1. Chapter 1: Two-Sector Models
Different Technologies: Ricardo
Different Factor Endowments: Heckscher-Ohlin
Assumptions:
Factors: Immobile between countries,
perfectly mobile between sectors.
Goods: Perfectly mobile between countries.
Identical homothetic utility functions relative quantities of
consumption goods depend only on relative prices,
not on income (distribution).

2. Topics of chapter 1 Ricardian model Equilibrium – factor prices
Gains from trade
Heckscher Ohlin model
Equilibrium – factor price equalization
Comparative Statics:
Change in product prices
Change in factor endowments
Factor intensity reversals

3. 1.1 Ricardian model 2 goods, 1 factor of production (labor). Notation:
yi…domestically produced quantity of good i, i=1,2
ai…labor needed per unit of good i, ai= Li/yi.
L…domestic labor force.
A * refers to the foreign country, hence
L*…labor force of the foreign country.
Production function:
yi = Li/ai,  marginal product of labor = 1/ai.

4. Equilibrium in autarky:
Both goods are produced only if wage w is the same in both sectors. In equilibrium wage equals value of marginal product, hence
w = p1/a1 = p2/a2.
Using good 2 as numeraire we get in equilibrium
p = p1/p2 = a1/a2.
The domestic country has a comparative advantage in producing good 1 if
pa < pa*  a1/a2 < a*1/a*2

5. Figure 1.1 Equilibria in Autarky

6. Free trade equilibrium:
Relative demand d = d1/d2 is a decreasing function of (world market) price p.
Relative supply y = (y1 + y*1)/(y2 + y*2) is an increasing step function of p:
p < pa  y1 = 0 [y2 = L/a2, y*2 = L*/a*2].
pa* > p > pa  y2 = 0, y1 = L/a1, y*2 = L*/a*2.
p > pa*  y1 = L/a1, y*1 = L*/a*1.
If the equilibrium price with free trade is strictly between the autarky prices then there is complete specialization in both countries and both countries gain from trade.

7. Trade patterns (and gains from trade) are determined by comparative advantage.
Absolute advantage matters for income determination: Suppose the domestic country has an absolute disadvantage in the production of both goods, i.e.
a*1 < a1, a*2 < a2.
w = p/a1 < p/a*1 < 1/ a*2 = w*.

8. Figure 1.2 Trade Equilibrium

9. Figure 1. 1a: Gains from trade of home country: let pa < p < pa
Figure 1.1a: Gains from trade of home country: let pa < p < pa*, compete specialization (production at point B, consumption at point above indifference through A (production = consumption in autarky))

10. 1.2 Two-Goods, Two-Factors Model
“Small Country“ Assumption: World market prices exogenous.
Two factors of production, capital K and labor L.
Production functions: concave and increasing in L and K, homogeneous of degree one (constant returns to scale):
yi = fi(Li,Ki),
L1 + L2  L
K1 + K2  K.

11. Production possibility frontier (PPF):
Maximum value function of the optimization problem:
max y2 = f2(L2,K2) s.t. y1 = f1(L1,K1) = y,
L1 + L2  L, K1 + K2  K.
Denote the solution as h(y1,L,K). h(…) is decreasing and concave in y1.
Gross domestic product (GDP) function:
G(p1,p2,L,K) = max p1y1 + p2y2 s.t y2 = h(y1,L,K).

12. This is equivalent to
max G(y1) = p1y1 + p2h(y1,L,K),
implying the FOC
p1 + p2[h/y1] = or
p = p1/p2 = -h/y1 = -y2/y1,
i.e. in the optimum the price ratio equals the slope of the PPF.
By the envelope theorem we get
G/pi = yi.

13. Equilibrium Conditions
Unit cost functions:
ci(w,r) = min wLi + rKi s.t. fi(Li,Ki)  1, Li,Ki  0.
ci(w,r) is non-decreasing and concave in w and r.
Denoting the optimal input-coefficients for labor and capital as aiL and aiK, it can be written as
ci(w,r) = waiL + raiK.
By the envelope theorem we get
ci/w = aiL, ci/r = aiK.

