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profit - 2 Profit maximization 2.1 Problem of the firm...

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2 Pro fi t maximization FIRMS MAXIMIZE PROFITS Revenue Cost max activities . Marginal analysis: ∂R ∂a i = ∂C ∂a i . Internal constraints: Technology. External constraints: Market conditions. Price-taking behavior: Small, competitive fi rm. 2.1 Problem of the fi rm Pro fi t function: π ( p ) = max y Y py , π ( p, w ) = pf ( x ) wx max x . Cost function: c ( w , y ) = min x V ( y ) wx . FOC (pro fi ts): p ∂f ( x ) ∂x i = w i , i = 1 , . . . , n, p D f ( x ) = w ; D f ( x ) = Ã ∂f ( x ) ∂x 1 , . . . , ∂f ( x ) ∂x n ! .
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SOC: h R n hD 2 f ( x ) h t 0 , D 2 f ( x ) = Ã 2 f ( x ) ∂x i ∂x j ! . Factor demand function: x ( p, w ) x ( p, w ). Supply function: y ( p, w ) = f ( x ( p, w )). 2.1.1 Di culties Non-di ff erentiability (Leontief); Boundary solutions = Kuhn-Tucker; Existence (CRtS, IRtS). 2.1.2 Cobb-Douglas Technology: f ( x 1 , x 2 ) = x a 1 x b 2 . Firm’s problem: π ( p, w 1 , w 2 ) = max x 1 ,x 2 pf ( x ) w 1 x 1 w 2 x 2 . FOC: p ∂f ( x ) ∂x 1 = pa f ( x ) x 1 = w 1 , p ∂f ∂x 2 = pb f ( x ) x 2 = w 2 .
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