International Trade The Gravity Equation and its Applications

Transcription

1 International Trade The Gravity Equation and its Applications Prof. Dr. Tobias Seidel University of Duisburg-Essen April 2014 Background Reading: Feenstra (2004) - Chapter 5

2 What Heckscher-Ohlin is (not) about Heckscher-Ohlin is about The pattern (not the volume) of trade Differences in structural characteristics of economies (like factor endowments) A country s trade with the rest of the world Heckscher-Ohlin is not (or less) about The volume of trade The relative size of countries Bilateral trade between single country pairs in a multi-country world Explaining these latter things requires different (or enriched) theory

3 3 An empirical observation: Intra-industry trade Measure: Grubel-Lloyd index Γ ij s for each industry (sector) s and countries i and j (Xs ij and Ms ij denote sectoral exports and imports, respectively) ij Γ ij Xs Ms ij s = 1 [0, 1] X ij s + M ij s Measure of total intra-industry trade between i and j Γ ij s = s Xs ij s X s ij + M ij s + M ij s Γ ij s Evidence: Large figures, sometimes well above 50%, in particular for trade between similar advanced countries (EU - US, say) Trade may be driven by things other than comparative advantage of factor endowments

4 Grubel-Lloyd indices of intra-industry trade, Source: WTO World Trade Report 2008

5 5 Outline of the talk In the previous lecture we checked whether factor endowments advantage shape the sectoral trade pattern according to theory A key assumption was constant returns to scale in production The gravity model relaxes that assumption and has proved a theory-grounded workhorse for empirical research 1 What drives the volume of world trade over time? Technology versus policy 2 What determines the geographical structure of trade? Borders, informal barriers to trade 3 What role for institutions, such as the WTO? We will first develop a theoretical framework And then discuss a number of empirical applications

6 Starting point: Models of perfect specialization Trade models which predict that countries specialize in distinct subsets of goods Heckscher-Ohlin framework with a continuum of goods (many more goods than factors) Monopolistic competition and increasing returns to scale Units of analysis are country pairs or dyads The explanandum: Volume of bilateral trade, e.g., exports from country i to country j: X ij (Note: X ij also the f.o.b. (free on board) value of imports of country j from country i); country index i, j = 1,..., C There are C(C 1)/2 independent bilateral trade relations With approximately 225 countries and territories, the potential number of independent trade relations amounts to 25, 200! Explanatory variables: Countries GDP, trade costs (tariffs, transport costs,...)

7 Frictionless trade Assumptions A1 Countries are specialized in distinct subsets of goods A2 Representative consumers have identical, homothetic preferences A3 Countries are identical except for their size A4 Free trade (goods price equalization holds exactly, no trade costs) Perfectly symmetric countries (identical symmetric preferences and technology) Goods prices (k = 1,..., N) can be normalized to unity value = volume

8 The Krugman (1979) model Since the monopolistic competition model is very well suited for this analysis, we briefly review the original Krugman model Love of variety in consumption Consumers demand similar but different varieties of a product Utility is increasing in the number of different varieties, ceteris paribus Increasing returns to scale More costly for a firm to produce different varieties than to specialize on a single variety due to fixed costs of production Gains from trade Trade allows a country to specialize on the production of few varieties while consuming many varieties

9 The Krugman (1979) model (cont d) Suppose there are i = 1,..., N varieties that are endogenously determined There is a fixed number of consumers L sharing love-of-variety preferences N U = v (c i ), v > 0, v < 0. i=1 Given the budget constraint w = N i=1 p ic i, each consumer maximizes utility to get v (c i ) = λp i where λ is the Lagrange multiplier. Totally differentiating the first-order condition yields the elasticity of demand for variety i ( ) v η i = c i v > 0, with dη i /dc i < 0

10 10 The Krugman (1979) model (cont d) On the production side, each firm requires the following labor to produce output y i, L i = α + βy i. This implies average costs AC i = wl i /y i and marginal costs βw. With symmetric countries, we apply the following two equilibrium conditions to solve the model (1) MR = MC (profit maximization) ( p 1 1 ) = wβ η (2) P = AC (free entry drives profits to zero) p = wα y where we have substituted y = Lc. + wβ

11 11 The Krugman (1979) model (cont d) Conditions (1) and (2) provide two equations in two unknowns, p/w and c From dη i /dc i < 0, (1) establishes a positive link between p/w and c (PP-schedule) (2) is simply the firm s average cost curve and thus establishes a negative link (ZZ-schedule) Finally, we can solve for the number of firms (varieties) by applying the labor-market-clearing condition L = N L i = i=1 N (α + βy i ) = N (α + βy) = N (α + βlc) i=1 This delivers N = 1 (α/l) + βc

12 12 The Krugman (1979) model: Trade p/w Z Z P (p/w) 0 (p/w) 1 Z P Z c 1 c 0 c Source: Krugman (1979)

13 13 Frictionless trade: The baby gravity equation Country i s GDP: Y i = N k=1 y k i World GDP: Y w = C N i=1 k=1 y k i = C i=1 Y i Trade is assumed to be balanced Country j s GDP equals its expenditure Country j s share in world expenditure is s j = Y j /Y w Perfect specialization and identical homothetic demand X ij k = sj yk i Simplest form of the gravity equation X ij = N s j yk i = sj Y i = Y j Y i /Y w = s j s i Y w k=1 I.e., bilateral exports from country i to country j are proportional to the product of their GDPs Empirical application: Trade within and outside the OECD (Helpman, 1987, and Debaere, 2005)

