_Lecture_09

# _Lecture_09 - Lecture 9 1-minute review Competitive firms...

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Unformatted text preview: Lecture 9 1-minute review Competitive firms • described by production function • buy arbitrary amount of inputs at market prices • sell arbitrary amount of outputs at market prices • maximize profit = revenue - cost 1 Production function f ( k,` ) Cost function: C ( p k ,p ` ,q ) = cost of q units of output when prices are p k ,p ` Minimize p k k + p ` ` subject to f ( k,` ) = q Solve by Lagrangian method 2 Lagrangian L = p k k + p ` `- λ [ f ( k,` )- q ] FOC p k- λ ∂f ∂k = 0 p `- λ ∂f ∂` = 0 f ( k,` )- q = 0 Solve the first two for λ and get ∂f ∂k p k = ∂f ∂` p ` RTS = ∂f ∂k ∂f ∂` = p k p ` = slope of price line 3 p = price of output Profit = Revenue - Cost Π( p,p k ,p ` ) = pq- C ( q ) Maximize: one-variable problem 4 Example: Cobb-Douglas f ( k,` ) = k 1 / 2 ` 1 / 4 p k = 20 ,p ` = 10 ,p = 100 • find cost function C ( p k ,p ` ,q ) • set profit function = Π( p,p k ,p ` ) = pq- C ( q ) • maximize 5 1) Find cost function RTS = slope (1 / 2) k- 1 / 2 ` 1 / 4 (1 / 4) k 1 / 2 `- 3 / 4 = 20 10 2 k ` = 2 k = ` 6 Plug into constraint k 1 / 2 k 1 / 4 = q k = q 4 / 3 ` = q 4 / 3 C ( q ) = 30 q 4 / 3 7 Profit function...
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_Lecture_09 - Lecture 9 1-minute review Competitive firms...

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