Indifference Curves

1
Indifference Curves

• An indifference curve shows a set of consumption
  bundles among which the individual is indifferent

Quantity of Y
Combinations (X1, Y1) and (X2, Y2) provide the
same level of utility
Y1
Y2
U1
Quantity of X
X1
X2
2
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
Y1
Y2
U1
Quantity of X
X1
X2
3
Marginal Rate of Substitution

• MRS changes as X and Y change
• reflects the individuals willingness to trade Y
  for X

Quantity of Y
Y1
Y2
U1
Quantity of X
X1
X2
4
Indifference Curve Map

• Each point must have an indifference curve
  through it

Quantity of Y
U1 lt U2 lt U3
U3
U2
U1
Quantity of X
5
Transitivity

• Can two of an individuals indifference curves
  intersect?

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
Quantity of Y
But B is preferred to A because B contains more X
and Y than A
C
B
U2
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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Utility and the MRS

• Suppose an individuals preferences for
  hamburgers (Y) and soft drinks (X) can be
  represented by

• Solving for Y, we get
• Y 100/X

• Solving for MRS -dY/dX
• MRS -dY/dX 100/X2

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Utility and the MRS

• MRS -dY/dX 100/X2
• Note that as X rises, MRS falls
• When X 5, MRS 4
• When X 20, MRS 0.25

10
Marginal Utility

• Suppose that an individual has a utility function
  of the form
• utility U(X1, X2,, Xn)
• We can define the marginal utility of good X1 by
• marginal utility of X1 MUX1 ?U/?X1
• The marginal utility is the extra utility
  obtained from slightly more X1 (all else constant)

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Marginal Utility

• The total differential of U is

• The extra utility obtainable from slightly more
  X1, X2,, Xn is the sum of the additional utility
  provided by each of these increments

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Deriving the MRS

• Suppose we change X and Y but keep utility
  constant (dU 0)
• dU 0 MUXdX MUYdY
• Rearranging, we get

• 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

• Intuitively, it seems that the assumption of
  decreasing marginal utility is related to the
  concept of a diminishing MRS
• Diminishing MRS requires that the utility
  function be quasi-concave
• This is independent of how utility is measured
• Diminishing marginal utility depends on how
  utility is measured
• Thus, these two concepts are different

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Marginal Utility and the MRS

• Again, we will use the utility function

• The marginal utility of a soft drink is
• marginal utility MUX ?U/?X 0.5X-0.5Y0.5
• The marginal utility of a hamburger is
• marginal utility MUY ?U/?Y 0.5X0.5Y-0.5

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Examples of Utility Functions

• Cobb-Douglas Utility
• utility U(X,Y) X?Y?
• where ? and ? are positive constants
• The relative sizes of ? and ? indicate the
  relative importance of the goods

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Examples of Utility Functions

• Perfect Substitutes
• utility U(X,Y) ?X ?Y

Quantity of Y
The indifference curves will be linear. The MRS
will be constant along the indifference curve.
Quantity of X
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Examples of Utility Functions

• Perfect Complements
• utility U(X,Y) min (?X, ?Y)

Quantity of Y
The indifference curves will be L-shaped. Only
by choosing more of the two goods together can
utility be increased.
Quantity of X
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Examples of Utility Functions

• CES Utility (Constant elasticity of substitution)
• utility U(X,Y) X?/? Y?/?
• when ? ? 0 and
• utility U(X,Y) ln X ln Y
• when ? 0
• Perfect substitutes ? ? 1
• Cobb-Douglas ? ? 0
• Perfect complements ? ? -?

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Examples of Utility Functions

• CES Utility (Constant elasticity of substitution)
• The elasticity of substitution (?) is equal to
  1/(1 - ?)
• Perfect substitutes ? ? ?
• Fixed proportions ? ? 0

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Homothetic Preferences

• If the MRS depends only on the ratio of the
  amounts of the two goods, not on the quantities
  of the goods, the utility function is homothetic
• Perfect substitutes ? MRS is the same at every
  point
• Perfect complements ? MRS ? if Y/X gt ?/?,
  undefined if Y/X ?/?, and MRS 0 if Y/X lt ?/?

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Nonhomothetic Preferences

• Some utility functions do not exhibit homothetic
  preferences
• utility U(X,Y) X ln Y
• MUY ?U/?Y 1/Y
• MUX ?U/?X 1
• MRS MUX / MUY Y
• Because the MRS depends on the amount of Y
  consumed, the utility function is not homothetic

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Important Points to Note

• If individuals obey certain behavioral
  postulates, they will be able to rank all
  commodity bundles
• The ranking can be represented by a utility
  function
• In making choices, individuals will act as if
  they were maximizing this function
• Utility functions for two goods can be
  illustrated by an indifference curve map

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Important Points to Note

• The negative of the slope of the indifference
  curve measures the marginal rate of substitution
  (MRS)
• This shows the rate at which an individual would
  trade an amount of one good (Y) for one more unit
  of another good (X)
• MRS decreases as X is substituted for Y
• This is consistent with the notion that
  individuals prefer some balance in their
  consumption choices

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Important Points to Note

• A few simple functional forms can capture
  important differences in individuals preferences
  for two (or more) goods
• Cobb-Douglas function
• linear function (perfect substitutes)
• fixed proportions function (perfect complements)
• CES function
• includes the other three as special cases