THE METHOD MORE SUITABLE FOR REPRESENTING FIXED CAPITAL WITHIN A SRAFFIAN FRAMEWORK

THE METHOD MORE SUITABLE FOR REPRESENTING FIXED CAPITAL WITHIN A SRAFFIAN FRAMEWORK

Felix Ibanez. Departamento de Analisis Economico II. UNED

Mariano Matilla. Depto. de Economia Aplicada Cuantitativa I. UNED

Ruben Osuna. Departamento de Analisis Economico I. UNED

Abstract

In this paper we show that the new method for representing Fixed Capital suits better than

the standard one to the main object of Sraffa’s price theory, centered in proving the role

performed by market prices and distributive variables in the reproduction of economic sys-

tem as a whole. The new method leads to market prices, along with distributive variables,

fulfilling the main equation that materializes the reproduction of economic system.

Whereas the Torrens rule additionally requires for the same purpose that the successive

book values of fixed capital stock used to produce each commodity being previously as-

certained. The referred reproduction of economic system consists of the following tasks: i)

replacement of consumed means of production for obtaining gross output of the economy;

and ii) distribution of net product between different categories of income (profits and

wages).

J.E.L. classification: D24.

Keywords: Reproduction of economic system; Fixed Capital; Sraffa; von Neumann.

1. INTRODUCTION The method commonly utilized in the literature on Fixed Capital within the Sraffian Theory of Prices

consists of treating machines at different stages of wear and tear as differentiated commodities with

their own price, which give rise to joint production processes. See for instance Sraffa (1960, chap. 10),

Morihisma (1969, Ch. 6), Garegnani (1970), Baldone (1974), Varri (1974), Schefold (1978a, 1980) and

Kurz and Salvadori (1995, chap. 7 and 9), among other authors. This treatment of Fixed Capital stems

from the Classical tradition, as reflected in Sraffa (1960, Appendix D), and, at the same time, corre-

sponds to the way that von Neumann (1945) deals with this topic in his pioneering work. Henceforth,

this treatment of Fixed Capital in terms of joint production is referred to as the Torrens rule, or, indis-

tinctively, as the standard method.

On the other hand, the new method for representing Fixed Capital1 uses always the value of fixed

capital stock when new, though modified by some non-negative parameters, to calculate the profit rate

in each period of time; whereupon machines at different stages of wear and tear are not required

being priced, and so, joint production processes never arise. Then, the new method deals only with

the price of machines brand-new, which are actually exchanged in markets, and not with the book

values of fixed capital stock within a firm, which are mere accounting values.

In this order of things, the core of this paper is to argue that the representation of Fixed Capital that

suits better to the main object of Sraffa’s Price Theory, centered in establishing the role performed by

the price of commodities within a market economy, is precisely the new method; since this method

allows us to provide a more straightforward proof than the standard one to this respect. The standard

method, though not wrong, as long as it requires for the same purpose the book values of fixed capital

stock being previously ascertained, seems to be roundabout.

In another order of things, it is a well-known fact that Sraffa’s Circulating Capital model shows that

the price of commodities along with the distributive variables are able to perform the reproduction of

economic system, i.e. the reposition of consumed means of production for obtaining the gross output of

an economy, and the distribution of net product between the income categories (wages and profits). This

proposition is implemented by means of a well-known main equation displaying that the value of net

product at ruling prices equals the lump sum of wages and profit accruing to capitalists calculated from

the value of advanced circulating capital stock.

Likewise, in this paper we provide a similar main equation for the new method to represent Fixed

Capital, which being satisfied by positive market prices along with non-negative value for distributive

variables, allows us to show that both latter items alone are able to perform the same role as that played

within the Sraffian standard model of Circulating Capital. Whereas the standard method for representing

Fixed Capital additionally requires for the same purpose previous knowledge of the successive book

values of fixed capital stock used to produce each commodity over time.

1 This new method has been put forward in Ibanez and Matilla (2004), and Ibanez, Matilla and Osuna (2004a-

b).

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The content of this paper is structured as follows. The second section tackles the Sraffian framework

for the analysis. The third and fourth sections respectively contain a brief exposition of the new method

for representing Fixed Capital and the standard one. The fifth section deals with the integrated price

system and the main equation displaying the reproduction of economic system when the new method is

adopted. The sixth section deals with the same both items referred to the standard method for represent-

ing Fixed Capital. In section seven we discuss on the most suitable method to be adopted for showing

the reproduction of economic system performed by the market prices and the distributive variables.

Finally some concluding remarks are added.

2. ANALYTICAL FRAMEWORK The analysis of Fixed Capital is developed within the following Sraffian framework:

a) Absence of joint production. Then, a technique as a whole consists of one production process to

obtain each commodity separately. We consider commodities; hence there are n produc-

tion processes. The h first commodities are machines (1

2≥n

h n≤ < ), the rest are circulating

capital goods.

b) Production processes are point/flow input-point output.

c) By definition, the duration of a production process is the time elapsed from when the first input is

incorporated until the corresponding output is obtained. We consider that all production proc-

esses are of the same duration; and we take the latter as one unit of time.

d) Homogeneous labor is the sole primary input.

e) Wages are paid ex post, i.e. at the end of each production process. The time period for wage

payments coincides then with the duration of production processes.

f) To this unit of time is referred the profit rate, and calculations for periodic allowance for deprecia-

tion of machines.

g) Wage and profit rates are uniform throughout the whole economy.

h) Machines at different stages of wear and tear are re-used for obtaining the same commodity over

time until they are scrapped, i.e. they cannot be transferred between or within sectors. It means

that scrapped machines in a production process have no residual value, i.e. they are not de-

manded by consumers, or by firms obtaining the same or a different commodity. If the contrary,

joint production processes, in strict sense, arise2.

Mathematical notationLet us focus, for instance, on the production process for obtaining the n-th within the technique as a

whole that we are considering:

2 See Ibanez, Matilla and Osuna (2004b, Appendix 1) where this possibility is contemplated.

3/24

a) Suppose that the h machines referred above are used in this process. Let

be the economic lifetime of each machine respectively. stands for

the number of production cycles (the duration of a production cycle as one unit of time) in which

the j-th machine is used.

