Paradoxical dilemma of economic growth

Masahiko ITAKI1(itaki@ir.ritsumei.ac.jp)

Ritsumeikan University, Kyoto, Japan

Abstract
We shall examine the paradoxical dilemma of growth, building upon the theoretical construction of the dual
equation between income distribution and resource allocation: whilst higher GDP per capita is generally
welcome, higher growth rates incur a reduction of the real wage rate and the consumption rate, and exacerbate
income discrepancy in the long run.
The dual equation encapsulates the intrinsic input-output relations and transforms them into a simple“nutshell economy.” Within this analytical framework, we examine the symmetry breaking between the profit
rate and the growth rate, and between the wage rate and the consumption rate, resulting in the theorem of
discrepancy between income distribution and resource allocation and the theorem of parallel traverse of thesefour variables with a widening symmetry breaking as the profit rate and the growth rate increase in tandem.

Key words: economic growth, dual equation, von Neumann, symmetry breaking, parallel traverse
JEL Classification
B51

Socialist • Marxian • Sraffian

C67

Input–Output Models

O41One, Two, and Multisector Growth Models

1

Masahiko ITAKI: 781-17-206 Mukaino-cho, Kawachinagano, Osaka, Japan 586-0013
1

1 Introduction
We shall examine the paradoxical dilemma of growth, building upon the theoretical construction of thedual equation between income distribution and resource allocation: whilst higher GDP per capita is generally
welcome, higher growth rates incur a reduction of the real wage rate and the consumption rate, and exacerbate
income discrepancy in the long run.
For this purpose, this article begins with an analysis of necessary relations and conditions to achievebalanced reproduction of a growing capitalist economy in the long term. It examines an economy as a dual
system of price and quantity: the former concerns income distribution between the profit rate r and the real
wage rate w and the latter concerns resource allocation between the growth rate g and the real consumption
rate c. We shall construct the dual equation between the price system and the quantity system.
The price and quantity systems have long been acknowledged to be in duality since von Neumann [1937](1945-46), just like in a mirror image, in which replacement of r by g and w by c produces exactly the samemathematical relationship. The dual equation integrates them with the help of the system converter, which
serves as a kind of “vehicle” which analytically transfers us from one system to the other. These two systems,
however, are not identical but contain symmetry breaking, i.e., 𝑟 ≠ 𝑔 and 𝑤 ≠ 𝑐. We are going to examinesome ramifications from the symmetry breaking in the analytical framework of what we refer to as the"nutshell economy" which is encapsulated in the dual equation. It reduces the intricate multi-sector inputoutput system into a concise primitive “food-gathering economy” without being supported by means of
production. The "nutshell economy" reminds us of Ricardo’s “old corn-ratio model” and can be regarded as
its modern resurrection (Sraffa’s introduction to Ricardo 1951). We shall examine similarities and differences
between our model and Pasinetti’s “vertically hyper-integrated” model (Pasinetti, 1980, 1981, 1988) during
the course of the discussion.
The examination hereafter leads us to two key findings: firstly, the theorem of discrepancy between
income distribution and resource allocation; secondly, the theorem of parallel traverse of those four variables
with widening symmetry breaking as r and g increase in tandem, and vice versa, in the long term.
The structure of the paper is as follows: first, a brief literature review on duality and a choice of minimum
subsistence basket of consumption as the numéraire; second, the dual equation; third, the “nutshell economy”
in the context of the classical political economy; and fourth, symmetry breaking, the theorem of discrepancy,
and the theorem of parallel traverse in the long term.
2 A brief literature review on duality: its origins and subsequent developments
The initial unveiling of the duality between the price system and the quantity system can be attributed to
von Neumann [1937] (1945-46)2, who referred to it as “the remarkable duality (symmetry) of the monetary2

Georg von Charasoff (1910) may be regarded as a forerunner of von Neumann. See Kurz and Salvadori
2variables (prices yj, interest factor β) and the technical variables (intensities of production xi, coefficient ofexpansion of the economy α )” (p.1). The theoretical significance of von Neumann's work has been
discussed in Bruckman and Weber (1971), Los and Los (1974), and Dore, Chakravarty, and Goodwin (1989).
Morishima (1969) particularly praised its treatment of joint production and fixed capital, deeming it the "vonNeumann Revolution" in the history of economic theories. Kurz and Salvadori (2000) traces the origins of
the idea of duality back to the long prehistory since the time of physiocrats and classical economists. Kurz
and Salvadori (2001) further explores the relationship between Sraffa's theory and von Neumann's, using
Sraffa's unpublished papers and letters.
Subsequent scrutiny and development of the concept of duality have been undertaken by Bruno (1969),
Spaventa (1970), Weizsäcker (1971), Craven (1979), and Schefold (1980). Mainwaring (1984) and Fujimoto(1977) employ a utility index instead of a consumption basket to demonstrate the duality. Morishima (1964,
pp.138-139) (1969, pp.97-101) (1976, pp.255-260) make attempts to incorporate capitalists' and workers'
consumption functions into the von Neumann model.

