Gross Stocks Estimated from Past Installations

Gross Stocks Estimated from Past InstallationsAuthor(s): Eric SchiffSource: The Review of Economics and Statistics, Vol. 40, No. 2 (May, 1958), pp. 174-177Published by: The MIT PressStable URL: http://www.jstor.org/stable/1925031 .

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I74 THE REVIEW OF ECONOMICS AND STATISTICS I74 THE REVIEW OF ECONOMICS AND STATISTICS

foreign supply curve AC and demand curve JM are unaffected by devaluation, while, upon devaluation to the extent indicated by AB(=JK), the home supply curve in terms of foreign currency shifts downward from its initial position (not drawn), passing through J, to KM, and the home demand curve in terms of foreign currency shifts downward from its initial position (not drawn), passing through A, to BC. Thus M is the new equilibrium position in the export market, and C that in the

CHART I E F A

I "~~~~ (s)/ \f d) Logarithm ( / (fd) of Price C

(hd) (hs) '

B G R Logorithm of Quantity

foreign supply curve AC and demand curve JM are unaffected by devaluation, while, upon devaluation to the extent indicated by AB(=JK), the home supply curve in terms of foreign currency shifts downward from its initial position (not drawn), passing through J, to KM, and the home demand curve in terms of foreign currency shifts downward from its initial position (not drawn), passing through A, to BC. Thus M is the new equilibrium position in the export market, and C that in the

CHART I E F A

I "~~~~ (s)/ \f d) Logarithm ( / (fd) of Price C

(hd) (hs) '

B G R Logorithm of Quantity

import market, while the balance of trade after devaluation in foreign currency is represented by EP, and that in terms of home currency by GR.

I2. In the case where the home supply curve of exports and the foreign supply curve of imports are infinitely elastic, the points E and C, D and H, N and L, and M and R merge, and Chart 2

becomes similar to the diagram presented by A. C. L. Day in "A Geometrical Demonstration of Sta- bility Conditions in International Trade," Eco- nomia Internazionale (February I954).

CHART 2

E F A N P J

CL Logarithm M of Price D X

B G H K Q R

Logorithm of Quantity

import market, while the balance of trade after devaluation in foreign currency is represented by EP, and that in terms of home currency by GR.

I2. In the case where the home supply curve of exports and the foreign supply curve of imports are infinitely elastic, the points E and C, D and H, N and L, and M and R merge, and Chart 2

becomes similar to the diagram presented by A. C. L. Day in "A Geometrical Demonstration of Sta- bility Conditions in International Trade," Eco- nomia Internazionale (February I954).

CHART 2

E F A N P J

CL Logarithm M of Price D X

B G H K Q R

Logorithm of Quantity

GROSS STOCKS ESTIMATED FROM PAST INSTALLATIONS Eric Schiff

GROSS STOCKS ESTIMATED FROM PAST INSTALLATIONS Eric Schiff

Statisticians trying to estimate gross stocks of durable assets sometimes find themselves in the following situation. No direct balance sheet data on stocks exist, but information on past installations is available for a substantial number of years. The assets installed during this period are characterized by an average service life (which may be a weighted combination of several subgroup life averages) and a retirement distribution (possibly likewise a weighted compound) around this average. The retirement distribution is unknown but may be assumed to have remained approximately constant over time. The average service life, which under this assumption has also remained approximately constant during the period, is known or can be estimated.' Then the value of the gross stock in place at some point in time may be approximated by cumulating past installations back over the period indicated by the life average (if the installation data are available that far back). Admittedly, this

Statisticians trying to estimate gross stocks of durable assets sometimes find themselves in the following situation. No direct balance sheet data on stocks exist, but information on past installations is available for a substantial number of years. The assets installed during this period are characterized by an average service life (which may be a weighted combination of several subgroup life averages) and a retirement distribution (possibly likewise a weighted compound) around this average. The retirement distribution is unknown but may be assumed to have remained approximately constant over time. The average service life, which under this assumption has also remained approximately constant during the period, is known or can be estimated.' Then the value of the gross stock in place at some point in time may be approximated by cumulating past installations back over the period indicated by the life average (if the installation data are available that far back). Admittedly, this

is a crude method whose results will, in general, deviate from what would be obtained if the distribution of retirements around the life average were known and the corresponding gross survival distribution were applied to past installations in reverse chronological order over the period indicated by the full range of the survival distribution (again assuming that installation figures are available that far back). In this case, a breakdown of survivors from all past installations by years of origin would be obtained, and the sum of all these year-of-origin survivors could be regarded as a correct estimate of the total survivorship in the terminal year. The question naturally arises: by how much, under typical conditions, must the gross stock value computed by the crude cumulation method be expected to deviate from what the correct but often im- practicable survival distribution method would yield?