14 Theorem (Helpman, 1987) Recall that X ij = X ji = s j s i Y w Total bilateral trade volume X ij + X ji = 2s j s i Y w Let i and j be part of the world region A : Y i + Y j = Y A Denote s ia = Y i /Y A, and s A = Y A /Y w We can rewrite the total bilateral trade volume V A = X ij + X ji V A Y A = s i Y w 2sj Y A = 2 Y j Y w Y i Y A = 2 Y j Y A Y i Y A Y A Since s ia + s ja = 1, we have ( s ia + s ja) 2 = ( s ia ) 2 + ( s ja ) 2 + 2s ja s ia = 1 A theorem follows V A [ Y A = sa 1 (s ia) 2 (s ja) ] 2 Y w = 2sjA s ia s A Bracketed expression is a size dispersion index that we call disp A for future reference

15 15 Size dispersion index We have derived the Theorem for the case of two countries in a region Helpman (1987) shows that it holds for a region A of many countries Then, disp A = 1 ( i A s ia ) 2 The index is maximized for countries of the same relative size 1/N disp A = 1 1/N 2 = 1 1/N = (N 1)/N i A As any country has a share approaching unity, disp A approaches zero The theorem says that the volume of trade in region A is related to the relative size of countries as measured by the dispersion index

16 16 Debaere (2005) - Setup Let us specify the region A as any pair of countries, A = i, j We can rewrite above theorem in natural logs ( X ij + X ji ) ln Y i + Y j = ln ( s i + s j) + ln disp A Debaere runs the following regression ( ) Xt ij + Xt ji ln Yt i + Yt j ( ) = α ij + γ ln st i + st j + β ln dispt A, where α ij is a dyadic fixed effect Econometric issues If st i + st j were constant over time, α ij would capture their information Debaere measures GDP in nominal USD and in PPP GDP can be instrumented by population size

17 Debaere (2005) - Results Note258that the dispersion index shows up as similarity P. Debaere / Journal of International Economics 66 (2005) Table 1 Eq. (2a) in logs with varying shares panel data Fixed effects regression with time-specific effects Dependent variable OECD Non-OECD lnvt/y lnvt/y lnvt/y lnvt/y lnvt/y lnvt/y lnvt/y lnvt/y Penn IMF Penn IMF Penn IMF Penn IMF ln(similarity) S.E. 0.11* 0.056* ln(world share) S.E. 0.13* 0.05* 0.95* 0.22* ln(similarity) IV S.E. 0.06* 0.06* 0.6* 0.26* ln(world share) IV S.E. 0.25* 0.22* * R Observ Standard errors under the estimated coefficients. * Significant at 95%. 17 not only affects the dependent variable in the regression, but also all the regressors that are a function of countries GDP (and thus, of their trade), there is potentially an endogeneity problem. As discussed, I therefore instrument for the measures e ij and sim ij Source: Debaere (2005)

18 18 Debaere (2005) - Results 1 Theoretical prediction that β = 1 has a chance to be met only in the OECD subsample ( ) 2 However, regardless of the exact specification (including ln st i + st j or not) and regardless of the data (IMF versus PWT), β > 0 for the OECD countries 3 For non-oecd countries, estimates of β either have the wrong sign, or are insignificant = Bottom line: The gravity model works for OECD countries, but less so for developing countries. This is in line with intuition

19 19 Estimating border effects (McCallum, 1995) A second interesting application of this free-trade gravity equation compares intranational trade with international trade between Canada and the US What is the effect of the border on Canada-U.S. trade? Sample of bilateral trade relations of Canadian provinces between each other and with US states (trade between US states is not present in McCallum s original regressions)

20 20 Estimating border effects (McCallum, 1995) McCallum estimates a variant of the gravity equation using data for 1988 ln(x ij ) = α + β 1 ln Y i + β 2 ln Y j + γδ ij + ρ ln d ij + ε ij δ ij is an indicator variable that takes value 1 if the trade relation involves Canadian provinces, and value 0 if it involves a Canadian province and a US state. δ ij measures the border effect d ij is the distance between any two provinces or states. It measures distance related trade costs Note that this equation proxies T ij by distance and border, but does not relate those trade costs to prices

21 21 McCallum s (1995) border puzzle Estimates of coefficients to Y i, Y j and d ij are in line with theoretical expectations and of reasonable size Very large coefficient on within-canadian trade in interval [2.75, 3.09] Interpretation: Within-Canadian trade is by a factor [exp (2.75) = , exp (3.09) = ] larger than trade between Canada and the US These extraordinarily high effects remain if Data for 1993 are used (in 1990 NAFTA agreement was signed) Trade between U.S. states are considered as well Border effect puzzle

22 Borders asymmetrically affect countries of different size McCallum s result is biased, since the size of border effects depends on the size of the country in question Example Assume that the US has a GDP 10 times as large as Canada Without border effects (and transport costs), Canada exports 90% of GDP to the US, it sells 10% internally Suppose the border effect reduces cross-border trade by a factor of one-half Then, Canada exports 45% of GDP to the US and trades 55% internally So the border effect reduces external trade by the factor 2 and increases internal trade by the factor 5.5. Total effect: internal trade is 11 times higher than external trade Conversely, in the US, the border effect reduces external trade from 10% of GDP to 5% (factor 2) and increases internal trade from 90% of GDP to 95% (factor 1.06). Total effect: internal trade is 2.1 times higher than external trade

