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

International Monetary Fund

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Language:: English

Print Publication Date:: 17 Dec 2009

Keywords:: M&G; trade price; value aggregate; final demand; times quantity component; quantity index; Producer price indexes; Price indexes; Producer prices; Import price indexes; Export fluctuations

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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{matrix} O + M + (t - s) = I C + C + I + G + X, & (20.1) \end{matrix}

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{matrix} O - I C + (t - s) = C + I + G + (X - M) . & (20.2) \end{matrix}

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 $G, p_{G}^{G S} y_{G}^{G S} + p_{G}^{G T} y_{G}^{G T} + p_{G}^{G F} y_{G}^{G F}$ , corresponds to the revenues received by the goods producing sector from its sales of good G to the service sector, $p_{G}^{G S} y_{G}^{G S}$ , where $p_{G}^{G S}$ is the price of sales of good G to sector S and $y_{G}^{G S}$ 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, $p_{G}^{G T} y_{G}^{G T}$ , where $p_{G}^{G T}$ is the price of sales of good G to sector T and $y_{G}^{G T}$ 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, $p_{G}^{G F} y_{G}^{G F}$ , where $p_{G}^{G F}$ is the price of sales of good G to sector F and $y_{G}^{G F}$ is the corresponding quantity sold. Similarly, the value sum in row and column S, $p_{S}^{S G} y_{S}^{S G} + p_{S}^{S T} y_{S}^{S T} + p_{S}^{S F} y_{S}^{S F}$ , 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, $p_{T}^{T G} y_{T}^{T G} + p_{T}^{T S} y_{T}^{T S} + p_{T}^{T F} y_{T}^{T F}$ , 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 $p_{G}^{G S}$ will usually not equal $p_{G}^{G T}$ or $p_{G}^{G F}$ ; 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, $p_{G}^{G S} y_{G}^{G S}$ , 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, $p_{G x}^{G R} x_{G}^{G R}$ , corresponds to the revenues received by the goods producing sector from its sales of good G to the ROW sector, where $p_{G x}^{G R}$ is the price of sales of good G to sector R and $x_{G}^{G R}$ 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, $p_{S x}^{S R} x_{S}^{S R}$ is the value of exports of services produced by the services sector and $p_{T x}^{T R} x_{T}^{T R}$ 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, $p_{G m}^{G R} m_{G}^{G R}$ , corresponds to the payments to the rest of the world by the goods producing sector for its imports of goods, where $p_{G m}^{G R}$ is the price of imports of good G to industry G and $m_{G}^{G R}$ is the corresponding quantity purchased, or more simply, it is the cost of imports of goods to the goods producing sector. Similarly, $p_{S m}^{G R} m_{S}^{G R}$ is the value of imported services that are used in the goods producing sector and $p_{T m}^{G R} m_{T}^{G R}$ 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{matrix} v f_{G} \equiv p_{G}^{G F} y_{G}^{G F} + P_{G x}^{G R} x_{G}^{G R} - p_{G m}^{G R} m_{G}^{G R} \\ - p_{G m}^{S R} m_{G}^{S R} - p_{G m}^{T R} m_{G}^{T R}; & (20.3) \end{matrix}

\begin{matrix} v f_{S} \equiv p_{S}^{S F} y_{S}^{S F} + P_{S x}^{S R} x_{S}^{S R} - p_{S m}^{S R} m_{S}^{S R} \\ - p_{S m}^{S R} m_{S}^{S R} - p_{S m}^{T R} m_{S}^{T R}; & (20.4) \end{matrix}

\begin{matrix} v f_{T} \equiv p_{T}^{T F} y_{T}^{T F} + P_{T x}^{T R} x_{T}^{T R} - p_{T m}^{G R} m_{T}^{G R} \\ - p_{T m}^{S R} m_{T}^{S R} - p_{T m}^{T R} m_{T}^{T R} . & (20.5) \end{matrix}

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

\begin{matrix} V a^{G} \equiv p_{G}^{G S} y_{G}^{G S} + p_{G}^{G T} y_{G}^{G T} + p_{G}^{G F} y_{G}^{G F} - p_{S}^{S G} y_{S}^{S G} \\ - p_{T}^{T G} y_{T}^{T G} + p_{G x}^{G R} x_{G}^{G R} - p_{G m}^{G R} m_{G}^{G R} \\ - p_{S m}^{G R} m_{S}^{G R} - p_{T m}^{G R} m_{T}^{G R}; & \begin{matrix} (20.6) \end{matrix} \end{matrix}

\begin{matrix} V a^{S} \equiv p_{S}^{S G} y_{S}^{S G} + p_{S}^{S T} y_{S}^{S T} + p_{S}^{S F} y_{S}^{S F} - p_{G}^{G S} y_{G}^{G S} \\ - p_{T}^{T S} y_{T}^{T S} + p_{S x}^{S R} x_{S}^{S R} - p_{G m}^{S R} m_{G}^{S R} \\ - p_{S m}^{S R} m_{S}^{S R} - p_{T m}^{S R} m_{T}^{S R}; & (20.7) \end{matrix}

\begin{matrix} V a^{T} \equiv p_{T}^{T G} y_{T}^{T G} + p_{T}^{T S} y_{T}^{T S} + p_{T}^{T F} y_{T}^{T F} - p_{G}^{G T} y_{G}^{G T} \\ - p_{S}^{S T} y_{S}^{S T} + p_{T x}^{T R} x_{T}^{T R} - p_{G m}^{T R} m_{G}^{T R} \\ - p_{S m}^{T R} m_{S}^{T R} - p_{T m}^{T R} m_{T}^{T R} . & (20.8) \end{matrix}

20.35 Note that each commodity final demand value aggregate, vf_G, vf_S, and vf_T, 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, vf_G equal to the value of vf_G divided by the corresponding net deliveries of commodity G to final demand, $y_{G}^{G S} + y_{G}^{G T} + y_{S}^{S G} - y_{T}^{T G} + x_{G}^{G R} - m_{G}^{G R} - m_{G}^{G R} - m_{S}^{G R} - m_{T}^{G R}$ , 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, va^G, va^S, and va^T, 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{matrix} G D P \equiv v f_{G} + v f_{S} + v f_{T} = v a^{G} + v a^{S} + v a^{T} . & (20.9) \end{matrix}

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{matrix} GDP & = v f_{G} + v f_{S} + V f_{T} \\ = [p_{G}^{G F} y_{G}^{G F} + p_{S}^{S F} y_{S}^{S F} + p_{T}^{T F} y_{T}^{T F}] + [p_{G x}^{G R} x_{G}^{G R} \\ + p_{S x}^{S R} x_{S}^{S R} + p_{T x}^{T R} x_{T}^{T R}] - [p_{G m}^{G R} m_{G}^{G R} + p_{G m}^{S R} m_{G}^{S R} \\ + p_{G m}^{T R} m_{G}^{T R} + p_{S m}^{G R} m_{S}^{G R} + p_{S m}^{S R} m_{S m}^{S R} m_{s}^{T R} \\ + p_{T m}^{G R} m_{T}^{G R} + p_{T m}^{S R} m_{T}^{S R} + p_{T m}^{T R} m_{T}^{T R}] \\ = [C + I + G] + [X - M] . (20.10) \end{matrix}

20.38 Note that the value aggregate, $p_{G}^{G F} y_{G}^{G F} + p_{S}^{S F} y_{S}^{S F} + p_{T}^{T F} y_{T}^{T F}$ , corresponds to the value of domestic final demand, the value aggregate $p_{G x}^{G R} x_{G}^{G R} + p_{S x}^{S R} x_{S}^{S R} + p_{T x}^{T R} x_{T}^{T R}$ corresponds to the value of exports, and the value aggregate $P_{G m}^{G R} m_{G}^{G R} + p_{G m}^{S R} m_{G}^{S R} + p_{G m}^{T R} m_{G}^{T R} + p_{S m}^{G R} m_{S}^{G R} + p_{S m}^{S R} m_{S}^{S R} + p_{S m}^{T R} m_{S}^{T R} + p_{T m}^{G R} m_{T}^{G R} + p_{T m}^{S R} m_{T}^{S R} + p_{T m}^{T R} m_{T}^{T R}$ 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 p^va and q^va respectively and denote the 15 dimensional p and q vectors that correspond to the third aggregation method over final demand components as p^fd and q^fd 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{matrix} P^{v a t} \cdot q^{v a t} = p^{f d t} \cdot q^{f dt}; t = 0, 1. & (20.11) \end{matrix}

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{matrix} P^{v a 1} \cdot q^{v a 0} = p^{f d 1} \cdot q^{f d 0}; & (20.12) \end{matrix}

P^{v a 0} \cdot q^{v a 1} = p^{f d 0} \cdot q^{f d 1} . (20.13)

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{matrix} {Q_{L}}^{v a} (p^{v a 0}, p^{v a 1}, q^{v a 0}, q^{v a 1}) \equiv p^{v a 0} \cdot q^{v a 1} / p^{v a 0} \cdot q^{v a 0}; \\ {Q_{L}}^{f d} (p^{f d 0} \cdot p^{f d 1}, q^{f d 0}, q^{f d 1}) \equiv p^{f d 0} \cdot q^{f d 1} / p^{f d 0} \cdot q^{f d 0}; \\ (20.14) \end{matrix}

\begin{array}{l} {Q_{p}}^{v a} (p^{v a 0}, p^{v a 1}, q^{v a 0}, q^{v a 1}) \equiv p^{v a 1} \cdot q^{v a 1} / p^{v a 1} \cdot q^{v a 0}; \\ {Q_{p}}^{f d} (P^{f d 0}, p^{f d 1}, q^{f d 0}, q^{f d 1}) \equiv p^{f d 1} \cdot q z^{f d 1} / p^{f d 1} \cdot q^{f d 0} . \\ (20.15) \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{matrix} Q_{L}^{v a} (p^{v a 0}, P^{v a 1}, q^{v a 0}, q^{v a 1}) = Q_{L}^{f d} (P^{f d 0}, p^{f d 1}, q^{f d 0}, q^{f d 1}) . \\ (20.16) \end{matrix}

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{matrix} Q_{p}^{v a} (p^{v a 0}, P^{v a 1}, q^{v a 0}, q^{v a 1}) = Q_{p}^{f d} (P^{f d 0}, p^{f d 1}, q^{f d 0}, q^{f d 1}) . \\ (20.17) \end{matrix}

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}^{v a} (p^{v a 0}, P^{v a 1}, q^{v a 0}, q^{v a 1}) = Q_{F}^{f d} (p^{f d 0}, p^{f d 1}, q^{f d 0}, q^{f d 1}) . \\ (20.18) \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 $p_{T}^{v a} (p^{v a 0}, p^{v a 1}, q^{v a 0}, q^{v a 1})$ , the Törnqvist price index using the 27 price times quantity components in the value-added aggregate, is equal to $p_{T}^{f d} (p^{f d 0}, p^{f d 1}, q^{f d 0}, q^{f d 1})$ , 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, $Q_{L}^{v a} (p^{v a 0}, p^{v a 1}, q^{v a 0}, q^{v a 1})$ . 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, $Q_{p}^{v a} (p^{v a 0}, p^{v a 1}, q^{v a 0}, q^{v a 1})$ .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, $Q_{F}^{v a} (p^{v a 0}, p^{v a 1}, q^{v a 0}, q^{v a 1})$ .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 $y_{G}^{G S}$ as before and the service sector pays industry G the price $p_{G}^{G S}$ for each unit of G that was delivered. However, industry G must remit the per unit²⁵ commodity tax²⁶ $t_{G}^{G S}$ of the per unit revenue $p_{G}^{G S}$ to the government sector and so industry G receives only the revenue $p_{G}^{G S} - t_{G}^{G S}$ 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 $p_{G x}^{G R} x_{G}^{G R}$ for its deliveries of goods to foreign purchasers from these purchases but if the government sector imposes a specific export tax equal to $t_{G x}^{G R}$ per unit of exports, then industry G gets to keep only the amount $p_{G x}^{G R} - t_{G x}^{G R}$ per unit sale instead of the full final demander price $p_{G x}^{G R}$ . If, however, $t_{G x}^{G R}$ is negative, then the government is subsidizing the export of goods and hence the subsidized price that the producer faces, $p_{G x}^{G R} - t_{G x}^{G R}$ , is actually higher than the final demander price $p_{G x}^{G R}$ . 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 $m_{G}^{G R}$ units of goods from foreign suppliers and pays these foreign suppliers the amount $p_{G m}^{G R} m_{G}^{G R}$ . However, if $t_{G m}^{G R}$ is positive (the usual case), then the government imposes a specific set of tariffs and indirect taxes on each unit imported equal to $t_{G m}^{G R}$ and hence industry G faces the higher price $p_{G m}^{G R} + t_{G m}^{G R}$ 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{matrix}  \end{matrix} \begin{matrix} v f_{G} & \equiv p_{G}^{G F} y_{G}^{G F} + p_{G x}^{G R} x_{G}^{G R} - p_{G m}^{G R} m_{G}^{G R} - p_{G m}^{S R} m_{G}^{S R} \\ - p_{G m}^{T R} m_{G}^{T R} - [t_{G}^{G S} y_{S}^{G S} + t_{G}^{G T} y_{G}^{G T} + t_{G}^{G F} y_{G}^{G F} \\ + t_{G x}^{G R} x_{G}^{G R} + t_{G m}^{G R} m_{G}^{G R} + t_{G m}^{S R} m_{G}^{S R} + t_{G m}^{T R} m_{G}^{T R}]; \\ (20.19) \end{matrix}

