Measuring the Unequal Gains from Trade *

Transcription

1 Measuring the Unequal Gains from Trade * Pablo D. Fajgelbaum Amit K. Khandelwal First Draft: September 2013 This Draft: November 2013 Abstract Individuals that consume different baskets of goods are differentially affected by the relative price changes caused by trade. We develop a methodology to measure the unequal gains from trade across consumers for many countries over time. The approach is based on aggregate statistics and model parameters that can be estimated from bilateral trade data. It exploits that changes in aggregate expenditures reflect changes in the relative prices of high- versus lowincome elastic goods, resulting in different welfare implications across consumers. We estimate the model and find that the unequalizing effects of trade vary considerably across countries depending on their pattern of specialization and that of their trade partners. In the majority of countries that we analyze, consumers at the 90th percentile of the expenditure distribution gain relatively more from trade than consumers at the the 10th percentile. The gains from trade are typically U-shaped with individual income, but in some countries that specialize in low-income elastic goods, such as India, or that are proximate to producers of high-income elastic goods, such as Mexico, the gains from trade are monotonically increasing with individual income. We also find pro-poor welfare changes in most countries in recent decades, a period of rapid increase in China s global exports. JEL Classification: D63, F10, F60 * We thank Arnaud Costinot, Esteban Rossi-Hansberg, Jonathan Vogel, and seminar participants at Harvard, Princeton, and UCLA for helpful comments. We acknowledge funding from the Jerome A. Chazen Institute of International Business at Columbia Business School. UCLA, Department of Economics, 8283 Bunche Hall, Los Angeles, CA pfajgelbaum@econ.ucla.edu Columbia Business School, Uris Hall, 3022 Broadway, New York, NY ak2796@columbia.edu

2 1 Introduction Understanding the distributional impact of international trade is one of the central tasks pursued by international economists. A vast amount of research has examined this question through the effect of trade on the earnings of individuals that vary by skill level (e.g., Stolper and Samuelson (1941)) or by some other characteristic. A second channel for the effect of trade on inequality, which has received considerably less attention, operates through the cost of the consumption basket across individuals with different consumption patterns. For example, trade-induced increases in the price of food negatively affects low-income consumers who typically have larger food expenditure shares than richer consumers. How important are the distributional effects of trade through this expenditure channel? Does it tend to favor high- or low- income consumers within a country? Naturally, the answers to these questions will vary across countries depending on what goods they produce and who they trade with. In this paper we develop a methodology to measure the distribution of welfare changes across individuals through the expenditure channel for many countries over time. The broad applicability of the approach is a result that it is based on aggregate statistics, such as expenditure shares across goods, and model parameters that can be estimated from bilateral trade data. We estimate the model and identify the effect of trade through counterfactual changes in trade costs. We find that trade has relatively adverse effects for low-income consumers in more than half of the countries that we consider and that the distributional effects of trade are often large relative to the aggregate effects. We also find that, in most countries, the distribution of welfare changes between 1994 and 2005, a period of increased participation of China in world trade, has been pro-poor. The intuition behind our approach is straightforward. Conditioning on moments of the income distribution, observed changes in the aggregate expenditure shares across goods reflect changes in the relative prices of high- versus low-income elastic goods. In an international trade context, this means that changes in aggregate import shares across sectors and origins convey information about not only the aggregate gains from trade, as is known, but also their distribution among consumers with different expenditure patterns. For example, to the extent that low-income consumers in the U.S. are more likely to purchase goods imported from China, an increase in Chinese exports to the U.S. may indicate an increase in welfare for low-income Americans. Implementing this approach requires a non-homothetic demand system that allows for goodspecific Engel curves. That is, the elasticity of the expenditure share with respect to the total expenditures made by an individual consumer must be allowed to vary across goods. The Almost- Ideal Demand System (AIDS) introduced by Deaton and Muellbauer (1980a) is a natural choice. The AIDS is a first-order approximation to any demand system, and it is widely used in applied work because it generally provides a good fit of individual expenditure data. Importantly for our purposes, it is flexible enough to satisfy the key requirement of good-specific income elasticities. 1 1 If good-specific income elasticities are neutralized, the AIDS collapses to the translog demand system studied in an international trade context by Feenstra and Weinstein (2010), Arkolakis et al. (2010) and Novy (2012). Feenstra and Reinsdorf (2000) shows how prices and aggregate expenditures relate to the Divisia index in the AIDS, and 1

