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* Department of Public Administration, Panteion University, Athens, Greece; E-mail:
mariolis@hotmail.gr
BULLETIN OF POLITICAL ECONOMY
9:2 (2015): 161-170
Testing Bienenfeld’s Second-Order
Approximation for the Wage-Profit Curve
THEODORE MARIOLIS*
This paper constructs Bienenfeld’s second-order approximation for the
wage-profit curve and tests it using data from ten symmetric input-output
tables of the Greek economy. The results suggest that there is room for
using low-dimensional models as surrogates for actual single-product
economies.
INTRODUCTION
Typical findings in many empirical studies of single-product systems
are that, in the economically relevant interval of the profit rate, (i) the
approximation of the production prices through Bienenfeld’s (1988)
linear and, a fortiori, quadratic formulae works pretty well; and (ii) the
wage-profit curves (WPCs) are almost linear, i.e. the correlation
coefficients between the distributive variables tend to be above 99%,
and their second derivatives change sign no more than once or, very
rarely, twice, irrespective of the numeraire chosen. As it has recently
been argued, these findings could be connected to the skew distribution
of the eigenvalues of the matrices of vertically integrated technical
coefficients (Schefold, 2008, 2013; Mariolis and Tsoulfidis, 2009, 2011,
2014, 2015; Iliadi et al., 2014).
The present paper, drawing on Bienefeld’s (1988) polynomial
approximation of prices, constructs a ‘proper rational’ approximation
for the WPC and tests its second-order form using data from the 19 x
19 Symmetric Input-Output Tables (SIOTs) of the Greek economy,
spanning the period 1988-1997.1 The selection of this dataset was based
on its extensive use in a number of closely related studies (see Mariolis
and Tsoulfidis, 2015a, chs 3 to 5, and the references therein).
The remainder of the paper is structured as follows. Section 2
presen ts the ne cessary pre limin arie s. Se ction 3 de riv es the
approximation for the WPC. Section 4 brings in the empirical evidence.
Finally, Section 5 concludes.
162 / THEODORE MARIOLIS
PRELIMINARIES
Consider a closed, linear and viable system involving only single
products, basic commodities (in the sense of Sraffa, 1960) and
circulating capital. Under the usual assumptions, the wage-price-profit
rate relations for the system may be written as
(1 )
w r
p l pA
(1)
where p denotes a 1×n vector of production prices, w the money wage
rate, and 1 (>0) the 1×n vector of direct labour coefficients, r the uniform
profit rate, and A the n×n matrix of direct technical coefficients.2 After
rearrangement, equation (1) becomes
w
p v pJ
(2)
or, if < 1,
1
[ ]
w
p v I J
(2a)
where
1
[ ]
v l I A
denotes the vector of vertically integrated labour
coefficients or labour values, and
1
[ ]
H A I A
(>0) the vertically
integrated technical coefficients matrix. Moreover, rR–1, 0 1,
denotes the ‘relative (or normalized) profit rate’, which equals the share
of profits in the Sraffian Standard system (SSS), and
1 1
1 1
1
A H
R
the maximum profit rate, i.e. the profit rate corresponding to [w = 0, p
> 0], which equals the ratio of the net product to the means of
production in the SSS. Finally, J RH denotes the normalized vertically
integrated technical coefficients matrix, where 1 1
1.
J H
R
If commodity zT, with vzT = 1, is chosen as the numeraire, i.e. pzT =
1, then equation (2a) implies
Z 1 T 1
( [ ] )
w
v I J z (3)
and
Z 1 T 1 1
( [ ] ) [ ]
p v I J z v I J
(4)
Post-multiplying equation (2) by SSC, i.e.
T T
1
[ ]
A
s I A x
, with
T
1
1,
A
lx and rearranging terms, gives
Z Z T
(1 )w
p s
(3a)
TESTING BIENENFELD’S SECOND-ORDER APPROXIMATION... / 163
where pzsT represents the price of net output, measured in terms of
commodity zT, of the SSS. Equations (3) and (4) gives the WPC and the
production prices, measured in terms of commodity zT, as functions of
, respectively. It then follows that:
(i) At = 0 we obtain wz(0) = 1 and pz(0) = v, while in the other
extreme case, i.e. at = 1, we obtain wz(1) = 1 and Z T 1
1 1
(1) ( )
J J
p y z y
.
