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Chapter 4. Consumer Theory II
Two essential tools of analysis in the modern treatment of consumer theory:
 The Indirect Utility Function
 The Expenditure Function
4.1 Indirect Utility and Expenditure
4.1.1 The Indirect Utility Function
The direct Utility Function: The ordinary utility function, u(x), is defined over the
consumption set X and represents the consumer’s preferences directly.
The indirect Utility Function:
 The problem: max u ( x) s.t. y  p.x, x  0
x
 the solution x* =x*(p,y)  The Indirect Utility Function: u(x*) = u*(p,y) = v(p,y)
 The indirect Utility Function represent the relation between prices, income and the
highest level of utility achieved.
 v : R n 1  R defined as: v( p, y )  max u ( x) s.t. y  p.x  0, x  0.
x
 v(p,y) is called indirect utility function.
 This function is clearly well-defined since, when preferences are monotonic and
strictly convex, a unique solution x(p,y) to the consumer’s problem exists.
 In the maximization problems max u ( x) s.t. y  p.x, x  0 , continuity of the
x
constraint function in the parameters is sufficient to guarantee that v(p,y) will be
continuous in p and y.
Theorem 4.1.1 Properties of the Indirect Utility Function
Let preferences be monotonic and differentiable, and let p>>0 and y>0. Then v(p,y) has
these properties:
1. Homogeneous of degree zero in p and y
2. Increasing in y
3. Non-increasing in p
4. Quansiconvex in p
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Proof:
1. Homogeneous of degree zero in p and y:
Equiproportionate changes in all p and y leave the consumer’s budget set unchanged.
y  p1 x1  p2 x2  ty  tp1 x1  tp2 x2
(t  0)
The set of feasible choices, and so the maximal level of utility the consumer can achieve,
must therefore also remain the same.
Changing all p an y by proportion t>0 must leave the maximal utilitly unchanged.
2. Increasing in y and Non-increasing in p
Considering the Lagrangian for the utility maximization problem: L( x,  )  u ( x)  [ y  p.x]
By the Envelop Theorem, we have:
v( p, y ) L

pi
pi
 x*  0
( x*, )
v( p, y ) L

 0
y
y ( x*, )
as the proof.
3. Quansiconvex in p
The Lagrangian multiplier (  ) will measure the sensitivity of the objective function u(x) to
changes in the constraint constant (y). (See the Exercise 2.29).
Thus the value of the Lagrangian multiplier at the solution measure the marginal utility of
income.
Let B1, B2, and Bt be the budget sets available when the consumer has income y and faces
prices p1, p2, and p t  tp1  (1  t ) p 2 , then:
B 1  {x | p 1 .x  y}
B 2  {x | p 2 .x  y}
B t  {x | p t .x  y}.
We need to show that: v( p t , y)  max[ v( p1 , y), v( p 2 , y)]
So we will show that every choice the consumer can possibly make when she faces budget B t
is a choice which could have been made when she faced either budget B 1 or budget B 2 . It
would be the case that every level of utility she can achieve faving B t is a level she could
have achieved either when facing B 1 or when facing B 2 . Then the maximum level of utility
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that she can achieve over B t could be no larger than at least one of the maximum level of
utility that she can achieve over B 1 or the one she can achieve over B 2 .
We want to show that if x  B t , then x  B 1 or x  B 2 for all t  [0,1] .
It is easy to realize that if t=1 or t=0  x  B t , then x  B 1 or x  B 2 .
For t  (0,1) . Suppose that if x  B t , then x  B 1 and x  B 2 , then
x. p1  y and x. p 2  y . Cos’s t  (0,1)  t>0 and (1-t)>0
 t.x. p1  t. y and (1  t ).x. p 2  (1  t ) y  t.x. p1  (1  t ).x. p 2  y  x. p t  y
 x  B t  contradicting our orginal assumption.
 if x  B t , then x  B 1 or x  B 2 for all t  [0,1] .
 v( p t , y)  max[ v( p1 , y), v( p 2 , y)]  v(p,y) is quansiconcave function in p ./.
 The indirect utility function tells us the maximal level of utility the consumer can
achieve facing different prices and incomes.
 The demand functions give us the utility maximizing choices of each commodity he
will make facing different prices and incomes.
 To get the indirect utility function, we simply substitute the demand functions into the
direct utility function.
Theorem 4.1.2 Roy’s Identity
 To get the indirect utility function, we simply substitute the demand functions into the
direct utility function.
 There is a question that how to derive the direct utility function from the indirect
utility function ? This theorem will answer this question.
Theorem: Let v(p,y) be any indirect utility function satisfying the conditions of Theorem
4.1.1. Then,
xi ( p, y )  
v( p, y ) / pi
v( p, y ) / y
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Fifth Property of an indirect utility function:
 An indirect utility function is demand – generating.: any demand function can be
generated from the indirect utility function.
Proof:
Let x* and  solve the Kuhn – Tucker conditions. The solution    ( p, y ) gives us the
marginal utility of income at the consumer equilibrium and xi *  xi ( p, y) gives the
consumer’s demand function for good i.
By the Envelope Theorem, we have:
v( p, y ) L

