Gains from Trade: The Role of Composition

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1 Gains from Trade: The Role of Composition Wyatt Brooks University of Notre Dame Pau Pujolas McMaster University February, 2015 Abstract In this paper we use production and trade data to measure gains from trade in a nonhomothetic model with sector-level heterogeneity in demand and supply elasticities in such a way that is directly comparable to the results from Arkolakis, Costinot and Rodriguez-Clare (2012). We provide an explicit formula for the marginal welfare gains from trade (the increase in real income from a small increase in import penetration), and show that, unlike in CES environments, gains from trade depend on the composition of trade and production in a country. Using sectorlevel estimates from Caron, Fally and Markusen (2015) and detailed sector-level expenditure data from 69 countries, we then show three important results. First, even though the average marginal trade elasticity is -7.5 (in line with the empirical gravity literature), the marginal gains from trade vary considerably across countries depending on their sectoral composition and import penetration. Second, the marginal welfare gains from trade are decreasing in import penetration both in the cross-section of countries, and as countries increase their trade costs. This fact implies a third result: total gains from trade (the increase in real income from autarky to observed trade) are biased significantly downward if one were to incorrectly assume that the trade elasticity was constant, as in the CES case. For the United States, gains from trade in the non-homothetic model are three times larger than in the CES model (2.2 vs. 0.7). This demonstrates that sector-level trade elasticities and taking into account differences in sectoral composition across countries provides significantly different predictions for gains from trade than do CES models. Keywords: Gains from Trade, Sectoral Composition, Non-Homotheticities JEL Codes: F10, F12, F13, F14 1

2 1 Introduction General equilibrium CES trade models, such as Krugman (1980), Eaton and Kortum (2002) and Melitz (2003), imply gains from trade that depend only on aggregate trade volumes and not on industrial composition. This is because, as Arkolakis, Costinot and Rodriguez-Clare (2012) points out (hereafter referred to as ACR), gains from trade depend only on import penetration (the ratio of imports to gross output) and a single trade elasticity. In CES models, the trade elasticity is a constant and is, in particular, independent of industrial composition. However, there is a large part of the empirical trade literature that considers sector-level heterogeneity in supply and demand elasticities, such as Bergstrand (1980), Feenstra (1993), Soderbery (2014), and Carol, Fally and Markusen (2015). In these models, instead of having a single, aggregate trade elasticity there are elasticities in each sector that may not be equal. In this paper, we show how to use these sector-level elasticity estimates from Caron, Fally and Markusen (2015), which will hereafter be referred to as CFM, in a very tractable way to measure gains from trade. Here we use a special case of the results developed in Brooks and Pujolas (2014) to compute trade and welfare elasticities in closed form in non-homothetic trade models. In particular, the formula derived for the marginal gains from trade ɛ W (that is, the increase in real income from a small increase in import penetration) is: 1 ɛ W = k K [ (θ k + 1 σ k ) X 01k X 00k X 01 X 00 X 00k +X 01k X 00+X 01 ( ] ( ) + (1 σ k ) X 01k k K ( k K X 01 k K σ k X 00k X 00 ) σ k X 00k + X 01k X 00 + X 01 ) In this formula, K is the set of sectors, {θ k } are the sector-level supply elasticities, {σ k } are the sector-level demand elasticities, X 00 terms are home country expenditures (with k is at the sector level) and X 01 are import expenditures. Although the formula is complicated, notice that it only depends on sector-level elasticities and expenditure shares. In this paper we apply this formula to sector-level data from 69 countries and combine it with the sector-level elasticity estimates from CFM to obtain model-implied gains from trade in this non-homothetic model. Importantly, if in the model k, σ k = σ and k, θ k = θ, then this would be a CES model and the ACR formula would apply. Moreover, as in Brooks and Pujolas (2014) we also have a closed form for the trade elasticity (the marginal increase in import penetration for a small change in trade costs), which is the 2