\begin{matrix} v f_{S} & \equiv p_{S}^{S F} y_{S}^{S F} + p_{S x}^{S R} x_{S}^{S R} - p_{S m}^{G R} m_{S}^{G R} - p_{S m}^{S R} m_{S}^{S R} \\ - p_{S m}^{T R} m_{S}^{T R} - [t_{S}^{S G} y_{S}^{S G} + t_{S}^{S T} y_{S}^{S T} + t_{S}^{S F} y_{S}^{S F} \\ + t_{S x}^{S R} x_{S}^{S R} + t_{S m}^{G R} m_{S}^{G R} + t_{G}^{S R} m_{S}^{S R} + t_{S m}^{T R} m_{S}^{T R}]; \\ (20.20) \end{matrix}

\begin{matrix} v f_{T} & \equiv p_{T}^{T F} y_{T}^{T F} + p_{T x}^{T R} x_{S}^{T R} - p_{T m}^{G R} m_{T}^{G R} - p_{T m}^{S R} m_{T}^{S R} \\ - p_{T m}^{T R} m_{T}^{T R} - [p t_{T}^{T G} y_{T}^{T G} + t_{T}^{T S} y_{T}^{T S} + t_{T}^{T F} y_{T}^{T F} \\ + t_{T x}^{T R} x_{T}^{T R} + t_{T m}^{G R} m_{T}^{G R} + t_{T m}^{S R} m_{T}^{S R} + t_{T m}^{T R} m_{T}^{T R}] . \\ (20.21) \end{matrix}

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 $p_{G}^{G F} y_{G}^{G F} + p_{G x}^{G R} x_{G}^{G R} - p_{G m}^{G R} m_{G}^{G R} - p_{G m}^{S R} m_{G}^{S R} - p_{G m}^{T R} m_{G}^{T R}$ , 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 vf_S and vf_T are similar.

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

\begin{matrix} v a^{G} & \equiv (p_{G}^{G S} - t_{G}^{G S}) y_{G}^{G S} + (p_{G}^{G T} - t_{G}^{G T}) y_{G}^{G T} \\ + (p_{G}^{G F} - t_{G}^{G F}) y_{G}^{G F} - p_{S}^{S G} y_{S}^{S G} - p_{T}^{T G} y_{T}^{T G} \\ + (p_{G x}^{G R} - t_{G x}^{G R}) x_{G}^{G R} - p_{G m}^{G R} + t_{G m}^{G R}) m_{G}^{G R} \\ - (p_{S m}^{G R} + t_{S m}^{G R}) m_{S}^{G R} - (p_{T m}^{G R} + t_{T m}^{G R}) m_{T}^{G R}; \\ (20.22) \end{matrix}

\begin{matrix} v a^{S} & \equiv (p_{S}^{S G} - t_{S}^{S G}) y_{S}^{S G} + (p_{S}^{S T} - t_{S}^{S T}) y_{S}^{S T} \\ + (p_{S}^{S F} - t_{S}^{S F}) y_{S}^{S F} - p_{G}^{G S} y_{S}^{S G} - p_{T}^{T G} y_{T}^{T S} \\ + (p_{S x}^{S R} - t_{S x}^{S R}) x_{S}^{S R} - (p_{G m}^{S R} + t_{G m}^{G R}) m_{G}^{S R} \\ - (p_{S m}^{S R} + t_{S m}^{S R}) m_{S}^{S R} - (p_{T m}^{S R} + t_{T m}^{S R}) m_{T}^{S R}; \\ (20.23) \end{matrix}

\begin{matrix} v a^{T} & \equiv (p_{T}^{T G} - t_{T}^{T G}) y_{T}^{T G} + (p_{T}^{T S} - t_{T}^{T S}) y_{T}^{T S} \\ + (p_{T}^{T F} - t_{T}^{T F}) y_{T}^{T F} - p_{G}^{G T} y_{G}^{G T} - p_{S}^{S T} y_{S}^{S T} \\ + (p_{T x}^{T R} - t_{T x}^{T R}) x_{T}^{T R} - (p_{G m}^{T R} + t_{G m}^{T R}) m_{G}^{T R} \\ - (p_{S m}^{T R} + t_{S m}^{T R}) m_{S}^{T R} - (p_{T m}^{T R} + t_{T m}^{T R}) m_{T}^{T R} . \\ (20.24) \end{matrix}

20.67 Looking at equation (20.22), it can be seen that the value added produced by industry G at producer prices, va^G, is equal to the value of deliveries of good G to industry S, $(p_{G}^{G S} - t_{G}^{G S}) y_{G}^{G S}$ ,²⁹ plus the value of deliveries of good G to industry T, $(p_{G}^{G T} - t_{G}^{G T}) y_{G}^{G T}$ , plus the value of deliveries of finished goods, less payments to industry S for service intermediate inputs, $- p_{S}^{S G} y_{S}^{S G}$ , less payments to industry T for transportation service intermediate inputs, $- p_{T}^{T G} y_{T}^{T G}$ , plus the value of exports delivered to the ROW sector, $(p_{G x}^{G R} - t_{G x}^{G R}) x_{G}^{G R}$ ,³⁰ less payments to the ROW for imports of goods G used by industry G, $- (p_{G m}^{G R} + t_{G m}^{G R}) m_{G}^{G R}$ , less payments to the ROW for imports of services S used by industry G, $- (p_{S m}^{G R} + t_{S m}^{G R}) m_{S}^{G R}$ , less payments to the ROW for imports of transportation services T used by industry G, $- (p_{T m}^{G R} + t_{T m}^{G R}) m_{T}^{G R}$ . The decompositions for the value added produced by industries S and T, va^S and va^T, 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, $p_{G x}^{G R} x_{G}^{G R} + p_{S x}^{S R} x_{S}^{S R} + p_{T x}^{T R} x_{S}^{T R}$ . 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 $(p_{G x}^{G R} - t_{G x}^{G R}) x_{G}^{G R}$ for industry G, $(p_{S x}^{S R} - t_{S x}^{S R}) x_{S}^{S R}$ for industry S, and $(p_{T x}^{T R} - t_{T x}^{T R}) x_{T}^{T R}$ 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, $(p_{G x}^{G R} - t_{G x}^{G R}) x_{G}^{G R} + (p_{S x}^{S R} - t_{S x}^{S R}) x_{S}^{S R} + (p_{T x}^{T R} - t_{T x}^{T R}) x_{T}^{T R}$ . 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 GDP_P 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 GDP_F. 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 GDP_P because the two methods for forming estimates of GDP_P are simply alternative methods for summing over the elements of the net supply matrix.³² Thus the following equation must hold:

\begin{matrix} {GDP}_{p} \equiv v f_{G} + v f_{S} + v f_{T} = v a^{G} + v a^{S} + v a^{T} . & (20.25) \end{matrix}

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{matrix} G D P_{p} = & v f_{G} + v f_{S} + v f_{T} \\ = & G D P_{F} - T \\ = & [C + I + G] + X - M - T, (20.26) \end{matrix}

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

\begin{matrix} T & \equiv t_{G}^{G S} y_{G}^{G S} + t_{G}^{G T} y_{G}^{G T} + t_{G}^{G F} y_{G}^{G F} + t_{G x}^{G R} x_{G}^{G R} \\ + t_{G m}^{G R} m_{G}^{G R} + t_{G m}^{S R} m_{G}^{S R} + t_{G m}^{T R} m_{G}^{T R} + t_{S}^{S G} y_{S}^{S G} \\ + t_{S}^{S T} y_{S}^{S T} + t_{S}^{S F} y_{S}^{S F} + t_{S x}^{S R} x_{S}^{S R} + t_{S m}^{G R} m_{S}^{G R} + t_{S m}^{S R} m_{S}^{S R} \\ + t_{S m}^{T R} m_{S}^{T R} + P_{T}^{T G} y_{T}^{T G} + p_{T}^{T S} y_{T}^{T S} + t_{T}^{T F} y_{T}^{T F} + t_{T x}^{T R} x_{T}^{T R} \\ + t_{T m}^{G R} m_{T}^{G R} + t_{T m}^{S R} m_{T}^{S R} + t_{T m}^{T R} m_{T}^{T R}; (20.27) \end{matrix}

and GDP_F 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{matrix} G D P_{F} & \equiv [P_{G}^{G F} y_{G}^{G F} + p_{S}^{S F} y_{S}^{S F} + p_{T}^{T F} y_{T}^{T F}] \\ + [t_{G x}^{G R} x_{G}^{G R} + t_{S x}^{S R} m_{G}^{S R} + t_{T x}^{T R} x_{T}^{T R}] \\ - [p_{G m}^{G R} m_{G}^{G R} + p_{G m}^{S R} m_{G}^{S R} + p_{G m}^{T R} m_{G}^{T R} \\ + p_{S m}^{G R} m_{S}^{G R} + p_{S m}^{S R} m_{S}^{S R} + p_{S m}^{T R} m_{S}^{T R} \\ + p_{T m}^{G R} m_{T}^{G R} + p_{T m}^{S R} m_{T}^{S R} + P_{T m}^{T R} m_{T}^{T R}] \\ = [C + G + I] + [X] - [M] . & (20.28) \end{matrix}

20.72 Note that the 15 terms that do not involve taxes on the right-hand side of equation (20.26), which define GDP_F, 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 GDP_P 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{matrix} {GDP}_{P} = {GDP}_{P} + T . & (20.29) \end{matrix}

20.73 Thus the value of production at final demand prices, GDP_F, is equal to the value of production at producer prices, GDP_P, 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 $P_{T}^{v a} (p^{v a 0,} p^{v a 1}, q^{v a 0}, q^{v a 1})$ , the Törnqvist price index using the 48 price times quantity components in the value-added aggregate, is equal to $P_{T}^{f d} (p^{f d 0,} p^{f d 1}, q^{f d 0}, q^{f d 1})$ , 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, GDP_P 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 GDP_P 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, GDP_F, with volume estimates for GDP at producer prices, GDP_P. Recall equation (20.29), which said that GDP_F equals GDP_P plus I. Suppose that data are available for two periods that respect equation (20.29) in each period and a quantity index, GDP_F, is constructed, defined by equation (20.28) with 15 separate price times quantity components. Then noting that GDP_P 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 GDP_P 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 GDP_F and GDP_P + T will coincide if the Laspeyres, Paasche, or Fisher formulas are used. For the GDP_P + T aggregate, two-stage aggregation could be used where the first-stage value aggregates are GDP_P, 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 GDP_F to volume growth in GDP_P at producer prices plus commodity tax effects. More generally, the identity (20.29) can be used to estimate GDP_F if the statistical agency is able to estimate GDP_P 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

