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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 Heckscher-Ohlin 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.
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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.
( )
( ) ( ) ( )
( )
0'' ,0'
,1
0 ,0 ,0 ,0 0
,2
2
2
2
2
<>
≡
=
=
>∂∂
∂<
∂∂
<∂∂
>∂∂
>∂∂
=
iii
ii
ii
ii
iiii
ii
i
i
i
i
i
i
i
i
iii
ii
ffkn
l
lfkkn
Fkx
nkF
nF
kF
nF
kF
nkFx
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:
( ) ( )
( ) ( ) ( ) ( ) ( )ii
liii
i
ii
ilii
i
i
iii
i
i
ii
li
i
ii
i
iii
i
i
lfllfkn
lfklfk
nkFkx
lffk
fkn
nkFnx
−=
−+=
∂∂
=∂∂
===∂
∂=
∂∂
2
,
'1',
Above the notation fi
l 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:
kkknnn
=+=+
21
21
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:
( ) ( )111
12121222
2 , , subject to ,max ,nkFxkkknnnnkFx ==+=+= This problem can be solved by substituting the resource constraint into the objective, setting up the Lagrange function and maximise over n2 and k2:
( ) ( )( )
( )( )
( ) ( )( ) ( )1
111
12
222
2
1
2
11
22
1
2
12
1
1
2
2
1
12
1
1
2
2
1
1111
112
0
0
,,
lfllflfllf
FF
lflf
FF
FFkF
kF
kL
FFnF
nF
nL
xnkFnnkkFL
l
l
k
k
l
l
n
n
kk
nn
−−
−=−=−=−=
=−−=∂∂
−∂∂
−=∂∂
=−−=∂∂
−∂∂
−=∂∂
−−−−=
λ
λλ
λλ
λ
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,
4
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: ( ) { }iiiiii rkwnxpvpr −−= max,
5
Above, vi denotes the vector of factor use in industry i (ni,ki). The profit maximised revenues satisfy the first order conditions:
( )
( ) ( )( )( ) rwllfp
rlfllfprkx
p
wlfpwnx
p
iii
i
ii
liii
ii
ii
ii
lii
ii
=−→
=−→=∂∂
=→=∂∂
Note that the first order conditions are fulfilled for both industries (when there is production of both goods). This implies that:
( ) ( )( ) ( )( ) ( ) ( )( )
( )( )
( ) ( )( ) ( )1
111
12
222
2
11
22
2
1
22
222
211
111
1
22
211
1
lfllflfllf
lflf
pp
lfllfprlfllfplfpwlfp
l
l
l
l
ll
ll
−−
==→
−==−
==
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.
6
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.: ( ) ( ) ( )knxxxxpxpknppG
xxx
,, s.t. max),,, 1222211,212
=+=
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.
( )( )
1
2
2
1
1
221
12211
0
,,max
xx
pp
xxpp
knxxpxpxx
∂∂
−=
→
=∂∂
+
+
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:
∂∂
+∂∂
+=∂∂
iii
i pxp
pxpx
pG 2
21
1
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.
7
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:
kkknnn
=+=+
21
21
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).
( )( ) ( )
( )[ ]
( ) 0
0
,
>==∂∂
=
<−=−=−−=∂∂
=
−=
=−
i
ii
i
i
iip
i
iii
i
ii
il
iiw
iii
iii
iii
i
kx
lfpr
r
kn
lldwdl
wlpfwr
r
wllfpwprrwllfp
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: ( ) ( )
( )
( ) nkklkl
nkkknk
kn
kkknnn
wprwpr
=−+
=−+
→=+=+=
1211
12
21
1
1
21
21
22
11 ,,
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
( )[ ]i
wi
ii
i
iii
i
ii
il
iiw
rkn
l
kn
lldwdl
wlpfwr
r
−==
<−=−=−−=∂∂
=
~
0
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
9
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.
10
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:
knll >> 21
~~
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:
21~~ ll
kn
>>
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.
