Advanced Microeconomics I: Consumers, Firms
and Markets
Chapters 1+2
Prof. Dr. Oliver Gürtler
Winter Term 2012/2013
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Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
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1. Introduction
Microeconomic theory covers the analysis of individual economic behavior
(e.g., a consumer or …rm) and the aggregation of individuals’actions in an institutional framework (e.g., a price mechanism in an impersonal market place).
Doing this, we intend to get a better understanding of economic activity and
outcomes. This is useful in two distinct senses:
– positive sense: we obtain a better understanding of individual behavior
in certain situations.
– normative sense: we understand when to intervene, both at the government level and at the institutional level.
However, the models we will analyze are highly simpli…ed and sometimes too
simple to be realistic.
Still, they have some general predictive power and represent the building blocks
of more complex and realistic testable models.
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Structure of the course:
2. Consumer Theory
3. Theory of the Firm
4. Partial Equilibrium
5. General Equilibrium
6. Social Choice and Welfare
Literature:
Jehle, Reny (2011): Advanced Microeconomic Theory, Pearson Education.
MasColell, Whinston, Green (1995): Microeconomic Theory, Oxford University
Press.
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Organizational matters:
The course consists of a lecture (10am) and an integrated exercise course
(2pm; starting in the …fth week).
Slides and exercise sets are available at ILIAS.
Students are expected to prepare the exercises and to present their solutions.
Some mathematical concepts are needed, most of which should be known to
everyone. The concepts that are probably not known to everyone are brie‡y
addressed when they are encountered for the …rst time.
The …nal exam takes place on Jan 31, 2013 at 10 am.
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2. Consumer Theory
In consumer theory, we focus on an individual’s (consumer’s) decision to consume a set of commodities (goods and services).
There are four building blocks in any such model:
– consumption set
– feasible set
– preference relation
– behavioral assumption
By specifying each of these in a given problem, many di¤erent situations involving choice can be formally analyzed.
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We focus on n commodities i = 1; :::; n.
De…nition 1 The consumption set X represents the set of all consumption bundles
that the consumer can conceive.
De…nition 2 A consumption bundle (or consumption plan) x = (x1 ; :::; xn ) is a
vector specifying the amounts of each of the di¤erent commodities.
Note: Time and location are included in the de…nition of a commodity.
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Assumption 1 The consumption set X satis…es:
1. X Rn+ (nonnegativity)
2. X is closed, it includes its own boundary.
3. X is convex: if x 2 X and y 2 X then z =
2 [0; 1]).
4. 0 2 X:
x + (1
) y 2 X (for every
De…nition 3 The feasible set B X represents those consumption plans that the
consumer can conceive and a¤ord.
The behavioral assumption speci…es the ultimate objectives in choice (typically utility maximization).
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2.1 Preferences and Utility
Each consumer is endowed with a binary relation, %, de…ned on the consumption set X.
The expression x % y means that “x is at least as good as y”for the consumer.
This relation gives us information about the consumer’s tastes for the di¤erent
objects of choice.
We impose several axioms that the relation should ful…ll.
These axioms formalize the view that the consumer can choose and that
choices are consistent in a particular way. Moreover, they serve to characterize
the consumer’s tastes over the consumption bundles.
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Axiom 1 Completeness. For all x; y 2 X, either x % y or y % x (or both).
! Consumer can always compare two consumption bundles and decide which is
(weakly) preferred.
Axiom 2 Transitivity. For any three elements x; y; z 2 X, if x % y and y % z,
then x % z.
! Axiom links pairwise comparisons in a consistent way.
! Consumer can rank any …nite number of elements in X from best to worst
(ties are possible).
De…nition 4 The binary relation % on the consumption set X is called a preference
relation if it satis…es Axioms 1 and 2.
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De…nition 5 The binary relation on the consumption set X is de…ned as follows:
x y if and only if x % y and not y % x.
The relation
is called the strict preference relation induced by %.
De…nition 6 The binary relation on the consumption set X is de…ned as follows:
x y if and only if x % y and y % x.
The relation
is called the indi¤erence relation induced by %.
Note: and do not necessarily have the same properties as %. For instance,
while both are transitive, neither is complete.
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De…nition 7 Let x0 be any point in X. Relative to this point, we can de…ne the
following subsets of X:
1. % x0
2. - x0
3.
x0
4.
x0
5.
x0
=
=
=
=
=
x
x
x
x
x
x 2 X; x % x0
x 2 X; x0 % x
x 2 X; x0 x
x 2 X; x x0
x 2 X; x x0
,
,
,
,
,
called
called
called
called
called
the
the
the
the
the
Note: For any bundle x0 , the three sets
X.
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"at least as good as" set.
"no better than" set.
"worse than" set.
"preferred to" set.
"indi¤erence" set.
x0 ,
x0 and
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x0 partition
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We introduce some additional axioms to put more structure on preferences
(from now on X = Rn+ ).
Axiom 3 Continuity. For all x 2 Rn+ , the sets % (x) and - (x) are closed in Rn+ .