S1, traditional services, and
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 $y_{G 4}^{G S}$ denotes the quantity of good G4 that is delivered by the goods producing industry G to the services industry S, $y_{G 4}^{G T}$ denotes the quantity of good G4 that is delivered by the goods producing industry G to the transportation industry T, $y_{G 4}^{G F}$ denotes the quantity of good G4 that is delivered by the goods producing industry G to the domestic final demand sector F, $y_{G 1}^{S F}$ 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, $y_{G 2}^{S F}$ 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, $y_{S 1}^{S G}$ denotes the quantity of traditional services S1 that is delivered by the services industry S to the goods producing industry G, $y_{S 2}^{S G}$ denotes the quantity of high-tech services S2 that is delivered by the services industry S to the goods producing industry G, $y_{T}^{T G}$ 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 $y_{G 4}^{G S}$ units of this good to the service industry S to be used as an intermediate input there, delivers $y_{G 4}^{G T}$ units of this good to the transportation industry T to be used as an intermediate input there, and delivers $y_{G 4}^{G F}$ units of this good to the domestic final demand sector F. Similarly, industry S produces the general service commodity S1 and delivers $y_{S 1}^{S G}$ units of this commodity to the goods producing industry G to be used as an intermediate input there, delivers $y_{S 1}^{S T}$ units of this service to the transportation industry T to be used as an intermediate input there, and delivers $y_{S 1}^{S F}$ units of this service to the domestic final demand sector F. Industry S also produces the high-technology service commodity S2 and delivers $y_{S 2}^{S G}$ units of this commodity to the goods producing industry G to be used as an intermediate input there, delivers $y_{S 2}^{S T}$ units of this service to the transportation industry T to be used as an intermediate input there, and delivers $y_{S 2}^{S F}$ 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 $y_{S G 1}^{S F}$ and $y_{S G 2}^{S F}$ .³⁷ Finally, industry T produces the transportation services commodity T and delivers $y_{T}^{T G}$ units of this commodity to the goods producing industry G to be used as an intermediate input there, delivers $y_{T}^{T S}$ units of these transport services to the service industry S to be used as an intermediate input there, and delivers $y_{T}^{T F}$ 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 $x_{G 4}^{G R}$ units of good G4, industry S exports $x_{S 1}^{S R}$ units of traditional services S1 and no units of high-tech services, and industry T exports $x_{T}^{T R}$ 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 $m_{G 1}^{G R}$ units of agricultural imports, $m_{G 2}^{G R}$ units of crude oil imports, $m_{G 3}^{G R}$ units of a pure imported intermediate good, and $m_{S 1}^{G R}$ 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 $m_{G 1}^{S R}$ units of agricultural goods (for distribution to domestic households), $m_{G 2}^{S R}$ units of crude oil (for distribution to households and own use), $m_{S 1}^{G R}$ units of foreign general services, and $m_{T}^{S R}$ units of foreign transportation services. Industry T imports $m_{G 2}^{T R}$ units of crude oil and $m_{T}^{T R}$ 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 $(p_{G 4}^{G S} - t_{G 4}^{G S}) y_{G 4}^{G S}$ for its sales of commodity G4 to industry S, whereas in Table 20.17, industry S pays the amount $p_{G 4}^{G S} y_{G 4}^{G S}$ for these purchases of intermediate inputs from industry G. The difference between these two intersectoral value flows is $t_{G 4}^{G S} y_{G 4}^{G S}$ , 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 $t_{S 1}^{S G} t_{G 4}^{G S}$ , 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 $x_{G 4}^{G R}$ units of good G4 and the foreign final demander pays the price $p_{G 4 x}^{G R}$ per unit but the exporting industry gets only the amount $p_{G 4 x}^{G R} - t_{G 4 x}^{G R}$ per unit; that is, the government gets the per unit (net) revenue $t_{G 4 x}^{G R}$ on these sales if it imposes a (net) export tax equal to $t_{G 4 x}^{G R}$ . 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, $p_{G 4}^{G S}$ , to the transportation services industry, $p_{G 4}^{G T}$ , and for exports, $p_{G 4 x}^{G R}$ , 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, $p_{G 4}^{G F}$ , is somewhat higher than the corresponding prices for deliveries of G4 to the service sector S, $p_{G 4}^{G S}$ , and to the transportation sector T, $p_{G 4}^{G T}$ , owing to higher commodity taxes on deliveries to sector F. The price of traditional domestic services used as an intermediate input by industry G, $p_{S 1}^{S G}$ , 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, $p_{S 2}^{S G}$ , drops rapidly throughout the sample period. The price of transportation services used as an intermediate input by industry G, $p_{T}^{T G}$ , 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, $p_{G 1 m}^{G R}$ , 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, $p_{G 2 m}^{G R}$ , 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, $p_{G 3 m}^{G R}$ , steadily drops at a rapid pace over the five periods.⁴¹ Finally, the price of the imported services commodity, $p_{S m}^{G R}$ , 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, $t_{S 1}^{S G}$ , $t_{S 2}^{S G}$ and $t_{T}^{T G}$ , 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 $t_{G 4 x}^{G R}$ 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, $y_{G 4}^{G S}, y_{G 4}^{G T}, y_{G 4}^{G F},$ , 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, $x_{G 4 x}^{G R}$ , 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, $y_{S 1}^{S G}$ , more than doubles over the five periods but the quantity of high-tech services used as an intermediate input, $y_{S 1}^{S G}$ , 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, $y_{T}^{T G}$ , exactly doubles over the five periods. The quantity of agricultural imports used by industry G, $m_{G 1 m}^{G R}$ , 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, $m_{G 3 m}^{G R}$ , 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, $p_{S 1}^{S G}$ and $p_{S 2}^{S G}$ , the prices of deliveries of good G4 from industry G to industry S, $p_{G 4}^{G S}$ , 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 $p_{S 1}^{S G}$ and $p_{S 2}^{S G}$ for sales to industry G and $p_{S 1}^{S T}$ and $p_{S 2}^{S T}$ for sales to industry T) and it sells commodities S1 and S2 to the domestic final demand sector F at prices $p_{S 1}^{S F}$ and $p_{S 2}^{S F}$ and it sells S1 to the rest of the world R as an export at the price $p_{S 1 x}^{S R}$ . 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 $p_{G 1 m}^{S R}, p_{G 2 m}^{S R}, p_{S 1 m}^{S R}$ , and $p_{T m}^{S R}$ , 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, $p_{T m}^{S R}$ , is the same as the price of domestic transportation services provided to industry S, $p_{T}^{T S}$ . Note that the service sector selling prices of goods G1 and G2 to the domestic final demand sector, $p_{G 1}^{S F}$ and $p_{G 2}^{S F}$ , are somewhat higher than the corresponding import purchase prices for these goods, $p_{G 1 m}^{S R}$ and $p_{G 2}^{S F}$ , 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

20.105 The quantities of industry S deliveries to industry G, $y_{S 1}^{S G}$ and $y_{S 2}^{S G}$ , and the quantities of deliveries of good G4 from industry G to industry S, $y_{G 4}^{G S}$ , are exactly the same in Tables 20.22 and 20.25.

20.106 Note that $y_{G 1}^{S F}$ , the quantity of imported food G1 sold by industry S to domestic final demanders F, is exactly equal to $m_{G 1 m}^{S R}$ , imports of food into industry S. However, $y_{G 2}^{S F}$ , the quantity of imported energy products G2 sold by industry S to domestic final demanders, is less than the quantity of energy imported by industry S, $m_{G 2 m}^{S R}$ . The reason for this difference is that industry S uses some of the imported energy for heat and other purposes as it supplies services to other sectors of the economy. Sales by industry S of traditional services S1 to industry G, industry T, domestic final demand F, and the rest of the world R, $y_{S 1}^{S G}, y_{S 1}^{S T}, y_{S 1}^{S F}$ , and $x_{S 1 x}^{S R}$ , respectively, all increase quite rapidly, doubling or tripling over the five periods. Imports of traditional services S1 into industry S, $m_{S 1 m}^{S R}$ , increase even more rapidly, growing from 3 to 13 over the five periods. The sales of high-tech services by industry S to industries G and T and to the domestic final demand sector F, $y_{S 2}^{S G}, y_{S 2}^{S T}$ , and $y_{S 2}^{S F}$ , respectively, all increase very rapidly, growing between 5- and 10-fold over the five periods. The quantities of domestic intermediate inputs of good G4, $y_{G 4}^{G S}$ , and of the transportation service, $y_{T}^{T S}$ , used by industry S grew fairly slowly over the five periods. Imported transportation services, $m_{T m}^{S R}$ , associated with the importation of G1 and G2 by industry S, doubled over the five periods.

C.5 Industry T prices and quantities

20.107 The nine final demand price series that form part of the industry T data in Tables 20.16 through 20.19 are listed for five periods in Table 20.26.

Table 20.26.

Industry T Final Demand Prices

20.108 The entries for $p_{T}^{T S}, p_{S 1}^{S T}$ , and $p_{S 2}^{S T}$ in Tables 20.26 and 20.23 are the same series as are the entries for $p_{T}^{T G}$ and $p_{G 4}^{G T}$ in Tables 20.26 and 20.20. Again, this reflects the fact that the sellers and purchasers of domestic intermediate inputs pay and receive the same amounts of money for their cross-industry purchases and sales.

20.109 The industry selling prices for transportation services show much the same trends across all destinations. The selling prices of transportation services to domestic final demand, $p_{T}^{T F}$ , are higher than the other selling prices owing to higher commodity taxation for deliveries to final demand.

20.110 The commodity tax series that form part of the industry T taxes listed in Tables 20.16 through 20.19 are listed for five periods in Table 20.27.

Table 20.27.

Industry T Commodity Taxes

20.111 The commodity taxes on deliveries of transportation services to industries G and S, $t_{T}^{T G}$ and $t_{T}^{T S}$ respectively, are small but the taxes on deliveries to the final demand sector, $t_{T}^{T F}$ , are fairly substantial, as are the taxes on the transportation sector’s imports of energy products, $t_{G 2 m}^{T R}$ . By convention, any taxes on industry T’s use of domestic intermediate inputs are paid by the selling industry so $t_{G 4}^{G T}, t_{S 1}^{S T}$ , and $t_{S 2}^{S T}$ are all zero. There are no taxes on the export of transportation services in this economy so that $t_{T x}^{T R}$ is zero as well. There are small taxes on industry T’s importation of transport services, $t_{T m}^{T R}$ .

20.112 The nine final demand quantity series that form part of the industry T data in Tables 20.16 through 20.19 are listed for five periods in Table 20.28.

20.113 The entries for $y_{T}^{T S}, y_{S 1}^{S T}$ , and $y_{S 2}^{S T}$ in Tables 20.28 and 20.25 are the same series as are the entries for $y_{T}^{T G}$ and $y_{G 4}^{G T}$ in Tables 20.28 and 20.22. All transportation service inputs and outputs grow relatively smoothly and roughly double over the five periods.

Table 20.28.

Industry T Quantities of Outputs and Inputs

20.114 This completes the listing of the basic price, tax, and quantity data that are used in subsequent sections of this chapter in order to illustrate how various index number formulas differ and how consistent sets of producer price indices can be formed in a set of production accounts that are roughly equivalent to the production accounts that are described in Chapter 15 of the 2008 SNA.

D. The Artificial Data Set for Domestic Final Demand

D.1 The final demand data set

20.115 In order to illustrate what kind of differences can result from the choice of different index number formulas, the price and quantity data that correspond to domestic deliveries to final demand that were listed in the previous section are used as a test data set in this section. The six final demand price series are listed in Table 20.29 and the corresponding quantity series are listed in Table 20.30.

Table 20.29.

Prices for Six Domestic Final Demand Commodities

Table 20.30.

Quantities for Six Domestic Final Demand Commodities

20.116 The prices $p_{1}^{t}, p_{2}^{t}, p_{3}^{t}, p_{4}^{t}, p_{5}^{t}$ , and $p_{6}^{t}$ in Table 20.30 correspond to the final demand prices $p_{G 1}^{S F}, P_{G 2}^{S F}, P_{G 4}^{G F}, p_{S 1}^{S F}, p_{S 2}^{S F}$ , and $p_{T}^{T F}$ , respectively, which are listed in Tables 20.20, 20.23, and 20.26.

20.117 The quantities $q_{1}^{t}, q_{2}^{t}, q_{3}^{t}, q_{4}^{t}, q_{5}^{t}$ , and $q_{6}^{t}$ in Table 20.30 correspond to the final demand quantities $y_{G 1}^{S F}, y_{G 2}^{S F}, y_{G 4}^{G F}, y_{S 1}^{S F}, y_{S 2}^{S F}$ , and $y_{T}^{T F}$ , respectively, which are listed in Tables 20.22, 20.25, and 20.28.

20.118 It is useful to also list the period T expenditures on all six domestic finally demanded commodities, p^t · q^t, along with the corresponding expenditure shares, $s_{1}^{t}, ..., s_{6}^{t}$ ; see Table 20.31.

Table 20.31.

Total Expenditures and Expenditure Shares for Six Domestic Final Demand Commodities

Table 20.31.

Total Expenditures and Expenditure Shares for Six Domestic Final Demand Commodities

	Expenditures	Food	Energy	Goods	Services	High-Tech Services	Transport Services
Period t	p^t·q^t	$s_{1}^{t}$	$s_{2}^{t}$	$s_{3}^{t}$	$s_{4}^{t}$	$s_{5}^{t}$	$s_{6}^{t}$
1	87.150	0.1377	0.1285	0.4016	0.2238	0.0396	0.0688
2	142.742	0.1156	0.1765	0.3643	0.2522	0.0283	0.0631
3	176.080	0.0818	0.1124	0.4089	0.3124	0.0266	0.0579
4	211.775	0.0982	0.0708	0.3818	0.3740	0.0223	0.0529
5	273.150	0.0871	0.1208	0.3423	0.3807	0.0126	0.0564

Table 20.31.

Total Expenditures and Expenditure Shares for Six Domestic Final Demand Commodities

	Expenditures	Food	Energy	Goods	Services	High-Tech Services	Transport Services
Period t	p^t·q^t	$s_{1}^{t}$	$s_{2}^{t}$	$s_{3}^{t}$	$s_{4}^{t}$	$s_{5}^{t}$	$s_{6}^{t}$
1	87.150	0.1377	0.1285	0.4016	0.2238	0.0396	0.0688
2	142.742	0.1156	0.1765	0.3643	0.2522	0.0283	0.0631
3	176.080	0.0818	0.1124	0.4089	0.3124	0.0266	0.0579
4	211.775	0.0982	0.0708	0.3818	0.3740	0.0223	0.0529
5	273.150	0.0871	0.1208	0.3423	0.3807	0.0126	0.0564

20.119 The expenditure shares for food, goods, and high-tech services decrease markedly over the five periods, the share for transport services decreases somewhat, the share of energy stays roughly constant but with substantial period-to-period fluctuations, and the share of general services increases substantially.

