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An introduction to the Heckscher-Ohlin model
Introduction
The Ricardian models can be described for cases where there is only one factor of production.
A natural extension of trade models with one factor of production is to introduce more factors.
In this note we will describe the so called Heckscher-Ohlin-Samuelson model with two
factors, two goods and two countries (it is also denoted the 2x2x2 model). Since two factors
of production are assumed, one needs to describe the assumed relationships between factor
use and production. These are described in section 1 below. In section 2 we discuss the
autarky equilibrium in the model. Thereafter the effects of trade are discussed.
Four noteworthy theorems from the Heckscher-Ohlin-Samuelson model are the HeckscherOhlin theorem, the Rybczynski theorem, the Stolper-Samuelson theorem and the Factor-Price
equalisation theorem.
Shortly, these say the following:
Heckscher-Ohlin theorem: A country will export the good which is intensive in the factor that
the country is relatively well endowed with and import the good which is intensive in the
factor that the country is relatively scarcely endowed with.
Rybczynski theorem: If a country receives one more unit of a factor of production, production
of the good which uses this factor intensively in production will increase, while production of
the other good decreases.
Stolper-Samuelson theorem: If the price of one good increases, the factor price of the factor
that is used intensively in production of this good increases and the factor price of other factor
decreases.
Factor price equalisation theorem: International trade equalises goods prices and therefore
also factor prices.
The HOS model emphasises resource endowments as the source of comparative advantages
and therefore the source of gains from trade. Thus the model is often referred to as the factor
proportions theory. In its purest form, the model predicts production patterns in both
countries, trade between the two countries, goods prices and factor prices.
Assumptions:
•
•
•
•
•
2 goods, x1 and x2.
2 factors of production, capital and labour, k, n.
2 countries, the home country, H, and the foreign country, F.
Factor endowments in the two countries are fixed. Thus, there is no factor mobility.
There are constant returns to scale and production of both goods are increasing,
concave and homogenous of degree one in each factor.
• There are identical technologies across countries.
• Since we have a 2x2x2 model, the number of goods equals the number of factors of
production.
2
Production technologies
Production technologies are assumed to be characterised by constant returns to scale. The
production functions are assumed to be homogenous of degree of one and they are increasing
and concave in each factor of production.
Therefore, production of each good i, can be described with the following production
functions.
xi = F i (k i , ni )
∂F i
∂F i
∂2F i
∂2F i
< 0,
>0
> 0,
0
,
<
∂k i
∂ni
(∂k i )2
(∂ni )2
 n
xi = k i F i 1, i
 ki
n
li ≡ i
ki
f i ' > 0,
∂2F i
>0
(∂k i ∂ni )

 = k i f i (li )

f i ''< 0
Above, the first equation is the production function. The second line describes the derivatives
of the production function. The third set of equations explores the fact that the production
function is homogenous of degree one. This implies that multiplying use of each factor with
any constant λ increases production with the same factor λ. Dividing with k therefore implies
that production is (1/k). Thus, the third expression gives another expression for total
production. The labour-capital ratio is defined as l (=n/k). The fifth line describes the
derivatives of the production per unit of capital as functions of the labour-capital ratio. The
sign of these follow from the derivatives of the production function. 1
The marginal products of labour and capital can be expressed as:
∂xi ∂F i (k i , ni )
1
=
= k i f i ' = f i ' = f l i (li )
ki
∂ni
∂ni
 n
∂xi ∂F i (k i , ni )
=
= f i (l i ) + k i f l i (l i ) − 2i
∂k i
∂k i
 ki

 = f i (l i ) − l i f l i (li )

