Eaton Kortum Model (2002)

Transcription

1 Eaton Kortum Model (2002) Seyed Ali Madanizadeh Sharif U. of Tech. November 20, 2015 Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

2 Introduction Eaton and Kortum (2002) Econometrica Alvarez and Lucas (2007) JME Caliendo and Parro (2015) The Review of Economic Studies Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

3 Questions What are the gains from trade? That have already been realized, relative to autarky. That could potentially be realized with costless trade. What is the role of trade in spreading the gains from new technology? What is the role of geography in determining patterns of specialization? How important is trade diversion following regional integration? Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

4 Eaton Kortum Model (a simple version) Suppose productivities in countries are independently distributed with Frechet distribution and parameters θ and T i ; So F Zi (z) = Pr (Z i z) = exp [ T i z θ] Refer to the parameter T i as country i s state of technology (absolute advantage across the continuum). A bigger T i, implies that a high effi ciency draw for any good j is more likely. Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

5 Eaton Kortum Model (a simple version) The parameter θ (which we treat as common to all countries) reflects the amount of variation within the distribution. A bigger θ implies less variability. Specifically, Z i (effi ciency) ( has geometric mean e γ/θ T 1/θ i and its log has standard deviation π/ θ ) 6. Here γ = (Euler s constant) and π = The parameter θ regulates heterogeneity across goods in countries relative effi ciencies. In a trade context θ governs comparative advantage within this continuum. A lower value of θ, generating more heterogeneity, means that comparative advantage exerts a stronger force for trade against the resistance imposed by the geographic barriers d ni. Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

6 Eaton Kortum Model (a simple version) The probability that country i is the lowest cost supplier to country n [ { }] X ni wi d ni wk d = π ni Pr nk min X n k =i = T i (w i d ni ) θ Φ n where Φ n = N k=1 T k (w k d nk ) θ z i z k Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

7 Eaton Kortum Model (a simple version) The sensitivity of trade to costs and geographic barriers depends on the technological parameter θ (reflecting the heterogeneity of goods in production) rather than the preference parameter σ (reflecting the heterogeneity of goods in consumption). Trade shares respond to costs and geographic barriers at the extensive margin: As a source becomes more expensive or remote it exports a narrower range of goods. In contrast, in models that invoke Armington or (with some caveats) monopolistic competition, adjustment is at the intensive margin: Higher costs or geographic barriers leave the set of goods that are traded unaffected, but less is spent on each imported good. Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

8 Gravity Equation Gravity Equation where Ξ i = N n=1 Y i = d θ ni Φ n X n = = = N X ni n=1 N X ni X n n=1 X n N π ni X n n=1 N n=1 = T i wi θ Ξ i T i (w i d ni ) θ Φ n X n Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

9 Gravity Equation Therefore: X ni = T i (w i d ni ) θ Φ n Y i X n = Ξ i Φ n dni θ X n Prices (can be shown): P n = γφ 1/θ n Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

10 Eaton Kortum Model (a simple version) So π ii = T i w θ i (P i /γ) θ Therefore c i = w i P i = 1 γ ( ) 1/θ Ti π ii In Autarky π ii = 1 Welfare gain from Autarky to d : c d c d = = (w/p) d = π 1/θ ii (w/p) Autarky Where θ is the trade elasticity. (refer to Arkolakis 2008). Trade gains are greater the smaller θ.(interpretation) Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

11 Eaton Kortum Model (a simple version) Equilibrium Conditions: Y i = If no deficit (D i = 0); then N X ni = n=1 N n=1 X n = Y n T i (w i d ni ) θ Φ n X n Also Equilibrium condition Y n = w n L n w i L i = N n=1 T i (w i d ni ) θ w n L n Φ n Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

12 Eaton Kortum Model (a simple version) It s a fixed point problem: Can be solved numerically for w i. Algorithm (Alvarez Lucas 2007): Define the set as : = { w R N + : } N w i L i = 1 i =1 Define the vector operator Z (w) on the set such that [Z (w)] i = w i 1 + v 1 N n=1 T i (w i d ni ) θ Φ n w i L i w n L n Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

13 Eaton Kortum Model (a simple version) Z (w) since: N [Z (w)] i L i i=1 N w i 1 + v 1 N n=1 i=1 T i (w i d ni ) θ Φ = n w n L n L i w i L i N N = w i L i + w i v 1 N T i (w i d ni ) θ n=1 Φ n w n L n L i i=1 i=1 w i L i ( ) T i (w i d ni ) θ = 1 + v w i L i w n L n N i=1 N n=1 Φ n Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

