Theory of Consumer Choice
V. Hajko
2014
V. Hajko (FSS MU) Introduction to Economics 1 / 19
1 Consumer behavior
2 Preferences
3 Utility functions and indifference curves
4 Consumer’s optimization problem
V. Hajko (FSS MU) Introduction to Economics 2 / 19
Consumer behavior
The principal interests of this lecture
How can we represent the consumer’s preferences?
How can we represent the choices a consumer can afford?
What determines how a consumer divides his or her resources between
two goods?
V. Hajko (FSS MU) Introduction to Economics 3 / 19
Consumer behavior
Rational behavior
The rational consumer behavior is driven by the effort to achieve the
highest level of satisfaction
In the economics, the consumers (sic!) are being satisfied by the
consumption of goods and services
The highest level of satisfaction is seen according to the consumer’s
preferences
With limited resources, people face tradeoffs
if you go swimming, you will have less time for jogging
if you buy more of one good, you will have less money to buy
something else
if you work more, you will earn more, but you will have less of the
leisure time
The consumers seek to consume the "best" bundle of goods they can
afford:
they maximize their utility functions subject to their budgetary
constrains
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Preferences
The preference relations
In order to represent the what is “better” or "worse", the consumer
has to be able to compare the choices.
This comparison can be represented by the preference relations.
This logic of comparison is used in the construction of the so-called
"utility function" - a representation of the consumer’s preferences.
The preference relation allows for the comparison of pairs of
alternatives, x and y
x y means x is at least as good as y; (x y means x is strictly
preffered to y)
x y means y is at least as good as x; (x y means y is strictly
preffered to x)
x ∼ y means x is indifferent to y
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Preferences
The rational preference relations
The preference relation is rational, if it fulfills two basic assumptions:
completeness and transitivity
Completeness: for any pair of x, y ∈ X we have x y, x y or both
(i.e. x ∼ y).
consumer is able to compare any two pairs and decide whether x is
preferred to y, y is preferred to x, or whether he or she is indifferent
between x and y
Transitivity: for all alternatives x, y.z ∈ X , if x y and y z then
x z.
rational behavior rules out a preference cycle
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Utility functions and indifference curves
Utility functions
A function u : X → R is a utility function representing preference
relation , if for all x, y ∈ X we have x y ⇔u (x) ≥ u (y)
The utility function is used to represent the preferences (the logical
relationships) with a numerical representation
The utility function is strictly individual
A preference relation can be represented by a utility function only if
it is rational
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Figure : The visual representations, function f (x, y) = z
(a) We need 3-D graphs to visually capture
the value of the function with two inputs
0
5
10
15
20
25
30
35
0
5
10
15
20
25
30
35
0
1
2
3
4
5
6
7
x 10
−3
(b) Contour plot for f (x, y) = z .
5 10 15 20 25 30
5
10
15
20
25
30
Utility functions and indifference curves
Utility function
The contour lines of the utility function represent all points, where
the various combinations of amounts of x and y lead to the same
value of the utility function
Assume the utility function U = X2Y 2; for the given value of utility
function, e.g. U = 100, we can write down the specific contour line
(which is called the indifference curve) as 100 = X2Y 2, or Y = 100
X2
The indifference curve shows all combination of bundles of x and y
that bring the same utility. Therefore the consumer is indifferent
between all bundles lying on the indifference curve.
Properties of the well behaved utility function
monotonicity (or its weaker version, local nonsatiation)
in essence, the larger amount of consumed goods or services will not
decrease the utility (the goods are desirable)
convexity
a formal expression of a basic inclination of economic agents for
diversification (the assumption that consumers prefer variety)
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Figure : The visual representation of the convex utility function with nonsatiation
(a) The utility function:
0
20
40
60
80
100
120
0
20
40
60
80
100
120
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
(b) The indifference map is the contour
plot of the utility function:
30 40 50 60 70 80 90 100 110
30
40
50
60
70
80
90
100
110
Utility functions and indifference curves
Indifference curves
For well behaved utility functions, the higher indifference curves show
the contours of higher values of the utility function
The higher indifference curves are preferred to lower ones
Indifference curves do not cross (this would violate the transitivity of
the preference relation)
Indifference curves are bowed inward (becase of the convexity
property)
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Utility functions and indifference curves
Marginal rate of substitution in consumption
Marginal rate of substitution in consumption (MRSC ) is the rate at
which a consumer is willing to trade one good for another
It is represented by the slope of the indifference curve
MRSC = MUX
MUY
, also MRSC = −∆Y
∆X
C
B
A
X
Y
IC1
IC2
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Consumer’s optimization problem
Budget constraint
Due to the limited resources, some of the consumption bundles are
unavailable to the consumer.
