Journal of Applied Input-Output Analysis, Vol. 10, 2004

Endogenising Capital:

A comparison of Two Methods

By
Manfred Lenzen* and Graham J Treloar*

Abstract
Two methods for endogenising capital transactions into a static input-output system — theaugmentation method and the flow matrix method — are compared. While the first method
treats all types of capital as one homogeneous commodity, the latter distinguishes capitaltransactions by origin and destination industry. Multipliers including capital requirements
are calculated for the Australian economy of 1996-97, for the factors energy, CO2, water,
and intermediate demand. In these cases, the augmentation method leads to a systematic
overestimation of low- and mid-range multipliers, and to a substantial underestimation ofhigh-range multipliers. The magnitude of this error is factor-dependent, and increases withincreasing variance and range of the corresponding factor intensities. On the other hand, theflow matrix method avoids these systematic errors, but has the disadvantage of high data requirements.

1. Introduction
It is general knowledge amongst practitioners of input-output analysis, that there is awhole range of input-output systems, tailored to answer particular questions. One dis

tinction between different systems is their degree and type of closure. A system can beclosed by endogenising particular parts of primary inputs and final demand into current
intermediate transactions (the use matrix). Examples for such closures are the endogenisation of trade with foreign regions, or of household expenditure and income. Anykind of closure leads to an increase in magnitude of the multipliers.
In this work, we are concerned with closing a static input-output system with respect

to fixed capital: production of fixed capital by industries and inputs of fixed capital

into industries are transferred from final demand and primary inputs respectively, andadded to current intermediate demand. In this case, the increase in multipliers reflects atransition from a short-term to a long-term perspective: Current transactions as pub

lished in use matrices represent a more or less constant stream of inputs that is turned
over at least annually. In contrast, capital demand often fluctuates highly between
Received September 2003,

final version received July 2004.

* School of Physics, A28, The University of Sydney NSW 2006, Australia.

** School of Architecture and Building, Geelong Waterfront Campus, Deakin University, Geelong Vic
3217, Australia.

2

Journal of Applied Input-Output Analysis, Vol. 10, 2004

years, and is effective as an input into production until many years after purchase.

Input-output analysis can address questions of the likely impact of (1) final demand shocks on quantities demanded throughout the economy (the Leontief model), or(2) primary input price shocks on consumer prices (the Ghosh model). In both models,
impacts can be determined as including or excluding the effects of capital. In this pa
per, we examine the effect that closing the input-output system with respect to capitalhas on static, ex-post Leontief-type multipliers, in monetary terms as well as in termsof energy consumption, CO2 emissions, water use, and intermediate demand. We exam
ine two approaches for endogenising capital: the augmentation technique, and the flow
matrix method. The following sections explain the methodology, the data preparation,and the results of the comparison. The paper is then concluded.

2. Methodology
It is a convention in National Accounts that purchases of capital goods are counted as

primary inputs and final outputs of the economic system rather than as intermediate inputs into production. In the Australian input-output tables, the primary input category'gross operating surplus' contains inputs of fixed capital into production, while final
demand includes gross fixed capital expenditure as a category. As a consequence, theuse matrix only describes "current" intermediate, but not "capital" transactions, and themultipliers in the corresponding Leontief or Ghosh inverses do not include effects ofcapital. Since capital investment ultimately occurs to facilitate production (for infra
structure replacement and capacity expansion), it can be regarded as an intermediate input (compare Bullard and Herendeen 1975). Accordingly, transactions involving fixed
capital can be separated from primary inputs and final demand, and endogenised into

intermediate transactions. In the input-output literature this process is referred to as"closing an open input-output system". Not many input-output studies incorporate capi
tal effects. Within those that do, there appear to be two dominant approaches: the aug

mentation technique, and the flow matrix method (see also Lee 1971, Casler 1983,
Wolff 1985, Gowdy 1992, Hohmeyer 1992, Wenzel and Pick 1997).

