Trade under Imperfect Competition and the Gravity Equation Guest lecture by Geoffrey Barrows
ECO 434: Economie Internationale Ecole Polytechnique, 2`eme Ann´ee http://gregory.corcos.free.fr/ECO434/ECO434.html
Outline
• Introduction: trade in similar products between similar countries • The Krugman Model - Assumptions - Autarky and open-economy equilibria • Empirical Evidence: the Gravity Equation - Microfoundations - Main lessons
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Introduction
• Ricardo/HOS predict trade in different goods by different
countries: - differences in productivity (Ricardo) or factor endowments (HOS) - specialization in comparative advantage goods • These models do not explain - trade between similar countries - trade in similar goods - the ’gravity equation’ (seen in Chapter 1) • We need a new, imperfect competition trade model - similar countries trade differentiated varieties of the same goods - consumers gain from additional product diversity
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0
Trade Value (billions USD) 2 4
6
World Trade by Region
1970
1980
1990 North−North South−South
2000
2010
North−South South−North
Most of world trade occurs between similar countries. Source: UN ComTrade
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Table: Intra-industry trade (IIT) shares in bilateral trade flows (top 10)
Source: Fontagn´ e, Freudenberg and Gaulier (2006). Intra-Industry Trade is defined as simultaneous trade of narrowly defined goods in both directions between two countries. This Table reports the 10 country pairs with the highest IIT shares.
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The Krugman Model
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Paul Krugman (1953- ) Scale Economies, Product Differentiation and the Pattern of Trade, American Economic Review, 1980
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Overview
• Ingredients: • Countries have identical endowments, technology and preferences. • Economies of scale • Differentiated (imperfectly substitutable) products. • Monopolistic competition: firms have market power on their variety. • Free entry: the number of firms is endogenous • All the neoclassical motives for trade are absent! • Trade occurs because consumers have demand for foreign varieties,
and large trade partners offer more varieties.
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Consumption • One factor: labor (L). National income: wL. • Utility defined over a continuum of varieties Ω:
Z max U = max q(ω)
q(ω)
q(ω)
σ−1 σ
σ σ−1
dω
Z s.t.
Ω
p(ω)q(ω)dω = wL Ω
with σ > 1 the elasticity of substitution between varieties. • Utility maximization yields the demand function:
q(ω) = with P =
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R Ω
p(ω)1−σ dω
1 1−σ
p(ω) P
−σ
wL P
(1)
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• P is the “ideal price index”: 1 unit of utility costs P units of income. • To see this, plug demand functions into the utility function Z U
=
q(ω)
σ−1 σ
σ σ−1
dω
Z =
Ω
wL σ P P
=
Ω
Z p(ω)
1−σ
dω
Ω
σ σ−1
=
p(ω) P
−σ σ−1 σ
wL P
σ ! σ−1
σ−1 σ
dω
wL P
• P is lower than the simple average of prices p(ω):
P=
R Ω
p(ω)1−σ dω
1 1−σ
1 prices
are lower in the large country: fewer varieties bear shipping costs. Price indices, wages
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Gains From Trade Price Levels (Home is the large country) 1.5 OE Home OE Foreign Aut Home Aut Foreign
1.4
1.3
1.2
1.1
1
0.9
0.8
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0
0.1
0.2
0.3
0.4
0.5 τ1−σ
0.6
0.7
0.8
0.9
1
Wages
• Substituting for prices, quantities and numbers of firms in the trade
balance equation np X q X = n∗ p X ∗ q X ∗ yields w w∗
=
P P∗
1−σ σ
1−σ
=
Lw 1−σ + L∗ (τ w ∗ ) L (τ w )
1−σ
! σ1
+ L∗ w ∗ 1−σ
• When τ = 1 wages are equal in both countries.
1 → LL∗ 2σ−1 . Thanks to better market access the large country’s firms sell and hire disproportionately more.
• When τ → +∞,
w w∗
• An extended model with mobile workers can explain economic
agglomeration in some countries or regions.
