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Function Optimization & Application of the Envelope Theorem

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See the attached file as equations are contained within the Word file.

Consider the following function:
(see file)
where a > 0 is a parameter.

1. Find the first order condition for a critical point of this function.
2. Is this a maximum or a minimum or an inflection point?
3. Solve for x* (a), the maximizer of the function f (x; a) :Also find y* (a), the maximized value of y as a function of a
4. Find (see file) and (see file)
5. Now use the FOC from 1 and the implicit function theorem to find
6. Use the envelope theorem to find . Why does this theorem allow you to simplify your calculations with respect to point 4?

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Solution Summary

This solution shows the steps required in order to optimize a specific function. This function is parameterized and we show how to find the maximizer - optimal solution in terms of the parameter. Finally, implicit function theorem and the envelope theorem are used to double-check the answers.

While a specific function is used, a diligent student should be able to follow the steps and apply them to a similar function. Of course, this is not a substitute for a textbook and the theory behind each step must be reviewed using class material.

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The solution uses Microsoft Equation to display properly written formulas, so please refer to the attached Word file for complete solution. The text below is contained within the file:

Consider the following function:

where a > 0 is a parameter.

1. Find the first order ...

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