1. Motivation
2. Description of Consumer Preferences
3. Indifference Curves
4. The Marginal Rate of Substitution
5. The Utility Function
Consumer Preferences and the Concept of
Utility
•
Marginal Utility and Diminishing Marginal Utility
6. Some Special Functional Forms
•
Marginal Utility and the Marginal Rate of Substitution
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• Equilibrium/comparative statics studies may
predict the direction of change but…
Consumer Preferences tell us how the consumer
would rank (that is, compare the desirability of) any two
combinations of goods, assuming these were available to
the consumer at no cost.
Over what price range?
How much?
• Elasticity good descriptive measure of demand
and supply but…
Ö
These goods are referred to as baskets or bundles. These
baskets are assumed to be available for consumption at a
particular time, place and under particular physical
circumstances.
Not predictive
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Complete:
Preferences are complete if the consumer can
rank any two baskets of goods (A preferred to B;
B preferred to A; or indifferent between A and B)
Transitive: Preferences are transitive if a consumer who
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An Indifference Curve or Indifference Set: is the set of all
baskets for which the consumer is indifferent
An Indifference Map : illustrates a set of indifference curves for
a consumer
prefers basket A to basket B, and basket B to
basket C also prefers basket A to basket C
A ⎬ B; B ⎬ C => A ⎬ C
Monotonic/Free Disposal: Preferences are monotonic if a basket with
more of at least one good and no less of
any good is preferred to the original
basket.
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6
1
One more assumption usually is made:
1). Monotonicity => indifference curves have negative
slope …and… indifference curves are not “thick”
Averages preferred to extremes => indifference curves
are bowed toward the origin (convex to the origin).
2). Transitivity => indifference curves do not cross
3). Completeness => each basket lies on only one
indifference curve
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y
8
y
Preference direction
IC2
IC1
IC1
x
x
9
y
10
y
Preferred to A
•A
•A
IC1
IC1
x
11
x
12
2
y
y
Suppose that B preferred to A.
but..by definition of IC,
B indifferent to C
A indifferent to C => B indifferent
to A by transitivity.
Contradiction.
IC1
Preferred to A
Less
preferred
•A
•A
IC1
x
x
13
y
14
y
IC1
Suppose that B preferred to A.
but..by definition of IC,
B indifferent to C
A indifferent to C => B indifferent
to C by transitivity.
Contradiction.
IC2
B
•
•A
A
•
C
•
•
B
x
IC1
x
15
y
16
y
A
•
A
•
(.5A, .5B)
•
(.5A, .5B)
•
IC2
•
B
IC2
•
IC1
B
x
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IC1
x
18
3
Example: For the indifference curves graphed
below, are the underlying preferences:
IC1 IC2
IC3
IC4
y
Complete?
Transitive?
Monotonic?
Are averages preferred to extremes?
It is the increase in good x that the consumer would require in
exchange for a small decrease in good y in order to leave the
consumer just indifferent between consuming the old basket or
the new basket…or…
Preference direction
0
x
“Neutrals”
The marginal rate of substitution: is the maximum rate at which the
consumer would be willing to substitute a little more of good x for a little
less of good y…or…
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20
An indifference curve exhibits a diminishing rate of substitution:
if the more of good x you have, the more
you are willing to give up to get a little of
good y…or…
The indifference curves get flatter as we move out along the
horizontal axis and steeper as we move up along the vertical axis.
It is the rate of exchange between goods x and y that does not
affect the consumer’s welfare…or…
It is the negative of the slope of the indifference curve:
Example: The Diminishing Marginal Rate of Substitution
MRSx,y = -∆y/∆x
(for a constant level of preference)
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y
The utility function:
assigns a number to each basket so
that more preferred baskets get a higher number than less
preferred baskets.
IC2
Utility is an ordinal concept: the precise magnitude of the
number that the function assigns has no significance.
IC1
0
x
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4
Example:
Students take an exam. After the exam, the students are
ranked according to their performance. An ordinal ranking
lists the students in order of their performance (i.e., Maria
did best, John did second best, Betty did third best, and so
on). A cardinal ranking gives the mark of the exam, based
on an absolute marking standard (i.e., Maria got 80, John
got 75, Betty got 74 and so on). Alternatively, if the exam
were graded on a curve, the marks would be an ordinal
ranking.
