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Fabio Petri - Microeconomics for the critical mind - provisional textbook
CHAPTER 2
CLASSICAL VALUE THEORY: PROOFS AND GENERALIZATIONS FOR
SINGLE PRODUCTION AND FIXED CAPITAL
2.1. This chapter presents a more mathematical treatment of the theory of long-period
prices, supplying more rigorous support for part of what was explained in the previous
chapter, and adding a discussion of other important issues: Leontief models, choice of
techniques, reswitching, simple fixed capital as a special case of joint production (the
general case of joint production, and land rent in multisector models, are discussed in
chapter 10). [Some recent Marxist approaches to the labour theory of value, discussed in
Appendix to Chapter 1, are formalized in an Appendix.]
The analysis presented in this chapter is not relevant only for the classical approach.
The notion of long-period prices is important independently of whether one adopts the
classical or some other approach. The thesis that relative product prices gravitate toward
long-period levels characterized by a uniform rate of return on the supply price of capital
goods is to be found not only in the classical authors, but also in Marshall, Jevons, Walras,
Wicksell, Samuelson.... ; indeed, although somewhat obscured by recent general
equilibrium theory, this notion continues to dominate applied economics, and it is more and
more re-asserting its centrality to-day also in theoretical work.
This chapter uses matrices; the reader must know the basic elements of linear
algebra. A review is available in the Mathematical Appendix, with some exercises. Readers
who need to study the present chapter before completing attendance to a mathematics
course where vectors and matrices are explained, should be able to obtain from that section
of the Mathematical Appendix a provisional level of familiarity with linear algebra
sufficient to understand the contents of this chapter (although a better grasp will be of
course obtained when the mathematics course is completed). Unless otherwise explicitly
indicated, vectors are to be intended as column vectors, but once a vector is defined as a row
vector then its symbol is not accompanied by a transposition apex.
We initially consider an economy where production is in yearly cycles, and where all
produced means of production are circulating capital goods, i.e. goods which when used as
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inputs disappear in the course of a single production cycle. In modern terminology they
would probably be called intermediate goods, but there is some ambiguity about the
meaning of the latter term (as we will see when we come to the non-substitution theorem),
so we avoid it. We also initially assume no joint production (and therefore we assume that
all capital is circulating capital: durable capital generates a form of joint production, as we
will explain). There is only one type of labour. We initially assume that land is
overabundant, hence a free good which we need not consider it explicitly among the inputs.
We initially take as given the productive methods of the several industries,
represented by vectors of technical coefficients i.e. amounts of inputs per unit of output.
There are n products, and to produce one unit of the j-th commodity it is necessary to use aij
units of the i-th commodity as means of production, i=1,...,n, and also aLj units of labour.
The technical coefficients of the j-th industry are represented by the column vector
(a1j,...,anj, aLj)T. The technical coefficients of produced inputs form an n×n square matrix
A=[aij], i,j=1,...,n. Labour technical coefficients form a row vector aL=(aL1,...,aLn). In the
A
matrix   , formed by adding to matrix A the vector aL as one more row, each column is
a L 
the set of technical coefficients of one industry.
Let us indicate the wage rate as w, prices as p1,...,pn, the rate of profit as r. Let us
initially treat the wage as advanced, i.e. paid at the beginning of the year while the products
come out at the end of the year. The equations determining relative prices of production are:
(1+r)(p1a11+...+pnan1+waL1) = p1
[2.1] .......................................................
(1+r)(p1a1n+...+pnann+waLn) = pn
If (r’, w’, p1’,..., pn’) satisfy these equations, then also (r’, tw’, tp1’,..., tpn’) satisfy
them, with t a positive scalar. An equation fixing the numéraire must therefore be added if
one wants to fix the absolute values of prices and of w, but for the moment we are
unconcerned by this issue. Using vectors and matrices, equations [2.1] can be written as
[2.2] (1+r)(pA+waL) = p
where r and w are scalars, and p=(p1,...,pn) is the row vector of prices. Readers not
thoroughly familiar with the matrix representation of systems of equations should check, by
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applying the rules about matrix and vector multiplication to [2.2], that one indeed obtains
the system of equations [2.1]. If one assumes wages paid at the end of the year, one obtains
[2.3] (1+r)pA+waL = p.
Let now x=(x1,...,xn)T be the column vector of quantities produced (this is often
called the vector of activity levels of the productive processes) and let y=(y1,...,yn)T be the
column vector of net products; the connection between the two is given by
[2.4] y = x – Ax = (I – A)x
where I is the identity matrix. For example if n = 2, system [2.4] is
y1=x1–a11x1–a12x2,
y2=x2–a21x1–a22x2,
which says that the net product of the i-th commodity is the quantity produced of it, minus
the quantity consumed of that commodity in the entire economy as means of production (not
as subsistence of workers).
Let now z = (z1,...,zn)T be the column vector of “subsistence”, or average physical
consumptions, per unit of labour. Then the surplus product is the vector s defined as[1]:
[2.5]
s := y–z(aLx) ≡ y–Nz = x – Ax – Nz
where N:=aLx is total labour employment, a scalar. The surplus product is obtained by
subtracting from the quantities produced (the social product) both the means of production
used up, Ax, and the subsistence consumption of workers, Nz.
Classical authors considered the subsistence consumption of workers to be as
indispensable to production as the means of production, and, having in mind the yearly
production cycle of agriculture, treated that subsistence consumption as necessarily
advanced by society, because workers must subsist during the year. Define the n×n matrix C
of technical coefficients of inputs inclusive of the subsistence inputs required by labour,
through the rule cij := aij + ziaLj i.e.:
The symbol ‘:=’ means ‘is defined as equal to’ and serves to introduce a new notion. The
symbol ‘≡’ means ‘is identically equal to’ and stresses that the two notions on the two sides of it
cannot take different values. Of course if A:=B then A≡B, but the converse is not generally true.
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C := A + zaL
(The reader not very familiar with matrices should make sure she is clear that zaL is an n×n
matrix.) With this notation, equation [2.5] becomes:
[2.7] s := y – zaLx = x – Ax – zaLx = x – Cx
If the wage rate only allows the purchase of the subsistence bundle, i.e. if
w = p1z1+...+pnzn = pz,
and if wages are advanced, then one can re-write equation [2.2] as (1+r)(pA+pzaL) = p, or,
since pA+pzaL = p(A+zaL) = pC:
[2.8] (1+r)pC = p .
2.2. Let us now remember that a real or complex number λ is said eigenvalue of a
square n×n matrix A if it satisfies the equation
Ax = x
for some real or complex scalar λ and some n×1 real or complex vector x different from the
null vector (0,...,0). Since this is equivalent to Ax=Ix where I is the identity matrix,
another form for the same equation is
(I–A)x=0
that makes it clear that we are dealing with a system of equations which is linear and
homogeneous in x. It is well known that a linear and homogeneous system of n equations in
n variables has a solution different from the null solution only if the determinant of the
matrix of coefficients is zero. An eigenvalue λ is a real or complex number that renders zero
the determinant of the matrix I–A.
Suppose that λ is an eigenvector of A and the column vector of real or complex
numbers x≠0 satisfies the equation Ax = x ; the vector x is called a right eigenvector of
matrix A, and the equation is called a right-eigenvector problem. For a given eigenvalue, an
eigenvector is only defined up to a multiplicative scalar, because if it is multiplied by a
scalar it still solves the equation. Thus two equi-proportional eigenvectors are generally
considered to be the same eigenvector. Therefore when we will say that two eigenvectors
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are different, we will mean (unless explicitly indicating otherwise) that they are not
equiproportional, i.e. that they are linearly independent.
If the equation is yA=λy, then a row vector y≠0 solution of this equation is called a
left eigenvector of A, and the equation is called a left-eigenvector problem. The reader not
very familiar with matrices is invited to check, by writing the explicit system of equations,
that, with AT the transpose of A, if y is a left eigenvector of A, then yT is a right eigenvector
of AT. (It is not in general the case that if y is a left eigenvector of A, then yT is a right
eigenvector of A.)
We need the following elementary facts about eigenvalues. An eigenvalue of a
matrix A renders the determinant of the matrix I–A equal to zero, and is therefore a root of
the equation that puts this determinant equal to zero; this equation is a polynomial equation
of degree n if A is an n×n matrix; by the fundamental theorem of algebra, the equation {A’s
determinant = 0} has n real or complex solutions, possibly repeated. To each eigenvalue is
associated an eigenvector; an eigenvalue repeated t times is generally associated with t
different (i.e. linearly independent) eigenvectors. The eigenvectors can be real or complex;
clearly, if the eigenvalue is complex, the associated eigenvector(s) must be complex. If AT
is the transpose of the square matrix A, then I–AT is the transpose of I–A (check it on a
2×2 matrix); the determinant of a matrix is the same as the determinant of its transpose
(check it as an exercise on a 3×3 matrix); hence a matrix and its transpose have the same
eigenvalues. Thus if Ax=λx has a solution with eigenvalue λ* and column vector x≠0, then
there exists a row vector y≠0 that solves yA=λ*y.
2.3. If we define
[2.9]
C* := 1/(1+r),
we can re-write equation [2.8] as
[2.8’]
pC=C*p ,
which highlights the fact that equation [2.8] is a left-eigenvector problem, and that the
vector p of prices of production is a left eigenvector of C. The problem is, that C has many
eigenvalues and left eigenvectors. In a realistic economy, the number n of different
commodities might be hundreds of thousands, so C would have hundreds of thousands of
eigenvalues. Does this mean that the classical economists were wrong in believing that the
rate of profit and the prices of production were uniquely determined? What helps us on this
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issue is that only real and non-negative solutions for the prices are economically acceptable:
complex or negative prices would have no economic interpretation. We are also helped by
the fact that C is non-negative. We can use a series of results on eigenvalues and
eigenvectors of non-negative matrices, generally collected under the name of PerronFrobenius theorem on non-negative matrices.
2.4. As a preliminary, we need the notion of indecomposable matrix. A square matrix
A is called decomposable (also reducible) if it is possible, by a series of exchanges of place
of rows and of the corresponding columns, to give the matrix a form
~
~ A
A   11
0
~
~
A12 
~ 
A22 
~
with A11 , A22 square submatrices, and 0 representing a matrix of zeroes.
If A is not decomposable, it is called indecomposable or irreducible.
Decomposability has a clear and important economic meaning. Exchanging the place
of two rows and of the corresponding columns of a matrix of technical coefficients, say
between places 1 and 3, simply means re-numbering commodities, giving number 3 to the
commodity previously numbered 1, and number 1 to the commodity previously numbered 3
(the reader is invited to check with a 3×3 matrix). Let us then suppose that commodities
have been re-numbered such that the matrix of technical coefficients A has taken the form
A11 A12 
 0 A  with A11 a square indecomposable s×s submatrix, with s  n and as small as
22 

possible. This means that the first s commodities[2] are inputs, directly or indirectly, in the
production of all commodities[3], while the other n–s commodities are not inputs (neither
directly nor indirectly) in the production of the first s commodities. (An indirect input of a
commodity is, directly or indirectly, necessary for the production of some of its direct
inputs.) The first type of commodities are called basic commodities, and the industries
producing them are called basic industries. The other commodities are called non-basic. The
From now on we will generally use for produced goods the term ‘commodities’,
traditionally used to indicate goods that are sold and therefore have a price.
3
We are excluding the case of economies which can be divided into two totally independent
sub-economies (in the sense that each one of them uses no commodity produced by the other subeconomy); we are also excluding the case of no commodity directly or indirectly used for the
production of all other commodities. The latter, ‘Austrian’, case will be discussed later.
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production of all commodities would have sooner or later to stop if even only one basic
commodity were no longer produced. On the contrary, the production of basic commodities
does not need the production of non-basics. In chapter 1, iron in equation [1.12], and iron
and cloth in equations [1.14-1.16] in footnote 55??, were non-basics.
Which commodities are basic depends on whether one intends the technical
coefficients as also including the subsistence of workers, or not. If one does (i.e. if one uses
matrix C), then as long as labour is directly or indirectly required for the production of all
commodities, all commodities included in the subsistence bundle z are necessarily basic,
and the ensemble of basic industries can be called the wage-industries: it collects the
industries that produce commodities included in the subsistence bundle or directly or
indirectly necessary for their production. The other industries can be called non-wage
industries or luxury-goods industries. Some wage-industries may fall among the non-basic
industries if one utilizes the matrix A, whose technical coefficients only indicate the means
of production. For example, let us assume that the economy produces, numbered in this
order: corn, flour, bread, and brioches. Corn is used as means of production in the
production of corn and of flour. Flour is a means of production of bread and of brioches.
Subsistence consists of bread only, that is, z = (0,0,z3,0)T.
a11 a12
0
0
Then A  
0
0

0
0
0
a23
0
0
0
a24 
and with this matrix, only corn is basic.
0

0
On the contrary in matrix C bread is an input in all industries if labour is required in
all production. Matrix C has the form:
c11 c12
0
0
C= 
c31 c32

0
0
0
c23
c33
0
0   a11
c24   0

c34   z3 a L1
 
0  0
a12
0
z3 a L 2
0
a23
z3 a L 3
0
0
0 
a24 
,
z3 a L 4 

0 
thus only brioches are non-basic.
It seems impossible that a commodity does not need labour to be produced, at least
indirectly (the production process may need no direct labour − it may for example consist of
simply waiting for time to pass, as for 12-years-old whiskey − but it will still need some
inputs whose production has required labour). Therefore in the C-representation of technical
coefficients we assume that basic commodities do exist. On the contrary, with the A7
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representation, basic commodities may not exist in two cases. The first is when there are
commodities produced by labour alone, and by going backwards from the inputs of a
commodity to the inputs of those inputs and so on one always ends up sooner or later with
inputs produced by labour alone[4]. This is logically possible and it has been often
postulated by theorists because it makes life simpler for some problems, but it is clearly
~
implausible in modern economies. The second case is when A12 =0: the economy consists of
two (and possibly more, in the general case) entirely disconnected sub-economies, each one
of which needs no input from the other(s). This case appears applicable only to economies
totally separated by geographical or political barriers, but then each economy should be
studied separately and there is no reason to try to determine prices and the rate of profit
simultaneously for all of them[5]. So we exclude this second case too. Thus in this chapter
we assume that there is at least one basic commodity.
In what follows we will often assume that C, or A, is indecomposable. This can be
interpreted in either of two ways:
1) the economy does not produce non-basic commodities;
2) the economy also produces non-basics but we are restricting attention to the sole
subset of basic industries and prices of basic commodities.
The legitimacy of the second interpretation derives from the fact that – as already
argued in chapter 1 – the prices of non-basics do not enter the price equations of basic
commodities, which can therefore be examined in isolation.
4
If there is some commodity that uses itself, directly or indirectly, for its production, then
the process of going from its inputs to the inputs of those inputs and so on would never end. We
will assume that no commodity is produced by labour alone. We do not deny that some paid
productive activities that produce services, such as private maths lessons, or massage, come close to
being produced by labour alone, but we choose to neglect them for the following reason: with
wages paid in arrears, the price of these services essentially coincides with their wage costs; thus
these prices are immediately determined once the real wage rate is given; and if these services enter
into the production of other goods, one can replace them with the labour that produces them, and
therefore there is no need to make them appear in the list of the commodities for whose price
determination we need a system of equations.
5
Capital mobility between the two technically separated economies is abstractly conceivable
but it requires at least a rate of exchange between currencies, which will generally imply the
possibility of trade, and then also the possibility of trade of inputs, which will put an end to the
separation of the economies.
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2.5. A and C are non-negative matrices. Thus we can utilize the results of the PerronFrobenius theorem (in fact a collection of theorems) on non-negative matrices. Below is a
statement of the more important results contained in this theorem (proofs in the Math.App.).
Perron-Frobenius theorem.
Let A be a square non-negative indecomposable matrix. Then:
(i) A has a real eigenvalue * > 0, not repeated, and dominant (that is, not smaller
in modulus than any other eigenvalue), and to it and only to it is associated a real nonnegative, and in fact positive, eigenvector[6] x*; for each other eigenvalue  of A, it is
   , and    if A is positive[7];
(ii) (ρI–A)–1 > 0 (where ρ is a real scalar) if and only if ρ>* ;
(iii) * is an increasing function of each element aij of A;
(iv) if s is the smallest, and S the greatest, of the sums of the elements of a row of A,
then s<*<S , unless s=S in which case s=*=S; the same holds for the sums of column
elements of A.
If A is decomposable the previous results are weakened as follows:
(i') A has at least one non-negative real eigenvalue; to the highest non-negative real
eigenvalue λ* is associated a semi-positive[8] eigenvector; if  is an eigenvalue of A, then
it is    ;
(ii')  I  A 1  0 if and only if    ;
(iii')   is a non-decreasing function of each element a ij of A.
2.6. Let us apply these results to the study of equation [2.8].
6
We do not specify whether it is a right or left eigenvector because the statement, although
traditionally intended for right eigenvectors, in fact applies to either, since a left eigenvector is a
right eigenvector of AT, again a square non-negative indecomposable matrix, and with the same
eigenvalues as A.
7
A matrix having a non-repeated eigenvalue of modulus greater than the modulus of all
other eigenvalues is called primitive. The last result in (i) can be expressed as: every positive
indecomposable matrix is primitive. But a matrix of technical coefficients is normally not positive,
it contains zeroes. An imprimitive nonnegative indecomposable matrix A, that is, having a second
eigenvalue equal in modulus to λ*, can, by renumbering rows and columns, be brought to this
structure: all elements are zero except a12, a23, a34, ... , an-1,n, an1 .
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Let us start in the same way as Sraffa’s Production of commodities by means of
commodities, by considering first an economy where the surplus product is zero. That is,
[2.10]
x – Cx = 0
or Cx = x
or (I – C)x = 0.
For a solution x>0 to exist simultaneously with a solution p>0 to equation [2.8], C
must be a special matrix. In order to grasp its nature, let us change the units in which we
measure each commodity (the choice of units is always arbitrary) and let us choose as unit
of measurement for each commodity the total quantity produced of it, so that xi=1 for each
i=1,...,n. Let us call the matrix, that corresponds to C in these new units, C*. Then by
assumption c*i1+...+c*in=1, the sum of the elements of each row of C* is 1. A non-negative
matrix with all row sums (or with all column sums[9]) equal to 1 is called a stochastic
matrix. This will be useful. But first let us understand the change that a matrix of
(subsistence-inclusive) technical coefficients C undergoes if we change the units in which
commodities are measured. Suppose we want to change the unit in which commodity 1 is
measured, passing to a unit that consists of x1 old units; for example, if the old unit is
kilograms and the new unit is tons, then x1=1000. To produce a ton of good 1 you need to
multiply by 1000 the inputs that produce 1 kg of good 1; while if another production process
uses 100 kg. of good 1, it uses 0.1 tons of good 1. So the first column of C must be
multiplied by x1, and the first row of C must be divided by x1. As a result of the two
operations, c11 does not change; and indeed, if, say, to produce 1 kg. of corn one needs 0.1
kg. of seed corn, then to produce 1 ton of corn one needs 0.1 tons of seed corn.
If x1 is the total quantity produced of good 1 in the old units, and one chooses that
quantity as the new unit, then one obtains that in the new unit the economy is producing 1
unit of good 1, and matrix C is altered as indicated. If one does the same for each
1 
commodity, then x becomes the unit vector e= ... and C is altered as follows: we must
 1 
‘Semipositive’ means non-negative and with at least one positive element.
If we were to consider the transpose of C, it would have the column sums equal to one.
This is what one obtains if one prefers to have the rows of the matrix of coefficients represent the
technology of each industry, i.e. if cij indicates the technical coefficient of input j in industry i. Then
prices are a column vector, the price equation is (1+r)Cp=p, and the quantities produced are a row
vector. For example Kurz and Salvadori (1995) prefer this indexing to the one adopted in this book.
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 x1 0 . 0 
0 . . 0
 with on the main
postmultiply C by the diagonal matrix X  diag  xi   
. . . .


 0 . 0 xn 
diagonal, for each commodity i, the number of old units xi corresponding to one new unit;
and we must pre-multiply C by the diagonal matrix X 1
0 
1 / x1 0 .
 0
. .
0 

,
 diag 1 / xi  
 .
. .
. 


