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Solving the cost minimization problem graphically

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Solving the cost minimization problem graphically • Move to lowest possible isocost line that still has a point common with the relevant isoquant FIGURE 10-10 The Isocost Line For given input prices ( r = 2 and w = 4 in the diagram), the isocost line is the locus of all input bundles that can be purchased for a given level of total expenditure C ($200 in the diagram). The slope of the isocost line is– w / r . Chapter 10: Costs Slide 6 Deriving the cost function Solving the cost minimization problem graphically (cont.) FIGURE 10-12 The Minimum Cost for a Given Level of Output A firm that is trying to produce a given level of output, Q 0 , at the lowest possible cost will select the input combination at which an isocost line is tangent to the Q 0 isoquant. Q1 Q2 Q C Q2 C1 Q0 C2 C3 Q1
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Chapter 10: Costs Slide 7 The Cost Function: relating costs and output Solving the cost minimization problem mathematically • Scheme below works if isoquants are convex ‘enough’ • Step (a) In optimum: ¾ “In the optimum an extra dollar invested in labour must raise production by just as much as an extra dollar invested in capital” • Step (b) In optimum we also have q=F(K,L) • Step (c) Combine conditions of steps (a) and (b) Æ Two equations + two unknowns Æ Find K* and L*. implies This curve isocost slope isoquant Slope KL KL K K L L K L K L K L K L w MP w MP w w MP MP w w MRTS w w MRTS = = = = = Chapter 10: Costs Slide 8 Special technologies and their cost functions Example: the cost min. problem: Cobb-Douglas technology • Let q=2K 1/2 L 1/2 • Let w L =2 and w K =4 The cost minimization problem for this example Step (a): Step (b): Step (c): 2 / 1 2 / 1 , 2 such that 2 4 min L K q L K L K = + K L w MP w MP K K L L 2 = = 0 2 2 / 1 2 / 1 = q L K q K L q K q K q K K 2 1 2 2 2 1 0 2 2 0 ) 2 ( 2 * * * 2 / 1 2 / 1 = = = = =
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Chapter 10: Costs Slide 9 Deriving the cost function From cost minimization to the cost function Example (cont.): the Cobb-Douglas production technology In steps a-c we have found L * (q) and K * (q) Cost function: Example: the Leontief technology • q = min(1/6L,K) • Solution cost minimization problem is now obtained by common sense, not by mathematical recipe (why not?) ¾ L * (q) = 6q and K * (q)=q • Cost function ¾ C(q) = w L L * (q)+ w K K * (q) = w L 6q +w K q=(w L 6 +w K )q ¾ For example, if w L =w K =1, then C(q)=7q q 2 2 2 2 1 4 2 1 2 ) ( ) ( ) ( * * = + = + = q q q K w q L w q C K L Chapter 10: Costs
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