# Chapter 03_Rational consumer choice

CHAPTER 3
McGraw-Hill/Irwin
RATIONAL CONSUMER
CHOICE
Chapter Outline
The opportunity set or budget constraint
Budget shifts due to price or income changes
Consumer preferences
Completeness
More-is-better
Transitivity
Convexity
Indifference curves
The best feasible bundle
Corner solution
The utility function approach to consumer choice
Generating Indifference Curves Algebraically
Summary
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An Overview
Theory of rational consumer choice – underlies all
Begins with the assumption that consumer enter
the market with well-defined preferences.
Given prices, their task is to allocate their incomes
to best serve these preferences.
2 steps:
describe various combinations of goods the consumer is
able to buy (depend on Y & P)
Select from among the feasible combinations the
particular one that the consumer prefers to all others
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The Opportunity Set or Budget
Constraint
Budget Constraint/ Budget Line
The set of all bundles that exactly exhaust the
consumer’s income at given prices.
Assume only 2 goods ( food and shelter) are
consumed, so we do not consider savings (all
income are spent).
Bundle A :
5 square yards/week of shelter & 7 pounds per week
of food
Bundle B:
3 sq yd/wk of shelter & 8 lb /wk of food
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Two bundles of goods
3-5
The Budget Constraint or Budget Line
Suppose consumer’s income, M = \$100/wk (all
spent on some combination of S & F)
Price of shelter, Ps = \$5/sq yd
Price of food, Pf = \$10/lb
If spent all income on S  could buy M/Ps =
(\$100/wk)  (\$5/sq yd) = 20 sq yd/wk (20,0).
If spent all income on F  could buy M/Pf =
(\$100/wk)  (\$10/lb) = 10 lb/wk (0, 10).
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The Budget Constraint or Budget Line
3-7
Slope of budget line = - Ps / Pf = - 1 / 2
It is the rate at which food can be exchanged for
shelter ( 1 lb of food can be exchanged for 2 sq yd
of shelter).
Feasible set or affordable set - Consumer can buy
any bundles along the budget line and also bundles
that lies within the budget triangle bounded by it
and two axes (e.g. D )
Bundle that lie outside the budget triangle (e.g. E)
are said to be infeasible or unaffordable
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Budget constraint must satisfy the following
equation:
Ps S + Pf F = M
(1)
To represent the formula for a straight line,
solve equation 1 for F in terms of S:
F = M – Ps S
Pf Pf
slope
intercept
F = 10 – ½ S
3-9
Budget shifts due to Price or Income
Changes
Slope and position of BL are fully
determined by the consumer’s income &
prices.
Assuming price of shelter increase from Ps1
= \$5 to Ps2 = \$10.
Since both income & Pf unchanged, the
vertical intercept of the BL stays the same.
The ↑Ps rotates the BL inward.
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The Effect of a Rise in the Price of
Shelter
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Income Changes
The effect of a change in income is much like
the effect of an equal proportional change in
all prices.
Assuming income decreased from \$100/wk to
\$50/wk – the horizontal intercept of the BL
falls from 20 sq yd/wk to 10 sq yd/wk, and
the vertical intercept falls from 10 lb/wk to 5
lb/wk.
There is a parallel shifts inward & slope of
BL remain unchanged
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The Effect of Decrease in Income
3-13
Consumer Preferences
Preference ordering is a scheme that enables
consumer rank different bundles of goods in
terms of their desirability.
For any two bundles of goods (e.g. A and B),
the preference ordering enables the consumer
to rank different bundles but not to make
their desirability.
E.g. consumer might be able to say that he prefers
A to B but not that A provides twice as much
satisfaction as B.
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Properties of preferences
Completeness
if consumer able to rank all possible combinations of goods
and services.
More-Is-Better
more of a good is preferred to less.
Transitivity
for any 3 bundles A, B, and C, if A preferred to B, and B
preferred to C, then A is always preferred to C. Transitivity
is a consistency property.
Convexity
mixtures of goods are preferable to extremes. Suppose
consumer are indifferent between A (4, 0) and B (0, 4). If
preferences are convex, consumer will prefer bundle C (2,2).
Implies that consumer like balance in mix of consumption
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goods.
Generating Equally Preferred Bundles
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An Indifference Curve
Is a set of bundles that the consumer considers equally
attractive.