. 0 1 / xn 
 0
obtaining the matrix of subsistence-inclusive technical coefficients C* = X–1CX that
expresses the same technical relations but in the new units[10].
If this change of units is applied to the no-surplus economy of equation [2.10], that
equation becomes e = C*e, and C* has all row sums equal to 1, hence it is a stochastic
matrix, with dominant real eigenvalue λ*=1, and it is indecomposable if C is
indecomposable because it has zeroes in the same places as C. Now, C* and C are similar
matrices (two matrices A, B are called similar if there is a diagonal matrix X such that
B=X−1AX); and it is a theorem of matrix algebra that similar matrices have the same
eigenvalues (cf. Math.App.). Conclusion: a change in the units in which commodites are
measured does not change the eigenvalues.
Hence, if the economy does not produce a surplus, the matrix of technical
coefficients C has a real dominant eigenvalue equal to 1 whichever the units in which
commodities are measured.
Does this guarantee that there are prices capable of making this economy function,
i.e. such that each industry is capable of purchasing its inputs with its output? If C is
indecomposable, then the answer is yes because, by property (iv) of the Perron-Frobenius
theorem, 1 is the dominant eigenvalue, and to it, and only to it, is associated both a real and
positive right eigenvector x* and a real and positive left eigenvector p*; by equation [2.9]
the rate of profit is zero; thus at prices p* each industry makes no profit but no loss either,
i.e. its output suffices to purchase the inputs it needs to go on functioning. Now, economic
10
Sraffa does not adopt this change of units, and prefers to adopt a representation of the
economy in terms, not of technical coefficients, but of total quantity produced, and total inputs
used, by each industry (and this, not only for the economy without surplus but also in subsequent
chapters). Therefore Sraffa’s matrix of input uses corresponds to CX, where X is the diagonal
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reasoning shows that C must be assumed indecomposable in this case of zero surplus. If C
is decomposable, it includes non-basic industries, that is to say, industries whose output
does not appear among the inputs of the other group of industries. The sole way to have a
zero surplus in this case is that these industries, taken as a whole, must absorb as inputs their
entire production; but this would mean that the ensemble of these industries is using as
inputs more than it produces, because it uses its entire production and furthermore some
basic commodities. Since its productions are not used for subsistence (subsistence goods are
all basic), these industries are totally senseless, they absorb inputs from outside industries
without yielding anything in exchange. Closing them down would only improve things,
because the ensemble of basic industries, if considered in isolation, is producing a surplus,
which is wasted by the non-basic industries; this surplus would become available for
consumption or growth, if the non-basic industries were closed down. The price system will
indicate this fact in the following way: if basic commodities have positive prices, then the
ensemble of non-basic industries necessarily makes a loss, because its product will be worth
less than its inputs; it cannot go on existing[11]. In conclusion, a zero surplus only makes
economic sense if all industries are basic.
The uniqueness (apart from scale) and positivity of the left eigenvector of C confirms
the correctness of Sraffa’s statement (1960, p. 4) that in an economy without surplus “There
is a unique set of exchange-values which if adopted by the market restores the original
distribution of the products and makes it possible for the process to be repeated; such values
spring directly from the methods of production.” These exchange-values (relative prices)
are associated with a zero rate of profit, because, the dominant eigenvalue being 1, from
[2.9] it is r=0 and the price equation is p=pC.
2.7. What makes profits possible is the existence of a surplus product. This can be
seen as follows. If, in the economy without surplus, one of the technical coefficients of
matrix defined in the text. Therefore it suffices to pre-multiply it by X-1 to obtain C*.
11
In this case the sole possibility for the ensemble of non-basic industries to avoid losses is
that the basic commodities they use (but then all basic commodities − because directly or indirectly
the price of each basic commodity enters the cost of all basic commodities) have price zero relative
to non-basic commodities, and that the rate of profit be zero. The solution associated with the
positive rate of profit determined by the subset of basic industries will entail negative prices of the
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matrix C decreases (either because of technical progress, or because the real wage rate
decreases) without going to zero, at the old prices there will arise in that industry a profit,
and no positive price vector can cause profit to disappear without reappearing in any other
industry, because now the vector of quantities produced is greater in only one component
than the vector of quantities used up (inclusive of the subsistence of workers). The sole way
to make the values of these two vectors equal, and thus to make profit disappear, would be
to give price zero to the commodity of which there is a surplus. But C being still
indecomposable, property (i) of the Perron-Frobenius theorem implies that prices are all
positive. Therefore there must be a profit somewhere; but then owing to competition the rate
of profit must be uniform, so there must be profit in all industries.
Mathematically, property (iii) of Perron-Frobenius theorem causes λC* to decrease if
any technical coefficient decreases. Since C is indecomposable, its dominant eigenvector is
the sole one associated with meaningful (non-negative) prices, in fact positive prices. Thus
we can put λC*=1/(1+r), and its decrease means an increase of r, that passes from zero to
positive.
2.8. We show now that the rate of profit r=(1/C*)–1 can also be interpreted as the
growth rate of the economy, if the entire surplus is re-invested and all industries grow at the
same rate. Let g be this uniform rate of growth. Then for each product its production must
n
be 1+g times its total utilization as input in the entire economy, or: xi   cij x j 1  g 
j 1
which, using matrices and vectors, becomes:
[2.11] (1+g*)Cx^ = x^.
Again we obtain that it must be C* = 1/(1+g*) because the sole economically meaningful
(i.e. real and nonnegative) right eigenvector x^ is the one associated with the dominant
eigenvalue. Thus g*=r. We can conclude that the rate of profit is positive if and only if, once
non-basic commodities. The reader is invited to check it for C =

0.5 0.5
.
0
1
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the real wage rate is given and included in the technical coefficients, the economy is capable
of positive uniform growth.
The rate of growth g*=(1/C*)–1 can be shown to be the maximum common rate of
growth of all industries. To prove it, let us suppose that the industries are initially in the
proportions x^: the common rate of growth of all industries in these proportions cannot be
greater than g* because growth at rate g* already uses all produced quantities. Now what
happens if we change the proportions among industries? Let us define the rate of surplus of
commodity i as
σi ≡ (xi - ∑jcijxj)/ ∑jcijxj.
Suppose this rate is 10%. It means that next period the total utilization of commodity i can
be at most 10% higher than in the current period. The rate of surplus of commodity i is an
increasing function of xi and a decreasing function of xj for j≠i; in fact it only depends on
the ratios xj/xi, and it is a decreasing function of these ratios. This means that if one industry
expands while the other industries remain unaltered, all but one of the rates of surplus
decrease. Thus, if one starts from the proportions associated with g*, which are such that all
rates of surplus are the same, any change in the proportion among industries causes at least
one rate of surplus to decrease. Now, the maximum common rate of growth of all
industries, if the proportions among outputs are given, is obviously equal to the minimum of
the rates of surplus. Therefore for all proportions among industries different from the ones
█
associated with g*, the maximum common rate of growth is less than g*.
Let us now note that the unique proportions among outputs, associated with growth at
rate g*, cause the proportions in which commodities appear among the total inputs to be the
n
same as among outputs: for each i=1,...,n one has xi  1  g * cij x j where
j 1
n
c
j 1
ij
x j is the
quantity of commodity i used as input in the entire economy; therefore the output vector is
the vector of total inputs, multiplied by (1+g*). Then total inputs, and outputs, can be seen
as different quantities of the same composite commodity; and the surplus too − output
vector minus input vector − can be seen as a quantity of that same composite commodity.
This commodity, of composition determined by the eigenvector x^ solution of [2.11], has
been called standard commodity by Sraffa, who has noticed that it allows the determination
of the rate of profit r=g* as a ratio between physically homogeneous quantities, i.e. without
having to determine relative prices − Ricardo’s dream.
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2.9. In modern analysis, it is more common not to include the workers’ consumptions
in the technical coefficients, and to assume that wages are paid at the end of the production
cycle rather than at the beginning. This is partly explained by the lesser importance of the
agricultural yearly cycle in modern industrialized economies. Wages are normally paid after
work has been performed, but if workers’ consumption consists essentially of agricultural
goods that come out once a year, like corn, then in between harvests the wages, even if paid
at the end of each week of work, are used to buy goods which must be there since the last
harvest, so the price at which wage goods are sold must include a rate of profit also for the
time in between the harvest and the sale of the wage goods; then to assume that wages are
advanced is a natural assumption, a simple way to approximate this situation (cf. Steedman,
Marx after Sraffa, ch. ??). When workers’ consumption includes a majority of industrially
produced goods, for which often the production cycle is even shorter than the interval
between two wage payments, then to assume that wages are paid at the end of the
production cycle is a more natural and realistic assumption. This means a representation of
technology through matrix A and the row vector of labour inputs aL. The price equations
will then take the form
[2.12] (1+r)pA + waL = p
and, beside some condition fixing the absolute levels of prices and of the wage, they must
be supplemented by some condition fixing the purchasing power of the wage rate, e.g.
w=pz=p1z1+...+pnzn where z is the average basket of goods purchased with the wage rate.
Since we have raised the issue of the length of the production cycle, arguing that it
can often be quite short, it is opportune now briefly to stop on this issue. For most
commodities it is unrealistic to assume a yearly production cycle, because production takes
much less. Then, if the purpose is the determination of relative prices and of the rate of
profit, the solution is to take as unit production period a period shorter than a year, possibly
very short, and to break up longer production processes into a number of processes, each
one of them taking one production period and producing an intermediate product which then
enters as one of the inputs into the subsequent production process. Thus the production of
corn can be broken up into a succession of production processes of, say, one week length,
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each one producing a different ‘product’, consisting of a more and more advanced stage of
ripeness of corn plants. The introduction of each one of these additional products also
introduces an additional price equation, so the determinateness properties of system [2.12]
are not altered. If the length of the production period is chosen sufficiently short, then even
for continuous production flows the assumption that production is of the point-input pointoutput type, with the product of a period coming all out at the end of the period, is
acceptable because the difference this assumption makes to revenues and profits will be
negligible.
We will study system [2.12] after concerning ourselves with another important
system of equations, the Leontief input-output model, that derives from an interest in the
quantity side of the economy, in particular in the relationship between quantities of
commodities produced as net products, and quantities used up as inputs. This will permit an
interesting clarification of the meaning of the notion of labour embodied.
But before, we complete on the standard commodity by noting that it can also be
defined for the A-representation of technology: it will be the composite commodity with the
same composition as the vector x solution of Ax=λA*x. The previous reasonings apply
unchanged, except for ‘net product’ in place of ‘surplus’, A in place of C, and λA in place of
λC. The common proportion between net product and total input of each commodity now
represents both the rate of profit if the rate of wages is zero, and the maximum rate of
growth if workers receive no wage. Note that Sraffa suggests that actually matrix A might
include the basic subsistences of workers (i.e. might actually be matrix C), in which case w
would indicate the part of the wage which is a share of the surplus[12]: with such an
12
It may be interesting to read how Sraffa, who initially adopts the C-representation,
justifies passing to the A-representation at a certain point: “We have up to this point regarded wages
as consisting of the necessary subsistence of the workers and thus entering the system on the same
footing as the fuel for the engines or the feed for the cattle. We must now take into account the
other aspect of wages since, besides the ever-present element of subsistence, they may include a
share of the surplus product. In view of this double character of the wage it would be appropriate,
when we come to consider the division of the surplus between capitalists and workers, to separate
the two component parts of the wage and regard only the ‘surplus’ part as variable; whereas the
goods necessary for the subsistence of the workers would continue to appear, with the fuel, etc.,
among the means of production. We shall, nevertheless, refrain in this book from tampering with
the traditional wage concept and shall follow the usual practice of treating the whole of the wage as
variable. The drawback of this course is that it involves relegating the necessaries of consumption to
the limbo of non-basic products. This is due to their no longer appearing among the means of
production on the left-hand side of the equations: so that an improvement in the methods of
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interpretation, a zero wage would not be an absurd assumption (but it would simply reduce
the system of equations [2.12] to the system [2.8]).
Non-basic commodities.
2.10. We have already met the difference made by non-basic commodities in a
number of instances. Let us explore some of their implications.
Let us remember that the matrix of technical coefficients of basic commodities is
indecomposable, but that we can distinguish what may be called C-basics and C-non-basics
from A-basics and A-non-basics depending on whether technical coefficients include (case
C) or not (case A) the necessary consumptions of workers. Some authors call wage
commodites the C-basic commodities and luxury commodities the C-non-basics. (As we will
see, in the Leontief problem the non-basic commodities are A-non-basic.)
The existence of non-basics can create problems to the determination of prices of
production. We use now the C-representation. Let us re-number commodities so that the
m<n basic commodities are the first ones, and let C11 represent the indecomposable
submatrix of technical coefficients of the basic commodities. The system
[2.18] (1+r)pβC11 = pβ
where pβ=(p1,...,pm) is the vector of prices of basic commodities, has been already
discussed. It determines positive relative prices of basics, and a rate of profit which is
positive if the dominant eigenvalue of C11 is less than one − as it must be, if it must be
possible to produce non-basics and yet have non-negative net products. The question is
whether we can be certain that there exist positive prices of non-basics that guarantee the
same rate of profit.
production of necessaries of life will no longer directly affect the rate of profits and the prices of
other products. Necessaries however are essentially basic and if they are prevented from exerting
their influence on prices and profits under that label, they must do so in devious ways (e.g. by
setting a limit below which the wage cannot fall; a limit which would itself fall with any
improvement in the methods of production of necessaries, carrying with it a rise in the rate of
profits and a change in the prices of other products.) In any case the discussion which follows can
easily be adapted to the more appropriate, if unconventional, interpretation of the wage suggested
above.” (1960 p. 9-10) Here Sraffa suggests that one might also assume that the coefficients of
matrix A in fact include the subsistence consumptions, and w only measures the surplus part of the
wage. However, an asymmetry would then arise, in that one part of the wage would be advanced
(the rate of profit would be computed on it) and another part would be paid in arrears, and it is
unclear why this division should coincide with the subsistence-surplus division of the real wage.
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We are dividing C in four sub-matrices, C11, C12, C21=0, C22. The m×(n–m) submatrix C12 collects the basic inputs of non-basic industries. The (n–m)×(n–m) square matrix
C22 collects the non-basic inputs of non-basic industries. If C22=0, positive prices of nonbasics certainly exist, they equal the costs of the inputs, that are all basic, augmented of the
given rate of profit: in the typical equation (1+r)(p1c1j+...+pmcmj)=pj, where j=m+1,...,n,
once a numéraire is chosen that determines the absolute values of basic prices, all variables
on the left-hand side are given.
If C22≠0 but no non-basic commodity is directly or indirectly an input to itself, then,
by re-numbering the non-basic commodities, C22 can be made to become an upper
triangular matrix with zero main diagonal:
.
0 c m 1,m  2
0 0
c m  2,m 3

C *22   .
.
0
.
.
.

0
.
0

... c m 1,n 
...
. 
.  . (The
...
c n 1,n 

...
0 
reader is invited to make sure she understands why.) In this case too no problem arises in
determining the prices of non-basics: commodity m+1 only uses basics so its price is
determined by the given prices of basics and the given rate of profit; thus the prices entering
the cost of commodity m+2 are all given; and so on.
Complications can only arise if there are non-basic goods that directly or indirectly
are inputs to themselves. In this case by re-numbering non-basics we can transform matrix
C into an upper quasi-triangular matrix, e.g.:
C11 C12 C13 C14 
0 C
C23 C24 
22
C*  
0
0 C33 C34 


0
0 C44 
0
called quasi-triangular because it has square indecomposable sub-matrices Cjj on its main
diagonal. For such matrices we have the theorem:
The eigenvalues of a decomposable square matrix in quasi-triangular form coincide
with the eigenvalues of the square sub-matrices on its main diagonal.
Indeed it is easy to see that the determinant of a quasi-triangular matrix is the product
of the determinants of its square matrices on the main diagonal. Therefore for the
determinant of a quasi-triangular matrix to be zero, it suffices that one of the determinants
of the square sub-matrices on its main diagonal be zero. Now, if C is quasi-triangular, λI−C
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is quasi-triangular too; for λ to be an eigenvalue of C, it suffices that it renders zero one of
the determinants det [λI−Cjj], where I is the identity matrix of opportune size.
The dominant eigenvalue of C is the greatest of the dominant eigenvalues of those, of
the matrices Cjj, that are not all zeroes. This dominant eigenvalue might be different from
the dominant eigenvalue λ1* of the sub-matrix C11 (the sub-matrix of basic commodities). In
this case a difficulty may arise. We know that the price equations of basic commodities:
(1+r) pβC11 = pβ
suffice to determine their relative prices and the rate of profit r 
1  1


1

1
1
 1 (and that on
this issue Ricardo was right, and Marx wrong: the rate of profit only depends on the real
wage and on the conditions of production of the industries directly or indirectly producing
the commodities appearing in the real wage basket). But if for example λ1*<λ2*, then
1