23 23 Towards a gravity equation that allows for trade costs To avoid these huge biases, we need to carefully derive a gravity equation that accounts for trade barriers This introduces additional complications because prices are no longer equalized across countries Patterns of trade are more complex than in the baby gravity equation

24 24 A refresher of the monopolistic competition model Special case: Dixit-Stiglitz preferences (Krugman, 1980) Constant elasticity of substitution (CES) utility function U = N l=1 c (σ 1)/σ l, σ > 1 Elasticity of substitution between products σ equals elasticity of demand if N is large Representative consumer in country j maximizes U subject to her budget constraint Y j = N l=1 p lc l Lagrange function ( L = N l=1 c (σ 1)/σ l + λ Y j ) N p l c l l=1

25 Deriving optimal demand Maximization of L with respect to c l and c k delivers the following first order conditions σ 1 Dividing leads to l=1 c 1/σ l = λp l σ σ 1 σ c 1/σ k = λp k c l ( ) σ pl = c k p k Substituting into the budget constraint yields N ( ) σ pl Y = c k p l = c k pk σ p k Hence, optimal demand is given by c k = p σ k Y N l=1 p1 σ l N l=1 p 1 σ l

26 Simplifying the expression for demand Introduce the exact price index P (the expenditure needed to purchase one unit of utility) N N ( ) 1 σ 1 = c (σ 1)/σ l = c (σ 1)/σ pl N k = c (σ 1)/σ k l=1 1 = c k [ N c k = P = [ N l=1 ( pl p k l=1 ( pl p k N p l c l = l=1 l=1 ) 1 σ ] σ σ 1 ) 1 σ ] σ 1 σ N l=1 p 1 σ l = p σ k p k [ N p 1 σ l l=1 ] σ 1 σ p 1 σ l l=1 [ N This gives rise to demand functions c l = p σ l P 1 σ Y ] σ 1 σ = ( N l=1 l=1 p 1 σ l ( ) 1 σ pl p k ) 1 1 σ

27 Production side Labor is the only factor of production, with conditional labor demand of firm l given by L l (y l ) = α + βy l I.e., the production function exhibits increasing returns to scale y l (L l ) = (L l α)/β y k (L k ) = (L l α)/β y l /y k = (L l α)/(l k α) > L l /L k With wage rate w, the cost function is K l (y l ) = wα + wβy l This cost function is identical across sectors Constant marginal costs MC = wβ Decreasing average costs AC = wα/yl + wβ

28 28 Characterizing the equilibrium Firms behave as monopolists: they set marginal cost (MC) equal to marginal revenue (MR) ( wβ = p l 1 1 ) p l σ w = β σ σ 1 Free entry forces prices to be equal to average costs (AC) (zero profits) p l = wα + wβ p l y l w = α + β Lc l Note that we have substituted y l by Lc l (identical households demand c l units of each variety) These two conditions determine Equilibrium firm size y = α β (σ 1) Consumption per capita c = y/l = α Lβ (σ 1)

29 29 Characterizing the equilibrium (cont d) Labor market clearing L = N l=1 (α + βy l ) = N (α + βy l ) N = L ασ Substituting this into the utility function, we get indirect utility U = ( ) 1 L α σ 1 σ ( σ 1 β ) σ 1 σ Increasing in L: love of variety Variety represents the only source of gains from trade liberalization (increase in L) This is different to Krugman (1979) Decreasing in fixed costs α and marginal costs β

30 30 Shortcomings IO issues No strategic interactions of firms Static model Empirical issues Output per firm/variety is constant no scale effect Firms are symmetric. Selection (if occurring) is unmodeled, all firms are exporters, only margin of adjustment to trade liberalization is number of varieties consumed while number of varieties produced remains unchanged no selection effect But: Scale and selection effects reappear if firms are heterogeneous in their productivity levels (Melitz, 2003)

31 1 Modeling trade costs Iceberg trade costs and demand A2 Iceberg trade costs (Samuelson, 1952): p ij = T ij p i ; T ij 1, T ii = 1 We no longer have p ij = p ii for all i, j = 1,..., C. Therefore need to model demand A4 Specific utility function: Constant elasticity of substitution (CES) Now, we need to distinguish between quantities and values (as we can no longer normalize p ij = 1 for all i, j = 1,..., C Let s denote the quantity of exports from i to j of good k by c ij k. Perfect specialization implies that c ij k is equal to the consumption of good k in country j Assuming that country i produces N i goods (varieties), country js representative consumer has the utility U j = C N i ( ) σ/(1 σ) c ij k, σ > 1 i=1 k=1

32 32 Modeling trade costs Simplifications To simplify, further assume that all exports of country i sell at the same price p ij (c.i.f.) in country j. Then, by symmetry we have c ij k = cij and C U j = N i ( c ij) σ/(1 σ), σ > 1 i=1 The representative consumer in country j maximizes U j subject to her budget constraint Y j = C i=1 Ni p ij c ij. This gives rise to demand functions ( p c ij ij ) σ = (P j ) 1 σ Y j [ C where P j = ( i=1 Ni p ij) ] (1 σ) 1/(1 σ) is country j s overall price index.

