20. Exports and Imports from Production and Expenditure Approaches and Associated Price Indices Using a Simplified Example and an Artificial Data Set

Abstract

20.1 Chapters 16 to 18 outlined alternative price index number formulas, the factors that determine the nature and extent of differences between their results, and the criteria for choosing among them. The criteria for choosing among the formulas included the fixed-basket, axiomatic, stochastic, and economic theoretic approaches. The first purpose of this chapter is to give the reader some idea of how much the major indices defined in the previous chapters differ using an artificial data set consisting of prices and quantities for six commodities over five periods. The period can be thought of as somewhere between a year and five years. The trends in the data are generally more pronounced than one would see in the course of a year. The six commodities can be thought of as the deliveries to the domestic final demand sector of all industries in the economy.

A. Introduction

20.1 Chapters 16 to 18 outlined alternative price index number formulas, the factors that determine the nature and extent of differences between their results, and the criteria for choosing among them. The criteria for choosing among the formulas included the fixed-basket, axiomatic, stochastic, and economic theoretic approaches. The first purpose of this chapter is to give the reader some idea of how much the major indices defined in the previous chapters differ using an artificial data set consisting of prices and quantities for six commodities over five periods. The period can be thought of as somewhere between a year and five years. The trends in the data are generally more pronounced than one would see in the course of a year. The six commodities can be thought of as the deliveries to the domestic final demand sector of all industries in the economy.

20.2 Chapter 15 showed how the nominal values of, and thus price indices for, exports and imports fit into the 2008 System of National Accounts (2008 SNA).1 Particular emphasis was given to the role of price indices as deflators for estimating volume changes in GDP by the expenditure approach. The second purpose of this chapter is to outline how price indices for exports and imports can be defined and reconciled from the expenditure and production approaches to estimating GDP. Indeed, the illustrative data used to outline and demonstrate differences in the results from different index number formulas are applied not only to export and import price indices (XMPIs) but also to price indices for the constituent aggregates of GDP from both the expenditure and production approaches.

20.3 There is a clear relationship in the 2008 SNA between GDP estimates from these two approaches that derives from the well-known identity between the sources and uses of goods and services as depicted in the 2008 SNA’s goods and services account. On the left-hand side of the account the total amount of resources available to the domestic economy consists of the sum of outputs and imports and this is equal, on the right-hand side, to the total amount used for consumption, investment, and exports; that is,

$\begin{array}{cc}O+M+\left(t-s\right)=IC+C+I+G+X,& \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.1\right)\end{array}$

where O is the value of output of goods and services, M is the value of imports of goods and services, IC is the value of goods and services used in the production process (intermediate consumption), C is final consumption expenditure of households and nonprofit institutions serving households (NPISHs),2 I is gross capital formation, G is final consumption expenditure of government, X is the value of exports of goods and services, and t and s are taxes and subsidies on products. Goods and services emanate from their original producers, either resident producers or producers abroad, for use by either resident users or users abroad.

20.4 Moving intermediate consumption from the right-hand side of the account to the left, as a negative resource, while moving imports from the left to the right as a negative use, results in both sides now summing to GDP. The left-hand side presents the production approach and the right-hand side presents the expenditure approach.

$\begin{array}{cc}O-IC+\left(t-s\right)=C+I+G+\left(X-M\right).& \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.2\right)\end{array}$

20.5 Exports and imports are explicitly identified in the expenditure approach, but this is not the case in the production approach. The production account in the 2008 SNA does not break O and IC down into output to the domestic market and the rest of the world.

20.6 GDP estimated from the production approach is based on the value added to the value of goods and services used in the production process (intermediate consumption), IC, to generate the value of output, O. GDP can be thought of as being equal to the sum of the value added produced by all institutional units resident in the domestic economy. The output is valued at basic prices to exclude taxes and subsidies, t and s, on products, while intermediate consumption and all other aggregates in the above equations are valued at purchasers’ prices to include them. Taxes less subsidies on products need to be added back to value added to ensure that the values of what are supplied and used are equal. GDP is defined from the production approach on the right-hand side of equation (20.2), therefore, as the sum of value added by resident producers plus the value of taxes less subsidies on products.

20.7 The expenditure approach involves summing the values of final consumption and gross capital formation (i.e., gross fixed capital formation, changes in inventories, and net acquisition of valuables3). These final expenditures do not properly represent all domestic economic activity because they exclude that directed to nonresidents, that is, exports, and include that arising from of nonresidents, that is, imports: Exports and imports are respectively added to and subtracted from final consumption expenditure and capital formation on the right-hand side of equation (20.2) to estimate GDP.

20.8 It was noted above that a second purpose of this chapter is to outline how price indices for exports and imports can be defined and reconciled from the expenditure and production approaches to estimating GDP. Further, the illustrative data used in this chapter to outline and demonstrate differences in the results from different formulas are applied not only to export and import price indices, but also to the constituent aggregates of GDP from both expenditure and production approaches, as indicated in equations (20.1) and (20.2). The representation of the two approaches in equation (20.2) is simplistic because the aggregates are not broken down by commodity detail. Chapter 6 of the 2008 SNA provides details of the production account but does not elaborate on which industries are actually using the imports or on which industries are actually doing the exporting by commodity. Table 14.12 in the 2008 SNA is an illustration of the supply and use tables and includes detailed information on output and intermediate consumption. Although Table 14.15 provides details on the amount of imports going to intermediate consumption, final consumption, and capital formation, there is no similar analysis of the amount of output going to exports. Hence, the main additions to the 2008 SNA Chapters 6 and 14 for XMPI Manual purposes are to add tables to the main production accounts that provide industry by commodity detail on exports and imports. With these additional tables on the industry by commodity allocation of exports and imports, the resident’s approach to collecting export and import price indices can be embedded in the SNA framework.

20.9 The focus of Section B is to outline an expanded production account that includes its constituent commodity detail, in Section B.1, and, in Section B.2 expanded input output tables and a reconciliation of the expanded production and expenditure GDP estimates, with the effects of taxes and subsidies included in Section B.3. The account given is of a simplified economy for illustration and Chapter 15 of this Manual sets out the framework more formally.

20.10 Section C provides illustrative data for the framework in Section B. The data are used not only to illustrate how XMPIs differ according to the index number formula used, but to embed the price index numbers for exports and imports into an illustrative framework for the deflation of the constituent aggregates of GDP and GDP as a whole from both the expenditure and production frameworks.

20.11 Illustrative data on domestic final demand deliveries are provided for a model of production. There are three industries in the economy and in principle, each industry could produce and use combinations of the six final demand commodities plus an additional imported “pure” intermediate input that is not delivered to the domestic final demand sector. In Section C, the basic industry data are listed in the input output framework that was explained in Section B; that is, there are separate supply and use matrices for domestically produced and used commodities and for internationally traded commodities.

20.12 To summarize: Price and quantity data for three industrial sectors of the economy are presented in Section C. This industrial data set is consistent with the domestic final demand data set outlined in Section D. A wide variety of indices are computed in Section D using this final demand data set.

20.13 Section E constructs domestic gross output, export, domestic intermediate input, and import price indices for the aggregate production sector. Only the Laspeyres, Paasche, Fisher, and Törnqvist fixed-base and chained formulas are considered in Section E and subsequent sections because these are the formulas that are likely to be used in practice. The data used in Sections E, F, and G are at producer prices; this means that basic prices are used for domestic outputs and exports and purchasers’ prices are used for imports and domestic intermediate inputs.

20.14 In Sections F.1 through F.3, value-added price deflators are constructed for each of the three industries. A national value-added deflator is constructed in Section F.4.

20.15 Section G compares alternative two-stage methods for constructing the national value-added deflator. This deflator can be constructed in a single stage by aggregating the detailed industry data (and this is done in Section F.4) or it can be constructed in two stages by either aggregating the three industry value-added deflators (see Section F.1) or aggregating the gross output, export, intermediate input, and import price indices that were constructed in Section F (see Section G.2). These two-stage national value-added deflators are compared with each other and their single-stage counterpart.

20.16 Finally, in Section H, final demand purchasers’ prices are used in order to construct domestic final demand price indices (Section H.1), export price indices (Section H.2), and import price indices (Section H.3). In Section H.4, national GDP price deflators are constructed using final demand prices. Finally, in Section H.5, the national value-added deflator, which is constructed using producer prices, is compared to the national GDP deflator, which is constructed using final demand prices. This section also shows how these two national deflators can be reconciled with each other, provided that detailed industry by commodity data on commodity taxes and subsidies are available.

B. Expanded Production Accounts for the Treatment of International Trade Flows

B.1 Introduction

20.17 In order to set the stage for the economic approaches to the XMPIs from the resident’s perspective, it is necessary to provide a set of satellite accounts for the production accounts in the 2008 SNA. It turns out that the 2008 SNA treatment of the production accounts is not able to provide an adequate framework for introducing a producer-based economic theory of the XMPIs that would be analogous to the economic producer price indices (PPIs) that were introduced in the PPI Manual (ILO and others, 2004b).

20.18 There is an extensive national income accounting literature on how to measure the effects of changes in the terms of trade (the export price index divided by the import price index) on national welfare.4 However, Kohli (1978 and 1991)5 observed that most international trade flows through the production sector of the economy and hence a natural starting point, useful for the illustrative needs of this chapter, for developing XMPIs is to embed exports and imports in the production accounts of an economy.

20.19 There are two main differences between the production accounts that are introduced in this chapter and the production accounts that are described in the 2008 SNA:

• The commodity classification is expanded to distinguish between domestically used and produced goods and services and internationally traded goods and services that flow through the production sector.

• The single supply of products and single use of products matrices (the supply and use matrices) that appear in the 2008 SNA6 are in principle replaced by a series of supply and use matrices so that the bilateral transactions of each industry with each one of the remaining industries can be distinguished.7

There is also some discussion of the role of transport in the input output tables because imports and exports of goods necessarily involve some use of transportation services.

B.2 Expanded input output accounts with no commodity taxation

20.20 In this section, a set of production accounts is developed for the production sector of an economy that engages in international trade. In order to simplify the notation, there are only three industries and three commodities in the commodity classification. Industry G (the goods producing industry) produces a composite good (commodity G), industry S produces a composite service that excludes transportation services (commodity S), and industry T provides transportation services (commodity T). In addition to trading goods and services between themselves, the three industries also engage in transactions with two final demand sectors:

• Sector F, the domestic final demand sector and

• Sector R, the rest of the world sector.

