Review questions 4
Due Oct 21
only questions 2,5,7,8 will be graded.
1. Suppose that y is a differentiable function of x that satisfies the equation 2
6xy
18 Find an expression for / by implicit diffe
Review Set 1
1. Find the derivatives of the following functions, where a, p, q and b are constants:
1
a) y 2
( x x 1) 5
b) y x x x
c) y x a ( px q) b
2. If a(t) and b(t) are positive-valued differenti
Review Question Set #5
due 10/31, Monday
1. Given the utility function u(x1 , x2 ) =
x1 +
x2 ,
(a) Is it quasiconcave for x1 > 0, x2 > 0?
(b) Is it concave for x1 > 0, x2 > 0?
2. Is f (x, y ) = ln(x +
Review Set 6
November 2, 2012
c1 d1 9
9
1. Consider A = c2 d2 4 and b = 3 where ci , di are the ith elements of the follwoing vectors:
2
c3 d3 1
2
4
c = 5 and d = a .
3
6
1.1. Determine for which val
E520 Optimization Theory in Economics
Fall 2011
Class meets: 9:30 10:45 a.m. MW in BH 208
Professor:
Michael Kaganovich
Office:
Wylie Hall 341, phone: 855-6967, e-mail: [email protected]
Office hou
Review Set 6
November 2, 2012
c1 d1 9
9
1. Consider A = c2 d2 4 and b = 3 where ci , di are the ith elements of the follwoing vectors:
2
c3 d3 1
2
4
c = 5 and d = a .
3
6
1.1. Determine for which val
Review Question Set 3
Due Friday, Sep 23.
l. Check which of these pairs of vectors are orthogonal.
a. (1, 2) and (-2, 1)
b. (1, -1, 1) and (-1, 1, -1)
c. (a, -b, l) and (b, a, 0)
2. A firm has two pla
Review Question Set #4 : Answer Keys
1. Dene G(x, y ) = 2x2 + 6xy + y 2 18 then we have G(x, y ) = 0. Applying the implicit
function theorem, we have
y
x
(1,2)
=
Gx
4x + 6y
=
Gy
2y + 6x
(1,2)
=
8
5
2.
Optimal Control Problems
Consider the consumption-saving problem in stock-and-ow formulation:
t0 +T
tt0 u(ct )
max
t=t0
yt+1 yt = g (yt , ct ), t = t0 , . . . , to + T 1(t+1 )
Lagrange multipliers of
Review Question Set #2 : Answer Keys
1. See Simon & Blume 379-383.
Theorem 16.1 A nn symmetric matrix is (a) positive denite (p.d.) if and only
if all of its n leading principal minors (l.p.m.) are po
Chapter 10
Some Examples of Stability Analysis of Steady States in Discrete Time
Models
Example 1. Solow model with complete depreciation
Kt+1 = St = s AKt L1 , (10.1)
where L is constant supply of la
Chapter 8
Parametric optimization and Envelope Theorems
8.1. Value Functions: Examples
Consider the general non-linear programming problem that depends on
exogenous
parameters:
V () = max f (x, )
X
s.
Chapter 9
Dynamic Programming
9.1. Finite Horizon. Deterministic Case
Consider the finite horizon Intertemporal Choice problem
T
V0 ( y 0 ) = max t u (ct )
t =0
s.t.
ct + xt y t , t = 0,., T
y t f (
E520 Optimization Theory in Economics
Fall 2012
Class meets: 8:00 9:15 a.m. MW in BH 233
Professor:
Michael Kaganovich
Office:
Wylie Hall 341, phone: 855-6967, e-mail: [email protected]
Office hour
E520 Optimization Theory in Economics
Fall 2007
Class meets: 11:15 a.m. 12:30 p.m. MW in WY 015
Professor:
Michael Kaganovich
Office:
254 Wylie Hall, phone: 855-6967, e-mail: [email protected]
Offi
Chapter 6
Some Concepts Related to Kuhn-Tucker Theorem:
Separation of Convex Sets, Duality in Concave Programming, and their
Applications in Economics
6.1. Separation of Convex Sets
Theorem (Separatio
1.3. Taylor Formula for Functions of a Single Variable
Let f be a function of single variable x. Let f be dierentiable in a
neighborhood of x0 and
let f (x0 ), and the value of its derivative f (x0 )
Chapter 7
Basic Techniques of Comparative Statics Analysis
All of the rst order conditions results for optimization problems we have
studied can be
formulated as a system of N equations with N unknown
Chapter 4
Quasi-Convex and Quasi-Concave Functions
Consider f : Rn R, a real-valued function of n variables.
Denition: f is called quasi concave (-q uasi-convex) on a convex set S ,
ior any x, y S and
OPTIMIZATION THEORY IN ECONOMICS
LECTURE NOTES
MICHAEL KAGANOVICH
INDIANA UNIVERSITY
c Michael Kaganovich-2000, 2006, 2010
Chapter 1
Introduction
Any economic activity, such as production or exchange,
1.3. Taylor Formula for Functions of a Single Variable
Let f be a function of single variable x. Let f be differentiable in a neighborhood of x 0 and
let f ( x 0 ), and the value of its derivative f (
Chapter 4
Quasi-Convex and Quasi-Concave Functions
Consider f : R n R , a real-valued function of n variables.
Definition: f is called quasi-concave (quasi-convex) on a convex set S, if for any x, y S
Chapter 3
Optimization Subject to Equality Constraints: Lagrange Principle
3.1. Geometric Introduction: One Constraint
Consider the problem of optimization subject to an equality constraint:
max f ( x
OPTIMIZATION THEORY IN ECONOMICS
LECTURE NOTES
MICHAEL KAGANOVICH
INDIANA UNIVERSITY
Michael Kaganovich - 2000, 2006, 2010
Chapter 1
Introduction
Any economic activity, such as production or exchange
E520
Michael Kaganovich
Homework #5 (due in class on November 2)
1. (30 points). Let a consumer be endowed at t=0 with quantity y0 > 0 of a composite good. He has to allocate
consumption over a period