General equilibrium analysis of the Eaton-Kortum model of international trade

Transcription

1 General equilibrium analysis of the Eaton-Kortum model of international trade Fernando Alvarez, Robert E. Lucas, Jr. Peter Karadi 12/04/2007

2 Outline of the paper Take the Eaton-Kortum (2002) model a static, multi-country general equilibrium model of technology, geography and international trade, with perfect competition, and constant-returns-to-scale technology, estimated for quantitative analysis.

3 Outline of the paper Take the Eaton-Kortum (2002) model a static, multi-country general equilibrium model of technology, geography and international trade, with perfect competition, and constant-returns-to-scale technology, estimated for quantitative analysis. Restate it to show that it is fairly standard Standard existence proof Uniqueness proof using the gross complementarity property

4 Outline of the paper Take the Eaton-Kortum (2002) model a static, multi-country general equilibrium model of technology, geography and international trade, with perfect competition, and constant-returns-to-scale technology, estimated for quantitative analysis. Restate it to show that it is fairly standard Standard existence proof Uniqueness proof using the gross complementarity property Some improvements Add a final good producing sector (services): using relative tradable/non-tradable prices for calibration Close it by assuming balanced trade (closer to an equilibrium model) Solution for optimal tariff

5 Motivation Two competing trade models Dornbusch, Fischer, Samuelson, 1977; Eaton-Kortum, 2002 Krugman, 1980; Melitz, 2003

6 Motivation Two competing trade models Dornbusch, Fischer, Samuelson, 1977; Eaton-Kortum, 2002 Krugman, 1980; Melitz, 2003 Similarities Both with heterogenous firms and nice aggregation results Both explain intra-industry trade across countries Both explain effects of geography (EK stronger) Lower trade barriers (special macro shock) higher welfare through increased competition increased TFP through reallocation towards more productive firms

7 Motivation Two competing trade models Dornbusch, Fischer, Samuelson, 1977; Eaton-Kortum, 2002 Krugman, 1980; Melitz, 2003 Similarities Both with heterogenous firms and nice aggregation results Both explain intra-industry trade across countries Both explain effects of geography (EK stronger) Lower trade barriers (special macro shock) higher welfare through increased competition increased TFP through reallocation towards more productive firms Main differences KM model: monopolistic competition and fixed costs EK model: perfect competition and iceberg costs

8 Motivation Two competing trade models Dornbusch, Fischer, Samuelson, 1977; Eaton-Kortum, 2002 Krugman, 1980; Melitz, 2003 Similarities Both with heterogenous firms and nice aggregation results Both explain intra-industry trade across countries Both explain effects of geography (EK stronger) Lower trade barriers (special macro shock) higher welfare through increased competition increased TFP through reallocation towards more productive firms Main differences KM model: monopolistic competition and fixed costs EK model: perfect competition and iceberg costs The paper seems to take sides.

9 The production Non-traded final good, using labor s, and intermediates q c = s α f q1 α f (1)

10 The production Non-traded final good, using labor s, and intermediates q c = s α f q1 α f (1) Continuum of traded intermediate goods with costs (inverse TFP) x [0, ), and cost distribution φ(x)

11 The production Non-traded final good, using labor s, and intermediates q c = s α f q1 α f (1) Continuum of traded intermediate goods with costs (inverse TFP) x [0, ), and cost distribution φ(x) Aggregated as CES [ η/(η 1) q = q(x) φ(x)dx] (η 1)/η (2) 0

12 The production Non-traded final good, using labor s, and intermediates q c = s α f q1 α f (1) Continuum of traded intermediate goods with costs (inverse TFP) x [0, ), and cost distribution φ(x) Aggregated as CES [ η/(η 1) q = q(x) φ(x)dx] (η 1)/η (2) Intermediate goods production: 0 q(x) = x θ s(x) β q m (x) 1 β, (3)