3 commonly measured elasticity from the empirical literature. It is equal to the reciprocal of the welfare elasticity given above. With both an easily measurable trade elasticity and welfare elasticity from the model, we can directly compare our results to those from ACR. Doing so yields a number of interesting results. Our first and most basic exercise is to simply input data from 69 countries into the formula given above (together with the CFM estimates) to compute their implied trade elasticities. The results are compared to country-level import penetration in Figure 1. The average trade elasticity is equal to -7.5, and if the model was CES, Figure 1 would be a flat line at However, the results displayed there differ in two important ways. First, there is considerable variation. Trade elasticities have a standard deviation of 2.9, and range from -3.2 for Peru to for Botswana. Second, they are decreasing in import penetration with a correlation of This implies that the gains from trade at the margin for the most open countries (e.g., Singapore, Malaysia or Ireland) are much smaller than for closed countries (e.g., Peru, the United States and Argentina). Again, CES models predict that these marginal gains (increase in real income from a small increase in import penetration) are the same. Our second main exercise is to compute the effects of a large change in trade costs. Following the literature (such as ACR) our exercise is to compare the economy in autarky to one operating at the observed level of trade. The analytical formula derived above is only accurate at the margin, so in this part we must calibrate the model and simulate the change in trade. The important result we find here is that the negative correlation between the trade elasticity and import penetration is true also as countries change their trade costs. That is, as trade costs increase and import penetration decline, the marginal trade elasticity also increases. This implies that the gains from trade moving from autarky to observed trade is substantially higher than without this effect (that is, if the trade elasticity were a constant). To summarize, the results of this paper imply that using sector-level estimates of trade elasticities has import insights for gains from trade. First, that composition and country characteristics (such as patterns of production and trade) can indeed substantially affect gains from trade. Second, they imply that incorrectly assuming that marginal trade elasticities are constant introduces important, downward bias into the measurement of gains from large changes in trade costs. Both of these extend our understanding of the determinants of gains from trade in ways that are impossible in usual CES models. 3

4 2 Model The model follows directly from CFM, which allows us to use their elasticity estimates to measure gains from trade with some minor modification that does not affect their estimation strategy. 1 It is a standard Eaton and Kortum (2002) model, but with sector-level heterogeneity in both the consumer s problem, and on the production side. Specifically, we consider a static model with two asymmetric countries, labeled 0 and 1. We interpret country 0 as our country of analysis and 1 as the rest of the world. Households in country j {0, 1} have non-homothetic preferences. In particular, they consume goods from K different sectors, each of which has its own income elasticity. Therefore, as a country s real income changes, they shift their consumption profile from low income elasticity goods to high income elasticity goods. The representative household in country j solves the following problem: max {Q jk } k K α k Q 1 1/σ k jk s.t. : k K P jk Q jk w j For all k, σ k > 1. Labor is inelastically supplied in each country, and there is a single wage rate w j. There are a measure of L j such households in each country, but they cannot be aggregated due to non-homotheticities. We assume that the wage in country 0, our country of analysis, is numeraire. The fact that there is heterogeneity in σ k means that the model cannot, in general, be solved in closed form. 2 Measurement of gains from trade, and its variation across countries, depends crucially on this heterogeneity. The production side of the economy follows Eaton and Kortum (2002) within each sector. Specifically, in each country-sector pair there is a competitive final goods sector. The final good in each sector is produced using intermediate goods from the set M jk, which has measure 1. Each country produces each intermediate good, though each with a different destinationspecific price so that country n sells good i M jk for price p n jk (i). The final goods producers then purchase each good from the country selling it to them at the lowest price. Hence, the final good producer in country j and sector k buys good i for price: p jk (i) = min{p 0 jk(i), p 1 jk(i)} 1 To be exact, we consider what they refer to as the theta-driven version of their model. 2 A notable exception is when K = {1, 2} and σ 1 = 2σ 2, so that the household s problem can be solved with the quadratic formula. 4

5 These intermediates are aggregated with constant elasticity of substitution ξ k into the final good in each sector. The final good is sold competitively for price P jk. Therefore the final goods producers minimize costs as follows: 3 P jk = min p jk (i)q jk (i) {q jk (i)} i M jk s.t. : 1 = ( i M jk q jk (i) 1 1/ξ k ) ξ k ξ k 1 The competitive intermediate goods producers of i M jk in country n draw productivity zjk n (i) from distribution F k n, with each such competitor receiving the same draw. As the notation indicates, the productivity distributions will differ by country and sector. Following Eaton and Kortum (2002) we assume that F n k is Frechet with cumulative density: F n k (z) = e Tnz θ k Here, T n controls the average productivity of firms in country n, and in the quantitative exercise discussed in later sections, it will be tightly linked to per capita output at the country level. The θ k parameter determines the shape of the distribution and, in particular, how different the draws in different countries are from one another. Hence, it determines comparative advantage and, as will be shown later, is the crucial determinant of gains from trade on the production side. These intermediate goods producers then use labor as the only factor of production to produce their intermediate good. They face an iceberg trade cost so that they must produce τjk n to sell one unit of their product in the country j. We assume that τ nk n =1. Therefore, the price they charge in each country is the solution to the following cost minimization problem: p n jk(i)q n jk(i) = min τ jkw n n ljk(i) n {ljk n (i)} s.t. : q n jk(i) = z n jk(i)l n jk(i) 3 Here we write their problem where their objective is to produce one unit. Since their production function is constant returns to scale, this is without loss of generality. 5

