D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 1
Sapienza University of Rome. Ph.D. Program in Economics a.y. 2014-2015
Microeconomics 1 – Lecture notes (*)
LN 1. Rev 2.0 - Preferences and Utility
1.1 Definition and properties of the preference relation ” · ”
1.1.1 The commodity space
1.1.2 Definition of the preference relation “ · ”. Rational preferences
1.1.3. Desirability of commodities. Monotonicity of preferences
1.1.4. Continuity of preferences
1.1.5 Inclination for diversification. Convexity of preferences
1.2 Representation of preferences by a utility function
1.3. From properties of preferences to properties of the representing utility function
1.4 Properties of a differentiable utility function representing monotone and convex
preferences
1.5 Smooth preferences
1.6 Summing up
1.7 Lexicographic preferences
1.7.1 Definition
1.7.2Properties
1.7.3 Non existence of a utility representation
We begin in this first Lecture Note the study of the classical, preference-based approach to
consumer demand. We assume here, as a primitive of the entire analytical construction, that
every consumer has a binary preference ordering · over bundles of commodities;1 we endow
(*)As the program of the course indicates, Mas-Colell, Winston and Green, Microeconomic Theory (henceforth MWG)
is the basic, but not the only, reference book for the course. These Lecture Notes aim, without the pretence of offering a
complete presentation of the subject matter of the course, to clarify some points that, on the basis of my experience,
may present particular difficulties for the students attending the course without a specific undergraduate preparation in
Microeconomics. Particular attention is devoted to the presentation of mathematical notions generally confined to the
appendix of the advanced textbooks in Microeconomics, with detailed explanations and ample recourse to diagrams in
an effort – perhaps at time excessive - of simplification and clarification.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 2
this preference ordering of various properties and examine the associated representation by
means of a numerical function, the utility function. Since the maximization of a utility
function subject to constraints is the standard, highly convenient tool for the determination
and the study of the properties of demand functions (correspondences), specific attention is
dedicated to establish a strict connection between properties of preferences and properties of
the utility functions representing them. The role of continuity of preferences is emphasized:
when preferences are not continuous, as in the well-known case of lexicographic preferences,
no standard utility representation is possible.
We start with the basic definition of rational preferences (Section 1.1) and move on to show
(Section 1.2) that if rational preferences are continuous they can be represented by a
continuous utility function, defined up to an increasing monotonic transformation. The
notions of ordinal and cardinal utility are accordingly clarified. Sections 1.3 and 1.4 introduce
more structure in the preference relation: using order and algebraic properties of the
commodity space, the notions of monotonic and convex preferences are defined. Particularly
important are the implications of convexity of preferences for the properties of the associated
utility functions. The notion of smooth preferences is outlined in section 1.5 as an extension
and a refinement of strict convexity and defined directly in terms of the differential properties
of utility functions.
Lexicographic preferences are a typical and amply studied example of a preference order that
fails to be continuous. Lexicographic preferences are defined in section 2.1 and their
properties analyzed in section 2.2. Section 2.3 offers a proof of the impossibility of
representing these preferences by a utility function.
1.1 Definition and properties of the preference relation “ · ”
1.1.1 The commodity space
The decision problem of the consumer is to determine the consumption demand/supply (for
instance, of labor services, land and real estate property rental services) of the various
commodities, given the constraints that determine the feasible set of his choices and the
benefit that he receives from alternative choices. The structure of the commodity space L is
defined by the following assumptions:
(i) the number of commodities is finite and equal to L, indexed by 1,...,l L ;2
1 We will later examine the choice-based approach to demand theory, which relies on the notion of revealed
preferences. 2 With a typical abuse of notation, L denotes both the set and the number of elements in the set.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 3
(ii) a commodity bundle is a specification of the quantities of each of the L commodities
purchased/supplied by a consumer: analytically a column vector 1,..., LLx x x , i.e. an
element (point) of the commodity space L ;
(iii) commodities are perfectly divisible; lx indicates the quantity of commodity l , a real
number that can take any value in ; a commodity bundle will therefore contain in general
positive, zero and negative terms;
(iv) the commodity space L is a real vector space; the Euclidean norm 1
22 21 ... Lx x x
determines the length of the vectors and the derived Euclidean metric ,d x y x y
determines the distance between any two vectors.
These assumptions deserve a brief comment.
1) Commodities are distinguished by quality, location, date and, in studies of behavior under
uncertainty, state of nature in which they are available.3 This implies that the number of
commodities may be very large. The critical assumption is that it is finite. In problems
concerning consumers’ choices over an infinite horizon, the commodity space is infinite even
the simplest case of a single commodity model. This raises analytical problems that are not
dealt with in these Notes.
2) Positive elements in the commodity bundle reflect commodities the consumer desires to
have inasmuch as they increase his well being, while negative entries indicate commodities
that reduce the consumer’s well being. The former can be properly termed goods, the latter
bads. In this latter category are included not only negative externalities - such as smoke,
noise, congestion, ect – but also the supply of services and possibly of commodities. We can,
for instance, consider the supply of labor services of some type as a negative consumption of
an otherwise available leisure time. We will soon return to this assumption and conveniently
redefine the commodity space as the non negative orthant L of the Euclidean L-dimensional
space.
3) The standard assumption that commodities are perfectly divisible leads to the analytical
implication that the individual demand functions of all commodities are continuous and thus
susceptible of study by calculus techniques. It is a quite strong assumption. Typically many
commodities are available in well defined units. Consumers can buy, for instance, a
refrigerator and not a fraction of it, let alone an infinitesimal fraction of it as theoretically
admissible under the assumption of perfect divisibility. When the analysis moves, however,
from the individual to the market level, aggregation may produce useful regularizing effects.
