Symmetric Off-Diagonals in the Pasinetti
Pure Labour Model
Andrew B. Trigg
May 2013
Department of Economics
Faculty of Social Sciences
The Open University
Walton Hall
Milton Keynes
MK7 6AA
UK
1
1. Introduction
In Pasinetti’s pioneering work on structural economic dynamics, a key contribution is
in the modelling of disproportional growth in the composition of industrial sectors.
Differential change in productivity and consumption patterns leads to uneven
development and structural change in the long term: all captured in a multisectoral
framework in which there is a social division of labour and specialization between
sectors. In the most abstract pure labour version of this approach (Pasinetti 1993)
labour is the only factor of a production, a simplification that allows a clear focus on
structural change, together with foundational insights into the structure of savings,
consumption, and debt.
The purpose of this paper is to introduce a new result concerning the intersectoral
flows of commodities between sectors in the Pasinetti framework. It will be shown
that the matrix of intersectoral flows is symmetric. The flow of commodities from
sector i to j is identical to the flow in the opposite direction between j and i . A
structural constraint is hence identified in the Pasinetti system that is specific to the
intersectoral relationships between sectors.
In Section 1, this symmetry result is derived from the Pasinetti pure labour model, and
illustrated using a three sector numerical example. Section 2 then examines the
implications of this result for Pasinetti’s structural dynamics, followed in Section 3 by
a consideration of savings and consumption; Section 4 offers some conclusions.
2. Symmetric Off-Diagonals
Consider the pure labour model, as developed by Pasinetti (1993, pp. 17-18):
Qi  ciQn
(1)
Qn  l1Q1  l2Q2    lmQm
(2)
pi  wli
(3)
w  p1c1  p2c2   pmcm
(4)
The first set of two equations represent the quantity system for an economy in which
labour is the sole factor of production and only consumption goods are produced –
a closed input-output model in which there is no exogenous component of final
demand. In equation (1), the total physical quantity of output produced (Qi ) for sector
i is related to the total volume of labour employed (Qn ) , according to the size of the
consumption coefficient (ci ) . Equation (2) shows, using direct labour coefficients
(li ) , that all labour is directed to the production of output in m sectors: Q1...Qm .
The second set of equations represents a price system. A pure labour theory of value
is shown in (3), with money prices ( pi ) proportional to labour coefficients according
to the scalar money wage rate (w) . Equation (4) shows that this wage is all spent on
consumption goods.
2
We can use this model to explore the structure of total flows in a multisectoral
economy. First, let each Cij represent the flow of physical consumption goods from
sector i to j , which in Table 1 are valued in labour units.
Table 1 Total Labour Unit Flows in a Multisectoral Economy
Buyer
Seller
1
2
3

m
1
2
3

m
l1C11
l2C21
l3C31

lmCm1
l1C12
l2C22
l3C32

lmCm 2
l1C13
l2C23
l3C33


l1C1m
l2C2 m
l3C3m

lm C m 3



lmCmm

Second, the total physical flows of consumption goods can then be related to the
quantity of output labour employed in each sector, such that:
Cij  cil j Q j
(5)
For each quantity of output Q j a volume of labour activity l j Q j is required; and this
induces a flow Cij of consumption goods from i to j , subject to the size of the
consumption coefficient c i . Substituting (5) into Table 1, a new intersectoral matrix
(Table 2) is derived.
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Table 2 Detailed Structure of Total Labour Unit Flows in a Multisectoral
Economy
Buyer
Seller
1
2
3