Technical assumption ensuring that there are no sudden preference reversals.
! Rules out "open" area in the indi¤erence set.
Axiom 4 Strict Monotonicity. For all x; y 2 Rn+ , if x
x
y, then x y.
y, then x % y, while if
Consumer always prefers a consumption bundle involving more to one involving
less.
! Rules out the possibility of having "zones of indi¤erence".
! Ensures that indi¤erence sets have negative slope.
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Axiom 5 Strict Convexity. If x 6= y and x % y, tx+(1
t) y
y for all t 2 (0; 1).
The consumer prefers "balanced" bundles to more extreme ones.
! The (absolute value of the) slope of the indi¤erence curve (the marginal rate
of substitution) is decreasing.
(Weak) convexity is de…ned analogously.
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De…nition 8 A real-valued function u : Rn+ ! R is called a utility function representing the preference relation % if for all x; y 2 Rn+ , u (x) u (y) , x % y.
Utility function represents consumer’s preference relation if it assigns higher
numbers to preferred bundles.
Utility function is a device for summarizing the information contained in the
preference relation.
Is it always possible to represent the preference relation by a continuous realvalued function?
Theorem 1 If the binary relation % is complete, transitive, continuous and strictly
monotonic, there exists a continuous real-valued function, u : Rn+ ! R, which
represents %.
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Proof.
Idea: For given preferences, …nd one function representing these preferences!
Let e = (1; :::; 1) 2 Rn+ and consider the mapping u : Rn+ ! R de…ned by
u (x) e
x
In words: assign to any bundle x the number u (x) such that the consumer is
indi¤erent between x and a bundle with u (x) units of every commodity.
Does such number always exist? If so, is it uniquely determined?
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To tackle the …rst question, de…ne
A
B
ft
ft
0 jte % x g
0 jte - x g
We have to show that A \ B is always nonempty.
By continuity of %, both A and B are closed in R+ .
By strict monotonicity, t 2 A ) t0 2 A for all t0
t.
! A is a closed interval of the form t^; 1 .
Similarly, B is a closed interval of the form 0; t~ .
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By completeness, te - x or te % x; i.e., t 2 A [ B.
! A [ B = R+ .
Since A [ B = 0; t~ [ t^; 1 , it must be that t^
t~.
! A \ B is nonempty.
To tackle the second question, suppose there were two numbers t1 and t2
satisfying t1 e x and t2 e x.
By transitivity, we have t1 e
t2 e.
By strict monotonicity, t1 = t2 .
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It remains to show that the function we have constructed represents % and is
continuous.
Consider two bundles x and y and their associated numbers u (x) and u (y).
Let x % y.
By the de…nition of u, u (x) e
By transitivity, u (x) e
x and u (y) e
y.
u (y) e.
By strict monotonicity, u (x)
u (y).
! The function we have constructed represents % :
To show that the function is continuous, we show that the inverse image under
u of every open ball in R is open in Rn+ (by Theorem A1.6 in the mathematical
appendix of Jehle/Reny).
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Since open balls in R are open intervals, we must show that u
open in Rn+ (for a < b).
By de…nition, the inverse image u
x 2 Rn+ jae
u (x) e
x 2 Rn+ jae
x
(ae) \
(ae) and
u
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1
((a; b)) is
((a; b)) is x 2 Rn+ ja < u (x) < b , or
(by strict monotonicity), or
(by u (x) e
x), or
(be)
By continuity of
!
be
be
1
1
, the sets % (be) and - (ae) are closed in Rn+ .
(be), being the complements of closed sets, are open.
((a; b)) is open. Q.E.D.
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According to the theorem, we can describe the consumer’s preferences in two
ways, either by the preference relation or a continuous utility function.
The latter approach is often more convenient.
The preference relation ranks consumption bundles.
! There is more than one utility function representing a particular ranking.
Theorem 2 Let
be a preference relation on Rn+ and suppose u (x) is a utility
function that represents it. Then v (x) also represents
if and only if v (x) =
f (u (x)) for every x, where f : R ! R is strictly increasing on the set of values
taken on by u.
Utility function is invariant to positive monotonic transforms.
Intuition: positive monotonic transform does not change the ranking of bundles.
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Theorem 3 Let
be represented by u : Rn+ ! R. Then
1. u (x) is strictly increasing if and only if is strictly monotonic.
2. u (x) is quasiconcave if and only if is convex.
3. u (x) is strictly quasiconcave if and only if is strictly convex.
The proof follows easily from the de…nitions involved. Consider part 2, for
example.
By de…nition a function u ( ) is quasiconcave if and only if the set
y 2 Rn+ ju (y) k is convex for every k 2 R.
If x is chosen such that u (x) = k, the de…nition of quasiconcavity of u ( )
coincides with the de…nition of convexity of %.
In the following we additionally assume u ( ) to be di¤erentiable.
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De…nition 9 The …rst-order partial derivative of u (x) with respect to xi is called
the marginal utility of good i.
Assume n = 2.
Let x2 = f (x1 ) be the function describing an indi¤erence curve.