6 Following the derivation in Eaton and Kortum (2002), this implies that: ( ) 1 θk ξ + 1 ξ k k 1 (T0 P jk = Γ (w 0 τjk) 0 θ k + T 1 (w 1 τjk) 1 θ ) 1/θk k θ k Also, the above assumption on the productivity distribution across intermediate goods allows one to solve for imports and consumption out of domestic production at the sector level. First we define the following notation: X jnk = p n jk(i)qjk(i) n i M jk X jn = k K X jnk λ jn = X jn m {0,1} X jm Then we can write: X jnk X j0k + X j1k = T n (w n τ n jk ) θ k T 0 (w 0 τ 0 jk ) θ k + T1 (w 1 τ 1 jk ) θ k Finally, equilibrium in this economy is given by the solutions to the problems of households and firms in each country, trade balance, and the following labor market clearing condition in each country: L j = k K n {0,1} i M jk l n jk(i) 3 Trade and Welfare Elasticities To measure gains from trade in this model, we apply the results from Brooks and Pujolas (2014). In that paper, we show how to solve for aggregate trade elasticities in non-homothetic models, and we show how to use that aggregate trade elasticity to solve for gains from trade. In particular, we define the aggregate trade elasticity as: 4 ɛ T = ) λ01 log( λ 00 log(τ) 1 + log(w1) log(τ) This is a real exchange rate-adjusted counterpart of the term commonly estimated at the aggre- 4 Here we assume all changes in trade costs are uniform across sectors. That is, all trade costs have a sector-specific component and a common component, and all comparisons concern changes in the common component. Hence, the definition of τ is j, k, n j, τ n jk = τ τ n jk 6

7 gate level in the gravity literature, where it is a constant. In the model described in Section 2, however, this term is not a constant. It varies as real income, trade costs, or other parameters of the model change. The main result in Brooks and Pujolas (2014) allows us to solve for this function as a closed form of sector-level elasticities in the model, and observed sector-level imports and production. There we show that: ɛ T = k K [ (θ k + 1 σ k ) X 01k X 00k X 01 X 00 X 00k +X 01k X 00+X 01 ( ] ( ) + (1 σ k ) X 01k k K ( k K X 01 k K σ k X 00k X 00 ) σ k X 00k + X 01k X 00 + X 01 This formula shows that if one has sector-level production data, import data, and sector-level elasticity estimates, one can measure the marginal aggregate trade elasticity for any country. In this paper, we will make use of the estimates of {θ k } and {σ k } in CFM and show what they imply for gains from trade. Interestingly, there are many parameters that one does not need in this formula, such as the preference parameters {α k }, the productivity parameters T j or the elasticity of substitution across intermediate goods {ξ k }. The informational content of these parameters is already summarized in the observed expenditure shares. That is, if one were to change any of these parameters, the trade elasticity would change only through changes in expenditures. Therefore, the exact values of those parameters are irrelevant, so long as they match observed expenditure patterns in the data. Besides being able to solve for an aggregate marginal trade elasticity, Brooks and Pujolas (2014) also shows how to solve for the increase in real income associated with a marginal increase in imports, which we will call marginal gains from trade. The second main result in that paper is similar to that of ACR, but extended to non-homothetic environments: the marginal gains from trade are equal to the reciprocal of the trade elasticity. Therefore, if one uses the formula above to solve for the trade elasticity, that number is sufficient to compute the marginal gains from trade. This can be written as: ) log(w ) log(λ 00 ) = 1 ɛ T (τ) These results are sufficient to study small changes in trade costs, but tell us nothing about large changes. For that case, we will focus on one particular case: increasing trade costs infinitely 7

