Product Variety and the Gains from International Trade [1 ed.] 9780262289375, 9780262062800

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Product Variety and the Gains from International Trade

Zeuthen Lecture Book Series Karl Gunnar Persson, editor Modeling Bounded Rationality Ariel Rubinstein Forecasting Non-stationary Economic Time Series Michael P. Clements and David E. Hendry Political Economics: Explaining Economic Policy Torsten Persson and Guido Tabellini Wage Dispersion: Why Are Similar Workers Paid Differently? Dale T. Mortensen Competition and Growth: Reconciling Theory and Evidence Philippe Aghion and Rachel Griffith Product Variety and the Gains from International Trade Robert C. Feenstra

Product Variety and the Gains from International Trade

Robert C. Feenstra

The MIT Press Cambridge, Massachusetts London, England

© 2010 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email special_sales@mitpress .mit.edu This book was set in Palatino by Toppan Best-set Premedia Limited. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Feenstra, Robert C. Product variety and the gains from international trade / Robert C. Feenstra. p. cm. — (Zeuthen lecture book series) Includes bibliographical references and index. ISBN 978-0-262-06280-0 (hbk. : alk. paper) 1. International trade—Econometric models. 2. Imports—Econometric models. 3. Exports—Econometric models. 4. Commercial products—Econometric models. I. Title. HF1379.F444 2010 382.01'5195—dc22 2009054150 10

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Contents

Series Foreword Preface ix

vii

1

Introduction

2

Consumer Benefits from Import Variety

3

Producer Benefits from Export Variety

4

The Extensive Margin of Trade and Country Productivity 57

5

Product Variety and the Measurement of Real GDP

6

Conclusions Notes 119 References 123 Index 129

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Series Foreword

The Zeuthen Lectures offer a forum for leading scholars to develop and synthesize novel results in theoretical and applied economics. They aim to present advances in knowledge in a form accessible to a wide audience of economists and advanced students of economics. The choice of topics will range from abstract theorizing to economic history. Regardless of the topic the emphasis in the lecture series is on originality and relevance. The Zeuthen Lectures are organized by the Institute of Economics, University of Copenhagen. The lecture series is named after Frederik Zeuthen, a former professor at the Institute of Economics. Karl Gunnar Persson

Preface

The contents of this manuscript were delivered as the Zeuthen Lectures on April 25 to 27, 2007, at the University of Copenhagen. I would like to thank Karl Gunnar Persson for the generous invitation to present these lectures, along with his colleagues at the Department of Economics for their hospitality. These lectures were also presented at the University of Nottingham on April 14 to 16, 2008. The comments of participants at both locations have benefited the research discussed here. In preparing this manuscript, I have received invaluable assistance from Hong Ma, now at Tsinghua University, to whom I am most grateful. Some time as passed since the lectures were given, so it is inevitable that the ideas have been further developed and benefited from closely related publications. In particular, portions of chapter 2 in this manuscript are drawn from Feenstra (2006) in the Review of World Economics /Weltwirtschaftliches Archiv (Springer) and Feenstra (1994) in the American Economic Review; portions of chapter 3 are drawn from Feenstra (2010) in the Canadian Journal of Economics (Blackwell Publishing); portions of chapter 4 are drawn from Feenstra and Kee (2008) in the Journal of International Economics (Elsevier); and portions of chapters 5 and 6 are drawn from Feenstra, Heston, Timmer, and Deng (2009) in the Review of Economics and Statistics (MIT Press). It is hoped that by integrating the material from these various sources into this manuscript I will have achieved more than the sum of the parts: that as a result, the contribution of import and export variety to the gains from trade will be more apparent, as well as directions for further research.

1

Introduction

One of the great achievements of international trade theory in the latter part of the twentieth century was the incorporation of the monopolistic competition model. The early articles by Helpman (1981), Krugman (1979, 1980, 1981), and Lancaster (1980) paved the way for the incorporation of increasing returns to scale and product variety into many aspects of international trade. On the theoretical side, the impact of these ideas has been very great indeed, with the static models of the 1980s giving rise to the dynamic models of endogenous growth in the 1990s, and in the opening years of the twenty-first century, to models with heterogeneous firms (Melitz 2003). But on the empirical side, the contribution of these models has developed more slowly. One reason for this is that the monopolistic competition models have required new empirical methods to implement their theoretical insights. My goal in these lectures is to describe the methods that have been developed to measure the product variety of imports and exports, and the gains from trade due to product variety. The monopolistic competition model predicts three sources of gains from trade not available in traditional models: first, a fall in prices after tariff reductions due to greater competition between firms and a reduction their markups; second, an increase in the variety of products available to consumers; and third, an improvement in industry productivity due to increasing returns to scale, or with heterogeneous firms, due to self-selection with only the more efficient firms surviving after trade liberalization. Let us consider each of these in turn. The first source of gains from trade, due to the reduction in firm markups, was stressed in Krugman (1979) but has been absent from much of the later literature. There have been estimates of reduced markups due to trade for several countries: Levinsohn (1993) for Turkey, Harrison (1994) for the Ivory Coast, and Badinger (2007a) for

2

Chapter 1

European countries. But these cases rely on dramatic liberalizations to identify the change in markups and are not tied in theory to the monopolistic competition model. The reason that this model is not used to estimate the change in markups is because of the prominence of the constant elasticity of substitution (CES) system, with its implied constant markups charged by firms. Under this framework we simply cannot evaluate the reduction in markups that are expected to accompany an increase in imports. In these lectures I will maintain the assumption of CES preferences and therefore will not address this first source of gains from trade.1 The second source of gains from trade is the consumer gains from having access to new import varieties of differentiated products. Those gains were emphasized by Helpman, Krugman, and Lancaster as being a major source of consumer benefits that are not available in traditional models. But the early empirical work on monopolistic competition and trade found it difficult to put a value onto these gains. For example, Harris (1984a, b) constructed a simulation model based on monopolistic competition to evaluate the gains to Canada from free trade with the United States.2 While Harris used engineering estimates of plant economies of scale within the model, he was reluctant to build in the assumption of product differentiation. The reason, I believe, was that Harris realized that the calculated gains from trade would be very sensitive to the extent of differentiation across products, namely on the elasticity of substitution. For technical reasons I will discuss in chapter 2, the empirical estimates for the elasticity that were available in the 1980s were quite poor. Often the estimated elasticities were too low, and as such would result in exaggerated estimates of the consumer benefits from product variety. Harris realized this potential for bias in his simulation results. So, whereas he always made use of economies of scale, he added product differentiation only a secondary feature to the simulation model. Fortunately, methods are now available—from Feenstra (1994)—to estimate the elasticity of substitution between varieties of traded goods and to use that information to construct the gains from import variety. These methods have recently been applied by Broda and Weinstein (2006) to measure the gains from import variety over 1972 to 2001 for the United States. The formulas they use depend on the elasticity of substitution and also on a direct measure of import variety, which is also called the extensive margin of imports (Hummels and Klenow 2005). Broda and Weinstein obtain an estimate of the gains from trade

Introduction

3

for the United States due to the expansion of import varieties over 1972 to 2001 that amounts to 2.6 percent of GDP in 2001. In chapter 2, we discuss the theory and empirical details of this approach, including STATA code for measuring import variety and for estimating the elasticities of substitution. A maintained assumption in chapter 2 (that will be relaxed in chapter 3) is that increased import variety does not lead to any reduction in domestic varieties. That outcome arises in the model of Krugman (1980), for example. In that case there is a simple theoretical formula to measure the gains from trade as compared with autarky, as noted by Arkolakis et al. (2008a): the ratio of real wages under free trade and under autarky trade equals (1 − import share)−1/(σ −1), where σ is the elasticity of substitution. As the import share rises or as goods are more differentiated, so that the elasticity σ falls, then there are greater consumer gains from import varieties. We apply this simple formula to measure the gain from import variety for 146 countries in 1996 and find that these vary between 9.4 and 15.4 percent of world GDP depending on the value of the elasticity used. As expected, smaller countries tend to have the largest import shares and correspondingly the greatest gains. The third source of gains from trade shifts the focus from the consumer side of the economy to the producer side, and asks whether international trade leads to an improvement in the productivity of firms. Such an improvement in productivity can arise from several sources. It might be that firms rely on imported intermediate inputs and so benefit from increased variety of those inputs. That is the case assumed in much of the endogenous growth literature (Romer 1990; Grossman and Helpman 1991), where the increased range of intermediate inputs fuels growth. One the empirical side, papers that assess the importance of differentiated intermediate inputs include Broda, Greenfield, and Weinstein (2006), and Goldberg et al. (2010), and the earlier contributions of Feenstra et al. (1999) and Funke and Ruhwedel (2000a, b, 2001). These benefits to producers are conceptually the same as the benefits that consumers enjoy from increased product variety. An alternative source of productivity gains arises if openness to trade leads firms to move down their average costs curves, taking greater advantage of economies of scale. Surprisingly, however, there is little evidence to support this hypothesis. In the Canadian case, for example, work by Head and Ries (1999, 2001) finds no systematic indication that Canadian firms grew more in the industries with the

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Chapter 1

greatest tariff reductions under the Canada–US free trade agreement. That that negative finding is confirmed by more recent work by Trefler (2004), who also finds little evidence of changes in firm scale. Furthermore, when we look at episodes of tariff liberalization in developing countries such as Chile and Mexico, as done by Tybout et al. (1991, 1995), there is again little indication that a fall in tariffs leads to an expansion in firm scale. So, based on this evidence, we cannot look to expansions of scale at the firm level to argue that trade liberalization leads to higher productivity. But there is another way that increases in industry productivity can occur, which is strongly supported by the data. In the extension of the monopolistic competition model due to Melitz (2003), firms are heterogeneous in their productivities, with only the more efficient firms becoming exporters. As trade is liberalized in the Melitz (2003) model, less efficient firms are forced to exit the market, and this raises overall productivity in the industry. That prediction is borne out in the experience of Canada following the free trade agreement with the United States. Trefler (2004) finds that while low-productivity Canadian plants shut down, high-productivity firms expanded into the United States. These results provide strong evidence that the Canada–US free trade agreement resulted in the self-selection of Canadian firms, with only the more productive firms surviving. Additional evidence for Europe is provided by Badinger (2007b, 2008). In chapter 3, we argue that this self-selection can still be interpreted as a gain from product variety, but now on the export side of the economy rather than for imports. Surprisingly, the consumer gains from import variety in the Melitz model cancel out with the reduction in domestic varieties when trade is opened. This finding helps to explain the theoretical results of Arkolakis et al. (2008b), where the gains from trade depend on the import share but are otherwise independent of the elasticity of substitution in consumption. They show that the ratio of real wages under free trade and autarky equals (1 − import share)−1/θ , where θ is the Pareto parameter of productivity draws. This formula comes from the production side of the economy, where the self-selection of firms leads to a constant-elasticity transformation curve between domestic and export varieties. Because θ > σ – 1 is required in equilibrium, then the gains from trade in this case are less than in the model of Krugman (1980), but the formula for the gains is similar.3 The reduction in the gains comes from the exit of domestic firm following trade liberalization. Applying this formula to measure

Introduction

5

the gains from trade for 146 countries in 1996, we find that these vary between 3.5 and 8.5 percent of world GDP, depending on the values used for θ. In chapter 4, we explore in more detail the measurement of product variety in trade. Hummels and Klenow (2005) proposed a measure they called the extensive margin of exports and imports that is fully consistent with product variety for a CES function. We show how to construct the the extensive margin of exports, measuring export variety but constructed using the import data for a country from its partners and the world; and conversely, the extensive margin of imports, measuring import variety but constructed with the export data for a country to its partners and the world. The extensive margin of exports is applied by Feenstra and Kee (2008) to estimate the gains from variety growth for 48 countries exporting to the United States over 1980 to 2000. They find that average export variety to the United States increases by 3.3 percent per year, so it nearly doubles over these two decades. That total increase in export variety is associated with a cumulative 3.3 percent productivity improvement for exporting countries;that is, after two decades, GDP is 3.3 percent higher than otherwise due to growth in export variety. In chapter 5, we take an alternative approach to quantify the world gains from trade due to product variety, drawing on the Penn World Table (PWT). As recently argued by Feenstra et al. (2009), the PWT can be used to measure both the standard of living of countries and the real production of countries. The standard of living can be called real GDP on the expenditure side, or RGDE for short, and measures the consumption of countries at a set of common “reference” prices. Alternatively, the real output of countries can be called real GDP on the output side, or RGDO for short, and measures the production of countries at a set of common reference prices. The difference between RGDE and RGDO reflects the trading opportunities that countries have, or more precisely, their terms of trade as measured by their ratio of export prices to import prices. Countries having high export prices or low import prices will benefit more from trade than other countries, and will have a higher ratio of RGDE/RGDO. By choosing a country with very poor terms of trade as the “reference” country, and setting RGDE = RGDO for that country, we can then interpret the difference between RGDE and RGDO, which is positive for all other countries, as a measure of the gains from having better terms of trade than the reference country.

6

Chapter 1

The calculation of RGDO can be made in two ways: by using observed export and import prices for countries, or by adjusting the export and import prices for product variety, namely for the extensive margin of exports and imports. In this second case we are including the impact of trade variety on the productivity of countries. The difference between these two calculations therefore gives a measure of the impact of product variety in trade on productivity, or the gains from trade due to product variety. This estimate can be compared to the simple formulas described above. We find that the worldwide gains from export and import variety amount to 9.4 percent of world GDP. Surprisingly, we do not find any evidence that these gains are higher in smaller countries. Rather, the higher trade shares in small countries are offset by the fact that these countries have lower extensive margins of export and imports; namely they trade fewer varieties of goods. As a result we find no correlation at all between country size and gains from trade due to variety. But there is another source of gains in the PWT data that arises from the differing terms of trade across countries (using observed prices, not adjusted for variety). We find that larger countries receive higher prices for their exports and therefore have better terms of trade than smaller countries. This finding leads to an additional source of gains from trade, which amounts to another 21.4 percent of worldwide GDP, or even greater than the gains due to variety. And in this case the terms of trade gains are positively correlated with country size. These additional gains can arise due to the benefits of country proximity, leading to lower transport costs and higher export prices, or due to lower trade barriers having the same effect. Alternatively, higher export prices might be an indication of product quality, or vertical differentiation in trade, which we do not consider in this book. These results from chapter 5 point to the importance of extending the monopolistic competition model to also incorporate product quality, which has already begun. Additional conclusions and directions for further research are discussed in chapter 6.

