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: 12 1 1 22 1 11 21 1 2 2 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 2 max 0 0 1 y x y y y y y y pxx wx wx y pw xx xw w y w w px x w w ww w w xp p αβ β α π βα +− −− =− = ⇒= = = ⎛⎞ ⎜⎟ ⎝⎠ 1 1 1 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 1 1 max , 1 0 y y y y y py Cwy py wy wy py y w w py w w yw w + ++ + + + + ⎡⎤ ⎢⎥ ⎣⎦ + + = ∂+ 1 1 1 y yp w w + + + + 1
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AEB 6184 – Production Economics Lecture XXI Professor Charles Moss Fall 2005 B. However, like our discussion regarding the cost function, we can bypass the primal form and derive the economic concepts of the dual profit function. II. Properties of the Profit Function A. Properties of () , p w Π : 1. ; ,0 pw Π≥ 2. If 12 p p , then ( ) ( ) 1 , Π 2 , 2 (profit is nondecreasing in ); p 3. If , then 1 ww ( ) ( ) 1 , Π≤ Π 2 , ) (profit is nonincreasing in w ); 4. ( , p w Π is convex and continuous in ( ) , p w ; and 5. ( ) ,, tp tw t p w Π= Π 0 > , t (positive linear homogeneity). B. Property 1 basically states that a producer faced with losing money from production will simply choose not to produce. Specifically, letting 0 x = implies , and in the absence of fixed cost 0 y = ( ) Π = .
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This note was uploaded on 07/15/2011 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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

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