Microeconomics I
c
Leopold S¨ogner
Department of Economics and Finance
Institute for Advanced Studies
Stumpergasse 56
1060 Wien
Tel: +43-1-59991 182
soegner@ihs.ac.at
http://www.ihs.ac.at/∼soegner
September, 2011
Course Outline (1)
Micro I
Learning Objectives:
• This course covers key concepts of microeconomic theory. The
main goal of this course is to provide students with both, a basic
understanding and analytical traceability of these concepts.
• The main concepts are discussed in detail during the lectures. In
addition students have to work through the textbooks and have
to solve problems to improve their understanding and to acquire
skills to apply these tools to related problems.
1
Course Outline (2)
Micro I
Literature:
• Andreu Mas-Colell, A., Whinston, M.D., Green, J.R.,
Microeconomic Theory, Oxford University Press, 1995.
Supplementary Literature:
• Gilboa, I., Theroy of Decision under Uncertainty, Cambridge University Press,
2009.
• Gollier C., The Economics of Risk and Time, Mit Press, 2004.
• Jehle G.A. and P. J. Reny, Advanced Microeconomic Theory, Addison-Wesley
Series in Economics, Longman, Amsterdam, 2000.
• Ritzberger, K., Foundations of Non-Cooperative Game Theory, Oxford
University Press, 2002.
2
Consumer Theory (1)
Rationality and Preferences
Micro I
• Consumption set X
• Rationality
• Preference relations and utility
• Choice correspondences and the weak axiom of revealed
preference
• Relationship between the axiomatic approach and the revealed
preference approach
Mas-Colell, chapter 1
3
Consumer Theory (1)
Rationality (1)
Micro I
• We consider agents/individuals and goods that are available for
purchase in the market.
• Definition: The set X of all possible mutually exclusive
alternatives (complete consumption plans) is called
consumption set or choice set.
• ”Simplest form of a consumption set”: We assume that each
good, xl ∈ x, l = 1, . . . , L can be consumed in infinitely divisible
units, i.e. xl ∈ R. With L goods we get the commodity vector
x and in the commodity space RL
.
4
Consumer Theory (1)
Rationality (2)
Micro I
• The elements of X are e.g. x and y. We assume that each good,
xl ∈ x, l = 1, . . . , L can be consumed in infinitely divisible units,
i.e. xl ∈ R.
• In most parts of the course we assume xl ≥ 0, i.e. only positive
amounts of xi can be consumed. xi = 0 is possible. I.e. X = RL
+.
5
Consumer Theory (1)
Rationality (3)
Micro I
• Approach I: describe behavior by means of preference relations;
preference relation is the primitive characteristic of the individual.
• Approach II: the choice behavior is the primitive behavior of an
individual.
6
Consumer Theory 1
Rationality (4)
Micro I
• Consider the binary relation “at least good as“, abbreviated by
the symbol .
• For x, y ∈ X, x y implies that from a particular consumer’s
point of view x is preferred to y or that he/she is indifferent
between consuming x and y.
• From we derive the strict preference relation : x y if
x y but not y x and the indifference relation ∼ where
x y and y x.
7
Consumer Theory 1
Rationality (5)
Micro I
• Often we require that pair-wise comparisons of consumption
bundles are possible for all elements of X.
• Completeness: For all x, y ∈ X either x y, y x or both.
• Transitivity: For the elements x, y, z ∈ X: If x y and y z,
then x z
• Definition[D 1.B.1]: The preference relation is called rational
if is complete and transitive.
• Remark: Reflexive x x follows from completeness [D 1.B.1];
alternative terms weak order, complete preorder.
8
Consumer Theory
Rationality (6) - Remarks (1)
Micro I
• Definition - Relation: A subset of a product set Y × Y is called
relation.
Properties of a Relation:
• reflexive: xRx for all x ∈ X.
• irreflexive: there is no x ∈ X such at xRx.
• symmetric: if xRy then yRx for all x, y ∈ X.
• asymmetric: there are no x, y ∈ X such that xRy and yRx.
• transitive: if xRy and yRz then xRz.
• complete: for all x, y ∈ X either xRy, xRy or both.
complete = comparable; irreflexive = nonreflexive.
9
Consumer Theory 1
Rationality (7) - Remarks (2)
Micro I
• Example: Equality relation =. For S ⊆ R and the product set
S × S we define xRy if x = y and xR/y if x = y.
• For each x ∈ S we get the pair (x, x) ⊆ S × S.
• For all x the equality relation generates the subset
{(x, x)|x ∈ S}.
• For preferences: for each pair x, y in X × X a consumer is able
to assign one of the following statements: xRy or yR/x
• The assignments of the one of these statements to any pair
(x, y) is called binary relation on X.
• All these pairs x, y constitute a subset of the product set of X.
10
Consumer Theory 1
Rationality (8) - Remarks (3)
Micro I
Partial Orders
• Strict Partial Order: A relation is called strict partial ordering if it
is irreflexive and transitive.
• Weak Partial Order: A relation is called non-strict or weak partial
ordering if it is reflexive and transitive.
• Order Relation: A relation is called strict ordering if it is
complete, irreflexive and transitive.
• Weak Order: A relation is a weak order if it is complete, reflexive
and transitive.
• Equivalence Relation: A relation is called equivalence relation if is
reflexive, symmetric and transitive. 11
Consumer Theory 1
Rationality (9) - Remarks (4)
Micro I
Partial Orders
• is irreflexive and transitive such that it fulfills the requirements
of a strict partial order.
• is reflexive and transitive and fulfills the requirements of a
weak partial Order.
• ∼ is reflexive and transitive and fulfills the requirement of an
equivalence relation.
Chains: all elements of X are comparable, i.e. xRy or yRx.
12
Consumer Theory 1
Rationality (10) - Remarks (5)
Micro I
Partial Orders and Partitions:
• Definition - Partition of a Set S: A decomposition of S into
nonempty and disjoint subsets such that each element is exactly
in one subset is called partition. These subsets are called cells.
• Theorem - Partition and Equivalence Relation: If S is not
empty and ∼ is an equivalence relation on S, then ∼ yields a
partition with cells ∼ (x0) = {x|x ∈ S , x ∼ x0}.
• Proof: Since ∼ is reflexive, every element x is at least contained in one cell,
e.g. ∼ (xA). We have to show that if x ∈∼ (xA) and x ∈∼ (xB) then
∼ (xA) =∼ (xB). If x ∈∼ (xA) and x ∈∼ (xB), transitivity results in
x ∼ xA ∼ xB, for all x in ∼ (xA). Therefore ∼ (xA) ⊆∼ (xB).
∼ (xB) ⊆∼ (xA) is derived in the same way.
13
Consumer Theory 1
Rationality (11)
Micro I
• From these remarks we conclude [P 1.B.1]: If is rational then,
– is transitive and irreflexive.
– ∼ is transitive, reflexive and symmetric.
– If x y z then x z.
14
Consumer Theory 1
Utility (1)
Micro I
• Definition : A function X → R is a utility function representing
if for all x, y ∈ X x y ⇔ u(x) ≥ u(y). [D 1.B.2]
• Does a rational consumer imply that the preferences can be
represented by means of a utility function and vice versa?
• Theorem: If there is a utility function representing , then
must be complete and transitive. [P 1.B.2]
• The other direction requires more assumptions - this comes later!
15
Consumer Theory 1
Choice Rules (1)
Micro I
• Now we follow the behavioral approach (vs. axiomatic approach),
where choice behavior is represented by means of a choice
structure.
• Definition - Choice structure: A choice structure (B, C(.))
consists: (i) of a family of nonempty subsets of X; with B ⊂ X.
The elements of B are the budget sets B. and (ii) a choice rule
C(.) that assigns a nonempty set of chosen elements C(B) ∈ B
for every B ∈ B.
• B is an exhaustive listing of all choice experiments that a
restricted situation can pose on the decision maker. It need not
include all subsets of X.
16
Consumer Theory 1
Choice Rules (2)
Micro I
• The choice rule assigns a set C(B) to every element B, i.e. it is
a correspondence.
• Put structure on choice rule - weak axiom of revealed preference.
17
Consumer Theory 1
Choice Rules (3)
Micro I
• Definition - Weak Axiom of Revealed Preference: A choice
structure (B, C(.)) satisfies the weak axiom of revealed
preference if for some B ∈ B with x, y ∈ B we have x ∈ C(B),
then for any B ∈ B with x, y ∈ B and y ∈ C(B ) , we must
also have x ∈ C(B ). [D 1.C.1]
• I.e. if we have a budget set where both are available and x is
chosen, then there cannot be a budget set containing both
bundles, for which y is chosen and x is not. Example:
C((x, y)) = x, then C((x, y, z)) = y is not possible.
18
Consumer Theory 1
Choice Rules (4)
Micro I
• Definition - Revealed Preference Relation ∗ based on the
Weak Axiom: Given a choice structure (B, C(.)), ∗ is defined
by:
x ∗y ⇔ there is some B ∈ B such that x, y ∈ B and
x ∈ C(B)). [D 1.C.2]
19
Consumer Theory 1
Choice Rules (5)
Micro I
• If a decision maker has a rational preference ordering , does the
choice structure satisfy the weak axiom of reveal preferences?