2.1 Augmentation method
Within most published input-output tables, gross fixed capital expenditure is available

as a column vector yic) as part of final demand y, while gross fixed capital input (forcapacity expansion or replacement) appears as a row vector v(c) as part of primary inputs v. It is therefore in most cases straightforward to move these vectors as an addi
tional column and row into the intermediate demand field, thus augmenting the existingarray of sectors by one (see Fig. 1). This artificially created additional sector is as

sumed to produce one homogeneous commodity - 'capital' - which is produced using

inputs according to y(c\ and consumed by other sectors according to v(c).

Endogenising capital: a comparison of two methods

3

Figure 1: Endogenising capital expenditure y{c) and capital investment v(c) into theintermediate use matrix U, to form an augmented use matrix U(a).

ye)

u

; yc)

y

v<c)

'

0

v

| 0

0

A Leontief inverse L(a) is then calculated using the augmented use matrix U(a):where I is a suitable unity matrix. S(a)t is a transposed augmented market share matrix(resulting from an augmented supply matrix V(a) =

(v
q

^

V<a)ij / 2jiV(a)ij, so that S(a) = (o )y (o according to S(a)y =i'j, where V and S are the original make and market

share matrices, respectively). The diagonal-only augmentation of V shows that the additional capital sector is assumed to have no by-products, and that no other sector di
rectly produces capital, but instead delivers into the "capital sector". Augmented total

output is jc(a)=

(y
V

/

x)
.

A

§(a)tlJ(a)f£(a)-i = [ (o—i

=

' /

V

v(c)v(c)"'\

q

( x)

, with x(a) being its diagonalised version. A(a) =// A ay{c) \J = ((c) a ) ^s ^ augmented direct requirements ma

trix.

This formulation is "blind" with regard to the type of capital, meaning that live
stock, machinery, buildings or artwork are not distinguished as different commodities.
While this is less of a problem in a purely monetary impact study operating in value
terms, different types of capital can exhibit significantly different factor intensities q interms of physical factors, for example energy intensities qe, CO2 intensities qc, or water
use intensities qw. Therefore, determining factor multipliersm(a) = q{a) L(a)(2)

in a generalised input-output study, an indiscriminate allocation of different types ofcapital can lead to distortions of factor multipliers m{a\ because particular types ofcapital are generally only used by some, but not all sectors (for example livestock is

4Journal of Applied Input-Output Analysis, Vol. 10, 2004

used as capital input only within agriculture).

Following Miyazawa et al. (1963), an analytical formulation of Eq. 1 can be writ
ten as_ ( I - A -a\c)\_ [ B (I + «r Kalc)B) Ba?K)

L ~ [ ^(00 )- {

Ka?B

K )

(3)

with B = (I-A)"1 and the scalar K=(\-a[c) BaJT1. The upper left term can be transformed into

B (I + a lc) Ka <c) B ) = (I + B< Kd?) B = [ B"'( I + Ba ? Ka t
J J

L

-fly

"'

Aflv (1 + tkly

= [l - A - a yc) Aaic)(I + Ba? W1]"'

Aflv )(4)

2.2 Flow matrix method
An obvious solution to this shortcoming is to disaggregate capital input v(c) by
supplying sector, and to disaggregate capital expenditure yic) by using sector. This disaggregation results in a capital flow matrix K that, as the current flow matrix A=S'U

x"1, shows producing sectors in its rows and using sectors in its columns, and thatmaps the flow of capital commodities according to their type and origin. Factor multipliers are then(5)

While in the augmentation method the dimension of all matrices increases by 1, it remains unchanged in the flow matrix method. Comparing Eqs. 4 and 5, one can interpret that the term a f Kaic)(l + Ba^Ka?)'1 in Eq. 4 is an approximation of K in Eq. 5.
In Section 4, we present results of Leontief-type multipliers iw(a) and m{T), in termsof energy, CO2 emissions, water use, and intermediate demand. We compare sets ofmultipliers from open and semi-closed models, as well as sets of semi-closed multipli
ers resulting from different closure methods. Note that the formalism holds equally fora Ghosh-type calculus.