pi
pi
  ( p, y ) x i ( p, y )
( x*, )
v( p, y ) L

  ( p, y )
y
y ( x*, )

xi ( p, y )  
v( p, y ) / pi
v( p, y ) / y
4.1.2 The Expenditure Function
 What is the minimum level of money expenditure, or outlay, which the consumer
must make facing a given set of prices in order to achieve a given level of utility?
 In this construction, we ignore any limitations imposed by the consumer’s income
and simply ask what the consumer would have to spend in order to achieve some
particular level of utility.
X2
X1
 Iso – expenditure curve: e  p1 x1  p2 x2
p.x
 The problem: e( p, u )  min
x
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s.t. u  u ( x)  0,
x0
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 The solution is x h ( p, u) that depends on prices and utility level.
 If preferences are monotonic and strictly convex, the solution will be unique.
 The lowest expenditure necessary to achieve utility u at prices p will be equal to cost
of the bundle x h ( p, u) : e( p, u )  p.x h ( p, u ) .
 We can use the consumer’s expenditure minimization problem to explore a very
different kind of “demand behavior”, this one entirely unobservable or hypothetical.
It is different from the Marshallian demand which is observable.
 If we fix the level of utility the consumer is permitted to achieve at some arbitrary
level u, how will his purchases of each good behave as we change the prices he faces?
 Utility – constant demand functions
Hicksian Demand Functions:
 Fix the utility level and solve the problem:
e( p, u )  min p.x s.t. u  u ( x)  0,
x
x0
to
have
consumption
bundle
will
change:
h
x1 ( p, u ) : x1h ( p10 , p 20 , u ) , x 2h ( p10 , p 20 , u ) .
 Change
price
p1,
the
optimal
choice
h
x 2 ( p, u ) : x1h ( p11 , p 20 , u ) , x 2h ( p11 , p 20 , u ) .
Theorem 4.1.3 Properties of the Expenditure Function
Let preferences be monotonic and let p>>0. Let u>u(0) and let e(p,u) be defined as in
4.1.3. Then e(p,u) is:
1. Increasing in u
2. Non-decreasing in p
3. Homogeneous of degree 1 in p
4. Concave in p
5. Also, the price partial derivatives of e(p,u) are the Hicksian demand functions
e( p, u )
 xih ( p, u )
pi
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Proof:
p.x
The problem: e( p, u )  min
x
s.t. u  u ( x)  0,
x0
Lagrangian for this problem: L( x,  )  p.x  [u  u ( x)]
If x h  x h ( p, u )  0 solves the minimization problem, then x h and  satisfy the KuhnTucker conditions:
L
u ( x h )
 pi  
0
xi
xi
xih
L
u ( x h )
 xih [ pi  
]0
xi
xi
L
 u  u( x h )  0

L

 [u  u ( x h )]  0

u ( x h )
 0 for at least one
From the Kuhn-Tucker conditions we must therefore have: p j  
x j
j.
By the monotonicity of preference, we have:
pj
u ( x h )
0  
 0.
x j
u ( x h )
x j
Prove the property 1:
By the Envelop theorem:
e( p, u ) L

u
u
 0
( x , )
h
 e(p,u) is increasing in u.
Prove the properties 2 and 5:
By the Envelop theorem we have:
e( p, u ) L

