1. Chapter 3: Many Goods and Factors
3.1 Equilibrium conditions
3.2 Gains from trade
3.3 Same number of goods and factors
3.4 More factors than goods (Ricardo-Viner)
3.5 More goods than factors

2. 3.1 Equilibrium conditions
Notation: i = 1,…,N goods, j = 1,…,M factors, vi = fi(vi), vi = (vi1,….,viM) vector of factor inputs. Production functions are positive, increasing, concave, homogeneous of degree one for all vi0. wi...factor prices. ci(w)  minvi0{wvifi(vi)1}, unit cost functions with ci/w = ai(w) = vector of factor inputs per unit of output of good i.

3. Zero-profit conditions: pi = ci(w), i = 1,…,N (1) Factor-market clearing: , j = 1,…,M, (2) or AY = V (2)

4. Gross domestic product function: G(p,V) = maxpifi(vi) s. t
Gross domestic product function: G(p,V) = maxpifi(vi) s.t. vi  V (3) By the properties of a maximum value function: G/pi = fi(vi) = yi, (4) G/Vj = pifi/vij = wj. (5) Young‘s theorem: 2G/piVj = 2G/Vjpi, hence 2G/piVj = yi/Vj = 2G/Vjpi = wj/pi: (6) „reciprocity relations“: Rybczynksi derivatives are identical to Samuelson-Stolper derivatives.

5. 3.2 Gains from trade
3.2.1 Fixed factor supply
One consumer
3.2.2 Variable factor supplies
One consumer
Many consumers

6. 3.2.1 Fixed factor supply, one consumer
Expenditure function:
e(p,u) = min{pCu(C)u},
C is the consumption vector. Clearly, eu  0.
Equilibrium in autarky:
e(p,u) = G(p,V) (7)
ep(p,u) = Gp(p,V) (8)
i.e. total income equals total planned expenditures and demand equals supply.

7. Gains from trade:
Let superscript I denote variables in the free trade equilibrium and superscript a denote variables in the autarky equilibrium. We get
e(pI,ua)  pICa  definition of expenditure function
= pIYa  condition of autarky equilibrium
 G(pI,V)  definition of GDP- function
= e(pI,uI)  equilibrium condition (7).
Since e is increasing in u it follows that ua  uI.

8. 3.2.2 Variable factor supply 3.2.2.1 One consumer
Factor use becomes argument of utility function  modified expenditure function:
e(p,V,u) = min{pCu(C,V)  u}, (9)
implying
eV(p,V,u) = w, (10)
which yields the factor supplies V(p,w,u). Substituting this into (9) we get a general expenditure function e*(p,V,u) which gives the minimum lump-sum transfer needed to achieve utility u at product prices p and factor prices w.

9. Consequently
e*(p,w,u) = min{pC – wVu(C,V)  u}; (11) e* is increasing in p, decreasing in w, concave and homogeneous of degree one in {p,w} jointly.
Equilibrium in autarky:
e*(p,w,u) = (12)
e*p(p,w,u) = Gp(p,V) (13)
–e*w(p,w,u) = V (14)
w = GV(p,V) (15)

10. Gains from trade e*(pI,wI,ua)  pICa – wIVa  definition of e*
= pIYa – wIVa  autarky equilibrium
 pIYI – wIVI  profit maximization
= e*(pI,wI,uI)  income = expenditure.
Since e* is increasing in u it follows that ua  uI.

11. Many consumers
Problem: Free trade changes factor prices (Samuelson-Stolper), if consumers differ with respect to factor endowments/supply functions some of them may be worse off than in autarky.
Question: Could they be fully compensated by the „winners“ of free trade, leaving the latter with a net gain?

12. Lump-sum transfers
Autarky equilibrium:
cha = consumption vector of consumer h in autarky.
vha = vector of factor supplies of consumer h in autarky.
Equilibrium conditions:
 cha = Ya,
 vha = Va.
uha = utility of consumer h in autarky equilibrium

13. Denote as Lh the lump sum transfer (positive or negative) consumer h needs in order to get utility uha at prices p and w. By definition of e*:
Lh = e*(p,w,uha)
Total disbursement = net revenue of government:
Rh = –Lh.
If in the free trade equilibrium Rh  0 then there are gains of trade.
Suppose the government spends Rh on goods, spending the same amount of money on each good. This implies the following demand function:

14. gj(p,w) = –he*(p,w,uha)/npj.
Recall that e* is homogeneous of degree one in (p,w)  gj is homogeneous of degree zero in (p,w).
Demand for goods and supply of factors of consumer h are given by
ch = ep*(p,w,uha)
vh = – ew*(p,w,uha)
Net demand of the rest of the world T is a function of p, and balanced trade implies pT(p) = 0. Domestic production Y and factor demand V are determined by the profit maximization of firms.

15. Free trade equilibrium:
ch + g +T – Y = 0
vh – V = 0
Revealed preference arguments show g  0:
Note that by construction u(ch,vh) = u(cha,vha), but the former is optimal at prices (p,w), hence
Lh = pch – wvh  pcha – wvha (16)
(Y,V) and (Ya,Va) are both technically feasible, but the former is optimal at prices (p,w), and yields zero profits:
0 = pY – wV  pYa – wVa (17)

16. Adding (16) for all consumers and subtracting (17) from the result yields
Lh = p(ch – Y) – w(Vh – V)
 p(cha – Ya) – w(Vha – Va)
= p(Ca – Ya) – w(Va – Va)
= 0
Consequently,
R = – Lh  0 Q.E.D.
An analogous result holds if commodity and factor taxes are used instead of lump sum transfers.