3 We begin by studying theoretically how changes in the prices of final consumption goods impact the distribution of welfare changes through the cost of expenditures across consumers that vary by expenditure level. We use basic properties of demand to demonstrate that, in the AIDS, the firstorder approximation to the welfare change (i.e., the compensating variation) of a given consumer caused by the impact of any distribution of price changes on the cost of the consumption basket can be recovered using demand parameters and aggregate statistics. These statistics include levels and changes in aggregate expenditure shares and moments of the distribution of expenditure levels across consumers. We then embed these demand-side results in a model of international trade. A natural benchmark that is also convenient as empirical framework is the canonical Armington model, in which products are differentiated by country of origin. In this context, the levels and changes in national import shares and demand parameters suffice to measure exactly (rather than to a first-order approximation) the distributional effect of any foreign-trade shock through the expenditure channel. The model allows the use of aggregate trade shares to exactly characterize the welfare change experienced by consumers at any income level through the expenditure channel. The compensating variation of a high-income consumer relative to a low-income consumer depends on how import price changes correlate with goods income elasticities. In turn, this direction in the price changes can be partly inferred from the bias of aggregate trade shares toward high income elastic goods, or aggregate beta of the economy. In the AIDS, each good i is characterized by an income elasticity β i ; the aggregate β is an aggregate import-share weighted average of these good-specific income elasticities. An increase in the aggregate beta occurs when the national import basket tilts towards goods consumed mostly by the rich, which may reveal a fall in the relative price of these goods and a relative welfare improvement for high-income consumers. The model delivers a non-homothetic gravity equation that we take to the data. Goods originating from each country have a distinct Engel slope from the perspective of the consumer, so that the estimation determines which countries produce goods that are relatively more valued by rich consumers. The estimation projects importer budget shares on terms that capture standard gravity forces (e.g., distance) and an importer s real income term whose elasticity can potentially vary across exporters. Consistent with the existing literature, such as Khandelwal (2010), Hallak and Schott (2011) and Feenstra and Romalis (2012), we find that richer countries export goods with higher income elasticities. Using the estimated parameters, we apply the results from the theory to ask: who are the winners and losers of trade within countries, and how large are the distributional effects relative to the aggregate effects? For that, we perform the counterfactual exercise of bringing each country from their current trade shares to autarky. Although the representative consumer loses when moving to autarky, the estimated parameters imply that for more than half of the countries we analyze, moving to autarky reduces welfare for an individual at the 10th expenditure percentile by less than for an individual at the 90th expenditure percentile. There is large variation in this difference suggests that this demand system could be useful for welfare evaluation in a trade context. See also Feenstra (2010). 2

4 across countries. For example, consumers at the 10th percentile in India gain 0.8 percentage points more than the 90th by moving to autarky, while in Mexico they gain 8.4 percentage points more. Relative to the welfare change of the representative consumer, these differences are large (50, and 29 percent in each country, respectively). In contrast, in the United States moving to autarky makes consumers at the 90th percentile 8 percentage points better off relative to the 10th. When examining the entire distribution, we find many countries where extremely rich (i.e., the 99th percentile) and poor (i.e., the 1st percentile) consumers fare better than the middle class (i.e., the 50th percentile). These results uncover the importance of country specialization patterns across goods with varying income elasticity of demand as a determinant of the potentially unequalizing effect of trade through the expenditure channel. In countries that specialize in low income-elastic goods, such as India, the reduction in the relative price of these goods caused by shutting down trade has a relatively positive impact on low-income consumers; that is, the gains from trade are pro-rich. In contrast, in countries like the U.S. or Japan that specialize in high income-elastic goods the gains from trade are pro-poor, while in countries that specialize in intermediate income-elastic goods, the welfare changes caused by trade are U-shaped with the level of individual expenditures. Our results also demonstrate the importance of geography in shaping the unequalizing effects of trade. In countries that are proximate, in a gravity sense, to exporters of high income elastic goods, the increase in the relative prices of these goods caused by shutting down trade tends to hurt high-income consumers. For example, we find that geographic proximity to the U.S., an exporter of high income elastic goods, is a strong predictor of pro-rich effects of trade. Finally, we perform welfare accounting using the actual changes in aggregate expenditure shares between 1994 and This period in particular spans the rise of China as a major exporter in the global economy. For almost all countries, China s rise has benefited the poor relative to the rich according to this expenditure channel; for example, the welfare of the poor in the U.S. rose 4.6 percentage points more than the rich over this period owing to the fact that relative prices of low-income elastic goods, an in particular of China s exports, declined. This inference from the aggregate shares is close to the finding of Broda and Romalis (2009), who show using detailed scanner-level data and a different methodology that inflation was 7.6 percentage points lower for the poor than the rich. Our approach to measure welfare gains from trade using aggregate statistics resembles Arkolakis et al. (2012), who show that, in a certain class of models, easily available data on aggregate expenditure shares and model parameters are sufficient to measure the aggregate gains from trade. The literature related to this approach, summarized by Costinot and Rodriguez-Clare (2013), is designed to measure only aggregate gains rather than distributional consequences. However, much of the public opposition towards increased openness stems from the belief that welfare changes are unevenly distributed. This paper is motivated by the belief that an approach that similarly relies on aggregate data to measure the (potentially) unequal gains from trade would be useful in assessing the implications of trade. 3