pi
pi
 xih ( p, u )  0
( x h , )
 e(p,u) is in-decreasing in p.
Prove the property 3: e(p,u) is homogeneous of degree 1
e( p, u )  p1 .x1h  p 2 .x 2h    p n .x nh  e(t. p, u )  t.e( p, u )
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 e(p,u) is homogeneous of degree 1.
Prove the property 4: e(p,u) is concave in p.
We need to prove that: e( p t , u)  t.e( p1 , u)  (1  t ).e( p 2 , u)
Let x1 , x 2 , x t minimize expenditure to achieve u when prices are p1 , p 2 , p t respectively.
By the definition of e(p,u), we have:
p 1 .x1  p 1 .x t  tp1 .x 1  tp1 .x t
p 2 .x 2  p 2 .xt  (1  t ) p 2 .x 2  (1  t ) p 2 .x t
 t.e( p 1 , u )  (1  t ).e( p 2 , u )  e( p t , u )
4.1.3 Relations between the Two
From the definitions of the expenditure function:
e( p, v( p, y ))  min p.x s.t. u ( x)  v( p, y )
e( p, v( p, y ))  min p.x s.t. u ( x)  max u ( x' ) s.t.
x
x
p.x'  y
We must of course have: e( p, v( p, y ))  y .
From the definition of the indirect utility function:
v( p, e( p, u ))  max u ( x) s.t.
x
p.x  e( p, u )
Substituting from the definition of the expenditure function, we have:
v( p, e( p, u ))  max u ( x) s.t.
x
p.x  min p.x' s.t. u ( x' )  u
x'
We must have: v( p, e( p, u ))  u
Theorem 4.1.4 Identities Relating Indirect Utility and Expenditure Functions
Let v(p,y) and e(p,u) be the indirect utility function and expenditure function for some
consumer. Then the following relations between the two obtain for all prices p, incomes y,
and utility level u:
e( p, v( p, y ))  y
v( p, e( p, u ))  u
 This theorem points us to an easy way to derive either one directly from knowledge of
the other, thus requiring us to solve only one optimization problem and giving us the
choice of which one we care to solve.
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 v(p,u) is strictly increasing in y.
v( p, e( p, u ))  u . Invert the indirect utility function in its income variable, we have:
 e( p, u)  v 1 ( p : u)
e( p, u ) is strictly increasing in u.
e( p, v( p, u ))  u . Invert the expenditure function in its utility variable, we have:
v( p, u)  e 1 ( p : y)
 Example: The CES direct utility function gives the indirect utility function:
v( p, y)  ( p1r  p2r ) 1 / r . y
For an income level equal to e(p,u), we must have: v( p, e( p, u))  ( p1r  p2r ) 1 / r .e( p, u)
By the theorem, we have ( p1r  p2r ) 1 / r .e( p, u)  u  e( p, u )  ( p1r  p2r )1 / r .u
Theorem 4.1.5 Identical Relations Between Marshallian and Hicksian Demand
Functions
For any prices p, income y, and utility level u, the following identical relations hold
between the consumer’s Hicksian and Marshallian demand functions:
xi ( p, y)  xih ( p, v( p, y))
for all p, y and i  1,2,, n
xih ( p, u )  xi ( p, e( p, u ))
for all p, y and i  1,2,, n
Example:
The Hicksian demands are: xih ( p, u )  ( p1r  p2r )1/ r )1 pir 1u
r
r 1 / r
y
The indirect utility function is: v( p, y )  ( p1  p 2 )
We have: xi ( p, y )  xih ( p, v( p, y )  ( p1r  p 2r ) (1 / r )1 pir 1 ( p1r  p 2r ) 1 / r y 
pir 1 y
p1r  p 2r
4.2 Properties of Consumer Demand
In statistically estimating consumer demand systems, characteristics of demand behavior
predicted by the theory are used to provide restrictions on the values which estimated
parameters are allowed to take, thereby ensuring that the empirical estimates are at least
logically consistent with the underlying theory from which they are constructed.
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 Relative Price and Real Income
 Economists generally prefer to measure important variables in real.
 Relative prices and real income are two such real measures
 Relative price: By the relative price of some good, we mean the number of units of
some other good which must be foregone in order to acquire 1 unit of the good in
question.
pi
$ / unit i unit j


p j $ / unit j unit i measure the units of good j forgone per unit of good i acquired.
 Consumer’s real income: We mean the total number of units of some commodity
which could be acquired if the consumer spent his entire money income on that
commodity.
Real income interm of good j:
y
$

 units of j
p j $ / unit of j
Theorem 4.2.1 Homogeneity
The consumer’s demand function xi ( p, y ) , are homogeneous of degree zero in all
prices and income.
Proof:
Equiproportionate changes in all p and y leave the consumer’s budget set unchanged.
y  p1 x1  p2 x2  ty  tp1 x1  tp2 x2
(t  0)
So the optimal point is unchanged.
 xi ( p, y)  xi (t. p, t. y)  the consumer’s demand function xi ( p, y ) is homogeneous of
degree zero.
Application:
With t 
1
0
pn
We have:
xi ( p, y)  xi (tp, ty)  xi (
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p
p1
y
,, n1 ,1, )
pn
pn
pn
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Demand for each of the n goods depends only on (n-1) relative prices and the consumer’s
real income.
4.2.2 Income and Substitution Effects
Ordinarily, a consumer will buy more of a good when its price declines, and less when its
price increases. However, these cases are not necessarily true.
Substitution effect:
 Since all goods are taken to be desirable by the consumer, even if the consumer’s total
command over goods were unchanged, we would expect him to substitute more of the
good which has become relatively cheapter for less of the goods which are now
relatively more expensive.
 The substitution effect is that (hypothetical) change in consumption which would
occur if relative prices were to change to their new level but the maximum level of
utility the consumer can achieve were kept the same as before the price changes.
 Change price of good 1 from x10 to x11 , the problem is that:
min p11 x1  p20 x2
s.t. u( x)  u 0
Income effect:
 When the price of any one good declines, the consumer’s total command over all
resources is effectively increased, allowing him to change his purchases of all
goods in any way he sees fit. The effect on quantity demanded of this generalized
increase in purchasing power is called the income effect.
 The income effect is defined as the residual out of the total effect which is left
after the substitution effect.
Total effect:
 Change price of good 1 from x10 to x11 , the problem is that:
max u( x) s.t.
p11 x1  p20 x2  y
Slutsky Equation – Fundamental Equation of Demand Theory: the general analytical
relationships between total effect, substitution effect, and income effect are summarized
by the Slutsky Equation.
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Theorem 4.2.2 The Slutsky Equation
Let x(p,y) be the consumer’s Marshallian demand system. Let u* be the level of utility the
consumer achieves at prices p and income y. Then,
x j ( p, y )
p
i


