Abstract

24.1 A terms of trade index is generally defined as an economy’s index of export prices divided by an index of import prices. The implementation of this definition would not warrant a chapter in a textbook or in a manual: It is more or less straightforward. However, many economists over the years have observed that an improvement in an economy’s terms of trade has effects that are very similar to an improvement in total factor productivity or multifactor productivity.1 Economists have also been interested in quantifying the effects of changing international prices on the real income generated by an economy. Once the discussion of changes in the terms of trade is broadened to include these topics, the original simplicity of the terms of trade index vanishes. Thus the purpose of this chapter is to address the effects of changing international prices on the real income of an economy or a sector of an economy. In order to narrow the topic, only an approach to these measurement problems that is based on economic approaches to producer and consumer theory is considered.

A. Introduction

A.1 Chapter overview

24.1 A terms of trade index is generally defined as an economy’s index of export prices divided by an index of import prices. The implementation of this definition would not warrant a chapter in a textbook or in a manual: It is more or less straightforward. However, many economists over the years have observed that an improvement in an economy’s terms of trade has effects that are very similar to an improvement in total factor productivity or multifactor productivity.1 Economists have also been interested in quantifying the effects of changing international prices on the real income generated by an economy. Once the discussion of changes in the terms of trade is broadened to include these topics, the original simplicity of the terms of trade index vanishes. Thus the purpose of this chapter is to address the effects of changing international prices on the real income of an economy or a sector of an economy. In order to narrow the topic, only an approach to these measurement problems that is based on economic approaches to producer and consumer theory is considered.

24.2 In Section A.2 of this chapter, a technical introduction to the effects of changing international prices on the growth of an economy’s real income is undertaken. A production theory framework is laid out and some preliminary definitions are made.2

24.3 Section B considers the effects of a change in the real export price facing the economy on the real income generated by the market-oriented production sector of the economy and Section C considers the effects of a change in the real import price. Various theoretical definitions for these effects are considered and empirical approximations to these theoretical indices are defined and analyzed. In Section D, the combined effects of changes in real import and export prices on the real income generated by the production sector are considered. These combined effects indices are then related to the partial indices defined in Sections B and C.

24.4 Some goods and services are imported directly into the household sector. An important example of such expenditures is tourism expenditures abroad. The production theory approach developed in Sections B through D is not applicable for these classes of household imported goods and services so in Section E, a consumer theory approach is developed. It turns out that the structure of the producer theory methodology can readily be adapted to deal with this situation with a few key changes.

24.5 There are also certain goods and services that are directly exported by households. For example, self-employed consultants can directly export business services to customers around the world. Also, small-scale household manufacturers of clothing and other goods can advertise on the Internet and sell their products abroad rather easily. Thus there is a need to model household exports, as well as goods and services that are directly imported by households. However, in principle, household exports can be treated using the production theory methodology developed in Sections B through D: All that needs to be done is to create a set of household production accounts.3 Thus these household production units will use various capital inputs (machines, parts of the structures that they inhabit), intermediate inputs, and their own labor in order to produce commodities for sale in their domestic and foreign markets. This household production sector is much the same as “regular” incorporated production units except that it will usually be difficult to get accurate measures of the capital employed and the labor used by these household production units. However, as the reader will note, when the producer theory approach to exports and imports is developed in Sections B through D, it is not necessary to know what inputs of labor and capital are actually used by the production units in order to implement the terms of trade adjustment factors that are developed in these sections. Thus there is no need to develop a separate theory for directly exported goods and services by households. Section E concludes.

A.2 Technical introduction

24.6 Let PXt be the price index for exports in an economy in period t and let PMt be the corresponding import price index. Then the period t terms of trade index, Tt, is defined as an export price index divided by an import price index:

TtPXt/PMt;t=0,1.(24.1)

24.7 A country’s terms of trade is said to have improved going from period 0 to 1 if T1/T0 is greater than one and to have deteriorated if T1/T0 is less than one. For an improvement, the export price index has increased more rapidly than the import price index.

24.8 Thus the definition of a terms of trade index is very straightforward and relatively easy to implement: Only the exact form of the export and import price index needs to be determined. Presumably, preliminary versions of a terms of trade index would use Laspeyres-type indices whereas a retrospective, historical version, compiled when current-period weights become available, would use a superlative index. However, the definition of a terms of trade index is not the end of the story as is explained below.

24.9 It has been well known for a long time that an improvement in a country’s terms of trade is beneficial for a country and has effects that are similar to an improvement in the country’s total factor productivity or multifactor productivity.4 However, determining how to measure precisely the degree of improvement owing to a change in a country’s terms of trade has proven to be a difficult question.

24.10 The measurement question addressed in this chapter is the following one: Can the effects of changes in the price of exports and imports on the growth of real income in the economy be determined? Thus at the outset, the focus is on the measurement of the real income generated by the economy, and then the effects of changes in international prices on the chosen real income measure are considered.

24.11 To begin the analysis, consider the following definition for the net domestic product of a country in period t, NDPt, as the sum of the usual macroeconomic aggregates:

NDPt=PCtCt+PItIt+PGtGt+PXtXtPMtMt;t=0,1,(24.2)

where NDPt is the net domestic product produced by the economy in period t; Ct, It, Gt, Xt, and Mt are the period t quantities of consumption, net investment,5 government final consumption, exports, and imports, respectively; and PCtPItPGtPXt and PMt are the corresponding period t final demand prices. Using the usual circular flow arguments employed by national income accountants, net domestic product is produced by the production sector in the economy and the value of this production generates a flow of income received by primary inputs used in the economy. Growth in this flow of income (which is also equal to NDPt) is analyzed in this chapter.6

24.12 The rate of growth in the flow of nominal net product going from period 0 to 1, NDP1/NDP0 (or more accurately, one plus this rate of growth), is of limited interest to policy analysts and the public as an indicator of welfare growth because it includes the effects of general inflation. Thus it is necessary to deflate the nominal net domestic product in period t, NDPt, by a “reasonable” period t deflator or price index, say PDt. The first problem that needs to be addressed is: What is a “reasonable” deflator?

24.13 Three choices have been suggested in the literature:

  • The price of consumption, PCt;

  • The price of domestic goods or the price of absorption, PAt (an aggregate of PCtPItandPGt, or

  • The net domestic product deflator, PNt (an aggregate of PCtPItPGtPXt andPMt where PMt has negative weights).

24.14 The consumption price deflator, PCt, and the absorption deflator, PAt, can be justified. Diewert and Lawrence (2006) and Diewert (2008) preferred the first deflator while Kohli (2006) preferred the second one. However, these authors do not recommend the use of either the GDP deflator or the net domestic product (NDP) deflator in the present context because they maintain that because virtually all internationally traded goods are intermediate goods and hence are not directly consumed by households, the prices of these goods are not needed to deflate nominal income flows into real income flows.7 The case for using the price of consumption as a deflator for the nominal income that is generated by the production side of the economy is very simple: The deflated amount, NDPt PCt, is the potential amount of consumption that could be purchased by the owners of primary inputs in period t if they chose to buy zero units of net investment and government outputs. If the price of domestic absorption is used as the deflator, then NDPt/PAt is the number of units of a (constant utility) aggregate of C, I, and G that could be purchased by the suppliers of primary inputs to the production sector of the economy in period t.

24.15 Suppose that a choice of the nominal income deflator, PDt, has been made. It is now desirable to look at the growth of the real income generated by the production sector in the economy, that is, look at the growth of NDPtPDt

  NDPt/PDt=[PCtCt+PItIt+PGtGt+PXtXtPMtMt]/PDt;t=0,1  =PCtCt+pItIt+pGtGt+pXtXtpMtMt(24.3)

where the real prices of consumption, net investment, government consumption, exports, and imports are defined as the nominal prices divided by the chosen income deflator PDt:8

   PCtPCt/PDt;pItPIt/PDt;pGtPGt/PDt;pXtPXt/PDt;pMtPMt/PDt.(24.4)

24.16 Using equations (24.3) and definitions (24.4), (one plus) the rate of growth of real income over the two periods under consideration can be defined as follows:

[NDP1/PD1]/[NDP0/PD0]=[pC1C1+pI1I1+pG1G1+pX1X1pM1M1]/[pC0C0+pI0I0+pG0G0+pX0X0pM0M0].(24.5)

24.17 Looking at equation (24.5), it can be seen that, holding all else constant, an increase in the period 1 real price of exports PX1 will increase real income growth generated by the production sector of the economy. Conversely, an increase in the period 1 real price of imports PM1 will decrease real income growth.

24.18 Equation (24.5) indicates the complexity of trying to determine the effects of changes in real import and export prices on the growth of real income: pX and pM change but so do the real prices of consumption, net investment, and government consumption. In addition, the quantities of C, I, G, X, and M are changing and in the background, there are also changes in the amount of labor L and capital K that is being utilized by the economy’s production sector. It is evident that some measure of the effect on real income growth of the changes in the real prices of exports and imports is desired, holding constant the rest of the economic environment. But if export and import prices change, producers will be induced to change the composition of their exports and imports. Thus a careful specification of what is exogenous and what is endogenous is needed in order to isolate the effects of changes in real export and import prices.

24.19 In the following section, a production theory framework is used in order to specify more precisely exactly what is being held fixed and what is being allowed to vary as real export and import prices change. Other approaches to modeling the effects on production and welfare of changes in the prices of exports and imports are reviewed in Diewert and Morrison (1986), Silver and Mahdavy (1989), and Kohli (2006).9

B. The Effects of Changes in the Real Price of Exports

B.1 Theoretical measures of the effects of changes in the real price of exports

24.20 Kohli (1978 and 1991) has long argued that because most internationally traded goods are intermediate products and services, it is natural to model the effects of international trade using production theory.10 Kohli’s example is followed in this section and in subsequent sections and a production theory framework is used with exports as outputs of the production sector and imports as intermediate inputs into the production sector.

