The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Lecture 3: International Trade Under Imperfect Competition Gregory Corcos Eco572: International Economics
October 1, 2014
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Introduction
Neoclassical theories fail to explain important aspects of international trade: most trade occurs between similar countries a large part of trade occurs within broadly defined sectors bilateral trade flows exhibit ’gravity’ patterns
New theories were designed in the 1970’s and 1980’s to address these shortcomings. These theories feature: Increasing returns to scale (fixed costs) and imperfect competition Products differentiation, either horizontally (varieties) or vertical (quality differences) Preference for product diversity
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Intra-Industry Trade
Decomposition of trade (% total)
Top-10 country pairs (% of bilateral trade, 2000)
Source: Fontagn´ e, Freudenberg & Gaulier (2006)
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Outline
1. The Krugman model 2. Specialization in the Helpman-Krugman model 3. Limits and extensions
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
The Krugman (1980) Model
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Consumption (Dixit-Stiglitz) Representative household supplying L units of labor and owning all firms. CES preferences over a continuum of varieties Ω: Z q(ω)
max U = max q(ω)
q(ω)
σ−1 σ
σ σ−1
dω
Ω
with σ > 1 the elasticity of substitution between varieties. R Budget constraint: Ω p(ω)q(ω)dω = wL (no profits in equilibrium). Utility maximization function: −σ yields the demand 1 R p(ω) wL 1−σ dω 1−σ q(ω) = P with P = p(ω) P Ω
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Interpretation of P
P is the “ideal price index” in the sense that an increase in real income (computed with that index) translates into an increase in utility. Proof: 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
1−σ
p(ω) Ω
σ σ−1
dω
=
p(ω) P
−σ σ−1 σ
wL P
σ ! σ−1
σ−1 σ dω
wL P
If both nominal income wL and the price index P increase by x%, utility remains unchanged.
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Interpretation of P (2)
Since σ > 1, the ideal price index P is lower than the simple average of prices p(ω): 1 R R P = Ω p(ω)1−σ dω 1−σ < Ω p(ω)dω This comes from consumer’s preference for diversity: at the optimal bundle real income and utility are higher than if the agent were consuming a single variety at the average price. For a given nominal income wL and average price, increased product diversity lowers price index P and increases welfare.
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Production - Each firm has monopoly over a variety ω which is imperfectly substitutable with other varieties (monopolistic competition). - Fixed cost: to produce q(ω) firms use f + - Optimal price: p =
q(ω) ϕ
labor units
σ w σ−1 ϕ
- Profit: π(ω) ≡ p(ω)q(ω) − w f +
q(ω) ϕ
=w
q(ω) (σ−1)ϕ
−f
- Free entry: π(ω) = 0 ⇒ q(ω) = (σ − 1)ϕf ⇒ All firms have the same quantity and price (ω now omitted) q - Labor market equilibrium: n such that n f + ϕ = L ⇒ n=
L σf
⇒ The number of firms increases with market size (L) and decreases with fixed costs (f ) and competition (σ).
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Back to the price index
- Equilibrium price index: Z P= Ω
σ w σ−1ϕ
!
1−σ
1 1−σ
dω
=
1 σ w 1−σ n σ−1ϕ
is decreasing in the number of varieties - At the equilibrium value of n: σ w P= σ−1ϕ
L σf
1 1−σ
- Larger economies have lower P’s and higher welfare in autarky.
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Opening the economy
- Consider two identical countries except for their size: L, L∗ . - Transport costs are of the ”iceberg” type: when 1 unit is shipped, 1/τ units is received, with τ > 1. The rest has ”melted away”. - Optimal prices σ w Domestic market: p D = σ−1 ϕ ≡p σ w X Foreign market: p = τ σ−1 ϕ = τ p
The price before transport (FOB price) is the same on both markets because the elasticity of substitution is the same and is constant. The price in the destination market (CIF price) is multiplied by τ , ie the transport cost is fully passed on the consumer.
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Two countries (2)
- Total production: q = q D + τ q X - Total profit: π = (p − wϕ )q D + (τ p − τ wϕ )q X − wf = w q − wf pq − w f + ϕq = (σ−1)ϕ - Free entry: π = 0 ⇒ q = (σ − 1)ϕf - Labor market equilibrium: n f + ϕq = L ⇒ n =
L σf
, n∗ =
L∗ σf
In some monopolistic competition trade models, such as Krugman (1979), n falls after opening to trade. Here the constant number of firms is an ’artefact’ of special assumptions on preferences and transport costs.