• Given a choice structure satisfying the weak axiom, does this
result in a rational preference relation?
20
Consumer Theory 1
Choice Rules (6)
Micro I
• Definition: Choice structure implied by a rational preference
relation : C∗
(B, ) = {x ∈ B|x y for every y ∈ B}. Assume
that C∗
(B, ) is nonempty.
• Proposition: If is a rational preference relation, then the
choice structure C∗
(B, ) satisfies the weak axiom. [P 1.D.1]
21
Consumer Theory 1
Choice Rules (7)
Micro I
• Other direction. We want to know whether the choice rule C(B)
is equal to C∗
(B, ) for all B ∈ B.
• Definition: Given a choice rule C(B). rationalizes the
choice rule if C(B) = C∗
(B, ) for all B ∈ B. [D 1.D.1]
• Proposition: If a choice structure (B, C(.)) satisfies the weak
axiom of revealed preference and B includes all subsets of X of
up to three elements, then C(B) = C∗
(B, ). The rational
preference relation is the only preference relation that does
so.[P 1.D.2]
22
Consumer Theory 1
Choice Rules (8)
Micro I
• Example 1.D.1: counterexample that choice structure cannot be
rationalized.
• Remark: The more subsets of X we consider, the stronger the
restrictions implied by the weak axiom.
• Remark: If the choice structure is defined for all subsets of X,
then the approaches based on the preference relation and on the
weak axiom are equivalent. To consider all budget sets is a strong
requirement. Another way to maintain this problem is the strong
axiom of revealed preference.
23
Consumer Theory 2
Walrasian Demand (1)
Micro I
• Feasible set
• Competitive budget sets.
• Walrasian/Marshallian Demand
• Walras’ law, Cournot and Engel aggregation.
Mas-Colell: Chapter 2
24
Consumer Theory 2
Consumption Set (1)
Micro I
• We have already defined the consumption set: The set of all
alternatives (complete consumption plans). The consumption set
is a subset of the commodity space RL
. We assume X = RL
+.
• Each x represents a different consumption plan.
• Physical restrictions: divisibility, time constraints, survival needs,
etc.
25
Consumer Theory 2
Consumption Set (2)
Micro I
Useful properties of a Consumption Set: X
1. ∅ = X ⊆ Rn
+
2. X is closed
3. X is a convex set
4. 0 ∈ X
26
Consumer Theory 2
Feasible Set (1)
Micro I
Definition - Feasible Set: B
• Due to constraints (e.g. income) we cannot afford all elements in
X, problem of scarcity.
• The feasible set B is defined by the elements of X which are
achievable given the economic realities.
• B ⊂ X.
27
Consumer Theory 2
Feasible Set (2)
Micro I
• By the consumption set and the feasible set we can describe a
consumer’s alternatives of choice.
• These sets do not tell us what x is going to be chosen by the
consumer.
• To describe the choice of the consumer we need a theory to
model or describe the preferences of a consumer.
• We start with choice under certainty, under some assumptions
(restrictive/not restrictive ?) we can derive a function capable to
describe the preferences of a consumer (utility function).
28
Consumer Theory 2
Competitive Budgets (1)
Micro I
• Assumption: All L goods are traded at the market (principle of
completeness), the prices are given by the price vector p, pl > 0
for all l = 1, . . . , L. Notation: p 0. Assumption - the prices are
constant and not affected by the consumer.
• Given a wealth level w, the set of affordable bundles is described
by
p · x = p1x1 + · · · + pLxL ≤ w.
• Definition - Walrasian Budget Set: The set
Bp,w = {x ∈ RL
+|p · · · x ≤ w} is called Walrasian or
competitive budget set. [D 2.D.1]
• Definition - Consumer’s problem: Given p and w choose the
optimal bundle x from Bp,w.
29
Consumer Theory 2
Competitive Budgets (2)
Micro I
• Definition - Relative Price: The ratios of prices pj/pi are called
Relative Prices.
• Here the price of good j is expressed in terms of pi. In other
words: The price of good xi is expressed in the units of good xj.
• Since the dxi/dxj = −pj/pi = constant, the relative price
measures the number of units of good i in the number of untis of
good j. xi/xj = pj/pi.
• On the market we receive for one unit of xj, pj/pi · 1 units of xi.
30
Consumer Theory 2
Competitive Budgets (3)
Micro I
• The budget set B describes the goods a consumer is able to buy
given nominal income/ wealth level w.
• The ratio w/pi is the maximum number a consumer can buy of
good i. This ratio is called Real Income in terms of good i.
• Definition - Numeraire Good: If all prices pj are expressed in
the prices of good n, then this good is called numeraire. pj/pn,
j = 1, . . . , n. The relative price of the numeraire is 1.
• There are n − 1 relative prices.
31
Consumer Theory 2
Competitive Budgets (4)
Micro I
• The set {x ∈ RL
+|p · x = w} is called budget hyperplane, for
L = 2 it is called budget line.
• Given x and x in the budget hyperplane, p · x = p · x = w
holds. This results in p(x − x ) = 0, i.e. p and (x − x ) are
orthogonal - see Figure 2.D.3 page 22.
• The budget hyperplane is a convex set. In addition it is closed
and bounded ⇒ compact. 0 ∈ Bp,w.
32
Consumer Theory 2
Demand Functions (1)
Micro I
• Definition - Walrasian demand correspondence:
Correspondence assigning a pair (p, w) a set of consumption
bundles is called Walrasian demand correspondence x(p, w); i.e.
(p, w) x(w, p). If x(p, w) is single valued it is called demand
function.
• Definition - Homogeneity of degree zero: x(p, w) is
homogeneous of degree zero if x(αp, αw) = x(p, w) for any p, w
and α > 0. [D 2.E.1]
• Definition - Walras law, budget balancedness: x(p, w)
satisfies Walras law if for every p 0 and w > 0, we get
p · x = w for all x ∈ x(p, w). I.e. the consumer spends all income
w with her/his optimal consumption decision. [D 2.E.2]
33
Consumer Theory 2
Demand Functions (2)
Micro I
• Assume that x(p, w) is a function:
• With p fixed at ¯p, the function x(¯p, w) is called Engel function.
• If the demand function is differentiable we can derive the
gradient vector: Dwx(p, w) = (∂x1(p, w)/∂w, . . . , xL(p, w)/∂w).
If ∂xl(p, w)/∂w ≥ 0, xl is called normal or superior, otherwise it
is inferior.
• see Figure 2.E.1, page 25
34
Consumer Theory 2
Demand Functions (3)
Micro I
• With w fixed, we can derive the L × L matrix of partial
derivatives with respect to the prices: Dpx(p, w).
• ∂xl(p, w)/∂pk is called the price effect.
• A Giffen good is a good where the own price effect is positive,
i.e. ∂xl(p, w)/∂pl > 0
• see Figure 2.E.2-2.E.4, page 26
35
Consumer Theory 2
Demand Functions (4)
Micro I
• Proposition: If a Walrasian demand function x(p, w) is
homogeneous of degree zero and differentiable, then for all p and
w:
L
k=1
∂xl(p, w)
∂pk
pk +
∂xl(p, w)
∂w
w = 0
for l = 1, . . . , L;, or in matrix notation
Dpx(p, w)p + Dwx(p, w) = 0. [P 2.E.1]
• Proof: By the Euler theorem (if g(x) is homogeneous of degree r,
then ∂g(x)/∂x · x = rg(x), [P M.B.2]), the result follows
directly when using the stacked vector (p, w) .
36
Consumer Theory 2
Demand Functions (5)
Micro I
• Definition - Price Elasticity of Demand: ηij = ∂xi(p,w)
∂pj
pj
xi(p,w) .
• Definition - Income Elasticity: ηiw = ∂xi(p,w)
∂w
w
xi(p,w) .
• Definition - Income Share:
si =
pixi(p, w)
w
,
where si ≥ 0 and
n
i=1 si = 1.
37
Consumer Theory 2
Demand Functions (6)
Micro I
• Between the income elasticity, income shares and elasticities, two
nice relationships are available:
• Engel aggregation: The income share weighted sum of income
elasticities adds up to one.
• Cournot aggregation: The income share weighted sum of price
elasticities (with respect to pj, i = 1, . . . , L) adds up to minus
the income share of good xj.
38
Consumer Theory 2
Demand Functions (7)
Micro I
• Theorem - Relationship between Elasticities and Income
Shares: For a Walrasian demand function x(p, w) fulfilling
budget balancedness (Walras’ law), the following relationships
have to hold: [P 2.E.2, P 2.E.3]
Engel aggregation:
L
i=1 siηiw = 1.
Cournot aggregation:
L
i=1 siηij = −sj for j = 1, . . . , L.
39
Consumer Theory 2
Demand Functions (8)
Micro I
Proof:
• Engel aggregation: Walras’ law implies w =
L
l=1 plxl(p, w).
Taking the partial derivative with respect to w results in
1 =
L
l=1
pl
∂xl(p, w)
∂w
.
