Math Review

Dušan Drabik

de Leeuwenborch 2105

Dusan.Drabik@wur.nl

The material contained in these slides draws heavily on:

Geoffrey A. Jehle and Philip J. Reny (2011). Advanced Microeconomic

Theory (3rd Edition). Prentice Hall, 672 p.

Basic definitions

2

Convex Sets

3

Open and Closed ε-Balls

4

5

6

Continuity

7

Weierstrass Theorem

8

Real-Valued Functions

9

Level Sets

10

Concave Functions

11

12

13

14

15

Summary

16

Calculus

17

Functions of a single variable

18

Functions of several variables

The Hessian of a function

of several variables

The Hessian is

symmetric

19

Homogeneous Functions

20

Unconstrained Optimization

21

Unconstrained Optimization

22

Constrained Optimization w/

Equality Constrains

23

Envelope TheoremDescribes how the optimal value of the objective function in a parametrized

optimization problem changes as one of the parameters changes

1 1 1 * * * *

1 1 2

*

Let , ,...,: be a functions. Let , ,...,

denote the solution of the problem of:

max ,

. .

, 0; i = 1, ...,k

for any fixed choice of the paramter .

Suppose that and

n

n

i

f h R R R C a x a x a x a

f a

s t

h a

a

a

x

x

x

x

1

1

* *

the Lagrange multipliers ,..., are functions and that

the NDCQ holds. Then:

, , , ,

where L is teh natural Lagrangian for this problem.

na a C

d Lf a a a a a

da a

x x

24

Comparative Statics

, , 0

, , 0

f x y t

g x y t

Consider a system of equations with 2 unknowns (x and y) and a

parameter t:

Determine

dx

dt

25

Inverse Function Theorem

1 1

1

Let be a defined on an interval I in R . If ' 0

for all , then

.) is invertible on

.) its inverse is a function on the interval ( )

.) for all z in the domain of the inverse functio

f C f x

x I

a f I

b g C f I

c

n

1'

'

g

g zf g z