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Cost and Production

Cost and Production. Chapters 6 and 7. 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

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Cost and Production

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  1. Cost and Production Chapters 6 and 7

  2. 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

  3. 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)

  4. 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) = MPL/MPK = w/r • Ratio of marginal products equals price ratio • Cost line is tangent to an isoquant

  5. Solution Characteristics • MPL/MPK = w/r = MRTS = -∆K/∆L • Slope of cost line = slope of isoquant • -MPL/MPK = -w/r • MPL/w = MPK/r • Marginal product per dollar equal across inputs • w/MPL = r/MPK • Cost of additional output equal across inputs

  6. 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 • MPL = ∆Q/∆L, K = K*; APL = Q/L, K=K*

  7. Finding Costs from Production Functions • Suppose Q = 10L0.2K0.8, w=$10, r=$20 • What is the cost of producing 100 units? • Use MPL/MPK = w/r • MPL/MPK = [(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(L0.2)(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

  8. Cost Exercise • Q = 10L0.2K0.8 w=$10, r=$20 • Suppose Q = 174 • Find the cost of producing 174 units • Cost = wL + rK

  9. Expansion Path • We now have two points on the expansion path for • Q = 10L0.2K0.8 • w = $10, r = $20 • Q1 = 100, Q2 = 174 • Calculate Average Costs for each Q • What is the implied relationship between cost and quantity produced? • Returns to Scale

  10. Long Run Cost Exercise • Q = 10L0.2K0.8 w=$5 r=$20 • Q = 200 • Find Long Run Total Cost • Find Long Run Average Cost

  11. Short Run Cost • Suppose K is fixed: K* = 20 • Q = 10L0.2K0.8 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)

  12. Short Run Cost Exercise • Q = 10L0.2K0.8 , 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 (MPL) • Average Product of Labor (APL) • Marginal Cost, MC = w/ MPL • Draw graphs to illustrate

  13. 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 • APL = MPL • AVC = MC • Short Run Cost and Long Run Cost

  14. 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/MPL • So P = MC implies • P = w/MPL • OR w = P(MPL) • π = PQ – C = Q(P – ATC)

  15. Example: Profit Max • Q = 10L0.2K0.8, w= $5, r = $20, K* = 20 • Price = $10.00 • P = MC = w/MPL • MPL = ∆Q/∆L = 10(0.2)L(0.2-1)K0.8 • MPL = 10(0.2)L(-0.8)(20)0.8 = 2(10.986)L-0.8 • MPL = 22L-0.8 • $10.00 = $5.00/(22L-0.8) • 2 = 1/(22L-0.8) or 44 = L0.8 • 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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