2 1, ,jt j =≥ … h

)

t

t

jt

b) Let be the least common multiple of machines’ economic lifetime

jointly used for producing the n-th commodity.

( 1. . . , , ht lc m t t= …

c) stands for the output level of this commodity produced over t successive

time periods. This output is not necessarily constant due to machines’ variable efficiency in

general.

1, ,jb j t= …

d) kn stands for a row vector with n components, the first h of which are positive and the rest are

zero. This vector represents the number of new machines of each class employed for produc-

ing the n-th commodity. In other words, this vector denotes the fixed capital stock when new

used to produce this commodity.

e) denotes periodic allowance for depreciation corresponding to this fixed

capital stock.

1, ,j j∆ = …

f) stands for a row vector with n components. The first h of which are zero, the 1, ,j =ja …

1≥m remaining ones ( hnm −= ) represent: i) consumed material inputs to obtain the out-

put of n-th commodity over t production periods; ii) material inputs corre-

sponding to maintenance, repair and partial reposition expenses associated to the wear of ma-

chines; and iii) corresponding material inputs to eventual elimination costs for worn-out ma-

chines.

1, ,jb j t= …

g) denotes vector of circulating capital corresponding to preceding consumed

material inputs in each period. This vector consists of the same zero components as those of

, as we argued in Ibanez, Matilla and Osuna (2004a).

1, ,j =*ja … t

t1, ,j =ja …

h) stands for the labor input employed over time to produce the n-th commod-

ity. Likewise in the preceding point, l

1, ,jl j t= …

j may also include direct labor employed in maintenance

and repair of machines, and in elimination of scrapped ones.

i) Wage rate is represented by w, and r stands for the profit rate, referred to the unit of time that

we have assumed above.

j) Let p be a column vector of n components. It stands for the price vector of marketed commodi-

ties. The first h components correspond respectively to the price of machines when new, and

the n-th component corresponds to the price for the n-th commodity.

4/24

3. THE NEW METHOD FOR REPRESENTING FIXED CAPITAL3

As we showed in Ibanez, Matilla and Osuna (2004b), the following equation system materializes this

new method:

where parameters jθ and 1, ,j j tρ = … are non-negative, and have been already explained in

the referred paper.

Integrated process

( )( )

( )

1 1 1 1 1

2 2 2 2 2

n

n

t t t t t n

r w

r w

r w

ρ θ

ρ θ

ρ θ

+ + ∆ + + =

+ + ∆ + + =

=

+ + ∆ + + =

*n 1 1

*n 2 2

*n t t

k p a p a p

k p a p a p

k p a p a p

l b p

l b p

l b p

(1)

It is obtained by simple summation of the preceding equations, as follows:

Thus, integrated process (2) turns out to be a linear combination of system (1).

By definition, it holds that:

1 1

1 1t t

j j j n n n nj j

r wt t

ρ θ ρ= =

⎛ ⎞+ + ∆ + +⎜ ⎟

⎝ ⎠∑ ∑ *

n j nk a p a p l b p= (2)

where 1 1 1

1 1 1t t t

n j n j nj j j

l l bt t t= = =

∆ ≡ ∆ ≡ ≡ ≡∑ ∑ ∑n ja a1

1 t

jj

bt =∑

1 2

1 1 2

0 0t

n n nhj

j h

k k kt

t t t=

⎛ ⎞∆ ≡ ⎜ ⎟

⎝ ⎠∑ p (3)

i.e. accumulated depreciation over t periods equals the value of machines when new that are replaced

in this span of time.

Hence, average allowance for depreciation displayed in integrated process (2) consists of a pro-

portional depreciation for each machine separately, as follows:

1 2

1 2

0 0n n nhn

h

k k kt t t

⎛ ⎞∆ = ≡⎜ ⎟

⎝ ⎠np δ p (4)

3 Consult Ibanez, Matilla and Osuna (2004b) where a comprehensive exposition of this new method is pro-

vided. This section only contains a brief summary.

5/24

Given this, integrated process (2) can be rewritten as follows:

1 1

1 1t t

j j j n n nj j

r wt t

ρ θ ρ= =

⎛ ⎞+ + + +⎜ ⎟

⎝ ⎠∑ ∑ *

n j n nk a p δ p a p l b p= (5)

4. THE TORRENS RULE Let be a row vector with h components, which represents the h classes of ma-

chines in successive phases of wear and tear employed for producing the n-th commodity. These ma-

chines are considered independent commodities, qualitatively differentiated from one another, in this

standard method for representing Fixed Capital in terms of joint production. stands for worn ma-

chines at the end of the first period, and for worn machines at the end of the t-th period.

1, ,j =jhk … t

t

1hk

thk

Let be a column vector of h components, which denotes the price of the re-

spective machines at different stages of wear and tear.

1, ,j =jhp …

The book value of fixed capital stock at the end of successive periods can be expressed as fol-

lows:

(6) 1,2, ,jVT j t≡ =j jh hk p …

By definition, allowance for depreciation of fixed capital stock in a single production period equals the

book value of this stock at the beginning minus the book value at the end of the considered period; thus,

one can write that:

(7) * *11, ,j j jVT VT j t VT∆ ≡ − = ≡ nk p…

where stands for the book value of fixed capital stock at the beginning of first period. *1VT

However, book value at the beginning of j-th period *jVT in the preceding identity can be ex-

pressed in terms of the book value at the end of the preceding period, as follows:

*1j jVT VT RE−= + j

where denotes the value of new machines that replace those scrapped at the beginning of j-th

period

jRE

4.

4 Variable RE , standing for replacement of machines at the beginning of each period, has been defined in

Ibanez, Matilla and Osuna (2004b, Appendix 2), where machines of different economic lifetime were considered.