Kemeny, Morgenstern, and Thompson (1956)

modifies von Neumann's strict assumptions and examined sub-economies with different rates of profit andgrowth. Burmeister and Kuga (1970) analyzes the case of joint production, while Morishima (1971) extends
their ideas and explores the general case of decomposability. Morishima (1976) suggests that the idea of
duality “was not entirely alien to Marx” (p.239)3. Pasinetti (1977, ch.7) examines von Neumann’s duality inclose comparison with the Sraffian standard system and Pasinetti (1980) (1981) (1988) extend it andconstructs the “vertically hyper-integrated model.”Building upon the extensive research conducted on the duality between the price and quantity systems,
this paper combines the two systems using the system converter and leads to the reduction of the complex
multi-sector input-output system into the dual equation, which encapsulates the simple "nutshell economy."
Prior to examining the dual equation, it is necessary to address the issue of choosing a numéraire.
3 Choice of numéraire and some literature relating to the minimum subsistence basket
The selection of a set of consumption goods and services as the numéraire is necessary for the duality tohold. This is because, as we soon discuss, both the real wage rate w in the price system and the realconsumption rate c in the quantity system are measured in terms of the same certain set of consumption goods
and services as long as they are economically meaningful as an indicator of the standard of living. Thissuggests that the duality does not derive either from the Sraffian standard system based on his standard
commodity (Sraffa 1960) or from the Walrasian system of general equilibrium which allows arbitrary choice(1995, pp.387-390).
3 In this aspect of Marx, see also Kurz and Salvadori (1995, pp.383-384).
3

of numéraire (Walras [1874])4.
We adopt a minimum subsistence basket of consumption goods and services as the numéraire,encompassing those that are physiologically and socially necessary for reproducing one unit of labor force
exhausted in a single production period. This specific selection offers several advantages: first, the utilizationof a scalar multiple of the basket allows us to measure the absolute magnitude of workers' real consumption,
thereby enabling an assessment of their real wages that exceed mere subsistence levels; second, we canreasonably assume the relative stability of the numéraire in terms of its content and proportion over a
specified period, as long as social norms and institutions remain unchanged; and third, despite its stability,
we can deal with the possibility of its gradual transformation over time when historically comparing
economies. Built on these advantages cliometricians recently devised the welfare ratio, equivalent to our real
wage rate, which is computed as “average annual earnings divided by the cost of a poverty line consumption
bundle for a family.” (Allen 2001, p.425)5 They conduct wide-raged international comparison of the ratios
over many centuries with a distinct set of the minimum consumption baskets specifically constructed to eachcity in each era in history.
On the other hand, we recognize that our numéraire hinges upon the strong assumptions with respect toconsumption behavior of workers and capitalists: workers are not allowed for consumer choice apart fromconsuming multiples of the minimum basket; and capitalists consume the same but a larger number of theminimum basket6. Although the introduction of certain appropriate converters between different consumption
baskets could easily remove the constraints such as these, we prefer to leave it to future articles.
It is not a surprise to find resonance of our idea of the subsistence minimum basket in the tradition of theclassical political economy. In Ricardo [1817] he acknowledged, contrary to the widely held perception,
variations in the minimum subsistence basket over time and across countries:“The natural price of labour is that price which is necessary to enable the labourers, one with another,to subsist and to perpetuate their race, without either increase or diminution” (Ricardo [1817], p.93)
and“It is not to be understood that the natural price of labour, estimated even in food and necessaries, isabsolutely fixed and constant. It varies at different times in the same country, and very materiallydiffers in different countries. It essentially depends on the habits and customs of the people.” (ibid.,

The lack of argument about the numéraire is a feature of the post-Keynesians, except Keynes himself (1936,ch.4), which is demonstrated by the fact that there is no entry for “numéraire” in the index of Setterfield(2010), King (2012), Harcourt and Kriesler (2013a, b). Michal Kalecki, deemed to be a forerunner of Keynes,
was indifferent to the problem of a measuring unit (Kalecki 1935). Kregel (1971, p.101) shares the sameassessment of Kalecki as ours.
5 See also Bassino and Ma (2005) and Allen et al. (2011).
6 As for criticism against adopting the unit of consumption as the numéraire in reference to Sraffa (1960),
see Kurz and Salvadori (1995, pp.120 and 422-423).
44

pp.96-97)
Adam Smith even referred almost exactly to our method of measuring w in international perspective:“The difference between the money price of labour in China and Europe is still greater than thatbetween the money price of subsistence; because the real recompence of labour is higher in Europe
than in China.” (Smith [1776], p.189)7
While the unit of consumption has been widely adopted as the numéraire by modern scholars such as
Spaventa (1970), Weizsäcker (1971), and Harris (1973), Kurz and Salvadori (1995, pp.113-114) and Zalai(2004, pp.16-18) closely align with our minimum consumption basket. The former introduced the vector b,
representing the unit of consumption as the numéraire, and the scalar c, denoting the number of units ofconsumption per unit of labor. The latter introduced the vector s, referring to "the unit consumption bundle
as numeraire" (p.18), and γ, representing "a scalar variable to measure the level of necessary consumption"(p.16). All these scholars require, in addition, a proper definition of b and s in terms of their qualitative
substance. In his examination of substitutional technical change, Okishio (1987, p.4) introduced bi as an
approximation of our minimum basket, although he did not explicitly specify its substance.
4 The dual equation and the net economic resource
We start analyzing necessary relations and conditions to achieve balanced reproduction of a growing
capitalist economy and to construct the dual equation between the price and quantity systems. The model
presented here aligns with the normative analysis adopted by Pasinetti (1962, p.267) first in the Keynesian
aggregate model and later extended by Pasinetti (1977, p.192), (1981), and (1988) to the multi-sector inputoutput model8.
In our analysis, we postulate two fundamental premises concerning the general economy, including thecapitalist economy. First, building upon the work of von Neumann [1937] (1945-46), we recognize that goods
and services are produced within circular reproduction processes9. This approach traces its roots back to
French and British classical economists, such as Quesnay [1759] and Smith [1776], inherited by Marx [1885]
and further developed by Leontief (1951) and Sraffa (1960). Second, echoing the distinctive emphasis ofMarx [1867, 1885, 1894], we understand an economy as encompassing reproduction processes of social
relationships among classes alongside those of goods and services for these classes. These two intertwined