20.120 Note that the price of food and energy fluctuates considerably from period to period but the quantities demanded tend to trend upwards at a fairly smooth rate, reflecting the low elasticity of price demand for these products. The fluctuations in energy prices tend to produce similar fluctuations in the price of domestic transportation services but the fluctuations in price are more damped in the case of transport services. The price of goods trends up fairly rapidly in periods 2 and 3 but then the rate of increase falls off. The corresponding quantity trends upwards fairly steadily. The price of traditional services, $p_{4}^{t}$ , increases rapidly in periods 2 and 3 and then increases more slowly. Overall, the price of traditional services increases more rapidly than the price of goods but the quantity of services demanded, $q_{4}^{t}$ , increases more rapidly than the quantity of goods, $q_{3}^{t}$ . The price of high-technology services, $p_{5}^{t}$ , decreases rapidly over the five periods, falling to about one-fifth of the initial price level. The corresponding quantities demanded, $q_{5}^{t}$ , increase rapidly, growing five-fold over the sample period. Thus overall, the data set exhibits a rather wide variety of trends in prices and quantities but these trends are not unrealistic in today’s world economy. The movements of prices and quantities in this artificial data set are more pronounced than the year-to-year movements that would be encountered in a typical country but they do illustrate the problem that is facing compilers of producer and consumer price indices: Namely, year-to-year price and quantity movements are far from being proportional across commodities so the choice of index number formula will matter.

D.2 Some familiar index number formulas

20.121 Every price statistician is familiar with the Laspeyres index, P_L, and the Paasche index, P_P, defined in Chapter 15. These indices are listed in Table 20.32 along with the two main unweighted indices that were considered in Chapters 10, 17, and 21: the Carli index and the Jevons index. The indices in Table 20.31 compare the prices in period t with the prices in period 1; that is, they are fixed-base indices. Thus the period t entry for the Carli index, P_C, is simply the arithmetic mean of the eight price relatives, $Σ_{i = 1}^{6} (1 / 6) (p_{i}^{t} / p_{i}^{1})$ , while the period t entry for the Jevons index, P_J, is the geometric mean of the six long-term price relatives, $Π_{i = 1}^{6} (p_{i}^{t} / p_{i}^{1}) 0^{1 / 6}$ .

Table 20.34.

Asymmetrically Weighted Fixed-Base Indices

20.122 Note that by period 5, the spread between the fixed-base Laspeyres and Paasche price indices is not negligible: P_L is equal to 1.7348 while P_P is 1.6570, a spread of about 4.7 percent. Because both of these indices have exactly the same theoretical justification, it can be seen that the choice of index number formula matters. There is also a substantial spread between the two unweighted indices by period 5: The fixed-base Carli index is equal to 1.5488, while the fixed-base Jevons index is 1.2483, a spread of about 24 percent. However, more troublesome than this spread is the fact that the unweighted indices are well below both the Paasche and Laspeyres indices by period 5. Thus when there are divergent trends in both prices and quantities, it will usually be the case that unweighted price indices will give very different answers than their weighted counterparts. Because none of the index number theories considered in previous chapters supported the use of unweighted indices, their use is not recommended for aggregation at the “higher level,” that is, when data on weights are available. However, in Chapter 21 aggregation at the “lower level” is considered for when weights are unavailable, and the use of unweighted index number formulas will be revisited. Finally, note that the Jevons index is always considerably below the corresponding Carli index. This will always be the case (unless prices are proportional in the two periods under consideration) because a geometric mean is always equal to or less than the corresponding arithmetic mean.⁴⁴

Table 20.32.

Fixed-Base Laspeyres, Paasche, Carli, and Jevons Indices

Table 20.33.

Chained Laspeyres, Paasche, Carli, and Jevons Indices

20.123 It is of interest to recalculate the four indices listed in Table 20.32 using the chain principle rather than the fixed-base principle. Our expectation is that the spread between the Paasche and Laspeyres indices will be reduced by using the chain principle. These chained indices are listed in Table 20.33.

20.124 It can be seen comparing Tables 20.32 and 20.33 that chaining eliminated most of the spread between the fixed-base Paasche and Laspeyres indices for period 5; that is, the spread between the chained Laspeyres and Paasche indices has dropped from 4.7 percent to 0.6 percent. Note that chaining did not affect the Jevons index. This is an advantage of the index but the lack of weighting is a fatal flaw. The “truth” would be expected to lie between the Paasche and Laspeyres indices but from Tables 20.32 and 20.33, the unweighted Jevons index is far below this acceptable range. The fixed-base and chained Carli indices also lie outside this acceptable range.

D.3 Asymmetrically weighted index number formulas

20.125 A systematic comparison of all of the asymmetrically weighted price indices is now undertaken. The fixed-base indices are listed in Table 20.34. The fixed-base Laspeyres and Paasche indices, P_L and P_P, are the same as those indices listed in Table 20.32. The Palgrave index, P_PAL, is defined by equation (16.55) in Chapter 16. The indices denoted by P_GL and P_GP are the geometric Laspeyres and geometric Paasche indices,⁴⁵ which were defined in Chapter 16. For the geometric Laspeyres index, P_GL, the weights for the price relatives are the base-period expenditure shares, $s_{i}^{1}$ . This index should be considered an alternative to the fixed-base Laspeyres index because each of these indices makes use of the same information set. For the geometric Paasche index, PGP, the weights for the price relatives are the current-period expenditure shares, $s_{i}^{t}$ . Finally, the index P_HL is the harmonic Laspeyres index that was defined by equation (16.59).

20.126 By looking at the period 5 entries in Table 20.34, we can see that the spread between all of these fixed-base asymmetrically weighted indices has increased to be much larger than our earlier spread of 4.7 percent between the fixed-base Paasche and Laspeyres indices. In Table 20.34, the period 5 Palgrave index is about 1.36 times as big as the period 5 harmonic Laspeyres index, P_HL. Again, this illustrates the point that owing to the nonproportional growth of prices and quantities in most economies today, the choice of index number formula is very important.⁴⁶

20.127 It is possible to explain why certain of the indices in Table 20.34 are bigger than others. When all weights are positive, it can be shown that a weighted arithmetic mean of N numbers is equal to or greater than the corresponding weighted geometric mean of the same N numbers which in turn is equal to or greater than the corresponding weighted harmonic mean of the same N numbers.⁴⁷ It can be seen that the three indices P_PAL, P_GP, and P_P all use the current-period expenditure shares $s_{i}^{t}$ to weight the price relatives $(p_{i}^{t} / p_{i}^{1})$ but P_PAL is a weighted arithmetic mean of these price relatives, P_GP is a weighted geometric mean of these price relatives, and P_P is a weighted harmonic mean of these price relatives. Thus because there are no negative components in final demand, by Schlömilch’s (1858) inequality,⁴⁸

\begin{matrix} P_{P A L} \geq P_{G P} \geq P_{p .} & (20.30) \end{matrix}

20.128 Viewing Table 20.34, we can see that the inequalities (20.30) hold for all periods. It can also be verified that the three indices P_L, P_GL, and P_HL all use the base-period expenditure shares $s_{i}^{1}$ to weight the price relatives $(p_{i}^{t} / p_{i}^{1})$ but P_L is a weighted arithmetic mean of these price relatives, P_GL is a weighted geometric mean of these price relatives, and P_HL is a weighted harmonic mean of these price relatives. Because all of the expenditure shares are positive, then by Schlömilch’s inequality,⁴⁹

\begin{matrix} P_{L} \geq P_{G L} \geq P_{H L} . & (20.31) \end{matrix}

Viewing Table 20.34, we can see that the inequalities (20.31) hold for all periods.

20.129 Now continue with the systematic comparison of all of the asymmetrically weighted price indices. These indices using the chain principle are listed in Table 20.35.

Table 20.35.

Asymmetrically Weighted Chained Indices

20.130 Viewing Table 20.35, we can see that the use of the chain principle only marginally reduced the spread between all of the asymmetrically weighted indices compared to their fixed-base counterparts in Table 20.34. For period 5, the spread between the smallest and largest asymmetrically weighted fixed-base index was 35.7 percent but for the period 5 chained indices, this spread was marginally reduced to 33.9 percent.

D.4 Symmetrically weighted index number formulas

20.131 Symmetrically weighted indices can be decomposed into two classes: superlative indices and other symmetrically weighted indices. Superlative indices have a close connection to economic theory; that is, as was seen in Chapter 18, a superlative index is exact for a representation of the producer’s production function or the corresponding unit revenue function that can provide a second order approximation to arbitrary technologies that satisfy certain regularity conditions. In Chapter 18 four primary superlative indices were considered:

The Fisher ideal price index, P_F, defined by equation (16.12);
The Walsh price index, P_W, defined by equation (16.19) (this price index also corresponds to the quantity index Q¹ defined by equation (18.60));
The Törnqvist Theil price index, P_T, defined by equation (18.11); and
The implicit Walsh price index, P_IW, that corresponds to the Walsh quantity index QW defined by equation (17.34) (this is also the index P¹ defined by equation (18.60)).

20.132 These four symmetrically weighted superlative price indices are listed in Table 20.36 using the fixed-base principle. Also listed in this table are two symmetrically weighted price indices:⁵⁰

Table 20.36.

Symmetrically Weighted Fixed-Base Indices

The Marshall Edgeworth price index, P_ME, defined by equation (15.18), and
The Drobisch price index, P_D, the arithmetic average of the Paasche and Laspeyres price indices.

20.133 Note that the Drobisch index P_D is always equal to or greater than the corresponding Fisher index P_i. This follows from the fact that the Fisher index is the geometric mean of the Paasche and Laspeyres indices while the Drobisch index is the arithmetic mean of the Paasche and Laspeyres indices and an arithmetic mean is always equal to or greater than the corresponding geometric mean. Comparing the fixed-base asymmetrically weighted indices, Table 20.34, with the symmetrically weighted indices, Table 20.36, we can see that the spread between the lowest and highest index in period 5 is much less for the symmetrically weighted indices. The spread was 1.8316/1.3499 = 1.357 for the asymmetrically weighted indices but only 1.72041/1.68389 = 1.022 for the symmetrically weighted indices. If the analysis is restricted to the superlative indices listed for period 5 in Table 20.19, then this spread is further reduced to 1.72041/1.69545 = 1.015; that is, the spread between the fixed-base superlative indices is only 1.5 percent compared to the fixed-base spread between the Palgrave and harmonic Laspeyres indices of 35.7 percent (1.8316/1.3499 = 1.357). The spread between the superlative indices can be expected to be further reduced if the chain principle is used.

20.134 The symmetrically weighted indices are recomputed using the chain principle. The results may be found in Table 20.37. A quick glance at Table 20.20 shows that the combined effect of using both the chain principle and symmetrically weighted indices is to dramatically reduce the spread between all indices constructed using these two principles. The spread between all of the symmetrically weighted indices in period 5 is only 1.7127/1.7116 = 1.0006 or 0.06 percent, which is negligible.

Table 20.37.

Symmetrically Weighted Chained Indices

20.135 The results listed in Table 20.37 reinforce the numerical results tabled in Robert Hill (2006) and Diewert (1978, p. 894): The most commonly used chained superlative indices will generally give approximately the same numerical results.⁵¹ This numerical approximation property holds in spite of the erratic nature of the fluctuations in the data in Tables 20.29 through 20.31. In particular, the chained Fisher, Törnqvist, and Walsh indices generally approximate each other very closely.

D.5 Superlative indices and two-stage aggregation

20.136 Attention is now turned to the differences between superlative indices and their counterparts that are constructed in two stages of aggregation; see Section E.6 of Chapter 18 for a discussion of the issues and a listing of the formulas used. In the artificial data set for domestic final demand, the first three commodities are aggregated into a goods aggregate and the final three commodities are aggregated into a services aggregate. In the second stage of aggregation, the good and services components will be aggregated into a domestic final demand price index.

20.137 The results of single-stage and two-stage aggregation are reported in Table 20.38 using period 1 as the fixed base for the Fisher index P_F, the Törnqvist index P_T, and the Walsh and implicit Walsh indices, P_W and P_IW.

Table 20.38.

Single-Stage and Two-Stage Fixed-Base Superlative Indices

20.138 Viewing Table 20.38, it can be seen that the fixed-base single-stage superlative indices generally approximate their fixed-base two-stage counterparts fairly closely. The divergence between the single-stage Törnqvist index P_T and its two-stage counterpart P_T2S in period 5 is 1.7108/1.7007 = 1.006 or 0.6 percent. The other divergences are even less.