2

Consumer Benefits from Import Variety

We begin with the gains from trade for consumers in the monopolistic competition model due to expansion in the variety of goods available through trade. These gains are based on the idea that each country produces products that are somewhat different from other countries. Whether we are talking about automobiles, consumer electronics, or food products, or nearly any other industry, it is very plausible that firms will differentiate their products and that cross-country trade allows consumer to purchase more varieties. So this cornerstone of the monopolistic competition framework seems plausible at face value. From a technical point of view, measuring the benefits of new import varieties is equivalent to the “new goods” problem in index number theory that arises because the price for a product before it is available is not observed, so we don’t know what price to enter in an index number formula. The answer given many years ago by Hicks (1940) was that the relevant price of a product before it is available is the “reservation price” for consumers, namely a price so high that their demand would be zero. Once the product appears on the market, it will have a lower price, determined by supply and demand. Then the fall in the price from its reservation level to the actual price can be used to measure the consumer gains from the appearance of that new good. This idea of Hicks has been applied to new products by Hausman (1997, 1999), who analyzes the appearance of cellular telephones or a new breakfast cereal. The empirical method that Hausman uses requires that we estimate a reservation price for each new product. But we run into difficulty when we try to apply this idea to the appearance of new products varieties from many countries due to trade liberalization. If we assume that each supplying country is providing a different variety from each other country, then we potentially have hundreds if not thousands of new product varieties through trade, and it is impractical

8

Chapter 2

to estimate the reservation price of each. So while the method recommended by Hicks is absolutely correct in theory, it is not that useful in practice when there are many new varieties. We resolve this difficulty by adopting a constant elasticity of substitution (CES) utility function. With many goods, the elasticity of demand is approximately equal to the elasticity of substitution, or σ. So a typical demand curve for this utility function is of the form q = kp −σ , where q denotes quantity, p denotes price, and k > 0 is a constant. This demand curve is illustrated in figure 2.1, and approaches the vertical axis as the price approaches infinity, the reservation prices of the good then being infinite. But provided that the elasticity of substitution is greater than unity, the area under the demand curve is bounded above, and the ratio of areas A/B in figure 2.1 is easily calculated as A/B = 1/(σ – 1) . Thus, even with an infinite reservation price, there is a well-defined area of consumer surplus from having the new good available, and measuring this area depends on having an estimate of the elasticity of substitution. The challenge for this chapter is to generalize this one-good example to a case where many new goods are potentially available from trade. To address that case, we do not rely on consumer surplus to measure the welfare gain, as in figure 2.1, but rather take the ratio of the CES expenditure functions—dual to the utility function—to derive an exact

p

A

1 A = B (σ –1)

B

q Figure 2.1 CES demand

Consumer Benefits from Import Variety

9

cost of living index for the consumer. By determining how new goods affect the cost of living index, we will have obtained an expression for the welfare gain from the new products. After solving this problem, we then apply the results to the monopolistic competition model of Krugman (1980). CES Utility Function We will work with the nonsymmetric CES function, ⎡ ( ) ⎤ Ut = U ( qt , It ) = ⎢∑ ait qitσ −1 σ ⎥ ⎣ i∈It ⎦

σ (σ −1)

, σ > 1,

(2.1)

where ait > 0 are taste parameters that vary with time and It ⊆ {1, . . ., N } denotes the set of goods available in period t at the prices pit. N is the maximum number of goods available. The minimum expenditure to obtain one unit of utility is ⎡ ⎤ e ( pt , It ) = ⎢ ∑ bit pit1−σ ⎥ ⎣ i∈It ⎦

1 (1−σ )

, σ > 1, bit ≡ aitσ .

(2.2)

For simplicity, first consider the case where It−1 = It = I, so there is no change in the set of goods, and also bit−1 = bit, so there is no change in tastes. We assume that the observed purchases qit are optimal for the prices and utility, that is, qit = Ut (∂e / ∂pit ). Then the index number due to Sato (1976) and Vartia (1976) shows us how to measure the ratio of unit-expenditure, or the change in the cost of living for the representative consumer: Theorem 2.1 (Sato 1976; Vartia 1976) If the set of goods available is fixed at It−1 = It = I, taste parameters are constant, bit−1 = bit, and observed quantities are optimal, then e ( pt , I ) ⎛ p ⎞ = PSV ( pt −1 , pt , qt −1 , qt I ) ≡ ∏ ⎜ it ⎟ ⎝ pit −1 ⎠ e ( pt −1 , I ) i ∈I

wi ( I )

,

(2.3)

where the weights wi(I) are constructed from the expenditure shares sit ( I ) ≡ pit qit ∑ i∈I pit qit as wi ( I ) ≡

( sit ( I ) − sit −1 ( I )) (ln sit ( I ) − ln sit −1 ( I )) . ∑ i∈I [( sit (I ) − sit−1 ( I )) (ln sit ( I ) − ln sit−1 ( I ))]

(2.4)

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Chapter 2

The numerator in (2.4) is the “logarithmic mean” of the shares sit (I ) and sit−1 (I ), and lies in between these two shares, while the denominator ensures that the weights wi (I ) sum to unity. The special formula for these weights in (2.4) is needed to precisely measure the ratio of unit-expenditures in (2.3), but in practice the Sato–Vartia formula will give very similar results to using other weights, such as wi (I ) = 12 [sit (I ) + sit −1 (I )], as used for the Törnqvist price index, for example. In both cases the geometric mean formula in (2.3) applies. The important point from theorem 2.1 is that goods with high taste parameters bit−1 = bit will also tend to have high weights. So, even without our knowing the true values of bit−1 = bit, the exact ratio of unit-expenditures is obtained. Now consider the case where the set of goods is changing over time, but some of the goods are available in both periods, so that It −1 ∩ I t ≠ ∅. We again let e(p, I) denote the unit-expenditure function defined over the goods within the set I, which is a nonempty subset of those goods available both periods, I ⊆ It −1 ∩ It ≠ ∅. We sometimes refer to the set I as the “common” set of goods. Then the ratio e( pt , I )/ e( pt −1 , I ) is still measured by the Sato–Vartia index in the theorem above. Our interest is in the ratio e( pt , It )/ e( pt −1 , I t −1 ), which can be measured as follows: Theorem 2.2 (Feenstra 1994) Assume that bit−1 = bit for i ∈ I ⊆ It −1 ∩ I t ≠ ∅, and that the observed quantities are optimal. Then for σ > 1, e ( pt , It ) ⎛ λ (I ) ⎞ = PSV ( pt −1 , pt , qt −1 , qt I ) ⎜ t ⎝ λt −1 ( I ) ⎟⎠ e ( pt −1 , It −1 )

1 (σ −1)

,

(2.5)

where the weights wi(I) are constructed from the expenditure shares sit ( I ) ≡ pit qit ∑ i∈I pit qit as in (2.4), and the values λt(I) and λt−1(I) are constructed as ⎛ ∑ i∈I ,i∉I piτ qiτ ⎞ ⎛ ∑ piτ qiτ ⎞ i∈I τ λτ ( I ) = ⎜ ⎟ , τ = t − 1, t. ⎟ = 1− ⎜ ⎜⎝ ∑ piτ qiτ ⎟⎠ ⎜⎝ ∑ piτ qiτ ⎟⎠ i∈Iτ i∈Iτ

(2.6)

Each of the terms λτ(I) < 1 can be interpreted as the period τ expenditure on the good in the common set I, relative to the period τ total expenditure. Alternatively, this can be interpreted as one minus the period τ expenditure on “new” goods (not in the set I), relative to the period τ total expenditure. When there are more new goods in period t, this will tend to lower the

Consumer Benefits from Import Variety

11

value of λt(I), which leads to a greater fall in the ratio of unit costs in (2.5), by an amount that depends on the elasticity of substitution. The importance of the elasticity of substitution can be understood from figure 2.2, where the consumer minimizes the expenditure needed to obtain utility along the indifference curve AD. If initially only good 1 is available, then the consumer chooses point A with the budget line AB. When good 2 becomes available, the same level of utility can be obtained with consumption at point C. Then the drop in the cost of living is measured by the inward movement of the budget line from AB to the line through C, and this shift depends on the convexity of the indifference curve, or the elasticity of substitution. Monopolistic Competition Model To illustrate the usefulness of the results above, we consider the monopolistic competition model of Krugman (1980). We will suppose that the utility function in (2.1) applies to the purchases of a good from various source countries i ∈ It . That is, the elasticity of substitution we are interested in is the Armington (1969) elasticity between the source countries for imports. We refer to the source countries as providing varieties of the differentiated good, so the gains being measured in (2.5) are the gains from import variety. In this case we can compare the formula in (2.5) with the gains from trade obtained in the model of Krugman (1980), as analyzed by Arkolakis et al. (2008a). q2 D B

C

A Figure 2.2 CES indifference curve

q1

12

Chapter 2

In particular, suppose that there are a number of countries, where the representative consumer in each has a CES utility function with elasticity σ > 1. Labor is the only factor of production and there is a single monopolistically competitive sector, with no other goods.1 Firms face a fixed cost of f to manufacture any good, and an iceberg transport cost to sell it abroad, but no other fixed cost for exports. Then it is well known that with profit-maximization and zero profits through free entry, the output of each firm is fixed at the amount 2 q = (σ −1) fϕ ,

(2.7)

where ϕ is the productivity of the firm, namely the number of units of output per unit of labor. With the population of L, the full-employment condition is then ⎡q ⎤ L = N ⎢ + f ⎥ = Nσ f , ⎣ϕ ⎦

(2.8)

which determines the number of product varieties produced in equilibrium as N = L /σ f . This condition holds under autarky or trade, so opening a country to trade has no impact on the number of varieties produced within a country. The gains from opening trade can be measured by the ratio of real wages under free trade and autarky. With labor as the only factor of production we can normalize wages at unity, so that the rise in real wages is measured by the drop in the cost of living, which is the inverse of (2.5). The “common” set of goods is those domestic varieties that are available both in autarky and under trade. Then the Sato–Vartia index PSV is just the change in the price of the domestic varieties, and with constant markups that equals the change in home wages, which we have normalized to unity. So the gains from trade amount to (λt / λt−1 )−1/(σ −1), as in (2.5). The denominator of that ratio reflects the disappearance of domestic varieties, namely those varieties available in period t − 1 but not in period t. As we have shown above, there are no disappearing domestic varieties in this model, so λt−1 = 1. The numerator λt measures the expenditure on the domestic varieties relative to total expenditure with trade, or one minus the import share. The gains from trade are therefore (1 − import share)−1/(σ −1), which is precisely the formula obtained by Arkolakis et al. (2008a). To implement this formula, we need to have reliable estimates of the elasticity of substitution for each product, as discussed next.

Consumer Benefits from Import Variety

13

Measuring the Elasticity of Substitution Recall from our discussion in chapter 1 that Harris (1984a, b) was reluctant to choose a particular value for the elasticity of substitution when simulating Canada–US free trade: existing estimates at that time tended to be too low, which would lead to exaggerated estimates of the gains from trade. The reason that these estimates were so low, I believe, was due to the standard simultaneous equations bias: the elasticity of demand cannot be estimated in a demand and supply system without instrumental variables that are orthogonal to the error terms. But in international trade we are interested in estimating the elasticity of substitution between source countries for each good; in other words, we want to measure the Armington (1969) elasticity between source countries. It is difficult if not impossible to find instruments that can be used in every market and country. Feenstra (1994) proposed a method to resolve this problem that makes use of the panel nature of datasets in international trade, namely having time-series observations on the amount imported from multiple source countries. To motivate this method, we begin with an interesting historical discussion of the identification problem drawn from Leamer (1981). The Identification Problem Leamer (1981) poses the identification problem in the following way. Suppose that we have collected the data on prices and quantities for a particular good over time, but that we do not have any additional information on the shocks to supply or demand. Using just the price and quantity data, and assuming normally distributed errors on the supply and demand curves, we can still ask what the maximum likelihood estimates of the supply and demand elasticities are. Leamer shows that the maximum likelihood estimates are not unique: the estimates can be anywhere along a hyperbolic curve, as illustrated in figure 2.3, between the demand elasticity σ and the supply elasticity ω. The fact that the estimates are not unique is just another way of saying that we cannot identify the supply and demand elasticities without further information. There is a fascinating paragraph in Leamer ’s article where he describes a historical debate between Leontief (1929) and Frisch (1933) concerning the identification problem. Since the maximum likelihood estimates of the supply and demand elasticities are not

14

Chapter 2

ω

σ Figure 2.3 Maximum likelihood estimates

unique, Leontief made the suggestion that we split the sample in half. The first half of the sample could give us estimates of the supply and demand elasticities over one hyperbolic curve, and the second half of the sample could give us estimates along a second curve. It would then seem that we could take the intersection of these two curves to obtain unique estimates of the supply and demand elasticities, and thereby overcome the identification problem. Leamer (1981, p. 321) reports that: “This procedure brought down upon [Leontief] the wrath of Frisch’s 1933 book, which is devoted almost completely to debunking the method.” The reason the idea does not appear to work is that by just splitting a sample in half, there is no reason to expect the two curves obtain to be different from each other. If the first half of the sample is drawn from the same statistical population as the second half, then the maximum likelihood estimates of the supply and demand elasticities would lie along the same hyperbolic curve in both cases. If the two curves are just the same, then their intersection is still the same curve, and we have not made any headway at all! But Leontief may have been right after all, if we just add another feature to the data. Rather than having the price and quantity of just one variety of a differentiated good over time, suppose that we have the price and quantity for that good exported from multiple countries over time. So, in addition to the time dimension of the dataset, we have

Consumer Benefits from Import Variety

15

a country dimension, making it a panel. We continue to assume that the elasticity of substitution between the goods from each country is constant over time, and also the same across countries. In other words, the variety supplied by one country is different from that supplied by any other country, but a German variety is just as different from a French variety as it is from an American variety. This assumption of a constant elasticity of substitution over time and across countries is a simplification, of course, but it allows us to make great progress on the identification problem. For now we can use the price and quantity exported of the German variety to get one curve of maximum likelihood estimates of the supply and demand elasticities, and then the French data on price and quantity exported to get a second curve, and then the American data to get a third curve, and so on, as illustrated in figure 2.4. The elasticity of substitution of demand σ is the same across countries, and we might assume the same is true for the elasticity of supply ω. Then a point near to the intersection of these multiple curves, as shown by point A, gives us an estimate of the supply and demand elasticities. Furthermore, in contrast to the proposal of Leontief, there are very good reasons to expect that these hyperbolic curves for each country will differ from each other. As we now show, the curve for each country depends on the variances and covariances of the supply and demand shocks, which can depend on the variance of exchange rates and other ω

A

σ Figure 2.4 Maximum likelihood estimates with panel

16

Chapter 2

macroeconomic variables. Provided that our panel of countries includes those with differing variances and covariances of shocks, then this method should result in reliable estimates for the supply and demand elasticities, even though we do not have instrumental variables in the conventional sense. Estimation with Panel of Countries Let us now describe this procedure formally. By differentiating the expenditure function in (2.2), we can calculate that the share of expenditure on each variety i is ⎡ pit ⎤ sit = bit ⎢ ⎥ ⎣ c ( pt , I t ) ⎦

1−σ

.