m
1
2
3

m
l1c1l1Q1
l2c2l1Q1
l3c3l1Q1

lmcml1Q1
l1c1l2Q2
l2c2l2Q2
l3c3l2Q2

lmcml2Q2
l1c1l3Q3
l2c2l3Q3
l3c3l3Q3


l1c1lmQm
l2c2lmQm
l3c3lmQm

lmcml3Q3



lmcmlmQm

We can now examine the relationship between the off-diagonals. It can first be stated,
from equation (1) of the Pasinetti model, that the ratio of physical quantities between
any two sectors i and j is governed by the relative magnitude of consumption
coefficients:
Qi
c
 i
Qj cj
(6)
Multiplying throughout (6) by a multiple of the labour coefficients for these two
sectors (lil j ) , and re-arranging, it follows that
l j c j liQi  li cil j Q j
(7)
This is a remarkable result. It shows that the off-diagonals in Table 3 are symmetrical.
The flow of total value, measured in labour units, from sector 2 to sector 1, for
example, are identical to the flow in the opposite direction, from sector 1 to sector 2:
l2c2l1Q1  l1c1l2Q2
(8)
This pairwise identity between off-diagonals also applies to flows from sectors 1 to 3
in symmetry with 3 to 1, sectors 2 to 3 in symmetry with 3 to 2, and so on and so
forth.
An economic reasoning for this symmetry is straightforward. Consider on the righthand side of (8) the effect of l2Q2 , the volume of labour employed in sector 2. Via
l1c1 , this sector 2 employment leads to consumption (and output) of good 1, and hence
employment of workers producing good 1. This employment of sector 1 workers also
appears on the left hand side of (8), in the form of l1Q1 . There, via l2c2 , sector 1
employment translates into expenditure on good 2, and hence employment in sector 2.
Since this is a consumption-only system, employment in sector 1 generates sufficient
consumer expenditure to generate matching employment in sector 2. There is a
mutual dependence of each sector on their respective levels of employment and
consumption.
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The symmetry of intersectoral flows can illustrated using a three sector example, in
which 40 units of labour employed. Coefficients are assumed to take values
c1  1.5, c2  1, c3  1.25,
l1  0.25, l2  0.5, and l3  0.1.
The solution of this system yields volumes of physical output
Q1  60, Q2  40, and Q3  50 .
Table 3 shows the flows between these three sectors measured in labour units. As
expected, the off-diagonals are symmetric, with for example 7.5 units flowing from
workers in sector 2 to workers in sector 1; this is matched by 7.5 units flowing in the
opposite direction, from workers in sector 2 to sector 1. There are two other pairwise
identities, of 1.875 (flows between sectors 1 and 3) and 2.5 (between 2 and 3). This
pattern is discerned despite specialisation of workers, and the different labour and
consumption coefficients between sectors.
Table 3 A Three Sector Example of Intersectoral Flows
Period 0
Sector 1
Sector 2
Sector 3
Sector 1
5.625
7.5
1.875
Sector 2
7.5
10
2.5
Sector 3
1.875
2.5
0.625
In the following and subsequent parts of the paper some implications of this result
will be explored.
3. Structural Economic Dynamics
In contrast to the usual one-good assumptions adopted in modern growth theory,
Pasinetti develops a model of structural dynamics in which proportions vary between
sectors. In the pure labour approach, disproportional changes in labour and
consumption coefficients are allowed for. First, ri is defined as the rate of change of
the per capita consumption coefficient for each sector i (Pasinetti 1993, p. 40). Rates
of change can be different between sectors such that
ri  r j
(9)
Using an exponential term to model growth, the consumption coefficient for good i at
time t then takes the form
ci (t )  ci (0)e ri t ,
(10)
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where ci (0) is the consumption coefficient for good i in the base period (period 0).
Second, labour productivity is posited to improve over time at a rate i for each sector
i , as expressed in declining labour coefficients:
li (t )  li (0)e  t
(11)
Under structural economic dynamics, the rates of change of consumption and labour
coefficients can vary between sectors:
ri  i
(12)
This approach can be illustrated by considering two periods (t  0, 1) in which our
earlier three sector example, as represented in Table 3, represents period 0. Allowing
total employment to increase by 10 per cent from 40 to 44 units, a change in the
proportions between sectors can be simulated: with one additional unit of labour in
sector 1, four additional units in sector 2, and one less unit in sector 3. In period 1,
new volumes of physical output are also established of 70 for sector 1, 60 for sector 2
and 65 for sector 3. To achieve this outcome, the following rates of change for
consumption and labour coefficients are specified:
r1  6%, r2  36% , r3  18%,
1  9%, 2  20% , 3  38%
The three-sector economy in period 1 is hence reported in Table 4.
Table 4 Symmetrical Flows under Structural Dynamics (Period 1)
Period 1
Sector 1
Sector 2
Sector 3
Sector 1
5.818
8.727
1.455
Sector 2
8.727
13.091
2.182
Sector 3
1.455
2.182
0.364
Despite the structural dynamics simulated between periods 0 and 1, with varying
consumption patterns and productivity, the same symmetric structure as Table 1 is
sustained. There remain three pairwise identities on the off-diagonals. For example,
the flows between 1 and 2 (and in the opposite direction between 2 and 1) increase to
8.727 in period 1, compared to the previous level of 7.5 in period 0. But note that the
other two pairwise identities decrease in size, from 1.875 to 1.455 (between sectors 1
and 3) and from 2.5 to 2.182 (between sectors 2 and 3). There is disproportional
structural change in the off diagonal elements of the intersectoral flows matrix, within
the confines of its symmetric structure.
The implications of this insight for our understanding of structural economic
dynamics can be explored by re-expressing the two-period simulation in coefficient
form. To do this per capita output coefficients
6
Qi
Qn
qi 
(13)
are required. By dividing (7) by Qn (total employment) a new expression for the
symmetry between off-diagonals is then derived:
l j c j li qi  li ci l j q j
(14)
Hence in Table 5 a symmetric table of intersectoral flows for m sectors can be
expressed in coefficient form (by dividing throughout Table 2 by Qn ). Each element
represents the intersectoral flow of labour units per unit of aggregate employment.
Table 5 Intersectoral Flows in Coefficient Form
Buyer
Seller
1
2
3

