Chapter 3: Marginal Analysis for Optimal Decision

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

3-2

Locating a shopping mall in a coastal area

West East15 15

Number of Customers per Week (Thousands)

Distance between Towns (Miles)

AH

B C D E F G

10 1010 10205

x

3.0 3.5 2.5 4.5 4.52.0 2.0

•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).

3-3

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

3-4

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).

3-5

Illustrating the power of marginal analysis

1. 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.

2. Now let’s move in one direction or another (We will move East, but you could move West).

3. Let’s move from location X to town C. The key question: what is the change in TTM as the result of the move?

4. Notice that the move reduces travel by one mile for everyone living in town C or further east.

5. Notice also that the move increases travel by one mile for everyone living at or to the west of point X..

3-6

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.

3-7

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.

3-8

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

3-9

Optimization

• Maximization problem• An optimization problem that involves

maximizing the objective function

• Minimization problem• An optimization problem that involves

minimizing the objective function

3-10

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

3-11

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

3-12

Choice Variables

• Continuous variables• Can choose from uninterrupted span of

variables

• Discrete variables• Must choose from a span of variables that is

interrupted by gaps

3-13

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

3-14

Optimal Level of Activity (Figure 3.1)

NB

TB

TC

1,000

Level of activity

2,000

4,000

3,000

A

0 1,000600200

Tota

l b

en

efit

an

d t

ota

l co

st

(dolla

rs)

Panel A – Total benefit and total cost curves

A

0 1,000600200

Level of activity

Net

ben

efit

(dolla

rs)

Panel B – Net benefit curve

•G

700

•F

••

D’

D

•

•C’

C

•

•

B

B’

2,310

1,085

NB* = $1,225

•f’’

350 = A*

350 = A*

•M

1,225 •

c’’1,000

•d’’600

3-15

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

3-16

Marginal Benefit & Marginal Cost

TBMB

A

Change in total benefit

Change in activity

Change in total benefit

Change in activity

TCMC

A

3-17

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

3-18

Relating Marginals to Totals (Figure 3.2)

MC (= slope of TC)

MB (= slope of TB)

TB

TC

•F

••

D’

D

•

•C’

C

Level of activity

800

1,000

Level of activity

2,000

4,000

3,000

A

0 1,000600200

Tota

l b

en

efit

an

d t

ota

l co

st

(dolla

rs)

Panel A – Measuring slopes along TB and TC

A

0 1,000600200

Marg

inal b

en

efit

an

d

marg

inal co

st (

dolla

rs)

Panel B – Marginals give slopes of totals

800

2

4

6

8

350 = A*

100

520

100

520

350 = A*

•

•

B

B’

b•

•G

•g

100

320

100

820

•

•

d’ (600, $8.20)

d (600, $3.20)

100

640

100

340

•

•c’ (200, $3.40)

c (200, $6.40)

5.20

3-19

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

3-20

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

3-21

Using Marginal Analysis to Find A* (Figure 3.3)

NB

A

0 1,000

600200

Level of activity

Net

benefit

(dolla

rs)

800

•c’’

•d’’

100

300 100

500

350 = A*

MB = MC

MB > MC MB < MC

•M

3-22

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

3-23

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

3-24

Irrelevance of Sunk, Fixed, and Average Costs

• These costs do not affect marginal cost & are irrelevant for optimal decisions

3-25

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

3-26

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

...A B Z

A B Z

MB MB MB

P P P