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Optimization using Lagrangian

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Kaidu consumes two goods, x_1 and x_2, with respective prices p_1 and p_2. His utility function is given by U_Kaidu (x_1,x_2 )=x_1^(1/4) x_2^(3/4) . Let p_1=5 and p_2=4. Kaidu's income is 80.

- Suppose the government imposes a $1 tax each unit consumption of x_2, solve for the new optimal consumption bundle.

How do i obtain this? Is the Lagrangian Method used and if so, then how?

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Solution Summary

Contained a detailed two page illustration of how to set up and solve this problem using the lagrangian multiplier. Optimal consumption bundle using the Lagrangian method is examined.

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Indeed, the easiest way to tackle this problem is using Lagrange optimization.
This type of optimization will provide you with the maximum amount of "happiness" (utility) that the consumer can obtain with some level of income.
Without getting technical, the Lagrangian we will construct will only have one maximum. Think of a hill. If you find the point where all first derivatives are zero, you will have found the top of the hill.
Consumer's Utility is given by:

In order to solve for the optimal (original) consumption bundle, we also need Kaidu's budget constraint. Often times, setting up a budget constraint is the most difficult part of word problems, ...

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