11
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:
21~~ l
knl >>
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 ( ) ( )
( )
( ) nkklkl
nkkknk
kn
nnnwprwpr
=−+
=−+
→=+=
1211
12
21
1
1
21
22
11 ,,
Therefore:
21
21 ll
klnk−−
=
An implication is that
12
( ) ( )
( )
( )
0
0,
1
01,
,,
2
2
1
1
21
21
11
21
1
211
11
21
21
111
11
<−
=
<−
−=
>−
=
>−
=
−−
==
nklkl
xk
dkdx
lllwpr
dkdx
klnn
xn
dndx
llwpr
dndx
llklnwprkwprx
p
p
pp
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
13
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
14
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. ( ) ( )
( ) ( )0 since
0
,,
21
12
1
1
12
1
211
1
1
21
11
12
21
1
>−
>−
−=−
−=
∂∂−∂
∂∂
∂−=
∂∂
=∂∂
+∂∂
=
ll
llk
x
llr
wr
wr
pr
dpdw
dwwrdw
wrdp
pr
wprwpr
p
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:
0 since
01
21
21
1
1
1
12
1
1
1
1
1
12
1
1
11
11
1
11
1
1
1
1
>−
<
−
−
=
−
+=
−−−=−=−=
−=+=
llll
lkx
lll
kx
kx
llk
x
lkx
dpdwl
kx
dpdwl
kx
dpdr
dwkn
dpkx
dwrdprdri
ii
i
iiwi
ipi
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:
( )[ ]
nnn
rkwr
kn
rkn
l
kn
lldwdl
wlpfwr
r
iwi
iii
iw
i
ii
i
iii
i
ii
il
iiw
=+
−=∂∂
=
−==
<−=−=−−=∂∂
=
21
~
0
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=pa
1/pa2. In order to prove the
Heckscher-Ohlin theorem we will now demonstrate that pah<pa
f. 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 pa
h. 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 pa
h. 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, pa
h cannot be the autarky price in the foreign country. A price pa
h 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, pa
h<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 pa
h<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, pa
h<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.
( ) ( ) 0, s.t. min xnkFrkwn =+ 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:
( ) ( )( ) ( )( )
k
n
FF
rw
kFr
kL
nFw
nL
nkFrkwnLnkFrkwn
=
→
=∂∂
−=∂∂
=∂∂
−=∂∂
−−+==+
0
0
1.1, s.t. min
λ
λ
λ
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 ( ) ( ){ }1| min, =+= k,n Frkwn rwc
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
( ) kn rawarwc +=,
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:
( )
∂∂
+∂∂
+=∂
∂wa
rwa
waw
rwc knn
,
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. ( ) ( )
k
n
nk
nn
kk
nn
kk
nknnkk
nk
arc
a
waw
war
wapF
wapF
waF
waF
nFF
kFFdaFdaF
aaFnkF
=∂∂
=∂∂→
=∂∂
+∂∂
=∂∂
+∂∂
=∂∂
+∂∂
∂∂
=∂∂
==+
==
andwc
0
0
0
, 0
1,
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
( )(w,r) cpw,rcp
22
11
==
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:
k xaxanxaxa
kkknnn
kk
nn
=+=+
=+=+
2211
2211
21
21
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:
rdr
cra
wdw
cwa
pdra
pdwa
pdp
dradwadrrc
dwwc
dp
i
ik
i
in
i
ik
i
in
i
i
ikinii
i
+=+=
→
+=∂∂
+∂∂
=
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:
rwp ikini ˆˆˆ θθ +=
And generally we have:
rwprwp
kn
kn
ˆˆˆˆˆˆ
222
111
θθθθ+=+=
25
But now we can solve for relative changes in wages and rents as the result of relative price changes. We get:
rpw
wpr
n
k
n
k
n
k
ˆˆˆ
ˆˆˆ
2
2
2
2
1
1
1
1
θθ
θ
θθ
θ
−=
−=
The first expression was obtained from the first equation above, the second from the second. Insert the second result into the first to obtain:
−−= rppr
n
k
nk
n
k
ˆˆˆˆ2
2
2
2
1
1
1
1
θθ
θθθ
θ
Now solve for relative change in rents:
−−
=
−
−
=
−
−
=
−=
−
−=
−
knnk
nn
nk
n
nk
n
knnk
nk
nk
n
kknnk
nk
nk
n
knk
knnk
nk
n
kn
k
k
n
pp
ppppr
ppr
pprr
2121
2112
21
21
21
12
2121
21
2
2
1
1
1
1
2121
21
2
2
1
1
1
1
21
2121
2
2
1
1
1
1
2
2
1
1
ˆˆ
ˆˆˆˆˆ
ˆˆˆ
ˆˆˆˆ
θθθθθθ
θθθ
θθθ
θθθθθθ
θθθ
θθθθθθθ
θθθ
θθθθθθθ
θθθ
θθθ
θθ
Now consider the denominator in the last expression. This can be rewritten as follows:
( ) ( )( ) ( ) nnnnnnknnk
kkkkkkknnk
1221212121
2121212121
1111
θθθθθθθθθθθθθθθθθθθθ
−=−−−=−−=−−−=−
Therefore, our above equation can be rewritten as:
( ) ( )
−
−−−
=
−−
=
−−
=
−−
=
nn
nnn
nn
nn
nn
nn
knnk
nn
ppp
ppppppr
21
212221
21
1221
12
2112
2121
2112
ˆˆˆ
ˆˆˆˆˆˆˆ
θθθθθ
θθθθ
θθθθ
θθθθθθ
We do the same exercise for changes in relative wages:
26
( ) ( )
−
−+−=
−−
=
−−
=
−−
=
−
−
=
−
−
=
−=
−
−=
−
−−=
kk
kkk
kk
kk
kk
kk
nkkn
kk
kn
kk
nkkn
kn
kn
k
nnkkn
kn
kn
k
nkn
nkkn
kn
k
nkn
nk
k
n
kn
k
n
ppppppppp
ppppw
ppw
ppw
wppw
12
211112
12
2112
21
1221
1212
1221
12
1221
1212
12
1
1
2
2
2
2
1212
12
1
1
2
2
2
2
12
1212
1
1
2
2
2
2
12
12
1
1
1
1
2
2
2
2
ˆˆˆˆˆˆˆˆˆ
ˆˆˆˆˆ
ˆˆˆ
ˆˆ1ˆ
ˆˆˆˆ
θθθθθ
θθθθ
θθθθ
θθθθθθ
θθθθ
θθθθθθ
θθθ
θθθθθθθ
θθθ
θθθθθθθ
θθθ
θθθθθ
θθ
θθθ
θ
Thus, we have found:
( ) ( )
−
−−−=
nn
nnn pppr
21
212221 ˆˆˆˆ
θθθθθ
and
( ) ( )
−
−+−=
kk
kkk pppw
12
211112 ˆˆˆˆ
θθθθθ
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
21 ˆˆ pp > What are the implications of these price changes for factor prices? We can immediately conclude that the following inequalities hold:
27
( ) ( )
( ) ( )2
21
212221
112
211112
ˆˆˆˆ
ˆ
ˆˆˆˆ
ˆ
pppp
r
pppp
w
nn
nnn
kk
kkk
<
−
−−−=
>
−
−+−=
θθθθθ
θθθθθ
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:
( ) ( )
( )0
ˆˆ
ˆˆˆˆ
ˆ
21
12
112
12
12
11112
<
−
−=
>
−
=
−+−
=
nn
n
kk
k
kk
kkk
pr
pppp
w
θθθ
θθθ
θθθθθ
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:
k xaxanxaxa
kkknnn
kk
nn
=+=+
=+=+
2211
2211
21
21
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:
dk dxadxadndxadxa
kk
nn
=+=+
2211
2211
28
This can further be rewritten as:
kdk
xdx
kax
xdx
kax
ndn
xdx
nax
xdx
nax
kk
nn
=+
=+
2
222
1
111
2
222
1
111
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:
kxx
nxx
kk
nn
ˆˆˆ
ˆˆˆ
2211
2211
=+
=+
λλ
λλ
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:
n
n
n
xnx1
22
11
ˆˆˆλλ
λ−=
Insert this expression for relative change in x1 into the second expression and solve for relative change in x2 through the sequences below.
