Download In particular, the assumptions about the wage

2x2x2 Heckscher-Ohlin-Samuelson (H-O-S) model with no factor substitution
In the H-O-S model we will focus entirely on the role of factor supplies and assume away
differences in technology. Therefore, unlike in the simple Ricardian model (but like in the
modified Ricardian model!), in the H-O-S model trade is based on differences in factor
endowments and not on differences in technologies (although like in the previous models trade
in the H-O-S model is also related to differences in supply conditions and not differences in
consumer preferences).
The H-O-S approach to international trade is based on two main suppositions:
 Goods differ in their factor requirements and can be ranked by their factor intensity, for
example, cars require more capital per worker than furniture (i.e. cars are capital intensive
while furniture is labor intensive)
 Countries differ in their factor endowments and can be ranked by their factor abundance,
for example, England has more capital per worker than Portugal.
These two suppositions lead to the fundamental theorem of the H-O-S model that characterizes
the pattern of international specialization and trade:
1
A CAPITAL ABUNDANT COUNTRY WILL TEND TO SPECIALIZE IN CAPITAL
INTENSIVE GOODS AND THEREFORE WILL EXPORT CAPITAL-INTENSIVE GOODS
IN EXCHANGE FOR LABOR-INTENSIVE GOODS.
Alternatively, this theorem can be also expressed in more general terms:
TRADE IS BASED ON DIFFERENCES IN RELATIVE FACTOR ENDOWMENTS AND
REDUCES THE PRINCIPAL EFFECTS OF THESE DIFFERENCES.
The following three issues will be addressed:
 How differences in factor endowments contribute to differences in supply conditions
(production possibility frontiers)?
 How differences in factor endowments are reflected in factor and product prices (closed
economy: factor endowments → factor prices → goods prices)?
 How trade affects factor prices and income distribution (open economy: trade → goods
prices → factor prices)?
2
Basic predictions of the H-O-S model can be summarized in four fundamental theorems:
1. Factor price equalization theorem that deals with the impact of international trade on factor
prices
 w   w*
 

 r   r*
2. Stolper-Samuelson theorem that deals with the connection between goods prices and factor
prices
 pF

 pM
  w
   
 r
3. Rybczyński theorem that deals with the connection between factor supplies and output
K , L  (QM , QF )
4. Heckscher-Ohlin theorem that deals with factor abundance and the pattern of trade
K K *
 the factor proportions version  ,

 L L* 
 w   w*
 the relative price version   , 

 r A  r * A
3
CLOSED ECONOMY
Demand Side
(Consumers):
Identical, homothetic
preferences everywhere
Supply Side
(Producers):
Neoclassical firms,
Perfect competition,
Constant returns to scale (CRS)
Homogenous products
Demand for final commodities
Fixed supply of factors of
production
(different in each country)
Derived demand for factors of
production
Factor prices
Commodity prices (autarky)
INTERNATIONAL TRADE
4
Model Assumptions
Two countries: Home (England) and Foreign (Portugal)
Two goods: Manufactures (cloth) and food (wine)
Two factors of production: Labor (L), Capital (K)
Both countries have the same technologies (production functions), but have different
endowments of capital and labor. In particular, England has more capital than Portugal, while
Portugal has more labor than England.
Both manufactures and food are produced using capital and labor. The output of each good
depends on how much capital and labor are used. This relationship is summarized by the
neoclassical production function:
Qi  Qi ( K i , Li ) , for i = F, M.
There are many variants of the H-O-S model. To keep things as simple as possible we will start
with the easiest version of the H-O-S model with no factor substitution (i.e. fixed factor
requirements per unit of output, independent of relative factor prices).
Capital and labor are mobile factors which can move freely between sectors within a country.
5
Production Possibilities
Recall that in the simplest Ricardian model the shape of the production possibility frontier
depended on unit labor requirements and the total supply of labor. In the H-O-S model it will
depend on both capital and labor requirements and the supplies of both labor and capital.
In the i-th industry one unit of output requires aLi units of labor and aKi units of capital so the
total amounts of labor and capital employed in the i-th sector are:
Li  a Li Qi
K i  a Ki Qi
Leontief production technology:
L K 
Qi  min  i , i 
 aLi aKi 
 LM K M 
 LF K F 
,
,
Q

min
,
 F

 , and
a
a
a
a
 LM KM 
 LF KF 
More specifically, in each sector we have: QM  min 
aKM aKF
.

