Reflections on Input-Output-Coefficients
Christian Lager
University of Graz, Department of Economics
Paper to be presented at the
III Spanish Conference on Input-Output Analysis
Albacete, 2009
(1) Introduction
Besides Wicksteeds production function which has (unfortunately) entered all economic
textbooks there are many other, usually more useful means to describe the process of
production. Different ideas and concepts can be found in the writings of John Hicks, Piero
Sraffa, John von Neumann, Wassily Leontief and Nicholas Georgescu-Roegen1. All of these
scholars were developing their concepts and tools with different backgrounds and were
aiming at special goals.
Sraffa (1960) was primarily concerned with a revival of the concepts of the old classical
economists; his investigation is concerned exclusively with stationary economic systems.
Consequently, the effects of changes in the scale of production as well as the utilization of
exhaustible or renewable resources are set aside. However, Sraffa dealt with the theory of
rent. Seeking for an equilibrium solution of an economic system expanding with a uniform
rate, von Neumann (1937) had to define all scarce resources away and postulate constant
returns. In order to ensure non negative prices he assumed the rule of free goods. Neither the
former nor the latter assumption can be found in Sraffa (1960). Scrutinizing the work of both
authors one will, however, find some unifying concepts and methods which are (i) the concept
of a ‘long run position’, (ii) the view that production is a circular process where commodities
are produced by means of commodities, and (iii), a totally vertically disintegrated
representation of production, by using point input - point output processes with uniform
duration.
With regard to vertical integration Hicks (1970, 1973) proposed the other extreme: a totally
vertically integrated representation of production. His ‘neo-Austrian’ approach, is based on
the ‘Austrian theory of capital’ founded by Böhm-Bawerk and elaborated by Wicksell, Hayek
and others. The ‘Austrians’ were primarily interested in the time structure of production and
conceived a process as a series of dated original factors, such as land or labour, which, finally,
produce a quantity of a consumer good. Hence, production is viewed as a one way avenue,
leading from primary inputs to consumption goods.
With regard to the two extremes of total vertically integration or disintegration GeorgescuRoegen takes a somehow intermediate position and defines elementary processes as flows of
inputs and flows of outputs. This route will be also taken in that paper.
1
For a detailed discussion and comparison of different concepts to represent production see Lager (2000).
In contrast to the scholars mentioned above, Wassily Leontief was primarily concerned with
the empirical implementation and the application of multisectoral models of production.
Therefore his models, as well as input-output models in general, are based on restrictive
assumptions concerning joint production, natural resources, the treatment of durable capital,
and, in particular, the time profile of inputs and outputs.
This paper aims at bridging the gap between theoretical reasoning and empirical application
by demonstrating that it is possible to apply some of the above mentioned theoretical concepts
to empirical input output data. In particular the simplifying assumptions with regard to the
time structure of production made in input output models will be abandoned for the benefit of
introducing a more general concept of production and time. In section 2 the concept of a flow
input – flow output process as a general description of production is explained. This concept
is applied in section 3 to derive prices of production which prevail in the long run provided
there is free competition. Using centre coefficients which are functions of the rate of profit,
the system of price equations may be displayed in the well known elegant matrix format. A
corresponding model for quantities is derived in section 4. This model retains all complex
features of the most general flow input – flow output approach and, thus, is able to deal with
processes of non-uniform duration, differing production and construction lags and wear and
tear of fixed capital goods with differing life. Using centre coefficients, which are now
functions of the rate of growth, the quantity model appears in the elegant and simple matrix
format of the open Leontief model. To simplify the empirical applications some
straightforward assumptions concerning the temporal structure of inputs and outputs are
discussed in section 5. A simple procedure of calculating centre coefficients on the basis of
empirical data is described in section 6. Finally some results are presented.
(2) Flow input – flow output processes: A general description of production
Any process of production can be, in general, described by a flow of inputs and a flow of
outputs. One should view a process of production not independent of the creation of its
capacities, buildings, machinery and other fixed capital items. It starts, in principle, with
some inputs, amongst them some fixed capital inputs. Then, after machines and buildings are
constructed and set in place, the actual production phase can take place and by means of
energy, raw material and labour some outputs are produced. Besides those “normal” inputs
and outputs, which are used and produced frequently, fixed capital items are to be replaced,
spare parts will be installed and repair work has to be done from time to time. That process of
production of outputs, maintenance and replacement of fixed capital items will continue until
all fixed capital items reach the end of their lives and will be disposed of by means of some
inputs or, if there is a market for scrapped machines, sold to some other producers or to
households.
To facilitate the analysis the following assumptions are made
(i) There is no general joint production, i.e. each process produces only one type of
commodity, i.e. all by-products including scrap or pollutants are assumed away.
(ii) There is only one primary factor which is available in abundance (labour)
(iii)
There is constant returns to scale
(iv)There is no technical change
(v) The income distribution is constant, i.e. the wage rate and the rate of profit remain
constant
Given these assumptions we may describe a single production process to produce some
quantities of commodity j by the tupel
  a0   a1   a 2 
  0j  ,  1j  ,  2j  ,
  l j   l j   l j 

