Measurement of Capital Stock, Depreciation and
Wealth
T.Rajeswari
National Accounts Division
Central Statistical Organisation
Abstract
In this paper an attempt has been made to generate estimates of capital stock and depreciation for
the manufacturing industry based on retirement functions. In doing so, the estimates of the growth in capital
stock, capital wealth and depreciation have been compared with the estimates published in the National
Accounts Statistics (NAS)-2008. In the first instance, estimates of retirements using a discard function for
assets have been worked out. This is then used to estimate the gross fixed capital stock. Then a depreciation
model is used to produce estimates of economic depreciation and wealth stock. Aim of this paper is to show
that assuming a mortality function and taking into account of the retirements of a cohort of assets would
reduce the values of consumption of fixed capital presently used in the NAS, thereby impacting the
GDP/NDP.
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I. Introduction
1.1
According to System of National Accounts (SNA, 1993), the stock of fixed assets surviving from
past investment and revalued at the purchasers' prices of the current period is described as the gross capital
stock. Capital stock features in two places in the SNA. It is needed to compile the balance sheets and as a
tool to derive an estimate of consumption of fixed capital. Operating surplus has been recognized as the
income derived from the use of capital in production just as compensation of employees is income derived
from the use of labour in the 1993 SNA. Thus there are two basic roles of capital, one as a measure of
wealth and the other as a measure of the contribution of capital to production. Thus gross capital stock gives
a measure of the contribution of capital to production while the net capital stock is a measure of wealth.
1.2
Gross capital stock is the value of all fixed assets still in use at the end of accounting period, at the
actual or estimated current purchaser’s prices for new assets of same type , irrespective of age of assets.
Capital stock comprise assets in the form of residential and non-residential buildings, dams, irrigational flood
control projects, other construction works, transport equipments, machinery and equipments, breeding stock,
draught animals, dairy cattle, capital expenditure on land improvement, plantation, orchard developments
and afforestation. The stock(fixed assets) includes uncompleted construction assets also. Durable goods in
the hands of households which are not used for further production of goods and services such as automobiles,
refrigerators, washing machines etc, as well as fixed assets mainly meant for defence purposes such as
warships, fighter aircrafts, transport vehicles and war materials do not form part of fixed capital stock as
these are assumed to have been consumed as soon as they are purchased. However, the construction works
undertaken by the households including buildings and capital expenditure on residential dwellings for
defence personnel, border roads, ordnance factories etc., form part of the fixed capital stock. As per the 1993
SNA, intangible assets like expenditure on mineral exploration and computer software development have
also been included in the boundary of Fixed Capital Stock.
1.3
The effects of aging on the value of an asset are explained by two events. First, the asset simply has
fewer service years to produce income, and second, its ability to render current and future services declines
as a result of lower productive capacity relative to that of newer assets. The decline in productive capacity
may result from both deterioration (or physical wear and tear) and obsolescence. Obsolescence affects
depreciation when older assets are less productive than newer and more technologically advanced assets. The
decline, in the current value of the stock of fixed assets owned and used by a producer as a result of physical
deterioration, normal obsolescence or normal accidental damage during the course of the accounting period
is called Consumption of fixed capital. It excludes the value of fixed assets destroyed by acts of war or
exceptional events such as major natural disasters which occur very infrequently. SNA 93 recommends
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independent estimates of consumption of fixed capital should be compiled in conjunction with estimates of
the capital stock. It further states that these can be built up from data on gross fixed capital formation in the
past combined with estimates of the rates at which the efficiency of fixed assets decline over their service
lives. From the initial estimate of capital stock subsequent changes in its value can then be deduced
analytically from information or assumptions about the rate at which its efficiency in production declines
over time. This method of building up estimates of the capital stock and changes in the capital stock over
time is known as the perpetual inventory method, or PIM.
