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Economics of Input and Product Substitution Chapter 7.

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1 Economics of Input and Product Substitution 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, energy use, 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) Variable Input Fixed Inputs Variable Inputs 5

6 Use of Multiple Inputs Output (i.e. Corn Yield) 250 Nitrogen Fert. Phos. 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 the same level of output  There is one isoquant for each output level 7 250

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

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

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 Pages Labor Capital Q=Q* L* K* 10 A Slope of an isoquant = Slope of the line tangent at a point Slope of an isoquant = Slope of the line tangent at a point

11 Page 107 MRTS KL here is – 4 ÷ 1 = – 4 MRTS KL here is – 4 ÷ 1 = – 4 11 MRTS KL = ∆K/∆L

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

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

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

15 Slope of an Isoquant Lets derive the slope of the isoquant like we did for the indifference curve under consumer theory ∆Q = 0 along an isoquant → Pages

16 Slope of an Isoquant Pages Labor Capital Q=Q* L* K* MRTS KL = –MPP L* /MPP K*

17 Slope of an Isoquant Pages Labor Capital Q = Q* L* K* A B K** L** What is the impact on the MRTS as input combination changes from A to B? Why? What is the impact on the MRTS as input combination changes from A to B? Why?

18 Introducing Input Prices 18

19 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-Cost Line when we are talking about production Pages

20 Plotting the Iso-Cost Line Page 109 Labor Capital Firm can afford 100 hour of labor at a wage rate of $10/hour for a budget of $1,000 Firm can afford 10 hours of capital at a rental rate of $100/hr with a budget of $1,000 Firm can afford 10 hours of capital at a rental rate of $100/hr with a budget of $1,000 Combination of Capital and Labor costing $1,000  Referred to as the $1,000 Iso-Cost Line Combination of Capital and Labor costing $1,000  Referred to as the $1,000 Iso-Cost Line 20

21 Plotting the Iso-Cost Line Page 109 How can we define the equation of this iso-cost line?  Given a $1000 total cost we have: $1000 = P K x Capital + P L x Labor → Capital = (1000÷P K ) – (P L ÷ P K ) x Labor →The slope of an iso-cost in our example is given by: Slope = –P L ÷ P K (i.e., the negative of the ratio of the prices of the two inputs) 21

22 Plotting the Iso-Cost Line Page 109 Labor Capital Doubling of Cost Note: Parallel cost lines given constant prices Original Cost Line 2,000÷P K 500 ÷ P K Halving of Cost 500 ÷ P L 2000 ÷ P L 22

23 Plotting the Iso-Cost Line Page 109 Labor Capital $1,000 Iso-Cost Line Iso-Cost Slope = – P L ÷ P K P L = $5 P L = $10 P L = $20 23

24 Plotting the Iso-Cost Line Page 109 Labor Capital $1,000 Iso-Cost Line Iso-Cost Slope = – P L ÷ P K 20 5 P K = $200 P K = $100 P K = $50 24

25 Least Cost Combination of Inputs 25

26 Least Cost Input Combination Page 109 Labor Capital TVC are predefined Iso-Cost Lines TVC * TVC ** TVC *** TVC *** > TVC ** > TVC * A B C 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 * Q*Q* 26

27 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

28 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 –MPP L ÷ MPP K = – (P L ÷ P K ) We can rearrange this equality to the following Page 111 Isoquant Slope Iso-cost Line Slope Iso-cost Line Slope 28

29 Least Cost Decision Rule Page 111 = MPP per dollar spent on labor MPP per dollar spent on labor MPP per dollar spent on capital MPP per dollar spent on capital 29

30 Least Cost Decision Rule Page 111 The above decision rule holds for all variable inputs For example, with 5 inputs we would have the following MPP 1 per $ spent on Input 1 = MPP 2 per $ spent on Input 2 = …… = MPP 5 per $ spent on Input 5 = 30

31 Page 111 Least Cost Input Choice for 100 Units of Output 7 60 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) 31

32 Page 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 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 = Least Cost Input Choice for 100 Units of Output 32

33 Page Capital rental rate is $100/hr  → A* represents 13 hrs of capital, $1,300  $100 = 13 Capital rental rate is $100/hr  → A* represents 13 hrs of capital, $1,300  $100 = Least Cost Input Choice for 100 Units of Output 33

34 What Happens if the Price of an Input Changes? 34

35 Page 112 What Happens if Wage Rate Declines? Assume initial wage rate and cost of capital result in iso-cost line AB 35

36 Page 112 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 36

37 Page 112 What Happens if Wage Rate Declines? The new point of tangency occurs at H rather than G 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 37 What is the minimum cost of producing 100 units of output? What is the minimum cost of producing 100 units of output?

