Volumes

• 16.11. A volume index is an average of the proportionate changes in the quantities of a specified set of goods or services between two periods of time. The quantities compared must be homogeneous, while the changes for different goods and services must be weighted by their economic importance as measured by their values in one or other, or both, periods. The concept of a volume index may be illustrated by a simple example. Consider an industry that produces two different models of automobile, one selling for twice the price of the other. From an economic point of view these are two quite different products even though described by the same generic term “automobile”. Suppose that between two periods of time:

  • (a) The price of each model remains constant;

  • (b) The total number of automobiles produced remains constant;

  • (c) The proportion of higher priced models produced increases from 50 per cent to 80 per cent.

  It follows that the total value of the output produced increases by 20 per cent because of the increase in the proportion of higher-priced models. This constitutes a volume increase of 20 per cent. As each higher-priced automobile constitutes twice as much output as each lower-priced automobile, a switch in production from low- to high-priced models increases the volume of output even though the total number of automobiles produced remains unchanged. The fact that the value increase is entirely attributable to an increase in volume also follows from the fact that no price change occurs for either model. The price index must remain constant in these circumstances.

• 16.12. The term “volume increase” is used in preference to “quantity increase” because there is a possible ambiguity about the use of the term “quantity increase”. It is sometimes argued that the situation described in the example is one in which the quantities remain unchanged (because the total number of automobiles remain unchanged), whereas the average quality of the automobiles produced increases (because of the increase in the proportion of higher-priced models). However, such an interpretation is based on a semantic confusion due to the fact that the same generic term, “automobile”, is applied to two products that are actually quite different from an economic point of view. It is not legitimate to add together quantities that are not identical with each other, even though they may be measured in the same kind of physical units. Adding together quite different models of “automobiles” is no more meaningful than adding together tons of different kinds of “foods”—for example, adding tons of rice to tons of apples or beef. In general, it is not possible to decompose a volume change into a quantity change and a change in average quality. The so-called “quantity index” has no meaning from an economic point of view if it involves adding quantities that are not commensurate. For quite different purposes, however, such as loading aircraft, ships or vehicles, adding quantities may provide useful information. Similarly, for purposes of traffic control or pollution, it may be useful to know the increase in the total numbers of vehicles produced or imported, irrespective of their price. However, such measures are not volume measures in an economic sense.

1. Introduction

• 16.14. A price index is an average of the proportionate changes in the prices of a specified set of goods and services between two periods of time. Similarly, a volume index is an average of proportionate changes in the quantities of a specified set of goods and services. As already emphasized, the price and quantity changes refer to individual goods or services as distinct from groups of similar products. Different qualities of the same kind of product must be treated as separate goods or services in this context.

• 16.15. In line with normal conventions, the period that serves as the reference point will be designated as period o and the period which is compared with it designated as period t. The two periods may be consecutive or be separated by intervening periods. The ratio of the price, or quantity, of a specific product in period t to the price, or quantity, of the same product in period o, is described as a price relative, or quantity relative: namely, p_t/p_o or q_t/q_o. Price and quantity relatives are pure numbers that are independent of the units in which the quantities are measured and the prices are quoted. Most index numbers can be expressed as, or derived from, weighted averages of these price or quantity relatives, the various formulas differing from each other mainly in the weights which they attach to the individual price or quantity relatives and the particular form of averages used—arithmetic, geometric, harmonic, etc.

2. Laspeyres and Paasche indices

• 16.16. The two most commonly used indices are the Laspeyres and Paasche indices. Both may be defined as weighted averages of price or quantity relatives, the weights being the values of the individual goods or services in one or other of the two periods being compared.

  • Let v_ij = p_ij q_ij: the value of the ith product in period j

  The Laspeyres price index (Lp) is defined as a weighted arithmetic average of the price relatives using the values of the earlier period o as weights:

  $\begin{array}{cc}\text{Lp}=\underset{\text{i}}{\mathrm{\Sigma }}\frac{v_{\text{io}}·p_{\text{it}}/p_{\text{io}}}{\underset{\text{i}}{\mathrm{\Sigma }}{V}_{\text{io}}}& \left(1\right)\end{array}$
  where the summation takes place over different goods and services. The Laspeyres volume index (Lq) is a similar weighted average of the quantity relatives, that is:

  $\begin{array}{cc}\begin{array}{cc}\text{Lq}=& \underset{\text{i}}{\mathrm{\Sigma }}\end{array}\frac{v_{io}·q_{it}/q_{io}}{\underset{i}{\mathrm{\Sigma }}{V}_{io}}& \left(2\right)\end{array}$
  The period that provides the weights for an index is described as the “base” period. It usually (but not always) coincides with the reference period to which the comparisons relate. As the summation always takes place over the same set of goods and services it is possible to dispense with the subscript / in expressions such as (1) and (2). As V_j is equal to p_jq_j by definition, it is also possible to substitute for V_j: in (1) and (2) to obtain:

  $\begin{array}{cc}\text{Lp}=\frac{{\mathrm{\Sigma }p}_t{q}_o}{\mathrm{\Sigma }{p}_o{q}_o}& \left(3\right)\end{array}$
  and

  $\begin{array}{cc}\text{Lq}=\frac{{\mathrm{\Sigma }p}_o{q}_t}{\mathrm{\Sigma }{p}_o{q}_o}& \left(4\right)\end{array}$

  Expressions (1) and (3) are algebraically identical with each other, as are (2) and (4).

• 16.17. Paasche price and volume indices are defined reciprocally to Laspeyres indices by using the values of the later period t as weights and a harmonic average of the relatives instead of an arithmetic average. A Paasche index (P_p or P_q) is defined as follows:

  $\begin{array}{cc}{\text{P}}_{\text{p}}=\frac{\mathrm{\Sigma }{v}_t}{\mathrm{\Sigma }{v}_t·p_o/p_t}=\frac{\mathrm{\Sigma }{p}_t{q}_t}{\mathrm{\Sigma }{p}_o{q}_t}& \left(5\right)\end{array}$
• and

  $\begin{array}{cc}{\text{P}}_{\text{q}}=\frac{\mathrm{\Sigma }{v}_t}{\mathrm{\Sigma }{v}_t·q_o/q_t}=\frac{\mathrm{\Sigma }{p}_t{q}_t}{\mathrm{\Sigma }{p}_t{q}_o}& \left(6\right)\end{array}$

  When a time series of Paasche indices is compiled, the weights therefore vary from one period to the next.