5 We are not the first to exploit differences in income elasticities across goods within the international trade literature. Fajgelbaum et al. (2011) is a theoretical study of the cross-country patterns of specialization that result from non-homothetic preferences in vertically differentiated products and within-country inequality. Recent papers by Hallak (2006), Hallak (2010), Fieler (2011) and Caron et al. (2012) find an important role for non-homothetic preferences in explaining trade patterns in the data. This role of non-homothetic demand in explaining trade data is an important ingredient of our approach, but in contrast to these papers our focus is on the distribution of welfare gains across individuals through the expenditure channel. A few recent empirical studies also analyze the effects of trade on consumer welfare through heterogeneity of tastes. Porto (2006) studies the effect of price changes implied by a tariff reform on the distribution of welfare using consumer survey data from Argentina. More recently, Atkin (2013) studies the effect of trade on consumers with heterogeneous preferences through habit formation. Broda and Romalis (2009) use scanner data to measure the price index of high- and low-income consumers in the U.S., while Faber (2012) studies the effect of input tariff reductions in Mexico on the price changes of final goods of different quality. In contrast to these papers that utilize detailed micro data for specific countries, our approach provides a framework to quantify the unequal gains from trade across consumers over a large set of countries over time using cross-country aggregate data that is easily available to researchers and policymakers. Within our framework we are able to show theoretically how either observed or counterfactual changes in aggregate expenditure shares map to the welfare changes of individuals in each point of the expenditure distribution. Finally, there is of course a large literature that examines trade and inequality through the earnings channel. A dominant theme in this literature, as summarized by Goldberg and Pavcnik (2007) and Harrison et al. (2010), is the poor performance of the Stolper-Samuelson mechanism, which predicts that trade increases the relative wages of low-skill workers in low-income countries, in rationalizing patterns from developing countries. Several recent studies, such as Feenstra and Hanson (1996), Helpman et al. (2012), Brambilla and Porto (2012), Frias and Verhoogen (2012), and Burstein et al. (2013) study different channels through which trade affects the distribution of earnings such as outsourcing, labor market frictions, quality upgrading, or capital-skill complementarity. We complement these and other studies that focus on the earnings channel by examining the implications of trade through the expenditure channel. An appealing feature of our approach is that expenditure data (e.g., household survey data) is widely available across countries and time, and the AIDS is often imposed on these data to estimate preferences. Our aggregate approach can take these model estimates, along with aggregate expenditures on these items, to measure the distribution of welfare changes across consumers in contexts beyond international trade. The remaining of the paper is divided into five sections. Section 2 uses standard consumer theory to derive generic expressions for the distribution of welfare changes across consumers, and applies these expression to the AIDS. Section 3 embeds these results in a standard trade framework. Section 4 estimates the parameters and quantifies the unequal gains from trade. Section 5 concludes. 4

6 2 Consumers We start by developing generic expressions for the distribution of welfare changes across consumers that vary in their total expenditures. We only use properties of demand implied by standard demand theory. The results from this section correspond to changes in prices and expenditures exogenously taken as given by consumers. In Section 3, we link these results to a standard model of trade in general equilibrium. 2.1 Basic Decomposition of Welfare Changes We study an economy with i = 1,..., I goods for final consumption with price vector p = {p i } I i=1 taken as given by h = 1,.., H consumers. Consumer h has indirect utility v h and total expenditures x h. Let X = H h=1 x h be the aggregate expenditure level. We denote by x (v h, p) the expenditure function with associated indirect utility function v (x h, p). We also let s i,h s i (x h, p) be the share of good i in the total expenditures of individual h and S i H h=1 (x h/x) s i,h be the share of good i in aggregate expenditures. Consider log-changes { p i } I i=1 in prices and { x h} H h=1 in individual expenditures.2 Because these are sufficient to characterize welfare changes, we do not yet need to make specific assumptions regarding the supply side of the economy. In the next section, we embed the demand-side results into a standard model of trade. individual h associated with these changes is From Roy s identity, the change v h in the indirect utility of v h = ln v (x h, p) ln x h [ ] ( p i ) s i,h + x h. (1) i=1 As is standard, we define the equivalent variation of consumer h associated with the changes {{ p i }, x h } as the percent increase in individual expenditures, ŵ h, that would lead to the welfare change v h if prices were kept constant. Using (1) and the definition of ŵ h we reach 3 ŵ h = ( p i ) s i,h + x h. (2) i=1 This is a well-known formula for the equivalent variation. 4 Henceforth, we refer to ŵ h as the welfare change of individual h, acknowledging that by this we specifically mean the first-order approximation to the compensating variation, expressed as share of the initial level of expenditures, associated with an infinitesimal change in prices or in the expenditure level. 2 We use ẑ d ln (z) to denote the log-change in a variable z. 3 To derive expression (2), start with the total change in the indirect utility associated with the changes {{ p i}, x h }, ˆv h = [ ] I ln v(x h,p) i=1 ln p i ˆp i + ln v(x h,p) ln x h ˆx h. Using Roy s identity, s i (x h, p) = ln v(x h,p) ln v(xh,p) 1, ln p i ln x h together with the definition of ŵ h, v h = ln v(x h,p) ln x h ŵ h, gives the result. 4 As a first order approximation to changes in prices or total expenditures, this is the same as the compensating variation. See Theil (1975). 5

7 We can decompose the welfare change of consumer h into three economically distinct terms: ŵ h = Ŵ + ψ ( h + x h ˆX ), (3) where is the aggregate welfare change, and Ŵ ( p i ) S i + ˆX, (4) i=1 ψ h ( p i ) (s i,h S i ) (5) i=1 is the individual welfare change associated with the composition of the expenditure basket. The first term in Ŵ captures the welfare change due to relative price changes in the absence of within-country inequality or when consumer preferences are homothetic. It also corresponds to the welfare gains of a representative consumer through the cost of the expenditure basket, as long as the individual demands can be aggregated. The individual welfare change ψ h captures that heterogeneous consumers may be differently affected by the same price changes due to differences in their expenditure basket. It is different from zero for some consumers only if there is variation across consumers in how they allocate expenditure shares across goods. The final term, ( x h ˆX) captures heterogeneity in the growth of expenditure levels. 2.2 Almost-Ideal Demand The Almost-Ideal Demand System (AIDS) introduced by Deaton and Muellbauer (1980a) is a particular case of the Log Price-Independent Generalized Preferences of Muellbauer (1975). The latter are in general defined by the indirect utility function v (x h, p) = F [ ( ) ] 1/b(p) xh, (6) a (p) where a (p) and b (p) are price aggregators and F [ ] is a well-behaved increasing function. The AIDS is the special case that satisfies ln a (p) = α + ln b (p) = α i ln p i i=1 i=1 j=1 γ ij ln p i ln p j, (7) β i ln p i. (8) i=1 The adding up, homogeneity, and symmetry constraints lead to the parameter restrictions I i=1 α i = 1, I i=1 β i = I j=1 γ ij = 0, and γ ij = γ ji for all i, j. 6