m
1
2
3

m
l1c1l1q1
l2c2l1q1
l3c3l1q1

lmcml1q1
l1c1l2q2
l2c2l2q2
l3c3l2 q2

lmcml2q2
l1c1l3q3

l1c1lm qm
l2c2l3q3

l2c2lm qm
l3c3l3q3

l3c3lm qm

lmcml3q3



lmcmlm qm
Two points can be made about this table of coefficients. First, the row and column
sums represent the shares of each sector out of total employment. To see this, by
using (13) and Pasinetti’s condition for full employment,
m
l c
i 1
i i
 1 (Pasinetti 1993, p.
18), the column sum for any sector i in Table 5 takes the form:
m
li qi  li ci  li qi 
i
liQi
Qn
(15)
The column sum captures the share of employment in sector i out of total
employment. Since under symmetry row and column sums are identical, it follows
that the row sums also represent employment shares.
Second, the column and row sums of the table of coefficients can be interpreted as
backward and forward linkages, which have been considered important in the inputoutput literature for understanding structural change. Backward linkages represent the
purchases by a particular industry of products from other industries; forward linkages
7
represent the destination of a particular industry’s outputs to other industries. This
approach was originally pioneered by Chenery and Watanabe (1958), where the
interindustry coefficients were used to interpret direct and indirect linkages between
industries. Since then more sophisticated linkage analysis has been developed for
open input-output models using multiplier decomposition (e.g. Sonis, Hewings and
Guo, 1996 ). For our purposes, however, under the confines of a closed input-output
framework, the analysis of linkages is confined to interindustry coefficients.
Reading down a typical column in Table 5, for any sector i , the coefficients show for
a given level of employment, Qn , the amount of output (measured in labour units)
purchased from each sector. These are the backward linkages for sector i . Now
reading along a typical row for sector i , the coefficients show, for a given level of
employment, the destination of its product to other sectors. These are the forward
linkages for sector i . The total backward linkages for each sector are represented by
the column sums of Table 5; the total forward linkages are represented by the row
sums. Since under symmetry the row and column sums are identical, it follows that
the total forward and backward linkages, as here defined, are also identical.
To summarise, the identical row and column sums of the intersectoral flows matrix (in
coefficient form) represent both (a) shares in employment of each sector; and (b)
backward and forward linkages for each sector.
Using these two insights, further consideration can be given of our two-period
simulation of structural dynamics. Dividing throughout Tables 3 and 4 by their
respective employment aggregates (40 and 44), structural change between the two
periods can be considered in coefficient form, as in Table 6.
Table 6 Two Period Structural Dynamics in Coefficient Form
Period 0
Sector 1
Sector 2
Sector 3
Sector 1
Sector 2
Sector 3
Row Sum
0.141
0.188
0.047
0.375
0.188
0.25
0.063
0.50
0.047
0.063
0.016
0.125
Column
Sum
0.375
0.5
0.125
1
Period 1
Sector 1
Sector 2
Sector 3
Row Sum
0.132
0.198
0.033
0.364
0.198
0.298
0.050
0.545
0.033
0.050
0.008
0.091
0.364
0.545
0.091
1
Comparison of row (and column) sums shows that under our simulation of structural
dynamics sector 2 increases its share of total employment from 0.5 to 0.545. The
other two sectors undergo reductions in their employment sectors, from 0.375 to
0.364 for sector 1, and from 0.125 to 0.091 for sector 3. By varying productivity and
consumption patterns, Pasinetti’s framework can be used to model disproportional
changes in employment between sectors.
8
It can, however, be argued that the symmetric structure of the Pasinetti framework
limits the degree of disproportional growth that can be modelled. Pairwise identities