• 16.18. The Paasche index may also be interpreted as the reciprocal of a “backward looking” Laspeyres: that is, the reciprocal of a “Laspeyres” index for period o that uses period t as the base period. Because of this reciprocity between Laspeyres and Paasche indices there are important symmetries between them. In particular, the product of a Laspeyres price (volume) index and the corresponding Paasche volume (price) index is identical with the proportionate change in the total value of the flow of goods or services in question, that is:

  $\begin{array}{cc}\text{Lp}·\text{Pq =}\frac{\mathrm{\Sigma }{p}_t{q}_o}{\mathrm{\Sigma }{p}_o{q}_o}\frac{\mathrm{\Sigma }{p}_t{q}_t}{\mathrm{\Sigma }{p}_t{q}_o}=\frac{\mathrm{\Sigma }{v}_t}{\mathrm{\Sigma }{v}_o}& \left(7\right)\end{array}$
• and

  $\begin{array}{cc}\text{Lq}·\text{Pp =}\frac{\mathrm{\Sigma }{p}_o{q}_t}{\mathrm{\Sigma }{p}_o{q}_o}\frac{\mathrm{\Sigma }{p}_t{q}_t}{\mathrm{\Sigma }{p}_o{q}_t}=\frac{\mathrm{\Sigma }{v}_t}{\mathrm{\Sigma }{v}_o}& \left(8\right)\end{array}$
  This relationship can be exploited whenever the total values for both periods are known. When both Σv_t and Σv_o are known, one or the other out of a complementary pair of Laspeyres and Paasche indices can be derived indirectly. For example,

  $\begin{array}{cc}\text{Lq=}\frac{\mathrm{\Sigma }{v}_t/\mathrm{\Sigma }{v}_o}{\text{Pp}}& \left(9\right)\end{array}$
• and

  $\begin{array}{cc}\text{P}\text{q =}\frac{\mathrm{\Sigma }{v}_t/\mathrm{\Sigma }{v}_o}{\text{L}\text{p}}& \left(10\right)\end{array}$

  Thus, the Laspeyres volume index can be derived indirectly by dividing the proportionate change in values by the Paasche price index, a procedure described as price deflation. As it is usually easier, and less costly, to calculate direct price than direct volume indices, it is common to obtain volume measures indirectly both in national accounts and economic statistics generally.

3. The relationship between Laspeyres and Paasche indices

• 16.20. Before considering other possible formulas, it is necessary to establish the behaviour of Laspeyres and Paasche indices vis-à-vis each other. In general, a Laspeyres index tends to register a larger increase over time than a Paasche index:

  • that is, in general

    $\begin{array}{cc}\text{both}{\text{L}}_{\text{p}}>{\text{P}}_{\text{p}}\text{and}{\text{L}}_{\text{q}}>{\text{P}}_{\text{q}}& \left(13\right)\end{array}$

  • It can be shown that relationship (13) holds whenever the price and quantity relatives (weighted by values) are negatively correlated. Such negative correlation is to be expected for price takers who react to changes in relative prices by substituting goods and services that have become relatively less expensive for those that have become relatively more expensive. In the vast majority of situations covered by index numbers, the price and quantity relatives turn out to be negatively correlated so that Laspeyres indices tend systematically to record greater increases than Paasche with the gap between them tending to widen with the passage of time.

1. The rebasing and linking of indices

• 16.31. It is convenient to start by considering the example of a time series of Laspeyres volume indices on a fixed base period and its associated series of values at constant prices. In the course of time, the pattern of relative prices in the base period tends to become progressively less relevant to the economic situations of later periods to the point at which it becomes unacceptable to continue using them to measure volume measures from one period to the next. It may then be necessary to update the base period and to link the old series to the series on the new base period.

• 16.32. For a single index taken in isolation linking is a simple arithmetic operation. However, within an accounting framework it is not possible to preserve the accounting relationships between an aggregate and its components while at the same time linking the aggregate and its components separately. The difficulties involved are best explained by referring to the numerical example given in table 16.1.

  Table 16.1.

  The rebasing and linking of volume indices and series at constant prices: a numerical example

  I. The basic data

  

  

  

  

  

  

  

  II. Laspeyres

  III. Linking by scaling down values from periods 10 to 15 by ratio $\mathrm{\Sigma }{p}_0{q}_{10}/\mathrm{\Sigma }{p}_{10}{q}_{10}$ = 116/218

  IV. Linking by scaling up values from periods 0 to 10 by ratio $\mathrm{\Sigma }{p}_{10}{q}_{10}/\mathrm{\Sigma }{p}_0{q}_{10}=218/116$

  V. Linking individual series at prices of period 0

  VI. Linking individual series at prices of period 10

  Table 16.1.

  The rebasing and linking of volume indices and series at constant prices: a numerical example

  I. The basic data

  Product	Period 0			Period 10			Period 15
  	p0	q0	v0	p10	q10	v10	p15	q15	v15
  A	6	5	30	9	12	108	11	15	165
  B	4	8	32	10	11	110	14	11	154
  A+B	-	-	62			218			319

  II. Laspeyres

  	Base year 0		Base year 10
  Product	Period 0	Period 10	Period 10	Period 15
  	p0q0	p0q10	p10q10	p10q15
  A	30	72	108	135
  B	32	44	110	110
  A+B	62	116	218	245
  Index	100	187.1	100	112.4
  Linked index(0)	100	187.1	187.1	210.3
  Linked index (10)	53.4	100	100	112.4

  III. Linking by scaling down values from periods 10 to 15 by ratio $\mathrm{\Sigma }{p}_0{q}_{10}/\mathrm{\Sigma }{p}_{10}{q}_{10}$ = 116/218

  Product	Actual values		Scaled down to 0 price level
  	p0q0	p0q10	p10q10	p10q15
  A	30	72	57.5	71.9
  B	32	44	58.5	58.5
  A + B	62	116	116	130.4
  Linked index	100	187.1	187.1	210.3

  IV. Linking by scaling up values from periods 0 to 10 by ratio $\mathrm{\Sigma }{p}_{10}{q}_{10}/\mathrm{\Sigma }{p}_0{q}_{10}=218/116$

  Product	Scaled up to 10 price level		Actual values
  	p0q0	p0q10	p10q10	p10q15
  A	56.4	135.3	108	135
  B	60.1	82.7	110	110
  A+B	116.5	218	218	245
  Linked index	53.4	100	100	112.4

  V. Linking individual series at prices of period 0

  Product	p0q0	p0q10	p0q15 (linked)
  A	30	72	90
  B	32	44	44
  A+B	62	116	130.4
  Linked index	100	187.1	210.3

  VI. Linking individual series at prices of period 10

  Product	p10q0 (linked)	p10q10	p10q15
  A	45	108	135
  B	80	110	110
  A+B	116.5	218	245
  Linked index	53.4	100	112.4

  View Table

  Table 16.1.

  The rebasing and linking of volume indices and series at constant prices: a numerical example

  I. The basic data

  Product	Period 0			Period 10			Period 15
  	p0	q0	v0	p10	q10	v10	p15	q15	v15
  A	6	5	30	9	12	108	11	15	165
  B	4	8	32	10	11	110	14	11	154
  A+B	-	-	62			218			319

  II. Laspeyres

  	Base year 0		Base year 10
  Product	Period 0	Period 10	Period 10	Period 15
  	p0q0	p0q10	p10q10	p10q15
  A	30	72	108	135
  B	32	44	110	110
  A+B	62	116	218	245
  Index	100	187.1	100	112.4
  Linked index(0)	100	187.1	187.1	210.3
  Linked index (10)	53.4	100	100	112.4

  III. Linking by scaling down values from periods 10 to 15 by ratio $\mathrm{\Sigma }{p}_0{q}_{10}/\mathrm{\Sigma }{p}_{10}{q}_{10}$ = 116/218

  Product	Actual values		Scaled down to 0 price level
  	p0q0	p0q10	p10q10	p10q15
  A	30	72	57.5	71.9
  B	32	44	58.5	58.5
  A + B	62	116	116	130.4
  Linked index	100	187.1	187.1	210.3

  IV. Linking by scaling up values from periods 0 to 10 by ratio $\mathrm{\Sigma }{p}_{10}{q}_{10}/\mathrm{\Sigma }{p}_0{q}_{10}=218/116$

