economics of input and product substitution chapter 7

Economics of Inputand ProductSubstitution

Chapter 7

Topics of DiscussionConcept of isoquant curveConcept of an iso-cost lineLeast-cost use of inputs Long-run expansion path of input useEconomics of business expansion and

contractionProduction possibilities frontierProfit maximizing combination of

products

Physical Relationships

Use of Multiple InputsIn Ch. 6 we finished by examining profit

maximizing use of a single input

Lets extend this model to where we have multiple variable inputsLabor, machinery rental, fertilizer

application, pesticide application, energy use, etc.

Use of Multiple InputsOur 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 InputFixed Inputs

Fixed InputsVariable Inputs

Use of Multiple Inputs

Output (i.e. Corn Yield)

Nitrogen Fert.

Phos. Fert.

Use of Multiple InputsIf we take a slice at a level

of output we obtain what is referred as an isoquantSimilar 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

Isoquant means “equal quantity”Isoquant means “equal quantity”

Two inputs

Output isidentical alongan isoquant and different across isoquants

Slope of an IsoquantThe 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 106-1079

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 106-107

Capital

Slope of an isoquant =Slope of the linetangent at a point

MRTSKL here is– 4 ÷ 1 = – 4

MRTSKL = ∆K/∆L

What is the slope overrange B?

MRTS here is–1 ÷ 1 = –1

What is the slope overrange C?

MRTS here is–.5 ÷ 1 = –.5

Slope of an IsoquantSince the MRTS is the slope of the

isoquant, the MRTS typically changes as you move along a particular isoquantMRTS becomes less negative as shown

above as you move down an isoquant

Pages 106-10714

Slope of an IsoquantLets derive the slope of the isoquant

like we did for the indifference curve under consumer theory

∆Q = 0 along an isoquant →

Pages 106-10715

ΔQ ΔQQ f L,K ΔQ ΔL + ΔK

ΔL ΔK

ΔQ ΔQ0 ΔL + ΔK

ΔL ΔKΔQ ΔQ ΔK ΔQ ΔQ

ΔK ΔLΔL ΔKΔK ΔL ΔL

Pages 106-10716

MPPΔK ΔQ ΔQMRTS

ΔL ΔKΔL MPP

Capital

MRTSKL = –MPPL*/MPPK*

Pages 106-10717

Capital

Q = Q*

What is the impact on theMRTS as input combinationchanges from A to B? Why?

Introducing Input Prices

Plotting the Iso-Cost LineLets 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 106-10719

Plotting the Iso-Cost Line

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 ofcapital 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 Line20

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)

Capital

Doubling of Cost

Note: Parallel cost linesgiven constant prices

Original Cost Line

2,000÷PK

500 ÷ PK

Halving of Cost

500 ÷ PL 2000 ÷ PL22

Capital

10050 200

Iso-Cost Slope = – PL ÷ PK

PL = $5PL = $10

PL = $20

Capital

10050 200

Iso-Cost Slope = – PL ÷ PK

PK = $200

PK = $100

PK = $50

Least Cost Combinationof Inputs

Least Cost Input Combination

CapitalTVC are predefined Iso-Cost Lines

TVC***

TVC*** > TVC** > TVC*

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*

Least Cost Decision RuleThe 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

Least Cost Decision RuleWhen 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

Least Cost Decision Rule

MPP MPP

MPP per dollar spent on labor

MPP per dollar spent on capital

Least Cost Decision Rule

The above decision rule holds for all variable inputs• For example, with 5 inputs we would have the

following

3 51 2 4

1 2 3 4 5

MPP MPPMPP MPP MPP

P P P P P

MPP1 per $ spent on Input 1

= MPP2 per $ spent on Input 2

MPP2 per $ spent on Input 2

= …… =MPP5 per $ spent on Input 5

MPP5 per $ spent on Input 5=

Least Cost Input Choice for 100 Units of OutputLeast 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)

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 = 130

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 = 1313

What Happens if the Price of an Input

Changes?

What Happens if Wage Rate Declines?What Happens if Wage Rate Declines?

