1. Marginal Analysis for Optimal Decision Making
Chapter 3
Marginal Analysis for Optimal Decision Making

2. 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. Choice Variables
Choice variables determine the value of the objective function
Continuous variables
Can choose from uninterrupted span of variables
Discrete variables
Must choose from a span of variables that is interrupted by gaps

4. 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

5. 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’’

6. 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

7. Marginal Benefit & Marginal Cost

8. 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

9. 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

10. 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
Optimal level of activity
When no further increases in net benefit are possible
Occurs when MB = MC

11. 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’’

12. 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

13. Irrelevance of Sunk, Fixed, & 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
These costs do not affect marginal cost & are irrelevant for optimal decisions

14. 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

15. 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