ECO2704 Lecture Notes: Eaton-Kortum...

Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

ECO2704 Lecture Notes: Eaton-Kortum Model

Xiaodong Zhu

University of Toronto

October 7, 2010

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

This paper develops a general equilibrium model of internationaltrade that explains the following stylized facts within a singleunified framework

• Trade diminishes dramatically with distance

• Prices vary across locations with greater differences betweenplaces further apart

• Factor rewards are far from equal across countries

• Countries relative productivities vary substantially acrossindustries

A good framework for empirical analysis:

• The model developed in this paper can easily be estimatedfrom trade flows data

• The model also provides a way to estimate cross countrydifferences in productivity of tradable sectors

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

• N countries, a continuum variety of goods produced bycompetitive firms

• Trade is subject to an iceberg cost: the cost of delivering oneunit of good from country i to country n is dni , dii = 1 anddni > 1 for n 6= i

• For simplificaiton, linear production technologies with labouras the only input . (The model in the paper has CRStechnology with intermediate input as well)

• Labour can move freely across firms

• Each firm’s productivity is randomly drawn from a distribution

• Productivities are independent across firms and countries

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Preferences and demand

Identical preferences for consumers in all countries n=1,...N

Un =

[ˆ 1

0

Qn(k)σ−1σ dk

]

σ

σ−1

Let Xn be the total expenditures of the representative consumer incountry n. Then,

Qn(k) =Pn(k)

−σ

p1−σn

Xn

Here Pn(k) is the price of variety k and pn is the price index facedby the consumer in country n,

pn =

[ˆ 1

0

Pn(k)1−σdk

]

11−σ

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

TechnologyTo produce variety k in country i, the firm solves the followingproblem:

maxLi (k) Pii(k)Zi (k)Li (k)− wiLi (k)

Pii(k) =wi

Zi (k)

Here Zi (k) is the productivity and wi is wage in country i.

If a consumer in country n buys variety k from country i, the priceshe faces is

Pni (k) = dniPii(k) =dniwi

Zi(k)

The actual price of variety k in country i is

Pn(k) = min Pni (k); i = 1, ...,N

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Productivity distributions

It is assumed that the distribution of productivities in country i isgiven by a Frechet distribution:

Pr [k ;Zi(k) ≤ z] = Fi (z) = exp

(

−

(

z

Ai

)−θ)

, θ > 1

which has the following properties:

E [Zi ] = AiΓ(1 − θ−1)

E [log(Zi )] = γθ−1 + log(Ai )

(γ is the Euler’s constant) and

Var [log(Zi )] =π

6θ2

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Potential price distribution: Pni(k)

Cross variety distribution of potential price in country n of goodsfrom country i:

Gni (p) = Pr [k ;Pni(k) ≤ p] = Pr

[

k ;dniwi

Zi (k)≤ p

]

Gni (p) = Pr

[

k ;Zi (k) ≥dniwi

p

]

= 1 − exp

(

−

(

dniwi

Ai

)−θ

pθ

)

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Actual price distribution: Pn(k)

Cross variety distribution of actual price of all goods in country n:

Gn(p) = Pr [k ;Pn(k) ≤ p] = 1 − Pr [k ;Pn(k) ≥ p]

Gn(p) = 1 −

N∏

i=1

Pr [k ;Pni(k) ≥ p] = 1 − exp(

−Φnpθ)

Here

Φn =

N∑

i=1

(

dniwi

Ai

)−θ

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Distribution of import variety by countryThe range of varieties that will be imported by country n fromcountry i:

πni = Pr [k ;Pn(k) = Pni (k)]

πni =

ˆ ∞

0

Pr [k ;Pn(k) = Pni(k)|Pni (k) = p] dGni (p)

=

ˆ ∞

0

Pr [k ;Pnj(k) ≥ p, j = 1, ...,N, j 6= i |Pni (k) = p] dGni (p)

=

ˆ ∞

0

∏

j 6=i

(1 − Gnj(p)) dGni (p)

=⇒ πni =

(

dniwi

Ai

)−θ

Φ−1n =

(

dniwi

Ai

)−θ

∑Nj=1

(

dnjwj

Aj

)−θ

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Conditional distribution of import prices: P∗ni

Distribution of prices across the range of varieties that country iactually exported to country n:

G ∗ni (p) = Pr [k ;Pn(k) ≤ p|k ;Pn(k) = Pni(k)]

=Pr [k ;Pn(k) ≤ p,Pn(k) = Pni (k)]

Pr [k ;Pn(k) = Pni(k)]

=1

πni

ˆ p

0

Pr [k ;Pnj(k) ≥ q, j = 1, ...,N, j 6= i ] dGni (q)

=1

πni

ˆ p

0

∏

j 6=i

(1 − Gnj(q)) dGni (q)

=⇒ G ∗ni (p) = Gn(p)

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Price Index

pn =

[ˆ 1

0

Pn(k)1−σdk

]

11−σ

=

[ˆ ∞

0

p1−σdGn(p)

]1

1−σ

pn = δΦ−1/θn

δ =

[

Γ

(

θ + 1 − σ

θ

)]1/(1−σ)