20.139 The results reported in Table 20.39 use chained versions of these indices for the two-stage aggregation procedure. Again, the single-stage and their two-stage counterparts are listed for the Fisher index P_F, the Törnqvist index P_T, and the Walsh and implicit Walsh indices, P_W and P_IW.

Table 20.39.

Single-Stage and Two-Stage Chained Superlative Indices

20.140 Viewing Table 20.39, we can see that the chained single-stage superlative indices generally approximate their fixed-base two-stage counterparts quite closely. The divergence between the chained Törnqvist index P_T and its two-stage counterpart P_T2S in period 5 is 1.7136/1.7122 = 1.0008 or 0.08 percent. The other divergences are all less than this. Given the large dispersion in period-to-period price movements, these two-stage aggregation errors are not large. However, the important point that emerges from Table 20.39 is that the use of the chain principle has reduced the spread between all eight single-stage and two-stage superlative indices compared to their fixed-base counterparts in Table 20.38. The maximum spread for the period 5 chained index values is 0.09 percent while the maximum spread for the period 5 fixed-base index values is 1.5 percent.

20.141 The final formulas that is illustrated using the artificial final expenditures data set are the additive percentage change decompositions for the Fisher ideal index that were discussed in Section C.8 of Chapter 17. The chain links for the Fisher price index will first be decomposed using the Diewert (2002a) decomposition formulas (16.41) to (16.43). The results of the decomposition are listed in Table 20.40. Thus P_F – 1 is the percentage change in the Fisher ideal chain link going from period t – 1 to t and the decomposition factor $v_{F i} Δ p i = v_{F i} (p_{i}^{t} - p_{i}^{t - 1})$ is the contribution to the total percentage change of the change in the ith price from $p_{i}^{t - 1}$ to $p_{i}^{t}$ to for i = 1, 2, . . . , 6.

20.142 Viewing Table 20.40, we can see that the price index going from period 1 to 2 grew 39.30 percent and the contributors to this change were the increases in the price of commodity 1, finally demanded agricultural products (3.31 percentage points); commodity 2, finally demanded energy (12.53 percentage points); commodity 3, finally demanded goods (11.85 percentage points); commodity 4, traditional services (9.28 percentage points); and commodity 6, transportation services (3.14 percentage points). High-technology services, commodity 5, decreased in price and this fall in prices subtracted 0.82 percentage points from the overall Fisher price index going from period 1 to 2. The sum of the last six entries for period 2 in Table 20.40 is equal to 0.3930, the percentage increase in the Fisher price index going from period 1 to 2. It can be seen that a big price change in a particular component i combined with a big expenditure share in the two periods under consideration will lead to a big decomposition factor, v_FiΔpi.

Table 20.40.

Diewert Additive Percentage Change Decomposition of the Fisher Index

20.143 Our final set of computations illustrates the additive percentage change decomposition for the Fisher ideal index that is due to Van Ijzeren (1987, p. 6) that was mentioned in Section C.8 of Chapter 17.⁵² First, the Fisher price index going from period t – 1 to t is written in the following form:

\begin{matrix} P_{F} (p^{t - 1}, p^{t}, q^{t - 1}, q^{t}) = \frac{Σ_{i = 1}^{6} q_{F i}^{*} p_{i}^{t}}{Σ_{i = 1}^{6} q_{F i}^{*} p_{i}^{t - 1}}, & (20.32) \end{matrix}

where the reference quantities need to be defined somehow. Van Ijzeren (1987, p. 6) showed that the following reference weights provided an exact additive representation for the Fisher ideal price index:

\begin{matrix} q_{F i} * \equiv (1 / 2) q_{i}^{t - 1} + [(1 / 2) Q_{F} (p^{t - 1}, p^{t}, q^{t - 1}, q^{t})]; \\ i = 1, 2, . . . ., 6, (20.33) \end{matrix}

where Q_F is the overall Fisher quantity index. Thus using the Van Ijzeren quantity weights q_Fi^*, we obtain the following Van Ijzeren additive percentage change decomposition for the Fisher price index:

\begin{matrix} P_{F} (p^{0}, p^{1}, q^{0}, q^{1}) - 1 = & \frac{Σ_{i = 1}^{6} q_{F i}^{*} p_{i}^{t}}{Σ_{i = 1}^{6} q_{F i}^{*} p_{i}^{t - 1}} - 1 \\ = & Σ_{i = 1}^{6} v_{F i}^{*} (p_{i}^{t} - p_{i}^{t - 1}), (20.34) \end{matrix}

where the Van Ijzeren weight for commodity i, $v_{F i}^{t *}$ , is defined as

v_{F i}^{t *} \equiv \frac{Σ_{i = 1}^{6} q_{F i}^{*}}{Σ_{i = 1}^{6} q_{F i}^{*} p_{i}^{t - 1}}; i = 1, 2, . . ., 6. (20.35)

20.144 The chain links for the Fisher price index are decomposed using formulas (20.32) to (20.35). The results of the decomposition are listed in Table 20.41. Thus P_F – 1 is the percentage change in the Fisher ideal chain link going from period t – 1 to t, and the Van Ijzeren decomposition factor $v_{F i}^{t *} Δ p_{i}^{t}$ is the contribution to the total percentage change of the change in the ith price from $p_{i}^{t - 1}$ to $p_{i}^{t}$ for i = 1, 2, . . . , 6.

Table 20.41.

Van Ijzeren Additive Percentage Change Decomposition of the Fisher Index

20.145 Comparing the entries in Tables 20.40 and 20.41, we can see that the differences between the Diewert and Van Ijzeren decompositions of the Fisher price index are very small.⁵³ This is somewhat surprising given the very different nature of the two decompositions.⁵⁴ As was mentioned in Section C.8 of Chapter 17, the Van Ijzeren decomposition of the chain Fisher quantity index is used by the Bureau of Economic Analysis in the United States.⁵⁵

E. National Producer Price Indices

E.1 The national gross domestic output price index at producer prices

20.146 In this subsection and the following three subsections, national domestic gross output, export, domestic intermediate input, and import price indices at producer prices (i.e., at basic prices for outputs and purchaser’s prices for intermediate inputs) will be calculated using the data for each of the three industrial sectors listed in Section B. Only fixed-base and chained Laspeyres, Paasche, Fisher, and Törnqvist indices will be computed because these are the ones most likely to be used in practice.

20.147 It should be noted that the price indices computed in this section are appropriate ones to use for the calculation of business sector labor or multifactor productivity purposes.

20.148 The data listed in Tables 20.20 through 20.28 for industries G, S, and T are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist price indices for domestic outputs (at producer prices or basic prices in this case) for periods t equal 1 to 5, $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , respectively. Producer prices (as opposed to final demand prices) are used in these computations. There are three domestic output deliveries from industry G, eight domestic output deliveries from industry S, and three domestic output deliveries from industry T so that each index is an aggregate of 14 separate series. The fixed-base results are listed in Table 20.42.

Table 20.42.

Fixed-Base National Domestic Gross Output Price Indices at Producer Prices

20.149 By period 5, the spread between the fixed-base Laspeyres and Paasche national domestic output price indices is 1.7017/1.5424 = 1.103 or 10.3 percent and the spread between the Fisher and Törnqvist indices is 1.6581/1.6201 = 1.023 or 2.3 percent. In Table 20.43, the four indices are recomputed using the chain principle. It is expected that the use of the chain principle will narrow the spreads between the various indices.

Table 20.43.

Chained National Domestic Gross Output Price Indices at Producer Prices

20.150 An examination of the entries in Table 20.43 shows that chaining did indeed reduce the spread between the various index numbers. In period 5, the spread between the chained Laspeyres and Paasche national domestic output price indices is 1.6644/1.6328 = 1.019 or 1.9 percent and the spread between the chained Fisher and Törnqvist indices is 1.6500/1.6485 = 1.0009 or 0.09 percent, which is negligible considering the variation in the underlying data.

E.2 The national export price index at producer prices

20.151 The data listed in Tables 20.20 through 20.28 for industries G, S, and T are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist price indices for all exported outputs (at producer prices or basic prices in this case), $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , respectively. There is one exported good from each of the three industries so that each export price index is an aggregate of three separate series. The fixed-base results are listed in Table 20.44.

Table 20.44.

National Fixed-Base Export Price Indices at Producer Prices

20.152 There is very little difference in any of the fixed-base series listed in Table 20.44. The corresponding chained indices in Table 20.45 are also very close to each other.

Table 20.45.

National Chained Export Price Indices at Producer Prices

E.3 The national domestic intermediate input price index at producer prices

20.153 The data listed in Tables 20.20 through 20.28 for industries G, S, and T are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist price indices for all domestic intermediate inputs (at producer prices or purchase prices in this case), $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , respectively. There are three domestic intermediate inputs used in each of industries G, S, and T so that each domestic intermediate input price index is an aggregate of eight separate series. The fixed-base results are listed in Table 20.46.

Table 20.46.

Fixed-Base National Domestic Intermediate Input Price Indices at Producer Prices

20.154 The spread between the Laspeyres and Paasche fixed-base indices is very large by period 5, equaling 1.5887/1.1306 = 1.405 or 40.5 percent. The spread between the Fisher and Törnqvist fixed-base indices is not negligible either, equaling 1.4268/1.3402 = 1.065 or 6.5 percent in period 5. These relatively large spreads are due to the fact that the price of high-tech services plummets over the sample period with corresponding large increases in quantities while the other prices increase substantially. As usual, we expect these spreads to diminish if the chained indices are used. The corresponding chained indices are listed in Table 20.47.

Table 20.47.

Chained National Domestic Intermediate Input Price Indices at Producer Prices

20.155 Chaining reduces the period 5 spread between Laspeyres and Paasche to 1.4573/1.3398 = 1.088 or 8.8 percent and between the Fisher and Törnqvist to 1.4015/1.3973 = 1.003 or 0.3 percent, which is an acceptable degree of divergence considering the volatility of the underlying data.

E.4 The national import price index at producer prices

20.156 The data listed in Tables 20.20 through 20.28 for industries G, S, and T are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist price indices for all imported intermediate inputs (at producer prices or purchase prices in this case), $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , respectively. There are four imported intermediate inputs used in industry G, four imported intermediate inputs used in industry S, and two imported intermediate inputs used in industry T so that each import input price index is an aggregate of 10 separate series. The fixed-base results are listed in Table 20.48.

Table 20.48.

Fixed-Base National Import Price Indices at Producer Prices

20.157 The spread between the Laspeyres and Paasche fixed-base import price indices is fairly large by period 5, equaling 1.5776/1.3596 = 1.160 or 16.0 percent. The spread between the Fisher and Törnqvist fixed-base indices is much smaller, equaling 1.4736/1.4645 = 1.006 or 0.6 percent in period 5. Note that each import price index has relatively large period-to-period fluctuations owing to the large fluctuations in the price of imported energy. As usual, we expect the fixed-base spreads to diminish if the chained indices are used. The corresponding chained indices are listed in Table 20.49.

Table 20.49.

Chained National Import Price Indices at Producer Prices

20.158 Chaining reduces the period 5 spread between Laspeyres and Paasche to 1.5128/1.4236 = 1.063 or 6.3 percent and between the Fisher and Törnqvist to 1.4680/1.4675 = 1.0003 or 0.03 percent, a negligible amount.

20.159 The domestic output price index and the domestic export index can be regarded as subindices of an overall gross output price index of the type that was described in the PPI Manual (ILO and others, 2004b). Similarly, the domestic intermediate input price index and the import price index can be regarded as subindices of the overall intermediate input price index that was described in the PPI Manual. All of these subindices can be thought of as aggregations of the same commodity (or group of commodities) across industries. At a second stage of aggregation, it is possible to aggregate over the domestic output price index and the export price index and to also aggregate over the domestic intermediate input price index and the import price index (with quantities indexed with negative signs) in order to form an economy-wide value-added price index. In the following section, the first stage of aggregation is across commodities within an industry; that is, in the following section, industry value-added price indices are constructed. A national value-added price index is also constructed in Section F. In Section F, the industry value-added deflators constructed in Section F are aggregated in order to form a two-stage economy-wide value-added price index. This two-stage aggregate value-added deflator will be compared with the two-stage aggregation method that aggregates over the domestic output price index, the export price index, the domestic intermediate input price index, and the import price index. These two methods of two-stage aggregation are compared in Section F along with the corresponding single-stage national value-added deflator.

F. Value-Added Price Deflators

F.1 Value-added price deflators for the goods producing industry

20.160 The data listed in Tables 20.20 through 20.22 for industry G are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist value-added price indices or deflators 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. The quantities of domestic intermediate inputs and imports are indexed with negative signs. Fixed-base and chained value-added Laspeyres, Paasche, Fisher, and Törnqvist price indices, $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , respectively, will be constructed. There are three domestic outputs and one export produced by industry G, and three domestic intermediate inputs and four imported commodities used as inputs by industry G, so that each value-added price index is an aggregate of 11 separate series. The fixed-base results are listed in Table 20.50.

