working paper
department
of economics
The Theory of Wage Differentials:
Production Response and Fact« jr Price Equalisation
by
Jagdi sh N. Bhagwati and T. N. Srinivasan
Number 51
February 1970
massachusetts
institute of
technology
MAS?
:
MAR 10 1970
DEWEY
The Theory of Wage Differentials:
Production Response and Factor Price Equalisation
Jagdish N. Bhagwati and T. N. Srinivasan
Number 51
February 1970
*The views expressed in this paper are the authors' responsibility
and do not reflect those of the Department of Economics nor of the
Massachusetts Institute of Technology.
Lll
R,
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/theoryofwagediffOObhag
The Theory of Wage Differentials:
Production Response and Facto r Price Equalisation
Previous analyses of the case where there is a distortionary wage
differential between different activities, by writers such as Fishlow and David
[2], Hagen [3], Bhagwati and Ramaswami
[1], and Johnson [4], have led to the
discovery of the following pathologies:
(i)
the production possibility curve may become convex to the origin,
instead of concave [1]
(ii)
[2
J
[4]
the feasible production possibility curve will shrink inside the
"best" production possibility curve [3]; and
(iii)
the commodity price-ratio will not be tangential (at points of
incomplete specialisation in production) to the feasible pro-
duction possibility curve.
However, this is not the end of the story*
It can be further shown
that (i) the shift in the production of a commodity, as its relative price
changes, may be either positive or negative; and that the shift is not
necessarily predictable from the convexity or concavity of the feasible
production possibility curve; and (ii) given the commodity price-ratio, we
cannot necessarily have unique capital- lab our ratios in the two activities
or unique factor price-ratios:
an important and interesting implication
of which result is that the factor price-equilisation theorem breaks down,
despite all the Samuelson conditions being met, even if there is an identical
wage differential in the same sector in both countries.
*
We would like to note that research in this field is independently
being conducted by Steve Magee, P. J. Lloyd, and by Murray Kemp and Horst
Herberg.
-3-
1
Ql
- Lf
*2
= (l-L)f (R
(R
1
(1)
)
2
LR
f
{f
l
1
+ (1-L)R
X
(
2
V
- R
x
factor allocations
,
.
(2)
)
- R
(3)
- P f^(R )
2
(4)
fJ(R 1 )}y = P (f
Equations (1)
2
(2)
,
2
- R
2
f
2
(R )>
2
(5)
and (3) represent the production functions and
Equation (4) states that the reward of the first factor
(i.e. its marginal value product) is the same in the production of either
commodity.
Equation
(5)
states that the reward of the second factor in the
production of the second commodity is Y times its reward in the production
of the first.
II:
The Comparative Statics of Equilibrium Outputs
In order now to investigate the response of output of either of the
as
two commodities. the commodity price-ratio changes, it is convenient to work
in terms of the variable w representing the ratio of the reward of the second
factor to that of the first in the production of the first commodity.
we can write:
w
yw
<1
(6)
Then
-4-
1
Given our concavity assumptions we can solve
to obtain
(6)
and (7) uniquely
and R_ as functions R.(w) and R-(w) of w.
R..
that R, (w) is an increasing function of w.
X
It is easily seen
Given R, let w
(R)
be the unique
A
solution of
R. (w)
interval [w
,
Then the relevant ramge of values for w is the
R.
•»
w] where w(w) is the smaller (larger) of w. (R) and w_(R)
(see Figure 1)
The value of L corresponding to any given w in this
.
interval is obtained from equation (3).
The equilibrium value (or values)
of w corresponding to a given p is (are) obtained from equation (4).
Let us examine this equation more closely »
(R
f
i D
—
-5—
f
Let us first rewrite it as!
i
(
- P
V
(8)
The left-hand side of equation (8) is a function of w alone.