aLM aLF
6
aKM,
aKF
aKM K M

 kM
aLM
LM
manufactures
aKF K F

 kF
aLF LF
food
aLM, aLF
Figure 1. Unit isoquants for manufactures and food.
7
When a country has fixed supplies of labor and capital and they are fully employed the sum of
demand for capital and labor must equal their total supplies:
LM  LF  aLM QM  aLF QF  L
K M  K F  aKM QM  aKF QF  K
Solving for QM in terms of QF we obtain so-called Rybczyński lines (labor and capital
constraints):
a 
L
QM   LF QF 
aLM
 aLM 
a 
K
QM   KF QF 
aKM
 aKM 
When
aLF aKF
the labor constraint is steeper than the capital constraint. This corresponds to

aLM aKM
our initial assumption that the production of manufactures is relatively more capital intensive
than production of food:
aKM aKF
(kM > kF).

aLM aLF
8
If the country had an unlimited supply of capital its output would depend only on unit labor
requirements and on the supply of labor – just like in the simple Ricardian model. Alternatively,
if the country had an unlimited supply of labor its output would depend only on capital
requirements and the supply of capital.
9
QM
L
aLM
L
a LF
QF
Figure 2. Labor constraint
10
QM
K
a KM
K
aKF
QF
Figure 3. Capital constraint
11
QM
L
aLM
K
a KM
E
L
a LF
K
aKF
QF
Figure 4. Production possibility frontier
12
The labor and capital constraints operate together to determine a country’s production possibility
frontier. Both constraints are binding when supplies of labor and capital are both limited.
What happens to the production possibility frontier when the supply of one factor increases?
The Rybczyński theorem can be used to study the impact of changes in factor supplies:
WHEN FACTORS ARE FULLY EMPLOYED AND FACTOR REQUIREMENTS ARE
GIVEN AN INCREASE IN THE SUPPLY OF ONE FACTOR OF PRODUCTION RAISES
THE OUTPUT OF THE GOOD THAT USES THE FACTOR INTENSIVELY AND
REDUCES THE SUPPLY OF THE OTHER GOOD.
Proof. Differentiate totally (assuming that factor requirements do not change) and write down
the full employment conditions in the matrix form, apply Cramer’s rule and solve for changes in
output produced in each sector:
 aLF
a
 KF
aLM   dQF   dL 

aKM  dQM  dK 
detA = aLF aKM  aKF aLM  0
(since by assumption manufactures are more capital intensive compared to food)
13
dQM 
det AM
a dK  aKF dL
 LF
det A aLF aKM  aKF aLM
dQF 
det AF
a dL  aLM dK
 KM
det A aLF aKM  aKF aLM
Assume that only the supply of capital increase (dK > 0) while the supply of labor (dL = 0)
remains unchanged, then we can easily see that an increase in K raises QM and reduces QF. This
proves the Rybczyński theorem.
The Rybczyński theorem can be illustrated graphically in two different ways:
 In the product space (QM, QF)
 In the factor space (K, L)
14
QM
K'
a KM
E’
1
QM
K
a KM
E
QM0
QF
Q1F
QF0
L
a LF
K
aKF
K'
aKF
Figure 5. Rybczyński theorem and production possibility frontier
15
kM
K
E’
K'
M’
K
E
M
kF
F’
O
F
L
L
Figure 6. Rybczyński theorem and Edgeworth box
16
INTERNATIONAL TRADE IN HECKSCHER-OHLIN MODEL
The Rybczyński theorem is fundamental to the functioning the Heckscher-Ohlin model. We will
use it to show how international differences in factor endowments determine the pattern of trade.
To see how differences in relative factor endowments lead to international trade let us consider
two countries: England (capital abundant) and Portugal (labor abundant).
Both countries have the same technologies (production functions), which means that the slopes
of labor and capital constraints are the same in these countries.
Moreover, countries have the same demand preferences represented by the set of indifference
curves.
Countries differ only in terms of their relative factor endowments. England has a lot of capital
and little labor while Portugal has a lot of labor and little capital.
17
QM
QET
QEA  CEA
C
CT
QPA  CPA
UT
QPT
UA
QET QEA  CEA
CT
QPA  CPA QPT
QF
Figure 7. Trade in the Heckscher-Ohlin model
18
Differences in factor endowments lead to differences in transformation curves between
countries. When England has more capital than Portugal and Portugal has more labor than
England the relative price of food (expressed in terms of manufactures) under autarky is higher
in England than in Portugal. In England the relative price of food is determined slope of the
labor constraint (idle capital), while in Portugal by the slope of the capital constraint (idle
labor).
 pF