  Product	Scaled up to 10 price level		Actual values
  	p0q0	p0q10	p10q10	p10q15
  A	56.4	135.3	108	135
  B	60.1	82.7	110	110
  A+B	116.5	218	218	245
  Linked index	53.4	100	100	112.4

  V. Linking individual series at prices of period 0

  Product	p0q0	p0q10	p0q15 (linked)
  A	30	72	90
  B	32	44	44
  A+B	62	116	130.4
  Linked index	100	187.1	210.3

  VI. Linking individual series at prices of period 10

  Product	p10q0 (linked)	p10q10	p10q15
  A	45	108	135
  B	80	110	110
  A+B	116.5	218	245
  Linked index	53.4	100	112.4

• 16.33. Part I of the table presents the underlying price, quantity and value data for two products, A and B, and the aggregate (A+B). It is assumed that constant price series are calculated for periods 0 to 10 using period 0 prices, with a change of base year in period 10. Constant price data for periods 10 onwards are calculated at period 10 prices. The resulting constant price data, and the Laspeyres volume indices for the aggregate, are shown in part II of the table. The question to be addressed is the best way to link these two sets of data as a whole.

• 16.34. Assuming it is desired to present a continuous run of “constant price” data from period 0 to period 15, there are several ways in which such data could be compiled. One possibility is to scale down the constant price data from periods 10 to 15 (calculated at period 10 prices) to the general level of prices in period 0 by multiplying through by a constant equal to $\mathrm{\Sigma }{p}_0{q}_{10}/\mathrm{\Sigma }{p}_{10}{q}_{10}$. This ensures that there is no break in continuity for the aggregate when the weights are switched from period 0 prices to period 10 prices. It yields a set of data which, from period 10 onwards, is expressed at the general price level of period 0 but at the relative prices of period 10. This solution is illustrated in part III of the table.

• 16.35. The difficulty with this solution is apparent from the table. In period 10 in which the link occurs two different sets of relative prices have to be used. As a result, discontinuities are introduced into the “constant price” series for A and B at the point at which the switch is made from one set of relative prices to the other. For this reason, the linked measures for A and B do not reflect the underlying volume movements. For example, the ratio of the “constant price” figure for A in period 15 to the corresponding value in period 0, namely 71.9/30 = 2.40, is very different from the actual change in the quantity of A, namely 15/5 = 3.

• 16.36. The same difficulties would arise if the series before the link were to be scaled up to the general price level of period 10. As illustrated in part IV of table 16.1 the constant price data valued at the prices of period 0 can be scaled up to the prices of period 10 by multiplying by the constant $\mathrm{\Sigma }{p}_0{q}_{10}/\mathrm{\Sigma }{p}_0{q}_0$. Discontinuities are again created for A and B at the point at which the switch is made from one set of relative prices to the other. The ratio of the “constant price” for A in period 15 to that in period 0 is 135/56.4 - 2.39 which again is very different from the actual quantity change, 3.

• 16.37. In order to preserve the volume movements at each level of aggregation, components have to be linked as well as the aggregates. This procedure is followed in parts V and VI of table 16.1. In part V the linked values are at the constant prices of period 0 while in part VI they are at the constant prices of period 10. The linked volume movements for Aand B reflect the underlying quantity changes, while the linked volume movements for the aggregate A + B take account of the change in weights in period 10. The problem that emerges with this method is that the constant price values for the components do not add up to the constant price values of the aggregates after the series have been linked. This can be seen in part V for the linked values of period 15 at prices of 0 and in part VI for the linked values of period 0 at the prices of 15. In other words when every series at each level of aggregation is individually linked, the resulting constant price data are not additively consistent after the linking has taken place.

• 16.38. When data are not additively consistent, as in the last column of part V and the first column of part VI, there is a discrepancy between the sum of the components and the corresponding aggregate at each individual level of aggregation. One way of eliminating the discrepancy would be to distribute it in proportion to the components. For example, the figures of 45 and 80 in the first column could be scaled down 42.9 and 74.6 to make them add to 116.5, the required total. However, this would automatically distort the volume comparisons for both A and B in period 0 as compared with periods 10 and 15. Alternatively, the total for (A+B) could be adjusted to make it equal to the sum of the components, i.e., 125 instead of 116.5. By distorting the volume comparisons at the aggregate level, however, this would defeat the main purpose of the exercise.

• 16.39. A choice has to be made between the two different methods illustrated in the table. The first approach, using the scalar adjustment as illustrated in parts III and IV of the table, preserves additive consistency at the expense of distorting the linked comparisons at a detailed level. The second approach, illustrated in parts V and VI, preserves the validity of the linked comparisons at each level of aggregation at the cost of destroying additive consistency. The volume movements for the overall aggregate are the same in both cases. On balance, the second method seems preferable, given that the main purpose is to obtain good price and volume measures.

• 16.40. When the base year is updated for constant price series in national accounts, the problem is how to present data for years prior to the new base year. In practice, the method illustrated in part VI of the table is usually followed which preserves the integrity of the volume movements at each level of aggregation at the cost of destroying additivity for years prior to the new base year. The question of how to deal with the resulting discrepancies is considered further below.

2. Rebasing and linking each period

1. Chain indices for value added and GDP

• 16.63. In order to derive balancing items such as gross value added residually the various elements involved must be additive. Consider the following example:

Let

• O_o = the value of output in period o;

  I_o = the value of intermediate consumption in period o;

  C_t = the chain volume index for output in period t;

  B_t = the chain volume index for intermediate consumption in period t.

  One possibility would be to measure the change in the volume of value added between periods O and t by extrapolating the base period values of both output and intermediate input by the relevant chain indices, as follows:

  $\begin{array}{cc}\frac{O_o{C}_t-I_{\mathrm{o}}{B}_t}{O_o-I_o}& \left(20\right)\end{array}$

  However, an index such as (20) would have no clear meaning because the chain indices C_t and B_t are not additive. In addition, its behaviour could be unpredictable and erratic, especially if the difference between O_o and I_o is small compared with their absolute levels. This method must be rejected on both conceptual and practical grounds.

• 16.64. When chain indices are used for output and intermediate consumption, an additional chain index must be compiled for value added itself. Suppose chain Laspeyres type volume indices are calculated for output and intermediate consumption. ALaspeyres type chain volume index for value added can then be calculated, each link in the chain being defined as follows:

  ${\text{L}}_{\text{q}}^{\text{VA}}\begin{array}{cc}=\frac{{\mathrm{\Sigma }p}_{t-1}{Q}_t-\mathrm{\Sigma }{p}_{t-1}{1q}_t}{{\mathrm{\Sigma }p}_{t-1}{Q}_{t-1}-\mathrm{\Sigma }{p}_{t-1}{q}_{t-1}}& \left(21\right)\end{array}$

  where the capital letters refer to outputs and the small letters to intermediate inputs. The denominator in (21) is value added in period t-1 while the numerator is obtained by revaluing the outputs and inputs in period t at t-1 prices. Expression (21) can be interpreted as measuring the change in value added between t-1 and t at the prices of t-1. As constant prices are used, the resulting measures are additively consistent.

