1. Classic Trade Theory
Ricardian Model - Technological Comparative Advantage:
Basic 2 Good Ricardian model (Feenstra, Chapter 1)
Continuum of Goods [Dornbush, Fischer and Samuelson (1977)]
Heckscher-Ohlin Factor Endowment Model:
2 Good 2 Factor Model (Feenstra, Chapters 1)
Stolper Samuelson
Leontief Paradox
Heckscher-Ohlin-Vanek Model and Tests of HO theory

2. Basic Ricardian 2 Good Model
We assume country specific technologies in the form of labor requirements: Home [ no *] : ai
Foreign [*] : ai*
The basic model allows 2 goods/ industries, i = 1, 2
Labor is exogenous, and mobile across industries: L, L*
Markets clear so workers are paid value of marginal product:
𝑝 𝑖 = 𝑤 𝑖 𝑎 𝑖
The mobility of workers implies that wages equalize across industries:
𝑝 1 𝑎 1 = 𝑝 2 𝑎 2
Set p2 as the numeraire and denote relative price p:
𝑝= 𝑝 1 𝑝 2
𝑝= 𝑎 1 𝑎 2

3. Economic Output under Autarchy (no trade)
Home PPF
Foreign PPF y2
y2*
L*/a2*
L/a2 A* A pa
pa*
L/a1 y1
L*/a1*
y1*
L is labor endowment
A is consumption point
𝑦 𝑖 is the output of industry I
𝑝 𝑎 is the autarchy price

4. Technology determines Comparative Advantage
Define Relative home productivity: 𝑅 1 = 𝑎 1 ∗ 𝑎 1 Suppose Home has a comparative advantage in good 1: 𝑎 1 𝑎 2 < 𝑎 1 ∗ 𝑎 2 ∗ 𝑝 𝑎 < 𝑝 𝑎∗ 𝑎 1 𝑤 1 < 𝑎 1 ∗ 𝑤 1 ∗ 𝑤 1 𝑤 1 ∗ < 𝑅 1 Comparative advantage is a function of relative wages and technology, but wages are endogenous.

5. World Relative Supply and Demand
pa* pe pa
Relative Demand
(𝐿/ 𝑎 1 )/( 𝐿 ∗ / 𝑎 2 ∗ )
(𝑦 1 + 𝑦 1 ∗ )/( 𝑦 2 + 𝑦 2 ∗ )

6. p
Relative Supply
pe = pa* pa
Relative Demand
(𝐿/ 𝑎 1 )/( 𝐿 ∗ / 𝑎 2 ∗ )
(𝑦 1 + 𝑦 1 ∗ )/( 𝑦 2 + 𝑦 2 ∗ )
Here the demand for good one is high enough that the equilibrium price of trade is equal to the autarchy foreign price
Foreign is no better off with trade.
When might this occur?

7. Gains from Trade p Relative Supply pa* pe pa Relative Demand
(𝐿/ 𝑎 1 )/( 𝐿 ∗ / 𝑎 2 ∗ )
(𝑦 1 + 𝑦 1 ∗ )/( 𝑦 2 + 𝑦 2 ∗ )
Here the trade price of good one is higher than in autarchy at home, and lower than in autarchy in foreign.
Both countries can gain from trade, and will fully specialize in their comparative advantage.

8. y2
y2*
L*/a2* B* p
L/a2 C C* A* A pa p B
pa*
L/a1 y1
L*/a1*
y1*
Points B and B* represent the level of production of goods 1 and 2 in each country under specialization.
C and C* are consumption in home and foreign.

