Envelope Theorem Extended Example
1
Motivation
In economics, we are often concerned with the following type of maximization
problem:
max (q; p),
q
where p > 0 is the parameter of the problem.
In other words, we are choosing q to maximize an objective func
Eigenvalue Test
The eigenvalues of a matrix characterize whether a symmetric matrix
A is positive denite, negative denite, positive semidenite, or negative
semidenite. In particular, we have the following result.
Theorem 1 The symmetric matrix A is
1. pos
Econ 519 (Part I)
Problem Set 11: Concave and Convex
Functions
1. Do the following exercises Simon and Blume (1994): Chapter 21:
21.2, 21.8, 21.23
2. Use the determinant test to check the concavity of the function f (x; y ) =
ln y x2 on R2 .
+
3. Use the
For the problem of maximizing x2 + y 2 subject to x
3x + y 5. If we set-up the problem by using Lagrangian
0, y
= x2 + y 2
+y
L(x; y;
1;
2;
3)
1[
x]
2[
y]
3 [3x
0, and
5]
and solve, the unique solution is (x; y; 1 ; 2 ; 3 ) = (0; 5; 30; 0; 10). Now
consid
Econ 519 (Part I)
Final Exam (2012) (50 Points)
Instructions: Neatly write your name on the top right hand side of the
exam. There are 7 questions. You will have until 11:00 am to complete as
many questions as possible. Please use a seperate piece of pape
Econ 519 (Part I)
Final Exam (2012) Solution
1. (5 Points) Is the function f (x; y ) = lnfln(x + y )g quasi-concave on the
set of strictly positive x and y values? You must justify your answer.
Solution:
This solution was a little tricky. First, the funct
Econ 519 (Part I)
Final Exam (2011) (50 Points)
Instructions: Neatly write your name on the top right hand side of the
exam. There are 7 questions. You will have until 11:00 am to complete as
many questions as possible. Please use a seperate piece of pape
Econ 519 (Part I)
Final Exam (2011) Solution
1. Suppose
z = f (x; y; t) = yex + t2 ,
x = g (y; t) = ln(y + t), and
y = h(t) = t3 9:
Use the chain rule to nd
dz
dt
at t = 2?
Soution: Re-write these three equations as a function of t to get z (t) =
f (g (h(
Constrained Optimization 2: Inequality
Constraints and Equality Constraints
Inequality constraints are dealt with by introducing a new variable s
0 into the problem called a slack variable. The role of s is to turn an
inequality constraint into an equalit
Constrained Optimization 1: Equality
Constraints
1
Examples
1. Maximize the function f (x; y ) = xy subject to the constraint x + y = 1
First, we set-up the Lagrangian:
L(x; y; ) = xy
(x + y
1):
The FOC for the problem are:
@L
=
(x + y 1) = 0,
@
@L
=y
= 0
Completing the Square
Suppose you have a quadratic of the form
ax2 + bx + c,
where a 6= 0.
There is an algebra trick called completing the squarewhich is good to
know. It goes like this:
ax2 + bx + c
b
= a x2 + x +
a
b
= a x2 + x +
a
c
a
c
a
b2
.
4a2
Now