See Allen et al. (2011) for the historical validity of Smith’s proposition.
See Garbellini and Wirkierman (2014) for the “normative rather than positive character” (p.234) of
Pasinetti’s models. The normative approach in the analytical framework of multi-sector input-output modelseems to have begun with Leontief [1928] who emphasized the need of an analysis on “the general conditions
which must be fulfilled within the framework of a circular flow” (1991, p.193). Pasinetti and Leontief
inherited the approach from Marx [1885] which dealt with the simple reproduction scheme (1978, pp.468471).
9 See Kurz and Salvadori (1995, ch.13).
5
7
8

premises apply to any historical mode of production including a growing capitalist economy.
In line with von Neumann (ibid., p.2-3), we adopt the following modified and/or simplified assumptions:(a) Single-product processes of production: the number of production processes matches the number of
products produced.
(b) Constant returns to scale.
(c) Natural resources are abundantly available and thus, are provided free of charge during production
periods based on the principle of free goods. Additionally, there exists a pool of unemployed workers,
ensuring the availability of necessary wage workers in response to production growth. However, therule of free goods does not apply to labor force10.
(d) Household consumption of wage workers, capitalists, and other non-working classes of society takes
place as part of reproduction processes. Wage workers' consumption is not necessarily fixed at theminimum subsistence level, allowing for the possibility of savings. Capitalists and other non-working
classes do not allocate all their income to investment but also engage in consumption.
(e) Means of production are limited to circulating capital, which is fully utilized within a single production
period.
(f) Production spans a single production period.
(g) Joint production is not permitted.
In addition to the assumptions above, we also assume, as von Neumann did, free competition among firms
for the uniform rates of profit and growth to be established among all production processes. Furthermore,aligning with von Neumann, classical economists, and Marx, we assume that wages are paid ante factum atthe beginning of each production period: wages are advanced to workers, and capitalists seek profits ascompensation for these advanced wages. This stands in sharp contrast to Sraffa's wages paid post factum atthe end of each production period.
4.1 The price system
Assuming no extra profits or costs in production processes 11 , the price system of a single degree of
freedom can be described by the following equations:𝒑 = (1 + 𝑟)(𝒑𝑨 + 𝑤𝒍)
𝒑𝒃 = 1(1)(2)

in which
p: a row vector (1×n) representing prices of products produced,
A: a square matrix (n×n) denoting material inputs per unit of output,

10
11

See Kurz and Salvadori (1995, p.403) and Steedman (1987, p.4930).
See Kurz and Salvadori (1995, pp.403-407) for the use of inequalities instead of equations.
6b: a column vector (n×1) representing a set of consumption goods and services chosen as thenuméraire in physical units of wage rates and consumption rates,
l: a row vector (1×n) representing direct labor inputs per unit of output,
I: an identity matrix (n×n),
r: the uniform profit rate across sectors (scalar), andw: the real wage rate (scalar) paid ante factum for unit of labor, measured in terms of numéraireb.
From equation (1), we can derive:𝒑[𝑰 − (1 + 𝑟)𝑨] = (1 + 𝑟)𝑤𝒍
in which when matrix A is indecomposable and non-negative, the positive inverse matrix 𝟎 <[𝑰 − (1 + 𝑟)𝑨]−𝟏 exists due to the Perron-Frobenius theorems (Pasinetti, 1977, pp.273-274) Thus, we
obtain:𝒑 = (1 + 𝑟)𝑤𝒍[𝑰 − (1 + 𝑟)𝑨]−𝟏
As it is well-known, given r = 0, 𝒍[𝑰 − 𝑨]−1 represents “labor-embodied value” of products (Ricardo[1817] ), which is proportional to relative prices p, and when 0 < r, (1 + 𝑟)𝒍[𝑰 − (1 + 𝑟)𝑨]−1 signifies“labor-commanded value” of products (Smith [1776], Book I Ch. V), i.e., the amount of labor force that canbe exchanged for each product. As recognized by Smith, when r = 0, “labor-embodied value” is identicalwith “labor-commanded value” in their relative terms (Smith [1776], Book I Ch. VI pp.47-48) 12 .
[𝑰 − (1 + 𝑟)𝑨]−𝟏 acts as the converter when 0 < 𝑟, transforming direct labor input l into a vector with thesame composition of elements as that of the relative prices p. We refer to it as the labor converter, and theresulting vector 𝒍[𝑰 − (1 + 𝑟)𝑨]−1 as labor converted in the price system13.
Shifting our analytical focus to the price of the numéraire, given that 𝒑𝒃 = 1, we have:𝒑𝒃 = 1 = (1 + 𝑟)𝑤𝒍[𝑰 − (1 + 𝑟)𝑨]−𝟏 𝒃
in which 𝒍[𝑰 − (1 + 𝑟)𝑨]−𝟏 𝒃 denoting the scalar value of labor converted for producing one unit of
numéraire.
We can derive the following equation for income distribution between wages and profits:

See Pasinetti (1988, pp.131-132) for this identity with respect to his vertically hyper-integrated sectoral
analysis.
13 Our labor converted corresponds to Pasinetti’s “vertically hyper-integrated labor coefficient”, whichcontains the quantity of labor necessary directly, indirectly, and hyper-indirectly to production (1981) (1988,
p.130). However, it should have been (1 + 𝑟)𝒍[𝑰 − (1 + 𝑟)𝑨]−𝟏 instead of 𝒍[𝑰 − (1 + 𝑟)𝑨]−𝟏 if he had
adopted the assumption of wages paid ante factum instead of post factum. He regarded it as providing “acomplete generalization of Marx’s ‘transformation problem’” (1988, p.131). Pasinetti (1977, pp.140-142),
this time with the assumption of wages paid ante factum, referred to (1 + 𝑟)𝒍[𝑰 − (1 + 𝑟)𝑨]−𝟏 as “the linear
operator which transforms the value into the prices of production.” See Harcourt (2006, pp.123-133) for an
evaluation of Pasinetti’s model with a close reference to Sraffa (1960).
712

(1 + 𝑟)𝑤 =1𝒍[𝑰 − (1 + 𝑟)𝑨]−1 𝒃(3)

the right-hand side of which signifies how many units of b can be produced by unit of labor converted,reflecting productivity in terms of labor converted, which is divided between the profit rate and the real wagerate.
4.2 The quantity system
Assuming no excess supply or demand in the physical terms among production processes, the quantity
system of a single degree of freedom can be described by the following equations:𝒒 = (1 + 𝑔)(𝑨𝒒 + 𝑐𝒃)
𝒍𝒒 = 1(4)(5)

in which
q: a column vector (n×1) representing outputs per unit of labor,
g: the uniform growth rate of output across sectors (scalar), andc: the real consumption rate (scalar), indicating the units of a set of subsistence consumption
goods and services consumed per unit of labor, measured in terms of numéraire b.
The growth rate and the consumption rate are defined without reference to prices, which is in a sharp contrast
to, and sometimes an advantage over, the post-Keynesian models in terms of value.
It should be noted that c includes the goods and services consumed per unit of labor by capitalists, rentiers,
bureaucrats, and other individuals in addition to workers (Pasinetti, 1977, chap. VII). These goods andservices are necessary for the maintenance and reproduction of society and the social relations amongclasses14. The consumption per unit of labor, growing at the rate of g and allocated in a largely constant
proportion among the members, ensures stable class relations and governance institutions, which are essential
for the sustainability of society in the long term. This reflects our second premise based on Marx. In summary,workers are not solely reproduced physiologically through their own consumption but rather socially
reproduced alongside other non-working classes through the consumption of society's all members. It should
be further noted that consumption of unemployed and retired workers, provided by non-capitalist activities,15
is also included in c. This is a necessary condition for not applying the rule of free goods to workers under
the premise of general existence of unemployment. It goes, however, without saying that the working powerof unemployed workers unused and wasted in each production period loses its use value as well as exchange

In our model, landlords are excluded from the members of society due to the absence of rent as an income
category in our price system.
14“Some form of non-capitalist activities – subsistence agriculture as in the Lewis model, reliance on theinformal sector or some kind of petty commodity production, the family, or the welfare state” (Evans 1989,
p.167).
815

value at the end of each production period. In addition, we adopt in this paper the strong assumption thatworkers, capitalists, and other non-working classes consume the same minimum basket in terms of its content
and proportion.
We derive the following equation from equation (4):[𝑰 − (1 + 𝑔)𝑨]𝒒 = (1 + 𝑔)𝑐𝒃
When matrix A is indecomposable and non-negative, we have:𝒒 = (1 + 𝑔)𝑐[𝑰 − (1 + 𝑔)𝑨]−1 𝒃
in which [𝑰 − (1 + 𝑔)𝑨]−𝟏 acts as the converter of numéraire b, transforming it into the vector with thesame composition of elements as that of output q. We refer to it as the numéraire converter and hence,[𝑰 − (1 + 𝑔)𝑨]−1 𝒃 represents the numéraire converted. The numéraire, thus transformed to match actual
output, is laid in the basis of the quantity system.
Shifting our analytical focus from outputs to the total labor force, and assuming the total labor force to beunity, i.e., 𝒍𝒒 = 1, we obtain:
𝒍𝒒 = 1 = (1 + 𝑔)𝑐𝒍[𝑰 − (1 + 𝑔)𝑨]−1 𝒃
in which 𝒍[𝑰 − (1 + 𝑔)𝑨]−1 𝒃 represents the scalar value of the total units of labor for producing one unitof the numéraire converted. Consequently, we derive the following equation for resource allocation between
consumption and growth:(1 + 𝑔)𝑐 =1𝒍[𝑰 − (1 + 𝑔)𝑨]−1𝒃(6)

4.3 The double dualities and the dual equation
Equations (3) and (6) highlight the presence of double dualities between the R-w frontier (where R = 1 +r) and the G-c frontier (where G = 1 + g), as well as between the R-Rw frontier and the G-Gc frontier. It isnoticeable that even if we replace g and c with r and w, respectively, the mathematical properties of theseequations remain unchanged. These double dualities extend von Neumann's "remarkable duality (symmetry)"([1937] (1945-46), p.1) by allowing the real wage rate to exceed the subsistence level and incorporating theconsumption of capitalists and other non-working classes. Previous studies, as discussed in the literature
review, have only recognized the first duality16. The inverse relationships between R and w (G and c), as well
as between R and Rw (G and Gc), hold (see Appendix).
Thus far, the two systems have not explicitly differentiated between production processes (or sectors) formeans of production and those for means of consumption. They are both produced in each process (or sector).