9. Comparative advantage and Wages (Feenstra pg. 4 note 2)
Suppose Home has a comparative advantage in good 1.
a1/a2 < a1*/a2*
and an absolute disadvantage in both goods.
a1 > a1*
a2 > a2*
In Free Trade suppose each country specializes (pa < p < pa*)
-workers at home produce good 1 and earn w = p / a1
-workers in foreign produce good 2 earn w* = 1 / a2*
p = a1/a2 < a1*/a2*, rearranging: / a2* > p / a1*
We assumed a1 > a1*, resulting in p / a1* > p / a1
Wages in Foreign, w* = 1 / a2* > p / a1*> p / a1 = w

10. Beyond 2 Goods, Dornbush, Fischer and Samuelson (1977)
Generalization of Ricardian model to a continuum of goods, 𝑧∈[0,1] to allow integration.
Again 2 countries, one factor of production, labor, in fixed supply (L and L*)
Country-specific constant unit labor requirements: a (z) and a*(z)
With relative home productivity which can be written as before:
𝐴 𝑧 = 𝑎 ∗ 𝑧 𝑎 𝑧
Assume that we can rank all goods so that A(z) is continuous and decreasing in z.
price = marginal cost: P(z) = a(z)w and P*(z) = a*(z)w*
Assume home production: P (z) < P*(z) or A(z) > w/w*

11. Strong Structure on Demand
Homothetic Demand: after winning the lottery I take my grocery list and double the purchases of every item.
Constant expenditure Share b, demand C, total income Y
Identical tastes across countries
Many products:
𝑏 𝑖 = 𝑃 𝑖 𝐶 𝑖 /𝑌
1 𝑛 𝑏 𝑖 =1
Continuum of products:
𝑏 𝑧 = 𝑃 𝑧 𝐶 𝑧 𝑌 >0
b(z)=b*(z)
0 1 𝑏 𝑧 𝑑𝑧=1
Production at home:
𝑣 𝑧 = 0 𝑧 𝑏 𝑧 𝑑𝑧

12. Trade Equilibrium (DFS 1977)
Domestic wages must equal world spending on domestic goods
𝑤𝐿=𝑣( 𝑧 )(𝑤𝐿+ 𝑤 ∗ 𝐿 ∗ )
Rewriting in terms of relative wages:
w w ∗ = 𝑣 𝑧 1−𝑣 𝑧 L ∗ L =B 𝑧 ; L ∗ L
Combine with the conditions for Home Production:
𝐴 𝑧 = 𝑤 𝑤 ∗ =𝜔

13. Gains from Trade (DFS 1977)
Rest of the World

14. Hypothetical DFS Population Growth
Suppose that the foreign country increases their population (possibly by integrating with another country).
B(L*/L) is increasing in L*.
Home will produce fewer goods, but wages rise. Home is better off, Real income rises
Workers in F lose, wages decline in terms of goods produced abroad.
This is a Terms of Trade effect. The terms of trade are said to improve if the index of the price of a country's exports in terms of its imports rises.

15. DFS and a Gravity Equation
Note from equilibrium conditions
𝑣 𝑧 = 𝑤𝐿 𝑤 ∗ 𝐿 ∗ +𝑤𝐿
1−𝑣 𝑧 = 𝑤 ∗ 𝐿 ∗ 𝑤 ∗ 𝐿 ∗ +𝑤𝐿
Trade Volume is
𝑤 ∗ 𝐿 ∗ 𝑣 𝑧 +𝑤𝐿 1−𝑣 𝑧 = 2 𝑤 ∗ 𝐿 ∗ 𝑤𝐿 𝑤 ∗ 𝐿 ∗ +𝑤𝐿 = 2𝑌 𝑌 ∗ 𝑌 𝑊
Notice similarity to Newton's law of universal gravitation
Image © The Whipple Museum.

16. Criticisms of DFS/ Ricardian Model
Does not generalize easily to more than two countries
Comparative Advantage is generated by an exogenous technology
Assuming that costs are different across countries makes prediction difficult.
Why wouldn’t technology spill across borders?

