Lagrangian_Optimization_Example

Lagrangian_Optimization_Example - Lagrangian Example...

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Lagrangian Example 1 Constrained Optimization with the Lagrangian Multiplier The Lagrangian multiplier method allows us to solve for the utility maximizing bundle for consumers with a wide variety of utility functions. The utility functions may include several goods, goods which are independent, complements, substitutes, as well as goods consumed in a certain proportion, or goods which have declining marginal utility, goods which become bads if the quantity reaches a certain point, as well as arguments which are not goods at all, such as labor which gives disutility but which is necessary to earn income. The general method we will use to solve these problems is the following: 1. We construct the Lagrangian problem. We wish to maximize total utility, subject to the constraint that the quantity of all of the goods, times their prices, cannot exceed the budget: Max L = (Utility Function) – λ (PxQx + PyQy + PzQz – Budget) Here we show three goods, X, Y and Z, but this applies to utility functions with any number of goods. 2. We then take the first order conditions. This means that we take the derivative of the utility function with respect to a small change in the quantity of the first good (giving us the marginal utility of the first good), and the derivative of the budget constraint with respect to a small change in the quantity
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Lagrangian_Optimization_Example - Lagrangian Example...

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