 pM
A
T
A

p 
p 
a
a

 LF   F    F 
 KF
 England aLM  pM 
 pM  Portugal aKM
With free trade the relative price of manufactures is the same in two countries which means that
the relative price of manufactures increases in England and falls in Portugal. As a result of the
change in relative prices of manufactures output of manufactures increases in England and falls
in Portugal and output of agricultural goods increases in Portugal and falls in England.
As a result of opening to international trade both countries end up at their full employment
output points.
Recall that there are two types of gains from trade:
i) gains from international exchange due to the change in relative prices,
ii) gains from international specialization due to the movement to the full employment.
19
We have just demonstrated the factor-proportions version of the Heckscher-Ohlin theorem,
linking the pattern of trade to factor endowments:
THE LABOR ABUNDANT COUNTRY WILL ALWAYS EXPORT THE LABORINTENSIVE GOOD (AND INCREASES ITS PRODUCTION OF THAT GOOD, IF IT DID
NOT START AT ITS FULL EMPLOYMENT POINT), THE CAPITAL ABUNDANT
COUNTRY ALWAYS EXPORTS THE CAPITAL-INTENSIVE GOOD (AND INCREASES
ITS PRODUCTION, IF IT DID NOT START AT ITS FULL EMPLOYMENT POINT).
Alternative (factor price) version of the Heckscher-Ohlin theorem
You can notice that there is a strong relationship between goods prices and factor prices in the
Heckscher-Ohlin model. An increase in the relative price of labor-intensive good (pM) raises the
relative price of labor (w/r).
This finding can be illustrated graphically using the graph depicting the production possibility
frontier.
20
QM
L
aLM
K
a KM
E1
1
QM
E0
QM0
E2
QM2
Q1F
QF0
QF2
L
a LF
K
aKF
QF
Figure 8. Factor prices and goods prices.
21
The relationship between goods prices and factor prices can be summarized in the
following table:
PPF Point
E1
(Labor constraint is not binding)
E0
(Both constraints are binding)
E2
(Capital constraint is not binding)
Relative goods price Relative factor price
pF aKF

pM aKM
pF  aKF aLF

,
pM  aKM aLM
pF aLF

pM aLM



w
 0 labor is a free factor
r
w
 0,  
r
w
  capital is a free factor
r
The relationship between goods prices and factor prices can be summarized in the following
figure which tells us that the relative price of labor rises with the relative price of the labor
intensive good (food):
22
Relative
price of
labor
(w/r)
relative price of
food pF/pM
0
aKF
aLF
a KM
a LM
Figure 9. The relationship between goods prices and factor prices.
23
The relationship between goods prices and factor prices can be also derived algebraically.
Recall that in the Heckscher-Ohlin model we assumed perfect competition which means that
goods prices equal total unit costs.
pF  aLF w  aKF r
pM  aLM w  aKM r
Let’s rewrite this system of equations in a matrix form to obtain factor prices as functions of
goods prices.
 aLF
a
 LM
aKF   w  pF 

aKM   r   pM 
detA = aLF aKM  aKF aLM  0
w
det Aw
p a  pM aKF  pM aKM
 F KM

det A aLF aKM  aKF aLM  A
r
det Ar
p a  pF aLM
p a
 M LF
  M LM
det A aLF aKM  aKF aLM  A
 pF aKF


 pM aKM

  0

p 
 aLF
 F   0

 aLM pM 
(assuming full employment of both factors of production)
24
To derive the general relationship between relative factor prices and relative goods prices let’s
divide one equation by the other which yields:
w aKM

r aLM
  pF   aKF
  
 
p
  M   aKM
  aLF   pF
  
 
  aLM   pM

 
 

