Lecture Note 04
Consumer Theory
서울대학교 경영대학
오정석 교수
1
Learning Objectives

Preference




Utility, Preference and Indifference Curve
Marginal Rate of Substitution
Budget Constraint
Utility Maximization
2
Consumer Behavior
Theory of consumer behavior


The explanation of how consumers allocate
income to the purchase of different goods and
services
Three steps involved in the study of
consumer behavior




Consumer Preferences
Budget Constraints
Given preferences and limited incomes, what
amount and type of goods will be purchased?
3
Consumer Preferences – Basic
Assumptions
A market basket is a collection of one or more
commodities.
Individuals can choose between market baskets
containing different goods
Assumptions



1.

2.

3.
Completeness.
Consumers can rank market baskets
Transitivity.
If prefer A to B, and B to C, the must prefer A to C
More is better
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Utility


Given these assumptions, it is possible to show
that people are able to rank in order all possible
situations from least desirable to most
Utility - a numerical score representing the
satisfaction that a consumer gets from a given
market basket.

if A is preferred to B, then the utility assigned to A
exceeds the utility assigned to B
U(A) > U(B)
5
Utility



Although we numerically rank baskets and
indifference curves, numbers are ONLY for ranking
A utility of 4 is not necessarily twice as good as
utility of 2
There are two types of ranking


Ordinal Utility Function
 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 Utility Function
 Utility function describing the extent to which one market
basket is preferred to another.
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Economic Goods - More Is Better

In the utility function, the x’s are assumed to
be “goods”
more is preferred to less

Quantity of y
Preferred to x*, y*
?
y*
Worse
than
x*, y*
?
Quantity of x
x*
7
Indifference Curves



Baskets for each level of utility can be plotted to get an
indifference curve
An indifference curve shows a set of consumption bundles
among which the individual is indifferent
Slopes downward – more is better
Quantity of y
Combinations (x1, y1) and (x2, y2)
provide the same level of utility
y1
y2
U1
Quantity of x
x1
x2
8
Indifference Map

Each point must have an indifference curve
through it
Quantity of y
Increasing utility
U3
U2
U1 < U2 < U3
U1
Quantity of x
9
Transitivity

Can any two of an individual’s indifference
curves intersect?
Quantity of y
The individual is indifferent between A and C.
The individual is indifferent between B and C.
Transitivity suggests that the individual
should be indifferent between A and B
C
B
A
U2
But B is preferred to A
because B contains more
x and y than A
U1
Quantity of x
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Convexity

A set of points is convex if any two points
can be joined by a straight line that is
contained completely within the set
Quantity of y
The assumption of a diminishing MRS is
equivalent to the assumption that all
combinations of x and y which are
preferred to x* and y* form a convex set
y*
U1
Quantity of x
x*
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Convexity

If the indifference curve is convex, then the
combination (x1 + x2)/2, (y1 + y2)/2 will be
preferred to either (x1,y1) or (x2,y2)
Quantity of y
This implies that “well-balanced” bundles are preferred
to bundles that are heavily weighted toward one
commodity
y1
(y1 + y2)/2
y2
U1
Quantity of x
x1
(x1 + x2)/2
x2
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Marginal Rate of Substitution

The negative of the slope of the indifference
curve at any point is called the marginal rate
of substitution (MRS)
Quantity of y
dy
MRS  
dx U U1
y1
y2
U1
Quantity of x
x1
x2
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Marginal Rate of Substitution
A
Clothing 16
MRS = 6
14
12
-6
10
B
1
8
-4
6
D
1
-2
4
MRS = 2
E
1 -1
2
1
1
2
3
4
5
G
Food
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Marginal Rate of Substitution

MRS changes as x and y change

reflects the individual’s willingness to trade y for
x
Quantity of y
At (x1, y1), the indifference curve is steeper.
The person would be willing to give up more
y to gain additional units of x
At (x2, y2), the indifference curve
is flatter. The person would be
willing to give up less y to gain
additional units of x
y1
y2
U1
Quantity of x
x1
x2
15
Marginal Utility

Suppose that an individual has a utility
function of the form
utility = U(x,y)

The total differential of U is
U
U
dU 
dx 
dy
x
y

Along any indifference curve, utility is
constant (dU = 0)
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Deriving the MRS


Therefore, we get:
U
dy
MRS  
 x
dx Uconstant U
y
MRS is the ratio of the marginal utility of x
to the marginal utility of y
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Diminishing Marginal Utility
and the MRS

The MRS decreases as we move down the
indifference curve


Along an indifference curve there is a
diminishing marginal rate of substitution.
diminishing MRS requires that the utility function
be quasi-concave
18
The Budget Constraint



Preferences do not explain all of consumer
behavior.
Budget constraints also limit an individual’s
ability to consume in light of the prices they
must pay for various goods and services.
The Budget Line


Indicates all combinations of two commodities for
which total money spent equals total income.
We assume only 2 goods are consumed, so we
do not consider savings
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The Budget Constraint

Assume that an individual has I dollars to
allocate between good x and good y
pxx + pyy  I
Quantity of y
I
py
If all income is spent
on y, this is the amount
of y that can be purchased
The individual can afford
to choose only combinations
of x and y in the shaded
triangle
If all income is spent
on x, this is the amount
of x that can be purchased
I
px
Quantity of x
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The Budget Constraint

The Effects of Changes in Income

An increase in income causes the budget line to
shift outward, parallel to the original line (holding
prices constant).


