Microeconomics 1 Lecture notes 2 - uniroma1.it. 2... · D. Tosato – Appunti di Microeconomica –...

D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17 1

Sapienza University of Rome. Ph.D. Program in Economics a.y. 2012-2013

Microeconomics 1 – Lecture notes 2

LN 2 - Rev. 2.0 - Concavity and quasi concavity of the utility function u

2.1 Concave and quasiconcave utility functions: definition and properties

2.1.A. Concavity

2.1.B. Quasiconcavity

2.1.C The upper contour sets of quasiconcave (quasiconvex) functions: an extension

2.2 Characterization of concavity and quasiconcavity for a twice differentiable utility

function

2.3 Determinant rules for (strict) concavity and (strict) quasiconcavity

2.3.A.1 u x concave and strictly concave

2.3.A.2 Example with the Cobb-Douglas utility function 1 2u x x x

2.3.B.1 u x quasiconcave and strictly quasiconcave

2.3.B.2 Example with the Cobb-Douglas utility function 1 2u x x x

We have explored in Lecture Note 1 the connection between the weak preference relation ·

and its numerical representation u . We have first shown that rational and continuous

preferences can be represented by a continuous numerical function; we have then derived

further properties of the utility function, when more structure is assumed for the preference

order. More specifically, we have seen that monotone preferences are associated with a non

decreasing utility index and have asserted that convex preferences can be represented by

concave and quasiconcave utility functions. Assuming differentiability of the utility function,

we were able to characterize monotone preferences by the property of positive first order

partial derivatives of u - in economic terms, positive marginal utilities of all commodities.

We were further able to characterize convex preferences by the properties, in economic terms,

of diminishing marginal utilities of all commodities and diminishing marginal rates of

substitution between any pair of commodities. As we have seen, smooth preferences are

identified by even more stringent properties of the utility function.


Quasiconcavity is a generalization of the notion of concavity. A quasiconcave utility function

shares with a concave function the fundamental property of representing convex preferences.

Quasiconcavity of the utility function has, therefore, become the standard and less restrictive

assumption in the study of demand theory. The aim of this Lecture Note is to provide

definitions of concavity and quasiconcavity with reference first to functions of a single

variable and subsequently of several variables. The connection with the axiom of convexity of

preferences is graphically illustrated for the two variables case. The further assumption of

differentiability of the utility functions leads to important analytical characterizations of

concave and quasiconcave functions.

The present Lecture Note focuses on the task of giving precise definitions of these notions for

a 2C utility function u x . Convexity and quasiconvexity have a similar crucial role in the

study of minimization problems. As apparent from the title, the presentation is here centered

on the definition of the properties of concavity and quasiconcavity. Convexity and

quasiconvexity are defined residually, with a reversal of sign in the appropriate definitions

and with the indication of a different sequence of signs in the study of the properties of

Hessian and Bordered Hessian matrices. To mark the difference and diminish the risk of

confusion, we indicate as h x the 2C function considered and, somewhat paradoxically,

write in italic the terms convexity and quasiconvexity in the presentation of their properties.

The plan of the Lecture is the following. The definition of concave and quasiconcave

functions and their relation with convex preferences are presented in Section 2.1. The

characterization of concavity and quasiconcavity for twice differentiable utility functions is

examined in Section 2.2. Section 2.3 presents the determinant rules for concavity and

quasiconcavity of the utility function, namely negative definiteness and semidefiniteness of

the Hessian matrix of second order partial derivatives of the utility function. A summing up

table concludes this section. Definitions and properties of convex and quasiconvex functions

are indicated all along. A description of the connection between the conditions for concavity

(convexity) and quasiconcavity (quasiconvexity), on one hand, and the second order

conditions for the solution of maximization (minimization) problems, on the other, concludes

the study of concavity and quasiconcavity in Section 2.4. In each of these sections part A

deals with concavity, while part B examines quasiconcavity.

The role of the properties of concavity (convexity) and quasiconcavity (quasiconvexity) of

the relevant objective functions in determining the nature of their unconstrained or

constrained critical points is considered in Lecture Note 3, Section 3.6.

