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Maximizing Utility: Lagrange Multiplier Method Explained

Learn about the Lagrange multiplier method, an alternative to traditional substitution in solving optimization problems. This method involves creating a Lagrangian function that incorporates the objective function, constraints, and the Lagrange multiplier. Discover how to set up and solve Lagrangian functions for utility maximization examples. Interpret the Lagrange multiplier's role in optimizing objective functions. Explore practical examples and proofs to better grasp this optimization technique.

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Maximizing Utility: Lagrange Multiplier Method Explained

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  1. Economics 214 Lecture 37 Constrained Optimization

  2. Lagrange Multiplier Method • An alternative to substituting a equality constraint into an objective function as a method for solving optimization problems is the Lagrange multiplier method. • This involves forming a Lagrangian function that includes the objective function, the constraint, and a variable called a Lagrange multiplier. • Problems are often easier to solve with the Lagrange multiplier method than with the substitution method.

  3. Setting up and solving Lagrangian Functions

  4. Solving Lagrangian Functions

  5. Utility Maximization Example

  6. Utility Max example continued

  7. Interpreting Lagrange Multiplier • With our Lagrangian function, we have a new variable, , the Lagrange multiplier. • The Lagrange multiplier, , represents the effect of a small change in the constraint on the optimal value of the objective function.

  8. Proof of Interpretation

  9. Proof Continued.

  10. Utility Max Example

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