Chapter 3 Marginal Analysis for Optimal Decision Making Economics Economía M. En C. Eduardo Bustos Farías 3 Objective function • This defines the measure of effectiveness of the

Copyright © 2005 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Managerial Economics

Chapter 3Marginal Analysis for Optimal Decision Making

M. En C. Eduardo Bustos Farías


Economía M. En C. Eduardo Bustos Farías 2

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



Objective function• This defines the measure of effectiveness

of the system as a mathematical function of its decision variables

• The optimum solution to the model has been obtained if the corresponding values of the decision variables yield the best value of the objective function while satisfying all the constraints



Choice Variables• Choice variables determine the

value of the objective function• Continuous variables

• Can choose from uninterrupted span of variables

• Discrete variables• Must choose from a span of variables

that is interrupted by gaps



General ApproachFormulating and solving an LP model requires:• Optimizing (maximizing or minimizing)

a linear function of variables called the “objective function”

• Subject to a set of linear equalities and/or inequalities called “constraints”



Mathematical Expression of an LP Model

Max (or Min) Subject to:

and , j = 1, 2, ... , n

Z c x c x c x

a x a x a x ba x a x a x b

a x a x a x bx

n n

n n

n n

m m mn n m

j

= + + +

+ + + ≤ = ≥+ + + ≤ = ≥

+ + + ≤ = ≥≥ ∀

1 1 2 2

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

0

...

... ( , , )... ( , , )

... ( , , )M M M M

Con

stra

ints

Obj

eciv

eFu

nctio

n



Parameters and VariablesIn the preceding formulation,

• cj,bi, and aij, for (i=1,2,...,m; j=1,2,...,n) are constants which are determined depending on the technology of the problem

• the xj’s are the decision variables• only one of the signs (<, =, >) holds for

each constraint



Canonical Form of an LP Model

Max (or Min)

S. T.

, i = 1,2,... ,m

, j = 1,2,..., n

Z c x

a x b

x

j jj

n

ij jj

n

i

j

=

≤=≥

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

≥

=

=

∑

∑

1

1

0



Matrix Form of LP Model

Max

Z cx

s t Ax bx

=

≤

≥

v v

v v

v v. .

0

vc c c cn=[ , ,..., ]1 2v

Mx

xx

xn

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

2

v

Mb

bb

bm

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

2

A

a a aa a a

a a a

n

n

m m mn

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

11 12 1

21 22 2

1 2

L

L

M M M M

L

v

M0

00

0

=

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥



Observations• cj is the increase or decrease in the

overall measure of effectiveness (Z) that results from each unit increase or decrease in xj

• The number of relevant scarce resources is m, so that each of the first m linear inequalities corresponds to a constraint on the availability of one of these resources



Observations

• bi is the amount of resource i available to the n activities

• aij is the amount of resource i consumed by each unit of activity j

• The left side of the constraint inequalities is the total usage of the respective resource

• The non-negativity constraints rule out the possibility of negative activity levels



Rational people think at the margin.

Marginal changes are small, incremental adjustments to an existing plan of action.

People make decisions by comparing costs and benefits at the margin.



The Economic Way of Thinking

• Choosing at the Margin• People make choices at the margin, which

means that they evaluate the consequences of making incremental changes in the use of their resources.

• The benefit from pursuing an incremental increase in an activity is its marginal benefit.

• The opportunity cost of pursuing an incremental increase in an activity is its marginal cost.



Marginal Benefit• Marginal benefit is the benefit a person

receives from consuming one more unit of a good or service.• We can measure the marginal benefit from a

good or service by the dollar value of other goods and services that a person is willing to give up to get one more unit of it.

• The concept of decreasing marginal benefit implies that as more of a good or service is consumed, its marginal benefit decreases.



• Figure shows the decreasing marginal benefit from each additional slice of pizza, measured in dollars per slice.



Marginal Cost• Marginal cost is the opportunity cost of

producing one more unit of a good or service. The measure of marginal cost is the value of the best alternative forgone to obtain the last unit of the good.• We can measure the marginal cost of a good or

service by the dollar value of other goods and services that a person is must give up to get one more unit of it.

• The concept of increasing marginal cost implies that as more of a good or service is produced, its marginal cost increases.



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


Economía M. En C. Eduardo Bustos Farías 18NB

TB

TC

Optimal Level of Activity

1,000

Level of activity

2,000

4,000

3,000

A

0 1,000600200

Tota

l be

nefit

and

tota

l co

st (d

olla

rs)

Panel A – Total benefit and total cost curves

A

0 1,000600200

Level of activity

Ne

t be

nefit

(do

llars

)

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



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

TBMBA

Δ= =

ΔChange in total benefit

Change in activity

TCMCA

Δ= =

ΔChange in total costChange in activity



Relating Marginals to Totals• Marginal variables measure rates

of change in corresponding totalvariables• Marginal benefit & marginal cost are

also slopes of total benefit & total cost curves, respectively



MC (= slope of TC)

MB (= slope of TB)

TB

TC

Relating Marginals to Totals

•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 be

nefit

and

tota

l co

st (d

olla

rs)

Panel A – Measuring slopes along TB and TC

A

0 1,000600200

Ma

rgin

al b

ene

fit a

nd

ma

rgin

al c

ost

(do

llars

)

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



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

• Optimal level of activity• When no further increases in net benefit

are possible• Occurs when MB = MC



Using Resources Efficiently • It is a general principle that the more we

have of any good or service, the smaller is its marginal benefit and the less we are willing to pay for an additional unit of it.