3. Data preparation and estimation of flow matrices
A disadvantage of the flow matrix method is that most statistical bureaux do not esti
mate capital flow matrices regularly, or at all. Therefore, in our work, a 1996-97 capitalflow matrix for Australia was estimated in a semi-survey approach, combining spot
samples of industry survey data with the RAS balancing technique. This estimation involvedEndogenising capital: a comparison of two methods

51) separating total capital input v(c) by industry from gross operating surplus in theAustralian input-output tables (Australian Bureau of Statistics 2001), using auxil
iary data (Australian Bureau of Statistics 1996). Since the model is generalised tophysical quantities, capital input includes only physical capital; the sum over v isactually equal to the sum over y. The remaining part of gross operating surplus
contains profits that represent income of economic agents, and may be invested insome monetary form or other;

2) inserting spot samples of survey data (from Australian Bureau of Statistics 1994;
1995; 1996; 1999a; c; b) into a preliminary flow matrix;3) adding the remaining elements based on informed judgment: for example manufac
turing and services do not use livestock, hence the respective values are set to zero;4) netting the preliminary flow matrix as well as row and column totals y{c) and v(c) offixed survey-based elements;5) RAS-balancing the net K matrix, adjusting to net row and column totals; and6)adding fixed survey-based elements to balanced matrix.

There are other approaches to constructing a capital flow matrix. For example,
Casler (1983) suggests determining a representative mix of capital stock held by indus
tries through time, and calculating a capital corrections matrix from the depreciation

rates of capital stock items. This approach has the advantage of evening out fluctua
tions in rather sporadical capital purchases, and hence avoids atypically small or largecoefficients for any one year. The resulting capital flow matrices, however, would notcomprise a growth component of investment.

In the case of the augmentation method, only step 1) was performed, and v(c) andyc) were then simply added as a separate row and column to the use matrix. Multipliers
were then calculated according to Equations 2 and 5.
The factor data for q for energy consumption, CO2 emissions and water use weretaken from national energy statistics (Australian Bureau of Agricultural and Resource

Economics 1997, see also Lenzen 1998), the national greenhouse gas inventory (National Greenhouse Gas Inventory Committee 1998, see also Lenzen 1998), and the na
tional water accounts (Australian Bureau of Statistics 2000, see also Lenzen and Foran
2001). In the case of multipliers for total intermediate demand, qx = {1, 1, ..., 1}.

4. Results: CCh, energy, water and intermediate demand multipliers
including capital
Closing the input-output system by endogenising transactions involving fixed capital
into intermediate demand increases the size of matrix elements in the direct requirements matrix A, and therefore the internal feedback in the system. Multipliers are

therefore larger in this closed system than in the open one.
In the case of the 1996-97 Australian input-output system, and for the example ofCO2 emissions, multipliers increase on average by about 10%-15% when capital is endogenised (Fig. 2). This magnitude is reasonable, considering that in Australia in 1996-

97 capital expenditure constituted about 10% of total output. The relative frequency ofdifferences between CO2 multipliers ml = q (I - A)"1 excluding capital flow and mc including capital flow depends however on which method was used: In general, multipli-

6

Journal of Applied Input-Output Analysis, Vol. 10, 2004

Figure 2: Relative frequency of differences between CO2 multipliers ml excluding
capital flow, and mc including capital flow obtained fromthe flow matrix and the augmentation method.
25%■ Flow matrix method
20%

E3 Augmentation method|15%

s 2 10%
.3

n

5%

0%

T;
0%

1 II1 i 1 ■ n 1
15%

30%

45%

60%

nI n

75%

90%

Difference d = (mc- mc*)/mc(%)

ers that were obtained using the augmentation method tend to be larger than those obtained by the flow matrix method. This phenomenon is examined further in the follow
ing.

