PREFERENCES AND UTILITY

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Part 2

• PREFERENCES AND UTILITY

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Objectives of the chapter

• Study a way to represent consumers preferences
  about bundles of goods
• What are bundles of goods? combinations of
  goods. For instance
• Xslices of pizza
• Yglasses of juice
• Bundles
• P X1, Y1
• Q X3, Y0
• R Y3, X0
• S X2, Y1

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Objectives of the chapter

• Johns preferences are such that
• P is preferred to both Q and R
• S is preferred to P
• This way of representing preferences would be
  very messy if we have many bundles
• In this chapter we study a simple way of
  representing preferences over bundles of goods
• This is useful because in reality there are many
  bundles of goods

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Axioms of rational choice

• Before describing this simple method to represent
  preferences over bundles, we will study what
  requirements must the preferences satisfy in
  order for the method to work
• These requirements are the axioms of rational
  choice
• Without these requirements, it would be very
  difficult to come up with a simple method to
  represent preferences over many bundles of goods
• It is easy to read a tube map, but not so much to
  read a tube-bus-and rail map !!!!

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Axioms of Rational Choice

• Completeness
• if A and B are any two bundles, an individual can
  always specify exactly one of these
  possibilities
• A is preferred to B
• B is preferred to A
• A and B are equally attractive
• In other words, preferences must exist in order
  to be able to describe them through a simple
  method

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Axioms of Rational Choice

• Transitivity
• if A is preferred to B, and B is preferred to C,
  then A is preferred to C
• assumes that the individuals choices are
  internally consistent
• If transitivity does not hold, we would need a
  very complicated method to describe preferences
  over many bundles of goods

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Axioms of Rational Choice

• Continuity
• if A is preferred to B, then bundles suitably
  close to A must also be preferred to B
• If this does not hold, we would need a very
  complicated method to describe individuals
  preferences

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Utility

• Given these assumptions, it is possible to show
  that people are able to rank all possible
  bundles from least desirable to most
• Economists call this ranking utility
• if A is preferred to B, then the utility assigned
  to A exceeds the utility assigned to B
• U(A) gt U(B)

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Utility

• Game
• Someone state the preferences using numbers from
  1 to 10
• Can someone use different numbers from 1 to 10
  but state the same ordering?
• Can someone use numbers 1 to 100 and state the
  same preferences?

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Utility

• Game
• Clearly, the numbers are arbitrary
• The only consistent thing is the ranking that we
  obtain

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Utility

• Utility could be represented by a Table

Bundles Example Utility
P 1
Q 0
R 0
S 2
U(P)1gtU(Q)0 because we said that P was
preferred to Q U(B)U(C) because Q and R are
equally preferred
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Utility

• Notice that several tables of utility can
  represent the same ranking

Bundles Another example Example Utility
P 1 1
Q 0 0
R 0 0
S 4 2

• We can think that the rankings are real. They are
  in anyones mind. However, utility numbers are an
  economists invention
• The difference (2-1, 4-1) in the utility numbers
  is meaningless. The only important thing about
  the numbers is that they can be used to represent
  rankings (orderings)

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Utility

• Utility rankings are ordinal in nature
• they record the relative desirability of
  commodity bundles
• Because utility measures are not unique, it makes
  no sense to consider how much more utility is
  gained from A than from B. This gain in utility
  will depend on the scale which is arbitrary
• It is also impossible to compare utilities
  between people. They might be using different
  scales.

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Utility

• If we have many bundles of goods, a Table is not
  a convenient way to represent an ordering. The
  table would have to be too long.
• Economist prefer to use a mathematical function
  to assign numbers to consumption bundles
• This is called a utility function
• utility U(X,Y)
• Check that the previous example of the three
  columns table is obtained with the following
  utility functions
• UXY,
• and U(XY)2

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Utility

• Clearly, for an economist it is the same to use
  UXY than to use U(XY)2 because both
  represent the same ranking (see the table), so
  both functions will give us the same answer in
  terms of which bundles of good are preferred to
  others
• Any transformation that preserves the ordering
  (multiply by a positive number, take it at a
  power of a positive number, take ln) will give
  us the same ordering and hence the same answer
• We can use this property to simplify some
  mathematical computations that we will see in the
  future