24.21 For simplicity, it is assumed that C, I, G, and X (consumption, net investment, government consumption, and exports) are outputs of the production sector and M, L, and K (imports, labor, and capital) are inputs into the production sector.11 In period t, there is a feasible set of (C, I, G, X, M, L, K) outputs and inputs, which is denoted by the set St for periods t equal to 0 and 1. It will prove useful to define the economy’s period t real net domestic product function, nt(pC, pI, pX, pM, L, K) for t = 0, 1:

nt(pc,pI,pG,pX,pM,L,K)maxC,I,G,X,M{pCC+pII+pGG+pXXpMM:(C,I,G,X,M,L,K)belongstoSt}.(24.6)

24.22 Thus the real net product nt(pC, pI, pG, pX, pM, L, K) is the maximum amount of (net) real value added that the economy can produce if producers face the real price pC for consumption, the real price pI for net investment, the real price pG for government consumption, the real price pX for exports, and the real price pM for imports and given that producers have at their disposal the period t production possibilities set St as well as the amount L of labor services and the amount K of capital (waiting) services.12

24.23 It is reasonable to assume that the actual period t amounts of outputs produced and inputs used in period t, Ct, It, Gt, Xt, Mt, Lt, Kt, belong to the corresponding period t production possibilities set, St, for t = 0, 1. It is a stronger assumption to assume that producers are competitively profit maximizing in periods 0 and 1 so that the following equalities are valid:

nt(pCt,pIt,pGt,pXt,pMt,Lt,Kt)=pCtCt+pItIt+pGtGt+pXtXtpMtMt;t=0,1,(24.7)

where PCt,PIt,PGt,PXt,PMt are the real prices for consumption, net investment, government consumption, exports, and imports that producers face in period t13 and Lt and Kt are the amounts of labor and capital used by producers in period t. In what follows, it is assumed that equations (24.7) hold. Basically, these equations rest on the assumption that producers in the economy are competitively maximizing net domestic product in periods 0 and 1 subject to the technological constraints on the economy for each period.

24.24 In a first attempt to measure the effects of changing real export prices over the two periods under consideration, a hypothetical net domestic product maximization problem is considered where producers have at their disposal the period 0 technology set S0, and the period 0 actual labor and capital inputs, L0 and K0, respectively, and they face the period 0 real prices for consumption, net investment, government consumption, and imports, PC0,PI0,PG0, and PM0 respectively, but they face the period 1 real export price, and PM0. The solution to this hypothetical net product maximization problem is n0(PC0,PI0,PG0,PX1,PM0,L0,K0). Using this hypothetical net product or net income, a theoretical Laspeyres type measure αLX of the effects on real income growth of changes in real export prices from the period 0 level, PX0, to the period 1 level, PX1, can be defined as the ratio of the hypothetical net real income n0(PC0,PI0,PG0,PX1,PM0,L0,K0) to the actual period 0 net real income n0(PC0,PI0,PG0,PX0,PM0,L0,K0):14:14

αLXn0(pC0,pI0,pG0,pX1,pM0,L0,K0)/n0(pC0,pI0,pG0,pX0,pM0,L0K0).(24.8)

24.25 The index αLX of the effects of the change in the real price of exports is termed a Laspeyres type index because it holds constant all exogenous prices and quantities at their period 0 levels except for the two real export prices, pX0 and pX1, and the index also holds technology constant at the base-period level.

24.26 Using assumption (24.7) for t = 0, the denominator on the right-hand side of (24.8) is equal to period 0 observed real net product, PC0C0+PI0I0+PG0G0+PX0X0PM0M0. Using definition (24.6), it can be seen that C0, I0, G0, X0, and M0 is a feasible solution for the net product maximization problem defined by the numerator on the right-hand side of (24.8), n0(PC0,PI0,PG0,PX1,PM0,L0,K0). These facts mean that there is the following observable lower bound to the theoretical index αLX defined by (24.8):15

αLX[pC0C0+pI0I0+pG0G0+pX1X0pM0M0]/[pC0C0+pI0I0+pG0G0+pX0X0pM0M0]PLX(24.9)

where PLX is an observable Laspeyres type index of the effects on real income of a change in real export prices going from period 0 to 1. PLX generally understates the hypothetical change in the real income generated by the economy which is defined by the theoretical index αLX owing to substitution bias; that is, the change in the real price of exports will induce producers to substitute away from their base-period production decisions in order to take advantage of the change in real export prices from PX0 to PX1. Note that the numerator and denominator on the right-hand side of (24.9) are identical except that PX1 appears in the numerator and PX0 appears in the denominator.

24.27 It is possible to show that the Laspeyres type observable index PLX is a first order Taylor series approximation to the theoretical Laspeyres type index αLX (see below). A first order Taylor series approximation to the hypothetical net real income defined by n0(PC0,PI0,PG0,PX1,PM0,L0,K0) is given by the first line of equation (24.10):16

n0(pC0,pI0,pG0,pX1,pM0,L0,K0)n0(pC0,pI0,pG0,pX0,pM0,L0,K0)/px][px1px0]+[n0(pC0,pI0,pG0,,pX0,pM0,L0,K0)]=n0(pC0,pI0,pG0,pX0,pM0,L0,K0)+X0[pX1pX0]usingHotellingsLemma17=pC0C0+pI0I0+pG0G0+pX0X0pM0M0+X0[pX1pX0]using(24.7)fort = 0=pC0C0+pI0I0+pG0G0+pX1X0pM0M0,(24.10)

where pC0C0+pI0I0+pG0G0+pX1X0pM0M0 is the numerator on the right-hand side of equation (24.9). Because the denominator on the right-hand side of equation (24.9) is equal to pC0C0+pI0I0+pG0G0+pX0X0pMoM0, which in turn is equal to n0(pC0pI0pG0pX0pM0L0,K0),, it can be seen that PLX is indeed a first order approximation to the theoretical index αLX defined by (24.8).18

24.28 In a second attempt to measure the effects of changing real export prices over the two periods under consideration, a hypothetical net domestic product maximization problem is considered where producers have at their disposal the period 1 technology set S1, and the period 1 actual labor and capital inputs, L1 and K1 respectively, and they face the period 1 real prices for consumption, net investment, government consumption, and imports, PC1,PI1,PG1,, and PM1 respectively, but they face the period 0 real export price, PX0. The solution to this hypothetical (real) net product maximization problem is n1(PC1,PI1,PG1,PX0,PM1,L1,K1). Using this hypothetical net product or net income, a theoretical Paasche type measure αPX of the effects on real income growth of changes in real export prices from the period 0 level, pX0, to the period 1 level, pX1, can be defined as the ratio of the actual period 1 net real income n1(PC1,PI1,PG1,PX1,PM1,L1,K1) to the hypothetical net real income n1(PC1,PI1,PG1,PX0,PM1L1,K1):19:19

αPXn1(pC1,pI1,pG1,pX1,pM1,L1,K1)/n1(pC1,pI1,pG1,pX0,pM1,L1,K1).(24.11)

24.29 The index αPX of the effects of the change in the real price of exports is termed a Paasche type index because it holds constant all exogenous prices and quantities at their period 1 levels except for the two real export prices, pX0 and pX1, and the index also holds technology constant at the period 1 level.

24.30 Using assumption (24.7) for t = 1, the numerator on the right-hand side of equation (24.11) is equal to period 1 observed real net product, pC1C1+pI1I1+pG1G1+pX1X1pM1M1. Using definition (24.6), it can be seen that C1, I1, G1, X1, and M1 is a feasible solution for the net product maximization problem defined by the denominator on the right-hand side of equation (24.11), n1(pC1,pI1,pG1,pX0,pM1,L1,K1). These facts mean that there is the following observable upper bound to the theoretical index LX defined by equation (24.11):

αPX[pC1C1+pI1I1+pG1G1+pX1X1pM1M1]/[pC1C1+pI1I1+pG1G1+pX0X1pM1M1]PPX,(24.12)

where PPX is an observable Paasche type index of the effects on real income of a change in real export prices going from period 0 to 1. PPX generally overstates the hypothetical change in the real income generated by the economy which is defined by the theoretical index αPX owing to substitution bias; that is, the change in the real price of exports from pX1 to pX0 will induce producers to substitute away from their period 1 production decisions so that n1(pC1,pI1,pG1,pX0,pM1,L1,K1) will generally be greater than [pC1C1+pI1I1+pG1G1+pX0X1pM1M1] so that 1/n1(pC1,pI1,pG1,pX0,pM1,L1,K1) will generally be less than 1/[pC1C1+PI1I1+pG1G1pX0X1pM1M1] and the inequality in equation (24.12) follows. Note that the numerator and denominator on the right-hand side of equation (24.12) are identical except that pX1 appears in the numerator and pX0 appears in the denominator.

24.31 It is possible to show that the Paasche type observable index PPX is a first order Taylor series approximation to the theoretical Paasche type index αPX as is shown below. The proof is entirely analogous to the derivation of equation (24.10). A first order Taylor series approximation to the hypothetical net real income defined by n1(pC1,pI1,pG1,pX0,pM1,L1,K1) is given by the first line of equation (24.13) below:

n1(pC1,pI1,pG1,pX0,pM1,L1,K1)n1(pC1,pI1,pG1,pX0,pM1,L1,K1)+[n1(pC1,pI1,pG1,pX1,pM1,L1,K1)/pX][pX0pX1]=n1(pC1,pI1,pG1,pX1,pM1,L1,K1)+X1[pX0pX1]usingHotellingsLemma=pC1C1+pI1I1+pG1G1+pX1X1pM1M1+X1[pX0pX1]using(24.7)fort=1=pC1C1+pI1I1+pG1G1+pX1X1pM1M1,(24.13)

where pC1C1+pI1I1+pG1G1+pX0X1pM1M1 is the denominator on the right-hand side of equation (24.12). Because the numerator on the right-hand side of equation (24.12) is equal to pC1C1+pI1I1+pG1G1+pX1X1pM1M1, which in turn is equal to n1(pC1,pI1,pG1,pX1,pM1,L1,K1) it can be seen that PPX is indeed a first order approximation to the theoretical index αPX defined by equation (24.11).20

24.32 Note that both the Laspeyres and Paasche theoretical indices of the effects on real income generated by the production sector of a change in the (real) price of exports are equally plausible and there is no reason to use one or the other of these two indices. Thus if it is desired to have a single theoretical measure of the effects of a change in real export prices, αLX and αPX should be averaged in a symmetric fashion to form a single target index that would summarize the effects on real income growth of a change in real export prices. Two obvious choices for the symmetric average are the arithmetic or geometric means of αLX and αPX. Following Diewert (1997) and Chapter 16, it seems preferable to use the geometric mean of αLX and αPX as the “best” single theoretical estimator of the effects of a change in real export prices on real income growth, because the resulting Fisher-like (1922) theoretical index satisfies the time reversal test so that if the ordering of the two periods is switched, the resulting index is the reciprocal of the original index.21 Thus define the theoretical Fisher type measure αFX of the effects on real income growth of changes in real export prices as the geometric mean of the Laspeyres and Paasche type theoretical measures:

αFX[αLXαPX]1/2(24.14)

With the target index defined by equation (24.14) in mind, in the following section we consider the problem of finding empirical approximations to this theoretical index.