• 16.65. A chain volume index for value added can be compiled in this way using Laspeyres type volume indices for each link in the chain. However, in common with all chain indices, it should be noted that the three indices involved—the output index, the input index and the value added index—are not additively consistent among themselves. This can produce counter intuitive and unacceptable results in the long run. For example, for each individual link in the chain it is impossible for the output index to lie outside the range spanned by the intermediate consumption and value added indices. However, because chain indices are not additively consistent, in the long run the chain index for output may drift outside the range spanned by the other two chain indices. Such cases have been observed and documented.

• 16.66. It is equally possible, of course, to compile a chain volume index for value added using Paasche type volume indices linking successive periods, each link being defined as follows:

  ${\text{P}}_{\text{q}}^{\text{VA}}\begin{array}{cc}=\frac{{\mathrm{\Sigma }p}_t{Q}_t-\mathrm{\Sigma }{p}_t{q}_t}{{\mathrm{\Sigma }p}_t{Q}_{t-1}-\mathrm{\Sigma }{p}_t{q}_{t-1}}& \left(22\right)\end{array}$

  Each link provides an economically meaningful measure of the volume change in value added by using the prices of period t to value output and intermediate consumption in both periods.

• 16.67. A third possibility is to compile a chain volume index for value added that uses a Fisher volume index for each link—i.e., the geometric mean of the Laspeyres and Paasche indices given by (21) and (22). Such an index may provide the best volume measure of value added from a theoretical point of view. However, the chain Laspeyres index should provide a very close approximation to the chain Fisher in situations in which it is too difficult or time consuming to calculate the Fisher.

2. Single indicators

• 16.68. As value added at constant prices is equal to the difference between output at constant prices and intermediate consumption at constant prices it is affected by errors of measurement in both series. Assuming that such errors are at least partly random, the errors will tend to be cumulative, making value added extremely sensitive to error, especially in industries or sectors where value added accounts for only a relatively small proportion of the value of the total output. In some cases, it may be better to abandon the attempt to measure value added as the difference between two series subject to error and to try to estimate the volume movements of value added directly using only one time series—i.e., a “single indicator” instead of double deflation. Although single indicators may be biassed, they are much less sensitive to error. Over the short run the potential bias involved in using single indicators may be negligible compared with the potential errors in the double deflation estimates.

• 16.69. If there are good data on gross value added at current prices, one alternative to double deflation is to deflate current value added directly by a price index for gross output. This procedure can be described as single deflation. It is likely to yield a close approximation to the change in value added at constant prices, at least in the short run. Another possible procedure is to extrapolate value added in the base year by a volume index for output. This method is likely to yield similar results to the first method and can be used when data are not available for value added at current prices. The volume index used to extrapolate base year value added can itself be calculated either directly from quantity data or by deflating the current value of output by an appropriate price index. If the data on output at current prices are comprehensive and reliable, the latter method is likely to yield the better estimates.

• 16.70. The estimation of changes in value added at constant prices by deflating current value added by an output price index or extrapolating base year value added by an output volume index is an acceptable second-best solution when the data available are not sufficiently reliable and robust to permit the use of double deflation. Unfortunately, however, it is sometimes not even possible to obtain satisfactory estimates of price or volume changes for output—for example, in certain market and non-market service industries such as finance, business services, education or defence. In these cases it may be necessary to resort to third-best solutions by estimating movements of value added at constant prices on the basis of the estimated volume changes of the inputs into the industries. The inputs may be total inputs, labour inputs on their own or intermediate inputs on their own. For example, it is not uncommon to find the movement of value added at constant prices estimated by means of changes in compensation of employees at constant wage rates, or even simply by changes in numbers employed, in both market and non-market service industries. Compilers of data may be forced to adopt such expedients, even when there is no good reason to assume that labour productivity remains unchanged in the short- or long-term. Sometimes, volume changes for intermediate inputs may be used: for example, short-term movements of value added at constant prices for the construction industry may be estimated from changes in the volume of building materials consumed—cement, bricks, timber, etc. The use of indicators of this kind may be the only way in which to estimate short-term movements in output or value added, but they are not acceptable over long time periods.

3. GDP volume

• 16.71. Movements in the volume of GDP are always calculated by recalculating the values of the various components of GDP at the constant prices either of the previous year or of some fixed base year. Thus, the volume measure of GDP is frequently referred to as “GDP at constant prices”. When time series are constructed by multiplying the values of the base year by fixed base Laspeyres volume indices, it is appropriate to describe the resulting series as being at the constant prices of the base year. However, when the values of the base year are extrapolated by multiplying them by annual chain volume indices it is no longer strictly correct to describe them in this way. This is reflected by the non-additivity of the resulting data. Nevertheless, the series of values are expressed at the general price level of the base year and it is convenient to continue to describe them as being “at constant prices”. It is preferable to avoid the term “real GDP” as this may suggest the deflation of GDP by some general price not necessarily that of GDP itself.

• 16.72. Changes in the volume of GDP for the total economy may be calculated from the expenditure side from data on final expenditures and imports. The double deflation method used to measure gross value added at the level of an industry or sector may be applied at the level of the total economy by replacing output and intermediate consumption by final expenditures and imports.

• 16.73. The conclusions reached above with regard to the measurement of real value added by industry or sector apply equally at the level of the total economy and may be summarized as follows:

  • (a) The preferred measure of year to year movements of GDP volume is a Fisher volume index; changes over longer periods being obtained by chaining: i.e., by cumulating the year to year movements;

  • (b) The preferred measure of year to year inflation for GDP is, therefore, a Fisher price index; price changes over long periods being obtained by chaining the year to year price movements: the measurement of inflation is accorded equal priority with the volume movements;

  • (c) Chain indices that use Laspeyres volume indices to measure year to year movements in the volume of GDP and Paasche price indices to measure year to year inflation provide acceptable alternatives to Fisher indices;

  • (d) The chain indices for total final expenditures, imports and GDP cannot be additively consistent whichever formula is used, but this need not prevent time series of values being compiled by extrapolating base year values by the appropriate chain indices;

  • (e) Chain indices should only be used to measure year-to-year movements and not quarter to quarter movements.

• 16.74. Two further advantages of using chain indices for GDP may be noted. The quality of the inflation measures is greatly improved compared with the year-to-year movements in the implicit Paasche type deflators calculated on a reference period. A second advantage is that chaining avoids introducing apparent changes in growth or inflation as a result of changing the base year. When the base year for a time series of fixed weight Laspeyres type volume indices is brought forward, the underlying trend rate of growth may appear to slow down if the previous base has become very out of date. This slowing down is difficult to explain to users and may bring the credibility of the measures into question.

4. The publication of alternative volume and price series

• 16.75. Although the preferred measure of real growth and inflation for GDP is a chain Fisher index, or alternatively a chain Laspeyres or Paasche index, it must be recognized that the lack of additive consistency can be a serious disadvantage for many types of analysis in which the interrelationships between various flows in the economy are the main focus of interest. Most macroeconometric models fall into this category. It is therefore recommended that disaggregated constant price data should be compiled and published in addition to the chain indices for the main aggregates. The need to publish two sets of data that may appear to conflict with each other should be readily appreciated by analysts engaged in macroeconometric modelling and forecasting. Users whose interests are confined to a few global measures of real growth and inflation can be advised to utilize the chain indices and ignore the more detailed constant price estimates.

• 16.76. Constant price series have nevertheless to be rebased in the course of time. In general, constant price series should not be allowed to run for more than five, or at the most, ten years without rebasing. It is therefore recommended that disaggregated constant price data should be published for as many of the flows of goods and services in the System as possible, with a change of base year about every five years. When the base year is changed it is customary to link the data on the old base to the data on the new base rather than to carry the rebasing backwards.