! u (x1 ; f (x1 )) = const:
Di¤erentiating with respect to x1 yields:
@u 0
@u
+
f (x1 ) = 0
@x1
@x2
, M RS =
f 0 (x1 ) =
@u
@x1
@u
@x2
The marginal rate of substitution equals the ratio of the marginal utilities of
the two goods.
As indicated before, convexity of % may be interpreted as diminishing M RS.
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2.2 The Consumer’s Problem
Formally, the consumer seeks
x 2 B such that x % x for all x 2 B
Consumer is assumed to operate within a market economy.
His impact on the market is negligible.
! The vector of prices p = (p1 ; :::; pn )
of view.
0 is …xed from the consumer’s point
The feasible set (or budget set) B is de…ned by
B = x 2 Rn+ jpx
where m
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m
0 denotes the consumer’s …xed money income.
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The consumer’s maximization problem can be stated equivalently as
max u (x) s.t. px
x2Rn
+
m
Under the assumptions made u (x) is continuous and real-valued, while B is
nonempty and compact (closed and bounded).
! By the Weierstrass theorem, a solution to the maximization problem always
exists.
Since B is convex and u (x) strictly quasiconcave, the solution is unique.
The solution vector x depends on p and m:
The optimal quantities, viewed as functions of p and m, are known as ordinal
or Marshallian demand functions.
We write as x (p; m) the vector of these quantities.
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Derivation of the solution:
To solve this nonlinear programming problem with inequality constraint, we
form the Lagrangian
L = u (x) + (m px)
We assume x
conditions:
0. Then the solution satis…es the following optimality
@u (x )
@L
=
pi = 0, i = 1; :::; n
@xi
@xi
m px
0
(m px ) = 0
0
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Suppose
!
= 0. Then
> 0 and, hence, m
@u(x )
@xi
= 0; 8i, contradicting strict monotonicity.
px = 0.
Combining the conditions for di¤erent commodities i 6= j yields
@u(x )
@xi
@u(x )
@xj
=
pi
pj
! At the optimum, the marginal rate of substitution between any two goods
must be equal to the ratio of the goods’prices.
But: the conditions we stated are merely necessary. What about su¢ cient
conditions?
Under the assumptions imposed, the conditions are also su¢ cient for optimality.
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Proof:
Because u ( ) is quasiconcave, we have ru (x) (y
u (x) and x; y 0.
x)
0 whenever u (y)
The optimality conditions can be restated as
ru (x ) = p
px = m
If x is not utility-maximizing, there must be some z
0 such that
u (z) > u (x )
pz m
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Because u ( ) is continuous and m > 0, there exists some t 2 [0; 1] close
enough to one such that
u (tz) > u (x )
ptz < m
Setting y = tz, it follows
ru (x ) (y x ) = p (y
= (py px ) < 0
x )
This condition contradicts the condition from the beginning of the proof.
Q.E.D.
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2.3 Indirect Utility and Expenditure
2.3.1 The Indirect Utility Function
The ordinary utility function u (x) represents the consumer’s preferences directly.
! It is called the direct utility function.
De…nition 10 The function obtained by substituting the Marshallian demands in the
consumer’s utility function is the indirect utility function: v(p; m) = u(x(p; m))
Alternatively: v(p; m) = maxn u (x) s:t: px
x2R+
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Theorem 4 If u (x) is continuous and strictly increasing on Rn+ , then v (p; m) is
1. continuous on Rn++ R+ ;
2. homogeneous of degree zero in (p; m) ;
3. strictly increasing in m;
4. decreasing in p,
5. quasiconvex in (p; m).
Moreover, it satis…es Roy’s identity: If v (p; m) is di¤erentiable at p0 ; m0 and
@v (p0 ;m0 )
6= 0, then
@m
xi p0 ; m0 =
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@v (p0 ;m0 )
@pi
@v(p0 ;m0 )
@m
, i = 1; :::; n:
Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
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Proof of parts 2, 3, 5, 6:
To prove 2, we must show that v (p; m) = v (tp; tm) for all t > 0.
By de…nition,
v(tp; tm) = maxn u (x) s:t: tpx
x2R+
= maxn u (x) s:t: px
x2R+
tm
m = v (p; m)
To prove 3, note that
@v (p; m)
@u (x (p; m)) X @u @xi
=
=
@m
@m
@xi @m
i
@u
Using the optimality conditions @x
= pi and px (p; m) = m )
i
1, we obtain
@v (p; m)
=
>0
@m
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P
i
i
pi @x
@m =
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The proof of part 5 is more elaborate.
Let us consider the following three sets
with pt
tp1 + (1
B 1 = x p1 x
m1
B 2 = x p2 x
m2
B t = x pt x
mt
t) p2 and mt
tm1 + (1
t) m2 , t 2 [0; 1].
v (p; m) is quasiconvex in (p; m) if and only if
v pt ; mt
max v p1 ; m1 ; v p2 ; m2
; 8t 2 [0; 1]
It su¢ ces to show that every choice a consumer can make when facing B t
could also be made when facing either B 1 or B 2 .
