1. Chapter 3 Marginal Analysis for Optimal Decisions

2. Learning Objectives
Define several key concepts and terminology related to marginal analysis
Use marginal analysis to find optimal activity levels in unconstrained maximization problems and explain why sunk costs, fixed costs, and average costs are irrelevant for decision making
Employ marginal analysis to find the optimal levels of two or more activities in constrained maximization and minimization problems

3. 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

4. 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

5. 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

6. Choice Variables
Activities or choice variables determine the value of the objective function
Discrete choice variables
Can only take specific integer values
Continuous choice variables
Can take any value between two end points

7. Marginal Analysis
Analytical techniques for solving optimization problems that involves changing values of choice variables by small amounts to see if the objective function can be further improved

8. 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

9. 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’’

10. 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

11. Marginal Benefit & Marginal Cost

12. Relating Marginals to Totals
Marginal variables measure rates of change in corresponding total variables
Marginal benefit (marginal cost) of a unit of activity can be measured by the slope of the line tangent to the total benefit (total cost) curve at that point of activity

13. 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

14. 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

15. 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

16. 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’’

17. 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

18. 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 activity

19. Irrelevance of Sunk, Fixed, and Average Costs
Decision makers wishing to maximize the net benefit of an activity should ignore these costs, because none of these costs affect the marginal cost of the activity and so are irrelevant for optimal decisions

20. 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

21. 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

22. Summary
Marginal analysis is an analytical technique for solving optimization problems by changing the value of a choice variable by a small amount to see if the objective function can be further improved
The optimal level of the activity (A*) is that which maximizes net benefit, and occurs where marginal benefit equals marginal cost (MB = MC)
Sunk costs have previously been paid and cannot be recovered; Fixed costs are constant and must be paid no matter the level of activity; Average (or unit) cost is the cost per unit of activity; these 3 types of costs are irrelevant for optimal decision making
The ratio MB/P denotes the additional benefit of that activity per additional dollar spent (“bang per buck”)