• 16.77. In effect, the underlying issue is not whether to chain or not but how often to rebase. Sooner or later the base year for fixed weight Laspeyres volume indices and their associated constant price series has to be updated because the prices of the base year become increasingly irrelevant. When the base year is updated, series on the old base have to be linked to those on the new base. Thus, sooner or later additivity is lost as a result of linking (assuming the rebasing is not carried backwards). Long runs of data, therefore, almost inevitably involve some form of chain indices. Annual chaining is simply the limiting case in which rebasing is carried out each year instead of every five or ten years.

1. Binary comparisons

• 16.80. Price and volume indices may be compiled between pairs of countries using the same kinds of index numbers as those used to measure changes between time periods. A Laspeyres type volume index for country B based on country A may be defined as follows:

  ${}_{\mathrm{a}}{\mathrm{L}}_{\mathrm{b}}^{\mathrm{q}}\begin{array}{cc}=\frac{\mathrm{\Sigma }{v}_a·qb/qa}{\mathrm{\Sigma }{v}_a}=\frac{\mathrm{\Sigma }{p}_{a}qb}{\mathrm{\Sigma }{p}_a{q}_a}& \left(23\right)\end{array}$
  The fact that the prices in the two countries may be expressed in different currency units is immaterial as only the prices and values in country A are used. The exchange rate between the two countries is irrelevant and plays no part in the comparison. Volume indices such as (23) may be calculated for any flow of goods and services but in practice they have been calculated mainly for total final domestic expenditure and its components—household final consumption expenditure, government final consumption expenditure, and gross fixed capital formation. It is, of course, equally possible to utilize the prices and values in country B as weights by defining a Paasche type volume index for B based on A as follows:

  $\begin{array}{cc}{}_{\mathrm{a}}{\mathrm{P}}_{\mathrm{b}}^{\mathrm{q}}=\frac{\mathrm{\Sigma }{v}_{\mathrm{b}}}{\mathrm{\Sigma }{v}_{\mathrm{b}}\cdot q\mathrm{a}/q\mathrm{b}}=\frac{\mathrm{\Sigma }{p}_{\mathrm{b}\mathit{q}\mathrm{b}}}{\mathrm{\Sigma }{p}_{\mathrm{b}q\mathrm{a}}}& \left(24\right)\end{array}$
• 16.81. Given the complementary relationships between Laspeyres and Paasche price and volume indices noted earlier, it follows that the Laspeyres volume index for B based on A can be derived indirectly by deflating the ratio of the values in B and A, each expressed in their own currencies, by the Paasche price index for B based on A. This Paasche index is defined as follows:

  $\begin{array}{cc}{}_{\mathrm{a}}{\mathrm{P}}_{\mathrm{b}}^{\mathrm{q}}=\frac{\mathrm{\Sigma }{v}_{\mathrm{b}}}{\mathrm{\Sigma }{v}_{\mathrm{b}}\cdot p\mathrm{a}/p\mathrm{b}}=\frac{\mathrm{\Sigma }{p}_{\mathrm{b}\mathit{q}\mathrm{b}}}{\mathrm{\Sigma }{p}_{\mathrm{a}q\mathrm{b}}}& \left(25\right)\end{array}$

  The individual price relatives, p_a/p_b, that enter into the calculation of (25) are ratios of the prices of the same products in the two countries. It is much less easy to ensure that the same product is being priced in different countries than to collect prices for the same product in different time periods within the same country. By careful specification and identification of products, however, price relatives can be calculated directly from information collected in price surveys in different countries, but there is little doubt that it is more difficult to compile reliable international than intertemporal price relatives.

2. Multilateral comparisons

• 16.87. The need for multilateral international comparisons may arise for various reasons: for example, when an international organization needs to know the relative sizes of the GDPs of all of its member countries or when aggregates for blocks of countries are needed. Such aggregates can also serve as norms by which the situations of individual member countries can be appraised. In these cases the block constitutes an entity in its own right with its own characteristics. Unique, objective rankings of the volumes of GDP, or per capita GDP, for all the countries in the block require multilateral measures.

1. Quality differences, price variation and price discrimination

• 16.105. In general, most types of goods or services, whether simple food products such as potatoes or high technology products such as computers, are available on the market in many different qualities whose physical characteristics differ from each other. For example, potatoes may be old or new, red or white, washed or unwashed, loose or prepacked, graded or ungraded, etc. Unwashed, loose, old red potatoes are clearly different qualities of potatoes from washed, prepacked, new white potatoes. Consumers recognize and appreciate the differences and are prepared to pay different prices.

• 16.106. The expression “different qualities” is used to cover sets of goods or services whose characteristics are sufficiently different to make them distinguishable from each other from an economic point of view but which are sufficiently similar to each other to be described by the same generic term, such as potato, computer or transportation. Different prices are charged for different qualities of the same kinds of goods or services in much the same way that different prices are charged for goods or services which are generically different from each other and described by different names. Different qualities have to be treated in exactly the same way as different kinds of goods or services.

• 16.107. Differences in quality that are attributable to differences in the physical characteristics of the goods or services concerned are easily recognized, but not all differences in quality are of this kind. There are other factors which can give rise to differences in quality. For example, goods or services delivered in different locations, or at different times, must be treated as different qualities even if they are otherwise physically identical. These differences stem from the fact that the marginal utility of a particular kind of good or service for purchasers or consumers situated in one location may be very different from that for purchasers in other locations, while the costs of delivering goods or services in different locations also vary. Transporting a good to a location in which it is in greater demand is a process of production in its own right in which the good is transformed into a higher quality good.

• 16.108. Similarly, goods or services provided at different times of the day or at different periods of the year must be treated as different qualities even if they are otherwise identical. For example, electricity or transport provided at peak times must be treated as being of higher quality than the same amount of electricity or transport provided at off-peak times. The fact that peaks exist shows that purchasers or users attach greater utility to the services at these times, while the marginal costs of production are usually higher at peak times. The different prices or rates charged at peak and off-peak times provide measures of these differences in quality. Similarly, fruit and vegetables supplied out of season must be treated as higher qualities than the same fruit and vegetables in season which are cheaper to produce and of which consumers may be satiated.

• 16.109. Apart from differences in location or timing there are other factors which may contribute to quality differences. For example, the conditions of sale, or circumstances or environment in which the goods or services are supplied or delivered can make an important contribution to differences in quality. A restaurant meal provided with more attentive service in more luxurious or pleasant surroundings is a higher quality meal than exactly the same food and drink provided with less service in a less pleasant environment. A durable good sold with a guarantee, or free after-sales service, is higher quality than the same good sold without guarantee or service. Thus, the same goods or services sold by different kinds of retailers, such as local shops, specialist shops, department stores or supermarkets may have to be treated as different qualities for these kinds of reasons. Alternatively, the goods may be envisaged as being parts of different composite products which incorporate different amounts, or kinds, of retail services. Purchasers in large supermarkets may have to find, select and transport their own purchases to check-outs, but they are also offered the advantage of greater choice and the opportunity to reduce the amount of time spent shopping by making purchases in bulk. All these kinds of factors introduce qualitative differences.