1.4
The Gross Fixed Capital Stock (GFCS) and CFC for produced fixed assets are calculated by CSO
using the Perpetual Inventory Method (PIM). The essence of the perpetual inventory method is to add each
year’s gross investment (gross fixed capital formation) to the capital stock of the previous year. If the value
of assets which cease to exist each year is subtracted from this accumulated investment, then a gross measure
of the capital stock is obtained. If yearly deductions for depreciation are made, then a net measure of the
capital stock is the result. In the method followed by CSO, an estimate of GFCS is obtained in the first step.
This method generates an estimate of wealth stock of fixed capital accumulated by past purchases of assets.
To apply the PIM, assumptions are made about the average length of life of each class of separately
distinguishable assets. GFCF is then estimated for each class of assets for 'L' years prior to 'T', where 'L' is
the average life of an asset and 'T' is the year for which capital consumption and gross stock are to be
estimated. Appropriate price indices are then identified and applied to the estimates of GFCF to convert
them to constant prices. The estimates of GFCF at constant prices are then aggregated for 'L' years to obtain
the estimates of GFCS at constant prices at the end of the year. Then assuming straight-line depreciation, GFCS
of an asset is divided by 'L' to obtain the estimate of capital consumption at constant prices. The price
indices are used to convert the estimates of capital consumption to current prices. The estimates of NFCS
(i.e. GFCS for the year 'T' minus accrued capital consumption during 'L' years) for the year 'T' are first
calculated at constant prices and then converted to current prices using appropriate price indicators. In
the standard application of PIM adopted by CSO, the main assumption is that the total investment of a
particular asset does not deteriorate during expected service life of that asset and is discarded as a whole after
that time. That is the present procedure does not take into account the decay of the old capital stock but
excludes the stock that has expired the lifetime. This amounts to a simultaneous exit of assets. According to
the OECD Manual “Measuring Capital”, simultaneous exit is clearly unrealistic and capital stock is to be
estimated by cumulating gross fixed capital formation (GFCF) year by year and deducting retirements.
Retirements are to be calculated by postulating a retirement function that is applied to investment flows.
When these investment flows, corrected for retirements are cumulated, one obtains the gross capital stock.
Consumption of fixed capital or depreciation is calculated by superimposing a pattern of decline in value
over this time.
1.5
This paper attempts to compile estimates of capital stock, capital consumption and wealth stock in
the manufacturing sector of the economy by using survival function to account for retirements or
obsolescence/accidental damage of the assets. Appropriate price indices as used in the NAS have been
applied to the estimates of GFCF to convert them to constant prices. This new series of estimates has been
compared with the estimates published in the National Accounts Statistics 2008 ( referred to as old series in
this paper).
2. Assumed Life of Assets and Price Indices
2.1
PIM necessitates the availability of reliable estimates of average age of various types of fixed assets
in different industries. The average life for each type of assets presently used in NAS is based on extensive
discussions held by CSO with the concerned agencies like Directorate General of Technical Development;
Ministry of Industry; Railway Board; Bureau of Industrial Costs & Prices; National Productivity Council;
Departments of Posts & Tele-communications; Central Road Research Institute; Central Water Commission;
Ministry of Road Transport & Shipping and Indian Roads Congress etc. the requisite information on average
age of various assets have been obtained. Depreciation provision under Income-Tax Rules as well as in the
Companies (Amendment) Act, 1988 have also been considered. Appropriate price indices as used in the
NAS have been applied to the estimates of GFCF to convert them to constant prices.
