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Lecture 2

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Department
Economics
Course
ECON 2X03
Professor
James Bruce
Semester
Winter

Description
Lecture 2 Long Run Cost Minimization - Long run: all inputs variable - Cost minimization: selecting quantities of the inputs to minimize costs given a particular output - Zi= input i - Y = output - W i cost per unit of input i - Min(z1,z2) w1z1 +w2z2 o Subject to F(z1,z2) = y - Consider firms with diminishing MRTS: o Assume that from the firm;s perspective, the cost per unit of inputs is fixed, the cost (c) of some input bundle is C = w1z1 + w2z2 o Isocost: a line along which the input bundles have the same cost (FIXED) o C = w1z1 + w2z2 o Z2 = c/w2 + w1*z1/w2 o Cost-min where:  slope of isoquant = slope of isocost  MRTS = w1/w2 - MP1 = Δy/ Δz1 Δz1 = Δy1/MP1 - MP2 = Δy/ Δz2 Δz2 = Δy2/MP2 - MRTS = -Δz2/Δz1 = (-Δy2/MP2)/( Δy1/MP1) - MRTS = (-Δy2/MP2)*(MP1/Δy) = - (Δy2/Δy1)*(MP1/MP2) - Since Δy2 = -Δy1 ( since back to same isoquant): o MRTS = MP1/MP2 With diminishing MRTS: - Cost-min condition is MP1/MP2 = w1/w2 1/2 1/2 Eg: a firm has the production function y = z1 z2 . The cost per unit of input is 2; the cost per unit of input two is 8. If the firm wants to produce 6 units of output, what should it use? - Given dim. MRTS: MP1/MP2 = w1/w2 - MP1 = dF(z1,z2)/d(z1) = 0.5z1 -1/z2 1/2 1/2 -1/2 - MP2 = dF(z1,z2)/d(z2) = 0.5z1 z2 - (0.5z1 -1/z2 )/ (0.5z1 z21/2 -1/) = w1/w2 - z2/z1 = w1/w2 (w1 =2, w2 = 8) - z2/z1 = 1/4 - z2 = z1/4 - For isoquant where y = 6 - 6 = z1 z22 1/2 - Combine the cost-min condition with the isoquant - 6 = z1 z22 1/2= z1 (z1/4 ) 1/2 - 6 = 0.5z1 - z1 = 12 - z2 = z1/4 - z2 = 3 (Long Run) Total Costs (TC): costs of production when costs are minimized, expressed as a function of output (and input costs) - Conditional Input Demands: the cost-minimizing quantities of inputs expressed as a function of output (and input costs) - TC = w1z1* + w2z2* Long-Run Average Cost (LAC): LAC = TC/y Long-Run Marginal Cost (LMC): rate of change of TC as output changes - LMC = dTC/dy 1/3 2/3 Eg. A firm has the production function y = z1 z2 and input costs of w1 and w2. What are: 1. Its conditional input demands? 2. TC 3. LAC 4. LMC -- 1. w/ dim MRTS: MP1/MP2 = w1/w2 -2/3 2/3 1/3 -1/3 - (z1 z2 /3)/(2z1 z2 /3) = w1/w2 - z2/2z1 = w1/w2 - z2 = 2w1/31/2/3(cost1/3n condition) 2/3 - y = z1 z2 = z1 (2w1*z1/w2) - y = z1(w1/w2) 2/3 2/3 - z1*= (w2/2w1) y – conditional input demand #1 - z2 = 2w1*z1/w2 =(2w1/w2)*(w2/2w1) y 2/3 1/3 - z2* = (2w1/w2) y – conditional input demand #2 2. TC = w1z1* + w2z2* 2/3 1/3 - TC = w1*-2/32w11/3 + w21/3w1/2/3 y - TC = (2 +2 )*w1 w2 y 3, 4 LAC = LMC = (2 -2/+2 )*w1 w2 1/3 2/3 Cost-Min w/ Perfect Complements: Given the production function: - y = min{zb/2,zc} (cheese sandwi
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