1
1 
1
2
 1 : the right-hand side of this inequality is the rate of profit − let us indicate it
as r22 − that the non-basic commodities appearing in C22 are capable of yielding if one
imagines that all basic commodities (including wage goods) are free. This rate of profit is
the maximum one obtainable by raising indefinitely the prices of the non-basics in C22
relative to the prices of basics, and if the above inequality holds, it is less than r. (Note that
whether this is the case may depend on the level of the real wages: if real wages decrease,
this decreases some of the coefficients in C11 and therefore it lowers λ1 and raises r, while
r22 remains unaffected.) This means that r cannot be obtained in the production of those nonbasics, unless their prices become negative, which is economically unacceptable.
In this case, either those non-basics are not produced, or they are produced by people
who remain content with a lower rate of profit. The latter possibility (suggested by Sraffa,
1960, end of Appendix B) has been dismissed by some authors, who have argued that it will
be more convenient to sell the non-basics rather than use them as inputs, since the money
thus obtained can be re-invested in the production of basics, thus earning a higher rate of
profit; however, suppose the basic is a kind of beans, produced by farmers who risk
remaining unemployed if they turn to different jobs: for them, continuing the production of
the ‘unprofitable’ non-basic may be the best choice. Anyway non-basic commodities
entering as inputs into their own production are generally animals or plants (mink, orchids,
exotic beans, etc.); nature generally permits enormous maximum rates of increase of
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animals and of plants if these are protected from rival species; now, r22 is this maximum rate
of physical increase, assuming all the basics needed are available; so r22 can be safely
assumed to be greater than r. Accordingly we assume in what follows, when we admit nonbasic commodities, that the dominant eigenvalue of the sub-matrix of basic commodities is
also the dominant eigenvalue of C. (When we use the A representation of coefficients, we
make the same assumption, adapted to that representation.)
2.11. Let us now see the role of non-basics in determining the maximum rate of
growth, and the standard commodity.
It is intuitive that the maximum rate of growth of the economy requires that the
production of non-basics be zero. Any production of non-basics would absorb some of the
surplus (for the moment we adopt the C-representation of technology) of the sub-sector of
the economy consisting of the basic industries, thus reducing their maximum rate of
expansion.
We have argued, for indecomposable economies, that the composition of the
composite output commodity associated with the maximum rate of growth is the one and the
sole one that makes this composition equal to the composition of the composite total input
commodity necessary for its production, i.e. that satisfies the definition of the standard
commodity. We want to argue that the same is true in the presence of non-basics because
the standard commodity cannot include non-basics.
The reason is obvious for a first type of non-basics, the ones that are not used as
inputs in any industry: their presence among the inputs is zero and therefore also their
presence among the outputs must be zero. Then we can also exclude a second type, those
non-basics whose only use as means of production is for the direct or indirect production of
non-basics of the first type: their presence among the inputs becomes zero if the production
of the non-basics of the first type is zero. A third type of non-basics uses itself, directly or
indirectly, for its production. These are the non-basics to which correspond square submatrices Cjj not all zero, on the main diagonal of the quasi-triangular form of C (or of A).
For these, a composition in the output vector equal to their composition in the total inputs
vector is only achievable if their rate of growth equals (1/λj*)−1 where λj* is the dominant
eigenvalue of the matrix Cjj to which they belong. It would be a fluke if λj* were equal to
the dominant eigenvalue of the basics’ submatrix, in which case the sole solution is that
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these non-basics not be produced; but even if it were equal, then their positive production
would make it impossible to have the same composition in the output and in the input vector
for the basic commodities, since for these the only way to satisfy that condition is to let
them grow at their maximal rate, which absorbs their entire production. Thus this third type
of non-basics cannot be part of the standard commodity. A fourth and last type of nonbasics, the ones only required for the direct or indirect production of non-basics of the third
type, is then also excluded. These four types cover all possible non-basics, so our
demonstration is achieved[13].
These considerations developed for C-nonbasics also apply, mutatis mutandis, to Anonbasics. Actually with the A-representation of technology the possibility, that the
maximum rate of self-reproduction of some nonbasics (and therefore the maximum rate of
profit compatible with positive prices of all nonbasics) be inferior to the maximum rate of
profit (and of growth) of basics is more likely than with the C-representation, owing to the
absence of the necessary consumption of workers from the inputs. Thus imagine that the
sole basic commodity is corn. With the A-representation, R equals the maximum rate of
self-reproduction of corn if labour lives on air; R will be very high, in all likelihood superior
to the maximum rate of self-reproduction of many non-basic products, e.g. cattle. The
reason why we need not be worried by this possibility is that realistically the rate of profit
will always remain very far from such a high R, because labour cannot live on air. As Sraffa
puts it[14]: “Necessaries however are essentially basic and if they are prevented from
exerting their influence on prices and profits under that label, they must do so in devious
ways, e.g. by setting a limit below which the wage cannot fall”. Thus we can rather safely
assume that the rate of profit will always be below the minimum among the maximum rates
of self-reproduction of non-basics that use themselves in their production.
Leontief’s open model, and the Hawkins-Simon condition.
2.12. In order to understand Leontief’s model and its applications, the reader must
make an effort to abandon the assumption of a yearly production cycle, with all products
13
. ??Esercizio: In questa dimostrazione della non presenza di merci non-base nella mercetipo abbiamo distinto quattro varietà di merci-tipo: le si caratterizzi in termini di elementi positivi o
nulli nella forma quasi-triangolare [2.19] della matrice C. Si costruisca un esempio numerico in cui
la sottomatrice Cjj di merci non-base del terzo tipo ha diagonale principale nulla.
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coming out at the end of the year. The Leontief model is perfectly compatible with this case,
but it is more general and in its concrete applications it does not rest on such an assumption.
Let us start by observing that nothing in the definitions of the vector of total
productions x, or of the vector of net products y or of the surplus vector s, obliges us to
assume that the length of the production period or production cycle assumed in the price
equations is the same as the length of the accounting period used to calculate x, y or s. For
simplicity up to now we have identified the two periods through the assumption that the
year is both the accounting period and the production period. Now we must distinguish the
two, and admit for example that in a year − the accounting period − for at least some
commodities many production cycles may be performed. This is no impediment to
determining x as the vector of the total quantities produced during the year, nor to
determining y as x minus the non-subsistence inputs used up in order to produce x, nor to
determining s as y minus the subsistence consumption of the employed workers. For
example if for all commodities the production cycle takes one month, and in a year twelve
production cycles of identical dimension are carried through, x will be twelve times the
vector of the amounts produced each month, and y will be twelve times the net output of
each month. The relationships y=x−Ax and s = y−zaLx , where all vectors refer to the same
accounting period, remain valid. The main difference relative to our earlier assumption of a
yearly agricultural production cycle is the following: with a yearly production cycle and all
output coming out together at the end of the year, the production of one cycle/year is only
physically available at the end of the year, so it is not available for use as input, or for
consumption, during the year: the goods used as inputs or for consumption during the year
must necessarily be goods already available at the beginning of the year. On the contrary,
now that, more realistically, we admit that output is coming out during the entire year, most
of the commodities entering x are produced inside the year, and therefore inputs and
consumption during the year can and will often utilize commodities produced during the
year. Thus Ax may well for the most part consist of goods produced during the year, i.e.
goods that also appear in x. The same is true for consumption: when we assume a yearly
production cycle with all output coming out at the end of the year, the goods consumed
14
Cf. footnote 9??.
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during the year have to be already available at the beginning of the year; now that we admit
production during the year, most consumption will utilize that production.
This difference can also be visualized in terms of what goods one finds in inventories
at the beginning and at the end of the year. At the end of the year, the economy with yearly
production cycles finds in its inventories the entire production of that year, that is x, not xAx; and in order to have produced x, it had to have at least Ax in its inventories at the
beginning of the year[15]. If we admit many production cycles during the year, the economy
may need much smaller inventories than Ax at the beginning of the year, and may find itself
with much smaller inventories of produced goods than x at the end of the year, because
much of x disappears during the year, being used not only as input for further production
but also for consumption. For example, let us assume that all goods are produced in short
production cycles, a year encompassing 100 of them, and that all production cycles during
the given year produce identical quantities. Then each cycle produces the hundredth part of
x, and uses the hundredth part of Ax as inputs. In order to produce y=x−Ax as net product
during the year, the economy need have at its disposal as means of production at the
beginning of the year only the hundredth part of Ax; the production cycles after the first one
can use the products of the preceding cycle.
Thus the shorter the average production cycle, the smaller the amount of goods
necessary at the beginning of the year to realize a given y during the year. However, since
inputs must be available some time before the output comes out, some positive inventories
of inputs must always be available at the beginning of the year in order for y to be produced:
the need for some initial inventories would disappear only if y could be produced with
inputs directly or indirectly producible by labour alone (an ‘Austrian’, or ‘Smithian’,
structure of production): then production could start with labour alone, and its products,
together with additional labour, would be then used to produce the inputs directly or
indirectly necessary for the production of the net products. But this is not the case in real
economies.
We say ‘at least Ax’, in order to allow for the (totally unrealistic but logically admissible)
possibility that there is no consumption during the year, all consumption being effected the day
itself of the harvest. More realistically, at least zaLx must also be available at the beginning of the
year.
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Leontief’s open model is based on the above considerations and is formulated to
answer the question: if technical coefficients are given, what is the vector of total outputs
that must be produced during a year (or other accounting period) in order to obtain a certain
amount of goods available for consumption and for net investment? in other words, what
must x be in order to obtain a given y?
Its basis, under our assumption that all inputs (other than labour, of course) are
circulating capital goods, is equation [2.4], y=x–Ax , where now x is the unknown to be
determined, while y is a given vector[16]. (As an aside, in Leontief-type analyses the vector
of net products is generally called ‘final goods’. The word ‘final’ intends to convey that
these goods do not re-enter the production process during the period considered, differently
from the part Ax of x, which is called ‘intermediate’ goods and is viewed as ploughed back
and disappearing into the productive process; but this is misleading if not downright
mistaken, because it suggests that the inputs Ax utilized in the given accounting period
consist entirely of goods produced within the period, which is impossible unless production
can start with labour alone. Yet this mistake appears to be frequently made and it influences
the interpretation of the ‘non-substitution theorem’, cf. Appendix 5 of ch. 8.)
2.13. An economically significant, i.e. non-negative, solution to the Leontief
problem: “find x such that a given semipositive net product vector y is obtained”, will exist
if the matrix (I – A)–1, called the Leontief inverse, is non-negative, because then from y=(I–
A)x and y non-negative one obtains:
[2.13] x = (I–A)–1y ≥ 0.
Matrix A is not assumed indecomposable, but obviously it is non-negative; then by
property (ii’) of the Perron-Frobenius theorem, it is (ρI–A)–1 ≥ 0 if and only if ρ > A*,
where A* is the dominant eigenvector of A; the Leontief inverse can be intepreted as
Leontief’s closed model treats the net products as inputs to other sectors, e.g. families or
net investment or the state, so that all production is an input into something and there is no net
output; and these sectors contribute to the other sectors their services (labour services, capital
services, land services, government services etc.). This model has been very important in the
development of national accounting systems, but it is not relevant for the issues discussed in this
book.
16
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assuming ρ=1 , therefore it must be A* < 1. In the same way as the condition C*<1
implies that the economy is capable of producing a surplus product, so the condition A*<1
implies that the economy is capable of producing a positive net product, because it implies
that the equation x=(1+g)Ax has a non-negative solution: in other words, if the real wage
rate were zero, the economy would be capable of growth, and the rate of profit would be
positive. This condition is usually expressed by saying that the economy or the matrix A is
viable.[17] (This definition of viability can be criticized. The notion of viability intends to
express the capacity of the economy to go on functioning, but then a more appropriate
characterization would be λC<1, since x cannot go on being produced if the workers
producing it cannot obtain their subsistence. But this is the dominant terminology.)
In conclusion, the Leontief problem has solution if λA<1, or equivalently, if the
uniform rate of profit is positive when real wages are zero. If A is indecomposable, it is x>0
always: since y is semipositive and all industries are basic, all industries must be activated at
a positive level if a net product consisting even only of a single commodity is to be
produced. The reason why [2.13] admits the possibility that some elements of x be zero (if
A is decomposable) is that some non-basic industries may not need to be activated at all, if y
does not include positive quantities of their products nor of the non-basic commodities that
need their products as direct or indirect inputs. But if we restrict attention to the sole submatrix of technical coefficients of basic commodities, let us call it Aβ, then the
corresponding Leontief inverse is positive, implying that it is always necessary to produce a
positive vector xβ of basic commodities in order to obtain any semipositive vector of net
products of basics. And since the production of non-basics necessarily requires the input of
some basic commodity, xβ will always be positive, even if in y no basic commodity appears
with a positive entry. It is only the other elements of x that need not be positive, for example
they will all be zero if y has zeroes in all places corresponding to non-basics.
2.14. Historically, the formal study of Leontief models started (in the 1940s) when
economists were not yet conscious of the Perron-Frobenius theorem (which, although
proved by 1912, remained unnoticed by economists essentially up to the translation in
17
Exercise: assume a two-goods economy where corn and iron are produced by corn, iron
and labour. Show the viability conditions in terms of the coefficients. If you are unable, go on to
read §2.14.
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English, in 1959, of the treatise on matrix theory of the Russian mathematician
Gantmacher). For many years, the condition used to guarantee the existence of a positive
Leontief inverse was another one, the Hawkins-Simon condition, proved in 1949[18].
This condition is stated for a linear system Dx=c where x,c  Rn and D=[dij] is an
n×n matrix such that dij≤0 if i≠j. Matrix I–A of the Leontief problem satisfies this
condition. The Hawkins-Simon condition is that all naturally ordered (or leading, or NorthWest) principal minors of D are positive[19]. It can be proved (we omit the demonstration)
that the inverse of D is non-negative if and only if the Hawkins-Simon condition holds.
Putting D=I−A where A is semipositive, the Hawkins-Simon condition guarantees
that the Leontief inverse is semipositive. For D=I−A, the Hawkins-Simon condition is
therefore equivalent to the Perron-Frobenius condition that λA<1 for A semipositive and not
necessarily indecomposable.
It is useful to understand the economic interpretation of the Hawkins-Simon
condition applied to the Leontief problem[20]. We will see that it implies that it is possible to
produce a positive net output of any commodity.
The leading principal minor of order 1 of I−A is simply 1−a11, hence the HawkinsSimon condition, coupled with the condition that A≥0, requires that 0≤a11<1: commodity 1
must include among its direct inputs less than a unit of itself per unit produced. Since which
commodity is considered commodity 1 is arbitrary, the Hawkins-Simon condition implies
that technology is such that each commodity, if there were no constraint on the availability
of the inputs other than itself (for example, because they are supplied for free by other
nations), would be capable of producing a net output of itself.
Let us now consider commodities 1 and 2, assuming again that the availability of
inputs other than these two commodities poses no constraint, and also assuming 0≤a11<1.
18
Thus, in spite of a 1953 Econometrica article by Debreu and Herstein that had presented
the Perron-Frobenius results, in the influential 1958 book by Dorfman, Samuelson and Solow,
Linear Programming and Economic Analysis, no mention was made of those results and the
existence of economically acceptable solutions to the Leontief problem was demonstrated via the
Hawkins-Simon condition. Sraffa too did not know the Perron-Frobenius theorem, nor did the
Cambridge mathematicians he consulted for help.
19
We remind the reader that the leading (i.e. upper-left corner) principal minor of order k of
a matrix A is the determinant of the square sub-matrix formed by the common elements of the first
k rows and columns of A.
20
In the Leontief problem, to the Hawkins-Simon condition the assumption is added that
aii≥0 for all i, a condition not implied by the Hawkins-Simon condition which only implies 1–aii>0.
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We want to know under what conditions on the coefficients a11, a12, a21, a22 this economy is
capable of producing a positive net output of commodity 1, i.e. y1>0 with y2=0 for
nonnegative values of x1 and x2. It is
y1 = x1–a11x1–a12x2
y2 = x2–a21x1–a22x2
Suppose x1=1 and y2=0; then the above system of two equations simplifies to
y1 = 1–a11–a12x2
x2 = a22x2+a21.
From the second equation, x2=a21/(1–a22), so y1>0 if 1 > a11–a12a21/(1–a22). This can
be re-written as 1–a22–a11+a11a22–a12a21>0; the first three terms on the left-hand side are
equivalent to (1–a11)(1–a22); so we obtain that y1>0 if
(1–a11)(1–a22) – a12a21 > 0.
But this is the condition that the leading principal minor of order 2 of I–A be positive. Note
that this condition implies that it must be a22<1: if a22≥1 the minor is negative (this also
shows that x2≥0, so outputs are nonnegative). Thus the minor, if positive, remains positive if
the two commodites are re-numbered, commodity 1 becoming commodity 2 and vice-versa.
Which is as it must be, since the numbering of commodities is arbitrary and therefore should
not affect the results. Therefore the condition implies that (neglecting the other inputs) it is
also possible to produce a positive net output of the sole commodity 2; and therefore it is
possible to produce any net output vector (y1, y2)T, by producing the total output vector
given by the sum of the two vectors of total outputs corresponding to (y1, 0)T and to (0, y2)T.
Note that in a 2-goods economy the Hawkins-Simon condition imposes more than is
strictly necessary for the economy to be capable of producing a positive net output of
commodity 1: this would be still possible if a22≥1 (then a positive net output of commodity 2
is impossible) as long as a11<1 and a21=0 (i.e. commodity 2 is not needed for the production
of commodity 1). The Hawkins-Simon condition, by imposing a symmetry between
commodities, guarantees that it must be possible to produce a net output of commodity 1
whichever commodity is made to become commodity 1 by renumbering.
Let us finally explicitly note why, in a 2-goods economy, both leading principal
minors must be positive: if only the minor of order 2 were assumed to be positive, this
would leave open the possibility that both a11 and a22 are greater than 1, economically
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nonsensical and no longer guaranteeing nonnegative total outputs when net products are
positive[21].
It could be analogously shown − but we omit it because much more cumbersome,
and adding nothing new − that in a 3-goods economy the condition that also the leading
principal minor of order 3 be positive is equivalent to the condition that it must be possible
to produce a net output of commodity 1, whichever commodity is made number 1 by
renumbering, and therefore that it is possible to obtain a positive net output consisting of
either one of the three commodities − but then also of all three simultaneously.
Nowadays the Hawkins-Simon condition is seldom used because the PerronFrobenius theorem makes the connection with eigenvalues clearer and is richer in results,
but it was necessary to explain it because it occurs in many older papers and it is still
occasionally referred to.
The interpretation of the Leontief inverse.
2.15. The reason why the Leontief problem is soluble if the economy is viable
becomes clearer once we understand the economic meaning of the Leontief inverse (I−A)−1.
As a preliminary, we mention that the reason why, for a square non-negative matrix A, the
condition *<1 implies (I–A)–1 > 0 , can be clarified as follows. For any square matrix the
following useful theorem holds (cf. Heal, Hughes, Tarling, theorems 61, 62 p. 113) that
permits us to see the Leontief inverse as a matrix power series:

Let A be a square matrix. It is I  A 1   A t = I+A+A2+A3+... if and only if
t 0
lim A t  0 , and it is lim A t  0 if and only if all eigenvalues of A are less than 1in module.
t 
t 
(A corollary of this theorem, central in the study of dynamical systems described by
systems of linear difference equations, is that x(t)=Ax(t–1) converges to zero as t → +∞ if
and only if all eigenvalues of A are less than 1 in module.)
If A has dominant eigenvalue λ*<1, the theorem’s condition is satisfied. If
furthermore A is non-negative then I+A+A2+A3+... is non-negative; thus (I–A)–1 ≥ 0. If
Assume a11=2, a22=2, a12=1/2, a21=1/2: the condition (1–a11)(1–a22) – a12a21 > 0 is
satisfied; but for y1 and y2 positive, x1 and x2 are negative.
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furthermore A, an n×n matrix, is indecomposable, it can be proved that A+...+An is strictly
positive[22]; it follows that in this case (I–A)–1 > 0.
An immediate application of the result (I–A)−1 = I+A+A2+A3+... is the following.
The vector of labours embodied m = (m1,...,mn), a row vector, is defined by:
m = mA + aL
hence
[2.15] m = aL(I–A)–1 = aL+aLA+aLA2+aLA3+...
i.e. the labour embodied in a commodity is the sum of the direct labour employed in its
production, plus the direct labour necessary to produce its direct means of production, plus
the direct labour employed to produce the direct means of production of those means of
production, and so on. Labours embodied are well defined if this infinite sum converges,
which requires that λA*<1 or equivalently that the Hawkins-Simon condition holds.
Let us now intepret the coefficients of the Leontief inverse. We will derive
confirmation that the labour embodied in a commodity indicates the overall labour
employment (assuming homogeneous labour) necessary to produce a net product consisting
of one unit of that commodity.
Let y = (y1,...,yn)T be the vector of net products. We know that x=(I–A)–1y. It is
convenient at this point to indicate the Leontief inverse with the symbol Q=[qij]:
Q := [I−A]−1.
I supply the proof of this result by appealing – in order to help intuition – to the meaning
of an indecomposable matrix of technical coefficients: all commodities are basic. Consider such an
n×n matrix A and indicate as a(t)ij the element (i,j) of At. Then a(1)ij=aij; the element a(2)ij= ai1a1j+
ai2a2j+ ... + ainanj of matrix A2 is the amount of commodity i needed for the production of the means
of production of 1 unit of commodity j; the element a(3)ij of A3 is the amount of commodity i needed
for the production of the means of production of the means of production of 1 unit of commodity j;
and so forth. Since each commodity is directly or indirectly necessary for the production of each
commodity, a(t)ij cannot remain zero for ever; in fact at least one of aij, a(2)ij, ... , a(n)ij will be
certainly positive, hence for each (i,j) the sum aij+a(2)ij+ ... +a(n)ij is positive. The reason is that, if all
commodities are basic, the column vector aj=(a1j a2j ... anj)T of means of production of commodity
j must contain a positive quantity of at least one commodity different from j, otherwise j would need
only itself for its production so the other commodities would not be basic; and if some element of aj
is zero, the vector Aaj of means of production of the means of production of commodity j must
contain a positive quantity of at least one of the commodities not appearing in positive amount in aj,
otherwise those commodities would not be basic; for the same reason, at least one of the
commodities not appearing in aj nor in Aaj must appear in positive amount in the vector A2aj, and
at least one commodity not appearing in positive amount in any of these vectors up to Ataj must
appear in positive amount in At+1aj; so by the time one reaches Anaj all commodities including
commodity j will have appeared at least once among the direct and indirect means of production of
22
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Its columns are q1, q2,...,qn. Assume now that y has all elements zero except the first
one, which is equal to 1. Then
1
 
[2.16] x=(I–A)–1y=Q  0:  =q1.
0
 
Had the 1 been in the i-th place, the result would have been x=qi. We have found the
following interpretation of the elements of the Leontief inverse: its i-th column indicates the
total quantities that must be produced of the several goods if the net product is to consist of
one unit of good i.
This clarifies why the Leontief inverse generally contains zeroes: only the rows
referring to basic commodities will be certainly entirely positive, because in order to obtain
a positive net product of any commodity, it is necessary to produce a positive amount of all
basic commodities. (Exercise: prove rigorously this statement.) In a row referring to a nonbasic commodity i at least the elements qij where j refers to a basic commodity are zero.
Subsystems (vertically integrated sectors); labours embodied as employment
multipliers.
2.16. Let ui be the i-th base vector, that is, the i-th column of the identity matrix I, a
column vector with 1 in the i-th place and zero elsewhere. In equation [2.16] we actually
 y1 
 