20.21 The three industries deliver goods and services to the domestic final demand sector F.8 They also deliver goods and services to the rest of the world sector R9 and they utilize deliveries from the rest of the world sector as inputs into their production processes.10

20.22 The structure of the flows of goods and services between the three production sectors and the two final demand sectors is shown by four value flow matrices in Tables 20.1 through 20.4.

Table 20.1.

Domestic Supply Matrix in Current Period Values

20.23 Table 20.1 shows the value of the gross output deliveries to the domestic final demand sector F as well as the deliveries of each industry to the remaining two industries: It is the domestic supply matrix or domestic gross output by industry and commodity matrix for a particular period of time. The industry G, S, and T columns list the sales of goods and services to all domestic demanders for each of the three commodities.

20.24 The value sum in row and column $G,{p}_{G}^{GS}{y}_{G}^{GS}+{p}_{G}^{GT}{y}_{G}^{GT}+{p}_{G}^{GF}{y}_{G}^{GF}$, corresponds to the revenues received by the goods producing sector from its sales of good G to the service sector, ${p}_{G}^{GS}{y}_{G}^{GS}$, where ${p}_{G}^{GS}$ is the price of sales of good G to sector S and ${y}_{G}^{GS}$ is the corresponding quantity sold,11 plus the revenues received by the goods producing sector from its sales of good G to the transportation sector, ${p}_{G}^{GT}{y}_{G}^{GT}$, where ${p}_{G}^{GT}$ is the price of sales of good G to sector T and ${y}_{G}^{GT}$ is the corresponding quantity sold, plus the revenues received by the goods producing sector from its sales of good G to the domestic final demand sector, ${p}_{G}^{GF}{y}_{G}^{GF}$, where ${p}_{G}^{GF}$ is the price of sales of good G to sector F and ${y}_{G}^{GF}$ is the corresponding quantity sold. Similarly, the value sum in row and column S, ${p}_{S}^{SG}{y}_{S}^{SG}+{p}_{S}^{ST}{y}_{S}^{ST}+{p}_{S}^{SF}{y}_{S}^{SF}$, corresponds to the revenues received by the service sector from its sales of service S to the goods producing sector, the transportation sector, and the domestic final demand sector. Finally, the value sum in row and column T, ${p}_{T}^{TG}{y}_{T}^{TG}+{p}_{T}^{TS}{y}_{T}^{TS}+{p}_{T}^{TF}{y}_{T}^{TF}$, corresponds to the revenues received by the transportation sector from its sales of transportation services T to the goods producing sector, the general services sector, and the domestic final demand sector. It should be mentioned that these transportation prices are margin-type prices; that is, they are the prices for delivering goods from one point to another.12 Note also that ${p}_{G}^{GS}$ will usually not equal ${p}_{G}^{GT}$ or ${p}_{G}^{GF}$; that is, for a variety of reasons, the average selling price of the domestic good to the three sectors that demand the good will usually be different and a similar comment applies to the other commodity prices.13 Unfortunately, this means that we cannot use a common price for a commodity across sectors to deflate the value flows in 2008 SNA Tables A1 and A2 into volume quantity flows by commodity; that is, basic prices that are constant across sectors will usually not exist. This is another reason why it is useful to extend the 2008 SNA production accounts.

20.25 Table 20.2 shows the value of the purchases of intermediate inputs for each industry from domestic suppliers; it is the domestic use matrix or domestic intermediate input by industry and commodity matrix. Note that the value of purchases of goods from industry G by industry S, ${p}_{G}^{GS}{y}_{G}^{GS}$, is exactly equal to the value of sales of goods by industry G to industry S and this value appeared in the value of sales of goods G by industry G in Table 20.1. In fact, all of the domestic purchases of intermediate inputs listed in Table 20.2 have their domestic sales counterpart entries in Table 20.1.

Table 20.2.

Domestic Use Matrix in Current Period Values

20.26 Table 20.3 shows the value of the gross output deliveries to the rest of the world (ROW) final demand sector R; it is the ROW supply matrix or, more simply, the export by industry and commodity matrix.

Table 20.3.

Export or ROW Supply Matrix in Current Period Values

Note: ROW denotes rest of the world.

20.27 The value sum in row and column G, ${p}_{Gx}^{GR}{x}_{G}^{GR}$, corresponds to the revenues received by the goods producing sector from its sales of good G to the ROW sector, where ${p}_{Gx}^{GR}$ is the price of sales of good G to sector R and ${x}_{G}^{GR}$ is the corresponding quantity sold, or more simply, it is the value of exports by the goods producing sector to the rest of the world.14 Similarly, ${p}_{Sx}^{SR}{x}_{S}^{SR}$ is the value of exports of services produced by the services sector and ${p}_{Tx}^{TR}{x}_{T}^{TR}$ is the value of exports of transportation services produced by the transportation sector. The assumption is made here as a simplification that transportation services are separately invoiced and thus not included in the basic price of the good.15Note that not all of these transportation sector export revenues need be associated with the importation of goods into the domestic economy: Some portion of these revenues may be due to the shipment of goods between two or more foreign countries.

20.28 Table 20.4 shows the value of the purchases of intermediate inputs or imports from the rest of the world for each industry by commodity; it is the import or ROW use matrix or ROW intermediate input by industry and commodity matrix.

Table 20.4.

Import or ROW Use Matrix in Current Period Values

20.29 The value of imports in row and column G, ${p}_{Gm}^{GR}{m}_{G}^{GR}$, corresponds to the payments to the rest of the world by the goods producing sector for its imports of goods, where ${p}_{Gm}^{GR}$ is the price of imports of good G to industry G and ${m}_{G}^{GR}$ is the corresponding quantity purchased, or more simply, it is the cost of imports of goods to the goods producing sector. Similarly, ${p}_{Sm}^{GR}{m}_{S}^{GR}$ is the value of imported services that are used in the goods producing sector and ${p}_{Tm}^{GR}{m}_{T}^{GR}$ is the value of imported transportation services that are used in the goods producing sector. Note that industry G may purchase and separately invoice transportation services from domestic or foreign suppliers and a similar comment applies to the purchases of transportation services by industries S and T. The imported value flows for industries S and T are similar to the corresponding import values for the goods producing industry.

20.30 The above four matrices are in terms of current-period values. The corresponding constant period values or volume matrices can readily be derived from the matrices listed in Tables 20.1 through 20.4: Simply drop all of the prices from the above matrices and the resulting matrices, which will have only quantities as entries in each cell, will be the corresponding constant dollar input output matrices. However, note that unless all prices are identical for each entry in each cell of a row, the correct volume entries will not be obtained in general by deflating each row of each matrix by a common price deflator. This observation means that statistical agencies that use the common deflator method to obtain volume input output tables from corresponding nominal input output tables may be introducing substantial errors into their estimates of volume value added by sector. In principle, each cell in a nominal use or make matrix will require a separate deflator in order to recover the corresponding correct volume entry.

20.31 The nominal value flow matrices defined by Tables 20.1 through 20.4 and their volume counterparts can be used to derive the traditional supply and use matrices that appear in Table 14.12 of the 2008 SNA: The conventional supply matrix is the sum of the matrices in Tables 20.1 and 20.3 (the domestic and ROW supply matrices) and the conventional use matrix is the sum of the matrices in Tables 20.2 and 20.4 (the domestic and ROW use matrices). As outlined in Chapters 4 and 15 of this Manual and, in more detail, Chapter 14 of the 2008 SNA, the derivation of supply and use tables in volume terms at the product group level provides a framework that not only facilitates the application of appropriate deflators to the product groups comprising exports and imports, but also enables a reconciliation of the deflators and volume estimates used across all supply and use aggregates at a product group level.

20.32 The matrix that is needed for XMPIs in this illustration can be obtained by adding entries in Tables 20.1 and 20.3 and then subtracting the corresponding entries in Tables 20.2 and 20.4 in order to obtain a net supply matrix that gives the value of net commodity supply by commodity and by industry of origin. The net supply matrix can be aggregated in two ways:

• By summing over columns along each row; the resulting value aggregates are net supplies by commodity, which are equal to domestic final demands plus exports less imports (net final demands by commodity), or

• By summing over rows down each column; the resulting value aggregates are equal to value added by industry.

20.33 It will be useful to list the aggregates that result by implementing the above two methods of aggregation using the entries in Tables 20.1 through 20.4. The three commodity final demand aggregates turn out to be the following value aggregates:16

$\begin{array}{cc}v{f}_{G}\equiv {p}_{G}^{GF}{y}_{G}^{GF}+{P}_{Gx}^{GR}{x}_{G}^{GR}-{p}_{Gm}^{GR}{m}_{G}^{GR}\hfill & \\ \phantom{\rule[-0.0ex]{2.5em}{0.0ex}}-{p}_{Gm}^{SR}{m}_{G}^{SR}-{p}_{Gm}^{TR}{m}_{G}^{TR};\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.3\right)\end{array}$
$\begin{array}{cc}v{f}_{S}\equiv {p}_{S}^{SF}{y}_{S}^{SF}+{P}_{Sx}^{SR}{x}_{S}^{SR}-{p}_{Sm}^{SR}{m}_{S}^{SR}\hfill & \\ \phantom{\rule[-0.0ex]{2.5em}{0.0ex}}-{p}_{Sm}^{SR}{m}_{S}^{SR}-{p}_{Sm}^{TR}{m}_{S}^{TR};\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.4\right)\end{array}$
$\begin{array}{cc}v{f}_{T}\equiv {p}_{T}^{TF}{y}_{T}^{TF}+{P}_{Tx}^{TR}{x}_{T}^{TR}-{p}_{Tm}^{GR}{m}_{T}^{GR}\hfill & \\ \phantom{\rule[-0.0ex]{2.5em}{0.0ex}}-{p}_{Tm}^{SR}{m}_{T}^{SR}-{p}_{Tm}^{TR}{m}_{T}^{TR}.\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.5\right)\end{array}$