 

p 
f  F 
 pM 
Now we can easily see that an increase in the relative price of food raises the relative wage since
it increases the numerator and decreases the denominator of the above expression. (This proof
shows that the relationship demonstrated in our figure holds when both countries have the same
production technologies).
This relationship has three uses:
i)
ii)
To prove the relative-price version of the Heckscher-Ohlin theorem
To show how trade affects income distribution in each country (Stolper-Samuelson
theorem)
iii) To show when trade will equalize the two countries’ factor prices (Factor price
equalization theorem)
25
The relative-price version of the Heckscher-Ohlin theorem
Before opening to international trade the relative price of food was lower in Portugal than in
England, therefore the curve (in figure 9) tells us that the relative price of labor in Portugal had
to be lower before trade was opened. This leads us directly to the relative-price version of the
Heckscher-Ohlin theorem:
IF THE RELATIVE PRICE OF LABOR IS LOWER IN ONE COUNTRY BEFORE TRADE
WAS OPENED, THE RELATIVE PRICE OF THE LABOR-INTENSIVE GOOD MUST
ALSO BE LOWER, AND THE COUNTRY WILL EXPORT THE LABOR INTENSIVE
GOOD.
The Stolper-Samuelson theorem
The effects of international trade on factor prices and the distribution of income within a country
are described by the Stolper-Samuelson theorem which can be stated as follows:
THE OPENING TO INTERNATIONAL TRADE RAISES THE RELATIVE PRICE OF
LABOR IN THE LABOR ABUNDANT COUNTRY AND REDUCES IT IN THE CAPITAL
ABUNDANT COUNTRY.
26
Income redistribution effects:
Trade will raise the share of labor in the national income of the labor abundant country and
reduce the share of capital (it will have the opposite effects on income distribution in the capital
abundant country).
Intuition:
An increase in the relative price of food encourages food production and discourages the
production of manufactures. But remember that food production is labor intensive! Hence, the
resulting increase in food production raises the demand for labor by more than the decreases of
manufactures production reduces it. This drives the relative wage up. At the same time the
decrease of manufactures production reduces the demand for capital by more than the increase in
of food production raises it. This drives down the reward to capital.
The Stolper-Samuelson theorem can be put in a stronger form:
THE INCREASE IN THE RELATIVE PRICE OF FOOD THAT OCCURS IN THE LABOR
ABUNDANT COUNTRY RAISES REAL WAGE OF LABOR AND REDUCES REAL
RETURN TO CAPITAL IN TERMS OF BOTH GOODS. (ALTERNATIVELY, THE
DECREASE THAT OCCURS IN THE CAPITAL ABUNDANT COUNTRY REDUCES THE
REAL WAGE AND RAISES THE REAL RETURN TO CAPITAL).
27
These assertions hold for both definitions of real wage and real return to capital expressed both
in terms of food and manufactures.
w  aKM  pF aKF 




pM  A  pM aKM 
r
p 
 a  a
  LM  LF  F 
pM  A  aLM pM 
We can easily see that an increase in the relative price of food (labor intensive good) raises the
real wage and reduces the real return to capital when they are measured in terms of manufactures
(capital intensive good).
Now express the real wage and the real return to capital in terms of food:
w  aKM  pM


pF  A  pF
r  aLM

pF  A
 pM

 pF
 pF aKF


 pM aKM
  aKM
  
  A
 aLF
p  a

 F    LM
 aLM pM   A
   aKF
 1  
   aKM
  aLF
 
  aLM



 pM  

  1
p
 F  
 pM

 pF
An increase in the relative price of the labor intensive good also raises the real wage and reduces
the real return to capital when measured in terms of food.
28
Factor Price Equalization (FPE) theorem
FPE theorem is a special case of the Stolper-Samuelson theorem:
IF THERE ARE NO IMPEDIMENTS TO TRADE (TRADE BARRIERS OR TRANSPORT
COSTS) TRADE EQUALIZES COUNTRIES’ FACTOR PRICES, NOT ONLY REDUCES
THE DIFFERENCE BETWEEN THEM.
(in other words, there is complete compensation for the effects of differences in factor
endowments).
When the relative price of goods is the same in both countries real wages and real returns to
capital are the same in both countries (assuming that both goods are produced in both countries).
This is obvious since previously we derived the relationship between goods prices and factor
prices:
w aKM

r a LM
  pF   aKF
  
 
p
  M   a KM
  aLF   pF
  
 
a
  LM   pM

 


