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ch 8_Econ 281_Fall_20100

# ch 8_Econ 281_Fall_20100 - Chapter8CostCurves 1...

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1 Chapter 8  Cost Curves Chapter 8  Cost Curves

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2 Definition:   The Total Cost function (TC) relates  minimized total cost to output, Q, to the factor  prices (w and r) where: L* and K* are the  long run  input demand  functions (that is the optimal input-choice for  Labor and Capital for a targeted level of  production Q)  TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r) LONG RUN TOTAL COST FUNCTION Definition:  The  long run total cost curve  plots the  Total Cost function against output,  holding input  prices (w and r) constant
3 Q \$/yr TC(Q) Q 0 TC curve TC (Q 0 )

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4 Example 1 Let the input demands be L=Q(r/w) and K=Q(w/r). Assume r=2 and w=4 and find the TC function Solution: Plug r=2 and w=4 in the demands and find L=Q(2/4)=0.5Q and K=Q(4/2)=2Q. Then recall: TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r) Hence TC(Q,4,2)=4(0.5Q)+2(2Q)=6Q. Done.
5 What is the long run total cost function for production function  Q = 50L 1/2 K 1/2 ? Solution: Step 1: Find the input demand functions (done in Ch 7 for this prod.  function) 1. Tangency condition gives you: K/L = w/r 2. Plug  K = (w/r)L in the isoquant:   Q = 50L 1/2 K 1/2  and  get The long run input demand functions: L*(Q,w,r) = (Q/50)(r/w) 1/2 K*(Q,w,r) = (Q/50)(w/r) 1/2 Example 2

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6 Step 2 TC(Q,w,r) =wL*(Q,w,r) + rK*(Q,w,r)      = w  [(Q/50)(r/w) 1/2 ] + r  [(Q/50)(w/r) 1/2 ] =  (Q/50)(wr) 1/2  + (Q/50)(wr) 1/2 = (Q/25)(wr) 1/2 Another question: What is the long run total cost curve when  w = 25 and r = 100? TC(Q) = 2Q
7 Q TC TC(Q) = 2Q 1 M. 2 M. \$2M. \$4M. Q = 50L 1/2 K 1/2, w = 25, r = 100 TC curve

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8 Q TC (\$/yr) TC(Q) before TC(Q) after Q 0 TC after (Q)= 2  TC before (Q) If all input prices double TC af * (Q 0 ) TC bef (Q 0 )
9 Long Run Average Cost and Marginal Cost

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10 Definition:    the long run average cost function (AC) is the  long run total cost function  (TC) divided by output, Q. That is, the AC function tells us the firm’s average cost per  unit of output… AC(Q,w,r) = TC(Q,w,r)/Q Average total cost can be measured as the slope of the ray  from the origin to the total cost curve. Long run average cost (AC)
11 Q 0 Q 0 TC(Q 0 ) Average total cost can be measured as the slope of the ray  from the origin to the total cost curve. AC(Q 0 ) = TC(Q 0 )/Q 0

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12 Marginal Cost (MC) Long run marginal cost is the change in TOTAL COST that results from a small increase in the quantity of a output, keeping constant inputs’ prices The (Long run) Marginal Cost is the slope of TC curve If you plot MC against output you get the (long run) Marginal Cost curve
13 Q 0 Q 0 TC(Q 0 ) Long run Marginal Cost is the slope of TC curve

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14 Q AC, MC 0 MC AC “typical” shape of AC, MC Example:   Average and  Marginal Cost Curves
15 Recall that, for the production function  Q = 50L 1/2 K 1/2 , the total cost function was  TC(Q,w,r) = 1/25*(wr) 1/2  Q  Q TC

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ch 8_Econ 281_Fall_20100 - Chapter8CostCurves 1...

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