20.34 The three industry value-added aggregates are defined as follows:

$\begin{array}{cc}V{a}^{G}\equiv {p}_{G}^{GS}{y}_{G}^{GS}+{p}_{G}^{GT}{y}_{G}^{GT}+{p}_{G}^{GF}{y}_{G}^{GF}-{p}_{S}^{SG}{y}_{S}^{SG}\hfill & \\ \phantom{\rule[-0.0ex]{3.0em}{0.0ex}}-{p}_{T}^{TG}{y}_{T}^{TG}+{p}_{Gx}^{GR}{x}_{G}^{GR}-{p}_{Gm}^{GR}{m}_{G}^{GR}\hfill & \\ \phantom{\rule[-0.0ex]{3.0em}{0.0ex}}-{p}_{Sm}^{GR}{m}_{S}^{GR}-{p}_{Tm}^{GR}{m}_{T}^{GR};\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\begin{array}{c}\left(20.6\right)\end{array}\end{array}$
$\begin{array}{cc}V{a}^{S}\equiv {p}_{S}^{SG}{y}_{S}^{SG}+{p}_{S}^{ST}{y}_{S}^{ST}+{p}_{S}^{SF}{y}_{S}^{SF}-{p}_{G}^{GS}{y}_{G}^{GS}\hfill & \\ \phantom{\rule[-0.0ex]{3.0em}{0.0ex}}-{p}_{T}^{TS}{y}_{T}^{TS}+{p}_{Sx}^{SR}{x}_{S}^{SR}-{p}_{Gm}^{SR}{m}_{G}^{SR}\hfill & \\ \phantom{\rule[-0.0ex]{3.0em}{0.0ex}}-{p}_{Sm}^{SR}{m}_{S}^{SR}-{p}_{Tm}^{SR}{m}_{T}^{SR};\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.7\right)\end{array}$
$\begin{array}{cc}V{a}^{T}\equiv {p}_{T}^{TG}{y}_{T}^{TG}+{p}_{T}^{TS}{y}_{T}^{TS}+{p}_{T}^{TF}{y}_{T}^{TF}-{p}_{G}^{GT}{y}_{G}^{GT}\hfill & \\ \phantom{\rule[-0.0ex]{3.0em}{0.0ex}}-{p}_{S}^{ST}{y}_{S}^{ST}+{p}_{Tx}^{TR}{x}_{T}^{TR}-{p}_{Gm}^{TR}{m}_{G}^{TR}\hfill & \\ \phantom{\rule[-0.0ex]{3.0em}{0.0ex}}-{p}_{Sm}^{TR}{m}_{S}^{TR}-{p}_{Tm}^{TR}{m}_{T}^{TR}.\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.8\right)\end{array}$

20.35 Note that each commodity final demand value aggregate, vfG, vfS, and vfT, is equal to the value of industry deliveries of each of the three commodities plus export deliveries less imports of the commodity to each of the three industrial sectors. Note also that it is not in general appropriate to set the price of, say, vfG equal to the value of vfG divided by the corresponding net deliveries of commodity G to final demand, ${y}_{G}^{GS}+{y}_{G}^{GT}+{y}_{S}^{SG}-{y}_{T}^{TG}+{x}_{G}^{GR}-{m}_{G}^{GR}-{m}_{G}^{GR}-{m}_{S}^{GR}-{m}_{T}^{GR}$, because differences in the prices that are attached to these quantities imply that there are implicit quality differences between these quantities. Thus index number theory should be used to aggregate the value flows on the right-hand sides of equations (20.6) through (20.8). It is clear that index number theory must be used to construct a price and quantity for each of the value-added aggregates, vaG, vaS, and vaT, because by inspecting (20.6) through (20.8), it can be seen that each value-added aggregate is a sum over heterogeneous commodities, some with positive signs associated with their quantities (these are the gross outputs produced by the industry) and some with negative signs (these are the foreign-sourced and domestic intermediate inputs used by the industry).

20.36 The three final demand value aggregates defined by equations (20.6) through (20.8) can be summed and the resulting value aggregate is the GDP generated by the economy’s production sector. Alternatively, the value-added aggregates defined by (20.6) through (20.8) can also be summed and this sum will also equal GDP because these two methods of aggregation are simply alternative methods for summing over the elements of the net supply matrix. Thus the following equation must hold:

$\begin{array}{cc}GDP\equiv v{f}_{G}+v{f}_{S}+v{f}_{T}=v{a}^{G}+v{a}^{S}+v{a}^{T}.& \phantom{\rule[-0.0ex]{8.0em}{0.0ex}}\left(20.9\right)\end{array}\phantom{\rule{0ex}{0ex}}$

20.37 It is useful to use equation (20.9), which defines GDP as the sum of the value of final demands, and substitute equations (20.6) through (20.8) into this definition in order to obtain the following expression for GDP after some rearrangement of terms:

$\begin{array}{cc}\mathrm{GDP}& =v{f}_{G}+v{f}_{S}+V{f}_{T}\hfill \\ & =\left[{p}_{G}^{GF}{y}_{G}^{GF}+{p}_{S}^{SF}{y}_{S}^{SF}+{p}_{T}^{TF}{y}_{T}^{TF}\right]+\left[{p}_{Gx}^{GR}{x}_{G}^{GR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{p}_{Sx}^{SR}{x}_{S}^{SR}+{p}_{Tx}^{TR}{x}_{T}^{TR}\right]-\left[{p}_{Gm}^{GR}{m}_{G}^{GR}+{p}_{Gm}^{SR}{m}_{G}^{SR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{p}_{Gm}^{TR}{m}_{G}^{TR}+{p}_{Sm}^{GR}{m}_{S}^{GR}+{p}_{Sm}^{SR}{m}_{Sm}^{SR}{m}_{s}^{TR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{p}_{Tm}^{GR}{m}_{T}^{GR}+{p}_{Tm}^{SR}{m}_{T}^{SR}+{p}_{Tm}^{TR}{m}_{T}^{TR}\right]\hfill \\ & =\left[C+I+G\right]+\left[X-M\right].\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{5.5em}{0.0ex}}\left(20.10\right)\hfill \end{array}$

20.38 Note that the value aggregate, ${p}_{G}^{GF}{y}_{G}^{GF}+{p}_{S}^{SF}{y}_{S}^{SF}+{p}_{T}^{TF}{y}_{T}^{TF}$, corresponds to the value of domestic final demand, the value aggregate ${p}_{Gx}^{GR}{x}_{G}^{GR}+{p}_{Sx}^{SR}{x}_{S}^{SR}+{p}_{Tx}^{TR}{x}_{T}^{TR}$ corresponds to the value of exports, and the value aggregate ${P}_{Gm}^{GR}{m}_{G}^{GR}+{p}_{Gm}^{SR}{m}_{G}^{SR}+{p}_{Gm}^{TR}{m}_{G}^{TR}+{p}_{Sm}^{GR}{m}_{S}^{GR}+{p}_{Sm}^{SR}{m}_{S}^{SR}+{p}_{Sm}^{TR}{m}_{S}^{TR}+{p}_{Tm}^{GR}{m}_{T}^{GR}+{p}_{Tm}^{SR}{m}_{T}^{SR}+{p}_{Tm}^{TR}{m}_{T}^{TR}$ corresponds to the value of imports. Thus definition (20.10) corresponds to the traditional final demand definition of GDP.17

20.39 Equation (20.9) shows that there are two alternative ways that data on transactions between the domestic production sector and the rest of the world could be captured:

• In the final demand method, information on the price and quantity for each category of import (export) would be obtained from the foreign supplier (demander). This is the nonresident point of view.

• In the value-added method, information on the price and quantity of each type of import used by each industry and the price and quantity of each type of export produced by each industry would be obtained from the domestic producer. This is the resident point of view.

20.40 It is apparent that the practical compilation of trade price indices can be facilitated by developing the existing PPI methodology:18 The PPI methodology can be adapted to the XMPI case to expand the commodity classification in order to make the distinction between a domestically sourced intermediate input and a foreign import and make the distinction between an output that is delivered to a domestic demander versus an output that is delivered to a foreign demander, which is an export. Of course, in practice, it may be difficult to make these distinctions. But distinct advantages of building on existing PPI computer routines and data collection and verification methods exist though there the need will remain to extend the sample of establishments and commodities to be representative of buyers and sellers from/to domestic and foreign markets.

20.41 At this point, it is useful to consider alternative methods for constructing volume measures for GDP originating in the domestic production sector. Thus suppose that data on production sector transactions are available for periods 0 and 1 and that price and quantity information is available for these two periods so that the data in Tables 20.1 through 20.4 are available and hence net supply matrices for the production sector can be calculated for periods 0 and 1. It can be seen that there are three ways that a volume or quantity index of net outputs for the production sector of economy could be calculated:

• Change the signs of the nonzero entries in the domestic use matrix defined by Table 20.2 and change signs of the nonzero entries in the ROW use matrix defined by Table 20.4. Look at the nonzero cells in these two three-by-three matrices as well as the cells in the supply matrices defined in Tables 20.1 and 20.3. Collecting all of these nonzero transactions, one can see that there are 27 distinct price times quantity transactions. If there is a negative sign associated with any one of these terms, that negative sign is attached to the quantity. Now apply normal index number theory to these 27 price times quantity components of the aggregate.

• Sum up the value-added aggregates defined by (20.6) through (20.8). The resulting value-added aggregate will have 27 separate price times quantity components. If a value component has a negative sign associated with it, then attach the negative sign to the quantity (so that all prices will always be positive). Now apply normal price index number formulas theory to these 27 price times quantity components of the aggregate.

• Sum up the final demand value aggregates defined by (20.3) through (20.5). The resulting value of final demand aggregate will have 15 separate price times quantity components. If a value component has a negative sign associated with it, then attach the negative sign to the quantity. Now apply normal index number theory to these 15 price times quantity components of the aggregate.

20.42 It is evident that the quantity index or the volume estimate for GDP will be the same using methods 1 and 2 listed above because the two methods generate exactly the same set of 27 separate price times quantity components in the value aggregate. However, it is not evident that volume estimates for GDP based on method 3 will coincide with those generated using methods 1 and 2 because there are 27 price times quantity components to be aggregated when we use methods 1 or 2 compared to only 15 components when we use method 3.