If the flow of past installations, or the distribution of retirements, or both, is entirely irregular, no general answer is possible. However, complete ir- regularity is the exception rather than the rule in this field. The empirical distribution of retirements around the average service life of a homogeneous group of assets has long been known to approxi- mate in most cases some regular pattern fairly closely. The patterns most frequently observed in practice can be represented by bell-shaped distribu-

is a crude method whose results will, in general, deviate from what would be obtained if the distribution of retirements around the life average were known and the corresponding gross survival distribution were applied to past installations in reverse chronological order over the period indicated by the full range of the survival distribution (again assuming that installation figures are available that far back). In this case, a breakdown of survivors from all past installations by years of origin would be obtained, and the sum of all these year-of-origin survivors could be regarded as a correct estimate of the total survivorship in the terminal year. The question naturally arises: by how much, under typical conditions, must the gross stock value computed by the crude cumulation method be expected to deviate from what the correct but often im- practicable survival distribution method would yield?

If the flow of past installations, or the distribution of retirements, or both, is entirely irregular, no general answer is possible. However, complete ir- regularity is the exception rather than the rule in this field. The empirical distribution of retirements around the average service life of a homogeneous group of assets has long been known to approxi- mate in most cases some regular pattern fairly closely. The patterns most frequently observed in practice can be represented by bell-shaped distribu-

'If the installation and the stock are composites of as- set subgroups having different retirement distributions and life averages, the assumption of constancy of the weighted compound life average and retirement distribution is, of course, only legitimate if it may be assumed that the com- position of the aggregate has remained unchanged in the successive installations. In agreement with what is often premised in this case we suppose here that this additional assumption is also permissible, at least for a first approximation.

'If the installation and the stock are composites of as- set subgroups having different retirement distributions and life averages, the assumption of constancy of the weighted compound life average and retirement distribution is, of course, only legitimate if it may be assumed that the com- position of the aggregate has remained unchanged in the successive installations. In agreement with what is often premised in this case we suppose here that this additional assumption is also permissible, at least for a first approximation.



NOTES AND BOOK REVIEWS

tion curves, symmetrical or skew, with various de- grees of dispersion. The flow of past installations is often found to follow one of three patterns with an appreciable degree of approximation: growth at a constant relative rate (exponential growth); growth at a declining rate; and cyclical fluctuations around one or the other of these two growth trends. Hence, analysis of our problem on the basis of a few "models" reflecting some or all of these empirically important patterns is not mere algebraic play, but should be of some help and interest to those now struggling with the problem statistically.2

Let the distribution of retirements be represented by a continuous function f (x), extending over a range of s time units, where f (x) = o at the two points x = o and x = s, and the area under the distribution curve between these two points is i. At the end point s, assets installed at time t are (s-t) time-units old. The fraction of this installation which by time s has been retired is

s-t

f (x) dx 0

and the fraction still surviving at time s is therefore

s-t

I - f (x) dx 0

which equals

f (x) dx.

Let the flow of installations prior to s be represented by some function g (t), where the time t is expressed in the same units as x in the distribution and survival functions. Then, if the initial installation (dated s time-units back) is given unit value,

the total gross survivor value at time s from all past installations is

s s

Si= g (t) [Jf (x) dx ]dt. 0 s-t

The gross stock value derived by simply cumulating installations backward from s over the average service life, denoted by n, is

S2= g (t) dt. s-n

Let us work out the ratio S2/S1 for one of the empirically plausible assumptions about f (x) and g (t) listed above. For the retirement distribution, let us assume a symmetrical bell-shaped curve of the Pearson II type; e.g., the quartic

ax2 f (x) = X2- 4 (x2 n4x+4fn2) (I) n4

where n and all x-values are expressed in absolute time-units, measured from the origin,3 and a is a parameter to be determined by the stipulation that the area under the curve between x = o and x = 2n is unity:

2n

4 x2 (x2 - 4 n x + 4 n2) dx = i. n

0

This gives a = 0.9375/n, so that our retirement curve reads

f (x) = .9375X2 (X2-4 n x + 4n2). (2)

The function is shown as curve A on Chart i for x-values expressed in years and n = io years. Its

CHART I

JO .10

.04 S04

0 a 3 4 S 6 7 8 9 10 I1 12 13 14 5 6 18 19 20 Yar

contour is very similar to one of the symmetrical retirement curves fitted to empirical data by the Iowa Experiment Station (type S1).4 For compari-

2If the statistician is willing, on the basis of whatever considerations, to apply some specified distribution of retirements and survivals to his installation data, he is of course free to compute total gross survivors by the correct method. Since, however, computation of year-of-origin survivors by applying annual survival percentages to past installations over many years back is usually a tedious and time-consuming procedure which many statisticians wish to avoid even when it is feasible, the question of whether the results of the simple cumulation method re- main within acceptable limits of error maintains its interest in any case. The answer obtainable by using both methods on some empirical installation data and comput- ing the relative margin between the results is valid only for this particular empirical material. Our model analysis at- tempts to provide the basis for a more general answer by tracing the functional dependence of the margin on such variables as the relative dispersion of the retirements, the rate of growth in the trend of installations, etc.

'Equation (i) is a transform of the more frequently used writing f (x) = yo (i - x2/ioo)2, where yo is the ordinate associated with the mean, and x is expressed in tenths of, and is measured from, the mean. The form in the text appears more convenient for our purpose.

' Cf. Robley Winfrey, Statistical Analyses of Industrial Property Retirements, Bulletin I25 (Iowa Experiment Sta-



I76 THE REVIEW OF ECONOMICS AND STATISTICS

son, this curve has been added on the chart (Curve B).

Regarding the survival function, it is clear that in the present case of a strictly symmetrical retirement distribution curve extending from o to s,

s t

fI(x)dx= f(x)dx. s-t 0

Hence we have for our survival function

f (x) dx = s-t t

09375 fx2 (x2 - 4 n x + 4 n2) dx n 5

0

which gives s

f (x) dx = s-t

0.062 5 t3 o.65 P(3 t2 _ 15 n t+ 2o n2) (3) n5

As for the function to represent the flow of past installations, let us assume an exponential type of growth at a (continuous) rate r per time unit:

g (t) = ert. (4)

Applying the above survival pattern to this flow, the total gross survivor value at point 2n as derived by the survival distribution method is

2n

Si= t3ert (3t2- 15 n t+ 20 n2)dt. 0

Working this out, we obtain e2rn

S1 =-- r

7.5 [(r2n2 + 3) (e2rn - I) - 3 rn (e2rn + I)]

6 5~~~ (5) r6 n5

The gross stock value as derived by simply cumulating past installations up to 2n over a period equal to the life average, that is, from t = n to t = 2 n, is

2n

S2 = fert dt n

which gives ern

S2 = - (ern - I). (6) r

The ratio between the stock values derived by the two alternative methods can now be written in various ways. The most convenient form is perhaps S2

Si (rn)-5 (ie- rn)

(rn)5 - 7.5 [ (r2n2+3 ) (i -e-2rn) -3rn(I+e-2rn)]

(7) The result is interesting in several respects: i. Under the stated assumptions, the percentage

deviation between the two S depends on the product of r and n. (For r = o the two gross survivor values are equal, as expected.) The deviation is the same for, say, 2 per cent growth and io years life average, 4 per cent growth and 5 years life average, i per cent growth and 20 years life average, etc. Approxi- mately, this may be assumed to hold also if annual rather than continuous functions are used.

The percentage divergence between the two S, i. e., 100 (S2/S1 -I), has been plotted on Chart 2

CHART 2 Pu W Pe, C<rwS

3 3

2 2

0 3 3 2 4 5 6 7 0 9 0 Volvo$ of rn

as a function of the product r n. It will be seen that the gross stock value derived by the short-cut method exceeds the value derived by the correct method for all rn-products except rn = o.