• We call this general principle the principle of decreasing marginal benefit.

• The marginal benefit curve shows the relationship between the marginal benefit of a good and the quantity of that good consumed.



Using Resources Efficiently • Figure shows a

marginal benefit curve.

• The curve slopes downward to reflect the principle of decreasing marginal benefit.

• At point A, with pizza production at 0.5 million, people are willing to pay 5 CDs per pizza.



Using Resources Efficiently • At point B, with pizza

production at 1.5 million, people are willing to pay 4 CDs per pizza.

• At point E, with pizza production at 4.5 million, people are willing to pay 1 CD per pizza.



• Figure shows the increasing marginal cost of each additional slice of pizza, measured in dollars per slice.



Efficiency

• Efficiency and Inefficiency• If the marginal

benefit from a good exceeds its marginal cost, producing and consuming moreof the good uses resources more efficiently.



Efficiency

• If the marginal cost of a good exceeds its marginal benefit, producing and consuming less of the good uses resources more efficiently.



Efficiency

• If the marginal cost of a good equals its marginal benefit, resources are being use efficiently.



NB

A

0 1,000600200

Level of activity

Ne

t be

nefit

(do

llars

)

Using Marginal Analysis to Find A*

800

•c’’

•d’’

100

300 100

500

350 = A*

MB = MC

MB > MC MB < MC

•M



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, & Average Costs

• Sunk costs• Previously paid & cannot be recovered

• Sunk cost - cost that does not vary across management decision• option on land purchase• Fixed assets



Irrelevance of Sunk, Fixed, & Average Costs

• 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

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



Maximizing Utility

• Equalizing Marginal Utility per Dollar Spent• Using marginal analysis, a consumer’s total

utility is maximized by following the rule:• Spend all available income and equalize the marginal utility per dollar spent on all goods.

• The marginal utility per dollar spent is the marginal utility from a good divided by its price.



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



Example

C > B > A



Decision rule



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= = =



Maximizing Utility• Call the marginal utility of movies MUM• Call the marginal utility of soda MUS• Call the price of movies PM• Call the price of soda PS• The marginal utility per dollar spent on movies

is MUM/PM• The marginal utility per dollar spent on soda is

MUS/PS.• Total utility is maximized when:

MUM/PM = MUS/PS



Maximizing Utility• If MUM/PM > MUS/PS,

then moving a dollar from soda to movies increases the total utility from movies by more than it decreases the total utility from soda, so total utility increases.

• Only when MUM/PM = MUS/PS, is it not possible to reallocate the budget and increase total utility.



Maximizing Utility• Similarly, if MUS/PS

> MUM/PM, then moving a dollar from movies to soda increases the total utility from soda by more than it decreases the total utility from movies, so total utility increases.

• Again, only when MUM/PM = MUS/PS, is it not possible to reallocate the budget and increase total utility.



Value, Price, and Consumer Surplus

• Consumer Surplus• Consumer surplus is the value of a good

minus the price paid for it, summed over the quantity bought.

• It is measured by the area under the demand curve and above the price paid, up to the quantity bought.

• Figure on the next slide shows the consumer surplus for pizza for an individual consumer.




• The price paid is the market price, which is the same for each unit bought.

• The quantity bought is determined by the demand curve and the blue rectangle shows the amount paid for pizza.

• The green triangle shows the consumer surplus from pizza.




• The consumer surplus on the 10th slice is the $2 that the consumer is willing to pay minus the $1.50 that she does pay, which is 50 cents a slice.



Cost, Price, and Producer Surplus

• Producer Surplus• Producer surplus is the price of a good

minus the marginal cost of producing it, summed over the quantity sold.

• Producer surplus is measured by the area below the price and above the supply curve, up to the quantity sold.

• Figure on the next slide shows the producer surplus for pizza for an individual producer.




• The price is the market price, which is the same for each unit sold.

• The quantity sold is determined by the supply curve and the red area shows the total cost of producing pizza.

• The blue triangle shows the producer surplus from pizza.




• The producer surplus on the 50th pizza is the $15 that the producer receives minus the $10 that it cost to produce, which is $5 a pizza.



Is the Competitive Market Efficient?

• Efficiency of Competitive Equilibrium• Figure shows that a

competitive market creates an efficient allocation of resources at equilibrium.

• In equilibrium, the quantity demanded equals the quantity supplied.



Is the Competitive Market Efficient?

• At the equilibrium quantity, marginal benefit equals marginal cost, so the quantity is the efficient quantity.

• The sum of consumer and producer surplus is maximized at this efficient level of output.

Documents

Chapter 3 Marginal Analysis for Optimal Decision Making Economics Economía M. En C. Eduardo Bustos Farías 3 Objective function • This defines the measure of effectiveness of the