(2.9)

Taking natural logs and taking the difference over time, we can write this demand equation as Δ ln sit = φt − (σ −1) Δ ln pit + ε it ,

(2.10)

where φt ≡ (σ − 1)Δ ln c( pt , It ) is a time fixed-effect, and ε it ≡ Δ ln bit is an error term reflecting taste shocks. To this demand curve we add a supply curve, which is assumed to be Δ ln pit = βΔ ln qit + ξit ,

(2.11)

where ξit is the random error in supply. It is inconvenient to have the quantity appearing in the supply curve (2.11) and the share of expenditure appearing in the demand curve (2.10). To resolve this slight inconsistency, we combine (2.10) and (2.11) to eliminate the quantity from the supply curve, obtaining Δ ln pit = ψ it +

ρε it + δ it , σ −1

(2.12)

where the parameters appearing in this “reduced form” supply curve are

ψ it =

β (φt + Δ ln Et ) β (σ −1) and ρ = , 0 < ρ < 1, 1 + βσ 1 + βσ

(2.13)

where Et = ∑ i∈I pit qit is total expenditure, and the error term is t

δ it =

ξit . 1 + βσ

(2.14)

Consumer Benefits from Import Variety

17

Notice that the error term ε it ≡ Δ ln bit in the demand equation (2.10) is correlated with the price in (2.12). This illustrates the simultaneous equations problem, whereby an outward shift in the demand curve (and increase in ε it ≡ Δ ln bit ) will lead to a rise in price along the supply curve. This fact means that ordinary least squares cannot be used to estimate the demand equation (2.10). The typical procedure is to use instead some instruments that are correlated with the expenditure share and price but uncorrelated with the errors in these equations. But as we have mentioned, it is difficult to obtain such instruments for every international market. We now show how the simultaneous equation bias can in fact be avoided without conventional instruments, by exploiting the panel nature of the dataset. The key assumption we make is as follows: Assumption 2.1 The error terms ε it, i = 1, . . . , N and δ jt , j = 1, . . . , N are all independent, with mean zero and variances σ εi and σ δ i, respectively. This assumption means that the error terms in the demand and supply equations are uncorrelated, for all countries i, j = 1, . . . , N in the sample. To make use of this assumption, we begin by differencing the demand and supply equations with respect to some county k, as

εit = ε it − ε kt = ( Δ ln sit − Δ ln skt ) + (σ −1) ( Δ ln pit − Δ ln pkt ) , ρε δit = δ it − δ kt = ( Δ ln pit − Δ ln pkt ) − it σ −1 = (1 − ρ ) ( Δ ln pit − Δ ln pkt ) −

( σ ρ−1)(Δ ln s − Δ ln s ). it

kt

Multiply these two equations together and dividing by (1 – ρ)(σ – 1), we obtain Yit = θ1X1it + θ 2 X 2 it + uit ,

(2.15)

where Yit = ( Δ ln pit − Δ ln pkt ) ,

(2.16)

X1it = ( Δ ln sit − Δ ln skt )2 ,

(2.17)

X 2 it = ( Δ ln pit − Δ ln pkt ) ( Δ ln sit − Δ ln skt ) ,

(2.18)

2

θ1 =

ρ (σ −1)2 (1 − ρ )

, θ2 =

2ρ − 1 (σ −1) (1 − ρ )

,

(2.19)

18

Chapter 2

and uit =

εitδit . (σ −1) (1 − ρ )

(2.20)

Finally, let us average the variables appearing in (2.15) through (2.18) over time, and obtain the following equation that is estimated as a cross-country regression: Yi = θ1X1i + θ 2 X 2 i + ui.

(2.21)

Notice that the variables appearing in (2.21) are second moments of the data, that is, essentially the variances and covariance of log differences of prices and shares. Furthermore, as T → ∞ , the error term in (2.21) vanishes in its probability limit, since this error term is the crossmoment of the errors in the supply and demand curves, which are uncorrelated by assumption 2.1. Therefore, as shown in appendix 2.1 to this chapter, this procedure gives consistent estimates θ1 and θ 2 provided that the two right-hand side variables in (2.21) are not co-linear. That will be the case provided that there exist countries i and j such that

σ ε2i + σ ε2k σ δ2i + σ δ2k . ≠ σ ε2j + σ ε2k σ δ2j + σ δ2k

(2.22)

Condition (2.22) simply states the there is some countries i and j for which the relative variance of errors in supply and demand are different. In other words, there is some heteroskedasticity in the error terms for supply and demand across countries. That is a very plausible assumption, since the error terms in supply depend on country-specific shocks to wages, productivity, and exchange rates, while the error terms in demand depend on changes to the quality of goods supplied by each country. Also we show in appendix 2.1 that there are implicitly some instrumental variables used to estimate (2.15), and these are the country fixed effects. The estimation method is described at an application of the generalized method of moments (GMM), since it exploits the condition that the first moment of the residual in (2.15) or (2.21) is zero in expected value. While condition (2.22) is enough to ensure that ordinary least squares (OLS) estimates of (2.21) are consistent, efficient estimates require that weighted least squares (WLS) is used. To achieve that, a preliminary estimates of (2.21) is made with OLS, and then the inverse of the standard deviation of the residuals from that regression is used to re-

Consumer Benefits from Import Variety

19

weight the regression, running WLS to obtain efficient estimates. The STATA code to achieve this is included in appendix 2.2 of this chapter. Given the efficient estimates θˆ1, θˆ 2, a quadratic equation obtained from (2.19) is solved to obtain σˆ , ρˆ . This solution is the subject of the next theorem: Theorem 2.3 (Feenstra 1994) So long as θˆ1 > 0, then the estimates of σ and ρ are as follows: 12

⎤ 1 ⎡1 1 a. if θˆ 2 > 0, then ρˆ = + ⎢ − ⎥ , 2 2 ⎢⎣ 4 4 + θˆ 2 θˆ1 ⎦⎥

(

)

12

⎤ 1 ⎡1 1 b. if θˆ 2 < 0, then ρˆ = − ⎢ − ⎥ , 2 ˆ ˆ 2 ⎢⎣ 4 4 + θ 2 θ1 ⎥⎦

(

)

and in either case, ⎛ 2 ρˆ −1⎞ 1 > 1. σˆ = 1+ ⎜ ⎝ 1− ρˆ ⎟⎠ θˆ 2 1 As θˆ 2 → 0 , then ρˆ → and σˆ → 1 + θˆ1−1/ 2 . 2 In the event that θˆ1 is negative, then the formulas in theorem 2.3 fail to provide estimates for σ and ρ that are both in the ranges σˆ > 1 and 0 ≤ ρˆ < 1. For example, if θˆ1 is only slightly less than zero (−θˆ 22 / 4 < θˆ1 < 0 ), then ρˆ ∉[0 , 1], and (σˆ − 1) has the same sign as −θˆ 2 . In this case it is still possible that a value of σˆ exceeding unity is obtained. However, if θˆ1 is too low (θˆ1 < −θˆ 22 / 4), then imaginary values for ρˆ and σˆ are obtained. In general, σˆ must be greater than unity in order to apply theorem 2.2. In practice, the elasticities of substitution obtained from this method are very plausible indeed. Feenstra (1994) considered products like men’s leather athletic shoes, or cotton knit shirts, or various types of steel, and obtained estimates of the elasticity between 3 and 8. These were much higher values than obtained previously, and more in line with what trade economists would expect. Feenstra even added gold bullion and silver bullion as additional test cases, and obtained estimates of the elasticity of substitution for each of these products of 25 and 40, respectively. Those high estimates are essentially infinite, indicating that there is perfect substitution between country sources of

20

Chapter 2

gold or silver. To conclude, for the six products that Feenstra (1994) analyzed, the new method for estimating the elasticity of substitution worked very well indeed. Furthermore the method of splitting a sample to obtain identification also works in other contexts, such as modern finance, as shown by Rigobon (2003). He calls this method “identification through heteroskedasticity.” Like Feenstra (1994), Rigobon essentially re-discovered and justified the method proposed more than seventy-five years ago by Leontief. Gains from Import Variety for the United States Broda and Weinstein (2006) apply the above methods to measure the gains from trade for the United States. They define a good as either a 10-digit harmonized system (HS) category, over the period 1990 to 2001, or as a 7-digit tariff schedule of the United States (TSUSA) category, for the earlier period 1972 to 1988.3 The imports from various source countries are the varieties available for each good. The ratio (λt / λt−1 ) is constructed for each good, using the expenditure on new and disappearing source countries. In addition they estimate σ for each good, extending the methods above to perform a grid search over values of σ greater than unity when theorem 2.3 above results in imaginary values for σ. For the TSUSA categories of goods, they estimate roughly 12,000 values for σ, with a median value of 3.6. For the HS system, they estimate about 14,500 values for σ, with a median value of 2.9. In both cases the distribution of elasticities is highly skewed toward the right (so the mean values are much larger than the medians). Putting together the ratios (λt / λt−1 ) for each good with the elasticity of substitution, they obtain (λt / λt−1 )−1/(σ −1), which is aggregated over all goods. For the TSUSA data, they used 1972 as the base year and measured the gains from new supplying countries up to 1988, and then for the HS data, they used 1990 as the base year and measured the gains from new supplying countries up to 2001. Summing these, they obtain an estimate of the gains from trade for the United States due to the expansion of import varieties, which amount to 2.6 percent of GDP in 2001. Two features of Broda and Weinstein’s methods deserve special mention. First, by measuring the expenditure on new supplying countries relative to a base year, they are following the hypothesis of theorem 2.2 that the “common” set of countries should be those with constant taste parameters. In contrast, when countries first start exporting goods,

Consumer Benefits from Import Variety

21

it is reasonable to expect that the demand curve in the importing country shifts out over some number of years as consumers become informed about the product. Broda and Weinstein are allowing for such shifts for new and disappearing countries after the base year, and all such changes in demand for these countries are incorporated into the λτ terms in theorem 2.2. That is the correct way to measure the gains from new import varieties.4 Second, Broda and Weinstein (2006) did not incorporate any changes in the number of US varieties into their estimation, nor include the United States as a source country in the estimation of the elasticity of substitution for each good. That is the correct approach only under the limited case where the number of US varieties is constant. While that is true under our assumptions in the model of Krugman (1980), it is certainly not the case in more general models: we could expect that increases in import variety would result in some reduction in domestic varieties. In that case the gains from import varieties would be offset by the welfare loss from reduced domestic varieties. That potential loss was addressed only briefly by Broda and Weinstein (2006), and we will address it more fully in the next chapter. Worldwide Gains from Import Variety We conclude this chapter with a calculation of the worldwide gain from trade, where we apply the formula (1 − import share)−1/(σ −1) to a broad cross section of countries. For this purpose we use data from the Penn World Table (PWT), which provides measures of real GDP for many countries. We use version 6.1 of PWT, which has 1996 as the benchmark year, and focus our calculations on that year. We started with 151 countries from PWT as described in Feenstra et al. (2009). Five of these countries (including Hong Kong and Singapore) had either imports or exports that exceed their value of GDP, which is impossible in the monopolistic competition model because it does not consider imports as intermediate inputs. We did not attempt to extend the model to incorporate intermediate inputs, though this would be a valuable exercise. Instead, we simply omitted these five countries. With the remaining sample of 146 countries, we computed the import share as nominal imports relative to nominal GDP. In order to apply the gains from trade formula, we need to use a value for the elasticity of substitution, which we draw from Broda and Weinstein (2006).

22

Chapter 2

Notice that we are not applying the gains from trade formula to each sector, which might be more accurate but would again involve additional considerations of whether imports are used as intermediate inputs or final goods. So we simply make an aggregate calculation, and for this purpose, we need to decide on a single value to use for the elasticity of substitution. Recall that the distribution of elasticity estimates from Broda and Weinstein (2006) is highly skewed, with some very high estimates leading to a mean value for the elasticity that is considerably higher than its median. Ideally we would like to use a median estimate of the elasticity of substitution for each country that reflects its pattern of trade, namely which goods it actually imports. To this end we merged the harmonized system elasticity estimates from Broda and Weinstein (2006) with the 4-digit Standard International Trade Classification (SITC) import data for each country. For each 4-digit SITC product there are multiple HS codes and multiple elasticities, and so we choose the median elasticity within each of these product groups, denoted by σi. Some of the 4-digit SITC products are not imported by a country, in which case no elasticity is used. Then across all the 4-digit SITC industries actually imported by each country, we take the median value of σi. It turns out that this median estimate is very tightly distributed around 2.9 (which is the overall median for the HS data), lying between 2.82 and 3.08 for all countries in our sample. Accordingly we will use σ = 2.9 as the benchmark value for our calculation of the gains from trade. Alternatively, we can start with the median elasticity within each product group, σi, and then consider taking the mean value of 1/(σ i − 1) across all SITC products for each country. This gives us the mean value 1/(σ i − 1), from which we can recover the implied mean value of the −1 elasticity as σ = [1/(σ i − 1)] + 1. This mean value lies between 2.47 and 2.65 for all countries in our sample, so we will use 2.5 as a second value for the elasticity. The third value we will use is 3.6, which is the median value for the elasticity in the pre-1989 period considered by Broda and Weinstein (2006), using TSUSA rather than HS data. We then compute the gains from trade in the monopolistic competition model of Krugman (1980) as −1 (σ −1)

gains from import variety = ⎡ (1 − import share) ( − 1 ⎤ , ⎢ (1 − import share)−1 σ −1) ⎥ ⎦ ⎣ using the values σ = 2.5, 2.9, and 3.6. This formula expresses the gains from import variety as a percent of the free trade real wages, which will be convenient for comparison with later calculations.

Consumer Benefits from Import Variety

23

In table 2.1 we record the gains for a selection of 20 countries in our sample, over a range of real GDP per capita and the total gains over all 146 countries.5 Included in table 2.1 are the countries with the highest gains due to imports: Tajikistan, Equatorial Guinea, Malaysia, Malta, and Ireland. These countries have the highest import shares in the sample, and correspondingly the highest gains due to trade. The United States has among the smallest import share and percentage gains, but some countries with low per-capita income also have small import shares and gains. China, for example, has a relatively low import share in 1996, and correspondingly modest gains. Over all the 146 countries in the sample, the gains from trade in the Krugman (1980) model amount to 12.5 percent of real GDP in the benchmark case, and vary between 9.4 and 15.4 percent of real GDP with the alternative values for σ that we consider. In figure 2.5 we graph the gains from import variety against the real GDP of all countries in the sample, using the benchmark value σ = 2.9. Several of the countries with the highest gains are indicated, as well as the United States at the low end. As shown by the trend line in this figure, there is a negative (and statistically significant) relationship between gains due to import variety and real GDP, which simply reflects the negative relationship between import shares and GDP: large countries trade less as a share of GDP.6 While this negative relationship is not that surprising, it will serve as a comparison as we will continue to investigate the relationship between country size and alternative measures of the gains from trade in later chapters.

Gain from import variety (%)

75 Malaysia

Malta Equitorial Guinea

Tajikistan Ireland

50

25

0 100

United States 1,000

10,000

100,000

Real GDP, 1996 ($million) Figure 2.5 Gain from import variety and real GDP, 1996

1,000,000

10,000,000

24

Chapter 2

Table 2.1 Gains from import variety, 1996 Real GDP per capita (US$)

Import share (%)

Gains from import variety (%) Elasticity of substitution: σ = 2.5 σ = 2.9 σ = 3.6

Democratic Republic of Congo Tajikistan Nepal Equatorial Guinea Republic of Moldova China Guatemala Brazil Thailand Gabon Malaysia Saudi Arabia Malta Bahamas Israel Bermuda Ireland Germany Norway United States 146 countries

245

23.2

16.2

13.0

9.7

775 1,007 1,104 1,737 2,353 3,051 5,442 5,840 7,084 7,448 9,412 10,420 13,081 13,138 15,017 15,150 17,292 20,508 23,648

80.1 31.4 88.0 62.3 17.9 20.0 9.0 42.5 47.6 90.2 36.0 88.6 34.2 39.7 37.6 74.6 24.5 35.2 12.3

65.9 22.2 75.7 47.8 12.3 13.8 6.1 30.9 35.0 78.8 25.8 76.5 24.3 28.6 27.0 59.9 17.1 25.1 8.4

57.2 18.0 67.3 40.2 9.9 11.1 4.8 25.3 28.8 70.6 21.0 68.2 19.8 23.4 22.0 51.4 13.8 20.4 6.7

46.3 13.5 55.8 31.3 7.3 8.2 3.6 19.2 22.0 59.1 15.8 56.7 14.9 17.7 16.6 41.0 10.3 15.4 4.9

15.4

12.5

9.4

Source: Author ’s calculations. Note: Real GDP per capita is similar to that reported in PWT version 6.1, for 1996. Import share is nominal imports divided by nominal GDP, 1996 data. The gains from import variety are computed as [(1 − import share)−1/(σ −1) − 1] divided by (1 − import share)−1/(σ −1), and are expressed as a percentage of the free trade real GDP per capita.