This Lecture Note has no pretense of completeness and analytical sophistication, for which

we refer to the References at the end of the Lecture; the aim is more operational: to give tools

for the solutions of typical problems in economic analysis. With this goal in mind, analytical

derivations are worked out in detail considering only the two variable case.


2.1 Concave and quasiconcave utility functions: definition and properties

2.1.A. Concavity

Definition 2.1.1 The real-valued function :u x D , with D a convex subset of

L ,

2 is concave if, for all ,x y D ,

3 the utility of a convex combination

1z x y of x and y is no less than the weighted average of their separate

utilities, namely

(2.1) 1 1u z u x y u x u y for all 0,1

The real-valued function :u x D , with D a convex subset of L , is strictly

concave if, for all ,x y D ,

(2.2) 1 1u x u x y u x u y for all 0,1 .

The notion of convexity is correspondingly defined. If the function u x is concave (strictly

concave), then h x u x is convex (strictly convex).

Panels (a) and (b) of Fig. 2.1 illustrate the case of a concave and of a strictly concave utility

function for all x D in the one-variable case, i.e. for x a scalar. A function is concave

(strictly concave) if the line segment connecting u x and u x is everywhere on or below

the function (always below for strict concavity) so that

1 1u x u x x u x u x .4

An alternative description of concavity follows from the observation that, as the diagrams in

Fig. 2.1 show, the set of points , , ,S x y x D y y u x “on or below” the graph of

u x is convex.

Proposition 2.1. u x is concave if and only if the dashed areas in Fig. 2.1 are convex.5

1 The definitions of concavity and strict concavity, here formulated in terms of a utility function, obviously apply to

any function.

2 Nothing prevents D from being .

3 Note that, for comparison with x e , y is also a vector containing equal quantities of a composite commodity, for

instance y e with . 4 Different quantities of the single commodity x are indicated as x and x .

5 For a proof see JR (pp. 443-445).


Fig. 2.1 Panels (a) and (b) – Concave and strictly concave utility function

The graphical representation of a concave function in the two-commodity case, namely for the

utility function 1 2,u u x x , would obviously require the use of a three-dimensional

diagram. In this three-dimensional setting, it would appear as the rising part of an infinitely

extending dome over the commodity space 2 with a profile along any ray from the origin

analogous to that depicted in Fig. 2.1(b).6

We may, however, continue to use a two-dimensional diagram, in which the coordinate axes

measure the quantities of the two commodities and utility can be represented by a family of

nonintersecting convex indifference curves, with u x increasing along any vector pointing in

the north-east direction in the diagrams because of the assumed monotonicity of preferences.

As defined in Lecture Notes 1, preferences are convex, in particular strictly convex, if and

only if, assuming for convenience x y , their convex combination 1z x y is

preferred to both x and y .7 It follows that the convex combination 1z x y lies on

a higher indifference curve with the implication, in terms of the utility representation of

preferences, that 1 1u z u x y u x u x u y which coincides with

the definition of concavity of the utility function u x . This shows, as anticipated in Lecture

Notes 1, that convex preferences are represented by a concave utility function. We have thus

roved the following proposition.

6 We can in fact consider the one-commodity case as representing the case of a composite commodity of unchanging

composition. 7 ,x y and z represent now commodity bundles.


Proposition 2.2 Convex preferences admit of a numerical representation if only if the

utility function is (strictly) concave.


Definition 2.2 The real valued function :u x D , defined in the convex set

LD with values in , is quasiconcave if, for all , , 1x y z x y D , we

have

(2.3) 1 min ,u z u x y u x u y for all 0,1

u x is strictly quasiconcave when the weak inequality (2.3) is turned into a strict

inequality for all 0,1 .

The definition of quasiconvexity deserves a little attention.

Definition 2.3 The real valued function :h x D , defined in the convex set

D with values in , is quasiconvex if, for all ,x y D , we have

(2.4) 1 max ,h x y h x h y for all 0,1

h x is strictly quasiconvex when the weak inequality (2.4) is verified with the “less

than” sign for all 0,1 .