Table 20.50.

Fixed-Base Value-Added Price Deflators for Industry G

20.161 The spread between the Laspeyres and Paasche fixed-base value-added price indices is enormous by period 5, equaling 5.7905/1.7605 = 3.289 or 328.9 percent. The spread between the Fisher and Törnqvist fixed-base indices is large as well, equaling 3.1928/2.1276 = 1.501 or 50.1 percent in period 5. These very large spreads are due to the fact that the price of high-tech services plummets over the sample period with corresponding large increases in quantities while the other prices increase substantially. As well, because quantities have positive and negative weights in value-added price indices, the divergences between various index number formulas can become very large. As usual, we expect these spreads to diminish if the chained indices are used. The corresponding chained indices are listed in Table 20.51.

Table 20.51.

Chained Value-Added Price Deflators for Industry G

20.162 Chaining reduces the period 5 spread between the Laspeyres and Paasche to 2.9594/1.8066 = 1.638 or 63.8 percent and between the Fisher and Törnqvist to 2.3122/2.2720 = 1.018 or 1.8 percent, which is an acceptable degree of divergence considering the volatility of the underlying data. However, note that using the chained Laspeyres or Paasche value-added price indices for this industry will give rise to estimates of price change that are very far from the corresponding superlative index estimates. Thus the corresponding Laspeyres or Paasche estimates of real value added may be rather inaccurate, giving rise to inaccurate estimates of industry productivity growth.

F.2 Value-added price deflators for the services industry

20.163 The data listed in Tables 20.23 through 20.25 for industry S are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist value-added price indices at producer prices. There are eight domestic outputs and one export produced by industry S, and two domestic intermediate inputs and four imported commodities used as inputs by industry S, so that each value-added price index is an aggregate of 15 separate series. The fixed-base results are listed in Table 20.52. Producer prices are used in these computations.

Table 20.52.

Fixed-Base Value-Added Price Deflators for Industry S

20.164 The spread between the Laspeyres and Paasche fixed-base value-added price indices for industry S is 1.4913/1.2797 = 1.165 or 16.5 percent, which is a substantial gap. The spread between the Fisher and Törnqvist fixed-base indices is fairly small, equaling 1.3942/1.3814 = 1.009 or 0.9 percent in period 5. Note that the gap between the fixed-base Paasche and Laspeyres value-added price indices for the services industry is very much less than the corresponding gap for the fixed-base Paasche and Laspeyres value-added price indices for the goods producing industry. An explanation for this narrowing of the Paasche and Laspeyres gap is that while the services industry was subject to some very large fluctuations in the prices it faced, because most of the big fluctuations occurred for the food and energy imports, which are margin goods for the industry, these fluctuations were passed on to final demanders, leaving industry distribution margins largely intact. Thus the fluctuations in the value-added price indices for industry S turned out to be less severe than for industry G. As usual, the spreads between the Paasche and Laspeyres price indices should narrow when the chain principle is used; see Table 20.53.

Table 20.53.

Chained Value-Added Price Deflators for Industry S

20.165 Chaining reduces the period 5 spread between Laspeyres and Paasche to 1.4363/1.3863 = 1.036 or 3.6 percent in period 5 and between the Fisher and Törnqvist to 1.4145/1.4111 = 1.002, or 0.2 percent, which is negligible.

F.3 Value-added price deflators for the transportation industry

20.166 The data listed in Tables 20.26 through 20.28 for industry T are used to calculate fixed-base Laspeyres, Paasche, Fisher, and Törnqvist value-added price indices at producer prices. There are three domestic outputs and one export produced by industry T, and three domestic intermediate inputs and two imported commodities used as inputs by industry T, so that each value-added price index is an aggregate of nine separate series. The fixed-base results are listed in Table 20.54.

Table 20.54.

Fixed-Base Value-Added Price Deflators for Industry T

20.167 The spread between the Laspeyres and Paasche fixed-base value-added price indices is enormous by period 5, equaling 4.8128/1.8028 = 2.670 or 267.0 percent. The spread between the Fisher and Törnqvist fixed-base indices is fairly large as well, equaling 2.9456/2.2114 = 1.332 or 33.2 percent in period 5. These very large spreads are due to the fact that the price of high-tech services plummets over the sample period with corresponding large increases in quantities while the other prices increase substantially. As usual, we expect these spreads to diminish if the chained indices are used. The corresponding chained indices are listed in Table 20.55.

Table 20.55.

Chained Value-Added Price Deflators for Industry T

20.168 Chaining reduces the period 5 spread between Laspeyres and Paasche to 2.4248/1.9916 = 1.218 or 21.8 percent and between the Fisher and Törnqvist to 2.2389/2.1975 = 1.019 or 1.9 percent, which is an acceptable degree of divergence considering the volatility of the underlying data. However, note that using the chained Laspeyres or Paasche value-added price indices for this industry will give rise to estimates of price change that are fairly far from the corresponding chained superlative index estimates, a situation that is similar to what occurred for the industry G data. Thus whenever possible, it seems preferable to use chained superlative indices when constructing annual industry value-added deflators as opposed to using fixed-base or chained Paasche or Laspeyres indices. In the following section, all of the industry data are aggregated to form a national value-added deflator.

F.4 The national value-added price deflator

20.169 The data listed in Tables 20.20 through 20.28 for industries G, S, and T are used to calculate national Laspeyres, Paasche, Fisher, and Törnqvist value-added price indices at producer prices; that is, in this subsection, the national value-added deflator is constructed. Fixed-base and chained value-added Laspeyres, Paasche, Fisher, and Törnqvist price indices will be constructed, $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , respectively. There are 14 domestic outputs, 3 exported commodities, 8 domestic intermediate inputs, and 10 imported commodities so that each national value-added deflator is an aggregate of 35 separate series. The fixed-base results are listed in Table 20.56.

Table 20.56.

Fixed-Base National Value-Added Deflators

20.170 The spread between the national Laspeyres and Paasche fixed-base value-added price indices is fairly large by period 5, equaling 1.7555/1.6176 = 1.085 or 8.5 percent. The spread between the Fisher and Törnqvist fixed-base indices is small, equaling 1.6970/1.6851 = 1.007 or 0.7 percent in period 5. The corresponding chained indices are listed in Table 20.57.

Table 20.57.

Chained National Value-Added Deflators

20.171 The spread in period 5 between the national Laspeyres and Paasche chained value-added price indices equals 1.7612/1.6380 = 1.075 or 7.5 percent, which is slightly smaller than the corresponding 8.5 percent spread for the fixed-base Laspeyres and Paasche indices. The spread between the Fisher and Törnqvist chained indices in period 5 is 1.7156/1.6985 = 1.010 or 1.0 percent, which is slightly larger than the corresponding fixed-base spread of 0.7 percent. At the national level, the fixed-base and chained Fisher and Törnqvist indices all give much the same answer.

G. Two-Stage Value-Added Price Deflators

G.1 Two-stage national value-added price deflators: Aggregation over industries

20.172 In Section E.6 of Chapter 18, methods for constructing a price index by aggregating in two stages are discussed. It is pointed out that if a Laspeyres index is constructed in two stages of aggregation and the Laspeyres formula is used in each stage of aggregation, then the two-stage index will necessarily coincide with the corresponding single-stage index. A similar consistency in aggregation property holds if the Paasche formula is used at each stage of aggregation. Unfortunately, this consistency in aggregation property does not hold for superlative indices but it is pointed out in Chapter 18 that superlative indices should be approximately consistent in aggregation. In this section, the artificial data set is used in order to evaluate this approximate consistency in aggregation property of the Fisher and Törnqvist indices.

20.173 In the present context, there are two natural ways of aggregating in two stages. In method 1, the first stage of aggregation is the construction of a value-added deflator for each industry (along with the corresponding quantity indices) and in the second stage, the three industry value-added deflators are aggregated into a national value-added deflator. In method 2, the first stage of aggregation is the construction of national domestic output, domestic intermediate input, and export and import price indices (along with the corresponding quantity indices) and in the second stage, these four price indices are aggregated into a national value-added deflator.⁵⁶ The results for method 1 are listed in this subsection while the results for method 2 are listed in Section F.2.

20.174 In Table 20.58, the fixed-base single-stage Laspeyres, Paasche, Fisher, and Törnqvist indices are listed in the first four columns of the table⁵⁷ and the corresponding method 1 fixed-base two-stage indices are listed in the last four columns of the table.

Table 20.58.

Fixed-Base Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Industries Method

Table 20.58.

Fixed-Base Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Industries Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2276	1.2190
3	1.7776	1.8533	1.8151	1.8173	1.7776	1.8533	1.8915	1.8110
4	1.8743	1.9822	1.9275	1.9455	1.8743	1.9822	2.2616	1.9254
5	1.6176	1.7555	1.6851	1.6970	1.6176	1.7555	1.9488	1.6579

Table 20.58.

Fixed-Base Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Industries Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2276	1.2190
3	1.7776	1.8533	1.8151	1.8173	1.7776	1.8533	1.8915	1.8110
4	1.8743	1.9822	1.9275	1.9455	1.8743	1.9822	2.2616	1.9254
5	1.6176	1.7555	1.6851	1.6970	1.6176	1.7555	1.9488	1.6579

20.175 As is expected from the theory in Chapter 18, the single-stage Laspeyres and Paasche indices coincide exactly with their two-stage counterparts. What is not expected is how far the two-stage Fisher index, $P_{F 2 S}^{t}$ , is from its single-stage counterpart, $P_{F}^{t}$ , for periods 3 through 5. Obviously, the period-to-period changes in the Fisher industry value-added indices are so large that the two-stage approximation results discussed in Chapter 18 break down for this artificial data set. The spread between the fixed-base single-stage Fisher and Törnqvist indices in period 5 is 1.6970/1.6851 = 1.007 or 0.7 percent but the spread between the two-stage Fisher and Törnqvist indices in period 5 is 1.9488/1.6579 = 1.175 or 17.5 percent, a rather large deviation.

20.176 In Table 20.59, the chained single-stage Laspeyres, Paasche, Fisher, and Törnqvist indices are listed in the first four columns of the table⁵⁸ and the corresponding method 1 chained two-stage indices are listed in the last four columns of the table.

Table 20.59.

Chained Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Industries Method

Table 20.59.

Chained Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Industries Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2276	1.2190
3	1.7711	1.8336	1.8021	1.8098	1.7711	1.8336	1.8365	1.8124
4	1.8855	1.9530	1.9190	1.9315	1.8855	1.9530	1.9587	1.9326
5	1.6380	1.7612	1.6985	1.7156	1.6380	1.7612	1.7270	1.7137

Table 20.59.

Chained Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Industries Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2276	1.2190
3	1.7711	1.8336	1.8021	1.8098	1.7711	1.8336	1.8365	1.8124
4	1.8855	1.9530	1.9190	1.9315	1.8855	1.9530	1.9587	1.9326
5	1.6380	1.7612	1.6985	1.7156	1.6380	1.7612	1.7270	1.7137

20.177 It can be seen that chaining has reduced the spread between the two-stage superlative indices. The spread between the chained single-stage Fisher and Törnqvist indices in period 5 is 1.7156/1.6985 = 1.007 or 1.0 percent and the spread between the chained two-stage Fisher and Törnqvist indices in period 5 is 1.7270/1.7137 = 1.008 or 0.8 percent, a rather modest deviation. As is expected from the theory in Chapter 18, the single-stage chained Laspeyres and Paasche indices coincide exactly with their two-stage counterparts.

G.2 Two-stage national value-added price deflators: Aggregation over commodities

20.178 In this subsection, the national value-added price index is formed by an alternative two-stage aggregation procedure. In the first-stage aggregation, national domestic output, export, domestic intermediate input, and import price indices are calculated along with the corresponding quantity indices as was done in Section F. In the second stage of aggregation, the sign of the quantity indices that correspond to the domestic intermediate input and import indices is changed from positive to negative and the four price and quantity series are aggregated together to form an estimate for the national value-added deflator. The resulting two-stage fixed-base Laspeyres, Paasche, Fisher, and Törnqvist price indices, $P_{L 2 S}^{t}, P_{P 2 S}^{t}, P_{F 2 S}^{t}$ , and $P_{T 2 S}^{t}$ , are listed in the last four columns of Table 20.60 along with their fixed base single-stage counterparts, $P_{L}^{t}, P_{p}^{t}, p_{F}^{t}$ , and $P_{T}^{t}$ .

Table 20.60.

Fixed-Base Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Commodities Method

Table 20.60.

Fixed-Base Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Commodities Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2272	1.2294
3	1.7776	1.8533	1.8151	1.8173	1.7776	1.8533	1.8067	1.8094
4	1.8743	1.9822	1.9275	1.9455	1.8743	1.9822	1.9066	1.9269
5	1.6176	1.7555	1.6851	1.6970	1.6176	1.7555	1.6641	1.6822

Table 20.60.