 aTj j
 T
 lj j


   b0j , b1j , b 2j ,



T

bj j ,
where
aj  R n is a vector of quantities of produced inputs, i.e. a bundle of circulating and fixed
capital items, used in that process at time  , with elements aij ,
l j is a quantity of labour used in that process at time  ,
bj is a quantity of output of commodity j which appears as a product of that process at time
 and
T j is the duration of that process.
(3) Prices of production and centre coefficients depending on the rate of profit
If there is free competition neither extra profits nor losses will occur in any of those processes
which are actually utilized. Hence discounted costs are equal to discounted revenues, i.e.
j :
Tj



0
Tj

paj  wlj R   p j  bj R  ,
where R   1  r 

 0
is a discount factor, and r is the uniform rate of profit.
Using a concept introduced by Schefold (1989) we may specify that equation in terms of
centre coefficients and arrive at price equations
j : pa j  r   wl j  r   p j ,
1
Tj
 Tj
 
 

where the centre coefficients for capital inputs, a j  r     aj 1  r  
b
1

r


 ,

j
  0
  0


Tj
 Tj 
 
 

b
1

r
and for labour inputs, l j  r     l j 1  r  


 , are functions of the rate of

  0 j
  0


profit.
Hence we finally arrive at a system of price equations which may be displayed in the well
known elegant matrix format
pA  r   w l  r   p
where the matrix for produced inputs and the vector for labour inputs contain centre
coefficients which are not simple technical coefficients but are functions of the rate of profit.
A detailed discussion of that price model and conditions for positive prices can be found in
Schefold (1989), in Kurz and Salvadori (1995) or in Lager (1997, 2000).
(4) Quantities and centre coefficients depending on the rate of growth
The constant returns to scale assumption implies that for all positive intensities x j
  a0   a1   a 2 
  0j  ,  1j  ,  2j  ,
  l j   l j   l j 

 aTj j  
 T   x j  b0j , b1j , b 2j ,
 lj j 
 

T

bj j x j
is also a feasible process.
Let us assume that this process starts at point t in historical time at intensity xtj , and therefore
will generate quantities b0j xtj , b1j xtj , b2j xtj ,
b j j xtj of outputs at time t , t  1, t  2,
t  Tj .
T
If total outputs of commodity j are to grow at constant rate g, or at constant factor G  1  g ,
then this process will be (or has been) initiated at historical time t  s at intensity xtj G s ,
s  1, 2,
 or
s  1,  2,
.
The outputs of those overlapping processes are depicted in table 1.
Outputs
I
n
i
t
i
a
t
e
d
at
t-2
t-1
t
i
m
e
t
t+1
b0j xtj G
b1j xtj G
b0j xtj
b1j xtj
b2j xtj
b0j xtj G1
b1j xtj G 1
b2j xtj G1
b3j xtj G1
b1j xtj G2
b2j xtj G 2
b3j xtj G2
b3j xtj G2
t+1
t
t-2
b0j xtj G2
t-Tj
b j j xtj G
Table 1
t+2
b0j xtj G2
t+2
t-1
a
t
time
T 2
2 T j
T 1
1T j
b j j xtj G
T
b j j xtj G
T j
Adding outputs of all those overlapping processes which produce commodity j at time t and
t  s respectively, we obtain
 Tj