x hj ( p, u*)
x j ( p, y )
 x i ( p, y )
p
y


i 
TE
IE
SE
Proof:
The identity linking the Marshallian and Hicksian demand function:
x hj ( p, u*)  x j ( p, e( p, u*))

x hj ( p, u*)
pi

x j ( p, e( p, u*))
pi

x j ( p, e( p, u*)) e( p, u*)
e( p, u*)
pi
(1)
By the assumption , u* is the level of utility the consumer achieves facing prices p and
having income y (see the theorem)  u*=v(p,y)
 the minimum expenditure at prices p and utility u* will therefore be the same as the
minimum expenditure at price p and utility v(p,y)
e( p, u*)  e( p, v( p, y ))  y (2)
We have:
e( p, u*)
 xi ( p, y ) (3)
pi
Substitute (2) and (3) into (1), we have:
x j ( p, y )
p
i

TE

x hj ( p, u*)
x j ( p, y )
 x i ( p, y )
p
y


i 
SE
IE
Theorem 4.2.3 Negativity of Own-Substitution Terms:
Let xih ( p, u ) be the Hicksian demand for good i. Then,
xih ( p, u )
0
pi
Proof:
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e( p, u )
 xih ( p, u )
pi
 2 e( p , u )
pi
2
xih

pi
By the theorem 4.1.3, the expenditure function is a concave function of p./.
Norminal or superior goods: consumption increases as real income increases, holding
relative prices constant.
Inferior goods: consumption decreases as real income increases, holding relative prices
constant.
Theorem 4.2.4 The Law of Demand
Let preferences be complete, transitive, reflexive, monotonic, and strictly convex. If a good
is a normal good, then a decrease in price will cause an increase in quantity demanded. If a
decrease in price causes a decrease in quantity demanded, then the good must be an
inferior good.
Theorem 4.2.5 Symmetry of Substitution Terms
Let x h ( p, u) be the consumer’s system of Hicksian demands. Then,
h
xih ( p, u ) x j ( p, u )

p j
pi
Proof:
xih ( p, u )  2 e( p, u )

p j
pip j
By the Young’s Theorem:
 2 e( p , u )  2 e( p , u )

pi p j
p j pi
h
xih ( p, u ) x j ( p, u )


p j
pi
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Theorem 4.2.6 Negative Semi-Definiteness of the Substitution Matrix
 x h ( p, u ) 
Let x h ( p, u) be the consumer’s system of Hicksian demands, and let  i

 p j  i , j 1n
represent the entire matrix of Hicksian substitution terms. Then this matrix is negative
semi-definite.
Proof:
xih ( p, u )  2 e( p, u )

p j
pip j
The expenditure functions is concave in prices. From the theorem 2.1.3, the matrix of secondorder partials (the Hessian) of a concave function is negative semi-definite
Theorem 4.2.7 Negative Semi – definiteness of the Slutsky Matrix
Let x(p,y) be the consumer’s Marshallian demand system. Define the Slutsky matrix as the
n  n matrix of price and income responses given by:
 xi ( p, y )
x ( p, y ) 
 x j ( p, y ) i


y 
 p j
i , j 1,n
Then the theory of the preference-maximizing, atomistic consumer requires that the Slutsky
matrix be negative semi – definite.
4.2.3 Some Elasticity Relations
If preferences are monotonic, at least one good will be bought in a positive amount.
Definition 4.2.1 Demand Elasticities and Income Shares
Let xi ( p, y ) be the consumer’s Marshallian demand for good i. Then let
Income elasticity:  i 
Price elasticity:  ij 
xi ( p, y)
y
y
x i ( p, y )
xi ( p, y ) p j
p j
xi ( p, y )
And Income Share: si 
p i x i ( p, y )
so that si  0 and
y
n
s
i 1
i
 1.
Theorem 4.2.8 Aggregation in Consumer Demand
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Let x(p,y) be the consumer’s Marshallian demand system. Let  i ,  ij , and si be income
elasticity, cross-price elasticity and income share. Then the following relations must hold
between income shares, price, and income elasticities of demand:
n
Engel Aggregation:
s
i
i 1
i
1
n
Cournot Aggregation:
s 
i 1
i
ij
 s j
Proof:
Prove (1)
y  p.x( p, y )
n
 1   pi
i 1
n
n
xi ( p, y)
p x x ( p, y) y
 i i i
  si i
y
y
y
xi i 1
i 1
Prove (2)
y  p.x( p, y ) . Differentiating both sides with respect to p j
n
0   pi

i 1
n
xi ( p, y )
 xj
p j
 x j   pi
i 1
xi ( p, y )
p j
Multiply both sides of the equation by p j / y and get:

xj pj
y
n

i 1
n
n
pi xi ( p, y )
p x x ( p, y ) p j
pj   i i i
  si  ij
y
p j
y
p j
xi
i 1
i 1
n
  si  ij   s j
i 1
4.3 Duality In Consumer Theory
There is a question: Starting with an expenditure or an indirect utility function, can we “work
backwards” to discover the underlying direct utility function that would have generated it?
This question is answered by following the mathematical “duality” between various
optimization problems used to describe consumer behavior.
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4.3.1 Expenditure and Consumer Preferences
4.3.2 Indirect Utility and Consumer Preferences
 The duality between direct and indirect utility functions.
 Suppose we are given a continuous function v(p,y), homogeneous of degree zero
in p and y, increasing in y, non-increasing and quasiconvex in p. It can be shown
that there exists a non-decreasing, quasiconcave direct utility function u*(x) which
rationalizes v(p,y).
 v( p, y ) is homogeneous of degree zero in p and y  v(tp, ty)  v( p, y ) . Let
t  1 / y  v( p / y,1)  v( p, y )  v( ~
p )  v ( p, y )
p ) is called normalized indirect utility function, and depends on normaliz
 v( ~
price alone.
 With direct utility function u(x), the normalized indirect utility function is defined
as:
v( ~
p )  max u ( x) s.t. ~
p.x  1
x
Thetheorem 4.3.4 Duality Between Direct and Indirect Utility
Let v( p, y ) be any indirect utility function, and form the normalized indirect utility
p ) . Then the implied direct utility function is given by the following minimum
function v ( ~
value function:
u( x)  min
v( ~
p) s.t. ~
p.x  1 .
~
p
Proof:
Theorem 4.3.5 (Hotelling, Wold) Duality and the System of Inverse Demands
Let u(x) be the consumer’s direct utility function. Then the inverse demand function for
good i is given by:
~
pi ( x) 
u ( x) / xi
n
x
j 1
j
(u ( x) / x j )
Proof:
p)
From the normalized indirect utility function, v ( ~
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u( x)  min
v( ~
p) s.t. ~
p.x  1
~
p
The associated Lagrangian is:
L  v( ~
p )  [1  ~
p.x]
u ( x) L

Applying the Envelope Theorem:
x j
x j
  * ( x). ~
p j * ( x) (1)
~
p*( x ), *( x )
Multiplying both side by x j and summing from j  1,..., n gives:
n
u ( x)
xj
  * ( x )  ~
p j * ( x).x j   * ( x) (2)

x j
j 1
j 1
n
From (1) and (2), we have:
~
p j * ( x) 
u ( x) / x j
n
 x (u( x) / x )
j 1
j
j
4.4 Uncertainty
Certainty: the consumer knows the prices of all commodities and knows that any feasible
consumption bundles can be obtained with certainty.
Many eoconomic decisions contain some element of uncertainty: future income, future
prices…
4.4.1 Preferences
 Beforem, the consumer was assumed to have a preference ordering over all
consumption bundles x in the consumption set X. Implicit in our statement that
“bundle xi is preferred to bundle xj” was the assumption the individual chooes between
xi with certainty and xj with certainty.
 Instead of ordering consumption bundles, the individual is assumed to have a
preference ordering over gambles.
 Let’s first define an outcome as a result of some uncertain situation. For example,
outcomes of betting are win and loss.
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 A  {a1 , a2 ,..., an } : the set of all mutually exclusive ultimate outcomes that an
individual could endup with.
a i could be an m-dimensional commodity vector, or alternatively, a scalar. The way we
characterize A depend upon the nature of the particular problem we wish to address.
 Gambles: G  [ p1oa1p2 oa2  pn an ] denotes the entire gamble involving a1 with
probability p1 and a 2 with probability p 2 and so forth.
Where:
p1oa1 denotes the outcome a1 with its probability of occurrence p1 .
 denotes “and” – the logical symbol.
Definition 4.4.1 The Space of Gambles, g(A)
The space g(A) is the set of all possible gamble which can be constructed from the outcome
set A by varying the probabilities 0  pi  1 of each ai  A while ensuring that

