Detailed explanation of Jones (1965)
1 Outline
2 Notation
3 Basic relations
4 Equations of change (comparative statics)
5 An alternative derivation of the equations of change.
6 Factor intensities
7 Optimal use of inputs
8 Substitution between factors.
9 Endogenous technology
10 Magnification effects:
10.1 Rybzcynski Theorem
10.2 Stolper Samuelson Theorem
11 Demand
2 Notation
L – labor endowment
T – land endowment
M – manufacturing output
F – food output
w – wage
r – rent
pF, pW – competitive market prices
 LM 
aLM w
- labor’s share in the value of manufacturing output:  LM  TM  1 and  LF  TF  1
pM
1
 

   LM TM  is the matrix of factor shares in the value of output
  LF TF 
ij - share of factor i used in the production of good j.
LM - share of labor used in manufacturing: LF  LM  1 and TF  TM  1
 LM
 
 TM
xˆ 
LF 
is the matrix of sector shares in the factor endowments
TF 
dx
- infinitesimal percentage change in x (rate of change in x)
x
M 
aˆTM  aˆ LM
aˆ  aˆ LF
,  F  TF
- elasticities of substitution between land and labor in manufacturing
wˆ  rˆ
wˆ  rˆ
and food.
3 Basic relations
Technology is defined in terms of the amounts of land and labor used to produce one unit of
each good:
aij - quantity of factor i required to produce a unit of commodity j. So, aTF is the amount of land
required to produce one unit of Food. The technology matrix is defined as
a
A   LM
 aTM
aLF 

aTF 
The dimensions of the technology matrix is (number of factors)X(number of goods). The
technology matrix ties together the vector of endowments and the vector of outputs:
 M   aLM
  a
 F   TM
1
aLF   L 

aTF   T 
or
 L   aLM
  a
 T   TM
aLF   M 

aTF   F 
The equation above corresponds to the following two input market clearing conditions
(1)
aLM M  aLF F  L
aTM M  aTF F  T
The two equations above just say that the labor and land are completely used up in production.
2
Perfect competition will guarantee that the prices equal costs:
 pM

 pF
  aLM

  aTM
aLF   w 

aTF   r 
or
(2)
aLM w  aTM r  pM
aLF w  aTF r  pF
The input coefficients are a function of the input prices
aij  f  w, r  , i, j
A micro reminder: unit cost minimization is dual to the profit maximization
min aLM w  aTM r
aLM , aTM
s.t.M  aLM , aTM   1
or
max pM M  aLM , aTM 
aLM , aTM
s.t.aLM w  aTM r  pM
4 Equations of change (comparative statics)
In what follows we denote percentage changes in variables (also called rates of change) rather
than changes in levels. xˆ 
dx
- infinitesimal percentage change in x (rate of change in x).
x
Using percentage changes allows us for example to make judgments about the real changes in
incomes. If the percentage increase in wage is larger than percentage increases in prices of all goods
then the real wage unambiguously increases.
Rates of change are also a natural way to describe a magnification effect, when one variable
changes more than proportionately to the change of another variable. For example, output of
manufacturing changes more than proportionately to the changes in labor endowment. Magnification
would just mean that the rate of change in one variable, manufacturing output, is larger than the rate of
change in another variable, input endowment. Difference in rates of change of prices is a change in
relative price.
The derivation of equations [1.1] through [4.1] follows the same steps. We will perform one
derivation in detail leaving the rest to the homework. Let’s derive the first equation of change
(equation [1.1] in Jones, 1965) by starting with the first input market clearing condition (equation [1]
3
in Jones,1965) which states that all labor is allocated between production of manufactures, M, or food,
F:
(3)
aLM M  aLF F  L
Total differentiation of the above equation gives us
(4)
d  aLM M   d  aLF F   dL
A gentle calculus reminder about total differentiation of a sum: x  y  z  dx  dy  dz
Endowment of labor (L), outputs (M and F), and the input requirements ( aLM and aLF ) can all change.
So the terms in round brackets can be expanded as
(5)
d  aLM M   MdaLM  aLM dM
d  aLF F   FdaLF  aLF dF
A gentle calculus reminder about total differentiation of a product: xy  z  xdy  ydx  dz
Substituting expanded terms from (5) back into (4) gives us:
 MdaLM  aLM dM    FdaLF  aLF dF   dL
Now we need to transform this expression from changes to the rates of change. First, divide both sides
by L to get the rate of change in labor endowment,
dL
, on the right hand side:
L
 MdaLM aLM dM   FdaLF aLF dF  dL





