Chapter 12 Optimization With Equality Constraints

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Chapter 12 Optimization With Equality
Constraints

• Econ 130 Class Notes from
• Alpha Chiang, Fundamentals of Mathematical
  Economics, 3rd Edition

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Introduction

• Previously, all choice variables were independent
  of each other.
• However if we are to observe the restriction
  Q1Q2 1000, the independence between choice
  variables is lost.
• The new optimum satisfying the production quota
  constitutes a constrained optimum.

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Effects of a constraint
are positive for all positive levels of x1 and x2.
Budget constraint
Such renders x1 and x2 mutually
dependent. Problem How to maximize U subject
to the given constraint.
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Lagrange Multiplier Method
The symbol ? is called a Lagrange multiplier. It
is treated as an additional variable
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Lagrange Multiplier Method
In general
? measures the sensitivity of Z to changes in
the constraint
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n-Variables Case
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Multi-constraint case
Suppose there are two constraints
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Second Order Conditions
For a constrained extremum of
subject to
Second order necessary and sufficient condition
revolves around the algebraic sign of the second
order differential evaluated at a stationary
point.
We shall be concerned with the sign definiteness
or semidefiniteness of
for those dx and dy values (not both zero)
satisfying the linear constraint

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Second Order Conditions
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The Bordered Hessian

• Plain Hessian

• Bordered Hessian borders will be

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Second order condition
Determinantal Criterion for sign definiteness
min
max
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Second order condition
Conclusion
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Examples
Example 1. Find the extremum of
First, form the Lagrangian function
By Cramers rule or some other method, we can
find
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Examples
Example 1. contd
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Examples
Example 2. Find the extremum of
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Examples
Example 2. contd
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n-Variable Case
Objective function
subject to
with
Given a bordered Hessian
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n-Variable Case
bordered principal minors are
.
with the last one being
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n-Variable Case
.
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Example Utility Maximization and Consumer Demand
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Example Least cost combination of inputs
Minimize
subject to
First Order Condition
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Second order condition
Therefore, since Hlt0, we have a minimum.