1. Chapter 3: Marginal Analysis for Optimal Decision
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

2. 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) 15 10 10 10 5 20 10 15
West
East x A B C D E F G H
3.0
3.5
2.0
2.5
4.5
2.0
4.5
Distance between Towns (Miles)

3. 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

4. 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).

5. Illustrating the power of marginal analysis
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.
Now let’s move in one direction or another (We will move East, but you could move West).
Let’s move from location X to town C. The key question: what is the change in TTM as the result of the move?
Notice that the move reduces travel by one mile for everyone living in town C or further east.
Notice also that the move increases travel by one mile for everyone living at or to the west of point X..

6. 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.

7. 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.

8. 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

9. Optimization Maximization problem Minimization problem
An optimization problem that involves maximizing the objective function
Minimization problem
An optimization problem that involves minimizing the objective function

10. Optimization Unconstrained optimization Constrained 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

11. 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

12. Choice Variables Continuous variables Discrete variables
Can choose from uninterrupted span of variables
Discrete variables
Must choose from a span of variables that is interrupted by gaps

13. 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

14. Optimal Level of Activity (Figure 3.1)
1,000
Level of activity
2,000
4,000
3,000 A
600
200
Total benefit and total cost (dollars)
Panel A – Total benefit and total cost curves TB TC • G
700 • F • D’ D
2,310
1,085
NB* = $1,225 • B B’ • C’ C
350 = A* A
1,000
600
200
Level of activity
Net benefit (dollars)
Panel B – Net benefit curve • M
1,225 •
c’’
1,000 NB •
d’’
600 •
f’’

15. 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

16. Marginal Benefit & Marginal Cost

17. 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

18. Relating Marginals to Totals (Figure 3.2)
Level of activity
800
1,000
2,000
4,000
3,000 A
600
200
Total benefit and total cost (dollars)
Panel A – Measuring slopes along TB and TC
Marginal benefit and marginal cost (dollars)
Panel B – Marginals give slopes of totals 2 4 6 8 TB TC • G g
100
320
820 •
d’ (600, $8.20)
d (600, $3.20) • F • D’ D
350 = A*
100
520 • B B’ b
100
640
340 •
c’ (200, $3.40)
c (200, $6.40) • C’ C
MC (= slope of TC)
MB (= slope of TB)
5.20

19. 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

20. 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

21. Using Marginal Analysis to Find A* (Figure 3.3)
1,000
600
200
Level of activity
Net benefit (dollars)
800
350 = A*
MB = MC
MB > MC
MB < MC
100
300 • M NB •
c’’
100
500 •
d’’

22. 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

23. 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

24. Irrelevance of Sunk, Fixed, and Average Costs
These costs do not affect marginal cost & are irrelevant for optimal decisions

25. 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

26. 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