29
( ) ( )
nk
kn
nknnkk
kn
nknk
kn
nknk
kn
nknk
k
nknk
n
n
k
nknk
n
n
k
n
nknk
n
k
n
nkk
n
kk
n
nk
kn
nk
n
k
kn
n
nk
n
n
n
nkx
nk
λλnknkx
nkx
nkx
nkx
nkx
nkxx
kxxn
kxxn
xnx
22
112
222222
11
2222
11
2112
112
2112
1
2112
12
1
1
2112
12
1
1
1
21122
1
1
1
2122
1
122
1
221
221
221
1
1
221
22
11
1
22
11
ˆˆˆ
ˆˆ11
ˆˆˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆˆ
ˆˆˆˆ
ˆˆˆˆ
ˆˆˆ
λλλλ
λλλλλλλλ
λλλλ
λλλλλλ
λλλλλ
λλλλλ
λλ
λλλλλ
λλ
λλλλλ
λλ
λλλλ
λλλ
λλλ
λλλλ
λλ
λλλ
λλ
λλ
λ
−−
=
+−−−
=
−−−−
=−−
=
−−
−=
−
−=
−=
−
−=
−
−=+−
=+−
=+
−
−=
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
kxx
nxx
kk
nn
ˆˆˆ
ˆˆˆ
2211
2211
=+
=+
λλ
λλ
From the second of these we obtain:
k
k
k
xkx2
11
22
ˆˆˆ
λλ
λ−=
Insert this expression for relative change in production in industry 2 into the first expression above:
nxkxk
k
knn ˆˆˆ
ˆ2
11
2211 =
−+
λλ
λλλ
Use the above expression to solve for relative change in x1 through the following steps:
30
( ) ( )
kn
nk
knkknn
nk
knkn
nk
knkn
nk
knkn
nk
knkn
k
k
n
knkn
k
k
n
k
knkn
k
n
k
kn
k
kn
k
n
k
knn
k
n
k
knn
knx
kn
knknx
knnx
knx
knx
knx
knx
knxx
11
221
111111
22
1111
22
1221
221
1221
22
1221
21
2
2
1221
21
2
2
2
12211
2
2
2
12
2
211
2
2
2
1211
2
2
2
11211
ˆˆˆ
ˆˆ
11
ˆˆˆˆˆ
ˆˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆ
ˆˆˆˆ
λλλλ
λλλλλλλλ
λλλλλλ
λλλλλλ
λλλλλλ
λλλλλ
λλ
λλλλλ
λλ
λλλλλ
λλ
λλλ
λλλ
λλ
λλλλ
λλ
λλλλ
−−
=
−−−−
=
−−−−
=−−
=
−−
=−
=
−
−=
−=
−
−=
−
−=
−
−=−
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:
LL
KK
KK
LL
KL
KL
KL
22
11
2
2
1
1
>
>
→
>>
Our results were:
31
kn
nk
nk
kn
knx
nkx
11
221
22
112
ˆˆˆ
ˆˆˆ
λλλλ
λλλλ
−−
=
−−
=
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:
( ) 0ˆˆ1ˆˆ
0ˆˆ
11
1
11
21
22
12
>>−
−=
−=
<−
−=
nnnx
nx
kn
k
kn
k
nk
k
λλλ
λλλ
λλλ
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.