L   L
L  L
 L
To complete the transition from changes (such as daLM and dM ) to rates of changes (such as
and
daLM
aLM
dM
respectively) we divide and multiply every term in the above equation by the necessary
M
variable. We also drop the brackets at this time.
MdaLM aLM aLM dM M FdaLF aLF aLF dF F dL




L aLM
L
M
L aLF
L F
L
Rearranging multipliers in the above equation produces a slightly better looking expression
(6)
aLM M  daLM  aLM M  dM



L  aLM 
L  M
 aLF F  daLF  aLF F  dF   dL 




 
L  aLF 
L  F   L 

Every term on the left hand side consists of two parts: (a) share of industry in the labor endowment
and (b) rate of change. Take the first term
4
aLM M  daLM 


L  aLM 
Consider each of the two parts of this term:
(a) The first part of this term is
aLM M
. The numerator, aLM M , is the amount of labor used in the
L
production of manufactures:

 amount of labor

 amount of labor   amount   used in the

 

production
 used to produce   of

 one unit of food   food  
of food

 
 
aLM
F






The denominator is the amount of labor, so that the ratio is the share of labor used in the production of
food. Denote it as LM 
aLM M
. More generally
L
ij - share of factor i used in the production of good j.
Note that the relative endowment is bounded by the ratio of inputs in both industries
LF L LM
 
TF T TM
 da
(b) The second part of the term is  LM
 aLM

 . This is the rate of change in aLM . Denote rates of change

in a variable with a hat over that variable: aˆ LM 
daLM
dM
, Mˆ 
etc.
M
aLM
Now equation (6) can be rewritten as
LM aˆLM  LM Mˆ  LF aˆLF  LF Fˆ  Lˆ
rearranging the above equation produces equation (1.1) from Jones (1965)
LM Mˆ  LF Fˆ  Lˆ   LM aˆLM  LF aˆLF 
Following a similar procedure we can derive the rest of the equations.
5. An alternative derivation of the equations of change
5
There is a more elegant alternative to derive the rate of change which makes use of some
properties of total differentiation in terms of rate of change. These properties are outlined in the
“gentle calculus reminders” below. Start from the same labor market clearing condition
aLM M  aLF F  L
Using the rule below totally differentiate the equation in terms of rate of change
(7)
aLM M d  aLM M  aLF F d  aLF F  dL


L
aLM M
L
aLF F
L
A gentle calculus reminder: x  y  z 
x dx
y dy dz
. A useful interpretation of this


x y x x y y
z
relation is that the partial elasticity (holding y constant) of the sum, which is z, with respect to one of
the summands, say x, is the share of x in z. Two intuitive examples: (1) if every part of some total
increases by the same percentage the total will increase by the same percentage; (2) if one percent of a
total doubles, increases by 100%, the total will increase only by 1%. Convince yourself, take 2+3=5, 3
is 60% of the total, a 20% increase in 3 should increase the total by 20%x60%=12%. Does it?
The rates of changes on the left hand side can be further decomposed as
(8)
d  aLM M  daLM dM


aLM M
aLM
M
d  aLF F  daLF dF


aLF F
aLF
F
A gentle calculus reminder: xy  z 
dz dx dy

 . A percentage change in a multiplier changes the
z
x
y
product the same percentage. Convince yourself, take 20x5=100, if 5 increases by 20% to 6 the
product will increase by 20% as well: 20x5x1.2=120
Proof.
1) Total differentiation yields: ydx  xdy  dz . Divide both sides by z:
we get
ydx xdy dz


. Since z  xy ,
z
z
z
ydx xdy ydx xdy
dx dy dz



 

.
z
z
xy
xy
x
y
z
2) Another way to show the above rule of differentiation in terms of rate of change is to use logdifferentiation. Taking natural logarithms of each side of xy  z gives us ln x  ln y  ln z . Total log-
6
differentiation gives d ln x  d ln y  d ln z . Remember that d ln x 
1
dx dy dz
dx . So,
. This


x
x
y
z
will come in useful later when we derive elasticity of substitution between factors.
Substituting (8) into (7), using our definition of  ’s ( LM 
aLM M
a F
and LF  LF ), and denoting
L
L
rates of change with hats we get