As we showed in the referred paper, 1, ,jRE j t= … depends only on the price vector p of marketed

commodities. Obviously, in case that all machines have the same economic lifetime (t), it holds that

. 0 1,jRE j t= = …,

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Hence, identity (7) can be rewritten as follows:

(8) 1 01, ,j j j jVT RE VT j t VT−∆ ≡ + − = ≡ nk p…

where stands for the book value of fixed capital stock at the beginning of first period. 0VT

Given this, one can easily infer that:

(9) 1 1

1, ,h h

j j hj j

RE VT h t= =

∆ = + − =∑ ∑nk p …

A particularization of this latter equation referred to the last period would be as follows:

Given this, it turns out that , i.e. the book value of fixed capital stock at the end of last pe-

riod vanishes. Hence, it holds that

0tVT =

=thp O

. This is so, because accumulated depreciation (the term

of left hand side of (10)) equals, according to (3), the value of new machines that replace those

scrapped over the whole of t periods (the lump sum of the first two terms of right hand side of (10)).

The first of these latter terms represents the value of new machines that replace those scrapped at the

end of last period; and the second term represents the corresponding value of new machines that

replace those scrapped at the beginning of the t considered periods.

Therefore, according to this standard method for representing Fixed Capital, equations displaying

successive utilization of fixed capital stock for producing the n-th commodity adopt the following shape:

According to this equation system, this method for representing Fixed Capital calculates the profit

rate using the fixed capital stock book value at the beginning of each period (the term within parenthe-

ses in left hand side of (11)); which does not normally coincide with the book value corresponding to

the end of the preceding period in the general case in which all machines do not have the same eco-

nomic lifetime.

t

(10) 1 1

t t

j jj j

RE VT= =

∆ = + −∑ ∑nk p

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

1 1 1

1 2 2 2 2

1

1 1

1 1

1 1 0

n

n

t t t t n

r RE r wl VT b p

r VT RE r wl VT b p

r VT RE r wl b p−

+ + + + + = +

+ + + + + = +

=

+ + + + + = +

n 1

2

t

k p a p

a p

a p

1

(11)

7/24

Integrated process As we can see, in this latter equation system the successive book values of fixed capital stock do ap-

pear, and they are expressed in this standard method, according to (6), in terms of the price of machines

at different stages of wear and tear.

These additional unknowns to the price vector of marketed commodities p and the distributive vari-

ables can be eliminated by means of a well-known procedure within the Sraffian literature. It consists of

obtaining a linear combination of system (11) by multiplying each equation, respectively, by

and then, by summing up all the equations. ( )1 1t jr j−+ = …, ,t

Doing so, one obtains the following equation:

(12)

( ) ( ) ( ) ( ) ( )

( ) ( )

1 11

1 1

1 11 1

1 1

1 1 1 1 1

1 1

t tt j

t jj j

t tj j

t j t j nj j

r r RE r r r

l r w b r p

− −− +

= =

− −− + − +

= =

⎡ ⎤+ + + + + + +⎢ ⎥

⎣ ⎦⎡ ⎤ ⎡ ⎤

+ + = +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑

∑ ∑

n t-j+1k p a pj +

This equation can be considered as the integrated process corresponding to the standard method, a

counterpart of (5) for the new one.

Now, dividing both sides of this latter equation by

( ) ( )1

1

1 11

ttj

j

rr

r−

=

+ −+ =∑

one obtains:

( )( )

( )( )

( )

( )( )

( )

( )( )

( )( )

11

1

1

1

1 11 1

1 1

11 1

1 1 1 1

1 11 1

1 11 1 1 1

t tj

t jt tj

tj

tj

t tj j

t j t j nt tj j

r r r r RE r

r r

rr r

r

r rl r w b r

r r

−− +

=

−

=

− −− + − +

= =

++ + + +

+ − + −

⎡ ⎤+ + + +⎢ ⎥

+ − ⎣ ⎦⎡ ⎤ ⎡

+ + =⎢ ⎥ ⎢+ − + −⎣ ⎦ ⎣

∑

∑

∑ ∑

n

t-j+1

k p

a p

p⎤

+ ⎥⎦

(13)

As is evident, the following expression

( )( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−+ ∑=

−+−

t

j

jjtt

rbr

r

1

11 1

11

turns out to be a weighted average of the output obtained over t periods.

The same thing occurs with the material inputs:

( )( ) 1

1

11 1

tj

tj

rr

r−

=

⎡ ⎤+⎢ ⎥

+ − ⎣ ⎦∑ t-j+1a

and with the labor input:

8/24

( )( ) 1

11

11 1

tj

t jtj

rl r

r−

− +=

⎡ ⎤+⎢ ⎥

+ − ⎣ ⎦∑

Therefore, integrated process (13) can be interpreted as follows:

a) The right hand side corresponds to the value of a weighted average of the output obtained over t

periods.

b) The fourth term of left hand side represents the wage cost corresponding to the same weighted

average for labor input.

c) The third term is the corresponding cost to the same weighted average for consumed material

inputs, plus profit associated to circulating capital engaged in anticipated acquisition of these in-

puts.

d) Then, the first two terms represent the annual charge for machines, i.e. depreciation expense

plus profit corresponding to fixed capital stock.