 

29
2x2x2 Heckscher-Ohlin-Samuelson (H-O-S) model with factor substitution
The “HAT ALGEBRA” of the Heckscher-Ohlin model with factor substitution
So far we were dealing with the easiest possible version of the H-O-S model with no factor
substitution (i.e. with fixed factor requirements per unit of output, independent of relative factor
prices). Now we will allow for factor substitution in response to changes in relative factor
rewards. The appealing feature of this simple general equilibrium model is its ability to show
how easily some famous theorems can be derived from a simple model.
We will start with a small open economy that takes the relative price ratio as given, and later
discuss a large open economy case.
If the technology is given and factor endowments and commodity prices are treated as
parameters, the model serves to determine 8 unknowns:
 The level of commodity outputs (QM,QF)
 The factor allocation to each industry (LM,LF,KM,KF)
 Factor prices (w,r)
We need 8 equations to be able to solve the model analytically.
30
These equations can be given by:
 the production functions (2),
 the requirement that each factor receives the value of its marginal product (4),
 the requirements that each factor is fully employed (2).
The requirement that both factors are fully employed is given by equations:
LM  LF  L  aLM QM  aLF QF
K M  K F  K  aKM QM  aKF QF
above relationships emphasize the dual relationships between factor endowments and goods
outputs.
Unit costs of production in each industry are given by the perfect competition conditions:
pM  cM  a KM r  aLM w
p F  cF  aKF r  aLF w
The above relationships emphasize the dual relationships between factor prices and goods prices.
31
Is it enough to solve the model? No, not in the general case when unit factor requirements
change in response to relative factor price changes! Therefore, we must supplements the above
equations by four additional relationships determining the input coefficients. These are provided
by the requirement that in a competitive equilibrium each aij depends solely on the ratio of factor
prices (w/r).
Let’s use the cost minimization condition of a typical entrepreneur (we saw in the factor specific
model). In the manufacturing sector the unit production costs are given by:
CM = aKMr + aLMw
The entrepreneur treats factor prices as fixed and varies a’s so as to set the derivative of costs
equal to zero:
dCM = 0 = daKMr + daLMw
Now, express it in terms of the rates of changes (dividing by sides by cM):


 




 


dC
a
r
da
a
w
da
M
Cˆ M 
  KM  KM    LM  LM    KM aˆ KM   LM aˆ LM  0
 CM  aKM   CM  aLM 
CM

        

% change
  KM  aˆ KM     LM   aˆ LM 

rateofgrowth
32
Similarly, in the food producing sector we have:
Cˆ F   KF aˆ KF   LF aˆ LF  0
Alterations in factor proportions must balance out such that the Θ–weighted average of the
changes in input coefficients in each industry is zero. This implies that the relationship between
changes in factor prices and changes in goods prices is identical in the variable and fixed
coefficient cases (Wong-Viner theorem). To see it differentiate totally the perfect competition
conditions and then express them in the rates of changes.
In the manufacturing sector we have:
dpM  dcM  daKM r  aKM dr  daLM w  aLM dw
dpM daKM a KM r aKM r dr daLM aLM w aLM w dw




pM
aKM pM
pM r
aLM pM
pM w
pˆ M  aˆ KM  KM   KM rˆ  aˆ LM  LM   LM wˆ
Similarly, in the food sector we have:
pˆ F  aˆ KF  KF   KF rˆ  aˆ LF  LF   LF wˆ
33
The relationships between changes in goods prices and changes in factor prices can be written in
the matrix form:
 LM

