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7/30/2019 Lecture 1. Preferences and Utility
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Nicholson and Snyder, Copyright 2008 by Thomson South-Western. All rights reserved.
Preferences and Utility
Chapter 3
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Axioms of Rational Choice Completeness
if A and B are any two situations, an
individual can always specify exactly one ofthese possibilities:
A is preferred to B
B is preferred to A
A and B are equally attractive
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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
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Axioms of Rational Choice Continuity
if A is preferred to B, then situations suitably
close to A must also be preferred to B The assumption ensures the continuity of
the consumers preferences and rules out
jumps.
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Utility Given these assumptions, it is possible to
show that people are able to rank 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) > U(B)
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Utility Utility is affected by
the consumption of physical commodities
psychological attitudes peer group pressures
personal experiences
the general cultural environment
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Utility Economists generally devote attention to
quantifiable options while holding
constant the other things that affect utility ceteris paribusassumption
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Utility Assume that an individual must chooseamong consumption goods x1, x2,, xn
We can show his rankings using a utilityfunction of the form:
utility = U(x1, x2,, xn; other things)
Often other things are held constant
utility = U(x1, x2,, xn)
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Utility We can assume the individual isconsidering two goods, xand y
utility = U(x,y)
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Indifference Curves An indifference curve shows a set of
consumption bundles among which the
individual is indifferent
Quantity of x
Quantity of y
x1
y1
y2
x2
U1
Combinations (x1, y1) and (x2, y2)
provide the same level of utility
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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 x
Quantity of y
x1
y1
y2
x2
U1
1
UUdx
dyMRS
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Marginal Rate of Substitution MRS changes as xand ychange
reflects the individuals willingness to trade yforx
Quantity of x
Quantity of y
x1
y1
y2
x2
U1
At (x1, y1), the indifference curve is steeper.The person would be willing to give up more
yto gain additional units ofx
At (x2, y2), the indifference curveis flatter. The person would be
willing to give up less yto gainadditional units ofx
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Indifference Curve Map Each point must have an indifference
curve through it
Quantity of x
Quantity of y
U1 < U2 < U3
U1
U2
U3
Increasing utility
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Transitivity Can two of an individuals indifference
curves intersect?
Quantity of x
Quantity of y
U1
U2
A
BC
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
But B is preferred to A
because B contains more
xand ythan A
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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 x
Quantity of y
U1
The assumption of a diminishing MRS is
equivalent to the assumption that all
combinations ofxand ywhich are
preferred to x* and y* form a convex set
x*
y*
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Utility and the MRS Suppose an individuals preferences forhamburgers (y) and soft drinks (x) canbe represented by
yx 10utility
Solving fory, we get
y= 100/x
Solving for MRS = -dy/dx(along U1):MRS= -dy/dx= 100/x2
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Utility and the MRSMRS= -dy/dx= 100/x2
Note that as xrises, MRSfalls
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(x,y)
The total differential ofUis
dyy
Udx
x
UdU
Along any indifference curve, utility is
constant (dU= 0)
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Deriving the MRS Therefore, we get:
yUxU
dxdyMRS
constantU
MRSis the ratio of the marginal utility ofxto the marginal utility ofy
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Convexity of Indifference
Curves Suppose that the utility function is
yxutility
We can simplify the algebra by taking the
logarithm of this function
U*(x,y) = ln[U(x,y)] = 0.5 ln x+ 0.5 ln y
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Convexity of IndifferenceCurves
x
y
y
x
yU
xUMRS
5.0
5.0
*
*
Thus,
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Convexity of IndifferenceCurves
If the utility function is
U(x,y) = x + xy + y
There is no advantage to transforming
this utility function, so
x
y
yU
xUMRS
1
1
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Examples of Utility Functions
Cobb-Douglas Utility
utility = U(x,y) = xy
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 x
Quantity of y
U1U2
U3
The indifference curves will be linear.
The MRS will be constant along the
indifference curve.
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Examples of Utility Functions
Perfect Complements
utility = U(x,y) = min (x, y)
Quantity of x
Quantity of yThe indifference curves will be
L-shaped. Only by choosing more
of the two goods together can utility
be increased.
U1
U2
U3
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Examples of Utility Functions CES Utility (Constant elasticity of
substitution)
when
1,
0
utility = U(x,y) = x/ + y/
Perfect substitutes = 1
Cobb-Douglas = 0
Perfect complements = -
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Important Points to Note: If individuals obey certain behavioralpostulates, 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)
the rate at which an individual would trade
an amount of one good (y) for one more unitof another good (x)
MRSdecreases as xis substituted fory 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
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