33 33 Modeling trade costs Deriving the core gravity equation Total value of exports from country i to j is X ij = N i p ij c ij Substituting from above and using A2, we get X ij = N i Y j ( T ij p i P j ) 1 σ Recall that firm sizes are fixed by parameters Hence, GDP in country i is Y i = N i p i ȳ Combining both equations delivers X ij = Y j Y i (p i ) σ ȳ ( T ij P j ) 1 σ

34 How do we estimate this gravity equation? We will study three approaches to estimate the resulting gravity equation 1 The use of price indexes (Baier & Bergstrand, 2001) 2 The use of estimated border effects (Anderson & van Wincoop, 2003) 3 The use of fixed effects (e.g. Redding & Venables, 2000; Rose & van Wincoop, 2001)

35 Baier & Bergstrand (2001): Dissecting world trade growth Specification issues We have a panel data set with observations for Xt ij variables (T 2) Logarithms and first differences helps eliminate ȳ: and the other ln X ij = ln ( Y j Y i) σ ln p i + (1 σ) ln T ij (1 σ) ln P j Using s i = Y i / ( Y i + Y j) and s j = Y j / ( Y i + Y j), the equation can be rewritten as ln X ij = 2 ln ( Y j + Y i) + ln ( s i s j) σ ln p i + (1 σ) ln T ij (1 σ) ln P j The above equation attributes growth in bilateral trade to income growth, ln ( Y j + Y i), convergence in countries incomes, ln ( s i s j), changes in prices, ln p i and ln P j, and finally, changes in trade costs, ln T ij

36 Baier & Bergstrand (2001): Dissecting world trade growth Data issues To disentangle the different elements in T ij, Baier and Bergstrand introduce two variables Gross c.i.f./f.o.b. factors published by the IMF, meant to measure trade costs different than tariffs (i.e., transportation costs): 1 + a ij Gross tariff rate: 1 + tij Data: Bilateral trade data averages for two points in time: and Log differences are interpreted as growth rates

37 37 Baier & Bergstrand (2001): Data Real trade flows grow by 148 percentage points Table : Statistics for the growth rates and the log-levels of selected variables Source: Baier and Bergstrand (2001)

38 38 Baier & Bergstrand (2001): Results

39 39 Baier & Bergstrand (2001): Results 1 Income variables are the main drivers behind the growth of world trade Mean growth of the sum of the countries real incomes was 1.05 (or 105 percentage points); multiplying this by its coefficient of 2.37 yields a contribution of 2.49 Mean growth in importer income (1.03) has dampening effect on trade growth by a factor of 0.68; the product of 1.03 and 0.68 yields 0.70 The constant is related to the negative of the logarithmic change in world per capita real GDP; combining the intercept estimate (0.05) with the effect from the lagged trade flow ( 0.84 = ) yields 0.79 The estimate of the overall effect of income growth on trade growth is = 1 (100 percentage points); hence, income growth explains 100/148 = 2/3 of the trade growth 2 Tariff reduction explains ( )/148 = 26% of trade growth. Transport cost reductions explain 8% of trade growth 3 Income convergence does not play any substantial role

40 An alternative to using price data: Estimating border effects Using price measures has disadvantages: (i) difficult to compare levels across countries and (ii) unlikely to capture border effect properly So instead of using data to measure prices, we can model the difference between c.i.f prices p ij form f.o.b. prices p i as a function of distance and other factors ln T ij = τ ij + ρ ln d ij + ε ij where d ij is geographical distance and τ ij measures the border effect (legal/monetary system, language, culture,...) and ε ij is a random error We would need to estimate τ ij ; this is tricky: substitution of T ij from our gravity equation leads to a system of non-linear equations estimation of which requires simulation

41 41 Anderson and van Wincoop s (2003) gravity equation Bilateral exports are given by ( ) p X ij ij 1 σ = Y j P j Market-clearing implies that the sum of exports (including domestic sales) equals total expenditure ( ) T ij 1 σ Y j Y i = j = X ij = ( p i) 1 σ j P j Define Y = j Y j, s i = Y i /Y = ( p i) 1 σ j ( Π i ) 1 σ ( ) = T ij 1 σ j P s j j ( T ij P j ) 1 σ s j and

42 Anderson and van Wincoop s (2003) gravity equation Using these equations we get ( p i ) 1 σ s i = (Π i ) 1 σ ( ) T X ij ij 1 σ = Π i P j s i Y j = Y i Y j Y ( T ij Π i P j ) 1 σ This implies ( P j ) 1 σ ( ) T ij 1 σ = s i Π i i Under the assumption T ij = T ji it is clear that Π i = P i.

43 Empirical implementation The estimation strategy is to move the GDP terms to the left side, take logs and substitute transport costs by ln T ij = τ ij + ρ ln d ij + ε ij Dropping the constant term Y w yields ( ) X ij ln Y i Y j = ρ(1 σ) ln d ij +(1 σ)τ ij +ln(p i ) σ 1 +ln(p j ) σ 1 +(1 σ)ɛ ij The multilateral resistance terms can be solved from (P i ) 1 σ = ( ) C T j=1 si ij 1 σ P once we know transport costs j The transport costs, in turn, are obtained from the lnt ij -equation using the estimated value of ρ(1 σ) ln d ij + (1 σ)τ ij which comes from the gravity equation above As both sets of equations have to be used simultaneously, the approach requires customized programming

44 Empirical implementation A&vW work with an indicator 1 δ ij, or unity for trade between the U.S. and Canada, and zero otherwise Introducing γ on this variable, they replace (1 σ)τ ij with γ(1 δ ij ) Let α = ρ(1 σ) Gravity equation becomes ( ) X ij ln Y i Y j = α ln d ij + γ(1 δ ij ) + ln(p i ) σ 1 + ln(p j ) σ 1 + (1 σ)ɛ ij Note that provincial and state GDP terms have their coefficients constraint at unity

