Mathematical Economics:
Lecture 10
Yu Ren
WISE, Xiamen University
October 22, 2012
Chapter 15: Implicit Functions and Their Derivatives
Outline
1
Chapter 15: Implicit Functions and Their
Derivatives
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
New Section
Chapter 15: Implicit
Functions and Their
Derivatives
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Implicit Function
Explicit function: y = F (x1 , x2 , · · · , xn )
Implicit function G(x1 , x2 , · · · , xn , y ) = 0
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Examples
Example 15.1 the equation 4x + 2y = 5 or
4x + 2y − 5 = 0 express y as an implicit
function of x.
write y as an explicit function of x: y = 2.5 − 2x
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Examples
Example 15.2 Consider the equation:
y 2 − 5xy + 4x 2 = 0
convert it into an explicit function:
√
4x
5x ± 25x 2 − 16x 2
1
y=
= (5x ± 3x) =
x
2
2
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Questions
The fact that we can write down an implicit
function G(x, y ) = 0 does not mean that this
equation automatically defines y as a function of
x. example: x 2 + y 2 = 1
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Questions
two questions:
(a) Given the implicit equation G(x, y ) = c
and a point (x0 , y0 ) such that G(x0 , y0 ) = c,
does there exist a continuous function
y = y (x) defined on the interval I s.t.
G(x, y(x)) = c for all x in I and y(x0 ) = y0
(b) if y(x) exists and differentiable, what is
y 0 (x0 )?
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Questions
two questions:
(a) Given the implicit equation G(x, y ) = c
and a point (x0 , y0 ) such that G(x0 , y0 ) = c,
does there exist a continuous function
y = y (x) defined on the interval I s.t.
G(x, y(x)) = c for all x in I and y(x0 ) = y0
(b) if y(x) exists and differentiable, what is
y 0 (x0 )?
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Implicit Function Theorem
Theorem 15.1 Let G(x, y ) be a C 1 function on a
ball about (x0 , y0 ) in R 2 . Suppose that
G(x0 , y0 ) = c and consider the expression
G(x, y ) = c. If (∂G/∂y )(x0 , y0 ) 6= 0, then there
exists a C 1 function y = y(x) defined on an
interval I about the point x0 s.t.
(a) G(x, y (x)) ≡ c for all x in I
(b) y (x0 ) = y0
∂G
(x ,y )
0 0
∂x
(c) y 0 (x0 ) = − ∂G
(x ,y )
∂y
0
0
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Example
Example 15.7 Consider the equation
G(x, y ) ≡ x 2 − 3xy + y 3 − 7 = 0
one computes that
∂G
= 2x − 3y = −1 at(4,3)
∂x
∂G
= −3x + 3y 2 = 15 at(4,3)
∂y
∂G
1
0
∂x (x0 , y0 )
y (x0 ) = − ∂G
=
.
15
(x
,
y
)
0
0
∂y
1
) · 3 = 3.02
15
Mathematical Economics: Lecture 10
y1 ≈ y0 + y 0 (x0 )∆x = 3 + (
Yu Ren
math
Chapter 15: Implicit Functions and Their Derivatives
Example
Example 15.8 the equation
x2 + y2 = 1
First note that
y 0 (x)|x=0 = −
∂G/∂x
2x
0
=−
=− =0
∂G/∂y
2y
2
an explicit formula
p
1 − x2
−x
y 0 (x) = √
1 − x2
y (x) =
Yu Ren
Mathematical Economics: Lecture 10
math
Chapter 15: Implicit Functions and Their Derivatives
Higher Order derivatives & Hessian
Theorem 15.2 Let G(x1 , · · · , xk , y ) be a C 1
function on a ball about (x1∗ , · · · , xk∗ , y ∗ ).
Suppose
G(x1∗ , · · · , xk∗ , y ∗ ) = c
∂G ∗
(x1 , · · · , xk∗ , y ∗ ) 6= 0
∂y
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Higher Order derivatives & Hessian
Theorem 15.2 Then, there is a C 1 function