LM aˆLM  Mˆ  LF aˆLF  Fˆ  Lˆ ,
which can be rearranged to obtain equation (1.1) in Jones (1965)
Homework:
1. Using the same procedure please derive equations (2.1) – (4.1) in Jones (1965).
2. Rewrite equations of change (1.1) – (4.1) in Jones (1965) for the case of constant, Leontieff, input
coefficients.
The equations of change are therefore
LM Mˆ  LF Fˆ  Lˆ   LM aˆ LM  LF aˆLF 
(9)
TM Mˆ  TF Fˆ  Tˆ   TM aˆTM  TF aˆTF 
 LM wˆ  TM rˆ  pˆ M   LM aˆ LM  TM aˆTM 
 LF wˆ  TF rˆ  pˆ F   LF aˆ LF  TF aˆTF 
Where  LM 
aLM w
- labor’s share in the value of manufacturing output
pM
To gain some intuition for the equations of change consider a graphical representation of
change in endowment. First, consider the change in factor allocation. Note that the length of the factor
usage vectors connecting the origin with the endowment point is proportional to the output. When
endowment of labor increases (from E to E’) and the input coefficients do not change the output of M
increases and the output of F decreases. This can be seen by reduced length of a the factor use vector
(parallel to F through E’). The change in outputs does not have to be so dramatic if the input
requirements change (both industries will use more labor). The length of the factor use vector sloped
F’ that goes through E’ increases as the slope increases from F to F’.
7
Intuition behind LM Mˆ  LF Fˆ  Lˆ   aˆLM LM  aˆLF LF 
L
M’
M
E’
F’
F
LF L
E
LM L
T
Second, consider the zero profit condition.
ˆ  TM rˆ  pˆ M  LM aˆLM  TM aˆTM 
Intuition behind LM w
L
T
As a reminder, full employment of labor and land implies
8
aLM M  aLF F  L
aTM M  aTF F  T
dividing each equation by the left hand side and using definition of  ’s give us
(10)
LM  LF  1
TM  TF  1
Zero profit conditions imply
aLM w  aTM r  pM
aLF w  aTF r  pF
dividing each equation by the left hand side and using definition of  ’s give us
(11)
 LM  TM  1
 LF  TF  1
6 Factor intensities
Factor intensities distinguish the two industries. Manufacturing is arbitrarily chosen to be labor
intensive. It means that the amount of labor relative to the amount of land is greater in manufacturing:
(12)
aLM aLF

aTM aTF
One implication of the factor intensity is that labor’s share in manufacturing is greater than
labor’s share in food
(13)
 LM   LF
Proof.
Using the definitions of  LM and  LF inequality (13) can be rewritten as
aLM w aLF w

pM
pF
Using the fact that under perfect competition the price equals cost the above inequality becomes
aLF w
aLM w

aLM w  aTM r aLF w  aTF r
Divide each side by the numerator
9
1
1

a r
a r
1  TM
1  TF
aLM w
aLF w
Invert the inequality changing the inequality sign
1
aTM r
a r
 1  TF
aLM w
aLF w
Subtract 1 from each side, divide both sides by
r
, and invert both side of the resulting inequality
w
aLM aLF

aTM aTF
This is the definition of factor intensity given in (12), Q.E.D.
Another implication of the definition of the factor intensity is that the percentage of labor force
used in manufacturing must exceed the percentage of total land that is used in manufacturing.
(14)
LM  TM
Proof. Using the definitions of  ’s the above inequality becomes
aLM M aTM M

L
T
Using factor market clearing conditions in (1) the inequality becomes
aLM M
aTM M

aLM M  aLF F aTM M  aTF F
Divide each side by the numerator, invert the obtained inequality, subtract 1 from both sides, divide
both sides by
a
F
, and multiply both sides by LM
M
aTF
a F
a F a a
a a
a a
a a
1
1

; 1  LF  1  TF ; LF LM  TF LM ; LF LM  TF LM ;
a F
a F
aLM M
aTM M aLM aTF aTM aTF aLM aTF aTM aTF
1  LF
1  TF
aLM M
aTM M
aLF aLM

aTF aTM
The last equation is the definition of factor intensity in (12), Q.E.D.
This result looks very intuitive on the graph
10
M
L
45o-line
F
LF
E=1
LM
T
TM
The implications of the factors intensity can be also summarized in the matrix form. Define
two matrices
 

   LM TM 
  LF TF 
and
 LM
 
 TM
LF 
TF 
The determinants of both matrices is given by
   LMTF   LFTM and
  LM TF  LF TM
Applying restrictions imposed by factor clearing and perfect competition in equations (10) and (11).
The determinants can be simplified to
   LM 1   LF    LFTM
(15)
  LM   LF  LM  TM 
  LM   LF
and
  LM 1  TM   LF TM
(16)
 LM  TM  LM  LF 
 LM  TM
Both determinants are positive by the inequalities (13) and (14) implied by factor intensity.
11
7 Optimal use of inputs
Producers take prices of inputs and prices of the outputs as given and chooses inputs in such
combination that the unit cost is minimized (the first derivative is equal to zero).
d  waLM  raTM 
0
waLM  raTM
d  waLF  raTF 
0
waLF  raTF
Zero profit condition guarantees that the denominator of each equation equals the price. The two
equations above can be further transformed (see the previous gentle calculus reminders)
d  waLM  raTM  waLM d  waLM  raTM d  raTM 