Above the notation fil is used to indicate the derivate of fi with respect to l.
We will assume that the two goods differ in their factor intensities. Below this assumption
will be explained in detail. For now it suffices to note that this assumption implies that for any
given relative factor prices, the relative use of the two factors will differ between the two
industries.
1
In this note n/k is written as l. In the note by Ultveit-Moe (2010), it is written as a. Since Feenstra (2004) uses a
differently, we prefer to write n/k as l.
3
It is of interest to explore the production possibilities in the countries in the model. To do
this, one can maximise the production of one good for any given amount of the other good
and at the same time taking account of the economy’s resource constraints:
n1 + n2 = n
k1 + k 2 = k
The maximisation problem is therefore a constrained maximisation problem where one
maximises the production of one good, e.g. good 2, given the production of good 1 and the
resource constraints:
max x 2 = F 2 (k 2 , n2 ) subject to n1 + n2 = n, k1 + k 2 = k , x1 = F 1 (k1 ,n1 )
This problem can be solved by substituting the resource constraint into the objective, setting
up the Lagrange function and maximise over n2 and k2:
(
L = F 2 (k − k1 , n − n1 ) − λ F 1 (k1 , n1 ) − x1
)
∂L
∂F 2
∂F 1
=−
−λ
= − Fn2 − λFn1 = 0
∂n1
∂n2
∂n1
∂L
∂F 2
∂F 1
=−
−λ
= − Fk2 − λFk1 = 0
∂k1
∂k 2
∂k1
λ=−
Fn2
f l 2 (l 2 )
Fk2
f 2 (l 2 ) − l 2 f l 2 (l 2 )
−
=
−
=
−
=
Fn1
f l1 (l1 )
Fk1
f 1 (l1 ) − l1 f l1 (l1 )
The second and the third lines give the first order conditions. The fourth line gives the results
that the ratios of the marginal products in the two industries are equal for both factors (so that
MPL2/MPL1=MPK2/MPK1) . Also note that these results do not depend on absolute use of the
factors but simply their labour to capital ratios, l. The solution to the maximisation problem
gives the production of one good, good 2, as a function of the production of the other good,
good 1 and the factor endowments, so that x2=x2(x1,k,n). This expression is the production
possibility frontier for the economy. It is often denoted as the transformation function.
The Lagrange multiplier, λ, expresses the change in the objective when the constraint (here
the value of x1) is relaxed. λ is the shadow value of the constraint. Therefore, it gives the
increase in production of x2 when the production of x1 changes. This is the slope of the
production possibility frontier. The fourth line therefore says that the slope of the production
possibility frontier, the marginal rate of transformation, MRT, is equal to the ratio of marginal
productivities for both factors of production.
We are interested in the shape of the production possibility frontier. Generally, the production
possibility frontier is inverted from origo, i.e. it is a concave function. The production
possibility frontier can generally be graphed as in figure 1.
Now, assume that all factors are used to produce one good only, e.g. good 2. In that case the
economy is located at point 1 in the graph, along the vertical axis. If, on the other hand, all
resources are used to produced the other good, good 1, the economy is located where the
production possibility frontier crosses the horizontal axis. Note that in both these two cases,
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the factor ratios are similar in the two industries. Assume now that half of all resources are
used to produce each good, so that half of the labour and half of the capita available is used to
produced good 1 and good 2, respectively. In this case, the economy is on point 2 in the
graph. Point 2 is obviously on the 45 degree line Also in this case the factor ratios are similar
in the two industries. This will only be on the production possibility frontier in rare instances,
i.e. in those cases where the production possibility frontier is this straight line. The reason is
that substitution possibilities allow increases of production of one good without decreasing
production of other goods. In that case the production possibility frontier is outside the
straight line combining full specialisation in either of the two goods. This is what is illustrated
in the figure.
Figure 1
Above we found that the first order conditions from the maximisation problem gave the
results that ratio of the marginal productivities of the two factors for both goods were equal.
We also noted that this ratio depended on the relative factor use in both industries, li, and not
on their absolute use. Now assume that the endowments of both factors increase
proportionally. In this case the production possibility frontier shifts outward. We will see that
given that prices are the same, the shift in the production function increase production of both
goods proportionally.
Autarky equilibrium
Profit maximisation
Firms in the model are expected to maximise their profits. Their resulting revenues can be
written:
r ( pi , vi ) = max{pi xi − wni − rk i }
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Above, vi denotes the vector of factor use in industry i (ni,ki). The profit maximised revenues
satisfy the first order conditions:
pi
∂xi
= w → pi f l i (l i ) = w
∂ni
pi
∂xi
= r → pi f i (l i ) − l i f l i (l i ) = r
∂k i
(
)
→ pi f i (li ) − wl i = r
Note that the first order conditions are fulfilled for both industries (when there is production
of both goods). This implies that:
p1 f l1 (l1 ) = w = p 2 f l 2 (l 2 )
(
)
( (l ) − l
p1 f 1 (l1 ) − l1 f l1 (l1 ) = r = p 2 f
→
2
2
f (l ) f (l ) − l 2 f l (l 2 )
p1
= l1 2 = 1 2
p2
f l (l1 )
f (l1 ) − l1 f l1 (l1 )
2
2
2
)
f l 2 (l 2 )
2
We find that the first order conditions give as result that the ratio of marginal productivities in
both industries are equal to prices. This is in accordance with intuition and implies that labour
and capital are allocated so as to equate their returns in both industries. Since the ratios of
marginal productivities are equal for labour and capital, the above result imply that the
economy is on the production possibility frontier and therefore that production is maximised
(cfr the result from the constrained maximisation above). The fact that ratios of marginal
productivities equal the ratio of goods prices is also in line with the constrained maximisation
above: The ratio of goods prices equal the shadow price expressed in λ above. Also note that
first order conditions imply that the values of marginal products of labour and capital
respectively, are equal to factor prices.
The market solution therefore determines that relative production of the two goods will be
where the relative prices equal the production possibility frontier. The solution is graphed in
figure 2.
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Figure 2
GDP
We have seen that conditions for maximised production are fulfilled with profit seeking firms.
It is easy to understand that this also implies maximisation of GDP.:
G ( p1 , p 2 , n, k ) ) = max( p1 x1 + p 2 x 2 ) s.t. x 2 = x 2 ( x1 , n, k )
x x , x2
GDP maximisation is left as an exercise. However, note that the above problem can be solved
by substituting the constraint into the objective and deriving the first order conditions.
max( p1 x1 + p 2 x 2 (x1 , n, k ))
xx
p1 + p 2
∂x 2
=0
∂x1
→
p1
∂x
=− 2
p2
∂x1
This again shows that the economy will produce where the relative price of good 1 is equal to
the slope of the production possibility frontier.
Now, derivate the GDP function with respect to the price of good i. This gives:
 ∂x
∂x
∂G
= xi +  p1 1 + p 2 2
∂pi
∂pi
 ∂pi