2
3. Mortality functions
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3.1
As per 1993 SNA , regular expenditures on repair and maintenance do not change the fixed asset or
change its performance. It only puts a particular asset back into working order. But when we are considering
a cohort of assets say of machinery and equipments, some assets in a cohort may retire much earlier than the
estimated average service life of the entire cohort of assets. It is from this point of view that a retirement
function has to be used while estimating capital stock. An important issue, especially in relation to the
calculation of the capital stock, is the mortality function applied. It may be expected that the retirement of a
certain vintage of capital goods will not take place at once. A spread around the average expected service life
is normal. On the one hand, some capital goods will be retired prematurely because of obsolescence or
accidental damage; on the other, some capital goods may have a service life longer than average because of
careful use. By specifying a mortality function, this spread can be taken into account. To take into account
that similar assets are discarded at different ages, a bell-shaped distribution of discards was adopted instead
of the simultaneous exit method used at present. With the simultaneous exit discard function, all assets of a
given vintage are retained in the capital stock for their full service life (L years), and then discarded. The
main difference between the two methods is that the bell-shaped function produces a smoother growth
pattern. In this study, the distribution adopted is a bell-shaped (normal) distribution which has been truncated
so that all retirements occur between 40% and 160% of the average service life.
3.2
In this study it has been assumed that the actual service life of an asset is a random variable which
is normally distributed around the average service life, L. This means that, in general, when real capital
formation is growing, the bell-shaped function will produce lower estimates of capital stock than the
simultaneous exit function. Conversely, when real capital formation is falling, the bell-shaped function will
produce higher estimates because some older assets will be retained in the capital stock for more than L
years.
Normal Distribution
0.10000
0.09000
0.08000
0.07000
0.06000
0.05000
0.04000
0.03000
0.02000
0.01000
0.00000
Mean
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43
AGE Æ
4.
Methodology
4.1
Assuming that the discard pattern can be described using a probability distribution function, the
capital stock and discards can be related within a theoretical probability framework. Let f(x) denote a
probability density function i.e., the probability a capital good is discarded at age x. Then the retirement
3
coefficients ‘R’ which measure retired portion during the year n of a past or present investment can be
obtained by means of a survival function which is complement of normal mortality function, whose density
is expressed as follows:
f(x)=1/σ√2πe-1/2σ2[(x-L)/σ)]2 , where x denotes the number of years of existence of the asset in question, L
denotes its average service life, σ denotes the standard deviation and Si=1-f(i). The coefficients R are
calculated as follows: Ro =0 and Ri =1- Si for i=(1,m)( Ri=0 for i < 0 and i > m), where m is the
maximum service life of cohort of assets in question in the branch being considered ( regarded as
approximately equal to 2L). The annual retirements can then be calculated as
m
RETn= ∑GFCFn-i * Ri .
i=0
The estimates of Gross Fixed Capital Stock at constant prices can then be calculated as GFCS (n) =
GFCS(n-1) + GFCF(n) - RET(n) , where GFCS denotes gross capital stocks, GFCF, Gross fixed capital
formation at constant prices and RET retirements.
4.2
It may be seen from Table No.1 that cohort of assets of 1951 vintage with an average service life of
25 years and value 8717 when a normal distribution (truncated distribution so that all retirements occur
between 40% and 160% of the average service life) is applied will be discarded completely by 1991.
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TABLE No.1 – RETIREMENTS OF ASSETS
YEAR
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
AGE
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
Ri
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
RETIREMENTS
4
6
12
22
39
65
105
159
230
318
417
521
619
700
754
773
754
700
619
521
417
318
230
159
105
65
39
22
12
6
4
0.001
0.001
0.001
0.003
0.004
0.008
0.012
0.018
0.026
0.036
0.048
0.060
0.071
0.080
0.086
0.089
0.086
0.080
0.071
0.060
0.048
0.036
0.026
0.018
0.012
0.008
0.004
0.003
0.001
0.001
0.001
4
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Table No.2 shows GFCS figures for machinery and equipment. In the graph presented below it may be seen
that the old series show a steep fall in the growth rate at the point ‘X’ where simultaneous exit of assets took
place, while the new series based on retirement function is less volatile. This is because by specifying a
mortality function, discards of similar assets at different ages can be taken into account.