Pasinetti claims “formal symmetry” between the price and quantity systems (1988, pp.130 and 134). It is
doubtful, however, because in his “vertically hyper-integrated sectoral model” consumption is treated as
exogenously given rather than endogenously determined in the quantity system and his numéraire in the pricesystem does not necessarily consist of a set of consumption goods and services (ibid., p.128).
9
16

Now, we introduce the integrated consumer-goods-and-services sector, similar to the "integrated wage-goods
sector" (Garegnani, 1984, 1987) originating from Ricardo's old corn-ratio theory (Sraffa’s introduction to
Ricardo [1817] and Dmitriev [1904]).
Given the total output vector Q (n×1), the total labor force E (scalar) is:
𝐸 = 𝒍𝑸We then collect the outputs of consumer goods and services from all sectors and construct the column vector
Qc (n×1), which contains the consumer goods and services not only for workers but also for capitalists andother non-working classes of society17. Then, we acquire:
𝑸𝒄 = 𝐺𝐸𝑐𝒃
𝒒𝒄 = 𝐺𝑐𝒃
in which qc denotes Qc/E. It is important to note that the integrated sector of means of consumption also
grows at the same uniform rate g. Since qc is a product of a scalar multiple qc and b, we have:𝑞𝑐 𝒃 = 𝐺𝑐𝒃
𝑞𝑐 = 𝐺𝑐
This relationship holds regardless of specific values of G and c, as long as 0 < G and 0 < c. Although qc
may appear to be equal to Rw as well as Gc due to the duality between the R-Rw and G-Gc frontiers, theconnection between the quantity system and the price system requires a converter. From equations (3) and(6), we obtain:
1 = 𝐺𝑐𝒍(𝑰 − 𝐺𝑨)−𝟏 𝒃 = 𝑅𝑤𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃
𝐺𝑐 = 𝑅𝑤𝒍(𝑰 − 𝑅𝑨)−1 𝒃𝒍(𝑰 − 𝐺𝑨)−1 𝒃

Since 𝑞𝑐 = 𝐺𝑐, we can derive the dual equation that serves as the bridge between the quantity systemand the price system when 1 ≤ 𝐺:
1𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃
≥
𝑞
=
𝐺𝑐
=
𝑅𝑤
𝑐
𝒍(𝑰 − 𝑨)−1 𝒃𝒍(𝑰 − 𝐺𝑨)−1 𝒃(7)

in which the ratio 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃/𝒍(𝑰 − 𝐺𝑨)−𝟏 𝒃 functions as the system converter from the price system tothe quantity system. 𝒍(𝑰 − 𝑅𝑨)−1𝒃 represents the units of labor converted for producing one unit of
numéraire, while 𝒍(𝑰 − 𝐺𝑨)−1 𝒃 the units of direct labor for producing one unit of numéraire converted.
Alternatively, we can interpret the system converter as the ratio of labor commanded for producing one unit

It is important to highlight that the concept of the "integrated wage-goods sector" (Garegnani, 1984, 1987)
focuses exclusively on consumer goods for workers, without considering the consumption of capitalists andother non-working classes. In this regard, Dmitriev expanded Ricardo's original single wage-good (corn)model to incorporate a bundle of subsistence wage-goods, a development that Garegnani also acknowledged.
Dmitriev's work shed light on the inverse relationship between the real wage rate and the profit rate. For
further details, see Dmitriev [1904] (1974, pp.39-63) and D. M. Nuti's introduction.
17

10

of numéraire at the profit rate r compared to that at the profit rate g. When g = r, they converge, resulting in
c = w. The system converter reconciles the disparity between the labor converted and the numéraire converted
as the basis of the price system and the quantity system, respectively.
In the dual equation, qc represents the volume of means of consumption produced per unit of labor. It
encapsulates the relations within income distribution between profits and wages, as well as resource
allocation between growth and consumption in terms of physical units of numéraire b. We refer to it as thenet economic resource. Figures 1 and 2 illustrate the first and second duality and the dual equation, in which
G < R and thus, w < c.
[Figure 1][Figure 2]
When G or R = 1, i.e., zero growth rate or zero profit rate, the net economic resource qc reaches its
maximum value, which is equal to the maximum values of c and w. As G or R increases qc monotonically1decreases. The maximum value of qc, i.e., 𝒍(𝑰−𝑨)−𝟏𝒃 indicates the number of the numéraire basket that can beproduced per unit of labor since the denominator denotes the total labor units for producing one unit of
numéraire. This value signifies the general level of labor productivity for the entire economy in terms ofnuméraire. It remains constant regardless of the prevailing income distribution and resource allocation, as
long as the adopted production technique remains constant.
As seen in Fig.3, an upward shift of qc when G or R = 1, i.e.,

1, reflects an improvement in overall𝒍(𝑰−𝑨)−1 𝒃

labor productivity. However, this does not necessarily imply the outward expansion of the entire G-qc and Rqc frontiers; different frontiers such as αi, αj, and αk may intersect each other within the economicallymeaningful range of G and R. This occurrence is a result of the possibility of switching and reswitchingamong frontiers, exemplified by those at R1, R2, R3, and R4, driven by the choice of technique based on the
profit rate maximization criterion. In other words, productivity improvement does not necessarily guarantee
profit rate maximization, and vice versa18.
[Figure 3]
5 The “nutshell economy”
For switching and reswitching of technique, see Sraffa (1960, ch.12), Bharadwaj (1970), Pasinetti (1977,ch.6), and Kurz (2008). Schefold (2021) (2023) reveal, however, that the number of switching is relatively
small and that the probability of isolated reswitching tends to zero in large systems.
1118

