Kinh tế học - Chapter 3: Marginal analysis for optimal decision making

Chapter 3Marginal Analysis for Optimal Decision MakingOptimizationAn 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 variables2Choice VariablesChoice variables determine the value of the objective functionContinuous variablesCan choose from uninterrupted span of variablesDiscrete variablesMust choose from a span of variables that is interrupted by gaps3Net BenefitNet Benefit (NB)Difference between total benefit (TB) and total cost (TC) for the activityNB = TB – TCOptimal level of the activity (A*) is the level that maximizes net benefit4NBTBTCOptimal Level of Activity (Figure 3.1)1,000Level 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’’6005Marginal 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 activity6Marginal Benefit & Marginal Cost7Relating Marginals to TotalsMarginal variables measure rates of change in corresponding total variablesMarginal benefit & marginal cost are also slopes of total benefit & total cost curves, respectively8MC (= slope of TC)MB (= slope of TB)TBTCRelating Marginals to Totals (Figure 3.2)•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)5.209Using 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 = MC10NBUsing Marginal Analysis to Find A* (Figure 3.3)A01,000600200Level 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 MC12Irrelevance of Sunk, Fixed, & 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 activityThese costs do not affect marginal cost & are irrelevant for optimal decisions13Constrained 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 activities14Constrained OptimizationTo maximize or minimize an objective function subject to a constraintRatios of the marginal benefit to price must be equal for all activitiesConstraint must be met15