If individual preferences are sufficiently dispersed, aggregate demand may be nearly
3 With such an extensive definition of a commodity the theory of consumer’s behavior encompasses the more specific
theories of location and trade, of intertemporal choice and of the theory of decisions under uncertainty.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 4
continuous even if individual demands are discontinuous, as in the case of a commodity
available only in integer units (see MWG p. 122).
Discrete choice models describe an altogether different problem, that of choosing among
alternatives (binary or multiple); typical instances, largely studied in the literature, are the
choice of alternative transportation modes to reach the office, of the college to attend or of the
supermarket to shop at. Given the attributes of the alternative under consideration (quality,
price, distance, time, parking facility, costumer care) and the characteristics of the consumer
(age, family, income), a probabilistic model describes the chance that each specific alternative
will be chosen. The availability of survey data makes extensive econometric analysis possible.
A very brief introduction to discrete choice models is presented in the Appendix of Lecture
Note 4 – The utility maximization problem .
1.1.2 Definition of the preference relation “ · ”. Rational preferences
As a primitive of the construction of a theory of consumer’s choice we assume that every
consumer has a binary preference relation (or preference ordering) “ · ” over bundles of
commodities in L .4 Given , Lx y , the preference relation x · y defines a subset of the
space L L ; we can be read as x is preferred or indifferent to y, x is weakly preferred to y,
or x is at least as good as y, or x is no worse than y. We will generally adopt either one of the
first two readings.
From the weak preference relation · two further preference relations can be derived:
(i) the strict preference relation Lx y , when x · y and not y · x;
(ii) the indifference relation yx , when x · y and y · x. The indifference relation defines
an equivalence relation on L .
We impose a structure on the preference relation by means of a set of Axioms.
Axiom 1. Completeness of preferences
For all , Lx y , we have x · y or y · x or both;5
Axiom 2. Transitivity of preferences
For all , , Lx y z , if x · y and y · z, then x · y.
4 Different consumers will have different preference relations. For easier notation, we avoid indexing preferences with
a subscript indicating a specific consumer. 5 Connectedness is an alternative, though less frequently used term for completeness. More precisely, connectedness
implies that for all Lx y , there exists Lz such that x z y
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 5
We sum up these axioms in the following definition of rational preferences.6
Definition 1.1.1 The preference relation is rational if it is complete and transitive.7
Both axioms are very strong and demanding on the capabilities of a consumer to express
definite and non contradictory preferences. They are bound to be violated in practice, as many
experimental results show; they are, nonetheless, necessary to construct a theory of choice
capable of producing rich and sharp conclusions, which can be empirically tested.
From the assumption of completeness of preferences we can derive the definition of five
subsets of LR : the indifferent set I x and the upper and lower closed contour sets, I x
and I x and the upper and lower open contour sets I x and I x
(1.1)
R int
R
R for all R
R
R i
nt
L
L
L L
L
L
I x y y x I x
I x y y x
I x y y x x
I x y x y
I x y x y I x
·
·
I x is the subset of LR of commodity bundles y indifferent to x. The upper contour set
I x
defines the subset of LR containing commodity bundles preferred or indifferent to a
given bundle x, whereas the lower contour set I x defines the subset of LR containing
commodity bundles not strictly preferred to x.8 intI x I x
defines the subset of LR
containing commodity bundles strictly preferred to a given bundle x, whereas
intI x I x defines the subset of LR containing commodity bundles y to which the
bundle x is strictly preferred.
6 A further axiom that preferences are reflexive (defined as x · x) is added to the axioms of completeness and
transitivity, but is, in fact, implied by them. 7 These properties define a complete preordering, not a complete ordering since x y does not imply x y .
8 Closed sets as the upper and lower contour sets I x and I x
, contain their boundaries and have the property
that the limit of all sequences, whose elements belong to the set, is itself an element of the set.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 6
Fig. 1.1 – The indifferent set I x , the upper and lower contour sets I x and I x
Fig. 1.1 depicts the indifferent set I x and the upper and lower contour sets I x and
I x in a two-commodity space on the assumptions, to be later justified, that commodity
bundles with larger quantities of commodities are preferred to bundles with a lesser quantity
(monotone preferences) and that the indifference set is a continuous curve, convex in the
north-east direction (convex preferences).
1.1.3. Desirability of commodities. Monotonicity of preferences
The axioms of completeness and transitivity of preferences are not adequate to produce a
sensible theory of choice: they simply state that consumers will behave rationally and choose
what is best for them. In order to construct a meaningful and testable theory of consumer’s
choice we need to give further structure to the rational preference relation through a set of
axioms in line with the properties of the commodity space; we thus establish a condition of
consistency between these latter properties and those of the preference relation. These axioms
tend to reflect aspects of individual preferences that do appear to have a general relevance in
the description of consumption behavior. Departure from these axioms can and have been
considered and the consequences for the theory of consumer choice studied. The questions
involved are highly technical and will not be pursued here.
We use the order property of the commodity space to express the idea that commodities are
desirable.
With no sign restriction imposed on the commodity space, any commodity bundle Lx
may contain positive as well as negative values: the former represent commodities that a
x
1x
2x
xI
xI
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 7
consumer may wish to have in positive quantity, the latter commodities that the consumer
would rather not have to consume inasmuch as the reduce his well-being. While commodities
of the former group can properly be termed goods, those of the latter group can be better
called bads. The notion of desirability of commodities is the expressed by the following
axiom that states that in the neighborhood of any commodity bundle Lx there is a
commodity bundle Ly strictly preferred to x.