B.2 Empirical measures of the effects of changes in the real price of exports on the growth of real income generated by the production sector

24.33 Two empirical indices that provide estimates of the effects on the growth of real income of a change in real export prices have already been defined in Section B.1: the Laspeyres type index PLX defined on the right-hand side of equation (24.9) and the Paasche type index PPX defined on the right-hand side of equation (24.12). It was noted that PLX was a lower bound to the theoretical index αLX and PPX was an upper bound to the theoretical index αPX. Thus PLX will generally have a downward bias compared to its theoretical counterpart whereas PPX will generally have an upward bias compared to its theoretical counterpart. These inequalities suggest that the geometric mean of PLX and PPX is likely to be a reasonably good approximation to the target Fisher type index αFX defined as the geometric mean of αLX and αPX. Thus define the Diewert Lawrence index of the effects on real income of a change in real export prices going from period 0 to 1 as follows:22

PDLX[PLXPPX]1/2.(24.15)

It will be useful to develop some alternative expressions for the indices PLX, PPX, and PDLX.

24.34 As a preliminary step in developing these alternative expressions, recall definitions (24.7) which defined the production sector’s period t real net product, nt(pCt,pIt,pGt,pXt,pMt,Lt,Kt) for t = 0, 1, which will be abbreviated to nt. The period t shares of net product of C, I, G, X, and M are defined in the usual way as follows:

SCtpCtCt/nt;sItpItIt/nt;sGtpGtGt/nt;sXtpXtXt/nt;sMtpMtMt/nt;t=0,1.(24.16)

24.35 It can be seen that the shares defined by equation (24.16) sum up to unity for each period t but note that the period t “share” for imports, SMt, is negative whereas the other shares are positive.

24.36 Now consider the definition of PLX which occurred in equation (24.9) and subtract 1 from this expression:

PLX1=[PC0C0+pI0I0+pG0G0+pX1X0pM0M0]/[pC0C0++pI0I0+pG0G0+pX0X0pM0M0]1=[pX1pX0]X0/[pC0C0++pI0I0+pG0G0+pX0X0pM0M0]=[(pX1/pX0)1][pX0X0]/[pC0C0++pI0I0+pG0G0+pX0X0pM0M0]=sX0[rX1],(24.17)

where rX is (one plus) the rate of growth in the real price of exports going from 0 to 1; that is,

rXpX1/pX0.(24.18)

24.37 Thus PLX depends on only (rX – 1), the growth rate in the price of real exports going from period 0 to 1, and SX0, the share of exports in period 0 real net product; that is,

PLX=1+sX0(rX1).(24.19)

24.38 Using similar techniques, it can be shown that PPX depends only on SX1, the share of exports in period 1, and the real export price relative, rX, defined by equation (24.18):

PPX=[1+sX1(rX11)]1.(24.20)

Comparing equations (24.19) and (24.20), it can be seen that both PLX and PPX are increasing functions of rX so that as the real price of exports increases, both indices of growth in real income also increase as expected. It can also be seen that PLX is increasing (decreasing) in sX0 and PPX is increasing (decreasing) in sX1 if rX is more (less) than one. These properties are also intuitively sensible. Substituting expressions (24.19) and (24.20) into (24.15) leads to the following expression for the Diewert Lawrence export index:

PDLX={[1+sX0(rX1)]/[1+sX1(rX11)]}1/2.(24.21)

As indicated above, the Diewert Lawrence index PDLX defined by equation (24.21) is likely to be closer to the target Fisher index FX defined by equation (24.14) than the Laspeyres and Paasche type indices PLX and PPX defined by equations (24.19) and (24.20).

24.39 There is one additional empirically defined index that attempts to measure the effects of a change in real export prices on the growth of real income generated by the production sector and that is based on the work of Diewert and Morrison (1986). With the same notation that is used in equation (24.21), the logarithm of the Diewert Morrison index, PDMX, of the effects on real income of a change in real export prices going from period 0 to 1 is defined as follows:23

InPDMX(1/2)(sX0+sX1)InrX.(24.22)

24.40 It can be verified that PDMX satisfies the time reversal property that was mentioned earlier; that is, if the two time periods are switched, then the new PDMX index is equal to the reciprocal of the original PDMX index.

24.41 The interest in the Diewert Morrison index stems from the fact that it has a very direct connection with production theory; in fact this index is exactly equal to the target index αFX provided that the technology of the production sector can be represented by a general translog functional form in each period. This sentence is explained in more detail below.

24.42 In order to explain the above result, it is necessary to establish a general mathematical result. Thus let x[x1,,xN]andy[y1,,yM] be N and M dimensional vectors respectively and let f0 and f1 be two general quadratic functions defined as follows:

f0(x,y)a00+Σn=1Nan0xn+Σm=1Nbm0ym+(1/2)Σn=1NΣj=1Nanj0xnxj++(1/2)Σm=1MΣk=1Mbmk0ymyk+Σn=1NΣj=1Ncnm0xnym;(24.23)
f1(x,y)a01+Σn=1Nan1xn+Σm=1Nbm1ym+(1/2)Σn=1NΣj=1Nanj1xnxj++(1/2)Σm=1MΣk=1Mbmk1ymyk+Σn=1NΣj=1Ncnm1xnym;(24.24)

where the parameters anjt satisfy the symmetry restrictions anjt=ajnt for n, j = 1, … , N and t = 0, 1 and the parameters bmkt=bkmt for m, k = 1, … , M and t = 0, 1. It can be shown that if

anj0=anj1forn,j=1,...,N,(24.25)

then the following equation holds for all vectors x0, x1, y0, and y1:

f0(x1,y0)f0(x0,y0)+f1(x1,y1)f0(x0,y1)=Σn=1N[f0(x0,y0)/xn+f1(x1,y1)/xn][xn1xn0].(24.26)

24.43 The proof of the above proposition is very simple: Just use definitions (24.23) and (24.24), do the differentiation on the right-hand side of equation (24.26), and the result will emerge. The above result is a generalization of Diewert’s (1976, p. 118) quadratic identity. A logarithmic version of the above identity corresponds to the translog identity which was established in the appendix to Caves, Christensen, and Diewert (1982b, pp. 1412–13).

24.44 Recall the definition of the period t real net product function nt(pC, pI, pG, pX, pM, L, K) defined by equation (24.6). The notation will now be changed a bit. Let p[p1,...  ,p5] denote the vector of real output prices [pC, pI, pG, pX, pM] and let z[z1,z2] denote the vector of primary input quantities [L, K]. The example of Diewert and Morrison (1986, p. 663) is now followed and it is assumed that the log of the period t real net product function, nt(p, z), has the following translog functional form:24

Innt(p,z)a0t+Σn=15antInpn+(1/2)Σn=15Σj=15anjpnInpj+Σm=12bmtInzm+(1/2)Σm=12Σk=12bmktInzmInzk+Σn=15Σm=15cnmtInpnInzm;t=0,1.(24.27)

24.45 Note that the coefficients for the quadratic terms in the logarithms of prices are assumed to be constant over time; that is, it is assumed that anj0=anj1=anj. The coefficients must satisfy the following restrictions in order for nt to satisfy the linear homogeneity properties that are consistent with a constant returns to scale technology:25

Σn=15ant=1fort=0,1;(24.28)
Σm=12bmt=1fort=0,1;(24.29)
anj=ajnforalln,j;(24.30)
bmkt=bkmtforallm,kandt=0,1.(24.31)
Σk=1Mamk=0form=1,2;(24.32)
Σj=1Mbnjt=0forn=1,...,5andi=0,1;(24.33)
  Σj=1Ncmnt=0form=1,2andt=0,1;(24.34)
Σm=1Mcmnt=0forn=1,...,5andt=0,1.(24.35)

24.46 Note that using Hotelling’s Lemma, the logarithmic derivatives of nt(pCt,pIt,pGt,pXt,pMt,Lt,Kt) with respect to the logarithm of the export price are equal to the following expressions for t = 0, 1:

Innt(pCt,pIt,pGt,pXt,pMt,Lt,Kt)/InpX=[pXt/nt]nt(pCt,pIt,pGt,pXt,pMt,Lt,Kt)/pX=[pXt/nt]Xt=sXtusingequation(24.16).(24.36)

24.47 Because assumptions (24.27) imply that the logarithms of the net product functions are quadratic in the logarithms of prices and quantities, the result given by equation (24.26) can be applied to definitions (24.7), (24.8), (24.11), and (24.14) to imply the following result:

  2InαFX=InαLX+InαPX=[Inn0(pC0,pI0,pG0,pX0,pM0,L0,K0)/InpX+Inn1(pC1,pI1,pG1,pX1,pM1,L1,K1)/InpX]×[InpX1InpX0]=[sX0+sX1]In(pX1/pX0)usingequation(24.36).(24.37)

24.48 Thus using equations (24.22) and (24.37), one can see that under the assumptions made on the technology, the following exact equality holds:26

αFX=PDMX.(24.38)

Thus the Diewert Morrison index PDMX defined by equation (24.22) is exactly equal to the target theoretical index, αFX, under very weak assumptions on the technology.

24.49 Although the Diewert Morrison index gets a strong endorsement from the above result, the Diewert Lawrence index also had a reasonably strong justification and so the question arises: Which index should be used in empirical applications? Section B.3 shows that numerically these two indices will be quite close and so empirically, it will usually not matter which of these two alternative indices is chosen.

B.3 The numerical equivalence of the Diewert Lawrence and Diewert Morrison measures of the effects of changes in the real price of exports

24.50 Let p[p1,...,p5] denote the vector of real output prices [pC, pI, pG, pX, pM] and let q[q1,...,qp5] denote the corresponding vector of quantities [C, I, G, X, –M]. Thus the data pertaining to period t can be denoted by the vectors pt[pCt,pIt,pGt,pXt,pMt] and qt[Ct,It,Gt,Xt,Mt]. for t = 0, 1. Note that each of the four empirical indices PLX, PPX, PDLX, and PDMX defined in the previous section can be regarded as functions of the data pertaining to the two periods under consideration. Thus PLX should be more precisely be written as the function PLX(p0, p1, q0, q1), PPX should be written as PPX(p0, p1, q0, q1), and so on. In this section, it is desired to compare the numerical properties of the four indices PLX, PPX, PDLX, and PDMX.