assumed y=u1. If the single nonzero element in y had been different from 1, the vector  0 
 : 
0
could have been represented as y1u1, and the corresponding total output vector would have
been x=y1q1. Let us then use base vectors to write any vector y as:
y = y1u1+...+ynun.
Since in order to produce a net product which is the sum of two net products y and y’
the economy must produce a total output Qy+Qy’, it is clear that we can decompose the
output vector x=Qy as follows:
x = Qy = y1q1+...+ynqn.
Therefore we can visualize x as resulting from the sum of vectors of total productions
associated with net outputs consisting of only one commodity, net outputs whose sum yields
commodity j.
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y. Accordingly each industry can be, at least in imagination, decomposed into n subindustries, each one producing what is needed for the net product of only one commodity.
We can therefore view the economy as comprising n smaller sub-economies, the i-th one of
them including such a fraction of each industry of the original economy as is necessary to
produce yiqi and thus to obtain a net product yiui. Such sub-economies have been called
subsystems by Piero Sraffa; each subsystem produces all the means of production it needs; if
the i-th subsystem were installed on a separate island, one would observe only labour arrive
on the island to work every morning, no other arrival of inputs being necessary in order for
the subsystem to continue to operate and to send to the mainland the quantity yi of the i-th
commodity every period. Industry 1 produces a total output x1=y1q11+y2q12+...+ynq1n; its
fraction belonging to the i-th subsystem is the fraction producing yiq1i. Another term
sometimes used to describe the notion of subsystem is vertically integrated sector, a term
suggested by the notion of vertical integration in industrial economics, which refers to firms
which own plants that produce the means of production of their main final product, and
sometimes the means of production of those means of production and so on; however, the
present notion is different in that on the one hand it assumes a complete vertical integration
and dimensions of the several plants just right to ensure no need for outside input, and on
the other hand it does not assume a common property of the plants in the vertically
integrated sector; it is a purely theoretical notion.
The i-th column qi of the Leontief inverse is the vector of quantities produced by a
subsystem, or vertically integrated sector, whose net output is 1 unit of the sole i-th
commodity. Let us ask what the labour employment is in such a subsystem. The total
quantities produced in it are qi. If aL is the row vector of labour technical coefficients, total
employment in the subsystem is given by aLqi . But this is the labour embodied in the i-th
commodity because
[2.17]
m = aL(I–A)–1 = aLQ implies mi = aLqi .
Thus the labour contained or embodied in a commodity is the total employment
necessary to produce that commodity as net product. Labours embodied can thus be seen as
employment multipliers (relative to net products), in the sense that a net product y implies a
total employment N = my, and, if technical coefficients can be supposed constant, a
variation of the net product vector will imply a variation of total labour employment
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determinable in the same way. (However, this interpretation requires that labour is
homogeneous; if heterogeneous labours are reduced to homogeneity on the basis of relative
wages, then N does not indicate the actual quantity of labour time employed.)
This result has a corollary of some interest. We know from chapter 1 that, when the
rate of profit is positive, exchange ratios are generally not proportional to labours embodied.
If they were, variations in the total value of the net product would imply proportional
variations of total labour employment. Since they are not, it is possible to have variations of
the value of the net product, caused by changes in the composition of the net product with
no change in prices, that cause a variation of opposite sign in labour employment.
(Exercise: produce a numerical example of this last result.)
Exercise: study the condition under which the following growth-modified Leontief
problem has solution: it must be possible to produce a vector of consumption goods z, at the
same time guaranteeing a uniform rate of growth g of inputs.
Pricing with vertically integrated technical coefficients.
2.18. If we subtract the net product y from the total quantities produced x, what is left
is the (part of production that replaces the) means of production used-up in the entire
economy in order to produce x. Let us indicate it as k, defined as
k := x − y ≡ Ax.
Since x=[I−A]−1y, it is also
k=A[I−A]−1y=Hy,
where H := A[I−A]−1.
The meaning of matrix H can be grasped by again assuming y=ui. Then Hy=Hui is
the i-th column vector of H, which represents therefore the quantities of capital goods
employed in a subsystem having as net product 1 unit of good i, in other words the
quantities of capital goods directly or indirectly required to produce one net unit of good i.
Since a subsystem can be viewed as a vertically integrated industry, matrix H is called the
vertically integrated technical coefficients matrix: its i-th column can be interpreted as
indicating the technical coefficients of (that is, the consumption of inputs in) the production
of one unit of good i as net product (instead of as total product). It has a very simple
relationship with the Leontief inverse:
H=Q−I, i.e. hij=qij if i≠j, hij=qij−1 if i=j.
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(Exercise: explain this result on the basis of the economic interpretation of Q and H;
then prove it on the basis of the power series characterization of the Leontief inverse).
A vertically integrated sector that produces one unit of good j as net product employs
the vector Huj of commodities as means of production, and the quantity mj of labour, where
mj is the labour embodied in a unit of commodity j. Remembering that the vector of labours
embodied is m= aL(I–A)–1 , one can re-write the price equation [2.12] as follows. Postmultiply both sides of (1+r)pA+waL=p by (I–A)–1 to obtain (1+r)pH+wm=pQ, and use
H=Q−I to obtain
p = rpH + wm.
This shows that the price of each good can be seen as the sum of two components:
direct and indirect wages, and the rate of profit on the value of the direct and indirect capital
inputs. This is how the owner of an entire subsystem would determine the price of its net
product so as to earn the average rate of profit r on the capital invested. This result shows
from another viewpoint why prices are proportional to labours embodied if r=0.
The vertically integrated representation of conditions of production has been found
useful in several theoretical enquiries, surveyed in R. Scazzieri, “Vertical Integration”, in
Kurz-Salvadori, Elgar Companion to Classical Economics vol. II, 1998. For example, it
allows a better measurement of the impact of technical progress on the material as well as
on the labour inputs, and hence on the price, of a good, by taking into account in a more
explicit way the repercussions of technical change in one industry on costs through the
induced changes in all direct and indirect input requirements.
The relationship between rate of profit and rate of wages.
2.17. We now study in greater detail the problem that Ricardo and Marx were much
concerned with, the relationship between rate of wages and rate of profit.
Let us initally adopt the C representation, assuming wages are advanced; and let us
again suppose C indecomposable. This need not be intepreted as an assumption that no nonbasic commodity is produced, it can be intepreted as meaning that we concentrate on the
sole wage-industries, leaving for a subsequent study the determination of prices of nonbasics.
We know that the uniform rate of profit is determined by C*=1/(1+r) where C* is
the dominant eigenvalue of C. Property (iii) of the Perron-Frobenius theorem implies that, if
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any element of the subsistence vector increases, then some elements of C increase, therefore
λC* increases and r decreases. The conflict between wages and profits is confirmed.
Let us note that the wage rate need not be actually spent on purchasing the vector z
used to pass from matrix A to matrix C. If the money wage rate is just sufficient to purchase
z, then wage costs are correctly indicated by pz times the amount of labour employed,
independently of how the money wage is actually spent. This means that the real wage rate
might be determined by wage bargaining with reference to a basket of goods even
considerably different from the average basket purchased by the wage rate. For example, the
real wage rate might be fixed thus: the money wage must be ten times as much as necessary
to buy the bread needed by an average family of four persons. If this amount of bread is t
units, and bread is commodity j, then the wage rate is 10tpj, and C is obtained from A by
assuming zj=10t, zi=0 for i≠j. Given how C is derived from A and z, the rate of profit and
relative prices are the same if one writes
(1+r)pC = p
or if one writes
[2.19] (1+r)(pA+waL) = p,
w = pz
or, under the assumption that z=αv, if one writes
[2.20] (1+r)(pA+waL) = p,
w = αpv
where the scalar α indicates how many baskets v the real wage rate must be able to
purchase. The semipositive vector v is the basket of goods in terms of which the real wage
rate is fixed; in the example just given, it consists of one unit of bread and nothing of all
other goods, that is, v=(0,...,0,1,0,...,0)T with 1 in the j-th place. If the basket of goods in
terms of which the real wage rate is measured is precisely the one chosen as numéraire, i.e.
if one determines the absolute level of prices by putting pv=1, then w coincides with α and
α becomes unnecessary; one can write:
[2.21]
(1+r)(pA+waL) = p, pv=1 .
If one assumes that wages are paid at the end of the production period, the equations
become
[2.22]
(1+r)pA+waL = p, pv=1 .
Now that the real wage is measured in terms of a basket that does not necessarily
reflect how the wage is actually spent, a question arises: could it be that, when the rate of
profit changes, whether the real wage rate increases or decreases depends on the good or
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basket of goods that measures it? Could it be that an increase in the purchasing power of
wages in terms of bread goes together with a decrease in its purchasing power in terms of,
say, cloth? In this case, if workers consume both bread and cloth, it would be unclear
whether their lot has improved or not.
It can be proved that this ambiguity does not arise. When the rate of profit changes,
the purchasing power of the wage rate changes in the opposite direction whatever basket of
goods is chosen to measure it, and conversely, when the wage rate varies, the rate of profit
varies in the opposite direction, whichever the numéraire in terms of which the wage rate
varies. (Since relative prices will in general vary with distribution, the percentage variation
of the real wage will depend on the numéraire chosen, but not the sign.)
If one adopts the C representation and assumes advanced wages, this result can be
proved as follows. Since we want to admit variations of z, let C(z) indicate the matrix C
obtained from (A,aL,z). Assume that initially z=z^≡(0,...,0, z^j, 0,...0)T and that the
corresponding rate of profit r^>0 determined by (1+r^)p^C(z^)=p^ is positive. If we
increase zj, r decreases. Let us now choose a different z, call it z+, where the sole positive
element is another one, e.g. the i-th one. Let zi in z+ satisfy pi^zi=pj^zj. Then
p^z+aL=p^z^aL and therefore (1+r^)p^C(z+)=p^: the rate of profit is the same, and here
too if we increase z1 then r decreases. Thus to each higher and higher level of zj in z^ there
corresponds a higher and higher level of zi in z+ that obtains the same decrease in r. The
same reasoning applies to any other couple of single or composite commodities. Therefore
if r decreases (respectively, increases), the real wage rate increases (respectively, decreases)
in terms of any single or composite good.
■
Nowadays it is more common to assume wages paid at the end of the production
period and to adopt the A-representation of technology, and price equations such as [2.22].
Then let wadv indicate advanced wages, and w wages paid in arrears. With advanced wages
the equations are
p = (1+r)(pA + wadvaL) = (1+r)pA + (1+r)wadvaL.
By comparison with [2.22] we obtain
(1+r)wadv=w.
This permits to apply conclusions on the relationship between r and w also to the
relationship between r and wadv. Thus if we prove that as r decreases, w increases, it is
immediate that wadv increases too. Unfortunately it is not as easy to apply conclusions on the
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relationship between r and wadv to the relationship between r and w: we have proved that,
when wages are advanced, there is an inverse relationship between r and wadv, whichever the
basket in terms of which the wage rate is measured; but w=wadv(1+r), so an increase of r and
decrease of wadv has unclear effects on w. Therefore we study the relationship established by
[2.22] between r and w directly.
2.18. We first study the cases r=0 and w=0. With r=0, it is w=wadv and equation
[2.22] becomes
[2.23] p = pA + waL
or,
p = waL (I–A)–1.
Comparison with the equation that defines labours embodied, m=aL(I–A)–1, shows that p =
wm, i.e. that, when r=0, prices are proportional to labours embodied, thus relative prices are
equal to ratios between labours embodied: the strict labour theory of value holds true. If one
puts w=1, prices coincide with labours embodied. If A is vital (viable), the Leontief inverse
is non-negative, and in fact positive for all its elements denoting amounts of basic
commodities, so all labours embodied are positive, and prices too.
Note that if neither a condition pv=1 is added, nor the rate of profit is given, then
setting w=1 does not fix the purchasing power of the wage rate; it only amounts to
establishing labour as the numéraire, so that prices measure labours commanded. When r=0,
prices measured in labour commanded coincide with labours embodied.
Let us now consider the case w=0. Then equation [2.22] becomes p = (1+r)pA,
identical to p=(1+r)pC except that A has replaced C; so we know the mathematical
properties of this system of equations; it can be re-written pA = A*p , where A* is the
dominant eigenvalue of A (given our assumption that the dominant eigenvalue of the submatrix of coefficients of the basic commodities is also the dominant eigenvalue of A). This
maximum rate of profit r=(1/A*)–1 is also the maximum possible rate of growth if − in this
case − workers need not consume anything. This maximum rate of profit associated with the
A representation is generally indicated with R; below it will be useful to remember that
1+R=1/λA* .
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2.19. Let us now assume that A is indecomposable or that the dominant eigenvalue of
the sub-matrix of coefficients of the basic commodities of A is also the dominant eigenvalue
of A, and let us choose a semipositive vector of goods v as numéraire, i.e. we impose pv=1.
From p=(1+r)pA+waL we derive that, for r<R, it is
[2.24] p = waL [I–(1+r)A]–1 ,
where the inverse [I–(1+r)A]–1 is semipositive as long as r<R, because we can treat (1+r)A
as a new matrix, with dominant eigenvalue (1+r)λA*=(1+r)/(1+R)<1 as long as r<R.
 1

I  A , the inverse [I−(1+r)A]−1 can also be
1  r

Alternatively, from [I−(1+r)A] = (1  r ) 
re-written
1
1 r
1
 1

−1
1  r I  A and therefore it is positive if [ρI−A] is positive where
ρ=1/(1+r), which requires that ρ>λA* i.e. that r<R. Postmultiplying both sides of [2.24] by v
we obtain 1 = waL [I – (1+r)A]–1v, or
1
[2.25] w = −−−−−−−−−−−−− .
aL[I−(1+r)A]−1v
Equation [2.25] yields the real wage rate (measured in terms of v) as function of r.
It is possible directly to prove that this function has negative first derivative for w>0,
R>r>0. But it is economically more enlightening to reach this result in the following way.
Take equation [2.22] and on its right-hand side replace p with waL+(1+r)pA; in the
right-hand side of the resulting expression
p = waL+(1+r)[(1+r)pA + waL]A = waL[I+(1+r)A] +(1+r)2pA2
repeat the replacement of p with waL+(1+r)pA, and reiterate this replacement again and
again, obtaining
[2.26] p = waL [I+(1+r)A+(1+r)2A2+(1+r)3A3+...].
This equation shows that the price of each commodity can be seen as the sum of :
-
wages paid to the direct labour employed in its production;
-
wages paid to the direct labour employed in the production of its direct means of
production, multiplied by (1+r);
-
wages paid to the direct labour in the production of the means of production of its
means of production, multiplied by (1+r)2;
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and so on, ad infinitum (as long as there are basic commodities).
In this way the price of each commodity is expressed as the sum of the wages paid to a
series of quantities of ‘dated’ labour, each quantity being multiplied by the power of (1+r)
corresponding to how many periods have passed between the payment of that labour and the
sale of the commodity[23]. For example for price 1 we can write:
[2.26’] p1=waL1+w(1+r)L1(−1)+w(1+r)2L1(−2)+w(1+r)3L1(−3)+...
where L1(−t) is the quantity of labour employed in the indirect production of commodity 1
whose wage has been paid t periods before the product comes out; with a1 indicating the
first column of A (that is, the vector of inputs in industry 1), it is L1(−t)=aLAt−1a1.
If now in equation [2.26] we put w=1 i.e. if we measure prices in terms of labour
commanded, w disappears:
[2.26”] p = aL [I+(1+r)A+(1+r)2A2+(1+r)3A3+...],
and p is determined as a sum of non-negative vectors which are certainly positive at least
from a certain point on, and are (except for the first term) increasing functions of r. This
proves that all prices in terms of labour commanded are positive, and increase as r
increases (unless the good is produced by labour alone and wages are paid in arrears).
This means that all the reciprocals of labour-commanded prices, that indicate the purchasing
power of the real wage rate in terms of each commodity, decrease as r increases. This
proves that the function w(r) is decreasing whichever vector v is chosen as numéraire.
Of course this proof requires that [I+(1+r)A+(1+r)2A2+(1+r)3A3+...] converges, but
this is guaranteed as long as r<R because, as already noticed, the matrix (1+r)A has
dominant eigenvalue (1+r)λA* which is less than 1 as long as r<R, thus all eigenvalues of
(1+r)A are less than 1 in module and therefore
[I+(1+r)A+(1+r)2A2+(1+r)3A3+...] = [I–(1+r)A]–1.
Therefore equations [2.24] and [2.26] are equivalent and have economically significant
solution as long as r<R. When r=R, it is w=0 and these equations lose meaning, one must
use p=(1+R)pA.
23
Exercise: apply this same procedure to the price of a single commodity, e.g. commodity 1,
and determine in matrix terms the expression corresponding to equation [2.25] for this sole price.
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2.20. The function w(r) expressed by [2.25] is called wage-profit curve, also w-r
curve, or simply wage curve[24]. Note one thing that will be important later: the actual
quantities produced in the economy have no role in equation [2.25], the shape of the
function w(r) depends only on A, on aL and on the numéraire vector v.
If there is at least one basic commodity, and if the economy is viable (A*<1), the
w(r) curve drawn in a Cartesian diagram has positive horizontal intercept at r=R, is
decreasing[25] as long as r<R, and it has positive vertical intercept determinable as
follows[26]. If r=0 then p=wm; post-multiply both sides by the numéraire vector v; since
pv=1 we obtain 1=wmv. With W this maximum wage rate in terms of v:
[2.27] W = 1/(mv) > 0,
in other words, the wage rate corresponding to a zero rate of profit is the reciprocal of the
labour embodied in the numéraire basket of goods.
The shape of the w-r curve can therefore be represented as for example in Fig. 2.1.
Nothing can be said in general on the convexity or concavity of w(r); it might be
convex, concave, have inflexion points, because it is a polynomial function of degree n with
n generally very great. The function w(r) is a straight line only in two cases.
w
1/(mv) = W
Some neoclassical economists have called it ‘factor price frontier’, thus implying that w
and r are the prices of factors of production: w the price of labour, r the price of capital. However,
that capital can be considered a factor of production is hotly disputed (see chapter 7), so we avoid
this terminology.
25
For r > R, the function w(r) need not be decreasing, but r cannot become greater than R, it
can be proved that not only w but also at least one price would be negative.
26
??Vi è un caso in cui la curva w(r) non incrocia l’ascissa, ma tende solo asintoticamente
ad essa: ciò si ha quando risalendo ‘all’indietro’ dagli inputs ai loro inputs e agli inputs di questi e
così via, si arriva a inputs prodotti da solo lavoro (struttura ‘austriaca’ della produzione): allora al
tendere di w a zero il saggio di profitto tende a +∞. Ad esempio si abbiano due beni di cui il primo
prodotto solo da lavoro e il secondo dal primo e da lavoro; con salari posticipati si ha p1=waL1,
p2=(1+r)p1a12+waL2=(1+r)waL1a12+waL2 e basta porre p2=1 per vedere che al tendere di w a zero r
tende a crescere senza limiti. Il lettore può facilmente verificare che lo stesso vale anche con salari
anticipati. Si tratta però di un caso senza merci-base.
24
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w(r)
1   1
R
r
Fig. 2.1
The first case is when the labour theory of value is valid also for positive levels of the
rate of profit. This is a very restrictive case: p and m must be proportional for all admissible
levels of r, therefore also for r=R, which implies that it must be mA=λA*m, that is to say, m
must be a left eigenvector of A; this, together with m=mA+aL , implies m=A*m+aL , i.e.
aL=(1–A*)m , the vector of direct labour coefficients must itself be proportional to the
vector of labours embodied and must therefore be a left eigenvector of A, an extremely
restrictive condition. Marx’s condition of equal ‘organic composition of capital’ implies the
same thing: if c/v is uniform in all industries, since s/v is uniform also (s+v)/c is uniform, or
aLj /(a1jm1+...+anjmn) =α , all j;
this can be written in matrix terms as aL= αmA; from the definition of labours embodied it
is mA=m−aL; hence aL=α(m−aL) or aL=α/(1+α) · m, i.e. the vector of direct labour
coefficients must be proportional to the vector of labours embodied.
The second case in which w(r) is a straight line is when the numéraire is some
quantity of the standard commodity defined for matrix A. The standard commodity relative
to A is the composite commodity corresponding to a right-eigenvector of A associated with
its dominant eigenvalue, in other words it is the composite commodity produced if the
economy’s industries are in the proportions required for maximal growth rate with zero
wages. Assume that the economy produces this standard commodity and wages are paid in
standard commodity, then the rate of profit is determined in the same way as in an economy
where a single commodity corn is produced by corn and labour: in the latter economy, with
wages paid in arrears and the price of corn equal to 1, let a11 and aL1 be the seed and labour
technical coefficients, then the corn price equation is 1=(1+r)a11+waL1, so w=
1  (1  r )a 11
.
a L1
With an opportune choice of units this equation can be simplified. Choose the net product of
corn as the unit for corn, and total labour employment as the unit for labour. Then total corn
output is x such that x-a11x=1 so x=1/(1-a11); labour employment is aL1x=1; the amount of
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corn capital utilized is k=a11x=a11/(1-a11); if w=0 the rate of profit is the maximum one,
R=(1-a11)/a11=1/k; since net output goes either to wages or to profits and profits are rk, it is
1=rk+w, so if w=0 it is Rk=1, otherwise w=1–rk=1–r/R, a linear relationship. A similar
equation will hold with the standard commodity replacing corn, if wages (paid in arrears)
are measured in A’s standard commodity and we take as numéraire and unit what Sraffa
calls the Standard net product, the quantity of standard commodity that requires one unit of
total labour employment to be produced as net product[27]. This correspond to putting
pv*=1 where v* satisfies (i) Av*=λA*v* and (ii) aLx*=1 for x* solution of v*=x*−Ax*. The
quantity of standard commodity employed as capital, let us call it k*, satisfies Rk*=1 for the
same reason as for the corn economy, and we obtain again
w = 1–r/R.
Now the interesting thing is that it is not necessary that the economy be actually producing
standard commodity for this equation to hold true, all that is necessary is that the numéraire
be the Standard net product: this is because, as noticed earlier, the shape of the w(r) function
is independent of the quantities produced, it depends only on the numéraire (once the
technical coefficients are given).
This is the sole case in which w(r) is linear if the labour theory of value does not
hold. Indeed assume that relative prices vary with r and nonetheless the w(r) curve is linear,
i.e. of form w=α–βr with α and β positive constants: this means that one is choosing as
numéraire a commodity such that a unit of labour receives α units of it if r=0, so if the net
product consists only of that commodity, the net product is α units of it per unit of labour
employment; then by changing the unit in which this commodity is measured so that one
unit of labour employment produces one unit of this commodity as net product, α becomes 1
and β becomes β’=βα, so w=1+β’r. Having made such a choice of the unit of the numéraire
commodity, since the net product, of value 1, goes either to wages or to profits and r is the
rate of profit on the value of capital k, the following relation must hold:
1=w+rk=(1−β’r)+rk=1+r(k−β’),
which implies r(k–β’)=0, which can only remain true as r changes if k=β’; thus the value of
capital in terms of the net product must be a constant, it must not change as r varies, which
27
That is, the subsystem producing it as net product employs one unit of labour;
equivalently, it embodies one unit of labour.
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is only possible if the vector of capital goods is the same composite commodity as the net
product, so the net product must consist of standard commodity.
Thus Sraffa notices:
... if we make it a condition of the economic system that w and r should obey the
proportionality rule in question, the wage and commodity-prices are then ipso facto
expressed in Standard net product, without need of defining its composition, since with no
other unit can the proportionality rule be fulfilled ... And to find R ... we can find it as the
Maximum rate of profit from the production equations, by making w=0 ... And it is curious
that we should thus be enabled to use a standard without knowing what it consists of.
(Production of commodities by means of commodities, §43, pp. 31-32)
CHOICE OF TECHNIQUE
2.21. Until now we have assumed that there was no possibility to choose among
different methods to produce a commodity. Now we assume that, for each commodity, there
may be several alternative methods to produce it, each one represented by different
technical coefficients. We keep assuming that all capital goods are circulating capital goods,
that land is overabundant and a free good, that labour is homogeneous, and that there is no
joint production.
A technique (also called production system by other authors) is a set of methods of
production, one for each commodity. It is represented by a couple (A,aL) where A is square.
Two techniques are different if they differ in one or more methods. Thus the methods used
in two different techniques might coincide in all industries but one.
Note that different techniques need not include the same number of commodities.
Alternative techniques must of course be alternative ways to produce the goods in demand.
But an alternative way to produce a good may need a new good. Suppose for example that
in the given economy tomatoes are among the basic commodities produced, and call α the
dominant method with which they are produced and (α) the technique of which it is part;
then a new method β to produce tomatoes is discovered, which uses a specific new fertilizer
only needed for that purpose. Method β, for purposes of comparison with α, must then be
conceived as including the production methods of two industries, the new method for the
direct production of tomatoes, and the method for the production of the fertilizer, which is a
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new good not appearing in the currently dominant technique (α). Technique (β) includes one
more commodity and one more industry than technique (α). However, we postpone the
study of this case; for the moment we assume that the alternative techniques include the
same commodities.
We now study how competition selects the technique among the several available
ones, and how this choice may depend on income distribution.
The starting point is the following: at any given set of prices, real wage rate and rate
of profit, firms will compare the price, dependent on the method adopted, at which a given
commodity must be sold if it is to yield the ruling rate of profit, and will prefer, or will be
obliged by competition, to adopt the method associated with the lowest of these prices: if
this price is lower than the ruling price, the method yields extra profits, i.e. the firm can sell
the product at a price intermediate between its supply price with that method and the ruling
price, in this way it will subtract custom from its competitors, and earn a higher than
average rate of profit. If the ruling price is already the lowest price, no firm can adopt any
other method lest it incurs in ‘losses’ (i.e. in a lower-than-average rate of profit[28]). We will
call cost-minimizing (at the given prices and distributive variables w, r) the method that
yields the lowest such price. But, if it was not already the dominant method, the adoption of
the method that is cost-minimizing for the first commodity at the given prices and
distribution may have repercussions on the costs, and selling prices, of other commodities if
the first commodity is an input to other industries; if the first commodity is indirectly an
input to itself, these repercussions may end up altering the cost-minimizing method for its
production − as well as for other commodities. We must find out the final situation, if it can
be determined, toward which the choice of technique tends.
2.22. We start with the simplest case: only two different methods are known to
produce a non-basic commodity that only needs, with either method, basic commodities
(there are n of these) as means of production, and the real wage is fixed in terms of basic
commodities. In this case all prices entering the cost of producing the commodity − let it be
commodity q, q>n − with either method are known, once either the rate of profit, or the real
28
We will call loss the difference between actual earning and the earnings guaranteeing the
ruling rate of profit, when negative; the implicit assumption is that the ruling rate of profit is
included in the costs, either as an opportunity cost, or as a rate of interest.
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wage rate in terms of the chosen basic numéraire, is given. Let the two methods be α and β,
and the technical coefficients and price of production are distinguished by a superscript (α)
or (β) according to the method. Then:
n
 
