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 Ex., profit maximization in the long-run • Constrained optimization • An optimization problem in which the decision maker chooses values for the choice variables from a restricted set of values Ex., optimal combination of capital and labor given a cost constraint 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 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 3-8 NB( x ) B( x ) C ( x ) Maximize NB dNB( x ) dB( x ) dC ( x ) 0 dx dx dx dB( x ) dC ( x ) dx dx dB( x ) marginal benefit dx dC ( x ) marginal cost dx At Max : MB MC 3-9 Optimal Level of Activity (Figure 3.1) Total benefit and total cost (dollars) TC 4,000 • F D • •D’ 3,000 B • 2,310 G • TB 2,000 NB* = $1,225 C • 1,085 1,000 • B’ •C’ 0 200 A 350 = A* 600 700 1,000 Level of activity Net benefit (dollars) Panel A – Total benefit and total cost curves M 1,225 1,000 •c’’ • • 600 0 Panel B – Net benefit curve d’’ 200 350 = A* 600 f’’ A • Level of activity 1,000 NB 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) Total benefit and total cost (dollars) TC 4,000 100 320 3,000 100 •B 520 100 •C • B’ 1,000 C’ • • F • TB 820 100 2,000 640 •D D’• G 520 100 340 A 100 0 200 350 = A* 600 800 1,000 Level of activity Panel A – Measuring slopes along TB and TC Marginal benefit and marginal cost (dollars) MC (= slope of TC) 8 c (200, $6.40) 6 5.20 4 • •d’ (600, $8.20) b • •c’ (200, $3.40) d (600, $3.20) • 2 MB (= slope of TB) 0 • 1,000 g 200 350 = A* Panel B – Marginals give slopes of totals 600 Level of activity 800 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 • Optimal level of activity • When no further increases in net benefit are possible • Occurs when MB = MC 3-13 Using Marginal Analysis to Find A* (Figure 3.3) Net benefit (dollars) MB = MC MB > MC 100 300 • c’’ MB < MC M • 100 • d’’ 500 A 0 200 350 = A* 600 800 1,000 NB Level of activity 3-14 Optimization with Discrete Choice 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 do not affect marginal cost & are irrelevant for optimal decisions 3-18 Student Workbook 3-19 Student Workbook 3-20 Student Workbook • Suppose there were fixed costs of $10 that do not change with the level of activity, will this affect your previous answers? A TB 0 1 2 3 4 5 0 10 19 25 30 34 TC MB MC NB 10 -10 12 10 2 -2 15 9 3 4 19 6 4 6 25 5 6 5 32 4 7 2 3-21 Student Workbook 3-22 Constrained Optimization • Typical constrained maximization problem • Multiple beneficial activities • Constraint on the total resources available. The resources must be allocated efficiently among the activities. • Scarce resources must be allocated among various activities so as to maximize total benefit 3-23 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 3-24 Constrained Optimization • Expenditure on resources is optimally allocated when the last dollar spend on each activity provides identical marginal benefits MBA MBB MBZ ... PA PB PZ Proof is in Appendix to Chapter 3 3-25 Maximize Sales 3-26 Steps in Decision Process 1. Calculate marginal benefit of additional resources for all activities. 2. Calculate marginal benefit from last dollar spent on resources devoted to each activity. 3. Allocate each additional dollar of the resource to the activity that provides the greatest additional benefit 3-27 Ch. 3, Tech Problem 12 Total benefits of two activities. The price of X is $2 per unit The Price of Y is $10 per unit. Level of activity 0 1 2 3 4 5 6 TB of X 0 30 54 72 84 92 98 TB of Y 0 100 190 270 340 400 450 BC of $24, Maximize Total Benefits 3-28 Ch. 3, Tech Problem 12 The price of X is $2 per unit The Price of Y is $10 per unit. Level of TB of X MB of MBx/Px TB of Y MB of Y MBy/Py activity X 0 0 0 1 30 30 15 100 100 10 2 54 24 12 190 90 9 3 72 18 9 270 80 8 4 84 12 6 340 70 7 5 92 8 4 400 60 6 6 98 6 3 450 50 5 Original budget restraint $26 Alternative budget constraint of $58 3-29 Homework, Prob. 1 TB 1. MB A TC 2. MC A 3. NB TB TC NB TB TC 4. A A A NB 5. MB MC A 3-30 • Find a a is MB (2.5) • Find b b is MC NB MB MC A 4.5 2.5 MC MC 7 3-31