• Expand by xl/w for each element and arranging terms yields:
1 =
L
l=1
plxl
w
sl
∂xl(p, w)
∂w
w
xl
ηiw
.
40
Consumer Theory 2
Demand Functions (9)
Micro I
Proof:
• Cournot aggregation: Differentiate w =
L
l=1 plxl(p, w) by pk:
0 =
L
l=1
pl
∂xl(p, w)
∂pk
+ xk(p, w) .
• Arranging terms yields:
−xk =
L
l=1
pl
∂xl(p, w)
∂pk
.
41
Consumer Theory 2
Demand Functions (10)
Micro I
Proof:
• Expand both sides by pk/w:
−
pkxk
w
=
L
l=1
pl
∂xl(p, w)
∂pk
pk
w
.
• Multiply each element on the right hand side by xl/xl:
−
pkxk
w
−sk
=
L
l=1
plxl
w
sl
∂xl(p, w)
∂pk
pk
xl
ηlk
.
42
Consumer Theory 2
Demand Functions (11)
Micro I
• Discuss how demands have to interact by these relations.
• In the two good case: Can two normal goods fulfill these
relationships?
• In the two good case: Can two inferior goods fulfill these
relationships?
43
Consumer Theory 2
Demand Functions (12)
Micro I
• For L goods we can derive the matrix S(p, w) with own
substitution terms in the main diagonal and cross-substitution
terms off-diagonal; x(p, w) is a differentiable function.
• Definition - Substitution Matrix:
S(p, w) :=



∂x1(p,w)
∂p1
+ x1
∂x1(p,w)
∂w · · · ∂x1(p,w)
∂pL
+ xL
∂x1(p,w)
∂w
... ... ...
∂xL(p,w)
∂p1
+ x1
∂xL(p,w)
∂w · · · ∂xL(p,w)
∂pL
+ xL
∂xL(p,w)
∂w



= Dpx(p, w) + Dwx(p, w)x(p, w) .
44
Consumer Theory 2
Demand Functions (13)
Micro I
• Theorem - Walras’ Law and Symmetry of S(p, w) imply
Homogeneity: If a demand system x = x(p, w) satisfies budget
balancedness and the Slutsky matrix is symmetric, then it is
homogeneous of degree zero.
45
Consumer Theory 2
Demand Functions (14)
Micro I
Proof:
• To obtain Cournout and Engel aggregation we had (i)
−xi =
n
j=1 pj
∂xj(p,w)
∂pi
and (ii)
n
j=1 pj
∂xj(p,w)
∂w = 1.
• To proof to above theorem we have to show that
fi(ν) = xi(νp, νw) is constant in ν for ν > 0.
• Taking first derivatives with respect to ν yields (here w = νw
and pi = νpi)
dfi(ν)
dν
=


L
j=1
pj
∂xi(νp, νw)
∂pj

 +
∂xi(νp, νw)
∂w
w .
46
Consumer Theory 2
Demand Functions (15)
Micro I
Proof:
• First, budget balancedness results in: νp · x(νp, νw) = νw ⇒
n
j=1 pjxj(νp, νw) = w.
dfi(ν)
dν
=
L
j=1
pj
∂xi(νp, νw)
∂pj
+
∂xi(νp, νw)
∂w
xj(νp, νw) .
• The substitution matrix is assumed to be symmetric. Therefore
we are allowed to exchange i and j.
47
Consumer Theory 2
Demand Functions (16)
Micro I
Proof:
dfi(ν)
dν
=
L
j=1
pj
∂xj(νp, νw)
∂pi
+
∂xj(νp, νw)
∂w
xi(νp, νw)
=
L
j=1
pj
∂xj(νp, νw)
∂pi
+ xi(νp, νw)
L
j=1
pj
∂xj(νp, νw)
∂w
=
1
ν
L
j=1
pjν
∂xj(νp, νw)
∂pi
(i)
+
xi(νp, νw)
ν
n
j=1
νpj
∂xj(νp, νw)
∂w
(ii)
=
1
ν
(−xi(νp, νw) + xi(νp, νw)) = 0 .
48
Consumer Theory 3
The Axiomatic Approach (1)
Micro I
• Axioms on preferences
• Preference relations, behavioral assumptions and utility (axioms,
utility functions)
• The consumer’s problem
• Walrasian/Marshallian Demand
Mas-Colell, chapter 3.A-3.D
49
Consumer Theory 3
The Axiomatic Approach (2)
Micro I
• Axiom 1 - Completeness: For all x, y ∈ X either x y, y x
or both.
• Axiom 2 - Transitivity: For the elements x, y, z ∈ X: If x y
and y z, then x z.
• We have already defined a rational preference rations by
completeness and transitivity [D 3.B.1]
• If the number of elements is finite we can describe our
preferences by a function.
50
Consumer Theory 3
The Axiomatic Approach (3)
Micro I
Sets arising from the preference relations:
• (x) := {y|y ∈ X, y x} - at least as good (sub)set
• (x) := {y|y ∈ X, y x} - the no better set
• (x) := {y|y ∈ X, y x} - at preferred to set
• (x) := {y|y ∈ X, y x} - worse than set
• ∼ (x) := {y|y ∈ X, y ∼ x} - indifference set
• Plot these sets for the two good case.
51
Consumer Theory 3
The Axiomatic Approach (4)
Micro I
• Axiom 3.A - Local Nonsatiation: For all x ∈ X and for all
ε > 0 there exists some y ∈ X such that ||x − y|| ≤ ε and y x.
[D 3.B.3]
• This assumptions implies that for every open neighborhood of x
there must exist at least one y, which is preferred to x.
• Indifference “zones“ are excluded by this assumption.
52
Consumer Theory 3
The Axiomatic Approach (5)
Micro I
• Axiom 4.B - Monotonicity: For all x, y ∈ Rn
+: If x ≥ y then
x y while if x y then x y (weakly monotone). It is
strongly/strict monotone if x ≥ y and x = y imply x y.
[D 3.B.3]
• Here ≥ means that at least one element of x is larger than the
elements of y, while x y implies that all elements of x are
larger than the elements of y.
• Remark: Local nonsatiation vs. monotonicity: The latter implies
that more is always better, while Axiom 4.1 only implies that in
every neighborhood there has to exist a preferred alternative.
53
Consumer Theory 3
The Axiomatic Approach (6)
Micro I
• Discuss the differences of Axioms 4.1 and 4.2 (what are their
impacts on indifference sets?), e.g. by means Figures 3.B.1 and
3.B.2, page 43.
• Axiom 3 implies that we arrive at indifference curves.
54
Consumer Theory 3
The Axiomatic Approach (7)
Micro I
• Last assumption on taste - “mixtures are preferred to extreme
realizations“
• See Figure 3.B.3, page 44.
• Axiom 4.A - Convexity: If y x then νy + (1 − ν)x x for
ν ∈ (0, 1). [D 3.B.4]
• Axiom 4.B - Strict Convexity: If y = x and y x then
νy + (1 − ν)x x for ν ∈ (0, 1). [D 3.B.5]
• Indifference curves are therefore (strict) convex.
55
Consumer Theory 3
The Axiomatic Approach (8)
Micro I
• Equivalent definition:
• Axiom 4.A - Convexity: For all x ∈ X, the upper contour set
{y|y x} is convex. [D 3.B.4]
56
Consumer Theory 3
The Axiomatic Approach (9)
Micro I
• After we have arrived at our indifference sets we can describe a
consumer’s willingness to substitute good xi against xj (while
remaining on an equal level of satisfaction).
• Definition: Marginal rate of substitution: MRSij = |
dxj
dxi
| or
(MRSij =
dxj
dxi
) is an agent’s willingness to give up dxj units of
xj for receiving dxi of good xi.
• MRS corresponds to the slope of the indifference curve.
• By Axiom 4.B, the MRS is a strictly decreasing function, i.e. less
units of xj have to be given up for receiving an extra unit of xi,
the higher the level of xi (Principle of diminishing marginal
rate of substitution).
57
Consumer Theory 3
The Axiomatic Approach (10)
Micro I
• Definition - Homothetic Preferences: A monotone preference
on X is homothetic if all indifference sets are related by
proportional expansions along rays. I.e. x ∼ y then αx ∼ αy.
[D 3.B.6]
• Definition - Quasilinear Preferences: A monotone preference
on X = (−∞, ∞) × RL−1
is quasilinear with respect to
commodity one if : (i) all indifference sets are parallel
displacements of each other along the axis of commodity one.
I.e. x ∼ y then x + αe1 ∼ y + αe1 and e1 = (1, 0, . . . ) . (ii)
Good one is desirable: x + αe1 x for all α > 0. [D 3.B.7]
58
Consumer Theory 3
The Axiomatic Approach (11)
Micro I
• With the next axiom we regularize our preference order - we
make it smooth:
• Axiom 5 - Continuity: A preference order is continuous if it
is preserved under limits. For any sequence (xn, yn) with xn yn
for all n, and limits x, y (x = limn→∞ xn) we get x y.