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The latter condition holds trivially for t = 0 or t = 1. So, let us assume
t 2 (0; 1).
The proof is by contradiction. Therefore, assume that there exists some t 2
= B 1 and x 2
= B 2 . Then
(0; 1) and x 2 B t such that x 2
p1 x > m1 and p2 x > m2
Multiplying the …rst inequality by t and the second by (1
tp1 x > tm1 and (1
t) p2 x > (1
t) yields
t) m2
Adding the two conditions yields the desired contradiction x 2
= Bt:
! If x 2 B t , then x 2 B 1 or x 2 B 2 ; 8t 2 [0; 1].
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To prove the last part, note that
@v (p; m)
@u (x (p; m)) X @u @xj
=
=
@pi
@pi
@xj @pi
j
@u
Using the optimality conditions @x
=
pj and px (p; m) = m ) xi +
j
P
@xj
j pj @pi = 0 yields
@v (p; m)
=
xi
@pi
Roy’s identity is obtained by solving the condition for xi and inserting
for . Q.E.D.
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@v(p;m)
@m
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2.3.2 The Expenditure Function
The expenditure function is derived from a di¤erent kind of optimization problem.
We ask: What is the minimum level of money expenditure the consumer
must make to achieve a given level of utility?
Formally, the expenditure function is de…ned as
e (p; u) = minn px s:t: u (x)
x2R+
Let U = u (x) x 2 Rn+
denote the set of attainable utility levels.
! The domain of e ( ) is Rn++
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u
U.
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The solution to the minimization problem is denoted by xh (p; u).
! e (p; u) = pxh (p; u)
Idea: We imagine a process where the consumer’s income is changed to
compensate him for changes in prices such that he can achieve exactly the
same utility level as before.
Still, the optimal quantities of the goods will typically change.
! The optimal quantities, viewed as functions of p and u, are known as compensated or Hicksian demand functions.
xh (p; u) is the vector of these demands.
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Theorem 5 If u (x) is continuous and strictly increasing on Rn+ , then e (p; u) is
1. zero when u takes on the lowest level of utility in U ,
2. continuous on its domain Rn++ U;
3. for all p
0; strictly increasing and unbounded above in u;
4. strictly increasing in p;
5. homogeneous of degree one in p,
6. concave in p.
Moreover, if u (x) is strictly quasiconcave, we have Shephard’s lemma: e (p; u) is
0 and
di¤erentiable in p at p0 ; u0 with p0
@e p0 ; u0
= xhi p0 ; u0 , i = 1; :::; n:
@pi
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Proof of parts 1, 4, 6, 7:
To prove 1, note that the lowest value in U is u (0) (since u (x) is strictly
increasing). x = 0 requires an expenditure of 0, hence e (p; u (0)) = 0.
Part 4 follows from the proof of part 7.
To prove 6, note that the expenditure function will be concave in prices if
te p1 ; u + (1
t) e p2 ; u
e tp1 + (1
t) p2 ; u
for any two price vectors p1 and p2 and t 2 [0; 1].
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Denote the bundles minimizing expenditure to achieve u in the three situations
by x1 , x2 and xt .
By de…nition,
p1 x1
p1 x
p2 x2
p2 x
for any other bundle x achieving utility u:
In particular
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p1 x1
p1 xt
p2 x2
p2 xt
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Multiplying the …rst inequality by t and the second by (1
tp1 x1
tp1 xt and (1
t) p2 x2
t) yields
t) p2 xt
(1
Adding the two conditions, we obtain
tp1 x1 + (1
, tp1 x1 + (1
t) p2 x2
t) p2 x2
tp1 xt + (1
t) p2 xt
tp1 + (1
t) p2 xt
This is the same as
te p1 ; u + (1
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t) e p2 ; u
e tp1 + (1
t) p2 ; u
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To prove 7, we di¤erentiate e (p; u) with respect to pi .
X @xhj
@ pxh (p; u)
@e (p; u)
=
= xhi +
pj
@pi
@pi
@pi
j
From u (x) = u, we have
P
h
@u @xj
j @xj @pi
= 0.
Deriving the optimality conditions and combining them yields
Together, these conditions imply
7.
Since xhi
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P
j
pj
@xh
j
@pi
@u
@xj
=
pj @u
pi @xi .
= 0 completing the proof of part
0, part 4 follows immediately. Q.E.D.
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2.3.3 Relations Between The Two
There are close relations between the indirect utility function and the expenditure function.
From the de…nitions of e and v, we know
e (p; v (p; m))
v (p; e (p; u))
m; 8 (p; m)
0;
n
u; 8 (p; u) 2 R++
U:
One can further show that both of these inequalities, in fact, must be equalities.
This means that knowing the indirect utility function enables us to calculate
the expenditure function (and vice versa).
Accordingly, we do not have to solve both optimization problems to derive the
indirect utility function and the expenditure function.
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There are also close relations between the Marshallian and Hicksian demand
functions.