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Gravity Regressions
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From Theory to Gravity Regressions • In the model aggregate exports equal X = np X q X • We predict that exports from i to j equal
Xij = ni
σ τij wi σ − 1 ϕi Pj
1−σ Rj
or in logs: ln Xij
=
ln |
σ σ−1 {z
1−σ
constant
wi + ln ni + (1 − σ) ln ϕi {z } } | i−specific
+ ln Pjσ−1 |
+ ln Rj + (1 − σ) ln τij {z } {z } |
j−specific
transport cost
• ’Gravity’ empirical equation
ln Xij = κ + αXi + βMj + γ ln distij + δCij + εij 19/37
Trade between US States and Canadian Provinces
Source: Feenstra & Taylor (2011)
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Gravity Equation Estimates Variable dependante: ln Xij (2) (3) (4) (5) 1.185a
ln Population i
(1) 0.799a
ln GDP/capita i
1.072a
1.272a
a
0.896a
a
0.920a
a
-1.008
-1.511a
1970 No 9,035 0.583
2006 No 16,936 0.631
ln Population j ln GDP/capita j ln Distance
0.723 1.058
Trade Agreement GATT/WTO Common Currency Common Border Common Language Colonial Past Year Country Fixed Effects # Observations R2
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(6)
Gravity Equation Estimates ln Population i
(2) 0.823a
ln GDP/capita i
1.072a
1.110a
1.272a
1.265a
a
a
a
0.900a
a
0.912a
a
-1.199a
-1.619a
ln Population j ln GDP/capita j ln Distance
0.723
a
1.058
a
-1.008
Trade Agreement GATT/WTO
0.740
0.896
a
1.092
0.920 a
(6)
a
-0.838
-1.000
0.917a
0.643a
0.758a
0.493a
-0.011
0.038
a
0.811a
a
a
-1.511
0.306
Common Currency
1.470
1.460
-0.029
0.035
Common Border
0.588a
0.533a
1.152a
0.840a
Common Language
0.559a
0.535a
1.108a
0.909a
a
a
a
0.672
0.889a
2006 No 16,936 0.649
2006 Yes 16,936 0.741
Colonial Past Year Country Fixed Effects # Observations R2
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Dependent variable: ln Xij (3) (4) (5) 1.185a 1.191a
(1) 0.799a
1970 No 9,035 0.583
1.376
1.277
1970 No 9,035 0.607
1970 Yes 9,035 0.710
2006 No 16,936 0.631
.6
.8
Distance Elasticity 1 1.2 1.4
1.6
The Distance Elasticity Over Time
1945
1955
1965
1975
1985
1995
2005
• An elasticity of 1 means that doubling distance reduces trade by half. • Distance proxies transportation costs, time as a trade barrier,
cultural distance and informational frictions. • Increasing spatial concentration of trade over time... 23/37
Other Results
• Trade Agreements - Trade is 40% higher between partners of a trade agreement. • Currency Unions - Trade is twice as high between members of a currency union. • Cultural Links - Trade is higher between countries that share a language or historical links
Note: these are correlations. Stating causality requires further work...
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The Border Effect in the EU Effet Frontière : Flux Intranationaux / Flux Internationaux
30 Lancement du Marché Unique 25 20 15 10 5 0 1975
1980
1985 Année
Source : Head et Mayer (2000) Source: Head & Mayer (2000)
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1990
1995
Limits of the Gravity Equation
• The within-EU border effect remains an empirical puzzle. • ’Zeroes’: - trade flows are zero for over 50% of country pairs, even more when using sector-level trade flows - zeroes require special econometric techniques to estimate the gravity equation - zeroes require an extension of the Krugman model, where there is always some demand for each variety
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Conclusions
• Neoclassical trade models explain trade in different goods between
different countries. • The Krugman trade model explain trade in similar goods between
similar countries. • comparative advantage trade motives are absent • trade occurs because of demand for different varieties of the same
product • product diversity creates gains from trade
• The model is consistent with ’gravity’ regression results: trade flows
are higher between countries that are large, close, share the same language, history, currency or have signed trade agreements.
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Appendix
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How to derive the demand function • Lagrangian: L =
R Ω
q(ω)
σ−1 σ
dω
σ σ−1
−µ
R Ω
p(ω)q(ω)dω − wL
• First order conditions: −1 ∂L = q(ω) σ ∂q(ω) −1 σ
Z q(ω)
σ−1 σ
dω
1 σ−1
− µp(ω) = 0
Ω
1
⇔
q(ω)
⇔
q(ω) = Uµ−σ p(ω)−σ
U σ = µp(ω)
R p(ω)q(ω)dω = Uµ−σ Ω p(ω)1−σ dω 1 R • Define the price index P = Ω p(ω)1−σ dω 1−σ . The budget constraint is rewritten as wL = Uµ−σ P 1−σ
• Budget constraint: wL =
R
Ω
• Note that wL = PU. P is the monetary value of one unit of utility.