14 Eaton Kortum Model (a simple version) So N [Z (w)] i L i i=1 = 1 + v ( N i=1 = 1 + v ( N i=1 = 1 + v ( N = 1 i=1 w i L i w i L i w i L i N n=1 N n=1 N n=1 ( N i=1 ( w n L n Φ n ( wn L n Φ n )) T i (w i d ni ) θ w n L n N i=1 Φ n Φ n T i (w i d ni ) θ )) ) ) Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

15 Eaton Kortum Model (a simple version) Algorithm: Start with a guess point w 0 Iterate: w k +1 = Z (w ) k Continue until w k +1 w k < ε Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

16 Eaton Kortum Model 2002 In the paper: Final good production: a combination of labor and a CES aggregate of intermediate goods (Cobb-Douglas) Intermediate goods production:a combination of final goods and labor Consider non-tradeables Similar method and algorithm but more complicated equations Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

17 Extended models Alvarez and Lucas 2007 Same as EK 2002 but add capital Caliendo and Parro 2015 Same as EK but adds IO tables Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

18 Eaton Kortum Model: Special Cases Costless trade: d ni = 1 Identical countries: ( ) w i Ti /L 1/(1+θ) i = w j T j /L j π ii = π ni = (N 1) d θ d θ 1 + (N 1) d θ log C d =1 C d = = 1 θ log N Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

19 Estimation 1 From EK 2002 Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

20 Estimation 1 From the model of trade: So Define D ni = p i d ni p n D ni max j X ni = T i (w i d ni ) θ X n Φ n P n = γφn 1/θ ( ) X ni /X n pi d θ ni = X ii /X i p n and estimate: { log p i (j) p n (j) } mean j { log p } i (j) p n (j) Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

21 Estimation 1 Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

22 Estimation 1 ) Estimate θ by regressing log( Xni /X n X ii /X i on log ( ) D ni. To minimize the impact of errors in the right hand side variable, follow the theory and constrain the intercept to be 0. Results: θ = 8.28 Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

23 Estimation 1 Dummy Variable Approach to get ther parameters where log X ni X nn = log S i θ log d ni S n = log S i log S n θf (x ni ) θδ ni log d ni = f (x ni ) + δ ni log S i = log T i θ log w i x ni are variables used in gravity equations to proxy for d. Estimation results gives us the dummies and an estimate for θ log d ni. Using data on wages and an estimate of θ, we can find an estimate for ln d ni and log T i. Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

24 Estimation 2 Alternative method to estimate θ Proxy T i with R&D and Human Capital data as measured by years of schooling. S i = α 0 + α R log R i α H ( 1 H i ) θ log w i + τ i Labor market Equilibrium suggest that a country s wage increases with its level of Technology. So the technology shock τ i is correlated with w i. Endogeneity. Equilibrium condition suggests that wages and labor force are negatively correlated. So use labor force as an IV. Results: θ OLS = 2.84, θ IV = 3.60 Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

25 Calibration and estimation results α - fraction of employment in non-tradeables: 0.75 β - share of labor in value of tradeables: 0.5 η- SDS parameter: 2 (doesn t matter) 1/θ - productivity variance: 0.1 to 0.25 (matters a lot) Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

26 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

27 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

28 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

29 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

30 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

31 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

32 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

33 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

34 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

35 Results Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

36 Conclusion Answers to the original Questions!!! Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

37 Appendix: Ideas, Techniques, and Unit Costs The fundamental atom of technology is an idea. An idea is a recipe to produce some good j with some effi ciency q (which we call the quality of the idea) at some location i. Effi ciency is simply the amount of output that can be produced with a unit of input. At any moment a location i is characterized by the ideas available to it for production, and an input cost w i. Connecting an idea (for making a good j with effi ciency q) with a location gives rise to a technique for producing the good there at unit cost w i /q. Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

38 Ideas The quality of an idea is the realization of a random variable Q drawn independently from the Pareto distribution with parameter θ > 1. Pr [Q > q] = where q > 0 is the minimum quality level. { ( q q ) θ q > q 1 q < q Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

39 Ideas Time is continuous. Ideas for good j arrive at date t according to a Poisson process with intensity ār(t). We can think of R as reflecting research effort and ā (to be normalized shortly) as reflecting research productivity. for any q > q the arrival rate of effi ciency Q q ār(t) (q/ q) θ We normalize aq θ = 1 so the arrival rate of effi ciency Q q would be: R(t)q θ Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

40 Techniques We assume that there is no forgetting: So ideas accumulate: T (t) = t R (t) dt The number of ideas about good j with quality Q > q is distributed Poisson with parameter T (t)q θ. The distribution is: Pr [ q > q q > q ] = ( q /q ) θ Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41

41 Unit Costs Since a bundle of inputs costs w, the unit cost of producing good j with a technique of effi ciency Q is C = w/q Seyed Ali Madanizadeh (Sharif U. of Tech.) Eaton Kortum Model (2002) November 20, / 41