Theorem 6 We have the following relations between the Hicksian and Marshallian
demand functions for p
0; m 0, u 2 U and i = 1; :::; n :
1. xi (p; m) = xhi (p; v (p; m))
2. xhi (p; u) = xi (p; e (p; u))
If x solves the utility-maximization problem at (p; m), it also solves the
expenditure-minimization problem at (p; u), where u = u (x ).
Conversely, if x solves the expenditure-minimization problem at (p; u), it also
solves the utility-maximization problem at (p; m), where m = px .
In this sense x has a dual nature.
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2.4 Properties Of Consumer Demand
The theory developed so far leads to a number of predictions about behavior
in the marketplace.
In the following, we present some of these predictions.
Theorem 7 The consumer demand function xi (p; m) ; i = 1; :::; n; is homogeneous of degree zero in all prices and income, and it satis…es budget balancedness,
px (p; m) = m for all (p; m).
Proof:
We have shown before that v (p; m) is homogeneous of degree zero in (p; m),
i.e.,
v (p; m) = v (tp; tm) ; 8t > 0, or
u (x (p; m)) = u (x (tp; tm)) ; 8t > 0.
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Since budget sets at (p; m) and (tp; tm) are the same, x (p; m) was feasible
when x (tp; tm) was chosen (and vice versa).
From the last equality on the previous slide and strict quasiconcavity of u it
then follows that x (p; m) = x (tp; tm).
This condition says that xi (p; m) ; i = 1; :::; n; is homogeneous of degree zero
in prices and income.
The property of budget balancedness has been proven before. Q.E.D.
Homogeneity implies that we could focus on relative prices and real income.
For instance, if good n serves as numéraire, we have
x (p; m) = x (tp; tm) = x
45
p1
pn 1
m
; :::;
; 1;
pn
pn
pn
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An important consideration is the response in quantity demanded when prices
change.
Typically, we expect a consumer to buy more of a good when its price declines,
but this is not always true.
To get an understanding of the relevant e¤ects, we consider the Slutsky equation.
Theorem 8 Let x (p; m) be the consumer’s Marshallian demand system. Let u be
the level of utility the consumer achieves at prices p and income m. Then,
@xi (p; m)
@xhi (p; u )
=
@pj
@pj
xj (p; m)
@xi (p; m)
; i; j = 1; :::; n:
@m
The substitution e¤ect is the …rst, the income e¤ect the second term on
the RHS.
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Proof:
Start from the identity
xhi (p; u ) = xi (p; e (p; u ))
Di¤erentiating both sides with respect to pj , yields
@xhi (p; u )
@xi (p; e (p; u )) @xi (p; e (p; u )) @e (p; u )
=
+
@pj
@pj
@m
@pj
From Shephard’s lemma, we have
@e(p;u )
@pj
= xhj (p; u ) = xj (p; e (p; u )).
Moreover, we know u = v (p; m) and m = e (p; v (p; m)) = e (p; u ).
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Using these conditions, we obtain
@xhi (p; u )
@xi (p; m) @xi (p; m)
=
+
xj (p; m)
@pj
@pj
@m
()
@xi (p; m)
@xhi (p; u )
=
@pj
@pj
xj (p; m)
@xi (p; m)
@m
Q.E.D.
Intuition:
Substitution e¤ect: (hypothetical) change in consumption that would occur
after prices were set to the new level but where income was adapted such
the maximum level of utility the consumer can achieve were kept the same as
before the price change (Hicks).
Income e¤ect: change in consumption if prices were kept at the new level
but income was set back to its real value.
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The decomposition of the total e¤ect of a price change into the substitution
and income e¤ect is useful since we are able characterize the single e¤ects in
more detail.
Theorem 9 Let xhi (p; u) be the Hicksian demand for good i. Then
i = 1; :::; n:
@xh
i (p;u)
@pi
0,
In words, own-substitution terms are negative.
Proof:
From Shephard’s lemma, we have xhi (p; u) =
@e(p;u)
@pi .
Di¤erentiating with respect to pi , yields
@xhi (p; u)
@ 2 e (p; u)
=
@pi
@p2i
We have shown before that e (p; u) is concave in p, hence
Q.E.D.
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@ 2 e(p;u)
@p2i
0.
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De…nition 11 A good is called normal if consumption of it increases as income
increases, holding prices constant.
De…nition 12 A good is called inferior if consumption of it declines as income
increases, holding prices constant.
From the two theorems derived before and the de…nitions, it is easy to derive
the so-called "law of demand".
Theorem 10 A decrease in the own price of a normal good will cause quantity
demanded to increase. If an own price decrease causes a decrease in quantity demanded, the good must be inferior.
The law of demand connects concepts about the reactions of quantity demanded to income changes with concepts about the reactions of quantity
demanded to price changes.
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In the remainder of this section, we derive some useful elasticity relations.
These relations can be derived from the budget balancedness constraint,
px (p; m) = m.
De…nition 13 Let xi (p; m) be the consumer’s Marshallian demand for good i.
Denote as
1.
i
2.
ij
3. si =
good i.