(ii) The WPC is an improper rational function of degree n, and
strictly decreasing in 0 1 (see equations (3) and (3a)):
Z Z Z Z 1 T
/ [ ]w dw d w
p J I J z
�
Z Z T T
(0) (0)w
p Jz vJz
�
(3b)
Z Z T T 1 T
1 1
(1) (1) ( )w
J J
p s y z y s
�
(3c)
where vJzT equals the ratio of the capital-net output ratio (measured
in terms of labour values) in the vertically integrated industry
producing the numeraire commodity to the capital-net output ratio,
R–1, in the SSS. Finally, the WPC may admit up to 3n – 6 inflection
points (though it is not certain that they will all occur in 0 1;
Garegnani, 1970, p. 419).
(iii) If 1 (v) is the P-F eigenvector of A (J) or if SSC is chosen as the
numeraire, then
wS = 1 – (5)
i.e. the WPC is linear, as in a one-commodity world. The former case
corresponds to the Ricardo-Marx-Samuelson ‘equal value compositions
of capital’ case, and implies that p = v, i.e. the ‘pure labour theory of
value’ holds true. In the latter case,
S 1
0
(1 ) [ ] (1 )( )h
h
p v I J v J
(6)
which is the reduction of prices to ‘dated quantities of embodied labour’
(Kurz and Salvadori, 1995, p. 175) in terms of SSC. Differentiation of
equation (6) with respect to finally gives
S(0)
p v vJ
�
(6a)
APPROXIMATION FOR THE WAGE-PROFIT CURVE
Assume that prices are measured in terms of SSC. As is well known,
for any semi-positive n–vector y, yJh tends to the left P-F eigenvector
of J as h tends to infinity, i.e.
164 / THEODORE MARIOLIS
T T 1
1 1 1 1
lim [( )( ) ]
h
h
J J J J
yJ yx y x y
while an upper bound on the rate of convergence is given by
h
bc
K,
where
T 1 T
1 1 1 1
( )
h
J J J J
K J y x x y
the norm
K
denotes the maximum of the absolute values of the
elements of K, and b represents a positive constant, which depends on
J and c, for any c such that 2
1
c
J (see, e.g. Horn and Johnson,
1990, p. 501). Therefore, for a sufficiently large value of m such that
1 S
... (1)
m m
vJ vJ p (7)
it follows from equation (6) that
1
S 2 S
0
(1 )( ) (1 ) (1 ...) (1)
m
h m
h
p v J p
or
1
S A 1 S 1
1
( ) ( (1) )
m
h h h m m
h
p p v vJ vJ p vJ (8)
The vector pA is Bienenfeld’s (1988) polynomial approximation for
the price vector measured in terms of SSC:
(i) It is exact at the extreme, economically significant, values of .
(ii) For m 2, it gives, the correct slope of the actual pj – curves at
= 0 (consider equation (6a)).
(iii) Its accuracy is directly related to the rate of convergence in (7),
which in its turn is directly related to the magnitudes of
1
k
J. In the
extreme case where the non-labour conditions of production in all
industries are linearly dependent on each other, it holds that rank[J] =
1, which implies T 1 T
1 1 1 1
( )
J J J J
J y x x y
and 2
0
J. Thus, vJ = pS(1) and
Bienenfeld’s approximation becomes linear and exact for all , i.e.
S A S
( (1) ) p p v p v
Equations (3) to (6) imply
TESTING BIENENFELD’S SECOND-ORDER APPROXIMATION... / 165
Z 1 Z T S 1 S T
( ) ( ) ( ) ( )
w w
p z p z
or
Z S S T 1
( )
w w
p z
Thus, invoking approximation (8), we obtain
Z S A T 1
( )
w w
p z
or
1
Z A S 1 T S 1 T 1
1
{1 ( ) ( (1) ) }
m
h h h m m
h
w w w
vJ vJ z p vJ z (9)
The approximate wA curve has the following attributes:
(i) It is a proper rational function of degree m, and exact at the
extreme values of , while its accuracy is directly related to the
magnitudes of
1
k
J.