n
i 1
pi  1 .
Since each a i is the special gamble in g(A) where pi  1 and p j  0 , i  j , the set of all
ultimate outcomes A is itself a subset of g(A).
The problem of choice under uncertainty can be veiwed as a choice between alternative
gambles in g(A).
We can then define a binary relation  on g(A). Where the symbol  stands for the
statement “is atleast as well as”.
We again suppose that these preferences obey certain rules which we’ll lay down in the form
of axiom, called the “axioms of choice under uncertainty”.
Axiom G1: (Completeness) For any distinct gambles G1 and G2 in g(A), either G1  G2 or
G2  G1.
Axiom G2: (Reflexivity) For any gamble G  g ( A), G  G .
Axiom G3: (Transitivity) For any three gambles G1, G2, and G3 in g(A), if G1  G2 and
G2  G3 , then G1  G3 .
With the addition of Axiom G3,  gives a complete ordering of gambles.
One important consequence of this Axiom is that there must exist a best and a worst outcome
in A. Note that best and worst outcomes need not be unique.
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G  [ p1oa1p2 oa2  pn an ]  if all p j  0 and pi  0 ( i  j )  G  ai  By the
Axiom 3, there therefore must exist a best and a worst outcome in A. A best outcome aB  A
satistfies a B  a j for all a j  A . A worst outcome aW  A statisfies a w  a j for all a j  A .
Axiom G4: (Continuity) For any gamble G  g ( A) , there exists some probability z,
0  z  1, such that G ~ [ z o a B  (1  z ) o aW ] .
Indifference probability
Best – Worst gamble [ z o aB  (1  z ) o aW ]
For any gamble G there is some other gamble, involving only the best and the worst outcomes
in A, which the agent ranks indifferent to G.
Axiom G5. (Monotonicity) For any two best-worst gambles, G1  [ poa B  (1  p)oaW ] and
G2  [qoa B  (1  q)oaW ] , we have G1  G2 if and only if p  q .
Axiom G5 states that given the choice between any two best – worst gambles with different
probabilities attached to the same best outcome, an individual will never prefer the gamble
with the lower probability of the best outcome.
Together, Axiom G4 and G5 rule out some kinds of behavior which, at first glance, might
appear quite reasonable. Example, let A = {“death”, $10, $1000}; aW  death , a B  $1000
and a B  $10  aW . Consider the intermediate gamble G3 = $10. According to axiom G4,
there must be some best-worst gamble such that G3 ~ [ z o a B  (1  z ) o aW ] and (1-z)>0. If
there is no strictly positive probability of death at which you would prefer the gamble
[ z o $1000  (1  z ) o death] to $10 with certainty, then the preferences violate the combined
implications of Axiom G4 and G5.
Axiom G6. (Substitutability) For any outcome ai  A and any gamble G j  g ( A) , if
ai ~ G j , then
[ p1oa1  pi oai   pn oan ] ~ [ p1oa1  pi oG j   pn oan ]
Axiom G6 states that if the individual is indifferent between and outcome promised with
certainty and some gamble, then he mus also be indifferent between two otherwise identical
gambles offering each of these with the same probability.
Axiom G7. (Net Probability Rule) Let the gamble
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Gi  [ p1oa1   pi oG j   p n oa n ]
Where the gamble
G j  [q1oa1   qi oai   q n oa n ]
Then
Gi ~ [( p1  q1 pi )oa1   pi qi oG j   ( p n  q n pi )oa n ]
4.4.2 Von Neumann – Morgenstern Utility
Whether we can represent those preferences with a continuous real valued function? Say Yes!
Axiom G1, G2, G3 and some kind of continuity assumption should be sufficient to ensure the
existence of a simple numerical function representing  .
Instead of asking whether there is a certain kind of function, with a certain specific
mathematical property, representing  .
Let G  [ p1oa1   pi oai   p n oa n ] be any gamble in g(A), and suppose that the function
U : g ( A)  R represents the preference  by assigning larger numbers to preferred gambles
and equal numbers to indifferent gambles. Then, of course, U is a utility function in the usual
sence. But if, in addition, the numbers assigned by U to gambles G satisfy:
n
U (G )   p i U (ai ) , we say that the utility function U possesses the extra, expected
i 1
utility property.
A Utility Function possesses the Expected Utility Property if and only if the Utility number it
assigns to any gamble can be expressed as the Expected Value of the Utility numbers it
assigns to the Ultimate outcomes in that gamble.
Theorem 4.4.1 Existence of a Von Neumann – Morgenstern (VNM) Utility Function over
Gambles
Let preferences over gambles,  , satisfy Axiom G1 through G7. Then there exists a
function U : g ( A)  R such that, for all G1 and G2 in g(A), G1  G2 if and only if,
U (G1 )  U (G2 ) , and where, moreover, for any gamble G  [ p1oa1   pi oai   p n oa n ] ,
n
U (G )   p i U (ai ) .
i 1
Proof:
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Let G be any gamble in g(A), where: G  [ p1oa1 pi oai  pn oan ]
(P.1)
By the Axiom G1, G2, and G3, there is a complete ordering over g(A), and we can therefore
identify a best outcome a B and worst outcome a w in A.
By Axiom G4, there exists an “indifference probability”, 0  z i  1 , for each outcome a i
which satisfies ai ~ [ z i oaB  (1  z i )oaW ]
(P.2).
It is easy to prove that these indifference probabilities are unique (using Axiom G5).
According to Axiom G6, we can substitute from the right-hand side of (P.2) of each a i in
(P.1):
G ~  p1o[ z1oa B  (1  z1 )oaW ] pn o[ z n oa B  (1  z n )oaW ] (P.3)
Using transitivity and Axiom G7, we obtain
n
 n




