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Lecture 21-2005 - Profit Functions Lecture XXI I A Primal...

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Profit Functions Lecture XXI I. A Primal Approach to the Profit Function A. As was the case with the cost function, the profit function can be motivated from the primal approach. 1. Taking as a starting point, the Cobb-Douglas form, we could formulate the profit function as: 1 2 1 1 2 2 1 1 1 2 1 1 2 1 1 2 2 2 2 2 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 2 1 1 1 1 2 max 0 0 1 y x y y y y y y p x x w x w x y p w x x x w w x x y x w w p w x x w p x x w w w w w w x p p x p α β β α β β β β α β α β α β α β π π α α β π β α β β α α β α β α + + + = = = = = = = = = = 1 1 1 2 1 w w α α α β α β α β β α + + These results could be substituted back into the profit function to yield an optimum profit as a function of input and output prices. 2. Alternatively, the profit function could be motivated using the results of the cost function: ( ) 1 1 1 2 1 1 2 1 2 1 1 2 1 1 2 1 2 1 1 2 1 1 2 1 2 max , 1 0 y y y y y p y C w y w w p y w y w y w w w w p y y w w w w w w p y w w y w w β α α β α β α β α β β α α β α β α β β α α β α β α β α β π α β β α α β β α α β π α β β α + + + + + + + = = = + = + = + 1 1 1 2 1 1 2 1 2 y w w y p w w w w α β β α β α α β α β α β α β α β β α + + + +
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