It is left as an exercise to prove that the terms in the parenthesis equals zero. Therefore the
derivative of the GDP function with respect to a price equals the output of the good.
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Factor prices and factor market equilibrium
The first order conditions imply that the value of factors marginal productivities are equal to
unit factor costs. Factor market equilibrium requires that:
n1 + n2 = n
k1 + k 2 = k
A solution for the autarky equilibrium gives w, r, ni and ki. Product prices are assumed to be
given (so that the demand side for the economy for now is disregarded).
Now consider the first order condition for capital. This is rewritten as the first equation below.
It is obvious that returns to capital are a function of product prices and wages. This is written
explicitly in the second equation below. The derivatives of the returns for capital function
with respect to wage and with respect to prices are given in the third equation and the fourth.
Note that the terms within the brackets sum to zero (cfr. the first order condition for labour).
pi f i (li ) − wl i = r
r i ( pi , w) = pi f i (li ) − wl i
[
]
rwi =
∂ri
dl
n
= pf l i (li ) − w i − li = −li = − i < 0
∂w
dwi
ki
rpi =
∂ri
x
= f i (li ) = i > 0
∂pi
ki
Returns to capital decrease with wages and increase with product prices. Note also that the
derivative with respect to wages is the negative of l. This is a result of the envelope theorem
(and it follows from the first order conditions). The derivative with respect to prices is f.
Equilibrium in the economy implies that factor prices are equal in both industries and that
there is full employment of both factors of production. Therefore:
r 1 ( p1 , w) = r 2 ( p 2 , w)
n1 + n2 = n
k1 + k 2 = k
→
n1
n
k1 + 2 (k − k1 ) = n
k1
k2
l1 k1 + l 2 (k − k1 ) = n
Note that the above requires that rates of return are equal in both industries and that these
rates of return are functions of prices and the wage rates, r=r(p,w). These given important
characteristics of our economy. Rates of return are falling in wages. Their slope equal l (=n/k)
which are the relative uses of labour to capital in their respective industries.
From the above we get that for given goods and factor prices, relative factor demands are:
8
rwi =
[
]
∂ri
dl
n
= pf l i (li ) − w i − li = −li = − i < 0
∂w
dwi
ki
~ ni
li =
= −rwi
ki
An implication of this are negatively sloped relationships between returns to capital and
wages for both industries. These relationships, the factor returns curves, are graphed in figure
3.
From these considerations we get two important results.
The first is that we can now clearly define labour intensive and capital intensive industries. An
industry i is labour intensive (compared to the other industry) if its relative use of labour to
capital, li, is higher than the relative use of labour to capital in another industry j, lj for all
factor prices.
In figure 3 we see that industry 1 has everywhere a steeper rates of return curve. Therefore
industry 1’s use of labour to capital is higher than for industry 2 (remember that the slope is
equal to -n/k for all factor prices). When this is the case, the two curves cross only once. This
is a condition for a unique equilibrium. This is assumption that factor intensity reversals do
not occur.
The figure below is for the case when industry 1 is labour intensive and industry 2 is capital
intensive. This means that for all factor price combinations, industry 1 have a higher labour
capital ratio than does industry 2. We will continue to assume that this is the case throughout
this note.
Figure 3
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The other implication of the above is that figure 3 explains how labour is allocated between
the industries in an economy. The slope of the curves are the factor intensities. For the
equilibrium factor prices (i.e. for those which the rates of return are equal) there is only one
allocation of labour capital between the industries that is in accordance with full employment
of both capital and labour and therefore with factor market equilibrium (in the case of no
factor intensity reversal). This requires a somewhat more detailed explanation.
Figure 4 describes the use of resources in the economy. Along the horizontal axis, its use of
labour is measured. Use of capital is measured along the vertical axis. The lengths of the axes
measure the respective total endowments of labour (horizontal) and capital (vertical). Since
industry 1 is labour intensive, it has a higher labour capital ratio than industry 2 for all factor
prices. For the factor prices that are described in figure 3 (i.e. for fixed factor prices), labour
capital ratios in the two industries are fixed. That is, the two industries’ uses of both factors
are on the straight lines describing these factor uses in figure 4. The reason why the two
curves are straight lines is that we have assumed constant returns to scale. Proportional
increases in production give proportional increase in factor use. Note that the use of both
factors must sum to total endowments. This summation creates the parallelogram in the
figure.
Figure 4
In many presentations of the Heckscher-Ohlin model, the two graphs (figure 3 and figure 4)
are combined into one. This is possible since the rates of return functions give the labour
capital ratios for both industries. These are combined in the figures by drawing each
industries’ labour capital ratios 90 degrees from those given from figure 3. This is figure 5.
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Figure 5
We have implicitly made two assumptions about the characteristics in this economy. The first
was, as noted above, that the two factor rewards curves cross only once. This is the case when
there is no factor intensity reversal. The other assumption is that both products are actually
produced. This is the case only when the economy’s endowments is somewhere between the
two straight lines in figure 5. The area between the straight lines are denoted as the cone of
diversification. If the endowments are not in this cone, there will not be production of both
goods, diversification, but specialisation in one good only.
If factor endowments are not within the cone of diversification, only one good is produced. If
the endowments are to the left of the left straight line, the economy produces only the capital
intensive good. In this case relative factor endowments are:
~ ~ n
l1 > l2 >
k
That is, the economy’s endowments of capital is higher than what is required even in the
capital intensive industry at the given goods and factor prices. In this case only the capital
intensive good is produced and factor prices do depend on endowments. Returns to capital are
lower and wages are higher than what is in accordance with production of both goods.
If the endowments are to the right of the right straight line, the economy produces only the
labour intensive good. In this case relative factor endowments are:
n ~ ~
> l1 > l2
k
That is, the economy’s endowments of labour is higher than what is required even in the
labour intensive industry at the given goods and factor prices. In this case only the labour
intensive good is produced and factor prices do depend on endowments. Returns to capital are
higher and wages are lower than what is in accordance with production of both goods.
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When factor endowments are so as to allow production of both goods, that is, endowments
give a point within in the cone of diversification, they obey the inequalities:
~ n ~
l1 > > l2
k
Characteristics of autarky equilibrium
Here we will see that two of the four theorems, the Rybczynski theorem and the Stolper
Samuelson theorem, hold in the case of autarky equilibrium.
Rybczynski theorem: If a country a country receives one more unit of a factor, production of
the good which uses this factor intensively in production will increase, while production of
the other good decreases.
Stolper-Samuelson theorem: If the price of one good increases, the factor price of the factor
that is used intensively in production of this good increases and the factor price of other factor
decreases.
The Rybczynski theorem
We have assumed that industry 1 is the labour intensive. This implies that l1>l2. From the
above, had that
r 1 ( p1 , w) = r 2 ( p 2 , w)
n1 + n2 = n
→
n1
n
k1 + 2 (k − k1 ) = n
k2
k1
l1 k1 + l 2 (k − k1 ) = n
Therefore:
k1 =
n − l2 k
l1 − l 2
An implication is that
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x1 = rp1 ( p1 , w)k1 = rp1 ( p1 , w)
n − l2 k
l1 − l 2
dx1
1
= rp1 ( p1 , w)
>0
dn
l1 − l 2
dx1 n
n
=
>1
dn x1 n − l 2 k
dx1
l
= − rp1 ( p1 , w) 2 < 0
dk
l1 − l 2
dx1 k
l k
= 2
<0
dk x1 l 2 k − n
These are important results. They say that if the economy’s endowments of labour increases,
production of the labour intensive good increases more than proportionally with the increase
in labour. And, if the endowment of capital increases, production of the labour intensive good
decreases. It is a requirement for these results that the economy is within the cone of
diversification both before and after the shift. Also, it is assumed that the goods and factor
prices are the same before and after the shifts. For an autarky this is not realistic, but in a
traded equilibrium for a small country, it is of interest.
What happens?
To trace the effects in the economy, it can be useful to study figure 4 again, but to increase the
labour endowment in the economy. This is illustrated in figure 6.
Figure 6
Since we have assumed that prices do not change, the straight lines that describe relative
factor use are also fixed. The new labour resources can now be employed by increasing
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production of the labour intensive good. But for this to be possible along the given straight
line for this industry, this increase also requires some increase in use of capital in this
industry. Therefore the use of capital is reduced in the other industry. This is only possible
when production decreases. Therefore also labour from the capital intensive industry is
allocated to the labour intensive industry. The result is higher production and more use of
both factors in the labour intensive industry and less production and less use of both factor in
the capital intensive industry.
The economic mechanisms can be imagined as the following sequence: Larger labour supply
decreases wages. This increases returns to capital in both industries. But since labour is
relatively more important in the labour intensive industry, returns to capital increase more in
this industry (cfr figure 3). Therefore capital is reallocated from the capital intensive industry
to the labour intensive. This reallocation of capital increases wages since the increased capital
in the labour intensive industry has larger effects on demand for labour than the reduction in
capital in the capital intensive industry. The process goes on until rates of return are equalised.
Since nothing has happened to product prices, the process continues until wages are the same
as before the increase in the labour supply.
Figure 7 illustrates the Rybczynski effect with the use of the production possibility frontier. In
that figure, we assume an increase in the amount of labour in the economy. This increase
shifts the production possibility frontier out in both directions. The shift is biased so that
production possibilities shift more in the direction of the labour intensive good than in the
direction of the capital intensive good. For given goods prices, the result is an increase in the
production of the labour intensive good and a decrease in the production of the capital
intensive good. The line connecting these points is the Rybczynski-line.
Figure 7
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The Stolper-Samuelson theorem
We are also interested in the effects of a price increase on to the returns to both factors of
production. We note that equilibrium requires equal factor rewards. This is described in the
first equation below. In the second equation we totally differentiate these expressions with
respect to p1 and w (we will find the effect of a price increase good 1 on wages). Thereafter
we solve for the effect of a price increase on good 1 on wages. We then use our results from
above about effects of prices changes and factor price changes and insert the relevant
expressions into the last equation on the third line. The result that wages increase with an
increase in the price of good 1 follows.
r 1 ( p1 , w) = r 2 ( p2 , w)
∂r 1
∂r 1
∂r 2
dp1 +
dw =
dw
∂p1
∂w
∂w
dw
=
dp1 ∂r 1
(
since
− ∂r
x1
1
∂p1
2
− ∂r
)(
∂w
∂w
l1 − l2 > 0
)
=−
rp1
l2 − l1
=−
k1
>0
l2 − l1
We are also interested in the effects of a price increase on good 1 on the returns to capital. As
above we totally differentiate the returns to capital with respect to goods prices and wages
(first line). Thereafter we solve for the effect on returns to capital in industry of a price
increase on good 1 and insert the relevant expressions from above. We find:
dri = rpi dpi + rwi dw =
xi
n
dpi − i dw
ki
ki
 x1 

