Axioms of Rational Choice

1
Axioms of Rational Choice

• Completeness
• If A and B are any two situations, 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

2
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

3
Axioms of Rational Choice

• Continuity
• If A is preferred to B, then situations close
  to A must also be preferred to B
• Used to analyze individuals responses to
  relatively small changes in income and prices

4
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
• 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

• Utility rankings are ordinal in nature
• They record the relative desirability of
  commodity bundles
• Because utility measures are nonunique, it makes
  no sense to consider how much more utility is
  gained from A than from B
• It is also impossible to compare utilities
  between people

6
Utility

• Utility is affected by the consumption of
  physical commodities, psychological attitudes,
  peer group pressures, personal experiences, and
  the general cultural environment
• Economists generally devote attention to
  quantifiable options while holding constant the
  other things that affect utility
• ceteris paribus assumption

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Utility

• Assume that an individual must choose among
  consumption goods X1, X2,, Xn
• The individuals rankings can be shown by a
  utility function of the form
• utility U(X1, X2,, Xn)
• Keep in mind that everything is being held
  constant except X1, X2,, Xn

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Economic Goods

• In the utility function, the Xs are assumed to
  be goods
• more is preferred to less

Quantity of Y
Y
Quantity of X
X
9
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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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

• 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
12
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
13
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
15
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

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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