University of Toronto, Department of Economics, ECO 204 2009
‐
2010 S. Ajaz Hussain
ECO 204 2009
‐
2010
S. Ajaz Hussain
(Draft)
Chapter 1.1: Unconstrained Optimization, Equality Constrained Optimization & Envelope
Theorem
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‐
mail
about typos or suggestions for improvements
1: About Optimization Problems
In ECO 204 and upper level economics and finance courses, you will frequently solve
optimization problems. There are two categories of optimization problems in economics and
finance: static and dynamic. Static optimization is
at
a point in time whereas dynamic
optimization is
over
time. Here are some examples of each:
Static Optimization Examples:
•
A firm chooses output(s) to maximize revenues (or profits) over a period.
•
A firm chooses output(s) to maximize revenues (or profits) subject to capacity
constraints over a period.
•
A consumer chooses how much to consume to maximize utility subject to expenditure
equal to income over a period.
•
An investor chooses fraction of portfolio in risky and risk free assets to maximize
expected returns subject to a risk tolerance constraint for a period.
Dynamic Optimization Examples:
•
A firm chooses output(s) in each of T periods to maximize present value of revenues (or
profits).
•
A firm chooses output(s) and inventory quantities in each of T periods to maximize the
present value of revenues (or profits) subject to period
‐
specific capacity and inventory
related constraints
•
An investor chooses fraction of portfolio in risky and risk free assets in each of T periods
to maximize present value of expected returns subject to a period
‐
specific risk tolerance
constraint
1
ECO 204 (Draft) Chapter 1.1
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University of Toronto, Department of Economics, ECO 204 2009
‐
2010 S. Ajaz Hussain
•
An investor chooses the number of periods (T) to invest in risky and risk free assets.
Next, investor chooses fraction of portfolio in risky and risk free assets in
each
period to
maximize present value of expected returns subject to a period
‐
specific risk tolerance
constraint.
2: About Static Optimization Problems
In micro, you will primarily see static optimization problems. We’re going to examine 3 classes
of static optimization problems. Let’s first look at the general definition followed by a discussion
Unconstrained optimization problems
:
We choose
ݔ
to maximize (or minimize) the function
݂ሺݔሻ
. That is either:
max ݂ሺݔሻ
or:
min ݂ሺݔሻ
Note that any minimization problem can be transformed into a maximization problem (and vice
versa) since minimizing a function is the same as maximizing its negative:
min ݃ሺݔሻ ൌ
െ max ݃ሺݔሻ.
In the graph below, observe how the
ݔ
which minimizes
݂ሺݔሻ
also maximizes
െ ݂ሺݔሻ
.
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 Fall '08
 HUSSEIN
 Economics, Microeconomics, Optimization, S. Ajaz Hussain

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