Leonardo Felli
9 October, 2002
Microeconomics II
Lecture 1
Microeconomic Theory = the analysis of the behaviour
of individual economic agents;
and the aggregation of their actions in an institutional framework.
Four key elements are relevant in such definition:
1. individual agents: typically a consumer or a firm
(producer);
2. behaviour: traditionally utility maximization or
profit maximization;
1
Microeconomics II
2
3. the institutional framework: traditionally, the
price mechanism in an impersonal market place;
4. the mode of analysis, how the agents’ behaviour
is aggregated: equilibrium analysis.
What do we intend to get out?
A better understanding of economic activity and outcomes.
This is useful in two distinct senses:
• positive sense: a better understanding of individual agent’s behaviour in certain situations;
• normative sense: the ability to intervene or not,
both at the government level and at the institutional level.
Microeconomics II
3
Most of the models we shall analyze are highly simplified.
Hence, even though they have some general predictive power they may not be (directly) empirically
testable (they are too simple to be realistic).
Some of these models might be tested in a lab environment.
However, these models represent the building blocks
of more complex and realistic testable models.
Microeconomics II
4
Consumer Theory
agent = individual (consumer);
activity = consume one of a whole set of commodities
(goods and services). We focus on L commodities
l = 1, . . . , L
framework: consumption feasible set
X ⊂ RL
where x ∈ X is a consumption bundle which specifies
the amounts of the different commodities.
Time and location of a commodity may be included
in the definition of a commodity.
Microeconomics II
5
Let X be the set of commodity bundles that the individual can conceivably consume given the physical
constraints imposed by the environment.
Example of physical constraints: Impossibility to have
negative amounts of bread, water,. . . , indivisibility.
Constraints may be physical but also institutional
(legal requirements).
Example:
X = x ∈ R | xl ≥ 0, ∀l = 1, . . . , L = RL+
L
non negative orthant.
Microeconomics II
6
Properties of the consumption feasible set:
1. non-negativity;
2. it is a closed set: it includes its own boundary;
3. convexity: if x ∈ X and y ∈ X then define
x00 = αx + (1 − α)y ∈ X for every α ∈ [0, 1].
Each consumer is endowed with a preference relation
defined on the consumption feasible set X.
These preferences represent the primitive of our analysis.
Microeconomics II
7
The expression:
xy
means that “x is at least as good as y”.
From the weak preference relation two relevant binary relations may be derived:
• the strong preference relation defined as follows.
x y iff x y and not y x;
• the indifference preference relation ∼ defined as
follows.
x ∼ y iff x y and y x.
Microeconomics II
8
Axioms of choice:
1. Completeness: for every x, y ∈ X either x y
or y x, or both.
2. Transitivity: for every x, y, z ∈ X if x y and
y z then
x z.
3. Reflexivity: for every x ∈ X
x x.
A preference relation satisfying completeness, transitivity and reflexivity is termed rational.
4. Continuity: the preference relation in X is
continuous if it is preserved under the limit operation.
Microeconomics II
9
In other words, for every converging sequence of pairs
of commodity bundles {(xn, y n)}∞
n=0 such that
xn y n
∀n
where
x = lim xn
n→∞
y = lim y n
n→∞
then
x y.
An alternative formulation of such axiom is that:
given a bundle x both the upper contour set
{y ∈ X | y x}
and the lower contour set
{y ∈ X | x y}
are closed sets.
Microeconomics II
10
Finally, this axiom can be equivalently formulated in
the form: both the strict upper contour set
{y ∈ X | y x}
and the strict lower contour set
{y ∈ X | x y}
are open sets.
Define a utility function as a mapping u : X → R
which summarizes and represents the preference of a
consumer in an ordinal fashion.
One of the main results of consumer theory is the
following representation theorem.
Microeconomics II
11
Theorem: (Representation Theorem) If preferences
are
• rational (complete, reflexive and transitive);
• and continuous;
then there exists a continuous utility function that
represents such preferences.
A utility function represents a preference relation if the following holds:
xy
iff
u(x) ≥ u(y)
The proof of such theorem is rather lengthy. We
prove an easier theorem that makes the following extra assumption on the preference relation .
Microeconomics II
12
5. Strong monotonicity: for every x, y ∈ X if x ≥ y
(meaning xl ≥ yl for every l = 1, . . . , L) but x 6= y
(meaning that there exists an l such that xl > yl )
then
x y.
Theorem: (Easier Representation Theorem)
If preferences are
• rational (complete, reflexive and transitive);
• continuous;
• and strongly monotonic;
then there exists a continuous utility function that
represents them.
Microeconomics II
Proof:
Let
 