Assume initial wage rate and cost of capital result in iso-cost line AB

Wage rate ↓ means the firm can now afford B* instead of B amount of labor if all costs allocated to labor

The new point of tangencyoccurs 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 ofproducing 100 units of output?

Least Cost Combination of Inputs and Outputfor a Specific Budget

What Inputs to Use for a Specific Budget?What Inputs to Use for a Specific Budget?

Capital

An iso-cost line fora specific budget

A set of isoquants for different output levels

Firm can afford to produce 75 units of output using C3 units of capital and L3 units of labor

The firm’s budget not large enough to produce more than 75 units

On any point on this isoquant thefirm is not spending available budget here

Economics ofBusiness Expansion

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 variableEventually over a long-enough time

period all costs are variable

Long-Run Input Use

Fixed costs in short run ensure the U-Shaped SAC curves 3 different size firmsA is the smallest, C the

largest

A B CA*A firm wanting to minimize costOperate at size A if production

is in 0A rangeOperate at size B if production

is in AB range

Cost/unit

Output

The Planning Curve

The long run average cost (LAC) curvePoints of tangency with a series of short run average

total cost (SAC) curvesTangency not usually at minimum of each SAC curve

Output

Tangency Points

LAC sometimes referred to as Long Run Planning Curve

Economies of SizeTypical 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

Economies of SizeConstant returns to size

↑(↓) in output is proportional to the ↑(↓) in input use

i.e., double input use → doubling outputDecreasing 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 11449

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

Economies of SizeReasons 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 inputsPage 11651

Economies of SizeDecreasing returns of size

LRC is ↑ → the LRC is tangent to the collection of SAC curves to the right of their minimum

Output

nit SACD

Economies of SizeThe minimum point on the LRC is the

only point that is tangent to the minimum of a particular SAC

Output

C* is minimum point on SAC* and on LRC

Only plant size and quantity output where this occurs

The Planning Curve

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) SACi = SMCi when SACi is at its minimum

SAC4SMC4

The Planning Curve

Output

SAC4SMC4

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?

The Planning Curve

Output

Firm 1 would lose money with output price = P Produce where P = SMC1 → Q* At Q*, P < SAC1

The Planning Curve

Output

SAC4SMC4

SMC3SMC2

Firms of sizes 2, 3 and 4 would make a positive profit when output price is PP > SAC at profit maximizing levelP-SAC = per unit profit

Q2* Q3* Q4*

Per unit profit

The Planning Curve

Output

SAC4SMC4

SMC3SMC2

Firm 2’s total profit Per unit profit x Q2*

Q2* Q3* Q4*

Firm 2’s Total Profit

The Planning Curve

Output

SAC4SMC4

SMC3SMC2

Q2* Q3* Q4*

The Planning Curve

Output

SAC4SMC4

SMC3SMC2

Q2* Q3* Q4*

The Planning Curve

Output

SAC4SMC4

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

Optimal inputcombinationfor output=10

How to Expand Firm’s CapacityHow to Expand Firm’s Capacity

Two options: 1. Point B ?

How Can the Firm Expand Its Capacity?

Two options: 1. Point B?2. Point C?

Optimal inputcombinationfor output=10with budget DE

Optimal input combination for output = 20 with budget represented by FG

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

Producing More than One Output

Most agricultural operations produce more than one type of outputFor 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 inputsEach output is competing for the use of

limited inputs (e.g. labor, tractor time, etc)67

Lets first address the production decision from a technical perspective Similar to our examination of production

of a single output via the isoquant

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

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)

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

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

Points A → J are on the PPF Note axis labels What happens when firm changes

output mix from B to E?

Level of outputunattainable withwith firm’s existingresources

Inefficient use of firm’s existing resources

PPF represents maximum attainable products given fixed amount of inputs

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?

Canned FruitMRPT

Canned Veg.