between off-diagonal elements constrain the degree of variation between cells across
the industrial structure.
In their study of structural change in the US economy, Guo, J. and Planting, M.A.
(2000) have viewed this type of symmetric structure as an intermediate step in
developing a fully developed model of structural change. This argument is made by
providing a taxonomy of different types of input-output structure. These develop from
the most basic type, in which there is a complete absence of intersectoral linkages, to
the most advanced type in which there are asymmetrical variations between offdiagonal cells. Models with symmetrical off diagonals are placed in between,
providing an intermediate stage in the analysis of structural change. They argue that
under the structural change that characterised the US economy over the period 1972
to 1996 there were extensive variations between off-diagonal elements. This structural
change incorporated variations in forward and backward linkages of industrial sectors,
a feature that we have seen is not provided for within the constraints of the Pasinetti
pure labour model.
4. Consumption and Savings
The symmetric off-diagonals result can also throw some light on Pasinetti’s analysis
of consumption and savings. The starting point is the assumption, embedded in the
pure labour model, that savings, the difference between national income and overall
current consumption, must be zero. So long as there no capital goods, and no durable
consumption goods – Pasinetti assumes that all goods are perishable – income can be
earned only from the sale of current consumption goods; it is not possible for income
to exceed current consumption. There must be zero aggregate savings.
But within this macroeconomic constraint, the possibility is afforded for some
individuals to spend less than their income. Savings can then be lent to other
individuals who through dissaving spend more than their income. Individuals enter
into debt-credit relationships, in which for savers consumption is postponed, and for
debtors consumption is brought forward by borrowing out of future income. There is a
transfer of consumption between individuals over time. Under the zero aggregate
savings constraint, these savings and dissavings cancel out in the aggregate.
Pasinetti extents this consideration of debt-credit relationships to the sectoral level. He
writes of ‘the many ways in which savings and dissavings (and the possibilities of
inter-temporal transfers of consumption) become possible individually and sectorally,
even if they cancel out in the aggregate’ (Pasinetti 1993, p. 85, my emphasis). It is
possible, Pasinetti argues, for one sector to carry out savings that are cancelled out by
dissavings in another sector. The public sector may, for example, run a deficit that is
counterbalanced by savings in the private sector. Transfers of consumption ‘are
perfectly possible between individuals (and therefore also between the public sector
and the private sector of the economy), provided that – within the macro-economic
constraint concerning full employment – they compensate each other in the aggregate,
at any given moment in time’ (Pasinetti 1993, p. 101).
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The sectoral structure of savings and dissavings can be explored by transforming our
table of intersectoral flows into money units. Integral to Pasinetti’s price system, as
reported in equations (3) and (4), is the uniform money wage rate (w) . Using (3), this
can be used to transform each labour coefficient (li ) into a price coefficient ( pi ) . By
multiplying this scalar throughout Table 2, a new table of intersectoral flows can be
derived, expressed in money price units (Table 7).
Table 7 Total Money Flows in a Multisectoral Economy
Buyer
Seller
1
2
3