[D 3.C.1]
• Equivalently: For all x ∈ X the set “at least as good as“
( (x)) and the set “no better than“ ( (x)) are closed in X.
59
Consumer Theory 3
The Axiomatic Approach (12)
Micro I
• Topological property of the sets (important assumption in the
existence proof of a utility function).
• By this axiom the set (x) and (x) are open sets (the
complement of a closed set is open ...).
• The intersection of (x)∩ (x) is closed (intersection of closed
sets).
• Consider a sequence of bundles yn fulfilling yn x, for all n. For
yn converging to y, Axiom 3 imposes that y x.
60
Consumer Theory 3
The Axiomatic Approach (13)
Micro I
• Consider the preference relation (x) and a sequence of bundles
yn fulfilling yn x, for all n. For yn converging to y, Axiom 3
imposes that y x.
• The economic impact of Axiom 5: Consider the open
neighborhood Aε with x ∈ Aε. Aε cannot be an indifference set
by Axiom 5.
• ⇒ Indifference sets are closed sets.
• Provide a meaningful interpretation of this assumption from an
economic point of view (hint “jumps“ in preferences, addictive
behavior).
61
Consumer Theory 3
The Axiomatic Approach (14)
Micro I
Lexicographic order/dictionary order:
• Given two partially order sets X1 and X2, an order is called
lexicographical on X1 × X2 if (x1, x2) (x1, x2) if and only if
x1 < x1 (or x1 = x1, x2 < x2).
• Let x = (x1, x2), y = (x1, x2) and yn = (x1n, x2n). x1n
converges to x1. For all x1n < x1 we have yn x, while for any
x1n > x1 or x1n = x1 and x2 > x2 the order is reversed; here
yn x.
• Example in R2
+: x = (0, 1) and yn = (1/n, 0), y = (0, 0). For all
n, yn x, while for n → ∞: yn → y= (0, 0) (0, 1) = x.
• The lexicographic ordering is a rational (strict) preference
relation (we have to show completeness and transitivity). 62
Consumer Theory 3
The Axiomatic Approach (15)
Micro I
• Axioms 1 and 2 guarantee that an agent is able to make
consistent comparisons among all alternatives.
• Axiom 5 impose the restriction that preferences do not exhibit
“erratic behavior“ ; mathematically important
• Axioms 3 and 4 make assumptions on a consumer’s taste
(satiation, mixtures).
63
Consumer Theory 3
Utility Function (1)
Micro I
• Definition: Utility Function: A real-valued function
u : Rn
+ → R is called utility function representing the preference
relation if for all x, y ∈ Rn
+ u(x) ≥ u(y) if and only if x y.
• I.e. a utility function is a mathematical device to describe the
preferences of a consumer.
• Pair-wise comparisons are replaced by comparing real valued
functions evaluated for different consumption bundles.
• Function is of no economic substance (for its own).
64
Consumer Theory 3
Utility Function (2)
Micro I
• To apply calculus to “optimize “ utility a continuous
(differentiable) utility function becomes necessary.
• First of all we want to know if such a function exists.
• Theorem: Existence of a Utility Function: If a binary relation
is complete, transitive and continuous, then there exists a
continuous real valued function function u(x) representing the
preference ordering .
• Proof: be assuming monotonicity see page 47; Debreu’s (1959)
proof is more advanced.[P 3.C.1]
65
Consumer Theory 3
Utility Function (3)
Micro I
• Consider y = u(x) and the transformations v = g(u(x));
v = log y, v = y2
, v = a + by, v = −a − by. Do these
transformations fulfill the properties of a utility function?
• Theorem: Invariance to Positive Monotonic
Transformations: Consider the preference relation and the
utility function u(x) representing this relation. Then also v(x)
represents if and only if v(x) = g(u(x)) is strictly increasing
on the set of values taken by u(x).
66
Consumer Theory 3
Utility Function (4)
Micro I
Proof:
• ⇒ Assume that x y with u(x) ≥ u(y): A strictly monotone
transformation g(.) then results in g(u(x) ≥ g(u(y)). I.e. v(x) is
a utility function describing the preference ordering of a
consumer.
Assume that x y with v(x) ≥ v(y). g(.) is strictly positive
monotonic such that there exists an inverse function g−1
, this
inverse is positive monotone. Thus
u(x) = g−1
(v(x)) ≥ g−1
(v(y)) = u(y). Then u(x) ≥ u(y) if
v(x) ≥ v(y) and x y.
67
Consumer Theory 3
Utility Function (5)
Micro I
Proof:
• ⇐ Now assume that v(g(x)) is a utility representation, but g is
not strictly positive monotonic: Then g(u(x)) need not be ≥ to
g(u(y)) for some pair x, y where u(x) > u(y). I.e. is g(.) is not
monotone increasing we find a pair x, y where x y but
g(u(x)) ≤ g(u(y)). This contradicts the assumption that
v = g(u(x)) is a utility representation of .
68
Consumer Theory 3
Utility Function (6)
Micro I
• By Axioms 1,2,5 the existence of a utility function is guaranteed.
By the further Axioms the utility function exhibits the following
properties.
• Theorem: Preferences and Properties of the Utility
Function:
– u(x) is strictly increasing if and only if is strictly monotonic.
– u(x) is quasiconcave if and only if is convex:
u(xν
) ≥ min{u(x), u(y)}, where xν
= νx + (1 − ν)y.
– u(x) is strictly quasiconcave if and only if is strictly convex.
• First derivative - conditions see literature (Debreu).
69
Consumer Theory 3
Utility Function (7)
Micro I
• Definition: Indifference Curve: Bundles where utility is
constant (in R2
).
• Marginal rate of substitution and utility: Assume that u(x) is
differentiable, then
du(x1, x2) = 0
du(x1, x2) =
∂u(x1, x2)
∂x1
dx1 +
∂u(x1, x2)
∂x2
dx2 = 0
dx2
dx1
= −
∂u(x1, x2)/∂x1
∂u(x1, x2)/∂x2
MRS12 =
∂u(x1, x2)/∂x1
∂u(x1, x2)/∂x2
.
70
Consumer Theory 3
Utility Function (8)
Micro I
• If u(x) is differentiable and the preferences are strictly
monotonic, then marginal utility is strictly positive.
• With strictly convex preferences the marginal rate of substitution
is a strictly decreasing function (i.e. in R2
the slope of the
indifference curve diminishes).
• For a quasiconcave utility function (i.e.
u(xν
) ≥ min{u(x1
), u(x2
)}, with xν
= νx1
+ (1 − ν)x2
) and its
Hessian H(u(x)) we get: y H(u(x))y ≤ 0 for all vectors y,
where grad(u(x)) · y = 0.
• I.e. when moving from x to y that is tangent to the indifference
surface at x utility does not increase (decreases if the equality is
strict).
71
Consumer Theory 3
Consumer’s Problem (1)
Micro I
• The consumer is looking for a bundle x∗
such that x∗
∈ B and
x∗
x for all x in the feasible set B.
• Assume that the preferences are complete, transitive, continuous,
strictly monotonic and strictly convex. Then can be
represented by a continuous, strictly increasing and strictly
quasiconcave utility function. Moreover we can assume that we
can take first and second partial derivatives of u(x).
• Prices pi > 0, p is the vector of prices. We assume that the
prices are fixed from the consumer’s point of view. (Notation:
p 0 means that all elements of p are strictly larger than zero.)
• The consumer is equipped with wealth w.
72
Consumer Theory 3
Consumer’s Problem (2)
Micro I
• Budget set induced by w: Bp,w = {x|x ∈ Rn
+ ∧ p · x ≤ w}.
• With the constant expenditures w we get:
dw(x) = 0
dw = p · dx = 0
dx
dy
= −
py
px
with other prices constant .
• Budget line with two goods; slope −p1/p2.
73
Consumer Theory 3
Consumer’s Problem (3)
Micro I
• Definition - Relative Price: The ratios of prices pj/pi are called
Relative Prices.
• Here the price of good j is expressed in terms of pi. In other
words: The price of good xi is expressed in the units of good xj.
• Since the dxi/dxj = −pj/pi = constant, the relative price
measures the number of units of good i in the number of untis of
good j. xi/xj = pj/pi.
• On the market we receive for one unit of xj, pj/pi · 1 units of xi.
74
Consumer Theory 3
Consumer’s Problem (4)
Micro I
• The budget set Bp,w describes the goods a consumer is able to
buy given nominal income w.
• The ratio w/pi is the maximum number a consumer can buy of
good i. This ratio is called Real Income in terms of good i.
• Definition - Numeraire Good: If all prices pj are expressed in
the prices of good n, then this good is called numeraire. pj/pn,
j = 1, . . . , n. The relative price of the numeraire is 1.
• There are n − 1 relative prices.
75
Consumer Theory 3
Consumer’s Problem (5)
Micro I
• Definition - Utility Maximization Problem [UMP]: Find the
optimal solution for:
max
x
u(x) s.t. xi ≥ 0 , p · x ≤ w.
The solution x(p, w) is called Walrasian demand .
• Remark: Some textbooks call the UMP also Consumer’s
Problem.
