Chapter 9: OPTIMIZATION: A Special Variety of Equilibrium Analysis

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Chapter 9OPTIMIZATION A Special Variety of
Equilibrium Analysis

• Alpha Chiang, Fundamental Methods of Mathematical
  Economics
• 3rd edition

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Optimum Values and Extreme Values

• Economics is by and large a science of choice.
• common criterion of choice among alternative in
  economic is the goal of maximizing something or
  minimizing something
• optimization means quest for the best
• maximum and minimum as mathematical concepts
  collective term is extremum.
• Optimization
• delineate an objective function
• independent variables are choice variables
• eg. p is the object of maximization and Q is the
  choice variable.

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Relative Maximum and Minimum First Derivative
Test

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First Derivative Test
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Example
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Example
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Second and Higher Derivatives
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Example
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Example
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Interpretation of the 2nd Derivative

• f(x) measures the rate of change fo the function
  f.
• f(x) is the measure of the rate of change of the
  first derivative (rate of change of the rate of
  change).

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Interpretation of the 2nd Derivative
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Second Derivative Test
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Example

• Find the relative extremum of the function
• First and second derivatives
• Set first derivative to zero and solve
• Since second derivative is positive, the extremum
  is established as a minimum.

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Example