TABLE NO.2
Machinery
& Other
Equipment
Manufacturing UnRegistered
GROWTH
RATES
GFCS- Con
YEAR
AGE
old
New
1951
25
2289
2289
1952
25
2330
1953
25
1954
GFCS- Curr
old
New
GROWTH RATES
old
New
111
111
2330
1.8
1.8
123
123
11
11
2362
2362
1.4
1.4
141
141
14.4
14.4
25
2498
2498
5.8
5.8
148
148
4.7
4.7
1955
25
2726
2726
9.1
9.1
160
160
8.4
8.4
1956
25
2919
2919
7.1
7.1
177
177
10.3
10.3
1957
25
3359
3359
15.1
15.1
208
208
17.6
17.6
1958
25
3614
3613
7.6
7.6
230
230
10.6
10.6
1959
25
3826
3826
5.9
5.9
248
247
7.7
7.7
1960
25
4020
4020
5.1
5.1
266
266
7.5
7.4
1961
25
4267
4266
6.1
6.1
295
295
11
10.9
1962
25
4618
4614
8.2
8.2
329
329
11.4
11.4
1963
25
4930
4924
6.8
6.7
369
368
12.1
12.1
1964
25
2960
5237
-39.9
6.4
228
402
-38.3
9.3
1965
25
3390
5656
14.5
8
270
451
18.9
12.1
1966
25
3925
6172
15.8
9.1
329
517
21.5
14.5
1967
25
4902
7121
24.9
15.4
435
632
32.3
22.2
1968
25
6187
8362
26.2
17.4
575
777
32.2
22.9
1969
25
7733
9843
25
17.7
731
931
27.2
19.8
1970
25
9370
11388
21.2
15.7
928
1128
26.9
21.2
1971
25
10968
12864
17.1
13
1143
1341
23.2
18.9
1972
25
12774
14515
16.5
12.8
1401
1592
22.5
18.7
1973
25
14555
16105
13.9
11
1719
1902
22.7
19.5
1974
25
16355
17682
12.4
9.8
2141
2315
24.6
21.7
1975
25
18506
19581
13.2
10.7
2991
3164
39.7
36.7
5
old
New
1976
25
20179
21024
9.0
7.4
3685
3840
23.2
Manufacturing UnRegistered
GROWTH
RATES
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GFCS- Con
21.3
Machinery
& Other
Equipment
GFCS- Curr
old
New
GROWTH RATES
YEAR
AGE
old
New
old
New
1977
25
22256
22647
10.3
7.7
4076
4148
10.6
old
New
8
1978
25
24827
24543
11.6
8.4
4598
4545
12.8
9.6
1979
25
27182
26136
9.5
6.5
5355
5149
16.5
13.3
1980
25
29457
27556
8.4
5.4
6707
6274
25.3
21.9
1981
25
33797
30834
14.7
11.9
8659
7899
29.1
25.9
1982
25
40036
36089
18.5
17
11465
10335
32.4
30.8
1983
25
43798
38503
9.4
6.7
13284
11678
15.9
13
1984
25
47386
40517
8.2
5.2
15379
13150
15.8
12.6
1985
25
54386
45739
14.8
12.9
18376
15454
19.5
17.5
1986
25
61891
51331
13.8
12.2
23160
19209
26
24.3
1987
25
68869
56299
11.3
9.7
27255
22280
17.7
16
1988
25
72979
58141
6
3.3
30304
24142
11.2
8.4
1989
25
83820
66680
14.9
14.7
37679
29974
24.3
24.2
1990
25
95503
76108
13.9
14.1
48095
38328
27.6
27.9
1991
25
104098
82447
9
8.3
57570
45596
19.7
19
1992
25
109712
86092
5.4
4.4
70258
55132
22
20.9
1993
25
115516
89982
5.3
4.5
81846
63754
16.5
15.6
1994
25
123047
95491
6.5
6.1
90144
69957
10.1
9.7
1995
25
132463
102480
7.7
7.3
105388
81534
16.9
16.5
1996
25
150727
117654
13.8
14.8
128090
99984
21.5
22.6
1997
25
163702
127016
8.6
8
151589
117618
18.3
17.6
1998
25
176499
135329
7.8
6.5
168611
129281
11.2
9.9
1999
25
189545
142968
7.4
5.6
186351
140559
10.5
8.7
2000
25
203956
159530
7.6
11.6
203956
159530
9.4
13.5
2001
25
222376
179669
9
12.6
231716
187215
13.6
17.4
2002
25
232041
191452
4.3
6.6
258493
213277
11.6
13.9
2003
25
250090
212104
7.8
10.8
287354
243707
11.2
14.3
2004
25
277658
242163
11
14.2
330690
288417
15.1
18.3
2005
25
318272
285280
14.6
17.8
409298
366870
23.8
27.2
2006
25
358476
330017
12.6
15.7
494338
455093
20.8
24
2007
25
409103
387322
14.1
17.4
584199
553095
18.2
21.5
4.3 Consumption of Fixed Capital and Net Fixed Capital Stock
