Notes-Chapter12-Optimization2

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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 Q1+Q2 = 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 λ is called a Lagrange multiplier. It is treated as an additional variable:
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Lagrange Multiplier Method:
In general:
 λ 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:
max
min
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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 Cramer’s rule or some other method, we can find
	
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Examples
Example 1. cont’d: 
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Examples
Example 2. Find the extremum of 
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Examples
Example 2. cont’d: 
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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 |H|<0, we have a minimum.