Utility Maximization
Lecture 13
Dr. Jennifer P. Wissink
©2017 John M. Abowd and Jennifer P. Wissink, all rights reserved.
March 15, 2017
i>clicker question
James Bond like his martinis with equal parts Clearheart gin and Prairie Organic
vodka. Shaken not stirred. He considers gin and vodka to be perfect complements.
Which indifference curve map below would most likely be James Bond’s?
A
D
B
E
C
i>clicker question
Cathy and Susan consume only X and Y. Suppose Susan’s indifference curve is the
BLUE one and Cathy’s indifference curve is the RED one. Which one of the
following is the best choice?
A.
B.
C.
D.
E.
Susan likes X relatively more than Cathy likes X.
Susan likes Y relatively more that Cathy likes Y.
Susan has more income than Cathy.
Susan has less income than Cathy.
Susan and Cathy are irrational consumers since their indifference curves cross.
From Preferences to Utility

Utility is the way economists
describe and measure
preferences with a number.

Between two bundles...
– the better bundle gets a higher
utility number.
– the worse bundle gets a lower utility
number.
– if you are indifferent between the
two bundles, they get the same
utility number.

UtilityMaryclaire = u(B, C)
– This is a 3-dimensional function!

The indifference curve map
versus the utility function!
The Consumer Theory Problem:
Decisions, Decisions, Decisions!

So… for Maryclaire and Beans and Carrots,
what bundle is best?
– Ok, what bundle will she buy?



Let’s bring together what she is willing to do with what she
is able to do!
The optimal amount of beans and carrots to buy is the
amount that maximizes her utility subject to her budget set.
Maryclaire’s (i.e., the consumer’s) formal problem:
Choose a bundle of (Beans, Carrots) to
maximize UtilityMaryclaire = u(B, C)
subject to $PBB + $PCC ≤ $I
How to Find Maryclaire’s Best Bundle
When I=$40, PC=$2 & PB=$4
C
C
20
10
B
B
How to Find Maryclaire’s Best Bundle
When I=$40, PC=$2 & PB=$4
C
B
Maryclaire’s Best Bundle (B*, C*)
When I=$40, PC=$2 & PB=$4
C

Utility is at a maximum when:
– All income is allocated to the goods you derive
utility from AND…
– There is no way to transfer income from one
good to another and make yourself better off.
–  She is on the highest indifference curve
possible while still on the budget line.
C*
» (If Maryclaire is VERY nicely behaved…)
She is on an indifference curve that is tangent
to her budget line.
» At E* the slope of the indifference curve is
equal to the slope of the budget constraint.
E*
»  Maryclaire’s MRS = ERS at E*
Budget Line
B*
Indifference
Curve
B
Historical Note: What’s a Util?

20th Century Ordinalists

19th Century Cardinalists
Sir
John
Hicks
– Let U=u(B, C) be Maryclaire’s utility function from
consuming beans and carrots measured in utils.
– Then MUB = Maryclaire’s marginal utility of beans.
» It measures the change in utility as we change bean
consumption by an incremental unit while holding carrot
consumption constant.
» MUB = ∆U/ ∆B ceteris paribus
– And MUC = Maryclaire’s marginal utility of carrots.
» MUC = ∆U/ ∆C ceteris paribus
– Now, the “law of diminishing marginal utility”
would imply that, ceteris paribus, as good “i”
increases in the bundle (holding everything else
constant), eventually the MUi decreases.
William
Stanley
Jevons
The Cardinalists and the
“Bang per Buck” Story
 What’s
 So
the “bang per buck”?
what’s true at an optimal bundle?
– (1) Spend/Allocate all your income and
– (2) Equate the “bang per buck” across all
goods you consume.
i>clicker questions
a)
b)
c)
d)
e)
f)
Stan gets utility from only protein bars (B) and sports drinks (S).
His utility is currently 147 utils. He is spending all his income.
He is consuming B=100 bars and S=50 drinks.
Prices are PB=$2 and PS=$10.
At this bundle Stan’s marginal utilities are MUB=10utils & MUS =20utils.
The l.d.m.u. has already set in for Stan.
Stan’s income is
A. some value I can’t
compute.
B. $700
C. $150
D. $220
E. $30
Stan is successfully
solving the consumer
theory problem for himself.
A. Yes.
B. No.
C. Maybe so.
Stan should
If Stan moves $10 from S to
B his utility will increase by
A.
B.
C.
D.
E.
A.
B.
C.
D.
E.
buy more S & less B.
buy more B & less S.
buy more of both.
just buy more of S.
quit tennis.
30 utils.
20 utils.
3 utils.
5 utils.
2 utils.
Reconciling the “Bang per Buck”
Story with the MRS=ERS Story

Recall: The ERS = ($PB / $PC), when Beans are on the horizontal and
Carrots on the vertical.

Recognize (trust me): The MRS = (MUB / MUC), when Beans are on
the horizontal and Carrots on the vertical.

Recall for the ordinalist: At an optimal bundle E*
– (1) You allocate all your money to beans and carrots &
– (2) At E*, the MRS = ERS

Recall for the cardinalist: At an optimal bundle E*
– (1) You allocate all your money to beans and carrots &
– (2) At E*, you have MUB/$PB = MUC/$PC

Get same optimal bundle E* either way!
Now What? Use the Model to
Bake the Cake From Scratch
 What’s
the cake?
– The market demand curve
for beans.
 What’s
the recipe to bake
the cake?
– Use the BL/IC diagram to
get Maryclaire’s demand
curve for beans.
How to Find Maryclaire’s Demand for Beans When
I=$40, PC=$2 & PB Varies from $4 to $2 to $1
C
$PB
B
B
i>clicker question
So… are demand curves always necessarily
downward sloping?
That is, MUST they always satisfy the “law of
demand”?
Robert Giffen
A. Yes!
B. No.
A Giffen Good is a good that violates the
law of demand.
That means, when $Px increases, the
consumer buys more X. Or when the
$Px decreases the consumer buys less X!
Reacting To An Own-Price Change with Fixed Income:
Suppose the PX increases; what happens to QDX
INCOME EFFECT
SUBSTITUTION EFFECT
You feel poorer – your dollars now buy less
“X” normal
X now looks relatively
more expensive
“X” inferior
QDX increases
QDX decreases
QDX decreases
QDX decreases
QDX might increase OR decrease
i>clicker questions
Consider only “own price” changes for goods where the consumer has a fixed income.
If “X” is a normal good then
A.
B.
C.
D.
E.
its demand curve might be downward or upward sloping.
it will never satisfy the law of demand.
it will always satisfy the law of demand.
it can’t have a substitution effect.
it must only have an income effect.
Consider only “own price” changes for goods where the consumer has a fixed income.
If “Y” is an inferior good then
A.
B.
C.
D.
E.
its demand curve might be downward or upward sloping.
it will never satisfy the law of demand.
it will always satisfy the law of demand.
it can’t have a substitution effect.
it must only have an income effect.
Consider only “own price” changes for goods where the consumer has a fixed income.
If “W” is a good that violates the law of demand then
A.
B.
C.
D.
E.
it must be normal.
it must be inferior.
it might be inferior.
it’s stupid.
it’s hard to tell.