Chapter 15. Implicit function
theorem
Contour line and isogram
landform → contour line
utility function → indifference curve
product function → iso-quant curve
(equal output curve)
y  f ( x1, x2 ) → same rank set (contour line)
0
2
0
H ( y )  {( x1, x2 )   | y  f ( x1, x2 )}
1
Typical shape of contour lines
2
x2
0

f
(
x
)
0
1

f (x )  
0 
 f 2 (x ) 
x20
dx2
dx1
x10
x1
3
Slope of the tangency line of the
contour line
slope of tangency line at x  ( x , x )
0
0
1
0
2
Totally differentiate y  f (x)
0  f1dx1  f 2dx2
0
0
0
At x  x , 0  f1 (x )dx1  f 2 (x )dx2
0
0
0
( f1 (x ), f 2 (x )) is orthogonal to (dx1 , dx2 )
0
dx2
f1 ( x )

0
dx1
f2 ( x )
4
Gradient vector
f ( x 0 )  ( f1 ( x 0 ), f 2 ( x 0 ))
Orthogonal direction to the tangency line
of the contour line  ( f1 (x ), f 2 (x ))
0
0
0
f (x )  ( f1 (x ), f 2 (x )) is the direction that
0
0
maximizes the slope of the surface y  f (x)
at x  x
0
length of f (x 0 )
 size of the slope of the surface y  f (x)
5
Implicit function
z  f ( x, y )
At z  0 (or k ) relationship between
y and x ? y  h(x) ?
z  5x  y  2 z  0  y  5x  2
z  x  y 1 z  0  y   1 x
2
2
2
6
The implicit function
z  f ( x, y) explicit function
z is an explicit function of ( x, y)
F ( x, y, z)  0 implicit function
z is an implicit function of ( x, y)
In what case the implicit function is
expressed as an explicit function?
7
The implicit function theorem
f ( x, y)  0
implicit function
If there exist an explicit function y  h( x)
f ( x, h( x))  0
Then to each x the unique y corresponds
dh( x)
exists
dx
8
The implicit function theorem
dx
dy
f x ( x, h( x ))  f y ( x, h( x ))  0
dx
dx
dh( x )
f x ( x, h( x ))  f y ( x, h( x ))
0
dx
If f y ( x, h( x))  0,
dh( x )
f x ( x, h( x ))

dx
f y ( x, h( x ))
9
The implicit function theorem
For an implicit function f ( x, y)  0
If f y ( x, y)  0　x, y,
then there exists an explicit function
y  h( x )
f x ( x, h( x ))
h' ( x )  
f y ( x, h( x ))
10
The implicit function theorem
For the implicit function f ( x, y)  0,
if f y ( x , y )  0 at ( x , y ),
0
0
0
0
then there exists an explicit function y  h( x)
near that point
0
0
fx (x , y )
h' ( x )  
0
0
fy(x , y )
0
11
Application to economics:
marginal rate of substitution
iso - quant f ( x , x 2 )  y  f ( x1 , I ( x1 ))
0
1
0
0
explicit function x2  I ( x1 )
0
1
0
0
1
0
1
f1 ( x , x 2 )
f1 ( x , I ( x ))
I'(x )  

0
0
0
0
f 2 ( x 1, x 2 )
f 2 ( x 1 , I ( x 1 ))
0
1
0
1
0
d
f1 ( x , x 2 )
0
RTS12 ( x , x 2 )  
I ( x 1) 
0
0
dx1
f 2 ( x 1, x 2 )
0
1
0
marginal rate of technical substitution of the
1st factor w.r.t. the 2nd factor
12