Intermediate Micro
Sami Dakhlia
ECO 340
August 22, 2008
Calculus: a refresher
1
univariate calculus
1.1
Basic concepts:
Limits, continuity, differentiability
1.2
Definitions:
df
f (x0 + h) − f (x0 )
(x0 ) = f 0 (x0 ) ≡ lim
h→0
dx
h
(f + g)(x) ≡ f (x) + g(x)
(f · g)(x) ≡ f (x) · g(x)
f
f (x)
(x) ≡
g
g(x)
1.3
Theorems:
Let k ∈ IR and let f and g be differentiable functions at x = x0 . Then
(f ± g)0 (x0 ) = f 0 (x0 ) + g 0 (x0 ),
(kf 0 )(x0 ) = k f 0 (x0 ) ,
(f · g)0 (x) = f 0 (x0 )g(x0 ) + f (x0 )g 0 (x0 ),
f
f 0 (x0 )g(x0 ) − f (x0 )g 0 (x0 )
(x0 ) =
,
g
g(x0 )2
0
(f (x))k
xk
and
0
= n (f (x))k−1 · f 0 (x),
0
= kxk−1 . (Prove this for k ∈ IN .)
in particular,
composite function and chain rule: If
f (x) = h(g(x)) = (h ◦ g)(x),
then
f 0 (x) = h0 (g(x)) · g 0 (x).
1
1.4
Inverse Function Theorem:
Let f be a C 1 function defined on the interval I ⊆ IR. If f 0 (x) 6= 0 for all
x ∈ I, then:
1. f is invertible on I,
2. its inverse g is a C 1 function on the interval f (I), and
3. for all z in the domain of the inverse function g,
g 0 (z) =
1
f 0 (g(z))
.
This theorem is used to prove the power rule for k ∈ Q (and hence for k ∈ IR
by density argument).
1.5
Some functions:
exp, ln, sin, cos, tan
lim ln(1 + x) = x
x→0
(ln x)0 = 1/x
(ln f (x))0 = f 0 (x)/f (x)
df (x)
x
d ln f (x)
·
=
dx
f (x)
d ln x
x 0
x
(e ) = e
(eu (x))0 = u0 (x)eu (x)
(sin x)0 = cos x
(cos x)0 = sin x
π
π
) = − sin(x − )
2
2
π
π
sin x = − cos(x + ) = cos(x − )
2
2
sin x
tan x =
cos x
cos2 x − sin2 x
= 1 − tan2 x
(tan x)0 =
cos2 x
cos x = sin(x +
1.6
Taylor series linearization (univariate)
1. Approximation of order one:
R(h; a) ≡ f (a + h) − f (a) − f 0 (a)h,
where
R(h; a)
= 0,
h→0
h
lim
2
by definition of the derivative f 0 (a). When h is around 0.01, R(h; a)
is roughly 0.001.
2. Approximation of order two:
1
R2 (h; a) ≡ f (a + h) − f (a) − f 0 (a)h − f 00 (a)h2 ,
2
where
R2 (h; a)
= 0,
h2
by definition of the derivative f 00 (a). When h is around 0.01, R2 (h; a)
is roughly 0.00001.
lim
h→0
3. kth order Taylor polynomial of f at x = a:
Pk (a + h) = f (a) + f 0 (a)h +
2
1 00
1
f (a)h2 + · · · + f [k] (a)hk .
2!
k!
multivariate calculus
2.1
Basic concepts:
Limits, continuity, differentiability
2.2
Partial derivative:
Definition: Let f : IRn 7→ IR. Then for each variable xi at each point
x0 = (x01 , . . . , x0n ) in the domain of f ,
f (x01 , . . . , x0i + h, . . . , x0n ) − f (x01 , . . . , x0i , . . . , x0n )
∂f 0
(x1 , . . . , xn0 ) ≡ lim
,
h→0
∂xi
h
if this limit exists.
2.3
Total derivative:
2.4
Definitions:
Jacobian: Let F : IRn 7→ IRm be C 1 . Then the Jacobian derivative of F at
x∗ is the matrix

J = DF (x∗ ) = 


∂f1
∗
∂x2 (x )
∂f1
∗
∂x1 (x )
..
.
∂fm
∗
∂x1 (x )
···
..
.
∂fm
∗
∂x2 (x ) · · ·
∂f1
∗
∂xn (x )


..
.
.

∂fm
∗)
(x
∂xn
Hessian: Let f : IRn 7→ IR be C 2 . Then the Hessian is the square and
symmetric matrix




H = D2 f (x) = 



∂2f
(x)
∂x1 2
2
∂ f
∂x1 ∂x2 (x)
∂2f
∂x2 ∂x1 (x)
∂2f
(x)
∂x2 2
···
..
.
···
..
.
∂2f
∂x1 ∂xn (x)
∂2f
∂x2 ∂xn (x)
···
..
.
3
∂2f
∂xn ∂x1 (x)
∂2f
∂xn ∂x2 (x)
..
.
∂2f
(x)
∂xn 2




.



2.5
Chain rule:
df
d
∂f dx1
∂f dx2
∂f dxn
·
·
·
= (f (x(t)) =
+
+ ··· +
dt
dt
∂x1 dt
∂x2 dt
∂xn dt
Alternatively: Let F : IRn 7→ IRm and A : IRs 7→ IRn be C 1 functions.
Let s∗ ∈ IRs and x∗ = A(s∗ ) ∈ IRn . Consider the composite function
H = F ◦ A : IRr 7→ IRn .
Let DF (x∗ ) be the m × n Jacobian matrix of the partial derivatives of F at
x∗ and let DA(s∗ ) be the n × s Jacobian matrix of partial derivatives of A
at s∗ . Then, the Jacobian matrix DH(s∗ ) is given by the matrix product of
the Jacobians:
DH(s∗ ) = D(F ◦ A)(s∗ ) = DF (x∗ ) · DA(s∗ ).
2.6
convexity and quasi-convexity
2.7
Implicit functions and their derivatives
y = F (x1 , . . . , xn ) is an explicit function since the endogenous variable y
can be put on the left hand side. Often this is not possible as in (example)
or in more general terms: G(x1 , . . . , xn , y) = c.
Generalized Implicit Function Theorem: Let G(x1 , . . . , xn , y) be
a C 1 function around the point (x∗1 , . . . , x∗n , y ∗ ). Suppose further that this
point satisfies
G(x∗1 , . . . , x∗n , y ∗ ) = c
and that
∂G ∗
(x , . . . , x∗n , y ∗ ) 6= 0.
∂y 1
Then, there is a C 1 function y = y(x1 , . . . , xn , y) defined on an open ball B
about (x∗1 , . . . , x∗n , y ∗ ) so that:
1. G(x1 , . . . , xn , y(x1 , . . . , xn , y)) = c ∀(x1 , . . . , xn ) ∈ B,
2. y ∗ = y(x∗1 , . . . , x∗n ), and
3. for each index i,
∂G
(x∗1 , . . . , x∗n , y ∗ )
∂y ∗
i
(x1 , . . . , x∗n ) = − ∂x
.
∂G ∗
∗
∗
∂xi
∂y (x1 , . . . , xn , y )
4