G ~   pi z i oa B  1   pi z i oaW  (P.4) (remind that

 i 1

 i 1

p
i
 1 ).
We now propose a mapping from grambles to the real line.
n
We let (propose):
U (G )   piU (ai )
i 1
(P.5) where: U (ai )  z i ,
i  1,, n (P.6)
Take note that this mapping is indeed a function, since the indifference probabilities from
which it is constructed always exist and are unique.
We need to show that this function represents  .
Consider any two gambles in g(A) where:
G1  [q1oa1 qn oan ]
(P.7)
G2  [r1oa1  rn oan ]
(P.8)
Applying the mapping in (P.5) and (P.6) to the gamble G1, we obtain:
n
n
i 1
i 1
U (G1 )   qiU (ai )   qi zi
From the (P.4), we obtain:
G1 ~ U (G1 )oaB  1  U (G1 ) oaW  .
With completely analogous steps, we obtain that:
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G2 ~ U (G2 )oaB  1  U (G2 )oaW 
Note that 0  U (G1 )  1 and 0  U (G2 )  1 .
By the Axiom G5, we have:
G1  G2
 U (G1 )  U (G2 )
The theorem is proved./.
The proceduce to construct U(G).
1. Identify p i such that G  [ p1oa1 pi oai  pn oan ] .
2. Identify z i such that: ai ~ [ z i oaB  (1  z i )oaW ]
3. Let U (ai )  z i
4. U (G)   piU (ai )   pi zi
Example 4.4.1
Suppose that A = {$10,$4,-$2). We can reasonable suppose that a B = $10, aW = -$2.
Suppose we find that:
$10 ~ [1o$10  0o($2)]  U ($10)  1
(E.1)
$4 ~ [.6o$10  .4o($2)]  U ($4)  .6 (E.2)
 $2 ~ [0o$10  1o($2)]
 U ($2)  0
(E.3)
Under this mapping, the utility of the best outcome must be (identically) 1, and that of
the worst outcome must be (identically) 0. The utility assigned to intermediate outcomes
will depend on the individual’s attitude toward taking risks.
Consider two gambles:
G1  [.2o$4  .8o$10]
(E.4)
G2  [.07o  $2  .03o$4  .9o$10]
(E.5)
U (G1 )  .2U ($4)  .8U ($10)  .2 * .6  .8 *1  .92
U (G2 )  .07U ($2)  .03U ($4)  .9U ($10)  .918
 G1  G2
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We can rank any of the infinite number of gambles that could be constructed from the three
outcomes in A./.
 The VNM mapping does its assignment of numbers to gambles in two distinct stages:
1. First, all gambles in g(A) that offer one outcome with certainty are assigned
utility numbers that reflect th agent’s ordering of those alternatives with
certainty.
2. Then, all other gambles in g(A) are assigned utility numbers via the expected
utility calculation.
The VNM utility numbers assigned to ultimate outcomes must not only properly reflect the
agent’s ranking of those particular outcomes relative to each other, they must also be capable
of properly reflectig the agent’s ranking of gambles comprised of them through the (special)
expected utility calculation.
It should not, therefore, be terribly suprising that we are less free to transform VNM utility
functions if the ranking of every gamble is to be preserved.
Theorem 4.4.2 VNM Utility Functions are Unique Up to Positive Affine Tranformations
Let  satisfy Axiom G1 through G7, and suppose that the VNM utility function U(G)
represents  . Then the VNM utility function, V(G), represents those same preferences if,
and only if, V (G )    U (G ) , for some arbitrary scalar  and some scalar  >0.
Proof:
Sufficient condition is obvious
Necessary condition: We need to prove that if V(G) is another utility function 
V (G )    U (G ) with  ,   0 .
Let A  {a1 ,, an } and G  { p1oa1  pn oan } .
By the proof of Theorem 4.4.1 that if the VNM utility function U(G) represents  , it
possesses the expected utility property and so, for any G  G ( A) , we can write:
n
n
i 1
i 1
U (G )   piU (ai )   pi z i where z i satisfies: ai ~ [ z i oa B  (1  z i )oaW ] .
Suppose that V(G) is another VNM utility function which represents  , we must have the
following:
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V (G )   piV (ai )   pi [ z i oa B  (1  z i )oaW ]   pi z i V (a B )  1   pi z i V (aW )
n
n
i 1
i 1
 U (G ).V (a B )  (1  U (G )).V (aW )  V (aW )  V (a B )  V (aW )U (G )
For any outcome set A and VNM utility function V, the numbers V (a B ) and V (aW ) are
constants, with V (a B ) > V (aW ) . Setting   V (aW ) and   V (a B )  V (aW )  0 , so the
theorem is proved./.
 Theorem 4.4.2 tells us that VNM utility functions are not completely unique, nor are
they entirely ordinal. We can still find an infinite number of them that will rank
gambles in precisely the same order and also possess the expected utility property.
 However, unlike ordinary utility functions, here we must limit the posivite
transformation of VNM utility function under the form of V (G )    U (G ) with
,   0 .
 Yet the less than complete ordinality of the VNM utility function must not tempt us
into attaching undue significance to the absolute level of a gamble’s utility, or to the
differene in utility between one gamble and another. With what little weve required of