Fixed-Base Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Commodities Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2272	1.2294
3	1.7776	1.8533	1.8151	1.8173	1.7776	1.8533	1.8067	1.8094
4	1.8743	1.9822	1.9275	1.9455	1.8743	1.9822	1.9066	1.9269
5	1.6176	1.7555	1.6851	1.6970	1.6176	1.7555	1.6641	1.6822

20.179 Note that the single-stage fixed-base indices, $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ , listed in Table 20.60 coincide with the single-stage fixed-base indices $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ listed in Table 20.58. Note also that the single-stage Paasche and Laspeyres indices coincide with their two-stage counterparts in Table 20.60 as is expected from index number theory. Finally, note that the two-stage superlative indices, $P_{F 2 S}^{t}$ and $P_{T 2 S}^{t}$ , are reasonably close to their single-stage counterparts, $P_{F}^{t}$ and $P_{T}^{t}$ . The spread between the four superlative indices is 1.6970/1.6641 = 1.054 or 5.4 percent. It seems that the method 2 (aggregation over commodities method) two-stage aggregation procedure works more smoothly than the method 1 (aggregation over industry value-added method) two-stage aggregation procedure, leading to a reasonably close approximation between the single-stage and two-stage estimators for the national value-added deflator in the case of method 2.

20.180 In Table 20.61, the method 2 two-stage chained Laspeyres, Paasche, Fisher, and Törnqvist price indices, $P_{L 2 S}^{t}, P_{P 2 S}^{t}, P_{F 2 S}^{t}$ , and $P_{T 2 S}^{t}$ , are listed in the last four columns along with their fixed-base single-stage counterparts, $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ .

Table 20.61.

Chained Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Commodities Method

Table 20.61.

Chained Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Commodities Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2272	1.2294
3	1.7711	1.8336	1.8021	1.8098	1.7711	1.8336	1.8037	1.8150
4	1.8855	1.9530	1.9190	1.9315	1.8855	1.9530	1.9202	1.9318
5	1.6380	1.7612	1.6985	1.7156	1.6380	1.7612	1.7069	1.7186

Table 20.61.

Chained Single-Stage and Two-Stage National Value-Added Deflators: Aggregation over Commodities Method

Period t	$P_{L}^{t}$	$P_{P}^{t}$	$P_{F}^{t}$	$P_{T}^{t}$	$P_{L 2 S}^{t}$	$P_{P 2 S}^{t}$	$P_{F 2 S}^{t}$	$P_{T 2 S}^{t}$
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
2	1.2180	1.2353	1.2267	1.2261	1.2180	1.2353	1.2272	1.2294
3	1.7711	1.8336	1.8021	1.8098	1.7711	1.8336	1.8037	1.8150
4	1.8855	1.9530	1.9190	1.9315	1.8855	1.9530	1.9202	1.9318
5	1.6380	1.7612	1.6985	1.7156	1.6380	1.7612	1.7069	1.7186

20.181 As expected, chaining reduces the spread between the superlative indices. The spread between the four superlative indices is now 1.7186/1.6985 = 1.012 or 1.2 percent. Note also that the single-stage Paasche and Laspeyres chained indices coincide with their two-stage counterparts in Table 20.61. In the following section, the focus shifts from industry price indices to final demand price indices.

H. Final Demand Price Indices

H.1 Domestic final demand price indices

20.182 In this section, the standard fixed-base and chained Laspeyres, Paasche, Fisher, and Törnqvist price indices are listed for deliveries of commodities to the domestic final demand sector; see Table 20.62. Each index is an aggregate of six separate final demand series.

Table 20.62.

Fixed-Base and Chained Domestic Final Demand Deflators

20.183 The indices listed in Table 20.62 have already been listed in various tables in Section C above but for convenience, they are tabled again. Because these indices are discussed in Section C, the discussion is not repeated here.

H.2 Export price indices at final demand prices

20.184 In this subsection, the standard fixed-base and chained Laspeyres, Paasche, Fisher, and Törnqvist price indices are calculated for the three export series that are listed in Section B above. Final demand prices are used when calculating the indices listed in Table 20.63.

Table 20.63.

Fixed-Base and Chained Export Price Indices at Final Demand Prices

20.185 Because the three export price and quantity series have fairly smooth trends that are roughly proportional to each other, all of the indices listed above in Table 20.63 are quite close to each other.

H.3 Import price indices at final demand prices

20.186 In this subsection, the standard fixed-base and chained Laspeyres, Paasche, Fisher, and Törnqvist price indices are calculated for the 10 import series that are listed in Section B above. Final demand prices are used when calculating the indices listed in Table 20.64.

Table 20.64.

Fixed-Base and Chained Import Price Indices at Final Demand Prices

20.187 Because price and quantity trends for imports are far from being proportional, there are substantial differences between the Paasche and Laspeyres price indices. The spread between the fixed-base Paasche and Laspeyres is 1.5946/1.3726 = 1.162 or 16.2 percent while the spread between the chained Paasche and Laspeyres is 1.5257/1.4321 = 1.065 or 6.5 percent so that as usual, chaining reduces the spread. All of the superlative indices are close to each other.

H.4 GDP deflators

20.188 In this subsection, various GDP deflators are calculated; that is, the standard fixed-base and chained Laspeyres, Paasche, Fisher, and Törnqvist price indices are calculated for the 19 final demand series that are listed in Section B above. Final demand prices are used when calculating the indices listed in Table 20.65.

Table 20.65.

Fixed-Base and Chained GDP Deflators

20.189 The spread between the Paasche and Laspeyres fixed-base GDP deflators in period 5 is 1.7044/1.6591 = 1.027 or 2.7 percent while the spread between the Paasche and Laspeyres chained GDP deflators in period 5 is 1.7044/1.6591 = 1.048 or 4.8 percent. Thus in this case, chaining did not reduce the spread between the Paasche and Laspeyres indices. The superlative indices are all rather close to each other; in period 5, the spread between the four superlative indices was 1.7099/1.6816 = 1.017 or 1.7 percent and the spread between the two chained superlative indices was only 1.7099/1.6981 = 1.007 or 0.7 percent.

H.5 The reconciliation of the GDP deflator with the value-added deflator

20.190 The final set of tables for this chapter draws on the theory developed in Section B.3 of this Chapter. In that section, it was shown how volume estimates for GDP at final demand prices, GDP_F, could be reconciled with volume estimates for GDP at producer prices, GDP_P, using equation (20.26). Equation (20.26) said that GDP_F equals GDP_P plus a sum of tax terms, T. It was also shown that two-stage price and quantity indices for GDP_F could be constructed by aggregating over the 35 separate price and quantity series that are used to construct price and quantity indices for GDP_P plus aggregating over all of the tax series that make up the T aggregate. It was shown that the resulting price and volume estimates for GDP_F and GDP_P + T will coincide if the Laspeyres, Paasche, or Fisher formula is used. This methodology is tested out on the artificial data set for both fixed-base and chained Laspeyres, Paasche, Fisher, and Törnqvist price indices in Tables 20.66 (fixed base indices) and 20.67 (chained indices). The $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ indices reported in Table 20.66 are the fixed-base single-stage GDP deflators (for GDP_F) that were listed in the first four columns of Table 20.65 while the $P_{L 2 S}^{t}, P_{P 2 S}^{t}, P_{F 2 S}^{t}$ , and $P_{T 2 S}^{t}$ indices reported in Table 20.66 are the two-stage fixed-base price indices that result when we aggregate over the 35 component price and quantity series that make up GDP at producer prices, GDP_P, plus the nonzero tax series that are listed in Section B above and make up the tax aggregate T.

Table 20.66.

Fixed-Base GDP Deflators Calculated in Two Stages

20.191 As predicted by the theory presented in Chapter 18, the Laspeyres, Paasche, and Fisher single-stage estimates for the GDP deflator (the first three columns in Table 20.66) coincide exactly with the corresponding two-stage estimates that are built up by aggregating over GDP at producer prices plus aggregating over the tax series. The single-stage Törnqvist GDP deflator, $P_{T}^{t}$ , does not coincide with its two-stage counterpart, $P_{T 2 S}^{t}$ , but the correspondence is fairly close.

20.192 The $P_{L}^{t}, P_{P}^{t}, P_{F}^{t}$ , and $P_{T}^{t}$ indices reported in Table 20.67 are the chained single-stage GDP deflators (for GDP_F) that were listed in the last four columns of Table 20.65 while the $P_{L 2 S}^{t}, P_{P 2 S}^{t}, P_{F 2 S}^{t}$ , and $P_{T 2 S}^{t}$ indices reported in Table 20.67 are the two-stage chained price indices that result when we aggregate over the 35 component price and quantity series that make up GDP at producer prices, GDP_P, plus the nonzero tax series that are listed in Section B above and make up the tax aggregate T.

Table 20.67.

Chained GDP Deflators Calculated in Two Stages

20.193 Again as predicted by the theory presented in Chapter 18, the Laspeyres, Paasche, and Fisher single-stage estimates for the GDP deflator (the first three columns in Table 20.67) coincide exactly with the corresponding two-stage estimates that are built up by aggregating over GDP at producer prices plus aggregating over the tax series. The single-stage Törnqvist GDP deflator, $P_{T}^{t}$ , does not coincide with its two-stage counterpart, $P_{T 2 S}^{t}$ , but again, the correspondence is fairly close.

20.194 The equality of the single-stage and two-stage Laspeyres, Paasche, and Fisher GDP_F deflators in Tables 20.66 and 20.67 provides a very good check on the correctness of all of the various index number calculations that are associated with PPI programs and the production of GDP volume estimates.

I. Conclusion

20.195 Some tentative conclusions that can be drawn from the various indices that have been computed using the artificial data set are as follows:

It is risky to use fixed-base Paasche or Laspeyres indices in the sense that they can be rather far from the theoretically preferred superlative indices.
Chained indices seem preferable to the use of fixed-base indices in the sense that chaining generally reduces the spread between the Paasche and Laspeyres indices.
Chained Paasche and Laspeyres indices can be close to the theoretically preferred superlative indices, except in the value-added context; that is, chained Paasche and Laspeyres indices are often fairly close to each other (and the corresponding chained superlative indices) when constructing output, export, intermediate input, and import price indices. However, when constructing value-added indices, it seems preferable to use chained superlative indices.

In this Manual, the 2008 SNA (Commission of the European Communities and others, 2008) refers to the final draft of Volume 1 (Chapters 1–17) of the updated System of National Accounts adopted by the 39th session of the United Nations Statistical Commission, 26–29 February 2008, available at http://unstats.un.org/unsd/sna1993/draftingphase/ChapterList.asp. The 2008 SNA is an updated version of the 1993 SNA (Commission of the European Communities and others, 1993).

NPISHs are legal entities principally engaged in the production of nonmarket services for households whose main resources are voluntary contributions by households, such as charities and trade unions.

This third category of capital formation, net acquisition (i.e., acquisitions less disposals) of valuables, includes precious stones and metals and paintings used as “stores of value” and not for consumption or production.

See Diewert and Morrison (1986) for references to this early literature.

This production theory approach to modeling trade flows was also used by Diewert (1974a, pp. 142–46); Woodland (1982); Diewert and Morrison (1986); Alterman, Diewert, and Feenstra (1999); and Feenstra (2004, pp. 64–98).

See Table 14.12 in 2008 SNA.

This is what was done in Chapter 18 of the PPI Manual; see ILO and others (2004b, pp. 463–507).

These deliveries correspond to the familiar C + I + G final demand sectors.

These deliveries correspond to exports, X.

These deliveries correspond to imports, M.

Under the assumption that there are no quality differences between units of G, the appropriate price will be a unit value and the corresponding quantity will be the total quantity of G purchased by sector S during the period.

For a more detailed analysis of how transport quantities are tied to shipments of goods from one sector to another, see Diewert (2006).

Even if there is no price discrimination on the part of industry G at any point in time, the price of good G will usually vary over the reference period and hence if the proportion of daily sales varies between the three sectors, the corresponding period average prices for the three sectors will be different. The notation used here is unfortunately much more complicated than the notation that is typically used in explaining input output tables because it is not assumed that each commodity trades across demanders and suppliers at the same price. Thus the above notation distinguishes three superscripts or subscripts instead of the usual two: Two superscripts are required to distinguish the selling and purchasing sectors and one additional subscript is required to distinguish the commodity involved in each transaction. This type of setup was used in Chapter 19 of ILO and others (2004b).

We make the general convention that the last nontransportation domestic establishment that handles an exported good is regarded as the sector that exports the good. If we did not make this convention, virtually all exported goods would be credited to the transportation sector. This convention is consistent with our treatment of transportation services as a margin industry.

As outlined in Chapter 4 of this Manual and Chapter 14 of the 2008 SNA.

Each of the net supply aggregates defined by (20.3) through (20.5) does not have to be positive; for example, consider the case of an imported intermediate good that is not produced domestically. However, the sum of the net supply aggregates will be substantially positive.