Consumer Benefits from Import Variety

25

Appendix 2.1 In this appendix an instrumental variable (IV) interpretation is provided for the estimator for the elasticity of substitution. Let Ti ≤ T denote the number of periods for which the import price and quantity data for country i=1, . . . , N, i≠k, is available in first-differences (where we assume that the data for country k is available in all periods). Stacking (2.15) over time and then over countries, the total number of observations is L ≡ ∑ i ≠ k Ti. Letting Y denote the L × 1 vector with components Yit, X the L × 2 matrix with rows ( X1it , X 2 it ), u the L × 1 vector with components with uit, and θ the column vector (θ1 , θ 2 ), we can rewrite equation (2.15) as Y = Xθ + u.

(A2.1)

To control for the correlation of uit with X1it and X 2 it, we use the IV that are dummy variables for each country i ≠ k. Let  i denote a Ti × 1 vector of 1’s, i = 1, . . . , N, i ≠ k , and define Z as the L × ( N − 1) matrix: ⎡ 1 ⎢0 Z = ⎢⎢ 0 ⎢. ⎢⎣ 0

0 2 0 . 0

… 0⎤ … 0⎥ … 0 ⎥. ⎥ … . ⎥ …  N ⎥⎦

Then consider the usual IV estimator: −1 θˆ = [ X ′Z ( Z ′Z )−1 Z ′X ] X ′Z ( Z ′Z ) Z ′Y

= θ + [ X ′Z ( Z ′Z )−1 Z ′X ] X ′Z ( Z ′Z ) Z ′u. −1

(A2.2)

Consider taking the probability limit of (A2.2) as T → ∞, while holding the number of countries N fixed. Notice that plim ( Z ′u T ) is a ( N − 1) × 1 vector with components,

( )

T ⎡T ⎤ ⎤ δit εit δit εit T ⎡ i plim ⎢ ∑ = plim i ⎢ ∑ ⎥ ⎥, ( ( ) ) ( ( ) ) σ ρ σ T − 1 1 − T T − 1 1 − ρ ⎣ t =1 ⎦ ⎣ t =1 i ⎦

where in the last term we adopt the convention that country i is available in periods t = 1, . . . , Ti. Under assumption 2.1, this term equals zero from the assumed independence of δ it and ε it. Next, consider the inverted matrix appearing in (A2.2). By definition of the instruments, plim ( Z ′X / T ) is an ( N − 1) × 2 matrix with rows plim ⎡⎣ ∑ X1it / T ,∑ X 2 it / T ⎤⎦ . From (2.10), (2.17), and (2.18), we can compute t t

26

Chapter 2

∑( t =1

(1 − ρ ) T ⎛ εit2 ⎞ 2 (1 − ρ ) T ⎛ εitδit ⎞ 1 X1it = ∑⎜ ⎟ − β2 ∑ ⎜ ⎟ + β2 T β 2 t =1 ⎝ T ⎠ t =1 ⎝ T ⎠

∑(

X 2 it ρ (1 − ρ ) T ⎛ εit2 ⎞ (1 − 2 ρ ) T ⎛ εitδit ⎞ 1 T ⎛ δit2 ⎞ = ⎜⎝ ⎟⎠ + ∑ ∑ ⎜ ⎟ − β ∑ ⎜⎝ T ⎟⎠ , T β T β t =1 t =1 ⎝ T ⎠ t =1

T

T

t =1

)

⎛ δit2 ⎞ ∑ ⎜ ⎟, t =1 ⎝ T ⎠ T

)

The middle term in these expressions vanish in probability limit under assumption 2.1. The probability limit of these terms is therefore plim∑ t =1 T

plim∑ t =1 T

( ) ( ) ( XT ) = plim( TT ) ⎡⎢⎣ ρ (1β− ρ) (σ

X1it T ⎡ ( 1 − ρ )2 2 (σ δ2i + σ δ2k ) ⎤ 2 = plim i ⎢ + + σ σ ( ) ε ε i k ⎥⎦ , T T ⎣ β2 β2 2 it

i

2 εi

+ σ ε2k ) −

(σ δ2i + σ δ2k ) ⎤ β

⎥⎦ ,

(A2.3a) (A2.3b)

where we again use the fact that country i is available in Ti periods. The two terms in (A2.3) for a given country i≠k form one row of plim ( Z ’u / T ), while (A2.3) for another country j≠k form another row. We assume that there exist countries i and j for which plim(Ti / T ) ≠ 0 and plim(Tj / T ) ≠ 0, and that for these two countries, condition (2.22) holds. Then it can be shown that these two rows of the matrix plim ⎡⎣ ∑ t X1it / T ,∑ t X 2 it / T ⎤⎦ are independent, which ensures that the matrix plim [( X ’Z / T ) (Z ’Z / T )−1 ( Z ’X / T )] has full rank of two, and so is invertible. Then it follows immediately that plim θˆ = θ , so the IV estimator is consistent. To relate this estimator to (2.21) in the text, let Yi ≡ ∑ t Yit / Ti , X1t ≡ ∑ t X1it / Ti , X 2t ≡ ∑ t X 2 it / Ti , and ui ≡ ∑ t uit / Ti denote the means of the variables in (A2.1) over country i. Then after pre-multiplying (A2.1) by Z ( Z ′Z )−1 Z ′, L equations are obtained of the form Yi = θ1X1i + θ 2 X 2 i + ui ,

(A2.4)

where the equation for country i is repeated Ti times. Thus the IV estimate θˆ in (A2.2) can be equivalently obtained by running weighted least squares (WLS) on the observations i = 1, . . . , N, i ≠ k , in (A2.4), while using Ti as the weights. While the IV estimator in (A2.2) is consistent, it is not the most efficient. To see this, consider the error term uit in (A2.1). It has variance Euit2 = (σ ε2i + σ ε2k )(σ δ2i + σ δ2k )/[(1 − ρ)(σ − 1)]2 , which differs across countries i when (2.22) holds. To correct for this, weight all observations in (A2.1) for country i by the inverse of sˆi2 ≡ ∑ t uˆ it2 / Ti, where uˆ it ≡ Yit − θˆ1X1it − θˆ 2 X 2 it is the computed residual using the initial esti-

Consumer Benefits from Import Variety

27

mates θˆ . Letting sˆ denote the (L × L) diagonal matrix with sˆi2 repeated Ti times on the diagonal for i = 1, . . . , N, i ≠ k , and with Xˆ ≡ Z ( Z ′Z )−1 Z ′X , the weighted IV estimator is −1 θ * = ⎡⎣Xˆ ′Sˆ −1Xˆ ⎤⎦ Xˆ ′Sˆ −1Y

−1 = θ + ⎡⎣Xˆ ′Sˆ −1Xˆ ⎤⎦ Xˆ ′Sˆ −1u,

(A2.5)

where the second line follows since it can be shown that Xˆ ′Sˆ −1X = Xˆ ′Sˆ −1Xˆ . White (1982) demonstrates the consistency of θ * for an unbalanced panel when the errors uit are independent over i and t. In this case θ * is the efficient estimator (given the set of instruments), and its covari−1 ance matrix is consistently estimated by ⎡⎣ Xˆ ′Sˆ −1Xˆ ⎤⎦ . Finally, to recognize the measurement error because unit-values UVit that are used instead of true import prices, specify that Δ ln UVit = Δ ln pit + μit ,

(A2.6)

where pit are the true but unobserved prices and μit is the measurement error. It is assumed that μit is stationary with equal variance across supplying countries i, and that μit is independent of ε jt and δ jt . Then, using (A2.6) used to substitute for Δ ln pit, (2.15) can be rewritten as

( Δ ln UVit − Δ ln UVkt )2 = 2σ μ2 + θ1 ( Δ ln sit − Δ ln skt )2 + θ 2 ( Δ ln UVit − Δ ln UVkt ) ( Δ ln sit − Δ ln skt ) + ν it , (A2.7) where

ν it = uit + [( μit − μkt )2 − 2σ 2μ ] + 2 ( Δ ln pit − Δ ln pkt ) ( μit − μkt ) − θ 2 ( Δ ln sit − Δ ln skt ) ( μit − μkt ) . Because of the independence of the measurement error μit from ε jt and δ jt, the error ν it will have expected value of zero. Then it can be shown that the instruments Z are orthogonal to ν it but correlated with the right-hand side variables in (A2.7) provided that (2.22) holds. It follows that the IV estimator is consistent, where the term 2σ μ2 in (A2.7) is replaced by a constant θ 0. The efficient estimator and standard errors are constructed in the same manner as discussed above. Appendix 2.2: STATA Code for Estimating Elasticities of Substitution The author of this code is Anson Soderbery. The code was written for NBER working paper w14956, “Measuring the Benefits of Product

28

Chapter 2

Variety with an Accurate Variety Set” with Bruce Blonigen. Thanks are due to Ankur Patel for invaluable coding help. The code implements Feenstra (1994)’s method to estimate the elasticity of substitution for imports, which are then used to calculate the exact import price index for a particular good. The code is easily extendable to Broda and Weinstein (2006)’s procedure by looping the over goods (HS10s). *Contact: [email protected]; *Date: May 2009; set more off; capture clear; capture log close; capture estimates clear; set memory 500m; set matsize 2500; set linesize 200; *log using FILENAME.log, replace; /******************* DATA WORK *************************/; use “@@@@@@@@@@@@@.dta”; rename cvalue cusval; /* cusval = p_it * q_it */; rename cquan quantity; /* quantity = q_it */; * NARROW TO THE GOOD OF INTEREST; keep hs == ##########; /***** LOG UNIT VALUES AND SHARES *****/; *PRODUCTS ARE DISTINCT VARIETIES; egen product = group(country hs); *IN CASE THERE ARE MULTIPLE OBSERVATIONS OF A VARIETY IN A GIVEN YEAR; collapse (sum) quantity cusval, by(year product country); * WE LIMIT OUR SAMPLE TO OBSERVATIONS AFTER 1990; local minyear = 1990;

Consumer Benefits from Import Variety

drop if year=`maxt' & cusval>=`q_max'; xfill ref, i(product); xfill ref, i(t); local ref = ref;

29

30

Chapter 2

replace ref = product if ref == . & period>=`maxt' & usval>=`q_cutoff'; xfill ref, i(product); xfill ref, i(t); local ref = ref; replace ref = product if ref == . & period>=`maxt' ; xfill ref, i(product); xfill ref, i(t); local ref = ref; gsort -ref; * IN FEENSTRA 1994, REF IS ALWAYS “JAPAN”; local ref = ref; *** REFERENCE COUNTRY ***; keep if product==`ref'; summ; rename ls_dif h_ls_dif; rename lp_dif h_lp_dif; keep t h_ls_dif h_lp_dif; sort t; save “auto_cref.dta,” replace; use “auto_cusimport.dta,” clear; sort t; save, replace; merge t using “auto_cref.dta”; tab _merge; drop _merge; /******************* IV REGRESSION *******************/;

Consumer Benefits from Import Variety

31

gen y = (lp_dif-h_lp_dif)^2; gen x1 = (ls_dif-h_ls_dif)^2; gen x2 = (lp_dif-h_lp_dif)*(ls_dif-h_ls_dif); drop if y == . | x1 == . | x2 == . ; **** IV 1 ****; /* GENERATE DUMMIES. THEORY IMPLIES THE CODE SHOULD BE “tab product if product != `ref', gen(c_I_)” BUT RECREATING ORIGINAL ESTIMATES IN FEENSTRA (1994) REQUIRES THE FOLLOWING, WHICH KEEPS AN INDICATOR VARIABLE FOR THE REFERENCE COUNTRY */; qui tab product, gen(c_I_); foreach var of varlist x1 x2 {; regress `var' c_I_*, noc; gen `var'_F = e(F); predict `var'hat; }; * CONSISTENT, BUT NOT EFFICIENT ESTIMATES; reg y x1hat x2hat; /**** OPTIMAL WEIGHTS ****/; gen uhat = y - _b[_cons] - _b[x1hat]*x1 _b[x2hat]* x2; gen uhat2=uhat^2; drop if uhat2==.; qui regress uhat2 c_I_*, noc; predict uhat2hat; gen shat=sqrt(uhat2hat); /*********

WEIGHT DATA AND REESTIMATE

******/;

gen ones=1; foreach var of varlist y x1hat x2hat ones {; gen `var'star = `var'/shat; };

32

Chapter 2

/* CONSISTENT AND EFFICIENT ESTIMATES */ reg ystar x1hatstar x2hatstar onesstar, nocons ; gen theta1=_b[x1hatstar]; gen theta2=_b[x2hatstar]; * VAR-COV TO BE USED TO CALCULATE CI'S FOR SIGMA; estat vce; /*** CALCULATE SIGMA AND RHO ***/; gen rho1 = .5 + sqrt(.25–1/(4+theta2^2/theta1)) if theta2>0; replace rho1 = .5 - sqrt(.25–1/(4+theta2^2/theta1)) if theta2 uHat[`row',`col'] {; local MinU = uHat[`row',`col']; local rho_grid_hat = uHat[1,`col']; local sigma_grid_hat = uHat[`row',1]; }; local col = `col' + 1; }; local row = `row' + 1; }; foreach var in sigma rho {; capture replace `var'1 = ``var'_grid_hat' ; local `var'1_hat = ``var'_grid_hat' ; }; }; sort t; keep t sigma; merge t using “auto_cusimport.dta”; sort product t; tab _merge; drop _merge; save “auto_cusimport.dta,” replace; /***************** PRICE INDEX **************/; /* EXPAND THE DATA TO INCLUDE ZERO IMPORTS (to characterize entry and exit) */; qui summ t; local maxT = r(max); keep product; * FIRST WE NEED TO MAKE A UNIQUE LIST OF MAKES; duplicates drop product, force; * NOW EXPAND THE DATA TO MAKE A SKELETON FILE FOR MERGING ; expand `maxT'; sort product;