8 so that the economy is in autarky. For these large changes, our results are no longer analytical because expenditure shares are endogenous objects, hence one would need to know how they change (not just their levels) in order to measure welfare gains. However, we can learn something from some basic analytics. For real income W, we know that: Integrating with respect to log(τ): log(w ) log(τ) = 1 log(λ 00 ) ɛ T (τ) log(τ) log ( ) W (τ) W Aut = τ 1 log(λ 00 ) ɛ T (τ) log(τ) dlog(τ) = 1 ɛ T (τ) log(λ 00) τ log(λ 00 ) ɛ T (τ) 2 ɛ T log(τ) dlog(τ) The term on the left hand side is the exact increase in the log of real income from moving from autarky to the observed level of trade. The first term on the right hand side is known using the formula given above, and known expenditure levels. The second term is more difficult to compute because one would need to know how expenditure changes as τ changes, which cannot be computed in closed form. However, we can see that the sign of the second term on the right hand side is the same as the sign of the derivative of the trade elasticity. Hence, if we know how the marginal trade elasticity changes, we can at least provide a bound for large changes in trade cost. This observation is important for understanding the results in the next two sections. 4 Gains from Marginal Changes in Trade Costs 4.1 Country and Sector Data To compute the marginal aggregate trade elasticity we need the parameters θ k, σ k, and we need sector-level expenditure on both imports and domestic production. As explained before, we use the elasticity estimates directly from CFM, as the model described above is exactly the model they estimated. 5 For consistency, we use the same dataset that CFM used: the GTAP dataset. This dataset provides comparable expenditure data on 69 countries and 50 sectors. 6 5 Their identification strategy only provides ratios of σ k values. That is: k, η k = σ k σ 1 is known for all k, but not the level of σ 1. We choose σ 1 so that when the average value of ɛ T across countries is -7.5, the average of the estimates from the empirical literature. 6 Following CFM, we drop 7 sectors that are primarily raw materials, and dwellings. 8

9 4.2 Trade Elasticity by Country The results of this calculation are displayed in Figure 1. Each point is a country. The horizontal axis is λ 01, imports as a fraction of household expenditure, and the vertical axis is the marginal aggregate trade elasticity. Several things in this figure are noteworthy. First, values of the marginal trade elasticity are highly dispersed. Recall that this implies considerable dispersion in the marginal gains from trade, which is equal to the reciprocal of the trade elasticity. The average trade elasticity is (by construction), and the standard deviation is Botswana has the lowest marginal trade elasticity (equal to ), and Peru has the highest (equal to -3.24). This implies that the marginal gains from trade for Botswana, which equals 0.06, are less than half as much as for the average, which equals 0.13, while that of Peru is more than double, equal to The second notable feature of these marginal trade elasticities is their obvious relationship with import penetration. Indeed, the relationship between the two is strong, with a correlation of This implies that there is a systematic between marginal gains from trade and import openness in the cross-section of countries. If it s also true for each country as that country s trade costs change, it means that marginal gains from trade systematically understand the gains from large changes in trade costs. As we will discuss further in the next section, this is true. It is also noteworthy that dispersion in marginal trade elasticities implies that the marginal gains from trade are biased downward. That is, by Jensen s Inequality, the fact that the marginal trade elasticity is the reciprocal of the marginal gains from trade, and that the reciprocal is a convex function, implies that the average marginal gains from trade is higher than the marginal gains from trade evaluated at the average trade elasticity. Therefore, if one were to incorrectly assume in the model that the trade elasticity was a constant and equal to the average across countries, then they would have downward biased predictions for the marginal gains from trade. As we will show in the next section, this downward bias which exists at the margin is much greater when considering large gains from trade. 4.3 Decomposition and Comparison of Kenya and Bangladesh 5 Large Changes in Trade Costs For computing marginal gains from trade we were able to derive an analytical formula, which depended only on elasticity estimates and observed expenditure shares. No such formula exists for computing large gains, so for that we must calibrate the model and simulate changes in 9

10 tariffs. We calibrate the model as a single country (country 0) trading with the rest of the world (country 1). As the formula from Section 4 makes clear, we must match expenditure shares on imports and domestic goods exactly by sector. To do this, we use 2K parameters, the preference parameters {α k } and the trade costs {τ k }, to match 2K expenditures: imports by sector as a fraction of total spending, and domestic expenditure by sector as a fraction of total spending. This makes it so the marginal trade elasticity is exactly correct. Because the model is explicitly non-homothetic and has important income effects, we also match country 0 GDP per capita relative to the rest of the world. First we set T 1 = 1 and calibrate T 0 to match country 0 s share of world GDP. Second, we set L 0 and L 1 to match the relative population of country 0 and the rest of the world. Lastly we have {ξ k }, the elasticity of substitution of varieties within sectors. This term enters the marginal trade elasticity only indirectly through changes in prices. Its value restriction is that we require θ k + 1 > ξ k, so that prices are well-defined, and ξ k > 1, so that intermediate goods are substitutes. As k, θ k > 1, both of these restrictions are satisfied with k, ξ k = 2. We perform sensitivity on this value and find that it has no effects on our results. Given the calibrated model, we then increase trade costs infinitely to move the economy into autarky. 7 Table 1 lists the results from this exercise. The column labeled ACR is the result if one were to apply the ACR formula to the output from the model, assuming that the true, constant trade elasticity is -7.5, which is the average marginal trade elasticity in the model. The column labeled Actual is the increase in real income associated with moving the economy from autarky to the observed level of trade. 8 The main result to notice from this exercise is that gains from trade with large changes in trade costs are substantially higher than those implied by the marginal gains. For example, the United States has a marginal trade elasticity (evaluated at observed trade) that is less than one half of the average, implying a marginal gain from trade of 0.27 compared to 0.13 on average. This means that the US has twice as high gains from trade at the margin as the ACR formula predicts. However, as Table 1 shows, the gains from trade for this large change in trade costs is substantially larger than that: 2.2% compared to 0.7%, or three times larger. This is a general pattern across countries: their gains from trade for a large change in trade cost is bigger than 7 We do this particular exercise for comparability to the existing literature, such as ACR. Given that the model is calibrated, we could perform other exercises. 8 Because the model is non-homothetic, there is no perfect price index. Hence, we cannot use the change in the price level as our measure of real income. Instead, we use compensating variation: how much more real income one would need to be indifferent between the economy in autarky and that with observed levels of trade. In the homothetic case, these two measures coincide. 10

