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)
Variable Input
Fixed Inputs
Lets extend this to a two input production
function
 Output = f(labor, capital | land, energy, etc)
Variable Inputs
5
Fixed Inputs
Use of Multiple Inputs
Output (i.e. Corn Yield)
250
Nitrogen Fert.
6
Phos. Fert.
Use of Multiple Inputs
If we take a slice at a level
250
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
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 106-107
Slope of an Isoquant
The slope of an isoquant is
Capital
Q=Q*
Slope of an isoquant =
Slope of the line
tangent at a point
K*
A
L*
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
Labor move along an isoquant
Pages 106-107
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
12
Page 107
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 106-107
14
Slope of an Isoquant
Lets derive the slope of the isoquant
like we did for the indifference curve
under consumer theory
 Q  f  L,K   ΔQ  ΔQ ΔL + ΔQ ΔK
ΔL
ΔK
∆Q = 0 along an isoquant →
ΔQ
ΔQ
0
ΔL +
ΔK 
ΔL
ΔK
ΔQ
ΔQ
ΔK
ΔQ ΔQ
ΔK  
ΔL 

ΔL ΔK
ΔK
ΔL
ΔL
15
Pages 106-107
Slope of an Isoquant

MPPL
ΔK
ΔQ ΔQ

 MRTS KL  
ΔL ΔK
ΔL
MPPK
Capital
Q=Q*
MRTSKL = –MPPL*/MPPK*
K*
L*
16
Labor
Pages 106-107
Slope of an Isoquant
Capital
Q = Q*
K*
What is the impact on the
MRTS as input combination
changes from A to B? Why?
A
B
K**
L*
17
L**
Labor
Pages 106-107
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
19
Pages 106-107
Plotting the Iso-Cost Line
Capital
10
Firm can afford 10 hours of
capital at a rental rate of
$100/hr with a budget of $1,000
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
20
100
Labor
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
Halving of Cost
Labor
50
22
500 ÷ PL
100
200
2000 ÷ PL
Page 109
Plotting the Iso-Cost Line
Capital
$1,000 Iso-Cost Line
Iso-Cost Slope = – PL ÷ PK
10
PL = $10
PL = $20
23
50
100
PL = $5
200
Labor
Page 109
Plotting the Iso-Cost Line
Capital
20
$1,000 Iso-Cost Line
Iso-Cost Slope = – PL ÷ PK
PK = $50
10
PK = $100
5
PK = $200
50
24
100
200
Labor
Page 109
Least Cost Combination
of Inputs
25
Least Cost Input Combination
TVC are predefined Iso-Cost Lines
Capital
TVC*** > TVC** > TVC*
Q*
TVC***
A
TVC**
TVC*
B
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*
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
Isoquant
Slope
=
– (PL ÷ PK)
Iso-cost
Line Slope
We can rearrange this equality to the
following
28
Page 111
Least Cost Decision Rule
MPPL MPPK

PL
Pk
MPP per dollar
spent on labor
29
=
MPP per dollar
spent on capital
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 MPP2 MPP3 MPP4 MPP5




P1
P2
P3
P4
P5
MPP1 per $
spent on Input 1
30
=
MPP2 per $
spent on Input 2
=…… =
MPP5 per $
spent =
on Input 5
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
13
→ A* represents 13 hrs of
capital, $1,300  $100 = 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
35
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
Page 112
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?
37
Page 112
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
39
Page 113
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
42
Page 113
What Inputs to Use for a Specific Budget?
On any point on this isoquant the
firm is not spending available
budget here
43
Page 113
Economics of
Business Expansion
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
SACA
SACB
Fixed costs in short run
ensure the U-Shaped SAC
curves
SACC
A
A*
 3 different size firms
 A is the smallest, C the
largest
B
C
A firm wanting to minimize cost
Output
 Operate at size A if production
is in 0A range
 Operate at size B if production
is in AB range
46
Page 114
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
Cost/unit
SACA
SACB
47
SACC
LAC sometimes referred
to as Long Run Planning
Curve
LAC
Output
Page 114
Economies of Size
Cost/unit
Typical LAC curve
Output
What causes the LAC curve to decline,
48
become relatively flat and then increase?
Due to what economists refer to as
Page 114
economies of size
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
49
 ↑ (↓)in output is more than proportional to the
↑(↓) in input use
 i.e., double input use → more than double
output
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
Cost/unit
SACA
SACB
SACC
SACD
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
Cost/unit