waLM  raTM
pM
waLM
pM raTM
d  waLF  raTF  waLF d  waLF  raTF d  raTF 


waLF  raTF
pF
waLF
pF raTF
Using our definitions of  ’s and noting that the rate of change in  waLF  is equal to the rate of change
in aLF (because w is taken as given) we get two equations that describe optimal input requirements
(17)
 LM aˆ LM  TM aˆTM  0
 LF aˆLF  TF aˆTF  0
The two equations above correspond to equations (6) and (7) in Jones (1965). Alternatively, equations
(17) could have been obtained from the equations of change in (9) by noting that for a price taking
producer it is always the case that
pˆ M  pˆ F  wˆ  rˆ  0
8 Substitution between factors.
The elasticity of substitution between factors is a measure of responsiveness of the amounts of
factors used in the production to the changes in input prices. Elasticity of 2 means that if the price of
12
labor increases by 2% the labor to land ratio will decrease by 4%. The elasticity of substitution
between land and labor can be defined in one of two equivalent ways (the definition for Food is
obviously the same):
M
a 
d  TM  w
a
  LM  r
aTM
 w
d 
aLM
r
M
or
a 
d ln  TM 
 aLM 

 w
d ln  
r
These expressions can be now expressed in rates of change.
A gentle detour: A rate of change in a fraction is the difference between rates of change in the
numerator and the rate of change in the denominator
 y   dx  dy
d x
x
x
y
y
Proof. There are two ways to see it.
 y   dxy  xdyy  dx y  xdy y  dx  dy
d x
1)
x
y
2
x
y
y x
y2 x
x
y
2) The second method uses log differentiation (recall d ln x 
dx
)
x
 y   d ln  x   d ln x  d ln y  dx  dy
d x
x
 y
y
x
y
The elasticity of substitution between inputs in both industries are given by
(18)
M 
aˆTM  aˆ LM
wˆ  rˆ
and
F 
aˆTF  aˆ LF
wˆ  rˆ
Intuitively, these elasticities determine the shape of the isoquants
13
Perfect substitutes:
M  
Perfect complements:
M  0
TM
TM
Cobb-Douglas:
M 1
TM
LM
LM
LM
Homework:
1. What is the elasticity of substitution for a Cobb-Douglass production function given by
M  LM TM1 ,  0,1 where subscript M denotes the amount of factor used in the production
of manufacturing output?
2. What is this elasticity for a Constant Elasticity of Substitution (CES) production function given
by the following expression?
1
  
M
M   L  1    T

M

where
1   - to guarantee convex isoquants
1    1 - because  is a share parameter
A useful reminder: notice that Leontief (inputs are perfect complements), Cobb-Douglas (input cost
chares are constant), and linear (inputs are perfect substitutes) production functions are all special
cases of the CES production function:
If   1 then M   LM  1    TM
If   0 then M  LM TM1 (this could be shown using L’Hopital’s rule, see below)
If   0 then M  min  LM , TM  (this also can be shown applying L’Hopital’s rule)
14
These properties of the CES production (and similarly utility) function often come in useful in trade
models.
L’Hopital’s Rule is useful to calculate limits of indeterminate expressions. If lim f  x   lim g  x   0
x c
or lim f  x   lim g  x    then lim
x c
x c
x c
x c
f  x
f  x
 lim
.
g  x  x c g   x 
Application #1:
Let’s apply this rule to the CES production function to show that Cobb-Douglas is a special case. First
take natural logarithms of both sides.
lim ln M  
ln  LM  1    TM  

 0
Notice that both the numerator and the denominator converge to 0. Applying L’Hopital’s rule we get
 ln LM LM  1    ln TM TM 
 LM  1    TM 
lim ln M  
 0
1
lim ln M   ln LM  1    ln TM
 0
lim M  LM TM1
 0
Q.E.D.
Application #2:
Using the same procedure let’s show that Leontieff’s production function is a special case of CES.
lim ln M  
 
ln  LM  1    TM  

The numerator and the denominator both tend to infinity. Apply L’Hopital’s rule.
15
 ln LM LM  1    ln TM TM 
 LM  1    TM 
lim ln M  
 