Bundles that lie above an IC are all preferred to the bundles
that lie on it.
Bundles that lie on an IC are all preferred to those that lie
below it.
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Indifference Map
I1 < I2 < I3 < I4
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Properties of Indifference curve
Indifference curves are ubiquitous. Any bundle
has an indifference curve passing through it.
IC are downward sloping. An up-ward sloping IC
would violate the more-is-better property.
IC become less steep as we move downward and to
the right
IC cannot cross because they would have to violate
at least one of the assumed properties of
preference ordering.
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Why Two Indifference Curves Do not Cross
E is equally attractive as D
D is equally attractive as F
E is equally attractive as F (by the transitivity
assumption)
F is preferred to E (because more is better)
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An important property of a consumer’s preference is
the rate at which they are willing to exchange or
‘trade off' one good for another.
This is represented at any point on an IC by the
MRS (slope of IC).
MRS declines along the IC as we move downward to
the right – implying that consumers are willing to
give up goods that they already have a lot of to
obtain more of those goods they have few.
This is because the indifference curves are convex
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The Marginal Rates of Substitution (MRS)
More formally, IC may be expressed as a function,
Y = Y (X) and the MRS at point A is defined as the
absolute value of the derivative of the IC at that
point:
MRS = | dY (X) / dX |
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Diminishing Marginal
Rate of Substitution
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The Best Affordable Bundle
The indifference map tells how various
bundles are ranked in order of preference.
Budget line tells which bundles are affordable
Using these two tools – consumer choose the
most preferred or best affordable bundle
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The Best Affordable Bundle
Of the 5 labeled bundles – A, D, E, F and G  G is the most
preferred because it lies on the highest IC, however, it is not
affordable.
The best affordable bundle is F ( 6F : 8S)  MRS = Ps/Pf
(point of tangency).
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The Best Affordable Bundle
If MRS ≠ PF/PC then individuals can
If MRS > Ps/Pf
Will increase shelter and decrease food until
MRS = Ps/Pf
If MRS < Ps/Pf
Will increase food and decrease shelter until
MRS = Ps/Pf
3-26
A Corner Solution
A corner solution exists if a consumer buys in
extremes, and buys all of one category of good and
none of another.
MRS is not necessarily equal to slope of budget line
(MRS may be everywhere greater, or less, than the
slope of the BL)
Utility is maximized at a point on one axis where
the budget constraint intersects the highest
attainable indifference curve at zero consumption
for one good with all income used for the other
good.
The best affordable bundle need not always occur
at a point of tangency.
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A Corner Solution
At point A, MRS < Ps/Pf  the best the
consumer can do is to spend all his income
on food.
3-28
A Corner Solution
Frozen
Yogurt
(cups
monthly)
A
U1
U2
U3
B
A corner solution
exists at point B.
Ice Cream (cup/month)
3-29
Perfect Substitutes and Perfect
Complements
The shape of an indifference curve describes the
willingness of a consumer to substitute one good
to another
Indifference curves with different shapes imply a
different willingness to substitute
Two extreme cases are
Perfect substitutes
Perfect complements
3-30
Perfect Substitutes
 2 goods are perfect substitutes
when the MRS of one good for the
other is constant.
Example 1:
A person might consider apple juice
and orange juice perfect substitutes
They would always trade 1 glass
of OJ for 1 glass of AJ
Example 2:
• Amy likes M&M, plain and
peanut. For Amy the MRS between
plain and peanut M&M’s does not
vary with the quantities she
consumes.
Apple
4
Juice
(glasses)
3
2
1
0
1
2
3
4
Orange Juice
(glasses)
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Two goods are perfect
complements when the
IC for the goods are
shaped as right angles.
Example 1:
1 left shoe and 1 right
shoe- both must be used
at the same time.
Example 2:
Peter is very choosy
popcorn. He tops every
quart of popped corn with
exactly one quarter cup
of melted butter.
Perfect
Complements
Left
Shoes
4
3
2
1
0
1
2
3
4
Right Shoes
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Measuring Consumer Preferences
The theory of consumer behavior does not
required assigning a numerical value to the level of
satisfaction.
There are two types of ranking
Ordinal ranking
Places market baskets in the order of most preferred to
least preferred, but it does not indicate how much one
market basket is preferred to another.