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
From Theory to Gravity Regressions - Value of aggregate exports: X = nτ pq X (τ p) with: τ p −σ w ∗ L∗ P∗ P∗ σ w p = σ−1ϕ L n = σf 1−σ 1−σ 1 σ ∗ τw ⇒X = LL w∗ σf (σ − 1)ϕ P∗ q X (τ p) =
or in log: ln X = − ln(σf )+(1−σ) ln
τw σ +ln L+ln L∗ +(1−σ) ln ∗ +ln w ∗ (σ − 1)ϕ P
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
From Theory to Gravity Regressions (2) (see PC for more details)
Gravity regressions (Tinbergen, 1962) Bilateral trade flows follow a ’gravity law’ of the form Xij = G
(Li )α (Lj )β (Dij )θ
Li : size of country i; Dij : distance between i and j. The Krugman model is consistent with that finding if α = β = 1 and distance is a good proxy for bilateral transport costs τ (See PC). Transport costs affect trade along two ’margins’: increase in the number of available products (extensive margin) increase in the value of export per product (intensive margin)
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
International Trade: The gravity equation
Source: Head, Mayer and Ries (2008)
See details in PC ln Xij = Cst + ln Li + ln Lj + (1 − σ) ln τij + (1 − σ) ln wi − (1 − σ) ln Pj + ln wj
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Welfare gains 1
1
- Autarky: P = pn 1−σ and P ∗ = p ∗ n∗ 1−σ 1 - Open economy: P = p 1−σ n + (τ p ∗ )1−σ n∗ 1−σ and 1 P ∗ = p ∗ 1−σ n∗ + (τ p)1−σ n 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).
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Welfare Gains (2) 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
0.6
0.7
0.8
0.9
1
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Wages
Trade Balance:
1−σ τ w 1−σ τw∗ ∗ ∗ λ × L × L∗ × × w = λ × L × L × ×w ∗ P | {zP } | {z } X
X∗
⇒
w = w∗
Lw 1−σ + L∗ (τ w ∗ )1−σ L(τ w )1−σ + L∗ w ∗ 1−σ
1/σ
⇒ Without transport costs (τ = 1), wages are equalized across
countries ⇒ With high transport costs (τ → ∞): higher in the largest country
w w∗
→
L L∗
1 2σ−1
, ie wages are
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
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
0.6
0.7
0.8
0.9
1
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Interpretation
- Absent transport costs, all consumers have access to all varieties, prices converge and trade is balanced. - With a transport cost, the large country has lower prices (if L > L∗ , P < P ∗ ) and more varieties. Demand for imports is lower (increasing with P). - Balanced trade requires lower exports of the large country, thanks to a higher marginal cost: w > w ∗ Proof - Extension with mobile workers: migration towards the large country makes it larger... This is the foundation of the ’new economic geography’ (Krugman 1991).
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Specialization: The Helpman-Krugman Model
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Motivation
In the Krugman (1980) model, the only rationale for international trade is consumers’ preference for diversity Since there is a single (differentiated) good, there is no room for specialization patterns We now introduce a second sector to see how intra-industry trade affects specialization patterns
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Assumptions - Two sectors: differentiated good (as before) and homogeneous good. Cobb-Douglas utility function with fixed spending share µ in differentiated goods. - The homogeneous good is produced with labor using a 1:1 production function and is traded at no cost. By assumption both countries produce this good. Price is normalized to 1. Since labor is mobile across sectors, wages are equal to 1 in both countries. - The differentiated good sector is like in Krugman (1980), all firms produce q such that: q = qd + τ qX = µ
τ p −σ wL∗ p −σ wL + τµ P P P∗ P∗
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Specialization - The zero-profit condition implies q = q ∗ = (σ − 1)ϕf . - It can be shown that: L ∗ 1−σ 1−σ L =n −τ n 1−τ L∗ L∗ - Denoting sn ≡ n/(n + n∗ ) and sL ≡ L/(L + L∗ ): sn =
sL (1 + τ 1−σ ) − τ 1−σ , 1 − τ 1−σ
dsn >0 dsL
- sn ∈ [0, 1] describes specialization as a function of countries’ (exogenous) relative size sL : if sL < τ 1−σ /(1 + τ 1−σ ) then sn = 0 if sL > 1/(1 + τ 1−σ ) then sn = 1
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Specialization (2)
- When sL > 1/(1 + τ 1−σ ) and when sL < τ 1−σ /(1 + τ 1−σ ) one country fully specializes in differentiated goods. - The smaller transport costs, the narrower the size range with diversification. - Home market effect sn > sL whenever sL > 12 : the large country produces more than its share of differentiated goods dsn dsL > 1: the share in world output grows faster than the share in population a smaller transportation cost reinforces this effect 2-good extension: countries will be net exporters of goods for which they have high demand due to differences in preferences.