20.43 Denote the 27 dimensional p (price) and q (quantity) vectors that correspond to the first detailed cell and value-added methods for aggregating over commodities listed above as pva and qva respectively and denote the 15 dimensional p and q vectors that correspond to the third aggregation method over final demand components as pfd and qfd respectively.19 Add a superscript t to denote these vectors evaluated at the data pertaining to period t. Then using equation (20.9), we can see that the inner products of each of these period t price and quantity vectors are equal in the same period because they are each equal to period t nominal GDP:20

$\begin{array}{cc}{P}^{vat}\text{}\cdot \text{}{q}^{vat}={p}^{fdt}\text{}\cdot \text{}{q}^{f\mathit{dt}};\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}t=0,1.& \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.11\right)\end{array}$

20.44 What is not immediately obvious is that the inner products of the two sets of price and quantity vectors are also equal if the price vectors are evaluated at the prices of one period and the corresponding quantity vectors are evaluated at the quantities of another period; that is, for periods 0 and 1, the following equalities hold:21

$\begin{array}{cc}{P}^{va1}\cdot {q}^{va0}={p}^{fd1}\cdot {q}^{fd0};& \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.12\right)\end{array}$
${P}^{va0}\cdot {q}^{va1}={p}^{fd0}\cdot {q}^{fd1}.\phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.13\right)$

20.45 Laspeyres and Paasche quantity indices that compare the quantities of period 1 to those of period 0 can be defined as follows:

$\begin{array}{cc}{{Q}_{L}}^{va}\left({p}^{va0},{p}^{va1},{q}^{va0},{q}^{va1}\right)\equiv {p}^{va0}\cdot {q}^{va1}/{p}^{va0}\cdot {q}^{va0};\hfill & \\ {{Q}_{L}}^{fd}\left({p}^{fd0}\cdot {p}^{fd1},{q}^{fd0},{q}^{fd1}\right)\equiv {p}^{fd0}\cdot {q}^{fd1}/{p}^{fd0}\cdot {q}^{fd0};\hfill & \\ & \left(20.14\right)\end{array}$
$\begin{array}{ll}{{Q}_{p}}^{va}\left({p}^{va0},{p}^{va1},{q}^{va0},{q}^{va1}\right)\equiv {p}^{va1}\cdot {q}^{va1}/{p}^{va1}\cdot {q}^{va0};\hfill & \\ {{Q}_{p}}^{fd}\left({P}^{fd0},{p}^{fd1},{q}^{fd0},{q}^{fd1}\right)\equiv {p}^{fd1}\cdot q{z}^{fd1}/{p}^{fd1}\cdot {q}^{fd0}.\hfill & \\ \phantom{\rule[-0.0ex]{23.0em}{0.0ex}}\left(20.15\right)\hfill & \hfill \end{array}$

20.46 Using equations (20.11) and (20.13), and definitions (20.15), we can see that the two Laspeyres volume indices are equal:

$\begin{array}{c}{Q}_{L}^{va}\left({p}^{va0},{P}^{va1},{q}^{va0},{q}^{va1}\right)={Q}_{L}^{fd}\left({P}^{fd0},{p}^{fd1},{q}^{fd0},{q}^{fd1}\right).\\ \hfill \left(20.16\right)\end{array}$

20.47 Using equations (20.11) and (20.12) and definitions (20.16), we can see that the two Paasche volume indices are equal:

$\begin{array}{c}{Q}_{p}^{va}\left({p}^{va0},{P}^{va1},{q}^{va0},{q}^{va1}\right)={Q}_{p}^{fd}\left({P}^{fd0},{p}^{fd1},{q}^{fd0},{q}^{fd1}\right).\\ \hfill \left(20.17\right)\end{array}$

20.48 Because a Fisher ideal quantity index is the square root of the product of a Laspeyres and Paasche quantity index, it can be seen that equations (20.16) and (20.17) imply that all three Fisher quantity indices—constructed by aggregating over input output net supply table cells, by aggregating over industry value-added components (which is equivalent to aggregating over net supply table cells), or by aggregating over final demand components—are equal; that is, we have

$\begin{array}{r}{Q}_{F}^{va}\left({p}^{va0},{P}^{va1},{q}^{va0},{q}^{va1}\right)={Q}_{F}^{fd}\left({p}^{fd0},{p}^{fd1},{q}^{fd0},{q}^{fd1}\right).\\ \left(20.18\right)\end{array}$

20.49 The equality between the two methods for constructing volume estimates that is reflected in equations (20.16) through (20.18) could provide a potentially useful check on a statistical agency’s methods for constructing aggregate volume GDP measures.

20.50 The above results extend to more complex input output frameworks provided that all transactions between each pair of sectors in the model are accounted for in the model.

20.51 The equality (20.18) between the two methods for constructing an aggregate volume index for GDP using the Fisher quantity index as the index number formula can be extended to the case where the implicit Törnqvist quantity index is used as the index number formula. In this case, the value aggregates are deflated by the Törnqvist price index, and by writing out the formulae, it is straightforward to show that ${p}_{T}^{va}\left({p}^{va0},{p}^{va1},{q}^{va0},{q}^{va1}\right)$, the Törnqvist price index using the 27 price times quantity components in the value-added aggregate, is equal to ${p}_{T}^{fd}\left({p}^{fd0},{p}^{fd1},{q}^{fd0},{q}^{fd1}\right)$, the Törnqvist price index using the 15 price times quantity components in the final demand aggregate.22

20.52 It is well known that the Laspeyres and Paasche quantity indices are consistent in aggregation. Thus if Laspeyres indices of volume estimates of value added by industry are constructed in the first stage of aggregation and the resulting industry prices and quantities are used as inputs into a second stage of Laspeyres aggregation, then the resulting two-stage Laspeyres quantity index is equal to the corresponding single-stage index, ${Q}_{L}^{va}\left({p}^{va0},{p}^{va1},{q}^{va0},{q}^{va1}\right)$. Similarly, if Paasche volume indices of value added by industry are constructed in the first stage of aggregation and the resulting industry prices and quantities are used as inputs into a second stage of Paasche aggregation, then the resulting two-stage Paasche quantity index is equal to the corresponding single-stage index, ${Q}_{p}^{va}\left({p}^{va0},{p}^{va1},{q}^{va0},{q}^{va1}\right)$.23 Unfortunately, the corresponding result does not hold for the Fisher index. However, the two-stage Fisher quantity index usually will be quite close to the corresponding single-stage index, ${Q}_{F}^{va}\left({p}^{va0},{p}^{va1},{q}^{va0},{q}^{va1}\right)$.24 In the following section, commodity taxes are introduced into the supply and use matrices.

B.3 Input output accounts with commodity taxation and subsidization

20.53 Consider again the production model that corresponds to Tables 20.1 through 20.4 in the previous section but now assume that there is the possibility of a commodity tax (or subsidies) falling on the output of each industry and on the intermediate inputs used by each industry. Assume that the producing industry collects these commodity taxes and remits them to the appropriate level of government. These indirect commodity taxes will be introduced into each of the tables listed in the previous section. The counterpart to Table 20.1 is now Table 20.5.

Table 20.5.

Domestic Supply Matrix in Current Period Values with Commodity Taxes

20.54 The quantity of goods delivered to the service sector is ${y}_{G}^{GS}$ as before and the service sector pays industry G the price ${p}_{G}^{GS}$ for each unit of G that was delivered. However, industry G must remit the per unit25 commodity tax26 ${t}_{G}^{GS}$ of the per unit revenue ${p}_{G}^{GS}$ to the government sector and so industry G receives only the revenue ${p}_{G}^{GS}-{t}_{G}^{GS}$ for each unit of good G sold to industry S. The interpretation of the other prices and commodity taxes that occur in Table 20.5 is similar.

20.55 The domestic use matrix in current-period values is still defined by the entries in Table 20.2. This matrix remains unchanged with the introduction of commodity taxes and subsidies. This is because the domestic taxes and subsidies are assumed to be on the output of the producer. Had they been paid by the domestic purchaser on intermediate consumption they would appear here as part of the purchase price.

20.56 The ROW supply matrix or export by industry and commodity matrix defined earlier by Table 20.3 is now replaced by Table 20.6.

Table 20.6.

Export or ROW Supply Matrix in Current Period Values with Export Taxes

20.57 To interpret the entries in Table 20.6, consider the entries for commodity G and industry G. Industry G still gets the revenue ${p}_{Gx}^{GR}{x}_{G}^{GR}$ for its deliveries of goods to foreign purchasers from these purchases but if the government sector imposes a specific export tax equal to ${t}_{Gx}^{GR}$ per unit of exports, then industry G gets to keep only the amount ${p}_{Gx}^{GR}-{t}_{Gx}^{GR}$ per unit sale instead of the full final demander price ${p}_{Gx}^{GR}$. If, however, ${t}_{Gx}^{GR}$ is negative, then the government is subsidizing the export of goods and hence the subsidized price that the producer faces, ${p}_{Gx}^{GR}-{t}_{Gx}^{GR}$, is actually higher than the final demander price ${p}_{Gx}^{GR}$. The interpretation of the industry S and commodity S and industry T and commodity T entries are similar.

20.58 The ROW use matrix or the import matrix by industry and commodity defined by Table 20.4 in the previous section is now replaced by Table 20.7.

Table 20.7.

Import or ROW Use Matrix in Current Period Values with Import Taxes

20.59 As in Table 20.4, industry G imports ${m}_{G}^{GR}$ units of goods from foreign suppliers and pays these foreign suppliers the amount ${p}_{Gm}^{GR}{m}_{G}^{GR}$. However, if ${t}_{Gm}^{GR}$ is positive (the usual case), then the government imposes a specific set of tariffs and indirect taxes on each unit imported equal to ${t}_{Gm}^{GR}$ and hence industry G faces the higher price ${p}_{Gm}^{GR}+{t}_{Gm}^{GR}$ for each unit of good G that is imported.27 The interpretations of the industry S and commodity S and industry T and commodity T entries are similar.

20.60 The volume industry supply and use matrices that correspond to the nominal supply matrices defined by Tables 20.5 and 20.6 and nominal use matrices defined by Tables 20.2 and 20.7 can be obtained from their nominal counterparts after deleting all of the price and tax terms. For completeness, these volume supply and use matrices are listed below. These volume allocation of resources matrices apply to both the with and without commodity tax situations.

Table 20.8.

Constant Dollar Domestic Supply Matrix

Table 20.9.

Volume Domestic Use Matrix

Table 20.10.

Volume ROW Supply or Export by Industry and Commodity Matrix

Table 20.11.

Volume ROW Use or Import by Industry and Commodity Matrix

20.61 If we compare the volume allocation of resources matrices defined by Tables 20.8 through 20.11 with their monetary value at producer price counterparts, we can again see that it will generally be impossible to recover the true volume or quantity measures along any row by deflating the nominal values by a single price index for that commodity class; that is, price deflators that are common across industry will generally not exist. Thus the price statistician’s task is a rather daunting one: Appropriate specific price deflators or volume extrapolators will in principle be required for each nonzero cell in the system of nominal value input output matrices in order to recover the correct volume measures.28

20.62 As was shown in the previous section, the production sector’s nominal value net supply matrix that gives the value of net commodity supply by commodity and by industry of origin at the prices that producers face can be obtained by adding entries in Tables 20.5 and 20.6 and then subtracting corresponding entries in Tables 20.2 and 20.7. This new net supply matrix gives the value of net commodity supply by commodity and by industry of origin at prices that producers face.

