Bài giảng Managerial Economics - Chapter 03: Marginal Analysis for Optimal Decision

Chapter 3: Marginal Analysis for Optimal DecisionMcGraw-Hill/IrwinCopyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.OptimizationAn optimization problem involves the specification of three things:Objective function to be maximized or minimizedActivities or choice variables that determine the value of the objective functionAny constraints that may restrict the values of the choice variablesOptimizationMaximization problemAn optimization problem that involves maximizing the objective functionMinimization problemAn optimization problem that involves minimizing the objective functionOptimizationUnconstrained optimizationAn optimization problem in which the decision maker can choose the level of activity from an unrestricted set of valuesEx., profit maximization in the long-runConstrained optimizationAn optimization problem in which the decision maker chooses values for the choice variables from a restricted set of valuesEx., optimal combination of capital and labor given a cost constraintChoice VariablesChoice variables determine the value of the objective functionContinuous variablesDiscrete variablesChoice VariablesContinuous variablesCan choose from uninterrupted span of variablesDiscrete variablesMust choose from a span of variables that is interrupted by gapsNet BenefitNet Benefit (NB)Difference between total benefit (TB) and total cost (TC) for the activity NB = TB – TCOptimal level of the activity (A*) is the level that maximizes net benefitMarginal Benefit & Marginal CostMarginal benefit (MB)Change in total benefit (TB) caused by an incremental change in the level of the activityMarginal cost (MC)Change in total cost (TC) caused by an incremental change in the level of the activityOptimal Level of Activity (Figure 3.1)NBTBTC1,000Level of activity2,0004,0003,000A01,000600200Total benefit and total cost (dollars)Panel A – Total benefit and total cost curvesA01,000600200Level of activityNet benefit (dollars)Panel B – Net benefit curve•G700•F••D’D••C’C••BB’2,3101,085NB* = $1,225•f’’350 = A*350 = A*•M1,225•c’’1,000•d’’600Relating Marginals to TotalsMarginal variables measure rates of change in corresponding total variablesMarginal benefit & marginal cost are also slopes of total benefit & total cost curves, respectivelyRelating Marginals to Totals (Figure 3.2)MC (= slope of TC)MB (= slope of TB)TBTC•F••D’D••C’CLevel of activity8001,000Level of activity2,0004,0003,000A01,000600200Total benefit and total cost (dollars)Panel A – Measuring slopes along TB and TCA01,000600200Marginal benefit and marginal cost (dollars)Panel B – Marginals give slopes of totals8002468350 = A*100520100520350 = A*••BB’b••G•g100320100820••d’ (600, $8.20)d (600, $3.20)100640100340••c’ (200, $3.40)c (200, $6.40)5.20Using Marginal Analysis to Find Optimal Activity LevelsIf marginal benefit > marginal costActivity should be increased to reach highest net benefitIf marginal cost > marginal benefitActivity should be decreased to reach highest net benefitOptimal level of activityWhen no further increases in net benefit are possibleOccurs when MB = MCUsing Marginal Analysis to Find A* (Figure 3.3)NBA01,000600200Level of activityNet benefit (dollars)800•c’’•d’’100300100500350 = A*MB = MCMB > MCMB MCDecrease activity if MB < MCOptimal level of activityLast level for which MB exceeds MCIrrelevance of Sunk, Fixed, and Average CostsSunk costsPreviously paid & cannot be recoveredFixed costsConstant & must be paid no matter the level of activityAverage (or unit) costsComputed by dividing total cost by the number of units of the activityIrrelevance of Sunk, Fixed, and Average CostsThese costs do not affect marginal cost & are irrelevant for optimal decisionsStudent WorkbookStudent WorkbookStudent WorkbookSuppose there were fixed costs of $10 that do not change with the level of activity, will this affect your previous answers?ATBTCMBMCNB0010-1011012102-221915934325196464302556553432472Student WorkbookConstrained OptimizationTypical constrained maximization problemMultiple beneficial activitiesConstraint 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 benefitConstrained OptimizationThe ratio MB/P represents the additional benefit per additional dollar spent on the activityRatios of marginal benefits to prices of various activities are used to allocate a fixed number of dollars among activitiesConstrained OptimizationExpenditure on resources is optimally allocated when the last dollar spend on each activity provides identical marginal benefitsProof is in Appendix to Chapter 3Maximize SalesSteps in Decision ProcessCalculate marginal benefit of additional resources for all activities.Calculate marginal benefit from last dollar spent on resources devoted to each activity.Allocate each additional dollar of the resource to the activity that provides the greatest additional benefitCh. 3, Tech Problem 12Total benefits of two activities. The price of X is $2 per unitThe Price of Y is $10 per unit.Level of activityTB of XTB of Y000130100254190372270484340592400698450BC of $24, Maximize Total BenefitsCh. 3, Tech Problem 12The price of X is $2 per unitThe Price of Y is $10 per unit.Level of activityTB of XMB of XMBx/PxTB of YMB of YMBy/Py00013030151001001025424121909093721892708084841263407075928440060669863450505Original budget restraint $26Alternative budget constraint of $58Homework, Prob. 1Find aa is MB (2.5)Find bb is MC