The following Proposition establishes the relation between the notions of quasiconcavity and

quasiconvexity.

Proposition 2.3. If the real valued function :u x D , defined in the convex set

LD with values in , is quasiconcave, then the function h x u x is

quasiconvex.

Proof. Multiplying both sides of Definition (2.4) by minus one and reversing the inequality

sign, we have

(2.5) 1 min ,u x y u x u y

Noting that the left hand side of (2.5) is, by definition 1h x y and that in the right

hand side min 1 max 1u x y h x y , we obtain definition (2.4) of

quasiconvexity of a function.


Consider first the one-commodity case, 1L . Fig. 2.2 depicts two distinct functions, that

both meet the definition of quasiconcavity, as can be immediately checked. Panel (a) shows a

monotone function that in the interval ,x x is concave, whereas Panel (b) a function which

is initially convex and subsequently concave. This shows that a concave function is also

quasiconcave, but not vice versa.8

D

Fig. 2.2 Panels (a) and (b) – Examples of strictly quasiconcave utility functions

Note that, since a quasiconcave function may have concave as well as convex sections, there

exist no definition of quasiconcavity in terms of the property of the set of points lying “on or

below” the utility function u x as for a concave function.

Note that the utility functions depicted in the above mentioned Fig. 2.2, Panels (a) and ( b),

are both quasiconcave and quasiconvex, as can be immediately checked on the basis of the

definitions 2.2 and 2.3.

In the two-commodity case, the graphical representation of a quasiconcave utility function

would again require the use of a three-dimensional diagram. In this setting, it would appear as

the rising part of an infinitely extending bell over the commodity space 2 with a profile

along any ray from the origin analogous to that depicted in Fig. 2.2(b).9

We may, however, continue to use a two-dimensional diagram, in which the coordinate axes

measure the quantities of the two commodities and utility can be represented by a family of

8 Note that :u x D is quasiconcave if and only if it is either monotonic or first non decreasing and then non

increasing, as is the function depicted in Fig. 2.3, Panel (b). It is immediate to check that if the function is first

decreasing and then increasing, the function is not quasi concave. 9 We can in fact consider the one-commodity case as representing the case of a composite commodity of unchanging

composition.


nonintersecting convex indifference curves, with u x increasing along any vector pointing in

the north-east direction in the diagrams. But note once more that convex, in particular strictly

convex, preferences imply that the convex combination 1z y x of commodity bundles

x and y , for convenience assumed to be positioned on the same indifference curve I x , is

preferred to both and is therefore on a higher indifference curve with

min ,u z u x u y . This coincides with the definition of quasiconcavity and thus

shows, as anticipated in Lecture Notes 1, that convex preferences are also represented by a

quasiconcave utility function. Proposition 2.4 provides a formal proof.

Proposition 2.4 The utility function :u x D , with D a convex subset of +

L , is

quasiconcave if and only if the upper contour set I x is convex.

Proof. To prove the “if” part choose 0x

+

L , let 0I x be the upper contour set of 0x and

take 0,x y I x so that 0x x· and 0y x· . By representation we then have 0u x u x ,

0u y u x and by quasiconcavity

(2.6) 01 min ,u x y u x u y u x

We then have 01x y x · , which implies 01x y I x . Hence 0I x is

convex.

To prove the “only if” part, assume that 0I x is convex for all 0x

+

L . Take ,x y+

L

with u x u y and suppose 0x x . Hence, by construction 0,x y I x , by convexity

01x y I x and finally, by representation, we obtain quasiconcavity

01u x y u x .

The important conclusion is, therefore, that convex preferences do not require that the utility

function be concave, but can be represented by a quasiconcave function, which represents,

therefore, a generalization of the notion of a concave function. The important property of the

utility function corresponding to the axiom of convexity of preferences is then quasiconcavity

and not concavity. This connects with the difference between ordinal and cardinal properties

of utility functions. While concavity is a cardinal property, invariant only to affine positive

transformations, quasiconcavity is an ordinal property, invariant to positive monotonic

transformations of the utility function.


