PREFERENCES AND UTILITY
1. Axioms of Rational Choice
Choices is to state a basic set of postulates that characterize
‘rational’ behaviour
This begins with the concept of preferences.
Basic properties of preferences:
a. Completeness: For all x and y in X, either x y or y x or
both
(the individual can always specify exactly one of
three possibilities)
b. Transitivity: For all x, y, and z in X, if
then x z
x y and y z ,
c. Continuity: If an individual reports “A is preferred to
B” then situations suitably close to A must also be
preferred to B.
For all y in X, the sets {x : x y}and {x : x y} are closed sets.
It follows that {x : x y} and {x : x y} are open sets.
d. Weak Monotonicity: If x y then x y
(at least as much of everything is at least as good)
1
e. Strong Monotonicity: If x y and xy, then x y .
(at least as much of every good, and strictly more of
some good, is strictly better)
f. Convexity: Given x, y, and z in X such that x z and y z
then it follows that tx  (1  t ) y z for all 0 t  1.
(Convexity means an agent prefers averages to
extremes)
2. Utility
Nonuniqueness of utility measures
 U(A) = 5 and U(B) = 4 --> A is preferred to B,
 but this lack of uniqueness in the assignment of utility
numbers also implies that it is not possible to compare
utilities of different people
 If person A values the utility of dinner is 5 and person B
values the utility of dinner is 100, then we could not say
which individual values the dinner more, because they could
be using very different scales.
 No way of measuring whether a move from situation A to
situation B provides more utility to one person or another
The ceteris paribus assumption
 Utility refers to overall satisfaction --> affected by many
factors
2
 A common practice is to devote attention to choices among
quantifiable options (such as: numbers of food consumed,
number of hours worked, etc) while holding the other things
constant
 Ceteris paribus in mathematical expression:
utility  U ( x1 , x2 ,.....; other things)
constant
Therefore, we express it as: utility  U ( x1 , x2 ,.....)
What is utility function?
Utility function is used to indicate how an individual ranks certain
bundles of goods that might be purchased at a point of time.
Expression of some utility functions:
Utility  U (W )
W  real wealth
Utility  U (c, h)
c  consumption
h  hour of work
Utility  U (c1 , c2 )
c1  consumption of good 1
c2  consumption of good 2
3
UTILITY = individual’s preferences, which is represented by a
utility function of the form:
U ( x1 , x2 ,..., xn )
Economic goods: more of a good is preferred to less
Quantity of y
Preferred to
x*, y*
y*
worse
x*
Quantity of x
Trades and Substitution
Most economic activities involves voluntary trading between
individuals. When someone buys, a loaf of bread, she will give up
some money, as the bread has a greater value for her.
4
Indifference Curves
Quantity of y
y2
y1
U
x2
x1
Quantity of x
The curve U represents the combination of x dan y from which the
individual derives the same utility.
The slope of this curve represents:
= the rate at which the individiual is willing to trade x for y while
remaining equally well off.
= Marginal rate of substitution
Quantity of y
MRS  
MRS
dy
dx U U1
y2
y1
U
x2
x1
Quantity of x
5
The slope of indifference curve is negative, what does it mean?
 Individual has to give up something to get more of other
good
INDIFFERENCE Curve map
Quantity of y
Increasing utility
y2
U3
y1
U2
U1
x2
x1
Quantity of x
Question:
Can any two of an individual’s indifference curve intersect?
6
Quantity of y
C
D
E
A
U1
U2
B
Quantity of x
Assumption:
- Rational preference
- Nonsatiation (i.e more of goods always increases utility)
Convexity of Indifference Curves
An alternative way of stating the principle of a diminishing
marginal rate of substitution uses the mathematical notion of a
convex set.
The assumption of a diminishing MRS
= the assumption that all combinations of x and y that are preferred
or indifferent to a particular combination x*,y* form a convex set.
= any line joining two points above U1 is also above U1
7
What does it mean by convexity?
 Individuals prefer some balance in their
consumption --> the new combination will be
preferred to the initial combination
 A more well-balanced bundles of commodities are
preferred to bundles that are heavily weighted
toward one commodity
 Show Figure 3.6
Example:
1. utility  xy  100
Show:
- MRS
- is it diminishing?
2. Show the derivation of MRS from utility function of U(x,y)
8
3. Show the interpretation of MRS in this figure:
Quantity of y
U  xy  10
A
20
C
12,5
B
5
5
U=10
12,5
Utility Functions For Specific Preferrences
a. Cobb-Douglas Utility
Quantity2of y
0
U ( x, y)  x y 
U3
U2
 and  are:
U1
 positive constants
Quantity of x
 normalized to 1 ( +  =1)
 representing the relative importance of the two goods to
an individual
b. Perfect Substitutes
The linear function: U ( x, y)   x   y
9
Interpretation of this function:
 Constant MRS
 A person with these preferences would be willing to give up
the same amount of y to get one more of x, no matter how
much x was being consumed.
 Example: consumption of gasoline
Quantity2of y
0
U3
U2
U1
Quantity of x
c. Perfect Complements
U ( x, y )  min( x,  y )
Quantity2of y
0
U3
U2
U1
Quantity of x
Interpretation:
 Utility is given by the smaller of the two terms in the
parentheses
 More generally, neither of the two goods will be in
excess. The equilibrium will be at the corner of the
function; hence:  x   y or y  
x

