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Envelope Theorem.s05

Envelope Theorem.s05 - The Envelope Theorem The envelope...

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The Envelope Theorem The envelope theorem concerns how the optimal value for a particular function changes when a parameter of the function changes This is easiest to see by using an example
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The Envelope Theorem Suppose that y is a function of x y = - x 2 + ax For different values of a , this function represents a family of inverted parabolas If a is assigned a value, then y becomes a function of x only and the value of x that maximizes y can be calculated
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The Envelope Theorem Value of a Value of x * Value of y * 0 0 0 1 1/2 1/4 2 1 1 3 3/2 9/4 4 2 4 5 5/2 25/4 6 3 9 Optimal Values of x and y for alternative values of a
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The Envelope Theorem 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 a y* As a increases, the maximal value for y ( y *) increases The relationship between a and y is quadratic
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The Envelope Theorem Suppose we are interested in how y * changes as a changes There are two ways we can do this calculate the slope of y directly hold x constant at its optimal value and calculate y / a directly
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