2.2.C The upper contour sets of quasiconcave (quasiconvex) functions: an extension

Proposition 2.4 establishes the equivalence between the Definition 2.2 of quasiconcavity -

1 min ,u x y u x u y - and the convexity of the upper contour set – which, in

view of relation (1.10) of Lecture Note 1, we write as LI x y u y u x .

Because of the assumption of monotone preferences, the function u x is increasing as we

move north-east in the two- commodity diagrams of Fig. 2.2, Panels (a) and (b). The upper

contour set is therefore the subset of the commodity space bounded from below by a level set.

Consider now a generic function f x , which may be either monotonically increasing or

decreasing in the vector variable x . If f x is increasing in x , the situation is that of the

utility function: f x is quasiconcave if the upper contour set I x is convex. However, if

f x is decreasing in x , the larger the values of x , the smaller are the values of the

function. The upper contour set of a quasiconcave decreasing function is then represented by

the subset of the commodity space lying south-west of the level set. Fig. 2.3, Panels (a) and

(b) depict the upper contour sets respectively of an increasing and of a decreasing

quasiconcave function. We can conclude with the following proposition.

Proposition 2.5 The function f x is quasiconcave if only if the upper contour sets are

convex.

Panel (a) - f x increasing Panel (b) - f x decreasing

Fig. 2.3 Upper contour set of a quasiconcave function;


Consider now the quasiconvex function g(x). According to Definition 2.3 above, g(x) satisfies

the condition 1 max ,g x y g x g y for all 0,1 . Take x and y in the

same level set LI x y g y g x so that the upper and the lower contour sets are

respectively LI x y g y g x and LI x y g y g x

; then by

definition of quasiconvexity we have that g is convex if 1g x y I x In the

two-commodity diagram of Fig. 2.4, Panel (a) shows the case of an increasing g(x) and Panel

(b) the case of a decreasing g(x). In both instances the definition of quasiconvexity is

satisfied if the lower contour sets are convex.

Proposition 2.6 The function g(x) is quasiconvex if and only if the lower contour sets

are convex.

Panel (a) - g(x) increasing Panel (b) - g(x) decreasing

Fig. 2.4 – Lower contour sets of a quasiconvex function

2.2 Characterization of concavity and quasiconcavity for a twice differentiable utility

function

2.2.A. Concavity

Let us assume now the real valued function u x is twice continuously differentiable in the

open convex set int D . Definition 2.4 uses the first order derivative to determine the

best linear approximation to u x in a neighborhood of every xint D, while definition 2.5

uses the second order derivative to identify the curvature of the function in the neighborhood

of every xint D.


Definition 2.4. The continuously differentiable function u x is concave if and only if

(2.7) ( )u y u x u x y x

for every x and y in the open interval int D or equivalently

(2.7’) u x z u x u x z

and every z y x .10

This means, as shown in Panels (a) and (b) of Fig. 2.2, that the tangent line at every x -

defined as the set of y such that f y x u x u x y x - lies above the function

or at most on the function itself, if the latter is linear in the neighborhood of x . The function

u x is strictly concave at x , as in Fig. 2.5 Panel (b), when the inequality (2.7’) is verified

with the “greater than” sign.

Fig 2.5 – Panel (a) u x is concave; Panel (b) u x is strictly concave

The function h x is (strictly) convex if the sign in (2.7’) is turned from “ ” to

10

The motivation for the definition of the property of concavity of a function of a single variable in the form (2.3’) is

one of consistency with the notation commonly adopted for the definition of concavity in the case of a function of

several variables (see infra Definition 2.8 and inequality (2.9).


“ ”. Graphically, a function is convex at every intx D if only if the tangent plane lies

below or at most on the function itself.

Definition 2.5 The twice continuously differentiable function : intu x D is

concave for every intx D if only if the second derivative is non positive

(2.8)

2

20

d u xu x

dx

If this derivative is negative, the function u x is strictly concave.11

Condition (2.8) on the second derivative means that the first derivative must be non increasing

at every x . In economic terms, and in the case of strict concavity, this implies that the

marginal utility of the composite commodity x is decreasing. Fig. 2.1 Panel (b) illustrates this

case.