the agent’s binary comparisons between gambles in the underlying preference
ordering, we still cannot use VNM utility functions for interpersonal comparisons of
well – being, nor can we measure the “intensity” with which one gamble is perferred
to another.
4.4.3 Risk Aversion
The VNM utility function we created reflected some desire to avoid risk.
We shall assume that the VNM utility function is both increasing and differentiable over the
appropriate domain of wealth concerned.
We let the possible wealth outcomes be denoted A  {w1 ,, wn } .
n
The expected value of G: E[G ]   pi wi .
i 1
Now suppose that the agent is given a choice between accepting the gamble G on the one
hand, or receiving with certainty the expected value of G on the other. We can evaluate these
two alternative as follows:
The utility of the gamble: U (G)   piU (wi ) and
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The utility of the gamble’s expected value: U ( E[G])  U ( pi wi ) .
When someone would rather receive the expected value of a gamble with certainty than face
the risk inherent in the gamble itself, we say they are risk averse.
Definition 4.4.2 Risk Aversion, Risk Neutrality, and Risk Loving
Let A  {w1 ,, wn } . Then at G  g ( A) , an individual is said to be locally:
1. Risk averse whenever U ( E[G ])  U (G )
2. Risk neutral whenever U ( E[G ])  U (G )
3. Risk loving whenever U ( E[G ])  U (G )
If these relationships hold for all gambles G  g ( A) , these definitions apply globally.
 Each of these attitudes toward risk is equivalent to a particular property of the
VNM utility function.
 We will be asked to show that the agent is risk averse, risk neutral, or risk
loving over some subset of gambles if, and only if, their VNM utility function
is strictly concave, linear, or strictly convex, respectively, over the appropriate
domain of wealth.
 Consider a simple gamble involving two outcomes: G  [ p ow1 (1  p)ow2 ]
and E[G]  pw1  (1  p)w2 .
 The individual is offered a choice between receiving an amount of wealth
equal to E[G]  pw1  (1  p)w2 with certainty, or receiving the gamble G
itself.
We
can
compare
the
alternatives
as
follows:
U (G)  p.U (w1 )  (1  p)U (w2 ) and U ( E[G])  U ( pw1  (1  p)w2 ) .
U(w2)
U(E[G])
U(G)
U(w1)
P
w1
CE
E(G)
w2
Risk aversion and strict concavity of the VNM utility function
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 We can see that, strict concavity of the VNM utility function, U(E[G])>U(G),
so the individual is risk averse.
 Certainty Equivalent (CE): Amount of wealth we could offer with certainty
that would make him indifferent between accepting that wealth with certainty
and facing the gamble G.
 When a person is risk  CE < E(G).
 A risk – averse person will “pay” some positive amount of wealth in order to
avoid the gamble’s inherent risk. This willingness to pay to avoid risk is
measured by the Risk Premium.
Definition 4.4.3 Certainty Equivalent and Risk Premium
The Certainty Equivalent of any gamble G is an amount of wealth, CE, offered with
certainty, such that U (G )  U (CE ) . The Risk Premium is an amount of wealth, P, such
that U (G )  U ( E[G ]  P) . Clearly, the two are related, and P  E[G ]  CE .
Example 4.4.2
Suppose that U ( w)  log( w)  U is strictly concave in wealth, the individual is risk averse.
Let G offers 50-50 odds of winning or losing some amount of wealth, h, so that:
G  [.5o( w  h).5o( w  h)] where w is current wealth, and E[G] = w.
Log(CE) = (1/2)log(w+h) + (1/2)log(w-h) = log (w2 – h2)1/2.
Thus CE = (w2 – h2)1/2 < E[G] and P = w - (w2 – h2)1/2 > 0.
 Risk aversion and concavity of the VNM utility function in wealth are equivalent.
 The sign of the second derivative U’’(w) does tell us whether the individual is risk
averse, risk loving, or risk neutral, its size is entirely arbitrary.
 The size of U’’(w) depends on the positive affine transformations of U(w) and the
units in which the outcome is measured.
 Arrow (1965) and Pratt (1964) have proposed a measure a risk aversion which is
based on the second derivative, but which avoids these non-uniqueness problems.
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Definition 4.4.4 The Arrow – Pratt Measure of Absolute Risk Aversion
The Arrow – Pratt measure of absolute risk aversion is given by
Ra ( w) 
 U ' ' ( w)
U ' ( w)
Ra (w) is positive, negative, or zero as the agent is Risk Averse, Risk Loving, or Risk Neutral
repectively.
Any positive affine transformation of utility will leave the measure unchanged.
Changing units of measurement of outcome leaves Ra (w) unaffected
Ra (w) is only a local measure of risk aversion, so it need not be the same at every level
wealth.
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