# Economics of Input and Product Substitution

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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 Profit maximizing combination of products 2

Physical Relationships
3

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

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

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

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 250 7

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

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 9 Pages

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

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

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

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

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 14

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 → 15 Pages

Slope of an Isoquant Q=Q* MRTSKL = –MPPL*/MPPK* Pages 106-107 Capital
Labor L* 16 Pages

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

Introducing Input Prices
18

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 19

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 20 Page 109

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 prices of the two inputs) 21 Page 109

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 50 100 200 500 ÷ PL 2000 ÷ PL 22 Page 109

Plotting the Iso-Cost Line
Capital \$1,000 Iso-Cost Line Iso-Cost Slope = – PL ÷ PK 10 PL = \$5 PL = \$10 Labor 50 100 200 PL = \$20 23 Page 109

Plotting the Iso-Cost Line
Capital \$1,000 Iso-Cost Line 20 Iso-Cost Slope = – PL ÷ PK PK = \$50 10 PK = \$100 5 PK = \$200 Labor 50 100 200 24 Page 109

Least Cost Combination of Inputs
25

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 26 Page 109

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 27 Page 111

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 –MPPL ÷ MPPK = – (PL ÷ PK) We can rearrange this equality to the following Isoquant Slope Iso-cost Line Slope 28 Page 111

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

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 = = 30 Page 111

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 31 Page 111

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 60 130 Page 111 32

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 33

What Happens if the Price of an Input Changes?
34

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

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 36

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 What is the minimum cost of producing 100 units of output? Page 112 37

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

What Inputs to Use for a Specific Budget?
Capital M An iso-cost line for a specific budget N Labor Page 113 39

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

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 41

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

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

44

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 45 Page 114

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 A A* B C Output 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 46

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 SACB LAC sometimes referred to as Long Run Planning Curve SACC Cost/unit LAC Tangency Points Output 47 Page 114

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 48 Page 114

Economies of Size Constant returns to size Decreasing 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 49 Page 114

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 50 Page 115

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 51 Page 116

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 52 Page 116

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 53 Page 116

The Planning Curve 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) SACi = SMCi when SACi is at its minimum SMC1 SAC1 SMC4 SAC4 SMC2 Cost/unit SAC2 SMC3 SAC3 Output 54 Page 117

The Planning Curve Assume the market price for the product is P
Assume the firm is of size i The firm maximizes profit by producing where P=MCi What can you say about the performance of these 4 firms? SMC1 SAC1 SMC4 SAC4 SMC2 P SAC2 SMC3 SAC3 Output 55 Page 117

The Planning Curve Firm 1 would lose money with output price = P
Produce where P = SMC1 → Q* At Q*, P < SAC1 SMC1 SAC1 P Output Q* 56 Page 117

The Planning Curve 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 Per unit profit SMC4 SAC4 SAC2 SMC2 SMC3 P SAC3 Output Q2* Q3* Q4* 57 Page 117

The Planning Curve Firm 2’s total profit Per unit profit x Q2* P
SMC4 SAC4 SAC2 SMC2 SMC3 P Firm 2’s Total Profit SAC3 Output Q2* Q3* Q4* 58 Page 117

The Planning Curve Firm 3’s total profit Per unit profit x Q3* P
SMC4 SAC4 SAC2 SMC2 SMC3 P Firm 3’s Total Profit SAC3 Output Q2* Q3* Q4* 59 Page 117

The Planning Curve Firm 4’s total profit Per unit profit x Q4* P
SMC4 SAC4 SAC2 SMC2 SMC3 P Firm 4’s Total Profit SAC3 Output Q2* Q3* Q4* 60 Page 117

The Planning Curve Assume the product price falls to PLR
Only Firm 3 will not lose money It only breaks even as PLR=SAC3 (=MC3) For other firms, the price is less than any point on the other SAC curves Firm 4 would have to reduce its size SMC1 SAC1 SMC4 SAC4 SMC2 P SAC2 SMC3 SAC3 PLR Output 61 Page 117

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

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

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

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

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 Page 118

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

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

Producing More than One Output
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 69

Producing More than One Output
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) 70

Producing More than One Output
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 71

Producing More than One Output
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 72

Producing More than One Output
73 Page 120

Points A → J are on the PPF
Note axis labels What happens when firm changes output mix from B to E? 128 95 10 Page 120 74

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

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? Y2 PPF Y1 76 Page 119

Using slope definition MRPT = ∆Y2 ÷ ∆Y1
Slope between D and E is –1.30 = – 13  10 ↓ from 108 to 95 ↑ from 30 to 40 Page 120 77