38 Least Cost Combination of Inputs and Output for a Specific Budget 38

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

40 Page 113 What Inputs to Use for a Specific Budget? A set of isoquants for different output levels 40

41 Page 113 What Inputs to Use for a Specific Budget? Firm can afford to produce 75 units of output using C 3 units of capital and L 3 units of labor 41

42 Page 113 What Inputs to Use for a Specific Budget? The firm’s budget not large enough to produce more than 75 units 42

43 Page 113 On any point on this isoquant the firm is not spending available budget here On any point on this isoquant the firm is not spending available budget here What Inputs to Use for a Specific Budget? 43

44 Economics of Business Expansion 44

45 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

46 Long-Run Input Use Page 114 SAC A SAC B SAC C Fixed costs in short run ensure the U-Shaped SAC curves  3 different size firms  A is the smallest, C the largest A B C A* 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 Cost/unit Output 46

47 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 Page 114 SAC A SAC B SAC C Output Tangency Points LAC LAC sometimes referred to as Long Run Planning Curve Cost/unit 47

48 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 Page 114 Output Cost/unit 48

49 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 Page

50 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 Page

51 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 Page

52 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 Page 116 SAC A SAC B SAC C Output Cost/unit SAC D 52

53 Economies of Size The minimum point on the LRC is the only point that is tangent to the minimum of a particular SAC Page 116 SAC* Output Cost/unit LRC Q* C* C* is minimum point on SAC* and on LRC Only plant size and quantity output where this occurs 53

54 The Planning Curve Page 117 SAC 1 SAC 2 SAC 3 Cost/unit Output In the long run, the firm has time to expand or contract the size of their operation  Each SAC curve for each size plant has associated short run marginal cost curve (MC)  SAC i = SMC i when SAC i is at its minimum SAC 4 SMC 4 SMC 3 SMC 2 SMC 1 54

55 The Planning Curve Page 117 SAC 1 SAC 2 SAC 3 Output SAC 4 SMC 4 SMC 3 SMC 2 SMC 1 Assume the market price for the product is P  Assume the firm is of size i  The firm maximizes profit by producing where P=MC i  What can you say about the performance of these 4 firms? P 55

56 The Planning Curve Page 117 SAC 1 Output SMC 1 Firm 1 would lose money with output price = P  Produce where P = SMC 1 → Q*  At Q*, P < SAC 1 P Q* 56

57 The Planning Curve Page 117 SAC 2 SAC 3 Output SAC 4 SMC 4 SMC 3 SMC 2 Firms of sizes 2, 3 and 4 would make a positive profit when output price is P P > SAC at profit maximizing level P-SAC = per unit profit P Q2*Q2* Q3*Q3* Q4*Q4* Per unit profit 57

58 The Planning Curve Page 117 SAC 2 SAC 3 Output SAC 4 SMC 4 SMC 3 SMC 2 Firm 2’s total profit  Per unit profit x Q 2 * P Q2*Q2* Q3*Q3* Q4*Q4* Firm 2’s Total Profit 58

59 The Planning Curve Page 117 SAC 2 SAC 3 Output SAC 4 SMC 4 SMC 3 SMC 2 Firm 3’s total profit  Per unit profit x Q 3 * P Q2*Q2* Q3*Q3* Q4*Q4* Firm 3’s Total Profit 59

60 The Planning Curve Page 117 SAC 2 SAC 3 Output SAC 4 SMC 4 SMC 3 SMC 2 Firm 4’s total profit  Per unit profit x Q 4 * P Q2*Q2* Q3*Q3* Q4*Q4* 60 Firm 4’s Total Profit

61 The Planning Curve Page 117 SAC 1 SAC 2 SAC 3 Output SAC 4 SMC 4 SMC 3 SMC 2 SMC 1 Assume the product price falls to P LR  Only Firm 3 will not lose money  It only breaks even as P LR =SAC 3 (=MC 3 )  For other firms, the price is less than any point on the other SAC curves  Firm 4 would have to reduce its size P P LR 61

62 Page 118 Optimal input combination for output=10 Optimal input combination for output=10 How to Expand Firm’s Capacity 62

63 Page 118 Two options: 1. Point B ? Two options: 1. Point B ? How Can the Firm Expand Its Capacity? 63

64 Page 118 How Can the Firm Expand Its Capacity? Two options: 1. Point B? 2. Point C? Two options: 1. Point B? 2. Point C? 64

65 Page 118 Optimal input combination for output=10 with budget DE Optimal input combination for output=10 with budget DE Optimal input combination for output = 20 with budget represented by FG Optimal input combination for output = 20 with budget represented by FG How Can the Firm Expand Its Capacity? 65