13 The production Non-traded final good, using labor s, and intermediates q c = s α f q1 α f (1) Continuum of traded intermediate goods with costs (inverse TFP) x [0, ), and cost distribution φ(x) Aggregated as CES [ η/(η 1) q = q(x) φ(x)dx] (η 1)/η (2) Intermediate goods production: Endowment constraints s f q(x) = x θ s(x) β q m (x) 1 β, (3) s(x)φ(x)dx 1; q f + 0 q m (x)φ(x)dx q (4)

14 Probabilistic formulation Cost distribution: Φ c (x) = exp( λx)

15 Probabilistic formulation Cost distribution: Φ c (x) = exp( λx) Note: TFP parameter: y = x θ has a Fréchet distribution (type II extreme value distribution) Θ(y) = exp( λy 1/θ )

16 Probabilistic formulation Cost distribution: Φ c (x) = exp( λx) Note: TFP parameter: y = x θ has a Fréchet distribution (type II extreme value distribution) Θ(y) = exp( λy 1/θ ) Originally in EK: they start out by assuming Fréchet for the TFP terms Advantage: some justification: Innovations x arrive as a Poisson process, and individual innovations have Pareto distribution Ψ c = (q/ q) 1/θ, then the best ideas will have a Fréchet distribution with λ being the stock of knowledge Disadvantage: distribution less known: This paper uses an equivalent formulation Makes it more accessible by using the exponential.

17 Multi-country setup n countries

18 Multi-country setup n countries Consumption: c i, i = 1... n final goods, not traded

19 Multi-country setup n countries Consumption: c i, i = 1... n final goods, not traded Production: Labor endowments: L = (L1,..., L n ), not mobile Technology parameters λ = (λ1,..., λ n ) Same parameters: θ, β, α, η

20 Multi-country setup n countries Consumption: c i, i = 1... n final goods, not traded Production: Labor endowments: L = (L1,..., L n ), not mobile Technology parameters λ = (λ1,..., λ n ) Same parameters: θ, β, α, η Variable trade costs: Iceberg transportation costs: κij amount arrives to i from j (lost) Tariffs: ωij fraction of dollar gets to i from j (revenue generating)

21 Multi-country setup n countries Consumption: c i, i = 1... n final goods, not traded Production: Labor endowments: L = (L1,..., L n ), not mobile Technology parameters λ = (λ1,..., λ n ) Same parameters: θ, β, α, η Variable trade costs: Iceberg transportation costs: κij amount arrives to i from j (lost) Tariffs: ωij fraction of dollar gets to i from j (revenue generating) Commodity space: x = (x 1,..., x n ), independent random draws: φ(x) exp ( n i=1 λ i)

22 Prices Take w = (w 1,..., w n ) as given (we are going to solve for this)

23 Prices Take w = (w 1,..., w n ) as given (we are going to solve for this) CRS technology, and perfect competition, so cost minimization can be used for prices: Country aggregate ci p i = α α (1 α) (1 α) w α i p (1 α) mi (5)

24 Prices Take w = (w 1,..., w n ) as given (we are going to solve for this) CRS technology, and perfect competition, so cost minimization can be used for prices: Country aggregate ci p i = α α (1 α) (1 α) w α i p (1 α) mi (5) CES aggregate intermediate good qi p 1 η mi = p i (x) 1 η φ(x)dx = E [ p i (x) 1 η] (6) R n +

25 Prices Take w = (w 1,..., w n ) as given (we are going to solve for this) CRS technology, and perfect competition, so cost minimization can be used for prices: Country aggregate ci p i = α α (1 α) (1 α) w α i p (1 α) mi (5) CES aggregate intermediate good qi p 1 η mi = p i (x) 1 η φ(x)dx = E [ p i (x) 1 η] (6) R n + Each intermediate good x is bought in i from the lowest cost producer among countries j = 1,..., n [ ] w β j p i (x) = B min p1 β mj x θ j j κ ij ω ij where B = β β (1 β) (1 β). (7)