v j x tj    bj G   x tj
  0

and,
 Tj

v j xtj s    bj G   xtj G s  v j xtj G s
  0

The same reasoning lead to total inputs of commodity i at time t or t  s respectively
 Tj

uij xtj    aij G   x tj
  0

Tj
Note that v j   b j G

and,

 0
 Tj

uij xtj s    aij G   xtj G s  uij x tj G s
  0

Tj
and uij   aij G  are monotonically decreasing functions of the rate
 0
of growth but are constant over time provided there is growth with constant rates.
We may now apply the above developed theoretical concepts to observable data.
The well known accounting balance in national accounting statistics and input output analysis
is:
n
n
j 1
j 1
qi   X ijc   X ijf  fi
where
qi … gross output of commodity i (of sector i) produced during one calendar year
X ijc … intermediate consumption of circulating capital goods i in sector j during one calendar
year
X ijf … gross fixed capital formation of commodity i in sector j during one calendar year
f i … other final demand of commodity i (of sector i) during one calendar year
Applying the theoretical concept developed above to observable data, one should recall, that
national accounting or input output statistics do not record stocks of inputs or outputs at a
particular point in time but account for flows of inputs and outputs during an interval of time
which is usually taken to be one calendar year.
Assume that the calendar year consists of Y units of time and start at time t. Given the
assumptions specified above, we may set changes in prices aside. Assume furthermore that
there is long run growth at a constant rate g (or at a factor G). Hence observable annual flows
of outputs and inputs are given by
Y
qi   vi xit G s and
s 0
Y
X ijc  X ijf   uij x tj G s
s 0
where v j and u ij are functions of the rate of growth as defined above.
We may now derive input-output coefficients – just as in conventional input-output analysis –
by dividing inputs through outputs:
Tj
Y
aij 
X X
c
ij
f
ij
qj

 uij xtj G s
s 0
Y
v x G
s 0
j
t
j
s

uij
vj

a G 



0
Tj
ij
bG



0
.

j
Note that these observed input-output coefficients are independent of the length of the
calendar year and independent of the intensity of the process but are uniquely determined by
technical conditions and by the long run growth rate.
Note, furthermore, that these input-output coefficients have exactly the same functional form
as the centre coefficients derived for the price model, but are now functions of the rate of
growth.
Tj
While uij   aij G


 0
Tj
as well as v j   bj G  are decreasing functions of the growth rate (the
 0
rate of profit), center coefficients might increase or decrease with that rate. From
daij
dg