8 The first price aggregator, a (p), has the translog functional form. It is independent from non-homotheticities and can be interpreted as the cost of a subsistence basket of goods. The expenditure level, x h, must therefore be larger than this cost. The second price aggregator, b (p), has the Cobb-Douglas functional form and captures the relative cost of high-income elastic goods. For our purposes, a key feature of these preferences is that the larger is the consumer s expenditure level x h relative to the subsistence basket, the larger is the welfare gain from a reduction in the cost of high income-elastic goods, as captured by a reduction in b (p). Applying Shephard s Lemma to the indirect utility function defined by equations (6) to (8) generates an expenditure share in good i for an individual with expenditure level x h equal to s i (p, x h ) = α i + j=1 ( ) xh γ ij ln p j + β i ln a (p) (9) for i = 1,..., I. 5 These expenditure shares have two well-known features that suit our purposes. First, their elasticity with respect to the expenditure level is allowed to be good-specific. Goods for which β i > 0 are high-income elastic, while goods for which β i < 0 are low-income elastic. 6 Second, they admit aggregation, in the sense that market-level behavior can be exactly represented by the behavior of a single consumer. Specifically, the aggregate market share of good i is S i = s i (p, x), (10) where x is an inequality-adjusted mean of the distribution of expenditures across consumers, x = xe Σ, where x E [x h ] is the mean and Σ E [ x hx ln ( x hx )] is the Theil index of the expenditure distribution. 7 We identify x as the expenditure level of the representative consumer, so that the distribution of budget shares for the aggregate economy are the same as the distribution of budget shares for an individual with expenditure level x. To shorten notation, we let p be a column vector with the price changes and { } S, Ŝ be vectors with the levels and changes in aggregate expenditure shares, S i and Ŝi. We also let {α, β} be column vectors with the parameters α i and β i, and Γ be the matrix with element γ ij in row i, 5 Of course, expenditure shares must be restricted to be non-negative for all goods. Since expenditure shares add up to one, this guarantees that all expenditure shares are smaller than 1. For the rest of this section we assume that (9) predicts non-negative expenditure shares for all goods and consumers, so that the non-negativity restriction is not binding. We discuss how to incorporate this restriction explicitly for counterfactual analysis in section 3.3. At the estimated parameters this restriction is sometimes violated and we discuss how to deal with these cases in footnote Even though we define x h as the individual expenditure level, to follow standard terminology we refer to β i as income elasticity of the expenditure share in good i. In our application to trade below, individual income and expenditure are assumed to be equal. 7 The Theil index is a measure of inequality which takes the minimum Σ = 0 if the distribution is concentrated at a single point. Its maximum value equals the total number of consumers and is reached when a single consumer makes all expenditures. In the case of a lognormal expenditure distribution with variance σ 2, it is Σ = 1 2 σ2. 7

9 column j. With this notation, the Almost-Ideal Demand System is characterized by the parameters Θ = {α, α, β, Γ}, and the aggregate expenditure shares in (10) are represented by S = α + Γ ln p + βy, (11) where the term y = ln ( x/a (p)) denotes the ratio of the adjusted mean of the expenditure distribution to the homothetic price index. Henceforth, we follow Deaton and Muellbauer (1980a) and refer to y as the adjusted real income Recovering the Aggregate and the Individual Welfare Effects Combining (9) and (10) we can solve for the individual welfare change defined in (5), ( xh ) ψ (x h ) = ˆb ln, (12) x where ˆb is the log-change in the non-homothetic price index, b (p). Because the elasticities {β i } add up to zero, it can be expressed as the covariance between the good-specific income elasticities and the price changes across goods: ˆb = β p = COV [ ] {β i } I i=1, {ˆp i} I i=1. (13) The individual welfare effect is linear in the log-expenditure level of individual h. Therefore, the slope ˆb summarizes the unequal welfare changes in the economy. A positive (negative) value of ˆb reflects a relative price increase of high (low) income elastic goods, implying a relative welfare loss for rich (poor) consumers. 9 Next, we establish a property of this demand system which is key for our analysis. To a firstorder approximation, the welfare change (i.e., the compensating variation) of each consumer due to the change in the cost of expenditures caused by a change in prices can be recovered using demand parameters and aggregate statistics. } Proposition 1. The aggregate and unequal welfare effects through the expenditure channel {Ŵ, ˆb corresponding to arbitrary price changes p can{ be expressed } in closed form as function of the level and changes in aggregate expenditure shares S, Ŝ, the parameters {Γ,β}, the change in the adjusted mean and in total expenditures {ˆ x, ˆX} and the adjusted real income y. The key feature of the Proposition 1 is that, as long as that the substitution and expenditure elasticities parameters {Γ,β} are known, a researcher armed with a sequence of the aggregate 8 In the absence of within-country inequality, y can be interpreted as the expenditures of a representative consumer after paying for a subsistence-level consumption basket with price equal to a. 9 For example, ˆb < 0 reflects a reduction in the relative price of high-income elastic goods, so that the individual change is positive, ψ (x h ) > 0, for consumers whose expenditure level is above the representative consumer s, x h > x, and negative otherwise. 8

