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Lecture 04-2005

# Lecture 04-2005 - Some Simple Production Mechanics Lecture...

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1 Some Simple Production Mechanics Lecture IV I. Single Product Primal Optimization A. Profit Maximization 12 1 1 22 1 11 1 21 2 2 max 0 0 Y Y Y pxx wx wx Y pw xx w x x Y w αβ π α β =− = ⇒= = Substituting this result back into the first first-order condition yields: () 1 2 1 1 1 2 1 1 2 1 1 1 1 2 1 1 1 1 * 1 1 2 2 00 ,, YY Y Y Y Y p x x w x w px w w w w w w w w xpww p w +− −− −=⇒ −=  =   = = The second demand curve is then derived from the relationship between the two first-order conditions: 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 * 1 2 1 1 1 1 1 1 1 1 Y Y ww xp w p w w ββ αα αββ βα + +  =   = −−+− += = −−+ = 1 1 2 w By convention

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AEB 6184 Production Economics Lecture IV Professor Charles B. Moss Fall 2005 2 () 1 11 1 * 1 2 12 1 1 * 1 21 2 ,, YY xpww p ww ββ α βα β αβ αα −−  =   = Substituting the optimum levels of 1 x and 2 x into the production function can then derive the supply function. This yields production as a function of output and input prices. 1 1 * 1 1 1 1 1 1 1 Y Y Y Ypww p p p p ααβ αβ β αβ αβ −+ + +  =   = = 1 The dual profit function is simply the supply function and demand functions substituted into the original profit formulation * 1 1 1 1 1 1 1 1 2 Y Y Y pww p p wp π + =
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Lecture 04-2005 - Some Simple Production Mechanics Lecture...

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