17. The Heckscher-Ohlin (HO) Model
The law of comparative advantage says that countries trade when autarky prices are different from free trade prices
Assuming persistent differences in technology seemed unprincipled.
Variation in autarky prices can come from differences in factor endowments (capital, labor, skilled labor, land)
Heckscher-Ohlin model
At least two factors
Factors are stuck in countries
Technology is identical in all countries

18. H-O Two good, Two Factors (Labor and Capital) One Production Function (identical technology): 𝑦 𝑖 = 𝑓 𝑖 𝐿 𝑖 , 𝐾 𝑖 , 𝑖=1, 2 f is increasing, concave, and homogenous of degree 1 in L and K (Constant Returns to Scale) Resource Constraints: 𝐿 1 + 𝐿 2 ≤𝐿 𝐾 1 + 𝐾 2 ≤𝐾

19. Duality: Unit Cost Functions
Feenstra works with the dual of the production function:
𝑐 𝑖 𝑤,𝑟 = min 𝐿 𝑖 ,𝐾 𝑖 𝑤 𝐿 𝑖 +𝑟 𝐾 𝑖 𝑓 𝑖 𝐿 𝑖 , 𝐾 𝑖 ≥1
The solution of this function is written:
𝑐 𝑖 𝑤,𝑟 =𝑤 𝑎 𝑖𝐿 𝑤,𝑟 +𝑟 𝑎 𝑖𝐾 𝑤,𝑟
Note that using the envelope theorem (Shephard’s Lemma):
𝜕 𝑐 𝑖 𝑤,𝑟 𝜕𝑤 = 𝑎 𝑖𝐿

20. Equilibrium Conditions
Zero Profit Conditions: Free entry in perfectly competition determines prices
𝑝 1 = 𝑐 1 𝑤,𝑟 , 𝑝 2 = 𝑐 2 (𝑤,𝑟)
Full employment Conditions: All resources are consumed: 𝐿 𝑖 = 𝑦 𝑖 𝑎 𝑖𝐿 and 𝐿 1 + 𝐿 2 =𝐿
𝑎 1𝐿 𝑦 1 + 𝑎 2𝐿 𝑦 2 =𝐿, 𝑎 1𝐾 𝑦 1 + 𝑎 2𝐾 𝑦 2 =𝐾
Notice that fixing prices, ZP determine w and r.
The remaining equations determine 𝑎 𝑖𝐾 and 𝑎 𝑖𝐿 .
In equilibrium sectoral output levels (𝑦 1 , 𝑦 2 ) will depend on K and L but not factor intensities.

21. Factor-price Insensitivity (Feenstra Theorem 1)
With two goods and two factors, factor prices are uniquely determined by goods prices, irrespective of factor endowments.

22. Factor-price equalization (Feenstra Theorem 2)
Assume free trade, no transport cost, two goods, two countries, two factors, identical technologies, no factor-intensity reversals, and both countries produce both goods.
Then factor prices are equalized in both countries irrespective of factor endowments.

23. Stolper Samuelson Theorem
An increase in the price of a good will more than proportionally increase the returns to the factor intensively used in the production of that good.
Trade raises the real reward of a country’s abundant factor and reduces the real reward of its scarce factor.
Trade leads to a conflict of interest between the scarce and abundant inputs.
Trade leads to factor price convergence (Ohlin emphasized the tendency towards factor price convergence, but not equalization).

24. Mathematical Statement of the Stolper Samuelson Theorem
Suppose that: 𝑎 𝐿 1 𝑎 𝐾 1 > 𝑎 𝐿 2 𝑎 𝐾 2 then 𝜕ln 𝑤 𝜕 ln 𝑝 1 >1 Proof: Take total derivative of 𝐶 𝑖 𝑤,𝑟 = 𝑝 𝑖 for i=1,2 with respect to 𝑝 1 : 𝑎 𝐿 1 𝑑𝑤+ 𝑎 𝐾 1 𝑑𝑟=𝑑 𝑝 1 𝑎 𝐿 2 𝑑𝑤+ 𝑎 𝐾 2 𝑑𝑟=0