Let us normalize integrated process (13) for ulterior use. For this purpose, we can define a coefficient

nα as a ratio between the arithmetic mean and the weighted mean of output obtained over t periods, as

follows:

( )( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−+

=

∑

∑

=

−+−

=

t

j

jjtt

t

jj

n

rbr

r

bt

1

11

1

111

1

α (14)

Then, multiplying (13) by nα one obtains:

( )( )

( )( )

( )

( )( )

( )

( )( )

11

1

1

1

11

1 1

11 1

1 1 1 1

1 11 1

11

1 1

t tj

n n t jt tj

tj

n tj

t tj

n t j j ntj j

r r rr RE

r r

rr r

r

rl r w b p

tr

α α

α

α

−− +

=

−

=

−− +

= =

++ + +

+ − + −

⎡ ⎤+ + + +⎢ ⎥

+ − ⎣ ⎦⎡ ⎤ ⎡ ⎤

+ + =⎢ ⎥ ⎢ ⎥+ − ⎣ ⎦ ⎣ ⎦

∑

∑

∑ ∑

n

t-j+1

k p

a p

r +

(15)

5. THE NEW METHOD FOR REPRESENTING FIXED CAPITAL AND THE REPRODUCTION OF ECONOMIC SYSTEM Our technique as a whole referred at the outset consists of n processes of the same duration for

obtaining respectively n commodities. The first h commodities are machines, which, at the same time,

are exclusively used as inputs for producing the n-th commodity. Then, the first commodities

do not require machines in their corresponding production process, for the sake of simplification. Let

us represent these latter production processes as follows:

1n −

( )1 1, 1n nj jl b j n− −→ =n-1

ja …, , 1−

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where is the row vector of n components representing consumed material inputs and circulating

capital, the input of direct labor, and

n-1ja

1njl− 1n

jb− the output obtained in one production cycle.

The first h components of are zero, since they correspond to the use of

machines as inputs in these industries.

1, , 1j n=n-1ja … −

5.1 Integrated process and integrated price system corresponding to the new method Insofar as some machines are used in the production of the n-th commodity, the corresponding proc-

ess unfolds into t successive production processes, where t stands for the least common multiple of

machines’ economic lifetime used to produce the referred commodity. These t production processes

give rise to system (1), composed by t equations, which must be simultaneously satisfied by the price

vector p of marketed commodities and the distributive variables w and r.

However, the integrated price system associated to the technique as a whole for producing the n

commodities requires that only one equation can be introduced for representing the production proc-

ess of a commodity. Thus, it is necessary for the n-th commodity the use of a linear combination of (1),

that is, integrated process (5) obtained earlier:

where

1 2

1 11 2

1 10 0

t tn n nh

n j nj jh

k k kl l b

t t t t t t= =

⎛ ⎞≡ ≡ ≡⎜ ⎟

⎝ ⎠ 1

1 t

jj

b=

≡∑ ∑ ∑n n jδ a a

In this order of things, the corresponding integrated price system associated to the technique as a

whole for producing the n commodities can be written in matrix notation as follows:

1 1

1 1t t

j j j n n nj j

r wt t

ρ θ ρ= =

⎛ ⎞+ + + +⎜ ⎟

⎝ ⎠∑ ∑ *

n j n nk a p δ p a p l b p=

w

(16)

(17) ( ) ( )r + + + + =cK A p δ A p L Bp

where

1

1 t

j jj n n

tρ θ

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑ n

0

0

K0

k 1

1 t

jj n n

tρ

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑

n-11n-12

cn-1n-1

*j

a

a

Aa

a

n n×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦n

0

0

δ0

δ

10/24

1

1 t

j n nt = ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑

n-11n-12

n-1n-1

j

a

a

Aa

a

11

12

11

1 1

1

n

n

nn

t

jj n

l

l

l

lt

−

−

−−

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑

L

11

12

11

1

0 0 0

0 0 0

0 0 0

10 0 0

n

n

nn

t

jj n n

b

b

b

bt

−

−

−−

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑

B

5.2 The reproduction of economic system Multiplying (17) by t and e, the latter being a row vector of n unitary components, one obtains that:

( ) ( )rt t wt t+ + + + =ce K A p e δ A p eL eBp

Then, from this latter equation it turns out that:

where E, a row vector of n components, stands for the net product of our economy corresponding to t

periods of time.

This fundamental equation means that the net product E obtained in t periods, valued at market

prices, equals the lump sum of wages and profits accruing during this span of time over the whole econ-

omy.

This is so, because according to the last equation net product E is well defined as follows:

where: i) t means the vector of gross output obtained in our economy over t periods; ii) t repre-

sents a vector containing the number of machines of each class that has been replaced in our economy

during this span of time

eB eδ

5; and iii) t stands for the vector of material inputs consumed in our economy

over t periods.

eA

In summary, fundamental equation (18) materializes the reproduction of economic system in the

whole set of t periods of time, since it means that part of the gross product obtained in this span of time

has been used in the reposition of consumed means of production (machines and material inputs), and

remaining net product has been distributed between the income categories (wages and profits).

6. THE STANDARD METHOD FOR REPRESENTING FIXED CAPITAL AND THE REPRODUCTION OF ECONOMIC SYSTEM In this section we repeat the same argument as that contained in the preceding one, referred in this

case to the standard method for representing Fixed Capital.

wt (18) [ ]( ) ( )t rt= − + = + +cEp e B δ A p e K A p eL

( )t t t= − +⎡ ⎤⎣ ⎦E e B δ A (19)

5 This fact is evident according to the definition of vector in (4). nδ

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6.1 Integrated process and integrated price system corresponding to the Torrens rule when the book values of fixed capital stock are not ascertained

Now, integrated process corresponding to the n-th commodity is precisely (15) instead of (16):

( )( )

( )( )

( )

( )( )

( )

( )( )

11

1

1

1

11

1 1

11 1

1 1 1 1

1 11 1

11

1 1

t tj

n n t jt tj

tj

n tj

t tj

n t j j ntj j

r r rr RE

r r

rr r

r

rl r w b p

tr

α α

α

α

−− +

=

−

=

−− +

= =

++ + +

+ − + −

⎡ ⎤+ + + +⎢ ⎥

+ − ⎣ ⎦⎡ ⎤ ⎡ ⎤

+ + =⎢ ⎥ ⎢ ⎥+ − ⎣ ⎦ ⎣ ⎦

∑

∑

∑ ∑

n

t-j+1

k p

a p

r +

The n-1 first industries are the same as those considered in the preceding section.