24.51 Diewert (1978) undertook a similar comparison of all superlative indices that were known at that time. He showed that all known superlative indices approximated each other to the second order around when the derivatives were evaluated at a point where the period 0 price vector p0 was equal to the period 1 price vector p1 and where the period 0 quantity vector was equal to the period 1 quantity vector.27

24.52 A somewhat similar result holds in the present context; that is, it can be shown that the following equalities hold for the four indices PLX, PPX, PDLX, and PDMX:28

PLX(p,p,q,q)=PPX(p,p,q,q)=PDLX(p,p,q,q)=PDMX(p,p,q,q)=1;(24.39)
PLX(p,p,q,q)=PPX(p,p,q,q)=PDLX(p,p,q,q)=PDMX(p,p,q,q),(24.40)

where ∇PLX(p, p, q, q) is the 20-dimensional vector of first order partial derivatives of PLX(p0, p1, q0, q1) with respect to the components of p0, p1, q0, and q1 but evaluated at a point where p0=p1pandq0=q1q. The meaning of equations (24.39) and (24.40) is that the four indices approximate each other to the accuracy of a first order Taylor series approximation around a data point where the real prices are equal in each period and the net output quantities are also equal to each other across periods.

24.53 The second order derivatives of the Laspeyres and Paasche type indices, PLX and PPX, are not equal to each other when evaluated at an equal price and quantity point; that is,29

2PLX(p,p,q,q)2PPX(p,p,q,q),(24.41)

where ∇2PLX(p, p, q, q) is the 20 by 20 dimensional matrix of second order partial derivatives of PLX(p0, p1, q0, q1) with respect to the components of p0, p1, q0, and q1 but evaluated at a point where p0=p1pandq0=q1q.. Thus as might be expected, PLX and PPX do not approximate each other to the accuracy of a second order Taylor series approximation around an equal price and quantity point.

24.54 However, the second order derivatives of the Diewert Lawrence and Diewert Morrison indices, PDLX and PDMX, are equal to each other when evaluated at an equal price and quantity point; that is,29

2PDLX(p,p,q,q)=2PDMX(p,p,q,q).(24.42)

24.55 Thus PDLX and PDMX approximate each other to the accuracy of a second order Taylor series approximation around a data point where the real prices are equal in each period and the net output quantities are also equal to each other across periods. The practical significance of this result is that for normal time series data where adjacent periods are compared, the Diewert Lawrence and Diewert Morrison indices will give virtually identical results.30

B.4 Real-time approximations to the preferred measures

24.56 The Diewert Lawrence index of the effects on real income growth of a change in the real export price, PDLX defined by equation (24.21), depends on the real export price relative, rX, the period 0 real export share in net product, sX0,, and the corresponding period 1 real export share, sX1,. Our other preferred measure of the effects of a change in the real export price, PDMX defined by equation (24.22), also depends on these same three variables, rX, sX0,, and sX1,. However, the current-period export share sX1, is unlikely to be available to analysts until some time later than the current period. Thus the question arises: How can approximations be formed to the preferred indices defined by equations (24.21) and (24.22)? An answer to this question is as follows:

24.57 Suppose that it is suspected that quantities are relatively unresponsive to changes in relative prices so that the period 1 quantity vector [C1, I1, G1, X1, –M1] will be approximately proportional to the corresponding period 0 quantity vector [C0, I0, G0, X0, -M0]. Under these conditions, αLX will be close to the Laspeyres type index defined by (24.19), which is PLX = 1 + sX0 (rX – 1), and a close approximation to αPX can be obtained by using the formula [pC1C1+pI1I1+pG1G1+pX1X1pM1M1]/[pC1C1+pI1I1+pG1G1+pX0X1pM1M1]. Now multiply this last formula by PLX and take the positive square root in order to obtain a good approximation to the theoretical export price effects index αFX. Suppose that the share of exports in net product in period 1, sX1,, is expected to be approximately equal to the corresponding period 0 share, sX0, . Then simply use formula (24.22) with sX1, set equal to sX0,. If neither of the above conditions is expected to hold for the period 1 data, simply make an approximate forecast for the period 1 export share sX1, and use equation (24.22).

C. The Effects of Changes in the Real Price of Imports

24.58 The theory that was outlined in Section B can be repeated in the present section in order to measure the effects on real income generated by the production sector of a change in real import prices. Basically, all that needs to be done is to replace pX by pM and note that the import shares SMt defined in equation (24.16) are negative whereas the export shares SXt used in Section B were positive. Some of the definitions are listed here without much explanation. The reader should be able to work out the analogies with the export indices.

24.59 A theoretical Laspeyres type measure αLM of the effects on real income growth of changes in real import prices from the period 0 level, PM0, to the period 1 level, PM1 can be defined as the ratio of the hypothetical net real income n0(pC0,pI0,pG0,pX0,pM1,L0,K0) to the actual period 0 net real income n0(pC0,pI0,pG0,pX0,pM0,L0,K0):

αLMn0(pC0,pI0,pG0,pX0,pM1,L0,K0)/n0(pC0,pI0,pG0,pX0,pM0L0,K0).(24.43)

The index αLM of the effects of the change in the real price of imports is termed a Lapeyers type index because it holds constant all exogenous prices and quantities at their period 1 levels except for the two real import prices, PM0 and PM1, and the index also holds technology constant at the base-period level.

24.60 There is the following observable lower bound to the theoretical index αLM defined by equation (24.43):

αLM[pC0C0+pI0I0+pG0G0+pX0X0pM1M0]/[pC0C0+pI0I0+pG0G0+pX0X0pM0M0]PLM,(24.44)

where PLM is an observable Lapeyers type index of the effects on real income of a change in real import prices going from period 0 to 1. Note that the numerator and denominator on the right-hand side of equation (24.46) are identical except that PM1 appears in the numerator and PM0 appears in the denominator.

24.61 It is possible to show that the Laspeyres type observable index PMX is a first order Taylor series approximation to the theoretical Laspeyres type index MX; that is, it is possible to derive a counterpart to the approximation (24.10).

24.62 A theoretical Paasche type measure αPM of the effects on real income growth of changes in real import prices from the period 0 level, PM0, to the period 1 level, PM1, can be defined as the ratio of the actual period 1 net real income n1(pC1,pI1,pG1,pX1,pM1,L1,K1) to the hypothetical net real income n1(pC1,pI1,pG1,pX1,pM0,L1,K1)::

αPM(n1pC1,pI1,pG1,pX1,pM1,L1,K1)/n1(pC1,pI1,pG1,pX1,pM1,L1,K1).(24.45)

The index αPM of the effects of the change in the real price of imports is termed a Paasche type index because it holds constant all exogenous prices and quantities at their period 1 levels except for the two real import prices, PM0 and PM1, and the index also holds technology constant at the period 1 level.

24.63 Using assumption (24.7) for t = 1, the numerator on the right-hand side of equation (24.45) is equal to period 1 observed real net product, pC1C1+pI1I1+pG1G1+pX1X1pM1M1. Using definition (24.6), it can be seen that C1, I1, G1, X1, and M1 is a feasible solution for the net product maximization problem defined by the denominator on the right-hand side of equation (24.45), n1(pC1,pI1,pG1,pX1,pM0,L1,K1). These facts mean that there is the following observable upper bound to the theoretical index αLM defined by equation (24.45):

αPM[Pc1C1+PI1I1+PG1G1+PX1X1PM1M1]/[PC1C1+PI1I1+PG1G1+PX1X1PM0M0]PPM(24.46)

where PPM is an observable Paasche type index of the effects on real income of a change in real import prices going from period 0 to 1. Note that the numerator and denominator on the right-hand side of equation (24.46) are identical except that PM1 appears in the numerator and PM0 appears in the denominator.

24.64 It is possible to show that the Paasche type observable index PPM is a first order Taylor series approximation to the theoretical Paasche type index αPM; that is, a counterpart to the approximation (24.13) can be derived.

24.65 Note that both the Laspeyres and Paasche theoretical indices of the effects on real income generated by the production sector of a change in the (real) price of imports are equally plausible and there is no reason to use one or the other of these two indices. Thus if it is desired to have a single theoretical measure of the effects of a change in real import prices, αLM and αPM should be geometrically averaged. Thus define the theoretical Fisher type measure αFM of the effects on real income growth of changes in real import prices as the geometric mean of the Laspeyres and Paasche type theoretical measures:

αFM[αLMαPM]1/2.(24.47)

Now that a target export and import index has been defined by (24.65), the problem of finding empirical approximations to this theoretical index is considered.

24.66 Two empirical indices that provide estimates of the effects on the growth of real income of a change in real import prices have already been defined above: the Laspeyres type index PLM defined on the right-hand side of equation (24.44) and the Paasche type index PPM defined on the right-hand side of equation (24.46). It was noted that PLM was a lower bound to the theoretical index αLM and PPM was an upper bound to the theoretical index αPM. Thus PLM will generally have a downward bias compared to its theoretical counterpart whereas PPM will generally have an upward bias compared to its theoretical counterpart. These inequalities suggest that the geometric mean of PLM and PPM is likely to be a reasonably good approximation to the target Fisher type index αFM defined as the geometric mean of αLM and αPM. Thus define the Diewert Lawrence index of the effects on real income of a change in real import prices going from period 0 to 1 as follows:31

PDLM[PLMPPM]1/2.(24.48)

24.67 As in Section B, it will be useful to develop some alternative expressions for the indices PLM, PPM, and PDLM. Define the price relative rM for real import prices as

rMpM1/pM0.(24.49)

24.68 With the techniques described in Section B, the following alternative formulas for PLM, PPM, and PDLM can be derived:32

PLM=1+sM0(rM1);(24.50)
PPM=[1+sM1(rM11)]1.(24.51)

24.69 Noting that sM0 and sM1 are negative, one can see that both PLM and PPM are decreasing functions of rM so that as the real price of imports increases, both indices of growth in real income also decrease as expected.

24.70 Substituting expressions (24.50) and (24.51) into (24.48) leads to the following expression for the Diewert Lawrence import index:

PDLM={[1+sM0(rM1)]/[1+sM1(rM11)]}1/2.(24.52)

24.71 The Diewert Lawrence index PDLM defined by equation (24.52) is likely to be closer to the target Fisher index αFM defined by equation (24.47) than the Laspeyres and Paasche type indices PLM and PPM defined by equations (24.50) and (24.51).

24.72 Using the same notation that is used in equation (24.52), the logarithm of the Diewert Morrison index, PDMM, of the effects on real income of a change in real import prices going from period 0 to 1 is defined as follows:33

InPDMN(1/2)(sM0+sM1)InrM.(24.53)

24.73 It can be verified that PDMM satisfies the time reversal; that is, if the two time periods are switched, then the new PDMM index is equal to the reciprocal of the original PDMM index.

24.74 As in Section B, the interest in the Diewert Morrison index stems from the fact that it has a very direct connection with production theory; in fact, this index is exactly equal to the target index αFM provided that the technology of the production sector can be represented by a general translog functional form in each period. Again make the translog assumptions (24.27) through (24.35).