95,000 - 108,000 -13,000 40,000 30,000 10,000 ÷ = - 1.30 Page 148 78

Accounting for Product Prices
79

Economic Efficiency and Multiple Outputs
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 Page 122 80

Economic Efficiency and Multiple Outputs
Lets start with introducing output prices Assume we have two outputs: canned fruits (CF) and canned vegetables (CV) PCF and PCV = 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 Page 122

Plotting the Iso-Revenue Line
Assume PCF=\$33.33/case, PCV=\$25.00/case Cases of CF PCF = \$33.33/case → 30,000 cases of CF generates revenue of \$1 million 30,000 PCV = \$25.00/case → 40,000 cases of CV generates revenue of \$1 million Cases of CV 40,000 \$1 Mil Iso-Revenue Line Page 122 82

Plotting the Iso-Revenue Line
What is the equation that can be used to identify the R* iso-revenue line? We have 2 products (Y1, Y2) and associated product prices (PY1,PY2) The R* iso-revenue line is defined via: R* = PY1Y1 + PY2Y2 → PY2Y2 = R* – PY1Y1 → Y2 = (R*÷PY2) – (PY1÷PY2)Y1 Y2 General equation for the R* iso-revenue line Y1 Page 122 83

Plotting the Iso-Revenue Line
Line AB is original iso-revenue line PCF= \$33.33/case, PCV= \$25.00/case Combination of outputs that generate the same amount of revenue Slope = \$25.00 ÷ \$33.33 = 0.75 84 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 Note: Slope does not change Page 122 85

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

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

Determining the Profit Maximizing Combination of Products
88

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 89 Page 124

Remember we have a fixed amount of inputs available
What is the profit (π) maximizing combination of fruit and veg. to can given current PCF and PCV values? Remember we have a fixed amount of inputs available Determines location of the PPF → All costs are fixed → Maximizing revenue will maximize profit 140 120 100 Canned Fruit (1,000 Cases) 80 60 40 20 20 40 60 80 100 120 140 Canned Veg. (1,000 Cases) 90 Page 124

Profit Maximizing Combination of Products
140 120 Lets place on this PPF the \$1 Mil. iso-revenue line, AB 100 Canned Fruit (1,000 Cases) 80 60 40 A 20 B 20 40 60 80 100 120 140 Canned Veg. (1,000 Cases) 91 Page 124

Profit Maximizing Combination of Products
140 The further from the origin the iso-revenue line, the greater the level of revenue R*1<R*2<R*3 Why are the iso-revenue lines parallel in this model? 120 100 Canned Fruit (1,000 Cases) 80 60 R*2 40 20 R*3 R*1 20 40 60 80 100 120 140 Canned Veg. (1,000 Cases) 92 Page 124

Profit Maximizing Combination of Products
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 140 120 100 Canned Fruit (1,000 Cases) 80 60 R*2 40 20 R*3 R*1 20 40 60 80 100 120 140 Canned Veg. (1,000 Cases) 93 Page 124

Profit Maximizing Combination of Products
140 Shifting line AB out in a parallel fashion holds both prices constant M 120 100 Canned Fruit (1,000 Cases) 80 60 40 Slope of an PPF curve Slope of the Iso-cost line 20 20 40 60 80 100 120 140 Canned Veg. (1,000 Cases) 94 Page 124

Profit Maximizing Combination of Products
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 95 Page 124

Profit Maximizing Combination of Products
Price ratio = -(\$25.00 ÷ \$33.33) = 125,000 cases of fruit 18,000 cases of veg. MRPT equals -0.75 96 Page 120

Doing the Math… Let’s assume PCF is \$33.33 and PCV 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

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

Effects of a Change in the Price of One Product
99

Profit Maximizing Combination of Products
140 PCF reduced by 50% Firm must sell twice as many cases of CF to earn a particular level of revenue M 120 100 Canned Fruit (1,000 Cases) 80 C This gives us a new iso-revenue curve… line CB 60 40 A 20 B 20 40 60 80 100 120 140 Page 125 100 Canned Veg. (1,000 Cases)

Profit Maximizing Combination of Products
140 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 M 120 100 Canned Fruit (1,000 Cases) 80 C 60 40 A 20 B 20 40 60 80 100 120 140 Page 125 101 Canned Veg. (1,000 Cases)

Profit Maximizing Combination of Products
140 As a result of a ↓ in PCF →Firm would shift from M to N To maximize profit → firm would ↓ production of CF and ↑ production of CV M 120 100 N Canned Fruit (1,000 Cases) 80 C 60 40 A 20 B 20 40 60 80 100 120 140 Page 125 102 Canned Veg. (1,000 Cases)

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

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

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

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