In this context, corresponding integrated price system associated with the technique as a whole

can be represented in matrix notation as follows:

( ) ( )( ) ( ) ( ) ( )r r r r w r− + + + + =* * * *K p δp A p δ A p L Bp 6 (20)

where

( ) 11

1 1

0

0

( )0

(1 )1 (

(1 ) 1 (1 ) 1

t tj

n n t jt tj n

r

r r rr RE

r rα α −

− +=

1 )r×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥+

+ + +⎢ ⎥+ − + −⎢ ⎥⎣ ⎦

∑

*

n

K p

k p

1

1

( )

(1 )(1 ) 1

tj

n tj n n

r

rr

rα −

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+

+ −⎢ ⎥⎣ ⎦∑

n-11n-12

*n-1n-1

t-j+1

a

a

Aa

a

p6 We have subtracted vector δ from , since this latter vector represents the annual charge for ma-

chines (i.e. depreciation plus interest payment), and the first one represents depreciation of fixed capital stock. In

this way, we are trying to separate in two terms depreciation from interest payment corresponding to fixed capital

stock, instead of using one term for the lump sum of both items, as the annual charge for machines is.

p ( )r*K

12/24

11

12

11

1+1

1 1

( )

(1 )(1 ) 1

n

n

nn

tj

n t-jtj n

l

l

rl

rl r

rα

−

−

−−

−

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+

+ −⎢ ⎥⎣ ⎦∑

*L

Now, multiplying (20) by t and e one obtains:

( ) ( )( ) ( ) ( ) ( )t r r r t r wt r t− + + + + =* * * *e K p δp A p e δ A p eL eBp

Then, the following equation is satisfied for the economic system as a whole over t periods of time:

( ) ( )( ) ( ) ( ) ( )t t t r t r r r wt r⎡ ⎤− + = − + +⎣ ⎦* * *e B δ A p e K p δp A p eL* (21)

However, in this expression, counterpart of (18), ( )( )t t t r⎡ ⎤− +⎣ ⎦*e B δ A fails to be the net

product E obtained in our economy over t periods, according to (19), unless the following equation is

satisfied for all admissible values of r:

( )r =*A A

In other words, the following equation must necessarily hold for this purpose for all admissible val-

ues of r:

( )

( ) 1

1 1

11

1 1

t tj

n tj j

rr

trα −

= =

⎡ ⎤+ =⎢ ⎥

+ − ⎣ ⎦∑ ∑t-j+1 ja a

(22)

If this requirement were satisfied, not only ( )( )t t r⎡ ⎤− +⎣ ⎦*e B K A would be the net product E,

but also term , in the right hand side of (21), would be the profit distributed over t periods

corresponding to the circulating capital used within our economy.

( )rt r*eA p

At the same time, term in the right hand side of (21) is no longer the amount of wages

paid to the workers in our economy over t periods, unless, according to right hand side of (18), the

following equation is satisfied for all admissible values of r:

( )wt r*eL

( )r =*L L

In other words, the following equation must necessarily hold for this purpose for all admissible val-

ues of r:

( )

( ) 1+1

1 1

11

1 1

t tj

n t-jtj j

rl r

trα −

= =

⎡ ⎤+ =⎢ ⎥

+ − ⎣ ⎦jl∑ ∑

(23)

As obvious, equations (22) and (23) can only be satisfied in some particular cases, which are con-

templated in the appendix. However, in general, it is not possible warranting that both equalities al-

13/24

ways hold for all admissible values of r. Hence, (21) fails to be in general equivalent to fundamental

equation (18), which displays the reproduction of economic system.

Therefore, the fact of fixed capital stock book values not being ascertained causes in the standard

method a distortion so serious as to impede that market price vector p alone, along with distributive

variables w and r, perform the reproduction of economic system over t periods of time. Conclusively,

the book values of fixed capital stock are essential within the standard method for representing Fixed

Capital to perform the reproduction of economic system; that is, we cannot get rid of them without

causing serious troubles to this standard method.

6.2 Integrated process and integrated price system corresponding to the Torrens rule when the book values of fixed capital stock are ascertained

Integrated process (15), normally used in this standard method, is not the only linear combination of

system (11) to be considered.

In this sense, simple summation of the relevant equations of this latter system allows us obtaining

the following integrated process, as opposed to (15):

(24)

1

1 1 1

1 1 1 1

t t t

j jj j j

t t t t

j jj j j j

r RE VT

RE w l b p

−

= = =

= = = =

⎛ ⎞+ + + +⎜ ⎟

⎝ ⎠

+ + + + =

∑ ∑ ∑

∑ ∑ ∑ ∑

n j

n j

k p a p

k p a p j n

As we can see, this integrated process does not avoid the successive fixed capital stock book val-

ues in contrast to (15).

Equation (24) can be simplified. In effect, recalling (10), where 0tVT = , and then resorting to (3)

and to definition of in (4), one obtains that: nδ

(25) 1

t

jj

RE t=

+ =∑n nk p δ p

Now, introducing this latter equation in (24), we get the final shape of this integrated process be-

comes:

1

1 1

1 1

1 1

1 1 1

t t

jj j

t t t

j jj j j

r VTt t

w l bt t t

−

= =

= = =

⎛ ⎞+ + +⎜ ⎟

⎝

+ + + =

∑ ∑

∑ ∑ ∑

n j

n j

δ p a p

δ p a p1

np

⎠

(26)

As we can see, this integrated process corresponding to the standard method shares common fea-

tures with integrated process (16) for the new method, with respect to the inputs consumed (fixed

capital, material inputs and labor) to produce the n-th commodity. However, both integrated processes

differ from one another in the way of expressing the value of fixed and circulating capital stock from

which the profit rate is calculated (the term within parentheses in both expressions).