Consumer Benefits from Import Variety

35

* NOW WE CAN MAKE THE NEW TIME PERIOD; by product: gen t = _n; sort product t; * NOW MERGE TOGETHER THE FILES; merge product t using “auto_cusimport.dta”; tab _merge; drop _merge; * SO NOW WE CAN CHANGE THE NEW MISSING VALUES TO ZERO ; destring meanprice, replace; replace quantity = 0 if quantity == .; replace meanprice = 0 if meanprice == .; sort product t; /************ GENERATE ENTRY AND EXIT OF VARIETIES **************/; sort product t; capture by product: gen entry = quantity[_n-1]==0 & quantity[_n]>0; capture by product: gen exit = quantity[_n]>0 & quantity[_n+1]==0; by product: gen exit_1 = exit[_n-1]; gen entry_exp=entry*cusval; gen exit_exp=exit*cusval; qui by product: gen exit_exp_1=exit_exp[_n-1]; save “auto_cusimports2.dta,” replace; sort t product; collapse (sum) entry_exp exit_exp exit_exp_1, by(t); foreach r in entry_exp exit_exp exit_exp_1 {; rename `r' sum_`r'; }; sort t; save “ee_exp_cusimports.dta,” replace;

36

Chapter 2

use “auto_cusimports2.dta,” clear; sort t; save,replace; merge t using “ee_exp_cusimports.dta”; tab _merge; drop _merge; /*** COST SHARES ***/; sort t product; by t: egen sumtot = sum(cusval); sort product t; by product: gen sumtot_1 = sumtot[_n-1]; by product: gen cusval_1 = cusval[_n-1]; gen sum_t = sumtot—sum_entry_exp; gen sum_t1 = sumtot_1-sum_exit_exp_1; gen s_t = cusval/sum_t if entry!=1; gen s_t1 = cusval_1/sum_t1 if exit_1!=1; gen ls_t = ln(s_t); gen ls_t1 = ln(s_t1); gen w_n = (s_t-s_t1)/(ls_t-ls_t1); sort t product; by t: egen w_d = sum(w_n); /*NOTE that the w will already be missing for any non-overlap products, so we no longer need to worry about this as we next construct the pratio— it will automatically only calculate for overlap products. */; gen w = w_n/w_d; sort product t; qui by product: gen meanprice_1 = meanprice[_n-1]; gen pratio = meanprice/meanprice_1; gen pratio_w = (pratio)^w; gen lpratio_w = ln(pratio_w); sort t;

Consumer Benefits from Import Variety

qui by t: egen slpratio_w = sum(lpratio_w); gen Pindex = exp(slpratio_w); gen lambda = (sumtot-sum_entry_exp)/(sumtot); gen lambda_1 = (sumtot_1-sum_exit_exp_1)/ (sumtot_1); collapse (mean) entry_exp exit_exp entry exit sumtot sumtot_1 lambda lambda_1 Pindex (max) sigma, by(t); gen p_index_chg = Pindex*(lambda/lambda_1)^(1/ (sigma1-1)); gen lratio = lambda/lambda_1; gen lratio_corr = lratio^(1/(sigma1-1)); list, noobs clean; /* rm rm rm rm

CLEAN UP */; auto_cusimport.dta; auto_cusimports2.dta; ee_exp_cusimports.dta; auto_cref.dta;

37

3

Producer Benefits from Export Variety

Let me turn now to the second source of gains from trade in the monopolistic competition model, which is the self-selection of firms with only the more efficient firms surviving after trade liberalization. This prediction did not come from the original models of the early 1980s because those models used the simplifying assumption that all firms are the same, producing at the same scale and with the same costs. The number of firms surviving in the market still depends on the level of tariffs, but there is no essential difference between those that drop out and those that remain. So in the models of the 1980s there was no reason for the efficiency of surviving firms to differ from those that exit the market. That simplifying case has been generalized in different ways by Eaton and Kortum (2002), and by Melitz (2003). In extending the monopolistic competition model to allow for heterogeneous firms, Melitz has demonstrated a whole range of new and exciting results. For example, the opening of trade in a sector will bid up the wage and other factor prices, which forces the least efficient firms to exit the market. The more efficient firms will be able to cover the fixed costs of marketing overseas and will therefore begin exporting, while those firms in the middle range of efficiency will continue producing for just the domestic market. So the overall distribution of firm outputs shifts in favor of those that are most efficient, since they are producing for the domestic market and exporting, while the least efficient firms have exited. As a result average productivity in the industry rises due to trade. That outcome also occurs when there are reductions in tariffs, transport costs, or just a growth in the size of the export market: all these will improve average productivity calculated over domestic plus export sales.

40

Chapter 3

These predictions have received very strong support from empirical work utilizing firm-level datasets. Bernard, Eaton, Jensen and Kortum (2003) have shown that for the United States only a small fraction of the firms in any industry are exporters. But this small fraction of firms accounts for a very large amount of total sales within the industry. The implication is that these exporting firms are substantially more productive than other firms in the industry. That finding is reinforced using firm-level data for France by Eaton, Kortum and Kramarz (2004, 2008). These studies from the United States and France focus on the differences between firms in a single year, or over several years, but do not necessarily include a major trade liberalization. A recent study for Canada by Trefler (2004) does just that. Trefler uses firm-level data during the decades before and after the Canada–US free trade agreement, and is interested in the impact of the agreement on the selection and productivity of firms. He finds that while there was a fall in employment in the industries with the largest tariff cuts, these job losses were a short-term effect, and over a ten-year period, employment in Canadian manufacturing did not drop. While low-productivity plants shut down, high-productivity Canadian manufacturers expanded into the United States. In the formerly sheltered industries most affected by the tariff cuts, labor productivity jumped 15 percent, at least half from closing inefficient plants. That corresponds to a compound annual growth rate of 1.9 percent. Thus Trefler finds overwhelming evidence that the Canada–US free trade agreement resulted in the self-selection of Canadian firms, with the more productive firms expanding. These productivity gains translate directly into higher wages or lower prices, and are a welfare gain for consumers. Output Variety in the CES Case To theoretically demonstrate these gains in the Melitz (2003) model, we approach the problem in a somewhat unusual way: rather than focusing on consumer gains as in the last chapter, we now ask whether there exist any producer gains due to output variety. This question can be answered by extending the range of values for the elasticity of substitution that we considered in the previous chapter. There we restricted our attention to σ > 1 in the utility and expenditure functions (2.1) and (2.2), but a wider range of values for this elasticity can be considered. In particular, if σ < 0 then instead of obtaining convex indifference curves from (2.1) for a fixed level of Ut, we obtain a concave transfor-

Producer Benefits from Export Variety

41

mation curve as shown in figure 3.1.1 The parameter Ut in this case measures the resources devoted to production of the goods qit, i ∈ It, and the elasticity of the transformation curve (measured as a positive number) is −σ . To make this reinterpretation explicit, when σ < 0, we will denote its positive value by ω ≡ −σ , which is the elasticity of transformation. Then we will rewrite (2.1) using labor resources Lt to replace utility Ut, obtaining ⎛ ⎞ Lt = ⎜ ∑ ait qit(ω +1)/ω ⎟ ⎝ i∈It ⎠

ω /(ω +1)

, ait > 0, ω > 0.

(3.1)

The maximum revenue obtained using one unit of labor resources, dual to (2.9), is denoted by ⎡ ⎤ ψ ( pt , It ) = ⎢∑ bit pitω +1 ⎥ ⎣ i∈It ⎦

1/(ω + 1)

bit ≡ ait−ω , ω > 0.

,

(3.2)

With this reinterpretation, theorem 2.2 continues to hold as

ψ ( pt , It ) ⎛ λ (I ) ⎞ = PSV ( pt −1 , pt , qt −1 , qt , I ) ⎜ t ⎝ λt −1 (I ) ⎟⎠ ψ ( pt −1 , It −1 )

−1/(ω +1)

,

q2 B

C

A Figure 3.1 Constant-elasticity transformation curve

q1

(3.3)

42

Chapter 3

where the exponent appearing on (λt/λt−1) is now negative. In other words, the appearance of “new outputs,” so that λt < 1, will raise revenue on the producer side of the economy. To understand where this increase in revenue is coming from, consider the transformation curve in figure 3.1. If only good 1 is available, then the economy would be producing at the corner A, with revenue shown by the line AB. Then, if good 2 becomes available to producers, the new equilibrium will be at point C, with an increase in revenue. This illustrates the benefits of output variety. In figure 3.2 we illustrate the same idea in a partial equilibrium diagram, for a supply curve with constant elasticity ω. When the good becomes available for production, there is an effective price increase from the reservation price for producers (which is zero with a constant-elasticity supply curve) to the actual price. The gain in producer surplus is area C, and when it is measured relative to total sales C + D, we can readily compute that C/(C + D) = 1/(ω + 1). While this reinterpretation of our earlier consumer model is mathematically valid, there is a problem in its application to international trade: the transformation curve between two outputs is often taken to be linear rather than strictly concave. That is the case in the Ricardian model, for example, or in the transformation curve (1.8) of Krugman’s (1980) model. In that case the gains from output variety would vanish. So the question arises as to whether the strictly concave case we illustrate in figure 3.1 has any practical application? p

C 1 = C + D (ω +1)

C D q Figure 3.2 Constant-elasticity supply curve

Producer Benefits from Export Variety

43

We will now argue that a strictly concave transformation curve is indeed relevant, and in fact arises in the generalization of the monopolistic competition model due to Melitz (2003), which allows for heterogeneous firms that differ in their productivities ϕ . In the equilibrium with zero expected profits, only firms above some cutoff productivity ϕ * survive, and of these, only firms with productivities above ϕ x* > ϕ * actually export. We will argue that the endogenous determination of these cutoff productivities leads to a strictly concave constant-elasticity transformation curve between domestic and export varieties, adjusted for the quantity produced of each. Furthermore, this interpretation of the Melitz (2003) model will allow us to make a precise calculation of the producer gains due to export variety. Monopolistic Competition with Heterogeneous Firms We outline here a two-country version of the Melitz (2003) model that does not assume symmetry between the countries. We focus on the home country H, while denoting foreign variables with the superscript F. At home there is a mass of M firms operating in equilibrium. Each period, a fraction δ of these firms go bankrupt and are replaced by new entrants. Each new entrant pays a fixed cost of fe to receive a draw ϕ of productivity from a cumulative distribution G(ϕ ), which gives rise to the marginal cost of w/ϕ , where w is the wage and labor is the only factor of production. Only those firms with productivity above a cutoff level ϕ * find it profitable to actually produce (the cutoff level will be determined below). We let Me denote the mass of new entrants so that only [1 − G(ϕ *)]Me firms successfully produce. In a stationary equilibrium these should replace the firms going bankrupt: [1 − G(ϕ *)]Me = δ M.

(3.4)

Conditional on successful entry, the distribution of productivities for home firms is then ⎧ g(ϕ ) if ϕ ≥ ϕ * , ⎪⎪ 1 − G(ϕ *) μ(ϕ ) = ⎨ ⎪ otherwise , ⎪⎩0 where g(ϕ ) = ∂G(ϕ )/ ∂ϕ is the density function.

(3.5)

44

Chapter 3

Home and foreign consumers both have CES preferences that are symmetric over product varieties. Given home expenditure of wL, the revenue earned by a home firm from selling at the price p(ϕ ) is ⎡ p(ϕ ) ⎤ r(ϕ ) = p(ϕ )q(ϕ ) = ⎢ H ⎥ ⎣P ⎦

1− σ

wL , σ > 1,

(3.6)

where q(ϕ ) is the quantity sold and P H is the home CES price index. The profit-maximizing price from selling in the domestic market is the usual constant markup over marginal costs: p(ϕ ) =

( σσ− 1 ) wϕ .

(3.7)

Using this expression, we calculate variable profits from domestic sales as r(ϕ ) − (w /ϕ )q(ϕ ) = r(ϕ )/σ . The lowest productivity firm that just breaks even in the domestic market there satisfies the zero-cutoff-profit (ZCP) condition: r(ϕ *) = wf ⇒ q(ϕ *) = (σ − 1) fϕ * , σ

(3.8)

where f is the fixed labor cost. Note that this cutoff condition for the marginal firm is identical to what is obtained in Krugman’s (1980) model for all firms, as in (2.7). While firms with productivities ϕ ≥ ϕ * find it profitable to produce for the domestic market, only those with higher productivities ϕ ≥ ϕ x* > ϕ * find it profitable to export. A home exporting firm faces the iceberg transport costs of τ ≥ 1 , meaning that τ units must be sent in order for one unit to arrive in the foreign country. Letting px (ϕ ) and qx (ϕ ) denote the price received and quantity shipped at the factorygate, we find that the revenue earned by the exporter is ⎡ p (ϕ )τ ⎤ rx (ϕ ) = px (ϕ )qx (ϕ ) = ⎢ x F ⎥ ⎣ P ⎦

1− σ

w*L* ,

(3.9)

where P F is the CES price index in the foreign country and w*L* is foreign expenditure. Again, the optimal export price is a constant markup over marginal costs: px (ϕ ) =

( σσ− 1 ) wϕ .

(3.10)

Producer Benefits from Export Variety

45

The variable profits from export sales are therefore rx (ϕ ) − (w / ϕ )qx (ϕ ) = rx (ϕ )/ σ , so the ZCP condition for the exporting firm is rx (ϕ x* ) = wf x ⇒ qx (ϕ x* ) = (σ − 1) f xϕ x* , σ

(3.11)

where fx is the additional fixed labor cost for exporting. Provided that rx (ϕ )/ f x < r(ϕ )/ f , which we assume is the case, then the cutoff productivity for the exporting firm will exceed that for the domestic firm, ϕ x* > ϕ *. Then the mass of exporting firms is computed as ∞

Mx ≡

∫ Mμ(ϕ )dϕ < M.

(3.12)

ϕ x*

To close the model, we use the full-employment condition and also zero expected profits for any entrant. The labor needed for domestic sales for a firm with productivity ϕ is [q(ϕ )/ϕ ] + f , and for export sales is [qx (ϕ )/ϕ ] + f x, so the full employment condition is ∞

⎡ q(ϕ ) L = Me f e + M ∫ ⎢ + ϕ ϕ* ⎣

∞

⎤ ⎡ q (ϕ ) ⎤ f ⎥ μ(ϕ )dϕ + Mx ∫ ⎢ x + f x ⎥ μ x (ϕ )dϕ , ϕ ⎦ ⎣ ⎦ ϕ x*

(3.13)

where the distribution of productivities conditional on exporting is μ x (ϕ ) ≡ g(ϕ )/[1 − G(ϕ x* )] if ϕ ≥ ϕ x* , and zero otherwise. We can rewrite (3.13) by multiplying by w, and using the fact that (w /ϕ )q(ϕ ) = r(ϕ )(σ − 1)/σ , and likewise for exporters, to obtain

( ) ( )

∞ ∞ ⎤ σ −1 ⎡ ⎢ M ∫ r(ϕ )μ(ϕ )dϕ + Mx ∫ rx (ϕ )μ x (ϕ )dϕ ⎥ σ ⎢⎣ ϕ * ⎥⎦ ϕ x* σ −1 = w ( Me f e + Mf + Mx f x ) + wL , σ

wL = w ( Me f e + Mf + Mx f x ) +

where the second line is obtained using the definition of GDP, with zero expected profits. It follows immediately that there is a linear transformation curve between the mass of entering, domestic and exporting firms, that is L = σ ( Me f e + Mf + Mx f x ) .