There are some minor complications owing to the fact that small amounts of imports and exports may not pass through the domestic production sector; that is, some tourist expenditures made abroad would not be captured by transactions within the scope of the domestic production sector and a similar comment applies to government expenditures made abroad.

See ILO and others (2004b). Alterman, Diewert, and Feenstra (1999) also used the value-added methodology in their exposition of the economic approach to the XMPIs.

All prices are positive but if a quantity is an input, it is given a negative sign.

Notation: $p \cdot q \equiv Σ_{n = 1}^{N} p_{n} q_{n}$ where p and q are N-dimensional vectors with components p_n and q_n respectively.

The proof follows using the additivity of the inner products and the exact matching of a domestic intermediate input transaction to a corresponding domestic output transaction. Diewert (2006, pp. 293–94) used this method of proof, drawing on prior discussions on these issues with Kim Zieschang. Moyer, Reinsdorf, and Yuskavage (2006) derived similar results but under the assumption that commodity prices were constant across industries.

This observation was made in the PPI Manual and was confirmed by numerical computations; see ILO and others (2004b, pp. 505–06).

A word of warning is in order if two-stage aggregation is used: The value aggregates in the first stage of aggregation must be of the same sign. If they are not of the same sign, index number theory will fail. Thus it is not recommended that a first-stage aggregate equal to exports minus imports be constructed, because the value of net exports could be positive in period 0 and negative in period 1. A similar problem arises if one attempts to construct an index of real inventory change because the sign of the value aggregate can change from period to period. Diewert (2005a) provided some examples of index number failure in the inventory change context but his analysis is applicable more generally.

See Diewert (1978, p. 889) and Robert Hill (2006). However, using an artificial data set it will be shown in Section D of this chapter that the two-stage Fisher value-added index is not close to its singlestage counterpart so some caution must be used in aggregating value added across industries in a two-stage aggregation procedure.

Thus the commodity taxes are modeled as specific taxes rather than ad valorem taxes. This is not a restriction on the analysis because ad valorem taxes can be converted into equivalent specific taxes in each period.

If the sales of commodity G are being subsidized by the government sector, then the tax level per unit $t_{G}^{G S}$ will be negative instead of positive. It is assumed that the after-tax prices of the form $p_{G}^{G S} - t_{G}^{G S}$ are always positive. In a more detailed model, per unit commodity subsidies could be explicitly introduced instead of the present interpretation of $t_{G}^{G S}$ as a specific net (tax less subsidy) commodity tax.

If $t_{G m}^{G R}$ is negative, then the government subsidizes the importation of good G for use by industry G.

Of course, in practice, compromises with the theory will have to be made.

Note that industry S pays industry G the price $p_{G}^{G S}$ per unit of good G delivered to industry S, but industry G must remit the specific tax $t_{G}^{G S}$ out of this price to the government sector.

The foreign demanders for the exports of good G by industry G pay industry G the price $p_{G x}^{G R}$ per unit of the good exported but industry G must pay out of this amount the specific export $t_{G x}^{G R}$ to the government sector for each unit of good G that is exported.

It is also appropriate to use these tax-adjusted producer prices when constructing a PPI that is based on the economic approach to index number theory.

The first method sums entries along rows first and then sums down the sum column whereas the second method sums entries down columns first and then sums across the sum row.

However, in order to obtain this equality for the Törnqvist price index, it is necessary to treat each indirect tax as a separate price component for both the value-added and final demand methods of aggregation; that is, if the terms involving final demand prices and taxes are combined into single producer prices and then fed into the Törnqvist price index formula when using the value-added approach, then the resulting index value is not necessarily equal to the Törnqvist price index that directly aggregates the 36 components of final demand. Thus in Chapter 18, because the second method of aggregation was used, the exact equality of the two Törnqvist price indices does not hold.

See Jorgenson and Griliches (1967 and 1972) for an early exposition of how productivity accounts could be set up. The indirect tax conventions used in this chapter are consistent with the recommendations of Jorgenson and Griliches (1972, p. 85) on the treatment of indirect taxes in a set of productivity accounts.

Alternatively, the tax terms could be combined with the final demand prices and then there would be only 27 price times quantity value transactions in the aggregate.

Conversely, the identity GDP_P = GDP_F –T implies that if the statistical agency is able to estimate GDPF, and in addition, the statistical agency can make estimates of the 21 tax times quantity terms on the right-hand side of equation (20.27), then estimates of GDPP can be made. Thus the allocation of commodity taxes and subsidies to the correct cells of the system of supply and use matrices is important. These observations on the importance of commodity taxes in the input output framework are generalizations of observations made by Diewert (2006, pp. 303–04) in the context of a model of a closed economy.

This is not the only way the accounts could be set up. Note that the distribution services (in distributing G1 and G2) that the domestic service industry provides in this accounting framework is on a gross basis whereas the treatment of transportation services in industry T is on a net basis; that is, the present setup treats transportation services as a margin industry whereas the services associated with the direct distribution of imports to households are not treated in this way. This treatment of imports makes reconciliation of the production accounts with the final demand accounts fairly straightforward.

See Chapter 6, Section C, in the 2008 SNA and Chapter 4 of this Manual.

Note that the tax terms in Tables 20.5 through 20.8 are equal to per unit (or specific) commodity taxes less per unit commodity subsidies. These two effects could be distinguished separately at the cost of additional notational complexity. More generally, on basic price valuation, see Chapter 6, Section C, in the 2008 SNA and Chapter 4 of this Manual.

If the production accounts are to be used in order to measure total factor productivity growth using the economic approach suggested by Jorgenson and Griliches (1967 and 1972), it is important to use the prices that producers face in the accounting framework. The treatment of commodity taxes suggested in this Manual is consistent with the treatment suggested by Jorgenson and Griliches who advocated the following treatment of indirect taxes: “In our original estimates, we used gross product at market prices; we now employ gross product from the producers’ point of view, which includes indirect taxes levied on factor outlay, but excludes indirect taxes levied on output” (1972 p. 85).

Think of this pure imported intermediate as being high-tech equipment, which has been dropping in price owing to the computer chip revolution.

For theoretical treatments of the accounting problems associated with measuring the contribution of inventories to retailing and wholesaling production, see Paragraphs 6.57 through 6.79 of the 2008 SNA, Diewert and Smith (1994), Ehemann (2005), Diewert (2005a), and Peter Hill (2005).

The selling industry also receives any applicable commodity subsidies.

This is the Theorem of the Arithmetic and Geometric Mean; see Hardy, Littlewood, and Pólya (1934) and Chapter 20.

Vartia (1978, p. 272) used the terms logarithmic Laspeyres and logarithmic Paasche, respectively.

Allen and Diewert (1981) showed that the Paasche, Laspeyres, and Fisher indices will all be equal if either prices or quantities move in a proportional manner over time. Thus in order to get a spread between the Paasche and Laspeyres indices, it is required that both prices and quantities move in a nonproportional manner.

This follows from Schlömilch’s (1858) inequality; see Hardy, Littlewood, and Pólya (1934, chapter 11).

These inequalities were noted by Fisher (1922, p. 92) and Vartia (1978, p. 278).

These inequalities were also noted by Fisher (1922, p. 92) and Vartia (1978, p. 278).

Diewert (1978, p. 897) showed that the Drobisch Sidgwick Bowley price index approximates any superlative index to the second order around an equal price and quantity point; that is, PSB is a pseudo-superlative index. Straightforward computations show that the Marshall Edgeworth index PME is also pseudo-superlative.

More precisely, the superlative quadratic mean of order r price indices Pr defined by equation (17.84) and the implicit quadratic mean of order r price indices P^r^* defined by equation (17.81) will generally closely approximate each other provided that r is in the interval 0 ≤ r ≤ 2.

It was also independently derived by Dikhanov (1997) and used by Ehemann, Katz, and Moulton (2002).

The maximum difference between the two tables occurs in period 2 for the p4 contribution factor, which is 0.0928 in Table 20.40 and 0.0917 in Table 20.41.

The terms in Diewert’s decomposition can be given economic interpretations whereas the terms in the other decomposition are more difficult to interpret from the economic perspective. However, Reinsdorf, Diewert, and Ehemann (2002) showed that the terms in the two decompositions approximate each other to the second order around any point where the two price vectors are equal and where the two quantity vectors are equal.

Details of its use can be found in Ehemann, Katz, and Moulton (2002).

The domestic output and export quantities are positive numbers in this second stage of aggregation but the domestic intermediate input and import quantities are negative numbers in the second stage of aggregation.

These indices are the same as those listed in Table 20.56.

These indices are the same as those listed in Table 20.57.

	Industry G	Industry S	Industry T
G	0	$p_{G}^{G S} y_{G}^{G S}$	$p_{G}^{G T} y_{G}^{G T}$
S	$p_{S}^{SG} y_{S}^{SG}$	0	$p_{S}^{S T} y_{S}^{S T}$
T	$p_{T}^{T G} y_{T}^{T G}$	$p_{T}^{T S} y_{T}^{T S}$	0

	Industry G	Industry S	Industry T
G	0	$p_{G}^{G S} y_{G}^{G S}$	$p_{G}^{G T} y_{G}^{G T}$
S	$p_{S}^{SG} y_{S}^{SG}$	0	$p_{S}^{S T} y_{S}^{S T}$
T	$p_{T}^{T G} y_{T}^{T G}$	$p_{T}^{T S} y_{T}^{T S}$	0

	Industry G	Industry S	Industry T
G	$p_{G m}^{G R} m_{G}^{G R}$	$p_{G m}^{S R} m_{G}^{S R}$	$p_{G m}^{T R} m_{G}^{T R}$
S	$p_{S m}^{G R} m_{S}^{G R}$	$p_{S m}^{S R} m_{S}^{S R}$	$p_{S m}^{T R} m_{S}^{T R}$
T	$p_{T m}^{G R} m_{T}^{G R}$	$p_{T m}^{S R} m_{T}^{S R}$	$p_{T m}^{T R} m_{T}^{T R}$

	Industry G	Industry S	Industry T
G	$p_{G m}^{G R} m_{G}^{G R}$	$p_{G m}^{S R} m_{G}^{S R}$	$p_{G m}^{T R} m_{G}^{T R}$
S	$p_{S m}^{G R} m_{S}^{G R}$	$p_{S m}^{S R} m_{S}^{S R}$	$p_{S m}^{T R} m_{S}^{T R}$
T	$p_{T m}^{G R} m_{T}^{G R}$	$p_{T m}^{S R} m_{T}^{S R}$	$p_{T m}^{T R} m_{T}^{T R}$

	Industry G	Industry S	Industry T
G	$(p_{G m}^{G R} + t_{G m}^{G R}) m_{G}^{G R}$	$(p_{G m}^{S R} + t_{G m}^{S R}) m_{G}^{S R}$	$(p_{G m}^{T R} + t_{G m}^{T R}) m_{G}^{T R}$
S	$(p_{S m}^{G R} + t_{S m}^{G R}) m_{S}^{G R}$	$(p_{S m}^{S R} + t_{S m}^{S R}) m_{S}^{S R}$	$(p_{S m}^{T R} + t_{S m}^{T R}) m_{S}^{T R}$
T	$(p_{T m}^{G R} + t_{T m}^{G R}) m_{T}^{G R}$	$(p_{T m}^{S R} + t_{T m}^{S R}) m_{T}^{S R}$	$(p_{T m}^{T R} + t_{T m}^{T R}) m_{T}^{T R}$

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Abstract

A. Introduction

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

B.1 Introduction

B.2 Expanded input output accounts with no commodity taxation

B.3 Input output accounts with commodity taxation and subsidization

C. The Artificial Data Set

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

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

C.3 Industry G prices and quantities

C.4 Industry S prices and quantities

C.5 Industry T prices and quantities

D. The Artificial Data Set for Domestic Final Demand

D.1 The final demand data set

D.2 Some familiar index number formulas

D.3 Asymmetrically weighted index number formulas

D.4 Symmetrically weighted index number formulas

D.5 Superlative indices and two-stage aggregation

E. National Producer Price Indices

E.1 The national gross domestic output price index at producer prices

E.2 The national export price index at producer prices

E.3 The national domestic intermediate input price index at producer prices

E.4 The national import price index at producer prices

F. Value-Added Price Deflators

F.1 Value-added price deflators for the goods producing industry

F.2 Value-added price deflators for the services industry

F.3 Value-added price deflators for the transportation industry

F.4 The national value-added price deflator

G. Two-Stage Value-Added Price Deflators

G.1 Two-stage national value-added price deflators: Aggregation over industries

G.2 Two-stage national value-added price deflators: Aggregation over commodities

H. Final Demand Price Indices

H.1 Domestic final demand price indices

H.2 Export price indices at final demand prices

H.3 Import price indices at final demand prices

H.4 GDP deflators

H.5 The reconciliation of the GDP deflator with the value-added deflator

I. Conclusion

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