pq    1  r  aiq  pi  a Lq
w
i 1
n

pq    1  r  aiq  pi  a Lq
w
i 1
Competition will impose the adoption of the method with the lower pq, which can
depend on r. This is because, if for a moment we treat the commodity produced with either
method as two different commodities, their relative price will in general depend on r, and it
may happen that for a certain r it is pq(α)/pq(β)<1 and for another level of r it is pq(α)/pq(β)>1. If
one traces the curves pq(α)(r) and pq(β)(r) in the same diagram with r on the horizontal axis,
these curves can cross several times[29], cf. the drawing below, where method α is the more
convenient one for 0<r<r1 and for r2<r<r3, while method β is more convenient for r1<r<r2
and for r3<r<R, while at r1, r2, r3 the two methods are equally convenient and therefore can
co-exist.
pq
pq(α)
pq(β)
O r1
r2
Fig. 2.2
r3 R
The precise shape of curves pq(α)(r) and pq(β)(r) depends on the numéraire, but not
which curve is below the other one, since this only depends on the ratio between the two
prices, which is unaffected by the choice of numéraire.
29
The maximum number of intersections of the two curves in the positive quadrant is n,
where n is the number of commodities directly or indirectly entering the costs of the numéraire. We
will not prove this result (cf. Bharadwaj ??REF), but it is a straightforward implication of the fact
that a price is a polynomial function of the rate of profit, of degree equal to n.
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2.23. In the general case it is possible, and for basic commodities it is certain, that the
price of a commodity enters, directly or indirectly, into its own costs. In particular if one
changes the method of a basic commodity (the numéraire is given and is a basic single or
composite commodity) the system of equations determining w(r) changes; each technique
determines a different w(r) curve. Above all, when one wants to compare the convenience
of two methods to produce a basic commodity[30], the prices at which to compare the two
methods are not univocally determined, because the system of equations determining them
will depend on which one of the two methods appears in it. This will require some
additional notation.
For example, suppose that all commodities are basic, and two methods α and β for
commodity 1 are to be compared, while the methods of all other commodities are given.
Depending on which method is included in the matrix A and in the vector aL, there are two
techniques, (Aα,aLα) and (Aβ,aLβ), which generate different vectors of relative prices and
different values of the residual distributive variable when the rate of profit, or the real wage
rate, is given:
p     1  r  p    A  w   aL
p     1  r  p    A  w   aL
Therefore the situation is different for entrepreneurs if method β is discovered when
α is the generally adopted one, or vice-versa. Suppose that the numéraire is assigned, and
that initially only method α was known, hence prices and income distribuiton were (p(), r,
w()) − in this Section we treat the rate of profit as the given distributive variable that
remains unchanged across changes in methods, while the real wage rate is residually
determined and adapts to the technique. Now method β is discovered. Entrepreneurs will
find it convenient to adopt it in place of method α if at the current (p(), r, w()) method β
yields extra profits, i.e. permits to sell commodity 1 at a price inferior to p1(), that is if
n
[2.30]
1  r  a i1  p i    w    a L1  p1   .
i 1
30
The problem actually arises for all commodities whose price enters directly or indirectly
into its own costs.
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The expression on the left-hand side of this inequality requires attention: it is the
supply price[31] of commodity 1 produced with method  at the prices and wage associated
with technique . We need a way to indicate it: let it be p1() . The first superscript
indicates the method used to determine the supply price, the second superscript, included in
parentheses, indicates the technique determining the prices and wage used to determine the
costs that sum up to the supply price. When the supply price refers to the same method as
the one appearing in the technique determining the prices and distribution, then only one
superscript is used, in parentheses. Thus p1(α) is the price of production, i.e. the supply price,
of commodity 1 produced with method α at the prices of technique α.
In this case method  is found more convenient than α, and is introduced, if p1() <
p1(). The relevant comparison is between p1() and p1() . If on the contrary method β was
the one initially known and method α is discovered later, then the prices and distribution
used to determine which method is more convenient are (p(), r, w()), thus the relevant
n
comparison is between p1(β) and p1    1  r  a i1  p i    w    a L1  .
i 1
It is then conceivable that the following might happen. Suppose that, at the prices and
income distribution (α), method β yields extra profits; suppose that it replaces method α and
becomes the generally adopted one so that prices and distribution become (p(β), r, w(β)) or,
briefly, prices and wage (β); at these prices, it is discovered that method α yields extra
profits, and it is re-introduced; then no method is able to impose itself, and it is unclear what
would happen in the economy; perhaps it would indefinitely oscillate from adopting one
method to adopting the other one. If the theory is unable to exclude this case then it is in
trouble, because no such problem appears to occur in real economies. Luckily, for
sufficiently general situations this case can indeed be excluded. In particular, it can be
excluded when there is no joint production and no land, as we are now supposing. In this
case it is possible to prove that the more convenient method is the same both at prices and
wage (α) and at prices and wage (β), and it is the one whose technique generates the w(r)
curve that, for the given r, is higher, i.e. generates the higher w. Then, since competition
will tend to impose the more convenient method, even when the alternative methods are
31
The supply price of a commodity is the minimum price necessary to induce it to be
supplied; for produced commodities it is the price that just covers costs inclusive of the normal rate
of profit (in marginalist terminology, it is the price that yields zero profits).
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numerous and concern several commodities it can be assumed that the economy will
gravitate toward the technique yielding the highest w, when r is given, or the highest r,
when w is given (in terms of some numéraire common to all techniques). This dominant
technique may depend on the level of r, because the w(r) curves may cross, even several
times. The outer (or North-East) envelope of the w(r) curves indicates therefore the relevant
long-period relationship between r and w, when there is choice of techniques.
2.24. Suppose then two techniques that include the same commodities, all basic, and
differ in only one method, the method for commodity 1. We intend to prove that
Lemma 2.10. If at the given r it is p1   p1   , then w     w   .
In words: given the rate of profit, if at prices and distribution (α) method β yields
extra profits, then technique (β) generates the higher rate of wages, its w(r) curve is above
the other w(r) curve at the given r. (Of course the numéraire must be the same in
determining both w(r) curves.)
Proof. We use a proof based on the C-representation of production; this will show
that, for these issues, to assume that wages are paid at the end, or at the beginning, of the
production period makes no difference. We remember that if w stands for the wage rate paid
in arrears and wadv for the advanced wage rate, then for a given rate of profit r and a given
technique the two wage rates are connected by
[2.31] wadv=w/(1+r)
and relative prices of commodities do not change if one passes from advanced to paid-inarrears wages (or vice-versa) respecting this condition. Therefore, with r=r° given, if we
prove that technique (β) generates the higher wadv, we have also proved that it generates the
higher w. Let us then form the matrix Cα by assuming the subsistence vector to be the
numéraire vector v multiplied by the scalar w(α)/(1+r) (then the rate of profit determined by
p=(1+r)pCα is r°); and let us form the matrix Cβα in the same way, except with commodity 1
produced by method β [32]; the latter matrix differs from Cα only in its first column; the
32
Matrix Cβα is formed by assuming the advanced real wage rate to be w(α)/(1+r), which is
why we do not indicate it simply as Cβ, a symbol that might induce confusion by suggesting that the
real wage rate used in forming it is w(β)/(1+r).
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supply price p1β(α) is the same as with wages paid in arrears, and therefore by assumption it
is less than p1(α). Now imagine raising one technical coefficient of method β until at the
prices (α) the supply price of good 1 becomes equal to p1(α). The resulting matrix, call it C^,
that includes this hypothetical new method has the same dominant eigenvalue as matrix Cα
because it yields the same rate of profit. Since the matrix Cβα has one lower technical
coefficient than C^, by result (iii) of the Perron-Frobenius theorem its dominant eigenvalue
is lower, i.e. the rate of profit determined by p=(1+r)pCβ^ is higher than r°[33]. We can
therefore raise wadv, and thus all technical coefficients in Cβα which include a subsistence
component, until the rate of profit decreases back to its old level r°. Thus we have proved
█
that wadv(β)>wadv(α); but then it is also w(β)>w(α).
Note that this result implies that if at prices and distribution (α) it was p1    p1  ,
then it cannot be that at prices and distribution (β) it is p1    p1  : the latter inequality
would imply w    w    contradicting the result just reached that w    w    . Therefore it
cannot be that, once method β has replaced method α and prices and the real wage become
(β), method α comes out to be now the more convenient one; indefinite oscillations back
and forth between two methods are excluded.
w
I
w”
w’
II
O
r’
r”
r
Fig. 2.3. Suppose the discovery of a new, more convenient method shifts the w(r)
curve from I to II; there is room for both the original rate of profit and original real wage rate
(r’, w’) to increase, e.g. to r”, w”.
33
This result is usually called Okishio’s Theorem in the modern Marxist literature.
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This result proves that if at the prices and distribution of a technique there is a (basic)
method, not used in that technique, that yields extra profits, then if this method is introduced
the new technique yields a higher wage rate if the rate of profit stays fixed, or a higher rate
of profit if the real wage rate stays fixed. Graphically, if the new w(r) curve is above the old
one at the old rate of profit, since w(r) curves are decreasing and continuous for r≤R, the
new w(r) curve is also to the right of the old one at the old rate of wages. There is therefore
room for an increase both of r and of w, cf. Fig. 2.3.
This result is independent of the choice of numéraire: if at the given rate of profit
p1    p1   with one numéraire, then the same inequality holds with any other numéraire
because a change of numéraire changes all prices in the same proportion; and the proof of
Lemma 2.10 works whichever the numéraire vector v. This means that a change of
numéraire changes the shape of w(r) curves and the distance between the w(r) curves of two
techniques at the given rate of profit, but not which curve is the outer one.
2.25. Note that we have not proved the converse of Lemma 2.10, namely that, given
the rate of profit, if by introducing the new method in place of the old one the real wage rate
rises, then at the old prices and distribution the new method yields extra profits and
therefore it is convenient to introduce it. Therefore we have not excluded the possibility that
the outer envelope of the w(r) curves may not be reached because a method which, if
adopted, would raise the real wage rate may not be found more convenient at the prices and
distribution associated with another technique.
But this possibility is excluded by the following result:
Lemma 2.11. If two techniques (α) and (β) are such that at the given rate of
profit r it is p(α)≥0, p(β)≥0, and w(β)>w(α), then there is a method in technique
(β) which pays extra profits at the prices and distribution (α).
That is, it is not the case that p(α)≤pβ(α) (where, remember, vector pβ(α) is the vector of supply
prices of all commodities, produced with technique (β), at the prices and distribution (α)).
Proof. By contradiction. Now we can use the A-representation of technology.
Suppose w(β)>w(α) and p(α)≤pβ(α). The expression
p     p    1  r  p    A  w    a L
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can be re-written
p    I  1  r  A   w    a L
p     w    a L I  1  r  A  
or
1
.
Post-multiplying by the semipositive numéraire vector v, since p(α)v=1, it is
1  w    a L I  1  r  A   v  w    w    .
1
1
The last equality derives from the fact that (cf. [2.25]) wβ = −−−−−−−−−−−−− . But
aLβ[I−(1+r)Aβ]−1v
1≤w(α)/w(β) implies w     w   which contradicts the assumption w    w  . █
Thus, as long as there are non-adopted methods capable of raising the wage rate,
these methods will yield extra profits and therefore competition will force their introduction,
until a technique is reached whose w(r) curve is on the outer envelope of the w(r) curves.
Given r, w is maximized; given w, r is maximized.
So far we have assumed that the methods in use for the other commodities are given.
But the same process of choice of methods on the basis of cost minimization will be going
on for each commodity for which there is technical choice, and methods will be replaced as
long as other methods yield extra profits. This will push the economy finally to adopt some
technique on the outer envelope of the w(r) curves generated by all possible alternative
techniques. A technique with w(r) curve on the outer envelope for the given level of r is
cost-minimizing at that level of r, i.e. at its prices and wage rate there is no other technique
that generates a lower supply price or ‘cost of production’ for at least one commodity[34]. If
technique (α) is cost-minimizing, then for any other technique (β) it is
p(α)≤(1+r)p(α)Aβ+w(α)aLβ.
2.26. We now prove
The term ‘cost-minimizing’ can be criticized because it accepts the term ‘cost of
production’ to refer to the sum of value of means of production used up plus wages plus profits; this
terminology is avoided by Sraffa (1960, pp. 8-9), who prefers not to use the term ‘cost of
production’ because the marginalist tradition has tended to intend by ‘cost of production’ a quantity
independent of the price of the product and determining it, while (except for non-basics not entering
their own production) cost and price can only be determined simultaneously, because the price of
the commodity directly or indirectly enters its own cost. Other short expressions ae not easy to find,
however: one might perhaps speak, instead of cost-minimizing technique, of supply-priceminimizing technique; but my feeling is that, once the interdependence of costs and prices is
34
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Lemma 2.12. Two techniques (α) and (β) can co-exist if they are both cost-minimizing
at the given r, which implies that, in terms of any common numéraire v, it is w(α)=w(β)
and p(α)=p(β).
In view of the previous two lemmas this lemma means that at the given r both w(r) curves
are on the outer envelope (i.e. the two curves either cross or are tangent to each other on the
outer envelope).
By definition of cost-minimizing technique it is simultaneously true that
p(α)≤(1+r)p(α)Aβ+w(α)aLβ i.e. p(α)v = 1 ≤ w(α)aLβ[I−(1+r)Aβ]−1v = w(α)/w(β)
p(β)≤(1+r)p(β)Aα+w(β)aLα i.e. p(β)v = 1 ≤ w(β)aLα[I−(1+r)Aα]−1v = w(β)/w(α).
These two inequalities can be simultaneously satisfied only if w(α) = w(β) . On account of the
result just reached, the left-hand inequalities above can be written:
p(α) ≤ w(α)aLβ[I−(1+r)Aβ]−1 = w(β)aLβ[I−(1+r)Aβ]−1 = p(β)
p(β) ≤ w(β)aLα[I−(1+r)Aα]−1 = w(α)aLα[I−(1+r)Aα]−1 = p(α)
which can be simultaneously satisfied only if p(α) = p(β) .
Q.E.D.▌
This implies that if at the given r two w(r) curves corresponding to two different
techniques cross, or are tangent, on the outer envelope, then the two techniques generate the
same wage rate and relative prices, i.e. for each commodity for which the methods in the
two techniques differ, the two methods are equally profitable. Obviously, since already the
co-existence of two techniques differing in only one method is a fluke (their w(r) curves can
only have a finite number of points in common[35], hence the values of r for which the two
techniques can co-exist is a set of measure zero in the admissible set of values of r), it will
be an even more extremely improbable fluke that two techniques which can co-exist differ
in the method of more than one commodity.
In conclusion, if we are given all the w(r) curves associated with all possible
techniques, then their outer envelope will tell us which technique(s) will tend to be imposed
grasped, the term ‘cost’ does not induce in error and therefore can be used.
35
Expression [2.25] shows that the function w(r) is a polynomial function of r of degree n,
where n is the number of commodities. The equation establishing that two polynomial functions of
degree n yield the same value for the dependent variable is a degree-n polynomial equation, which
can have at most n distinct real solutions.
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by competition for each level of r, up to the maximum of the R’s associated with the
different techniques.
w
r
2.27. Another result of immediate practical relevance is the following: if r is given,
and if a more convenient method is discovered to produce a commodity, for example,
without loss of generality, commodity 1, then its introduction causes the price of commodity
1 to decrease relative to all other commodities.
Proof. Let r be given and put w=1, i.e. prices are measured in labour commanded. Let
p1(α) be the original price of commodity 1. At the old prices it is p1β(α)<p1(α), so the
introduction of method β reduces p1 and hence reduces the prices of all commodities into
whose costs p1 enters directly or indirectly. No other price will decrease if commodity 1 is a
non-basic not used as means of production in any other industry, in which case the
demonstration ends here. If some other price decreases (if commodity 1 is basic, all other
prices will decrease), we must prove that no price decreases by a greater percentage than p1.
Suppose that among the other commodities the price that proportionally decreases most is
ph. Let us consider whether it can have decreased by a greater percentage than p1 by
examining its price equation:
ph = (1+r) p1a1h +(1+r)(p2a2h+p3a3h+...+pnanh)+aLh .
On the right-hand side of the equality sign, the third addendum does not decrease; the
second addendum decreases at most by the same percentage as ph (this is the case if the only
positive technical coefficient in that addendum is ahh or if all other prices in it decrease by
the same percentage as ph; otherwise it decreases by a lesser percentage than ph because by
assumption ph is the price with the greatest percentage decrease); thus the second and third
addendum taken together certainly decrease by a lesser percentage than ph; the only way to
respect the equality is for p1 to have decreased by a greater percentage than ph.
█
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This result implies that, if r is given, and if the method concerns a basic commodity,
the purchasing power of the wage increases in terms of all goods but it increases most in
terms of the good whose production method has improved.
Now suppose that what is given is the real wage w in terms of a numéraire, and that a
new more convenient method is discovered for the production of the numéraire. The new
method is introduced and r rises. We want to show that all other prices rise relative to the
numéraire.
Proof. Suppose without loss of generality that the numéraire is good n. Consider its
price equation and the price equation of any other i-th commodity:
pn = 1 = (1+r)(p1a1n+p2a2n+p3a3n+...+pnann)+waLn .
pi = (1+r)(p1a1i+p2a2i+p3a3i+...+pnani)+waLi .
The first equation by itself does not exclude the possibility that no price changes, because
the reduction in costs might be compensated by the rise of r: but the second equation would
not be satisfied, because in it only r would have risen, so some price must change. And it
cannot be that pi decreases, because the second equation shows that, since r has increased, a
decrease of pi requires that (p1a1i+p2a2i+p3a3i+...+pnani) decreases by a greater percentage
than pi, this requires that some pj, j≠i, decreases by a greater percentage than pi, but this is
only possible if some pk, k≠j and k≠i, decreases by a still greater percentage, and so on until
one exhausts all remaining prices and there is no price left that can decrease even more than
the last one. The reasoning applies to any pi, i≠n, so all of them must increase.
█
Non-basics and choice of techniques
2.28. Up to now, except in the first example I have assumed 1) that all commodities
are basic, i.e. that the matrices Aα, Aβ, etc. of the alternative techniques are indecomposable,
and 2) that they include the same commodities. Let us discuss the implications of relaxing
the first of these two assumptions.
Choice of techniques for basic commodities is logically prior to choice of methods of
production for non-basics because the equations of basics are not influenced by the prices of
non-basics. Given a numéraire vector including positive quantities only of basic
commodities, we can determine the choice of techniques and hence the relative prices and
residual distributive variable once either the rate of profit, or the real wage rate (in terms of
a numéraire consisting of basics), is fixed (at economically acceptable levels). Then in the
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equations determining the prices of non-basics all other prices and the two distributive
variables are given; the choice of techniques for non-basics can then be studied in the
following way. Assume that there are non-basics and that commodities have been numbered
such that the first m commodities are basics, thus A is formed by four sub-matrices A11, A12,
0, A22, with A11 being the m×m matrix of coefficients of basic commodities in their
production, and A22 the (n−m)×(n−m) matrix of coefficients of non-basics in their
production. The price equation of the t-th non-basic commodity (m<t≤n) is
pt = (1+r)(p1a1t+...+pnant)+waLt.
In this equation the terms (1+r)(p1a1t+...+pmamt)+waLt are given. They can be summed
up into a single constant scalar bt (mnemonic for the basic component of cost). Therefore, if
we indicate with p^ the row vector of prices of non-basics, with b the semipositive row
vector of constant terms bt, and with A^ the matrix (1+r)A22, we can write
[2.33]
p^ = p^A^ + b.
It is [I − A^]−1 ≥ 0 because of our assumption that either the dominant eigenvalue of
the basics-sub-matrix of A is also the dominant eigenvalue of A, or anyway the rate of profit
remains below the minimum among the maximum rates of self-reproduction of non-basics
that use themselves in their production; hence p^ can be determined. It is strictly positive
because we assume that all non-basics use, directly or indirectly, some basic commodity. It
depends on A22 and on b. If at these prices some other method for some non-basic
commodity yields extra profits, then it will be introduced and it will lower the price of that
commodity and of all non-basics in whose production it enters directly or indirectly.
Therefore the choice of methods for non-basics can be analyzed in very simple terms: those
methods will be chosen, which minimize the prices of non-basics in terms of any numéraire
consisting of basics only.
Here too, for the case in which there are non-basics that directly or indirectly use
themselves in their production, we must prove that this choice of methods does not end up
in indefinite oscillations back and forth between two methods. Thus we must prove that if,
at the non-basic prices p^(α) associated with (A^α, bα), a method β for the t-th non-basic
commodity yields extra profits i.e. ptβ(α)<pt(α), then at the prices p^(β) associated with (A^β,
bβ), (all other methods having remained unchanged) method (α) does not yield extra profits
i.e. it is not the case that ptα(β)<pt(β). The proof is by contradiction. With symbols of meaning
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analogous to those used earlier, since A^α and A^β only differ in the t-th method we can
write
p^(α) ≥ p^β(α) = p^(α)A^β+bβ
p^(α)[I− A^β] ≥ bβ
p^(α) ≥ bβ[I− A^β]−1 = p^(β)
Thus ptβ(α)<pt(α) implies p^(α) ≥ p^(β). Analogously, ptα(β)<pt(β) implies p^(β) ≥ p^(α). These two
inequalities can be simultaneously satisfied only if p^(α) = p^(β), but this would imply that at
prices p^(α) it is ptβ(α)=pt(α), a contradiction with the assumed ptβ(α)<pt(α). QED
Let us notice that the introduction of non-basics does not change the basic principle
that had emerged about the choice of techniques in the case of basics. The tendency to
switch, at each set of prices and distribution, to methods (if such methods exist) that yield
extra profits because cost-reducing, will finally bring to choose a cost-minimizing
technique, i.e. a set of methods, one for each commodity, such that no alternative method
yields extra profits at the prices and distribution determined by that technique.
Now we can relax the assumption that the numéraire consists only of basic
commodities. The numéraire is irrelevant for the determination of relative prices; if the rate
of profit is given, the numéraire only fixes the absolute values of prices and of the rate of
wages: the adoption of a different numéraire, possibly including non-basics, only changes
all prices and w in the same proportion[36]. Of course the choice of numéraire influences the
position and the shape of the w(r) curve, and therefore if what is taken as given or as
exogenously varying is w rather than r, the choice of numéraire influences relative prices
and the rate of profit; but it does not influence the sign of the slope of the w(r) curve, which
is in all cases decreasing, nor, as already noticed, its position (above, below, or coinciding)
relative to the w(r) curves of other techniques. These considerations extend to w(r) curves
based on numéraires including, or even exclusively consisting of, non-basic commodities.
The intuitive reason is that if the real wage rate is measured in terms of a numéraire
including non-basics, then by adopting the C-representation and putting wadv=w/(1+r) one
renders those non-basics basic[37]. More formally, if the numéraire includes non-basic
36
If we pass from a numéraire v to another numéraire v^ such that it is pv^=bpv=1, i.e. if b
is the value of the new numéraire in terms of the old one, then passing from the old to the new
numéraire only means dividing all prices and w by b.
37
If r=r° is given, and determines the prices of basics and non-basics p(r°) and the wage rate
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commodities, the function w(r) is still given by [2.25], and the relationship between prices
and the real wage can still be based on equations [2.26] and [2.26”], with the sole difference
that now A, I, aL and v must be matrices and vectors of dimension greater than the number
of basic commodities, because they must also include, besides all basics, all non-basics
appearing in v and also all non-basics directly or indirectly required for the production of
those non-basics[38]: with this reinterpretation of those symbols, expressions [2.25] and
[2.26] are still valid, because they only require that [I−(1+r)A]−1 exists and is non-negative,
and this is the case as long as (1+r)<1/λA*, a reasonable assumption as argued earlier. The
analysis of choice of techniques for basic commodities can in the same way be generalized
to numéraire commodities[39] as the reader will have no difficulty in checking.
Techniques including different commodities
2.29. We can now discuss the case of choice between two alternative methods for the
production of a commodity, when a method requires the use of commodities not appearing
in the technique including the other method. (It is clear that the issue of choice of techniques
can always be reduced to a succession of choices between techniques which differ in the
method of producing only one common commodity.)
To fix ideas, suppose that technique (α) includes n commodities, and that technique
(β) has in common with technique (α) these n commodities, as well as the methods to
produce all of them except commodity 1; method β for commodity 1 requires as inputs, in
addition to some of the n commodities common to both techniques, some commodities
indexable as n+1, n+2, ..., n+h, which do not appear at all in technique (α). Technique (β)
must therefore also include industries producing these commodities, some of which might
use themselves directly or indirectly in their own production. Therefore the determination of
the supply price p1β(α) of commodity 1 at prices and distribution (α) requires the
w(r°)=αp(r°)v=α in terms of a numéraire v consisting of basics only, then the real wage rate in
terms of a new numéraire v^ including non-basics is given by w^(r°)=w(r°)/b=(α/b)p(r°)v^=α/b
where p(r°)v^=bp(r°)v. One can then determine C on the basis of the subsistence vector
(1+r°)−1(α/b)v^ so that p(r°)=(1+r°)p(r°)C; the system p(r)=(1+r)p(r)C+w+(r)aL, pv^=1, yields w(r)
in terms of the numéraire v^ (which now consists of basics) as w(r)=w+(r)+α/b.
38
All commodities included in the numéraire become basic if one adopts the B
representation of technology and one reasons in terms of the advanced wage rate wadv − from which,
as we know, it is often very easy to reach conclusions on the behaviour of w.
39
Where this term means commodities appearing in the numéraire basket plus commodities
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determination also of the supply prices of commodities n+1,..., n+h. This means that we
have h+1 equations to determine the h+1 supply prices of these commodities. Formally,
these h+1 equations are like price equations of non-basic commodities in technique (α), and
therefore they can be univocally solved. Thus p1β(α) is well determined and can be compared
with p1(α). The determination of p1α(β) does not even present this complication since all its
means of production appear in technique (β). The comparison of the convenience of the two
methods can therefore be effected. Now, the proof given above of Lemma 2.10 that if
p1    p1   , then at the given r it is w    w   , did not need A to be indecomposable: the
matrices Bα and Bβ^ were indecomposable anyway since all commodities need labour,
directly or indirectly, to be produced. The key to that proof was that the subsistence was
assumed to consist of numéraire. All that is needed to apply the same proof here is that
commodity 1, or one of the commodities which directly or indirectly use commodity 1 for
their production, be included in the numéraire. (If commodity 1 is non-basic and not
included in the numéraire not even indirectly, a method that decreases its cost has no effect
on w.) Lemmas 2.11 and 2.12 too do not need an assumption that A be indecomposable. We
conclude that those three lemmas are also valid for the case in which techniques differ in the
commodities they include; only, the comparison on the profitability of alternative methods
must be applied to the common commodities. Thus, the conclusion of Lemma 2.12 that if
two techniques are both on the outer envelope at the given r, then p(α)=p(β) , must be
intended to apply to the prices of the common commodities only.
More generally, if two techniques differ in the commodities they include, we can
always imagine to add, to the industries appearing in the first technique, new non-basic
industries producing the commodities only produced in the other technique; these new
industries will use the methods with which they appear in the second technique. The first
technique thus enlarged is called a ‘bordered’ technique. The price equations thus added to
the first technique will tell us what supply prices these added commodities would have if
produced in the economy as non-basics, using the first technique methods when possible.
Thus if two techniques differ in the production of tomatoes, and in the second
technique the method for the production of tomatoes uses a specific fertilizer only used for
the production of tomatoes and not used in the first technique, in the second technique there
directly or indirectly required for their production.
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will appear an industry producing this fertilizer, which will not be present in the first
technique. We can hypothetically imagine that a demand for this fertilizer arises from, say,
some research laboratory even when the economy uses the first technique, in which case
this fertilizer is produced in the first technique too, as a non-basic commodity. With this
enlargement, all techniques can be made to include the same commodities; the difference
will concern how these commodities are produced and which commodities are basic: some
commodities may be basic in one technique and not basic in another. Having completed or
‘bordered’ the techniques in this way, the criterion of choice of methods on the basis of cost
minimization and of comparison of techniques differing in only one method becomes
applicable. In the fertilizer example, the enlarged technique α and enlarged technique β
differ only in the method of direct production of tomatoes; they both produce the fertilizer
and all other goods (apart from tomatoes) with the same method; in α, tomato production
does not use the fertilizer; in β it does. We have shown that cost minimization brings to a
definite solution in this case too, unique in relative prices and the wage rate, not unique in
the methods only when alternative methods for the production of a commodity happen by a
fluke to be equally profitable, and such that, if measured in any commodity common to both
techniques, the wage rate is maximized for the given rate of profit.
The Samuelson-Hicks-Garegnani model
2.30. As an example of 'bordered' techniques, we study the model first proposed by
Samuelson in 1962 and then often utilized in the Cambridge controversy in capital theory. It
is a two-industries model, where there is a single consumption good, which can be produced
by alternative methods; each different method for the production of the consumption good
utilizes labour and a different circulating capital good, which is produced by itself and
labour via a method specific to it. Thus each technique uses a different capital good, the sole
commodity common to alternative techniques is the consumption good, and the comparison
of alternative methods can only concern the method of producing it; but the comparison
cannot be restricted to the sole method directly producing the consumption good because
the capital good utilized differs, so it must necessarily extend to the entire technique, and
therefore it always involves goods not produced in both techniques (unless these are
completed or ‘bordered’).
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Let the consumption good be commodity 2, and the capital good be commodity 1α,
or 1β, or 1γ, etc., depending on the method for the production of the consumptio good, i.e.
on the technique. Technique α is then a couple (Aα,aLα) where the 2×2 matrix Aα has a
positive first row a1α=(a11α, a12α) and a second row consisting of zeroes. We choose the
consumption good as numéraire. The price equations are
[2.34] p1α = (1+r)p1αa11α+w(α)aL1α
[2.35] p2 = 1 = (1+r)p1αa12α+w(α)aL2α .
Formally the consumption good is a non-basic, but being the numéraire its direct
method of production influences the shape of the w(r) curve, which is given by the function
(where of course the coefficients are those of the technique in use):
1−(1+r)a11
[2.36] w = ——————————— .
aL2+(1+r)(a12aL1−a11aL2)
If we want to compare the profitability of producing the consumption good with
method α or with method β, we must also consider capital good 1β and its method of
production. We take the rate of profit as given. Having determined w(α), the supply price
p2β(α) of the consumption good produced with method β at the prices and distribution (α) is
given by the system of equations
[2.37] p1β(α) = (1+r)p1β(α)a11β+w(α)aL1β
[2.38] p2β(α) = (1+r)p1β(α)a12β+w(α)aL2β.
which can be solved in succession: the first equation determines the supply price of the
capital good 1β, then the second equation determines p2β(α) . These two equations are
formally identical to the price equations of two non-basics, for example a particular bean
that needs itself to be produced, and is used to prepare a special flour.
In this case, the ‘bordered’ technique α has capital good 1α as good 1, the
consumption good (produced with method α) as good 2, and capital good 1β as good 3; its
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a11