Using slope definitionMRPT = ∆Y2 ÷ ∆Y1

Slope between D and E is –1.30 = – 13 10

Using slope definitionMRPT = ∆Y2 ÷ ∆Y1

Slope between D and E is –1.30 = – 13 10

↑ from30 to 40

↓ from 108 to 95

95,000- 108,000 -13,000

40,000- 30,000 10,000

÷ - 1.30 =

Accounting forProduct Prices

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 modelEnables us to discuss the concept of

economic efficiency in the context of multiple outputs

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

Plotting the Iso-Revenue Line

Cases of CF

Cases ofCV

PCF = $33.33/case → 30,000 cases of CF generates revenue of $1 million

30,000

40,000

Assume PCF=$33.33/case, PCV=$25.00/case

PCV = $25.00/case → 40,000 cases of CV generates revenue of $1 million

$1 Mil Iso-Revenue Line$1 Mil Iso-Revenue Line82

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 General equation for the

R* iso-revenue line

PSlope

Line AB is original iso-revenue linePCF= $33.33/case, PCV= $25.00/case Combination of outputs that generate the

same amount of revenue

Line AB is original iso-revenue linePCF= $33.33/case, PCV= $25.00/case Combination of outputs that generate the

same amount of revenue

Slope = $25.00 ÷ $33.33 = 0.75

PSlope

Plotting the Iso-Revenue LineIso-revenue line would shift out to EF

If the revenue target doubled orOutput prices decrease by 50%

The line would shift in to CD If revenue targets are halved orOutput prices are doubled

Iso-revenue line would shift out to EFIf the revenue target doubled orOutput prices decrease by 50%

The line would shift in to CD If revenue targets are halved orOutput prices are doubled

Note: Slopedoes not change

PSlope

Iso-revenue line would rotate:Out to line BC if PCF ↓ by 50%In to line BD if PCF doubled

Note: Slope is changing

PSlope

Iso-revenue line would rotate Out to line AD if PCV ↓ by 50% In to line AC if PCV doubled

Note: Slope is changing

PSlope

Determining the Profit Maximizing

Combination ofProducts

Profit MaximizingCombination of Products

In the cost minimization problem where we produce one productThe 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

isoquantLets develop a similar decision rule but

this time withMultiple outputsFixed supply of inputs

20 40 60 80 100 120 140

Canned Veg. (1,000 Cases)

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

20 40 60 80 100 120 140

) Lets place on this PPF the $1 Mil. iso-revenue line, AB

20 40 60 80 100 120 140

)Profit Maximizing

Combination of Products

The further from the origin the iso-revenue line, the greater the level of revenueR*

1<R*2<R*

Why are the iso-revenue lines parallel in this model?

20 40 60 80 100 120 140

)Profit Maximizing

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

20 40 60 80 100 120 140

)Profit Maximizing

Shifting line AB out in a parallel fashion holds both prices constant

PAt M MRPT =

Slope of an PPF curve

Slope of theIso-cost line

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

MRPTequals-0.75

125,000 cases of

Price ratio = -($25.00 ÷ $33.33) = - 0.75Price ratio = -($25.00 ÷ $33.33) = - 0.75

18,000 cases of

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 × $33.33 + 18,000 × $25.00 = $4,166,250 + $450,000 = $4,616,250

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

Effects of a Changein the Price of One Product

PCF reduced by 50% Firm must sell twice as many

cases of CF to earn a particular level of revenue

20 40 60 80 100 120 140

This gives us a new iso-revenue curve… line CB

20 40 60 80 100 120 140

MTo 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

20 40 60 80 100 120 140

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

Summary #1Concepts of iso-cost line and isoquantsMarginal rate of technical substitution

(MRTS)Least cost combination of inputs for a specific

output levelEffects of change in input priceLevel of output and combination of inputs for

a specific budgetKey decision rule …seek point where MRTS =

ratio of input prices, or where MPP per dollar spent on inputs are equal

Summary #2Concepts of iso-revenue line and the

production possibilities frontierMarginal rate of product transformation

(MRPT)Concept of profit maximizing

combination of productsEffects of change in product priceKey decision rule – maximize profits

where MRPT -ratio of the product prices

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

economics of input and product substitution chapter 7

isocost line isocost

capital labor

isocost line page

particular isoquant

cost use of inputs

labor capital q

isoquant similar

point slide

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