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# note6 - Microeconomic Theory Econ 101A Fall 2008 GSI Eva...

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Microeconomic Theory Econ 101A Fall 2008 GSI: Eva Vivalt Section Notes 6: The Neoclassical Firm 1 Perfect Competition - General Example Under perfect competition, the firm takes both the output price p and the input prices w 1 , w 2 , ... as fixed. 1.1 The Two-Stage Approach The firm’s profit maximization problem can be seen as a combination of two problems: 1) a cost mini- mization problem, and 2) a profit maximization problem with a cost function. Stage 1: the firm’s cost minimization problem: min x 1 ,x 2 w 1 x 1 + w 2 x 2 s.t. f ( x 1 , x 2 ) = y This can be solved using the Lagrangian: L = w 1 x 1 + w 2 x 2 - μ [ f ( x 1 , x 2 ) - y ] The solution to this problem yields the Input Requirement Functions (IRFs), also called Conditional Input Functions (CIFs) or Conditional Factor Demands (CFDs). x 1 = x c 1 ( w 1 , w 2 , y ) x 2 = x c 2 ( w 1 , w 2 , y ) Plugging these into the objective function yields the cost function: C ( w 1 , w 2 , y ) = w 1 x c 1 ( w 1 , w 2 , y ) + w 2 x c 2 ( w 1 , w 2 , y ) Stage 2: the firm’s profit maximization problem: max y py - C ( w 1 , w 2 , y ) FOC: p - C y ( w 1 , w 2 , y ) = 0 or p = MC ( w 1 , w 2 , y ) SOC: - C yy ( w 1 , w 2 , y ) < 0 or MC y ( w 1 , w 2 , y ) > 0 (Look back to the first week’s notes: py - C ( w 1 , w 2 , y ) is our function and so the second derivative must be < 0 for it to be a maximum.) 1

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The solution to the second stage yields the optimal level of output y = y * ( w 1 , w 2 , p ). Finally, the op- timal level of output can be substituted back into the conditional input demand functions to yield the unconditional input demand functions: x 1 ( w 1 , w 2 , p ) = x c 1 ( w 1 , w 2 , y * ( w 1 , w 2 , p )) x 2 ( w 1 , w 2 , p ) = x c 2 ( w 1 , w 2 , y * ( w 1 , w 2 , p )) 1.2 Direct Profit Maximization Alternatively, the profit maximization problem can be set up and solved as an unconstrained maximization problem in one step: max x 1 ,x 2 pf ( x 1 , x 2 ) - w 1 x 1 - w 2 x 2 This yields the FOCs: pf x 1 ( x 1 , x 2 ) - w 1 = 0 pf x 2 ( x 1 , x 2 ) - w 2
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