5.1 The dual equation in the context of the classical political economy
Our dual equation can be expressed as follows:1𝒍[𝑰 − (1 + 𝑟)𝑨]−1 𝒃
(𝑤
≥
𝑞
=
𝑐
+
𝑔𝑐
=
+
𝑟𝑤)
𝑐
𝒍(𝑰 − 𝑨)−1 𝒃𝒍[𝑰 − (1 + 𝑔)𝑨]−1𝒃(8)in which g and r are absolute numbers, denoting unitless ratios; thus, it is necessary for the dual equation to
hold that qc, w, and c share a common measurement unit, which is represented by a set of consumer goods
and services, denoted as b, functioning as the chosen numéraire in a physical unit. Therefore, unless b is thenuméraire, neither the double dualities nor the dual equation would hold, suggesting that these economic
properties are numéraire dependent. Sraffa (1960) adopted the standard commodity as his numéraire thatmeasures relative prices of commodities and the "real wage"19; consequently, the dual equation does notderive from the Sraffian standard system.
The equation presented above depicts an economic system in which qc units of the consumption basket,
produced per unit of labor, are entirely distributed between wages w and profits rw and allocated between
consumption c and growth gc, assuming as if intermediate inputs of means of production are absent. This
portrayal resembles that of a primitive economic society, where individuals rely solely on manual foodgathering without the aid of tools or machinery. Most of the gathered food is directly consumed, while a
portion is saved to augment the labor force for future food-gathering periods. By utilizing the system
converter, the complexity arising from the intricate input-output relationships among multiple sectors in an
actual economy is simplified into a concise framework.
This "nutshell economy," encapsulated within the dual equation, evokes the ideas of two prominentclassical economists, Adam Smith and David Ricardo. The opening sentence of Smith's seminal work, The
Wealth of Nations, resonates with this context:
The annual labour of every nation is the fund which originally supplies it with all the necessaries
and conveniencies of life which it annually consumes, and which consist always either in theimmediate produce of that labour, or in what is purchased with that produce from other nations.
(Smith [1776] (1937), p.lvii.)
Through this concise statement, Smith effectively dismantled the fallacy of mercantilism concerning theorigin of wealth. Notably, Smith does not refer to any means of production such as machinery, tools, or raw
materials, but to means of consumption as the ultimate embodiment of national wealth. In his view, meansof production still remain in the intermediate stage of production processes towards final consumption. Thisobservation gains particular significance when viewed in the context of the dual equation, in which the net

Sraffa (1960) did not intend to imply that workers physically "consume" a specific bundle of standard
commodities (p.23). It is crucial to note that Burmeister (1968, 1977, 1984) and Samuelson (1987)
misinterpreted Sraffa's views in this regard. Kurose and Yoshihara (2019) further highlighted themisunderstanding in its footnote 8.
1219

economic resource finds its expression in Qc rather than AQ. Smith's argument is firmly rooted in theframework of the quantity system, and his approach exhibits remarkable precision from the modern
perspective of the dual equation20.
Second, the methodology employed in our reduction to a primitive economy bears a striking resemblance
to Ricardo's early corn-ratio theory. This approach predates Ricardo's development of the labor-embodied
value theory, which aimed to determine income distribution between profits and wages through the physicalcorn-ratio in agricultural production:[I]n agriculture the same commodity, namely corn, forms both the capital (conceived as composedof the subsistence necessary for workers) and the product; so that the determination of profit by thedifference between total product and capital advanced, and also the determination of the ratio of thisprofit to the capital, is done directly between quantities of corn without any question of valuation.
(Sraffa in Ricardo [1817], “Introduction” p.xxxi.)
In this context, the dual equation can be seen as a contemporary "resurrection" of Ricardo's old corn-ratio
theory. It enables the reduction of the intricate structure of a real economy to a simple "food-gathering
economy," facilitating a clear and straightforward analysis of the inverse relationships between the growthrate and the consumption rate, as well as between the profit rate and the wage rate.
5.2 The burden of growth and profits
In Figure 2, each frontier exhibits a monotonic downward shift as R or G increases. The question arises
as to why the net economic resource, under constant technique, experiences a decline instead of remaining
unchanged. In the quantity side of the dual equation, reduction of the net economic resource 𝑞𝑐 (= 𝐺𝑐),
resulting from an increase in G and a decrease in c, reflects the fact that G does not proportionately expand
in response to the shrinking consumption rate; in other words, the elasticity of G with respect to c is less thanunity. Given 𝑐 ∗ = 𝑐 − 1, the equation is rewritten as 𝑞𝑐 = (1 + 𝑔)(1 + 𝑐 ∗ ), which implies that the elasticityof the growth rate g with respect to the newly defined consumption rate c* is less than unity. The rate c*represents the level of consumption beyond the minimum subsistence.
This observation suggests that economic growth disproportionately depletes economic resources andsacrifices consumption. This is because growth of output necessitates significant physical investment andadditional employment in each production process, leaving less resources for consumption. This dilemma of
economic growth happens under the assumption of choice of technique as well as that of a single constantMarx, quoting the same passage, criticized Smith in that “Smith’s first error, then, is to equate the valueof the annual product with the annual value product. The latter is simply the product of the current year’s
labour” (Marx [1885], p.453 with original italic), while the former includes means of production produced
in previous years. Marx was accurate in highlighting Smith’s confusion, though Smith’s reference to theultimate embodiment of national wealth remains valid.
13
20