Cardinal ranking
Utility function describing the extent to which one
market basket is preferred to another.
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Utility
Although ranking of market baskets are good
(where we use indifference curve to describe
graphically consumer preferences), sometimes
numerical value are useful
The concept is known as Utility
A numerical score representing the
satisfaction that a consumer gets from a
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Utility
Utility function
Formula that assigns a level of utility to
If the utility function is
U(F,C) = F + 2C
It tells the level of satisfaction obtained from
consuming F units of food and C units of clothing
o A market basket with 8 units of food and 3 units of
clothing gives a utility of
14 = 8 + 2(3)
3-35
Utility - Example
Market Food
A
8
B
6
C
4
Clothing
Utility
3
4
4
8 + 2(3) = 14
6 + 2(4) = 14
4 + 2(4) = 12
Consumer is indifferent between A & B because
the utility is same (14) and prefers both to C
because utility is smaller (12)
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Utility Example
If the utility function, U = FC
C
25 = 2.5(10)
A
25 = 5(5)
B
25 = 10(2.5)
Clothing
If the utility function, U (F, C) = 4FC
C
100 = 4 (2.5)(10)
A
100 = 4 (5) (5)
B
100 = 4 (10) (2.5)
15
C
10
U3 = 100
A
5
B
0
5
10
U2 = 50
15
U1 = 25
Food
3-37
The utility function approach to consumer choice
As consumption moves along an indifference curve:
Additional utility derived from an increase in the
consumption one good, food (F), must balance the loss of
utility from the decrease in the consumption in the other
good, clothing (C).
Formally:
0  MUF(F)  MUC(C)
No change in total utility along an indifference curve.
Trade off of one good to the other leaves the consumer just
as well off.
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Marginal Utility and Consumer
Choice
Rearranging:
 C / F   MU F / MU C
Since
 C / F   MRS of F for C
We can say
MRS  MUF/MU C
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Marginal Utility and Consumer
Choice
When consumers maximize satisfaction:
MRS  PF/PC
Since the MRS is also equal to the ratio of the
marginal utility of consuming F and C
MUF/MUC  PF/PC
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Marginal Utility and Consumer
Choice
Rearranging, gives the equation for utility
maximization:
MU F / PF  MU C / PC
Total utility is maximized when the budget is
allocated so that the marginal utility per dollar of
expenditure is the same for each good.
This is referred to as the equal marginal principle.
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Generating Indifference Curves
Algebraically
Using calculus to solve the consumer’s budget
allocation problem.
Let U (X, Y) be the consumer’s utility function
M, Px and Py = income, price of X and price of Y
Consumer’s allocation problem can be stated as:
Maximize U(X, Y) subject to PxX + PyY = M
X, Y
(Eq.1)
Find values of X and Y using method of Lagrangian
Multipliers (technique used to maximize or minimize a
function subject to one or more constraints)
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The Method of Lagrangian Multipliers
First transformed the constrained maximization
problem in equation (1) into unconstrained
maximization problem:
Maximize £ = U(X,Y) -  (PxX + PyY – M)
X, Y, 
(Eq.2)
Differentiate £ with respect to X, Y and  and then
equate the derivatives to zero.
£ = U - Px = 0
X X
(Eq.3)
£ = U - Py = 0
Y Y
(Eq.4)
£ = M – PxX - PyY= 0

(Eq.5)
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Next solve Eq.3-4 for X, Y and  (to guarantee that
the budget constraint is satisfied).
Optimal X and Y values can be obtained by dividing
Eq.3 by Eq.4 to get;
U/X = Px = Px
(Eq.6)
U/ Y Py Py
 Eq. 6 implies optimal values of X and Y must satisfy MRS
= Px/Py. The terms U/X and U/ Y is marginal utility
(MU) of X and Y.
Rearranging Eq.6 in the form;
U/X = U/ Y
Px
Py
 MUx = MUy
Px
Py
(Eq.7)
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An Example
Suppose that U (X, Y) = XY and that M = 40, Px = 4 and Py = 2.