(3.14)

To obtain further results, we assume a Pareto distribution for productivities:

46

Chapter 3

G(ϕ ) = 1 − ϕ −θ , with θ > σ − 1 > 0.

(3.15)

In that case, it can be shown (see appendix 3.1) that the number of entering firms is proportional to the labor force, Me = L(σ − 1)/σθ f e , which was assumed by Chaney (2008), for example. So the transformation curve between domestic and export varieties is further simplified as L=

σθ ( Mf + Mx fx ) . θ −σ +1

(3.16)

The fact that this transformation curve is linear between the mass of domestic and exported varieties is similar to that found in the Krugman (1980) model, in (2.8). But this fact does not tell us about the transformation curve between the economy’s outputs, since we also need to take into account the quantity produced of each variety. In Krugman’s model, the quantity produced by each firm is fixed, as in (2.7). But in the Melitz (2003) model, only the cutoff zero–profit firm has output identical to that in Krugman’s model, and the cutoff productivity ϕ * itself is endogenously determined. So, to determine the transformation curve for the economy, we first need to determine the correct measure of output used to adjust the varieties M and Mx. To determine the appropriate measure of quantity, it is convenient to invert the demand curve and treat revenue as a function of quantity. So from (3.6) we obtain 1σ

⎛ wL ⎞ r(ϕ ) = Ad q(ϕ )(σ −1) σ , where Ad ≡ P H ⎜ H ⎟ . ⎝P ⎠

(3.17)

We introduce the notation Ad as shift parameter in the demand curve facing home firms for their domestic sales. It depends on the CES price index P H , and also on domestic expenditure wL. Likewise export revenue can be written as: ⎛ P F ⎞ ⎛ τ w *L * ⎞ rx (ϕ ) = Ax qx (ϕ )(σ −1) σ , where Ax ≡ ⎜ ⎟ ⎜ i F ⎟ ⎝ τ ⎠⎝ P ⎠

1σ

.

(3.18)

Integrating domestic and export revenue over firms, we obtain GDP: ∞

∞

ϕ*

ϕ x*

wL = Ad M ∫ q(ϕ )(σ −1) σ μ(ϕ )dϕ + Ax Mx ∫ qx (ϕ )(σ −1) σ μ x (ϕ )dϕ .

(3.19)

Producer Benefits from Export Variety

47

Thus, in order to measure GDP, the mass of domestic and export varieties are multiplied by the quantities shown above. Feenstra and Kee (2008) demonstrate that the first-order conditions for maximizing GDP subject to the resource constraint for the economy, taking Ad and Ax as given, are precisely the monopolistic competition equilibrium conditions. So the quantities appearing in (3.19) are the “right” way to adjust the mass of domestic and export varieties. We can simplify these quantities by noting that CES demand, combined with constant markup prices in (3.7), imply that the quantity sold equals q(ϕ ) = (ϕ /ϕ )σ q(ϕ ) for any choice of reference productivity ϕ . We follow Melitz (2003) in specifying ϕ as average productivity, ⎡∞ ⎤ ϕ ≡ ⎢ ∫ ϕ (σ −1) μ(ϕ )dϕ ⎥ ⎢⎣ϕ * ⎥⎦

1/(σ − 1)

,

(3.20)

and likewise for the average productivity ϕ x for exporters, computed  +A M  ), after using ϕ x* and μ x. It follows that GDP simply equals ( Ad M x x using the adjusted mass of varieties:  ≡ Mq(ϕ )(σ −1)/σ M

 ≡ M q (ϕ )(σ −1)/σ . and M x x x x

(3.21)

To simplify the expression for GDP further, we note that a property of the Pareto distribution is that an integral like (3.20) is always a constant multiple of the lower bound of integration. That is, 1/(σ − 1) θ ⎤ ϕ = ⎡ ϕ *, ⎢⎣ θ − σ + 1 ⎥⎦

(3.22)

as obtained by evaluating the integral in (3.20), which is finite provided that θ > σ − 1. The cutoff productivity ϕ * is in turn related to the mass of firms by [1 − G(ϕ *)]Me = δ M , and based on the mass of entering firms Me = L(σ − 1)/σθ f e and the Pareto distribution, it follows that (ϕ *)−θ =

δσθ f e M. L(σ − 1)

(3.23)

Gathering together these results, we can compute the adjusted mass of domestic varieties as σ −1  = M ⎛ ϕ ⎞ q(ϕ *)(σ −1) σ = θ M [(σ − 1) fϕ *](σ −1) σ M ⎜⎝ ⎟⎠ ϕ* θ −σ +1 − [(σ − 1) θσ ] f , = k1 f (σ −1) σ M 1−[(σ −1) θσ ] e L

( )

48

Chapter 3

where the first equality follows from (3.21) and q(ϕ ) = (ϕ /ϕ *)σ q(ϕ *). The second equality uses (3.22) and the ZCP condition q(ϕ *) = (σ − 1) fϕ *, and the third follows from (3.23), where k1 > 0 depends on the parameters θ, σ, and δ. Thus the adjusted mass of domestic varieties is an increasing but nonlinear function of the mass M .  replaced A similar expression holds for exports, but with f, M, and M  by fx, Mx, and Mx . Solving for M and Mx and substituting these into the linear transformation curve (3.16), we obtain a concave transformation  , with elasticity ω ≡ [θσ (σ − 1)] − 1 > 0:  and M curve between M x  (ω +1) ω f 1+[(ω +1)(σ −1)] ωσ + M  (ω +1) ω f 1+[(ω +1)(σ −1)] ωσ ) ω /(ω +1) , (3.24) L = k 2 f e1/(ω +1) ( M x x where k2 > 0 again depends on the parameters θ, σ, and δ. Summing up, from the Melitz (2003) model we have obtained a constant-elasticity transformation curve, with elasticity ω ≡ [θσ (σ − 1)] −1 > 0, just like in (3.9) as we initially asserted. Our earlier results in theorems 2.1 and 2.2 continue to apply to this transformation curve. In  +A M  ) subject particular, consider the problem of maximizing ( Ad M x x to this transformation curve. This Lagrangian problem leads to the following solution, analogous to (3.10): Theorem 3.1 (Feenstra and Kee 2008) Assume that the distribution of firm productivity in Pareto, as in (3.15). Then maximizing GDP subject to the transformation curve (3.24) results in ψ ( Ad , Ax )L, where: w = ψ ( Ad , Ax ) ≡

1 1/(ω +1) 2 e

k f

ω +1

⎡⎣ Ad

ω +1

f 1−[θ (σ −1)] + Ax

f x1−[θ (σ −1)] ⎤⎦

1 (ω + 1)

.

(3.25)

The function ψ ( Ad , Ax ) is the revenue earned with L = 1 on the transformation curve, and equals wages. Note that the exponents appearing on the fixed costs f and fx in (3.25) are obtained as −{ω + (1 + ω )[(σ − 1) σ ]} = 1 − [θ (σ − 1)] < 0. This expression also appears as the exponent on fixed costs in the gravity equation of Chaney (2008). We can now apply theorem 2.2 to compute the gain from trade. Denoting autarky by t–1, the economy is at the corner of the transfor mation curve with Axt −1 = M xt − 1 = 0 , as illustrated by point A in figure 3.3. Using t to denote the trade situation, under free trade we have  > 0 , as at point C. We can therefore evaluate the gains Axt > 0 and M xt from trade as the ratio of real wages in trade and under autarky:

Producer Benefits from Export Variety

49

~ Mxt B

C

~ Mt

A Figure 3.3 Constant-elasticity transformation curve in Melitz (2003)

ψ ( Adt , Axt ) ⎛ PtH ⎞ wt / PtH = ⎜ ⎟ H wt −1 / Pt −1 ψ ( Adt −1 , 0) ⎝ PtH−1 ⎠ ⎛ A ⎞⎛ R ⎞ = ⎜ dt ⎟ ⎜ dt ⎟ ⎝ Adt −1 ⎠ ⎝ wt Lt ⎠ ⎛ w / PH ⎞ =⎜ t tH ⎟ ⎝ wt −1 / Pt −1 ⎠

1σ

−1

−1 (ω +1)

⎛ PtH ⎞ ⎜⎝ H ⎟⎠ Pt −1

⎛ Rdt ⎞ ⎜⎝ ⎟ wt Lt ⎠

−1

(3.26)

−1 (ω + 1)

,

where the first line follows from wages in theorem 2.1; the second line follows from theorem 2.2, using the domestic “price” Ad as the common good available both periods, with spending on domestic  ; and the third line follows directly from goods in period t of Rdt ≡ Adt M t the definition of Ad in (3.17). We use this equation to solve for the ratio of real wages, obtaining the result: Theorem 3.2 (Arkolakis et al. 2008b) The gains from trade in the Melitz (2003) model are wt / PtH ⎛ R ⎞ = ⎜ dt ⎟ H wt −1 / Pt −1 ⎝ wt Lt ⎠

− [1 (ω +1)][σ (σ −1)]

⎛ R ⎞ = ⎜ dt ⎟ ⎝ wt Lt ⎠

−1 θ

,

(3.27)

50

Chapter 3

where the final equality is obtained because ω ≡ [θσ (σ − 1)] − 1, so [1 (ω + 1)][σ (σ − 1)] = 1 θ . Note that the ratio of domestic expenditure Rdt to total income wt Lt is equal to one minus the import share, so this formula is identical to the gains from trade in the Krugman (1980) model, except that we replace the exponent −1/(σ − 1) in that case with −1/θ in (3.27). This finding is precisely the result derived by Arkolakis et al. (2008b), and remarkably the elasticity of substitution σ does not enter the formula at all (except insofar as it affects the import share). Our derivation gives some intuition as to where this simple formula comes from. Namely, the movement from a corner of the transformation curve A in figure 3.3, with exports equal to zero, to an interior position like C, gives rise to gains equal to one minus the import (or export) share with the exponent −1/(ω + 1), which is a straightforward application of theorem 2.2 on the production side of the economy. We might interpret these gains as due to export variety. These gains are shown in the second line of (3.26), and reflect the increase in wages due to the productivity improvement as the exporting firms drive out less productive domestic firms. But in addition this productivity improvement drives down prices, and therefore further increase real wages: that is shown as we substitute for the endogenous value of Ad, and thereby solve for real wages in (3.27). Through these two channels the gains equal one minus the import (or export) share with the exponent −1/θ , which exceeds −1/(ω + 1) = −(σ − 1)/θσ in absolute value. But what about any further gain due to import variety? Now we must be careful, because the Melitz model leads to the exit of domestic firms and therefore a reduction in domestic varieties, which must be weighted against the increase in import variety. Baldwin and Forslid (2004) argue that the total number of product varieties falls with trade liberalization, whereas Arkolakis et al. (2008b) show that it can rise or fall. But simply counting the total number of varieties is not the right way to evaluate the welfare gains: instead, we need to take the ratio (λt /λt −1 )−1/(σ −1) on the consumption side of the economy, as in theorem 2.2. As we now show, this ratio turns out to be unity: the gains due to new import varieties are exactly offset by reduced domestic varieties. Therefore the productionside gains we have already identified in theorem 3.2 are all that is available. To obtain this result, we use the CES price index for the Melitz model:

Producer Benefits from Export Variety

51

∞ ⎡∞ ⎤ P = ⎢ ∫ p(ϕ )1−σ Mμ(ϕ )dϕ + ∫ p F (ϕ )1−σ M F μ F (ϕ )dϕ ⎥ ⎢⎣ϕ * ⎥⎦ ϕ xF * H

1 ( 1− σ )

,

(3.28)

where ϕ xF* denotes the zero-profit cutoff for the foreign exporters, with prices p F (ϕ ). This CES price index is conceptually identical to what we referred to as the unit-expenditure function in (2.2). The average price of domestic goods appearing in (3.28) is ⎡∞ ⎤ 1− σ ⎢ ∫ p(ϕ ) Mμ(ϕ )dϕ ⎥ ⎢⎣ϕ * ⎥⎦

1 ( 1− σ )

=

( σσ− 1 ) ⎛⎜⎝ wϕ ⎞⎟⎠ M

−1 (σ −1)

,

(3.29)

which uses the prices (3.7) together with the definition of average productivity in (3.20). When comparing autarky (denoted by t – 1) with free trade (denoted by t), we need to take into account the changing price of domestic goods and their changing variety, as in (3.29), along with the fact the all imported goods are new. Applying theorem 2.2 gives rise to the following ratio of unit-expenditures: 1 ( σ − 1) PtH ⎛ wt /ϕ t ⎞ ⎛ Rdt / wt Lt ⎞ = . PtH−1 ⎜⎝ wt −1 /ϕ t −1 ⎟⎠ ⎜⎝ Mt / Mt −1 ⎟⎠

(3.30)

The first term appearing on the right of (3.30) is just the change in the average price of domestic goods, reflecting the change in wages and in average productivity. The aggregate domestic good is available in both periods, so the first term reflects the Sato–Vartia index PSV over the “common” domestic good in theorem 2.2. The numerator of the second term on the right is the spending on domestic goods relative to total spending in period t; this equals λt in theorem 2.2, or one minus the share of spending on new imported varieties. The denominator of the second term is λt−1 in theorem 2.2, and reflects the reduction in the number of domestic varieties, Mt < Mt−1. We now show that Mt / Mt −1= Rdt / wt Lt in (3.30), so the reduction in the number of domestic varieties just cancels with share of spending on new imported varieties, and there are no further consumption gains. This result is obtained from the ZCP condition for domestic firms, in (3.8). The second expression appearing in (3.8) is q(ϕ *) = (σ − 1) fϕ * , which is familiar from the Krugman model in (2.7). We will combine this with the first expression appearing in (3.8),

52

Chapter 3

r(ϕ *)/σ = wf , which can be rewritten using the inverse demand curve in (3.17), to obtain ⎡ Adt q(ϕ t* )(σ −1) σ ⎢ ( σ −1 ) σ ⎢⎣ Adt −1 q(ϕ t*−1 )

⎤ ⎛ wt ⎞ ⎥=⎜ ⎟. ⎥⎦ ⎝ wt −1 ⎠

Using the definition Ad ≡ P H (wL / P H )1/σ , we can simplify this expression as ⎡ q(ϕ t* ) ⎤ ⎛ wt / PtH ⎞ . ⎢ ⎥=⎜ H ⎟ ⎢⎣ q(ϕ t*−1 ) ⎥⎦ ⎝ wt −1 / Pt −1 ⎠ Now using the ZCP condition that q(ϕ *) = (σ − 1) fϕ *, we immediately obtain ⎛ ϕ t* ⎞ ⎛ wt / PtH ⎞ , ⎜ ⎟ =⎜ H ⎟ ⎝ ϕ t*−1 ⎠ ⎝ wt −1 / Pt −1 ⎠

(3.31)

so that the increase in real wages reflects the increase in the ZCP productivities. From (3.22) we know that the ratio of ZCP productivities equals the ratio of average productivities, (ϕ t /ϕ t−1 ). Then comparing (3.30) with (3.31), we immediately see that Mt / Mt −1= Rdt / wt Lt , as we intended to show. The finding that there are no additional consumption gains from variety in the Melitz (2003) model, which is implicit in Arkolakis et al. (2008b), is discussed explicitly by Di Giovanni and Levchenko (2009) and Arkolakis, Costinot, and Rodríguez-Clare (2009). Di Giovanni and Levchenko argue that if the distribution of firm size follows Zipf ’s law, then the extensive margin of imports accounts for a vanishing small portion of the total gains from trade. They assume a Pareto distribution for productivities that correspond to Zipf ’s law as θ → (σ − 1). In comparison, our results above are more general because we show that the extensive margin of imports has a welfare contribution that just cancels with the reduced extensive margin of domestic goods, and this result holds for all values of θ > (σ − 1). Then again, Arkolakis, Costinot, and Rodríguez-Clare (2009) use a framework that is considerably more general than ours, and still show that the extensive margin of imports has no additional gains from trade; that paper is recommended for further reading.

