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# Economics of Input and Product Substitution

Download Presentation ## Economics of Input and Product Substitution

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1. 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

2. 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

3. 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

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

5. 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

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

7. 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

8. 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

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

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

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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

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

23. 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

24. 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

25. 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

26. 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

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

28. 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

29. 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

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

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

32. 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

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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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