duij
dg
1
v j  uij v j
2
dv j  1
 duij

 aij
vj ¦ 0
dg  dg
dg 
dv j
follows that centre coefficients increase (decrease) with the rate of growth, if and only if the
percentage decrease of outputs with the rate of growth is larger (smaller) than the percentage
decrease of inputs with that rate, i.e.
daij
dg
¦ 0 
v 
dg
dv j
1
j
¦
u 
dg
duij
1
ij
If inputs and outputs are constant then the respective centre coefficient is independent of the
rate of growth. Whether centre coefficient are increasing or decreasing functions of the
growth rate depend crucially on the time profile of input and output flows. The larger the
outputs (inputs) at the beginning (end) of the process, the smaller (larger) is the effect of an
increase in the growth rate for v j  uij  , and therefore the larger is the likelihood of decreasing
input-output coefficients.
We may finally express the quantity model in matrix format, i.e.
q  Aq  f ,
where A is a matrix of centre coefficients which are functions of the rate of growth.
Note that this approach displays the same simple and elegant features as the Leontief model.
There are, however, remarkable differences:
 Conventional input output coefficients are based on a rather simple concept of
production, i.e. the view that no time elapses between circulating inputs and outputs
and therefore assumes instantaneous production, and, if fixed capital is taken into
account, usually infinite life and exponential depreciation (radioactive decay) is
supposed.
The concept of centre coefficients, on the contrary, which is based on a special Sraffavon Neumann approach2, is able to display (i) complex production and construction
lags and different life time of fixed capital, (ii) processes with different production
time, (iii) wear and tear of fixed capital, reinvestment, maintenance and repair
activities and, (iv) increasing and/or decreasing efficiency of production.
 Conventional input-output analysis has been primarily designed for modeling the
industrial interdependencies with regard to intermediate inputs. The inter-industry
flows arising from the deliveries of fixed capital are usually not taken into account.
2
For the limitations of the concept of centre coefficients and its relation to a more general Sraffa-von Neumann
type of model, see Kurz and Salvadori (1995) or Lager (1997, 2000)
The concept of centre coefficients treats circulating and fixed capital inputs basically
in the same consistent way. Differences in the duration of inputs and in the time
structure of outputs are considered by the well known economic principle of
discounting. Consequently the multipliers derived from a centre coefficient based
model are more comprehensive and, however, lager than the Leontief-multipliers.
 Multipliers are, in general, a useful tool for comparative static analysis. The concept of
centre coefficients presupposes that the economy prevails in a long run position (long
run growth) where all capacities are fully adjusted to demand such that production is
cost-minimizing, i.e. unit costs are minimal. Hence, centre coefficient based
multipliers should be used for long run analysis. In the short run, on the contrary,
demand might be met by adjusting the degree of utilization of capacities already in
place rather than by adjusting fixed capital stocks and, therefore, one cannot suppose
that fixed capital stocks are fully adjusted to demand. Hence for short run analyses of
temporary and transitory changes, centre coefficients would overestimate the effects.
In this case conventional Leontief multipliers might be more appropriate3.
(5) Simplifying assumptions for temporal output and input profiles
To simplify the analysis for empirical applications, we may assume some particular profiles
for input- and output flows.
Let us first subdivide the period of the process  0, T j  into a ‘construction’ period  0,  j  ,
in which capacities (buildings, machinery, etc.) are constructed, but no actual production
takes place, and a ‘productive’ period   j  1, T j  , where positive quantities of outputs j are
produced.
3
There remains, however, the short run choice of technique problem of finding the optimal method of utilization
of fixed capital in place. For a discussion of that problem see Kurz (????)
For the sake of simplicity it is assumed that outputs are constant during the ‘productive’
period4.5 Hence, bj  b j for     j  1, T j  and bj  0 for   0,  j  , and therefore the sum
of the ‘discounted’ flow of outputs is given by

1
2
v j  bj 1  G  G 
G


 T j   j 1
v j  b j T j   j 
G
  j 1

 T  j
1 G j
 bj
1  G 1

G
  j 1
for G  1, and
for G  1.
Let us turn now to some simplifying assumptions for inputs. We shall distinguish three groups
of inputs: (i) ‘basic materials’, (ii) ‘fixed capital’ and (iii) ‘spare parts’.
The first group of inputs, labeled ‘basic materials’, consist of all produced means of
production which are absorbed, consumed or used up during that process such as energy, fuel,
raw- or basic materials. For these inputs ‘strict proportionality’ between inputs and outputs is
assumed. Furthermore, because production is a time consuming process, a production lag of
 ij periods is proposed, i.e. inputs of commodity i used at time  are proportional to outputs
of commodity j produced at time    ij and, therefore,

