Multilateralism and the World Trade Organisation The Architecture and Extension of International Trade Regulation Routledge Advances in International Political Economy 1st Edition Rorde Wilkinson
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Feenstra, Robert C.
Cataloging-in-Publication
Data
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
Series Foreword vii
Preface ix
1 Introduction 1
2 Consumer Benefits from Import Variety 7
3 Producer Benefits from Export Variety 39 4 The Extensive Margin of Trade and Country Productivity 57
5 Product Variety and the Measurement of Real GDP 83 6 Conclusions 109 Notes 119 References 123 Index 129
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
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 ReviewofWorld Economics/WeltwirtschaftlichesArchiv (Springer) and Feenstra (1994) in the AmericanEconomicReview; portions of chapter 3 are drawn from Feenstra (2010) in the CanadianJournalofEconomics (Blackwell Publishing); portions of chapter 4 are drawn from Feenstra and Kee (2008) in the JournalofInternationalEconomics (Elsevier); and portions of chapters 5 and 6 are drawn from Feenstra, Heston, Timmer, and Deng (2009) in the ReviewofEconomicsandStatistics (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. Preface
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
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 extensivemargin of imports (Hummels and Klenow 2005). Broda and Weinstein obtain an estimate of the gains from trade
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 11 import share) σ , 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
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 1 import share) 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
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 extensivemargin 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 termsoftrade 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.
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
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 qkp = 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
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,
where ait > 0 are taste parameters that vary with time and IN t ⊆ {,...,} 1 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
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, qUep ittit =∂∂(/). 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
where the weights wi(I) are constructed from the expenditure shares sIpqpq ititititit iI ( ) ≡ ∈ ∑ as
The numerator in (2.4) is the “logarithmic mean” of the shares sI it () and sI it 1 (), 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 wIsIsI iitit ()[()()] =+ 1 21 , 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 II tt∩≠∅ 1 . 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, III tt ⊆∩≠∅ 1 . We sometimes refer to the set I as the “common” set of goods. Then the ratio epIepI tt (,)/(,) 1 is still measured by the Sato–Vartia index in the theorem above. Our interest is in the ratio epIepI tttt (,)/(,) 11 , which can be measured as follows:
Theorem 2.2 (Feenstra 1994) Assume that bit−1 = bit for iIII tt ∈⊆∩≠∅ 1 , and that the observed quantities are optimal. Then for σ > 1,
where the weights wi(I) are constructed from the expenditure shares sIpqpq ititititit iI ( ) ≡ ∈ ∑ as in (2.4), and the values λt(I) and λt−1(I) are constructed as
Each of the terms λτ(I) < 1 can be interpreted as the period τ expenditure onthegoodinthecommonsetI,relativetotheperiod τ totalexpenditure. Alternatively, this can be interpreted as oneminustheperiod τ expenditureon“new”goods(notinthesetI),relativetotheperiod τ totalexpenditure. When there are more new goods in period t, this will tend to lower the
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 iI t ∈ . 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).
2.2
Figure
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
qf=−1()σϕ,
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
which determines the number of product varieties produced in equilibrium as NLf = /σ . This condition holds under autarky or trade, so opening a country to trade has noimpact 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 (/) /() λλ σ tt 1 11 , 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 nodisappearingdomesticvarieties in this model, so λt−1 = 1. The numerator λt measures theexpenditureonthedomesticvarietiesrelativetototalexpenditurewithtrade,oroneminustheimportshare. The gains from trade are therefore () /() 1 11 import share σ , 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.
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 betweensourcecountriesforeachgood; 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
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
Figure 2.3
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 variancesandcovariances of the supply and demand shocks, which can depend on the variance of exchange rates and other
2.4
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
Taking natural logs and taking the difference over time, we can write this demand equation as ΔΔ lnln, sp ittitit =−−1 ( ) + φσε
where φ σ ttt cpI ≡−()ln(,) 1 Δ is a time fixed-effect, and εitit b ≡Δ ln is an error term reflecting taste shocks. To this demand curve we add a supply curve, which is assumed to be
lnln, pqititit =+βξ
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, pitit it
where the parameters appearing in this “reduced form” supply curve are
where Epq titit iI t = ∈ ∑ is total expenditure, and the error term is