11 those at the margin. When comparing to the usual ACR formula, this means there are two effects that determine if the implied gains from trade for any particular country are higher or lower. First is the country s marginal trade elasticity. Countries with low marginal trade elasticities have low gains from trade in the non-homothetic model, but have the same, average trade elasticity applied to them in the ACR formula, which is higher than their true elasticity. For such countries, this means that, at the margin, the ACR formula always predicts higher gains from trade than the non-homothetic model. The opposite is true for countries with high marginal trade elasticities. However, for all countries the marginal gains from trade increase as the country moves toward autarky. This is exemplified by Figure 2, which shows the gains from trade for the United States as the country moves toward autarky. As they approach autarky, the marginal trade elasticity gets larger and larger, from the value of at the observed level of trade it rises to close to autarky. As the formula discussed in Section 3 makes clear, the fact that the elasticity increases as trade costs increase explains this pattern. Therefore, there are two sources of bias in evaluating gains from trade in this model: heterogeneity in marginal trade elasticities, and the fact that marginal trade elasticities are increasing in trade costs. Both of these mean that if one were to incorrectly assume in the model that the average marginal trade elasticity were a constant (both across countries and as parameters change), gains from trade measures are biased downward. 6 Conclusion In this paper we use the results of Brooks and Pujolas (2014), data on production and imports, and sector-level elasticity estimates from CFM to measure how gains from trade vary across countries. We find a stark result, which is that gains from trade decline as import penetration rises. This is in contrast to standard CES models in which the marginal gains from trade are constant, and suggests that aggregate trade elasticities measured at the margin may cause systematically biased predictions for gains from trade if there is, in fact, substantial sector-level heterogeneity as found in the empirical literature. 11

12 7 References 1. Arkolakis, C., Costinot, A., and Rodriguez-Clare, A. (2012), New Trade Models, Same Old Gains, American Economic Review, 102(1): Bergstrand, J. (1990), The Hecksher-Ohlin-Samuelson Model, the Linder Hypothesis, and the Determinants of Bilateral Intra-Industry Trade, Economic Journal, 100(403): Brooks, W. and Pujolas, P. (2014), Nonlinear Gravity, Unpublished manuscript. 4. Caron, J., Fally T., and Markusen J. (2014), International Trade Puzzles: A Solution Linking Production and Preferences, Quarterly Journal of Economics, 129(3): Eaton, J. and Kortum, S. (2002), Technology, Geography and Trade, Econometrica, 70(5): Feenstra, R. (1994), New Product Varieties and the Measurement of International Prices, American Economic Review, 84(1): Krugman, P. (1980), Scale Economies, Product Differentiation and the Patterns of Trade, American Economic Review, 70(5): Melitz, M. (2003), The Impact of Trade on Aggregate Industry Productivity and Intra- Industry Reallocation, Econometrica, 71(6), Soderbery, A. (2014), Estimating Import Supply and Demand Elasticities: Analysis and Implications, Journal of International Economics, forthcoming. 12

13 8 Tables and Figures Figure 1: Elasticities Across Countries Vertical axis is the marginal trade elasticity for each country. Average elasticity is

14 Figure 2: Varying Openness for the US Dashed line is increase in real income given by the model; solid line is increase in real income given by the ACR formula. 14

15 Table 1: Gains from trade by country Actual is the percentage increase in real income going from autarky to observed trade. ACR is the gains from trade implied by the ACR formula with a constant elasticity equal to -7.5, the average of the marginal elasticities. 15

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