A matrix is  0
 0

a12
0
0
Adv Micro chapter 2
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0 

0  and its labour coefficients vector is
a11 

a L1
p.
a L 2
60
a L1
.
The ‘bordered’ technique β includes the same three goods and differs in the sole production
method of the consumption good, whose technical coefficients are now a12β and aL2β .
Capital good 1β is non-basic in the ‘bordered’ technique α, and capital good 1α is non-basic
in the ‘bordered’ technique β, and since neither appears in the numéraire, the shapes of the
w(α)(r) curve and of the w(β)(r) curve are unaffected by their addition.
Our previous results imply that p2β(α) is greater than, equal to, or smaller than 1
according as the rate of wages w(β) is less than, equal to, or greater than w(α)[40]. Therefore in
order to ascertain which method is more convenient in the long period, we need only trace
the w(r) curves and pick the technique whose curve w(r) lies above the other curve at the
given level of r[41].
2.31. In this very simple case, it is easy to show that two w(r) curves can cross more
than once. Let us make our study of the w(r) curve as simple as possible. We assume
0<a11<1; aL1>0; a12>0. Let us measure each capital good in such a unit that aL1α= aL1β =...=1.
Let us indicate the price of the capital good simply as pα, pβ etc. when produced in its
technique; let us further simplify the symbols by putting a11≡a, a12≡b, aL2=l. The price
equations of technique α are then
pα=(1+r)pαaα+wα
1=(1+r)pαbα+wlα
Expression [2.36] becomes
40
As an exercise, the reader can try to prove this result for the present case directly from
equations [2.34-2.38].
41
If the economy is using technique α and then technique β is discovered, in order to pass to
it one will have to use some provisional method to produce the first units of capital good 1β; a more
complete analysis would have to prove that, in spite of the need initially to use a provisional method
later discarded because less efficient and therefore more expensive (which is the reason why it is
not considered in the equations in the text), the greater cost of the production of the first units of
capital good 1β does not prevent the shift in method from being convenient. We omit a rigorous
analysis of this issue, but it should be clear that, as a longer and longer number of successive
periods is considered in which the consumption good is produced by capital good 1β, the initial
greater cost of producing the first units of that capital good becomes less and less relevant and, if
prices and distribution remain determined by α, the average rate of profit earned over the ensemble
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w(r)=[1−(1+r)a]/[l+(1+r)(b−al)].
We first study the intercepts with the axes. For r=0 it is w=W=[1−a]/[l+b−al]>0;
for w=0 it is r=R=(1-a)/a>0; within these values, w(r) is well defined, positive and
differentiable and the derivative is negative:
dw/dr = −b/[ l+(1+r)(b−al)]2 < 0 .
We further note that dw/dr is respectively a constant, a decreasing function of r (that
is, increasing in absolute value), or an increasing function of r, according as b−al is
respectively zero, negative or positive, i.e. according as the proportion between physical
capital and labour is respectively the same in the two industries (i.e a=b/l), greater in the
capital industry (a>b/l), or smaller in the capital industry (a<b/l). In these three cases w(r) is
respectively a straight line (with slope −b/l2 = −a/l and W=(1−a)/l), concave, convex.
Let us now show that two w(r) curves can cross twice in the positive orthant. Several
numerical examples have proved this possibility, but we want to reach a graphical intuition.
Two w(r) curves w(α)(r) and w(β)(r) can have the same intercepts and yet be, one concave and
the other convex or a straight line: the same R requires that aα=aβ; the same W requires that
lα(1−a)+bα=lβ(1-a)+bβ; then one can obtain different wage curves with the same intercepts
by assuming, for example, these two conditions and in addition that lβ>lα, that bα−alα=0
which implies w(α)(r) is a straight line, and that bβ−alβ<0 which implies w(β)(r) is concave.
Now consider a third technique γ with w(γ)(r) curve a straight line too but with just slightly
greater intercepts[42]: this curve will intersect w(β)(r) twice, cf. Fig. 2.4.
w
of periods by the use of the new method sooner or later rises above r.
42
This requires that aα decreases slightly, that (1−aα)/[ lα(1−aα)+bα] increases slightly, and
that bα−aαlα remains equal to zero; this is achievable, e.g. let us suppose that lα remains unchanged
and let us remember that initially bα=aαlα i.e. Wα=(1−aα)/lα; then a small decrease of aα
accompanied by a proportional decrease of bα suffices to obtain the desired result. If one wishes to
maintain the slope unchanged, it suffices that lα decreases in the same proportion as aα, and that bα
decreases by the same amount as aαlα, so as to simultaneously maintain aα/ lα unchanged and bα−aαlα
equal to zero. The increase in the intercepts can be made as small as one likes, so a double
intersection with the concave w(β)(r) curve can certainly be achieved.
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β
α
γ
r
Fig. 2.4
We have here one instance of the phenomenon called ‘reswitching of techniques’: as
r increases from zero, the choice of techniques at a certain point ‘switches’ from technique γ
to β, but at a higher rate of profit it switches back, or ‘reswitches’, to technique γ. (We will
see later (ch. 7) that this phenomenon creates problems to the marginalist approach.)
2.32. Let us go in greater depth into why reswitching of techniques can happen.
Consider two consumption goods, ‘a’ and ‘b’, let their prices be pa(r) and pb(r), and let us
indicate with La(−t), respectively Lb(−t) the quantities of dated labour one obtains, respectively
for good ‘a’ and for good ‘b’, if one performs the reduction to dated quantities of labour.
We know that either price, if measured in labour commanded (i.e. if we put w=1), can be
expressed as the sum of a generally infinite[43] series of dated quantities of labour, each one
of them multiplied by (1+r) raised to a power equal to the number of periods inbetween the
payment of wages to that quantity of labour, and the sale of the product (cf. equations
[2.26’, 2.26”]). The difference D(r) ≡ pa(r)−pb(r) is then given by
D(r) ≡ pa(r)−pb(r) =
= (aLa−aLb)+(1+r)(La(−1)−Lb(−1))+(1+r)2(La(−2)−Lb(−2))+...+(1+r)t(La(−t)−Lb(−t))+...
where each of the several terms La(−t)−Lb(−t) can be positive, negative, or zero. This means
that the function D(r) can have a complex behaviour: it can alternate upward- and
43
The reduction will generate a finite number of dated quantities of labour if, going
backwards from any product to its produced inputs and their produced inputs and so on, one reaches
after a finite number of steps capital goods produced by labour alone (or labour and land alone, if
one admits land). Then no capital good is basic, all capital goods are different from the capital
goods which directly or indirectly produce them. This type of technology is called ‘Austrian’
because assumed by Bohm-Bawerk and many other Austrian economists (and Wicksell).
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downward-sloping sections, and it can cross the horizontal axis (i.e. take value zero) for
several values of r. The possibility of such behaviour is easily grasped, by for example
supposing that after a certain t the terms are all zero, or are so small that their effect on
pa(r)−pb(r) is negligible; then D(r) is, or is sufficiently approximated by, a polynomial
function of r, of degree t; for t>2 the derivative of this function can change sign, and the
equation D(r)=0 can (but it need not) have up to t distinct real solutions. What remains to be
shown is that these phenomena can happen for economically acceptable values of r. Several
numerical examples have shown that it is indeed so. Sraffa (1960 p. 37) has produced the
following numerical example:
we may suppose two products which differ in three of their labour terms … , while
being identical in all the others. One of them, ‘a’, has an excess of 20 units of labour
applied 8 years before, whereas the excess of the other, ‘b’, consists of 19 units employed
in the current year and 1 unit bestowed 25 years earlier. (They are thus not unlike the
familiar instances, respectively, of the wine aged in the cellar and of the old oak made into
a chest.).
The equation of D(r) in this case is D(r) ≡ pa−pb = 20w(1+r)8 − [19w+w(1+r)25]; Sraffa
assumes that the numéraire is the Standard Commodity (this renders the real wage a linear
function of r), and that the maximum rate of profit is 25%; thus in the above expression it is
w=1−(r/0.25). Sraffa’s graphical representation of D(r) is reproduced below.
Grafico di Sraffa 1960 p. 38
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Fig. 2.5. Sraffa 1960 p. 38
Now, it suffices to re-interpret pa and pb as, not the prices of two different
commodities, but as p1α and p1β(α), i.e. as the supply price of commodity 1 according to
which one of two alternative methods α or β is used to produce it, and the sign of D(r) will
indicate which of the two methods is more convenient: method α will be preferred if D(r)<0.
Two or more changes in the sign of D(r) indicate reswitching. (In Sraffa’s own example, reinterpreted as referring to alternative methods for the production of the same commodity,
there isn’t reswitching, but it would suffice to reduce by a small amount one of the relevant
dated labour quantities of commodity a, and this, by slightly reducing pa(r), would cause a
downward shift of the curve D(r) which would then cross the horizontal axis twice.)
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2.33. An even simpler numerical example, due to Paul Samuelson (1966), is based on
'Austrian' production techniques: ‘reducing’ costs to dated quantities of wages with
compound profits on them, it is assumed that going backwards from the commmodity’s
means of production to their means of production and to the means of production of those
means of production and so on, after a finite number of steps one reaches commodities
produced by labour alone (there is no basic commodity), so the reduction to dated quantities
of labour has a finite number of terms. The example is the following: the production of 1
unit of champagne requires the payment of 7 wages two periods before the sale of the
product, while the production of whiskey requires the payment of 2 wages three periods
before, and of 6 wages one period before the product is sold. The long-period prices pc of
champagne and pw of whiskey are given by
pc = 7w(1+r)2
pw = 2w(1+r)3+6w(1+r).
Put w=1, then pc=7(1+r)2, while pw=2(1+r)3+6(1+r). The reader can check that:
pc=pw for r=50% and for r=100%,
pw<pc for 0.5<r<1,
pc<pw for 0<r<0.5 and for r>1.
At r=0 it is pc<pw because cost consists only of wages and a unit of champagne requires the
payment of 7 wages against 8 for a unit of whiskey; but the price difference decreases as r
increases, and is reversed as r becomes greater than 50%, because interest costs initially
increase faster in the production of champagne than of whiskey; however, as r continues to
increase, compound interest ends up by causing a greater increase of the cost of whiskey, so
the price of whiskey starts approaching the price of champagne and the ratio between the
two prices is reversed as r becomes greater than 100%[44].
44
Exercise: The reader is invited to re-write the production conditions of the two
commodities in these terms: assume all production processes take one period, and each final
commodity is produced by a process which is the last one of a series starting with a process where a
(single or composite) commodity is produced by unassisted labour; this commodity is then used,
alone (i.e. through aging) or together with living labour, to produce the next period another (single
or composite) commodity, in turn used with or without labour to produce another commodity, and
so on up to the final champagne, or whiskey, commodity. (You must decide whether wages are
advanced or paid at the end of the period. In this second case, the production of whiskey, for
example, requires considering 4 successive production processes.) The reader is invited to
determine the labour-commanded price of each one of the intermediate commodities thus implicitly
postulated, and to check that with such an ‘Austrian’ assumption of absence of basic commodities
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pc/pw
1
1/2
–1+ 3
1
7/8
O
r
Fig. 2.6
The reader can check that the derivative of the ratio pc/pw with respect to r is positive,
zero or negative according as r is less than, equal to, or more than –1+ 3 (45). Graphically
pc/pw behaves as shown in Fig. 2.6.
If now we interpret the production method of whiskey as in fact an alternative method
for the production of champagne, this alternative method is the more convenient one for
50%<r<100% while the original method is more convenient for r<50% and for r>100%: as r
rises from zero, at r=50% the optimal technical choice switches to the alternative
(‘whiskey’) method, but at r=100% there is reswitching back to the original method.
The possibility of reswitching derives from the possibility of reversals in the direction
of the movement of relative prices as the rate of profit or of interest rises. The latter
possibility had not been suspected before Sraffa, and it questions some fundamental theses
of the marginalist/neoclassical approach to capital, as will be explained in chapter 7.
there is no maximum rate of profit: when w tends to zero the rate of profit rises without limit,
tending to +∞.
45
The rate of interest is assumed to be non-negative, so the solution r = –1– 3 is excluded.
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FIXED CAPITAL
2.34. Up to now we have assumed that all produced inputs (capital goods) are
circulating capital goods i.e. are entirely used up in a single production cycle, disappearing
entirely into their product: examples are seed to produce corn, raw minerals used to make
metal sheets, wood and fabric that go to make a sofa, flour to make bread, parts assembled
to make a car. Many capital goods on the contrary are durable: a hammer, a tractor, a
computer can be used for years.
The way generally used nowadays to include these goods into long-period price
theory is to treat them as goods that enter the productive process and also come out of the
same productive process jointly with the main product, but one period older, therefore they
are not really the same durable capital good that entered the production process as an input.
Thus the productive process is visualized as jointly producing a main product and one-yearolder durable capital goods. (We call the length of the production period a ‘year’, but this is
only to help intuition, in fact the ‘year’ can be assumed to have any length.)
In this way durable capital is viewed as a particular case of joint production, i.e. of
productive processes that inseparably and necessarily produce different products together.
Cases of joint production different from the ones originated by the presence of durable
capital goods are, for example, wheat and straw; wool and mutton; oil refining; transport
processes that simultaneously involve different goods or goods (or persons) transported for
different lengths. For the purpose of comparison with the marginalist approach we do not
need to examine general joint production now. We will come back on joint production after
discussing the marginalist approach. Now we concentrate on pure, or simple, fixed capital,
which is the case in which a durable capital good (or ensemble of several capital goods
forming a fixed capital complex) is used in the production always of the same main product,
and there is no joint utilization of durable capital goods of different ages[46].
For example in an economy that produces corn, iron and tractors, where corn and
iron are circulating capital goods, while tractors are durable, last three years, and are used in
the production of corn, the productive methods will use and produce commodities as
follows (‘*’ stands for ‘together with’, ‘→’ stands for ‘produce’):
46
Unless the efficiency of older capital goods is independent of the age of the capital goods
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(1) corn * iron * labour
→ new tractors
(2) corn * iron * labour
→ iron
(3) corn * iron * new tractors * labour
→ corn * one-year-old tractors
(4) corn * iron * one-year-old tractors * labour → corn * two-years-old tractors
(5) corn * iron * two-years-old tractors * labour → corn (* three-years-old tractors
that are no longer of any use and are simply thrown away as scrap)
We avoid assuming that the scrap can be sold at a positive value because this would
generally mean that, through some productive process, it might be recycled and become
iron. Then we would have true joint production, the last tractor-using process would
produce jointly corn and a product used as input in another industry, and we postpone
treatment of this case to the general treatment of joint production (ch. 10). On the other
hand, getting rid of the useless three-year-old tractors might be costly, but we neglect this
possibility through an assumption of free disposal, namely, that it is always possible to
dispose costlessly of any amount of any product. Formally, this assumption means that this
economy also has at its disposal the following possible methods (in addition to methods that
use inputs and produce nothing at all):
(6) corn * iron * new tractors * labour
→ corn,
(7) corn * iron * one-year-old tractors * labour
→ corn,
(8) corn * iron * new tractors * labour
→ one-year-old tractors
(9) corn * iron * one-year-old tractors * labour
→ two-years-old tractors
(The inputs are respectively the same as in methods 3, 4, 3, 4.) This means that the
economy implicitly has a problem of choice of techniques on how to produce corn. The
economy will produce corn with method 6, and therefore will use methods 1, 2, and 6, if it
is convenient to use tractors only if new; corn will be produced with methods 3 and 7, and
therefore the activated methods will be 1, 2, 3, 7 if it is convenient to use one-year-old
tractors but not two-year-old tractors. Methods 8 and 9 on the contrary can be neglected
because they will always be less convenient than, respectively, methods 3 and 4, because
jointly used in the method from which they emerge − as will be explained later.
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the latter methods produce in addition another good, corn, which has positive value. For the
same reason we can neglect the processes − mathematically assumed available by the
assumption of free disposal − that use inputs and produce nothing, or that use the same
inputs as other methods but produce less of outputs with a positive value.
It is generally possible to establish a maximum number of periods of use of a durable
capital good, say k, after which the good is totally unusable and must be thrown away
(attempts to produce with the capital good of age k or greater produce zero amounts of the
main product). As that age approaches, the capital good might be usable only at the cost of
greater and greater expenses due to maintenance and repair, and might possibly be of lower
and lower efficiency. It might therefore be convenient to stop using the capital good before
it reaches age k. The problem of choice of techniques is whether to go on using the capital
good for k periods, or to stop before. All techniques in which the use of the capital good
stops earlier than at age k are called truncations (of the possible life of the capital good),
and for simplicity the longest possible use is also often called a truncation. In our example,
k=3 and, if we indicate a truncation with a number indicating the oldest age with which the
capital good enters an activated process, the possible truncations are 0, 1 and 2. The number
of processes in which the capital good is used is equal to the age at which it is thrown away,
so it is the truncation number plus one. The earliest possible truncation is always truncation
0, in which case the capital good is used only when new, and if this is the case for all fixed
capital goods used in the economy, the system is identical to a single-production system
with no fixed capital.
This problem of choice of techniques is the sole one we analyze now; after showing
that it is solved in a nice way, one will be able to add it to the usual problem of choice of
techniques (that consists in choosing between two alternative methods, while here we have
to choose whether to add an additional method or not).
To fix ideas, let us continue to suppose that the problem concerns the production of
corn (which we take as numéraire) and that the durable capital good is tractors. The steps to
arrive at the optimal truncation are as follows. (i) Solve the system of equations assuming
that truncation 0 is used, i.e. assigning price zero to the one-year-old tractors jointly
produced and assuming that only new tractors are used. (In our example, this would mean
solving the price equations associated with methods 1, 2, and 3 in which the price of oneyear-old tractors is set at zero; or, equivalently, the price equations associated with methods
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1, 2 and 6.) The resulting system of equations is identical to that of single production
because only new tractors are used. We assume that the given rate of profit r is not higher
than the maximum rate of profit associated with that truncation, hence prices are all positive
and the real wage rate is non-negative. (ii) At the prices and real wage rate determined in the
first step, calculate the rate of profit on the method using one-year-old tractors (method 4 in
our example), assuming these tractors to have price zero, and assuming two-year-old
tractors to have price zero. (iii) If this rate of profit is less than r, stop there: truncation 0 is
optimal[47]. (iv) If this rate of profit is greater than r, then the price of one-year-old tractors
cannot be zero because then it would be convenient to set up firms using method 4, and the
demand for one-year-old tractors would increase until it raised their price from zero; so their
price must be positive. But then to keep extraprofits at zero in the firms using method 3 the
price of corn must decrease, i.e. (since corn is the numéraire) its cost of production, i.e. the
real wage and the price of iron and of new tractors, must rise. The real wage must rise,
because otherwise the price of iron and of new tractors would not change and then the cost
of production of corn would not rise. I prove now that indeed the price of all inputs to
method 3 (other than corn, of course) increases when the real wage rises. The proof,
formulated so as to apply to the general case (and therefore applicable to any number of
inputs), is the following[48]: in the costs of any one of these inputs the real wage rate
increases; then the price of this input can decrease only if the price of at least one of its
inputs decreases by an even greater percentage[49]; considering the price of this second
commodity one reaches the same conclusion that at least one of its inputs (different from
both the first and the second commodity) must decrease in price by an even greater
