14.581 International Trade Lecture 16: Gravity Models (Theory) – – 14.581 Week 9 Spring 2013 14.581 (Week 9) Gravity Models (Theory) Spring 2013 1 / 44
Today’s Plan 1 The Simplest Gravity Model: Armington 2 Gravity Models and the Gains from Trade: ACR (2012) 3 Beyond ACR’s (2012) Equivalence Result: CR (2013) 14.581 (Week 9) Gravity Models (Theory) Spring 2013 2 / 44
1. The Simplest Gravity Model: Armington 14.581 (Week 9) Gravity Models (Theory) Spring 2013 3 / 44
The Armington Model Image courtesy of rdpeyton on flickr. CC NC-BY-SA 14.581 (Week 9) Gravity Models (Theory) Spring 2013 4 / 44 4
The Armington Model: Equilibrium Labor endowments L i for i = 1 , ... n CES utility ⇒ CES price index 1 − σ P 1 − σ = ∑ n i = 1 ( w i τ ij ) j Bilateral trade fiows follow gravity equation : ( w i τ ij ) 1 − σ X ij = 1 − σ w j L j ∑ n l = 1 ( w l τ lj ) d ln X ij / X jj In what follows ε ≡ − = σ − 1 denotes the trade elasticity d ln τ ij Trade balance ∑ X ji = w j L j i 14.581 (Week 9) Gravity Models (Theory) Spring 2013 5 / 44
The Armington Model: Welfare Analysis Question: Consider a foreign shock: L i → L ; for i = j and τ ij → τ ij ; for i = j. i How do foreign shocks affect real consumption, C j ≡ w j / P j ? Shephard’s Lemma implies d ln C j = d ln w j − d ln P j = − ∑ n = 1 λ ij ( d ln c ij − d ln c jj ) i with c ij ≡ w i τ ij and λ ij ≡ X ij / w j L j . Gravity implies d ln λ ij − d ln λ jj = − ε ( d ln c ij − d ln c jj ) . 14.581 (Week 9) Gravity Models (Theory) Spring 2013 6 / 44
The Armington Model: Welfare Analysis Combining these two equations yields ∑ n = 1 λ ij ( d ln λ ij − d ln λ jj ) i d ln C j = . ε Noting that ∑ i λ ij = 1 = ⇒ ∑ i λ ij d ln λ ij = 0 then d ln λ jj d ln C j = − . ε ˆ = x ; / x ) Integrating the previous expression yields ( x − 1 / ε . ˆ jj ˆ j = λ C 14.581 (Week 9) Gravity Models (Theory) Spring 2013 7 / 44
The Armington Model: Welfare Analysis ˆ jj requires (computer) work In general, predicting λ We can use exact hat algebra as in DEK (Lecture #3) i e Gravity equation + data λ ij , Y j , and ε But predicting how bad would it be to shut down trade is easy... In autarky, λ jj = 1. So A / C j = λ jj 1 / ( σ − 1 ) C j Thus gains from trade can be computed as GT j ≡ 1 − C A = 1 − λ 1 / ε j / C j jj 14.581 (Week 9) Gravity Models (Theory) Spring 2013 8 / 44
The Armington Model: Gains from Trade Suppose that we have estimated trade elasticity using gravity equation Central estimate in the literature is ε = 5 We can then estimate gains from trade: % GT λ jj j Canada 0 . 82 3 . 8 Denmark 0 . 74 5 . 8 France 0 . 86 3 . 0 Portugal 0 . 80 4 . 4 Slovakia 0 . 66 7 . 6 U.S. 0 . 91 1 . 8 14.581 (Week 9) Gravity Models (Theory) Spring 2013 9 / 44
2. Gravity Models and the Gains from Trade: ACR (2012) 14.581 (Week 9) Gravity Models (Theory) Spring 2013 10 / 44
Motivation New Trade Models Micro-level data have lead to new questions in international trade: How many firms export? How large are exporters? How many products do they export? New models highlight new margins of adjustment: From inter-industry to intra-industry to intra-firm reallocations Old question: How large are the gains from trade (GT)? ACR’s question: How do new trade models affect the magnitude of GT? 14.581 (Week 9) Gravity Models (Theory) Spring 2013 11 / 44
ACR’s Main Equivalence Result ACR focus on gravity models PC: Armington and Eaton & Kortum ’02 MC: Krugman ’80 and many variations of Melitz ’03 ˆ = x ; / x ) Within that class, welfare changes are ( x 1 / ε ˆ C = λ ˆ Two suffi cient statistics for welfare analysis are: Share of domestic expenditure, λ ; Trade elasticity, ε Two views on ACR’s result: Optimistic: welfare predictions of Armington model are more robust than you thought Pessimistic: within that class of models, micro-level data do not matter 14.581 (Week 9) Gravity Models (Theory) Spring 2013 12 / 44
Primitive Assumptions Preferences and Endowments CES utility Consumer price index, P 1 − σ = p i ( ω ) 1 − σ d ω , i ω ∈ Ω One factor of production: labor L i ≡ labor endowment in country i w i ≡ wage in country i 14.581 (Week 9) Gravity Models (Theory) Spring 2013 13 / 44
Primitive Assumptions Technology Linear cost function: 1 1 − β β C ij ( ω , t , q ) = qw i τ ij α ij ( ω ) t 1 − σ + w w j ξ ij φ ij ( ω ) m ij ( t ) , ii i , " , ii " variable cost fixed cost q : quantity, τ ij : iceberg transportation cost, α ij ( ω ) : good-specific heterogeneity in variable costs, ξ ij : fixed cost parameter, φ ij ( ω ) : good-specific heterogeneity in fixed costs. 14.581 (Week 9) Gravity Models (Theory) Spring 2013 14 / 44