dr1 x1
k1  x1 x1  l1 
dw x1
dw x1
 =
= − l1
= − l1
= − l1  −
 = + 
dp1 k1
dp k1
dp k1
 l 2 − l1  k1 k1  l 2 − l1 


x1 
l 
1 − 1  < 0
k1  l1 − l 2 
since
l1 − l 2 > 0
We have shown that an increase in the price of the labour intensive good increases wages and
reduces rates of return for capital. This is the Stolper-Samuelson theorem. The last result
should be qualified by noting that rates of returns to capital in both sectors are equal in
equilibrium.
What are the mechanisms behind the Stolper-Samuelson theorem? When there is an increase
in the price of the labour intensive good, producers want to increase the production of this
good. Therefore, they bid up factor prices to attract the needed factors of production. The
15
higher wages and the higher returns to capital in the labour intensive industry stimulate
reallocation of both labour and capital to the labour intensive industry. Production of the
capital intensive good decreases. But since the expanding industry is labour intensive, labour
becomes relatively scarcer in the economy. Thus, wages increase the most. The reduced
production in the capital intensive economy makes capital less scarce. Therefore, returns to
capital decreases as returns in the industries are equalised.
Above we noted that factor demands and labour market equilibrium were given by:
rwi =
[
]
∂ri
dl
n
= pf l i (li ) − w i − li = −li = − i < 0
∂w
dwi
ki
~ ni
= −rwi
li =
ki
∂ri
= −k i rwi
∂w
n1 + n2 = n
ni = k i
Demand for labour is decreasing in wages in both industries. In figure 8 (left hand part) this is
expressed as two negatively sloped demand curve from both sides in the figure. The
horizontal axis is the total labour force in the economy, and industry 1’s (industry 2’s)
demand for labour is measured from the left (right) hand side. Wages are measured along the
vertical axes. Equilibrium is where the two demand curves cross. The right hand side of the
figure is simply the factor reward curves as described in figure 3, but now with shifted axes
(so that wages are measured along the vertical axis and rates of return along the horizontal
axis). Note that since industry 1 is the labour intensive one, now the factor reward curve for
this industry slopes less than the one for the capital intensive industry. The two figures
therefore allow analysis of the consequences of a product price increase. Increased price of
the labour intensive good implies an upward shift in the labour demand and the factor reward
curve for this industry (to D1’ and r1’). Since the price on the capital intensive good is
unchanged, there is no shift in the factor rewards curve for this industry. Its labour demand
curve, on the other hand, starts shifting down. This is because capital is gradually allocated
towards the labour intensive industry. For the same reason, the labour demand curve in the
labour intensive industry continues to shift upwards. These shifts continue until factor rewards
in the two industries are equalised. Initial shifts are marked with ‘. Equilibrium values for
labour demands after capital reallocation in the two industries are marked with *
16
Figure 8
We have now arrived at the Rybczynski theorem and the Stolper-Samuelson theorem:
Rybczynski theorem: If a country a country receives one more unit of a factor of production,
production of the good which uses this factor intensively in production will increase, while
production of the other good decreases.
Stolper-Samuelson theorem: If the price of one good increases, the factor price of the factor
that is used intensively in production of this good increases and the factor price of other factor
decreases.
In the appendix, a derivation of the two theorems based on cost functions from cost
minimization is presented. That discussion follows closely the discussion in Feenstra’s text
book. An advantage of that method is that it allows discussions of relative changes in factor
prices as consequences of relative price changes. A conclusion from that discussion is that a
good price increase results in a more than proportionate increase in the factor price of the
factor that is used intensively in the production of that good and that a reduction in a good
price lead to a more than proportional reduction in the factor that is used intensively in the
production of that good. This is important. One implication of these results is that owner of
factors of production will definitely see their real income change when goods prices change,
independently of the consumption baskets. Even if workers only consume labour intensive
goods, an increase in the price of labour intensive goods will make workers better off.
Trade
We have now constructed an apparatus for analysing international trade based on differences
in factor endowments. In this section we will do this and investigate the effects of opening up
for trade between two countries that differ only in their relative factor endowments. From our
introduction we had the Heckscher-Ohlin theorem:
17
A country will export the good which is intensive in the factor that the country is relatively
well endowed with and import the good which is intensive in the factor that the country is
relatively scarcely endowed with.
Now assume that nh/kh=lh>lf=nf/kf. This implies that the home country is relatively labour
abundant while the foreign country is relatively capital abundant. Remember that good 1 is
the labour intensive good. Define relative autarky prices as pa=pa1/pa2. In order to prove the
Heckscher-Ohlin theorem we will now demonstrate that pah<paf.
We will need to introduce characteristics of the demand side in the economy in order to
proceed. We will assume that both countries are populated by identical consumers with
identical preferences. The preferences are assumed to be of the standard type with positive
and decreasing marginal utilities for both goods. Furthermore, preferences are assumed to be
homothetic. This implies that marginal rates of substitution are unaffected by proportional
changes in all quantities. Therefore income expansion paths are straight lines through the
origin. Budget shares are constant and income elasticities of demand are equal to one. 2 One
implication of homothetic preferences is that consumers’ indifference curves have the same
shapes outward in the goods space. Since consumers are identical, the preferences in the
economy can be represented by one set of indifference curves. Capitalists and workers have
identical preferences for goods and income distribution changes between groups of consumers
therefore do not result in aggregate demand changes.
In this economy, the autarky equilibrium can be illustrated as in figure 9.
Figure 9
2
Homothetic preferences imply that if the consumer is indifferent between the bundles x and y, they are also
indifferent between tx and ty for any t>0.
18
As indicated in the figure, autarky prices and production of both goods are determined where
the indifference curve of the representative consumer is tangent to the economy’s production
possibility frontier.
Now we will show that autarky relative price (of the labour intensive good to the capital
intensive good) is lower in the home country (which is well endowed with labour relative to
capital) than it is the foreign country. We will show that if this not the case, that is if autarky