technique.
This constraint within the quantity system manifests itself as a similar and interconnected constraint
within the price system: the profit rate does not proportionately increase in response to the declining wage
rate; namely, the elasticity of the profit rate r with respect to w*, defined as w – 1, is less than unity. This
imposes a significant burden on wages: even a marginal increase in profits results in a substantial reduction
in wages.
6 Symmetry breaking and related theorems
6.1 Symmetry breaking and the theorem of discrepancy
We have so far assumed w < c and drawn Figures accordingly. Let us examine the case in which 𝑐 ≤ 𝑤,
contrary to the assumption, and confirm its invalidity. Given that c encompasses the consumption ofcapitalists and other non-working classes in addition to workers', the condition 𝑐 ≤ 𝑤 implies that theformer's consumption is equal to or less than workers' savings. Such a scenario would be difficult to conceive
within capitalism, particularly in contemporary capitalism which is characterized by extravagant
consumption of capitalists and the presence of colossal state institutions. Von Neumann’s assumption ofcapitalists’ no consumption and workers’ no saving represents an extreme case in which c = w and thus, G =R (i.e., g = r), resulting in perfect symmetry. In the light of this, we hypothesize that the consumption ofcapitalists and other non-working classes exceeds workers' savings, thereby leading to the condition 𝑤 < 𝑐.
Referring to the dual equation, we have:1<𝑐
𝑅 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃=( )𝑤
𝐺 𝒍(𝑰 − 𝐺𝑨)−1 𝒃(9)which results in G < R (i.e., g < r). This is what we call the symmetry breaking. It suggests that since thesystem converter is greater than unity, the wage rate deficiency (w < c) exceeds the growth rate deficiency(G < R).
It follows that:𝑞𝑐 = 𝐺𝑐 > 𝑅𝑤

(10)

It represents the theorem of discrepancy between income distribution Rw and resource allocation Gc, which
directly arises from the symmetry breaking.
6.2 The theorem of parallel traverse with widening symmetry breaking
The construction of the dual equation does not alter the single degree of freedom either in the price systemor the quantity system, leaving either R or w undetermined in the former and either G or c undetermined in
the latter. It allows any specific combination of R and w, and G and c within the economically meaningful
range. Therefore, the values of either R or w and G or c are to be exogenously determined by certain external
factors outside our model so as to close the systems.
14

However, this does not mean that these two systems are completely independent and autonomous eachother21. Once exogenously given and depicted somewhere on the dual frontiers, these variables move intandem with widening symmetry breaking as R and G increase. It is because c contains workers’ consumption
and capitalists’ consumption which change in the opposite directions as R changes. This mechanism is what
we refer to as the parallel traverse22 with widening symmetry breaking, which is explained as follows:
Differentiate equation (11) with respect to R, and we acquire𝑐 ′𝑅 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃]( ) = [( )𝑤𝐺 𝒍(𝑰 − 𝐺𝑨)−1 𝒃′

Here we separate workers’ and capitalists’ consumption rates, cw and cc, respectively, assuming the former’s
somewhat stable and positive propensity to consume (βw), whilst not specifying the latter’s propensity toconsume. Substitute 𝛽𝑤 𝑤 + 𝑐𝑐 for c:′(

𝛽𝑤 𝑤 + 𝑐𝑐 ′𝑅 ′ 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃
𝑅 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃) =( )+ ( )[]−1𝑤
𝐺 𝒍(𝑰 − 𝐺𝑨) 𝒃
𝐺 𝒍(𝑰 − 𝐺𝑨)−1 𝒃

𝑐𝑐 ′𝑅 ′ 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃
𝑅 𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃
0<( ) =( )+ ( )[]−1𝑤
𝐺 𝒍(𝑰 − 𝐺𝑨) 𝒃
𝐺 𝒍(𝑰 − 𝐺𝑨)−1 𝒃

′

(11)

in which the derivative of cc/w is positive because cc is at least constant or increases as R increases and w
decreases. Since the following two inequalities hold on the right side,
0<
0<

𝑅
𝐺

𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃𝒍(𝑰 − 𝐺𝑨)−1 𝒃

the following two derivatives must be simultaneously positive.
𝑅 ′
0<( )
𝐺
𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃0<[]𝒍(𝑰 − 𝐺𝑨)−1 𝒃′

It follows that the system converter exhibits a monotonically increasing trend as R increases, leading to a𝑅 ′widening gap between R and G (i.e., G < R and 0 < (𝐺 ) ), which corresponds to the widening gap between

Pasinetti (1977, p.111) claims complete autonomy of these two systems. Lavoie (2022, p.43) and Bhaduri(2011, p.95) regard their separation as a commonly shared view between Sraffians and post-Keynesians.
22 In the tradition of the post-Keynesian economics, the traverse “defines the movement of the economy
outside equilibrium” (Halevi, Hart, and Kriesler 2013, p.175), distinct from our usage, which originated inHicks (1965) and Low (1976). See also Kriesler (2012).
15
21