10
 Therefore, there is the fixed proportion between the two
goods --> this is at the vertices of the indifference curve
d. CES Utility
The three specific utility functions illustrated above are special
cases of the CES utility, which is:
U ( x, y ) 
x


y

; when   1,   0
or
U ( x, y )  ln x  ln y; when   0
When  = 1 --> perfect substitutes
When  = 0 --> cobb-douglas
Note:
All of the utility functions described above are homothetic, that is,
the marginal rate of substitution for these functions depends only
the ratio of the amounts of the two goods, not on the total
quantities.
The importance of homothetic function:
 One indifference curve is much like another
 Slopes of the curves depend only on the ratio of y/x, not
on how far the curve is from the origin
 Hence, we can only study one indifference curve to
study the behaviour of an individual
11
Example of nonhomothetic case:
1. Consumer Preferences
X = consumption set
Asumsi: X = a closed and convex set
Note:
Convexity: if x and x’ are in V(y), then tx+(1-t)x’ is in V(y)
for all 0 t  1. Therefore, V(y) is a convex set.
Konsumen diasumsikan
consumption bundles X.
memiliki
preference
dalam
Simbol-simbol:
x y = the bundles x is at least as good as the bundle y
x y = x is strictly preferred to y.
x y = x and y is indifference, if and only if x y and y x
12
Beberapa asumsi yang sering digunakan:
Indiffirence curve = a set of all consumption bundles that
are indifferent to each other
Example:
Let u(x1,…., xk) be a utility function. Suppose that we
increase the amount of good i; how does the consumer have
to change his consumption of good j in order to keep utility
constant?
2. Consumer Behaviour
Basic hypothesis = a rational consumer will always choose
a most preffered bundle from the set of affordable
alternatives.
13
The set of affordable alternatives is just the set of all
bundles that satisfy the consumer’s budget constraints.
Let:
m= the fixed amount of money available
p=(p1, …., pk) = the vector of prices of goods 1,…, k
So: the set of affordable bundles, the budget set of the
consumer is given by:
B  x in X : px  m
The problem of preference maximization:
Max u(x)
Such that px m
x in X
Under an assumption of the local nonsatiation, the
consumer’s problem can be restated as:
v(p,m)=max u(x)
such that: px=m
v(p,m)  indirect utility function
x = then consumer’s demanded bundle
x(p,m) = the consumer’s demand function
14
HOW TO SOLVE THE CONSUMER’S PROBLEM?
 Use Lagrangian
The Lagrangian for the utility maximization problem:
L  u ( x)   ( px  m)
=the Langrange multiplier.
Solution:
FOC (differentiating the Lagrangian with respect to xi):
u ( x)
p  0
for i=1,…, k
xi
i
Divide the ith FOC by the jth FOC:
 ( x*)
 xi
 ( x*)
x j

pi
pj
for i,j = 1,…, k
Graphical illustration for two goods:
15
Budget line of the consumer: x : p1x1  p2 x2  m
3. Indirect Utility
Indicrect utility = v(p,m)  gives maximum utility as a
function of p and m
Properties of the indirect utility function:
(1)
v(p, m) is nonincreasing in p; that is, if p’ p,
v(p’, m)  v(p, m). Similarly, v(p, m) is nondecreasing
in m.
PROOF:
16
(2)
v(p,m) is homogenous of degree 0 in (p,m)
PROOF:
(3)
v(p,m) is quasiconvex in p; that is  p : v( p, m)  k is a
convex set for all k
PROOF:
(4)
v(p,m) is continous at all
p
0,
m>0
See Figure 7.2 (Price indifference curves)
17
See Figure 7.3 (Utility as a function of income)
Expenditure Function (The inverse of the indirect
utility function)
Expenditure function:
e(p, u) = min px
Such that: u(x)  u
Expenditure function = relates income and utility
= gives the minimum cost of
achieving a fixed level of utility
Properties of the expenditure function:
(1)
(2)
(3)
e(p,u) is nondecreasing in p.
e(p,u) is homogenouse of degree 1 in p
e(p,u) is concave in p.
18
(4)
e(p,u)is continous in p, for p>> 0.
(5)
If h(p,u) is the expenditure-minimizing bundle
necessary to achieve utility level u at prices p, then:
hi ( p, u ) 
e( p, u )
pi
for i= 1,…,k
 HICKSIAN
DEMAND FUNCTION
(assuming: derivatives exist and pi>0)
Hicksian demand function: h ( p, u)
 not directly
observable
Marshallian demand function: x( p, m)  ordinary market
demand function
i
4. Some Important Identities
UTILITY MAX PROBLEM:
EXPENDITURE MIN PR
v(p,m*) = max u(x)
such that: px m*
e(p,u*) = min px
such that: u(x) u*
FOUR IMPORTANT IDENTITIES:
19
(1)
e(p, v(p,m))  m
The minimum expenditerue necessary to reach utility
v(p,m) is m.
(2)
v(p, e(p,u))  u
The maximum utility from income e(p,u) is u
(3)
xi(p,m)  hi (p, v(p,m))
The Marshallian demand at income m is the same as
the Hicksian demand at utility v(p,m).
(4)
hi (p,u) xi(p,e(p,u))
The Hicksian demand at utility u is the same as the
Marshallian demand at income e(p,u)
Identity (4) shows that the Hicksian demand function—the
solution to the expenditure minimization problem—is equal
to the Marshallian demand function at an appropriate level
of income.
Roy’s Identity
If x(p,m) is the Marshallian demand function, then:
v( p, m)
pi
xi ( p, m)  
v( p, m)
m
for i=1, …, k
20
Proof:
5. The Money Metric Utility Functions
Min pz
Such that u(z) u(x)
Figure:
Money metric utility function = gives the minimum
expenditure at price p necessary to purchase a bundle at
least as good as x.
21
An alternative definition (minimum income function):
m( p, x)  e( p, u ( x))
Indirect money metric utility function:
 ( p; q, m)  e( p, v(q, m))
Figure:
Example (The Cobb-Douglas Utility Function):
22
23