51
@xi (p;m)
m
@m
xi (p;m) the income elasticity of demand for good i;
pj
@xi (p;m)
@pj
xi (p;m) the price elasticity of demand for good i and
Pn
pi xi (p;m)
0 (with i=1 si = 1) the income share spent on purchases
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of
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Theorem 11 Let x (p; m) be the consumer’s system of Marshallian demands. Then
the following relations must hold among income shares, price, and income elasticities
of demand:
Pn
1. Engel aggregation: i=1
Pnsi i = 1:
2. Cournot aggregation: i=1 si ij = sj ; j = 1; :::; n:
Proof:
Di¤erentiate both sides of the budget constraint px (p; m) = m with respect
to income to obtain
n
X
@xi
pi
=1
@m
i=1
The LHS can be rewritten as
n
X
pi xi m @xi
i=1
xi m @m
=
n
n
X
pi xi @xi m X
si
=
m @m xi
i=1
i=1
52
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Di¤erentiating both sides of the budget constraint px (p; m) = m with respect
to pj , yields
n
X
@xi
xj +
pi
=0
@pj
i=1
This condition is equivalent to
n
X
pi pj @xi
=
m @pj
i=1
,
pj xj
m
n
X
pi xi pj @xi
=
m xi @pj
i=1
pj xj
m
Using the respective de…nitions, gives us the second condition of the theorem.
Q.E.D.
Because of budget balancedness, all consumer demand responses to price and
income changes must add up in a way that preserves the budget constraint.
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2.5 Uncertainty
Until now, we have assumed that consumers act in a world of perfect certainty.
In what follows, we relax this assumption and consider uncertain situations.
! Consumers are assumed to have a preference relation over gambles (instead
of consumption bundles).
Let A = fa1 ; :::; an g denote a …nite set of outcomes.
The ai ’s involve no uncertainty, but it is not clear which of these outcomes is
realized.
Let pi denote the probability that outcome ai is realized.
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De…nition 14 Let A = fa1 ; :::; an g be the set of outcomes. Then GS , the set of
simple gambles (on A), is given by
(
)
n
X
GS
(p1 a1 ; :::; pn an ) pi 0;
pi = 1 :
i=1
Note: When one or more of the pi ’s is zero, we can drop those components
from the expression.
It is sometimes the case that gambles have prizes that are themselves gambles.
Such gambles are called compound gambles.
Example: In some lotteries, one can win monetary prizes, but also tickets for
the lottery.
For simplicity, we rule out in…nitely layered compound gambles.
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Let G denote the set of all (simple and compound) gambles.
A gamble g 2 G can then be written as g = p1 g 1 ; :::; pk g k , k
where g i might be a compound gamble, a simple gamble or an outcome.
1,
As indicated before, the consumer (or decision maker) is assumed to have
preferences, %, over G.
We impose certain axioms the preference relation is assumed to ful…ll, called
axioms of choice under uncertainty.
0
Axiom 6 Completeness. For any two gambles g; g 0 2 G, either g % g 0 or g % g (or
both).
Axiom 7 Transitivity. For any three gambles g; g 0 ; g 00 2 G, if g % g 0 and g 0 % g 00 ,
then g % g 00 .
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Note that each ai 2 A is represented in G as a degenerate gamble.
! The two axioms imply that the elements in A can be ordered by %.
Assume that these elements have been indexed such that a1 % a2 % ::: % an .
Axiom 8 Continuity. For any gamble g 2 G, there is some probability
such that g (
a1 ; (1
) an ).
Axiom 9 Monotonicity. For all probabilities
(
a1 ; (1
) an ) if and only if
.
Monotonicity implies a1
;
2 [0; 1], (
a1 ; (1
2 [0; 1]
) an ) %
an .
! Decision maker is not indi¤erent between all outcomes in A.
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Axiom 10 Substitution. If g = p1 g 1 ; :::; pk g k and h = p1 h1 ; :::; pk
are in G, and if g i hi for every i, then g h.
hk
If decision maker is indi¤erent between the realizations of one gamble and
another, and the realizations occur with the same probabilities, then he is
indi¤erent between the two gambles.
In a compound gamble, one can calculate the e¤ective probabilities of the
single outcomes in A:
For any gamble g 2 G, if pi denotes the e¤ective probability assigned to ai by
g, then we say that g induces the simple gamble (p1 a1 ; :::; pn an ) 2 GS .
The …nal axiom states that the decision maker cares only about the e¤ective
probabilities a gamble assigns to the single outcomes in A.
Axiom 11 Reduction to Simple Gambles. For any gamble g 2 G, if (p1 a1 ; :::; pn an )
is the simple gamble induced by g, then (p1 a1 ; :::; pn an ) g.
By this axiom and transitivity, a decision maker’s preferences over all gambles
are completely determined by the preferences over simple gambles.
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We can show that preferences, obeying the above axioms, can be represented
with a continuous, real-valued function.
Let u : G ! R describe a utility function representing % on G.
! For every g 2 G, u (g) denotes the utility number assigned to g.
Similarly, u (ai ) denotes the utility number assigned to the degenerate gamble
(1 ai ) (called the utility of outcome ai ).