14/24

Now, as we proceeded in 6.1, we introduce integrated process (26) instead of (15) as the n-th

equation of the integrated price system, leaving unaltered the rest of equations corresponding to the

first industries. Doing so, we get the integrated price system in matrix notation becomes: 1n −

(27) ( ) ( )r w+ + + + + =δp VT Ap δ A p L Bp

where

1

1 1

0

0

0

1 t

jj n

VTt

−

= ×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦∑

VT

Multiplying (27) by t and e one obtains:

( ) ( )rt t wt t+ + + + + =e δp VT Ap e δ A p eL eBp

From here, it turns out the fundamental equation satisfied by the net product E of our economy

corresponding to t periods of time, as defined in (19). This fundamental equation, similar to (18),

adopts in this case the following mathematical shape:

[ ]( ) ( )t rt wt= − + = + + +Ep e B δ A p e δp VT Ap eL (28)

As we can see, (18) and (28) only differ from the way of expressing the value of fixed and circulat-

ing capital stock, appearing in the right hand side of both equations, from which de profit rate is calcu-

lated. Hence, the main difference between both methods consists, in essence, that the standard one

uses the successive book values of fixed capital stock to calculate the profit rate, in contrast to the

new method, which only resorts to the price vector of marketed commodities p.

Fundamental equation (28) means, similarly as we interpreted (18), that the market prices, along

with the distributive variables, and in this case also along with the successive book values of fixed

capital stock: i) are able to distribute the net product of our economy between the income categories

profits and wages; and at the same time ii) allow the reposition of inputs consumed over t periods of

time, i.e. the reproduction of economic system.

Therefore, from the argument contained in this section, one can conclude that the book values of

fixed capital stock play an essential role in this standard method with respect to the reproduction of

economic system. Due to the fact that avoiding the book values provoke so deep distortion in the

fundamental equation to be satisfied by the net product of our economy, as one can easily see by

direct comparison of (21) and (28), as to impede the reproduction of economic system being per-

formed in most cases.

15/24

7. THE MORE SUITABLE METHOD FOR REPRESENTING FIXED CAPITAL WITHIN A SRAFFIAN FRAMEWORK In this section we will prove that the new method for representing Fixed Capital is better adapted than

the standard one for the purpose of Sraffa’s theory of prices, centered in the reproduction of economic

system as a whole.

In Sraffian circulating capital models, the market prices along with the distributive variables are suf-

ficient information in order to warrant the reposition of consumed means of production and the distribu-

tion of net product in an automatic way between the income categories (wages and profits). In other

words, the market prices along with the distributive variables are sufficient information in order to

perform the reproduction of economic system.

The same thing happens whenever we represent Fixed Capital according to the new method, as

we have shown in section 5.

The main feature of this new method is that it resorts to some parameters to express the value of

fixed capital stock changing over time, due to the fact of this stock suffering wear and tear deprecia-

tion. Instead of resorting for the same purpose to the book values of fixed capital stock, which have to

be ascertained, as the standard method does.

In this sense, the procedure of using some parameters θ , for distorting the value of fixed capital

stock when new over time, has been precisely introduced in the new method to need not machines at

different stages of wear and tear being priced, as the standard method requires, nor to resort to the

successive book values of fixed capital stock as a whole in each industry.

In addition, as we pointed out in Ibanez, Matilla and Osuna (2004a), it is to be underscored that the

new method has been devised to contemplate any use of the so-called Replacement Fund for Fixed

Capital as internal financial source, for a firm managing a production process that uses machines for

obtaining a commodity as output. And so, this method contemplates any way to finance the fixed

capital stock by a firm, and hence, any firm’s financial structure.

To complete the argument it only lacks an existence proof of positive prices for marketed commodi-

ties, associated with non-negative values for distributive variables and with any non-negative value for

parameters θ and with any positive value for parameters ρ , satisfying integrated price system (17); and

thereby fundamental equation (18) displaying the reproduction of economic system. This existence proof

will be brought in a subsequent paper, and, as one can guess, it would allow us embracing any efficiency

pattern of fixed capital stock over time, any use of the so-called Replacement Fund, and then, any firm’s

financial structure.

Furthermore, it can be guessed that this proof will be very simple, and standard and well-known as-

sumptions will be required, since integrated price system (17) resembles to some extent to the corre-

sponding to the well-known Sraffian Circulating Capital model. The main difference between both price

systems consists of that in (17) the value of consumed inputs (machines and material inputs) differ from

the value of fixed and circulating capital stock from which de profit rate is calculated.

On the other hand, according to (28), the standard method for representing Fixed Capital requires

that in general the book values of fixed capital stock in each industry have to be ascertained, along with

the price of marketed commodities, for being performed the reproduction of economic system. Since

16/24

market prices alone, along with the distributive variables, are unable to perform this role, according to

(21), except in some special cases contemplated in the appendix.

However, the price of old machines , associated according to (6) to the succes-

sive book values of fixed capital stock for producing the n-th commodity in our argument, do not regulate

any transaction in our economy. This is so, since actually there is no market for old machines, insofar as

we have assumed that old machines cannot be transferred between and within sectors for producing any

commodity

1, ,j =jhp … t

7. And so, old machines are never demanded: i) by the same firm to initiate a different produc-

tion process in parallel for obtaining the same commodity; or ii) by other firms for producing a different

commodity; or iii) by consumers8.

If these conditions were violated, and there effectively existed markets for all old machines, we

should be in fact dealing with circulating capital goods instead of proper machines. This is so, since in

this case each firm should have to replace each machine one-year-older at the end of every produc-

tion cycle for obtaining any commodity9.

For this reason, the book values of fixed capital stock are mere accounting values, and cannot be

considered at the same step as that of actual market price for traded commodities. These accounting

values reveal themselves to be important within the standard method due to the fact that the latter re-

quires the pricing of old machines to express the value of fixed capital stock changing over time in each

industry, since this stock is suffering wear and tear depreciation.

However, once the price of marketed commodities (vector p) is determined within this standard

method for the corresponding value of profit rate r; for instance, by resorting to integrated price system

(20), which gets rid of the book values. Then, we can always ascertain the successive book values of

fixed capital stock as a whole for producing the n-th commodity by solving recursively equation system

(11)10, since this commodity is the only one in our argument that requires machines as input.

t

7 This assumption is equivalent to the assumption of scrapped machines by a firm having zero residual value

in any case.