1
 
.. 
e=
 
1
and for given x ∈ X let
B(x) = {t ∈ R | (t e) x}
(upper contour set)
where
 
t
 
.. 
(t e) = 
 
t
and
W (x) = {t ∈ R | x (t e)}
(lower contour set)
13
Microeconomics II
By strong monotonicity:
• B(x) is non-empty;
• W (x) is non-empty since 0 ∈ W (x);
By continuity of :
• B(x) and W (x) are both closed.
By completeness
• B(x) ∪ W (x) = R
By connectedness of R (divisibility theorem):
• there exists a tx ∈ R such that
(tx e) ∼ x.
14
Microeconomics II
15
Define now
u(x) = tx.
Claim:
u(·) represents the preference relation . In other
words given x ∈ X and y ∈ X:
u(y) ≥ u(x)
iff
yx
Proof:
(Sufficiency:) Assume
u(y) ≥ u(x);
• by definition of u(·) it implies
ty ≥ tx;
• by strong monotonicity
(ty e) (tx e);
Microeconomics II
• by definition of u(·)
y ∼ (ty e)
(tx e) ∼ x;
• by transitivity:
y x.
(Necessity:) Assume
y x;
• by definition of ty and tx:
y ∼ (ty e)
(tx e) ∼ x;
• by transitivity:
(ty e) (tx e);
16
Microeconomics II
17
• by strong monotonicity:
ty ≥ tx;
• by definition of u(·):
u(y) ≥ u(x).
Question: did we really use strong monotonicity or
something weaker?
Answer: weaker.
The final step is the proof of the continuity of the
utility function u(·).
Continuity of u(·) means:
n
for any sequence {xn}∞
with
x
=
lim
x
we have
n=0
n→∞
lim u(xn) = u(x).
n→∞
Microeconomics II
18
Alternatively continuity can be stated as: given a
bundle y
{x | u(x) ≥ u(y)}
and
{x | u(x) ≤ u(y)}
are both closed.
Recall that continuity of preferences can be stated,
given y ∈ X as:
{x | x y}
and
{x | y x}
are both closed.
Since the last two sets are the same as the previous
two (by the Claim above) this concludes the proof.
Microeconomics II
19
Notice that there exists preferences that have no utility representation.
Consider for example the following lexicographic preferences:
(x1, x2) (y1, y2)
if and only if
• either x1 > y1;
• or x1 = y1 and x2 > y2.
Microeconomics II
20
Discontinuity follows from the fact that the upper
contour set and the lower contour set are both neither
closed nor open:
x1
6
{x | x x̂}
(x̂1, x̂2)
.............................................s
{x | x̂ x}
-
x2
Microeconomics II
21
For completeness, let us introduce a weaker assumption than strong monotonicity (usually assumed):
6. Local non-satiation:
A preference relation is locally non-satiated if for
every x ∈ X and every ε > 0, there exists y ∈ X
such that:
k y − x k≤ ε and
yx
where k y − x k denotes the Euclidean distance between points x and y in an L-dimensional vector
space:
"
k y − x k=
L
X
# 21
(xl − yl )2
.
l=1
Thick indifference curves violate local non-satiation
(however, there still exists a utility representation).