Lecture 13 - 1 Lecture 13: Cost Functions 1 Cost...

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Unformatted text preview: 1 Lecture 13: Cost Functions 1 Cost Minimization z Suppose that the production function is CES: q = ( k ρ + l ρ ) γ / ρ 2 q ( k l ) z The Lagrangian expression for cost minimization of producing q is L = vk + w l + λ [ q- ( k ρ + l ρ ) γ / ρ ] Cost Minimization z The first-order conditions for a minimum are ∂ L / ∂ k = v- λ ( γ / ρ )( k ρ + l ρ ) ( γ- ρ )/ ρ ( ρ ) k ρ-1 = 0 3 ∂ L / ∂ l = w- λ ( γ / ρ )( k ρ + l ρ ) ( γ- ρ )/ ρ ( ρ ) l ρ-1 = 0 ∂ L / ∂λ = q- ( k ρ + l ρ ) γ / ρ = 0 2 Cost Minimization z Dividing the first equation by the second gives us σ ρ ρ / 1 1 1 − − 4 σ ρ ρ / 1 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = l k l k k l v w • This production function is homothetic – the RTS depends only on the ratio of the two inputs – the expansion path is a straight line Cost Minimization z Suppose that the production function is Cobb- Douglas: 5 q = k α l β z The Lagrangian expression for cost minimization of producing q is L = vk + w l + λ ( q- k α l β ) Cost Minimization z The first-order conditions for a minimum are ∂ L / ∂ k = v- λα k α-1 l β = 0 6 ∂ L / ∂ l = w- λβ k α l β-1 = 0 ∂ L / ∂λ = q- k α l β = 0 3 Cost Minimization z Dividing the first equation by the second gives us k k w β β − β α 1 7 RTS k k k v w = ⋅ α β = α β = β − α l l l 1 • This production function is homothetic Total Cost Function z The total cost function shows that for any set of input costs and for any output level, the...
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This note was uploaded on 04/07/2008 for the course ECON 102 taught by Professor Akbulut during the Spring '08 term at Claremont.

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Lecture 13 - 1 Lecture 13: Cost Functions 1 Cost...

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