25. 𝑎 𝐿 1 𝑑𝑤+ 𝑎 𝐾 1 𝑑𝑟=𝑑 𝑝 1 𝑎 𝐿 2 𝑑𝑤+ 𝑎 𝐾 2 𝑑𝑟=0 Eliminate ⅆ𝑟 term: 𝑎 𝐿 1 − 𝑎 𝐾 1 𝑎 𝐿 2 / 𝑎 𝑘 2 = 𝑑 𝑝 1 𝑑𝑤 1 𝑎 𝐿 1 − 𝑎 𝐾 1 𝑎 𝐿 2 𝑎 𝑘 2 = 𝑑𝑤 𝑑 𝑝 1
(1)
1/ 𝑎 𝐾 𝑎 𝐿 1 𝑎 𝐾 1 − 𝑎 𝐿 2 𝑎 𝑘 2 = 𝑑𝑤 𝑑 𝑝 1

26. 𝑝 1 𝑤 = 𝑎 𝐿 1 𝑤+ 𝑎 𝐾 1 𝑟 𝑤 > 𝑎 𝐿 1 𝑤 𝑤 = 𝑎 𝐿 1
From homogeneity of cost function and Shephard’s lemma: 𝑝 1 = 𝑎 𝐿 1 𝑤+ 𝑎 𝐾 1 𝑟 Divide by w and see that
(2)
𝑝 1 𝑤 = 𝑎 𝐿 1 𝑤+ 𝑎 𝐾 1 𝑟 𝑤 > 𝑎 𝐿 1 𝑤 𝑤 = 𝑎 𝐿 1
Putting equations (1) and (2) together:
𝑝 1 𝑑𝑤 𝑤 𝑑 𝑝 1 > 𝑎 L 1 / 𝑎 𝐾 𝑎 𝐿 1 𝑎 𝐾 1 − 𝑎 𝐿 2 𝑎 𝑘 2 >1 ∎

27. Many Goods Heckscher-Ohlin-Vanek Model (Feenstra chapter 2)
M factors, world endowment vector 𝑽= ( 𝑉 1 ,…, 𝑉 𝑀 ), indexed by k
C countries with endowments 𝑽 𝒊 = ( 𝑉 1 𝑖 , … , 𝑉 𝑁 𝑖 ), indexed by i
factors are immobile across countries, perfectly mobile within countries
N industries produced with CRS as before.
𝐴= 𝑎 𝑗𝑘 ′ is the factor requirement of production. The transposition implies that columns are the industries, rows are the factors.

28. Factor Content of Trade
𝑌 𝑖 is the vector of outputs in each industry for country i.
𝐷 𝑖 is the vector of demands for goods in each country i.
𝑇 𝑖 = 𝑌 𝑖 − 𝐷 𝑖 is the net exports of country i.
Factor content of trade is 𝐹 𝑖 ≡𝐴 𝑇 𝑖
𝐴 𝑌 𝑖 is the demand for factors in i, set equal to endowment 𝑉 𝑖 .
𝐷 𝑖 is homothetic and identical across countries, and trade is balanced so must be the share of income times world demand.
𝐷 𝑖 = 𝑝 ′ 𝑌 𝑖 𝑝 ′ 𝑌 𝑊 𝐷 𝑊
World consumption equals world production:
𝑝 ′ 𝑌 𝑖 𝑝 ′ 𝑌 𝑊 𝐴𝐷 𝑊 = 𝑝 ′ 𝑌 𝑖 𝑝 ′ 𝑌 𝑊 𝐴 𝑌 𝑊 = 𝑝 ′ 𝑌 𝑖 𝑝 ′ 𝑌 𝑊 𝑉 𝑊
𝐹 𝑖 ≡𝐴 𝑇 𝑖 =𝐴 𝑌 𝑖 − 𝐷 𝑖 =𝐴 𝑌 𝑖 −𝐴 𝐷 𝑖 = 𝑉 𝑖 − 𝑝 ′ 𝑌 𝑖 𝑝 ′ 𝑌 𝑊 𝑉 𝑊