relative prices are equal, there will be a contradiction.
Assume that also the foreign autarky relative price is pah. Also assume that foreign’s relatively
capital abundance is because it has more capital, while we assume labour endowments to be
equal to those in the home country. Thus nh/kh=lh>lf=nf/kf since kh<kf. In this case the two
countries production possibility frontiers can be illustrated as in figure 10. Note that the
foreign country’s production possibility frontier is shifted out (it can produce more of both
goods) but in biased way (since higher endowments of capital mean that their production
possibility of the capital intensive goods is relatively higher vis-à-vis the home country than
their production possibility of the labour intensive good). If the two countries have equal
relative prices, the home country produces at point A and the foreign country at B
respectively. Point B must lie above and to the left of point A because of the Rybczynski
theorem: More capital leads to more production of good 2 and less production of good 1 when
goods prices (and therefore factor prices) are similar.
In the foreign country, however, consumers’ budget line is passing through B and has the
slope pah. Since preferences are homothetic foreign consumers demand goods in the same
proportion as consumers in the home country. Their desired consumption point is therefore at
point C. But since point B and point C do not coincide, pah cannot be the autarky price in the
foreign country.
A price pah there is excess demand for good 1 and excess supply of good 2 in the foreign
country and therefore prices have to adjust. The relative price of good 1 to good 2 is higher
under autarky in the foreign country than in the home country. In autarky, therefore, pah<paf.
Figure 10
19
We have demonstrated that autarky relative price of good 1 to good 2 is higher in the foreign
country than in the home country. If the two countries trade, prices converge. With perfectly
free trade, goods prices become similar (also see the discussion below about factor price
equalisation). Denote the common relative price for the two countries when there is trade as p.
We will have pah<p<paf.
Figure 11
In figure 11, the home country is illustrated to the left and the foreign country is illustrated to
the right. In the home country, production is a point B and consumption is at point C in figure
11. In the foreign country, production is a point B* and consumption is a point C*. The
difference between consumption and production of a good is imports. We see that the home
country exports good 1 and imports good 2 and that foreign country imports good 1 and
export good 2. We have demonstrated the validity of the Heckscher-Ohlin theorem:
A country will export the good which is intensive in the factor that country is relatively well
endowed with and import the good which is intensive in the factor that the country is
relatively scarcely endowed with.
In autarky there was lower relative price in the home country as compared to the foreign
country, pah<paf. In autarky therefore, we know that (wh/rh)a<(wf/rf)a.
Now, note that trade led to convergence of goods prices. By the Stolper Samuelson theorem
we know that:
If the price of one good increases, the factor price of the factor that is used intensively in
production of this good increases and the factor price of other factor decreases.
It follows that wages increase and capital rents decrease in the home country and that wages
decrease and capital rents increase in the foreign country. This is called factor price
convergence. How far does this convergence go? Under some assumptions the convergence
goes all the way. This is the factor price equalisation theorem:
20
International trade equalises goods prices and therefore also factor prices.
Factor price equalisation is the extreme outcome of factor price convergence. It depends on a
number of assumptions (some which have partly already been made explicit, while some has
been implicitly assumed). The main argument for factor price equalisation is that trade
equalises goods prices. For this to be the case the following assumptions have to be fulfilled:
•
•
•
•
There is perfect competition in products and factor markets.
There are zero transportation costs.
There are fixed factor endowments and no international factor mobility.
There are identical technologies across countries so that production functions are
identical.
• It is possible to rank goods unambiguously by factor intensities (so that there is no
factor intensity reversal).
• Production functions exhibit constant returns to scale.
• Relative endowments lie within the cone of diversification. This implies that both
countries endowments are so that when there is trade, the resulting common goods
prices lead to common equilibrium factor prices so that there is production of both
goods in both countries and full employment of both factors in both countries.
The last assumption implies that the traded equilibrium replicates the integrated
equilibrium. This is the equilibrium that would have been established if the two countries
were integrated into one. This is illustrated in figure 12.
In figure 12 world’s endowment of labour is measured along the horizontal axis and
world’s endowment of capital is measured along the vertical axis. At given factor prices
(the ones resulting from goods price equalisation) factor uses in both industries are given
by the straight expansion lines from the lower left hand corner and the upper right hand
corner. The home country is measured from the lower left corner. The foreign country is
measured from the upper right corner. For a given factor endowment within the
parallelogram, both countries’ endowments are within the cone of diversification (why?).
The home country is assumed to be the relatively labour abundant. Thus, the endowment
point is below the diagonal, for instance at point E in the diagram. Consumption, on the
other hand is on the diagonal (explain why).
This illustrates why the Heckscher-Ohlin model is sometimes interpreted as indirect trade
in factors of production. The home country produces at point E where its production
makes use of its factors of production. If consumes at point D which corresponds to a
different combination of factors of production. The difference between point E and D is
the factor content of trade. We can also say where on the diagonal point D is. The straight
line between E and D has the same slope as the ratio of factor prices.
21
Income distribution effects of trade
We have seen that factor price equalisation follows from trade in the Heckscher-Ohlin
model of trade. In the above example, workers in the home country are made better off
and capitalists are made worse off. In the foreign country it is the opposite. Workers are
made worse off and capitalists better off. Therefore:
Owners of a country’s abundant factors gain from trade while owners of a country’s scare
factors lose from trade.
In reality it may not always be clear what factors are abundant and what are scarce. In
recent decades for instance, on average US workers have seen their incomes weakly
deteriorate as share of national income in recent years. However, the median income of
American workers have deteriorated sharply and the median wage income in the USA was