m
1
2
3
p1c1l1Q1
p2c2l1Q1
p3c3l1Q1

pmcml1Q1
p1c1l3Q3
p1c1l2Q2
p2c2l2Q2 p2c2l3Q3
p3c3l2Q2 p3c3l3Q3


pmcml2Q2 pmcml3Q3

m

p1c1lmQm


p2c2lmQm
p3c3lmQm



pmcmlmQm
This transformation into money units, since it uses the uniform scalar wage rate, does
not alter the symmetrical structure of intersectoral flows. The money flows between
any sectors i and j are matched by identical flows in the opposite direction.
Furthermore, this means that the column totals in Table 7 are also identical to row
totals. For any sector i money its total outlays are matched by its total
money receipts.
This can be illustrated in our three sector example, by transforming Table 3 assuming
a uniform wage rate of £2 for each unit of labour. The total employment of 40 units
earns a national income of £80 that in Table 8 is distributed between sectors.
Table 8 Money Intersectoral Flows in Three Sector Example (£ units)
Period 0
Sector 1
Sector 2
Sector 3
Outlays
Sector 1
11.25
15
3.75
30
Sector 2
15
20
5
40
Sector 3
3.75
5
1.25
10
Receipts
30
40
10
This example shows the money outlays for each sector as represented by column
sums and the receipts of each sector represented by row totals. For sector 1, the total
outlays of £30 are precisely matched by receipts of £30; for sector 2 receipts and
outlays are £40; for sector 2 there is a £10 outlay/receipt. There is no possibility
afforded for sectors to have outlays that either exceed or undershoot their receipts due
to the symmetric structure of the intersectoral flows.
10
It would therefore be wrong to state that ‘the only constraint’ on the pure labour
system is the zero aggregate savings condition. Under symmetry, there is a zero
sectoral savings condition in which it is not possible for any sector to have outlays
that exceed its receipts. The system is not wide open to possibilities of transfers of
consumption between sectors over time.
4. Conclusions
This paper presents a new result concerning the structure of intersectoral flows
generated by the Pasinetti pure labour model. Off diagonal elements, representing
flows from one sector to another, are identical in both directions. These pairwise
identities can be attributed to the mutual dependence, via consumer expenditure, of
each sector’s employment on the employment of workers in other sectors.
Two main implications of this result are explored. First, the symmetric structure is
integral to Pasinetti’s model of structural dynamics, with varying consumption and
productivity patterns, and disproportional structural change. But Pasinetti’s pure
labour model is argued to provide an intermediate stage in the analysis of structural
dynamics; a more advanced treatment would allow for full variation in off-diagonal
cells. Second, it is shown that under symmetry there are restrictions on the savings
and dissavings that can be carried out at the sectoral level. Under this zero sectoral
savings constraint, deficits and surpluses cannot be established for either the private
or public sectors. Again, the symmetric structure of intersectoral flows suggests that
the model should be viewed as an intermediate stage, preparing the ground for a more
advanced treatment of imbalances between sectors.
It should be emphasised that such an unravelling of economic structures, on a step by
step basis, depends critically on the analytical clarity that is provided by Pasinetti’s
foundational insights. Only by boiling down the key features of production systems,
in their most abstract form, can we decompose the myriad of complex processes into
their component parts. By focusing on the importance of symmetric structures, this
paper offers a contribution to this Pasinetti-inspired research programme.
References
Chenery, H.B. and Watanabe, T (1958), "International Comparisons of the Structure
of Production", Econometrica, 26 (4), October, pp.487-521.
Guo, J. and Planting, M.A. (2000) ‘Using input-output analysis to measure U.S.
structural change over a 24 year period’, Paper presented to the 13th International
Conference on Input-Output Techniques, Macerata, Italy, August 21-28,
Pasinetti, L.L. (1993) Structural Economic Dynamics. Cambridge: Cambridge
University Press.
Sonis, M., Hewings, G.J.D., and J. Guo (1996) ‘Sources of Structural Change in
Input-Output Systems: A Field of Influence Approach’, Economic Systems Research,
Vol. 8, No. 1.
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