Can buy more of both goods with more income
A decrease in income causes the budget line to
shift inward, parallel to the original line (holding
prices constant).

Can buy less of both goods with less income
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The Budget Constraint
Clothing
(units
per week)
A increase in
income shifts
the budget line
outward
80
60
A decrease in
income shifts
the budget line
inward
40
20
L3
(I =
$40)
0
40
L1
L2
(I = $80)
80
120
(I = $160)
160
Food
(units per week)
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The Budget Constraint

The Effects of Changes in Prices



If the price of one good increases, the budget line
shifts inward, pivoting from the other good’s
intercept.
If price of food increases and you buy only food
(x-intercept), then can’t buy as much food. The
point shifts in
If buy only clothing (y-intercept), can buy the
same amount. No change
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The Budget Constraint
Clothing
(units
per week)
A decrease in the
price of food to
$.50 changes
the slope of the
budget line and
rotates it outward.
An increase in the
price of food to
$2.00 changes
the slope of the
budget line and
rotates it inward.
40
L3
(PF = 2)
L2
L1
(PF = 1/2)
(PF = 1)
40
80
120
160
Food
(units per week)
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Optimization Principle

To maximize utility, given a fixed amount of
income to spend, an individual will buy the
goods and services:

that exhaust his or her total income

for which the psychic rate of trade-off between
any goods (the MRS) is equal to the rate at
which goods can be traded for one another in
the marketplace
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A Numerical Illustration

Assume that the individual’s MRS = 1



willing to trade one unit of x for one unit of y
Suppose the price of x = $2 and the price
of y = $1
The individual can be made better off

trade 1 unit of x for 2 units of y in the
marketplace
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First-Order Conditions for a
Maximum

We can add the individual’s utility map to
show the utility-maximization process
Quantity of y
The individual can do better than point A
by reallocating his budget
A
The individual cannot have point C
because income is not large enough
C
B
U3
U2
U1
Point B is the point of utility
maximization
Quantity of x
27
First-Order Conditions for a
Maximum

Utility is maximized where the indifference
curve is tangent to the budget constraint
Quantity of y
slope of budget constraint  
B
px
py
slope of indifferen ce curve 
U2
dy
dx U  constant
px
dy
 MRS
py
dx U  constant
Quantity of x
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Second-Order Conditions for a
Maximum

The tangency rule is only necessary but not
sufficient unless we assume that MRS is
diminishing


if MRS is diminishing, then indifference curves are
strictly convex
If MRS is not diminishing, then we must check
second-order conditions to ensure that we are
at a maximum
29
Second-Order Conditions for a
Maximum

The tangency rule is only a necessary
condition

we need MRS to be diminishing
Quantity of y
There is a tangency at point A,
but the individual can reach a higher
level of utility at point B
B
A
U2
U1
Quantity of x
30
The n-Good Case

The individual’s objective is to maximize
utility = U(x1,x2,…,xn)
subject to the budget constraint
I = p1x1 + p2x2 +…+ pnxn

Set up the Lagrangian:
L = U(x1,x2,…,xn) + (I - p1x1 - p2x2 -…- pnxn)
31
The n-Good Case

First-order conditions for an interior maximum:
L/x1 = U/x1 - p1 = 0
L/x2 = U/x2 - p2 = 0
•
•
•
L/xn = U/xn - pn = 0
L/ = I - p1x1 - p2x2 - … - pnxn = 0
32
Implications of First-Order
Conditions

For any two goods,
U / xi
pi

U / x j p j

This implies that at the optimal allocation
of income
pi
MRS ( xi for x j ) 
pj
33
Interpreting the Lagrange
Multiplier
U / x1 U / x2
U / xn


 ... 
p1
p2
pn


MUx1
p1

MUx2
p2
 ... 
MUxn
pn
 is the marginal utility of an extra dollar of
consumption expenditure

the marginal utility of income
34
Interpreting the Lagrange
Multiplier

At the margin, the price of a good
represents the consumer’s evaluation of
the utility of the last unit consumed

how much the consumer is willing to pay for
the last unit
pi 
MUxi

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