10 { statistics S, Ŝ, x, ˆX, } y can account for Ŵ (i.e., the first-order approximation to the welfare change of the representative consumer) and for the deviation from that level through the expenditure channel corresponding to a consumer with expenditure level x, ψ (x). 10 The proof in the appendix presents the solution for {Ŵ, ˆb} as function of aggregate statistics and parameters. The intuition for this result follows from the fact that the aggregate welfare effect and the slope of the individual welfare effect both depend on weighted averages of price changes, Ŵ = S p + ˆX and ˆb = β p. In turn, unobserved price changes p can be backed out from aggregate expenditure shares inverting the demand system (11). The AIDS satisfies the key requirement that, once aggregate-expenditure moments { } are controlled for, an observed distribution of expenditure shares changes and levels, S, Ŝ, maps to a unique set of price changes p, as is needed for this result to hold. 2.4 Distribution of Welfare Changes The previous result allows to characterize the welfare change of consumers in each point of the expenditure distribution. It may also be useful to characterize the distribution of welfare changes. For that, we make the empirically plausible assumption that the expenditure distribution is lognormal, ln (x h ) N ( µ, σ 2). 11 In that case, considering just the aggregate and the individual welfare changes through the expenditure channel from equation (3) (i.e., assuming that x h ˆX = 0), the compensating variation of household h is normally distributed: ŵ h = Ŵ ˆb [ ln (x h ) ( µ + σ 2)] N (Ŵ + ˆbσ 2, (σˆb)²). (14) Holding Ŵ constant, under a log-normal distribution a higher value for ˆb not only implies that price changes are more strongly pro-poor, but also implies an increase in the mean of the distribution of welfare changes across consumers. This occurs because a larger ˆb denotes a relative increase in welfare for all consumers below the representative consumer, who has an expenditure level above the mean individual. Naturally, a higher ˆb also raises the dispersion in the distribution of welfare changes. Hence, the distribution of welfare changes can be characterized using the same aggregate statistics as in Proposition From Consumer Theory to the Unequal Gains From Trade We have used properties of demand to express the distribution of welfare changes across consumers as function of aggregate expenditure shares and demand parameters. Now, we embed these results 10 The aggregate expenditure level ˆX is only needed as it enters in the definition of the aggregate effect, but is not needed to compute the individual welfare change ψ (x). The adjusted real expenditure y can be obtained from national accounts data. Alternatively, if all the parameters Θ = {α, α, β, Γ} are known, it can be constructed using price levels from (7). If price levels are not observed, they can also be inferred from aggregate expenditure shares; see the proof of Proposition 1 in the appendix. 11 Log-normal expenditure distributions provide a good fit of empirical distributions. See Banks et al. (1997). 12 Measures of inequality that depend on both the first and second moment of the lognormal distribution may respond ambiguously to ˆb. For example, the coefficient of variation is increasing with ˆb only if the aggregate welfare change is sufficiently large. 9

11 in a standard model of trade. A natural benchmark is the canonical Armington model, in which products are differentiated by country of origin. Then, we use the model as an empirical framework to measure the importance of trade as a driver of inequality through the expenditure channel. 3.1 International Trade Framework The world economy consists of i = 1,..., I countries, each of them specialized in the production of a different good. From the perspective of an individual consumer, these goods can be demanded with different income elasticities. For example, expenditure shares on Chinese goods may decrease with total individual expenditures. We let p in be the price of goods from country n in country i and p i be the price vector in country i. We denote the local price in country i of domestically produced goods by p i. Bilateral iceberg trade costs τ in and perfect competition imply that p in = τ in p n. When it corresponds, we index variables by i to denote that they are specific to the importing country i. Labor is the only factor of production. We let z h be the efficiency units of individual h and we assume that individual income equals expenditures. Each country is characterized by a Theil index of its distribution of efficiency units, Σ i, which equals the Theil index of the expenditure distribution. Individual h in country i receives income of x h = p i z h and the mean of the expenditure distribution is x i = p i z i. All income is spent on traded goods. Using (10), the aggregate import share in country i for goods originated from the exporter n is S in = α n + γ nn ln (p in ) + β n y i, (15) n =1 where, letting a i = a (p i ) be the homothetic price index in country i, the term y i = ln ( x i /a i ) denotes as before the ratio of the adjusted mean of the expenditure distribution to the homothetic price index, x i = x i e Σ i. The richer is the importing country (higher x i ) or the more unequal it is (higher Σ i ), the larger is its import share from countries with high income elastic goods, β n > 0. The expenditure share of the individual consumer h is then where y i,h = y i + ln (x h / x i ). s in,h = α n + γ nn ln (p in ) + β n y i,h, (16) n =1 For cleaner analytic expressions we assume that cross-elasticities are symmetric, 13 γ nn = { γ I ( 1 1 I ) γ if n n if n = n. (17) While it simplifies the algebra, this assumption is not necessary to characterize the distribution of welfare changes in our framework The normalization by I in (17) is without loss of generality and only serves the purpose of shortening notation. 14 Proposition 1 is valid for the general asymmetric case and equally applicable in the context of the Armington 10