 LF
 KM   wˆ   pˆ M  ( LM aˆ LM   KM aˆ KM )  pˆ M 


 KF   rˆ   pˆ F  ( LF aˆ LF   KF aˆ KF )   pˆ F 
Interpretation: factor prices depend only on commodity prices – this is our factor price
equalization theorem!
Our equations prove the factor price equalization theorem (between countries), even though it to
show only one country. However, we can easily reinterpret each change “ˆ” as a rate of change
between countries rather than as a change over time. Two countries are the same in some key
respects. They have the same price ratio because they trade freely without transport costs.
Unfortunately, a similar kind of argument does not apply to the case of the factor market
clearing conditions (that do not simplify so easily).
Let us express our factor market clearing conditions using the rates of change.
First, take a total differential of the full employment condition for labor to obtain:
daLM QM  aLM dQM  daLF QF  aLF dQF  dL
34
Then divide both sides by L to get:
 daLM  aLM QM   a LM QM  dQM   daLF  aLF QF   aLF QF  dQF  dL


  

 




a
L
L
Q
a
L
L
Q
L









LM  
M  
LF  
  
 
 F 



 
 
Lˆ
aˆ LM
 LM
 LM
Qˆ M
aˆ LF
 LF
 LF
Qˆ F
LM  LF  1
Therefore, we have:
LM Qˆ M  LF Qˆ F  Lˆ  [aˆ LM LM  aˆ LF LF ]
In the same way we obtain:
KM Qˆ M  KF Qˆ F  Kˆ  [aˆ KM KM  aˆ KF KF ]
The relationships between changes in factor supplies and changes in output levels can be written
in the matrix form:
LM

 KM
LF  Qˆ M   Lˆ  (LM aˆ LM  LF aˆ LF ) 
 
KF  Qˆ F   Kˆ  (KM aˆ KM  KF aˆ KF )
35
The term (LM aˆ LM  LF aˆ LF ) shows the percentage change in the total quantity of labor required by
the economy as a result of changing factor proportions in each industry at unchanged outputs
(constant λs).
Crucial feature: If factor prices change, factor proportions alter in the same direction in both
industries. The extent of this change depends on the elasticities of substitution between factors in
each industry (assume constant elasticity of substitution between factors).
The elasticity of substitution between labor and capital in the manufacturing sector is defined as:
M 
d ( K M / LM ) /( K M / LM ) d (a KM / aLM ) /(aKM / a LM ) aˆ KM  aˆ LM


d ( w / r ) /( w / r )
d ( w / r ) /( w / r )
wˆ  rˆ
This elasticity tells us how the capital-labor ratio will change if relative wage (wage-rental ratio)
changes by 1 %.
Similarly, for the food sector we can write the elasticity of substitution between labor and capital
as:
F 
d ( K F / LF ) /( K F / LF ) d (aKF / aLF ) /(a KF / aLF ) aˆ KF  aˆ LF


d ( w / r ) /( w / r )
d ( w / r ) /( w / r )
wˆ  rˆ
36
Now, we need to find changes in unit factor requirements as functions of changes in factor
prices. To do so let us use the above definitions of the elasticity of substitution and combine
them with cost minimization conditions:

Cˆ M   KM aˆ KM   LM aˆ LM  0  aˆ LM   KM aˆ KM
 LM

Cˆ F   KF aˆ KF   LF aˆ LF  0  aˆ LF   KF aˆ KF
 LF
M 
aˆ KM  aˆ LM

  M ( wˆ  rˆ)  aˆ KM  aˆ LM  aˆ KM (1  KM )
wˆ  rˆ
 LM
F 
Hence, we get:
aˆ KF  aˆ LF

  F ( wˆ  rˆ)  aˆ KF  aˆ LF  aˆ KF (1  KF )
wˆ  rˆ
 LF
aˆ KM   LM  M ( wˆ  rˆ)
aˆ LM   KM  M ( wˆ  rˆ)
aˆ KF   LF  F ( wˆ  rˆ)
aˆ LF   KF  F ( wˆ  rˆ)
37
Substituting the expressions for the changes in unit factor requirements in response to changes in
factor prices into the set of equilibrium conditions we obtain:
 LM