20.63 As shown in the previous section, the net supply matrix can be aggregated by summing over columns along each row (the resulting value aggregates are the values of net supply by commodity at producer prices) or by summing over rows down each column (the resulting value aggregates are equal to value added by industry at producer prices).

20.64 The three value of commodity net supply aggregates at producer prices including taxes and subsidies on output (the counterparts to the aggregates defined by equations (20.6) through (20.8)) turn out to be the following value aggregates:

$\begin{array}{}\\ \\ \\ \end{array}\begin{array}{cc}v{f}_{G}& \equiv {p}_{G}^{GF}{y}_{G}^{GF}+{p}_{Gx}^{GR}{x}_{G}^{GR}-{p}_{Gm}^{GR}{m}_{G}^{GR}-{p}_{Gm}^{SR}{m}_{G}^{SR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}-{p}_{Gm}^{TR}{m}_{G}^{TR}-\left[{t}_{G}^{GS}{y}_{S}^{GS}+{t}_{G}^{GT}{y}_{G}^{GT}+{t}_{G}^{GF}{y}_{G}^{GF}\\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{Gx}^{GR}{x}_{G}^{GR}+{t}_{Gm}^{GR}{m}_{G}^{GR}+{t}_{Gm}^{SR}{m}_{G}^{SR}+{t}_{Gm}^{TR}{m}_{G}^{TR}\right];\\ & \phantom{\rule[-0.0ex]{10.0em}{0.0ex}}\phantom{\rule[-0.0ex]{10.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}\left(20.19\right)\hfill \end{array}$
$\begin{array}{cc}v{f}_{S}& \equiv {p}_{S}^{SF}{y}_{S}^{SF}+{p}_{Sx}^{SR}{x}_{S}^{SR}-{p}_{Sm}^{GR}{m}_{S}^{GR}-{p}_{Sm}^{SR}{m}_{S}^{SR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}-{p}_{Sm}^{TR}{m}_{S}^{TR}-\left[{t}_{S}^{SG}{y}_{S}^{SG}+{t}_{S}^{ST}{y}_{S}^{ST}+{t}_{S}^{SF}{y}_{S}^{SF}\\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{Sx}^{SR}{x}_{S}^{SR}+{t}_{Sm}^{GR}{m}_{S}^{GR}+{t}_{G}^{SR}{m}_{S}^{SR}+{t}_{Sm}^{TR}{m}_{S}^{TR}\right];\\ & \phantom{\rule[-0.0ex]{15.0em}{0.0ex}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\phantom{\rule[-0.0ex]{3.0em}{0.0ex}}\left(20.20\right)\end{array}$
$\begin{array}{cc}v{f}_{T}& \equiv {p}_{T}^{TF}{y}_{T}^{TF}+{p}_{Tx}^{TR}{x}_{S}^{TR}-{p}_{Tm}^{GR}{m}_{T}^{GR}-{p}_{Tm}^{SR}{m}_{T}^{SR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}-{p}_{Tm}^{TR}{m}_{T}^{TR}-\left[p{t}_{T}^{TG}{y}_{T}^{TG}+{t}_{T}^{TS}{y}_{T}^{TS}+{t}_{T}^{TF}{y}_{T}^{TF}\\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{Tx}^{TR}{x}_{T}^{TR}+{t}_{Tm}^{GR}{m}_{T}^{GR}+{t}_{Tm}^{SR}{m}_{T}^{SR}+{t}_{Tm}^{TR}{m}_{T}^{TR}\right].\\ & \phantom{\rule[-0.0ex]{15.0em}{0.0ex}}\phantom{\rule[-0.0ex]{3.0em}{0.0ex}}\left(20.21\right)\end{array}$

20.65 Looking at equation (20.19), it can be seen that the net value of production of good G at producer prices is equal to ${p}_{G}^{GF}{y}_{G}^{GF}+{p}_{Gx}^{GR}{x}_{G}^{GR}-{p}_{Gm}^{GR}{m}_{G}^{GR}-{p}_{Gm}^{SR}{m}_{G}^{SR}-{p}_{Gm}^{TR}{m}_{G}^{TR}$, which is the net value of production of commodity G, delivered to the domestic final demand and ROW sectors, at final demand prices, less a term in square brackets that represents the net revenue (commodity tax revenue less subsidies for commodity G) that the government sector collects by taxing (or subsidizing) transactions that involve commodity G. The interpretations for vfS and vfT are similar.

20.66 The three industry value-added aggregates at producer prices turn out to be the following value aggregates:

$\begin{array}{cc}v{a}^{G}& \equiv \left({p}_{G}^{GS}-{t}_{G}^{GS}\right){y}_{G}^{GS}+\left({p}_{G}^{GT}-{t}_{G}^{GT}\right){y}_{G}^{GT}\hfill \\ & \phantom{\rule[-0.0ex]{1.3em}{0.0ex}}+\left({p}_{G}^{GF}-{t}_{G}^{GF}\right){y}_{G}^{GF}-{p}_{S}^{SG}{y}_{S}^{SG}-{p}_{T}^{TG}{y}_{T}^{TG}\hfill \\ & \phantom{\rule[-0.0ex]{1.3em}{0.0ex}}+\left({p}_{Gx}^{GR}-{t}_{Gx}^{GR}\right){x}_{G}^{GR}-{p}_{Gm}^{GR}+{t}_{Gm}^{GR}\right){m}_{G}^{GR}\hfill \\ & \phantom{\rule[-0.0ex]{1.3em}{0.0ex}}-\left({p}_{Sm}^{GR}+{t}_{Sm}^{GR}\right){m}_{S}^{GR}-\left({p}_{Tm}^{GR}+{t}_{Tm}^{GR}\right){m}_{T}^{GR};\hfill \\ & \phantom{\rule[-0.0ex]{10.0em}{0.0ex}}\phantom{\rule[-0.0ex]{3.0em}{0.0ex}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\left(20.22\right)\end{array}$
$\begin{array}{cc}v{a}^{S}& \equiv \left({p}_{S}^{SG}-{t}_{S}^{SG}\right){y}_{S}^{SG}+\left({p}_{S}^{ST}-{t}_{S}^{ST}\right){y}_{S}^{ST}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+\left({p}_{S}^{SF}-{t}_{S}^{SF}\right){y}_{S}^{SF}-{p}_{G}^{GS}{y}_{S}^{SG}-{p}_{T}^{TG}{y}_{T}^{TS}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+\left({p}_{Sx}^{SR}-{t}_{Sx}^{SR}\right){x}_{S}^{SR}-\left({p}_{Gm}^{SR}+{t}_{Gm}^{GR}\right){m}_{G}^{SR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}-\left({p}_{Sm}^{SR}+{t}_{Sm}^{SR}\right){m}_{S}^{SR}-\left({p}_{Tm}^{SR}+{t}_{Tm}^{SR}\right){m}_{T}^{SR};\hfill \\ & \phantom{\rule[-0.0ex]{8.0em}{0.0ex}}\phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\left(20.23\right)\end{array}$
$\begin{array}{cc}v{a}^{T}& \equiv \left({p}_{T}^{TG}-{t}_{T}^{TG}\right){y}_{T}^{TG}+\left({p}_{T}^{TS}-{t}_{T}^{TS}\right){y}_{T}^{TS}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+\left({p}_{T}^{TF}-{t}_{T}^{TF}\right){y}_{T}^{TF}-{p}_{G}^{GT}{y}_{G}^{GT}-{p}_{S}^{ST}{y}_{S}^{ST}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+\left({p}_{Tx}^{TR}-{t}_{Tx}^{TR}\right){x}_{T}^{TR}-\left({p}_{Gm}^{TR}+{t}_{Gm}^{TR}\right){m}_{G}^{TR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}-\left({p}_{Sm}^{TR}+{t}_{Sm}^{TR}\right){m}_{S}^{TR}-\left({p}_{Tm}^{TR}+{t}_{Tm}^{TR}\right){m}_{T}^{TR}.\hfill \\ & \phantom{\rule[-0.0ex]{18.0em}{0.0ex}}\left(20.24\right)\end{array}$

20.67 Looking at equation (20.22), it can be seen that the value added produced by industry G at producer prices, vaG, is equal to the value of deliveries of good G to industry S, $\left({p}_{G}^{GS}-{t}_{G}^{GS}\right){y}_{G}^{GS}$,29 plus the value of deliveries of good G to industry T, $\left({p}_{G}^{GT}-{t}_{G}^{GT}\right){y}_{G}^{GT}$, plus the value of deliveries of finished goods, less payments to industry S for service intermediate inputs, $-{p}_{S}^{SG}{y}_{S}^{SG}$, less payments to industry T for transportation service intermediate inputs, $-{p}_{T}^{TG}{y}_{T}^{TG}$, plus the value of exports delivered to the ROW sector, $\left({p}_{Gx}^{GR}-{t}_{Gx}^{GR}\right){x}_{G}^{GR}$,30 less payments to the ROW for imports of goods G used by industry G, $-\left({p}_{Gm}^{GR}+{t}_{Gm}^{GR}\right){m}_{G}^{GR}$, less payments to the ROW for imports of services S used by industry G, $-\left({p}_{Sm}^{GR}+{t}_{Sm}^{GR}\right){m}_{S}^{GR}$, less payments to the ROW for imports of transportation services T used by industry G, $-\left({p}_{Tm}^{GR}+{t}_{Tm}^{GR}\right){m}_{T}^{GR}$. The decompositions for the value added produced by industries S and T, vaS and vaT, are similar.