The limitation of the consumption is called the budget constraint:
I ≥ px X + py Y , where I is consumers disposable income, px is the
price of one unit of X and X is the number of units of X consumed.
The slope of the budget line represents the so called Marginal rate of
substitution in exchange (MRSE = −pX
pY
) - this represents the ratio at
which the goods are valued by the market (e.g. “1 apple = 2
bananas” - this is called the relative price)
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Consumer’s optimization problem
Budget constraint
An increase in income shifts the budget constraint outward
A change in the prices causes the budget constraint to change its
slope ("rotate")
D
C
B
A
X
Y
IC1
IC2
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Consumer’s optimization problem
Consumer’s optimization problem
The consumer wants to attain the maximal utility
The higher level of utility is equivalent with higher value of the utility
function
Therefore, the consumer seeks to find such a combination that will
maximize the value of the utility function AND satisfy the budget
constraint
max U (X, Y ) subject to: I ≥ px X + py Y
since we are assuming monotonicity, we can simplify the budget
constraint to I = px X + py Y
we can write the Lagrangean for this optimization problem as:
L (X, Y , λ) = U (X, Y ) + λ (I − px X − py Y )
By differentiating L with respect to x,y and λ and setting the
derivatives equal to zero, we get the first order conditions:
∂U
∂X − λpx = 0
∂U
∂Y − λpy = 0
I − px X − py Y = 0
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Consumer’s optimization problem
The optimal consumption
The first order conditions imply the consumer will seek to consume
the goods in the ratio that fulfills MRSC = MRSE , i.e. MUX
MUY
= pX
pY
or
MUX
pX
= MUY
pY
The marginal utility MUX is the change in the total utility caused by
infinetisimaly small change in the consumption of X, i.e.
MUX = ∂U
∂X
This implies the optimum will occur at the point where the indifference
curve and the budget constraint are tangent.
The the attainable consumption, hence also the attainable utility, is
limited by the budget constraint.
Combined with the previous condition, this implies the optimum will
occur at the point where the highest indifference curve and the budget
constraint are tangent.
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Consumer’s optimization problem
Income and substitution effects
The income effect is the change in consumption that results when a
price change moves the consumer to a higher or lower indifference
curve.
the price change increases (decreases) the purchasing power of the
consumer (his relative wealth), which encourages (discourages) the
consumer to buy the goods in question
The substitution effect is the change in consumption that results
when a price change moves the consumer along an indifference curve
to a point with a different marginal rate of substitution.
the price change encourages greater (lower) consumption of the good,
since it has become relatively cheaper (more expensive)
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Consumer’s optimization problem
Normal goods vs. inferior goods
Based on the income elasticity of demand (εID =
∂QD
Q
∂I
I
) we can
distinguish normal and inferior goods:
With the increase in income, the consumer buys more of a normal good.
With the increase in income, the consumer buys less of an inferior good.
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Consumer’s optimization problem
Individual demand
The demand function QD = f (p, ·) represents the quantity of the
given goods demanded, dependent on the price (and possibly other
factors, e.g. QD
X = f (px , py , I)).
Consumer’s demand curve summarizes the solutions of the
optimization problems for the various levels of price (given no changes
in the other factors, e.g. for the given level of income and prices of
other goods).
Typically a demand curve slopes downward (with a decrease in price,
the consumer would buy more of the goods).
There is an exception (but very rare): the so-called Giffen goods - an
increase in the price raises the quantity demanded (the income effect
overwhelmes the substitution effect).
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