• 16.110. In general, therefore, it is necessary to pay attention to differences in the situation, or conditions, in which goods and services are supplied, as prima facie these may all be expected to introduce qualitative differences into the goods or services supplied. In economic theory it is generally assumed that whenever a difference in price is found between two goods and services which appear to be physically identical there must be some other factor, such as location, timing, conditions of sale, etc., which is introducing a difference in quality. Otherwise, it can be argued that the difference could not persist, as rational purchasers would always buy lower priced items and no sales would take place at higher prices. In most cases, therefore, differences in prices at the same moment of time must be taken as prima facie evidence that the goods or services concerned represent different qualities of the same general kind of good or service. As explained in the first section of this chapter, this implies that if there is a switch towards higher priced—i.e., higher quality—goods or services, this will be recorded as an increase in volume and not price.

• 16.111. Nevertheless, it must be questioned whether the existence of observed price differences always implies corresponding differences in quality. There are strong assumptions underlying the standard argument which are seldom made explicit and are often not satisfied in practice: for example, that purchasers are well informed and that they are free to choose between goods and services offered at different prices.

• 16.112. First, purchasers may not be properly informed about existing price differences and may therefore inadvertently buy at higher prices. While they may be expected to search out for the lowest prices, costs are incurred in the process. Given the uncertainty and lack of information, the potential costs incurred by searching for outlets in which there is only a possibility that the same goods and services may be sold at lower prices may be greater than the potential savings on expenditures, so that a rational purchaser may be prepared to accept the risk that he or she may not be buying at the lowest price. Situations in which the individual buyers or sellers negotiate, or bargain over prices, provide further examples in which purchasers may inadvertently buy at a higher price than may be found elsewhere. On the other hand, the difference between the average price of a good purchased in a market or bazaar in which individual purchasers bargain over the price and the price of the same good sold in a different type of retail outlet, such as a department store, should normally be treated as reflecting differences in quality attributable to the differing conditions under which the goods are sold.

• 16.113. Secondly, purchasers may not be free to choose the price at which they purchase because the seller may be in a position to charge different prices to different categories of purchasers for identical goods and services sold under exactly the same circumstances—in other words, to practise price discrimination. Economic theory shows that sellers have an incentive to practise price discrimination as it enables them to increase their revenues and profits. However, it is difficult to discriminate when purchasers can retrade amongst themselves; i.e., when purchasers buying at the lowest prices can resell the goods to other purchasers. While most goods can be retraded, it is usually impossible to retrade services, and for this reason price discrimination is extensively practised in industries such as transportation, finance, business services, health, education, etc., in most countries. Lower prices are typically charged to purchasers with low incomes, or low average incomes, such as pensioners or students. When governments practise or encourage the practice of price discrimination it is usually justified on welfare grounds, but market producers also have reasons to discriminate in favour of households with low incomes as this may enable them to increase their profits. Thus, when different prices are charged to different consumers it is essential to establish whether or not there are in fact any quality differences associated with the lower prices. For example, if senior citizens, students or schoolchildren are charged lower fares for travelling on planes, trains or buses, at whatever time they choose to travel, this must be treated as pure price discrimination. However, if they are charged lower fares on condition that they travel only at certain times, typically off-peak times, they are being offered lower quality transportation.

• 16.114. Thirdly, buyers may be unable to buy as much as they would like at a lower price because there is insufficient supply available at that price. This situation typically occurs when there are two parallel markets. There may be a primary, or official, market in which the quantities sold, and the prices at which they are sold are subject to government or official control, while there may be a secondary market—a free market or unofficial market—whose existence may or may not be recognized officially. If the quantities available at the price set in the official market are limited there may be excess demand so that supplies have to be allocated by rationing or some form of queuing. As a result, the price on the secondary or unofficial market will tend to be higher. It is also possible, but less likely, that lower prices are charged on the secondary or unofficial market, perhaps because the payment of taxes on products can be evaded in such a market.

• 16.115. For the three reasons just given, i.e., lack of information, price discrimination or the existence of parallel markets, identical goods or services may sometimes be sold to different purchasers at different prices. Thus, the existence of different prices does not always reflect corresponding differences in the qualities of the goods or services sold.

• 16.116. When there is price variation for the same quality of good or service, the price relatives used for index number calculation should be defined as the ratio of the weighted average price of that good or service in the two periods, the weights being the relative quantities sold at each price. Suppose, for example, that a certain quantity of a particular good or service is sold at a lower price to a particular category of purchaser without any difference whatsoever in the nature of the good or service offered, location, timing or conditions of sale, or other factors. Asubsequent decrease in the proportion sold at the lower price raises the average price paid by purchasers for quantities of a good or service whose quality is the same and remains unchanged, by assumption. It also raises the average price received by the seller without any change in quality. This must be recorded as a price and not a volume increase.

• 16.117. It may be difficult to distinguish genuine price discrimination from situations in which the different prices reflect differences in quality. Nevertheless, there may be situations in which large producers—especially large service producers in fields such as transportation, education or health—are able to make the distinction and provide the necessary information. If there is doubt as to whether the price differences constitute price discrimination, it seems preferable to assume that they reflect quality differences, as they have always been assumed to do so in the past.

2. Changes in quality over time

• 16.118. Goods and services, and the conditions under which they are marketed, are continually changing over time, with some goods or services disappearing from the market and new qualities or new goods or services replacing them. In principle, the price relatives that enter into the calculation of intertemporal price indices should measure pure price changes by comparing the prices of identical goods and services in different time periods. If the qualities of goods or services being compared are not identical, there are effectively four options:

  • (a) To ignore the change in quality and compile a price relative as if no difference in quality existed;

  • (b) To omit the items in question and not compile a price relative for them;

  • (c) To adjust the observed price of the new quality for the change in quality which has taken place;

  • (d) To treat the two qualities as if they were two separate goods and to estimate their prices in the periods in which they are not sold.

• 16.119. The first option should be avoided. Ignoring changes in quality is likely to introduce serious biases of unknown size, or even direction, into the measured price or volume indices. The second option is also not to be recommended. It will also tend to introduce bias. If goods or services are omitted from the calculation of price indices that are intended to be comprehensive in coverage, their omission is equivalent to assuming that their rate of price change is the same as the average for the items covered by the indices. However, items subject to quality change tend to be atypical and unrepresentative, so that assuming that their prices change at the same rate as for goods or services whose characteristics do not change is highly questionable.

• 16.120. The third and fourth options are clearly to be preferred although they are not always easy to implement in practice. Some methods that may be employed are explained in the following paragraphs.

• 16.121. The adjustment for quality change may be based on the relative prices of the two qualities on the market when there is at least one period when both qualities are on sale on the market at the same time. Suppose, for example, that one quality is replaced by another as follows:

  

  Period	Old quality	New quality
  0	po
  t	pt	pt*
  n		pn*

  View Table

  Period	Old quality	New quality
  0	po
  t	pt	pt*
  n		pn*

  In period 0, only the old quality is available while in period n only the new quality is available. If both are available simultaneously in some intervening period t, the ratio of their relative prices in period r, namely p_t/p_t*, may be used as a measure of the relative volumes of the old and the new qualities. Thus, the price relative connecting the new quality in period n with the old quality in period 0, after adjusting for the difference in their qualities, is given by:

  $P_{\mathrm{n}}*\left(P\mathrm{t}/P\mathrm{t}*\right)/p\mathrm{o}$

  This is equivalent to constructing a price relative connecting p_n* with p_o by splicing the price change for the new quality to the price change for the old quality, using period t as the link period: that is,

• 16.122. This procedure may be used to deal with quality changes in all those cases in which the new and the old qualities overlap on the market for a significant amount of time. If the relative prices of the two qualities do not remain constant during this time, some kind of average of their relative prices may be used to estimate their relative qualities.