66 Page 118 How Can the Firm Expand Its Capacity? This combination of inuts costs more to produce 20 units of output since budget HI exceeds budget FG 66

67 Producing More than One Output Most agricultural operations produce more than one type of output  For example a grain farm in Southern Wisconsin  Produces wheat, oats, barley and some alfalfa hay  Raises some cattle on the side Production of these outputs requires a set of inputs  Each output is competing for the use of limited inputs (e.g. labor, tractor time, etc) 67

68 Producing More than One Output Lets first address the production decision from a technical perspective  Similar to our examination of production of a single output via the isoquant 68

69 For a single output we defined an isoquant as the collection input combinations that has the same maximum output represented by that isoquant Lets now define the collection of output combinations that could be produced with a fixed supply of inputs Producing More than One Output 69

70 The collection of outputs technically feasible with a fixed amount of inputs is referred to as the production possibilities set The boundary of that set is referred to as the production possibilities frontier (PPF) Producing More than One Output 70

71 Output combinations within the frontier (boundary) are technically possible but inefficient  Can produce more of at least one of the outputs  Again remember that the amount of inputs available for production is assumed fixed Producing More than One Output 71

72 Output combinations on the frontier are technically efficient  Can not produce more of at least one output unless less is produced of at least one of the other outputs  Remember the assumption: The amount of inputs available for production is fixed Producing More than One Output 72

73 Page 120 Producing More than One Output 73

74 Page 120 Points A → J are on the PPF  Note axis labels  What happens when firm changes output mix from B to E?

75 Page 120 Level of output unattainable with with firm’s existing resources Level of output unattainable with with firm’s existing resources Inefficient use of firm’s existing resources K*K* PPF represents maximum attainable products given fixed amount of inputs 75

76 Slope of the PPF The slope of the production possibilities curve is referred to as the Marginal Rate of Product Transformation (MRPT)  In the above example, the MRPT is given by:  In general we have:  What sign will the MRPT possess? Page 119 Y1Y1 Y2Y2 PPF 76

77 Page 120 Using slope definition  MRPT = ∆Y 2 ÷ ∆Y 1  Slope between D and E is –1.30 = – 13  10 Using slope definition  MRPT = ∆Y 2 ÷ ∆Y 1  Slope between D and E is –1.30 = – 13  10 ↑ from 30 to 40 ↑ from 30 to 40 ↓ from 108 to 95 ↓ from 108 to 95 77

78 Page , , ,000 40, ,000 10,000 ÷ = 78

79 Accounting for Product Prices 79

80 Economic Efficiency and Multiple Outputs Page 122 Up to this point we have only considered technical efficiency, i.e., the PPF Lets now introduce prices (both output and input) to the model  Enables us to discuss the concept of economic efficiency in the context of multiple outputs 80

81 Economic Efficiency and Multiple Outputs Page 122 Lets start with introducing output prices  Assume we have two outputs: canned fruits (CF) and canned vegetables (CV)  P CF and P CV = the prices received for CV and CV, respectively What would be the combinations of CF and CV production that would generate $1 million in total revenue (TR)?  Collection of these combinations generates an iso-revenue line 81

82 Plotting the Iso-Revenue Line Cases of CF Cases of CV P CF = $33.33/case → 30,000 cases of CF generates revenue of $1 million Page ,000 40,000 Assume P CF =$33.33/case, P CV =$25.00/case P CV = $25.00/case → 40,000 cases of CV generates revenue of $1 million $1 Mil Iso-Revenue Line 82

83 Plotting the Iso-Revenue Line Page 122 Y2Y2 Y1Y1 What is the equation that can be used to identify the R* iso-revenue line?  We have 2 products (Y 1, Y 2 ) and associated product prices (P Y1,P Y2 )  The R* iso-revenue line is defined via: R* = P Y1 Y 1 + P Y2 Y 2 → P Y2 Y 2 = R * – P Y1 Y 1 → Y 2 = (R * ÷P Y2 ) – (P Y1 ÷P Y2 )Y 1 General equation for the R* iso-revenue line 83

84 Plotting the Iso-Revenue Line Line AB is original iso-revenue line  P CF = $33.33/case, P CV = $25.00/case  Combination of outputs that generate the same amount of revenue Line AB is original iso-revenue line  P CF = $33.33/case, P CV = $25.00/case  Combination of outputs that generate the same amount of revenue Slope = $25.00 ÷ $33.33 = 0.75 Page

85 Page 122 Plotting the Iso-Revenue Line Iso-revenue line would shift out to EF  If the revenue target doubled or  Output prices decrease by 50% The line would shift in to CD  If revenue targets are halved or  Output prices are doubled Iso-revenue line would shift out to EF  If the revenue target doubled or  Output prices decrease by 50% The line would shift in to CD  If revenue targets are halved or  Output prices are doubled Note: Slope does not change 85

