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Chapter 3 Marginal Analysis for Optimal Decisions McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. 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 3-2 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 3-3 Optimization • Maximization problem - An optimization problem that involves maximizing the objective function • Minimization problem - An optimization problem that involves minimizing the objective function 3-4 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 3-5 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 3-6 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 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 Marginal Benefit & Marginal Cost Change in total benefit TB MB Change in activity A Change in total benefit TC MC Change in activity A 3-9 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 3-10 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 3-11 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 3-12 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 3-13 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 3-14 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 3-15 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 3-16 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 MBC MBZ ... PA PB PC PZ 3-17 Example MBta MBct MBsp MBad = = = Pta Pct Psp Pad Pta = $15,000 Pct = $30,000 Psp = $35,000 Pad = $50,000 30,000 $15,000 2 = = ta – teacher aid ct – certified teacher sp – Specialist ad – Administrator MBta = 30,000 MBct = 70,000 MBsp = 70,000 MBad = 90,000 70,000 = $30,000 2.33 = 70,000 $35,000 2 = = 90,000 $50,000 1.8 The next person hired should be a: Certified Teacher 3-18