Math Review - DeDSpvmouche.deds.nl/pspdf/am1920-drabik-mathrev.pdf · Math Review Dušan Drabik de...

of 25/25
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.
  • date post

    21-Jul-2020
  • Category

    Documents

  • view

    1
  • download

    0

Embed Size (px)

Transcript of Math Review - DeDSpvmouche.deds.nl/pspdf/am1920-drabik-mathrev.pdf · Math Review Dušan Drabik de...

  • Math Review

    Dušan Drabik

    de Leeuwenborch 2105

    [email protected]

    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.

    mailto:[email protected]

  • 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