45 45 Results Coefficient on the indicator variable is estimated at ˆγ = 1.65 ] Interpretation: exp(τ ij ) = exp [ γ(1 δ ij ) 1 σ For cross-border trade we have δ ij = 0 Taking values for the elasticity of substitution of σ = 5, 10, and 20, we obtain estimates of exp(τ ij ) of 1.5, 1.2, and 1.09, indicating border barriers of between 9% and 50% in terms of their implied effect on price

46 How much more trade within than across the border? Let ( P i ) σ 1 be the multilateral resistance terms in the absence of the border effect (thus using only distance to compute T ij ) Comparing equations with and without border effects X ij X ij = [e ] ( ˆγ(1 δij ) P i P j ) σ 1 P i P j 46 Consider intra-canadian trade (i = j = CAN, δ CAN,CAN = 1) Intra-Canadian trade is 4.3 times larger with border effect than without σ = 5 (but results not sensitive to changes in σ) Intra-U.S. trade is 1.05 times larger with border effects than without Cross-border trade is 0.41 times smaller with border effect than without Thus, intra-canadian trade is 4.3/0.41 = 10.5 times higher than cross-border trade Intra-U.S. trade is 1.05/0.41 = 2.6 times higher than cross-border trade Smaller economies have a much larger impact of the border effects

47 47 A fixed effects procedure Drawback: Requires programming to solve unconstrained minimization problem Alternative procedure: Source and destination country fixed effects With fixed effects ν i, ν j, we may estimate the equation ( ) X ij ln Y i Y j = γ ( 1 δ ij) + α 1 ln d ij + ν i + ν j + u ij This approach yields ˆγ = 1.55 which is close to the 1.65 reported above The implied average border effect is exp(1.55) = 4.7

48 48 Comparing border effects estimates

49 49 Comparing border effects estimates Interestingly, the average border effect of the three approaches is very similar However, as McCallum ignores the price channels, his estimates are inconsistent This appears to have the effect that the border effect for Canada is overstated, while it is understated for the US

50 50 Questions [ 1 Why can we interpret 1 ( s ia) 2 ( + s ja ) ] 2 as a dispersion index? What does Helpman s theorem imply? 2 Why is it important to control for multilateral resistance terms in empirical gravity applications? 3 What are the advantages and disadvantages of the A&vW approach, the fixed effects approach, the Taylor-series approximation approach, and the ratio-of-ratios approach? 4 How does the gravity equation extend to panel data? Do first differencing or fixed effects (within) transformation suffice to control for multilateral resistance terms when T 2? 5 Modify the trade costs specification such that it accounts for common use of a currency! 6 What empirical applications of the gravity approach can you think of?

51 The Eaton-Kortum Model Prof. Dr. Tobias Seidel University of Duisburg-Essen April 2014 Background Reading: Eaton, Kortum (2002), Technology, Geography, and Trade, Econometrica 70,

52 2 What s the problem? Ricardo (1817) provided a mathematical example that countries can mutually benefit by specializing on their goods at which they have a comparative advantage. This changed the view in that also countries that were absolutely better in producing all goods could benefit from trading with technologically inferior countries. The 2x2 model is a standard part of each introductory trade book, but has played no influence in explaining the global pattern of international trade. Until recently! 200 years after its birth, the Ricardian theory experiences a revival. Why???

53 3 Why? Even in the most basic version of the Ricardo model, several equilibria can emerge that need to be analyzed separately. 1 England makes only cloth and Portugal only wine. 2 England makes both cloth and wine and Portugal only wine. 3 England makes only cloth and Portugal both cloth and wine. This is tedious as relative labor demand is kinky. As Eaton and Kortum (2012) put it:... stairways are trouble not only for wheeled vehicles but for comparative statics.

54 4 A continuum of goods... Dornbusch, Fischer, and Samuelson (1977) had the idea of adding a lot of goods to the list - ending up with a continuum. They replaced the stairway with a ramp! This smoothness makes the model very tractable. For example, it is straightforward to introduce trade costs to explain that countries buy more goods from themselves. One limitation remains: There are only two countries.

55 Adding more countries... Technically, we could add a continuum of countries (a finite integer number would cause a stairway again). While it might be justifiable to deal with a continuum of goods, it is less plausible to introduce a continuum of countries especially with respect to empirical work. Here s where Eaton and Kortum (2002) comes in. They turn a messy discreet problem into a tractable, continuous problem. With many goods and many countries, it does not help to construct chains of comparative advantage. Instead, they introduce a probabilistic approach: Labor input requirements a(j) are realizations of a random variable.

56 6 Probabilistic approach This way of thinking about technology has two advantages: 1 Distributions can be smooth (producing our ramp). 2 We do not need to keep track of all individual a(j) s, of which there are many, but only the parameters of which they are drawn, which can be small in number. Let s see how it works in detail...

57 7 Preliminaries Countries have differential access to technology, so efficiency varies across countries and varieties. Country i s efficiency in producing good j [0, 1] denoted by z i (j). Cost of a bundle of inputs c i identical across commodities within a country (mobile factors) for now taken as given. With CRS, the cost of producing good j in country i is c i /z i (j). Geographic barriers introduced by iceberg trade costs. Delivering one unit from country i to country n requires producing d ni units in i. Positive barriers imply d ni > 1 and for all i it is assumed d ii = 1. Also, it is imposed that d ni d nk d ki.