Consumption of fixed capital or depreciation is calculated by superimposing a pattern of decline in
value over this time. CSO employs the concept of economic depreciation which is consistent with the
definition of consumption of fixed capital in the System of National Accounts (SNA). This concept refers to
that part of gross product which is required to replace fixed capital used up in the production process during
an accounting period. Consumption of Fixed Capital represents the reduction in the value of fixed assets used
up in production process during the accounting period resulting from physical deterioration, normal
obsolescence or normal accidental damage. In a production process labour, capital and intermediate inputs
are combined to produce one or more outputs. The non-financial assets return capital service to the process
of production and decay over time. The decay is measured as CFC which differs from the concept of
depreciation in business accounting.
4.4
SNA 1993, mentions three possible profiles for measuring depreciation: Constant decline in
efficiency until the asset disintegrates; a linear decline in efficiency ; a constant geometric, or exponential,
6
decline in efficiency. The most familiar model of depreciation is the straight-line method in which equal
amounts are deducted from the stock every year. The rate of straight-line depreciation is given by 1/L where
L is the average service life of an asset. However, the choice of using a depreciation function in the PIM
model needs to be considered in conjunction with the choice of asset life assumptions. In this study , a
straight line depreciation function has been used. To calculate the coefficients of consumption of fixed
capital (Di) which gives the annual rate of depreciation of a past or present investment using retirement
coefficients R, the following matrix was used which is based on the assumption of straight line depreciation.
D1
D2
D3
D4
D5
.
.
.
Dm
1/1 1/ 2 1/3 …………………1/m
1/ 2 1/3…………………1/m
.
. .
1/3…………………1/m
.
.
==
.
.
.
. . .
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m
.
R1
R2
R3
R4
R5
.
.
.
Rm
.
.
. . . . . 1/m
.
m
where ∑Di = ∑Ri =1
i=0
i=0
The consumption of fixed capital for year n is then obtained using the following formula:
m
CFCn = ∑GFCFn-i * Di ,, where m is the maximum service life of the cohort of assets.
i=0
4.5
The Net Fixed Capital Stock (NFCS) (constant price) is then calculated as Gross Fixed capital stock
less accumulated depreciation. Both the gross and net capital stock estimates are then converted to current
prices by using appropriate price indices. This also gives the depreciation at current prices.
5. Conclusions
5.1
It may be seen from Table No.3 that the change in discard pattern from a simultaneous exit to normal
discards gave estimates of CFC which are lower than the old series. Also normal retirement function along
with straight line depreciation function gave estimates of consumption of fixed capital and wealth stocks
with similar growth pattern as that of old estimates (presented in NAS 2008). The objective of this paper
has been to discuss the impacts of using mortality functions on capital stock figures and depreciation.