Axiom 3’. Non satiation of preferences
For every Lx and 0 , there is a Ly such that y z and y x .
9
By turning bads such as smoke, noise, congestion, labor time into goods by taking their
opposite as clean air, silence, uncongested roads, leisure time, we can obtain a convenient
redefinition of commodities as all belonging to the group of goods, i.e. to the non negative
orthant 0RR xx LL of the commodity space. The notion of desirability of
commodities can now be formulated in stronger terms using the order property of R L in
terms of a preference for having a larger rather than a smaller quantity of commodities.
Axiom 3. Monotonicity of preferences.
The preference relation · is monotone if for all Lyx R, y x y x . In words, the
commodity bundle y is strictly preferred to the commodity bundle x if y contains a greater
quantity of all commodities: having more of all commodities is preferred to having less of
them.
The preference relation · is strongly (or strictly) monotone if for all Lyx R,
,y x y x y x In words, the bundle y is strictly preferred to the bundle x if y
contains a greater quantity of at least one commodity: having more of at least one commodity
is preferred to having less of that commodity and equal quantities of all the other, which
means that all goods are desirable.
The preference relation · is weakly monotone if for all Lyx R, yxy · x. In words,
the bundle y containing at least the same quantity of all commodities as the bundle x cannot,
by the completeness axiom, be less desirable than x .
Fig. 1.2 illustrates the difference between the properties of monotonicity and strong
monotonicity in two dimensions. Given the commodity bundle 2Rx , consider the subspace
9 An important implication of non satiation of preferences is that a consumer may prefer to a bundle x a bundle y with a
smaller quantity of all commodities, goods and bads. In other words, a consumer may be willing to sacrifice some
goods in order to reduce the quantity of some bads that he would have otherwise to consume if he chose bundle x.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 8
x2R with origin in
0x and coordinate axes BC and BA . Monotonicity of preferences
implies that all bundles xy 2R , with the coordinate axes BC and AB excluded, are
preferred to x . Strong monotonicity of preferences implies that all bundles xy 2R , with
the coordinate axes BC and AB included, are preferred to x .
It is immediate to verify that monotone preferences are locally non satiated but not vice versa.
We will assume in the sequence that preferences are monotone.
1.1.4. Continuity of preferences
We use the topological structure of the commodity space L to define the notion of
continuity.
Axiom 4. The rational preference relation · is continuous if:
(i) Given the sequences of commodity bundles nx x and ny y with ny ≿ nx , then
y ≿ x ;
(ii) Equivalently, if and only if the upper and lower contour sets I x and I x , are
closed.
1x
2x
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 9
Definition (i) of continuity thus requires that the preference relation between commodity
bundles 1
n
ny
and
1
n
nx
be preserved in the limit.
To show the equivalence between the two definitions consider first the “if” part of the
statement (ii). If the preference relation is continuous according to (i) of Axiom 4, all the
terms of the sequence ny ≿ nx , as well as the limiting values y ≿ x belong by definition in
the upper contour set of x. For the “only if” part of the proof, assume that the upper contour
set is closed and suppose, by contradiction, y x . This means inty I x , an open set.
Then there exists an open ball B y and an N such that for all n N ny B y ,
contradicting the assumption ny ≿ nx .
Continuity imposes a regularity condition on the preference field: it excludes the presence of
sudden jumps in preference associated with an infinitesimal change in the composition of the
commodity bundle. A typical violation of the continuity axiom is represented by
lexicographic preferences, which will be separately examined at the end of this Note.
A strengthening of the continuity assumption of preferences is needed when continuous
differentiability of demand functions is required. The related notion of smooth preferences
will be introduced after the representation of continuous preferences by a utility function and
defined by the properties of that function.
1.1.5 Inclination for diversification. Convexity of preferences
We use the algebraic property of the commodity space to express the notion that consumers
exhibit a basic inclination towards diversification, that they prefer a more balanced
commodity bundle to commodity bundles with a more extreme composition. This inclination
for diversification is captured by the axiom of convexity of preferences. It is this assumption
of convexity that makes it possible to establish the shape of the indifference sets.10
Axiom 5 - Convexity of preferences
The preference relation · is convex if for all Lx R the upper contour set
RLI x y y x · is convex, i.e. if, given y · x, their convex combination
1z y x , with 0,1 , is such that z · x.
10
Strictly convex preferences exclude in fact that the optimal choice be a commodity bundle of extreme composition.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 10
The preference relation · is strictly convex if, for all Lx R with xy , their strictly convex
combination 1z y x , with 0,1 , is such that z x .
Panels (a) and (b) of Fig. 1.3 show the difference between convexity and strict convexity of
preferences using indifference sets. In the first case the indifference set may contain a linear
segment, in the second this is exluded; in the first case the bundle z lies on the same
indifference set as x and y , in the second z lies on a higher indifference set. A mixture of x
and y cannot be worse than either x or y.