percentage; proceeding in this way one necessarily exhausts the available commodities, and
the last one of them cannot decrease in price because there is no commodity left whose price
can decrease even more; but then the next-to-last commodity cannot decrease in price either,
and so no one can. But if no commodity directly or indirectly used in the production of corn
47
In order to keep proofs simple I am assuming that the efficiency of older durable capital
goods does not increase with age.
48
This proof, inspired by an analogous proof by Sraffa, applies to all cases in which
technical progress increases the amount produced of numéraire while the inputs remain unchanged.
49
To fix ideas suppose the input is iron. Since one component (the wage bill) of the cost of
production of iron has increased, in order for its total cost and price to decrease by 1% the cost of
the other inputs must decrease by more than 1%, and this requires that at least one of these inputs
other than iron decreases in price by more than 1%.
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can decrease in price, then they must all rise in price since their costs have increased owing
to the wage rise. Thus a positive price of one-year-old tractors and a rise in the real wage
rate and in all other prices (except of course the price of corn) are finally reached at which
the rate of profit is r also in the production of corn with the use of one-year-old tractors.
Truncation 1 is feasible, it yields a higher wage rate than truncation 0, and at the prices of
truncation 0 it is convenient to pass to truncation 1. (v) Now we repeat the procedure for
method 5, that uses two-year-old tractors, taking as starting points the prices and wage rate
determined by truncation 1 and assuming a zero price of three-year-old tractors. If at these
prices (that include a price zero for two-year-old tractors) method 5 yields a rate of profit
less than r, then we do not introduce it and stop at truncation 1; if method 5 yields extra
profits, then for the same reasons discussed under step (iv) a positive price for two-years-old
tractors will exist that, together with a higher real wage and price of all other goods, makes
this method yield the rate of profit r; truncation 2 is feasible, it yields a higher rate of real
wage than truncation 1, and at the prices of truncation 1 it is convenient to pass to truncation
2. (vi) In our tractors example truncation 2 is the last possible one, but the procedure in the
general case will go on until either the truncations are exhausted (age k is reached), or the
method extending the use of the capital good by one more year is unable to yield the rate of
profit r even at a price zero of the old capital good it uses. If this next truncation were added
and a uniform rate of wages were imposed, the price of the oldest capital good used would
come out negative[50]. At the last feasible truncation, all prices are positive and the real
wage rate is maximized for the given r: the economy is on the outer envelope of the w(r)
curves.
Which truncation maximizes the real wage rate can depend on the level of the rate of
profit. Also, just like two techniques can co-exist on the outer envelope, so two truncations
50
Suppose truncation t is the optimal one according to the procedure outlined; at its prices
and wage rate the method using the machine of age t+1 yields a rate of profit lower than r when the
machine has price zero; in order to raise its rate of profit, one must give the t+1-years-old machine a
negative price; but this lowers the profits of the method using the t-years-old machine, and requires
therefore a lowering of its costs, which requires a lowering of the wage rate and of all other prices,
for the reasons symmetrical to the ones of the case in which the t+1-years-old machine obtains a
positive price; this symmetrical reasoning concludes that the price of the machine which is able to
restore a uniform rate of profit and wage rate is negative, and the wage rate is lower than in
truncation t.
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can be equally optimal – but in this case the oldest capital good used by the truncation of
higher number has price exactly zero.
Simple fixed capital confirms therefore that competition tends to select the methods
that, given one of the two distributive variables, maximizes the other one.
Assuming competition to have already performed this function and that the result in
our example is that truncation 2 is adopted, here are the price equations of that example,
specified without the use of matrices and vectors in order to help readers to understand what
is going on. Indices distinguishing commodities and industries can be kept reasonably
simple, in this case, as follows. Technical coefficients refer to methods that produce 1 unit
of main product, thus the three methods that produce corn all produce 1 unit of corn, the
first two of them also jointly produce tractors, in amounts respectively bT1 for the output of
one-year-old tractors, bT2 for the output of two-year-old tractors. The input technical
coefficients of corn and of iron have a first index indicating the input good: corn is good 1,
iron is good 3 [51]; the second index indicates the industry or method: the iron industry is 3,
the new tractor industry is T, the corn industry comprises three methods and accordingly has
index 1 but followed by a further index 0, 1 or 2 indicating the age of the tractor used to
produce corn. A new tractor is good T0, a one-year-old tractor is T1, a two-year-old tractor
is T2; since tractors of age h are only used as inputs in one method (the method that
produces corn using tractors of age h), and are only produced in one method (the method
that produces corn using tractors of age h−1), for their technical coefficients it is
unnecessary to use three indices: thus for example aT0 is the input coefficient of tractors in
the method that produces 1 unit of corn using new tractors, and produces jointly one-yearold tractors in amount bT1. (A natural choice of units makes bT1=aT0, and bT2=aT1.) Thus,
with ‘*’ standing for ‘together with’, and ‘→’ standing for ‘produce’ we have:
a110 * a310 * aT0 * aL10 → 1 unit of corn * bT1
a111 * a311 * aT1 * aL11 → 1 unit of corn * bT2
a112 * a312 * aT2 * aL12 → 1 unit of corn (* three-year-old tractors of value zero)
a13 * a33
* aL3 → 1 unit of iron
a1T * a3T
* aLT → 1 new tractor
51
There is no good 2; iron is given number 3 to avoid confusion with two-year-old tractors
which are indicated as T2.
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The price of corn is p1, the price of iron is p3, the price respectively of new, one-yearold and two-years-old tractors are pT0, pT1 and pT2. Three-years-old tractors have price zero
and do not appear explicitly. Wages are paid in arrears. The price equations are then as
follows:
(1+r)(a110p1+a310p2+aT0pT0) + waL10 = p1 + bT1pT1
(1+r)(a111p1+a311p2+aT1pT1) + waL11 = p1 + bT2pT2
[2.41]
(1+r)(a112p1+a312p2+aT2pT2) + waL12 = p1
(1+r)(a13p1+a33p3) + waL3 = p3
(1+r)(a1Tp1+a2Tp2) + waLT = pT0
p1 = 1
The last equation specifies the numéraire. The other equations are as many as the
goods produced, which include the older tractors: matrix A is square. With the inclusion of
the numéraire equation, there are six equations in seven variables: five prices, r, and w. If r
or w is given, the system is determinate. Earlier truncations eliminate one or more
equations but they also eliminate the same number of products and of prices (the prices of
the old machines no longer utilized) so the system of equations remains determinate. In
particular, matrix A remains square.
Another noteworthy result of the analysis of choice of truncation with simple fixed
capital is that the outer envelope of the w(r) curves is downward-sloping : if r increases, the
real wage rate associated with the optimal truncation (which can change as a result of the
change of r) decreases. The proof is as follows.
The analysis given above of how the optimal truncation is reached shows that since
prices were positive at truncation 0, they remain positive throughout, and also the prices of
the older capital goods that it is convenient to use are positive[52]. Thus we have proved that
on the outer envelope of the w(r) curves all prices are positive. Let us then assume that we
are on the outer envelope and that r rises (by an amount such that the truncation does not
52
The proof of this result given in the text assumes that the rate of profit is not higher than
the R of the first truncation. But one might take the real wage rate as given instead of the rate of
profit, and then at each passage to a more convenient truncation it would be the rate of profit that
would rise; the reasoning concerning the choice of truncation can easily be adapted to this case, and
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change), and let us change numéraire, choosing as numéraire the wage rate, w=1, i.e.
expressing prices in labour commanded. To show that the real wage rate decreases we have
to show that all prices thus measured increase. The proof is again based on the fact that if
one price does not increase, then some other price must decrease, and it can do so only if
some third price decreases by a greater percentage, which requires that some fourth price
decreases by a still greater percentage....but when all commodities have been thus accounted
for, the price of the last commodity cannot decrease because no other price can decrease by
a still greater percentage, hence no price can decrease. (The reader is invited to apply this
reasoning to the price equations of the corn & tractors example above, remembering that
w=1, and to verify that the presence of jointly produced older tractors does not undermine
the reasoning.)
■
It must be stressed that we have proved that the w(r) curve associated with a certain
trauncation is certainly downward-sloping provided all prices are positive; this is
guaranteed for the portion of the w(r) curve on the outer envelope, but otherwise it is not
guaranteed. We have seen above that, if at the rate of profit r and at the prices of truncation t
the method using the machine of age t+1 yields a rate of profit lower than r, then in
truncation t+1 that machine will have a negative price at the rate of profit r. Then if the rate
of profit rises, it is no longer guaranteed that all prices measured in labour commanded must
rise, because the negative price of the machine may increase in absolute value, reducing the
costs of production in the process in which it is used, as well as the profits in the process
that produces it as a joint product; thus the labour-commanded price of the final commodity
in whose production the machine is used may well decrease, indicating a rise of the real
wage rate in terms of that commodity; and the labour-commanded prices of other
commodities, which use that commodity as means of production, may then decrease too.
An implication of the previous considerations deserves notice. Suppose the rate of
profit is positive, and that at this r>0 the optimal truncation in the production of corn with
tractors is truncation 1, which is in fact the truncation the economy is using. It might be that
at r=0 the optimal truncation is another one and precisely truncation 0. This means that at
r=0 truncation 1 yields a negative price for one-year-old tractors. Now, we have seen that in
this shows that the final rate of profit can be above the initial R with all prices positive.
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simple production the prices at r=0 are proportional to labours embodied. Must we conclude
that in this economy, which is using truncation 1, one-year-old tractors embody a negative
quantity of labour? Is it possible to give a meaning to a negative quantity of labour
embodied? On this issue we stop here, we will examine it in some detail later, after
discussing general joint production.
[??CONTROLLA QUALI ALTRI RISULTATI DI MAINWARING SONO
IMPORTANTI]
The little difference between production with (simple) fixed capital and single
production from the point of view of pricing is highlighted by the following result: given r
and the corresponding truncation, it is always possible to derive, from the methods
producing a final good with new machines, one-year-old machines, etcetera, a composite
method that does not use old machines and determines the same price for the final good.
The procedure is the following. We illustrate it again for corn and tractors, but the
illustration is general. Suppose that corn is produced by k different methods, each producing
one unit of corn, and using respectively tractors of age 0, 1, 2, ... , k−1. Measure tractors of
age t and tractors of age t+1 in the same technical units; then aTh=bT,h+1. Suppose that a firm
runs simultaneously all k methods, at levels of activity such that the process that uses new
tractors produces x0 units of corn, the process using one-year-old tractors produces x1 units
of corn, ... , the process using the oldest (i.e. k−1-years-old) tractors produces xk−1 units of
corn, where these amounts x0, x1, ... , xk−1 are determined so as to satisfy two conditions:
1) x0+x1+...+xk−1=1;
2) for h=0,..., k−2 it is xh/xh+1=(1+r)aT,h+1/bT,h+1, or, bT,h+1xh=(1+r)aT,h+1.
(These conditions can always be satisfied for positive values of the xh’s if r > −1.) This firm
can be seen as producing 1 unit of corn with a composite method that uses, as input of each
good, the sum of the inputs of that good in the k processes. For example the technical
coefficient of labour in the composite method is aL10x0+aL11x1+...+aL1,k−1xk−1. The cost is
equal to the sum of the cost side of the respective price equations, each one multiplied by
the appropriate xh, and analogously for the revenue side. If one calculates the total cost and
the total revenue, one finds that in the sum on the total cost side there appear terms
(1+r)aT1pT1x1, (1+r)aT2pT2x2, ... , (1+r)aT,k−1pT,k−1xk−1, which are respectively equal to the
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terms appearing on the total revenue side bT1pT1x0, bT2pT2x1, ... , bT,k−1pT,k−1xk−2. The costs of
old tractors used as inputs exactly balance the revenues from the jointly produced old
tractors, so they can be both crossed out; but then the costs and revenue of the firm are the
same as if the corresponding input and output coefficients of old tractors were zero; the
remaining costs are the sole costs of the inputs of final goods (including new tractors), and
the remaining revenue is the sole revenue from the one unit of corn produced overall. Thus
in this composite method the cost of producing one unit of corn depends solely on the
prices, and on quantities of inputs, of final commodities, i.e. can be determined
independently of the prices of old tractors. If one replaces the original k price equations of
the k methods producing corn with the price equation of the single composite method thus
obtained, the resulting system of equations looks like that of single production. True, the
coefficients of this derived composite method depend on r and on the coefficients of the
production with old tractors, but the sole role of these magnitudes is to influence the
amounts of inputs of final commodities to which production with fixed capital is equivalent:
pricing of final commodities produced with fixed capital can be ‘reduced’ to pricing with
single production, and this explains why so many results about prices, choice of techniques,
shape of the w(r) envelope are the same as in single production.
It is perhaps worth stressing that this analysis does not need the assumption that the
durable capital good consists of a single good. It may well consist of a number of
commodities, even as vast as to form an oil refinery or a nuclear plant. As long as these
commodities start being jointly utilized all together and are not later used for the production
of a different output, they can be treated as a single commodity and hence as a single
durable capital good (sometimes called a ‘plant’).
Depreciation
Assuming again a given truncation, now we briefly inquire into the behaviour of
depreciation. In practice, depreciation is often estimated with very rough methods, usually
as a constant. Often it is also unscrupulously played upon (using difficult-to-criticize
estimates of future sales, future rates of utilization of plant, expected technical progress[53])
53
Technological obsolescence can be a cause of an economic life of a durable capital good
inferior to what it would be in the absence of technical progress. We cannot take account of this
extremely important aspect of real economies in this chapter. Non-normal degrees of utilization will
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to influence the appearance and prospects of the firm vis-à-vis investors, tax agents etc. We
here assume that the plant is normally utilized and that future prices can be reasonably
predicted to be the same as current prices. Then the future profits that can be earned by
processes that use older durable capital goods can be estimated with precision, and the value
of the older capital goods can be obtained, from the price equations that we have indicated.
Correct depreciation is generally not constant[54].
As the age of a durable capital good increases, its value will normally decrease; it
can initially increase only if its efficiency initially increases with use, e.g. because a period
of ‘running in’ is necessary in order to achieve full efficiency. Not much more can be said in
general, owing to the possible diversity of the pattern of efficiency, maintenance, normal
repairs etc. over time as the capital good ages. Something more can only be said in the case
of constant efficiency. In this case one must assume that the technical life of the capital
good is limited, i.e. that after a given number of years it suddenly falls to pieces.
It is common in the literature to refer for brevity to durable capital goods as
‘machines’, so now we do it too, in order to accustom the reader to such a usage. If a
machine has constant efficiency throughout its technical life (i.e. up to age k), then
machines of different age are interchangeable as long as they have not reached age k, and
therefore a producer will be ready to pay the same sum to rent a machine for one period,
independently of its age. The value of the machine will then vary with age simply because
the number of periods, for which that same yearly rent can be earned by the owner,
decreases with age. At age k the value of the machine is zero. When new, the value of the
machine equals its cost of production. If the machine can be used up to age k, i.e. for k
years, if the price of the new machine is p, and if the rental per year (paid at the end of the
year) is ρ, then the present value of the k rentals that a new machine will earn during its
economic life must equal its purchase price p, hence the following relationship must hold:
be discussed in a later chapter.
54
A non-constant depreciation is of course compatible with a constant amortization (i.e. a
constant sum set aside each period and employed so as to yield the market rate of interest, whose
cumulated value grows to have the same value as the new durable capital good by the time it must
be replaced). But for purposes like establishing the value of a partially worn-out fixed plant or lorry
that must be bought, it is depreciation that counts.
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(1+r)k − 1
[2.42] p = ρ/(1+r) + ρ/(1+r)2 + ... + ρ/(1+r)k = ρ · ————— if r>0 [= kρ if r=0][55].
r(1+r)k
(The first rental ρ is divided by (1+r), and so on, because paid at the end of the period). The
rental ρ thus obtained is also the constant annuity or amortization to be set aside each year
(and invested so as to earn the rate of profit r) in order to have at the end of k years the sum
p(1+r)k, that is the profit at rate r for k years on the capital p invested in the new machine,
plus the capital itself p, that can be used to replace the machine: this annuity is given by the
well-known formula
r (1  r ) k
[2.42bis] ρ = p∙
.
(1  r ) k  1
Let us use these formulas to study the evolution of the value of the constant-efficiency
machine with age. One year later, the value of the machine is
(1+r)k−1 − 1
[2.43] ρ/(1+r) + ρ/(1+r)2 + ... + ρ/(1+r)k−1 = ρ · —————
r(1+r)k−1
if r>0 [= (k−1)ρ if r=0]
The value of the machine has decreased from its initial value p by the amount
D1=ρ/(1+r)k;
From the end of the first to the end of the second year it has decreased by the amount
D2=ρ/(1+r)k−1=D1(1+r);
and so on. The loss of value − the depreciation − increases with the age of the machine,
being multiplied by (1+r) each year.
The sum of the k terms Dt must equal p, i.e. D1(1+(1+r)+(1+r)2+...+(1+r)k−1)=p. This
makes it possible to study the effect of changes of r on the fraction of the initial value p of a
machine represented by depreciation each year. To such an end let us assume p=1. If r
55
This is because
(1  r ) k 1  (1  r ) k 2  ...  1
(1  r ) k
and the numerator of the fraction on the right-hand side can be replaced by [(1+r)k–1]/r because of
the identity (xk-1+xk-2+...+1)(x–1)=xk–1, i.e. (xk-1+xk-2+...+1)=(xk–1)/(x–1), valid for x≠1.
1/(1+r) + 1/(1+r)2 + ... + 1/(1+r)k =
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increases to r’, the sole way for the sum D1(1+(1+r’)+(1+r’)2+...+(1+r’)k−1) to remain equal
to 1 is for D1 to decrease, i.e. it must be D1(r’)<D1(r). The growth of depreciation quotas is
faster than when the rate of profit was r, and an age t’ must be reached such that
Dt’(r’)>Dt’(r), otherwise the sum cannot be 1; obviously for t>t’ it is also Dt(r’)>Dt(r). Thus
the new series of depreciation quotas corresponding to a higher r must have the first
depreciation quotas smaller, and the last ones bigger, than when r was lower. This means
that at all intermediate ages, the value of the machine (as a fraction of its value when new) is
the higher, the higher the rate of profit. (The reader is invited to look at Sraffa 1960, p. 69,
for a graphical representation of how the level of the rate of profit affects the way the value
of a durable capital good changes with age.)
One implication of this result concerns the value of a constant stock of machines of
constant efficiency and uniformly distributed by age. Assume that a machine of constant
efficiency lasts k years and that k processes, each one using it at a different age, are run side
by side, all with the same dimension, e.g. each one of them using one machine. The
ensemble of these processes needs one new machine each year in order to leave the stock of
machines and therefore their aggregate value unchanged[56]. The previous result on the
behaviour of depreciation then means that, relative to the value of a new machine, the value
of the uniformly age-distributed stock of machines is the greater, the higher the rate of
profit; or looking at it from the opposite angle, the cost of maintaining intact the efficiency
of a uniformly age-distributed stock of machines is a smaller fraction of its value, the higher
the rate of profit. Exercise: determine the value of this stock of machines for given p, r, k.
The Leontief problem and fixed capital
We ask now whether the existence of simple fixed capital creates difficulties in the
Leontief problem. The answer is no, as long as the net product vector to be produced only
consists of new commodities. This is because it is possible to produce a net product
