StolPer Samuelson Theorem Note

8/9/2019 StolPer Samuelson Theorem Note

1/10

A note on the Stolper-Samuelson Theorem

May 3, 2005

1 Preliminaries

Lets start with some notation. Consider a variable z. Then a change in the variable z is given

by dz. A percentage change in z (e.g. Variable z has increased by 10%) is given by dz/z .

To simplify notation we set

z dzz

Obviously, both dz and z can take both positive and negative values. A rst question we can

ask ourselves is how will a sum change when its components change?

Suppose z = x + y then a change in z is given by dz = d(x + y) = dx + dy. Furthermore,

straightforward algebra gives us

dzz

= dx + dy

x + y

= dxx + y

+ dyx + y

= xx + y

dxx

+ yx + y

dyy

This means that

dzz

= z = xx + y

x + yx + y

y

We set

x = xx + y

and y = yx + y

1


2/10


3/10

This can be applied to a change in real wage:

dw p = dwp wdp p2=

dw p

w p

dp p

dw p w p

= dw p pw = dww dp p

w p = w pIn other words, a percentage change in the real wage will be the di ff erence between the per-

centage in the wage minus the percentage change in the prices. Note that this result can be

applied to any ratio (also change in relative price of a good).

2 Stolper-Samuelson Theorem

Lets consider two industries, producing good 1 and good 2 respectively. Both industries use

both capital and labour to produce their goods but indi ff erent intensities (factor intensities).

In order to produce one unit of the good industry i requires aLi units of labour and aKi units

of capital. The assumption that we will be making is that

aK 1aL 1

> aK 2aL 2

In other words, good 1 is relatively more capital intensive and good 2 is relatively more labour

intensive.

The cost of labour (the wage) is w and the cost of capital is r . Hence, the cost of producing

one unit of the good (the marginal cost) is equal to aLi w + aKi r . In perfect competition, the

price of a good is equal to the marginal cost of production. This means that in an industry i

the price will be given by:

P i = M C (w, r ) = aLi w + aKi r

Another question that we can ask ourselves is how the price of the good will change if

e.g. the wage increases by 10% and the cost of capital by 20%. This can be computed in the

3


4/10

following way:

P i = MC (w, r ) = aLi w + aKi r

dP i = dMC (w, r )dP iP i

= dMC (w, r )

MC

= 1MC MC w dw + MC r dr

= 1MC

(aLi dw + aKi dr )

= aLi

MC dw +

aKiMC

dr

= aLi wMC dww + aKi rMC drr

P i = Li w + Ki r (1)

where

Li = aLi w

aLi w + aKi r and Ki =

aKi raLi w + aKi r

In other words Li respresents the share of the labour cost in the marginal cost and Ki

represents the share of the capital cost in the marginal cost. Since both represent shares we

have

Li +

Ki = 1 . This implies that

Ki = 1 Li and Li = 1 Ki

Applying equation (1) to both industries we have

P 1 = L 1 w + K 1 r (2)

MC 1 = aL 1 w + aK 1 r

L 1 = aL 1 waL 1 w + aK 1 r

and K 1 = aK 1 r

aL 1 w + aK 1 rP 2 = L 2 w + K 2 r (3)

MC 2 = aL 2 w + aK 2 r

L 2 = aL 2 waL 2 w + aK 2 r

and K 2 = aK 2 r

aL 2 w + aK 2 r

Using equations (2) and (3), we can compute the di ff erence in percentage change of both prices

4


5/10

as follows:

P 1 P 2 = L 1 w + K 1 r L 2 w K 2 r

= L 1 w + (1 L 1 ) r L 2 w (1 L 2 ) r

= r + L 1 (w r ) r L 2 (w r )

= ( L 1 L 2 ) ( w r )

This last line can be rewritten as

w r = 1

L 1 L 2 P 1 P 2 In other words, the di ff erence in the percentage change in the income of labour and that of capital, depends on 1) the di ff erence in labour shares of both products ( L 1 L 2 ), and 2)

the di ff erence in percentage change of the prices of the two products. By assumption, we will

look at a case where the relative price of good 1 increases. In other words, a situation where

P 1 P 2 > 0. In order to determine the sign of w r we need to determine the sign of L 1 L 2 .

This is done as follows.

L 1 L 2 = aL 1 waL 1 w + aK 1 r

aL 2 w

aL 2 w + aK 2 r < 0

aL 1

aL1

w + aK 1

r

aL 2

aL2

w + aK 2

r < 0

aL 1aL 1 w + aK 1 r

< aL 2

aL 2 w + aK 2 r aL 1 aL 2 w + aL 1 aK 2 r < a L 1 aL 2 w + aK 1 aL 2 r

aL 1 aK 2 r < a K 1 aL 2 r

aK 2aL 2

< aK 1aL 1

L 1 < L 2

In words, this result tells us that the share of the labour cost in the marginal cost of production

the be the largest for the good which is labour intensive (here: good 2).

With this we can establish the rst result of the Stolper-Samuelson theorem:

P 1 P 2 > 0

w r < 0

r > w

5


6/10

In other words, if P 1 increases faster than P 2 then - because it is a capital intensive good

( L 1 < L 2 ) - the income (cost) of capital will increase faster than the wage.The Stolper-Samuelson theorem goes even further. It expresses the result not only in terms

of nominal returns (incomes) but also in terms of real incomes. In other words, the Stolper-

Samuelson theorem predicts that the owners of one factor of production will become richer

(i.e. will be able to buy more) while the owners of the other factor of production will become

poorer (i.e. will be able to buy less). This is shown in the following way:

P 1 = L 1 w + K 1 r

P 1 = L 1 w + r L 1 r

P 1 = L 1 ( w r ) + r (4)

P 1 < r

P 2 = L 2 w + K 2 r

= w K 2 w + K 2 r

= w K 2 (w r )

P 2 > w

Putting these di ff erent elements together we have

r > P 1 > P 2 > w

Change in the real income of factor owners of capital

- in terms of good 1: r P 1 > 0; Following an increase in relative price of good 1, owners

of capital can buy more of good 1.