Fig. 1.3 Panel (a) – Convex preferences Fig. 1.3 Panel (b) - Strictly convex
preferences
1.2 Representation of preferences by a utility function
The possibility of representing preferences by means of a numerical function (a utility
function) is a central result in classical demand theory, which is based on the assumption that
consumers behave optimally, in the sense that they choose a consumption bundle that is
optimal given their preferences and their budget constraint. As stated, the solution of this
problem is rather awkward as there are no analytical conditions capable of determining the
commodity bundle which is optimal with respect to preferences. The representation of
preferences by a utility function offers a neat way out of the difficulty: the problem of
determining an optimal consumption bundle becomes one of finding the solution to a standard
problem of maximization of a function subject to constraints, for which we have well defined
analytical techniques if the utility function is continuous, as generally assumed. Key to the
possibility of representing preferences by a numerical function is the Axiom 5 of continuity of
preferences.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 11
Proposition 1.2.1. If the rational preference relation · is continuous, there exists a
continuous utility function u x such that
(1.2) x · y u x u y
We refer to Debreu (1954 and 1959, pp. 55-59) for the difficult proof of this proposition. A
relatively simple, elegant and constructive proof of Proposition 1.2.1 is possible if we assume
that preferences are monotonic (Axiom 4) and convex (Axiom 6).11
In the proof of the
following proposition, which is constructed in three steps, we follow MWG (pp. 47-48) with
the help of Fig. 1.2.12
Proposition 1.2.2. If the rational preference relation · defined in the commodity space
L , is continuous and monotone, there exists a continuous utility function u x ,
defined up to a positive monotonic transformation, which verifies condition (1.2).
The proof is articulated in three steps.
Proof. Step 1 - Construction of a utility function. Let 1,1,...,1e be the unit vector, e
with 0 a commodity bundle containing equal quantities of all commodities and
0R eeZ L the set of all such commodity bundles. As indicated in Fig. 1.4, Z is the
subset of L
R represented by the 45° degree half line.
11
Convexity is not necessary; it merely reflects the usual way to depict indifference curves in the diagrams. 12
See also JR (pp. 14-16) and Varian (p. 97).
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 12
Fig. 1.4 – Construction of a utility function representing preferences
By the assumed monotonicity of preferences, x · 0 and for all such that e x , e x
as depicted in Fig. 1.4. Consider now the intersection of the indifference curve I x with the
45° degrees line: this determines the commodity bundle x x . We can then define the
sets eA
R · x and xA
R · e . By monotonicity, they are both non
empty. Since the upper and lower contour sets of x are closed, so are the sets A and A . By
construction they are connected and have a non empty interception, the real number . We
can conclude that there exists a scalar such that xe . In fact, there can be at most one
such scalar, since 1 2 would imply 1 2e e and thus 1 2e e . Let us call this scalar
x ; it is our numerical representation of preferences.
Proof. Step 2 - x verifies the defining property (1.2) of Proposition 1.2.1. For the if
part, assume x · y. By construction of , we have xex and yey . We then have
x e · y e and, therefore, x y . The only if part follows working our way
backward in the above statement. We can now set u x x as our utility function.
1x
2x
AA)()( xux
x
45
ex)(
e 0R eeZ L
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 13
Proof Step 3 - u x is continuous. We can now show that the utility function
: Lu x x that we have just constructed is continuous.13
We use for this purpose
the following definition of continuity.
Definition 1.2.1. – Let D be a subset of L . The function :f D is continuous if
the inverse image 1f B of every open (closed) ball B in , is open (closed) in D .
Since open balls in are intervals, consider the open interval of values of the utility function
,a b with a b and the inverse image of this interval 1 ,u a b . Fig. 1.5, similar to Fig.
1.4, clarifies the structure of the proof. Three indifference curves are now drawn in the
diagram: , and u a u x u b with a u x b . Let us first try to see what is the
intuition behind his very important result. The initial step is to associate to these three utility
levels a preference relation among three corresponding commodity bundles. Using to this end
the 45° degrees line – with the same procedure as used in step 1 of the proof of Proposition
1.2.2 - we can associate with the utility level a the commodity bundle e at the intersection
of the indifference curve with utility a and the 45° degrees line. We determine with the same
procedure the commodity bundle e , that has by construction utility level b . Using the
relations (1.1), we can define the upper contour set LI e y y e and the lower
contour set LI e y y e . These sets are both open and so is their intersection
depicted in Fig. 1.5 as the colored strip between the indifference curves associated to utility
levels a and b . The commodity bundles in this strip represent the inverse image of the open
interval of utility values ,a b .
13
We follow very closely the proof offered by JR (pp. 15-16). But see also Hildenbrand and Kirman (1988, p. 68).
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 14
Fig. 1.5 – Continuity of the utility function u x
Turning to the analytical derivation, we have
(1.3)
1 ,
L
L
L
L
u a b x a u x b
x ae u x e be
x ae x e be
x ae x be
The first equality follows from the definition of the inverse image, the second from
associating to the relation among the utility levels , and a u x b the corresponding preference
relation among the commodity bundles, containing equal quantities of all commodities; the
existence of a utility function representing preferences guaranties that the order of preference
is the same as the order of the utility levels. The third equality follows from the construction
of the utility function and the fourth from the indifference relation x e x . We can finally
write the last line of (1.3) as the intersection of the sets, defined in (1.1), containing the
commodity bundles preferred to ae - i.e. the set LI ae y R y ae - and the
commodity bundles not preferred to be - i.e. the set LI be y R y be :
(1.4) 1 , L Lu a b I ae y R y ae I be y R y be
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 15
By the assumption of continuity of preferences, the sets LI ae y aeR y · and
LI be y R y be · are closed in LR , while the sets I be and I ae are open as
complements of closed sets. 1 ,u a b is therefore also open, as the intersection of two open
sets: the open strip in fig. 1.5 between the indifference curves with utility respectively equal
to a and to b.