Krugman Model Limits and extensions theTheforeign economy. Specialization: Helpman-Krugman home economy is very large and the production of differentiated Specialization (3) the home country. s, the larger country hosts a higher proportion of output than its on. Denoting by sL the share of the home country in the global the output share sn writes: sn 1
(sL) is higher put grows e share in ffect). st reinforces
h countries good is portation
Lower transportation cost
0
International Economics Bénassy-Quéré & Coeuré 2009-2010
1
sL 18
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Limits of the model - Exporting firms are larger and more productive than strictly domestic firms ⇒ different ϕ, q, p Inequalities between firms, in terms of jobs and exports
Source: Crozet & Mayer (2007)
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Productivity of firms with more than 20 employees that enter the export markets
Source : Crozet & Mayer (2007)
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Limits of the model (2) - Define a ’firm extensive margin’: total trade flows equal the average value of exports per firm times the number of exporting firms. - Trade grows at both margins: Intensive and extensive (firm) margins of international trade, 2003
Source: Crozet & Mayer (2007) Interpretation: A point is a destination country for French exports. Closer/Bigger countries (in terms of GDP) are served by more firms and each firm export more, on average. “Cultural” proximity increases trade at the extensive margin.
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Extension: Melitz (2003)
- (To be thoroughly presented in the M2 course.) - Based on Krugman: Monopolistic competition, Iso-elastic preferences, Increasing returns to scale, Proportional transportation cost - with two additional assumptions: Fixed cost of entering export markets Heterogeneous firms (random productivity draws)
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Consequences
- Heterogeneity in terms of productivity transmits into an heterogeneity in terms of prices, quantities, profits: More productive firms sell more at a lower price - Only a subset of firms can sell positive quantities (those that are productive enough) (Selection) - With extra export costs, the subset of firms that do export is even smaller (exporters are even more productive) - Exogenous shocks induce adjustments at the extensive margin (Entries/exits into export markets) - New type of gains from trade: trade liberalization reallocates resources towards more productive firms.
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
• G(c) is a cumulative distribution function, Pr(C < c) = G(c).
– describes the cumulative probability of drawing a given marginal cost – g(c) proportional to the number of firms with a given productivity level
Mechanisms: Selection
• Firms with high marginal costs exit, firms with intermediate marginal costs serve only the domestic marmet, firms with low marginal costs also export. Exporters are rare. g(c)
Cost draw, c
0
cX Export and domestic markets
Serve domestic market only
cD
Exit
cM
8
G (c) is a cumulative distribution function (P(C < c) = G (c)) that describes the cumulative probability of drawing a given marginal cost. g (c) is proportional to the number of firms with a given productivity level
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
A model with heterogeneous firms: Trade liberalizations, productivity gains
Implications: Trade Liberalization
• New gains from trade: “Pro-competitive effects” of trade – – – – –
Weak firms are driven out, strong firms expand The industry’s average productivity increases Prices fall, expensive domestic varieties are replaced by cheaper imported ones But: The number of varieties may fall On net: Consumers better off from lower prices G(c)
Cost draw, c
0
cX
cX’
New exporters
cD
cD
Firms that exit
11
“Pro-competitive effect of trade”: Weak firms exit, strong firms expand. Average productivity increases as a consequence (Price level goes down).
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
The gravity equation with extensive margin Table: Decomposition of French aggregate exports (34 industries, 159 countries, 1986-1992)
ln GDPkj ln Distj contigj Colonyj Frenchj N R2
All firms Average Number of shipment shipments 0.461a 0.417a (.007) (.007) -0.325a -0.446a (.013) (.009) -0.064c -0.007 (.035) (.032) 0.100a 0.466a (.032) (.025) 0.213a 0.991a (.029) (.028) 23,553 23,553 0.480 0.591
Single-region firms Average Number of shipment shipments 0.421a 0.417a (.007) (.008) -0.363a -0.475a (.012) (.009) 0.002 0.190a (.038) (.036) 0.141a 0.442a (.035) (.027) 0.188a 1.015a (.032) (.028) 23,553 23,553 0.396 0.569
OLS estimates with year and industry dummies. Robust standard errors in parentheses.