consisting entirely of new commodities: having found the truncation that determines for
how many periods a durable capital good is used, say h periods, all that is necessary is that
the output of the final product for which that capital good is used be equally divided among
the h processes, so that the old capital goods are entirely used up and do not appear in the
56
The new machine to be added each year is then analogous to a repair and maintenance
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net product of the economy. Thus in the previous example of corn produced with tractors of
three ages, one third of the total amount of corn must be produced by each process; the total
production of new tractors, on the other hand, must be such as to replace the amount of new
tractors in use and to produce in addition the desired net product of new tractors. Therefore
there are no difficulties in determining subsystems producing positive net outputs of final
commodities; labour employment will then be certainly positive because the labour
embodied in a final commodity is positive.
Difficulties might arise if one wanted a net product consisting also of old capital
goods, and the assumption of free disposal were not made. Thus suppose that in our
example of corn and tractors, the production of 1 unit of corn requires 1 tractor, whichever
the age of the tractor in the truncation in use. Suppose now that corn is not used as input in
any industry, so any production of corn is production of a net output of corn; and suppose
that it is desired to have a net output consisting only of 1 one-year-old tractor. This is
impossible, because the production of a one-year-old tractor requires that a new tractor be
used to produce corn, so it is necessarily associated with the production of some corn. In the
absence of free disposal the sole way to have a zero net output of corn is to have some
process run at a negative level of activity. If production of the desired net output of oneyear-old tractors also entails a net output of x units of corn, this net output of corn can be
reduced to zero by adding negative industries in the dimensions which, if positive, would
form a subsystem producing x units of corn as its sole net product (we have seen that this is
always possible).
Interestingly, this solution can be interpreted in a way that makes sense. Negative
industries are of course impossible, but they can be interpreted as variations in the size of
already existing industries: a diminution of the quantity produced of a good is perfectly
conceivable, and it is equivalent to the addition of a firm of negative size. Thus if we
interpret a certain net output vector y as a vector of desired variations in net outputs, the
dimensions of the industries associated with it can be interpreted as the variations in the
sizes of industries necessary to obtain the desired variation of the net output vector, then
negative industries can be interpreted as decreases in activity levels and make perfect sense;
intervention capable of maintaining unaltered forever the efficiency of the stock of machines.
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and they can arise even independently of the existence of durable capital goods, simply
owing to negative components of y.
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APPENDIX 1 TO CHAPTER 2
NEW APPROACHES TO THE LABOUR THEORY OF VALUE
2A1.1. This Appendix presents more formally the attempts to reformulate the labour
theory of value, which have been discussed in Appendix 2 of Ch. 1.
Until the end of the 1970s it was universally accepted that the ‘labour value’ of an
aggregate of commodities was quantitatively identical with the labour embodied in it,
determined, in the case (to which we restrict ourselves for simplicity) of circulating capital
and no joint production (with m the row vector of labours embodied, aL the row vector of
direct labour coefficients, and A the matrix of the technical coefficients of produced means
of production of the dominant methods, each column a different industry), by:
m = aL + mA.
Comparison with the correct equations of prices of production had shown that it is
generally impossible to establish an equality between the labour value, and the value in
prices of production (let us call it production-price value), for two non-proportional vectors
of goods: for example if one normalizes prices so as to render the labour value of the social
product equal to its production-price value, then the labour value of the physical profits (the
subvector of non-wage goods in the net output vector) generally differs from their
production-price value (and therefore the labour value of capital too differs from its
production-price value). The hard-to-escape conclusion was, not only that in general the rate
of profit determined as the ratio of the labour value of profits to the labour value of capital is
not the correct one, but also that, if one argues that
a) the exchange value of the social product is produced by (in the sense that it is
proportional to) the labour embodied in it
then one cannot argue that
b) the exchange value of profits is produced by (in the sense that it is
proportional to) the surplus or unpaid labour.
This was feared to undermine the thesis that profits result from the exploitation of
labour. (The different analysis of the role of the labour theory of value in Marx presented
here in chapter 1 started to become available in English only with Garegnani (1984), and it
appears to be still largely unfamiliar to English-speaking economists.)
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The New Solution or New Interpretation proposed by Gérard Duménil (1980, 19834) and Duncan Foley (1982) does not require specific formalization since it consists of
choosing such a money unit that the exchange value (in terms of money) of the net product
is equal to the labour embodied in it, and then claiming that in this way money magnitudes
"express" labour time i.e. are actually amounts of labour value[57]. By assumption “money
represents social labor time”, and labour values are now defined to be the numerical
expression of relative prices when the net product vector per unit of labour is chosen as
numéraire. This definition is possible whatever the way prices are determined, which is why
Foley argues that “The New Interpretation....is completely general, in that it is consistent
with any theory of price formation” (2000, p. 23). It amounts simply to a choice of
numéraire for already given relative prices, so we do not need to write down equations. As
explained in 1A2.1 the (labour) value of labour-power is defined as its relative price in
terms of that numéraire; this definition of the value of labor-power identifies it numerically
with the share of wages in the monetary value of the net product. Assuming no land rents,
since prices now express labour time, profits express unpaid labour time, hence the rate of
exploitation (ratio of paid to unpaid labour time) is defined as the ratio between share of
profits and share of wages. Since labour values are defined as relative prices with the net
product per unit of labour as numéraire, the uniform rate of profit determined as a ratio
between labour values obviously coincides with the one determined on the basis of prices of
production, thus the two equalities a) and b) mentioned above are satisfied; in fact, they are
satisfied whatever the prices.
2A1.2. The approach of Wolff, Roberts and Callari (1982) argues (cf. §1A2.4) that
Marx’s thesis in volume 3 of Capital that prices of production are redistributed labour
values means that the price of production of the inputs in an industry is actually a quantity
of labour value; if we accept this, then the labour value of a product must be determined as
the price of production of inputs plus the living labour added in the production process.
Then the value of labour-power is, analogously, the price of production of the wage basket,
not the labour embodied in it according to the usual definition of labours embodied.
Here and in the remainder of this Appendix, ‘value’ without adjectives means labour
value; ‘exchange value’ means relative price, ‘monetary value’ or ‘money value’ means exchange
value in terms of money.
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Formally, assume circulating capital, and let z be the average physical wage basket, a
column vector. Wages are advanced. The price-of-production equations are, with z the
subsistence basket:
[2.45] p = (1+r)(pA+pzaL)
Labour values λ are now defined differently from m, and precisely as:
[2.46] λ = pA + aL
where p is determined by equation [2.45]. Let x be the vector of quantities produced. Then
aLx is labour employment L, and hence equals total labour value added. Subtracting from it
the labour value of wages, one obtains profits in labour values. Hence the rate of profit
determined via these redefined labour values is:
[2.47] r = (aLx–pzaLx)/(pAx+pzaLx).
Let us compare this with the prices-of-production determination of r:
[2.48] r = (px–pAx–pzaLx)/(pAx+pzaLx).
Only the numerators differ. But they become identical if one stipulates:
[2.49] px–pAx = aLx = L.
This equation sets the production-price value of the net product y=x–Ax equal to its
labour value and hence to labour employment; it amounts to choosing as numéraire for the
prices of production the physical net product vector per unit of labour, as in the New
Interpretation. It follows that the total price of the social product equals its total labour
value:
[2.50] px = pAx + aLx = (pA+aL)x = λx
Furthermore, since the value of labour-power is defined as pz, total profits π are:
[2.51] π = px–pAx–pzaLx = aLx – pzaLx = L – total labour value of necessary
labour = surplus labour value.
In this way, total value and total profits are the same in prices and in these re-defined
labour values. Both equalities a) and b) mentioned at the beginning of this Appendix are
satisfied.
This would appear to be the New Interpretation applied to prices of production, plus
the suggestion that the resulting labour values might reflect Marx’s idea of prices of
production as redistributed labour values more closely than Foley or Duménil perceived.
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2A1.3. The Temporal Single System (TSS) approach can be seen as a radicalization
of the approach of Wolff, Callari and Roberts, in that it argues that market prices too should
be seen as redistributed labour values. Prices are always, by definition, redistributed labours
embodied; commodities are viewed as if containers of amounts of a fluid produced by living
labour; this fluid gets (somewhat miraculously) redistributed among commodities so as to
render the amount of labour contained in each commodity proportional to the sale price; if
the commodity is used as means of production, the fluid ‘human labour’ thus contained in it
is then transferred to the product. Hence, supposing production to happen in separate oneperiod cycles, once for the previous period t–1 the price level is normalized so as to render
the total price of the product equal to the total labour embodied in it, the labour embodied
(before the redistribution operated by market prices) in a commodity produced in period t is
defined as the sum of the living labour expended in its direct production, plus the historical
market price of the capital inputs, determined by past technology and distribution and
market accidents; labours embodied before the redistribution operated by market prices are
determined by:
[2.52] λt = pt–1A+aL
where pt–1 is given, resulting from a ‘normalization’ of the given relative prices at t–1 (a
determination of the price level) such that the condition
[2.53]
pt–1xt–1=λt–1xt–1
is satisfied (where xt–1 is the given vector of quantities produced at t–1 and λt–1 is again
given), i.e. such that the total exchange value of the product at t–1 equals the total labour
embodied in it[58].
Constant capital C and variable capital V in the entire economy are accordingly
defined at (normalized) historical costs:
[2.54] Ct=pt–1Axt, Vt= pt–1zaLxt.
The price level is again normalized so as to make the total labour value of the social
product equal to its market price:
[2.55] ptxt = λtxt = Ct+Vt+St
and it is then argued that the total labour value Ct+Vt+St gets redistributed by the process of
exchange so that pt becomes a vector of labours embodied.
58
Let πt–1 be the vector of given money prices at t–1, and let pt–1 be defined as απt–1, with α
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In the above equation, the vector of relative market prices at time t is again among
the data, and the equation only determines their absolute levels so as to render the exchange
value of the total product equal to the labour embodied in it[59]. Surplus labour value S is
the difference between living labour Lt=aLxt and variable capital as defined by [2.54],
hence, with π standing for profits, and dropping for simplicity the index t from the current
quantities:
[2.56] π = px–pt–1Ax–pt–1zaLx = λx–pt–1Ax–pt–1zaLx = (C+V+S)–C–V = S.
Thus profits too are the same in prices and in labour values; again, equalities a) and
b) are both satisfied; and in fact, according to this approach, labours embodied get
redistributed so as to become proportional to nominal market prices; hence market prices,
once their level is normalized so as to satisfy equation [2.55], become quantities of labour
embodied; at the moment of exchange, labours embodied are no longer λt, they are pt.
This approach too adds nothing to our understanding of the forces at work in a
market economy, because it takes both input prices and output prices as given; indeed, it
tells us even less than the previous approach because the given prices are market prices,
which are not even explained as gravitating around normal, or long-period, relative levels.
2A1.4. It has been pointed out in chapter 1 that it is totally unclear what roles these
redefined labour values are to have in a better comprehension of the functioning of
capitalism; in none of the three approaches the determination of these ‘labour values’
contributes anything toward explaining prices or the rate of profit in the sense of clarifying
the forces and mechanisms that determine them[60]. The redefinition of paid (or necessary)
and unpaid (or surplus) labour time appears to have no use except to salvage by arbitrary
a positive scalar. Then pt–1xt–1=λt–1xt–1 determines pt–1 by determining α.
59
Analogously, as pointed out in Ch. 1 fn. 108?? relative prices at t–1, as well as λt–1, must
be data in equation [2.53]; this shows that there is an infinite regress in this theory, because the
determination of λt–1 requires the knowledge not only of relative prices at t–2 but also of λt–2, and so
on. In other words, the assumed redistribution of labours embodied due to the exchange process
renders relative labour values equal by assumption to relative prices, but the approach is unable to
determine the magnitudes of the labours embodied to be thus redistributed (or, as a consequence,
the magnitude of the final redistributed labour values) unless one assumes the magnitudes of the
labour values of their inputs to be known, and these require in turn the knowledge of the magnitudes
of previous labour values, ad infinitum. In conclusion, only relative labour values are determinable
because coinciding with relative market prices by assumption: truly a big gain of understanding!
60
When an explanation of normal prices and of the rate of profit is attempted, one has
recourse to ‘Sraffian’ equations, as shown for example by the Wolff-Callari-Roberts approach.
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redefinition Marx's determination of the rate of profit as a ratio between labour values. Paid
or necessary labour as traditionally defined has a practical content since it would become
visible if the two subsystems producing wage goods, and profit goods, were geographically
separated; paid labour as redefined by these approaches loses this meaning.
Nor can one find in these authors a discussion of the question, why prices should
‘represent’ or ‘express’ quantities of labour since prices can be logically conceived to exist
even in a science-fiction completely automated economy where machines (and possibly
animals) reproduce themselves, and human labour is totally absent from production
processes. The completely automated economy is a logical experiment only, no doubt, but
sufficient to show that the notion of normal relative price does not logically presuppose the
appearance of quantities of human labour among its determinants (and indeed it has been
seen that if in the Sraffian equations one replaces labour with the physical wages it gets,
then it is possible to define and determine prices on the basis of data where human labour
does not appear). True, changes of the rate of profit due to changes in the length of the
working day without changes in subsistence are less clear in their cause if one does not
make quantities of labour appear in the equations, but this is not sufficient to argue that
prices 'express' quantities of labour. Prices express the forces determining relative exchange
ratios.
Another grave deficiency of these approaches is that the question, in Marx’s
terminology, how ‘complex’ labour is ‘reduced’ to ‘simple’ labour, is not tackled; and yet a
convincing answer would appear to be an indispensable preliminary to the thesis that the
labour time of heterogeneous and differently paid workers produces – albeit in different
quantities per time unit – a homogeneous substance called 'labour value'. But perhaps it is
not accidental that the question has been little discussed: on this issue, it seems, these new
approaches cannot but run into indeterminacy. Labour heterogeneity did not prevent
Ricardo or Marx from talking of labour embodied as a homogeneous magnitude because,
owing to their analytical aim of determining the rate of profit, they ‘reduced’ labour to
homogeneity on the basis of relative wages; which is what they had to do because, as
illustrated in chapter 1, labours embodied had to be proportional to wages embodied in
order to be proportional to relative prices when the rate of profit was zero or the organic
composition of capital was uniform, the two cases from which takes off the 'compensation
of deviations' argument which in Marx as well as in Ricardo motivated starting from labour
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values in order to determine the rate of profit. Note that because this reduction of 'complex'
to 'simple' labour rested on given relative wages explained outside the 'core', the implied
uniform rate of surplus value or uniform 'rate of exploitation' in this analytical sense did not
mean that all categories of workers were equally badly treated (equally 'exploited' in a
common language sense) by capitalists and therefore had common interests; in order to
determine the rate of profit the reduction had to be proportional to relative wages even if,
owing e.g. to trade unions, in different industries exactly the same work were paid different
wages, or even if a category of workers was able to earn a high wage solely because of a
capitalist strategy of creating a labour aristocracy in order to divide and weaken the labour
movement.
But in the three approaches studied in this Appendix prices are determined before
labour ‘values’ and independently of them, so one loses the classical analytical criterion for
determining the relative ‘value’-creating capacities of different labours, and no other
criterion replaces it. Thus relative ‘value’-creating capacites remain undetermined[61]; one
might determine them totally arbitrarily (a racist might even deny any value-creating
capacity to a certain race or sex or caste) and one would still obtain the aggregate results of
the approaches and the validity of a) and b), because those results are imposed ex post by
the normalizations adopted.
Let us show this for the approach of Wolff, Roberts and Callari. Let there be two
types of labour ‘α’ and ‘β’ with respective physical column wage vectors zα and zβ and row
vectors of technical coefficients aLα and aLβ. Equation [2.45] becomes
[2.57] p = (1+r)(pA+pzαaLα+ pzβaLβ).
Assume that the amount of value created by a unit of type ‘α’ labour, respectively by
a unit of type ‘β’ labour, is respectively α and β. Then equation [2.46] determining labour
values before their redistribution becomes
61
This is implicitly admitted by Duménil, Foley and Lévy, 2008, who after stating in open
contradiction to Marx that “Wages are not necessarily proportional to the value productivity of
workers”, limit themselves to proposing how in the New Interpretation one should derive the rates
of exploitation of different workers from given relative value productivities of heterogeneous
labour, but on what determines these differences in value productivities they remain silent, saying
only that in order to determine them “some additional assumption about relative rates of
exploitation (which Marx often explicitly assumes to be equal) is required” (p. 560). Where the
‘often’ should not have appeared, and why Marx assumed equal rates of exploitation is left
mysterious.
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[2.58] λ = pA + αaLα + βaLβ.
Now ‘labour value added’, or labour value of the net product, is (αaLα + βaLβ)x.
Subtracting from it the labour value of wages (pzαaLα+ pzβaLβ)x, one obtains profits in
labour values. Hence the rate of profit determined via labour values is:
[2.59] r = ((αaLα + βaLβ)x – (pzαaLα+ pzβaLβ)x) / (pAx+(pzαaLα+ pzβaLβ)x).
Let us compare this with the prices-of-production determination of r:
[2.60] r = (px–pAx–(pzαaLα+ pzβaLβ)x)/(pAx+(pzαaLα+ pzβaLβ)x).
Again only the numerators differ. But they become identical if one stipulates the equality
between price-of-production value of the net product and its labour value:
[2.61] px–pAx = (αaLα + βaLβ)x.
Then the total production price of the social product equals its total labour value:
[2.62] px = pAx + (αaLα + βaLβ)x = (pA + αaLα + βaLβ)x = λx
and profits equal surplus value, given by total labour value added (αaLα + βaLβ)x minus the
labour value of wages i.e. their production price (pzαaLα+ pzβaLβ)x:
[2,63] π = px–pAx–(pzαaLα+ pzβaLβ)x = (αaLα + βaLβ)x – (pzαaLα+ pzβaLβ)x.
Thus the correct rate of profit and the validity of a) and b) are obtained for any
magnitudes of α and β: one is free to fix them arbitrarily.
But the absence of compelling criteria for determining the ‘value’-creating capacities
of different labour times would appear to radically question the notion itself of such a
‘value’ creation process[62].
62
Actually what seems to emerge in these authors is an insufficient awareness of the
fuzziness of the notion of ‘quantity of labour’ and consequent frequent difficulty of measuring it
independently of the costs it causes, even for a given type of work. For example in the same time
stretch a librarian paid by the hour can perform vastly different amounts of different work
operations (e.g. hand out books to readers, compile catalogues, research information on the Internet)
depending on the day and the hour, and there seems to be no evident criterion for summarizing
his/her activities into a ‘quantity of labour’ per hour: the amount of labour is measured in terms of
hours because its cost is fixed in terms of hours.
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