Denoting
function by p(w) we get:
p'(w)
p(w)
f
d P (w) =n
1
_
dw
p(w)
ll<*l>
d
V
11
f
(R.)
ii
is
V
2
£
j2fi(R.)
jr—
.
f
f2 (
ll
dw
l
f
where
w)
dR (w >
2
"
(
dw
}
From (6) and (7) we get:
.
dR
(
V
fl(R
i
dR (w)
)
l
d"
2
= 1
(9)
= y
(10)
4<v}
L
t
2
n »2>
f
2
(R
2
<1R
)
2
(»)
dw
{4«2>}
If we wish to ensure that a solution exists for all non-negative
values of w, we have to assume the Inada conditions:
f^R, f*(R
Lim
R-K>(«)
L
]
f*(R
±
)
X
)
= 0(~)
-5-
Hence
mD .!W
^)
p(w)
f
+
!pV
1
~+
w+R
2
f
(R
w
Jp
yw+R
2
)
[using (6) and (7)]
(11)
(wf R
x
)
(yw+R
2
)
Equation (11) points at once to an interesting set of possibilities.
First, if in the relevant interval [w
[yR, (w) - R (w)]
[w
,
w] of values of w,
changes sign, then p(w) also changes sign s since w+R.
and yw+R (w) are both non-negative.
function of w.
,
In other words, p(w) is not a montonic
Thus equation (8) can have more than one value of w in
w] as a solution.
This means that the same commodity price ratio p
can be consistent with more than one equilibrium combination of the outputs
of the two commodities.
Second, consider two countries with identical production functions
and the same type and degree of distortion (i.e. the second factor in the
production of the second commodity receives y times its reward in the production of the first commodity in both countries).
same commodity price-ratio p.
Suppose they face the
If yR- (w) - R„(w) changes sign in both coun-
tries within the respective interval of values of w, then one country's
equilibrium value w could be different from that of the other.
words, factor price equalisation will fail to take place.
In other
It is important
to note that this failure could take place, even though there is no factor-
intensity reversal in the usual sense:
even though [R.(w) - R„(w)] has the
same sign for all relevant values of w, for both countries, [yR^w) - R 2 (w)|
can still change its sign.
2
It should of course be kept in mind that R 2 (w) is the factor intensity in the production of the second commodity when the factor price-ratio
faced by producers of this commodity is yw.
-6-
The precise conditions under which such multiple-equilibria will arise
can be readily derived and related to the conditions defining the nature of
To do this, we proceed now to derive first
output response to price change.
the slope of the production possibility curve 9 given y.
and (3)
we get:
,
dQ
Using (1), (2),
l
dw
dLl
Tf l
dw
——2
dw
l
dw
1
dL-2
2
,. ,.,2
+ (1-L)f 1 dw
1 dw
'
= - -j-f
(
dL
dw
dR
V
dR
R)
dR
(R- R
dw- +
l>dw(R -R
2
2
1
)
Hence
dR
,
fa
{(R2" R)
f
dR.
dw^
+ (R
"V *T
dw
dR.
.
+
}
f
<V
l
R)(R 2- R 1>
1
dw
(R^)
{(f
1
-^)
+ R
2
(R -R)
f [}
2
(R
1
^
-$± +
dR
+ *2 >
1
dR,
-R
(R
dw
x
2~V
f -*-.*}
t(~
f
(V
R>
1+
dw
*1
f
dR
l
3
<
R" R 1>
1
dw
1
(R^)'
^
dR
dR
.
{(w+R )(R -R)
2
2
^± + (w+R^R-R^ ^-}
a2-\r
[using (6)
and
2
dQ_
2
dw
f
1
dR 1
{(>W+R )(R -R)
2
2
^+
dR
(yw+R^ (R- Rl )
2
(R - Rl )
2
^
?
