Chapter 3: Marginal Analysis

for Optimal Decision

McGraw-Hill/Irwin

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

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

3-2

Optimization

• Maximization problem

• An optimization problem that involves

maximizing the objective function

• Minimization problem

• An optimization problem that involves

minimizing the objective function

3-3

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

3-4

Choice Variables

• Choice variables determine the value of

the objective function

• Continuous variables

• Discrete variables

3-5

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

3-6

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

3-7

Optimal Level of Activity

(Figure 3.1)

TC

Total beneft and total cost

(dollars)

4,000

D

•

•D’

3,000

B

2,310

2,000

1,085

•

C

•

1,000

• B’

G

•

F

•

NB* =

$1,225

•C’

0

200

TB

A

350 = A*

600 700

1,000

Net beneft

(dollars)

Level of

Panel A – Total benefit and total costactivity

curves

M

1,22

1,00

5

0

600

0

•c’’

•

•

d’’

200

Panel B – Net benefit curve

350 = A*

Level of

activity

600

A

f’’

•

1,000

NB

3-8

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

3-9

Marginal Benefit & Marginal Cost

Change in total benefit ∆TB

MB =

=

Change in activity

∆A

Change in total benefit ∆TC

MC =

=

Change in activity

∆A

3-10

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

3-11

Relating Marginals to Totals

(Figure 3.2)

TC

Total beneft and total cost

(dollars)

4,000

G

•

100 F

320

3,000

100

•B

520

100

2,000

640

•C

•

B’

1,000

C’

•

•D

D’•

•

TB

820

100

520

100

340

A

100

0

200

350 = A*

600

Marginal beneft and

marginal cost (dollars)

Level of

Panel A – Measuring slopes along TB activity

and TC

8

c (200, $6.40)

6

5.2

0

4

2

•

800

1,000

MC (= slope of TC)

•d’ (600, $8.20)

b

•

•c’ (200, $3.40)

d (600, $3.20)

•

MB (= slope of TB)

g

0

200

Panel B – Marginals give slopes of

totals

350 = A*

Level of

activity

600

800

•

1,000

A

3-12

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

3-13

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

3-14

Using Marginal Analysis to Find A*

(Figure 3.3)

MB = MC

Net beneft

(dollars)

MB > MC

100

300

0

•

c’’

200

MB < MC

M

•

100

•

d’’

350 = A*

500

600

A

800

NB

1,00

0

Level of activity

3-15

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

3-16

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

3-17

Irrelevance of Sunk, Fixed, and

Average Costs

• These costs do not affect marginal cost & are

irrelevant for optimal decisions

3-18

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

3-19

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

MBA MBB

MBZ

=

= ... =

PA

PB

PZ

3-20

for Optimal Decision

McGraw-Hill/Irwin

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

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

3-2

Optimization

• Maximization problem

• An optimization problem that involves

maximizing the objective function

• Minimization problem

• An optimization problem that involves

minimizing the objective function

3-3

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

3-4

Choice Variables

• Choice variables determine the value of

the objective function

• Continuous variables

• Discrete variables

3-5

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

3-6

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

3-7

Optimal Level of Activity

(Figure 3.1)

TC

Total beneft and total cost

(dollars)

4,000

D

•

•D’

3,000

B

2,310

2,000

1,085

•

C

•

1,000

• B’

G

•

F

•

NB* =

$1,225

•C’

0

200

TB

A

350 = A*

600 700

1,000

Net beneft

(dollars)

Level of

Panel A – Total benefit and total costactivity

curves

M

1,22

1,00

5

0

600

0

•c’’

•

•

d’’

200

Panel B – Net benefit curve

350 = A*

Level of

activity

600

A

f’’

•

1,000

NB

3-8

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

3-9

Marginal Benefit & Marginal Cost

Change in total benefit ∆TB

MB =

=

Change in activity

∆A

Change in total benefit ∆TC

MC =

=

Change in activity

∆A

3-10

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

3-11

Relating Marginals to Totals

(Figure 3.2)

TC

Total beneft and total cost

(dollars)

4,000

G

•

100 F

320

3,000

100

•B

520

100

2,000

640

•C

•

B’

1,000

C’

•

•D

D’•

•

TB

820

100

520

100

340

A

100

0

200

350 = A*

600

Marginal beneft and

marginal cost (dollars)

Level of

Panel A – Measuring slopes along TB activity

and TC

8

c (200, $6.40)

6

5.2

0

4

2

•

800

1,000

MC (= slope of TC)

•d’ (600, $8.20)

b

•

•c’ (200, $3.40)

d (600, $3.20)

•

MB (= slope of TB)

g

0

200

Panel B – Marginals give slopes of

totals

350 = A*

Level of

activity

600

800

•

1,000

A

3-12

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

3-13

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

3-14

Using Marginal Analysis to Find A*

(Figure 3.3)

MB = MC

Net beneft

(dollars)

MB > MC

100

300

0

•

c’’

200

MB < MC

M

•

100

•

d’’

350 = A*

500

600

A

800

NB

1,00

0

Level of activity

3-15

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

3-16

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

3-17

Irrelevance of Sunk, Fixed, and

Average Costs

• These costs do not affect marginal cost & are

irrelevant for optimal decisions

3-18

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

3-19

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

MBA MBB

MBZ

=

= ... =

PA

PB

PZ

3-20