1

 LM ln LM  1    TM  ln TM

 LM  1    TM 
1    TM
 LM
 
ln LM   
ln TM

 LM  1    TM
 LM  1    TM 


1
1    TM 
1


  LM 

ln LM 
1
  LM 


1    TM 
ln TM

1
We need to consider two cases TM  LM and TM  LM . If TM  LM then the denominator of the first
term becomes 1 and the second term becomes 0. Similar is true for the other case.
ln L ,if TM  LM
lim ln M   M
 
ln TM ,if TM  LM
or in a more familiar form
lim M  min  LM , TM 
 
Q.E.D.
9 Endogenous technology
Up to this point, we have described relations between variables and between changes in
variables. In a way we have described the ways in which the economy will adjust to changes. Now we
can combine those relations to determine the effect of exogenous changes on the model.
Start by determining how the input coefficients change when prices of inputs change.
Intuitively the adjustment of input requirements to the changes in factor prices has to be determined by
two things: behavior of producers and technology limitations. The behavior of producers is driven by
profit maximization in a perfectly competitive environment. From introductory micro we know that
the producers set the ratio of marginal products equal to the ratio of input prices. The technology
limitations can be described by substitutability of inputs. Proceed by combining the relation between
16
changes of input requirements dictated by the producers’ profit maximization (equations (17) which
corresponds to equations [6] and [7] in Jones, 1965) and the elasticity of substitution (equation (18)
which corresponds to equations [8] and [9] in Jones, 1965). Solve
 LM aˆ LM  TM aˆTM  0


aˆTM  aˆ LM
 M  wˆ  rˆ
for aˆTM , aˆLM (the solution for aˆTF , aˆLF will follow a similar procedure). Express aˆTM from the second
equation
 LM aˆ LM  TM aˆTM  0

 M  wˆ  rˆ   aˆ LM  aˆTM
substitute for aˆTM into the first equation
LM aˆLM  TM  M  wˆ  rˆ   aˆLM   0
pull together terms with aˆTM
aˆLM LM  TM   TM  M  wˆ  rˆ 
use the fact that the cost shares add up to one,  LM  TM  1
aˆLM  TM  M  wˆ  rˆ 
The solution for labor input requirement in Food is identical
ˆ  rˆ 
aˆLF  TF F  w
ˆ 0,
Check whether these two equations make sense intuitively. An increase in wage rate, w
will decrease the amount of labor used in the production, aˆ LF  0 . An increase in the price of land
will have the opposite effect. The connection between the price of labor and use of labor is stronger if
technology exhibits greater substitutability between labor and land (larger  M ). Notice that in the
case of constant input shares (Leontieff’s production function,  M  0 ) prices have no effect on the
use of labor. Also, if labor takes a larger share in the final cost of production (smaller TF ) the
responsiveness to the change in price is smaller because any change is a smaller percentage.
Let me illustrate these points graphically:
17
The effect of an increase in wage:  wˆ  rˆ 
Unit isoquant
TM
Increase in
wage pivots
the isocost
Isocost with
w
slope 
r
aTM
aLM
LM
Increase in the use of labor
On the following graphs the solid isocost represents the initial situation and the broken line is the new
isocost line after the wage has increased.
The effect of an increase in wage for two different elasticities of substitution,  M .
TM
Technology with
less substitutability,
lower  M .
The effect on aLM
is smaller when
inputs are less
substitutable
LM
18
The effect of an increase in wage for two different labor cost shares,  LM .
TM
Technology with
lower labor share  LM .
LM
The magnitude of the
effect on aLM is the same
but it is a larger percentage
change if technology uses
less labor.
Following the same procedure we can derive the effect of changes in factor prices on the use of land
Homework: derive expressions for aˆTM and aˆTF .
The effect of factor prices on factor is given by the following relations
aˆ LM  TM  M  wˆ  rˆ 
(19)
aˆ LF  TF  F  wˆ  rˆ 
aˆTM   LM  M  wˆ  rˆ 
aˆTF   LF  F  wˆ  rˆ 
These correspond to equations [8] and [9] in Jones (1965).
Recall that in the equations of change in (9) the changes in factor prices, good prices, and
changes in endowments were all connected by endogenous adjustment in factor use in response to
changes in factor prices. Now that we know how input usage changes with factor prices (equations
(19)) we can substitute out a’s from (19) into (9) to derive connections between outputs and
endowments, and between factor prices and prices of the goods.
Start with the equation of change derived from the labor market clearing condition
LM Mˆ  LF Fˆ  Lˆ   LM aˆLM  LF aˆLF 
and substitute expressions for aˆ LM and aˆLF
19
LM Mˆ  LF Fˆ  Lˆ  LM TM  M  wˆ  rˆ   LFTF F  wˆ  rˆ 
collecting similar terms gives us
LM Mˆ  LF Fˆ  Lˆ  LMTM  M  LFTF F  wˆ  rˆ 
The term in curly brackets has an economically meaningful interpretation. Think of an
ˆ  rˆ   10% ).
increase in the relative wage by 10% (in the above equation it would correspond to  w
As labor gets relatively more expensive the producers of both manufactures and food will reduce use
of labor at the rate equal to the elasticity of substitution, respectively,  M and  F . The percentage
effect on the unit labor requirements can be obtained by further multiplication with the shares of land
in the respective cost functions, just like in (19). So the terms TM  M and TF F describe the effect of
an increase in relative wages on the unit labor requirements. If we further multiply the reduction in the
use of labor in each industry by their shares in the total use of labor we get a measure of aggregate
saving of labor, keeping output constant, in response to an increase in wage.
Performing the same transformations for land (the second equation in (9)) we get
TM Mˆ  TF Fˆ  Lˆ  TM  LM  M  TF LF F  wˆ  rˆ 
To simplify notation introduce a variable for the terms in curly brackets
 L  LM TM  M  LFTF F
T  TM  LM  M  TF LF F
Now turn to the equations of change implied by the zero profit condition. The price of manufacturing
changes according to
LM wˆ  TM rˆ  pˆ M  LM aˆLM  TM aˆTM 
substituting for a’s we get
LM wˆ  TM rˆ  pˆ M   LMTM  M  wˆ  rˆ   TM LM  M  wˆ  rˆ  
which simplifies to
 LM wˆ  TM rˆ  pˆ M
We are ready to summarize the links between variables. The following equations link endowments
and outputs
(20)
LM Mˆ  LF Fˆ  Lˆ   L  wˆ  rˆ 
TM Mˆ  TF Fˆ  Tˆ   T  wˆ  rˆ 
20
The factor prices and output prices are connected in the following way
(21)
 LM wˆ  TM rˆ  pˆ M
 LF wˆ  TF rˆ  pˆ F
There are several interesting points that have to be noted.
First, note that equations (21) look the same as equations of change derived from the zero
profit condition for constant input coefficients (Leontieff production), the last two equations in (9).
What does it mean? It just means that the changes in the factor allocation exactly offset the changes in
factor prices. Recall the logic of the Envelope theorem. When calculating changes in the optimized
value of a the objective function with respect to changes in parameters one can disregard the effect on
the choice variables (because the effect of those variables on the objective function is zero).
Second, it should be immediately obvious from (21) that the factor prices are uniquely
determined by the prices of output! This is the price equalization theorem. It also means that if the
prices of the output do not change the prices of inputs do not change either.
10 The Magnification Effects
Two relations in our model are particularly interesting. The relation between endowments and
outputs (this would determine the pattern of specialization) and the relation between prices and returns
to factors (this would determine who gains and who loses from trade).
10.1 Rybczynski Theorem
At some unchanging level of output prices consider the relation between outputs and
endowments implied by (20). Constant prices mean constant prices of inputs
ˆ  rˆ  0
w
solve the following system of equations for M̂ and F̂
LM Mˆ  LF Fˆ  Lˆ