2. For any assumed average service life, the deviation between the results of the two methods first increases, but later decreases, as we increase the assumed growth rate of the installations. The same is true for any assumed rate of installation growth as we increase the assumed life average. There is one rn-product at which the deviation reaches a maximum. It can be seen from the diagram that the maximum deviation is about + 5.26 per cent, associated with, approximately, rn = 2 (which may mean 4 per cent installation growth combined with 50 years life average, or IO per cent installation growth combined with 20 years life average, etc.). The maximum, as well as the rn with which it is associated, can of course be determined analytically.

tion, I936). The symmetrical retirement functions set up by Winfrey are also of the Pearson II type, but with multi-decimal fractional powers (used to obtain as good a fit as possible to the empirical retirement data), which makes them unnecessarily awkward to handle for our purpose.



NOTES AND BOOK REVIEWS NOTES AND BOOK REVIEWS

On the basis of this information we may make the flat statement: If the retirement distribution can be assumed to resemble the curve shown in Chart I, and the trend of installations over a past period of double the average service life of the assets can be roughly represented by a constant rate of growth, then, whatever that growth rate and whatever the life average, the gross survivor value computed by the "crude" method does not deviate by more than 6 per cent from what would be obtained by the correct method.5 In the case of a structurally similar retirement distribution with less relative dispersion the divergence for any given r n, hence also the maximum divergence, would be smaller.6

On the basis of this information we may make the flat statement: If the retirement distribution can be assumed to resemble the curve shown in Chart I, and the trend of installations over a past period of double the average service life of the assets can be roughly represented by a constant rate of growth, then, whatever that growth rate and whatever the life average, the gross survivor value computed by the "crude" method does not deviate by more than 6 per cent from what would be obtained by the correct method.5 In the case of a structurally similar retirement distribution with less relative dispersion the divergence for any given r n, hence also the maximum divergence, would be smaller.6

As far as they go, these findings indicate that the simple cumulation method does yield an ac- ceptably close approximation to the correct result. To secure a fully generalized answer, covering most cases likely to occur in practice, it would be neces- sary (and probably sufficient) to add a similar analysis for the other f (x) and g (t) types listed above. If survival rates based on a skew retirement distribution curve were applied to the g (t)- function underlying the preceding analysis, the divergence between the two S would have different values and different maxima, depending on the degree of skewness and dispersion of f (x). But a maximum divergence for some specifiable rn- product may again be expected to exist for any assumed f (x). If the assumed g (t)-pattern is anything other than the simple exponential growth function we have used, the percentage deviation between the two gross stock values must be expected to depend also on the specific contour of the installation flow during the past period indicated by the range s, hence on the length of that range and thus, in general, on the life average n as well.

As far as they go, these findings indicate that the simple cumulation method does yield an ac- ceptably close approximation to the correct result. To secure a fully generalized answer, covering most cases likely to occur in practice, it would be neces- sary (and probably sufficient) to add a similar analysis for the other f (x) and g (t) types listed above. If survival rates based on a skew retirement distribution curve were applied to the g (t)- function underlying the preceding analysis, the divergence between the two S would have different values and different maxima, depending on the degree of skewness and dispersion of f (x). But a maximum divergence for some specifiable rn- product may again be expected to exist for any assumed f (x). If the assumed g (t)-pattern is anything other than the simple exponential growth function we have used, the percentage deviation between the two gross stock values must be expected to depend also on the specific contour of the installation flow during the past period indicated by the range s, hence on the length of that range and thus, in general, on the life average n as well.

'The margin between 6 per cent and the 5.26 per cent we have derived is certainly sufficient to allow for any possible difference of our result from what would be obtained if annual rather than continuous functions were used.

8In our analysis we have purposely experimented with a retirement distribution curve having a fairly high co- efficient of variation, which of course tends to increase the relative disparity between Si and S2. If retirements are completely concentrated at the average service life, the gross survivor values obtained by the two methods are always equal.

In this as in any similar analysis, minor erratic oscil- lations of actual installations around a generally realistic g (t)-trend, or of actual retirements around a generally

'The margin between 6 per cent and the 5.26 per cent we have derived is certainly sufficient to allow for any possible difference of our result from what would be obtained if annual rather than continuous functions were used.