• 16.123. When the two qualities are not produced and sold on the market at the same time it becomes necessary to resort to indirect methods of quantifying the change in quality between the old and new qualities. Producers of many kinds of durable goods, such as automobiles, deliberately stop producing the old model when the new model is introduced on the market. Such producers typically vary the characteristics of their product—size, performance, style, etc.—in order to stimulate demand without changing the general nature of the product or the purposes for which it is used. In such cases it is necessary to estimate what would be the relative prices of the old and new models, or qualities, if they were produced and sold on the market at the same time and to use the estimated relative prices as measures of the relative qualities.

• 16.124. One possibility is to use the estimated relative costs of production as estimates of their relative prices and hence their relative qualities. It may often be feasible for producers to provide such estimates, The observed change in price between the old and the new qualities may then be adjusted as follows. Suppose the price of the new quality or model is x per cent higher than the price of the old, while it is estimated that the new quality would cost y per cent more to produce than the old, then the price relative connecting the old and new qualities, adjusted for the change in their qualities, is equal to

  $\frac{100+x}{100+y}$

• 16.125. The characteristics of buildings and other structures are so variable that it may be almost impossible to find identical buildings and structures being produced in successive periods of time. In these circumstances, instead of trying to compare the prices of actual buildings or structures in one period with those of other, possibly not very similar, buildings or structures in another period, it may be better to specify a small number of hypothetical and relatively simple standard buildings and structures and to estimate what their prices would be in each of the periods. The specifications of these standard buildings or structures are chosen on the advice of construction experts who are also asked to estimate what their prices would be in each of the periods. Alternatively, the experts can be asked to estimate the costs of construction in both periods, the price relatives being assumed to be equal to the ratios of the construction costs. Construction experts are frequently asked to make such estimates in practice. This method has been used successfully to make international comparisons of the prices of buildings and structures as well as for estimating price changes over time.

1. Compensation of employees

• 16.143. The quantity unit for compensation of employees may be considered to be an hour’s work of a given type and level of skill. As with goods and services, different qualities of work must be recognized and quantity relatives calculated for each separate type of work. The price associated with each type of work is the compensation paid per hour which may vary considerably, of course, between different types of work. A volume measure of work done may be calculated as a weighted average of the quantity relatives for different kinds of work weighted by the values of compensation of employees in the previous year or fixed base year. Alternatively, a “price” index may be calculated for work by calculating a weighted average of the proportionate changes in hourly rates of compensation for different types of work, again using compensation of employees as weights. If a Laspeyres type volume index is calculated indirectly by deflating the changes in compensation of employees at current values by an index of the average change in hourly rates of compensation, the latter should be a Paasche type index.

2. Consumption of fixed capital

• 16.144. Another cost of production and charge against gross value added, namely consumption of fixed capital, may also be measured at the prices of the previous year or some fixed base year. Indeed, when estimates of the gross and net capital stocks are compiled using the perpetual inventory method, an estimate of consumption of fixed capital at the constant prices of some base year is already built into such calculations.

3. Taxes and subsidies on products

• 16.145. It is also possible to factor taxes or subsidies on products into their own price and volume components because the quantity units on the basis of which such taxes or subsidies are payable can also be used as quantity units to define “tax prices”. A tax price is the amount of tax payable per unit of good or service, whether the tax is payable per unit of quantity or ad valorem. The tax prices can be used to construct tax price relatives which can then be averaged to obtain “tax price indices” using the values of the amounts paid in tax as weights. A “tax volume index” can be derived indirectly by deflating changes in the total amounts of taxes paid by the appropriate version of the tax price index. A tax volume index is essentially a volume index for a flow of goods and services in which the relative importance of the goods and services is measured not by their relative market prices but the relative amounts of tax paid per product. It shows by how much the receipts from taxes on products change purely in response to changes in the volumes of goods and services subject to tax. Factoring tax receipts into price and volume components may be useful for purposes of fiscal analysis and may also be useful in the context of the input-output table.

4. Net operating surplus

• 16.146. The net operating surplus is an accounting residual which does not possess quantity and price dimensions of its own. It may also be negative, of course. It is not possible, therefore, to decompose the net operating surplus into its own price and volume components. It is therefore also not possible to calculate the average volume change for all the uses of gross value added shown in the generation of income account.

• 16.147. Thus, the limit to a set of integrated price and volume measures within the accounting framework of the System is effectively reached with the net operating surplus. It is conceptually impossible to factor all the flows in the income accounts of the System, including current transfers, into their own price and volume components. Of course, any income flow can be deflated by a price index for a numeraire set of goods and services to measure the increase or decrease of the purchasing power of the income over the numeraire, but as explained at the beginning of this chapter, this is quite different from decomposing a flow into its own price and volume components.

1. Introduction

• 16.148. It is possible to deflate any income flow in the accounts, and even a balancing item such as saving, by a price index in order to measure the purchasing power of the item in question over a designated numeraire set of goods and services. By comparing the deflated value of the income with the actual value of the income in the base year, it is possible to determine by how much the real purchasing power of the income has increased or decreased. Income deflated in this way is generally described as “real income”.

• 16.149. In interpreting real incomes two points need to be borne in mind:

  • (a) Real incomes are measured with reference to the price level in some selected reference year. Real values cannot exist in isolation: they vary depending upon the choice of reference year;

  • (b) Real incomes measure changes in purchasing power over some selected numeraire; they thus also depend upon the choice of numeraire.

• 16.150. As there may often be no obvious, or uncontroversial choice of numeraire there has always been some reluctance to show real incomes in national accounts on the grounds that the choice of numeraire should be left to the user of the statistics and not the compiler. However, when major changes in prices occur, it can be argued that compilers of statistics are under an obligation to present at least some measures of real income. Not all users of the accounts have the opportunity, inclination or expertise to calculate the real incomes which may be most suited to their needs. Moreover, there is a demand from many users for multi-purpose measures of real income, at least at the level of the economy as a whole and the purpose of this section is to indicate how such measures may be compiled.

2. Trading gains and losses from changes in the terms of trade

• 16.151. Before considering the various different real income aggregates which may be defined for the total economy, it is necessary to explain the fundamental difference between gross domestic product at constant prices, and real gross domestic income, or real GDI. GDP at constant prices is essentially an output volume measure. It may be calculated at the level of the total economy by a form of double deflation in which imports valued at the basic prices of the previous year, or some fixed base year, are subtracted from total final expenditures valued at the purchasers’ prices of the previous year, or the fixed base year. A measure of this kind is essentially a volume measure of the output from domestic production, even though it includes taxes on imports and possibly some other taxes on products that are excluded from the gross values added of resident producers.

