for analysis of the distribution of income andcapital (wealth). According to the principle of marginal productivity of capital, the ratio of
net output to capital declines when the wage rate rises. Stiglitz (2016) considers the recent
rise in the US ratio of wealth to income without a substantial increase in the wage rate to bea paradox, and argues that the movement of the ratio cannot be explained by the standard
neo-classical growth theory with the well-behaved production function.
We show that the movements of the ratio of net output to capital and the capital intensity
when the rate of pro…t changes could be more complicated than those predicted by the models
assuming the neo-classical well-behaved production function. This is because not only thequantities but also prices of capital change when the rate of pro…t changes if capital is treatedas a bundle of reproducible and heterogeneous commodities. The changes in both quantities
and prices generate the complicated e¤ects on the aggregation of income and capital. If
capital is treated as the primary and homogeneous factor of production, as in the neoclassical production function, then the change in the rate of pro…t generates only a change inthe quantities. According to the lesson of the Cambridge capital controversy, the phenomena
indicated by Stiglitz are not necessarily a paradox, as we show in Sections 3 and 4. The
growth models assuming the neo-classical well-behaved production function are not usefulfor predicting what happens to the steady states of the two-class economies on which C3 is
imposed. This result implies that it is necessary to analyse the distribution of income andcapital (wealth) by the models with capital as a bundle of reproducible and heterogeneouscommodities.
Of course, our study leaves some open questions. We focus only on the properties ofsteady states. Signi…cant remaining questions are whether an economy starting from theinitial value converges toward the steady state and, if it does, what equilibria it is likely
to converge toward. Examination of these questions in the two-class model with capital
as a bundle of reproducible and heterogeneous commodities remains as our future research
agenda.

6

Appendix A

Here, we show the numerical example in which the rate of economic growth is a decreasing
function of that of pro…t:
Example 2:14 We assume a two-class economy with two commodities. Commodity 1 has
00
two alternative processes and : [a 1 ; L 1 ] =; 1 , [a 1 ; L 1 ] =; 2 . In ad0:5
0:125
0:5
dition, commodity 2 has only one process: [a2 ; L2 ] =; 1 . Therefore, the technology
0available in the economy as a whole is given as follows:
A=
14

0
0
0:5
;L = 1 2 1 .
0:5 0:125 0

The example is from Hosoda (1989).

25

Then, we have
[A ; L ] =0 0:5; (1; 1) ;0:5 0

[A ; L ] =0
0:5; (2; 1) .
0:125 0Here, the set= f ; g can be de…ned. Figure 5 depicts the wage curves of technique2
and the factor price frontier. It demonstrates that neither reswitching of technique
nor reverse capital deepening occur: the technique switches from to at r
0:143, andtechnique , which has lower capital intensity than technique , is chosen for r 2 [0:143; 3].
Note that the wage curve of technique is linear, since the so-called organic composition ofcapital is uniform in both sectors.
Insert Figure 5 here.
We assume that c = 0:7 and w = 0:3 as before. From (15) and (21), we obtain therate of economic growth as the function of the rate of pro…t in the case of (ILA, ILA) and(ILA, OLG). Figures 6 and 7 show the relationship between the rate of economic growth
and that of pro…t in the former case and that in the latter case, respectively. Note that theprice is normalised by w = 1 and the activity level by L x = 1 for 8r 2 [0; R).15 Figure 6(ILA, ILA) shows that the DE obtained under technique is indicated by g for r 2 [0; 0:143);
the DE obtained under technique is indicated by g for r 2 [0:143; 0:917], where r 0:917
is the intersection between g = 0:7r and g = g (r); and the PE is obtained under techniquefor r 2 (0:917; 3), where r = 3 is the maximum rate of pro…t obtainable under technique. Figure 7 (ILA, OLG) shows that the DE is obtained under technique for r 2 [0; 0:143);
and the DE is obtained for r 2 [0:143; 0:994], where r
0:994 is the intersection between
g = 0:7r and g = g (r); and the PE is obtained under technique for r 2 (0:994; 3).
Insert Figures 6 & 7 here.
Note that g in Figure 6 is constant independently of the level of r, since the ratio of net
output to capital is constant irrespective of the level of r if the so-called organic compositionof capital is uniform in all sectors (Pasinetti, 1977, p. 86). g in Figure 6 is not a monotonic
function; the rate of economic growth is minimised at r 1:067. Therefore, we obtain therate of economic growth as a decreasing function of the rate of pro…t for r 2 [0:143; 0:917].
In Figure 7, both g and g are decreasing functions of the rate of pro…t.
15

Hosoda (1989) also draws the …gure in the case of (ILA, ILA) based on the same numerical example.
Although function (II) of Figure 6 in his study corresponds to g in our Figure 6, the form of his function (II)is di¤erent from that of our g . We assume that this is because he does not normalise the prices by w = 1,
unlike in our study.

26

7

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29

Figure 1: Factor price frontier of Example 1

Figure 2: The rate of economic growth in (ILA, ILA) of Example 1
Note: Here,

∗

=

∗
∗ ∗ ∗

∗

for =

,

.

Figure 3: The rate of economic growth in (ILA, OLG) of Example 1
Note: Here,

=

∗ ∗ ∗

for =

,

.

Figure 4: The case of
Note: Here,>(= 0.35,=∗ ∗ ∗= 0.6) in Example 1

for =

,

.

Figure 5: Factor price frontier of Example 2

Figure 6: The rate of economic growth in (ILA, ILA) of Example 2
Note: Here,

∗

=

∗
∗ ∗ ∗

∗

for = ", # .

Figure 7: The rate of economic growth in (ILA, OLG) of Example 2
Note: Here,

=

∗ ∗ ∗

for = ", # .