Producer Benefits from Export Variety

53

Worldwide Gains from Export Variety We conclude this chapter with a calculation of the worldwide gain from trade, where we apply the formula in theorem 3.2 to a broad cross section of countries. We noted earlier that the ratio of domestic expenditure Rdt to total income wt Lt is equal to one minus the import share, at least when trade is balanced. With unbalanced trade, this ratio is equal to one minus the export share, since Rdt equals the total output of domestic goods, while wt Lt is total GDP and is equal to the value of domestic goods plus exports. So the formula for the gains from trade in theorem 3.2 is actually (1 − export share)−1/θ , which we measure now. As in chapter 2 we use data from version 6.1 of PWT, which has 1996 as the benchmark year, and focus our calculations on that year. Once again, we started with 151 countries as described in Feenstra et al. (2009), but had to omit 5 countries (including Hong Kong and Singapore) because either imports or exports exceeded their value of GDP, which is impossible in the monopolistic competition model. With the remaining sample of 146 countries, we compute the export share as nominal imports relative to nominal GDP. In order to apply the gains from trade formula, we then need to use a value for the Pareto parameter θ. We will apply the gains from trade formula over the entire economy, and not distinguish the export shares of various sectors (a productlevel analysis of the gains from trade will be undertaken in the next chapter). So we need to choose a benchmark value for θ, and some additional values for sensitivity analysis. In theory, the value for θ is bounded below by σ − 1, as specified in (3.15).2 Using this restriction, we will consider the value θ = σ = 2.9, obtained with the median estimate for σ. This estimate may very well understate the realistic value for θ, and by using “too low” a value for θ, we are exaggerating the potential gains from trade. A higher estimate for θ comes from the work of Eaton and Kortum (2002). They estimate the parameter of a Frechet distribution that is similar to θ, and find values in the range of 6 to 12, with the most robust estimates around 8, which we use as a second value for θ.3 We then choose θ = 5 as a third value lying inbetween 2.9 and 8. We next compute the gains from trade in the monopolistic competition model of Melitz (2003) as ⎡ (1 − export share)−1/θ − 1 ⎤ gains from export variety = ⎢ ⎥, −1/θ ⎣ (1 − export share) ⎦

54

Chapter 3

using the values θ = 2.9, 5.0, and 7.5. This formula expresses the gains from export variety as a percentage of the free trade real wages. In table 3.1 we record the gains for the same selection of countries as used in chapter 2. Included in table 3.1 are the countries with the highest gains due to exports: Tajikistan, Gabon, Malaysia, Malta, and Ireland. These countries have the highest export shares in the sample and, correspondingly, the highest gains due to trade. The United States once again has among the lowest export share and the lowest percentage gains. Over all the 146 countries in the sample, the gains from trade in the Melitz (2003) model vary between 3.5 and 8.5 percent of real GDP with θ = 7.5 and θ = 2.9, respectively. In figure 3.4 we graph the gains from export variety against the real GDP of all countries in the sample, using the middle value θ = 5. The trend line in this figure shows a negative relationship (and significant at the 5 percent level) between the gains due to export variety and real GDP. It is surprising that this relationship for exports in figure 3.4 is noticeably weaker than what we found for imports in figure 2.5; if trade were balanced, of course, the graphs would be nearly identical. But once again, an overall negative relationship between country size and the gains from trade is obtained. In chapter 5 we will re-visit this relationship between the gains from trade and the economic size of countries; there we suggest that larger countries may have an advantage in attracting a greater variety of both imports and exports, leading instead to a potentially positive relationship between size and trading gains.

Gain from export variety (%)

50 Malaysia Ireland Tajikistan 25

Gabon

United States 0 100

1,000

10,000

100,000

Real GDP, 1996 ($million) Figure 3.4 Gain from export variety and real GDP, 1996

1,000,000

10,000,000

Producer Benefits from Export Variety

55

Table 3.1 Gains from export variety, 1996 Real GDP per capita (US$)

Export Share (%)

Gains from export variety (%) Pareto parameter θ = 2.9

Democratic Republic of Congo Tajikistan Nepal Equatorial Guinea Republic of Moldova China Guatemala Brazil Thailand Gabon Malaysia Saudi Arabia Malta Bahamas Israel Bermuda Ireland Germany Norway United States 146 countries

θ=5

θ = 7.5

245

30.0

11.6

6.9

4.6

775 1,007 1,104 1,737 2,353 3,051 5,442 5,840 7,084 7,448 9,412 10,420 13,081 13,138 15,017 15,150 17,292 20,508 23,648

78.7 19.6 42.6 46.6 20.0 17.0 7.0 37.2 79.2 91.4 55.8 76.4 32.8 26.2 21.3 87.7 25.6 44.6 11.2

41.3 7.2 17.4 19.5 7.4 6.2 2.5 14.8 41.8 57.1 24.5 39.2 12.8 9.9 7.9 51.5 9.7 18.4 4.0

26.6 4.3 10.5 11.8 4.4 3.7 1.4 8.9 27.0 38.8 15.0 25.1 7.6 5.9 4.7 34.2 5.7 11.1 2.3

18.6 2.7 7.1 8.0 2.9 2.5 0.9 6.0 18.9 27.9 10.3 17.5 5.2 4.0 3.1 24.4 3.9 7.6 1.6

8.5

5.1

3.5

Source: Author ’s calculations. Notes: (1) Real GDP per capita is similar to that reported in PWT version 6.1, for 1996. (2) Export share is nominal exports divided by nominal GDP, 1996 data. (3) The gains from export variety are computed as [(1 − export share)−1/θ − 1] divided by (1 − export share)−1/θ , and are expressed as a percentage of the free trade real GDP per capita.

56

Chapter 3

Appendix 3.1 Using L = σ ( Me f e + Mf + Mx f x ) and the full-employment condition (3.13), we have that

( σσ− 1 ) L = M ∫ ⎡⎢⎣ q(ϕϕ ) ⎤⎥⎦ μ(ϕ )dϕ + M ∫ ⎡⎢⎣ q ϕ(ϕ ) ⎤⎥⎦ μ (ϕ )dϕ. ∞

∞

x

x

ϕ*

x

ϕ x*

Evaluating these integrals: ∞

σ

∞

q(ϕ *) ⎛ ϕ ⎞ ⎡ q(ϕ ) ⎤ ∫ϕ * ⎢⎣ ϕ ⎥⎦ μ(ϕ )dϕ = ϕ∫* ϕ ⎜⎝ ϕ * ⎟⎠ μ(ϕ )dϕ ∞

σ −1

=

q(ϕ *) ⎛ ϕ ⎞ ⎜ ⎟ ϕ * ϕ∫* ⎝ ϕ * ⎠

=

q(ϕ *) θ ⎛ ϕ ⎞ ⎜ ⎟ ϕ * σ −θ − 1 ⎝ ϕ *⎠

(σ − 1)θ , = f (θ − σ + 1)

θϕ −θ −1 dϕ (ϕ *)−θ σ −θ − 1 ∞

ϕ*

where the first line uses q(ϕ ) = (ϕ /ϕ *)σ q(ϕ *) and the last line uses q(ϕ *)/ϕ * = (σ − 1) f . Likewise ∞

(σ − 1)θ ⎡ qx (ϕ ) ⎤ μ x (ϕ )dϕ = f x . ⎥ ϕ ⎦ θ −σ +1 ϕ*

∫ ⎢⎣ x

Substituting these in to the full-employment condition above, we obtain L=

σθ ( Mf + Mx f x ), θ −σ +1

from which it follows that Me = L(σ − 1)/σθ f e .

4

The Extensive Margin of Trade and Country Productivity

The previous chapter has shown that the Melitz (2003) model gives rise to gains from trade on the production side of the economy. Intuitively, movements along the transformation curve in figure 3.3 due to greater export variety will be associated with higher GDP and productivity. That hypothesis is strongly confirmed empirically by Feenstra and Kee (2008), who analyze 48 countries exporting to the United States over 1980 to 2000. They find that average export variety to the United States increases by 3.3 percent per year, so it nearly doubles over these two decades. That total increase in export variety is associated with a cumulative 3.3 percent productivity improvement for exporting countries: namely, after two decades GDP is 3.3 percent higher than otherwise due to growth in export variety, on average. That estimate is greater than the welfare gains for the United States found by Broda and Weinstein (2006), which was that after 30 years, real GDP was 2.6 percent higher than otherwise due to growth in import variety. These results demonstrate that the gains on the production side of the economy can be substantial. We will review the results of Feenstra and Kee in this chapter, but before that, we first need to discuss the measurement of export variety. Hummels and Klenow (2005) coined the term the “extensive margin in trade” to refer to changes in imports or exports that are due to changes in the number of goods, namely, due to product variety rather than changes in the amount purchased of each good. They have shown how the extensive margin can be constructed across countries and correlate it with country size. It turns out that their measure of the extensive margin is completely consistent with the CES aggregator function we have used in (2.1) and (3.1). So we begin this chapter with a brief review of the exact price index for a CES function, and then discuss the measurement of the extensive margin of exports

58

Chapter 4

and imports. STATA programs to implement these measures are provided in the appendix to this chapter. Measuring the Extensive Margin in Trade Consider a world economy with h = 1, . . . , H countries, each of which produce many types of goods. For now we will not make any distinction between goods produced for the home market and those exported, and just suppose that all goods are exported. In each period t, let the set of goods produced and exported by country h be denoted by Ith ⊂ {1, 2, 3,...}. For i ∈ Ith the quantity of good i is qith > 0, and the vector of each type of good produced in country h in period t is denoted by qth > 0. The outputs of each country h are constrained by a CES transformation function with the aggregate resources Lht > 0: ⎛ ⎞ L = f (q , I ) = ⎜ ∑ ai (qith )(ω + 1)/ω ⎟ ⎝ i∈Ith ⎠ h t

h t

h t

− ω /(ω + 1)

, ai > 0 ,

(4.1)

where the elasticity of transformation between goods is ω > 0. This equation just repeats (3.1), while adding the explicit notation for producing countries h. At the beginning of chapters 2 and 3 we showed how the ratio of two CES functions over time can be evaluated. These results can equally well be applied to the ratio of two CES functions across countries rather than over time. Specifically, the ratio of the CES revenue functions for countries h and j in period t equals the product of the price index over common goods, It ≡ ( Ith ∩ Itj ) ≠ ∅ , multiplied by terms reflecting the revenue share of “unique” goods: ⎛ pith ⎞ ∏ ⎜ j⎟ i ∈It ⎝ pit ⎠

wit ( It )

⎛ λth ( I t ) ⎞ ⎜⎝ λ j ( I ) ⎟⎠ t t

−1/(ω + 1)

,

h , j = 1,..., H .

(4.2)

The weights wit(It) are constructed from the revenue shares in the two countries: wit ( It ) ≡

[sith ( It ) − sitj ( It )] [ln sith ( It ) − ln sitj ( It )] , ∑ i∈I [sith ( It ) − sitj ( It )] [ln sith ( It ) − ln sitj ( It )]

(4.3a)

t

sith ( It ) ≡

h h it it h h it it i∈I

p q , and likewise for country j , ∑ pq t

(4.3b)

The Extensive Margin of Trade and Country Productivity

59

and the terms λth (It ) and λtj (It ) are

λth ( It ) =

∑ ∑

i ∈It i ∈Ith

pith qith pith qith

= 1−

∑ ∑

i ∈Ith , i ∉It i ∈Ith

pith qith

piht qith

, and likewise for j.

(4.4)

Notice that the revenue shares in (4.3b), for each country, are measured relative to the common set of goods It. The weights in (4.3a) are the logarithmic mean of the shares sith (It ) and sitj (It ), and sum to unity over the set of goods i ∈ It . We have that λth ≤ 1 in (4.4) due to the differing summations in the numerator and denominator. This term will be strictly less than one if there are goods in the set Ith that are not found in the common set It. In other words, if country h is selling some goods in period t that are not sold by country j, this will make λth < 1. The greater the value of unique goods that are exported by country h and not country j, the lower is the value of λth/λtj, so it is an inverse measure of country h export variety relative to that of country j. Taking its inverse, then λtj/λth is a direct measure of the relative export variety of country h. Suppose that we choose the comparison country j as the world, which we denote by F. Then the set ItF = ∪ h I th is the total set of varieties exported by all countries in year t, and pitF qitF is the total value of exports for product i (summed over all source countries). Now compare country h = 1, . . . , H to country F, and it is immediate that the common set of goods exported is It ≡ Ith ∩ ItF = Ith, or simply the set of goods exported by country h. Therefore from (4.4) we have that λth = 1, and export variety by country h is measured by

λtF (Ith ) = λth (Ith )

∑ ∑

i ∈Ith i ∈ItF

piF qiF piF qiF

(4.5)

.

Hummels and Klenow (2005) further refine this term to measure export variety from each country h to its partner country j. With the world as the comparison, denoted by F, the extensive margin of exports from country h to j is defined as hj t ,exp

EM

∑ = ∑

hj

pitFj qitFj

Fj

pitFj qitFj

i ∈It i ∈It

,

(4.6)

60

Chapter 4

which extends (4.5) by adding the superscript j for the destination country. Hummels and Klenow also define the intensive margin of exports as hj t ,exp

IM

∑ = ∑

hj

pithj qithj

hj

pitFj qitFj

i ∈It i ∈It

(4.7)

.

It follows that the product of the extensive and intensive margins are hj t ,exp

EM

× IM

hj t ,exp

∑ = ∑

hj

pithj qithj

Fj

Fj Fj it it

i ∈It i ∈It

p q

=

coun ntry h exports to j = Sthj, imp . total imports by j

(4.8)

That is, the product of the extensive and intensive margin of exports from country h to j equals the import share of country h in country j, or the bilateral exports divided by country j total imports. Note that in order to calculate the extensive and intensive margin of exports from h to j, we actually use country j import data. For example, US import data can be used to calculate the extensive margin of exports for all countries selling to the United States (as is done by Feenstra and Kee 2008, discussed later in this chapter). In order to form the extensive margin of exports from country h to all destination countries, Hummels and Klenow take the geometric mean of (4.7),

EMthF,exp

Fj Fj ⎛ ∑ i ∈I hj pit qit ⎞ t = ∏⎜ ⎟ Fj Fj j≠ h ⎜ ⎝ ∑ i ∈Itwj pit qit ⎟⎠

wth

,

(4.9)

where wthj is the Sato-Vartia weight: wth ≡

(Sthj,exp − StFj,exp ) ( ln Sthj,exp − ln StFj,exp ) , ∑ j≠ h ⎡⎣(Sthj,exp − StFj,exp ) ( ln Sthj,exp − ln StFj,exp )⎤⎦

(4.10)

with Sthj, exp =

country h exports to j . total exports by h

The intensive margin of exports from country h to all destination countries, IMthF,exp, is constructed using the same geometric mean.