Proposition 1.2.2 is very important in the development of neoclassical demand analysis: it
opens the way to solve the problem of optimal consumption behavior determining the
commodity bundle which maximizes a continuous utility function rather than the commodity
bundle which is maximal with respect to the preference ordering. As established in the
following proposition, there is not a unique representation of a given preference ordering. In
effect there is an entire family of functions, obtained by applying an increasing monotonic
transformation of the utility assigned to the commodity bundle x , namely the transformation
f x . Increasing monotonic transformations preserve the preference order. This has
important implications for the meaning of the utility representation of preferences but not, as
it will become clear in the sequel, on the solution of the maximization problem.
Proposition 1.2.3. Let u x be a function representing the preference relation and
:f a strictly increasing function. Then the function v x f u x represents
the same preference ordering if and only if
(1.5) v x v y u x u y
Proof. Let x · y; then by representation u x u y and by the assumption on f
f u x f u y , whence v x v y . The only if part follows from the assumption that
f is a positive monotonic transformation, which can accordingly be inverted.
A well-behaved utility function is conveniently represented in 2-commodity space by a family
of indifference curves as in Fig. 1.6, where the shape of the curves and the direction of
increase of utility reflect the assumption of monotonicity and convexity used in the
construction of the function. The axioms defining rational preferences imply that the
commodity space is densely populated by indifference curves (axiom of completeness), no
two of which can intersect (axiom of transitivity).14
The actual number attributed to each
indifference curve is irrelevant, provided that it is an increasing number, so that to higher
(further displaced from the origin in the north-east direction) indifference curves a
progressively greater number is assigned.
14
The non intersection of any two indifference curve is easily derived by contradiction.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 16
Properties of preferences that are invariant under strictly increasing transformations are called
ordinal, and ordinal utility the corresponding representation of preferences. Properties that are
maintained under a positive affine transformation15
are called cardinal and cardinal utility a
utility function with such properties.16
While positive affine transformations preserve the
order of utility differences and thus make such differences comparables, increasing monotonic
transformation do not in general preserve this order and thus exclude the possibility to make
comparison of utility differences. Furthermore, whether defined up to a positive monotonic
transformation or up to a positive affine transformation, the utility function in no way offers a
measure of the intensity of individual preferences and thus the possibility of comparing the
utility scales of different consumers.17
Fig. 1.6 – A map of indifference curves
1.3. From properties of preferences to properties of the representing utility function
We will now establish a relation between the axioms of monotonicity and convexity of
preferences and the resulting properties of the utility function.
A function representing monotone preferences must assign to a bundle y - containing a
greater quantity of all commodities than bundle x - a utility greater than to bundle x :
15
The function :h is a positive affine transformation if h u x a bu x with 0b . 16
Contrary to positive monotonic transformations, positive affine transformations preserve also the order of differences
of utility. 17
A bit of history of utility theory from cardinal to ordinal representation of preferences is briefly given in a subsequent
Lecture Note dedicated to the revealed preference approach to demand analysis.
2x
1x
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 17
u y u x .18
In other words, the bundle y must lie on a higher indifference curve than the
bundle x.
The notion of monotone preferences has further important implications. First, it excludes the
possibility that there may be a saturation, or bliss point in the commodity space.19
It further
implies that indifference sets cannot be thick, because otherwise in the neighborhood of any
bundle x there would be a bundle y x and thus preferred to x. It finally points to the
possible form of the indifference set I x . Note, with reference to Fig. 1.2, that all points in
the interior of quadrant numbered 1 are preferred or indifferent to x , whereas all points in the
interior of quadrant numbered 3 are strictly less preferred than x . The indifference set I x
must therefore lie in quadrants 2 and 4, possibly coinciding with the axes BC and AB as in
the case the Leontief utility function,20
and is thus represented by a function decreasing from
left to right. Note that a concave indifference curve would meet this requirement; it is the
axiom of convexity that confers to the indifference curves the standard form with a
diminishing slope measured in absolute value moving in the diagram from left to right.
In order to establish the properties of a utility function representing convex preferences we
would need to introduce the definition of concave and quasiconcave functions which we defer
to Lecture Note 2 where we will examine at length the definitions and properties of these
functions. We merely assert here that both concave and quasiconcave functions represent
convex preferences.
We can at this point redefine the notions of indifference, upper and lower closed contour sets
presented in relation (1.1) and based on the preference relation · in terms of concave and
quasiconcave functions representing convex preferences as indicated in the following relation
(1.10)
L
L
L
L
x
xuyuyxI
xuyuyxI
xuyuyxI
R allfor
R
R
R
Let us now draw some important implications of convex (strictly convex) preferences for the
utility function representing them.
18
The Leontief utility function 1 2min ,u x x x represents monotone preferences but does not represent strongly
monotone preferences. Referring back to Fig. 1.2, the half lines AB e BC are an indifference curve of the Leontief utility
function relative to the commodity bundle 0x ; the vertical “axis” AB is the set of points 0 0
1 2 2, 0x x x x with
0u x u x . No increase in utility derives from an increase in the quantity of only one commodity, the quantity of
the other one remaining fixed. 19
If our definition of commodities should admit of the presence of both goods and bads, so that the reference
commodity space is L
rather than L , the conclusion of the non existence of a saturation point would equally
follow. 20
See ft note 19.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 18
1.4 Properties of a differentiable utility function representing monotone and convex
preferences
If we assume that the utility function is differentiable, we can give a better characterization of
monotone preferences, namely that monotone preferences are represented by a utility function
with strictly positive first order partial derivatives
Definition 1.4.1. A utility function u x representing monotone preferences has
positive first order partial derivative: 0u x , where the Nabla operator
(1.4) 1 ... ... T
l Lu x u x u x u x
denotes the transpose of the vector of first order partial derivatives of the utility function, that
is of the marginal utilities of the various commodities.