26 Prices II. Some amazingly useful properties of the exponential distribution: x exp(λ) and k > 0, then kx exp(λ/k) x exp(λ), y exp(µ), and independent, then z = min(x, y) is z exp(λ + µ) Pr{x y} = λ/(λ + µ)

27 Prices II. Some amazingly useful properties of the exponential distribution: x exp(λ) and k > 0, then kx exp(λ/k) x exp(λ), y exp(µ), and independent, then z = min(x, y) is z exp(λ + µ) Pr{x y} = λ/(λ + µ) Using the first two facts, after some algebra, we get that individual intermediates: pi (x) exp(µ(w)), where ( n w β µ(w) = B 1/θ j ψ ij (w), ψ ij (w) = p ) 1/θ mj(w) 1 β λ j κ ij ω ij j=1 (8)

28 Prices II. Some amazingly useful properties of the exponential distribution: x exp(λ) and k > 0, then kx exp(λ/k) x exp(λ), y exp(µ), and independent, then z = min(x, y) is z exp(λ + µ) Pr{x y} = λ/(λ + µ) Using the first two facts, after some algebra, we get that individual intermediates: pi (x) exp(µ(w)), where ( n w β µ(w) = B 1/θ j ψ ij (w), ψ ij (w) = p ) 1/θ mj(w) 1 β λ j κ ij ω ij j=1 and the aggregate intermediates: n equations for p m = (p m1,..., p mn ) as a function of w p mi (w) = AB nψ ij (w) j=1 θ (8), A = Γ(1 + θ(1 η)) (9)

29 Trade balance Let D ij be the probability that a good x in i is the cheapest in j By LOLN it is also the fraction of goods from j and, amazingly, it is the proportion of expenditure spent on goods from j, because the price distribution Gi (p) in country i is the same for all source countries j: higher productivity influences the range of products sold (D ij ) not their price distribution (differently from the Melitz model with fixed markups).

30 Trade balance Let D ij be the probability that a good x in i is the cheapest in j By LOLN it is also the fraction of goods from j and, amazingly, it is the proportion of expenditure spent on goods from j, because the price distribution Gi (p) in country i is the same for all source countries j: higher productivity influences the range of products sold (D ij ) not their price distribution (differently from the Melitz model with fixed markups). Using the 3. property of the exponential distribution, we can easily get that D ij = ψ ij n k=1 ψ ik (10)

31 Trade balance Let D ij be the probability that a good x in i is the cheapest in j By LOLN it is also the fraction of goods from j and, amazingly, it is the proportion of expenditure spent on goods from j, because the price distribution Gi (p) in country i is the same for all source countries j: higher productivity influences the range of products sold (D ij ) not their price distribution (differently from the Melitz model with fixed markups). Using the 3. property of the exponential distribution, we can easily get that D ij = ψ ij n k=1 ψ ik (10) Trade balance L i p mi q i n D ij ω ij = j=1 n L j p mj q j D ji ω ji. (11) j=1

32 Equilibrium wages Finding the equilibrium wages (w) reduces to finding the zeros of the following excess labor demand system: Z i (w) = 1 n w j (1 s fj (w)) L j D ji (w)ω ji L i w i (1 s fi (w)) w i F j (w) j=1 (12) where F i is the country s spending on tradeables that reaches producers n F i = D ij ω ij, (13) j=1 and s fi is the labor s share in the production of the final goods α [1 (1 β)f i ] s fi = (1 α)βf i + α[1 (1 β)f i ]. (14)

33 Existence and Uniqueness Theorem 1: for any w R n ++ there is a unique p m (w), which is for each p mi (w) is strictly increasing in w, strictly decreasing in κij, ω ij

34 Existence and Uniqueness Theorem 1: for any w R n ++ there is a unique p m (w), which is for each p mi (w) is strictly increasing in w, strictly decreasing in κij, ω ij Theorem 2: There is a w R n ++ s.t. Z(w) = 0, because Z(w) satisfies all the standard properties of an excess demand function