 KM
LF  Qˆ M   Lˆ   L ( wˆ  rˆ) 
 
KF  Qˆ F   Kˆ   K ( wˆ  rˆ)
where  L  LM  KM  M  LF  KF F is the aggregate percentage saving in labor inputs at
unchanged outputs associated with a 1% increase in the relative wage (the saving resulting from
the adjustment to less labor-intensive techniques in both industries as relative wages rise), and
similarly  K  KM  LM  M  KF  LF  F .
JONES (1965) MAGNIFICATION EFFECTS
If commodity prices are unchanged factor prices are constant and the system of equations tells us
that changes in commodity outputs are related to changes in factor endowments. If both
endowments expand at the same rate both commodity outputs expand at identical rates.
Qˆ F  Lˆ  Kˆ  Qˆ M
38
This can be demonstrated as follows:
 LM LF  Qˆ M   Lˆ 

 ˆ    ˆ 

 KM
KF   Q F 
K 
det   LM KF  KM LF 
LM (1  KM )  KM (1  LM ) 
LM  KM  0
(negative when M is capital intensive)
 Kˆ  KM Lˆ
Qˆ F  LM
 LM KM
KF Lˆ  LF Kˆ ( Lˆ  Kˆ )  (LM Kˆ  KM Lˆ )
ˆ
QM 

LM  KM
LM  KM
Now we can easily notice that when Lˆ  Kˆ then Qˆ F  Qˆ M .
However, if both factor endowments grow at different rates, the good intensive in the use of the
fastest growing factor expands at a greater rate than either factor, and the other commodity
grows (if at all) at a slower rate than either factor.
39
For example, suppose that labor expands more rapidly than capital. With M capital intensive
compared to F we have then:
Qˆ F  Lˆ  Kˆ  Qˆ M
This is called the MAGNIFICATION EFFECT of factor endowments on commodity outputs
at unchanged commodity prices.
For simplicity consider a special case when the endowment of only one factor increases, say
labor Lˆ  0 .
 LM LF  Qˆ M   Lˆ 

 ˆ    

 KM
KF   Q F 
0
det   LM KF  KM LF 
LM (1  KM )  KM (1  LM ) 
LM  KM  0
(negative when M is capital intensive)
40
 KM Lˆ
KM
Qˆ F 
 Lˆ since 1
 LM KM
LM  KM


0
Qˆ M 
KF Lˆ
(1  LM ) Lˆ ˆ

L
LM  KM LM  KM

0
(you can notice that the numerator is bigger than the denominator since by assumption 1>λLM)
Hence, we observe the following magnification effect:
ˆM
Qˆ F  
Lˆ  
Kˆ  Q

0
0
0
This is our Rybczyński theorem which can be restated as follows:
At the unchanged commodity prices an expansion in one factor results in an absolute
decline in the commodity intensive in the use of the other factor.
41
Similarly, the magnification effect is also the feature of the link between commodity prices and
factor prices. In the absence of technological change or taxes/subsidies, if the price of capital
intensive good M grows more rapidly than the price of the labor intensive good F, then the
reward to factor used intensively in the production of manufactures (capital) grows more than
the price of manufactures and we have:
rˆ  pˆ M  pˆ F  wˆ
Intuition: The source of the magnification effect is easy to detect. Since the relative change in
the price of either commodity is a positive weighted average of factor price changes it must be
bounded by these changes.
For simplicity consider a special case when the price of only one good increases, say pˆ M  0 . In
this case the increase in the price of M raises the return to the factor used intensively in M
(capital) by an even greater amount (and lower the return to the other factor).
42
Now we have:
 LM

 LF
 KM   wˆ   pˆ M 

 KF   rˆ   0 
wˆ 
 KF pˆ M
0
 LM   LF
rˆ 
1   LF
pˆ M  pˆ M
 LM   LF

1
Hence, we observe the following magnification effect:
rˆ  
pˆ M  
pˆ F  w
ˆ
0
0
0
This is our Stolper-Samuelson theorem which can be restated as follows:
An increase in the price of a capital intensive good M raises the return to the factor used
intensively in M (capital) by an even greater amount and lowers the price of the other
factor.
43
Finally, we are ready to study the large economy case.
Endogenous demand
To close the model we assume that consumer trade patterns are homothetic and ignore any
differences between the workers and capitalists. Thus, the ratio of quantities of goods consumed
depends only on the relative commodity price ratio:
p
QM
 f  M
QF
 pF