uijB  ijBb j G
 ij   j 1
G
 ij  j  2

G
 ij T j
.
Hence,
 T
1  G j j  ij   j 1
u   bj
G
for G j  1
1  G 1
B
ij
B
ij
and
uijB   ijB b j T j   j 
for G j  1
The corresponding centre coefficients for ‘basic materials’ are
4
It should be mentioned here that other, more interesting output profiles could be assumed. While increasing
outputs, particularly at the beginning of the ‘productive’ period may characterize ‘learning by doing’,
decreasing outputs will go hand in hand with a decay in the efficiency of elder fixed capital equipment.
5
Note that the widely used assumption of exponential depreciation of fixed capital with a constant rate,
presupposes exponential decay of an infinite flow of outputs. This implies that the largest absolute
reduction of outputs happens at the beginning of the process, and is, therefore, a rather crazy assumption.
a 
B
ij
uijB
vj
 ijBG
 ij
Note, that this center coefficients are equal to constant input-output coefficients if there is
either no growth, i.e. G j  1 , or there is instantaneous production without production lag, i.e.
 ij  0
The characteristic of a ‘fixed capital’ input, such as a machine, a building or a car, is that it is
not absorbed if it is utilized, but participates in the process of production for a length of time,
called its life, until it is worn out and is (perhaps) replaced by a new identical item.
The temporal input pattern of fixed capital item i participating in the production of
commodity j, is determined by its life, ij , and its construction period,  ij , which is the
length of time between the date of the input of that capital good and the date when it is ready
to be used productively.
In order that all machinery and equipment are ready for use to initiate production at the
beginning of the productive period of that process, i.e. at time  j  1 , a quantity  ijF of fixed
capital item i has to enter the process j the first time at date  j  ij  1 . Note that the capital
item with the largest construction lag is invested the first time at the very beginning of the
process, hence,  j  max  ij  . To maintain productive capacity for the duration of the
process, the capital item must then be replaced frequently after all ij units of time until it is
worn out, together with all other equipment at the end of the process, at time T j .
The duration of the entire process, T j , is determined by the length of the ‘productive period’,
 j , which is the lowest common multiple of the lives, ij , of all capital items participating in
that process, and the length of the ‘construction period’,
 j , which is the maximum
production lag. Hence we have T j   j   j .
Given these assumptions and definitions, the observed quantity of inputs of fixed capital item
i used in process j is

uijF   ijF 1  G j
where nij 
u  
F
ij
F
ij
 ij
 Gj
2 ij

 n 1 
  
 G j  ij  ij G j  j ij  ,

j
is the number of (re)investments. Hence we obtain
ij
1 Gj
 j
1 Gj
 ij
G

  j  ij

for G j  1
uijF   ijF nij
for G j  1
It is straightforward to calculate centre coefficients for fixed capital inputs:
a 
F
ij
uijF
vj

ijF 1  G 1
bj 1  G
 ij
G
ij 1

ijF G  1
bj 1  G
 ij

G ij .
The third category of inputs, labeled ‘spare parts’ consists of all commodities or services
which are used to repair or to maintain previously installed equipment. Spare parts, like fixed
capital, appear with constant frequency. But, in contrast to machines or buildings which must
be ready at the very beginning of the “productive” period, at time  j , spare parts enter the
process the first time after that date.
Once, fixed capital is in place and ready to support the production of commodity j, at time
 j , it must be maintained and/or repaired in periodic intervals of length ij by using some
quantities  ijS of input i. In that case ‘spare parts’ enter the process at time  j  ij then at
 j  2ij , … and finally for the last time at  j   nij  1 ij . The only difference between the
temporal profile between fixed capitals and spare parts is that the construction lag of the latter
is nil and the number of investments equals nij  1 , where nij is defined as above. Hence we
obtain for inputs of ‘spare parts’

uijS   ijS G
which is
 ij
G
2 ij

G


 nij 1 ij
G
 j
,

  

