Constrained Optimization Economics 214 Lecture 41

2 nd 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.

Bordered Hessian for Bivariate Function The Bordered Hessian for the Lagrangian function

Determinant Bordered Hessian

2 nd 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.

2 nd 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.

Utility Maximization Example

Utility Max example continued

2 nd Order Conditions

2 nd Utility Maximization Example

2 nd Example Continued

2 nd Order Conditions