• difference in magnitudes of utility have no
interpretation per se
• utility not comparable across individuals
• any transformation of a utility function that preserves
the original ranking of bundles is an equally good
representation of preferences.
e.g. U = xy vs. U = xy + 2 represent the same
preferences.
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y
Example: Utility and a single indifference curve
Example: U = xy
Check that underlying preferences are complete,
transitive, monotonic and averages are preferred to
extremes…
5
2
27
0
10 = xy
2
5
x
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y
Example: Utility and a single indifference curve
The marginal utility:of a good, x, is the additional utility that
the consumer gets from consuming a little more of x when the
consumption of all the other goods in the consumer’s basket
remain constant.
∆U/∆x (y held constant) = MUx
∆U/∆y (x held constant) = MUy
Preference direction
5
…or…the marginal utility of x is the slope of the utility function with respect to
x.
The principle of diminishing marginal utility: states that the marginal utility
falls as the consumer consumes more of a good
20 = xy
2
0
10 = xy
2
5
x
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Example: U = Ax2+By2;
MUx=2Ax; MUy=2By
(where: A and B positive)
MRSx,y = MUx/MUy = 2Ax/2By
MUx(∆x) + MUy(∆y) = 0 …along an IC…
MUx/MUy = -∆y/∆x = MRSx,y
= Ax/By
Marginal utilities are positive (for positive x and y)
Marginal utility of x increases in x;
Positive marginal utility implies the indifference
curve has a negative slope (implies monotonicity)
marginal utility of y increases in y
Diminishing marginal utility implies the
indifference curves are convex to the origin
(implies averages preferred to extremes)
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Example: U= (xy).5;
MUx=y.5/2x.5; MUy=x.5/2y.5
Implications of this…
a. Is more better for both goods? Yes, since
marginal utilities are positive for both.
Indifference curves are negatively-sloped,
bowed out from the origin, preference direction
is up and right
b. Are the marginal utility for x and y
diminishing? Yes. (For example, as x increases,
for y constant, MUx falls.)
Indifference curves intersect the axes
c. What is the marginal rate of substitution
of x for y? MRSx,y = MUx/MUy = y/x
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y
Example: Graphing Indifference Curves
Do the indifference curves intersect the axes?
A value of x = 0 or y = 0 is inconsistent with any
positive level of utility.
IC1
35
x
36
6
y
Example: Graphing Indifference Curves
1. Cobb-Douglas: U = Axαyβ
where: α + β = 1; A, α,β positive constants
MUX = αAxα-1yβ
Axα β-1
MUY = β y
Preference direction
MRSx,y = (αy)/(βx)
IC2
“Standard” case
IC1
x
37
y
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y
Example: Cobb-Douglas
Example: Cobb-Douglas
Preference direction
IC2
IC1
IC1
x
x
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y
Perfect Substitutes: U = Ax + By
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Example: Perfect Substitutes
Where: A, B positive constants
MUx = A
MUy = B
MRSx,y = A/B so that 1 unit of x is equal to
B/A units of y everywhere
(constant MRS).
Slope = -A/B
IC1
0
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IC2
IC3
x
42
7
y
3. Perfect Complements: U = Amin(x,y)
Example: Perfect Complements (nuts and bolts)
where: A is a positive constant.
MUx = 0 or A
MUy = 0 or A
MRSx,y is 0 or infinite or undefined
(corner)
IC1
0
x
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y
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U = v(x) + Ay
Example: Perfect Complements (nuts and bolts)
Where: A is a positive constant.
MUx = v’(x) = ∆V(x)/∆x, where ∆ small
MUy = A
IC2
"The only thing that determines your personal trade-off
between x and y is how much x you already have."
IC1
*can be used to "add up" utilities across individuals*
0
x
45
y
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y
Example: Quasi-linear Preferences
(consumption of beverages)
Example: Quasi-linear Preferences
(consumption of beverages)
IC2
•
•
•
0
IC’s have same slopes on any
vertical line
IC1
IC1
x
0
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x
48
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1. Described consumer preferences without any restrictions
imposed by budget
2. Minimal assumptions on preferences to get interesting
conclusions on demand…seem to be satisfied for most
people. (ordinal utility function)
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