86 Page 122 Iso-revenue line would rotate:  Out to line BC if P CF ↓ by 50%  In to line BD if P CF doubled Iso-revenue line would rotate:  Out to line BC if P CF ↓ by 50%  In to line BD if P CF doubled Plotting the Iso-Revenue Line Note: Slope is changing 86

87 Page 122 Plotting the Iso-Revenue Line Iso-revenue line would rotate  Out to line AD if P CV ↓ by 50%  In to line AC if P CV doubled Iso-revenue line would rotate  Out to line AD if P CV ↓ by 50%  In to line AC if P CV doubled Note: Slope is changing 87

88 Determining the Profit Maximizing Combination of Products 88

89 Profit Maximizing Combination of Products In the cost minimization problem where we produce one product  The input combination that minimizes the cost of producing a given output level is where  The slope of the isocost curve equals the slope of the isoquant  → the isocost curve is just tangent to the isoquant Lets develop a similar decision rule but this time with  Multiple outputs  Fixed supply of inputs Page

90 Page Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) What is the profit (π) maximizing combination of fruit and veg. to can given current P CF and P CV values? Remember we have a fixed amount of inputs available  Determines location of the PPF  → All costs are fixed  → Maximizing revenue will maximize profit 90

91 Page Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) Lets place on this PPF the $1 Mil. iso-revenue line, AB Profit Maximizing Combination of Products A B 91

92 Page Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) Profit Maximizing Combination of Products The further from the origin the iso-revenue line, the greater the level of revenue  R * 1

93 Page Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) Profit Maximizing Combination of Products R*2R*2 R*3R*3 R*1R*1 To find the maximum revenue attainable given available inputs  Lets find the iso-revenue line that is just tangent to the PPF  At the tangency point it is physically possible to produce that combination of outputs given our fixed input base 93

94 Page Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) Profit Maximizing Combination of Products Shifting line AB out in a parallel fashion holds both prices constant Slope of an PPF curve Slope of an PPF curve Slope of the Iso-cost line Slope of the Iso-cost line M 94

95 In summary: The profit maximizing combination of two products is found where the slope of the PPF is equal to the slope of the iso- revenue line and on the highest iso revenue curve possible given the limited inputs Page 124 Profit Maximizing Combination of Products 95

96 MRPT equals MRPT equals ,000 cases of fruit 125,000 cases of fruit Price ratio = -($25.00 ÷ $33.33) = Profit Maximizing Combination of Products Page ,000 cases of veg. 18,000 cases of veg. 96

97 Doing the Math… Let’s assume P CF is $33.33 and P CV is $25.00 If point M represents 125,000 cases of fruit and 18,000 cases of vegetables, then total revenue at point M is: Revenue = 125,000 × $ ,000 × $25.00 = $4,166,250 + $450,000 = $4,616,250 97

98 Doing the Math… At these same prices, if we instead produce 108,000 cases of fruit and and 30,000 cases of vegetables→ total revenue would fall Revenue = (108,000 × $33.33) + (30,000 × $25.00) = $3,599,640 + $750,000 = $4,349,640 $266,610 less than $4,616,250 earned at M 98

99 Effects of a Change in the Price of One Product 99

100 Page 125 Profit Maximizing Combination of Products P CF reduced by 50%  Firm must sell twice as many cases of CF to earn a particular level of revenue Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) A B This gives us a new iso- revenue curve… line CB C M 100

101 Page 125 Profit Maximizing Combination of Products Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) A B C M To determine the effects of this price change on the product mix  Shift out the new iso-revenue curve  Until it is just tangent to the PPF curve 101

102 Page 125 Profit Maximizing Combination of Products Canned Veg. (1,000 Cases) Canned Fruit (1,000 Cases) A B C M N As a result of a ↓ in P CF  →Firm would shift from M to N  To maximize profit → firm would ↓ production of CF and ↑ production of CV 102

103 Summary #1 Concepts of iso-cost line and isoquants Marginal rate of technical substitution (MRTS) Least cost combination of inputs for a specific output level Effects of change in input price Level of output and combination of inputs for a specific budget Key decision rule …seek point where MRTS = ratio of input prices, or where MPP per dollar spent on inputs are equal 103

104 Summary #2 Concepts of iso-revenue line and the production possibilities frontier Marginal rate of product transformation (MRPT) Concept of profit maximizing combination of products Effects of change in product price Key decision rule – maximize profits where MRPT -ratio of the product prices 104

105 Chapter 8 focuses on market equilibrium conditions under perfect competition…. 105


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