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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 Preview

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 ...

#### 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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