(ii) It gives the correct slope of the actual WPC at = 1 (consider
equations (9) and (3c)):
A A T 1 S T 1
(1) ( (1) ) ( (1) )
w
p z p z
�
or
A T 1 T Z
1 1
(1) ( ) (1)
w w
J J
y z y s
� �
For m 2, it also gives the correct slope at = 0 (consider equations
(9) and (3b)):
A S T
(0) (1)
w
p z
�
, for m = 1
A T Z
(0) (0)
w w vJz
� �
, for m 2
(iii) For m 2, it may admit up to 2m – 1 inflection points.
Setting m = 1, wA reduces to the homographic function
A S S T 1
[1 ( (1) 1)]
w w
p z (9a)
which has exactly the same algebraic form as the WPC for the
Samuelson-Hicks-Spaventa or ‘corn-tractor’ model (see Spaventa,
19 70). Moreove r, A
0
w
��
for
0 1
if f S T
(1) 1
p z or, since
S S
(1) (1)
p J p ,
S T 1
(1)
R
p Hz , i.e. at = 1 the vertically integrated
166 / THEODORE MARIOLIS
industry producing zT is labour intensive relative to the SSS. When
rank[J]=1, this approximation becomes exact for all .3
Finally, setting m = 2, wA reduces to
A S T 2 S T T 1
[1 ( 1) ( (1) )]
w w
vJz p z vJz (9b)
Let
, 1,2, 3
be the roots of equation A
0
w
��
, let be the
discriminant of the numerator of
A
w
��
, and set
T
vJz
and
S T
(1)
p z
,
. It is easily checked that
2 2
1 2 3
( )( )
1
1 2 1 3 2 3
3 ( )
1 2 3
3
2 2 4
432 [(1 ) 4 ]( )
It then follows that:
(i) All the roots are real and distinct iff
1 2
4 (1 )
, where
2 1 2
4 (1 )
for
0 1
, while
1 2 2
4 (1 )
for
1
.
(ii) The second-order approximate curve (9b) has at most one
inflection point in the interval
0 1
. More specifically, that inflection
point occurs when < (>) 1 and 1
( ) 2 (1 )
or
2
( )
.
EMPIRICAL EVIDENCE
The test of the second-order approximate WPC, measured in terms of
the ‘actual’ real wage rate, with data from the flow SIOTs of the Greek
economy, spanning the period 1988-1997, gave the results summarized
in Table 1. This table reports:
(i) The actual, *, and the estimated, e, values of , 0 1, at
which the inflection points occur.
(ii) The signs of the discriminant, , and of 1 –.
(iii) The mean of the absolute error in this approximation, i.e.
1
Z A
0
MAE w w d
(iv) The Euclidean angles (measured in degrees) between pS(1) and
vJm, m = 0, 1, 2, which are denoted by m.
TESTING BIENENFELD’S SECOND-ORDER APPROXIMATION... / 167
(v) The modulus of the subdominant eigenvalue, |J2|, and the
arithmetic (AM) and geometric (GM) means of the moduli of the non-
dominant eigenvalues.
