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 differentiation.
The point x, y)= 1, 2 lies on the graph of
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 differentiable functions of t, and A, , are constants, find the
x
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 + y ) concave and/or quasiconcave quasiconcave on the se
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 values of a vectors c and d are linearly independent.
1.2.
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: mkaganov@indiana.edu
Office hours: by appointment (to make one, please send an e-mail
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 values of a vectors c and d are linearly independent.
Answ
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 plants that produce outputs of three goods. Its total labo
Review Question Set #5 : Answer Keys
1. U (x1 , x2 ) =
x1 +
x2 .
Consider Hessian matrix
3
2
H=
1 x1
4
0
0
3
1
4 x2 2
For x1 > 0, x2 > 0, this Hessian is negative semi-denite. The function is concave,
thus it is quasiconcave as well.
2. f (x, y ) = ln(x
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. Substitute f (Y ), g (Y ) for C, M and dene H (X, I, Y
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 positive, or (b) negative denite
(n.d.) if and only if al
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 labor, s (O, 1) is the propensity to save,
A > 0, (0, 1).
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.t. gi (x, ) 0, i = 1, . . . , m (8.1)
x0
Here x is vect
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 ( xt 1 ), t = 1,., T 1
(9.1)
ct , xt 0, t = 0,., T
y 0 >
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: mkaganov@indiana.edu
Office hours: by appointment (to make one, please send an e-mail m
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: mkaganov@indiana.edu
Office hours: 3 4 p.m. MW or by appointment (to make one, p
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 (Separation of Convex Sets).
LetXand Y be convex sets in Rn such
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 ) at x0 , be given.
Consider p1 (x) = f (x0 ) + f (x0 )(x
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 unknowns (which include
variables of the
optimization problem
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 [0, 1]
f ( + (1 )y ) mincfw_f (x), f (y ) (4.1)
(f ( +
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, can be viewed
as reallocation of
resources. The motiva
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 ( x 0 ) at x 0 , be given.
Consider p1 ( x) = f ( x 0 )
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 and
[0,1]
f (x + (1 ) y ) mincfw_ f ( x), f ( y )
(4.
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)
subject to g ( x) = 0
(3.1)
where f and g are continu
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, can be viewed as reallocation of
resources. The motiv
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 of 3 years. In each year t he allocates the available