}
-7-
Now:
dR
-(fj)
dw
1
f
f
11
2
dR
and
1
-(f
2
dw
2 2
Y
)
2
f
2
f
11
Next, let a(R.) denote the elasticity of substitution of the factors in
the production of the ith commodity.
i
i
i
1
qcf -^
a(R
t
f
p
)
Using (6) and (7) therefore we can write:
.
h
dQ_3
f±
f
ll
afa +
fj {(w+R )(R -R)
2
2
dw
w(R
dQ
2
-t
2
- R
2
(w+R
2
x
)
(R-R
1
afa}
)
(12)
2
x
)
afa +
{(yw+R )(R -R)
1
It is well known that
(yw+Rj^) (R-R
x
afa)
)
(13)
dw
w^-R^'
Hence
(w+R
|
2
)
(yw+R
f
x
2
(
(R -R)
2
)
(R -R)
2
(w+R )(R -R)
2
2
- p
(yw+R
a^
+
1
afa +
afa
(w+R )(R-R
(yw+R
x
)
+ (w+R )(R-R
1
1
)
a R
2
2
(R-R-^ a R
2 2
1
afa
)
(14)
(R -R) a
2
2
fa + (yw+R 1 )(R-R 1 )a 2 R 2
)
It is seen from (14) that if there is no distortion, i.e. if y
1,
dQ x
-dQi
then -r~— = - p, showing that the domestic rate of transformation (
Q )
,
equals the commodity price ratio p.
However, if y + 1, we see from (1A) that:
r
-dQ
(R -R) a
n
p - (- --jq-) =
p(Y-Dw
2
\
fa
+ (R-R x )
(yw+R )(R -R)a R
2
2
1 1
cr
2
R
2
+ (yw+R.^ (R-R^ ofa
(14a)
The expression in the square parenthesis is always positive; hence
>
p - -
dQ
l
-rpr—
according as y -
1.
This means that the commodity price-ratio
-8-
will not equal and will indeed exceed (fall short of) the domestic rate
-dQ
1
according as the degree of distortion in the factor
of transformation (
)
p,
,
price ratio faced by the producers of the second commodity as compared to
those of the first, i.e. y» exceeds (falls short of) unity.
Note further that, in generals the degree of divergence between the
commodity price-ratio and the marginal rate of transformation can be expected
to vary with the equilibrium point on the production possibility curve at
which this divergence is being measured (although this is not inevitable)
3
It is also clear that this variation in the degree of divergence can obtain
under CES production functions (where
a.,
and a_ are constant) and even under
Cobb-Douglas production functions (where a. - a
= 1)
.
Furthermore, the
possibility of multiple equilibria, which we have already noted, also implies that, corresponding to the same commodity price-ratio, there could be
different divergences between the price-ratio and the marginal rate of trans-
formation at alternative equilibrium points on the production possibility
curve.
We are now in a position to examine the response of the equilibrium
output Q- of the second commodity to change in the international price-ratio
p.
—
From (13), it is clear that -
-
according as R„(w) ^
R. (w)
definition (except in the trivial case where w = w, either R„(w)
R„(w) < R n (w) for all w in [w
,
w]
.
also uniquely determined.
By
> Ri(w)
or
w] is determined
dQ
2
the sign of -r— is
Since the interval [w
uniquely once the aggregate factor endowment is known,
.
,
To get the response of Q 2 with respect to p we
This can be seen readily by dividing (14a) on both sides by 'p' , which
yields the formula for -p/dQ../dQ~, the relative degree of divergence. Note
also that, except for the multiple-equilibrium possibility discussed in the
text, any movement along the production possibility curve in equilibrium will
require a change in the commodity price-ratio.
-9-
have to evaluate
—
-:
=
dp
-—
dw
"
Remember also that
T"~ •
dp
sign as p'(w) in equation (11); and that p'(w) |
Using these arguments
,
—
will have
dp
the same
according as [yR. - R„] |
0,
we can now proceed to analyse the following six cases
(excluding the degenerate case of w = w)
Case
Given
I
R. (w)
R (w) > R„(w) and
;
> R (w)
,
yR-i (w)
> R~(w) for all
w in
[w
w
,
]
this case will arise when either y - 1 or when y is less
than unity but not sufficiently less than unity to make yR-j(w) less than R„(w)
for some w in [w
,
w].
case,
dQ
dw
specialisation
i n this
-r— > 0.
Hence
—
-=
2
> 0.
-:
—
and assuming incomplete
>
Thus, if we compare the equilibrium
output Q_ corresponding to two different international price ratios, the one
associated with the higher price of the second commodity in terms of the first
will be larger.
