Microeconomics 1 Lecture notes LN 1. Rev 2.0 - … · Sapienza University of Rome. Ph.D. Program in Economics a.y. 2014-2015 Microeconomics 1 – Lecture notes (*) LN 1. Rev 2.0 -

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.


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.


(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.


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


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.


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


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.


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


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.


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.


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).


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


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).


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


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.


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


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.


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 .


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.


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


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.


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).


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


(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


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.


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 .


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.

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