Chapter 3: Marginal Analysis
for Optimal Decision
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
Locating a shopping mall in a
coastal area
•Villages are located East to West
along the coast (Ocean to the North)
•Problem for the developer is to locate
the mall at a place which minimizes
total travel miles (TTM).
Number of Customers per Week (Thousands)
West
15
10
A
B
3.0
10 10
xC
3.5
D
2.0
2.5
5
20
10
E
F
G
4.5
2.0
15
East
H
4.5
Distance between Towns (Miles)
3-2
Minimizing TTM by enumeration
•The developer selects one site at a time, computes the
TTM, and selects the site with the lowest TTM.
•The TTM is found by multiplying the distance to the
mall by the number of trips for each town (beginning
with town A and ending with town H).
•For example, the TTM for site X (a mile west of town
C) is calculated as follow:
(5.5)(15) + (2.5)(10) + (1.0)(10) + (3.0)(10) + (5.5)(5) +
(10.0)(20) + (12.0)(10) + (16.5)(15) = 742.5
3-3
Marginal analysis is more
effective
Enumeration takes lots of
computation. We can find the
optimal location for the mall
easier using marginal analysis—
that is, by assessing whether
small changes at the margin will
improve the objective (reduce
TTM, in other words).
3-4
Illustrating the power of marginal
analysis
1. Let’s arbitrarily select a location—say, point X. We
know that TTM at point X is equal to 742.5—but we
don’t need to compute TTM first.
2. Now let’s move in one direction or another (We will
move East, but you could move West).
3. Let’s move from location X to town C. The key
question: what is the change in TTM as the result of
the move?
4. Notice that the move reduces travel by one mile for
everyone living in town C or further east.
5. Notice also that the move increases travel by one
mile for everyone living at or to the west of point X..
3-5
Computing the change in
TTM
To compute the change in total travel miles (TTM)
by moving from point X to C:
TTM = (-1)(70) + (1)(25) = - 45
Reduction in TTM for
those residing in and
to the East of town C
Increase in TTM for
those residing at or to
the west of point X.
Conclusion: The move to town C unambiguously
decreases TTM—so keep moving East so long as
TTM is decreasing.
3-6
Rule of Thumb
Make a “small” move to a nearby alternative if, and
only if, the move will improve one’s objective
(minimization of TTM, in this case). Keep moving,
always in the direction of an improved objective, and
stop when no further move will help.
•
Check to see if moving from town C to town D
will improve the objective.
•
Check to see if moving from town E to town F
will improve the objective.
3-7
Optimization
• An optimization problem involves the
specification of three things:
• Objective function to be maximized or
minimized
• Activities or choice variables that determine
the value of the objective function
• Any constraints that may restrict the values of
the choice variables
3-8
Optimization
• Maximization problem
• An optimization problem that involves
maximizing the objective function
• Minimization problem
• An optimization problem that involves
minimizing the objective function
3-9
Optimization
• Unconstrained optimization
• An optimization problem in which the decision
maker can choose the level of activity from an
unrestricted set of values
• Constrained optimization
• An optimization problem in which the decision
maker chooses values for the choice
variables from a restricted set of values
3-10
Choice Variables
• Choice variables determine the value of
the objective function
• Continuous variables: Can assume an infinite
number of values within a given range—
usually the result of measurement.
• Discrete variables
3-11
Choice Variables
• Continuous variables
• Can choose from uninterrupted span of
variables
• Discrete variables
• Must choose from a span of variables that is
interrupted by gaps
3-12
Net Benefit
• Net Benefit (NB)
• Difference between total benefit (TB) and total
cost (TC) for the activity
• NB = TB – TC
• Optimal level of the activity (A*) is the level
that maximizes net benefit
3-13
Optimal Level of Activity
(Figure 3.1)
Total benefit and total cost (dollars)
TC
4,000
•
F
D
•
•D’
3,000
B
•
2,310
G
•
TB
2,000
NB* = $1,225
C
•
1,085
1,000
• B’
•C’
0
200
A
350 = A*
600 700
1,000
Level of activity
Net benefit (dollars)
Panel A – Total benefit and total cost curves
M
1,225
1,000
•c’’
•
•
600
0
Panel B – Net benefit curve
d’’
200
350 = A*
600
f’’
A
•
Level of activity
1,000
NB
3-14
Marginal Benefit & Marginal Cost
• Marginal benefit (MB)
• Change in total benefit (TB) caused by an
incremental change in the level of the activity
• Marginal cost (MC)
• Change in total cost (TC) caused by an
incremental change in the level of the activity
3-15
Marginal Benefit & Marginal Cost
Change in total benefit TB
MB 

Change in activity
A
Change in total benefit TC
MC 

Change in activity
A
3-16
Relating Marginals to Totals
• Marginal variables measure rates of
change in corresponding total variables
• Marginal benefit & marginal cost are also
slopes of total benefit & total cost curves,
respectively
3-17
Relating Marginals to Totals
(Figure 3.2)
Total benefit and total cost (dollars)
TC
4,000
100
320
3,000
100
•B
520
100
•C
•
B’
1,000
C’
•
•
F
•
TB
820
100
2,000
640
•D
D’•
G
520
100
340
A
100
0
200
350 = A*
600
800
1,000
Level of activity
Panel A – Measuring slopes along TB and TC
Marginal benefit and
marginal cost (dollars)
MC (= slope of TC)
8
c (200, $6.40)
6
5.20
4
•
•d’ (600, $8.20)
b
•
•c’ (200, $3.40)
d (600, $3.20)
•
2
MB (= slope of TB)
0
•
1,000
g
200
350 = A*
Panel B – Marginals give slopes of totals
600
Level of activity
800
A
3-18
Using Marginal Analysis to Find
Optimal Activity Levels
• If marginal benefit > marginal cost
• Activity should be increased to reach highest net
benefit
• If marginal cost > marginal benefit
• Activity should be decreased to reach highest net
benefit
3-19
Using Marginal Analysis to Find
Optimal Activity Levels
• Optimal level of activity
• When no further increases in net benefit are
possible
• Occurs when MB = MC
3-20
Using Marginal Analysis to Find A*
(Figure 3.3)
Net benefit (dollars)
MB = MC
MB > MC
100
300
•
c’’
MB < MC
M
•
100
•
d’’
500
A
0
200
350 = A*
600
800
1,000
NB
Level of activity
3-21
Unconstrained Maximization with
Discrete Choice Variables
• Increase activity if MB > MC
• Decrease activity if MB < MC
• Optimal level of activity
• Last level for which MB exceeds MC
3-22
Irrelevance of Sunk, Fixed, and
Average Costs
• Sunk costs
• Previously paid & cannot be recovered
• Fixed costs
• Constant & must be paid no matter the level of
activity
• Average (or unit) costs
• Computed by dividing total cost by the number of
units of the activity
3-23
Irrelevance of Sunk, Fixed, and
Average Costs
• These costs do not affect marginal cost & are
irrelevant for optimal decisions
3-24
Constrained Optimization
• The ratio MB/P represents the additional
benefit per additional dollar spent on the
activity
• Ratios of marginal benefits to prices of
various activities are used to allocate a
fixed number of dollars among activities
3-25
Constrained Optimization
• To maximize or minimize an objective
function subject to a constraint
• Ratios of the marginal benefit to price must
be equal for all activities
• Constraint must be met
MBA MBB
MBZ

 ... 
PA
PB
PZ
3-26