8 Notwithstanding this, scrapped machines can actually be traded, i.e. they can be demanded by other firms

or by consumers through corresponding markets; and so, scrapped machines are considered to have a positive

residual value, which is in fact a market price, or a positive transfer value, which is nearly a market price. See for

instance Ibanez, Matilla and Osuna (2004b, Appendix 1). This possibility does not affect conclusions in our

argument; and so, it has not been considered in this paper for the sake of simplification.

9 In this case, each equation of (11) can be interpreted as the production process managed by an independ-

ent firm to obtain the n-th commodity, instead of the same firm managing successively over time each one of the t

production processes, as we have assumed.

10 We do not need to ascertain the price of old machines (vectors in our argument),

which would lead us to know the book value of each machine separately, and so its corresponding depreciation

pattern over time. It is sufficient, according to system (11), to know the book value of the fixed capital stock as a

whole. For this purpose, we can use integrated price system (20) to ascertain in first place vector p of marketed

commodities for a given value of profit rate r.

1, ,j =jhp …

17/24

Then, equation (26) corresponding to the integrated process for the n-th commodity is satisfied by

the market prices and the book values of fixed capital stock previously obtained and the associated

value for the distributive variables, since (26) is a linear combination of (11). Hence, integrated price

system (27) is also satisfied.

Therefore, in this state of things, we can always express in each industry the arithmetic mean of

the successive fixed capital stock book values, from which the profit rate is calculated in (27), as a

proportion of the value of fixed capital stock when new, as follows:

1

11 1

00

00

00

1 t

j n nj n

VTt

γ−

×= ×

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎢ ⎥⎣ ⎦

∑ n

VT

δ p

And so, we can make the book values of fixed capital stock disappear from equations (26), (27)

and (28) as follows:

a) The new n-th integrated process:

b) The new integrated price system:

where

( )1 n n nγ

×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥+⎣ ⎦

*

n

0

0

K0

δ

c) The new main equation displaying the reproduction of economic system:

( )

1

1 1

11

1 1 1

t

nj

t t t

j jj j j

rt

w l bt t t

γ=

= = =

⎡ ⎤+ + +⎢ ⎥

⎣ ⎦

+ + + =

∑

∑ ∑ ∑

n j

n j

δ p a p

δ p a p1

np

wt

(29)

(31) [ ]( ) ( )t rt= − + = + +*Ep e B δ A p e K A p eL

(30) ( ) ( )r w+ + + + =*K A p δ A p L Bp

Now, according to this latter equation very similar to (18), we can interpret that only the market

prices, along with the distributive variables, are able to perform the reproduction of economic system

18/24

when we use the standard method for representing Fixed Capital, even though the latter previously

requires ascertaining the book values of fixed capital stock in each industry.

As we can see, we have obtained this striking, though trivial, result by making vector VT of arith-

metic mean of fixed capital stock book values in each industry disappear from fundamental equation

(28). And this has been carried out by expressing this arithmetic mean in each industry, from which

the profit rate is calculated, as a proportion of the value of fixed capital stock when new by means of

parameters γ. This trivial procedure is always possible, once the price for marketed commodities

(vector p) and the book values of fixed capital stock in each industry are ascertained.

However, the new method uses from the very beginning in each equation of (1) a parameter θ for

avoiding the successive book values of fixed capital stock. This kind of parameter θ performs a similar

role as that of parameter γ in the standard method. The former has been precisely introduced to

express total asset value displayed in firm’s balance sheet at the beginning of each period in terms of

the value of fixed capital stock when new and that of circulating capital, i.e. in terms of the price of

marketed commodities alone, without resorting to the successive book values of fixed capital stock.

The main difference between parameters θ and γ is that the former can be hypothetically estab-

lished prior to price determination in devising an existence proof of positive market prices, in contrast

to the latter parameter, which requires the price of marketed commodities and the successive fixed

capital stock book values being previously ascertained.

The conclusion is evident. Neither the price of old machines, nor the successive book values of

fixed capital stock as a whole used in each industry are necessary being previously ascertained, as

the standard method requires, to show that actually the market prices along with the distributive vari-

ables are able to perform the reproduction of economic system according to (31). And so, the standard

method becomes roundabout for this latter purpose.

This is so, as is obvious, because the standard method requires additional information to market

prices and distributive variables in order to warrant the reproduction of economic system. This informa-

tion is precisely the t-1 successive book values of fixed capital stock, appearing in integrated process

(26) in our argument, corresponding to the n-th commodity.

Hence, the number of book values of fixed capital stock to be ascertained in general in the stan-

dard method for the whole economic system is:

( )1

1n

j

j

t=

−∑

where stands for the least common multiple of machines’ economic lifetime used to produce the j-

th commodity

jt11.

This large additional information is absolutely unnecessary when the new method for representing

Fixed Capital is utilized, since, as we argued above, the book values of fixed capital stock in each

11 In case that no machine is used to produce the j-th commodity, it holds that 1jt = .

19/24

industry play no role in order to warrant the existence of positive prices for marketed commodities

satisfying integrated price system (17), and then main equation (18).

Consequently, if we are trying to extend the Sraffian standard model of Circulating Capital, in which

only the market prices along with the distributive variables perform the reproduction of economic system,

to the more general context of Fixed Capital, it turns out to be more suitable employing the new method

than the standard one.

8. CONCLUSION According to the argument contained in this paper, we can assert that the standard method for repre-

senting Fixed Capital is able to show the reproduction of economic system being performed by the

market prices and the distributive variables, if, and only if, this method resorts in most cases to previ-

ous calculation of the price of old machines, or, at least, to previous calculation of the successive book

values of fixed capital stock as a whole used in each industry.

This procedure has been proven to be roundabout in comparison to the one adopted by the new

method for representing Fixed Capital of resorting to some parameters to distort the value of fixed

capital stock when new, to represent wear and tear depreciation of this stock over time. This proce-

dure dispenses us from the successive book values of this stock being ascertained, in order to show

the reproduction of economic system being performed by the market prices and the distributive vari-

ables.