We may now define a convex function by reversing the properties of a concave function.

Definition 2.6. A function h x is convex or strictly convex if (i) the line segment

joining any two points on the function lies on or above the function; (ii) the tangent

plane lies on or below the function; (iii) the second order partial derivative of the

function is non decreasing or strictly increasing.

This means, going back to the previous definition of a concave function, that the weak

inequality sign in the relations (2.1), (2.7’) and (2.8) is reversed and, with reference to

Proposition 2.1, that is now convex (strictly convex) the set above, and not below, the

function itself.12

Turning from the single to the several variables case, assume now that the real valued

function u x is twice continuously differentiable in the open convex set int LD .

Definition 2.4 of concavity of a function of a single variable is based on the relationship

between the tangent line and the function itself. In the multivariable case the same idea

applies to the relationship between the tangent (hyper)plane and the function.

Using the Nabla operator 1 ... ... T

l Lu x u x u x u x to represent the gradient of

u x , we have the following Definition.

11

Continuity will take care of the boundary points of the domain D. 12

This is a good point to stress the difference between convex sets and convex functions and to remark the dual relation

between, on the one hand, a concave function and the closed “below-the-function” convex set and, on the other,

between a convex function and the closed “above-the function” convex set.


Definition 2.7. The function u x is concave if and only if13,14

(2.9) u x z u x u x z

for all Lx and all Lz , with

Lx z . The function u x is strictly concave

if the inequality (2.9) holds strictly for all Lx and all 0z .

Concavity of the multivariable function u x requires, in perfect analogy with Definition 2.4,

that the tangent hyperplane to the utility function for all x be on or above the function itself.

Since in the two variable case the concavity of the utility function implies the convexity of the

indifference curve, this condition means that the tangent line to every point on the

indifference curve I x for all Lx , must be on or below the indifference curve as

illustrated in Fig. 2.6, in which only the case of a strictly convex indifference curve is

depicted. This requires that the slope of the tangent line be equal to the slope of the

indifference curve at 1 2,x x x . In analytical terms, for all 2y let z y x be the

vector representing deviations from the chosen point x. The slope of the line through the

points y and x is 2 2 2

1 1 1

y x z

y x z

, while the slope of the indifference curve is equal to the

marginal rate of substitution:

1

1,2

2

u xMRS x

u x . Equating these two slopes we obtain

1 1 1 2 2 2 0u x y x u x y x u x y x u x z . The last equality, which is

the standard general definition of the tangent hyperplane at point Lx , will be later used

in the determination of the property of quasiconcavity.

For completeness, the function h x is convex for all Lx if the inequality sign in (2.9) is

reversed.

13

Following MWG, the notation y x indicates the scalar product of the row vector y and the column vector x, i.e.

1

L

l ll

y x y x

14 Condition (2.9) is derived utilizing again the concavity of the auxiliary function g t u x tz with

Lz and

Lx tz . See JR, Theorem A2.4, p. 467.


Fig. 2.6– Tangent line to the indifference curve at x

We can ask again what condition corresponds, in the case of a multivariable function, to the

condition of nonpositive second order derivative in the case of a single variable function. The

answer is that the Hessian matrix H x of the second order partial derivatives of the function

u x be negative semidefinite. This condition is the generalization to the multivariable case

of the condition on the second order derivative of a single variable function.

Definition 2.8 The function u x is concave at every Lx and for all Lz if and

only if the quadratic form z H x z is negative semidefinite,15

i.e. if

(2.10) 0z H x z

The L L matrix of second order partial derivatives

(2.11)

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

L

L

L L LL

u u u

u u uH x

u u u

which satisfies condition (2.10) for a quadratic for to be negative semidefinite, is called

negative semidefinite.

15

To determine this condition we can use again the auxiliary function g t u x tz . The second order derivative of

g t at 0t coincides with the quadratic form z H x z ; (2.10) then follows from 0 0g t . See JR, pp.

