Back Matter
Author: Mika Saito1
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

APPENDIX I

The Cost Minimization Problem

The cost minimization problem of the representative agent in country i is as follows:

MinCosti=piDDi+piMMiwherepiMMi=jipijMMijsubject to X¯i=[δiDiσi1σi+(1δi)Miσi1σi]σiσi1where Mi=[jiϕijMijσsi1σsi]σsiσsi1(19)

where X¯i is industry-specific aggregate inputs; Di is industry-specific domestic inputs, computed in a similar manner as in the consumption good case; Mi is the industry-specific aggregate volume of imported inputs; Mij is industry-specific imported inputs from foreign country j; σi is the elasticity of technical substitution between domestic inputs and the aggregate of imported inputs; σsi is the elasticity of technical substitution among imported inputs from different countries; δi and φij are the industry-specific distribution parameters; piD is the f.o.b. price of domestic inputs in country i; piM is the c.i.f. (plus customs duties) price of the aggregate of imported inputs in country i; and pijM is country i’s c.i.f. (plus customs duties) import price of inputs from country j.

The first-order conditions provide the optimality conditions, which are identical in functional form to equations (5) and (6).

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1

I would like to thank Jeffrey Bergstrand, Robert Feenstra, Keith Head, George Jakubson, Jaime Marquez, Peter Pedroni, Clinton Shiells, Erik Thorbecke, Henry Wan, Jr., and participants at the Midwest Theory and International Economics Meeting in Minneapolis, and seminars at Cornell University, the U.S. International Trade Commission, and the IMF Institute for their support and comments. All remaining errors are mine.

2

For more details see Deaton and Muellbauer (1980).

3

An alternative product differentiation model in the literature is the increasing-returns model (Krugman, 1980). This model identifies varieties by individual firm instead of by individual country as in the Armington model. Head and Ries (2001), however, find empirical evidence that favors differentiated products of the Armington type rather than of the Krugman type.

4

Many other versions of Armington specifications are discussed in the literature besides the two addressed in this paper. For example, Kohli (1998) and Shiells, Roland-Holst, and Reinert (1993) explore more flexible functional forms than the ones discussed here.

5

See Pollak and Wales (1992) for definitions of weak separability and strong separability.

6

In the 1960s Uzawa (1962), McFadden (1963), and Sato (1967) developed n-factor production functions that retained the CES properties of the two-factor case of Arrow and others (1961). Among the three, the two-level CES of Sato (1967) is the most general and has empirical applicability to this paper.

7

The terms intergroup and intragroup elasticities are taken from Sato (1967).

8

The term distribution parameter is taken from Arrow and others (1961). In the trade literature, δi is sometimes referred to as the home bias parameter.

9

There are additional assumptions besides σi = σsi : we need δij = (1 − δi)ϕij to obtain equation (3) and Mi=[jiϕijMijσi1σi]σiσi1 to obtain equation (4).

10

Using these optimality conditions to estimate elasticities implies that we focus on the substitution effect; the income elasticities of demand for Mi, Mij and Di are assumed to be 1. We may, however, be treating some of the income effect imbedded in data as part of the substitution effect; see Marquez (2001) for a survey of different income elasticity estimates in the literature.

11

Section III.D discusses how we classify the two-digit ISIC industries into final (consumption) goods and intermediate inputs industries.

12

Explanations for the growth of intermediate inputs trade, however, differ among these papers: some attribute it to a fall in transport costs (Krugman and Venables, 1995), others to a fall in tariff rates (Yi, 2003), others to a fall in information costs (Jones, 2000), and so on.

13

This scenario does not necessarily imply that the United States and France never used to produce medium-size engines, or that Germany never used to produce large engines. Instead, it implies that the prices of these products in these countries used to be sufficiently high that cross-border demand was zero.

14

The estimates obtained in the first and the second stages in Feenstra (1994) are assumed equivalent to the intragroup and the intergroup elasticities, respectively.

15

Further research is needed to establish this claim.

16

The proximity of the producer price index of the United Nations’ Industrial Statistical Yearbook and the unit labor cost of the OECD’s International Sectoral Data Base is within the acceptable range; the correlation coefficients between the percentage change in the producer price index and the unit labor cost for the Group of Seven countries are 0.85 (in the machinery and equipment industry) and 0.84 (in the “other manufacturing products” industry) at the higher end, and 0.66 (in the food products industry) and 0.69 (in the nonmetallic minerals products industry) at the lower end.

17

Shephard’s lemma states that the cost-minimizing labor input is given by the derivative of the cost function with respect to the wage rate and hence implies that pitcit=1, where cit is the unit labor cost and pit is the unit total cost. Since γ1it=lnpitlncit=pit/pitcit/cit=1citpit, γ1it should equal citpit, which is the labor cost share.

18

The markup γ1i is indexed by the importer i rather than the exporter j, since the markup may be adjusted according to which markets the goods are sold in.

19
An alternative import price index can be derived using duality:
piM=(jiφijσsipijM(1σsi))11σsi,

but this price index cannot be computed since it requires distribution parameters such as φij, which cannot be separately identified from other time-invariant parameters such as τij (see more details in the next subsection). Equation (9), which defines the price of aggregate imports as a variant of the geometric mean, is used as a second-best approximation to this price index, which defines it as a variant of the harmonic mean.