24.75 Using Hotelling’s Lemma, the logarithmic derivatives of nt(pCt,pIt,pGt,pXt,pMt,Lt,Kt) with respect to the logarithm of the import price are equal to the following expressions for t = 0, 1:

In(ntpCt,pIt,pGt,pXt,pMt,Lt,Kt)InpM=[pMt/nt]nt(pCt,pIt,pGt,pXt,pMt,Lt,Kt)/pM=[pMt/nt][Mt]=sMtusing(24.16).(24.54)

24.76 Noting that assumptions (24.27) imply that the logarithms of the net product functions are quadratic in the logarithms of prices and quantities, one can apply the result given by (24.26) to definitions (24.43), (24.45), and (24.47) to imply the following result:

2InαFM=InαLM+InαPM=[Inn0(pC0,pI0,pG0,pX0,pM0,L0,K0)/InpM+Inn1(pC1,pI1,pG1,pX1,pM1,L1,K1)/InpM][InpM1InpM0]=[sM0+sM1]In(pM1/pM0) using (24.54)(24.55)

24.77 Thus using (24.53) and (24.55), one can see that under the assumptions made on the technology, the following exact equality holds:34

αFM=PDMM.(24.56)

Thus the Diewert Morrison import price effects on real income growth index PDMM defined by equation (24.53) are exactly equal to the target theoretical index, αFM, under very weak assumptions on the technology.

24.78 It can be shown that the following equalities hold for the four empirical indices PLM, PPM, PDLM, and PDMM:35

  PLM(p,p,q,q)=PPM(p,p,q,q)=PDLM(p,p,q,q)=PDMM(p,p,q,q)=1;(24.57)
PLM(p,p,q,q)=PDMM(p,p,q,q)=PDLM(p,p,q,q)=PDMM(p,p,q,q),(24.58)

where ∇PLM(p, p, q, q) is the 20 dimensional vector of first order partial derivatives of PLM(p0, p1, q0, q1) with respect to the components of p0, p1, q0, and q1 but evaluated at a point where p0 = p1 ≡ p and q0 = q1 ≡ q. As usual, the meaning of equations (24.57) and (24.58) is that the four indices approximate each other to the accuracy of a first order Taylor series approximation around an equal prices and quantities data point.

24.79 The second order derivatives of the Laspeyres and Paasche type indices, PLM and PPM, are not equal to each other when evaluated at an equal price and quantity point; that is,

2PLM(p,p,q,q)2PPM(p,p,q,q),(24.59)

where ∇2PLM(p, p, q, q) is the 20 by 20 dimensional matrix of second order partial derivatives of PLM(p0, p1, q0, q1) with respect to the components of p0, p1, q0, and q1 but evaluated at a point where p0 = p1 ≡ p and q0 = q1 = q. Thus as might be expected, PLM and PPM do not approximate each other to the accuracy of a second order Taylor series approximation around an equal price and quantity point.

24.80 However, the second order derivatives of the Diewert Lawrence and Diewert Morrison import indices, PDLM and PDMM, are equal to each other when evaluated at an equal price and quantity point; that is,36

2PDLM(p,p,q,q)=2PDMM(p,p,q,q).(24.60)

24.81 Thus PDLM and PDMM approximate each other to the accuracy of a second order Taylor series approximation around a data point where the real prices are equal in each period and the net output quantities are also equal to each other across periods. The practical significance of this result is that for normal time series data where adjacent periods are compared, the Diewert Lawrence and Diewert Morrison indices give virtually identical results.37

D. The Combined Effects of Changes in the Real Prices of Exports and Imports

24.82 In this section, instead of separately considering the effects of a change in real export or real import prices on the real income generated by the production sector, the effects of a combined change in real export and import prices are considered. It turns out that the same type of analysis that was used in the previous two sections can be used in the present section.

24.83 In a first attempt to measure the effects of changing real import and export prices over the two periods under consideration, a hypothetical period 1 net domestic product maximization problem is considered where producers have at their disposal the period 0 technology set S0 and the period 0 actual labor and capital inputs, L0 and K0 respectively, and they face the period 0 real prices for consumption, net invec0587-01ent, and government consumption, pC0, pI0, and pG0 respectively, but they face the period 1 real export and import prices, pX1 and pM1. The solution to this hypothetical net product maximization problem is n0(pC0,pI0,pG0,pX1,pM1,L0,K0). A theoretical Laspeyres type measure αLXM of the effects on real income growth of the combined changes in real export and import prices from their period 0 levels, pX0 and pM0, to their period 1 levels, pX1 and pM1, can be defined as the ratio of the hypothetical net real income n0(pC0,pI0,pG0,pX1,pM1,L0,K0) to the actual period 0 net real income n0(pC0,pI0,pG0,pX0,pM0,L0,K0):

αLXMn0(pC0,pI0,pG0,pX1,pM1,L0,K0)/n0(pC0,pI0,pG0,pX0,pM0,L0,K0).(24.61)

24.84 The index αLXM of the effects of the change in the real prices of exports and imports is termed a Laspeyres type index because it holds constant all exogenous prices and quantities at their period 0 levels except for the four real export and import prices, pX0,pX1,pM0,andpM1, and the index also holds technology constant at the base-period level.

24.85 As usual, a feasibility argument leads to the following observable lower bound to the theoretical index αLXM defined by equation (24.61):

αLXM[pC0C0+pI0I0+pG0G0+pX1X0pM1M0]/[pC0C0+pI0I0+pG0G0+pX0X0pM0M0]PLXM,(24.62)

where PLXM is an observable Laspeyres type index of the effects on real income of a change in real export and import prices going from period 0 to 1. Note that the numerator and denominator on the right-hand side of equation (24.62) are identical except that pX1 and pM1 appear in the numerator while pX0 and pM0 appear in the denominator. It is possible to show that the Laspeyres type observable index PLMX is a first order Taylor series approximation to the theoretical Laspeyres type index «LMX; that is, it is possible to derive a counterpart to the approximation (24.10).

24.86 A theoretical Paasche type measure αPM of the effects on real income growth of changes in real export and import prices from the period 0 levels, pX0 and pM0, to the period 1 levels, pX1 and pM1, can be defined as the ratio of the actual period 1 net real income n1(pC1,pI1,pG1,pX1,pM1,L1,K1) to the hypothetical net real income defined by n1(pC1,pI1,pG1,pX0,pM0,L1,K1)

αPXMn1(pC1,pI1,pG1,pX1,pM1,L1,K1)/n1(pC1,pI1,pG1,pX0,pM0,L1,K1).(24.63)

The index αPXM of the effects of the changes in the real prices of exports and imports is termed a Paasche type index because it holds constant all exogenous prices and quantities at their period 1 levels except for the real export and import prices, pX0, pX1, pM0, and pM1, and the index also holds technology constant at the period 1 level.

24.87 Using assumption (24.7) for t = 1, the numerator on the right-hand side of equation (24.63) is equal to the period 1 observed real net product, pC1C1+pI1I1+pG1G1+pX1X1pM1M1. Using definition (24.6), one can see that C1, I1, G1, X1, and M1 are a feasible solution for the net product maximization problem defined by the denominator on the right-hand side of equation (24.63), n1(pC1,pI1,pG1,pX0,pM0,L1,K1). These facts mean that there is the following observable upper bound to the theoretical index αLXM defined by equation (24.63):

αPXM[pC1C1+pI1I1+pG1G1+pX1X1pM1M1]/[pC1C1+pI1I1+pG1G1+pX0X1pM0M1]PPXM,(24.64)

where PPXM is an observable Paasche type index of the effects on real income of a change in real export and import prices going from period 0 to 1. Note that the numerator and denominator on the right-hand side of equation (24.64) are identical except that pX1 and pM1 appear in the numerator while pX0 and pM0 appear in the denominator.

24.88 As usual, it is possible to show that the Paasche type observable index PPXM is a first order Taylor series approximation to the theoretical Paasche type index PXM; that is, a counterpart to the approximation (24.13) can be derived.

24.89 Note that both the Laspeyres and Paasche theoretical indices of the effects on real income generated by the production sector of a change in the (real) prices of exports and imports are equally plausible and there is no reason to use one or the other of these two indices. Thus, as usual, αLXM and αPXM should be geometrically averaged. Hence define the theoretical Fisher type measure FXM of the effects on real income growth of changes in real export and import prices as the geometric mean of the Laspeyres and Paasche type theoretical measures:

αFXM[αLXMαPXM]1/2.(24.65)

Now that the target export and import index has been defined by (24.65), the problem of finding empirical approximations to this theoretical index is considered.

24.90 Two empirical indices that provide estimates of the effects on the growth of real income of a change in real export prices have already been defined above: the Laspeyres type index PLXM defined on the right-hand side of equation (24.44) and the Paasche type index PPXM defined on the right-hand side of equation (24.46). It was noted that PLXM was a lower bound to the theoretical index LXM and PPXM was an upper bound to the theoretical index PXM. Thus PLXM will generally have a downward bias compared to its theoretical counterpart whereas PPXM will generally have a upward bias compared to its theoretical counterpart These inequalities suggest that the geometric mean of PLXM and PPXM is likely to be a reasonably good approximation to the target Fisher type index FXM defined as the geometric mean of LXM and PXM. Thus define the Diewert Lawrence index of the effects on real income of a change in real export and import prices going from period 0 to 1 as follows:38

PDLXM[PLXMPPXM]1/2.(24.66)

24.91 As in Section B, it will be useful to develop some alternative expressions for the indices PLXM,

PPXM, and PDLXM. Using the techniques described in Sec tion B, one can derive the following alternative formulas for PLXM, PPXM, and PDLXM:

PLXM=1+sX0(rX1)+sM0(rM1);(24.67)
PPXM=[1+sX1(rX11)+sM1(rM11)]1,(24.68)

where the export shares SXt (positive) and import shares SMt (negative) are defined in (24.16), rXpX1/pX0 is the real export price relative, and rMpM1/pM0 is the real import price relative. It can be seen that both PLXM and PPXM are increasing functions of rX and decreasing functions of rM, because the SMt are negative as defined in equation (24.16), so that as the real price of exports increases, both indices of growth in real income increase and as the real price of imports increases, both indices of growth in real income decrease.

24.92 Substituting expressions (24.67) and (24.68) into (24.48) leads to the following expression for the Diewert Lawrence export and import index:

PDLXM={[1+sX0(rX1)+sM0(rM1)]/[1+sX1(rX11)+sM1(rM11)]}1/2.(24.69)

The Diewert Lawrence index PDLXM defined by equation (24.69) is likely to be closer to the target Fisher index FXM defined by equation (24.65) than the Laspeyres and Paasche type indices PLXM and PPXM defined by equations (24.67) and (24.68).

