Finance 510: Microeconomic Analysis

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Finance 510 Microeconomic Analysis

• Optimization

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Don't Panic!
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Functions
Optimization deals with functions. A function is
simply a mapping from one space to another.
(that is, a set of instructions describing how to
get from one location to another)
Is the range
Is a function
Is the domain
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Functions
For any
and
Note A function maps each value of x to one and
only one value for y
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For example
For
Range
Domain
6
20
For
Range
Y 14
5
Domain
0
5
X 3
7
20
Here, the optimum occurs at x 5 (y 20)
Range
5
Domain
0
5
Optimization involves finding the maximum value
for y over an allowable range.
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What is the solution to this optimization problem?
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There is no optimum because f(x) is discontinuous
at x 5
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What is the solution to this optimization problem?
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There is no optimum because the domain is open
(that is, the maximum occurs at x 6, but x 6
is NOT in the domain!)
0
6
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What is the solution to this optimization problem?
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There is no optimum because the domain is
unbounded (x is allowed to become arbitrarily
large)
0
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Necessary vs. Sufficient Conditions
Sufficient conditions guarantee a solution, but
are not required
Necessary conditions are required for a solution
to exist
Gas is a necessary condition to drive a car
A gun is a sufficient condition to kill an ant
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The Weierstrass Theorem
The Weierstrass Theorem provides sufficient
conditions for an optimum to exist, the
conditions are as follows
is continuous over the domain of
The domain for
is closed and bounded
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Derivatives
Formally, the derivative of
is defined as follows
All you need to remember is the derivative
represents a slope (a rate of change)
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Slope
0
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Example
0
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Useful derivatives
Linear Functions
Exponents
Logarithms
Products
Composites
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Practice Makes Perfect
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Unconstrained maximization
Strictly speaking, no problem is truly
unconstrained. However, sometimes the constraints
dont bite (the constraints dont influence the
maximum)
First Order Necessary Conditions
If
is a solution to the optimization problem
or

then
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An Example
Suppose that your company owns a corporate jet.
Your annual expenses are as follows

• You pay your flight crew (pilot, co-pilot, and
  navigator a combined annual salary of 500,000.
• Annual insurance costs on the jet are 250,000
• Fuel/Supplies cost 1,500 per flight hour
• Per hour maintenance costs on the jet are
  proportional to the number of hours flown per
  year.
• Maintenance costs (per flight hour)
  1.5(Annual Flight Hours)

If you would like to minimize the hourly cost of
your jet, how many hours should you use it per
year?
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An Example
Let x Number of Flight Hours
First Order Necessary Conditions
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An Example
Hourly Cost ()
Annual Flight Hours
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How can we be sure we are at a minimum?
Secondary Order Necessary Conditions
If
is a solution to the maximization problem
then

If
is a solution to the minimization problem
then

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The second derivative is the rate of change of
the first derivative
Slope is increasing
Slope is decreasing
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An Example
Let x Number of Flight Hours
First Order Necessary Conditions
Second Order Necessary Conditions
For Xgt0
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Multiple Variables
Suppose you know that demand for your product
depends on the price that you set and the level
of advertising expenditures.
Choose the level of advertising AND price to
maximize sales
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Partial Derivatives
When you have functions of multiple variables, a
partial derivative is the derivative with respect
to one variable, holding everything else constant
Example (One you will see a lot!!)
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Multiple Variables
First Order Necessary Conditions
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Multiple Variables
(2)
(1)
(1)
(2)
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Again, how can we be sure we are at a maximum?
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Recall, the second order condition requires that
For a function of more than one variable, its a
bit more complicated
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Actually, its generally sufficient to see if all
the second derivatives are negative
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Constrained optimizations attempt to
maximize/minimize a function subject to a series
of restrictions on the allowable domain
To solve these types of problems, we set up the
lagrangian
Function to be maximized
Constraint(s)
Multiplier
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Once you have set up the lagrangian, take the
derivatives and set them equal to zero
First Order Necessary Conditions
Now, we have the Multiplier conditions
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Constrained Optimization
Example Suppose you sell two products ( X and Y
). Your profits as a function of sales of X and
Y are as follows
Your production capacity is equal to 100 total
units. Choose X and Y to maximize profits
subject to your capacity constraints.
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Constrained Optimization
Multiplier
The first step is to create a Lagrangian
Constraint
Objective Function
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Constrained Optimization
First Order Necessary Conditions
Multiplier conditions
Note that this will always hold with equality
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Constrained Optimization
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The Multiplier
Lambda indicates the marginal value of relaxing
the constraint. In this case, suppose that our
capacity increased to 101 units of total
production.
Assuming we respond optimally, our profits
increase by 5
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Another Example
Suppose that you are able to produce output using
capital (k) and labor (l) according to the
following process
The prices of capital and labor are
and
respectively.
Union agreements obligate you to use at least one
unit of labor.
Assuming you need to produce
units of output, how would
you choose capital and labor to minimize costs?
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Minimizations need a minor adjustment
To solve these types of problems, we set up the
lagrangian
A negative sign instead of a positive sign!!
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Inequality Constraints
Just as in the previous problem, we set up the
lagrangian. This time we have two constraints.
Holds with equality
Doesnt necessarily hold with equality
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First Order Necessary Conditions
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Case 1
Constraint is non-binding
First Order Necessary Conditions
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Case 2
Constraint is binding
First Order Necessary Conditions
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Constraint is Binding
Constraint is Non-Binding
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Try this one
You have the choice between buying apples and
oranges. You utility (enjoyment) from eating
apples and bananas can be written as
The prices of Apples and Bananas are given by
and
Maximize your utility assuming that you have 100
available to spend
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(Objective)
(Income Constraint)
(You cant eat negative apples/oranges!!)
Objective
Non-Negative Consumption Constraint
Income Constraint
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First Order Necessary Conditions

• We can eliminate some of the multiplier
  conditions with a little reasoning
• You will always spend all your income
• You will always consume a positive amount of
  apples

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Case 1 Constraint is non-binding
First Order Necessary Conditions
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Case 1 Constraint is binding
First Order Necessary Conditions