The Extensive Margin of Trade and Country Productivity

61

Analogous definitions hold to calculate the extensive and intensive margins of imports from country h to j, which are constructed using the export data of country h:1 hj t , imp

EM

hj t , imp

IM

∑ = ∑ ∑ = ∑

hj

i ∈It

i ∈IthF

hj

i ∈It

hj

i ∈It

pithF qithF pithF qithF pithj qithj

,

(4.11)

.

pithF qithF

(4.12)

Multiplying the extensive and intensive margin of imports, we obtain hj t , imp

EM

× IM

hj t , imp

∑ = ∑

hj

i ∈It

i ∈IthF

pithj qithj hF hF it it

p q

=

coun ntry h exports to j = Sthj, exp . (4.13) total exports by h

Again, the overall margins of country j imports from the world F are computed by taking the geometric means across all partner countries h. Hummels and Klenow (2005) consider a cross section of countries in a single year, 1995, and contrast the trade of larger versus smaller countries. As countries grow, then so do their exports and imports, and the question is whether this growth is due to importing and exporting a more diverse range of products, on the extensive margin in trade, or due to trading more of the same products, on the intensive margin in trade. It turns out that about two-thirds of the differences in the amount of trade between countries is explained by importing and exporting a more diverse range of products, or the extensive margin, while onethird of the differences in trade is due to trading more of the same products, or the intensive margin. Table 4.1 reports the Hummels and Klenow (2005) benchmark regression. Using a sample of 126 exporting countries for which real GDP and employment data are available from the Penn World Table version 6.1, they examine the relative importance of the extensive margin versus the intensive margin in explaining a nation’s overall exports. To do this, they regress each margin on GDP per worker (Y/L) and number of workers (L) for each country, expressed relative to the world average:

62

Chapter 4

Table 4.1 Extensive and intensive margins of exports, 1995 Dependent variable

Estimation done by

Y/L

L

R2

Y

R2

Number of countries

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Extensive margin

Hummels and Klenow

0.85 (0.05) 66% 1.05 (0.05) 71%

0.53 (0.03) 59% 0.63 (0.03) 62%

0.79

0.61 (0.03) 62% 0.74 (0.03) 65%

0.74

126

0.8

126

0.44 (0.05) 34% 0.43 (0.05) 29%

0.36 (0.03) 41% 0.38 0.03 38%

0.6

0.38 (0.03) 38% 0.39 (0.03) 35%

0.6

126

0.64

126

Revised

Intensive margin

Hummels and Klenow Revised

0.86

0.63

Source: Hummels and Klenow (2005) and author ’s calculations. Notes: 1. Data sources: extensive margin and intensive margin are calculated from the import data from UNCTAD for 1995; employment and real GDP are from PWT version 6.1 for 1995. 2. All variables are in natural logs. Standard errors are in parentheses. Percentages describe the contribution of each margin relative to the sum of the coefficients from both regressions. 3. “Hummels and Klenow” indicates it is a replication for Hummels and Klenow (2005) benchmark regression table 2. “Revised” indicates it is from the author ’s calculation using the same dataset.

ln EMthF,exp = α 1 + β1 ln

Yh + γ 1 ln Lh + ε1h Lh

and ln IMthF,exp = α 2 + β2 ln

Yh + γ 2 ln Lh + ε 2 h. Lh

The relative magnitudes of the coefficients βi /(β1 + β 2 ) and γ i /(γ 1 + γ 2 ) indicate how country size affects the magnitude of trade along the extensive and intensive margins. The results are shown in table 4.1, where column 1 indicates the dependent variable and column 2 whether the estimates are from Hummels and Klenow (2005) or as revised here. The coefficients on GDP per worker and employment are shown in columns 3 and 4, with the adjusted R2 in column 5. They also regress the natural log of each margin on the exporter ’s log GDP relative to

The Extensive Margin of Trade and Country Productivity

63

rest-of-world log GDP, which is reported in column 6 and 7 of table 4.1. Column 9 gives the number of countries in both regressions. Hummels and Klenow’s regressions in table 4.1 indicate that 66 percent of the growth in exports associated with higher GDP per worker occurs on the extensive margin, while 34 percent occurs on the intensive margin. Similar relative percentages are obtained from the coefficients on the number of workers, or on total GDP. Thus the extensive margin plays a more prominent role: for those additional exports done by larger or wealthier economies, around two-thirds is attributed to the extensive margin, while the remaining one-third occurs on the intensive margin. We also report in table 4.1 our replication using the same data but slightly correcting the STATA code in computing the extensive margin of exports.2 The main difference between our calculation of the extensive margin of exports is that our estimates tend to be lower for all countries in the sample. But our updated regressions deliver exactly the same implication as that of Hummels and Klenow, and confirm the relatively dominant role of the extensive margin in explaining nations’ overall exports. In table 4.2 we follow the exercise of Hummels and Klenow but extend the regression to cover a larger sample of countries and examine more years. We repeat the regressions for the odd-numbered years 1993 to 2001, using nearly 160 countries drawn from the Penn World Table version 6.2, which has data on employment and real GDP.3 Again, all the regressions emphasize the importance of the extensive margin in explaining trade flows. On average, about 60 percent of the contribution of country size to its exports occurs on the extensive margin, with the remaining 40 percent occurring on the intensive margin. Finally, utilizing export data, we turn to calculating the extensive and intensive margins of imports as specified in equations (4.11) and (4.12). We then apply the Hummels and Klenow regressions to the import margins. The results are reported in table 4.3. In this case the extensive margin of imports plays a less important role than the intensive margin: only about 30 percent of the contribution of country size to its imports occurs on the extensive margin, with the remaining 70 percent occurring on the intensive margin. Export Variety and Country Productivity We turn now to the link between the extensive margin of exports and country productivity. Our analysis of the Melitz (2003) model in chapter

64

Chapter 4

Table 4.2 Extensive and intensive margins of exports, revised PWT 6.2 data Year

Dependent variable

Y/L

L

R2

Y

R2

Number of countries

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

1993

Extensive margin

1.06 (0.05) 60% 0.70 (0.05) 40%

0.50 (0.03) 51% 0.47 (0.03) 49%

0.79

0.65 (0.03) 55% 0.53 (0.03) 45%

0.67

159

1.03 (0.05) 60% 0.68 (0.05) 40%

0.52 (0.03) 51% 0.49 (0.03) 49%

0.82

0.66 (0.03) 54% 0.55 (0.03) 46%

0.71

1.01 (0.04) 60% 0.68 (0.04) 40%

0.52 (0.03) 51% 0.50 (0.03) 49%

0.84

0.66 (0.03) 55% 0.55 (0.03) 45%

0.73

0.99 (0.04) 57% 0.76 (0.05) 43%

0.50 (0.03) 48% 0.54 (0.03) 52%

0.83

0.64 (0.03) 51% 0.61 (0.03) 49%

0.71

1.01 (0.05) 56% 0.78 (0.05) 44%

0.49 (0.03) 47% 0.56 (0.03) 53%

0.83

0.65 (0.03) 51% 0.63 (0.03) 49%

0.70

Intensive margin

1995

Extensive margin

Intensive margin

1997

Extensive margin

Intensive margin

1999

Extensive margin

Intensive margin

2001

Extensive margin

Intensive margin

0.70

0.73

0.75

0.75

0.75

0.66

163

0.71

162

0.73

163

0.73

164

0.72

Source: Author ’s calculations. Notes: 1. Data sources: extensive margin and intensive margin are calculated from the import data from COMTRADE for various years; employment and real GDP are from PWT version 6.2. 2. All variables are in natural logs. Standard errors are in parentheses. Percentages describe the contribution of each margin relative to the sum of the coefficients from both regressions.

The Extensive Margin of Trade and Country Productivity

65

Table 4.3 Extensive and intensive margin of imports, revised PWT 6.2 data Year

Dependent variable

Y/L

L

R2

Y

R2

Number of countries

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

1993

Extensive margin

0.46 (0.03) 32% 0.98 (0.05) 68%

0.21 (0.02) 25% 0.60 (0.03) 75%

0.67

0.28 (0.02) 29% 0.70 (0.03) 71%

0.56

159

0.44 (0.02) 31% 0.96 (0.05) 69%

0.22 (0.02) 27% 0.61 (0.03) 73%

0.74

0.28 (0.02) 28% 0.71 (0.03) 72%

0.65

0.46 (0.02) 31% 1.02 (0.04) 69%

0.22 (0.02) 25% 0.63 (0.03) 75%

0.75

0.28 (0.02) 27% 0.74 (0.03) 73%

0.64

0.45 (0.04) 30% 1.04 (0.04) 70%

0.21 (0.03) 25% 0.63 (0.03) 75%

0.75

0.28 (0.02) 27% 0.74 (0.03) 73%

0.63

0.43 (0.02) 29% 1.03 (0.04) 71%

0.22 (0.02) 25% 0.66 (0.03) 75%

0.72

0.29 (0.02) 27% 0.77 (0.03) 73%

0.63

Intensive margin

1995

Extensive margin

Intensive margin

1997

Extensive margin

Intensive margin

1999

Extensive margin

Intensive margin

2001

Extensive margin

Intensive margin

0.80

0.82

0.88

0.86

0.85

0.75

163

0.77

162

0.82

163

0.79

164

0.79

Source: Author ’s calculations. Notes: 1. Data sources: extensive margin and intensive margin are calculated from the export data from COMTRADE for various years; employment and real GDP are from PWT version 6.2. 2. All variables are in natural logs. Standard errors are in parentheses. Percentages describe the contribution of each margin relative to the sum of the coefficients from both regressions.

66

Chapter 4

3 showed that countries with a greater share of exports will experience higher industry productivity. That outcome arises from the self-selection of firms, with the least productive domestic firms exiting as trade is opened, and the most productive firms becoming exporters. This relationship was captured by the transformation curve shown in figure 3.3, where the opening of trade moves the industry from a corner at A to an interior equilibrium at C. The strict concavity of the transformation curve follows from the logic of the Melitz model, and the elasticity of this curve was found to be ω ≡ [θσ (σ − 1)] − 1 > 0 , where θ is the Pareto parameter in the distribution of productivities. More formally, theorem 3.1 showed the revenue function ψ ( Ad , Ax ) that incorporates that elasticity, which we make use of now. To determine the empirical link between export variety and country productivity, Feenstra and Kee (2008) consider a multicountry, multi-sector model. They suppose that each sector i = 1, . . . , N in country h = 1, . . . , H and year t has the revenue function h h ψ ith = ψ i ( Adit , Axit ), as given by theorem 3.1.4 They assume a translog functional form for GDP across the sectors and then use the CES h h , Axit ) within each sector. Define the revenue function ψ ith = ψ i ( Adit h h h h vector ψ t = (ψ 1t ,..., ψ Nt , ψ N + 1,t ) to also include a price ψ Nh +1,t for the nontraded sector N + 1. Denoting factor endowments by the vector h Vth = (v1ht ,..., vKt ) , the translog GDP function is N +1

K

i =1

k =1

ln Rth (ψ th , Vth ) = α 0h + β0 t + ∑ α i ln ψ ith + ∑ βk ln vkth +

1 N +1 N +1 ∑ ∑ γ ij ln ψ ith ln ψ hjt 2 i =1 j =1

N +1 K 1 K K + ∑ ∑ δ k ln vkth ln vht + ∑ ∑ φik ln ψ ith ln vkth . 2 k =1  =1 i =1 k =1

(4.14)

Feenstra and Kee (2008) allow this function to differ across countries based on the fixed effects α 0h , which reflect exogenous technology differences, and also allow for the year fixed effects β0t, which are equal across countries. To satisfy homogeneity of degree one in prices and endowments, we need the restrictions

γ ij = γ ji , δ k = δ k ,

N +1

∑α i =1

K

n

= ∑ β k = 1, k =1

N +1

∑γ i =1

ij

N +1

K

K

i =1

k =1

k =1

= ∑ ϕ ik = ∑ δ k = ∑ ϕ ik = 0. (4.15)

The share of sector i in GDP then equals the derivative of ln Rth (ψ th , Vth ) with respect to ln ψ ith :

The Extensive Margin of Trade and Country Productivity

N +1

K

j =1

k =1

sith = α i + ∑ γ ij ln ψ jth + ∑ ϕ ik ln vkth ,

67

i = 1,..., N + 1.

(4.16)

We will use these share equations for each sector as estimating equations, but first we need to explain how the CES revenue functions are measured. CES Revenue Functions Key to the empirical work is to measure the CES revenue functions h h ψ ith = ψ i ( Aidt , Aixt ) in each sector. To this end we difference the GDP function and share equations with respect to a comparison country denoted by F. The CES revenue function in each sector will also be differenced with respect to country F in log form, which means we take the log of the ratio ψ ith /ψ itF . To evaluate the ratio of CES functions, we can apply the index number formula due to Sato (1976) and Vartia (1976). Assuming that the fixed costs fix and fi in each sector are the same across countries, we have that the CES ratio in sector i equals h h h ψ i ( Aidt , Aixt ) ⎛ Aidt ⎞ = F F F ⎟ ⎠ ψ i ( Aidt , Aixt ) ⎜⎝ Aidt

1−With

h ⎛ Aixt ⎞ ⎜⎝ A F ⎟⎠ ixt

With

h h / Aidt ⎛ A h ⎞ ⎛ Aixt ⎞ = ⎜ idt F ⎟⎜ F F ⎟ ⎝ Aidt ⎠ ⎝ Aixt ⎠ / Aidt

With

,

(4.17)

where With is the logarithmic mean of the export shares in countries h and F.5 The first equality in (4.17) follows directly from the Sato–Vartia formula, which allows us to evaluate the ratio of CES functions (3.25) without knowledge of the fixed costs, but using the data on export shares instead, and the second equality follows by algebra. To simplify this expression further, we now show how the ratio of export and h h /Aidt domestic shift parameters, Aixt , is closely related to export variety. To demonstrate this, we use (3.6) through (3.8) for the domestic sector and (3.9) through (3.11) for the export sector to compute rih (ϕ ih * ) and rixh (ϕ ixh* ), where we omit the time subscript for simplicity. Defining the expenditure in each sector as Eih at home and EiF abroad, we obtain rixh (ϕ ixh* ) ⎛ ϕ ixh* PiF/τ i ⎞ =⎜ ⎟ rih (ϕ ih * ) ⎝ ϕ ih * Pih ⎠

σ i −1

f EiF = ix fi Eih

where we treat the fixed costs as the same across countries. Raising this expression to the power (1/σi), using the definition of the demand shift

68

Chapter 4

parameters in (3.17) and (3.18), and noting that the Pareto distribution gives us Mixh/Mih = (ϕ ixh*/ϕ ih * )−θi from (3.12), we find that ⎛ Aixh ⎞ h (σ i −1) σ iθi ⎛ fix ⎞ ⎜⎝ h ⎟⎠ = ( χ i ) ⎜⎝ f ⎟⎠ Aid i

1/σ i

(4.18)

,

where

χ ih ≡

Mixh Mih

and 0