• 16.152. However, the total real income which residents derive from domestic production depends also on the rate at which exports may be traded against imports from the rest of the world. If the prices of a country’s exports rise faster (or fall more slowly) than the prices of its imports—i.e., if its terms of trade improve—less exports are needed to pay for a given volume of imports so that at a given level of domestic production goods and services can be reallocated from exports to consumption or capital formation. Thus, an improvement in the terms of trade makes it possible for an increased volume of goods and services to be purchased by residents out of the incomes generated by a given level of domestic production. Real GDI measures the purchasing power of the total incomes generated by domestic production so that when the terms of trade change there may be a significant divergence between the movements of GDP at constant prices and real GDI. The difference between the change in GDP at constant prices and real GDI is generally described as the “trading gain” (or loss). The differences between movements in GDP at constant prices and real GDI are not always small. If imports and exports are large relative to GDP, and if the commodity composition of the goods and services which make up imports and exports are very different, the scope for potential trading gains and losses may be large. This may happen, for example, when the exports of a country consist mainly of a small number of primary products, such as cocoa, sugar or oil, while its imports consist mainly of manufactured products. It can easily be shown that trading gains or losses, T, are measured by the following expression:

  $\begin{array}{cc}T=\frac{X-M}{P}=(\frac{X}{P_x}-\frac{M}{P_m})& \left(30\right)\end{array}$

  where

  • X = exports at current prices

    M = imports at current prices

    P_x = the price index for exports

    P_m = the price index for imports

    P = a price index based on some selected numeraire.

  P_x, P_m and P all equal to 1 in the base year. It can be seen that the term in brackets measures the trade balance calculated at the export and import prices of the reference year whereas the first term measures the actual current trade balance deflated by the numeraire price index. It is perfectly possible for one to have a different sign from the other.

• 16.153. There is one important choice to be made in the measurement of trading gains or losses, i.e., the selection of the price index P with which to deflate the current trade balance. There is a large but inconclusive literature on this topic, but one point on which there is general agreement is that the choice of P can sometimes make a substantial difference to the results. Thus, the measurement of real GDI can sometimes be sensitive to the choice of P and this has prevented a consensus being reached on this issue.

• 16.154. It is not necessary to try to summarize here all the various arguments in favour of one deflator rather than another, but it is useful to indicate what are the main alternatives which have been advocated for P. They can be grouped into three classes, as follows:

  • (a) One possibility is to deflate the current balance, X-M, either by the import price index (which has been strongly advocated) or by the export price index, with some authorities arguing that the choice between P_m and P_x should depend on whether the current trade balance is negative or positive;

  • (b) The second possibility is to deflate the current balance by an average of P_m and P_x: various different kinds of averages have been suggested—simple arithmetic or harmonic averages, or more complex trade weighted averages;

  • (c) The third possibility is to deflate the current balance by some general price index not derived from foreign trade: for example, the price index for gross domestic final expenditure, or the consumer price index.

• 16.155. The failure to agree on a single deflator reflects the fact that no one deflator is optimal in all circumstances. The choice of deflator may depend on factors such as whether the current balance of trade is in surplus or deficit, the size of imports and exports in relation to GDP, etc. On the other hand, there is general agreement that it is highly desirable, and for some countries vitally important, to calculate the trading gains and losses resulting from changes in the terms of trade. In order to resolve this deadlock it is recommended to proceed as follows:

  • (a) Trading gains or losses, as defined in equation (30) above, should be treated as an integral part of the System;

  • (b) The choice of appropriate deflator for the current trade balances should be left to the statistical authorities in a country, taking account of the particular circumstances of that country;

  • (c) If the statistical authorities within a country are uncertain what is the most appropriate general deflator P to be used, some average of the import and export price indices should be used, the simplest and most transparent average being an unweighted arithmetic average of the import and export price indices. (This is referred to in the specialist literature on the subject as the Geary method.)

• 16.156. These proposals are intended to ensure that the failure to agree on a common deflator does not prevent aggregate real income measures from being calculated. Some measure of the trading gain should always be calculated even if the same type of deflator is not employed by all countries. In those circumstances in which there is uncertainty about the choice of deflator an average of the import and the export price indices is likely to provide a suitable deflator.

3. The interrelationship between volume measures and real income aggregates

• 16.157. Assuming that measures of trading gains or losses are available, various different real income aggregates may be identified within the System. The links between them are displayed in the following list:

  • (a) Gross domestic product at constant prices: i.e., GDP in the current year valued at the prices, or price level, of the base year obtained by extrapolating (i.e., multiplying) the value of GDP in the base year by the volume index for GDP, whether a fixed weight or a chain index;

  • plus the trading gain or loss resulting from changes in the terms of trade:

  • (b) equals: real gross domestic income;

  • plus real primary incomes receivable from abroad

  • minus real primary incomes payable abroad;

  • (c) equals: real gross national income;

  • plus real current transfers receivable from abroad

  • minus real current transfers payable abroad;

  • (d) equals: real gross national disposable income;

  • minus consumption of fixed capital at constant prices;

  • (e) equals: real net national disposable income.

• 16.158. In the above accounting scheme, each aggregate is derived sequentially starting from gross domestic product at constant prices, the volume measure of GDP. The transition from (a) to (b) above has been explained in the previous section. The steps needed in order to move from (b) to (d) above involve the deflation of flows between resident and non-resident institutional units, namely, primary incomes and current transfers received from abroad and paid to abroad. There may be no automatic choice of price deflator, but it is recommended that the purchasing power of these flows should be expressed in terms of a broadly based numeraire, namely the set of goods and services that make up gross domestic final expenditure. In other words, primary incomes and current transfers should both be deflated by a price index for gross domestic final expenditures. This price index should, of course, be defined consistently with the volume and price indices for GDP.

• 16.159. A possible alternative which could be considered is to use the following accounting framework:

  • (a) Gross domestic product at constant prices as defined above

    plus imports at constant prices

    minus exports at constant prices;

  • (b) equals: gross domestic final expenditures at constant prices, i.e., final consumption expenditure plus gross capital formation at constant prices

    less consumption of fixed capital at constant prices;

  • (c) equals: net domestic final expenditures at constant prices

    plus net current receipts from abroad measured in real terms (i.e., deflated by the price index for net domestic final expenditure), namely:

  • (i) Deflated value of the current trade balance (exports minus imports);

  • plus (ii) Deflated values of primary incomes receivable minus primary incomes payable abroad;

    plus (iii) Deflated value of current transfers receivable minus current transfers payable abroad;

  • (d) equals: real net national disposable income.

• 16.160. In this framework, all current flows to and from abroad are deflated by a single deflator, that for item (c), i.e., net domestic final expenditures. In effect, this index acts as a general price deflator for all items from row (c) onwards. If there are no satisfactory estimates available for consumption of fixed capital at constant prices so that all expenditure measures have to remain gross, the price index for gross domestic final expenditures may be used instead.

• 16.161. This alternative framework has the advantage that the various components of real income are all measured with reference to a single numeraire, the set of goods making up net domestic final expenditures. It is easier, therefore, to grasp the significance of real net national disposable income as its deflator is explicit. The alternative framework measures the trading gain or loss by using the deflator for net domestic final expenditures as the general deflator P, whereas it can be argued that P ought always to be based on flows which enter into foreign trade. There is, for example, a possibility that the implicit trading gain may be negative when there is a positive change in the terms of trade, or vice versa, and this may be considered a serious disadvantage. On balance, therefore, the original framework presented above is to be preferred. However, the guidelines laid down by the System need to be somewhat more tolerant in the domain of real income measurement than elsewhere, and the use of alternative deflators, for what may be considered to be good statistical or economic reasons, is by no means incompatible with the basic philosophy underlying the System, which in certain other fields also (e.g., sub-sectoring the households sector) needs to be implemented flexibly and not rigidly.