17. 3.3 Same number of goods and factors
Equilibrium conditions:
pi = ci(w), i = 1,…,N (1)
A(w)Y = V (2)
Factor price equalization  no factor intensity reversals.
Sufficient condition: For all w > 0 matrix A satisfies:

18. Condition (18) states that all principal minors of the factor input coefficient matrix A are between two strictly positive bounds 0 < b < B.
Lemma (Samuelson , Nikaido 1972):
Let M = n and (18) hold for all w > 0. Then for all p > 0 the set of equations (1) has a unique solution for factor prices w > 0.
Factor endowments of all countries must be in the cone of diversification spanned by the columns of A(w).

19. Samuelson-Stolper theorem:
Differentiating (1) totally yields
dpi = ai1dw1 + … + aiMdwM (19)
Dividing both sides by pi = ci yields
dpi/pi = [w1ai1/ci][dw1/w1] +…+
+ [wMaiM/ci][dwM/wM] (19a)
Let ij = wjaij/ci = cost share of factor j in industry i, and ŵ = dw/w denote the percentage change of variable w. (19a) can be written as (“Jones’ Algebra”):
(19b)

20. Assume that each industry uses at least two inputs, each input is used in at least two industries and all goods are still produced after small price changes. If the price of good i is increased and the prices of all other goods remain unchanged then there must exist two factors j and k for which the following inequalities hold:
(19c)
For a change in the price of each good there will exist some factor that gains in real terms and another factor that loses.

21. Rybczynski theorem:
Differentiating (2) totally with respect to Vk, holding p and w fixed yields:
(20a)
(20b)
(20b)   good i s.t. dyi/dVk > 0, but then
(20a)   good j s.t. dyj/dVk < 0, thus for any increase in the endowment of each factor there must be a good whose output increases and another one whose output falls.

22. Enemies and friends Theorem (Jones and Scheinkman 1977): For each factor, there must be a good such that an increase in the price of that good will lower the return to the factor. Proof: Follows from (20) and the reciprocity relation (6): (20)  dyj/dVk < 0 for some good j, (6)  dwk/dpj < 0 for some good j. Since all other prices are fixed wk/pi has fallen (good j is the „enemy“ of factor k). But not every factor must have a „natural friend“.

23. 3.4 More factors than goods (Ricardo-Viner)
Samuelson-Stolper theorem:
Equation (19c) and thus the generalized version of the Samuelson-Stolper theorem also holds.
The generalized Rybczynski theorem and the Jones-Scheinkman theorem do NOT hold since a change of factor endowments changes factor prices and aij.

24. The specific factors model
Two goods, one mobile factor („labor“), and two sector specific factors (K1,K2).
GDP-function:
G(p,L,K1,K2) = max{p1f1(L1,K1) + p2f2(L – L1,K2)} (21)
In equilibrium:
w = p1f1L(L1,K1) = p2f2L(L – L1,K2) (22)
Exogenous increase of p1:
dw/dp1 = f1L + p1f1LL[dL1/dp1] < f1L = w/p (23)
dw/w < dp1/p1.
Total effect on real wage ambiguous (dw>0,dp2= 0)

25. Figure 1:
Changes of return to capital: shift of labor to good 1 increases value of marginal product of K1 further, reduces r2.

26. Changes in factor endowments:
dK1 > 0 = dK2: MPL increases in both sectors, but MPK1 and MPK2 go down since capital/labor ratios are increased.
Increase in labor endowment: capital/labor ratios are reduced in both sectors, hence w goes down, but r1 and r2 go up – no Rybczynski-like effect.
Comparative advantage: if two countries are identical except K1 > K*1 then the home country has a comparative advantage in good 1. Effect of different labor endowments is indeterminate.

27. Figure 2: increasing K1

28. Figure 3: Changes in labor endowment

29. 3.5 More goods than factors
For illustration: Two factors, three goods  zero profit condition (1) has two unknowns and three equations  solution only for „special“ prices. At such special prices all three goods are produced in strictly positive quantities  market clearing condition (2) has two equations in three unknowns  solution underdetermined.
FPE set exists, but for allocation within the FPE-set production in each country is not unique.

30. Figure 4: FPE-set for two factors, three goods; home country can reach E via 0P’1, 0P1 or appropriate convex combinations of the two.

31. Large differences in factor endowments – no FPE
Two factors, continuum of goods, no FIRs, range of goods = z[0,1], goods are ranked by increasing order of capital intensity, Cobb-Douglas utility function, implying demand
D = (z)[wL + rK]/c(w,r,z) = y(z) (24)
where y(z) denotes the supply function and c(.) the unit cost function. Equilibrium in the factor market is given by
(25)

32. Equations (24) and (25) are the equilibrium conditions for the home country in autarky. If the foreign country has identical tastes and technologies and the factor endowments are in the FPE set then in the free trade equilibrium production is indeterminate. Suppose next endowments are outside the FPE set and (w,r)  (w*,r*). Equilibrium prices are determined by p(z) = minc(w,r,z), c(w*,r*,z) (26) since goods will only be produced in the country with smaller unit costs.

33. Figure 5: Suppose L/K > L*/K* and w/r < w*/r*. There exists z* s.t. c(w,r,z*) = c(w*,r*,z*), and the home country produces all goods zz*, and the foreign country zz*.