Given that the flow matrix method provides a more accurate picture of the origin

and destination of different types of capital commodities, the relative difference be

tween CO2 multipliers ml? from the flow matrix and multipliers m? from the augmen
tation method can be regarded as a measure for the error that is associated with using
the augmentation method. The relative frequency distribution for this error centres

Figure 3: Relative frequency of differences between CO2 multipliers mf from theflow matrix method and multipliers mf from the augmentation method.
35%

30%

>> ^ 25%
20%£ <n 15%
n

" 10%
.3

5%
0%

1

n

H nSni,: • !l i fl n hi

ill

20%

30%

Difference d = (wc(a) - wc(f)) / m(%)-30% -20% -10%

0%

10%

Endogenising capital: a comparison of two methods

7around 2%, with the bulk of samples contained between -1% and +12% (Fig. 3). This
indicates once again that the augmentation method tends to overestimate most CO2
multipliers.

This behaviour can be explained by the fact that the augmentation method does

not distinguish between different types of capital. For example, because of associatedemissions from land clearing, Australian livestock embodies a larger amount of CO2
emissions than all other capital commodities. Correctly, these emissions should be allocated to a few agricultural industries that use livestock as a capital input for breeding.

However, in the augmentation method these livestock-related emissions get lumped together with those related to other capital commodities, and distributed across all indus

tries according to total capital input value, thus overstating the corresponding CO2 mul-

Table 1: CO2 multipliers (kg/A$) for the open and the semi-closed system, the flow
matrix and augmentation techniques, and relative changes (%) between
the two techniques, for selected Australian industries.
CO2 multipliers (kg/A$)
Industry

excludingflowaugmenta

relative

capitalflowmatrixtion

changetechniquetechnique(%)

Dairy cattle

0.96

1.39

1.16

-17.1

Meat and meat products

7.17

7.97

7.32

-8.1

Dairy products

0.87

1.13

1.04

-7.3

Beef cattle

27.56

29.4

27.78

-5.5

Textile products

0.69

0.81

0.79

-1.5

Electricity supply

11.31

11.47

11.53

0.5

Pulp, paper and paperboard

6.43

6.54

6.58

0.7

Accommodation, cafes and restaurants

0.76

0.88

0.89

0.8

Basic chemicals

2.39

2.49

2.51

1.1

Iron and steel

3.02

3.13

3.17

1.3

Residential building

0.76

0.92

0.94

2.0

Petroleum and coal products

1.23

1.32

1.36

2.5

Road transport

1.20

1.29

1.32

2.6

Ships and boats

0.84

0.92

0.95

3.3

Commercial fishing

0.83

0.93

0.96

3.3

Plastic products

0.77

0.85

0.88

3.7

Coal oil and gas

1.17

1.34

1.40

4.3

Electronic equipment

0.61

0.69

0.72

4.8

Motor vehicles and parts

0.59

0.67

0.70

4.9

Education

0.15

0.19

0.20

6.5

Iron ores

0.52

0.67

0.72

8.0

Communication services

0.31

0.42

0.47

10.0

Health services

0.11

0.16

0.18

11.8

Libraries, museums and the arts

0.14

0.20

0.23

15.2

Banking

0.14

0.24

0.28

17.2

Insurance

0.06

0.13

0.15

21.0

8

Journal of Applied Input-Output Analysis, Vol. 10, 2004

Figure 4: Relative frequency of differences between water multipliers ml0 fromthe flow matrix method and multipliers m™ from the augmentation method.
16%
14%§ r 10%t £8%

Ji *7 6%
•S

4%

n

JHL

0%
-30% -20% -10%

n
0%

10%

20%

30%

I

Difference d = (mJa)- W0) /mJf)(%)

Figure 5: Relative frequency of differences between energy multipliers m? fromthe flow matrix method and multipliers mf from the augmentation method.