TM Mˆ  TF Fˆ  Tˆ
21
Substituting for F̂ from the second equation into the first we get
Tˆ  
LM Mˆ  LF
TM
TF
Mˆ
  Lˆ
Solving for M̂ gives us
LM TF Mˆ  LF Tˆ  LF TM Mˆ  TF Lˆ
Mˆ  LM TF  LF TM   TF Lˆ  LF Tˆ
The term in brackets is the determinant of the  matrix.
 Lˆ  LF Tˆ
Mˆ  TF

It might be useful to rewrite this equation using the property of  ’s:
(22)
 Tˆ  TM Lˆ Lˆ  Tˆ
Mˆ  LM



The assumption of labor intensity of the manufacturing sector implies that   LM  TM  0 , see
equation (16). Since  ’s are defined as shares and cannot exceed 1 the determinant has to also be
smaller than 1.   LM  TM  1
Solving for F̂ in a similar way produces
(23)
 Tˆ  TM Lˆ
Fˆ  LM

It useful to substitute F̂ from (23) into equation (22)
(24)
Lˆ  Tˆ
Mˆ  Fˆ 

Consider a situation when the labor endowment expands faster than the endowment of land. In other
words the relative endowment of labor increases. This would mean that Lˆ  Tˆ  0 and Mˆ  Fˆ . In
other words an increase in the relative endowment of one factor increases the relative output of the
good that uses this factor intensively.
It turns out that we can also compare the changes in endowments to changes in output.
Rewrite equation (23) by adding and subtracting TM Tˆ in the numerator
22
 Tˆ  TM Lˆ  TM Tˆ  TM Tˆ
Fˆ  LM

further rearranging and using the expression for the determinant   LM  TM gives us
Fˆ 