Hence, it turns out from our argument that the method more suitable for the main object of Sraffa’s

Theory of Prices, centered in the reproduction of economic system, is precisely the new method to

represent Fixed Capital, instead of the standard one.

20/24

APPENDIX

PARTICULAR CASES ABOUT THE REPRODUCTION OF ECONOMIC SYSTEM WITHIN THE STANDARD METHOD

As we proved in 6.1, the reproduction of economic system cannot be performed in general when one

uses the standard method for representing Fixed Capital and at the same time one is avoiding the fixed

capital stock book values. This is so, since avoiding such book values prevents us from obtaining the

main equation to be satisfied by the net product of our economy that displays the reproduction of eco-

nomic system, except in some particular cases that are precisely contemplated in this appendix.

As we deduced in 6.1, the new method requires that (22) and (23) are simultaneously satisfied for

the reproduction of economic system. This entails the following requirements are fulfilled:

a) The arithmetic mean and the weighted mean of output obtained over t periods are to be equal:

( )( ) 1

11 1

11 1

t tj

t j jtj j

rb r b

r−

− += =

⎡ ⎤+ =⎢ ⎥

+ − ⎣ ⎦∑ ∑ t

Whereby it turns out that 1nα = according to (14).

b) The arithmetic mean and the weighted mean of material inputs consumed over t periods are to be

equal for being satisfied (22):

( )( ) 1

1 1

11 1

t tj

tj j

rr t

r−

= =

⎡ ⎤+ =⎢ ⎥

+ − ⎣ ⎦∑ ∑t-j+1 ja a

c) Finally, the arithmetic mean and the weighted mean of labor input employed over t periods are to

be equal for being satisfied (23):

( )( ) 1

11 1

11 1

t tj

t j jtj j

rl r

r−

− += =

⎡ ⎤+ =⎢ ⎥

+ − ⎣ ⎦∑ ∑ l t

t

t

The case of machines with constant physical efficiency Whatever the profit rate value is, one can easily deduce that requirements (22) and (23) are fulfilled

whenever machines used to produce the n-th commodity have constant physical efficiency throughout

their whole economic lifetime, as follows:

1, ,j n j nl l b b j= = = =j na a …

A special case of machines with variable efficiency Equations (22) and (23) also hold in a very special case of variable efficiency, whenever vectors

differ from one another by some scale factor. In other words, whenever

in each period of time if there is an alteration of output level b

( ), , 1,2, ,j jb l j =ja …

j, due to variable efficiency of machines

used to produce the n-th commodity, consumed material inputs aj and direct labor employed lj must

21/24

vary by the same proportion, i.e. by the same scale factor. Thus, we are dealing with a case very close

to the preceding one of constant physical efficiency.

In effect, by definition it holds in this special case that:

1, , 1j j t j j j tb b l l j tα α α≡ ≡ ≡ =j ta a … −

) 1⎞

+ ⎟⎠

Then, one can easily obtain the following expressions for the output:

11 1

1 11 1 1 2

1 (1 ) (1t t t t

j jj t j t j t t j

j j j j

b b b r b rα α−

− −− + − +

= = = =

⎛ ⎞ ⎛= + + = +⎜ ⎟ ⎜

⎝ ⎠ ⎝∑ ∑ ∑ ∑

Similar expressions can be obtained for material inputs and direct labor.

Under these circumstances, from (14) it turns out the following expression:

1

1

11

2

11

(1 ) 1(1 ) 1

t

jj

n tj

t jtj

t

rr

r

αα

α

−

=

−− +

=

⎛ ⎞+⎜ ⎟

⎝ ⎠=⎛ ⎞

+ +⎜ ⎟+ − ⎝ ⎠

∑

∑

Consequently, it holds that:

11

11 1

1(1 ) 1

(1 ) 1

t tj

n t j t jtj j

rl r l l

r tα α

−−

− += =

⎛ ⎞ ⎛ ⎞+ = + =⎜ ⎟ ⎜ ⎟+ − ⎝ ⎠ ⎝ ⎠

∑ ∑1

t

jj

t=∑

And a similar expression can be obtained for consumed material inputs. Whereupon equations

(22) and (23), as requirements for the reproduction of economic system, are automatically satisfied.

The case of zero profit rate Finally, we can also deduce that the standard method allows the reproduction of economic system,

without resorting to the book values of fixed capital stock, in case of 0=r , whatever the efficiency

pattern of machines.

In this case, direct addition of equations pertaining to system (11) leads us to obtain, once recalled

(25), the following integrated process:

1 1 1

1 1 1t t t

j jj j j

l w b pt t t= = =

+ + =∑ ∑ ∑n jδ p a p n

which is a particular case of (16) when 0=r .

Hence, in this particular case, the standard method is able to show the reproduction of economic

system without previous knowledge of the book values, since integrated price system (17) and main

equation (18) hold when , despite we are dealing with the standard method for representing

Fixed Capital. Thus, in this particular case, the reproduction of economic system is performed by the

market prices alone along with the distributive variables, dispensing the book values of fixed capital

stock from playing any role in the standard method.

0=r

22/24

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chap. 6, 88-137.

Garegnani, P. (1970): “Heterogeneous capital, the production function and the theory of distribution”.

Review of Economic Studies, 37, 407-436.

Ibanez, F.; Matilla, M. (2004): An alternative representation of fixed capital within a Sraffian frame-

work: constant efficiency. Working Paper. http://www.uned.es/dpto-analisis-

economico2/fichprof/fibanez/working_papers/Fixed_Capital1_constant_efficiency.pdf

Ibanez, F.; Matilla, M.; Osuna, R. (2004a): An alternative representation of fixed capital within a Sraf-

fian framework: variable efficiency. Working Paper. http://www.uned.es/dpto-analisis-

economico2/fichprof/fibanez/working_papers/Fixed_Capital2_variable_efficiency.pdf

Ibanez, F.; Matilla, M.; Osuna, R. (2004b): An alternative method for representing fixed capital within a

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