12 Before we proceed it is useful to define a measure of dispersion among the β n s, σ 2 β = as well as the aggregate beta for economy i, β i = βn, 2 (18) n=1 β n S in. (19) n=1 The parameter σ 2 β is proportional to the variance of the β n s and captures the strength of nonhomotheticities. A higher σβ 2 implies a steeper the slope of Engel curves for some goods. If preferences are homothetic then σβ 2 = 0. In turn, the aggregate beta β i measures the bias in the aggregate expenditure shares of country i to imports from high-β exporters. The larger is β i, the relatively more the economy as a whole spends in goods that are preferred by high-income consumers. Consider a shock to the prices faced by country i, { p in } I n=1. For the type of welfare accounting that we consider, the source of this shock does not need to be specified. 15 The compensating variation (as share of initial income) associated with this shock for an individual with income x h who lives in country i is given by (3), where now the term ( x h ˆX) equals zero. Using (4) the aggregate welfare change in country i is now Ŵ i S in (ˆp i ˆp in ). (20) n=1 As is well understood, the aggregate welfare change can be characterized by the distribution of terms-of-trade improvements vis-á-vis each trade partner, ˆp i ˆp in. 16 Totally differentiating (15) we can express these terms of trade improvements relative to exporter n as function of budget shares and aggregate expenditures, ˆp i ˆp in = 1 γ [ds in ds ii (β i β n ) dy]. (21) Combining (21) with (13) and (20) and using the definition of a i from (7) gives the following results. Proposition 2. The aggregate and the individual welfare effects {Ŵi, ˆb } i can be exactly recovered framework. In the empirical analysis in Section 4, this assumption simplifies the estimation because it restricts the number of parameters to be estimated, but it can be relaxed if there is enough variation to estimate asymmetric cross-elasticities. 15 For example, these price changes can be caused by shocks to foreign-country wages or to trade costs. 16 If preferences had the CES form with constant elasticity of substitution σ, so that ˆp i ˆp in = 1 1 σ (Ŝii Ŝin ), then (20) would readily yield the well-known expression Ŵ i CES = 1 Ŝii, which corresponds to equation (1) in 1 σ Arkolakis et al. (2012). 11

13 using the model parameters {γ, {β n }}, the levels and changes in aggregate trade shares, {{S in }, {ds in }}, and the adjusted real income y i. The aggregate welfare effect is Ŵ i ŴH,i + ŴNH,i (22) where ( ) Ŵ H,i = 1 S in ds in ds ii, γ n=1 (23) Ŵ NH,i = 1 ( βi γ ) i dyi. (24) The slope in the individual effect is ˆbi = 1 γ ( σ 2 β dy i d β i ), (25) where the change in the adjusted real income is dy i = I n=1 S inds in ds ii y i d β i β i β i + γ σβ 2y. (26) i This proposition presents a closed-form characterization of the welfare effects of a foreign-trade shock that includes three novel margins. First, preferences are non-homothetic with good-specific income elasticities. Second, the formulas accommodate within-country inequality through the Theil index of expenditure distribution Σ i, which enters through of y i. Third, and key for our purposes, the proposition characterizes the welfare change experienced by individuals at each income level through ˆb i (see 12), so that the entire distribution of welfare changes can be recovered. The aggregate welfare change, Ŵi, includes an aggregate homothetic part ŴH,i that is independent from the β i s and an aggregate non-homothetic component, ŴNH,i, which adjusts for the country s pattern of specialization in high- or low- income elastic goods and the change in adjusted real income. Assuming that γ > 0, which implies elastic demand, the richer or the more unequal country i becomes (the higher dy i is), the larger the aggregate non-homothetic term is when the country is relatively specialized in high income elastic goods (β i > β i ), and the more the aggregate welfare effect is understated by the homothetic term. Equation (25) is the application of equation (13) to the Armington framework. As we have established, when ˆb i > 0 the price changes favor low-income consumers. To understand expression (25), we note that changes in import shares reflect both changes in relative prices and in the aggregate real income of the importing country. Suppose that we observe d β i > 0, which means that aggregate trade shares have moved towards high-β exporters. This is the case, for example, if the U.S. exports goods that are mostly consumed by rich consumers and the importing country has increased its imports from the U.S. In this circumstance, if γ > 0 and the aggregate real income of the economy stayed constant (dy i = 0), then such bias must reflect a reduction in the relative 12

