Chapter 3: Marginal Analysis for Optimal Decision McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc All rights reserved 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 3-2 Optimization • Maximization problem • An optimization problem that involves maximizing the objective function • Minimization problem • An optimization problem that involves minimizing the objective function 3-3 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 3-4 Choice Variables • Choice variables determine the value of the objective function • Continuous variables • Discrete variables 3-5 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 3-6 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 3-7 Optimal Level of Activity (Figure 3.1) TC Total beneft and total cost (dollars) 4,000 D • •D’ 3,000 B 2,310 2,000 1,085 • C • 1,000 • B’ G • F • NB* = $1,225 •C’ 200 TB A 350 = A* 600 700 1,000 Net beneft (dollars) Level of Panel A – Total benefit and total costactivity curves M 1,22 1,00 600 •c’’ • • d’’ 200 Panel B – Net benefit curve 350 = A* Level of activity 600 A f’’ • 1,000 NB 3-8 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 3-9 Marginal Benefit & Marginal Cost Change in total benefit ∆TB MB = = Change in activity ∆A Change in total benefit ∆TC MC = = Change in activity ∆A 3-10 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 3-11 Relating Marginals to Totals (Figure 3.2) TC Total beneft and total cost (dollars) 4,000 G • 100 F 320 3,000 100 •B 520 100 2,000 640 •C • B’ 1,000 C’ • •D D’• • TB 820 100 520 100 340 A 100 200 350 = A* 600 Marginal beneft and marginal cost (dollars) Level of Panel A – Measuring slopes along TB activity and TC c (200, $6.40) 5.2 • 800 1,000 MC (= slope of TC) •d’ (600, $8.20) b • •c’ (200, $3.40) d (600, $3.20) • MB (= slope of TB) g 200 Panel B – Marginals give slopes of totals 350 = A* Level of activity 600 800 • 1,000 A 3-12 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 3-13 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 3-14 Using Marginal Analysis to Find A* (Figure 3.3) MB = MC Net beneft (dollars) MB > MC 100 300 • c’’ 200 MB < MC M • 100 • d’’ 350 = A* 500 600 A 800 NB 1,00 Level of activity 3-15 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 3-16 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 3-17 Irrelevance of Sunk, Fixed, and Average Costs • These costs not affect marginal cost & are irrelevant for optimal decisions 3-18 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 3-19 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 3-20