Lecture 9 _Feb 16_

Lecture 9 _Feb 16_ - Economics 100A Lecture #9: Tuesday,...

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Economics 100A Lecture #9: Tuesday, Feb. 16 1) Cost minimization (§7.2) 2) Returns to scale (§6.5) 3) Technical progress (§6.6) 4) Cost concepts (§7.1)

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(1) Cost minimization The firm’s problem: Find cheapest input combination that produces the desired output Why? Because failure to minimize cost will sacrifice some profits not profit maximizing Therefore, profit-max’ing firms want to solve: Minimize L,K Total Cost = rK + wL Subject to: f(L,K) = Q 0 “Constrained optimization” once again!
Cost min (cont’d) The parameters and constraints w = wage rate (e.g., \$12 per hour for worker) r = rental rate of capital (e.g., \$200 per hour for use of a backhoe)

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Iso-cost lines Input combinations that cost the same amount : wL + rK = TC (a constant) Solve for capital in terms of labor: K = (TC - wL) / r Slope of iso-cost line = Δ K/ Δ L = - w/r Compare with budget lines Budget line slope = -P X /P Y But a firm has no budget!! Intercepts: K = TC/r and L = TC/w
K , Units of capital per hour a b d e c \$150 isocost \$100 isocost \$50 isocost \$100 ——— \$5 = 20 \$150 ——— \$5 = 30 \$50 ——— \$ 5 = 10 \$100 ——— \$10 10 = \$50 ——— \$10 5 = \$150 ——— \$10 15 = L , Units of labor per hour Family of iso-cost lines

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Iso-cost meets iso-quant Lowest iso-cost on a given iso-quant Slope of iso-quant = -MRTS L,K = - MP L /MP K Slope of iso-cost = w / r Put the two together: MRTS L,K = MP L /MP K = w / r Another view: MP L /w = MP K /r 1/w = amount of labor that can be purchased for \$1 MP L = the amount of output that can be produced with last unit of labor MP L ×(1/w) = output derived from last \$ spent on labor Similar interpretation for capital Equate incremental output of last \$ spent on each input across inputs
L K TC 0 /w TC 1 /w TC 2 /w TC 2 /r TC 1 /r TC 0 /r Isoquant Q = Q 0 increase in cost decrease in cost Iso-cost meets iso-quant

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Extremes of technical substitution L K L K TC/w TC/r TC/w TC/r Perfect complements Perfect substitutes Q 0 Q 0
(2) Returns to scale When “scale up” all inputs, does output scale up same/more/less? Suppose that all inputs doubled … then

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This note was uploaded on 03/18/2010 for the course ECON 100A taught by Professor Woroch during the Spring '08 term at University of California, Berkeley.

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Lecture 9 _Feb 16_ - Economics 100A Lecture #9: Tuesday,...

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