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Unit 7.

Unit 7. Analyses of LR Production and Costs as Functions of Output (Ch. 5, 6, 8). LR  Max . 1. Produce Q where MR = MC 2. Minimize cost of producing Q  optimal input combination. Isoquant. The combinations of inputs (K, L) that yield the producer the same level of output.

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Unit 7.

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  1. Unit 7. • Analyses of LR Production and Costs as Functions of Output (Ch. 5, 6, 8)

  2. LR  Max  • 1. Produce Q where MR = MC • 2. Minimize cost of producing Q  optimal input combination

  3. Isoquant • The combinations of inputs (K, L) that yield the producer the same level of output. • The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.

  4. Typical Isoquant

  5. SR Production in LR Diagram

  6. MRTS and MP • MRTS = marginal rate of technical substitution = the rate at which a firm must substitute one input for another in order to keep production at a given level = - slope of isoquant = = the rate at which capital can be exchanged for 1 more (or less) unit of labor • MPK = the marginal product of K = • MPL = the marginal product of L = • Q = MPK K + MPL L • Q = 0 along a given isosquant  MPK K + MPL L = 0  = ‘inverse’ MP ratio

  7. Indifference Curve & Isoquant Slopes

  8. Cobb-Douglas Isoquants • Inputs are not perfectly substitutable • Diminishing marginal rate of technical substitution • Most production processes have isoquants of this shape

  9. Linear Isoquants • Capital and labor are perfect substitutes

  10. Leontief Isoquants • Capital and labor are perfect complements • Capital and labor are used in fixed-proportions

  11. Budget Line • = maximum combinations of 2 goods that can be bought given one’s income • = combinations of 2 goods whose cost equals one’s income

  12. Isocost Line • = maximum combinations of 2 inputs that can be purchased given a production ‘budget’ (cost level) • = combinations of 2 inputs that are equal in cost

  13. Isocost Line Equation • TC1 = rK + wL •  rK = TC1 – wL •  K = Note: slope = ‘inverse’ input price ratio = = rate at which capital can be exchanged for 1 unit of labor, while holding costs constant.

  14. Increasing Isocost

  15. Changing Input Prices

  16. Different Ways (Costs) of Producing q1

  17. Cost Minimization (graph)

  18. LR Cost Min (math) •  - slope of isoquant = - slope of isocost line     

  19. SR vs LR Production

  20. Assume a production process: • Q = 10K1/2L1/2 • Q = units of output • K = units of capital • L = units of labor • R = rental rate for K = $40 • W = wage rate for L = $10

  21. Given q = 10K1/2L1/2 * LR optimum for given q

  22. Given q = 10K1/2L1/2, w=10, r=40 • Minimum LR Cost Condition  inverse MP ratio = inverse input P ratio  (MP of L)/(MP of K) = w/r  (5K1/2L-1/2)/(5K-1/2L1/2) = 10/40  K/L = ¼  L = 4K

  23. Optimal K for q = 40?(Given L* = 4K*) • q = 40 = 10K1/2L1/2 •  40 = 10 K1/2(4K)1/2 •  40 = 20K •  K* = 2 •  L* = 8 •  min SR TC = 40K* + 10L* = 40(2) + 10(8) = 80 + 80 = $160

  24. SR TC for q = 40? (If K = 5) q = 40 = 10K1/2L1/2  40 = 10 (5)1/2(L)1/2  L = 16/5 = 3.2  SR TC = 40K + 10L = 40(5) + 10(3.2) = 200 + 32 = $232

  25. Optimal K for q = 100?(Given L* = 4K*) • Q = 100 = 10K1/2L1/2 •  100 = 10 K1/2(4K)1/2 •  100 = 20K •  K* = 5 •  L* = 20 •  min SR TC = 40K* + 10L* = 40(5) + 10(20) = 200 + 200 = $400

  26. SR TC for q = 100? (If K = 2) • Q = 100 = 10K1/2L1/2 •  100 = 10 (2)1/2(L)1/2 •  L = 100/2 = 50 •  SR TC = 40K + 10L = 40(2) + 10(50) = 80 + 500 = $580

  27. Two Different costs of q = 100

  28. LRTC Equation Derivation[i.e. LRTC=f(q)] •  LRTC = rk* + wL* = r(k* as fn of q) + w(L* as fn of q) • To find K* as fn q from equal-slopes condition L*=f(k), sub f(k) for L into production fn and solve for k* as fn q • To find L* as fn q from equal-slopes condition L*=f(k), sub k* as fn of q for f(k) deriving L* as fn q

  29. LRTC Calculation Example • Assume q = 10K1/2L1/2, r = 40, w = 10 L* = 4K (equal-slopes condition) • K* as fn q • q = 10K1/2(4K)1/2 = 10K1/22K1/2 = 20K  • LR TC = rk* + wL* = 40(.05q)+10(.2q) = 2q + 2q = 4q • L* as fn q • L* = 4K* = 4(.05 q) L* = .2q

  30. Graph of SRTC and LRTC

  31. Expansion Path  LRTC

  33. Technological Progress

  34. Multiplant Production Strategy • Assume: P = output price = 70 - .5qT qT = total output (= q1+q2) q1 = output from plant #1 q2 = output from plant #2  MR = 70 – (q1+q2) TC1 = 100+1.5(q1)2 MC1 = 3q1 TC2 = 300+.5(q2)2 MC2 = q2

  35. Multiplant  Max • (#1) MR = MC1 • (#2) MR = MC2 • (#1) 70 – (q1 + q2) = 3q1 • (#2) 70 – (q1 + q2) = q2 from (#1), q2 = 70 – 4q1 • Sub into (#2),  70 – (q1 + 70 – 4q1) = 70 – 4q1  7q1 = 70  q1 = 10, q2 = 30 •  = TR – TC1 – TC2 = (50)(40) - [100 + 1.5(10)2] - [300 + .5(30)2] = 2000 – 250 – 750 = $1000

  36.  If q1 = q2 = 20? •  = TR - TC1 - TC2 = (50)(40) - [100 + 1.5(20)2] - [300 + .5(20)2] = 2000 – 700 – 500 = $800

  37. Multi Plant Profit Max(alternative solution procedure) • 1. Solve for MCT as fn of qT knowing cost min  MC1=MC2=MCT  MC1=3q1 q1 = 1/3 - MC1 = 1/3 MCT MC2 = q2 q2 = MC2 = MCT  q1+q2 = qT = 4/3 MCT  MCT = ¾ qT • 2. Solve for profit-max qT  MR=MCT  70-qT = ¾ qT 7/4 qT = 70  q*T = 40  MC*T = ¾ (40) = 30

  38. Multi Plant Profit Max(alternative solution procedure) • 3. Solve for q*1 where MC1 = MC*T  3q1 = 30 q*1 = 10 • 4. Solve for q*2 where MC2 = MC*T  q*2 = 30

  39. Graph  Max, 2 Plants (linear MCs)

  40.  Max (?), 2 Plants, nonlinear MCs

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