• Remark: Some textbooks call x(p, w) Marshallian demand.
76
Consumer Theory 3
Consumer’s Problem (6)
Micro I
• Proposition - Existence: If p 0, w > 0 and u(x) is
continuous, then the utility maximization problem has a solution.
• Proof: By the assumptions Bp,w is compact. u(x) is a continuous
function. By the Weierstraß theorem (MasColell p. 945,
maximum value theorem in calculus), there exists an x ∈ Bp,w
maximizing u(x).
77
Consumer Theory 3
Consumer’s Problem (7)
Micro I
• Suppose u(x) is differentiable. Find x∗
by means of Kuhn-Tucker
conditions for the Lagrangian:
L(x, λ) = u(x) + λ(w − p · x)
∂L
∂xi
=
∂u(x)
∂xi
− λpi ≤ 0
∂L
∂xi
xi = 0
∂L
∂λ
= w − p · x ≥ 0
∂L
∂λ
λ = 0
78
Consumer Theory 3
Consumer’s Problem (8)
Micro I
• If the p · x < w then λ = 0. Then the marginal utilities have to
be negative ⇒ p · x = w.
• If xi is consumed then ∂L/∂xi = 0.
• λ = (∂u(x)/∂xi)/pi for all the goods consumed.
• Envelope-theorem: ∂L/∂w = λ marginal utility of income.
• Sufficient conditions for the Kuhn-Tucker optimization problem:
since u(x) is quasiconcave the sufficient conditions for a
maximum are fulfilled. See e.g. [P M.K.3], page 961.
79
Consumer Theory 3
Consumer’s Problem (9)
Micro I
• By altering the price vector p and income w, the consumer’s
maximization provides us with the correspondence x∗ = x(p, w),
which is called Walrasian demand correspondence. If preferences
are strictly convex we get Walrasian demand functions
x = x(p, w).
• Graphical representation: prices p−i, w fixed. Plot xi as a
function of pi - usual convention - we plot price-demand
relationship (pi vertical axis, xi horizontal axis).
• What happens to the function if w or pj changes?
80
Consumer Theory 3
Consumer’s Problem (10)
Micro I
• In a general setting demand need not be a smooth function.
• Theorem - Differentiable Walrasian Demand Function: Let
x∗
0 solve the consumers maximization problem at price
p0 0 and w0 > 0. If u(x) is twice continuously differentiable,
∂u(x)/∂xi > 0 for some i = 1, . . . , n and the bordered Hessian
of u(x) has a non-zero determinant at x∗
then x(p, w) is
differentiable at p0, w0.
• See arguments at page 94 - use implicit function theorem. See
also P. Klinger Monteiroa, et al. (1996), On the differentiability
of the consumer demand function, Journal of Mathematical
Economics Vol. 25 (2), 1996, pages 247-261.
81
Consumer Theory 3
Consumer’s Problem (11)
Micro I
• From the first order conditions we know
u(x) − λpi = 0 , p · x − w = 0.
• Define the vectors of variables y = (x, λ) and parameters
β = (p|w). Consider the function f(y; β) described by the FOCs.
• We have L + 1 first order conditions, taking partial derivatives
with respect to y, results in the Jacobian of dimension
L + 1 × L + 1
Dy =
D2
u(x) p
p 0
.
82
Consumer Theory 3
Consumer’s Problem (12)
Micro I
• The implicit function theorem [T. M.E.1], page 941, tells us that
if the Jacobian Dy evaluated at some point (¯y, ¯β) is nonsingular,
then the system of equations at (¯y, ¯β) can be solved by
continuously differentiable functions ηL(β). In addition,
Dβη(β) = −[Dyf(¯y, ¯β)]−1
Dβf(¯y, ¯β).
83
Consumer Theory 3
Consumer’s Problem (13)
Micro I
• Theorem - Properties of x(p, w): Consider a utility function
u(x) representing a rational, continuous and locally nonsatiated
preference order. Then x(p, w) has the following properties:
[P 3.D.2]
– Homogeneity of degree zero in (p, w).
– Walras’ law: p · x = w.
– Convexity/uniqueness: If is convex, where u(x) is
quasiconcave, then x(p, w) is a convex set. If preferences are
strictly convex, where u(x) is strictly quasiconcave, then
x(p, w) is single valued, i.e. x(p, w) is a function.
84
Consumer Theory 3
Consumer’s Problem (14)
Micro I
Proof:
• Property 1 - Homogeneity in p, w: We have to show that
x(µp, µw) = µ0
x(p, w). Plug in µp and µw in the optimization
problem ⇒ Bp,w = Bµp,µw. The result follows immediately.
85
Consumer Theory 3
Consumer’s Problem (15)
Micro I
Proof:
• Property 2- Walras’ law: If x ∈ x(p, w) and p · x < w, then there
exists a y in the neighborhood of x, with y x and p · y < w by
local nonsatiation. Therefore x cannot be an optimal bundle.
This argument holds for all interior points of Bp,w.
86
Consumer Theory 3
Consumer’s Problem (16)
Micro I
Proof:
• Property 3 - x(p, w) is a convex set: If preferences are convex
then u(xν
) ≥ min{u(x), u(y)}, where xν
= νx + (1 − ν)y;
replace ≥ by > if is strictly convex. I.e. u(x) is quasiconcave.
We have to show that if x, y ∈ x(p, w), then xν
∈ x(p, w). From
the above property x, y and xν
have to be elements of the
budget hyperplane {x|x ∈ X and p · x = w}.
Since x and y solve the UMP we get u(x) = u(y), therefore
u(xν
) ≤ u(x) = u(y). By quasiconcavity of u(x) we get
u(xν
) ≥ u(x) = u(y), such that u(xν
) = u(x) = u(y) holds for
arbitrary x, y ∈ x(p, w). I.e. the set x(p, w) has to be convex.
87
Consumer Theory 3
Consumer’s Problem (17)
Micro I
Proof:
• Property 3 - x(p, w) is single valued if preferences are strictly
convex: Assume, like above, the x and y solve the UMP; x = y.
Then u(x) = u(y) ≥ u(z) for all z ∈ Bp,w. By the above result
x, y are elements of the budget hyperplane.
• Since preferences are strictly convex, u(x) is strictly quasiconcave
⇒ u(xν
> max{u(x), u(y)}. xν
= νx + (1 − ν)y and x , y are
some arbitrary elements of the budget hyperplane.
• Now u(xν
) > min{u(x), u(y)}, also for x, y. Therefore the pair
x, y cannot solve the UMP. Therefore, x(p, w) has to be single
valued.
88
Consumer Theory 3
Consumer’s Problem (18)
Micro I
Proof:
• Remark: If u(x) is twice continuously differentiable, the u(x) is
strictly quasiconcave if and only if, its Hessian is negative definite
(zD2
u(x)z < 0) in the subspace {z| u(x)z = 0}. [T M.C.4]. By
this property: in an optimal point we have u(x)z = 0 by the
Kuhn-Tucker conditions. The utility function has to decrease
when moving from the optimum to some vector z; where z is in
the budget hyperplane.
89
Consumer Theory 4
Duality
Micro I
• Instead of looking at u(x), we’ll have an alternative look on
utility via prices, income and the utility maximization problem ⇒
indirect utility
• Expenditure function, the dual problem and Hicksian demand
• Income- and substitution effects, Slutsky equation
Mas-Colell, chapter 3.D-3.H
90
Consumer Theory 4
Indirect Utility (1)
Micro I
• We have already considered the direct utility function u(x) in the
former parts.
• Start with the utility maximization problem
max
x
u(x) s.t. p · x ≤ w
• For element x∗
∈ x(p, w) solves this problem for any (p, w) 0.
By the highest levels of utility attainable with p, w, we define a
maximal value function. This function is called indirect utility
function v(p, w). It is the maximum value function
corresponding to the consumer’s optimization problem.
91
Consumer Theory 4
Indirect Utility (2)
Micro I
• v(p, w) is a function, by Berg’s theorem of the maximum x(p, w)
is upper hemicontinuous and v(p, w) is continuous (see
Mas-Colell, page 963, [M.K.6], more details in Micro II).
• If u(x) is strictly quasiconcave such that maximum x∗
is unique,
we drive the demand function x∗
= x(p, w).
• In this case the indirect utility function is the composition of the
direct utility function and the demand function x(p, w), i.e.
v(p, w) = u(x∗
) = u(x(p, w)).
92
Consumer Theory 4
Indirect Utility (3)
Micro I
• Theorem: Properties of the Indirect Utility Function
v(p, w): [P 3.D.3] Suppose that u(x) is a continuous utility
function representing a locally nonsatiated preference relation
on the consumption set X = RL
+. Then the indirect utility
function v(p, w) is
– Continuous in p and w.
– Homogeneous of degree zero in p, w.
– Strictly increasing in w.
– Nonincreasing in pl, l = 1, . . . , L.
– Quasiconvex in (p, w).
93
Consumer Theory 4
Indirect Utility (4)
Micro I
Proof:
• Property 1 - Continuity: follows from Berg’s theorem of the
maximum.