25%

n

20%

15%

n
-6%

n II n
-4%

-2%

0%

fllllln
2%

4%

nR PI
6%

Difference d = (we(a)- W0) / we(f) (%)
tipliers of all non-livestock-using industries. In contrast, the CO2 multipliers of agricul
tural industries are significantly underestimated (see Tab. 1). The more the augmentation technique underestimates the CO2 multiplier, the more CO2-intensive the capital input of the respective industry is compared to the average commodity "capital". Agricul
tural and agriculture-related industries receive — directly and indirectly — relatively

CCh-intensive capital inputs (livestock), while most other industries, especially services,
receive less CO2-intensive capital ( for example office equipment).

If instead of CO2, water use is examined, the deviation of multipliers //i(wa) obtained

from the augmentation from multipliers ml0 obtained using the flow matrix method iseven larger: The relative frequency distribution of the error associated with the aug-

Endogenising capital: a comparison of two methods

9

Figure 6: Relative frequency of differences between intermediate demand

multipliers m? from the flow matrix method and multipliers m!a) fromthe augmentation method.
50%
45%

■

_ 40%

£»
(U

O

I

35%
30%
25%

20%
.S

15%
10%
5%
0%
-6%

-4%

-2%

J

n

0%

2%

o

4%

6%

Difference d = (mi(a) - wi(f)) / wi(f) (%)

Table 2: Comparative measures for variation in factor multiplier anddifferences between flow matrix and augmentation method.
Var(q) / Mean(q)Max(q) / Min(q)Mean(d)Var(d)

Water

899.3

7,839,496

7.8%

2.45%

CO2

28.1

2,148,357

2.2%

1.53%

Energy38.4

339,299

1.1%

0.05%

0

1

1.1%

0.01%

Intermediate demandmentation method is centred around 8% and ranges roughly from -5% to +25% (Fig.

4). A few percent of multipliers are overestimated by more than 40%, and those ofwater-intensive industries (largely dairy cattle and products, rice growing, water supply
and electricity generation) are underestimated by more than 30%.
In contrast, both methods produce similar results if energy and intermediate demand are considered. Here, errors of the augmentation method range only from -0.5%
to 1.5%, and 0 to 1.5%, respectively. A summary of results from numerical experi
ments (Tab. 2) shows that the magnitude of the error d, and its variance across indus

tries, is related to the type of factor: Generally speaking, the larger the relative varianceof intensities q across industries, the larger the error of the augmentation method.
It is however also intuitively clear that the range of intensities must play a role forthis error too, since in the augmentation method factor content from intensive indus
tries is distributed across all industries. A comparison of CO2 and energy multipliers
demonstrates that the larger the factor range (expressed as the ratio between the largest

and the smallest factor intensity), the larger the error. An example of two factors thatfeature the same range but different variances could not be found.

The error does however not disappear for a factor with zero variance and range 1,
as shown in the example of intermediate demand. The remaining "baseline" error of

10

Journal of Applied Input-Output Analysis, Vol. 10, 2004

about 1.1% is factor-independent, and must be attributed entirely to the differences in

the coefficients of A(a) and A + K.3

5. Conclusions
Comparing the two methods for endogenising capital introduced in this paper, it can beconcluded that the augmentation method leads to a systematic overestimation of lowand mid-range multipliers, and to a substantial underestimation of high-range multipli
ers. The magnitude of this error is factor-dependent, and increases with increasing variance and range of the corresponding factor intensities.
In order to calculate interindustry effects of capital expenditure, it is therefore un
der most circumstances preferable to employ the flow matrix method. However, adrawback of this approach is its relatively high data requirements: In many countries,

capital flow matrices are not estimated by statistical bureaux at all, so that these haveto be constructed based on disparate, and often incomplete and inconsistent data

sources. If capital transactions data are not available at all, the augmentation method

has to be used. In this case, some estimate of the magnitude of the error in the multiplier values can be obtained by computing the variance and range of the factor intensi
ties under consideration.

Acknowledgements
The authors would like to thank one anonymous reviewer for a thorough examinationof our manuscript. Also, the text surrounding Eqs. 3 and 4 as well as footnote 3 isbased on this reviewer's idea.

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11

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