Thus the (comparative static) response of equilibrium output
to a price change is 'normal'.
Case II
Given
R.,(w)
R.(w) < R (w) and yR (w) < R (w) for all w in [w
:
<
?
w]
,
R»(w), this case will arise when either Y - 1 or when Y is
greater than unity but not sufficiently greater to make YR,(w) exceed R (w)
2
dQ
2
<
for some w in [w , w].
and again assuming incomplete
In this case dQ_
< 0.
> 0.
specialisation
Thus the output response is again
Hence -
—
—
—
,
dp
normal
1
.
Case III
Given
R..
:
R
(w)
> R„(w)
but y R n w )
<
R (w) for all w in [w
2
,
w]
R_(w) this can happen only when y is sufficiently less than
dQ
dw
2
,
,
,,
n
In this case >
and, assuming incomplete specialisation, -j- < U.
dW
(w)
>
—
unity.
dQ
Hence
dp
-=
—
2
< 0.
Thus if we compare the equilibrium outputs corresponding to
two different international prices for the second commodity in terms of the
-10-
first, then the output corresponding to a higher price will be smaller
Case IV
s
R.(w) but yR-(w) > R (w) for all w in [w
?
,
w]
this can arise only when Y is sufficiently greater than
'perverse'
— ——
Case
— V
w in
[w
<
R.,(w)
In this case also the output response in a comparative-static sense
unity.
is
;
< R_(w)
R, (w)
This
'perverse' comparative-static response.
is a case of
Given
.
,
[w
w]
,
w]
R„(w) but yR (w) - R„(w) changes sign at one or more
R, (w) >
x
:
Z
1
We saw earlier than when
.
z
yR..
(w) - R~(w)
changes sign in
the same international price ratio p may correspond to more than one
,
equilibrium value for w and hence for the outputs Q, and Q„.
l
Thus the
1
dw
—
will be
different depending on the particular equilibrium
dQ
2
will depend on
value of w at which it is evaluated. Hence the sign of -z
derivative
—
dp
the particular equilibrium point at which it is evaluated.
It is easy to see
that if we order the equilibrium points in increasing order of the value of
dQ
2
will alternate in sign as we move from one equilibrium point to
Q„ then Z
dp
—
the next.
is
Thus if at one equilibrium point the comparative static response
'normal', then at the next it will be 'perverse'.
Case VI
w in [w
,
w].
:
R
(w)
<
R (w) but yR (w) - R_(w) changes sign at one or more
Here again the possibility of multiple equilibria arises and
the conclusions in Case V apply to this case also.
A
4
We may note that the possibility of multiple w-values corresponding
to a single p-value can be readily illustrated, using the well-known Larner
technique.
Figure 2 shows how, consistent with commodity 1 remaining intensive
in the use of factor 2 in two alternative equilibria (i.e. OM 2 is steeper than
OM 1 , and so is ON 2 steeper than ON 1 ), two alternative values of the factor-price
ratio are possible (at AC and DF respectively for commodity 1 and at AB and DE
for commodity 2 which has to pay more for its use of factor 2 than commodity 1
has to) when the commodity price-ratio involves exchange of T for 1 units of
the two commodities.
-11-
III:
Relationship of Output Response to Shape of the
Production Possibility Curve
The possibility of "perverse" production response to change in the
commodity price-ratio , in the presence of the wage differential, raises in
turn the question as to whether the "perverse" response will arise if and
ortly
if the production possibility curve is convex to the origin.
Such an inference is implicit in the earlier literature [1] [4]
s
in the
way the diagrams are drawn, for example, to show that the output of a commodity increases with its relative price when the production possibility
However, such an inference is logically
curve is concave to the origin.
valid only when there is no wage differential*.
In the absence of such a
differential the commodity price-ratio will be tangential to the production
possibility curve and hence the output response to price change depends entirely on the curvature of this curve.
But, once the differential is presents,
the commodity price-ratio no longer equals the domestic rate of transformation
and hence there is no a priori reason to expect any necessary connection bet-
ween output response and the curvature of the production possibility curve.