58 8 Preliminaries Delivering an unit of good j produced in country i to country n costs ( ) ci p ni (j) = d ni. z i (j) With perfect competition, p ni is what consumers in n would have to pay for good j imported from i. But consumers would choose to buy the good from the country that offers the lowest price, that is p n (j) = min{p ni (j); i = 1,..., N}, where N is the number of countries.

59 9 Preliminaries Delivering an unit of good j produced in country i to country n costs ( ) ci p ni (j) = d ni. z i (j) With perfect competition, p ni is what consumers in n would have to pay for good j imported from i. But consumers would choose to buy the good from the country that offers the lowest price, that is p n (j) = min{p ni (j); i = 1,..., N}, where N is the number of countries.

60 10 Preliminaries Buyers (final consumers or firms) purchase quantities Q(j) to maximize a CES objective [ 1 U = 0 ] σ Q(j) σ 1 σ 1 σ dj, where σ > 0 represents the elasticity of substitution between goods. Note: DFS (1977) order goods according to z 1 (j)/z 2 (j) where relative wages determine the breakpoint in this chain of comparative advantage. With more than two countries, there is no natural ordering of commodities. Solution: Probabilistic approach!

61 11 Technology Country i s efficiency in producing good j is the realization of a random variable Z i (drawn independently for each j) from its country-specific probability distribution F i (z) = Pr[Z i z] By the law of large numbers, F i (z) captures the fraction of goods for which country i s efficiency is below z. The likelihood that country i supplies a particular good to country n is the probability π ni that i s price turns out to be the lowest. The probability theory of extremes provides a form for F i (z) that yields a simple expression for π ni and the resulting distribution of prices. EK assume Frà c chet (type II extreme value distribution): F i (z) = e T i z θ, where T i > 0 (absolute advantage) and θ > 0 (inverse of variability, comparative advantage).

62 12 Why Frà c chet? Central limit theorem implies (among other things): The highest or lowest value in a large sample drawn from a well-behaved distribution follows an extreme value distribution. If technologies for making a good are the results of inventions that occur over time and if the output per worker delivered by an invention is drawn from a Pareto distribution, then output per worker using the most efficient technology follow a Frà c chet distribution (see Eaton and Kortum, 1997, 1999). Only for Frà c chet does the distribution of prices inherit an extreme value distribution.

63 13 Prices What does this imply for the distribution of prices? Country i presents country n with a distribution of prices G ni (p) = Pr[P ni p] = 1 F i (c i d ni /p) or G ni (p) = 1 e [T i (c i d ni ) θ ]p θ. Hence, the distribution G n (p) for what country n actually buys is G n (p) = 1 Π N i=1 [1 G ni (p)].

64 14 Prices Inserting G ni (p) delivers G n (p) = 1 e Φnpθ, where Φ n = N T i (c i d ni ) θ. i=1 The price parameter Φ n summarizes how (i) states of technology around the world, (ii) input costs around the world, and (iii) geographic barriers govern prices in each country n.

65 15 Prices Three useful properties of price distributions: 1 Probability that country i provides a good at the lowest price in country n given by π ni = T i(c i d ni ) θ Φ n With a continuum of goods, π ni is also the fraction of goods that country n buys from country i. 2 The price of a good that country n actually buys from any country i also has the distribution G n (p). 3 The exact price index for the CES objective (assuming σ < 1 + θ) is p n = γφ 1/θ n, where γ = [ Γ ( θ + 1 σ θ )] 1 1 σ

66 16 A gravity equation A corollary of property (2) is that country n s average expenditure per good does not vary by source. Hence, X ni X n = T i(c i d ni ) θ Φ n = T i (c i d ni ) θ N k=1 T k(c k d nk ) θ, This expression resembles the standard gravity equation! Let Q i be total export sales of i to write Q i = N m=1 X mi = T i c θ i N m=1 d θ mi X m Φ m Solving for T i c θ i and substituting in the above expression delivers X ni = N m=1 ( ) θ dni p Xn n ( dmi p m ) θ Xm Q i.

67 17 Confronting the model with data Model exhibits a direct link between trade flows and price differences: X ni /X n X ii /X i = Φ i Φ n d θ ni = ( pi d ni p n ) θ This equation can be used to estimate θ. But let us first plot normalized import shares against distance (as a crude measure of geographical barriers) based on bilateral trade in manufactures among 19 OECD countries...

68 18 Trade and geography 1752 J. EATON AND S. KORTUM 0.1~ ~~+ x~~ ~ * *. e A,..Ue * ',.,..,... ** 00 v *0 t; t E * *.. * c....v v. 5. * distance (in miles) between countries n and i FIGURE 1.-Trade and geography.

69 19 Identifying θ EK measure the term (p i d ni )/p n by using retail prices for 50 manufactured products in 19 OECD countries. The price measure reflects what the price index in destination n would be for a buyer there who insisted on purchasing everything from source i, relative to the actual price index in n (based on the cheapest source). Insights: (i)the cheapest foreign source is usually nearby and the most expensive far away. (ii) Large countries usually suffer the most if required to buy everything from a given foreign source. From plotting the data, the implied θ is 8.28.