1  G j ij  ij   j
G

1  G ij
 ij 
uijS   ijS   1
 j



uijS   ijS
for G  1
for G  1
Centre coefficients for ‘spare parts’ are given by
aijS 
uijS
v Sj

  G  G 1
1  G 1  G 
 ijS G
bj
 ij
 j
 ij
 j
(6) Estimating centre coefficients using Austrian input-output tables
Each of the derived center coefficients for

 ‘basic products’,
aijB  ijBG ij ,
 ‘spare parts’,
a 
S
ij
 ‘fixed capital’, a 
F
ij
  G  G 1 , and
1  G 1  G 
 ijS G
bj
ijF G  1
bj 1  G
 ij
 ij
 j
 ij
 j

G ij ,
is the product of a constant coefficient and a function of the rate of growth.
Given some assumptions on lags and life of capital items, in a first step the functions of the
long run growth rates6 were calculated and thereafter the fixed coefficients were estimated by
using observable centre coefficients.
Two sources were used: The matrix of intermediate inputs of dimension commodity by
commodity7 and the matrix of gross fixed capital formation of dimension commodity by
industry. Both were derived from input-output tables for Austria for the year 2000, 2002 and
2005 at current basic prices8.
6
Long run growth rates, differing by sectors, were calculated as five year moving averages of actual growth
rates of gross outputs in constant prices.
7
To avoid to deal with joint production symmetric input output tables were used where by-products are defined
away by ‘technology’ assumptions.
8
Note that data at constant prices would have been more appropriate but were not available.
All deliveries recorded as gross fixed capital formation were considered as ‘fixed capital’
inputs (see table 2). Given some assumptions about the life time of fixed capital (see annex 1)
which are based on OECD (2009) and, furthermore, assumptions on construction lags, which
are generally taken to be one month, except for the commodity NACE 45 (construction)
where the lag is taken to be half a year, for each cell of the matrices functions of growth
where calculated for the year 2000, 2002 and 2005 using long run rates of growth for those
years. These functions where then used to estimate the constant factors.
"Basic
"Spare
"Fixed
Materials"
Parts"
Capital"
Intermediate Consumption
X
inexpensive small tools
X
ordinary maintainance and repair services
costs of using rented fixed assets (rent for buildings,
leasing of machines or cars)
all other products recorded as intermediate goods
X
X
Gross fixed capital formation
dwellings and other buildings and structures
X
machinery and equipment
X
cultivated assets such as trees and livestock
X
computer software
X
improvements to existing fixed assets that go well beyond the
X
requirements of ordinary maintenance and repairs
Table 2
Table 2 shows that intermediate uses contain all three types of inputs. Hence, this matrix had
to be subdivided into three matrices ‘basic products’, ‘spare parts’ and ‘fixed capital’.
The percentage shares of ‘spare parts’ and ‘fixed capital’ inputs in total intermediate inputs
are represented in annex 2 and 3. Because of lacking of adequate empirical information, most
of these shares are derived by ‘educated guesses’. Only the share of ‘spare parts’, i.e. the
maintenance component, for commodity NACE 50 (Sale, maintenance and repair of motor
vehicles and motorcycles; retail sale of automotive fuel) and NACE 52 (Retail trade, except of
motor vehicles and motorcycles; repair of personal and household goods) could be derived
from statistical data.
Given the life of ‘spare parts’, i.e. basically the interval of maintenance inputs, which is taken
to be 3 years, except for buildings where it is assumed to be 5 years, first, the functions of the
rate of growth were calculated, and then the constant parameters were estimated.
The same procedure has been applied to ‘fixed capital’ intermediate inputs, which are small
and inexpensive tools and devices as well as the cost of rented fixed assets. The life of these
items (the interval of payments) is assumed to be generally 3 years. The uniform construction
lag is taken to be one week.
The temporal structure of ‘basic products’ is determined by the production lag, i.e. the time