20.68 Looking at equations (20.19) through (20.21), we can see that it is natural to ignore the commodity tax transactions and to sum the remaining transactions involving exports into an aggregate that is the value of exports at final demand prices, ${p}_{Gx}^{GR}{x}_{G}^{GR}+{p}_{Sx}^{SR}{x}_{S}^{SR}+{p}_{Tx}^{TR}{x}_{S}^{TR}$. It is this value aggregate that is equal to the value of X in GDP, valued at final demand prices. However, looking at the industry value-added aggregates defined by equations (20.22) through (20.24), we can see that it is natural to work with the net revenues received by the industries for their exports, which are $\left({p}_{Gx}^{GR}-{t}_{Gx}^{GR}\right){x}_{G}^{GR}$ for industry G, $\left({p}_{Sx}^{SR}-{t}_{Sx}^{SR}\right){x}_{S}^{SR}$ for industry S, and $\left({p}_{Tx}^{TR}-{t}_{Tx}^{TR}\right){x}_{T}^{TR}$ for industry T. Thus from the viewpoint of industry accounts, it is natural to aggregate these export revenues across industries in order to obtain the value of exports aggregate at producer prices, $\left({p}_{Gx}^{GR}-{t}_{Gx}^{GR}\right){x}_{G}^{GR}+\left({p}_{Sx}^{SR}-{t}_{Sx}^{SR}\right){x}_{S}^{SR}+\left({p}_{Tx}^{TR}-{t}_{Tx}^{TR}\right){x}_{T}^{TR}$. Thus for production accounts that are based on the economic approach to index number theory, it is more appropriate to use tax-adjusted producer prices as the pricing concept rather than final demand prices.31 Similar comments apply to the treatment of imports. Later in this section, it will be shown how final demand based estimates for volume GDP can be reconciled with production-based estimates of volume GDP originating in the production sector at producer prices.

20.69 The three final demand value aggregates defined by equations (20.19) through (20.21) can be summed and the resulting value aggregate is the GDPP generated by the economy’s production sector at producer prices. Note that we have added the subscript P to this GDP concept at producer prices to distinguish it from the more traditional concept of GDP at final demand prices, which we denote by GDPF. The two GDP concepts will be reconciled later.

20.70 The value-added aggregates at producer prices defined by equations (20.22) through (20.24) can also be summed and this sum will also equal GDPP because the two methods for forming estimates of GDPP are simply alternative methods for summing over the elements of the net supply matrix.32 Thus the following equation must hold:

$\begin{array}{cc}{\mathrm{GDP}}_{\text{p}}\equiv v{f}_{G}+v{f}_{S}+v{f}_{T}=v{a}^{G}+v{a}^{S}+v{a}^{T}.& \phantom{\rule[-0.0ex]{8.0em}{0.0ex}}\left(20.25\right)\end{array}$

20.71 It is useful to explicitly write out GDPP as the sum of the three final demand aggregates defined in equations (20.22) through (20.24). After some rearrangement of terms the following equation is obtained:

$\begin{array}{cc}\hfill \mathrm{G}\mathrm{D}{\mathrm{P}}_{\mathrm{p}}=& v{f}_{G}+v{f}_{S}+v{f}_{T}\hfill \\ \hfill =& \mathrm{G}\mathrm{D}{\mathrm{P}}_{\mathrm{F}}-T\hfill \\ \hfill =& \left[C+I+G\right]+X-M-T,\phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.26\right)\hfill \end{array}$

where T is the value of commodity tax net revenues (taxes less subsidies) defined as the sum of the following terms:

$\begin{array}{cc}T& \equiv {t}_{G}^{GS}{y}_{G}^{GS}+{t}_{G}^{GT}{y}_{G}^{GT}+{t}_{G}^{GF}{y}_{G}^{GF}+{t}_{Gx}^{GR}{x}_{G}^{GR}\hfill \\ & \phantom{\rule{0ex}{0ex}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{Gm}^{GR}{m}_{G}^{GR}+{t}_{Gm}^{SR}{m}_{G}^{SR}+{t}_{Gm}^{TR}{m}_{G}^{TR}+{t}_{S}^{SG}{y}_{S}^{SG}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{S}^{ST}{y}_{S}^{ST}+{t}_{S}^{SF}{y}_{S}^{SF}+{t}_{Sx}^{SR}{x}_{S}^{SR}+{t}_{Sm}^{GR}{m}_{S}^{GR}+{t}_{Sm}^{SR}{m}_{S}^{SR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{Sm}^{TR}{m}_{S}^{TR}+{P}_{T}^{TG}{y}_{T}^{TG}+{p}_{T}^{TS}{y}_{T}^{TS}+{t}_{T}^{TF}{y}_{T}^{TF}+{t}_{Tx}^{TR}{x}_{T}^{TR}\hfill \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{t}_{Tm}^{GR}{m}_{T}^{GR}+{t}_{Tm}^{SR}{m}_{T}^{SR}+{t}_{Tm}^{TR}{m}_{T}^{TR};\phantom{\rule[-0.0ex]{6.0em}{0.0ex}}\left(20.27\right)\hfill \end{array}$

and GDPF is the value of GDP at final demand prices defined as the sum of the following components of final demand at final demand prices:

$\begin{array}{ccc}GD{P}_{F}& \equiv \left[{P}_{G}^{GF}{y}_{G}^{GF}+{p}_{S}^{SF}{y}_{S}^{SF}+{p}_{T}^{TF}{y}_{T}^{TF}\right]\hfill & \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+\left[{t}_{Gx}^{GR}{x}_{G}^{GR}+{t}_{Sx}^{SR}{m}_{G}^{SR}+{t}_{Tx}^{TR}{x}_{T}^{TR}\right]\hfill & \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}-\left[{p}_{Gm}^{GR}{m}_{G}^{GR}+{p}_{Gm}^{SR}{m}_{G}^{SR}+{p}_{Gm}^{TR}{m}_{G}^{TR}\hfill & \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{p}_{Sm}^{GR}{m}_{S}^{GR}+{p}_{Sm}^{SR}{m}_{S}^{SR}+{p}_{Sm}^{TR}{m}_{S}^{TR}\hfill & \\ & \phantom{\rule[-0.0ex]{1.0em}{0.0ex}}+{p}_{Tm}^{GR}{m}_{T}^{GR}+{p}_{Tm}^{SR}{m}_{T}^{SR}+{P}_{Tm}^{TR}{m}_{T}^{TR}\right]\hfill & \\ & =\left[C+G+I\right]+\left[X\right]-\left[M\right].\phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\hfill & \phantom{\rule[-0.0ex]{5.0em}{0.0ex}}\left(20.28\right)\end{array}$

20.72 Note that the 15 terms that do not involve taxes on the right-hand side of equation (20.26), which define GDPF, correspond to the 15 terms on the right-hand side of equation (20.10), which provided our initial decomposition of GDP when there were no commodity taxes. However, when there are commodity taxes (and commodity subsidies), the new decomposition of GDPP requires that the 21 tax terms defined by equation (20.27) be subtracted from the right-hand side of equation (20.26). Note that using definition (20.28), we can rewrite the identity (20.26) in the following form:

$\begin{array}{cc}{\text{GDP}}_{\text{P}}={\text{GDP}}_{\text{P}}+T.& \phantom{\rule[-0.0ex]{8.0em}{0.0ex}}\left(20.29\right)\end{array}$

20.73 Thus the value of production at final demand prices, GDPF, is equal to the value of production at producer prices, GDPP, plus commodity tax revenues less commodity tax subsidies, T, which is a traditional national income accounting identity.

20.74 As was discussed in the previous section, three methods can be used to construct a volume or quantity index of net outputs (at producer prices) produced by the production sector:

• Sum the two supply matrices and subtract the two use matrices and look at the cell entries in the resulting matrix. Collecting all of the nonzero transactions, we can see that there are 48 distinct price times quantity transactions. If a negative sign is associated with any one of these terms, that negative sign is attached to the quantity. Now apply normal index number theory to these 48 price times quantity components of the aggregate.

• Sum up the value-added aggregates defined by equations (20.22) through (20.24). The resulting value-added aggregate will have the same 48 separate price times quantity components that occurred in the first method of aggregation. If a value component has a negative sign associated with it, then attach the negative sign to the quantity (so that all prices will always be positive). Now apply normal index number theory to these 48 price times quantity components of the aggregate. This method will generate the same results as the first method listed above.

• Sum up the final demand value aggregates defined by equations (20.19) through (20.21). The resulting value of final demand aggregate will have 36 separate price times quantity components. If a value component has a negative sign associated with it, then attach the negative sign to the quantity. Now apply normal index number theory to these 36 price times quantity components of the aggregate.

20.75 It is evident that the quantity index or the volume estimate for GDP will be the same using methods 1 and 2 listed above because the two methods generate exactly the same set of 48 separate price times quantity components in the value aggregate. However, it is not evident that volume estimates for GDP based on method 3 will coincide with those generated using methods 1 and 2 because there are 48 price times quantity components to be aggregated when we use methods 1 or 2 compared to only 36 components when we use method 3. However, equations (20.11) through (20.18) in the previous section (with some obvious changes in notation) continue to hold in this new framework with commodity taxes and subsidies. Thus value-added (at producer prices) Laspeyres, Paasche, and Fisher quantity indices will be equal to their final demand counterparts, where the 21 terms involving taxes are used in the formulas. Note that the specific tax terms play the role of prices in these index number formulas and the associated quantities have negative signs attached to them when calculating these final demand (at producer prices) index numbers.

20.76 The equality (20.18) between the two methods for constructing an aggregate volume index for GDP using the Fisher quantity index as the index number formula can be extended to the case where the implicit Törnqvist quantity index is used as the index number formula. In this case, the value aggregates are deflated by the Törnqvist price index and by writing out the formulas, it is straightforward to show that ${P}_{T}^{va}\left({p}^{va0,}{p}^{va1},{q}^{va0},{q}^{va1}\right)$, the Törnqvist price index using the 48 price times quantity components in the value-added aggregate, is equal to ${P}_{T}^{fd}\left({p}^{fd0,}{p}^{fd1},{q}^{fd0},{q}^{fd1}\right)$, the Törnqvist price index using the 36 price times quantity components in the final demand aggregate.33

20.77 As noted in the previous section, GDPP can be calculated using two-stage aggregation where the first stage calculates volume value added (at producer prices) by industry. The two-stage estimates of GDPP will coincide exactly with their single-stage counterparts if the Laspeyres or Paasche formulas are used and will approximately coincide if the Fisher formula is used. It should be noted that the value added at producer prices approach for the calculation of industry aggregates is suitable for productivity analysis purposes.34 It should be emphasized that in order to construct accurate productivity statistics for each industry, it generally will be necessary to construct separate price deflators for each nonzero cell in the augmented input output tables that have been suggested in this chapter.