Our numerical example in Section IV does in fact show that there is no such
However it is nevertheless of interest to derive analytically
connection.
To this we now turn.
the curvature of the production possibility curve.
We showed [equation (14)] that:
(w+R )(R -R) a
dQ
2
£i
f2i
" "
t\
2
(YW+R
d
We can then derive
dQ
.
-rr- = 1/-
dQ
2
(13)
—
dw
.
\
.
2
r
d(
2
2
*2
)
_R
]
(R -R)
2
1
+ (w+R )(R-R
1
1
OjR^Cyw+RjKR-R^ a fa
d
by using the relation
\
r =
dQ
.
a^
)
d
-j—
/
j
2
for
We have already
r
y derived the expression
*
dQ
dw
l\
"nr"!
2)
•
~Ifr~
dQ
"
2
-r—
dw
Let us denote the numerator of the detailed expression for
2
in equation
dQ
rjg-,
wi£h
-12-
the negative sign, by N(w) and the denominator by D(w).
d_
«±
dw
dQ.
Then
-
dN
dD
„,
„,
D <»>
- N(w)
.
.
g
=
2
Now
dN
dw
si
dR
— f.1
•
l
(w+R
dw
11
dR
- f
+ dw~
(1
V
)(
2
*r
i
)
(R~R
2
}
)
x
dN
dw
g R
l
l
f
w
- ft
i
l
°i d^
+
}
(1
+
)(M i
djr
2 2
i i
=
2
>< R 2" R)
}
°2 R 2
dR
1
)
we get:
w
°
2
(R -R)
2
da
+ (w+R
a R
ofa +
(w+R )(R -R)
11
W+R
a R
—
dw
-:
(
dR.
(-™)
dR
Using the relations
a^
)
°1 R 1 +
d*
-dR
+ (wfR
1
dR
,
+
+ (wfR
+ (W+R
R) °1 R
1
do
(R
OR
(R -R)
)
2
o^
2
+ (R-Rp a R
2 2
h}
—
°2 R2
a ? R2
4-
a R
{(R -R)
l l
2
+
2
)a R
+-^-i^ (w+R )(R -R) +(R-R ) a *
+ a (wfR )(R-R
ai
2 2
1
2
2
1
2
1 )J
do
do
(w+R
x
(a R
)
2 2
)V+
(w+R )(R -R)
2
2
R^~
+ (w+R^ (R-R^
dw
Therefore
dN
dw
Wll
w
(w+R )(R -R)
2
2
OR
+ (w+RjXR-Rj^)
R
2 2
a-KJ
-
f!
(R -R)
2
R
1 1
+ (R-R
x
)
a R
2
2
+
(—^»)
(continued)
l l
-13-
a^
(w+R )(R -R)
2
2
+ (w+R )(R-R
1
O
)
1
oo
+
do
-
w
R.-wR)
M(R.12
1
(R_-R.
-
v
2
do
+ R
(irt.R
2
1
+
)
)(R-R
)
1
R, (w+R„)
v
/v(R„-R)'
l
2
2
dw
2
dw
Therefore
dN
dw
2
1
O^
(w+R )(R -R)
c
2
+ (w+R^KR-R^
2
R
G R
f.