29. Pattern Of Trade in Each Factor
𝐹 𝑘 𝑖 = 𝑉 𝑘 𝑗 − 𝑝’ 𝑌 𝑖 𝑝’ 𝑌 𝑊 𝑉 𝑘 𝑊 is the Heckscher-Ohlin-Vanek Theorem for each factor 𝑘.
Comparative advantage in each factor k can be determined by the share of world endowment of factor k over the share of world GDP.
If 𝑉 𝑘 𝑖 𝑉 𝑘 𝑊 > 𝑝 ′ 𝑌 𝑖 𝑝 ′ 𝑌 𝑊 then country 𝑖 is abundant in factor 𝑘.
This implies that 𝐹 𝑘 𝑖 >0, or that the abundant factor is exported.

30. The Leontief Paradox Using U.S. input-output tables from 1947:
The capital labor ratio in U.S. exports was $13,700 of capital per man-year
The capital labor ratio in U.S. imports was $18,200 of capital per man-year
The U.S. is relatively capital abundant after WWII and should export capital intensive products and import labor intensive products.

31. Leamer 1980 Observation
Leontief studied the factor content of exports and imports, but what matters is the relative factor abundance of production and consumption.
𝐾 𝑖 − 𝐹 𝐾 𝐿 is the factor content of consumption of capital, while 𝐿 𝑖 − 𝐹 𝑙 𝑖 is the factor content of consumption of labor.
For a capital abundant country
𝐾 𝑖 𝐾 𝑖 − 𝐹 𝑘 𝑖 > 𝐿 𝑖 𝐿 𝑖 − 𝐹 𝑙 𝑖
We would expect U.S. production to be more capital intensive than its consumption, which according to Feenstra Table 2.2, it is (slightly).

32. Edgeworth Box of Factor content of Trade
𝑂 2
𝑄 is employment in industry 1.
Good 2
𝑉 𝑖
𝐹 𝑘 1
𝐴𝐷 𝑖
𝑲 𝑾 = 𝑲 𝟏 + 𝑲 𝟐
𝐹 𝑘 2
𝑄 1
Good 1
𝑂 1
𝑳 𝑾 = 𝑳 𝟏 + 𝑳 𝟐

33. Edgeworth Box and Share of trade
𝑂 2
w/r
𝑉 𝑖
𝐴𝐷 𝑖
𝑲 𝑾 = 𝑲 𝟏 + 𝑲 𝟐
X 2 𝐻
𝐶 2 𝐻
𝐶 1 𝐻
𝑋 1 𝐻
𝑂 1
𝑳 𝑾 = 𝑳 𝟏 + 𝑳 𝟐

34. Integrated Equilibrium: Imagine there are no countries
Zero profit conditions
𝑝 𝑖 = 𝑐 𝑖 𝒘 𝑖=1,…𝑁
Factor markets clear
𝑖∈𝑁 𝑎 𝑛𝑖 𝒘 𝑦 𝑖 = 𝑉 𝑗 𝑗=1,…𝑀
Where 𝑦 𝑖 is the quantity of good i.
Goods market clear
𝑏 𝑖 𝐩 𝑗∈I 𝑝 𝑗 𝑥 𝑗 = 𝑝 𝑖 𝑥 𝑖 ∀𝑖∈𝐼
Notice that the number of equations is equal to the number of unknown parameters.

35. Return to Factor Price Equalization
Goal is to replicate the equilibrium in the integrated equilibrium using trade in goods.
Assume:
every country uses the same input mix as in the integrated equilibrium.
Full employment of factors in all countries.
Because every country is using the same technologies, the factor prices will be the same. Because the factor prices are the same the goods prices p will be the same.
Full employment and identical (p, w) imply the same income and production as the integrated equilibrium.