Table 1
Indicators and determinants of the accuracy of the second-order
approximate wage-profit curve
1988 19 89 1990 1991 1992 1993 1994 1995 1996 1997
* 0.269 0.300 0.360 0.302 0.534 0.385 0.336 0.110 0.200 –
( 0)
z
w
��
e0.275 0.298 0.311 0.299 0.410 0.325 0.302 0.216 0.247 0.136
<0 <0 <0 <0 <0 <0 <0 <0 <0 <0
1–>0 >0 >0 >0 >0 >0 >0 >0 >0 >0
MAE 0.007 0.009 0.008 0.007 0.004 0.007 0.008 0.006 0.006 0.005
029.3 o 31.1 o 30.4 o 30.0 o 25.3 o 26.5 o 27.8 o 28.1 o 27.3 o 26.4 o
115.0 o 16.6 o 15.7 o 14.8 o 10.3 o 13.3 o 13.7 o 13.1 o 12.5 o 10.8 o
26.4 o 8.3 o 7.7 o 7.2 o 4.3 o 6.7 o 6.9 o 5.9 o 5.7 o 4.7 o
|J2| 0.643 0.683 0.675 0.657 0.624 0.667 0.678 0.655 0.664 0.641
AM 0.168 0.175 0.176 0.176 0.185 0.174 0.171 0.161 0.167 0.157
GM 0.086 0.086 0.088 0.087 0.089 0.081 0.088 0.074 0.074 0.073
From this table, the associated numerical results and the hitherto
analysis we arrive at the following conclusions:
(i) Although the systems deviate considerably from the equal value
compositions of capital case (see the values of 0), the approximation
works pretty well. With the exception of the year 1997, both curves
switch from convex to concave at ‘adjacent’ values of the relative profit
rate (the error |e – *| is in the range of 0.002 to 0.124), and the MAE
is no greater than 0.010. The graphs in Figure 1 are sufficiently
representative and display the actual (depicted by a solid line) and the
approximate (depicted by a dotted line) WPCs for the years 1989 (where
MAE, m and
2
J
exhibit their highest values) and 1997 (where
Z
0
w
��
).
(ii) In all systems, vJm tend quickly to S
(1)
p, and the moduli of the
first non-dominant eigenvalues fall quite rapidly, whereas the rest
constellate in much lower values forming a ‘long tail’. In fact, as it has
already been pointed out (Mariolis and Tsoulfidis, 2011, pp. 104-105),
the moduli of the eigenvalues of each system matrix follow an
exponential pattern of the form
0.2
0 1
exp( )
y x
168 / THEODORE MARIOLIS
where 0
1.738 1.827
,1
0.986 1.040
and the R – squared
is in the range of 0.975 to 0.991. These findings, which are in absolute
accordance with those detected in other studies of quite diverse actual
economies (see Mariolis and Tsoulfidis, 2015a, chs 5-6, and the
references therein), indicate that the accuracy of this second-order
approximation would not be so sensitive to the numeraire choice.
1989
0.5
1.0
0.5
0.5
1.0
1997
0.5
1.0
0.5
1.0
Figure 1: Actual and approximate wage-relative profit rate curves; years 1989
and 1997
-
TESTING BIENENFELD’S SECOND-ORDER APPROXIMATION... / 169
CONCLUDING REMARKS
Using data from ten symmetric input-output tables of the Greek
econo my, it has been argued that Bienen feld’s se cond-order
approximate wage-profit curve, which has at most one inflection point
in the economically relevant interval of the profit rate, is accurate
enough. This argument reduces to the skew distribution of the
eigenvalues of the matrices of vertically integrated tech nic al
coefficients, and suggests that low-dimensional models, with no more
than three industries, can be profitably used as surrogates for actual
single-product economies.
Future research efforts should be directed to the ‘inverse problem’,
that is, to (i) determine the conditions that notional production systems
should fulfil in order to generate Bienenfeld’s-like low-order wage-
price-profit rate curves; and (ii) explore the structural interfaces of those
systems with the actual ones.
Acknowledgements
I am indebted to two anonymous referees for comments and hints. A first draft of
this paper was presented at a Work sh op of the ‘Stud y Group on Sraffia n
Economics’ at the Panteion University, in December 2014: I am grateful to Nikolaos
Rodousakis and George Soklis for remarks. Furthermore, I would like to thank
Lefteris Tsoulfidis for suggestions and discussions. The usual disclaimer applies.
Notes
1. For the available data and the construction of the relevant variables, see
Tsoulfidis and Mariolis (2007, pp. 435-437).
2. The transpose of a 1×n vector
[ ]
j
y
y is denoted by yT. Furthermore, A1
denotes the Perron-Frobenius (P-F) eigenvalue of a semi-positive n×n matrix
A, and T
1 1
( , )
A A
x y
denote the corresponding eigenvectors, while Ak, k = 2, ...,
n and 2 3 ...
n
A A A , denote the non-dominant eigenvalues.
3. For the derivation of an alt er nati ve homographic approximati on, i.e .
S T 1
[1 ( 1)]
w
vJz , see Mariolis (2015).
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