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Math Review
Dušan Drabik
de Leeuwenborch 2105
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