20.78 The final topic for this section is how to reconcile volume estimates for GDP at final demand prices, GDPF, with volume estimates for GDP at producer prices, GDPP. Recall equation (20.29), which said that GDPF equals GDPP plus I. Suppose that data are available for two periods that respect equation (20.29) in each period and a quantity index, GDPF, is constructed, defined by equation (20.28) with 15 separate price times quantity components. Then noting that GDPP is defined by the sum of equations (20.22) through (20.24) with 48 price times quantity components,35 and T is defined by equations (20.27) with 21 price times quantity components, we could combine these transactions and construct an alternative quantity index for this sum of GDPP and T value aggregate using the same index number formula. Using the same method of proof as was used in the previous section, we can show that the resulting volume estimates for GDPF and GDPP + T will coincide if the Laspeyres, Paasche, or Fisher formulas are used. For the GDPP + T aggregate, two-stage aggregation could be used where the first-stage value aggregates are GDPP, GDP at producer prices, and T, commodity tax revenue less commodity subsidies. The two-stage estimates will be exactly equal to the corresponding single-stage estimates if the Laspeyres or Paasche formulas are used for the quantity index and will be approximately equal if the Fisher formula is used. This type of decomposition will enable analysts to relate volume growth in final demand GDPF to volume growth in GDPP at producer prices plus commodity tax effects. More generally, the identity (20.29) can be used to estimate GDPF if the statistical agency is able to estimate GDPP and in addition, the statistical agency can form estimates of the 21 tax times quantity terms on the right-hand side of equation (20.27).36

C. The Artificial Data Set

C.1 The artificial data set framework: Real supply and use matrices

20.79 An artificial data set is presented in this section for the supply and use tables outlined in the previous section. It is useful to expand the commodity classification from one good G to four goods, G1, G2, G3, and G4, and from one service to two services, S1 and S2. The four goods are

• G1, agricultural products or food good;

• G2, crude oil or, more generally, energy products;

• G3, an imported pure intermediate good that is used by the domestic goods producing industry, and

• G4, a general consumption nonenergy, nonfood good.

The two services are

• S2, high-technology services such as telecommunications and Internet access.

The remaining commodity in the commodity classification is T, transportation services.

20.80 The constant dollar table counterparts to Tables 20.8 through 20.11 are now modified into Tables 20.12 through 20.15. The counterpart to Table 20.8 is Table 20.12. This matrix shows the production by commodity and by industry that is delivered to domestic demanders. Thus ${y}_{G4}^{GS}$ denotes the quantity of good G4 that is delivered by the goods producing industry G to the services industry S, ${y}_{G4}^{GT}$ denotes the quantity of good G4 that is delivered by the goods producing industry G to the transportation industry T, ${y}_{G4}^{GF}$ denotes the quantity of good G4 that is delivered by the goods producing industry G to the domestic final demand sector F, ${y}_{G1}^{SF}$ denotes the quantity of good G1 (food imports) delivered by the services industry S (which includes retailing and wholesaling) to the domestic final demand sector F, ${y}_{G2}^{SF}$ denotes the quantity of good G2 (energy imports) delivered by the services industry S (which includes retailing and wholesaling) to the domestic final demand sector F, ${y}_{S1}^{SG}$ denotes the quantity of traditional services S1 that is delivered by the services industry S to the goods producing industry G, ${y}_{S2}^{SG}$ denotes the quantity of high-tech services S2 that is delivered by the services industry S to the goods producing industry G, ${y}_{T}^{TG}$ denotes the quantity of transportation services T that is delivered by the transportation industry T to the goods producing industry G, and so on.

Table 20.12.

Real Domestic Supply Matrix

20.81 Looking at the entries in Table 20.13, we can see that there is no domestic production of goods G1 (agricultural products) and G2 (crude oil) by industries G and T and no domestic production of G3 (the imported intermediate good used by the goods producing industry G) by any of the industries. Industry G produces good G4 and delivers ${y}_{G4}^{GS}$ units of this good to the service industry S to be used as an intermediate input there, delivers ${y}_{G4}^{GT}$ units of this good to the transportation industry T to be used as an intermediate input there, and delivers ${y}_{G4}^{GF}$ units of this good to the domestic final demand sector F. Similarly, industry S produces the general service commodity S1 and delivers ${y}_{S1}^{SG}$ units of this commodity to the goods producing industry G to be used as an intermediate input there, delivers ${y}_{S1}^{ST}$ units of this service to the transportation industry T to be used as an intermediate input there, and delivers ${y}_{S1}^{SF}$ units of this service to the domestic final demand sector F. Industry S also produces the high-technology service commodity S2 and delivers ${y}_{S2}^{SG}$ units of this commodity to the goods producing industry G to be used as an intermediate input there, delivers ${y}_{S2}^{ST}$ units of this service to the transportation industry T to be used as an intermediate input there, and delivers ${y}_{S2}^{SF}$ units of this service to the domestic final demand sector F. It is also assumed that the service industry imports G1 (agricultural produce) and G2 (crude oil) and stores and distributes these imports to the household sector; these are the deliveries ${y}_{SG1}^{SF}$ and ${y}_{SG2}^{SF}$.37 Finally, industry T produces the transportation services commodity T and delivers ${y}_{T}^{TG}$ units of this commodity to the goods producing industry G to be used as an intermediate input there, delivers ${y}_{T}^{TS}$ units of these transport services to the service industry S to be used as an intermediate input there, and delivers ${y}_{T}^{TF}$ units of transport services directly to the domestic final demand sector F.

Table 20.13.

Real Domestic Use Matrix

20.82 The counterpart to Table 20.9 is now Table 20.13. This matrix lists the industry demands for commodities that originate from domestic sources; that is, it shows the industry by commodity intermediate input demands for commodities that are supplied from domestic sources.

20.83 Because there is no domestic production of goods G1 through G3, the rows that correspond to these commodities in Table 20.13 all have zero entries. The remainder of the table is the same as Table 20.9. Note that the domestic intersectoral transfers of goods and services in Tables 20.12 and 20.13 match up exactly; that is, the eight nonzero quantities in Table 20.13 are exactly equal to the corresponding entries in Table 20.12. The counterpart to Table 20.10 is now Table 20.14.

Table 20.14.

Real ROW Supply or Export by Industry and Commodity Matrix

20.84 Because there is no exportation of goods G1 through G3, the rows that correspond to these commodities in Table 20.14 all have zero entries. The remainder of the table is the same as Table 20.10. Thus industry G exports ${x}_{G4}^{GR}$ units of good G4, industry S exports ${x}_{S1}^{SR}$ units of traditional services S1 and no units of high-tech services, and industry T exports ${x}_{T}^{TR}$ units of transportation services to the rest of the world.

20.85 The counterpart to Table 20.11 is now Table 20.15. This matrix lists the industry demands for commodities that originate from foreign sources; that is, it shows the industry by commodity intermediate input demands for intermediate inputs from foreign sources.

Table 20.15.

Real ROW Use or Import by Industry and Commodity Matrix

20.86 From Table 20.15, it can be seen that the goods producing industry uses ${m}_{G1}^{GR}$ units of agricultural imports, ${m}_{G2}^{GR}$ units of crude oil imports, ${m}_{G3}^{GR}$ units of a pure imported intermediate good, and ${m}_{S1}^{GR}$ units of imported service inputs. Industry G does not import the domestically produced good, G4, nor does it import transportation services in this simplified example. Industry S imports ${m}_{G1}^{SR}$ units of agricultural goods (for distribution to domestic households), ${m}_{G2}^{SR}$ units of crude oil (for distribution to households and own use), ${m}_{S1}^{GR}$ units of foreign general services, and ${m}_{T}^{SR}$ units of foreign transportation services. Industry T imports ${m}_{G2}^{TR}$ units of crude oil and ${m}_{T}^{TR}$ units of foreign-sourced transportation services.

C.2 The artificial data set framework: Value supply and use matrices

20.87 The value matrix counterparts to the two supply and two use matrices listed in Section B.1 above are listed in this section. Table 20.16 is the counterpart to Table 20.5.

Table 20.16.

Nominal Value Domestic Supply Matrix with Commodity Taxes

20.88 All of the prices that begin with the letter p are the prices that domestic final demanders pay for a unit of the commodity (except for minor complications with respect to the treatment of export prices). In Table 20.16, these prices correspond to purchasers’ prices in the 2008 SNA.38 However, the industry sellers of these commodities do not generally receive the full final demand price: Commodity taxes less commodity subsidies must be subtracted from these final demand prices in order to obtain the net prices that are listed in Table 20.16. These net selling prices are the prices that the industrial producers actually receive for their sales of outputs to domestic demanders. In Table 20.16, these prices correspond to basic prices in the 2008 SNA.39 The notation used for prices in Table 20.16 matches the notation used for quantities in Table 20.12.

20.89 The reader should note that in this chapter, for convenience, the p prices will be referred to as final demand prices and the p – t prices will be referred to as producer prices. Conceptually, the final demand prices are the prices that domestic final demanders pay per unit for their purchases of commodities delivered to final demand categories. However, for an exported commodity, the final demand price is not the total purchase price (including transportation services provided by foreign establishments, import duties, and other applicable commodity taxes) that the foreign importer pays for the commodity; rather, in this case, the final demand price is only the payment made to the domestic producer by the foreign importer. Conceptually, producer prices are the prices that domestic producers receive per unit of output produced that is sold or the prices that domestic producers pay per unit of input that is purchased (including applicable commodity taxes and all transportation margins).40 Table 20.17 is the counterpart to Table 20.2. It is also the value counterpart to Table 20.13.

Table 20.17.

Nominal Value Domestic Use Matrix

20.90 Note that in Table 20.16, industry G receives only the revenue $\left({p}_{G4}^{GS}-{t}_{G4}^{GS}\right){y}_{G4}^{GS}$ for its sales of commodity G4 to industry S, whereas in Table 20.17, industry S pays the amount ${p}_{G4}^{GS}{y}_{G4}^{GS}$ for these purchases of intermediate inputs from industry G. The difference between these two intersectoral value flows is ${t}_{G4}^{GS}{y}_{G4}^{GS}$, the tax (less subsidy) payments made by industry G to the government on this intersectoral value flow. Thus the values of domestic intersectoral transfers of goods and services in Tables 20.16 and 20.17 do not match up exactly unless the commodity tax less subsidy terms ${t}_{S1}^{SG}{t}_{G4}^{GS}$, and so on are all zero. The counterpart to Table 20.6 is now Table 20.18, which in turn is the value counterpart to the real Table 20.14.

Table 20.18.