24.93 As in Sections B and C, there is an alternative to the Diewert Lawrence index PDLXM defined by equation (24.69), namely the Diewert Morrison index PDMMX. Using the same notation that is used in equation (24.69) above, the logarithm of the Diewert Morrison index, PDMXM, of the effects on real income of changes in real export and import prices going from period 0 to 1 is defined as follows:39

InPDMXM(1/2)(sX0+sX1)InrX+(1/2)(sM0+sM1)InrM.(24.70)

24.94 It can be verified that PDMXM satisfies the time reversal; that is, if the two time periods are switched, then the new PDMXM index is equal to the reciprocal of the original PDMXM index.

24.95 As in Section B, the Diewert Morrison index is exactly equal to the target index FXM provided that the technology of the production sector can be represented by a general translog functional form in each period. Thus again make the translog assumptions (24.27) through (24.35). Using the Hotelling’s Lemma results (24.36) and (24.54) and noting that assumptions (24.27) imply that the logarithms of the net product functions are quadratic in the logarithms of prices and quantities, one can apply the result given by equation (24.26) to definitions (24.61), (24.63), and (24.65) to imply the following result:

2InαFXM=InαLXM+InαPXM=[Inn0(pC0,pI0,pG0,pX0,pM0,L0,K0)/InpX+Inn1(pC1,pI1,pG1,pX1,pM1,L1,K1)/InpX]×[InpX1InpX0]+[Inn0(pC0,pI0,pG0,pX0,pM0,L0,K0)/InpM+Inn1(pC1,pI1,pG1,pX1,pM1,L1,K1)/InpM]×[InpM1InpM0]=[sX0+sX1]In(pX1/pX0)+[sM0+sM1]In(pM1/pM0).(24.71)

24.96 Thus using equations (24.70) and (24.71), one can see that under the assumptions made on the technology, the following exact equality holds:40

αFM=PDMM.(24.72)

24.97 Thus the Diewert Morrison combined export and import price effects on real income growth index PDMXM defined by equation (24.70) is exactly equal to the target theoretical index FXM defined by equation (24.65) under very weak assumptions on the technology.

24.98 As in Sections B and C above, it can be shown that the four empirical indices PLXM, PPXM, PDLXM, and PDMXM numerically approximate each other to the first order around an equal price and quantity point:

  PLXM(p,p,q,q)=PPXM(p,p,q,q)=pDLXM(p,p,q,q)=PDMXM(p,p,q,q)=1;(24.73)
PLXM(p,p,q,q)=PPXM(p,p,q,q)=PDLXM(p,p,q,q)=PDMXM(p,p,q,q),(24.74)

where ∇αPLXM(p, p, q, q) is the 20 dimensional vector of first order partial derivatives of PLXM(p0, p1, q0, q1) with respect to the components of p0, p1, q0, and q1 but evaluated at a point where p0 = p1 ≡ p and q0 = q1 ≡q.

24.99 The second order derivatives of the Laspeyres and Paasche type indices, PLXM and PPxM, are not equal to each other when evaluated at an equal price and quantity point; that is,

2PLXM(p,p,q,q)2PPXM(p,p,q,q).(24.75)

However, the second order derivatives of the Diewert Lawrence and Diewert Morrison combined export and import indices, PDLXM and PDMXM, are equal to each other when evaluated at an equal price and quantity point; that is,41

2PDLXM(p,p,q,q)=2PDMXM(p,p,q,q).(24.76)

24.100 Thus PDLXM and PDMXM approximate each other to the accuracy of a second order Taylor series approximation around a data point where the real prices are equal in each period and the net output quantities are also equal to each other across periods. The practical significance of this result is that for normal time series data where adjacent periods are compared, the Diewert Lawrence and Diewert Morrison combined effects indices will give virtually identical results.

24.101 The above material is very similar to the results derived in Sections B and C. But at this point, some new results can be derived. In Section B, measures of the effects on real income growth of a change in real export prices were derived; in Section C, measures of the effects on real income growth of a change in real import prices were derived; and finally, in this section, measures of the combined effects on real income growth of a change in both real export prices and real import prices were derived. A natural question to ask at this point is: How do the partial measures considered in Sections B and C compare to the combined effects measures considered in the present section?

24.102 Using the Diewert Morrison measures, the answer to the above question is very simple. Recalling the expressions (24.22), (24.53), and (24.70), which defined the Diewert Morrison index of the effects on real income growth of a change in real export prices PDMX, in real import prices PDMM, and in the combined effects of changes in real export and import prices PDMXM, respectively, one can see the following simple multiplicative relationship between these three indices:

PDMXM=PDMXPDMM;(24.77)

that is, the combined price effects index PDMXM is exactly equal to the product of the export price effect index PDMX and the import price effect index PDMM.42 Thus when the Diewert Morrison indices are used, the product of the partial effects is equal to the combined effect.43

24.103 The exact decomposition given by equation (24.77) for the Diewert Morrison indices translates into the following approximate decomposition for the Diewert Lawrence indices:

PDLXMPDLXPDLM.(24.78)

24.104 The meaning of the approximate equality is this: From the approximation results derived in this section and the previous sections, it is known that the Diewert Morrison combined effects index PDMXM approximates the Diewert Lawrence combined effects index PDLXM to the second order around an equal price and quantity point and the Diewert Morrison separate effects indices PDMX and PDMM similarly approximate the corresponding Diewert Lawrence separate effects indices PDMX and PDMM. Using these approximation results and the exact identity (24.77) means that the right-hand side of equation (24.78) will approximate the left-hand side of equation (24.78) to the second order around an equal price and quantity point.

24.105 The decomposition results derived above should be useful when dealing with disaggregated export and import data. The reader should be able to use the techniques explained in this chapter to extend the analysis to the case where there are a large number of export and import categories. The corresponding analogues to equations (24.77) and (24.78) will enable the analyst to decompose the overall effects on real income growth owing to changes in the prices of internationally traded goods into separate effects in each category. These separate effects multiply together to give the overall effect on real income growth of changes in the real prices of exports and imports.

E. The Effects on Household Cost-of-Living Indices of Changes in the Prices of Directly Imported Goods and Services

E.1 The case of a single household: Basic framework

24.106 As was mentioned in the introduction to this chapter, households frequently directly import consumer goods and services from abroad without these goods and services passing through the production sector of the economy. Examples of such commodities are tourist expenditures abroad and the direct importation of automobiles. Thus it would be useful to have a framework for modeling the effects of changes in the prices of these directly imported products on household welfare.

24.107 The case of a single household that imports a product is considered here. Let Ct and Mt denote the quantities of a domestic and foreign commodity consumed by the household in period t and let pCt and pMt denote the corresponding period t nominal prices for T0, 1.44 The period thousehold nominal expenditure on all goods and services or period t household “income,” Y t, is defined as the total value of consumer expenditures on consumption products provided by the domestic production sector and by directly imported products:

  YtPCtCt+PMtMt;t=0,1.(24.79)

24.108 As opposed to the generation of real income approach taken in previous sections in this chapter, in this section, a more traditional cost-of-living approach to changes in the prices of directly imported goods and services is taken. Thus the objective of the present section is to derive measures of the effects on the household’s cost-of-living index of a change in import prices from the period 0 nominal level, PM0, to the period 1 nominal level, PM1.

24.109 At this point, household preferences over different combinations of C and M are brought into the picture. It is assumed that in period T0, 1, the household’s preferences are defined by the period t utility function, Ut(C, M), where the function Ut is increasing, continuous, and quasiconcave in its two variables C and M. It will prove useful to define the household’s period t expenditure function, et(pC, PM, u) for periods T0, 1, positive (nominal) prices PC and PM, and utility level u belonging to the range of Ut:

et(PC,PM,u)minC,M{PCC+PMM:Ut(C,M)u};t=0,1.(24.80)

Thus et(pC, PM, u) is the minimum income that the household needs in period t in order to attain the utility level u, given that it faces the prices PC and PM for domestically and foreign supplied goods and services respectively.45

24.110 In what follows, it is assumed that the household minimizes the cost of achieving its utility level

ut Ut(Ct, Mt) in each period t so that the following equalities hold:

et(PCt,PMt,ut)=PCtCt+PMtMt;t=0,1.(24.81)

Assumptions (24.81) are the household counterparts to the producer equalities (24.7). The household expenditure functions defined in this subsection play a key role in the remainder of Section E.

E.2 Theoretical measures of the effects on income of changes in household import prices

24.111 Note that e0(PC0 , PM 1, u0) is the amount of income that the household would need, using the household preferences of period 0, to be able to attain the same level of utility that it attained in period 0 (which is u0) if it faced the period 0 domestic consumption price PC0 but the period 0 household import price was changed from PM0 to the period 1 import price PM1. This hypothetical amount of expenditure could be compared to the period 0 actual expenditure level, e0(pC0,pM0,u0). Thus a theoretical Konüs (1924) Laspeyres partial cost-of-living index that measures the effects of changes in the price of imports that the household faces going from the period 0 level, PM0, to the period 1 level, PM1, can be defined as the ratio of the hypothetical expenditure e0(pC0,pM1,u0) to the actual period 0 expenditure:

KLMe0(PC0,PM1,u0)/e0(PC0,PM0,u0).(24.82)

24.112 The index KLM of the effects of the change in the price of imports is termed a (partial) Laspeyres type index because it holds constant all exogenous prices and utility levels at their period 0 levels except for the import price, PM1.46

24.113 Note that as PM1 increases, KLM defined by equation (24.82) also increases. This is quite different from the properties of the corresponding producer index αLM defined by equation (24.43) where LM decreased as the (real) price of imports increased. But there is a difference in perspective between the previous sections and the present one: In the previous sections, growth of real income owing to changes in international prices was positive for the providers of primary input services whereas in the present section, growth in cost owing to changes in international prices is negative for households.

24.114 There is the following observable upper bound to the theoretical index κLM defined by equation (24.82):47

KLM[PC0C0+PM1M0]/[PC0C0+PM0M0]PLM*(24.83)

where PLM* is an observable Laspeyres partial import price index of the effects on the cost-of-living of a change in import prices going from period 0 to 1, holding the price of consumption constant at the period 0 level. Note that the numerator and denominator on the right-hand side of (24.83) are identical except that PM1 appears in the numerator and PM0 appears in the denominator.