f
l l ll
—i-ii
2
+
(0i
a
R
(R -R)
X X
2
+ (R-R
1
R
)
2 2
+
l°2
-—=
do
(R -R )
2
1
(R^-wR)
do
—^
+ Rl (w+R )(R -R) --± + R (w+R )(R-R )
2
2
2
1
1
Next,
i°dw
2
f
^i
(Yw+R
11 dw
2
)
(R -R)
2
O^ +
(Yw+Rj^)
(R-R^
dR
dR
+
f
+
(Y
dw">
+ (YwfR
(R 2" R) a R
i l
dR
f
+ (Yw+R )(R -R)
2
0l
2
R
dw~ °1 1
d(J
-~
1
2
)
+ R
l
lSr
dR
+
(
Y +
R
2 2
-dR.,
^XM
R
>
2
2
+ (VrfR
dR
(YwfR )(R-R
1
1
<o
)
-
2iw
)(-_)
o R
2
2
dO
+ R
2
^-
Therefore
R f
dD m °2 2 ll
dw
w
+
f
Y
(yw+R
{
2
)
2
(R _R) a R
2
o^ +
(R -R)
i i
+
(
R- R
x)
(yw+R^ (R-R^ a R
2 2
a R
2 2
(continued)
+ a
2
,
-14-
°2 R 2
+
f
~w~~ \
(R 2~ R)
CT
+ (YwfR
R
1 1
2
)
°l
R
l
+ <Yw+R
)
1
(R-R^ O
^
I
a R
+
i l
f
"IT"
|(Yw+R
2
)
(R -R) a
2
~
+ (r-r
x
x
)
a R
2
do
+ R (YwfR 2 )(R
1
~R)
2
(Yw+R^ a R\
-
2 2
da
+ R (Y^R )(R-R
1
2
1
)^£ ]
Therefore
2
f
dD
dw
a R
li
2
2
(YwfR
w
+
f
Y <R 2- R)
L {
l
a
)
(R -R)
2
a^ +
Vl+^V
(yw+R
1
a R
2
)
(R-R^ a R
2 2
2
}
-kj
('—)
+
2
^(YwfR )(R -R)
2
CJjRj
+ (Yw+R^ (R-R^ a R
2 2
2
}
a
°i
dC7
2
M 12
(Ro-R-,
v
)
2
1
w
(R,R -YwR) + R. (Yw+Rj
XI
/x(R -R)
**2
1
2
'
l
dw
da,
+ R
(YvH-R
2
1
)(R-R
1
)^
Therefore
2
dD
dw
(Yw+R
2
)
(R -R) o R
2
1 1
+
(Yw+Rj^)
(R-R^ a R
2
£>
2
_„2
~
G R
f
h
2 2
a a (R -r.)
fj
y((R -R) a R
2
i 1
+ R (yw+R
(R -R)
+ (R-R
x
a R
)
2
l
Z
+
w
2
2
)
2
1 2
da
da
x
(R R -wR)
-£± + R (Yw+R )(R-R
1
2
1
)
^~
Kence
_.
D
dN
dw
—
dw
DN
„ dD
=
N
"lVii
°2
R
2
£
n
w
«i
l
-
1
(continued)
15-
Df
+ YNf
I
«R 2 -R)
V
I
R
V
—
—
—
w
a a R R
(
l Z
X
2
2
l Z
1
l
-
~
A
a
+ (R- Rl ) a R - Ra a (R -R )
2 2
i 2
2
1
11
)
1
(Df ;
2
+ Nff)
da
- R (R -R)
1
2
|(w+R )Df
2
da
(
+ (y-DRjRjfJfJ ^(Rjj-R)
(R -R)
2
a^ +
(Y-l)f Jf
J
2
- -
dQ
1
-
!
2
'
1
+a 2 (R-R1 )|
(R-R^ a R
2 2
I(R -R)
2
a^
- R0 O (R -R
1 2
+
l
l>
-D
dW
2
w(R - Rl )
2
2
^2^
1
1
dQ,
using
2
d Q
2
+ (yw+R,)Nf
1'
(Y-l)(R -R )w(R»R )(R 2 »R)R 1 R 2 fJf J
-
Hence
+ (yi^-R^Nf
yw
yw+R 2
-w
w-HL
DN
w
1
2
1(w+R,)Df:r
i—
^-y»1
dw
V
R (R-.
-RJ
2
J
1
)
"l^WV*-
jg^ - a^
we get:
J
-w^-R^ 2
as
>
-— .. „
2
D
2
-
-
-
2
Z
.1,2
N(yR,-R,)
(m+R
±
)
(y-DRnRjf^f,
/r
D
\i<
+
(yw+R
2
)
R 2~ R) a R
l l
+ (R" R
1
}
a R
2 2
A.