14 price of imports from the U.S., implying a positive welfare impact on sufficiently rich consumers (ˆbi < 0). However, the increase in imports from the U.S. captured by d β i > 0 may also reflect an increase in aggregate real income, dy i > 0. Hence, the change in the aggregate beta is adjusted by the change in adjusted real income to infer the bias in relative price changes. 17 It is worth contrasting these results with the CES and the (homothetic) translog preferences as these are common demand systems used in international trade. The main contrast is, of course, that with CES or translog preferences, as with any demand system in which the distribution of expenditure shares does not vary with individual income, the individual tastes effect is absent ( ψ h = 0). For the same reason, the aggregate welfare effect with either CES or translog preferences does not include a dependence on aggregate real income, while such dependence is captured in our context by the aggregate non-homothetic component, ŴNH,i. In turn, when non-homotheticities are shut down, the aggregate welfare effect Ŵi in our context collapses to ŴH,i, which corresponds to the aggregate gains with translog demand. 18 This term includes the entire distribution of levels and changes in expenditure shares, {S in, ds in }, while with CES preferences the aggregate gains only depend on the own trade share, S ii Exact Welfare Changes The results in Proposition 2 express changes in individual welfare as the compensating variation of a consumer (relative to initial income) that corresponds to an infinitesimal change in prices caused by an arbitrary trade shock, and can be readily applied to obtain first-order approximations to exact welfare changes. In this subsection, we use these results to measure exact welfare changes that correspond to discrete changes in the vector of prices. Consider two scenarios, A and B, with associated distributions of prices { p A i, } pb i and aggregate trade shares { S A i, } SB i. Integrating (3), we obtain the exact compensating variation experienced by an individual with expenditure level x h in country i when conditions change from A to B: wh B wh A = ( W B i W A i ) ( xh x i ) ln ( b B i b A i ). (27) 17 Suppose, for example, that σβdy 2 i > d β i > 0. The second inequality means that aggregate expenditure shares have become more biased toward high-income elastic goods, but the first one indicates that such bias was caused by an increase in income and that the relative price of imports from the U.S. actually increased. In this case, even though we observe d β i > 0, the distribution of price changes favors low-income consumers, ˆb i > These results hold under perfect competition. Feenstra and Weinstein (2010) measures the aggregate gains from trade in the U.S. under translog preferences stemming from competitive effects, and Arkolakis et al. (2010) study the aggregate gains from trade with competitive effects under homothetic translog demand and Pareto distribution of productivity. The non-homothetic AIDS that we study nests the demand system studied in these papers in the case that β n = 0 for all n, but we abstract from competitive effects. 19 We note, however, that the terms Ŵ i CES and ŴH,i are approximately the same if budget shares are not too dispersed. In that case, the homothetic-symmetric AIDS becomes equivalent to the demand generated by CES preferences with demand elasticity σ = 1 + γi. To see why, note that the aggregate budget share with CES demand is S in = e An p 1 σ in for some constant A n so that for two exporters j, k such that S ij S ik it predicts S ij/s ik 1 ln (S ij/s ik ) = A j A k (σ 1) ln (p ik /p ij). This implies S ij S ik S ik [A j A k (σ 1) ln (p ij/p ik )], while from (15), S ij S ik = (α j α k ) γ ln (p ij/p ik ). Therefore, in this situation, CES demand is equivalent to AIDS by setting A j = α j/s ik, A k = α k /S ik, and σ 1 = γ/s ik. 13

15 The term w B h /wa h is the fraction of the initial expenditure level x h that, if given to an individual with that expenditure level in the scenario A, would leave her indifferent with moving to scenario B. Measuring this exact individual-level change in welfare requires the change in the non-homothetic price index between the two scenarios, b B i /ba i, and in the aggregate welfare effect, W B i /Wi A. By construction, the latter equals the welfare change of the representative consumer in country i, which from (22) can be expressed as W B i W A i ( ) ( ) W B H,i W B NH,i =. (28) W A H,i W A NH,i Th aggregate welfare change is the product of the homothetic and non-homothetic welfare terms. The former is obtained by integrating (23), 20 W B H,i W A H,i = e 1 γ ( 1 I 2 n=1(sin) B 2 I n=1(sin) A 2) 1 γ (Sii B SA ii), (29) while integrating the non-homothetic component of aggregate welfare in (24) gives W B NH,i W A NH,i = e 1 γ β i(y B i ya i ) 1 γ B A β i dy i. (30) Computing (30) requires knowing the change in the adjusted real income of the economy, yi B yi A. A closed-form solution for the adjusted real income in the new scenario, yi B, can be obtained solving the differential equation in (26): 21 yi B = 1 σβ 2 γ + β i B β i ± ( γ + β B i β i ) 2 2σ 2 β [ (γ + βa i β i ) y A i σ2 β 2 ( y A i ) 2 + γ ln ( W B H,i W A H,i )]. (31) The integral in the second term of the exponent of (30) does not have a closed-form characterization, but can be numerically solved between the scenarios A and B using (19) and (26). Finally, as shown in (27), the exact distributional effects are function of the change in the non-homothetic price index. Using (25), the closed-form expression is ln ( b B i b A i ) = 1 [( y B γ i yi A ) σ 2 β ( βb i β i A )], (32) where β B i and β A i are function of trade shares from (19) and the change in aggregate real income follows from (31). 20 If we set the β n s to zero, the exact aggregate welfare change W B i /W A i collapses to W B H,i/W A H,i. The solution for this term in (29) is identical to the exact aggregate welfare change with translog demand in equation (10) of Feenstra and Weinstein (2010) fixing the number of varieties over time (proof of this equivalence is available upon request). 21 The larger root of y B i in (31) must be chosen whenever γ + β A i β i σ 2 βy A A < 0 and the smaller root must be chosen otherwise. See the derivation in the appendix. 14