24.115 It is possible to show that the Laspeyres type observable index pLM* is a first order Taylor series approximation to the theoretical index KLM; that is, it is possible to derive a counterpart to the approximation (24.10).48

24.116 The above theoretical measure of the effects of a change in the price of imports used the period 0 preferences for the consumer. It is possible to develop a parallel measure of price change using the consumer’s period 1 preferences. Note that e1(pC1,pM0,u1) is the amount of income that the household would need, using the household preferences of period 1, to be able to attain the same level of utility that it attained in period 1 (which is u1) if it faced the period 1 domestic consumption price PC1 and the period 0 import price PM0. This hypothetical amount of expenditure could be compared to the period 1 actual expenditure level, e1(PC1,PM1,u1). Thus a theoretical Konüs Paasche partial cost-of-living index that measures the effects of changes in the price of imports that the household faces going from the period 0 level, PM0, to the period 1 level, PM1, can be defined as the ratio of the actual period 1 expenditure e1(PC1,PM1,u1) to the hypothetical expenditure e1(pC1,pM0,u1):

KPMe1(PC1,PM1,u1)/e1(PC1,PM0,u1).(24.84)

The index κPM of the effects of the change in the price of imports is termed a (partial) Paasche type index because it holds constant all exogenous prices and utility levels at their period 1 levels except for the two import prices, PM0 and PM1.49

24.117 There is the following observable lower bound to the theoretical index κPM defined by equation (24.84):

KPM[PC1C1+PM1M1]/[PC1C1+PM0M1]PPM*,(24.85)

where PPM* is an observable Paasche partial import price index of the effects on the cost of living of a change in real import prices going from period 0 to 1, holding the price of consumption constant at the period 1 level. Note that as usual, the numerator and denominator on the right-hand side of equation (24.85) are identical except that PM1 appears in the numerator and PM0 appears in the denominator.

24.118 It is possible to show that the Paasche type observable index PPM* is a first order Taylor series approximation to the theoretical index κLM; that is, it is possible to derive a counterpart to the approximation (24.13).

24.119 Note that both the Konüs Laspeyres and Paasche theoretical partial cost-of-living indices of the effects generated by a change in the price of imports are equally plausible and there is no reason to use one or the other of these two indices. Thus if it is desired to have a single theoretical measure of the effects of a change in import prices on the household’s cost of living, κLM and κPM should be geometrically averaged. Hence define the theoretical Fisher type partial cost-of-living index κFM of the effects of changes in household import prices as the geometric mean of the Konüs Laspeyres and Paasche theoretical measures:

KFM[KLMKPM]1/2.(24.86)

With the target import index (24.86) defined, the problem of finding empirical approximations to this theoretical index will now be considered.

E.3 Empirical measures of the effects on income of changes in household import prices

24.120 Two empirical indices that provide estimates of the effects on the cost of living of a household have been defined above: the Laspeyres partial import price index pLM* defined on the right-hand side of equation (24.83) and the Paasche partial import price index PPM* defined on the right-hand side of equation (24.85). It was noted that pLM* was an upper bound to the theoretical index κLM and pLM* was a lower bound to the theoretical index κPM. Thus pLM* will generally have a upward bias compared to its theoretical counterpart while PPM* will generally have a downward bias compared to its theoretical counterpart These inequalities suggest that the geometric mean of pLM* and PPM* is likely to be a reasonably good approximation to the target Fisher type index κFM defined as the geometric mean of κLM and κPM. Thus define the Diewert Lawrence partial cost-of-living index of the effects of a change in import prices going from period 0 to 1 as follows:50

PDLM*[PLM*PPM*]1/2.(24.87)

24.121 As in Section B, it will be useful to develop some alternative expressions for the indices pLM*, PPM*, and PDLM*. Define the household’s period t share of directly imported commodities SMt for t=0, 1 and the price relative for nominal import prices RM as follows:

SMtPMtMt/PCtCt+PMtMt,t=0,1;RMPM1/PM0.(24.88)

24.122 Using the techniques described in Section B, we can derive the following alternative formulas for pLM*, PPM*, and PDLM*:

PLM*=1+SM0(RM1);(24.89)
PPM*=[1+SM1(RM11)]1.(24.90)

24.123 Because SM0 and SM1 are positive, it can be seen that both pLM* and PPM* are increasing functions of RM so that as the nominal price of imports increases, both partial cost-of-living indices increase as expected.

24.124 Substituting expressions (24.89) and (24.90) into (24.87) leads to the following expression for the Diewert Lawrence import index:

PDLM*={[1+SM0(RM1)]/[1+SM1(RM11)]}1/2.(24.91)

The Diewert Lawrence index PDLM* defined by equation (24.91) is likely to be closer to the target Fisher index κFM defined by equation (24.86) than the Laspeyres and Paasche type indices pLM* and PPM* defined by equations (24.89) and (24.90).

24.125 Using the same notation that is defined in equation (24.88), the logarithm of the Diewert Morrison partial cost-of-living index, PDMM*, of the effects of a change in import prices going from period 0 to 1 is defined as follows:51

InPDMM*(1/2)(SM0+SM1)InRM.(24.92)

It can be verified that PDMM* satisfies the time reversal; that is, if the two time periods are switched, then the new PDMM* index is equal to the reciprocal of the original PDMM* index.52

24.126 As in Section B, the interest in the Diewert Morrison index stems from the fact that it has a very direct connection with consumer theory; in fact this index is exactly equal to the target index κFM provided that the preferences of the consumer are translog in each period with certain quadratic coefficients equal to each other. The assumptions made on the consumer’s expenditure functions et for each period are the following general translog counterparts to the translog assumptions (24.27) through (24.35) that were made in earlier sections, letting p [PC, PM]:

Inet(p,u)a0t+Σn=12antInpn+(1/2)Σn=12Σj=12anjInpnInpj+b1tInu+(1/2)b11t(Inu)2+Σn=12CntInpnInu;t=0,1.(24.93)

24.127 Note that as before, the coefficients for the quadratic terms in the logarithms of prices are assumed to be constant over time; that is, it is assumed that anj0=anj1=anj.. The coefficients must satisfy the following restrictions in order for et to be linearly homogeneous in the prices p:

Σn=12ant=1fort=0,1;(24.94)
anj=ajnforalln,j;(24.25)
Σk=12ank=0forn=1,2;(24.96)
Σn=12cnt=0fort=0,1.(24.97)

24.128 Note that using Shephard’s Lemma, the logarithmic derivatives of et(PCt,PMt,ut) with respect to the logarithm of the import price are equal to the following expressions:

Inet(PCt,PMt,ut)/InPM=[PMt/et]et(PCt,PMt,ut)/PM;t=0,1=[PMt/et]Mt=SMtusingequation(24.88).(24.98)

24.129 Noting that assumptions (24.93) imply that the logarithms of the expenditure functions are quadratic in the logarithms of prices and utility, one can apply the result given by equation (24.26) to definitions (24.82), (24.84), and (24.86) to imply the following result:

2InKFM=InKLM+InKPM=[Ine0(PC0,PM0,u0)/InPM+Ine1(PC1,PM1,u1)/InPM][InPM1InPM0]=[SM0+SM1]In(PM1/PM0)usingequation(24.98)(24.99)

Thus using equations (24.92) and (24.99), it can be seen that under the assumptions made on the technology, the following exact equality holds:53

KFM=PDMM*.(24.100)

Thus the Diewert Morrison partial cost-of-living PDMM* defined by equation (24.92) is exactly equal to the target theoretical index, κFM, under very weak assumptions on the technology.

24.130 It can be shown that counterparts to the equalities (24.57), (24.58), and (24.60) hold for the four empirical partial cost-of-living indices pLM*, PPM*, PDLM*, and PDMM*.54 Thus PDLM* and PDMM* approximate each other to the accuracy of a second order Taylor series approximation around a data point where the prices are equal in each period and the consumer demands are also equal to each other across periods. The practical significance of this result is that for normal time series data where adjacent periods are compared, the Diewert Lawrence and Diewert Morrison indices will give virtually identical results.

24.131 Obviously, the above analysis can be repeated to develop theoretical and empirical partial indices that measure the effects on the cost of living of a change in domestic consumer prices pCt. Thus define the period t share of consumer expenditures on domestic goods as SCt and the consumption price relative RC as follows:

SCtPCtCt/[PCtCt+PMtMt],t=0,1;RCPC1/PC0.(24.101)

24.132 Using this notation, we can define the logarithm of the Diewert Morrison partial cost-of-living index, PDMC*, of the effects of a change in consumption prices going from period 0 to 1 as follows:

InPDMC*(1/2)(SC0+SC1)InRC.(24.102)

24.133 The Diewert Morrison complete cost-of-living index can also be defined and this index, PDMCM*, is simply the usual Törnqvist Theil price index which is defined as follows:

lnPDMCM*(1/2)(SC0+SC1)lnRC+(1/2)(SM0+SM1)lnRM.(24.103)

24.134 Using equations (24.92), (24.102), and (24.103), we have the following counterpart to the earlier production theory multiplicative result (24.77):

PDMCM*=PDMC*PDMM*;(24.104)

that is, the overall cost-of-living index PDMCM* is exactly equal to the product of the partial domestic consumption price cost-of-living index PDMC* and the partial import price cost-of-living index PDMM*. Thus when the Diewert Morrison indices are used, the product of the partial effects is equal to the combined effect.55

24.135 The exact decomposition given by equation (24.104) for the Diewert Morrison indices translates into the following approximate decomposition for the counterpart Diewert Lawrence indices:

PDLCM*PDLC*PDLM*(24.105)

F. Conclusion

24.136 There are a large number of approaches that have been suggested over the years that attempt to determine the welfare effects of changes in the prices of exports and imports. The approach taken in this chapter is rather narrow in scope in that only approaches to the measurement of the effects on real income of changes in international prices that are based on producer theory (Sections BD) or consumer theory (Section E) have been considered. However, the approaches outlined in this chapter should prove to be useful in an environment where large fluctuations in food and energy prices are taking place.

24.137 The production theory approach outlined in this chapter can be extended to provide a more complete description of the factors that determine the growth in the real income generated by a production sector. In addition to changes in the real prices of exports and imports, other determinants include changes in real domestic prices, changes in the utilization of primary inputs, and changes in productivity.56 For additional materials on how these additional explanatory factors can be added to the export and import price change factors, see Diewert, Mizobuchi, and Nomura (2005); Diewert and Lawrence (2006); and Kohli (2006).57

1

For materials on these productivity concepts, see the pioneering articles by Jorgenson and Griliches (1967 and 1972) and the excellent Organization for Economic Cooperation and Development manuals, OECD (2001) and Schreyer (2007).

2

Background material on producer theory approaches to production theory can be found in Caves, Christensen, and Diewert (1982b); Diewert (1983a); Balk (1998a); Alterman, Diewert, and Feenstra (1999); and Chapter 18 of the present Manual.

3

In practice, this is not an easy task.

5

Note that when the focus is on income flows generated by an economy, it is necessary to deduct depreciation of capital from gross investment because depreciation is not a sustainable income flow. Thus in this chapter, the target macroeconomic aggregate is (deflated) net domestic product rather than gross domestic product.