14. Two sets of equilibrium conditions:
1. Zero-profit condition:
p1 = c1(w,r) (E1)
p2 = c2(w,r) (E2)
2. Full-employment condition:
a1Ly1 + a2Ly2 = L, (E3)
a1Ky1 + a2Ky2 = K (E4)

15. Figure 1.3 Production Possibility Frontier

16. Factor prices
Lemma: So long as both goods are produced and factor intensity reversals do not occur, then each price vector (p1, p2) corresponds to unique factor prices (w,r).
Factor intensity reversals:  s such that
a1K/a1L > a2K/a2L for r/w > s and
a1K/a1L < a2K/a2L for r/w < s.
Equilibrium conditions (E1,E2) have more than one intersection in (r,w)-space: total differential of zero-profit condition: 0 = aiLdw + aiKdr 
dr/dw = -aiL/aiK.

17. Figure 1.4 Factor Price Insensitivity

18. Figure 1.5 Factor Intensity Reversal

19. Factor Price Equalization Theorem (Samuelson)
2 countries, two goods, both countries produce both goods in a free trade equilibrium, identical technologies, different factor endowments, no factor intensity reversals (FIRs)  factor prices (w,r) are equalized across countries.
Proof: Figure 1.4 is the same for both countries  same equilibrium prices in both countries imply identical factor prices.

20. Stolper-Samuelson Theorem
An increase in the relative price of a good will increase the real return to the factor used intensively in that good, and reduce the real return to the other factor.
Proof: Let p = p1/p2,  = w/r. From the equilibrium conditions we get
p = [a1L + a1K]/[a2L + a2K]
implying

21. Rybczynski Theorem
An increase in a factor endowment will increase the output of the industry using it intensively, and decrease the output of the other industry.
Proof: Factor market clearing implies
L = y1a1L + y2a2L
K = y1a1K + y2a2K
Define k  K/L und y  y2/y1, hence
k = [a1K + ya2K]/[a1L + ya2L] implying

22. Interpretation: There is a unique relationship between relative factor endowments and relative outputs, the share of the capital intensive good increases when the capital endowment is increased. With identical homothetic preferences this implies that the capital intensive good must be relatively cheaper in the capital rich country.
Absolute changes: totally differentiating the market clearing conditions yields:
dL = a1Ldy1 + a2Ldy2
dK = a1Kdy1 + a2Kdy2

23. By Cramer‘s rule we get
dy1 = [a2KdL - a2LdK]/[a1La2K - a2La1K]
dy2 = [a1LdK - a1KdL]/[a1La2K - a2La1K]
From the first equation follows
dy1/dL  0  a1La2K - a2La1K  0  a1L/a1K  a2L/a2K
Interpretation: An increase in the labor endowment induces an increase in the output of good 1 if and only if its production is more labor intensive than the production of good 2.
Empirical Example: „Dutch disease“: discovery of oil moved resources away from less oil intensive export industries.

24. Alternative Illustration of the Rybczynski Theorem
Recall the full employment conditions:
L = y1a1L + y2a2L
K = y1a1K + y2a2K
Multiplying the vectors (a1L,a1K) and (a2L,a2K) by y1 and y2 respectively and summing them up yields the total vector endowments (L,K). Conversely, for each factor endowment (L,K) and vectors (aiL,aiK) there exist outputs yi, i=1,2 s.t. the factor markets are cleared. Figure 1.6 shows that outputs are positive iff endowment vector lies between factor requirement vectors.

25. Figure 1.6: Cone of diversification

26. Suppose next that L is increased to L‘. From Figure 1
Suppose next that L is increased to L‘. From Figure 1.7 it can be seen that full employment of factors can only be maintained if the quantity of the capital intensive good 2 is reduced and that of the labor intensive good 1 is increased.

27. Figure 1.7: Increase of Labor Endowment

28. Rybczynski line:
As L (or K) is increased the PPF is shifted out. The Rybczynski line connects all equilibrium points for K (L) constant and L (K) changed. Its slope is obtained as follows:
K = y1a1K + y2a2K a1Kdy1 + a2Kdy2 = 0 
dy2/dy1 = -a1K/a2K< 0. (See Figure 1.8).

29. Figure 1.8: Rybczynski line

30. Integrated World Equilibrium
Thought experiment: Consider a world with perfect mobility of all factors. Compute its equilibrium (Integrated World Equilibrium). How could factor endowments be distributed between two countries between which there is no factor mobility such that the IWE would be reproduced as a free trade equilibrium? Figure 1.9 shows the set of all such factor endowments (0A10*A2:  y,y* s.t. sum of input vectors equals world factor endowment).

31. Figure 1.9: Factor Price Equalization Set