12. Cost Min

12. Cost Min - Cost Minimization Professor John Diamond...

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Cost Minimization Professor John Diamond ECON 370: Microeconomic Theory Lecture 12
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2 Cost Minimization: Introduction Cost-minimizing firm produces y 0 at min total cost Min cost for each y yields: Total cost function - c(y) With input prices w = (w 1 ,w 2 ,…,w n ), Total cost function is c(w ,y)
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3 The Cost-Minimization Problem Consider a firm using 2 inputs to make 1 output Production function is y = f(x 1 ,x 2 ) Take output level y 0 as given. Given input prices w 1 and w 2 , Total cost of input bundle (x 1 ,x 2 ) is w 1 x 1 + w 2 x 2
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4 The Cost-Minimization Problem So, for given w 1 , w 2 and y, firm’s cost - minimization problem is to solve 2 2 1 1 0 x , x x w x w min 2 1 subject to y ) x , x ( f 2 1
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5 The Cost-Minimization Problem Solution x 1 *(w ,w 2 ,y)and x ,y) are firm’s conditional demands for inputs 1, 2 Smallest possible total cost for producing y is To solve for conditional input demands ) y , w , w ( x w ) y , w , w ( x w ) y , w , w ( c 2 1 * 2 2 2 1 * 1 1 2 1
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6 Iso-cost Lines: Introduction Iso-cost curve contains all input bundles that cost same amount E.g., given w 1 and w 2 , the $100 iso-cost line has equation 100 x w x w 2 2 1 1
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7 Iso-cost Lines: Equation Generally, given w 1 and w 2 , the equation of the “c” iso -cost line is 2 1 2 1 2 w c x w w x c x w x w 2 2 1 1 Slope is - w 1 /w 2 Intercept is c/w 2
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8 Iso-cost Lines: Graph c’ w 1 x 1 +w 2 x 2 c” w 1 x 1 +w 2 x 2 c’ < c” x 1 x 2 Slopes = -w 1 /w 2
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9 The Cost-Minimization Problem: Graph x 1 x 2 Cost minimizing bundle that will produce y’: f(x 1 ,x 2 ) y’ x 1 * x 2 *
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10 The Cost-Minimization Problem: Graph x 1 x 2 f(x 1 ,x 2 ) y’ x 1 * x 2 * At an interior cost-min input bundle: (a) f(x 1 *,x 2 *) = y’ (b) slope of isocost = slope of isoquant ) x , x ( at MP MP TRS w w * 2 * 1 2 1 2 1
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11 Cobb-Douglas Example of Cost Min.
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This note was uploaded on 04/10/2010 for the course ECON 370 taught by Professor Diamond during the Spring '08 term at Rice.

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12. Cost Min - Cost Minimization Professor John Diamond...

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