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Economics of Input and Product Substitution

Economics of Input and Product Substitution. Chapter 7. Topics of Discussion. Concept of isoquant curve Concept of an iso-cost line Least-cost use of inputs Long-run expansion path of input use Economics of business expansion and contraction Production possibilities frontier

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Economics of Input and Product Substitution

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  1. Economics of Inputand ProductSubstitution Chapter 7

  2. Topics of Discussion • Concept of isoquant curve • Concept of an iso-cost line • Least-cost use of inputs • Long-run expansion path of input use • Economics of business expansion and contraction • Production possibilities frontier • Profit maximizing combination of products 2

  3. Physical Relationships 3

  4. Use of Multiple Inputs • In Ch. 6 we finished by examining profit maximizing use of a single input • Lets extend this model to where we have multiple variable inputs • Labor, machinery rental, fertilizer application, pesticide application, etc. 4

  5. Use of Multiple Inputs • Our general single input production function looked like the following: • Output = f(labor| capital, land, energy, etc) • Lets extend this to a two input production function • Output = f(labor, capital| land, energy, etc) Fixed Inputs Variable Input Variable Inputs Fixed Inputs 5

  6. Use of Multiple Inputs Output (i.e. Corn Yield) Phos. Fert. 250 Nitrogen Fert. 6

  7. Use of Multiple Inputs • If we take a slice at a level of output we obtain what is referred as an isoquant • Similar to the indifference curve we covered when we reviewed consumer theory • Shows collection of multiple inputs that generates a particular output level • There is one isoquant for each output level 250 7

  8. Isoquant means “equal quantity” Output is identical along an isoquant and different across isoquants Two inputs Page 107 8

  9. Slope of an Isoquant • The slope of an isoquant is referred to as the Marginal Rate of Technical Substitution (MRTS) • Similar in concept to the MRS we talked about in consumer theory • The value of the MRTS in our example is given by: MRTS = Capital ÷ Labor • Provides a quantitative measure of the changes in input use as one moves along a particular isoquant Pages 106-107 9

  10. Slope of an Isoquant • The slope of an isoquant is the Marginal Rate of Technical Substitution (MRTS) • Output remains unchanged along an isoquant • The ↓ in output from decreasing labor must be identical to the ↑ in output from adding capital as you move along an isoquant Capital Q=Q* K* Labor L* Pages 106-107 10

  11. MRTS here is – 4 ÷ 1 = – 4 Page 107 11

  12. What is the slope over range B? MRTS here is –1 ÷ 1 = –1 Page 107 12

  13. What is the slope over range C? MRTS here is –.5 ÷ 1 = –.5 13 Page 107

  14. Slope of an Isoquant • Since the MRTS is the slope of the isoquant, the MRTS typically changes as you move along a particular isoquant • MRTS becomes less negative as shown above as you move down an isoquant Pages 106-107 14

  15. Introducing Input Prices 15

  16. Plotting the Iso-Cost Line • Lets assume we have the following • Wage Rate is $10/hour • Capital Rental Rate is $100/hour • What are the combinations of Labor and Capital that can be purchased for $1000 • Similar to the Budget Line in consumer theory • Referred to as the Iso-CostLine when we are talking about production Pages 106-107 16

  17. Plotting the Iso-Cost Line Capital Firm can afford 10 hours of capital at a rental rate of $100/hr with a budget of $1,000 10 Firm can afford 100 hour of labor at a wage rate of $10/hour for a budget of $1,000 • Combination of Capital and Labor costing $1,000 • Referred to as the $1,000 Iso-Cost Line Labor 100 Page 109 17

  18. Plotting the Iso-Cost Line • How can we define the equation of this iso-cost line? • Given a $1000 total cost we have: $1000 = PK x Capital + PL x Labor → Capital = (1000÷PK) – (PL÷ PK) x Labor • →The slope of an iso-cost in our example is given by: • Slope = –PL ÷ PK • (i.e., the negative of the ratio of the price • of the two inputs) Page 109 18

  19. Plotting the Iso-Cost Line Capital 2,000÷PK 20 Doubling of Cost Original Cost Line Note: Parallel cost lines given constant prices 10 500 ÷ PK 5 Labor Halving of Cost 50 200 100 Page 109 500 ÷ PL 2000 ÷ PL 19

  20. Plotting the Iso-Cost Line Capital $1,000 Iso-Cost Line Iso-Cost Slope = – PK÷ PL 10 PL = $5 PL = $10 Labor 100 50 200 PL = $20 Page 109 20

  21. Plotting the Iso-Cost Line Capital $1,000 Iso-Cost Line 20 Iso-Cost Slope = – PK÷ PL PK = $50 10 PK = $100 5 PK = $200 Labor 100 50 200 Page 109 21

  22. Least Cost Combinationof Inputs 22

  23. Least Cost Input Combination TVC are predefined Iso-Cost Lines Capital TVC*** > TVC** > TVC* Q* Pt. C: Combination of inputs that cannot produce Q* Pt. A: Combination of inputs that have the highest of the two costs of producing Q* Pt. B: Least cost combination of inputs to produce Q* TVC*** A TVC** B TVC* C Labor Page 109 23

