stolpersamuelsontheorem

A note on the Stolper-Samuelson Theorem

May 3, 2005

1

Preliminaries

Let’s 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≡
ˆ

dz
z

Obviously, both dz and z can take both positive and negative values. A firstquestion we can
ˆ
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
dz
z

=
=
=

dx + dy
x+y
dx
dy
+
x+y x+y
x dx
y dy
+
x+y x
x+y y

This means that
x
y
dz
=z=
ˆ
x+
ˆ
y
ˆ
z
x+y
x+y
We set
θx =

x
y
and θy =
x+y
x+y
1

Inother words, θx and θy represent the shares (i.e. the importance) in the sum z of x and y.
This means that the equation can be rewritten as
z = θx x + θy y
ˆ
ˆ
ˆ
In words, the percentage change of a sum=weighted sum of percentage changes of its components where the weights are the shares of the component in the sum.
A second question we can ask ourselves is how will a ratio change when itscomponents (i.e.
numerator and denominator) change.
Suppose that
z=

x
y

then a change in z is given by
dz

µ ¶
x
y
dxy − xdy
y2
dx x dy
−
y
y y

= d
=
=

and a percentage change in z is given by
d
z =
ˆ
=
=
=
=
=

³ ´
x
y
x
y

µ ¶
x y
d
y x
µ
¶
dx x dy y
−
y
y y x
x dy y
dx y
−
y x y y x
dx dy
−
x
y
x−y

z = x−y
ˆ
ˆ ˆ
In words, apercentage change in a ratio is simply given (the percentage in the numerator) (the percentage change in the denominator).
2

This can be applied to a change in real wage:
µ ¶
w
dwp − wdp
d
=
p
p2
dw w dp
=
−
p
p p
³ ´
w
µ ¶
d p
w p
dw dp
= d
=
−
w
p w
w
p
p
µˆ¶
w
= w−p
ˆ ˆ
p
In other words, a percentage change in the real wage will be the difference between thepercentage 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

Let’s consider two industries, producing good 1 and good 2 respectively. Both industries use
both capital and labour to produce their goods but indifferent intensities (factor intensities).
In order toproduce 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
aK2
aK1
>
aL1
aL2
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 (themarginal 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:
Pi = 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 inthe
3

following way:
Pi
dPi
dPi
Pi

ˆ
Pi

= M C (w, r) = aLi w + aKi r
= dM C (w, r)
dM C (w, r)
=
MC
µ
¶
1
∂M C
∂M C
=
dw +
dr
MC
∂w
∂r
1
(aLi dw + aKi dr)
=
MC
aLi
aKi
=
dw +
dr
MC
MC
aLi w dw aKi r dr
=
+
MC w
MC r
= θLi w + θKi r
ˆ
ˆ

(1)

where
θLi =

aLi w
aLi w + aKi r

and θKi =

aKi r
aLi w + aKi r

In other words θLirespresents the share of the labour cost in the marginal cost and θKi
represents the share of the capital cost in the marginal cost. Since both represent shares we
have θLi + θKi = 1. This implies that
θKi = 1 − θLi

and θLi = 1 − θKi

Applying equation (1) to both industries we have
ˆ
P1

= θL1 w + θK1 r
ˆ
ˆ

M C1

= aL1 w + aK1 r
aL1 w
=
aL1 w + aK1 r
= θL2 w + θK2 r
ˆ
ˆ

θL1
ˆ...