文档介绍：Math Review for . Students
Eco 1011H (Fall, 2002)
by Shouyong Shi
Lecture I: Optimization
Lagrangian Method
Reading materials:
. Dixit (2nd edition), chapters 2, 3 and 8.
1. Mathematical method
(1) Why use mathematical methods?
Formulate: plicated economics concepts down into simpler mathematical
² ideas that are widely understood; ., marginal productivity.
Communicate: Create mon language that di®erent areas can all exploit; .,
² utility maximization, pro¯t maximization, dynamic programming.
Generate: Mathematical models that capture particular economic concepts often
²
{ reveal ings in economic intuition.
{ yield additional results that are not anticipated.
(2) Applications:
Microeconomics
²
{ utility maximization subject to the budget constraint;
{ pro¯t maximization, or cost minimization;
{ choice between ¯nancing methods;
{ incentive problems;
{ search and unemployment.
Macroeconomics (dynamics)
²
{ business cycles;
{ economic growth;
{ ary theory;
{ competitive asset pricing.
Game theory
²
{ duopoly pricing;
{ coordination games;
{ auctions.
2. Maximization with Equality Constraints
(1) Without constraints
(i) Single variable objective function: max U(x)
Graph;
²
The maximum x¤: U(x) U(x¤) for all x = x¤;
² · 6
Taylor expansion (approximation) around x¤:
²
1 2
U (x) U(x¤) + U (x¤)(x x¤) + U (x¤) (x x¤) ;
¼ x ¡ 2 xx ¡
Maximization conditions:
²
¯rst-order: Ux(x¤) = 0
second-order: U (x¤) 0:
xx ·
T
(ii) Two-variable objective function: max U(x), x = (x1; x2)
T
Similar Taylor expansion around x¤ = (x¤; x¤) , but with
² 1 2
à !
U1(x)
Ux(x) = ;
U2(x)
à !
U11(x) U21(x)
Uxx(x) = ;
U12(x) U22(x)
Maximization conditions:
²
¯rst-order: Ux(x¤) = 0
second-order: Uxx(x¤) negative semi-de¯nite.
(iii) Generalized to an objective function with n variables
First-order condition: U (x¤) = 0 where
² x
0 1
U1(x)
B C
B . C
Ux(x) = @ . A ;