• Property 2 - Homogeneous in (p, w): We have to show that
v(µp, µw) = µ0
v(p, w) = v(p, w); µ > 0. Plug in µp and µw in
the optimization problem ⇒
v(µp, µw) = {maxx u(x) s.t. µp · x ≤ µw} ⇔
{maxx u(x) s.t. p · x ≤ w} = v(p, w).
94
Consumer Theory 4
Indirect Utility (5)
Micro I
Proof:
• Property 3 - increasing in w: Given the solutions if the UMP with
p and w, w , where w > w: x(p, w) and x(p, w ). The
corresponding budget sets are Bp,w and Bp,w , by assumption
Bp,w ⊂ Bp,w . By local nonsatiation there are bundles in
Bp,w \ Bp,w that are strictly better. Therefore utility has to
increase for some bundles in Bp,w \ Bp,w. Since v(p, w) is a
maximal value function, it has to increase if w increases.
95
Consumer Theory 4
Indirect Utility (6)
Micro I
Proof:
• In addition we know: By local nonsatiation Walras law has to
hold, i.e. x(p, w) and x(p, w ) are subsets of the budget
hyperplanes {x|x ∈ X and p · x = w},
{x|x ∈ X and p · x = w }, respectively. We know where we find
the optimal bundles. The hyperplane for w is a subset of Bp,w
and Bp,w (while the hyperplane for w is not contained in Bp,w).
Interior points cannot be an optimum under local nonsatiation.
• If v(p, w) is differentiable this result can be obtained by means of
the envelope theorem.
96
Consumer Theory 4
Indirect Utility (7)
Micro I
Proof:
• Property 4 - non-increasing in pl: W.l.g. pl > pl, then we get
Bp,w and Bp ,w, where Bp ,w ⊆ Bp,w. The rest is similar to
Property 3.
97
Consumer Theory 4
Indirect Utility (8)
Micro I
Proof:
• Property 5 - Quasiconvex: Consider two arbitrary pairs p1
, x1
and
p2
, x2
and the convex combinations pν
= νp1
+ (1 − ν)p2
and
wν = νw1 + (1 − ν)w2; ν ∈ [0, 1].
• v(p, w) would be quasiconvex if
v(pν
, wν) ≤ max{v(p1
, w1), v(p2
, w2)}.
• Define the consumption sets: Bj = {x|p(j)
· x ≤ wj} for
j = 1, 2, ν.
98
Consumer Theory 4
Indirect Utility (9)
Micro I
Proof:
• First we show: If the optimal x ∈ Bν, then x ∈ B1 or x ∈ B2.
This statement trivially holds for ν equal to 0 or 1.
For ν ∈ (0, 1) we get: Suppose that x ∈ Bν but x ∈ B1 or
x ∈ B2 is not true (then x ∈ B1 and x ∈ B2), i.e.
p1
· x > w1 ∧ p2
· x > w2
Multiplying the first term with ν and the second with 1 − ν
results in
νp1
· x > νw1 ∧ (1 − ν)p2
· x > (1 − ν)w2 99
Consumer Theory 4
Indirect Utility (10)
Micro I
Proof:
• Summing up both terms results in:
(νp1
+ (1 − ν)p2
) · x = pν
· x > νw1 + (1 − ν)w2 = wν
which contradicts our assumption that x ∈ Bν.
• From the fact that xν, pν
is either ∈ B1 or ∈ B2, it follows that
v(pν
, wν) ≤ max{v(p1
, w1), v(p2
, w2)}.
100
Consumer Theory 4
Expenditure Function (1)
Micro I
• With indirect utility we looked at maximized utility levels given
prices and income.
• Now we raise the question a little bit different: what expenditures
e are necessary to attain an utility level u given prices p.
• Expenditures e can be described by the function e = p · x.
101
Consumer Theory 4
Expenditure Function (2)
Micro I
• Definition - Expenditure Minimization Problem [EMP]:
minx p · x s.t. u(x) ≥ u, x ∈ X = RL
+, p 0. (We only look at
u ≥ u(0).)
• It is the dual problem of the utility maximization problem. The
solution of the EMP h(p, u) will be called Hicksian demand
correspondence.
• Definition - Expenditure Function: The minimum value
function e(p, u) solving the expenditure minimization problem
minx p · x s.t. u(x) ≥ u, p 0, is called expenditure function.
• Existence: The Weierstraß theorem guarantees the existence of
an x∗
s.t. p · x∗
are the minimal expenditures necessary to attain
an utility level u.
102
Consumer Theory 4
Expenditure Function (3)
Micro I
• Theorem: Properties of the Expenditure Function e(p, u):
[P 3.E.2]
If u(x) is continuous continuous utility function representing a
locally nonsatiated preference relation. Then the expenditure
function e(p, u) is
– Continuous in p, u domain Rn
++ × U.
– ∀p 0 strictly increasing in u.
– Non-decreasing in pl for all l = 1, . . . , L.
– Concave in p.
– Homogeneous of degree one in p.
103
Consumer Theory 4
Expenditure Function (5)
Micro I
Proof:
• Property 1 - continuous: Apply the theorem of the maximum.
104
Consumer Theory 4
Expenditure Function (6)
Micro I
Proof:
• Property 2 - increasing in u: Suppose u2 > u1, the set
{x|u(x) ≥ u1} has to be a proper subset {x|u(x) ≥ u2} by the
properties of the utility function. We show this be showing that
the contrapositive is correct. Assume that expenditures do not
increase: 0 ≤ p · h2 ≤ p · h1. h(p, u1) solves the EMP with utility
level u1 and p. Then by continuity of u(x) and local nonsatiation
we can find an α ∈ (0, 1) such that αh2 is preferred to h1 with
expenditures αp · h2 < p · h1. This contradicts that h1 solves the
EMP for p, u1. Therefore, expenditures induced by the first set
have to be higher than in in second one. Since u(x) is increasing
in x, we can find arbitrary pairs of x1
, x2
fulfilling the above
properties. Therefore e(p, u) is unbounded in u.
105
Consumer Theory 4
Expenditure Function (7)
Micro I
Proof:
• Property 2 - with calculus: From
minx p · x s.t. u(x) ≥ u , x ≥ 0 we derive the Lagrangian:
L(x, λ) = p · x + λ(u − u(x))
.
• From this Kuhn-Tucker problem we get:
∂L
∂xi
= pi − λ
∂u(x)
∂xi
≥ 0 ,
∂L
∂xi
xi = 0
∂L
∂λ
= u − u(x) ≤ 0 ,
∂L
∂λ
λ = 0
106
Consumer Theory 4
Expenditure Function (8)
Micro I
Proof:
• λ = 0 would imply that utility could be increased without
increasing the expenditures (in an optimum) ⇒ u = u(x) and
λ > 0.
• Good xi is demanded if the price does not exceed λ∂u(x)
∂xi
for all
xi > 0.
• The envelope theorem tells us that
∂e(p, u)
∂u
=
∂L(x, u)
∂u
= λ > 0
• Since u(x) is continuous and increasing the expenditure function
has to be unbounded.
107
Consumer Theory 4
Expenditure Function (9)
Micro I
Proof:
• Property 3 - non-increasing in pl: similar to property 3.
108
Consumer Theory 4
Expenditure Function (10)
Micro I
Proof:
• Property 4 - concave in p: Consider an arbitrary pair p1
and p2
and the convex combination pν = νp1
+ (1 − ν)p2
. The
expenditure function is concave if
e(pν, u) ≥ νe(p1
, u) + (1 − ν)e(p2
, u).
• For minimized expenditures it has to hold that p1
x1
≤ p1
x and
p2
x2
≤ p2
x for all x fulfilling u(x) ≥ u.
• x∗
ν minimizes expenditure at a convex combination of p1
and p2
.
• Then p1
x1
≤ p1
x∗
ν and p2
x2
≤ p2
x∗
ν have to hold.
109
Consumer Theory 4
Expenditure Function (11)
Micro I
Proof:
• Multiplying the first term with ν and the second with 1 − ν and
taking the sum results in νp1
x1
+ (1 − ν)p2
x2
≤ pνx∗
ν.
• Therefore the expenditure function is concave in p.
110
Consumer Theory 4
Expenditure Function (12)
Micro I
Proof:
• Property 5 - homogeneous of degree one in p: We have to show
that e(µp, u) = µ1
e(p, u); µ > 0. Plug in µp in the optimization
problem ⇒ e(µp, u) = {minx µp · x s.t. u(x) ≥ u}. Objective
function is linear in µ, the constraint is not affected by µ. With
calculus we immediately see the µ cancels out in the first order
conditions ⇒ xh
remains the same ⇒
µ{minx p · x s.t. u(x) ≥ u} = µe(p, u).
111
Consumer Theory 4
Hicksian Demand (1)
Micro I
• Theorem: Hicksian demand: [P 3.E.3] Let u(x) be continuous
utility function representing a locally nonsatiatated preference
order; p 0. Then the Hicksian demand correspondence has the
following properties:
– Homogeneous of degree zero.
– No excess utility u(x) = u.