TM Tˆ  Lˆ

  Tˆ
Since we consider the case when Lˆ  Tˆ  0 , it follows that Tˆ  Fˆ . Similarly, rewriting (22) by adding
and subtracting LM Lˆ in the numerator of the first term
 Tˆ  TM Lˆ  LM Lˆ  LM Lˆ Lˆ  Tˆ
Mˆ  LM



collecting terms, and using the expression for the determinant   LM  TM produces
Mˆ 



LM Lˆ  Tˆ  Lˆ 


Lˆ  Tˆ

1  LM ˆ ˆ
L  T  Lˆ





Recall that we started by considering an increase in the relative endowment of labor Lˆ  Tˆ  0 . So
now we have that Mˆ  Lˆ . Combining all inequalities we get the magnification effect of a change in
labor endowment on the output
Mˆ  Lˆ  Tˆ  Fˆ
This is a formal statement of the Rybzcynski theorem. Isn’t it beautiful?!?
Some special cases are illustrative. If the amount of land does not change, Tˆ  0 , then
Mˆ  Lˆ  0  Fˆ . In other words any change in a factor endowment not only shifts the relative outputs,
Mˆ  Fˆ , but also reduces the output of the good that does not use the factor intensively.
Intuition checkpoint: What would be different if production of Food did not require labor at all?
Describe changes in output of food and manufacturing.
Answer: It means that LF  1  LM  0 , it implies that Mˆ  Lˆ . In other words the change in labor
endowment will increase the output of the labor intensive manufacturing at the same rate. The only
way it is possible is by attracting away some capital so that 0  Tˆ
23
To get an intuitive feel for the Rybzcynski theorem. Let’s demonstrate this effect on a standard
production possibilities frontier (PPF). An increase in the amount of labor expands the PPF more
proportionately in the direction of the labor intensive product. Note that the PPF expands in both
directions because the economy can produce more of both goods. It is just that most of the increase in
labor goes to the manufacturing. Extra labor in manufacturing will require additional land and the only
way to get is to take it away from production of food.
M
F
10.2 Stolper Samuelson Theorem
Now turn to the changes in prices and their relation to the factor returns. Trade policy often
works through affecting prices. Trade liberalization lowers the price of the goods that can be produced
cheaper at the foreign countries.
Start by solving (21) for ŵ and r̂ as a function of changes in prices pˆ M and pˆ F . We will
make use of the previously derived implication of the factor intensity:
  LMTF  TMLF  LM  LF  0 . Also, recall that since  ‘s are the cost shares they have to add to
1 for each industry:  LM  TM   LF  TF  1 .
 LM wˆ  TM rˆ  pˆ M

 LF wˆ  TF rˆ  pˆ F
Substitute for r̂ from the first equation into the second
24
 LF wˆ  TF
pˆ M   LM wˆ
TM
 pˆ F
Collect the terms with ŵ on the right hand side
TF pˆ M  TM pˆ F  wˆ TFLM  TM LF 
Use the definition of the determinant of the  matrix,    LMTF  TM LF
(25)
wˆ 
TF pˆ M  TM pˆ F

Using the fact that the cost shares add to 1, equation (25) can be rewritten as
(26)
wˆ 
 LM pˆ F   LF pˆ M pˆ M  pˆ F



The same steps can be performed to solve for r̂ . Substitute for ŵ into the first equation
pˆ F  TF rˆ
 LM
 LF
 TM rˆ  pˆ M
Transform, collect terms with r̂ , use the definition of the determinant of the  matrix.
(27)
rˆ 
 LM pˆ F   LF pˆ M

Notice that the first term in equation (26) above is r̂ , so that
(28)
wˆ  rˆ 
pˆ M  pˆ F

Consider an increase in the relative price of manufacturing, the labor intensive sector
pˆ M  pˆ F  0
From (28) we immediately get that
wˆ  rˆ
That is the increase in the relative price of manufacturing increases relative returns to labor. This
conclusion is not sufficient to determine the effect on real returns. Changes in returns relative to
changes in prices of the final good. Rewrite (26) by adding and subtracting  LM pˆ M in the numerator of
the first term.
wˆ 
 LM pˆ F   LF pˆ M   LM pˆ M   LM pˆ M pˆ M  pˆ F



Recall that   LM   LF .
25
wˆ  1   LM 
pˆ M  pˆ F

 pˆ M
Since the first term on the right hand side is greater than zero
wˆ  pˆ M
The increase in the price of manufacturing is multiplied in the returns to labor, which is used
intensively in the production of manufacturing. By now we expect a similar multiplication to occur in
the returns to land. Rewrite equation (27) by adding and subtracting  LF pˆ F in the numerator
rˆ 
 LM pˆ F   LF pˆ F   LF pˆ F   LF pˆ M

