Chapter 3 Marginal Analysis for Optimal Decisions
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
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
Optimization Maximization problem Minimization problem An optimization problem that involves maximizing the objective function Minimization problem An optimization problem that involves minimizing the objective function
Optimization Unconstrained optimization Constrained 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
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
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
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
Optimal Level of Activity (Figure 3.1) 1,000 Level of activity 2,000 4,000 3,000 A 600 200 Total benefit and total cost (dollars) Panel A – Total benefit and total cost curves TB TC • G 700 • F • D’ D 2,310 1,085 NB* = $1,225 • B B’ • C’ C 350 = A* A 1,000 600 200 Level of activity Net benefit (dollars) Panel B – Net benefit curve • M 1,225 • c’’ 1,000 NB • d’’ 600 • f’’
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
Marginal Benefit & Marginal Cost
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
Relating Marginals to Totals (Figure 3.2) Level of activity 800 1,000 2,000 4,000 3,000 A 600 200 Total benefit and total cost (dollars) Panel A – Measuring slopes along TB and TC Marginal benefit and marginal cost (dollars) Panel B – Marginals give slopes of totals 2 4 6 8 TB TC • G g 100 320 820 • d’ (600, $8.20) d (600, $3.20) • F • D’ D 350 = A* 100 520 • B B’ b 100 640 340 • c’ (200, $3.40) c (200, $6.40) • C’ C MC (= slope of TC) MB (= slope of TB) 5.20
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
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
Using Marginal Analysis to Find A* (Figure 3.3) 1,000 600 200 Level of activity Net benefit (dollars) 800 350 = A* MB = MC MB > MC MB < MC 100 300 • M NB • c’’ 100 500 • d’’
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
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
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
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
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
Summary Marginal analysis is an analytical technique for solving optimization problems by changing the value of a choice variable by a small amount to see if the objective function can be further improved The optimal level of the activity (A*) is that which maximizes net benefit, and occurs where marginal benefit equals marginal cost (MB = MC) Sunk costs have previously been paid and cannot be recovered; Fixed costs are constant and must be paid no matter the level of activity; Average (or unit) cost is the cost per unit of activity; these 3 types of costs are irrelevant for optimal decision making The ratio MB/P denotes the additional benefit of that activity per additional dollar spent (“bang per buck”)