– Convexity/uniqueness: If is convex, then h(p, u) is a convex
set. If is strictly convex, then h(p, u) is single valued.
112
Consumer Theory 4
Hicksian Demand (2)
Micro I
Proof:
• Homogeneity follows directly from the EMP.
min{p · x s.t. u(x) ≥ u} ⇔ α min{p · x s.t. u(x) ≥ u} ⇔
min{αp · x s.t. u(x) ≥ u} for α > 0.
• Suppose that there is an x ∈ h(p, u) with u(x) > u. By the
continuity of u we find an α ∈ (0, 1) such that x = αx and
u(x ) > u. But with x we get p · x < p · x. A contradiction that
x solves the EMP.
• For the last property see the theorem on Walrasian demand or
apply the forthcoming theorem.
113
Consumer Theory 4
Expenditure vs. Indirect Utility (1)
Micro I
• With (p, w) the indirect utility function provides us with the
maximum of utility u. Suppose w = e(p, u). By this definition
v(p, e(p, u)) ≥ u.
• Given p, u and an the expenditure function, we must derive
e(p, v(p, w)) ≤ w.
• Given an x∗
solving the utility maximization problem, i.e.
x∗
∈ x(p, w). Does x∗
solve the EMP if u = v(p, w)?
• Given an h∗
solving the EMP, i.e. h∗
∈ h(p, u). Does h∗
solve
the UMP if w = e(p, u)?
114
Consumer Theory 4
Expenditure vs. Indirect Utility (2)
Micro I
• Theorem: Equivalence between Indirect Utility and
Expenditure Function: [P 3.E.1] Let u(x) be continuous utility
function representing a locally nonsatiatated preference order;
p 0.
– If x∗
is optimal in the UMP with w > 0, then x∗
is optimal in
the EMP when u = u(x∗
). e(p, u(x∗
)) = w.
– If h∗
is optimal in the EMP with u > u(0), then h∗
is optimal
in the UMP when w = e(p, u). v(p, e(p, u)) = u.
115
Consumer Theory 4
Expenditure vs. Indirect Utility (3)
Micro I
Proof:
• We prove e(p, v(p, w)) = w by means of a contradiction. p, w
∈ Rn
++ × R+. By the definition of the expenditure function we get
e(p, v(p, w)) ≤ w. In addition h∗
∈ h(p, u).
To show equality assume that e(p, u) < w, where u = v(p, w) and x∗
solves the UMP: e(p, u) is continuous in u. Choose ε such that
e(p, u + ε) < w and e(p, u + ε) =: wε.
The properties of the indirect utility function imply v(p, wε) ≥ u + ε. Since
wε < w and v(p, w) is strictly increasing in w (by local nonstatiation) we
get: v(p, w) > v(p, wε) ≥ u + ε but u = v(p, w), which is a
contradiction. Therefore e(p, v(p, w)) = w and x∗
also solves the EMP,
such that x∗
∈ h(p, u) when u = v(p, u).
116
Consumer Theory 4
Expenditure vs. Indirect Utility (4)
Micro I
Proof:
• Next we prove v(p, e(p, u)) = u in the same way. p, u
∈ Rn
++ × U. By the definition of the indirect utility function we
get v(p, e(p, u)) ≥ u.
Assume that v(p, w) > u, where w = e(p, u) and h∗
solves the
EMP: v(p, w) is continuous in w. Choose ε such that
v(p, w − ε) > u and v(p, w − ε) =: uε.
The properties of the expenditure function imply
e(p, uε) ≤ w − ε. Since uε > u and e(p, u) is strictly increasing
in u we get: e(p, u) < e(p, uε) ≤ w − ε but w = e(p, u), which is
a contradiction. Therefore v(p, e(p, u)) = u. In addition h∗
also
solves the UMP.
117
Consumer Theory 4
Hicksian Demand (3)
Micro I
• Theorem: Hicksian/ Compensated law of demand: [P 3.E.4]
Let u(x) be continuous utility function representing a locally
nonsatiatated preference order and h(p, u) consists of a single
element for all p 0. Then the Hicksian demand function
satisfies the compensated law of demand: For all p and p :
(p − p )[h(p , u) − h(p , u)] ≤ 0.
118
Consumer Theory 4
Hicksian Demand (4)
Micro I
Proof:
• By the EMP: p · h(p , u) − p · h(p , u) ≤ 0 and
p · h(p , u) − p · h(p , u) ≤ 0 have to hold.
• Adding up the inequalities yields the result.
119
Consumer Theory 4
Shephard’s Lemma (1)
Micro I
• Investigate the relationship between a Hicksian demand function
and the expenditure function.
• Theorem - Shephard’s Lemma: [P 3.G.1] Let u(x) be
continuous utility function representing a locally nonsatiatated
preference order and h(p, u) consists of a single element. Then
for all p and u, the gradient vector of the expenditure function
with respect to p gives Hicksian demand.
pe(p, u) = h(p, u)
120
Consumer Theory 4
Shephard’s Lemma (2)
Micro I
Proof with calculus:
• Suppose that the envelope theorem can be applied (see e.g.
Mas-Colell [M.L.1], page 965):
• Then the Lagrangian is given by: L(x, λ) = p · x + λ(u − u(x)).121
• The Kuhn-Tucker conditions are:
∂L
∂xi
= pi − λ
∂u(x)
∂xi
≥ 0
∂L
∂xi
xi = 0
∂L
∂λ
= u − u(x) ≥ 0
∂L
∂λ
λ = 0
• λ = 0 would imply that utility could be increased without
increasing the expenditures (in an optimum) ⇒ u = u(x) and
λ > 0.
122
Consumer Theory 4
Shephard’s Lemma (3)
Micro I
Proof with calculus:
• Good xi is demanded if the price does not exceed λ∂u(x)
∂xi
for all
xi > 0.
• The envelope theorem tells us that
∂e(p, u)
∂pl
=
∂L(x, u)
∂pl
= hl(p, u)
for l = 1, . . . , N.
123
Consumer Theory 4
Shephard’s Lemma (4)
Micro I
Proof:
• The expenditure function is the support function µk of the
non-empty and closed set K = {x|u(x) ≥ u}. Since the solution
is unique by assumption, µK(p) = pe(p, u) = h(p, u) has to
hold by the Duality theorem.
• Alternatively: Assume differentiability and apply the envelope
theorem.
124
Consumer Theory 4
Expenditure F. and Hicksian Demand (1)
Micro I
• Furthermore, investigate the relationship between a Hicksian
demand function and the expenditure function.
• Theorem:: [P 3.E.5] Let u(x) be continuous utility function
representing a locally nonsatiatated and strictly convex
preference relation on X = RL
+. Suppose that h(p, u) is
continuously differentiable, then
– Dph(p, u) = D2
pe(p, u)
– Dph(p, u) is negative semidefinite
– Dph(p, u) is symmetric.
– Dph(p, u)p = 0.
125
Consumer Theory 4
Expenditure F. and Hicksian Demand (2)
Micro I
Proof:
• To show Dph(p, u)p = 0, we can use the fact that h(p, u) is
homogeneous of degree zero in prices (r = 0).
• By the Euler theorem [M.B.2] we get
L
l=1
∂h(p, u)
∂pl
pl = rh(p, u).
126
Consumer Theory 4
Walrasian vs. Hicksian Demand (1)
Micro I
• Here we want to analyze what happens if income w changes:
normal vs. inferior good.
• How is demand effected by prices changes: change in relative
prices - substitution effect, change in real income - income effect
• Properties of the demand and the law of demand.
• How does a price change of good i affect demand of good j.
• Although utility is continuous and strictly increasing, there might
be goods where demand declines while the price falls.
127
Consumer Theory 4
Walrasian vs. Hicksian Demand (2)
Micro I
• Definition - Substitution Effect, Income Effect: We split up
the total effect of a price change into
– an effect accounting for the change in the relative prices pi/pj
(with constant utility or real income) ⇒ Substitution Effect.
Here the consumer will substitute the relatively more expensive
good by the cheaper one.
– an effect induced by a change in real income (with constant
relative prices) ⇒ Income Effect.
128
Consumer Theory 4
Walrasian vs. Hicksian Demand (3)
Micro I
• Hicksian decomposition - keeps utility level constant to identify
the substitution effect.
• The residual between the total effect and the substitution effect
is the income effect.
• See Figures in Chapter 2
129
Consumer Theory 4
Walrasian vs. Hicksian Demand (4)
Micro I
• Here we observe that the Hicksian demand function exactly
accounts for the substitution effect.
• The difference between the change in Walrasian (total effect)
demand induced by a price change and the change in Hicksian
demand (substitution effect) results in the income effect.
• Note that the income effect need not be positive.
130
Consumer Theory 4
Walrasian vs. Hicksian Demand (5)
Micro I
• Formal description of these effects is given by the Slutsky
equation.
• Theorem - Slutsky Equation: [p 3.G.3] Assume that the
consumer’s preference relation is complete, transitive,
continuous, locally nonsatiated and strictly convex defined on
X = RL
+. Then for all (p, w) and u = v(p, w) we have
∂xl(p, w)
∂pj
T E
=
∂hl(p, u)
∂pj
SE
−xj(p, w)
∂xl(p, w)
∂w
IE
l, j = 1, . . . , L.