70 20 Figure 2 graphs our measure of normalized import share (in logarithms) against Dni. Observe that, while the scatter is fat, there is an obvious negative relationship, as the theory predicts. The correlation is The relationship in Figure 2 thus confirms the connection between trade and prices predicted by our Price measure 1754 statisticsj. EATON AND S. KORTUM TABLE II PRICE MEASURE STATISTICS Foreign Sources Foreign Destinations Country Minimum Maximum Minimum Maximum Australia (AL) NE (1.44) PO (2.25) BE (1.41) US (2.03) Austria (AS) SW (1.39) NZ (2.16) UK (1.47) JP (1.97) Belgium (BE) GE (1.25) JP (2.02) GE (1.35) SW (1.77) Canada (CA) US (1.58) NZ (2.57) AS (1.57) US (2.14) Denmark (DK) Fl (1.36) PO (2.21) NE (1.48) US (2.41) Finland (FI) SW (1.38) PO (2.61) DK (1.36) US (2.87) France (FR) GE (1.33) NZ (2.42) BE (1.40) JP (2.40) Germany (GE) BE (1.35) NZ (2.28) BE (1.25) US (2.22) Greece (GR) SP (1.61) NZ (2.71) NE (1.48) US (2.27) Italy (IT) FR (1.45) NZ (2.19) AS (1.46) JP (2.10) Japan (JP) BE (1.62) PO (3.25) AL (1.72) US (3.08) Netherlands (NE) GE (1.30) NZ (2.17) DK (1.39) NZ (2.01) New Zealand (NZ) CA (1.60) PO (2.08) AL (1.64) GR (2.71) Norway (NO) Fl (1.45) JP (2.84) SW (1.36) US (2.31) Portugal (PO) BE (1.49) JP (2.56) SP (1.59) JP (3.25) Spain (SP) BE (1.39) JP (2.47) NO (1.51) JP (3.05) Sweden (SW) NO (1.36) US (2.70) FI (1.38) US (2.01) United Kingdom (UK) NE (1.46) JP (2.37) FR (1.52) NZ (2.04) United States (US) FR (1.57) JP (3.08) CA (1.58) SW (2.70) Notes: The price measure Di is defined in equation (13). For destination country n, the minimum Foreign Source is mini#n exp D,i. For source country i, the minimum Foreign Destination is minn7i exp Dni.

71 21 Trade and geography TECHNOLOGY, GEOGRAPHY, AND TRADE X -2 - *- * - X * 0.2E u -8 * *% 0M 0~~~~~~~~~~~~~~~ -10 I E~~~~~ ~FGR 2. Trd an prices price measure: Dni FIGURE 2.-Trade and prices.

72 22 Endogenizing input costs To close the model, (i) the input bundle is decomposed into labor and intermediates. (ii) Then, EK determine prices given wages. (iii) Finally, wages are determined. Cobb-Douglas with labor cost share β. Intermediates comprise the CES-aggregated good with the overall price index as the appropriate index of intermediate goods prices. The cost of an input bundle in country i is thus c i = w β i pp 1 β Note: c i depends on prices in i and hence on Φ i which summarizes input costs in all countries!

73 23 Determining price levels Due to rich interactions of costs in prices via the price index, we obtain a system of equations that generally requires numerical methods: [ N ] ( ) 1/θ θ p n = γ T i d ni w β i pp 1 β (1) i=1 Expanding the trade-shares equation delivers X ni X n = π ni = T i ( ) γd ni w β i p 1 β θ i (2) p n The p i s are determined from the system of equations above. Finally, conditions for the labor market equilibrium are needed to determine wages.

74 Labor market Manufacturing labor income in country i is labor s share of country i s export around the world, including to itself (domestic sales). w i L i = β N π ni X n Denoting aggregate final expenditures as Y n with α the fraction spent on manufactures, total manufacturing expenditures are then n=1 X n = 1 β β w nl n + αy n, where the first term captures demand for manufactures as inputs by the manufacturing sector itself. Also, we have Y n = w n L n + Y O n

75 25 Case 1: Labor is mobile Workers can move freely between manufacturing and non-manufacturing. Wage w n is given by productivity in non-manufacturing and total income Y n is exogenous. Combining total labor income with total manufacturing expenditure delivers N w i L i = π ni [(1 β)w n L n + αβy n ], (3) n=1 determining manufacturing employment L i.

76 Case 2: Labor is immobile Number of manufacturing workers is fixed at L n. Non-manufacturing income Yn O is exogenous. We now get N w i L i = π ni [(1 β + α)w n L n + αβyn O n=1 ], (4) determining manufacturing wages w i.

77 Estimation Equations (1) and (2), along with either (3) or (4), comprise the full general equilibrium. Equation (2) allows us to learn about states of technology T i and geographic barriers d ni. Normalizing by the importer s home sales delivers X ni = T ( ) θβ ( ) θ(1 β) i wi pi d θ ni X nn T n w n p n

78 28 Estimation Further, we can use (2) for both country i and n to get p i = w ( ) 1/θβ ( ) i Ti Xi /X 1/θβ ii p n w n T n X n /X nn Plugging this into the previous equation yields ln X ni X nn = θln(d ni ) + 1 β ln T i T n θln w i w n where X ni lnx ni [(1 β)/β]ln(x i /X ii ).