35 Existence and Uniqueness Theorem 1: for any w R n ++ there is a unique p m (w), which is for each p mi (w) is strictly increasing in w, strictly decreasing in κij, ω ij Theorem 2: There is a w R n ++ s.t. Z(w) = 0, because Z(w) satisfies all the standard properties of an excess demand function Theorem 3: ω ij = ω i and α β, and trade costs not too large, then there is exactly one solution to Z(w) = 0 with n i=1 = 1, because Z has the gross substitute property Z i (w) w k > 0 (15)

36 Algorithm Discrete analogue of the tatonnement process { } n w = w R n + : w i L i = 1 i=1 (16) and mapping T : w w, T (w) i = w i (1 + νz i (w)/l i ), i = 1,..., n, ν (0, 1], (17) which is the percentage increase in country i s wage in proportion to a scaled version of its excess demand.

37 An easy example Zero gravity: κ ij = ω ij = 1 GDP; higher dispersion higher TFP L i w i = λ θ/(β+θ) i L β/(β+θ) i (18) Relative prices; higher productivity higher relative price of non-tradables p i p m = α α (1 α) (1 α) ( λi L i ) αθ/(β+θ) p α m (19)

38 An easy example Zero gravity: κ ij = ω ij = 1 GDP; higher dispersion higher TFP L i w i = λ θ/(β+θ) i L β/(β+θ) i (18) Relative prices; higher productivity higher relative price of non-tradables p i p m = α α (1 α) (1 α) ( λi L i ) αθ/(β+θ) p α m (19) Welfare gain to costless trade from autarky (κ ij = ω ij = 0) ( ) ( ci0 log = (1 α) θ n c i β log j=1 w ) jl j. (20) w i L i the smaller the country, the more it gains, because size matters through cost advantage at home.

39 Some results Calibration α = 0.75 β = 0.5 θ = {0.1, 0.15, 0.25} κ = 0.75 (κ ij using distance) [ω ij ] from tariff data [Li ] and ([λ i ]) are estimated from using GDP and relative tradable/non-tradable prices

40 Some results Calibration α = 0.75 β = 0.5 θ = {0.1, 0.15, 0.25} κ = 0.75 (κ ij using distance) [ω ij ] from tariff data [Li ] and ([λ i ]) are estimated from using GDP and relative tradable/non-tradable prices Explaining trade volumes: 0.69 correlation

41 Some results Calibration α = 0.75 β = 0.5 θ = {0.1, 0.15, 0.25} κ = 0.75 (κ ij using distance) [ω ij ] from tariff data [Li ] and ([λ i ]) are estimated from using GDP and relative tradable/non-tradable prices Explaining trade volumes: 0.69 correlation Optimal tariff Tariff Small, open limit economy: L r 1 0 with λ r 1/L r 1 = k > 0 Welfare maximizing best response ω to a given ˆω Optimal ω > 0, increasing in θ (elasticity of export demand) Reason: monopoly pricing by the government

42 Conclusion Companion paper to Eaton-Kortum, 2002 While EK works hard with empirics using the bilateral international manufacturing trade data to explain effects of geography and technology on trade and prices This paper works hard to make the theory stronger and more accessible

43 Conclusion Companion paper to Eaton-Kortum, 2002 While EK works hard with empirics using the bilateral international manufacturing trade data to explain effects of geography and technology on trade and prices This paper works hard to make the theory stronger and more accessible Significant value-added Presented the model in a standard form Closed it more elegantly Proved existence and uniqueness Calculated optimal tariffs in a small economy

44 ARTICLE IN PRESS Motivation The model Equilibrium Results Conclusion Figures F. Alvarez, R.E. Lucas, Jr. / Journal of Monetary Economics 54 (2007) θ = 0.25 WELFARE GAIN RELATIVE TO ZERO TARIFF IN PERCENT: log difference x Parameters α = 0.75 β = 0.5 θ = 0.1 θ = TARIFF IN PERCENT: (1 - ω) x 100 Fig. 5. Welfare effects of tariff factor o for a small open economy. Three values of y.