131
Consumer Theory 4
Walrasian vs. Hicksian Demand (6)
Micro I
• Equivalently:
Dph(p, u) = Dpx(p, w) + Dwx(p, w)x(p, w)
• Remark: In the following proof we shall assume that h(p, u) is
differentiable. Differentiability follows from duality theory
presented in Section 3.F. This is a topic of the Micro II course.
132
Consumer Theory 4
Walrasian vs. Hicksian Demand (7)
Micro I
Proof:
• First, we use the Duality result on demand:
hl(p, u) = xl(p, e(p, u)) and take partial derivatives with respect
to pj:
∂hl(p, u)
∂pj
=
∂xl(p, e(p, u))
∂pj
+
∂xl(p, e(p, u))
∂w
∂e(p, u)
∂pj
.
• Second: By the relationship between expenditure function and
indirect utility it follows that u = v(p, w) and
e(p, u) = e(p, v(p, w)) = w.
133
Consumer Theory 4
Walrasian vs. Hicksian Demand (8)
Micro I
Proof:
• Third: Shephard’s Lemma tells us that ∂e(p,u)
∂pj
= hj(p, u), this
gives
∂hl(p, u)
∂pj
=
∂xl(p, w)
∂pj
+
∂xl(p, w)
∂w
hj(p, u)
134
Consumer Theory 4
Walrasian vs. Hicksian Demand (9)
Micro I
Proof:
• Forth: Duality between Hicksian and Walrasian demand implies
that h(p, v(p, w)) = x(p, w) with v(p, w) = u. Thus
∂e(p,u)
∂pj
= xj(p, w).
• Arranging terms yields:
∂xl(p, w)
∂pj
=
∂h(p, u)
∂pj
− xj(p, w)
∂xl(p, w)
∂w
.
135
Consumer Theory 4
Walrasian vs. Hicksian Demand (10)
Micro I
• From the Sultsky equation we can construct the following matrix:
Definition - Slutsky Matrix:
S(p, w) :=





∂x1(p,w)
∂p1
+ x1(p, w)
∂x1(p,w)
∂w · · ·
∂x1(p,w)
∂pL
+ xL(p, w)
∂x1(p,w)
∂w
. . . ... . . .
∂xL(p,w)
∂p1
+ x1(p, w)
∂xL(p,w)
∂w · · ·
∂xL(p,w)
∂pL
+ xL(p, w)
∂xL(p,w)
∂w





136
Consumer Theory 4
Walrasian vs. Hicksian Demand (11)
Micro I
• Theorem Suppose that e(p, u) is twice continuously
differentiable. Then the Slutsky Matrix S(p, w) is negative
semidefinite, symmetric and satisfies S(p, w)p = 0.
137
Consumer Theory 4
Walrasian vs. Hicksian Demand (12)
Micro I
Proof:
• Negative semidefiniteness follows from the negative
semidefiniteness of Dph(p, u) which followed from the concavity
of the expenditure function.
• Symmetry follows from the existence of the expenditure function
and Young’s theorem.
• S(p, w) · p = 0 follows from an Euler theorem argument already
used in [P 3.G.2]
138
Consumer Theory 4
Roy’s Identity (1)
Micro I
• Goal is to connect Walrasian demand with the indirect utility
function.
• Theorem - Roy’s Identity: [P 3.G.4] Let u(x) be continuous
utility function representing a locally nonsatiatated and strictly
convex preference relation defined on X = RL
+. Suppose that
the indirect utility function v(p, w) is differentiable for any
p, w 0, then
x(p, w) = −
1
wv(p, w)
pv(p, w)
i.e.
xl(p, w) = −
∂v(p, w)/∂pl
∂v(p, w)/∂w
.
139
Consumer Theory 4
Roy’s Identity (1)
Micro I
Proof:
• Roy’s Identity: Assume that the envelope theorem can be applied
to v(p, w).
• Let (x∗
, λ∗
) maximize {maxx u(x) s.t. p · x ≤ w} then the
partial derivatives of the Lagrangian L(x, λ) with respect to pl
and w provide us with:
∂v(p, w)
∂pl
=
∂L(x∗
, λ∗
)
∂pl
= −λ∗
x∗
l , l = 1, . . . , L.
∂v(p, w)
∂w
=
∂L(x∗
, λ∗
)
∂w
= λ∗
.
140
Consumer Theory 2
Indirect Utility (11)
Micro I
Proof:
• Plug in −λ from the second equation results in
∂v(p, w)
∂pl
= −
∂v(p, w)
∂w
x∗
l
such that
−
∂v(p, w)/∂pl
∂v(p, w)/∂w
= xl(p, w).
• Note that ∂v(p, w)/∂w by our properties on the indirect utility
function.
141
Correspondences (1)
Micro I
• Generalized concept of a function.
• Definition - Correspondence: Given a set A ∈ Rn
, a
correspondence f : A Rk
is a rule that assigns a set f(x) to
every x ∈ A.
• If f(x) contains one element, then f is a function.
• A ⊆ Rn
and Y ⊆ Rn
are the domain and the codomain.
• Literture: e.g. Mas-Colell, chapter M.H, page 949.
142
Correspondences (2)
Micro I
• The set {(x, y)|x ∈ A , y ∈ Y , y ∈ f(x)} is called graph of the
correspondence.
• Definition - Closed Graph: A correspondence has a closed
graph if for any pair of sequences xm → x ∈ A, with xm ∈ A
and ym → y, with ym ∈ f(xm), we have y ∈ f(x).
143
Correspondences (3)
Micro I
• Regarding continuity there are two concepts with
correspondences.
• Definition - Upper Hemicontinuous: A correspondence is UHC
if the graph is closed and the images of compact sets are
bounded. I.e. for B compact, B ∈ A the set f(B) is bounded.
• Definition - Lower Hemicontinuous: A correspondence is LHC
if for every sequence xm → x, xm, x ∈ A and every y ∈ f(x), we
can find a sequence ym → y and an integer M such that
ym ∈ f(xm) for m > M.
144
Theorem of the Maximum (1)
Micro I
• Consider a constrained optimization problem:
max f(x) s.t. g(x, q) = 0
where q ∈ Q is a vector of parameters. Q ∈ RS
and x ∈ RN
.
f(x) is assumed to be continuous. C(q) is the constraint set
implied by g.
145
Theorem of the Maximum (1)
Micro I
• Definition: x(q) is the set of solutions of the problem, such that
x(q) ⊂ C(q) and v(q) is the maximum value function, i.e. f(x)
evaluated at an optimal x ∈ x(q).
• Theorem of the Maximum: Suppose that the constraint
correspondence is continuous the f is continuous. Then the
maximizer correspondence x : Q → RN
is upper hemicontinuous
and the value function v : Q → R is continuous. [T M.K.6], page
963.
146
Duality Theorem (1)
Micro I
• Until now we have not shown that c(w, y) or e(p, u) is
differentiable when u(x) is strictly quasiconcave.
• This property follows from the Duality Theorem.
• Mas-Colell, Chapter 3.F, page 63.
147
Duality Theorem (2)
Micro I
• A set is K ∈ Rn
is convex if αx + (1 − α)y ∈ K for all x, y ∈ K
and α ∈ [0, 1].
• A half space is a set of the form {x ∈ Rn
|p · x ≥ c}.
• p = 0 is called the normal vector: if x and x fulfill
p · x = p · x = c, then p · (x − x ) = 0.
• The boundary set {x ∈ Rn
|p · x = c} is called hyperplane. The
half-space and the hyperplane are convex.
148
Duality Theorem (3)
Micro I
• Assume that K is convex and closed. Consider ¯x /∈ K. Then
there exists a half-space containing K and excluding ¯x. There is
a p and a c such that p · ¯x < c ≤ p · x for all x ∈ K (separating
hyperplane theorem).
• Basic idea of duality theory: A closed convex set can be
equivalently (dually) described by the intersection of half-spaces
containing this set.
• Mas-Colell, figure 3.F.1 and 3.F.2 page 64.
149
Duality Theorem (4)
Micro I
• If K is not convex the intersection of the half-spaces that
contain K is the smallest, convex set containing K. (closed
convex hull of K, abbreviated by ¯K).
• For any closed (but not necessarily convex) set K we can define
the support function of K:
µK(p) = inf{p · x|x ∈ K}
• When K is convex the support function provides us with the dual
description of K.
• µK(p) is homogeneous of degree one and concave in p.
150
Duality Theorem (5)
Micro I
• Theorem - Duality Theorem: Let K be a nonempty closed set
and let µK(p) be its support function. Then there is a unique
¯x ∈ K such that ¯p · ¯x = µK(¯p) if and only if µK(p) is
differentiable at ¯p. In this case pµK(¯p) = ¯x.
• Proof see literature. E.g. see section 25 in R.T. Rockafellar,
Convex Analysis, Princeton University Press, New York 1970.
151