Other alternative price indices such as the log-change price index of Sato (1974) or the exact price index of Feenstra (1994) would not be applicable either, since they are expressed in terms of changes rather than levels.

20

The left-hand side of the equation is in quantities rather than in values. Since trade data are given in values, appropriate price adjustments are made for the actual regressions (details are available from the author upon request).

21

In fact, a common practice is to use the variation in tariff rates and transport costs over time to estimate elasticities of substitution (Baier and Bergstrand, 2001; Head and Ries, 2001; Hummels, 1999; and others).

22

Indeed, Hall (1988), Basu (1996), and others find evidence for increasing returns and price-cost markups.

23

Each of these is efficient under a particular set of assumptions. For example, the fixed effect estimator is more efficient if uit is serially uncorrelated, whereas the first differencing estimator is more efficient if uit follows a random walk. See Chapter 10 in Wooldridge (2002) for more details.

24

The GMM estimates in Erkel-Rousse and Mirza (2002), which range between -76.9 (wrong sign) and 29.37 at the three-digit ISIC, may be an indication of large off-diagonal entries in the weighting (variance-covariance) matrix caused by possible serial correlation.

25

The group-mean tests are used since they have an advantage over the pooled tests; the autoregressive parameter under the alternative hypothesis is not required to be the same across all countries in the panel in the group-mean tests, but this is required in the pooled tests.

26

Pollak and Wales (1992) also argue that to estimate the parameters of the demand systems corresponding to each group in the first stage and of the demand systems of the CES aggregators in the second stage would impose an implausible stochastic structure that prevents disturbances associated with demand functions for goods in one group from affecting the demand for goods in other groups.

27

Details are available from the author upon request.

28

The 14 industrialized countries used in this study are Australia, Belgium-Luxembourg, Canada, Denmark, France, Finland, Federal Republic of Germany, Italy, Japan, the Netherlands, Norway, Sweden, the United Kingdom, and the United States.

29

This approximation assumes constancy of the ratio between gross output and value added, or nonsubstitutability between materials and factors of production.

30

For example, the U.S. input-output table for 1998 shows that the value added for agriculture was approximately $105 billion, but the sum of consumption, investment, government spending, and net exports of agricultural products was only about $35 billion.

31

υi90 for Belgium, Finland, Norway, and Sweden are not available in the OECD Input-Output Database, and therefore the ratio for Denmark is used instead.

32

If we had used data for the textiles industry (ISIC code 32) disaggregated at the three-digit level in the OECD Input-Output Database, the classification of apparel (ISIC code 322) as a final goods industry and of textiles proper (ISIC code 321) as an intermediate inputs industry would have been more apparent. In this study, however, the textiles industry is treated as a final goods industry, despite the fact that this industry produces a large amount of intermediate inputs.

33

When we refer to final goods, we typically mean consumption goods. In the machinery and equipment industry (ISIC code 38), however, final goods are not necessarily consumption goods: for example, automobiles (in ISIC code 384) and televisions (in ISIC code 383) are durables demanded by consumers, but other final goods, such as office computing machinery (in ISIC code 385), are fixed investment goods demanded by producers. The relatively large share of fixed investment goods in this industry is another reason for not categorizing it as either an intermediate input or a final consumption goods industry.

34

For most industries, the panel is smaller because of missing data.

35

Before implementing the FMOLS estimator, the time series properties of the left-hand and right-hand-side variables and the cointegrating relationship between the two are tested: see Table 4 for the t-bar statistics for the unit root tests and the group augmented Dickey-Fuller (ADF) statistics for the cointegration tests. Also, the intergroup and intragroup elasticity estimates for the individual countries are reported in Table 5.

36

The nonmetallic mineral products industry is an exception, but without Canada (whose estimated elasticity takes the wrong sign, -5.43), the estimate is much closer to the estimates of other intermediate inputs industries.

37

Both Marshall (1925; Book V, Chapter VI), who first introduced the concepts of composite demand (or supply), and Hicks (1946; Chapters III and VII), who later made the mathematical refinement of Marshall’s analysis by the use of concepts such as the marginal rate of substitution and the elasticity of substitution, consider that the demand for intermediate inputs of firms is governed by a choice among complementary sets of factors of production, whereas the consumer’s budget tends to be a choice among mild substitutes.

38

Again, before implementing the FMOLS estimator, we check the time series properties of the left-hand and right-hand-side variables and the cointegrating relationship between the two: see Table 4 for the t-bar statistics for the unit root tests and the group ADF statistics for the cointegration tests. The elasticity estimates for the individual countries are reported in Table 5.

39

The food products industry is an exception, but without the United Kingdom (whose estimated elasticity takes the wrong sign, -2.11) the estimate becomes much closer to the estimates of other final (consumption) goods industries.

40

It is commonly observed that the Armington elasticities estimated from time-series variation tend to be lower (and around unity) than those estimated off of cross-sectional variation. The literature still has not established why this is the case; further investigation is needed, but is beyond the scope of this paper.

41

The Wald test statistics are also reported at the individual country level in Table 5.

Armington Elasticities in Intermediate Inputs Trade: A Problem in Using Multilateral Trade Data
Author: Mika Saito