Let us express this relationship in terms of the rates of change using the elasticity of substitution
between two commodities on the demand side σD:
Qˆ
M

 Qˆ F   D ( pˆ M  pˆ F )
since


d QM / QF  / QM / QF  Qˆ M  Qˆ F
D 

d  pM / pF  /( pM / pF )  pˆ M  pˆ F 
44
Previously we considered the effect of a change in factor endowments at unchanged commodity
prices. With the model closed by the demand relationship commodity prices will have to adjust
so as to clear the commodity markets.
Recall that in the general case when commodity prices change also factor prices change:
 LM

 KM
LF  Qˆ M   Lˆ   L ( wˆ  rˆ) 
 
KF  Qˆ F   Kˆ   K ( wˆ  rˆ)
Hence,
(Qˆ M  Qˆ F ) 
LM
1
1
( Lˆ  Kˆ ) 
(   )( wˆ  rˆ)
 KM
LM  KM L K
We can notice that on the supply side the change in the ratio of outputs produced depends on the
change in factor endowments and the change in factor prices.
Let us concentrate for the moment on the change in the relative factor prices which can be
obtained from:
 LM

 LF
 KM   wˆ   pˆ M 

 KF   rˆ   pˆ F 
45
Hence,
wˆ  rˆ 
 LM
1
 pˆ M  pˆ F 
  LF
Now we can substitute the relationship between the changes in the ratio of factor prices and the
changes in the ration of goods prices into our relationship between the change in the ratio of
output produced and the change in the ratio of factor prices:
(Qˆ M  Qˆ F ) 

LM
LM

1
1
1
( Lˆ  Kˆ ) 
( L   K )
( pˆ M  pˆ F )
 KM
LM  KM
 LM   LF

1
Lˆ  Kˆ   S  pˆ M  pˆ F 
 KM
where
S 
L  K
LM  KM  LM
  LF 
(the elasticity of substitution between the goods on the supply side – along the product
transformation curve)
46
Equilibrium
In equilibrium, the change in the ratio of output produced has to be equal to the change in the
ratio of output consumed. This allows us to determine the change in the commodity price ratio as
we can notice that this change is given by the mutual interaction of demand and supply.
(Qˆ M  Qˆ F ) 
LM


1
Lˆ  Kˆ   S  pˆ M  pˆ F    D  pˆ M  pˆ F 
 KM
Hence
 pˆ M
 pˆ F   
(LM

1
Lˆ  Kˆ
 KM )( S   D )

Having determined the change in the commodity price ratio we can determine the change in the
output ratio as a function of the change in the factor endowment ratio:
47
(Qˆ M  Qˆ F ) 

LM


1
1
1
Lˆ  Kˆ   S  pˆ M  pˆ F  
Lˆ  Kˆ   S  
( Lˆ  Kˆ ) 
 KM
LM  KM
 (LM  KM )( S   D )

D

(LM  KM )( S   D )



( Lˆ  Kˆ )
When commodity prices adjust to the initial changes in output brought about by the change in
factor endowments, the composition of outputs may in the end not change by as much as the
factor endowments. This depends whether the “elasticity” expression σD/(σS+σD) is smaller than
the “factor-intensity” expression (λLM - λKM). Large values of dampen the spread of output, small
values of work in the similar way.
These effects can be summarized in the table below:
 D

 D  S
 D

 D  S

  
(LM


  
(LM

 D

 D  S

1
  
(LM  KM )

1
 KM )
Less than 1:1 change
1
 KM )
1:1 change (complete dampening)
No magnification effect
More than 1:1 change
Magnification effect
dampened
exists
although
is
48
D 
  1 we observe the full magnification effect.
 D  S 

When 
Conclusion:
The only part of the Rybczyński theorem which is challenged by the introduction of the demand
side is the one that concerns the magnification effect.
49