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Cost and Production Chapters 6 and 7

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Production Functions Describe the technology available to the firm Q = f(K, L) ◦ where K and L are input quantities ◦ K is capital, L is labor Represented graphically as Isoquants ◦ Combinations of K and L that produce the same quantity of output ◦ Convex Downward sloping: K and L are substitutes Diminishing marginal product Marginal Rate of Technical Substitution MRTS = -∆K/∆L = negative of slope of isoquant

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Costs TC = rK + wL ◦ Where r is “price” of capital ◦ And w is the wage, or price of labor ◦ Prices should reflect opportunity costs Represented graphically by a cost line ◦ Combinations of K and L that cost the same ◦ K = (TC/r) – (w/r)L ◦ A straight line connects the end points K = (TC/r) and L = (TC/w)

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Producer’s Problem Choose input quantities to minimize the cost of producing a given quantity of output Min TC = rK + wL s.t. Q* = f(K, L) Same solution as max Q, s.t. TC ◦ (∆Q/∆L)/(∆Q/∆K) = MP L /MP K = w/r ◦ Ratio of marginal products equals price ratio ◦ Cost line is tangent to an isoquant

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Solution Characteristics MP L /MP K = w/r = MRTS = -∆K/∆L ◦ Slope of cost line = slope of isoquant ◦ -MP L /MP K = -w/r MP L /w = MP K /r ◦ Marginal product per dollar equal across inputs w/MP L = r/MP K ◦ Cost of additional output equal across inputs

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Long Run vs. Short Run Long Run: All Inputs Variable ◦ Expansion path implies long run cost function ◦ Combinations of K & L that minimize cost as output increases ◦ All costs are variable ◦ Returns to Scale: relative proportional changes in inputs and output Short Run: at least one input fixed ◦ S-R Expansion path implies short run cost f’n ◦ Combinations of L and K = K* as output increases ◦ Fixed and variable costs ◦ Marginal and Average Product of labor MP L = ∆Q/∆L, K = K*; AP L = Q/L, K=K*

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Finding Costs from Production Functions Suppose Q = 10L 0.2 K 0.8, w=$10, r=$20 What is the cost of producing 100 units? Use MP L /MP K = w/r ◦ MP L /MP K = [(0.2)/(0.8)](K/L) = ¼ (K/L) ◦ w/r = $10/$20 = ½ ◦ ¼ (K/L) = ½ or K = 2L Substitute into Production Function ◦ 100 = 10(L 0.2 )(2L) 0.8 = 10L(2) 0.8 = 10L(1.74) ◦ L = 100/17.4 = 5.75, K = 2L = 11.5 Cost = wL + rK = $10(5.75) + $20(11.5) = $287.40

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Cost Exercise Q = 10L 0.2 K 0.8 w=$10, r=$20 Suppose Q = 174 Find the cost of producing 174 units Cost = wL + rK

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Expansion Path We now have two points on the expansion path for Q = 10L 0.2 K 0.8 w = $10, r = $20 Q 1 = 100, Q 2 = 174 Calculate Average Costs for each Q What is the implied relationship between cost and quantity produced? Returns to Scale

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Long Run Cost Exercise Q = 10L 0.2 K 0.8 w=$5 r=$20 Q = 200 Find Long Run Total Cost Find Long Run Average Cost

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Short Run Cost Suppose K is fixed: K* = 20 Q = 10L 0.2 K 0.8 w=$5 r=$20 Find Short Run Total Cost for Q = 200 What are fixed costs? Variable costs? Calculate ◦ Average Total Cost (ATC) ◦ Average Variable Cost (AVC) ◦ Average Fixed Cost (AFC)

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Short Run Cost Exercise Q = 10L 0.2 K 0.8, w=$5 r=$20, K* = 20 Find Short Run Total Cost for Q = 500 What are fixed cost and variable cost? ATC, AVC, AFC? Marginal Product of Labor (MP L ) Average Product of Labor (AP L ) Marginal Cost, MC = w/ MP L Draw graphs to illustrate

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Short Run Production and Cost General Relationships ◦ Short Run Production Average and Marginal Products of Labor ◦ Short Run Costs ATC, AFC, AVC SRMC ◦ Short Run Production and Cost AP L = MP L AVC = MC ◦ Short Run Cost and Long Run Cost

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Simple Profit Maximization π = PQ – C, for a price-taker ◦ At a maximum ∆ π /∆Q = 0 = P - MC ◦ MC = ∆C/∆Q ◦ P = MC In the short run ◦ MC = w/MP L ◦ So P = MC implies ◦ P = w/MP L ◦ OR w = P(MP L ) π = PQ – C = Q(P – ATC)

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Example: Profit Max Q = 10L 0.2 K 0.8, w= $5, r = $20, K* = 20 Price = $10.00 P = MC = w/MP L ◦ MP L = ∆Q/∆L = 10(0.2)L (0.2-1) K 0.8 MP L = 10(0.2)L (-0.8) (20) 0.8 = 2(10.986)L -0.8 MP L = 22L -0.8 ◦ $10.00 = $5.00/(22L -0.8 ) 2 = 1/(22L -0.8 ) or 44 = L 0.8 L = (44) 1.25 = 113 Q = 10(113) 0.2 (20) 0.8 = 283 π = PQ – C = 10(283)-5(113)-20(20)=$1865

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