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• From the budget constraint wL = Uµ−σ P 1−σ and the f.o.c. q(ω) = Uµ−σ p(ω)−σ , one obtains the demand function: −σ p(ω) wL q(ω) = P P • Demand for variety ω increases with overall purchasing power wL/P, and decreases with the relative price of variety ω. • σ is equal to • the price elasticity: a 1% rise in p(ω) reduces demand by σ% • the elasticity of substitution: since
q(ω) q(ω 0 )
=
p(ω) p(ω 0 )
−σ
, increasing the
relative price of the ω variety by 1% reduces the relative demand for this variety by σ% Back to main text
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How to derive the optimal price • Start from the firm’s profit function: q(ω) π(ω) = p(ω)q(ω) − w f + ϕ • Maximize with respect to price given demand and price index P ⇒ First order condition: ∂π(ω) w = P σ−1 wL (1 − σ)p −σ + σp −σ−1 = 0 ∂p(ω) ϕ Or after rearranging: p=
σ σ−1 | {z }
w ϕ |{z}
Mark−up Marginal cost Back to main text
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Price indices in a two-country world economy • The price index is now written as:
X
P=
ω∈H
1−σ
p(ω)
+
X
∗
(τ p (ω))
1 1−σ
1−σ
ω∈F
• In the symmetric equilibrium, p(ω) = p, ∀ω ∈ H and p ∗ (ω) = p ∗ , ∀ω ∈ F ⇒ and
P = np 1−σ + n∗ (τ p ∗ )1−σ
1 1−σ
P ∗ = n(τ p)1−σ + n∗ p ∗ 1−σ
1 1−σ
• Absent transportation costs (τ = 1), with identical countries, the two indices are 1
equal: P = P ∗ = (2n) 1−σ p. Both countries have access to the same varieties in the same conditions. 1
• Both indices are lower than those in autarky, which are: P = n 1−σ p and P∗ = n
1 1−σ
p∗
• At given wages, opening up the economy has a positive impact on welfare (U = wL/P). This comes from consumers’ preference for diversity Back to main text 32/37
Wages • We have expressed optimal prices p(ω) and P as functions of nominal income wL.
• L is exogenous but w is endogenous • The wage is determined by the goods market equilibrium equation. Due to Walras’P law, it is equivalent to rely on (i) the domestic P market (wL = ω wl(ω)); (ii) the foreign market (w ∗ L∗ = ω w ∗ l ∗ (ω)); (iii) the trade balance (X = X ∗ )
• We use the trade balance: λ × L × L∗ × ⇒ with ⇒
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τ w 1−σ P∗
× w ∗ = λ × L × L∗ ×
1−σ σ w P = ∗ ∗ w P P np 1−σ + n∗ (τ p ∗ )1−σ = ∗ P n(τ p)1−σ + n∗ p ∗ 1−σ 1/σ w Lw 1−σ + L∗ (τ w ∗ )1−σ = w∗ L(τ w )1−σ + L∗ w ∗ 1−σ
τw∗ P
1−σ ×w
Relative imports: M n∗ = M∗ n
τ p ∗ /P τ p/P ∗
1−σ
w wL = ∗ ∗ ∗ w L w
• Starting from the symetric equilibrium:
L L∗
=
w ∗ /P w /P ∗
w w∗
=
1−σ
M M∗
• An increase in L/L∗ increases the relative number of firms in H
which reduces P/P ∗ . This makes foreign goods relatively more expansive → ↓ M/M ∗ • For trade to be balanced, needs to be compensated by an increase in
the relative wage w /w ∗ → ↑ M/M ∗ through an income effect (↑ aggregate demand) and a substitution effect (↓ relative competitiveness of domestically produced varieties) ⇒ Wages are relatively high in large countries
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Back to section 1
Wages (2) Relative wage in the large country, as a function of the “freeness” of trade τ 1−σ Relative Wage in the Large Country 1.09 1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1
0
0.1
0.2
0.3
0.4
0.5 τ1−σ
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0.6
0.7
0.8
0.9
1
Welfare gains
1
1
• Autarky: P = pn 1−σ and P ∗ = p ∗ n∗ 1−σ • Open economy: P = p 1−σ n + (τ p ∗ )1−σ n∗
∗
P = p
∗ 1−σ ∗
1−σ
n + (τ p)
n
1 1−σ
and
1 1−σ
• Without transportation costs: 1
1
P = P ∗ = (2np 1−σ ) 1−σ < (np 1−σ ) 1−σ since σ > 1 ⇒ Opening up the economy yields a welfare gain deriving from more diversity. • In Krugman (1979), trade has a pro-competitive effect too (fall in p
due to a rise in σ).
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Prices as a function of the “freeness” of trade τ 1−σ Price Levels (Home is the large country) 1.5 OE Home OE Foreign Aut Home Aut Foreign
1.4
1.3
1.2
1.1
1
0.9
0.8
0
0.1
0.2
0.3
0.4
0.5 τ1−σ
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0.6
0.7
0.8
0.9
1