y = y(x1 , · · · , xn ) defined on an open ball B
about (x1∗ , · · · , xk∗ )
(a) G(x1∗ , · · · , xk∗ , y (x1∗ , · · · , xk∗ )) ≡ c for all
(x1 , · · · , xk ) in B
(b) y ∗ = y (x1∗ , · · · , xk∗ )
(c)
∂y
∗
∂xi (x1 , · · ·
∂G
(x ∗ ,··· ,x ∗ ,y ∗ )
1
k
∂x
, xk∗ ) = − ∂G
(x ∗ ,··· ,x ∗ ,y ∗ )
∂y
1
k
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Level curves and their tangents
Definition: A point (x0 , y0 ) is called a regular
point of the C 1 function G(x, y ) if ∂G
∂x (x0 , y0 ) 6= 0
∂G
or ∂y (x0 , y0 ) 6= 0. If every point (x, y ) on the
locus G(x, y ) = c is a regular point of G, then
we call the level set a regular curve
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Level curves and their tangents
Theorem 15.3 Let (x0 , y0 ) be a point on the
locus of points G(x, y ) = c in the plane, where
G is a C 1 function of two variables. If
(∂G/∂y )(x0 , y0 ) 6= 0, then G(x, y ) = c defines a
smooth curve around (x0 , y0 ) which can be
thought of as the graph of a C 1 function
y = f (x). Furthermore, the slope of this curve is:
∂G
∂x (x0 , y0 )
− ∂G
∂y (x0 , y0 )
.
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Level curves and their tangents
Theorem 15.3 If ∂G/∂y (x0 , y0 ) = 0, but
∂G/∂x(x0 , y0 ) 6= 0, then the Implicit Function
Theorem tells us that the locus of point
G(x, y ) = c is a smooth curve about (x0 , y0 ),
which we can consider as defining x as a
function of y. It also tells us that the tangent line
to the curve at (x0 , y0 ) is parallel to the y − axis.
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Level curves and their tangents
Theorem 15.4: Let G be a C 1 function on a
neighborhood of (x0 , y0 ). Suppose that (x0 , y0 ) is
a regular point of G. Then the gradient vector
OG(x0 , y0 ) is perpendicular to the level set of G
at (x0 , y0 ).
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Level curves and their tangents
Definition: A point (x0 , y0 ) is called a regular
point of the C 1 function F (x1 , · · · , xn ) if
OF (x ∗ ) 6= 0, that is, if some (∂F /∂xi )(x ∗ ) is not
zero. If every point (x, y ) on the level set
Fc = {(x1 , · · · , xn ) : F (x1 , · · · , xn ) = c} is a
regular point of F , then we call the level set Fc a
regular surface.
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Level curves and their tangents
Theorem 15.6 If F : R n → R 1 is a C 1 function, if
x ∗ is a point in R n , and if some (∂F /∂xi )(x ∗ ) 6= 0
then: (a) the level set of F through x ∗
Fc = {(x1 , · · · , xn ) : F (x1 , · · · , xn ) = c} can be
viewed as the graph of a real valued C 1 function
of (n-1) variables in a neighborhood of x ∗ (b) the
gradient vector OF (x ∗ ), considered as a vector
at x ∗ , is perpendicular to the tangent hyperplane
of FF (x ∗ ) at x ∗
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Nonlinear Systems
F1 (y1 , y2 , · · · , ym , x1 , · · · , xn ) = c1
F2 (y1 , y2 , · · · , ym , x1 , · · · , xn ) = c2
..
.
. = ..
Fm (y1 , y2 , · · · , ym , x1 , · · · , xn ) = cm
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Nonlinear Systems
Question:
What is
∂yi
∗
∗
(x
,
y
)?
∂xj
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Nonlinear Systems
∂F1
∂F1
∂F1
∂F1
dy1 + · · ·
dym +
dx1 + · · ·
dxn = 0
∂y1
∂ym
∂x1
∂xn
..
.
. = ..
∂Fm
∂Fm
∂Fm
∂Fm
dy1 + · · ·
dym +
dx1 + · · ·
dxn = 0
∂y1
∂ym
∂x1
∂xn
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Nonlinear Systems

