MicroEconomics chap003

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