𝑤 ′

w and c (i.e., w < c and ( 𝑐 ) < 0).
We can understand this simple but significant phenomenon in the case in which the traverse is initiatedby an increase in R as follows. As R increases, workers' consumption rate (cw) decreases in proportion to thedecrease in w. Meanwhile, capitalists' consumption (cc) out of enhanced profits at least keeps constant andgenerally increases. However, it only partially offsets the decrease in cw even if capitalists’ propensity toconsume is the same as workers’. This is because the elasticity of R (or r) with respect to w (or w*) is lessthan unity. It is worth emphasizing, however, that capitalists' propensity to consume is generally lower thanthat of workers, further reinforcing our argument (Kaldor, 1955-56, p.95). It follows that c (i.e., cw + cc) does
not decrease in proportion to w, and thus, G does increase, but not in proportion to R, resulting in that the
downward traverse on the G-c frontier is parallel to that on the R-w frontier, with a widening symmetry
breaking between them23.
The theorem also suggests what normatively happens when other variables take the initiative: an increasein G accelerates an increase in R and causes a drastic drop of w and a somewhat mitigated decrease in c; anincrease in w causes a drop of R accordingly, but not as much drop of G or rise of c; and an increase in cprimarily comes from a rise of w and cw, and results in a corresponding decrease in G, accompanied by a
drastic drop of R. It is also suggested that in the downward traverse the symmetry breaking decelerates in the
concave part and accelerates in the convex part of the frontier.
7 Conclusions
For addressing the paradox of economic growth, this article combines the price system and the quantity
system through the system converter and formulates the dual equation, facilitated by adopting the minimum
subsistence basket of consumer goods and services as numéraire and introducing the integrated consumergoods-and-services sector. The resulting simple "nutshell economy," encapsulated in the dual equation,
unveils two insights. First, it reveals the symmetry breaking in the duality, signifying that G < R and w < c,leading to the theorem of discrepancy between income distribution Rw and resource allocation Gc, namely
Rw < Gc. Second, it establishes the theorem of parallel traverse of R and G, and w and c in tandem with theirwidening symmetry breaking as R and G increases, and vice versa, in the long term.
We have constructed our argument using a single and constant set of techniques for industries. However,
the monotonically declining relations of the R-qc and G-qc frontiers and the R-w and G-c frontiers can beconstructed by combining the outermost envelopes of techniques that exist and are available, such as αi, αj,and αk in Figure 3. Therefore, our entire argument so far generally holds true with the choice of technique.
As far as our long-term framework is concerned, the theorem of parallel traverse provides support for the
profit-led growth model instead of the wage-led growth model in the post-Keynesian economics. For thelatter, see, among others, Lavoie and Stockhammer (2013), Lavoie (2022, ch.6), and Skott (2016).
16
23

Now we understand that the paradox is not a paradox that overshadows the virtue of higher GDP per
capita, but a paradox that is inherent in investment and capital accumulation processes of capitalism. Unless
we can assume that technological development in terms of higher productivity is initiated by capitalists andresults in income distribution in favor of workers, it would also cause higher GDP at the expense of reductionof workers’ wages and consumption.
As for monotonically changing symmetry breaking in the parallel traverse, Lavoie, a prominent postKeynesian, opposes and states that “Once the variations in the rate of utilization of capacity are taken into
account, in the long run and not only in the short run, the inverse relationship between the real wage rate andthe rate of growth may not hold.” (Lavoie, 2022, p.381.)24 For meeting his challenge, we need to introduce
inventory into our circulating-capital model, instead of underutilized capacity, which enables us to analyze
the effects of short-term oscillations of investment demand within the long-term parallel traverse. This
research theme is to be scrutinized in the next paper.
Appendix:Differentiate equations (1) and (2) with respect to R, and we acquire:𝒑′ = 𝒑𝑨 + 𝑅𝒑′ 𝑨 + 𝑤𝒍 + 𝑅𝑤′𝒍𝒑′𝒃 = 0in which p’ and w’ denote the vector of derivatives of prices and the derivative of the wage rate, respectively.
It follows that:𝒑′ = (𝒑𝑨 + 𝑤𝒍 + 𝑅𝑤′𝒍)(𝑰 − 𝑅𝑨)−1
Since 𝒑′𝒃 = 0
𝒑′ 𝒃 = 0 = (𝒑𝑨 + 𝑤𝒍 + 𝑅𝑤′𝒍)(𝑰 − 𝑅𝑨)−1 𝒃
𝑤′ = −𝒑𝑨(𝑰 − 𝑅𝑨)−1 𝒃 + 𝑤𝒍(𝑰 − 𝑅𝑨)−1 𝒃
𝑅𝒍(𝑰 − 𝑅𝑨)−1 𝒃

Since the denominator and the numerator are positive,
𝑤′ < 0Because the R-w and the G-c frontiers are in duality, r and w, and g and c are both in the inverse relations
with each other in the economically meaningful range.
Next, the inverse relations between Rw and R are confirmed: from equation (1), we already acquired:𝑅𝑤𝒍 = 𝒑(𝑰 − 𝑅𝑨)
Differentiate it with respect to R, and it follows that:𝒑′ = (𝑤𝒍 + 𝑅𝑤 ′ 𝒍 + 𝒑𝑨)(𝑰 − 𝑅𝑨)−1

For surveys of the post-Keynesian theories on growth and distribution, see, in addition to Lavoie (2022,ch.6), Kurz and Salvadori (2010), Panico (2012), Keen (2012), and Dutt (2018).
17
24

And
𝒑′𝒃 = 0
Therefore𝒑′ 𝒃 = 0 = (𝑤𝒍 + 𝑅𝑤 ′ 𝒍 + 𝒑𝑨)(𝑰 − 𝑅𝑨)−1𝒃(𝑅𝑤)′ = 𝑤 + 𝑅𝑤 ′ = −𝒑𝑨(𝑰 − 𝑅𝑨)−𝟏 𝒃<0𝒍(𝑰 − 𝑅𝑨)−𝟏 𝒃

The R-Rw and the G-Gc frontiers are in duality, and thus, Gc and G are in the inverse relations as well.
The inverse relations can be directly proved by the Perron-Frobenius theorems: given a real number R
and 𝑅 < 1/𝜆𝑚 , in which 𝜆𝑚 denotes the eigenvalue of A, all the elements of matrix (𝑰 − 𝑅𝑨)−1 arecontinuous, increasing function of R (Pasinetti, 1977, p.273) (Pasinetti, 1980, p.23).

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