De…nition 15 The utility function u : G ! R has the expected utility property
if, for every g 2 G,
n
X
u (g) =
pi u (ai ) ;
i=1
where (p1 a1 ; :::; pn an ) is the simple gamble induced by g.
! u assigns to each gamble the expected value of utilities that might result!
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A utility function possessing the expected utility property is referred to as von
Neumann-Morgenstern (VNM) utility function.
Theorem 12 Let preferences % over gambles in G satisfy the above axioms. Then
there exists a utility function u : G ! R representing % on G, such that u has the
expected utility property.
Implication: Under the axioms imposed, one can assign utility numbers to the
outcomes in A such that the decision maker prefers one gamble over another
if and only if it has a higher expected utility.
Proof:
We construct a function and show that it possesses the desired properties.
Consider an arbitrary gamble g 2 G and let u (g) be the number satisfying
g
60
(u (g) a1 ; (1
u (g)) an )
Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
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By continuity, such number exists and, by monotonicity, it is unique.
! We have derived a real-valued function u on G.
To show that u represents %, consider two arbitrary gambles g; g 0 2 G and
suppose g % g 0 .
By transitivity and the de…nition of u, this is equivalent to
(u (g) a1 ; (1
u (g)) an ) % (u (g 0 ) a1 ; (1
By monotonicity, this is equivalent to u (g)
61
u (g 0 )) an )
u (g 0 ).
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It remains to show that u has the expected utility property.
To do this, let g 2 G be an arbitrary gamble and gs
GS the simple gamble it induces.
(p1 a1 ; :::; pn an ) 2
By reduction to simple gambles, we have g gs and, hence, u (g) = u (gs ).
Pn
We therefore must show that u (gs ) = i=1 pi u (ai ).
By de…nition, u (ai ) satis…es
ai
By substitution, gs
(u (ai ) a1 ; (1
u (ai )) an )
(p1 a1 ; :::; pn an )
The simple gamble induced by g 0 is
!
n
X
0
gs =
pi u (ai )
a1 ; 1
i=1
62
qi
p1 q 1 ; :::; pn q n
n
X
i=1
!
pi u (ai )
an
Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
g0 .
!
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By reduction to simple gambles and transitivity, gs g 0 gs0 , or
!
!
!
n
n
X
X
gs
pi u (ai )
a1 ; 1
pi u (ai )
an
i=1
i=1
By de…nition, u (gs ) is the unique number satisfying
gs
(u (gs ) a1 ; (1
u (gs )) an )
Comparing the two expressions, we obtain
u (gs ) =
n
X
pi u (ai )
i=1
Q.E.D.
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VNM utility functions are not unique.
However, because of the expected utility property not every positive monotonic
transformation is possible.
Theorem 13 Suppose the VNM utility function u ( ) represents %. Then the VNM
utility function v ( ) represents those same preferences if and only if for some scalar
and some scalar > 0
v (g) = + u (g) ;
for all gambles g.
In words: VNM utility functions are unique up to positive a¢ ne transformations.
Proof:
For simplicity, we suppose g
prove only necessity.
64
(p1 a1 ; :::; pn an ) is a simple gamble and
Advanced Microeconomics I: Consumers, Firms and Markets Chapters 1+2
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Since u ( ) represents %, we have
u (a1 )
u (a2 )
u (an ) and u (a1 ) > u (an ) :
:::
Hence, for any i there is a unique
u (ai ) =
i
2 [0; 1] such that
i u (a1 )
+ (1
i ) u (an ) :
Because of the expected utility property, it follows that
u (ai ) = u ( i a1 ; (1
ai ( i a1 ; (1
i ) an ) ; or
)
an ) :
i
If v ( ) represents % as well, we have
v (ai ) = v (
i
a1 ; (1
i)
an ) :
If v ( ) has the expected utility property, this condition can be transformed into
v (ai ) =
65
i v (a1 )
+ (1
i ) v (an ) :
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Combining the expressions for u (ai ) and v (ai ), yields
u (a1 )
u (ai )
for every i such that ai
u (ai )
1
=
u (an )
i
=
i
an and, thus,
i
v (a1 )
v (ai )
v (ai )
v (an )
> 0.
Rearranging, we obtain
(u (a1 )
with
66
u (ai )) (v (ai ) v (an )) = (v (a1 ) v (ai )) (u (ai )
, v (ai ) = + u (ai ) ;
u (a1 ) v (an )
u (a1 )
u (an ) v (a1 )
;
u (an )
v (a1 )
u (a1 )
u (an ))
v (an )
> 0.
u (an )
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Using this condition, it follows that
v (g) =
n
X
pi v (ai ) =
i=1
=
+
n
X
pi ( + u (ai ))
i=1
n
X
pi u (ai ) =
+ u (g) :
i=1
Q.E.D.
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VNM utility functions are not completely unique, nor are they entirely ordinal.
Still, we should not attach undue signi…cance to the absolute level of a gamble’s
utility.