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Expenditure share of imports from country iLet Ωi = k ;Pn(k) = Pni (k) be the set of varieties that country nimports from country i. Then, country n’s expenditures on importsfrom country i is

Xni =

ˆ

k∈Ωi

Pni(k)Qn(k)dk =

´

k∈ΩiP1−σ

ni (k)dk

p1−σn

Xn

Note thatˆ

k∈Ωi

P1−σni (k)dk = E

[

P1−σni |Ωi

]

Pr [Ωi ] = E[

P1−σni |Ωi

]

πni

From the previous slide, however, we know that the averageexpenditure per good on imports from i is

E[

P1−σni |Ωi

]

= E[

P1−σn

]

=

ˆ

P1−σn (k)dk = p1−σ

n

=⇒Xni

Xn

= πni

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Trade flowThe fraction of goods that country n buys from country i , πni , isalso the fraction of its expenditure on goods from country i:

Xni

Xn

=

(

dniwi

Ai

)−θ

Φ−1n ∝

(

Ai

wi

)θ (dni

pn

)−θ

The exporter i’s total sales are:

Ri =

N∑

m=1

Xmi ∝

(

Ai

wi

)θ N∑

m=1

(

dmi

pm

)−θ

Xm

Gravity equation:

Xni =

(

dni

pn

)−θXnRi

∑Nm=1

(

dmi

pm

)−θXm

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Real income and welfare

Previous slide shows that

πni ∝

(

Ai

wi

)θ (dni

pn

)−θ

or

wi ∝ Aiπ−1/θni d−1

ni pn

Thus, for i=n, we have

wi

pi

∝ Aiπ−1/θii

That is, the smaller the domestic expenditure share πnn is, the

higher the real income and welfare.

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

For a given vector of total expenditures (X1, ...,XN), we havealready shown how consumption, prices and trade flows aredetermined

In equilibrium country n’s total expenditures should also equal itstotal income:Xn = wnLn, n=1,...,N

So, the only endogenous variables that need to be solved for are thewages w1, ...,wN

Total income equals total exports and domestic sales:

wiLi =

N∑

n=1

Xni =

N∑

n=1

πniXn

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Equation for solving wagesFor i=1,...,N:

wiLi =

N∑

n=1

(

dniwi

Ai

)−θ

Φ−1n wnLn

Alveraz and Lucas (2007, JME) showed the existence anduniqueness of the solutions to the equation system above

The case of free trade: dni = 1 for all n and i

wiLi =

(

wi

Ai

)−θ N∑

n=1

Φ−1n wnLn

=⇒ wi ∝

(

Aθi

Li

)1/(1+θ)

Under free-trade, price levels are identical, but factor prices aredifferent across countries.

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Estimating θ

Note thatXni/Xn

Xii/Xi=

(

pidni

pn

)−θ

Equivalently,

log

(

Xni/Xn

Xii/Xi

)

= −θlog

(

pidni

pn

)

This equation can be used to estimate the parameter θ

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Estimating competitiveness

Also note that

Xni/Xn

Xnn/Xn=

(

Ai

wi

)θ (An

wn

)−θ

d−θni

or

log

(

Xni/Xn

Xnn/Xn

)

= θlogTi − θlogTn − θlogdni

This equation can be used to estimate the competitiveness measureTi = Ai/wi

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

• Alvarez and Lucas (2007) “General Equilibrium Analysis of theEaton-Kortum Model of International Trade,” Journal ofMonetary Economics, 54(6), 1726-1768.

• Closes the model without an outside sector to solve for

endogenous factor prices

• Rodriguez-Clare, Andres (2009) “Offshoring in a RicardianWorld,” American Economic Journal: Macroeconomics,forthcoming.

• Analyzes offshoring within the Eaton-Kortum framework

• Ramondo, Natalia and Andres Rodriguez-Clare (2008) “Trade,Multinational Production and the Gains from Openness,”University of Texas, mimeograph, http://www.eco.utexas.edu/nr3353/Trade

• Analyzes both trade and multinational production within the

Eaton-Kortum framework

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

• David Donaldson (2008) “Railroads of the Raj: Estimating theEconomic Impact of Transportation Infrastructure,” MITEconomics, mimeograph

• Uses the Eaton-Kortum framework to evaluate the impact of

the construction of the railway network in Colonial India

• Arkolakis, Costas, Arnaud Costinot and AndresRodriguez-Clare (2009) “New Theories, Same Old Gains?”Yale University, mimeograph

• Shows that in a class of international trade models, welfare can

be expressed in terms of the trade share as in the

Eaton-Kortum framework

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Introduction Model Price Distributions Trade Flows Equilibrium Empirical Analysis Subsequent Research

Multi-Sector Models:

• Costino and Donaldson and Kunmunjer (2010) “What goodsdo countries trade? A Quantitative Exploration of Ricardo’sIdeas,” unpublished manuscript, MIT Economics Department

• Kerr, W. R. (2009): “Heterogeneous Technology Diffusion andRicardian Trade Patterns”, unpublished manuscript, HarvardBusiness School

• Yi and Zhang (2010) “Structural Change in an OpenEconomy,” unpublished manuscript, University of Michigan

• Tombe, Trevor (2010) “The Missing Food Problem,” jobmarket paper, University of Toronto

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