  24. Least Cost Decision Rule • The least cost combination of two inputs (i.e., labor and capital) to produce a certain output level • Occurs where the iso-cost line is tangent to the isoquant • Lowest possible cost for producing that level of output represented by that isoquant • This tangency point implies the slope of the isoquant = the slope of that iso-cost curve at that combination of inputs Page 111 24

  25. Least Cost Decision Rule • When the slope of the iso-cost = slope of the isoquant and the iso-cost is just tangent to the isoquant • –MPPK÷ MPPL=– (PK÷ PL) • We can rearrange this equality to the following Isoquant Slope Iso-cost Line Slope Page 111 25

  26. Least Cost Decision Rule MPP per dollar spent on labor MPP per dollar spent on capital = Page 111 26

  27. Least Cost Decision Rule • The above decision rule holds for all variable inputs • For example, with 5 inputs we would have the following MPP1 per $ spent on Input 1 MPP2 per $ spent on Input 2 MPP5 per $ spent on Input 5 = = = = … … Page 111 27

  28. Least Cost Input Choice for 100 Units of Output • Point G represents 7 hrs of capital and 60 hrs of labor • Wage rate is $10/hr and rental rate is $100/hr • → at G cost is • $1,300 = (100×7) + (10×60) 7 60 Page 111 28

  29. Least Cost Input Choice for 100 Units of Output • G represents a total cost of $1,300 every input combination on the iso-cost line costs $1,300 • With $10 wage rate → B* represent 130 units of labor: $1,300$10 = 130 7 130 60 Page 111 29

  30. Least Cost Input Choice for 100 Units of Output • Capital rental rate is $100/hr • → A* represents 13 hrs of capital, $1,300  $100 = 13 13 130 Page 111 30

  31. What Happens if the Price of an Input Changes? 31

  32. What Happens if Wage Rate Declines? Assume initial wage rate and cost of capital result in iso-cost line AB Page 112 32

  33. What Happens if Wage Rate Declines? Wage rate ↓ means the firm can now afford B* instead of B amount of labor if all costs allocated to labor Page 112 33

  34. What Happens if Wage Rate Declines? The new point of tangency occurs at H rather than G The firm would desire to use more labor and less capital as labor became relatively less expensive Page 112 34

  35. Least Cost Combination of Inputs and Outputfor a Specific Budget 35

  36. What Inputs to Use for a Specific Budget? Capital M An iso-cost line for a specific budget N Labor Page 113 36

  37. What Inputs to Use for a Specific Budget? A set of isoquants for different output levels Page 113 37

  38. What Inputs to Use for a Specific Budget? Firm can afford to produce 75 units of output using C3 units of capital and L3 units of labor Page 113 38

  39. What Inputs to Use for a Specific Budget? The firm’s budget not large enough to produce more than 75 units Page 113 39

  40. What Inputs to Use for a Specific Budget? On any point on this isoquant the firm is not spending available budget here Page 113 40

  41. Economics ofBusiness Expansion 41

  42. Long-Run Input Use • During the short run some costs are fixed and other costs are variable • As you increase the planning horizon, more costs become variable • Eventually over a long-enough time period all costs are variable Page 114 42

  43. Long-Run Input Use Cost/unit • Fixed costs in short run ensure the U-Shaped SAC curves • 3 different size firms • A is the smallest, C the largest SACA SACB SACC Output A* A B C • A firm wanting to minimize cost • Operate at size A if production is in 0A range • Operate at size B if production is in AB range Page 114 43

  44. The Planning Curve • The long run average cost (LAC) curve • Points of tangency with a series of short run average total cost (SAC) curves • Tangency not usually at minimum of each SAC curve SACA LAC sometimes referred to as Long Run Planning Curve SACB SACC Cost/unit LAC Tangency Points Output 44 Page 114

  45. Economies of Size • Typical LAC curve • What causes the LAC curve to decline, become relatively flat and then increase? • Due to what economists refer to as economies of size Cost/unit Output 45 Page 114

  46. Economies of Size • Constant returns to size • ↑(↓) in output is proportional to the ↑(↓) in input use • i.e., double input use → doubling output • Decreasing returns to size • ↑ (↓)in output is less than proportional to the ↑(↓) in input use • i.e., double input use → less than double output • Increasing returns to size • ↑ (↓)in output is more than proportional to the ↑(↓) in input use • i.e., double input use → more than double output 46 Page 114

  47. Economies of Size • Decreasing returns to size →Firm’s LAC curve are increasing as firm is expanded • Increasing returns to size → Firm’s LAC curve are decreasing as firm is expanded 47 Page 115

  48. Economies of Size • Reasons for increasing returns of size • Dimensional in nature • Double cheese vat size • Eventually the gains are reduced • Indivisibility of inputs • Equipment available in fixed sizes • As firm gets larger can use larger more efficient equipment • Specialization of effort • Labor as well as equipment • Volume discounts on large purchases on productive inputs 48 Page 116

  49. Economies of Size • Decreasing returns of size • LRC is ↑ → the LRC is tangent to the collection of SAC curves to the right of their minimum SACA SACB SACD SACC Cost/unit Output 49 Page 116

  50. Economies of Size • The minimum point on the LRC is the only point that is tangent to the minimum of a particular SAC • C* is minimum point on SAC* and on LRC • Only plant size and quantity output where this occurs SAC* LRC Cost/unit C* Q* Output 50 Page 116

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