Assuming a mortality function and thereby taking into account of the retirements of a cohort of assets would
reduce the existing consumption of fixed capital figures used in the NAS, thereby impacting the GDP/NDP.
However, the choice of mortality functions is extremely crucial and more technical studies on the type of
mortality functions to be used for various categories of assets are required.
7
TABLE No.3
Year
Current
CFC(old)
56928
64679
72200
78258
87529
103782
122984
145966
2000
2001
2002
2003
2004
2005
2006
2007
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CFC(New)
51726
58955
65749
71122
74741
94011
105553
125654
Current
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Year
2000
2001
2002
2003
2004
2005
2006
2007
PERCENTAGE DIFFERENCE AND GROWTH RATES BETWEEN OLD AND NEW
ESTIMATES OF CFC AND NFCS
(Rs.Crores)
NFCS(old)
1361410
1513302
1639795
1740431
1948520
2351017
2806329
3377740
NFCS(New)
1441165
1595763
1722134
1825973
2038931
2444816
2905292
3485526
Constant
CFC(old)
56927
61485
65343
69668
75258
83016
93119
105581
CFC(New)
51726
56030
59507
63322
64246
75091
79837
90734
Constant
NFCS(old)
1361410
1438992
1488764
1556266
1658469
1839629
2076768
2368970
NFCS(New)
1441165
1524468
1580363
1654517
1764310
1953751
2200006
2502226
Current(Growth rates)
CFC(old)
13.6
11.6
8.4
11.8
18.6
18.5
18.7
CFC(New)
14.0
11.5
8.2
11.1
19.0
18.6
19.0
Current(Growth rates)
Constant(Growth Rates)
CFC(old)
8.0
6.3
6.6
8.0
10.3
12.2
13.4
CFC(New)
8.3
6.2
6.4
7.3
10.7
12.3
13.6
Constant(Growth Rates)
NFCS(old)
NFCS(New)
NFCS(old)
NFCS(New)
11.2
8.4
6.1
12.0
20.7
19.4
20.4
10.7
7.9
6.0
11.7
19.9
18.8
20.0
5.7
3.5
4.5
6.6
10.9
12.9
14.1
5.8
3.7
4.7
6.6
10.7
12.6
13.7
Percentage Diff
Current
-9.1
-8.8
-8.9
-9.1
-9.1
-9.4
-9.1
-9.1
Percentage Diff
Current
5.9
5.4
5.0
4.9
4.6
4.0
3.5
3.2
References
1. Central Statistical Organisation, India (2008): National Accounts Statistics, 2008
2. Central Statistical Organisation, India (2007): National Accounts Statistics: Sources and Methods
3. Commission of European Communities, International Monetary Fund, Organisation for Economic
Cooperation and Development, United Nations and World Bank (1993): System of National
Accounts 1993
4. Depreciation Rates for the Productivity Accounts, Statistics Canada,
http://www.statcan.ca
5. Economic Depreciation and Retirement of Canadian Assets: A Comprehensive Empirical Study,
Statistics Canada , http://www.statcan.ca
6. OECD Manual on Measuring Capital: Measurement of Capital Stocks, Consumption of Fixed
Capital and Capital Services, http://www.oecd.org
7. Perpetual Inventory Method ,Service lives, Discard patterns and Depreciation methods, Statistics
Netherlands, Department of National Accounts, http://www.oecd.org
8. Paper on “Retropolation of the GFCF Series and Calculation of Fixed Capital Stocks on the ESA95 basis in the French national accounts” presented by Department of National Accounts INSEE –
France at the International Workshop "Capital Stock Estimation: Recent Contributions" Valencia,
December 18th, 2000. http://www.insee.fr
8
Constant
-9.1
-8.9
-8.9
-9.1
-9.1
-9.5
-9.1
-9.1
Constant
5.9
5.9
6.2
6.3
6.4
6.2
5.9
5.6