T
- R a,a«(R--R,)L )a,(R -R) + a,(R-R,),\
2
2
2
12V
ri
l^V/
(continued)
-16-
(R _R
a
i
2
2
)
l
1
(R 2" R) °1 R + (R
1
~V
2
R
2
da
w (R - Rl ) (R -R) (R- Rl )
2
2
^
^
a
2
It should be obvious from (15)
- a
(15)
1 ciw
that it is difficult in general to
2
d Q
x
determine the sign of
j
.
One has therefore to consider special cases.
d ^2
,2
f
d /Q
If there is no differential (i.e. y = 1)
(1)
,
<
dQ 2
lengthy right-hand-side
is always positive.
1
then
since the
'
bracket in (15) cancels out and (R 1 -R«)N
terra in the
Thus we get the standard result that the production
possibility curve is concave to the origin*
Where, however, Y J 1» we can show that the production possibility
(2)
curve may have both convex and concave stretches.
We can do this by evaluating
2
d Q,
2
at two extreme points:
complete specialisation in Q.
(so that R.
R)
dQ 2
and in Q_ (so that R» = R)
and showing that the production possibility curve
do
i
is we 11is concave at one end and convex at the other.
Assuming that -j
dw
,
—
behaved for i = 1, 2, we can see from (15) that, quite generally:
d
2
2
-w(R -R)'
Q
= R) =
(given R
2
2
r
D
dQ,
N( Y R-R
2
(
)
_(w+R) (yw+R
2
Y-l)R 2 f^f
2
olR a-2a 2 )
i<V R)2
)
2
\
(given R
"
iKyRj-R)
(w+R
)
(yw+R.
= R) =
~ w(R" R
l
}
(y-DR^Jf^R-R^
)
~
D
2
2
a 2 R (l-2 o ± )
-17-
Suppose now that R (w)
R (w) for all
>
2
and R (w) > R > R (w) for all w in
(
2
w
,
w)
.
win
[w
w)
,
.
Then R (w)=R=R (w)
In this case, it follows that
I
and D < 0.
Assume further the specific values: Y > 1 and a., (w) - -r-.
d2Q
l
Then clearly
s- <
when R„ » R (i.e. w = w) . If it so happens that when
dQ 2
d2q
i
~ > 0. Thus the
R. * R (i.e. w « w)
YR < R (w) and a (w) ^ y, then
2
dQ 2
production possibility curve will be convex in the neighbourhood of one
N >
,
specialisation point and concave in the neighbourhood of the other.
We may finally consider the case where o = a equals a constant,
can
showing that this * lead to a production possibility curve which is smoothly
(3)
convex (to the origin) throughout.
5
In this case of CES production func-
tions, with identical elasticities for both industries, (15) reduces to:
2„
d q
dQ
„
N(yR -R
» ,[a. .
R.
f
i
/ ^l \
R (w) = (t^-)
1
w°
"
-e"
2
1/e
,
(where a s
,
.
—
rr)
£+1
then we get
This means that R
= nR,
2
/ex
•:
1
1
2
D
)
Y0t
\°
/
2
R (w) = hr^-J w
and
1 2
+
i
i
and 6 =
,
)
+ (1-a. )]
ot
where n = (By)
2
(w+R.^ (Yw+R
D^
we write
2,
„ v „ „
„,„ „ ,2,1,2.
a'(Y-l)R R R(R -R ) f^fj(l-2a)
„
;
1
2
2
If
r„, „
,„
„ ^2
-wd^-y
x
2
Z
a
/l
-.
2/
Since a
.
> 0,
n ^ 1
according as
1
1
It can be shown that N = -f^"(n-l) OR (wR+nR
By j 1.
)
and D =
d2Q
2
aR.(YwR+r|R
Substituting these into the expression for
).
1
l
j
t
(n-1)
we get:
dQ
2
2
d Q
dQ.
x
- wRi
2
4
(
n -l)
D
2
5
2
3
fj
(n-YXwR+n
(w+R 1"'"™
1 )(yw+n
-
R.
)
RJ
"V
+
(Y-D(i-2a)nRR ]
2
/_,„^
D
R.')