Value ROW Supply or Export by Industry and Commodity Matrix

20.91 Because there is no exportation of goods G1 through G3, the rows that correspond to these commodities in Table 20.18 all have zero entries. The remainder of the table is straightforward. Thus industry G exports ${x}_{G4}^{GR}$ units of good G4 and the foreign final demander pays the price ${p}_{G4x}^{GR}$ per unit but the exporting industry gets only the amount ${p}_{G4x}^{GR}-{t}_{G4x}^{GR}$ per unit; that is, the government gets the per unit (net) revenue ${t}_{G4x}^{GR}$ on these sales if it imposes a (net) export tax equal to ${t}_{G4x}^{GR}$. Similarly, net export taxes (if applicable) must be subtracted from the final demand prices for the other industries. In the numerical example that follows, it is assumed that the net export tax in industry G is negative (so that exports are subsidized in industry G) and that taxes in industries S and T are zero. The counterpart to Table 20.7 is now Table 20.19, which in turn is the value counterpart to the real Table 20.15.

20.92 It should be straightforward for the reader to interpret the final demand prices (these terms begin with p) and the accompanying import duties, excise duties, and other commodity taxes on imports (these terms begin with t). The quantities of imports (these terms begin with an m) are the same as the quantity terms in the corresponding real table, Table 20.15. From a practical point of view, governments have a tendency to tax imports (so that the tax terms in this table will tend to be positive) and to subsidize exports (so that the tax terms in the previous table will tend to be zero or negative).

C.3 Industry G prices and quantities

20.93 All of the price and quantity series used in this chapter are listed in the four nominal value supply and use matrices that are listed in Tables 20.16 through 20.19. The 11 final demand price series that form part of the industry G data in these matrices are listed for five periods in Table 20.20. The commodity that the price refers to is listed in the first row of the table.

Table 20.19.

Value ROW Use or Import by Industry and Commodity Matrix

Table 20.20.

Industry G Final Demand Prices for All Transactions

Table 20.21.

Industry G Commodity Taxes

20.94 Some points to note about the price entries in Table 20.20 are as follows. The industry G final demand prices that it faces for deliveries of commodity G4 to the service industry, ${p}_{G4}^{GS}$, to the transportation services industry, ${p}_{G4}^{GT}$, and for exports, ${p}_{G4x}^{GR}$, are all much the same: Prices trend up fairly rapidly for periods 2 and 3 and then level off for periods 4 and 5. However, the final demand price for deliveries of G4 to the domestic final demand sector F, ${p}_{G4}^{GF}$, is somewhat higher than the corresponding prices for deliveries of G4 to the service sector S, ${p}_{G4}^{GS}$, and to the transportation sector T, ${p}_{G4}^{GT}$, owing to higher commodity taxes on deliveries to sector F. The price of traditional domestic services used as an intermediate input by industry G, ${p}_{S1}^{SG}$, also increases rapidly initially and then levels off for the last two periods. However, the price of high-tech domestic services used as an intermediate input by industry G, ${p}_{S2}^{SG}$, drops rapidly throughout the sample period. The price of transportation services used as an intermediate input by industry G, ${p}_{T}^{TG}$, increases dramatically in period 2 owing to the increase of the price of imported oil and then decreases for the next two periods as the price of oil drops before increasing again in period 5. The price of agricultural imports into industry G, ${p}_{G1m}^{GR}$, fluctuates considerably from period to period but overall, agricultural import prices do not increase as rapidly as do many other prices. The price of oil imports into industry G, ${p}_{G2m}^{GR}$, fluctuates violently, doubling in period 2, then falling so that by period 4, the price is below the period 1 price but then the price more than doubles for period 5. The price of the imported intermediate good, ${p}_{G3m}^{GR}$, steadily drops at a rapid pace over the five periods.41 Finally, the price of the imported services commodity, ${p}_{Sm}^{GR}$, increases rapidly over periods 2 and 3 but then the rate of price increase slows down. Over the entire period, the price of services tends to increase somewhat more rapidly than the price of manufactured output, G4.

20.95 The 11 commodity tax series that form part of the industry G taxes listed in Tables 20.16 through 20.19 are listed for five periods in Table 20.21. Recall that by convention, the selling industry pays all commodity taxes so the taxes on industry G’s purchases of intermediate inputs from industries S and T, ${t}_{S1}^{SG}$, ${t}_{S2}^{SG}$ and ${t}_{T}^{TG}$, are all identically equal to zero. However, in Table 20.21, these tax rates are listed (with zero entries) so that Table 20.21 is dimensionally comparable to Table 20.20.

Table 20.22.

Industry G Quantities of Outputs and Intermediate Inputs

Table 20.23.

Industry S Final Demand Prices

20.96 Note that the taxes listed above are all positive or zero except that the exports of good G4 by industry G are subsidized so the taxes ${t}_{G4x}^{GR}$ have a negative sign attached to them instead of the usual positive sign.

20.97 The 11 quantity series that form part of the industry G data in Tables 20.16 through 20.19 are listed for five periods in Table 20.22.

20.98 The quantities of good G4 produced by industry G, ${y}_{G4}^{GS},{y}_{G4}^{GT},{y}_{G4}^{GF},$, which are deliveries to the domestic services industry, the domestic transportation industry, and the domestic final demand sector respectively, all grow at roughly the same rate. However, the quantities of G4 exported by industry G, ${x}_{G4x}^{GR}$, grow a bit more rapidly, particularly during the final two periods. The quantity of traditional domestic services used as an intermediate input by industry G, ${y}_{S1}^{SG}$, more than doubles over the five periods but the quantity of high-tech services used as an intermediate input, ${y}_{S1}^{SG}$, grows tenfold owing to the rapid price drop in this commodity. The quantity of domestic transportation services used as an intermediate input by industry G, ${y}_{T}^{TG}$, exactly doubles over the five periods. The quantity of agricultural imports used by industry G, ${m}_{G1m}^{GR}$, increases steadily from 5 units to 8 units while the quantity of oil imports increases from 10 in period 1 to 15 in period 3 but then the growth rate slows over the final two periods. Imports of the high-technology pure intermediate imported good, ${m}_{G3m}^{GR}$, increase rapidly from 10 to 35 units, reflecting the real-world tendency towards globalization and international outsourcing. Finally, imports of service inputs into industry G increase rapidly, growing from 2 units in period 1 to 6 units in period 5.

C.4 Industry S prices and quantities

20.99 The 15 final demand price series that form part of the industry S data in Tables 20.16 through 20.19 are listed for five periods in Table 20.23.

20.100 Some points to note about the price entries in Table 20.23 are as follows. The prices of service sector deliveries to industry G, ${p}_{S1}^{SG}$ and ${p}_{S2}^{SG}$, the prices of deliveries of good G4 from industry G to industry S, ${p}_{G4}^{GS}$, are exactly the same as in Tables 20.20 and 20.23. This reflects the bilateral nature of transactions between sectors. Industry S sells commodities S1 and S2 to industries G and T (these are the prices ${p}_{S1}^{SG}$ and ${p}_{S2}^{SG}$ for sales to industry G and ${p}_{S1}^{ST}$ and ${p}_{S2}^{ST}$ for sales to industry T) and it sells commodities S1 and S2 to the domestic final demand sector F at prices ${p}_{S1}^{SF}$ and ${p}_{S2}^{SF}$ and it sells S1 to the rest of the world R as an export at the price ${p}_{S1x}^{SR}$. The industry S final demand selling prices are much the same over these four destinations, except that export price for S1 falls off somewhat and the selling prices to the domestic final demand sector for the high-technology service S2 are somewhat higher, reflecting a higher level of final demand taxation. Industry S also imports G1 (agricultural or food imports for resale to domestic households) and G2 (oil imports for resale to domestic households) and it also imports some foreign general services S1 and some foreign transportation services T. These import prices are ${p}_{G1m}^{SR},{p}_{G2m}^{SR},{p}_{S1m}^{SR}$, and ${p}_{Tm}^{SR}$, respectively. The import prices for these first three classes of imports are much the same as the corresponding import prices that applied to the imports of these commodities by industry G. The price of imported transportation services, ${p}_{Tm}^{SR}$, is the same as the price of domestic transportation services provided to industry S, ${p}_{T}^{TS}$. Note that the service sector selling prices of goods G1 and G2 to the domestic final demand sector, ${p}_{G1}^{SF}$ and ${p}_{G2}^{SF}$, are somewhat higher than the corresponding import purchase prices for these goods, ${p}_{G1m}^{SR}$ and ${p}_{G2}^{SF}$, but this is natural: The service sector must make a positive margin on its trading in these commodities in order to cover the costs of storage and distribution.

20.101 The service industry obviously contains elements of the traditional storage, wholesaling, and retailing industries. The treatment of these industries that is followed in the artificial data example is a gross output treatment as opposed to a margin industry treatment. In the gross output treatment, goods for resale are purchased and the full purchase price times the amount purchased appears as an intermediate input cost and then the goods are sold subsequently at a higher price and this selling price times the amount sold appears as a contribution to gross output. In the margin treatment, it is assumed that the amount sold during the accounting period is at least roughly equal to the amount purchased, and the difference between the selling price and the purchase price (the margin) is multiplied by the amount bought and sold and is treated as a gross output with no corresponding intermediate input cost. Thus for the case of an imported good, if the margin treatment of wholesaling/retailing/storage (WRS) output is used, the margin would be credited to this WRS industry and the full import price plus the margin would appear as an intermediate input by the purchasing industry (or final demand sector). Thus the margin treatment of the WRS industry would be similar to the margin treatment that has been accorded to the transportation industry. However, there is a difference between the WRS industry and the transportation industry: For the transportation industry, one can be fairly certain that the goods “purchased” by the transport industry are equal to the goods “sold” by the industry and the margin treatment is perfectly justified. This is not necessarily the case for the WRS industry: Sales are not necessarily equal to purchases in any given accounting period. Thus it seems preferable to use the gross output treatment for these distributive industries over the margin approach, although individual countries may feel that sales are sufficiently close to purchases so that the margin approach is a reasonable approximation to the actual situation and hence can be used in their national accounts.42

20.102 The 15 commodity tax series that form part of the industry S taxes listed in Tables 20.16 through 20.19 are listed for five periods in Table 20.24.

Table 20.24.

Industry S Commodity Taxes

20.103 Note that the tax rates on domestic intermediate inputs used by industry S are all set equal to zero under the convention used in this chapter that the selling industry pays any applicable commodity taxes.43

20.104 The 15 quantity series that form part of the industry S data in Tables 20.16 through 20.19 are listed for five periods in Table 20.25.

Table 20.25.

Industry S Quantities of Outputs and Inputs