Using the Implicit Function Theorem
1
The Theorem
n
m
m
~
Suppose
 F : R → R is a differentiable function, and let k ∈ R . Suppose
a1
 .. 
~a =  .  ∈ Rn is a solution to F (~x) = ~k. Write JFI for the matrix formed
an
from the first n−m columns of the Jacobian JF , and JFD for the matrix formed
from the last m columns of JF . If the m × m matrix JFD (~a) is invertible, then
a1


n−m
there is an open set U ⊂ R
containing ~b =  ...  and a function
an−m
f : U → Rm so that


an−m+1
~x
 .. 
~
f (b) =  .  and F
= ~k for ~x ∈ U .
f (~x)
am
The Jacobian of f is
−1
~x
~x
D
I
Jf (~x) = − JF
JF
f (~x)
f (~x)
In particular,
Jf (~b) = −(JFD (~a))−1 JFI (~a).
1
for ~x ∈ U .
2
Two Examples
First Example. Let F : R4 → R be given by F (~x) = x1 x2 − x3 x4 , and
consider the solution
 
1
2

~a = 
−1
5
to the equation F (~x) = 7. Can we regard x4 as a function of x1 , x2 , x3 near
this point? In this case JFD is the scalar
∂F
= −x3 ,
∂x4
so JFD (~a) = 1 and indeed x4 is an implicit function of x1 , x2 , x3 near ~a. The
Jacobian of x4 = f (x1 , x2 , x3 ) is
∂x4 ∂x4 ∂x4 1 x
x
−x
=
2
1
4
∂x1
∂x2
∂x3
x3
near this point; in particular, at ~a it is −2 −1 5 .
Second Example. Let G : R3 → R2 be given by
 
2
x
2
x
+
xyz
+
y
G y  =
,
x2 + y 2 − z 2
z
 
0
1
and consider the solution ~a = 1 to the equation G(~x) =
. Are y, z
−3
2
functions of x near ~a? Here
2x + yz xz + 2y xy
xz + 2y xy
D
JG =
and JG =
2x
2y
−2z
2y
−2z
so that
JGD (~a)
2 0
=
.
2 −4
This matrix has determinant −8, and so is invertible. Thus y, z indeed are
functions of x near ~a, and their derivatives at x = 0 are
−1
1 dy −2 0 2
2 0
−1
I
dx = −
JG (~a) =
=
.
1
1
dz
2 −4
−4 4 0
− 21
dx
2
Hence if x increases by 0.2, then y must decrease by about 0.2 and z must
decrease by about 0.1.
3
An Economic Model
What follows is an economic model known as the “IS-LM” model; it was
developed in 1936 by John R. Hicks to represent ideas of John Maynard
Keynes. Let Y be the national income, C national consumption, I national
investment, G government expenditure, M the demand for money, M s the
money supply, T taxation, r the interest rate. We assume that C = f (Y −T ),
I = I(r) and M = M (Y, r) where
0 < f 0 (x) < 1
(1)
(an increase in the excess of income over tax increases consumption);
I 0 (r) < 0
(2)
(an increase in the interest rate decreases investment); and
∂M
∂M
> 0,
<0
(3)
∂Y
∂r
(monetary demand increases with higher income and decreases with higher
interest). In addition, it is assumed that
Y = C + I + G and M s = M
(4)
(income is the sum of consumption, investment and government expenditure,
and the money supply equals the demand for money). Thus, if we define
 
G
M s   
Y
−
C
−
I
−
G

F
,
T =
M − Ms
Y 
r
0
then we are looking for solutions of F =
. Now we ask whether Y, r are
0
functions of G, M s , T subject to the conditions (4). The Jacobian of F is
−1 0 f 0 (Y − T ) 1 − f 0 (Y − T ) −I 0 (r)
JF =
∂M
∂M
0 −1
0
∂Y
∂r
3
so
JFD
=
1 − f 0 (Y − T ) −I 0 (r)
∂M
∂Y
∂M
∂r
with determinant
det JFD = (1 − f 0 (Y − T ))
∂M
∂M
+ I 0 (r)
.
∂r
∂Y
Then the assumptions (1,2,3) imply that both terms are negative, so det JFD 6=
0. So Y, r can be thought of as functions of G, M s , T , and
∂Y ∂Y ∂Y ∂M
−1
−1 0 f 0 (Y − T )
I 0 (r)
∂G
∂M s
∂T
∂r
=
∂r
∂r
∂r
0 −1
0
1 − f 0 (Y − T )
det JFD − ∂M
∂G
∂M s
∂T
∂Y
∂M
∂M 0
−1
− ∂r
−I 0 (r)
f (Y − T )
∂r
=
.
∂M 0
∂M
0
D
(Y
−
T
)
−
1
−
f
f (Y − T )
det JF
∂Y
∂Y
Thus, for example,
∂Y
∂M
= −P
> 0,
∂G
∂r
where P is the positive quantity det−1J D .
F
4
Exercises
 
x1

−2x21 x2 + x3 + x24
x2 
4
2


. For each
1. Let H : R → R be defined by H   =
x1 x3 x4 − x2
x3
x4
of the following
points ~a, determine whether x3 , x4 subject to the condition
∂x3
∂x3 1
∂x1
∂x2
H(~x) =
are functions of x1 , x2 near ~a. If they are, find ∂x4 ∂x
at ~a.
4
0
∂x1
∂x2
 
1
2

a. ~a = 
1
2
 
1
0

b. ~a = 
1
0
 
1
0

c. ~a = 
0
1
2. In the IS-LM model above, what can you say about
4

0
0

d. ~a = 
 0 .
−1

∂Y
∂M s
and
∂Y
∂T
?