Constrained Optimization

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Constrained Optimization

• Economics 214
• Lecture 41

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2nd Order Conditions Constrained Optimization
Sufficient conditions in optimization problems
require determining The sign of the second total
differential. The sign of the second Total
differential of a Lagrangian function
Depends on the sign of the determinant of the
bordered Hessian of the Lagrangian function.
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Bordered Hessian for Bivariate Function
The Bordered Hessian for the Lagrangian function
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Determinant Bordered Hessian
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2nd Order Conditions for Maximum

• Sufficient Condition for a Maximum in the
  Bivariate Case with one Constraint A Lagrangian
  function is negative definite at a stationary
  point if the determinant of its bordered Hessian
  is positive when evaluated at that point. In
  this case the stationary point identified by the
  Lagrange multiplier method is a maximum.

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2nd Order Condition for Minimum

• Sufficient Condition for a minimum in the
  Bivariate Case with one Constraint A Lagrangian
  function is positive definite at a stationary
  point if the determinant of its bordered Hessian
  is negative when evaluated at that point. In
  this case the stationary point identified by the
  Lagrange multiplier method is a minimum.

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Utility Maximization Example
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Utility Max example continued
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2nd Order Conditions
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2nd Utility Maximization Example
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2nd Example Continued
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2nd Order Conditions