Chapter 3

1. 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. TC Total benefit and total cost (dollars) G • 4,000 TB F • D • 3,000 • D’ 2,310 B • 2,000 NB* = $1,225 C • 1,085 • 1,000 B’ • C’ A A 200 200 350 = A* 350 = A* 700 1,000 1,000 0 0 600 600 Level of activity Panel A – Total benefit and total cost curves Level of activity Net benefit (dollars) M • 1,225 • 1,000 c’’ • 600 d’’ f’’ • NB Panel B – Net benefit curve Optimal Level of Activity (Figure 3.1)

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. TC Total benefit and total cost (dollars) G • 4,000 TB 100 F • • 320 D 3,000 • 100 D’ 820 • 520 B 100 100 2,000 • 640 • C B’ 520 1,000 C’ • 100 340 A 100 200 350 = A* 800 1,000 0 600 Level of activity Panel A – Measuring slopes along TB and TC Marginal benefit and marginal cost (dollars) MC (= slope of TC) • 8 d’ (600, $8.20) c (200, $6.40) • 6 b • 5.20 4 d (600, $3.20) • • c’ (200, $3.40) 2 MB (= slope of TB) A g • 350 = A* 200 1,000 0 600 800 Level of activity Panel B – Marginals give slopes of totals Relating Marginals to Totals (Figure 3.2)

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. Net benefit (dollars) MB = MC M 100 • • 100 300 c’’ • 500 d’’ A 800 0 200 600 1,000 350 = A* NB Level of activity Using Marginal Analysis to Find A* (Figure 3.3) MB > MC MB < MC

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