6

Note that the flow of income of concern here is the income received by primary inputs used in the market sector of the economy and thus excludes the difference between real primary incomes and current transfers receivable and payable from abroad. Indeed the framework used by the Commission of the European Communities and others (2008), 2008 System of National Accounts (2008 SNA) and outlined in Silver and Mahdavy (1989) defines real net disposable national income as the volume of GDP, plus the trading gain or loss resulting from changes in the terms of trade, plus difference between real primary incomes and current transfers receivable and payable from abroad. The formulas for the terms of trade effect given in the 2008 SNA are, unlike the formal framework outlined here, heuristic in nature.

7

There are other reasons for not using the GDP or NDP deflators as measures of general inflation; see Kohli (1982, p. 211; and 1983, p. 142), Hill (1996, p. 95), and Diewert (2002c, pp. 556–60) for additional discussion.

8

If it is desired to explain nominal income growth generated by the production sector, then it is not necessary to deflate the period t data by PDt. In this case, it can be assumed that PDt equals one so that PC0 =PC0, and so forth.

9

Ulrich Kohli, the chief economist for the Swiss National Bank, has long had an interest in adjusting income measures for changes in a country’s terms of trade using production theory; see Kohli (1990, 2003, 2004a, 2004b, and 2006) and Fox and Kohli (1998). Kohli’s methodology is compared with the Diewert and Lawrence methodology that is used in this chapter and in Diewert (2008).

10

The recent textbook by Feenstra (2004) also takes this point of view.

11

These scalar quantities could be replaced by vectors but this extension is left to the reader.

12

Depreciation has been subtracted from gross investment so the user cost of capital in the present model excludes depreciation so the price of capital services is basically Rymes’ (1968 and 1983) waiting services; see also Cas and Rymes (1991).

13

Producers actually face the prices PCt,PIt,PGt,PXt,PMt rather than the deflated (by PDt) prices PDt. However, if producers maximize net product facing the prices PCt,PIt,PXt,PMt, they will also maximize net product facing the real prices PCt,PIt,PGt,PXt,PMt. There is one additional difficulty: The prices that producers face are different than the prices that consumers and other final demanders face because of commodity taxes. Thus strictly speaking, the theory that is developed in this section and subsequent sections that relies on production theory applies to producer prices (or basic prices) rather than final demand prices.

14

Definition (24.8) is similar to the Laspeyres output price effect defined by Diewert and Morrison (1986, p. 666) except that they used a GDP function instead of a net product function and they did not deflate their aggregate by a price index. Diewert, Mizobuchi, and Nomura (2005, pp. 19–20) and Diewert and Lawrence (2006, pp. 12–17) developed much of the theory used in this chapter.

15

The inequality (24.9) rests on a feasibility argument and this type of argument was first used by Konüs (1924) in the consumer price context.

16

This type of approximation was used by Diewert (1983a, pp. 1095–96) and Morrison and Diewert (1990, pp. 211–12) in the producer theory context but the basic technique (in the consumer theory context) is from Hicks (1942, pp. 127–34; and 1946, p. 331).

17

Hotelling’s (1932, p. 594) Lemma says that the first order partial derivatives of the net product function nt(PCt,PIt,PGt,PXt,PMt, Lt, Kt) with respect to the prices pC, pI, pG, pX, pM are equal to Ct, It, Gt, Xt and Mt respectively for t = 0, 1.

18

This result was established by Diewert and Lawrence (2006, p. 16).

19

Definition (24.11) is analogous to the Paasche output price effect defined by Diewert and Morrison (1986, p. 666) in the nominal GDP context. Diewert, Mizobuchi, and Nomura (2005, p. 19) and Diewert and Lawrence (2006, p. 13) used definitions (24.8), (24.11), and (24.14).

20

This result was established by Diewert and Lawrence (2006, p. 16) and is closely related to similar results derived by Morrison and Diewert (1990, pp. 211–13).

21

The arithmetic average of the Laspeyres and Paasche theoretical indices does not satisfy this time reversal test.

22

Diewert and Lawrence (2006, pp. 14–17) seem to have been the first to define and empirically estimate the indices defined by PLX, PPX, and (24.15) but the closely related work of Morrison and Diewert (1990, pp. 211–12) should also be noted.

23

Strictly speaking, Diewert and Morrison (1986, p. 666) defined their index in the context of a GDP function rather than a net product function and did not deflate prices by a price index. The first applications of formula (24.22) were made by Diewert, Mizobuchi, and Nomura (2005) and Diewert and Lawrence (2006), but the basic methodology is from Diewert and Morrison. Kohli (1990) independently developed the same methodology as that of Diewert and Morrison.

24

This functional form was first suggested by Diewert (1974a, p. 139) as a generalization of the translog functional form introduced by Christensen, Jorgenson, and Lau (1971). Diewert (1974a, p. 139) indicated that this functional form was flexible. Flexible functional forms can approximate arbitrary functions to the second order at any given point and hence it is desirable to assume that the technological production possibilities can be represented by a flexible functional form in each period. Flexible functional forms are discussed in more detail in Diewert (1974a).

25

There are additional restrictions on the parameters which are necessary to ensure that nt(p, z) is convex in p and concave in z. The restrictions (24.29), (24.33), and (24.34) are not required for the results in this chapter. However, they impose constant returns to scale on the technology which is useful if a complete decomposition of real income growth into explanatory factors is attempted as in Diewert and Lawrence (2006).

26

This result is a straightforward adaptation of the results of Diewert and Morrison (1986, p. 666).

27

Subsequent research by Robert Hill (2006) has shown that Diewert’s approximation results break down for the quadratic mean of order r superlative indices as r becomes large in magnitude.

28

The proof is a series of straightforward computations.

29

Again, a long series of routine computations establishes this result. Note that these second derivative matrices are not equal to ∇2PLX(p, p, q, q) or to ∇2PPX(p, p, q, q).

30

See Tables 5 and 9 in Diewert and Lawrence (2006), which establish the approximate equality of these indices (to four significant figures) using Australian data in a gross product framework and Tables 12 and 14 which establish the approximate equality of these indices in a net product framework for Australia.

31

Diewert and Lawrence (2006, pp. 14–17) seem to have been the first to define and empirically estimate the indices defined by PLM, PPM, and equation (24.48).

32

Remember that the period 0 “share” of imports in net real product, SM0, is negative whereas the period 0 share of exports which appeared in the counterpart result (24.19), SX0, was positive.

33

Strictly speaking, Diewert and Morrison (1986; p. 666) defined their index in the context of a GDP function rather than a net product function and did not deflate prices by a price index. The first applications of formula (24.22) were made by Diewert, Mizobuchi and Nomura (2005) and Diewert and Lawrence (2006).

34

This result is a straightforward adaptation of the results of Diewert and Morrison (1986, p. 666).

35

The proof is a series of straightforward computations.

36

Again a long series of routine computations establishes this result.

37

See Tables 5, 9, 12, and 14 in Diewert and Lawrence (2006) for a numerical illustration of this result.

38

Diewert and Lawrence (2006, p. 15) did not actually define the index on the right-hand side of equation (24.66); instead they defined a counterpart index that looked at the effects on real income growth of a change in all real output prices (rather than the effects of a change in just real export and import prices). However, the basic idea behind equation (24.48) is from Diewert and Lawrence.

39

As noted above, strictly speaking, Diewert and Morrison (1986, p. 666) defined their index in the context of a GDP function rather than a net product function and did not deflate prices by a price index. The first applications of formula (24.70) were made by Diewert, Mizobuchi, and Nomura (2005) and Diewert and Lawrence (2006).

40

This result is again a straightforward adaptation of the results of Diewert and Morrison (1986, p. 666).

41

Again a series of routine computations establishes this result.

42

This is a counterpart to a result obtained by Diewert and Morrison (1986, p. 666) and Kohli (1990).

43

This result generalizes to the case where there is a finer classification of exports and imports. In this case, the Diewert Morrison disaggregated effect indices can be multiplied together to obtain the overall effect of changes in all real export and import prices.

44

These price and quantity scalars can be replaced by vectors but for simplicity, only the scalar case is considered here. Note that the Ct in the present section matches up with the Ct that appeared in previous sections but that the Mt in this section is not equal to the production theoretic Mt that appeared in previous sections.

45

The expenditure function et can be used to represent consumer preferences under some assumptions on the utility function Ut; see Diewert (1974b) for the properties of et and references to the literature. An important property for present purposes is that et(pC, PM, u) is increasing in PC and PM.

46

The index defined by equation (24.82) is a partial price change counterpart to a cost-of-living index concept defined by Balk (1989, p. 159) in the context of changing consumer preferences. Balk used the idea of holding tastes constant to work out the effects of price changes. The measure defined by equation (24.84) is also related to various price compensating variations defined by Hicks (1945–46, p. 68; and 1946, pp. 331–32) except that Hicks used differences rather than ratios.

47

Note that assumption (24.81) for t = 0 implies that e0(PC0,PM0,u0)=PC0C0+PM0M0.. Because [C0, M0] is a feasible solution for the cost minimization problem defined by e(PC0,PM1,u0), it must be the case that e(PC0,PM1,u0) is equal to or less than PC0C0+PM1M0, which establishes the inequality in equation (24.83).

48

A first order Taylor series approximation to e(PC0,PM1,u0) is e0(PC0,PM0,u0)+[e0(PC0,PM0,u0)/PM][PM1PM0]=[PC0C0+PM0M0]+M0[PM1PM0]=PC0C0+PM1M0 using M0=e0(PC0,PM0,u0)/PM which is implied by Shephard’s (1953, p. 11) Lemma. This type of approximation is from Hicks (1942 and 1946, p. 331).

49

The index defined by equation (24.85) is also a partial price change counterpart to a cost-of-living index concept defined by Balk (1989, p. 159) in the context of changing consumer preferences.

50

Diewert and Lawrence did not actually suggest this index in the consumer context but it is the consumer theory counterpart to their producer context index discussed earlier.

51

Strictly speaking, Diewert and Morrison did not define this index in the consumer context but it obviously has the same structure as their partial index which was defined in the producer context.

52

The Diewert Lawrence index P*DLM also satisfies this time reversal property.

53

This result is a partial counterpart to results obtained by Caves, Christensen, and Diewert (1982b, p. 1410); and Balk (1989, pp. 165–66).

54

As usual, the proof is a series of straightforward computations.

55

This result generalizes to the case where there is a finer classification of domestic consumption and directly imported household imports.

56

Another factor which is important in explaining real income growth is tax and tariff policy; see Diewert (2001) and Feenstra, Reinsdorf, and Slaughter (2008) on this topic.

57

Diewert (2008) provides a reconciliation of the approaches of these authors.

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