In economic terms, the assumption that preferences are monotone implies, therefore, that the
marginal utilities of all commodities are positive. Since marginal utilities are not invariant to
increasing monotonic transformation of the utility function, they are cardinal properties.
We did noticed that monotone preferences give rise to indifference curves decreasing from
left to right in the two commodity case. Convexity of preferences adds the further property
that indifference curves are convex. This means that, ass we move along a convex
indifference curve from left to right, we increase the quantity of commodity 1 and reduce the
quantity of commodity 2 maintaining the same level of utility. Assuming that the utility
function is differentiable, we can then define the notion of marginal rate of substitution of
commodity 1 for commodity 2 as the quantity of commodity 2 that the consumer is willing to
give up to obtain an additional, “marginal” unit of commodity 1 with unchanged utility21
(1.11)
12
1,2 1 2
1 2
,u xdx
MRS x xdx u x
The marginal rate of substitution in consumption - MRS , for short – is, by definition (1.11),
positive. Fig. 1.3, Panel (b) associates with strict convexity of preferences the property that
the marginal rate of substitution is decreasing as we move from left to right along an
indifference curve. This property requires not only that marginal utilities be a diminishing
function of the quantity of every commodity, but also that the increase in the consumption of
21
The second equality in (1.11) follows from total differentiation of the indifference curve 1 2,u x x u with respect to
1x and 2x .
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 19
one commodity does not decrease the marginal utility of the other one.22
So from strict
convexity of preferences follows, under the stated conditions, the notion of diminishing
marginal utility of goods.23
The full implications of strongly monotone and strictly convex
preferences are further examined in the following section.
Note, furthermore, that the marginal rate of substitution is invariant to an increasing
monotonic transformations v x f u x of the utility function, while marginal utilities are
not. Definition (1.11) becomes
(1.12)
1 1 12
1,2 1 2
1 2 2 2
,v x f u x u xdx
MRS x xdx v x f u x u x
where f is the derivative of f with respect to its argument u x . The marginal rate of
substitution is, therefore, an ordinal property of the utility function.
1.5 Smooth preferences
While strict convexity of preferences excludes the presence of linear segments of the
indifference curves, it is nonetheless compatible with the situations described in Fig. 1.7. In
Panel (a) there is a kink in the indifference set at 0x . The marginal rate of substitution is
decreasing, but with a discontinuity. At all price ratios in the interval determined by the slopes
of the two limiting budget lines A B and A B - i.e. for 1 1 1
2 2 2
,p p p
p p p
- the optimal
consumption choice is always 0x . In Panel (b) there are two kinks at the points in which the
indifference curve intersect the axes; the marginal rate of substitution is again discontinuous
at these points. In both instances the resulting demand functions are not continuously
differentiable. In order to exclude these possibilities, the notion of strict convexity of
preferences must be further strengthened into that of smooth preferences which, in order to
22
The marginal rate of substitution is decreasing if its derivative with respect to 1x is negative. After various
rearrangements and indicating the second order partial derivatives of the utility function as ijx , 1,2i j , we have
3 2 21 22 1 2 12 2 11 1 22
1
,dMRS x xu u u u u u u u
dx
which is negative if the marginal utilities of both commodities
are decreasing 0iiu and the increase in the consumption of good i does not reduce the marginal utility of good j
0iju .
23 Note that the marginal rate of substitution is constant in the linear segments of the indifference curves representing
simply convex preferences, as in Panel (a) of Fig. 1.5. Linear segment represent situations in which commodities are
perfect substitutes.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 20
dodge the situation described by Panel (b) of Fig, 1.7, need in turn to be redefined with
respect to the positive orthant of the commodity space L .
Fig. 1.7 – Panels (a) and (b). Indifference curves with a kink
This development in the theory of preferences can be traced back to two problems that arose,
the first, in the theory of individual consumption and, the second, in the theory of general
equilibrium.
In the field of classical demand theory, the issue is referred to as the problem of integrability
of demand relations, that is of finding preferences that rationalize a (Walrasian) demand
function .24
This problem has a long history that dates back to a long ignored and constantly
referenced contribution by G.B. Antonelli (1886), who determined the integrability conditions
of the system of partial equations resulting from the indirect demand functions.25
24
As Hurwicz (1971) notes, by the end of the 1940’s the conditions necessary for single-valued demand relations (i.e.,
direct and indirect demand functions) to be generated by utility maximization subject to a budget constraint had been
clearly established. But what remained was the problem “about the sufficiency of these conditions? That is, what
properties of the demand relations guarantee the existence of a “generating” utility function (or, more generally, of a
“generating” preference ordering […]?” (Hurwicz, 1971, p. 176). 25
Perhaps surprisingly, although Antonelli’s paper was apparently intended to be the first chapter of a never completed
book, his only other work in economic theory is a book on the Compulsory Amortization of Capital. Antonelli was an
engineer with a deep knowledge of all problems pertaining to navigation, port and communications, as well as of
problems relating to the oil industry (see the Bibliographical Note in the appendix to the English translation of
Antonelli’s 1886 paper). He extensively published on these subject matters.
The integrability condition of indirect demand functions consists in the equality of the cross-partial derivatives which
reduces to requiring that the Slutsky matrix be symmetric (see MWG, pp. 75.80; Varian, pp. 125-129, 483-484).