Primitive Assumptions Technology Linear cost function: 1 β 1 − β C ij ( ω , t , q ) = qw i τ ij α ij ( ω ) t + w w j ξ ij φ ij ( ω ) m ij ( t ) 1 − σ i m ij ( t ) : cost for endogenous destination specific technology choice, t , t ∈ [ t , t ] , m ij > 0 , m ij ≥ 0 ; ;; 14.581 (Week 9) Gravity Models (Theory) Spring 2013 15 / 44
Primitive Assumptions Technology Linear cost function: 1 β 1 − β C ij ( ω , t , q ) = qw i τ ij α ij ( ω ) t 1 − σ + w j ξ ij φ ij ( ω ) m ij ( t ) w i Heterogeneity across goods � � G j ( α 1 , ..., α n , φ 1 , ..., φ ) ≡ ω ∈ Ω | α ij ( ω ) ≤ α i , φ ij ( ω ) ≤ φ i , ∀ i n 14.581 (Week 9) Gravity Models (Theory) Spring 2013 16 / 44
Primitive Assumptions Market Structure Perfect competition Firms can produce any good. No fixed exporting costs. Monopolistic competition Either firms in i can pay w i F i for monopoly power over a random good. Or exogenous measure of firms, N i < N , receive monopoly power. Let N i be the measure of goods that can be produced in i Perfect competition: N i = N Monopolistic competition: N i < N 14.581 (Week 9) Gravity Models (Theory) Spring 2013 17 / 44
Macro-Level Restrictions Trade is Balanced Bilateral trade fiows are X ij = x ij ( ω ) d ω ω ∈ Ω ij ⊂ Ω R1 For any country j, ∑ i = j X ij = ∑ i = j X ji Trivial if perfect competition or β = 0. Non trivial if β > 0. 14.581 (Week 9) Gravity Models (Theory) Spring 2013 18 / 44
Macro-Level Restrictions Profit Share is Constant R2 For any country j, Π j / ( ∑ n = 1 X ji ) is constant i where Π j : aggregate profits gross of entry costs, w j F j , (if any) Trivial under perfect competition. Direct from Dixit-Stiglitz preferences in Krugman (1980). Non-trivial in more general environments. 14.581 (Week 9) Gravity Models (Theory) Spring 2013 19 / 44
Macro-Level Restriction CES Import Demand System Import demand system ( w , N , τ ) → X R3 � i ; � ε < = = 0 i j ii ; ε j ≡ ∂ ln ( X ij / X jj ) ∂ ln τ i ; j = otherwise 0 Note: symmetry and separability. � � 14.581 (Week 9) Gravity Models (Theory) Spring 2013 / 44
Macro-Level Restriction CES Import Demand System The trade elasticity ε is an upper-level elasticity: it combines x ij ( ω ) ( intensive margin ) Ω ij ( extensive margin ). ⇒ complete specialization. R3 = R1-R3 are not necessarily independent If β = 0 then R3 = ⇒ R2. 14.581 (Week 9) Gravity Models (Theory) Spring 2013 21 / 44
Macro-Level Restriction Strong CES Import Demand System (AKA Gravity) R3’ The IDS satisfies χ ij · M i · ( w i τ ij ) ε · Y j X ij = ∑ n ε · M i · ( w i ; j ) ; = 1 χ i ; τ i ; j ; i where χ ij is independent of ( w , M , τ ) . ; as R3 but, but additional structural Same restriction on ε ii j relationships 14.581 (Week 9) Gravity Models (Theory) Spring 2013 22 / 44
Welfare results State of the world economy: Z ≡ ( L , τ , ξ ) Foreign shocks : a change from Z to Z ; with no domestic change. 14.581 (Week 9) Gravity Models (Theory) Spring 2013 23 / 44
Equivalence (I) Proposition 1: Suppose that R1-R3 hold. Then 1 / ε . � I jj W j = λ I jj and ε Implication: 2 suffi cient statistics for welfare analysis λ New margins affect structural interpretation of ε ...and composition of gains from trade (GT)... ... but size of GT is the same. 14.581 (Week 9) Gravity Models (Theory) Spring 2013 24 / 44
Gains from Trade Revisited Proposition 1 is an ex-post result... a simple ex-ante result: Corollary 1: Suppose that R1-R3 hold. Then � A j = λ − 1 / ε W . jj 14.581 (Week 9) Gravity Models (Theory) Spring 2013 25 / 44
Equivalence (II) A stronger ex-ante result for variable trade costs under R1- R3’ : Proposition 2: Suppose that R1-R3’ hold. Then I 1 / ε � W j = λ jj where � � − 1 , I jj = ε ∑ n λ i = 1 λ ij ( w ˆ i τ ˆ ij ) and ε λ ij w ˆ j Y j ( w ˆ i τ ˆ ij ) n w i = ∑ j = 1 I � ε and { λ ij } are suffi cient to predict W j (ex-ante) from τ ˆ ij , i = j . 14.581 (Week 9) Gravity Models (Theory) Spring 2013 26 / 44
Taking Stock ACR consider models featuring: ( i ) Dixit-Stiglitz preferences; ( ii ) one factor of production; ( iii ) linear cost functions; and ( iv ) perfect or monopolistic competition; with three macro-level restrictions: ( i ) trade is balanced; ( ii ) aggregate profits are a constant share of aggregate revenues; and ( iii ) a CES import demand system. Equivalence for ex-post welfare changes and GT under R3’ equivalence carries to ex-ante welfare changes 14.581 (Week 9) Gravity Models (Theory) Spring 2013 27 / 44
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