- in terms of good 2: r P 2 > 0; Following an increase in relative price of good 1, owners

of capital can buy more of good 2.

This means that following an increase in the relative price of good 1, the consumption

possibilities of capital owners increase ; they become richer .

Change in the real income of factor owners of labour (=workers)

- in terms of good 1: w P 1 < 0; Following an increase in relative price of good 1, owners

of labour can buy less of good 1.

6


7/10

- in terms of good 2: w P 2 < 0; Following an increase in relative price of good 1, owners

of labour can buy less of good 2.This means that following an increase in the relative price of good 1, the consumption

possibilities of labour owners decrease ; they become poorer .

3 Graph

In this section, we will show the Stolper-Samuelson theorem using a graph. As before we

assume that

aK 1aL 1> aK 2aL 2

Because both industries are in perfect competition, the price of each good is equal to its

marginal cost:

P 1 = aL 1 w + aK 1 r (5)

P 2 = aL 2 w + aK 2 r (6)

Notice that while the prices, the unit labour requirements and the unit capital requirements all

have indices (either 1 or 2), the factor prices, w and r, dont. This is because even though theindustries are di ff erent, both have to pay the same wage and the same cost of capital. These

prices are determined by the factor markets and cannot be in uenced by the rms.

Equations (5) and (6) can be plotted in the (w,r) space. To make things a little easier, well

rewrite these two equations as

r = P 1aK 1

aL 1aK 1

w

r = P 2aK 2

aL 2aK 2

w

7


8/10

A

B

Figure 1

The red line represents all the combinations of w and r for which industry 1 is in equilibrium.

This will be the case when there are no pro ts and hence no entry in industry 1. Similarly, the

green line represents the combinations for which industry 2 is in equilibrium. Obviously, from

the graph, it is clear that the whole economy, that is both industry 1 and 2, is in equilibrium

at the intersection of the two lines. Hence point A represents the equilibrium

The Stolper-Samuelson theorem makes a prediction about how factor prices change when

goods prices change. To simplify things, we will assume that only P 1 increases ( P 1 > 0) while

P 2 remains the same ( P 2 = 0). This means that we still have the relative price of good 1

increasing since P 1 P 2 > 0. The Stolper-Samuelson theorem predicts that

r > P 1 > 0 = P 2 > w

Consider Figure 2. This represents exactly the same as in Figure 1, except that here we are

8


9/10

focussing on changes of factor prices and goods prices. If we take point A as reference point,

we can draw in addition to the 45 line two other lines through point A, a horizontal blue lineand a vertical blue line. This divides the whole space in 6 di ff erent areas.

A

B

Figure 2

For example, the horizontal blue line represents the combinations of points were r has not

changed compared to point A. In addition, points on that blue line on the right of A represent

outcomes where r has not changed while w has increased. Similarly, points on the left of A

represent where r has not changed and w has decreased. The 45

line represents outcome whereboth r and w increase in the same proportions. Points on the 45 line above point A represent

outcome where r and w are identical and bigger than at point A. Below point A, there are still

the same, but smaller than at point A.

Using a similar reasoning, we can establish for each area how both factor prices have changed

9


10/10

compared to point A. Hence, we can easily determine on the graph, the impact of an increase

in the relative price of good 1 ( P 1 > 0, P 2 = 0): point B is located in the area where r > 0 > w.This is the rst result of the Stolper-Samuelson Theorem: an increase in the relative price of a

good leads to an increase in the price of the factor used intensively in production of that good

and leads to a decrease in the price of the factor not used intensively.

However, the Stolper-Samuelson theorem goes even further: not only does it predict a

change in the nominal incomes, but also a change in the real incomes. Hence, some groups

in the country will become richer while others become poorer. To establish that we need to

include product prices in the analysis also. Establishing that at point B workers have become

poorer, is easy: while prices of goods have not changed ( P 2 = 0) or even increased (

P 1 > 0),

they receive less money ( 0 > w). Hence, they can buy less of both goods; they become poorer.

What about the purchasing power of the capital owners? They do receive more money

(r > 0). This means that they can buy more of good 2, since its price has not changed

( P 2 = 0). For good 1, the story is slightly di ff erent: while they receive more money ( r > 0),

the price of good 1 has also increased ( P 1 > 0). They will be better o ff in point B compared

to point A if r P 1 > 0. To establish this, consider the inequalities in orange. On the 45

line above point A, we have r = w = P 1 . In words, on the the 45 line, the price of good 1

and the prices of both factors increase in the same proportions. To establish this result, look

at equation (4). Since on this line r = w, that equation becomes P 1 = r . Any point above the

45 line we have r > w Again using equation (4), we have r > P 1 : the percentage increase

of the price of good 1 is the percentage change r and something negative. This means that r

increases more that P 1 ( r > P 1 ). Point B is not only above the 45 line but also left of the

vertical blue line. This means that we have r > P 1 > 0 > w. And this is the second part of the

Stolper-Samuelson Theorem: when the price of good 1 increases, because this good is capital

intensive, the real income of capital owners increases while the real income of labour owners

decreases.

10

Documents

StolPer Samuelson Theorem Note