0x
0xI
2x
1x1x
2x
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 21
In the field of the theory of general competitive equilibrium, with the development of the
differentiable approach, the smoothness (differentiability) of demand functions assumes a key
role in the study of uniqueness and stability of equilibria. Since demand is a derived function,
it was natural to complete the preference approach to classical demand theory with finding
conditions on preferences and utility that would lead to differentiable demands. Debreu
(1972) approached the problem from the point of view of the required properties of the
indifference set. He thus showed that the weak preference relation · can be represented by a
2C utility function26
if and only if preferences are continuous, strongly monotone and the
indifference set I x is a 2C differential manifold for all
Lx . Hence the definition that
the preference ordering is smooth if the indifference set has the stated property.
This is not the place to go into the highly technical contribution by G. Debreu. Since the
classical approach to demand theory is based on utility maximization, the important point is
here to indicate the conditions which must be satisfied by utility functions representing
smooth preferences. Smooth preferences are therefore defined in terms of the properties of the
representing utility functions.
Axiom 6. Smooth preferences
The preference relation · on 0RR xx LL is smooth if it is represented by a utility
function u x such that:
i) RR:
Lxu is of class 2C on
L
R ;
ii) For each Lx R , the indifference set ;RR LL xyyxI
iii) For each Lx R , 0;u x
iv) For each Lx R , the Hessian matrix H x - the matrix of second order partial
derivatives of the utility function - is negative definite in the linear subspace
0R zxuzZ L . The properties of the leading principle minors of the bordered
Hessian
(1.12)
0
B
T
H x u xH x
u x
that have to be satisfied are described in Lecture Note 2.
The meaning of these conditions is the following: i) the utility function is twice continuously
differentiable; thus excluding the presence of a kink, implying a discontinuity of the second
26
In general by rC utility function, i.e. by a utility function with continuous derivatives up to the r order.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 22
order derivative, as in Fig. 1.7, panel (a); ii) all indifference curves belong to the strictly
positive orthant, thus excluding kinks as in Fig. 1.7, panel (b); iii) marginal utilities are
strictly positive due to the assumption that preferences are strongly monotone; iv) the utility
function is strictly quasiconcave. Actually point iv) strengthened the notion of strict
quasiconcavity excluding the possibility of an even infinitesimal flat region in the curvature
of u x at x .27
Smooth preferences add therefore the condition that the utility function must
be strictly quasiconcave in the strictly positive orthant of the commodity space at all x .
1.6 Summing up of properties of preferences and of the representing utility function
Preference order · Utility function u x
1. Preferences are continuous u x is continuous
2. Preferences on L
R are monotone if For all Lyx R, y x u y u x
for all Lyx R, y x y x
2’. Preferences on L
R are strongly For all Lyx R,
,y x y x u y u x
monotone if for all Lyx R, ,y x y x y x
2’’. Preferences on L
R are weakly monotone if For all Lyx R, y x u y u x
for all Lyx R, y x y x ·
3. Preferences on L
R are convex if u x is concave or quasiconcave
for all Lx R the upper contour set
xyyxI L ·R is convex, i. e. if
the convex combination 1x y x
with 0,1
3’. Preferences on L
R are strictly convex if u x is strictly concave or strictly
for all Lyx R, with xy 1x y x quasiconcave
27
See MWG (1995, p. 95).
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 23
with a Î 0,1( )
4. Preferences are smooth if indifference The Hessian matrix of u x is negative
curves defined in L are of class
2C definite in the linear subspace
0R zxuzZ L
1.7 Lexicographic preferences
1.7.1 Definition
The existence of a utility function representing preferences implies that the preference relation
“≽” is rational, i.e. complete and transitive. We may ask if the converse is true. The answer is
no: the property that preferences are rational need not guarantee that preferences can be
represented by a utility function.
Definition 2.1. Let 1 2,x x x and 1 2,y y y be any two commodity bundles in 2R .
We say that x is lexicographically preferred to y , and write x ≽𝐿 y , if and only if
either 1 1x y or 1 1x y and 2 2x y .
Preferences are thus ordered as are the words in a dictionary: first all the words starting with
the letter “a” and, among these words, first those starting with the letters “aa” and then those
with initial letters “ab” and so on. In other words, the quantity of the first commodity
commands the order of preference; in case of a tie, it is the quantity of the second commodity
that determines the order of preference. Extension to commodity bundles containing more
than just two goods is immediate; the two goods assumption is retained for purposes of
graphical representation.
1.7.2 Properties
As can be easily verified, lexicographic preferences are complete, transitive, monotone and
convex. The indifference sets and the upper and lower contour sets are instead very peculiar.
Commodity bundles 1 2,x x x and 1 2,y y y are indifferent – to be written as x L~ y -
if x ≽𝐿 y and y ≽𝐿 x . As a result, y is indifferent to x if and only if 1 1y x and 2 2y x ,
that is if and only if y x , in other words if the preference order is antisymmetric. The
indifference class of x is, therefore, a singleton
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 24
(1.13) 2 2I x y R x y R y x x y ~
The indifference set of x degenerates to a single point, so that every commodity bundle
constitutes a distinct indifference class.
The upper and lower contour sets of 1 2,x x x are illustrated in Fig. 1.7, where the
Cartesian axes measure, as usual, the quantities of commodity one and commodity two in the
bundles under comparison. The upper contour set 2I x y R y x · is the region of the
commodity space where 1 1y x - in the diagram to the right of the vertical line ABC - and the
solid line AB where 1 1y x and 2 2y x . Note, in particular, that the segment AB is part of the
upper contour set. The upper contour set is therefore neither closed, nor open. The same is
obviously true of the lower contour set – in the diagram to the left of the vertical line ABC
including the dashed part AC.
