🚢 International Trade Theories & Models

International trade theory explains why nations exchange goods and services, how trade patterns are determined across borders, and the distributional consequences of globalization on domestic factor returns.

---

1. 🌾 The Ricardian Model of Comparative Advantage

David Ricardo (1817) demonstrated that trade gains depend on differences in labor productivities (technology) rather than absolute cost advantages.

1.1 Opportunity Cost Criterion

Consider two nations (Home $H$, Foreign $F$) and two goods (Cloth $C$, Wine $W$) with unit labor requirements $a_{LC},a_{LW}$:

$$\text{Opportunity Cost of Cloth in terms of Wine at Home: }O{C}_C^H=\frac{a_{LC}}{a_{LW}}$$$$\text{Opportunity Cost of Cloth in terms of Wine at Foreign: }O{C}_C^F=\frac{a_{LC}^{\ast }}{a_{LW}^{\ast }}$$

• Home has a Comparative Advantage in Cloth if:$$\frac{a_{LC}}{a_{LW}}<\frac{a_{LC}^{\ast }}{a_{LW}^{\ast }}$$
• Both countries gain from trade if the terms of trade $P_C/P_W$ lie strictly between domestic autarky price ratios:$$\frac{a_{LC}}{a_{LW}}<\frac{P_C}{P_W}<\frac{a_{LC}^{\ast }}{a_{LW}^{\ast }}$$

---

2. 🏭 The Heckscher-Ohlin (H-O) Factor Proportions Model

The $2\times 2\times 2$ Heckscher-Ohlin model assumes identical technologies across countries, explaining trade patterns through differences in national factor endowments (Labor $L$ vs. Capital $K$).

                        THE FOUR CORE THEOREMS OF H-O
 ┌───────────────────────────┬─────────────────────────────────────────────────────────┐
 │ Theorem                   │ Statement & Economic Mechanism                          │
 ├───────────────────────────┼─────────────────────────────────────────────────────────┤
 │ 1. Heckscher-Ohlin        │ A country exports goods that intensively use its        │
 │    Theorem                │ relatively abundant factor of production.               │
 ├───────────────────────────┼─────────────────────────────────────────────────────────┤
 │ 2. Stolper-Samuelson      │ An increase in the relative price of a good increases   │
 │    Theorem                │ the real return to the factor used intensively in it,   │
 │                           │ and reduces the real return to the other factor.        │
 ├───────────────────────────┼─────────────────────────────────────────────────────────┤
 │ 3. Rybczynski             │ At constant commodity prices, an increase in a factor's │
 │    Theorem                │ endowment increases the output of the intensive good    │
 │                           │ more than proportionately and reduces other output.     │
 ├───────────────────────────┼─────────────────────────────────────────────────────────┤
 │ 4. Factor Price           │ Free trade in goods equalizes real wages (w) and real   │
 │    Equalization (FPE)     │ capital rental rates (r) across countries under CRS.    │
 └───────────────────────────┴─────────────────────────────────────────────────────────┘

---

3. 🌐 Krugman New Trade Theory (1979)

Classical models explain inter-industry trade between dissimilar countries. Paul Krugman introduced New Trade Theory to explain intra-industry trade (e.g. Germany and Japan trading cars) based on:

1. Monopolistic Competition (Dixit-Stiglitz): Consumers have a "love of variety" ($U=(\sum c_i^{\rho }{)}^{1/\rho }$).
2. Internal Economies of Scale: Increasing returns at the firm level allow larger global markets to offer more product varieties at lower average costs ($AC\downarrow$).

---

4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Ricardian Production Possibility Frontier & Trade Gains

Problem: Home has $L=1200$ hours of labor. Unit labor requirements are $a_{LC}=2$ hours/yard of cloth and $a_{LW}=4$ hours/bottle of wine. Foreign has $L^{\ast }=800$ hours with $a_{LC}^{\ast }=4$ hours/yard of cloth and $a_{LW}^{\ast }=2$ hours/bottle of wine.

1. Determine opportunity costs and identify comparative advantage for each country.
2. If world equilibrium terms of trade settle at $P_C/P_W=1$, calculate total world output and consumption gains relative to autarky where each country spent $50\mathrm{\%}$ of labor on each good.

Step-by-Step Rigorous Solution:

1. Calculate Opportunity Costs:

   • Home: $O{C}_C=\frac{a_{LC}}{a_{LW}}=\frac{2}{4}=0.5\text{ wine/cloth}$.
   • Foreign: $O{C}_C^{\ast }=\frac{a_{LC}^{\ast }}{a_{LW}^{\ast }}=\frac{4}{2}=2.0\text{ wine/cloth}$.
   • Since $0.5<2.0$, Home has comparative advantage in Cloth, and Foreign has comparative advantage in Wine.

2. Autarky Output (50% labor split):

   • Home: $Q_C=\frac{600}{2}=300\text{ cloth}$, $Q_W=\frac{600}{4}=150\text{ wine}$.
   • Foreign: $Q_C^{\ast }=\frac{400}{4}=100\text{ cloth}$, $Q_W^{\ast }=\frac{400}{2}=200\text{ wine}$.
   • World Total: $Q_C^{aut}=400\text{ cloth}$, $Q_W^{aut}=350\text{ wine}$.

3. Specialized Trade Output (100% specialization):

   • Home produces only Cloth: $Q_C=\frac{1200}{2}=600\text{ cloth},Q_W=0$.
   • Foreign produces only Wine: $Q_C^{\ast }=0,Q_W^{\ast }=\frac{800}{2}=400\text{ wine}$.
   • World Total: $Q_C^{trade}=600\text{ cloth}$, $Q_W^{trade}=400\text{ wine}$.
   $$\text{Net World Output Gain}=+200\text{ yards of cloth}\text{ and }+50\text{ bottles of wine}$$

   Result: Mutual specialization according to comparative advantage expands the global consumption possibility frontier for all trading partners.