It is, for instance, not possible to use VNM utility functions for interpersonal
comparisons of well-being.
In the remainder of this section, we analyze the relationship between a VNM
utility function and the decision maker’s attitude toward risk.
We con…ne attention to gambles whose outcomes consist of di¤erent amounts
of wealth w.
We assume A = R+ and, hence, wealth levels to be nonnegative.
Moreover, we consider only gambles giving …nitely many outcomes a strictly
positive e¤ective probability.
Finally, we assume u ( ) to be strictly increasing and twice di¤erentiable for all
wealth levels.
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De…nition 16 Let u ( ) be a decision maker’s VNM utility function for gambles over
nonnegative levels of wealth. Then for the simple gamble g = (p1 w1 ; :::; pn wn ) ;
the decision maker is said to be
1. risk averse at g if u (E (g)) > u (g) ;
2. risk neutral at g if u (E (g)) = u (g) ;
3. risk loving at g if u (E (g)) < u (g) :
If for every nondegenerate simple gamble, g, the decision maker is, for example, risk
averse at g, then he is said simply to be risk averse. Similarly, a decision maker can
be de…ned to be risk neutral and risk loving.
Note that
u (E (g)) = u
n
X
i=1
u (g) =
n
X
pi wi
!
and
pi u (wi )
i=1
Intuition: The gamble entails risk, while both options lead to the same expected value.
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A decision maker is risk averse (risk neutral, risk loving) if and only if his
VNM utility function is strictly concave (linear, strictly convex) over the
appropriate domain of wealth.
Illustration for two wealth levels w1 and w2 :
With two wealth levels, we have
u (E (g)) = u (p1 w1 + (1
p1 ) w2 ) and u (g) = p1 u (w1 ) + (1
p1 ) u (w2 )
Consider a graph, where w is measured on the abscissa and u ( ) on the ordinate.
If we connect the points (w1 ; u (w1 )) and (w2 ; u (w2 )), we obtain a line segment.
u (g) is the ordinate of a particular point on this line segment.
If the decision maker is risk averse, we have u (E (g)) > u (g).
! The utility function must proceed above the line segment and, thus, be strictly
concave.
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De…nition 17 The certainty equivalent of any simple gamble g over wealth levels
is an amount of wealth, CE, o¤ered 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,
P E (g) CE.
When a person is risk averse and prefers more money to less, we have CE <
E (g) and, thus, P > 0.
! A risk averse decision maker will "pay" some positive amount of wealth to
avoid the gamble’s inherent risk.
Similarly, we have CE = E (g) and CE > E (g) for risk neutral and risk
loving decision makers.
Often, we do not only want to know whether a decision maker is risk averse,
but also how risk averse he is.
De…nition 18 The Arrow-Pratt measure of absolute risk aversion is given by
00
(w)
Ra (w) = uu0 (w)
.
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Ra (w) is positive, zero or negative as the decision maker is risk averse, risk
neutral or risk loving.
Ra (w) is una¤ected by any positive a¢ ne transformation of the utility function
(u00 (w) is not).
Decision makers with a larger Arrow-Pratt measure have a lower certainty
equivalent and are willing to accept fewer gambles.
Proof of the last statement:
Consider two decision makers and denote their VNM utility functions by u (w)
and v (w), with u0 (w) ; v 0 (w) > 0.
Let Ra1 (w) =
u00 (w)
u0 (w)
>
v 00 (w)
v 0 (w)
= Ra2 (w), for all w
0.
Assume that v (w) takes on all values in [0; 1) and write
h (x) = u v
1
(x) , for all x
0.
Di¤erentiating h twice with respect to x, one can show that h0 (x) > 0 and
h00 (x) < 0.
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Consider a gamble (p1 w1 ; :::; pn wn ) over wealth levels and denote by CE1
and CE2 the two decision makers’certainty equivalents for the gamble.
By de…nition,
n
X
i=1
n
X
pi u (wi ) = u (CE1 ) ;
pi v (wi ) = v (CE2 ) :
i=1
Substituting v (w) by x and using h (x) = u v
u (CE1 ) =
n
X
1
(x) , we obtain
pi h (v (wi ))
i=1
By Jensen’s inequality (because h is strictly concave), this expression is strictly
smaller than
!
n
X
h
pi v (wi ) = h (v (CE2 )) = u (CE2 )
i=1
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Because u ( ) is strictly increasing, it follows that CE1 < CE2 .
If both decision makers have the same initial wealth, this …nding implies that
2 will accept any gamble that 1 will accept. Q.E.D.
Note that h (x) = u v
1
(x) implies u (w) = h (v (w)).
! u ( ) is more concave than v ( ), because it is a concave function of v ( ).
Arrow-Pratt measure of absolute risk aversion typically varies with wealth.
A VNM utility function is said to display constant, decreasing or increasing
absolute risk aversion over some domain of wealth if, over that interval,
Ra (w) remains constant, decreases or increases with an increase in wealth.
Decreasing absolute risk aversion (DARA) is often a sensible assumption.
! The greater wealth, the less averse one becomes to accepting the same gamble.
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