about the same as in the beginning of the 1970s in 2005. Wage differences in the USA has
therefore increased dramatically. Some analysts have interpreted this as the results of
imports of low skill labour intensive goods. While the US is abundant in high skilled
labour it is scarce in low skilled labour. As a result, wages for high skilled workers may
have increased and wages of low skilled workers may have decreased as imports of low
skilled labour intensive goods have increased.
Appendix
“Jones algebra” - The Rybczynski theorem and the Stolper-Samuelson theorem revisisted:
Below we will use results from cost minimisation to derive the Rybczynski theorem and
the Stolper-Samuelson theorem. The methodology is sometimes referred to as “Jones
algebra”, following Jones (1965) who was the first to apply it. The presentation in this
appendix follows the presentation in Feenstra (2004) and Krugman, Obstfeld and Melitz
(2011). The point of departure is firms’ cost minimisation. Cost minimisation determines
unit cost functions. Naturally cost minimisation is closely related to profit maximisation.
22
The cost minimisation problem for a firm is generally the problem to minimise costs for a
given level of production. We proceed below without using industry subscripts and
introduce them when needed.
min
(wn + rk )
s.t. F (k , n ) = x0
In our model we assume that there are constant returns to scale. Therefore, for constant
factor costs, minimising unit costs will be the same problem whatever the scale of
production. We can therefore set x0=1. Thus our problem is one a constrained
minimisation problem:
min (wn + rk ) s.t. F (k , n ) = 1
L = (wn + rk ) − λ (F (k .n ) − 1)
∂L
∂F
= w−λ
=0
∂n
∂n
∂L
∂F
= r −λ
=0
∂k
∂k
→
w Fn
=
r Fk
In terms of production theory, this equality says that cost minimisation requires that the
marginal rate of substitution equals the ratio of factor prices. Or: the slope of the isoquant
must be equal ratio of factor prices. Note that 1/Fi is the increase in the use of factor i for a
one unit increase in production since Fi is the marginal productivity of factor 1, i.e. the
increase in production for a one unit increase in the use of factor i. Therefore w/Fn is the
marginal cost from increasing production with the use of labour and r/Fk is the marginal
cost from increasing production with the use of capital. These are equal and equal to λ. λ
is the Lagrange multiplier, i.e. the value (reduction of costs) form relaxing the constraint.
The solution to the minimisation problem above is the cost function. It can be written as
c(w, r ) = min
{wn + rk | F (k,n ) = 1}
That is, c(w,r) is the minimum cost to produce one unit of output. There are constant
returns to scale. Therefore, the resulting unit cost functions are both marginal costs and
average costs. The c(w,r) functions satisfy the following characteristics:
• c(w,r) is increasing in w and r. This is intuitive. If one factor price increases, the unit
costs to produce a good either is constant (in the case the factor is not used) or
increasing for all positive uses of the factor.
• c(w,r) is homogenous in degree one. This is also intuitive. If one multiplies the factor
price vector with a constant, s, costs increase with the same factor: c(sw,sr))=sc(w,r).
The factor combination (n,k) that minimises costs (wn+rk) also minimises (wsn+rsk).
• It can be shown that c(w,r) is concave in w and r.
We shall write the solution to the cost minimisation problem as
23
c(w, r ) = wa n + ra k
Above an is the optimal quantity of n and ak is the optimal quantity of k. Thus, in line with
the notation above, factor intensity is given by l=(an/ak). It is important that these factor
choices depend on factor prices.
Differentiating the unit cost function with respect to wage gives the following results:
∂a 
∂c(w, r )
 ∂a
= an +  w n + r k 
∂w
∂w 
 ∂w
We will now show that the terms in the paranthesis equal zero. Therefore, the derivative
of the unit cost function with respect to wage is equal to the labour needed to produce one
unit. It follows that the derivative with respect to r is equal the capital needed to produce
one unit.
The above result can be seen by considering the constraint in the minimisation problem.
Differentiating this gives the second line below. This holds for all changes of labour and
capital, dn and dk, and also those that result from wage changes. Therefore the third
equation is valid. In the fourth equation we multiply through with product prices. In the
fifth we make use of the results from profit maximisation (see the main text) where the
value of the marginal product of labour and capital equal wages and rents, respectively.
We therefore obtain the result that the parenthesis above equals zero. So the derivative of
the cost function with respect to wages equals an and the derivative of the cost function
with respect to capital equals ak.
F (k , n ) = F (ak an ) = 1
Fk dak + Fn dan = 0
Fk =
∂F
∂F
, Fn =
∂k
∂n
∂ak
∂a
+ Fn n = 0
∂w
∂w
∂a
∂a
pFk k + pFn n = 0
∂w
∂w
∂a
∂a
r k +w n =0
∂w
∂w
→
∂c
= an
∂w
and
∂c
= ak
∂r
Fk
We have noted that the cost functions equal average costs. When there is perfect
competition there is no profits. We now introduce industry subscripts. For two industries
therefore, industry 1 and 2, we have that
24
p1 = c1 (w,r )
p 2 = c 2(w,r)
Note that an and ak denote the use of labour and capital to produce one unit. Therefore
n1=x1a1n and k1=x1a1k. Furthermore, from the resource constraints in the economy we
have:
n1 + n2 = n
k1 + k 2 = k
a1n x1 + a 2 n x 2 = n
a1k x1 + a 2 k x 2 = k
We have argued above (see page 7 and 8) that given factor endowments within the cone of
diversification and exogenous product prices, factor prices and factor intensities are given.
We will take this as given here (but see the discussion at pp 11-13 in Feenstra) and use the
above equations to find expressions for the effects of changes in product prices and in
endowments (for the Stolper-Samuelson and the Rybczynski theorems).
The Stolper-Samuelson theorem
What happens with factor prices if the product prices change? Totally differentiate the
cost function for industry i) above to find:
dpi =
∂c
∂ci
dw + i dr = ain dw + aik dr
∂r
∂w
→
dpi ain dw aik dr wa in dw raik dr
+
=
+
=
pi
pi
pi
ci w
ci r
Above we first used the result that the partial derivative of the cost function with respect
to factor prices equal unit factor uses. Thereafter we calculated effects of relative price
changes. In the last equation we used the fact that prices are equal to unit costs and
multiplied and divided by each factor price. Note that wain/ci is the cost share of labour
and that raik/ci is the cost share of capital. These will be denoted θin and θik respectively.
Thus θin=wain/ci and θik=raik/ci. It should be noted that these costs shares sum to unity:
θin+θik=1. Relative changes will be written with the superscript ^, i.e. with a hat above.
The above equation can therefore be rewritten as:
pˆ i = θ in wˆ + θ ik rˆ
And generally we have:
pˆ 1 = θ1n wˆ + θ1k rˆ
pˆ 2 = θ 2 n wˆ + θ 2 k rˆ
25
But now we can solve for relative changes in wages and rents as the result of relative price
changes. We get:
θ1n
wˆ
θ1k θ1k
θ
pˆ
wˆ = 2 − 2 k rˆ
θ 2n θ 2n
rˆ =
pˆ 1
−
The first expression was obtained from the first equation above, the second from the
second. Insert the second result into the first to obtain:

θ  pˆ
θ
pˆ
rˆ = 1 − 1n  2 − 2 k rˆ 
θ1k θ1k  θ 2 n θ 2 n 
Now solve for relative change in rents:
 pˆ
θ θ
θ  pˆ 
rˆ − 1n  2 k rˆ  = 1 − 1n  2 
θ 1k  θ 2 n  θ 1k θ 1k  θ 2 n 
 θ θ − θ1nθ 2 k
rˆ 1k 2 n
θ 1k θ 2 n

 pˆ 1 θ1n  pˆ 2
 =

−
θ
θ
1k
1k  θ 2 n


θ 1k θ 2 n
rˆ = 
 θ1k θ 2 n − θ1nθ 2 k
 θ 2 n pˆ 1 − θ1n pˆ 2

 θ1k θ 2 n − θ1nθ 2 k



 pˆ 1 θ1n  pˆ 2


θ − θ θ
1k  2 n
 1k
 
θ1k θ 2 n
  = 

   θ1k θ 2 n − θ1nθ 2 k
 θ 2 n pˆ 1 θ1n pˆ 2

−
θ
θ
 1k 2 n θ1k θ 2 n

 =




Now consider the denominator in the last expression. This can be rewritten as follows:
θ1k θ 2 n − θ1nθ 2 k = θ1k (1 − θ 2 k ) − (1 − θ1k )θ 2 k = θ1k − θ 2 k
θ1k θ 2 n − θ1nθ 2 k = (1 − θ1n )θ 2 n − θ1n (1 − θ 2 n ) = θ 2 n − θ1n
Therefore, our above equation can be rewritten as:
 θ pˆ − θ pˆ   θ pˆ − θ pˆ
rˆ =  2 n 1 1n 2  =  2 n 1 1n 2
 θ1k θ 2 n − θ1nθ 2 k   θ 2 n − θ1n
 (θ1n − θ 2 n ) pˆ 2 − θ 2 n ( pˆ 1 − pˆ 2 ) 


θ1n − θ 2 n


  θ1n pˆ 2 − θ 2 n pˆ 1 
 = 
 =
  θ1n − θ 2 n 
We do the same exercise for changes in relative wages:
26
 pˆ 1 θ1n 

wˆ 
−
θ 2n
 θ1k θ1k 
 θ θ  pˆ
θ  pˆ 
wˆ 1 − 2 k 1n  = 2 − 2 k  1 
 θ 2 nθ1k  θ 2 n θ 2 n  θ1k 
 θ θ − θ 2 k θ1n  pˆ 2 θ 2 k  pˆ 1
wˆ  2 n 1k
 = θ − θ  θ
θ 2 nθ1k
2n
2 n  1k


wˆ =
pˆ 2
−
θ 2k
θ 2n



 pˆ 2 θ 2 k  pˆ 1   
 θ1k pˆ 2 − θ 2 k pˆ 1 
θ 2 nθ1k



 =
 
 θ − θ  θ   =  θ θ − θ θ 
θ 2 nθ1k
2 n  1k  
2 k 1n 
 2 n 1k
 2 n

 θ1k pˆ 2 − θ 2 k pˆ 1   θ1k pˆ 2 − θ 2 k pˆ 1   θ 2 k pˆ 1 − θ1k pˆ 2   (θ 2 k − θ1k ) pˆ 1 + θ1k ( pˆ 1 − pˆ 2 ) 

 = 

 = 
 = 
−
−
−
−
θ
θ
θ
θ
θ
θ
θ
θ
θ
θ
2k
1k
2k
1k
2 k 1n 
1k
2k
 2 n 1k
 


 

θ 2 nθ1k
wˆ = 
 θ 2 nθ1k − θ 2 k θ1n
Thus, we have found:
 (θ − θ 2 n ) pˆ 2 − θ 2 n ( pˆ 1 − pˆ 2 ) 

rˆ =  1n
−
θ
θ
1n
2n


and
 (θ − θ1k ) pˆ 1 + θ1k ( pˆ 1 − pˆ 2 ) 

wˆ =  2 k
θ 2 k − θ1k


Now, we keep the assumption that the first industry is the labour intensive one and that the
second is the capital intensive one. This implies that θ1n>θ2n and θ1k<θ2k. In words: the
cost share of labour is higher in the labour intensive industry and the cost share of capital
is higher in the capital intensive industry. From this we conclude that the denominators in
the expressions above are positive.
Now assume that prices in the industries change. Assume in particular that the relative
price of good 1 increases so that the price increase for good 1 is higher compared to the
price increase of good 2. This implies that
pˆ 1 > pˆ 2
What are the implications of these price changes for factor prices? We can immediately
conclude that the following inequalities hold:
27
 (θ − θ1k ) pˆ 1 + θ1k ( pˆ 1 − pˆ 2 ) 
 > pˆ 1
wˆ =  2 k
θ 2 k − θ1k


 (θ − θ 2 n ) pˆ 2 − θ 2 n ( pˆ 1 − pˆ 2 ) 
 < pˆ 2
rˆ =  1n
θ1n − θ 2 n


Wages increase more than the price increase of the first good. Workers can therefore
afford to buy more of good 1 and of good 2 and real wages have therefore increased. For
rental incomes we note that rents change less than the price of good 2. Capital owners can
afford less of both goods and capitalists incomes have fallen.
Now assume that there is only a price increase in the price of the first good. We get the
following results:
 (θ − θ1k ) pˆ 1 + θ1k ( pˆ 1 )   θ 2 k pˆ 1
 = 
wˆ =  2 k
−
θ
θ
2k
1k