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Economic Goods

• In the utility function, the x and y are assumed
  to be goods
• more is preferred to less

Quantity of y
y
Quantity of x
x
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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
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Indifference Curve Map

• Each point must have an indifference curve
  through it

Quantity of y
U1 lt U2 lt U3
Quantity of x
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Transitivity

• Can any 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

• Economist believe that
• Balanced bundles of goods are preferred to
  extreme bundles
• This assumption is formally known as the
  assumption of convexity of preferences
• Using a graph, shows that if this assumption
  holds, then the indifference curves cannot be
  strictly concave, they must be strictly convex

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Convexity

• Formally, 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)

This means that well-balanced bundles are
preferred to bundles that are heavily weighted
toward one Commodity (extreme bundles). The
middle points are better than the Extremes, so
the middle is at a higher indifference Curve.
Quantity of y
y1
(y1 y2)/2
y2
U1
Quantity of x
x1
(x1 x2)/2
x2
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Marginal Rate of Substitution

• Important concept !!
• MRSYX is the number of units of good Y that a
  consumer is willing to give up in return for
  getting one more unit of X in order to keep her
  utility unchanged
• Lets do a graph in the whiteboard !!!
• MRSYX is the negative of the slope of the
  indiference curve (where Y is in the ordinates
  axis)

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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
y1
y2
U1
Quantity of x
x1
x2
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Marginal Rate of Substitution

• Notice that if indifference curves are strictly
  convex, then the MRS is decreasing (as x
  increases, the MRSyx decreases)
• See it in a graph As x increases, the amount
  of y that the consumer is gives up to stay in
  the same indifference curve (that is MRSyx)
  decreases
• If the assumption that balanced bundles are
  preferred to extreme bundles (convexity of
  preferences assumption holds then the MRSyx is
  decreasing!!

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Marginal Rate of Substitution

• MRS changes as x and y change
• and it is decreasing

Quantity of y
y1
y2
U1
Quantity of x
x1
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 the indifference curve for
level 10 y 100/x

• Taking derivatives, we get the MRS -dy/dx
• MRS -dy/dx 100/x2

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Utility and the MRS

• MRSyx -dy/dx 100/x2
• Note that as x rises, MRS falls
• when x 5, MRSyx 4
• when x 20, MRSyx 0.25
• When x20, then the individual does not value
  much an additional unit of x. He is only willing
  to give 0.25 units of y to get an additional unit
  of x.

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Another way of computing the MRS

• Suppose that an individual has a utility function
  of the form
• utility U(x,y)
• The total differential of U is

Along any indifference curve, utility is constant
(dU 0) dU/dy and dU/dx are the marginal
utility of y and x respectively
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Another way of computing the MRS

• Therefore, we get

MRS is the ratio of the marginal utility of x to
the marginal utility of y Marginal utilities are
generally positive (goods)
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Example of MRS

• Suppose that the utility function is

We can simplify the algebra by taking the
logarithm of this function (we have explained
before that taking the logarithm does not change
the result because it preserves the ordering,
though it can make algebra easier) U(x,y)
lnU(x,y) 0.5 ln x 0.5 ln y
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Deriving the MRS

• Thus,

Notice that the MRS is decreasing in x The MRS
falls when x increases
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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
• The algebra can usually be simplified by taking
  ln(). Lets do it in the blackboard.

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Examples of Utility Functions

• Perfect Substitutes
• U(x,y) U ?x ?y
• Y -(?/ ? )x (1/ ? )U , indifference curve for
  level U

The indifference curves will be linear. The MRS
(?/ ? ) is constant along the indifference
curve.
Quantity of y
Quantity of x
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Examples of Utility Functions

• Perfect Substitutes
• U(x,y) U ?x ?y
• dU ?dx ?dy
• Notice that the change in utility will be the
  same if (dx ? and dy 0) or if (dx 0 and dy
  ?). So x and y are exchanged at a fixed rate
  independently of how much x and y the consumer is
  consumed
• It is as if x and y were substitutes. That is why
  we call them like that

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Examples of Utility Functions

• Perfect Complements
• utility U(x,y) min (?x, ?y)

The indifference curves will be L-shaped. It is
called complements because if we Are in the
kink then utility does not increase by we
increase the quantity of only one good. The
quantity of both Goods must increase in order to
increase utility
Quantity of y
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