16 3.3 Allowing for Zeros at the Consumer Level in Counterfactual Scenarios Equations (27) to (31) can be used to make either ex post evaluations of the distribution of welfare changes (for an observed change in trade shares) or ex ante evaluations (for a counterfactual change in trade shares). Below, we explore welfare changes corresponding to movement to autarky as well as an ex post analysis over the the period { } We face the possibility that the restriction that expenditure shares s B in,h are non-negative binds for some consumers in scenario B. That is, it is possible that the demand system defined in (16) predicts a negative share for consumer h in some goods. We follow the approach in Feenstra (2010) and treat these goods as not consumed. This amounts to setting their price equal to the reservation value at which the expenditure share predicted by (16) equals zero. Feeding these reservation prices into the demand system delivers a set p B i,h of consumer-h specific prices that { } ensures expenditure shares s B in,h lie in [0, 1] for all goods and all consumers. Even though the actual prices faced by all consumers in scenario B are a common price vector, p B i, the welfare and the expenditure shares of consumer h in scenario B can be calculated using the counterfactual consumer-h specific prices p B i,h. We refer to pb i,h as the effective prices faced by consumer h in scenario B. 22 To measure the welfare change of a consumer { h who } has an initial expenditure { } level equal to x h and whose expenditure shares vary between s A in,h in scenario A and s B in,h in scenario B (with the restriction s B in,h 0 possibly binding in some goods), it suffices to set the counterfactual aggregate shares in scenario B equal to S B in,h = sb in,h β n ln ( xh x i ). (33) The fictitious aggregate shares Sin,h B are generated, by construction, using the set of effective prices p B i,h at which consumer h chooses the distribution of expenditure shares in scenario B, { } s B in,h. 23 Therefore, to obtain the welfare change of consumer h associated with the individual { } { } change in expenditure shares from s A in,h to s B in,h, we use use the shares defined in (33) in place of the share { Sin} B in the equations (27) to (31). Following this procedure, we find the consumer-h 22 The restriction that shares must be non-negative necessarily binds in autarky. For example, if (16) predicts that consumer h has expenditure share s in,h = 0 in a good with β n < 0, then the non-negative constraint binds in good n for every consumer h richer than h. Therefore, in autarky the effective prices must vary across consumers, even though actual prices are the same for all consumers. These actual prices in autarky in a country i with β i < 0 can be calculated from (16) by setting the prices {p in} such that the richest consumer in the economy has expenditure share equal to 1 in good i and equal to 0 in every good n i with β n 0. The prices of the remaining goods are found where the lowest-income consumer in the economy has expenditure share equal to Note that the shares S B in,h do not necessarily lie in [0, 1], as they formally correspond to the shares predicted by the demand system (15) when individual shares are not restricted to lie in [0, 1] and the prices are such that individual h chooses the expenditure shares { s B in,h}. We simply use these aggregate shares to read off the effective prices for household h. 15

17 effective change in the non-homothetic price index: ( ) b B i,h ln = 1 γ b A i,h [( y B i,h ya i + ln ( xh x i )) σ 2 β ( βb i,h β A i ) ], (34) where y B i,h results from evaluating (31) at the shares defined in (33) in place of the shares { S B in}, and where β B i,h = n β ns B in,h.24 4 Empirical Application 4.1 Non-Homothetic Gravity Equation We use the model from Section 3 to implement the empirical exercise. To measure the unequal gains from trade we first need to estimate the parameters of the demand system. The key parameters are the elasticity of substitution γ across exporters and the income elasticity of the good supplied by each exporter, {β n }. We estimate these parameters using the gravity equation generated by the model. Let X in be the value of exports from exporter n to importer i and let Y i be the total expenditures of the importer i. Combining (15) and the definition of y i gives where [ ( )] [ ( ) ] X in τin p n xi = α n γ ln + β n ln + Σ i, (35) Y i τ i p a (p i ) ln τ i = 1 I ln p = 1 I ln (τ in ), n=1 ln (p n ). n=1 In equilibrium, total income of each exporter n must equal sales, Y n = I i=1 X in. Using this condition we can solve for the first term in square brackets in (35). Letting Y W = I i=1 Y i stand for world income, we can express import shares in country i in gravity form, X in Y i = Y n Y W γt in + β n Ω i, (36) 24 Of course, for consumers such that the restriction s B in,h 0 does not bind for any n, the fictitious aggregate shares coincide with the actual ones, S B in,h = S B in, and the effective prices for consumer h coincide with the actual prices, p B i,h = p B i. In that case, the equations (27) to (31) remain unchanged. 16

18 where T in = ln Ω i = [ ln ( τin τ i ( xi a i ) j=1 ( Yj ) + Σ i ] Y W j=1 ) ln ( Yj ( τjn Y W τ j ), (37) ) [ ln ( xj a j ) + Σ j ]. (38) The first two terms in the right-hand side of (36) are standard gravity terms. They capture relative market size of the exporter, bilateral trade costs, and multilateral resistance through trade costs relative to third countries. The last term, β n Ω i,, is the non-homothetic component of the gravity equation, which includes the good-specific Engel curves that are needed to measure the unequal gains from trade across consumers. The larger Ω i is either because average income or inequality in the importing country i is high relative to the rest of the world the higher is the import share from exporter n in importer i when n is specialized in high income elastic goods (β n > 0). 4.2 Data and Empirical Implementation To implement the gravity equation specified in (36) we require bilateral trade flows and production data. We use the TradeProd database published by the Centre d Etudes Prospectives et d Informations Internationales (CEPII). These data merge bilateral trade from UN Comtrade with annual production data from UNIDO. The data are cleaned and harmonized across 26 industrial ISIC (revision 2) sectors for a large number of countries, and we use 1994 as the baseline year to implement our analysis since the coverage of the production data is broadest that year. 25 merge the bilateral and production information with CEPII s Gravity database to obtain bilateral distance and other gravity measures. Income per capita and population come from the World Bank and we obtain gini coefficients from the World Income Inequality Database (Version 2.0c, 2008) published by the World Institute for Development Research. 26 Introducing an additive error to the gravity equation gives the following estimating equation: S in Y n Y W = γt in + β n Ω i + ɛ in (39) In the theory, S in measures the exporter s share in country i s expenditures. We Since TradeProd only records manufacturing activity, empirically this term captures the share of manufacturing expenditure in country i on manufacturing exports from n. Similarly, we use country n s share in worldwide manufacturing expenditure to construct Yn Y W. The term T in captures bilateral trade costs between exporter n and importer i relative to the 25 We refer the reader to CEPII s website for detailed information about the database. In this database, spending on domestic goods is total production less total exports. 26 Since annual ginis are not available for all countries, we take the average gini between Data for Iran is not contained in the database, so we obtain its gini from the World Development Indicators database. 17