Fig. 1.7 - Upper and lower contour sets of x
As a result, lexicographic preferences are not continuous. Continuity of preferences requires
that the preference relation between commodity bundles 1
n
nx
and
1
n
ny
be preserved in
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 25
the limit. Let in this case 1
11 ,1n
ny
n
and 1
1,2n
nx
, with limits respectively
1,1y and 1, 2x .28
Then ny L·nx ; but, as n goes to infinity, x L· y . A discontinuity,
i.e. a preference reversal, occurs as we move from the upper contour set of x to its lower
contour set.
1.7.3 Non existence of a utility representation
The discontinuity of lexicographic preferences immediately signals the existence of a
insurmountable problem in the search for a utility function representing these preferences.
The reason for this negative result is worth looking into a little more deeply: some basic
mathematics is involved.
We have underlined the fact that each commodity bundle is an indifference set to itself,
meaning that there are as many indifference sets as there are commodity bundles. In a way,
we have two dimensions of change in preferences: a first dimension as we move horizontally
in Fig. 1.7, connected with increasing or decreasing quantities of commodity one with a given
quantity of commodity two, and a second dimension as we move vertically in Fig. 1.7,
connected with increasing or decreasing quantities of commodity two with a given quantity of
commodity one. An order preserving utility number from the one-dimensional real line should
be assigned to each of these two dimensions. In effect this is not possible as the following
highly technical argument by contradiction shows. But notice, first, that no such problem
arises with the usual non-degenerate indifference curves, which densely cover the commodity
space and can be put in a one-to-one relation with the one-dimensional real line.
Suppose by contradiction that there existed a utility function representing lexicographic
preferences. For each quantity 1x of commodity one, we have for instance 1 1, 2 ,1Lx x
and, from the assumption of the existence of a utility function, 1 1, 2 ,1u x u x . We can
therefore assign to 1x a non-degenerate interval of values satisfying the above inequality
(1.14) 1 1 1,1 , , 2R x u x u x
Take now the quantity 1 1x x and suppose 1 1, 2 ,1Lx x and, from the assumption of the
existence of a utility function, 1 1, 2 ,1u x u x . Following the previous procedure we can
assign to 1x a non-degenerate interval of values satisfying this new inequality
(1.15) 1 1 1,1 , , 2R x u x u x
28
For simplification, we have assumed that nx is a constant sequence.
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 26
Notice that all commodity bundles in the interval 1R x are strictly preferred to those in the
disjoint interval 1( )R x and should therefore be assigned a greater utility level. Then in each of
these interval we can pick a distinct rational number in increasing order to represent
preferences. But here is the problem. Since 1x R and the real numbers are uncountable, the
number of such intervals is equally uncountable and so should accordingly be the cardinality
of rational numbers. We know, however, that the set of rational numbers is countably, not
uncountably infinite. We have reached, therefore, due to the initial assumption of the
existence of a utility function representing lexicographic preferences, a contradiction.29
29
This is Debreu’s (1954,1959, pp. 72-73, fn. 2) line of proof, followed by MWG (1995, p. 46). Ellickson (1993,
pp.198-199) expressed the difficulty with the statement that there are at most only countably many non-degenerate
disjoint intervals in the real line, while we require uncountably many, one for each 1x R .
D. Tosato – Appunti di Microeconomia – Lecture Notes of Microeconomics – a.y. 2016-2017 27
References
Antonelli, G.B. (1886), Sulla Teoria Matematica della Economia Politica, Pisa, nella
Tipografia del Falchetto; riprodotto in Giornale degli Economisti e Annali di Economia,
Nuova Serie, 1951, vol. 10, pp. 233-263. Translation by J.S. Chipman and A.P. Kirman as On
the Mathematical Theory of Political Economy, with Biographical Notes and Bibliography by
S. Antonelli, in Chipman, J.S., Hurwicz, L., Richter, M.K. and Sonnenschein, H.F. (eds.),
Preferences, Utility, and Demand, New York, Harcourt Brace Jovanovich, Inc., ch. 16, pp.
333-364.
Debreu, G (1954), “Representation of a preference ordering by a numerical function”, in
Thrall, Davis and Coombs (eds.), New York, J. Wiley, pp. 159-165
------------- (1959), Theory of Value. An Axiomatic Analysis of General Equilibrium, New
York, J. Wiley and Sons
--------------- (1972), “Smooth Preferences”, Econometrica, vol. 40, pp. 603-615
Ellickson, B. (1993), Competitive Equilibrium. Theory and Applications, Cambridge,
Cambridge University Press
Hildenbrand, W. and A.P. Kirman (1988), Equilibrium Analysis, Amsterdam, North-Holland
Hurwicz, L. (1971), “On the Problem of Integrability of Demand Functions”, in Chipman,
J.S., Hurwicz, L., Richter, M.K. and Sonnenschein, H.F. (eds.), Preferences, Utility, and
Demand, New York, Harcourt Brace Jovanovich, Inc., ch. 9, pp. 174-214
Jehle, G.A. and P.J. Reny (JR) (2001), Advanced Microeconomic Theory,Boston, Addison
Wesley, 2nd ed.
Mas-Collell, A., Whinston, M.D. and J.R. Green (MWG) (1995), Microeconomic Theory,
New York, Oxford University Press
Varian, H.R. (1992), Microeconomic Analysis, New York, W.W. Norton & Company, 3rd
ed.