79 29 Estimation Defining S i 1 β ln(t i) θln(w i ), we arrive at the baseline equation that is estimated by GLS ln X ni X nn = θln(d ni ) + S i S n where we can think of S i as country i s competitiveness, its state of technology adjusted for labor costs. Taking β = 0.21 from the data, the LHS can be directly computed. S i and S n are captured by the coefficients on source-country dummies. Proxies for geographic barriers are distance d k (lying in the kth interval, k = 1,..., 6), border b, language l, trading area e h, an overall destination effect m n. δ ni captures all other factors such that ln(d ni ) = d k + b + l + e h + m n + δ ni

80 Results 1762 J. EATON AND S. KORTUM TABLE III BILATERAL TRADE EQUATION Variable est. s.e. Distance [0, 375) -0d (0.16) Distance [375, 750) -6d (0.11) Distance [750, 1500) -Od (0.10) Distance [1500,3000) -Od (0.16) Distance [3000, 6000) -Od (0.09) Distance [6000, maximum] - Od (0.10) Shared border -Ob 0.30 (0.14) Shared language (0.15) European Community -0e (0.13) EFTA -Oe (0.19) 30 Source Country Destination Country Country est. s.e. est. s.e. Australia S, 0.19 (0.15) -Om, 0.24 (0.27) Austria S (0.12) - m (0.21) Belgium S (0.11) - m (0.19) Canada S (0.14) - m (0.25) Denmark S (0.12) -6m (0.19)

81 Results Shared language (0.15) European Community -0e (0.13) EFTA -Oe (0.19) Source Country Destination Country Country est. s.e. est. s.e. Australia S, 0.19 (0.15) -Om, 0.24 (0.27) Austria S (0.12) - m (0.21) Belgium S (0.11) - m (0.19) Canada S (0.14) - m (0.25) Denmark S (0.12) -6m (0.19) Finland S (0.12) - m (0.22) France S (0.11) -6m (0.19) Germany S (0.12) - m (0.19) Greece S (0.12) -6m (0.20) Italy S1( 1.78 (0.11) -6m (0.19) Japan Sil 4.20 (0.13) -6m (0.22) Netherlands S (0.11) -6m (0.19) New Zealand S (0.15) -Om (0.27) Norway S (0.12) -Om (0.21) Portugal S (0.12) -6m (0.21) Spain S (0.12) -6m (0.19) Sweden S (0.12) -6m (0.22) United Kingdom S (0.12) -6m (0.19) United States S (0.14) -6m (0.25) Total Sum of squares 2937 Error Variance: Sum of squared residuals 71 Two-way (02o2) 0.05 Number of observations 342 One-way (02o-2) Notes: Estimated by generalized least squares using 1990 data. The specification is given in equation (30) of the paper. The parameter are normalized so that E!9 Si = 0 and E19 mn = 0. Standard errors are in parentheses.

82 32 Estimation EK use wage and price data to provide to alternative estimates for θ. They apply θ = 3.60, θ = 8.28, and θ = For each of these values, they derive estimates for T i and geographic barriers. Based on the definition of S i and using wages, one can back out T i. Geographic barriers can be obtained by dividing the coefficients in the results tables above by the respective value of θ and exponentiate. We are now ready to study a number of counterfactuals.

83 33 Counterfactuals Objective: Real GDP Y n /p α n, where the manufacturing sector share is α = Gains from trade: Autarky vs. Status Quo vs. Free Trade 2 How do technology and geography shape the pattern of specialization? 3 How does trade help spreading technological know-how? 4 What are the implications of tariff reductions?

84 TECHNOLOGY, GEOGRAPHY, AND TRADE Raising geographic barriers TABLE IX THE GAINS FROM TRADE: RAISING GEOGRAPHIC BARRIERS Mobile Labor Percentage Change from Baseline to Autarky Immobile Labor Country Welfare Mfg. Prices Mfg. Labor Welfare Mfg. Prices Mfg. Wages Australia Austria Belgium Canada Denmark Finland France Germany Greece Italy Japan Netherlands New Zealand Norway Portugal Spain Sweden United Kingdom United States Notes: All percentage changes are calculated as 100ln(x'/x) where x' is the outcome under autarky (d,j x is the outcome in the baseline. oo for n :A i) and 34 when trade is shut down could be seen as indicating their overall comparative

85 1770 J. EATON AND S. KORTUM 2. Lowering geographic barriers TABLE X THE GAINS FROM TRADE: LOWERING GEOGRAPHIC BARRIERS Baseline to Zero Gravity Percentage Changes in the Case of Mobile Labor Baseline to Doubled Trade Country Welfare Mfg. Prices Mfg. Labor Welfare Mfg. Prices Mfg. Labor Australia Austria Belgium Canada Denmark Finland France Germany Greece Italy Japan Netherlands New Zealand Norway Portugal Spain Sweden United Kingdom United States Notes: All percentage changes are calculated as 1001n(x'/x) where x' is the outcome under lower geographic barriers and x is the outcome in the baseline. 35 Three of the four countries we have identified as "natural manufacturers,"

86 3. Technology vs. geography 1772 J. EATON AND S. KORTUM o! c 0.7- E 0.6 o 0.5 o o / J 0 r ~~~~~~~~~~~~~~~~~~Denm Germany l l l l l l l l l l l l l l l l l l l l l l I (toward autarky) factor increase In geographic barriers (toward zero gravity) FIGURE 3.-Specialization, technology, and geography. 36 technology T1 by 20 percent, first for the United States and then for Germany.

87 37 Technology vs. geography For smaller countries, manufacturing shrinks as geographic barriers diminish. Production shifts to larger countries where inputs are cheaper. As geographic barriers continue to fall, however, the forces of technology take over and manufacturing employment grows.

88 38 Conclusions Eaton and Kortum (2002) develop a model with N countries and a continuum of goods incorporating realistic geographic features into general equilibrium. Their model can explain that trade diminishes dramatically with distance; prices vary across locations, with greater differences between places farther apart; factor rewards are far from equal across countries countries relative productivities vary substantially across industries. Shortcoming: Proper model of labor market. Alvarez and Lucas (2006, JME) add this aspect to the EK-model.