  θ 2 k − θ1k
 − θ ( pˆ ) 
rˆ =  2 n 1  < 0
 θ1n − θ 2 n 

 > pˆ 1

This is the Stolper-Samuelson theorem: If the price of one good increases, the factor price of
the factor that is used intensively in production of this good increases and the factor price of
other factor decreases.
The Rybczynski-theorem
Above we had the factor market equilibrium conditions:
n1 + n2 = n
k1 + k 2 = k
a1n x1 + a 2 n x 2 = n
a1k x1 + a 2 k x 2 = k
The first two equations imply that the use of labour and capital in both industries cannot
exceed the total endowments of labour and capital. The two next equations rewrite these
conditions in terms of unit requirements times total production for both factors. We are
interested in the effects of changes in endowments on production of both goods when we keep
prices fixed. When prices are fixed the unit factor requirements are also fixed. Therefore aji
can be treated as constants.
Differentiate the full employment conditions to obtain:
a1n dx1 + a 2 n dx 2 = dn
a1k dx1 + a 2 k dx 2 = dk
28
This can further be rewritten as:
x1 a1n dx1 x 2 a 2 n dx 2 dn
+
=
n x1
n x2
n
x1 a1k dx1 x 2 a 2 k dx 2 dk
+
=
k x1
k
x2
k
We have now obtained expressions for how relative changes in each endowment are related to
relative changes of production of both goods. Note that these depend on the terms (xiaij/J), or
the fraction of resource J (J=n,k) used in the production of good i. These will be defined as λij
so that λ1n=(x1a1n/n) denotes the fraction of the labour force used to produce good 1 etc. Note
that λ1n+λ2n=1 and that λ1k+λ2k=1. As above, relative changes will be written with a ^ above
each variable. We can therefore rewrite the above as:
λ1n xˆ1 + λ 2 n xˆ 2 = nˆ
λ1k xˆ1 + λ 2 k xˆ 2 = kˆ
We will now solve for relative changes in x1 and x2 as functions of relative changes in n and k.
From the first of these equations we obtain:
xˆ1 =
nˆ
λ1n
−
λ2 n xˆ 2
λ1n
Insert this expression for relative change in x1 into the second expression and solve for
relative change in x2 through the sequences below.
29
xˆ1 =
nˆ
λ1n
 nˆ
λ1k 
−
λ2 n xˆ 2
λ1n
−
λ2 n xˆ 2 
 + λ xˆ = kˆ
λ1n  2 k 2
 λ1n
λ1k nˆ λ1k λ2 n xˆ 2
−
+ λ2 k xˆ 2 = kˆ
λ1n
λ1n
λ λ xˆ
λ nˆ
− 1k 2 n 2 + λ2 k xˆ 2 = kˆ − 1k
λ1n
λ1n

λ λ 
λ nˆ
xˆ 2  λ2 k − 1k 2 n  = kˆ − 1k
λ1n 
λ1n

 λ λ − λ1k λ2 n  ˆ λ1k nˆ
 = k −
xˆ 2  2 k 1n
λ
λ1n
1n


 ˆ λ1k nˆ 
λ1n

k −
xˆ 2 =
λ2 k λ1n − λ1k λ2 n 
λ1n 
xˆ 2 =
λ1n kˆ
λ1k nˆ
−
λ2 k λ1n − λ1k λ2 n λ2 k λ1n − λ1k λ2 n
xˆ 2 =
λ1n kˆ − λ1k nˆ
λ1n kˆ − λ1k nˆ
=
λ2 k λ1n − λ1k λ2 n λ2 k (1 − λ2 n ) − (1 − λ2 k )λ2 n
=
λ1n kˆ − λ1k nˆ
λ2 k − λ2 k λ2 n − λ2 n + λ2 k λ2 n
xˆ 2 =
λ1n kˆ − λ1k nˆ
λ2 k − λ2 n
On the third last line we made use of the facts that 1=λ1n+λ2n and 1=λ1k+λ2k.
We will also solve for relative changes in x1. From the expressions
λ1n xˆ1 + λ 2 n xˆ 2 = nˆ
λ1k xˆ1 + λ 2 k xˆ 2 = kˆ
From the second of these we obtain:
xˆ 2 =
kˆ
λ2 k
−
λ1k xˆ1
λ2 k
Insert this expression for relative change in production in industry 2 into the first expression
above:
 kˆ λ1k xˆ1 
 = nˆ
λ1n xˆ1 + λ2 n 
−

λ
λ
2
k
2
k


Use the above expression to solve for relative change in x1 through the following steps:
30
λ1n xˆ1 −
λ2 n λ1k xˆ1
λ kˆ
= nˆ − 2 n
λ2 k
λ2 k

λ λ
xˆ1  λ1n − 2 n 1k
λ2 k


λ kˆ
 = nˆ − 2 n
λ2 k


λ kˆ
 = nˆ − 2 n
λ2 k

 λ λ − λ2 n λ1k 
λ kˆ
 = nˆ − 2 n
xˆ1  1n 2 k
λ2 k
λ2 k


ˆ

λ2 k
 nˆ − λ2 n k 
xˆ1 =
λ1n λ2 k − λ2 n λ1k 
λ2 k 
λ λ
λ λ
xˆ1  1n 2 k − 2 n 1k
λ2 k
 λ2 k
xˆ1 =
λ2 k nˆ
λ2 k nˆ − λ2 n kˆ
=
λ1n λ2 k − λ2 n λ1k λ1n λ2 k − λ2 n λ1k
xˆ1 =
λ2 k nˆ − λ2 n kˆ
λ2 k nˆ − λ2 n kˆ
=
λ1n λ2 k − λ2 n λ1k λ1n (1 − λ1k ) − (1 − λ1n )λ1k
=
λ2 k nˆ − λ2 n kˆ
λ1n − λ1n λ1k − λ1k − λ1n λ1k
xˆ1 =
λ2 k nˆ − λ2 n kˆ
λ1n − λ1k
Again we made use of the facts that 1=λ1n+λ2n and 1=λ1k+λ2k.
Throughout this note we have assumed that industry 1 is labour intensive so that industry 2 is
capital intensive. For given product and factor prices therefore, industry 1 uses more labour
relative to capital than does industry 2. An implication of this is that the share of labour used
in industry 1 exceeds the share of capital used in this industry and that the share of capital
used in industry 2 exceeds the share of labour used in this industry. This is easily seen from
the inequalities:
L1
L L
> > 2
K1 K K 2
→
L1 K1
>
L
K
K 2 L2
>
K
L
Our results were:
31
xˆ 2 =
λ1n kˆ − λ1k nˆ
λ2 k − λ2 n
xˆ1 =
λ2 k nˆ − λ2 n kˆ
λ1n − λ1k
Now assume that there is no change in the endowment in capital and that there is an increase
in the endowment of labour. In this case we have:
− λ1k nˆ
<0
λ2 k − λ2 n
(1 − λ1k )nˆ > nˆ > 0
λ2 k nˆ
=
xˆ1 =
λ1n − λ1k λ1n − λ1k
xˆ 2 =
This is the Rybczynski theorem: If a country a country receives one more unit of a factor of
production, production of the good which uses this factor intensively in production will
increase, while production of the other good decreases.