Source: Crozet and Koenig, 2010
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Appendix
Limits and extensions
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
How to derive the demand function
- Lagrangien: L =
R
q(ω) 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)
⇔ q(ω)
−1 σ
Z q(ω)
σ−1 σ
1 σ−1
− µp(ω) = 0
dω
Ω
1
U σ = µp(ω)
⇔ p(ω)q(ω) = Uµ−σ p(ω)1−σ R R - Integrate over Ω: Ω p(ω)q(ω)dω = Uµ−σ Ω p(ω)1−σ dω and σ R σ−1 σ R σ−1 = Uµ−σ Ω p(ω)1−σ dω σ−1 U ≡ C = Ω q(ω) σ dω - Remember that wL ≡ PC , this gives: P =
R Ω
p(ω)1−σ
1 1−σ
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
How to derive the demand function (2)
−1
1
- From: wL = Uµ−σ P 1−σ and q(ω) σ U σ = µp(ω), one obtains the demand function: −σ p(ω) wL q(ω) = P P ⇒ Everything else being equal, a 1% rise in p(ω) reduces demand q(ω) by σ% (ie σ measures the price-elasticity of demand) ⇒ The demand q(ω) also depends on the consumer’s purchasing power wL/P −σ p(ω) q(ω) = : Increasing the relative price of the ω variety by - q(ω 0) p(ω 0 ) 1% reduces the relative demand for this variety by σ% (elasticity of substitution
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
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 function: −σ wL q(ω) = p(ω) P P (Monopolistic competition → The firm considers aggregate prices as given ⇒ 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
Appendix
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Price indices in a two-country economy (Krugman) - The price index now writes: ! P=
X ω∈H
p(ω)
1−σ
+
X
∗
1 1−σ
1−σ
(τ p (ω))
ω∈F
- In the symmetric equilibrium, p(ω) = p, ∀ω ∈ H and p ∗ (ω) = p ∗ , ∀ω ∈ F 1 1−σ ⇒ P = np 1−σ + n∗ (τ p ∗ )1−σ 1 1−σ and P ∗ = n(τ p)1−σ + n∗ p ∗ 1−σ - Absent transportation costs (τ = 1), if marginal costs are equalized (p = p ∗ ), the two indices are equal whatever the relative size of the two 1 countries: 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 1 P ∗ = n 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
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Wages in the Krugman model - We have expressed prices p(ω) and P as functions of nominal income wL, based on consumer’s and firm’s optimization - L is exogenous but w is endogenous - In order to derive the wage level, you need to introduce one last equation: goods market equilibrium. Due to the Walras law, it is equivalent to rely on (i) the domestic market (wL = sumω wl(ω)); (ii) the foreign market (w ∗ L∗ = sumω w ∗ l ∗ (ω)); (iii) the trade balance (X = X ∗ ) - We used the trade balance: 1−σ τ w 1−σ τw∗ ∗ ∗ λ × L × L∗ × × w = λ × L × L × ×w P∗ P 1−σ σ w P ⇒ = w∗ P∗ np 1−σ + n∗ (τ p ∗ )1−σ P with = ∗ P n(τ p)1−σ + n∗ p ∗ 1−σ 1/σ Lw 1−σ + L∗ (τ w ∗ )1−σ w ⇒ = w∗ L(τ w )1−σ + L∗ w ∗ 1−σ
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Wages in the Krugman model (2) Relative imports: M n∗ = M∗ n
τ p ∗ /P τ p/P ∗
1−σ
wL w = ∗ ∗ ∗ 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
Back to section 1
The Krugman Model
Specialization: Helpman-Krugman
Limits and extensions
Appendix
Specialization (Helpman-Krugman) - All firms producing differentiated goods produce the same quantity q that they sell at the same price p: p −σ wL τ p −σ w ∗ L∗ q = qD + τ qX = µ + τµ P P P∗ P∗ - Replace price indices by their open-economy expressions: q=µ
np 1−σ
(τ p)−σ p −σ wL + τ µ w ∗ L∗ ∗ ∗ 1−σ 1−σ + n (τ p ) n(τ p) + n∗ p ∗ 1−σ
- Perfect labor mobility across sectors + Trade in homogeneous goods (same wage). Assume ϕ = σ/(σ − 1) so that p(ω) = w = 1 (normalization). The production of differentiated good writes, for each variety: L τ 1−σ L∗ τ 1−σ L L∗ ∗ q=µ + q = µ + n + n∗ τ 1−σ nτ 1−σ + n∗ n + n∗ τ 1−σ nτ 1−σ + n∗ - Since q = q ∗ , we have: L τ 1−σ L∗ τ 1−σ L L∗ + = + ∗ 1−σ 1−σ ∗ ∗ 1−σ 1−σ n+n τ nτ +n n+n τ nτ + n∗ or:
L L 1−σ n 1 − ∗ τ 1−σ = n∗ − τ L L∗