∂F1
∂y1
···
∂F1
∂ym


dy1
 ..
.
. 
 . · · · ..   ..  =
∂Fm
∂Fm
dym
·
·
·
∂y
∂ym 
1
1
Σni=1 ∂F
∂xi dxi


..
−

.
m
Σni=1 ∂F
∂xi dxi
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Nonlinear Systems


dy1
 ...  =
dym
 ∂F
1
∂y1 · · ·

−  ... · · ·
∂Fm
∂y1 · · ·
∂F1
∂ym
−1 
.. 
. 
∂Fm
∂ym

1
Σni=1 ∂F
dx
i
∂xi


..


.
∂F
Σni=1 ∂xmi dxi
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Example
Example 15.15 Consider the system of
equations
F1 (x, y , a) ≡ x 2 + axy + y 2 − 1 = 0
F2 (x, y, a) ≡ x 2 + y 2 − a2 + 3 = 0
the Jacobian of (F1 , F2 ) with respect to the
endogenous variable x and y at the point x = 0,
y = 1, a = 2:
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Example
∂F1
det
∂F1
∂x ∂y
∂F2 ∂F2
∂x ∂y
22
(0, 1, 2) = det
= 4 6= 0
02
we can solve the system for x and y as functions
of a near (0, 1, 2)
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Example
2x+ay xy 1 ,F2 )
det ∂(F
det
dy
∂(x,a)
2x -2a
=−
=−
∂(F
,F
)
1 2
2x+ay ax+2y da
det ∂(x,y)
det
2x 2y
20
det 0
dy
-4 = 8 = 2 > 0
=−
2
da
4
det 2
02
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Example
if a increases to 2.1, the corresponding y will
increase to about 1.2.
Let’s use another method to compute the effect
on x:
(2x + ay)dx + (ax + 2y )dy + xyda = 0
2xdx + 2ydy − 2ada = 0
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Example
plug in x = 0, y = 1, a = 2:
2xdx + 2ydy = 0da
0dx + 2ydy = 4da
so if a increases to 2.1, the corresponding x will
decrease roughly to -.2.
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Comparative Statics
Economic Environment: pure exchange
economy, two consumers 1 and 2, two
goods x and y, initial endowments: (e1 , 0),
(0, e2 ), utility functions: U1 , U2 :
Ui (xi , yi ) = αui (xi ) + (1 − α)ui (yi ), price
levels: p, q.
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Comparative Statics
Maximizing the utility functions, we have in
equilibrium
α
u 0 (x1 ) − pu10 (y1 ) = 0
1−α 1
px1 + y1 − pe1 = 0
α
u 0 (x2 ) − pu20 (y2 ) = 0
1−α 2
x1 + x2 − e1 = 0
y1 + y2 − e2 = 0
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Comparative Statics
Question:how a change in the initial endowment
e2 affects the equilibrium consumption bundles
while keeping e1 fixed
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Comparative Statics
Differentiate the equations
α
u100 (x1 )dx1 − pu100 (y1 )dy1 − u10 (y1 )dp = 0
1−α
pdx1 + dy1 − (x1 − 1)dp = 0
α
u200 (x2 )dx2 − pu200 (y2 )dy2 − u20 (y2 )dp = 0
1−α
dx1 + dx2 = 0
dy1 + dy2 = de2
math
Yu Ren
Mathematical Economics: Lecture 10
Chapter 15: Implicit Functions and Their Derivatives
Comparative Statics
Solve the above equations, we can get equation
(50) and (52) in page 363
math
Yu Ren
Mathematical Economics: Lecture 10