SAC*
SAC* and on LRC
Only plant size and quantity
output where this occurs
LRC
C*
Q*
53
Output
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
Cost/unit
SMC1
SAC1
SMC4
SMC2
SAC2
SMC3
SAC3
Output
54
SAC4
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
Q*
56
Output
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
SAC2
SMC2
SAC4
SMC3
P
SAC3
Output
Q2*
57
Q3 *
Q4*
Page 117
The Planning Curve
Firm 2’s total profit
 Per unit profit x Q2*
SMC4
SAC2
P
SMC2
SAC4
SMC3
Firm 2’s Total Profit
SAC3
Output
Q2*
58
Q3 *
Q4*
Page 117
The Planning Curve
Firm 3’s total profit
 Per unit profit x Q3*
SMC4
SAC2
SMC2
SAC4
SMC3
P
Firm 3’s Total Profit
SAC3
Output
Q2*
59
Q3 *
Q4*
Page 117
The Planning Curve
Firm 4’s total profit
 Per unit profit x Q4*
SMC4
SAC2
P
SMC2
SAC4
SMC3
Firm 4’s Total Profit
SAC3
Output
Q2*
60
Q3 *
Q4*
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
67
Each output is competing for the use of
limited inputs (e.g. labor, tractor time, etc)
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
74
Page 120
K*
Inefficient use of
firm’s existing
resources
Level of output
unattainable with
with firm’s existing
resources
PPF represents maximum
attainable products given fixed
amount of inputs
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:
Canned Fruit
MRPT 
Canned Veg.
Y2
Y2
In general we have: MRPT 
Y1
What sign will the MRPT possess?
76
PPF
Y1
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
77
Page 120
95,000
- 108,000
-13,000
78
÷
40,000
- 30,000
10,000
=
- 1.30
Page 148
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
$1 Mil Iso-Revenue Line
82
40,000
Cases of
CV
Page 122
Plotting the Iso-Revenue Line
What is the equation that can be used
to identify the R* iso-revenue line?
Y2
R*
PY2
Slope  
83
 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
PY1
PY2
*
R
PY1
Y1
General equation for the
R* iso-revenue line
Page 122
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
PCV
Slope  
PCF
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
PCV
Slope  
PCF
85
Page 122
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
PCV
Slope  
PCF
86
Page 122
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
PCV
Slope  
PCF
87
Page 122
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
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
Canned Fruit (1,000 Cases)
140
120
100
 Determines location of the PPF
 → All costs are fixed
 → Maximizing revenue will
maximize profit
80
60
40
20
20
90
40
60
80
100
Canned Veg. (1,000 Cases)
120
140
Page 124
Profit Maximizing
Combination of Products
Canned Fruit (1,000 Cases)
140
 Lets place on this PPF the $1 Mil.
120
iso-revenue line, AB
100
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
The further from the origin
Canned Fruit (1,000 Cases)
140
the iso-revenue line, the
greater the level of revenue
120
100
 R*1<R*2<R*3
 Why are the iso-revenue
lines parallel in this model?
80
60
R*2
40
20
R*
20
92
R*3
1
40
60
80
100
Canned Veg. (1,000 Cases)
120
140
Page 124
Profit Maximizing
Combination of Products
To find the maximum revenue
Canned Fruit (1,000 Cases)
140
attainable given available inputs
120
100
80
60
R*2
40
20
R*
20
93
 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
R*3
1
40
60
80
100
Canned Veg. (1,000 Cases)
120
140
Page 124
Profit Maximizing
Combination of Products
140
Shifting line AB out in a parallel
fashion holds both prices constant
Canned Fruit (1,000 Cases)
M
120
PCV
At M  MRPT = 
PCF
100
80
YCF
PCV

=
YCV
PCF
60
40
Slope of an
PPF curve
20
20
94
40
60
80
100
Canned Veg. (1,000 Cases)
120
Slope of the
Iso-cost line
140
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 isorevenue 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) = - 0.75
125,000
cases of
fruit
96
18,000
cases of
veg.
MRPT
equals
-0.75
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 × $33.33 + 18,000 × $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
•
98
$266,610 less than $4,616,250 earned at M
Effects of a Change
in the Price of
One Product
99
Profit Maximizing
Combination of Products
PCF reduced by 50%
M
Canned Fruit (1,000 Cases)
140
120
 Firm must sell twice as many
cases of CF to earn a
particular level of revenue
100
80
60
40
C
This gives us a new isorevenue curve… line CB
A
20
B
20
100
40
60
80
100
Canned Veg. (1,000 Cases)
120
140
Page 125
Profit Maximizing
Combination of Products
To determine the effects of this
M
Canned Fruit (1,000 Cases)
140
120
price change on the product mix
 Shift out the new iso-revenue
curve
 Until it is just tangent to the PPF
curve
100
80
60
40
C
A
20
B
20
101
40
60
80
100
Canned Veg. (1,000 Cases)
120
140
Page 125
Profit Maximizing
Combination of Products
Canned Fruit (1,000 Cases)
140
As a result of a ↓ in PCF
M
120
100
 →Firm would shift from M
to N
 To maximize profit → firm
would ↓ production of CF
and ↑ production of CV
N
80
C
60
40
A
20
B
20
102
40
60
80
100
Canned Veg. (1,000 Cases)
120
140
Page 125
Summary #1
Concepts of iso-cost line and isoquants
Marginal rate of technical substitution
103
(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
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