interval between inputs and outputs, which is taken to be 5 days for most industries.
Exceptions to that rule can be found in annex 4.
Given those parameters, one can easily calculate centre coefficients for all components of
intermediate inputs as well as for gross fixed capital inputs at different growth rates.
Column sums for centre coefficients at a zero growth rate for the Austrian economy for three
components of intermediate demand and for fixed capital are displayed in table 3. While
‘basic products’ account in the average for almost 50% of total outputs, small and inexpensive
tools and devices, which are treated as ‘fixed capital’ and ‘spare parts’ are much smaller an
account, in the average, only for 2.6 and 2.3 percent of gross output. New and more
interesting is the contribution of fixed capital formation. The column sums of centre
coefficients for ‘fixed capital’ vary from 60 percent (NACE 71, renting of machinery) to 0,3%
(NACE 30, office machinery and computers) of gross outputs.
Centre coefficients (column sums) for various rates of growth are shown in table 4. All centre
coefficients increase with the rate of growth. While intermediate inputs do not vary much with
that rate, for ‘fixed capital’ centre coefficients the impact of a change in long run growth is
much more significant. In the most relevant domain of growth, i.e. between -2% and 2%, a
one percent increase of the rate of growth results, in the average, in an increase of the column
sum of ‘fixed capital’ centre coefficients of one percent point. The impact of a change in the
overall growth rate on center coefficients vary between industries. The most important effects
can be found in NACE 70 (real estate) but also in NACE 1 and 2 (agriculture and forestry)
and in NACE 71 (renting service of machinery and equipment). Note, that the total column
for centre coefficients for intermediate inputs and fixed capital inputs for NACE 23 (mineral
oil products) exceeds unity for a growth rate of 2%. This violation of the Hawkins-Simons
condition indicates, that this economic system is not capable to grow with a rate of 2% or
more, or, on the other hand, does not allow for a uniform rate of profit of 2% or more.
In table 4 multipliers (column sums of the Leontief inverse) calculated with centre
coefficients for intermediate inputs and for total capital inputs at different rates of growth are
shown. Long run multipliers which account also for the adjustment of fixed capital inputs are
generally and significantly larger than short run multipliers which account for intermediate
inputs only. The effects of including ‘fixed capital’ inputs vary, in the average, between 17%
and 29% depending crucially on the rate of growth. The largest impacts of changes in long
run growth can be found in NACE sectors 70 (real estate), 90 (sewage and refuse disposal),
71 (renting of machinery), 1 to 5 (agriculture, forestry and fishing) but also in some transport
sectors.
Conclusion ??????????
References
Kurz, H.D. and Salvadori, N. (1995), Theory of Production - A Long Period Analysis,
Cambridge, Cambridge University Press.
Leontief, W. et. al. (1953), Studies in the Structure of the American Economy, New York:
Oxford University Press.
Lager, Ch. (1997), Treatment of Fixed Capital in the Sraffian Framework and in the Theory of
Dynamic Input-Output Models, Economic Systems Research, Vol. 9, No 4, 1997, pp.
357-373.
Lager, Ch. (2000), Production, Prices and Time: a Comparison of Some Alternative Concepts,
Economic Systems Research, Vol. 12, No 2, 2000, pp. 231-253.
Neumann, J. von (1937), "Über ein ökonomisches Gleichungssystem und eine
Verallgemeinerung
des
Brouwerschen
Fixpunktsatzes",
Ergebnisse
eines
mathematischen Kolloquiums, 8, pp. 73-83.
Schefold, B. (1971), Mr Sraffa on Joint Production, Ph. D. thesis, University of Basle,
mimeo.
Schefold, B. (1989), Mr Sraffa on Joint Production and other Essays, London: Unwin
Hyman.
Sraffa, P. (1960), Production of Commodities by Means of Commodities. Prelude to a
Critique of Economic Theory, Cambridge: Cambridge University Press.