(ywR+n
*'"
1
(16)
J
Kemp and Herberg have independently arrived at this conclusion for CES
production functions.
-18-
It is clear from (16)
a„ and
that if a.
-r
- a -
1
(i.e. when 6
1)
d2Q
- >
dQ
The reason is that, in this case, either
for all y ^ !•
1 >
n > Y
2
Thus the production possibility curve is convex throughout.
or Y > 1 > !•
IV:
A Numerical Example
The following numerical example demonstrates the possibilities of (a)
multiple equilibria corresponding to a given commodity price-ratio, (b)
perverse comparative-static response to changes in this price ratio and (c)
'normal
response being associated with 'perverse' curvature of the pro-
1
duction possibility curve.
Let f
R
=»
4.
1
=
£
1.
R
_1
l
-1
+ ±
and f
2
I RK
9
2
1/2
+
8
2
Let Y
9
2
8 and
It is easy to deduce 5 using equations (6) and (7), that R.(w) = w
R (w) = w
2
2
,
w =
is positive in
and w = 16.
2
w =
2 5
w
< 4,
Hence
r
"1
YR,(w) - R <w)
1/2
,
= 8w 1/2 - w 2 and thus
2
zero when w « 4 aad raegative in 4 < w - 16 » i.
The resulting production
Of course R (w) > R (w) for all w in (2, 16).
?
possibility curve is presented in Figure 3.
This curve is convex throughout,
as can be verified also by algebraic analysis.
In Figure 4, the function p(w) is plotted.
As was proved earlier
(recall equation (11)) and as is evident from the figure, p(w) increases with
w when YR,(w)
as
>
R (w), i.e. when w is in the interval
(2,
4)
and decreases
2
w increases when yRAw)
<
R (w), i.e. when w
is in the interval (4,
16).
2
Thus p(w) starts from a value of about 5.56 when w - 2, reaches a maximum of
6
when w =
4,'
and declines steadily to a value of 4.32 when w » 16„
Thus it
follows that if the commodity price-ratio happens to be anywhare in the range
5.56 < p < 6, there are two equilibrium values of w corresponding to each
-19-
such p, one in the interval (2, 4) and the other in the interval (4, 7.81).
If either 4.32 < p < 5.56 or p
6,
there is one and only one equilibrium
value of w.
dQ
Next, note that with R_(w) > R. (w) for all w in (2, 16),
/
Suppose we now increase p from p
l
4.32 upto p = 5.56.
—
-;
dw
< 0.
Then the equilibrium
value of w decreases steadily from w = 16 to w = 7.81 (approximately)
and Q_ increases steadily from Q
to Q
0.66 (approximately).
Thus
the response is 'normal,' i.e. the equilibrium output of the second com-
modity is larger when its relative price is higher, even though the production
possibility curve is convex to the origin.
References
[1]
Bhagwati, J. and V. K. Ramaswarai.
"Domestic Distortions, Tariffs and
the Theory of Optimum Subsidy,"" Journal of Political Economy
February 1963.
[2]
Fishlow, A. and P. David.
"Optimal Resource Allocation in an Imperfect Market Setting," Journal of Political Economy ,
December 1961.
[3]
Hagen, E.
[4]
Johnson, H.
"Factor Market Distortions and the Shape of the Transformation Curve," Econometrics s July 1966.
"An Economic Justification of Protectionism," Quarterly
Journal of Economics , November 1958.
,
YR (w)
1
FIGURE
1
Factor
1
v\
M
1
A*
\
\\ V
^
^^"^^^^w
w
E C
FIGURE
2
Comm.
1
Coram.
2
^^Sfc.
,
—
>>
Factor
2
w
OS
CCM
o
o
fM
o
o
•
o
•
CJ>
o
•
oo
o
O
•
•
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•
O
O
•
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o
o
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CN
•
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O
in
o
o
O
O
o
o
o
00
o
o
O
<n
O
o
o
-v^tA-^O
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o
o
o
o
00
O
O
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o
o
o
o
c-
vO
in
i/"t
o
o
o
m
o
o
oo
o
o
o
o
-*
o
o
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