Chapter 3: Marginal Analysis for Optimal Decision McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

Locating a shopping mall in a coastal area Villages are located East to West along the coast (Ocean to the North) Problem for the developer is to locate the mall at a place which minimizes total travel miles (TTM). Number of Customers per Week (Thousands) 15 10 10 10 5 20 10 15 West East x A B C D E F G H 3.0 3.5 2.0 2.5 4.5 2.0 4.5 Distance between Towns (Miles)

Minimizing TTM by enumeration The developer selects one site at a time, computes the TTM, and selects the site with the lowest TTM. The TTM is found by multiplying the distance to the mall by the number of trips for each town (beginning with town A and ending with town H). For example, the TTM for site X (a mile west of town C) is calculated as follow: (5.5)(15) + (2.5)(10) + (1.0)(10) + (3.0)(10) + (5.5)(5) + (10.0)(20) + (12.0)(10) + (16.5)(15) = 742.5

Marginal analysis is more effective Enumeration takes lots of computation. We can find the optimal location for the mall easier using marginal analysis—that is, by assessing whether small changes at the margin will improve the objective (reduce TTM, in other words).

Illustrating the power of marginal analysis Let’s arbitrarily select a location—say, point X. We know that TTM at point X is equal to 742.5—but we don’t need to compute TTM first. Now let’s move in one direction or another (We will move East, but you could move West). Let’s move from location X to town C. The key question: what is the change in TTM as the result of the move? Notice that the move reduces travel by one mile for everyone living in town C or further east. Notice also that the move increases travel by one mile for everyone living at or to the west of point X..

Computing the change in TTM To compute the change in total travel miles (TTM) by moving from point X to C: TTM = (-1)(70) + (1)(25) = - 45 Reduction in TTM for those residing in and to the East of town C Increase in TTM for those residing at or to the west of point X. Conclusion: The move to town C unambiguously decreases TTM—so keep moving East so long as TTM is decreasing.

Rule of Thumb Make a “small” move to a nearby alternative if, and only if, the move will improve one’s objective (minimization of TTM, in this case). Keep moving, always in the direction of an improved objective, and stop when no further move will help. Check to see if moving from town C to town D will improve the objective. Check to see if moving from town E to town F will improve the objective.

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 Choice variables determine the value of the objective function Continuous variables: Can assume an infinite number of values within a given range—usually the result of measurement. Discrete variables

Choice Variables Continuous variables Discrete variables Can choose from uninterrupted span of variables Discrete variables Must choose from a span of variables that is interrupted by gaps

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 are also slopes of total benefit & total cost curves, respectively

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 the activity

Irrelevance of Sunk, Fixed, and Average Costs These costs do not affect marginal cost & 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