As we did before, use the definition of the determinant to simplify
rˆ  pˆ F 
 LF  pˆ M  pˆ F 

Since the second term is smaller than zero we conclude that
pˆ F  rˆ
Combining the relations between changes in prices of goods and the returns to factors we get
(29)
wˆ  pˆ M  pˆ F  rˆ
This is the Stolper-Samuelson theorem. Isn’t she perfect? An increase in relative prices is magnified in
the changes in returns. Consider a situation when the price of food does not change, pˆ F  0 . Then
equation (29) becomes
wˆ  pˆ M  0  rˆ
An increase in the price of manufacturing product increases the returns to labor, which is used
intensively in manufacturing, and decreases the returns to the other factor. Why is the decrease? It
makes sense intuitively, as the price of the manufacturing product increases the returns to labor
increase because the value of the output increases but also because the marginal return to land
increases in manufacturing attracting land away from the production of food where it is more efficient.
11 Demand
26
The last missing link in the model is the demand. We have discussed in detail the supply side
of the economy without saying anything about the demand. Assume that consumers are all the same
and that the demand is homothetic. Then the relation between the quantity demanded and prices can be
usefully characterized by the elasticity of substitution between manufacturing good and food. The
relative demands depend on relative prices, or in terms of rates of change
(30)
 Mˆ  Fˆ   
D
 pˆ M  pˆ F 
Recall that when we discussed the connection between endowments and outputs (Rybzcynski
theorem) we had to assume that prices stay constant because we did not know which adjustments will
have to occur when prices are allowed to change. We can relax the assumption of constant prices.
Subtract the second equation in (20) from the first to get.
Mˆ  LM  TM   Fˆ  LF  TF   Lˆ  Tˆ   L  T  wˆ  rˆ 
Recall that   LM  TM and that LM  TM  1  LF   1  TF     LF  TF 
(31)
Lˆ  Tˆ  L  T 
Mˆ  Fˆ 

 wˆ  rˆ 


Recall that during or derivations of the Stolper-Samuelson relation we have derived a relation between
relative prices and prices of inputs. From equation (28) we know that
wˆ  rˆ 
pˆ M  pˆ F

ˆ  rˆ into (31) to get
Substitute for w
Lˆ  Tˆ  L  T 
Mˆ  Fˆ 

 pˆ M  pˆ F 


Denote
S 
 L   T 
 
This term is a measure of elasticity of substitution between commodities on the supply side. Substitute


for Mˆ  Fˆ from the definition of the demand elasticity
 D  pˆ M  pˆ F  
Lˆ  Tˆ

  S  pˆ M  pˆ F 
27
Rearranging and expressing the change in price in terms of other variables yields
pˆ M  pˆ F  
Lˆ  Tˆ
  D   S 
A change in factor endowment reduces relative prices of the commodity that uses the factor
intensively. Plug this expression into the expression for demand elasticity and we have a relation
between changes in endowments ad changes of outputs
(32)
 Mˆ  Fˆ         Lˆ  Tˆ 
D
D
S
The relation between output and endowments depend on the relative size of the supply and demand
elasticities of substitution.
The relation between output and endowments with unchanging prices can be shown on the
following graph
Production Possibilities
Frontier
Indifference curve
between M and F
M
Equilibrium
Isovalue line. The slope of this line is
p
 F . Every point on the isovalue
pM
line is affordable to the economy that
produces  M A , FA  and faces
MA
FA
F
prices  pM , pF 
Additional labor will cause the following changes
28
M
The isovalue line gets
steeper, so the relative
price of food increases.
F
Consider two extreme cases of (32). First, when the goods are perfect complements,  D  0 ,


relative quantity of the goods does not change Mˆ  Fˆ  0 when endowments change. For illustrative
purposes consider the case when the two goods are perfect complements in ratio 1 to 1, so that the
indifference map consists of the indifference curves with the kink moving along the 45 degree line.
M
45o-line
Indifference curves
when  D  0
F
Next, consider the effect of the increase in the labor endowment when the two commodities are perfect
substitutes,  D   . Notice that equation (32) becomes
 Mˆ  Fˆ   1  Lˆ  Tˆ 
29
The equation above is the same as equation (24) which connects the change in endowments to changes
in output under constant prices. Indeed, in case of perfect substitutes the indifference curve has the
same slope as the isovalue line for any bundle in the interior. So the graphical representation is the
same as in the case demonstrating Rybzcynski theorem.
M
Indifference curves when
 D   . The same lines are
also the isovalue lines in the
case of perfect substitutability
between goods.
F
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