Constrained maximization problem;
Maximize £ = XY -  (4X + 2Y – 40)
(Eq.1)
X, Y, 
First-order conditions for a maximum of £ are given;
£ =  (XY) - 4 = Y - 4 = 0
(Eq.2)
X
X
£ =  (XY) - 2 = X - 2 = 0
Y
Y
(Eq.3)
£ = 40 – 4X – 2Y = 0
(Eq.4)

Divide Eq. 2 by Eq.3 and solve for Y, get Y = 2X.
Substitute Y = 2X into Eq.4 and solve for X (X =5). Y = 2X = 10.
Thus (5, 10) is the utility-maximizing bundle.
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An Alternative Method
Solve the budget constraint for Y in terms of X and substitute
in the utility function. Then maximize by taking first
derivative with respect to X and equate to zero.
Suppose U (X, Y) = XY, with M= 40,Px = 4 and Py=2.
Budget constraint, 4X + 2Y = 40, which solves for Y = 20 – 2X.
Substitute this into utility function,
U(XY) = X (20 – 2X) = 20X – 2X2
Take first derivative of U with respect to X and equate with
zero,
dU = 20 – 4X = 0
dX
Solve for X = 5. Substitute X into budget constraint to get the
optimal value of Y = 10
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The Optimal Bundle when U=XY, Px=4, Py=2,
and M=40
3-47
Example Question 1
1. Sally consumes two goods, X and Y. Her utility
function is given by the expression U = 3  XY2. The
current market price for Y is \$5, while the market
price for X is \$10. (Note MUy= 6XY and MUx =
3Y2). Income = \$500.
a. Determine the X, Y combination which maximizes
Sally’s utility given her budget constraint.
b. Calculate the impact on Sally’s optimum market
basket of an increase in the price of X to 15. What
would happen to her utility due to price increase?
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a. To maximize utility,
MRS = Px / Py
MRS = MU x / MU y
Since MU x = 3Y2 and
MU y = 6XY
MRS = 3Y2 = Y
6XY
2X
Px / Py = 10 / 5 = 2
To
determine
exact
quantities, substitute Y = 4X
into
I = Px X + Py Y
500 = 10 X + 5 (4X)
500 = 30 X
X = 16.67
Y = 4 (16.67)
Y = 66.67
MRS = Px / Py  Y = 2
2X
Y = 4X
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b. Calculate the impact on Sally’s optimum market basket of
an increase in the price of X to 15. What would happen to her
utility due to price increase?
Solution:
MRS remains Y / 2X, and Px / Py becomes 15 / 5 = 3
Equating MRS to Px / Py we get
Y = 3  Y = 6X
2X
Substitute Y = 6X into I = Px X + Py Y
500 = 15 X + 5 Y
500 = 15 X + 5(6X)
500 = 45 X
X = 11.11 and Y = 66.67
Utility before price change: U = 3 (16.67)(66.67)2 = 222,289
Utility after price change: U = 3 (11.11)(66.67)2 = 148,148
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Example 2
An individual consumes products X and Y and
spends \$25 per time period. The prices of the two
goods are \$3 per unit for X and \$2 per unit Y. The
consumer in this case has a utility function
expressed as:
U (X, Y) = 0.5 XY,
MUx = 0.5 Y,
MUy = 0.5 X
a. Express the budget equation mathematically
I = Px X + Py Y
25 = 3X + 2 Y
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b. Determine the values of X and Y that will maximize utility
in the consumption of X and Y.
In equilibrium, utility is maximized when
MU x / Px = MU y/ Py
0.5 Y / 3 = 0.5 X / 2
2Y = 3 X
Y = (3/2) X
On the budget line , 25 = 3X + 2 Y
25 = 3X + 2 (3 X)
2
25 = 6X
X = 4.17 units, Y = 6.25 units
c. Determine the total utility that will be generated.
U (X, Y) = 0.5 X Y = 0.5 (4.17) (6.25)
U = 13.03
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Summary
Theory of consumer choice rests on the assumption that
people behave rationally in an attempt to maximize
satisfaction by purchasing a particular combination of goods
& services.
2 parts: consumer’s preferences & budget line.
4 properties of consumer’s preferences, indifference curves
and MRS.
Budget line – definition, slope and the effect changes in
prices and income.
Consumer maximization condition, MRS = ratio of prices.
Utility function approach & cardinal vs ordinal utility
Using calculus to maximize utility.
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