8In our analysis we have purposely experimented with a retirement distribution curve having a fairly high co- efficient of variation, which of course tends to increase the relative disparity between Si and S2. If retirements are completely concentrated at the average service life, the gross survivor values obtained by the two methods are always equal.

In this as in any similar analysis, minor erratic oscil- lations of actual installations around a generally realistic g (t)-trend, or of actual retirements around a generally

realistic f (x) -curve, will hardly affect the reliability of the results, except perhaps in the case of very short average service lHves.

realistic f (x) -curve, will hardly affect the reliability of the results, except perhaps in the case of very short average service lHves.

GEOGRAPHIC EARNINGS DIFFERENTIALS AND FOREIGN TRADE Philip J. Bourque

GEOGRAPHIC EARNINGS DIFFERENTIALS AND FOREIGN TRADE Philip J. Bourque

In the February I956 issue of this REVIEW 1 Ir- ving Kravis has observed that the export industries tend to pay higher earnings than the industries which produce outputs competitive with imports. The present note proposes to examine further the earnings differentials in the export and import- competing industries. It is argued here that, when geographic earnings differentials in the United States are considered, certain results of the wage comparisons Professor Kravis claims to have estab- lished must be modified.

Kravis' results may be partially summarized as follows: average hourly earnings in 328 manufacturing industries were $I.252 (weighted by man- hours) in I947; average hourly earnings in the 46 leading export industries were $I.32 I (weighted by exports), and in the 36 leading import-competing industries were $I.I43 (weighted by imports).2

Why do average hourly earnings in the export

In the February I956 issue of this REVIEW 1 Ir- ving Kravis has observed that the export industries tend to pay higher earnings than the industries which produce outputs competitive with imports. The present note proposes to examine further the earnings differentials in the export and import- competing industries. It is argued here that, when geographic earnings differentials in the United States are considered, certain results of the wage comparisons Professor Kravis claims to have estab- lished must be modified.

Kravis' results may be partially summarized as follows: average hourly earnings in 328 manufacturing industries were $I.252 (weighted by man- hours) in I947; average hourly earnings in the 46 leading export industries were $I.32 I (weighted by exports), and in the 36 leading import-competing industries were $I.I43 (weighted by imports).2

Why do average hourly earnings in the export

industries tend to be above the national average and those in import-competing industries below? To a considerable extent Professor Kravis attributes the difference to the predominance of durable goods industries in the export group. "It seems fair to conclude," he writes, "that the differences in earn- ing levels between leading export and leading import-competing industries in manufacturing is largely but not entirely the result of the varying mix of durable and nondurable goods industries in each group." 3He apparently derives his conclusion from the following observation: the difference between the export and the import-competing industries in respect to the trade-weighted average hourly earnings is i8 cents for I947; but if the leading export and import-competing industries are segre- gated according to nondurable and durable goods, the corresponding differences in average hourly earnings are ii cents for nondurables and I2 cents for durables. Similar results are found for I952.

However, from these data I would conclude that

industries tend to be above the national average and those in import-competing industries below? To a considerable extent Professor Kravis attributes the difference to the predominance of durable goods industries in the export group. "It seems fair to conclude," he writes, "that the differences in earn- ing levels between leading export and leading import-competing industries in manufacturing is largely but not entirely the result of the varying mix of durable and nondurable goods industries in each group." 3He apparently derives his conclusion from the following observation: the difference between the export and the import-competing industries in respect to the trade-weighted average hourly earnings is i8 cents for I947; but if the leading export and import-competing industries are segre- gated according to nondurable and durable goods, the corresponding differences in average hourly earnings are ii cents for nondurables and I2 cents for durables. Similar results are found for I952.

However, from these data I would conclude that Irving B. Kravis, "Wages and Foreign Trade," this

REvIEw, xxxvi (February I956). 'The 46 leading export industries and 36 leading import-

competing industries are defined by Kravis, ibid.

Irving B. Kravis, "Wages and Foreign Trade," this REvIEw, xxxvi (February I956).

'The 46 leading export industries and 36 leading import- competing industries are defined by Kravis, ibid. 3Ibid., 20. 3Ibid., 20.



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Gross Stocks Estimated from Past Installations