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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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