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ch20-21lec - ECON 306 Chapters 20-21 Cost Minimization And...

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ECON 306 Chapters 20-21: Cost Minimization And Cost Curves RUST, CHO, DIAZ
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Cost Minimization x Minimize cost to produce some given level of output y at given factor prices ) , ( 2 1 w w by choosing optimal input combination ) , ( 2 1 x x . x y x x f x w x w x x ± ) , ( s.t. min 2 1 2 2 1 1 2 , 1 x Geometric solution: slope of isoquant equals slope of isocost curve (Fig. 19.1). This is the tangency condition: 2 1 ) 2 . 1 ( w w x x TRS ² . a) Isocost curves – combinations of inputs that have the same cost: C x w x w ± 2 2 1 1 . Can write this as: C x w x w ± 2 2 1 1 . Slope of isocost curve 2 1 / w w ² . b) Slope of isoquant is 2 1 2 2 1 1 2 1 / ) , ( / ) , ( ) 2 , 1 ( MP MP x x x f x x x f x x TRS ² w w w w ² . x Optimal choice ) , , ( 2 1 1 y w w x and ) , , ( 2 1 2 y w w x are the conditional factor demand functions , they depend on the level of output. x Optimal cost is the cost function . x Examples: a) Fixed proportions – } , min{ ) , ( 2 1 2 1 x x x x f . To produce y units of output we require y x x 2 1 units of each factor. Then we have y w w y w y w y w w c ) ( ) , , ( 2 1 2 1 2 1 ± ± . b) Perfect substitutes – 2 1 2 1 ) , ( x x x x f ± . Since 1 x and 2 x are perfect substitutes, choose whichever is cheaper. Thus y w w y w y w y w w c } , min{ } , min{ ) , , ( 2 1 2 1 2 1 . c) Cobb-Douglas – b a x x x x f 2 1 2 1 ) , ( . We have b a b a b b a a y w w K y w w c ± ± ± 1 2 1 2 1 ) , , ( .
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Revealed Cost Minimization x The Weak Axiom of Cost Minimization (WACM) – conditional factor demand functions (the optimal choice) gives us the input combination that have the lower production cost.
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