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05. Averages and Marginals

05. Averages and Marginals - that is to say d A x d x = M x...

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ECON 201 Shomu Banerjee A N ERDY N OTE ON THE R ELATIONSHIP B ETWEEN A VERAGES AND M ARGINALS The aim of this note is to provide the technical details behind the relationship between averages and marginals that we covered in class. This material is once again purely optional and meant for the more technically minded and curious among you. Suppose f ( x ) is a function (such a total product function, where x would stand for a variable input). Then the average A ( x ) is given by A ( x ) = f ( x ) x = f ( x ) . x -1 while the marginal M ( x ) is given by M ( x ) = d f ( x ) d x . Then the derivative of the average function (using the product-of-functions rule) is d A ( x ) d x = d f ( x ) d x . x -1 + f ( x ) . (-1) x -2 = M ( x ) . x -1 - A ( x ) . x -1 , that is to say:
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Unformatted text preview: that is to say: d A ( x ) d x = M ( x ) x- A ( x ) x . Now we are ready to derive the results stated in class. (1) If the average is increasing, d A ( x ) d x > 0, i.e., M ( x ) x > A ( x ) x , or M ( x ) > A ( x ) — the marginal is greater than the average. (2) If the average is falling, d A ( x ) d x < 0, i.e., M ( x ) x < A ( x ) x , or M ( x ) < A ( x ) — the marginal is less than the average. (3) When the average is at a maximum (or a minimum), d A ( x ) d x = 0. Thus M ( x ) = A ( x ) — the marginal equals the average. In other words, the marginal cuts the average at its maximum (or minimum) point....
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