467-469. MWG (p. 933) offer a different approach considering a second order Taylor expansion around 0t .


If (2.10) holds with the strict inequality sign for all 0z , the function is strictly

concave. In this case the quadratic form is negative definite and the matrix H x is

called negative definite.

In words, u x is strictly concave if, departing from every x in any direction z in the upper

contour set of x , the function increases at a decreasing rate.

The Hessian matrix is symmetric - lm mlu u - because u x is, by assumption, twice

continuously differentiable. As made clear in the following paragraph 2.3 by the determinant

conditions for negative definiteness and semidefiniteness, all the terms on the principal

diagonal of H x must be non positive. This is easily verified setting 0lz and 0k lz ,

since in this case 2

ll lz H x z u z which is non positive only if 0llu . Again, the

function h x is convex at every Lx if and only if the inequality sign in (2.10) is

reversed.


Note that, since a quasiconcave function may have concave as well as convex sections, there

exists no definition of quasiconcavity in terms of the property of the set of points positioned

“on or below” the utility function u x analogous to the previous definition 2.1. Assuming

differentiability, a definition of a quasiconcave function, similar but not identical to the

previous definition 2.4 and based on the position of the tangent plane, is given in the

following Definition 2.9 Note, going back to the definition 2.2, that a function is

quasiconcave if taken any two point ,x y in the convex domain D , any convex combination

of them has utility greater than or equal to the minimum of the utilities of those points. The

definition, therefore, cannot be based on the properties of a single point in the domain, but

must take into consideration the relative position of x and y . We consider first the case of a

function of a single variable.

Definition 2.9 The real valued function :u x D , defined in the convex set

D with values in , is quasiconcave if and only if, for all ,x y D ,

(2.12) 0 whenever u x y x u y u x

If 0 whenever u x y x u y u x , then u x is strictly quasiconcave.


Condition (2.12) is verified, as indicated in the diagrams of both Panels (a) and (b) of Fig. 2.2.

The inequality (2.12) needs to be reversed whenever u y u x .

The definition of quasiconcavity involving the second order derivative must wait for the case

of a function of several variables.

While the Definition 2.9 of quasiconcavity continues to apply also in the case of a function of

several variables, with the obvious substitution of the derivative u x with the Nabla u x

, the analogue of the Definition 2.8 based on the properties of a quadratic form is now more

complex.

While the definitions of concavity and strict concavity require to check the properties of the

Hessian matrix in the whole space of definition of the variables, i.e. on L , the definitions

of quasiconcavity and strict quasiconcavity of a twice differentiable function require to check

the properties of the Hessian matrix in a linear subspace of L . The dimensions of this

subspace are determined by the number of the binding constraints that the choice variables

must satisfy. We consider here the case of a unique binding constraint, as is the case in the

standard utility maximization problem subject to the wealth constraint. The properties of the

Hessian matrix must then be ascertained in the linear subspace 0LZ z u x z .

The relevant matrix is now the Bordered Hessian BH x , obtained by bordering the L L

Hessian with a row and a column of the first order derivatives of u x . With just one binding

constraint the bordered Hessian is thus a 1 1L L matrix. There are two equivalent

ways to write, in the compact block notation, the bordered Hessian matrix

(2.13)

0

B

T

H x u xH x

u x

(2.14)

0T

B u xH x

u x H x

where the Hessian H x is the L L matrix of the second order derivatives of the function

u x , the 1L column vector u x is the vector of first derivatives and the 1 L row

vector T

u x is its transpose, and finally 0 is a 1 1 zero matrix.16

We will consider

formulation (2.13).

16

The format (2.13) is adopted by MWG (Appendix D, pp. 938-939); JR as well as Sundaran use format (2.14), while

Simon and Blume present both. As mentioned in the following section 2.3.B.1, the choice of presentation of the

Bordered Hessian has implications for the formulation of the determinant conditions for semidefiniteness of a function.


Definition 2.10. u x is quasiconcave if and only if the quadratic form z H x z is

negative semidefinite in the subspace 0LZ z u x z ; if the quadratic

form z H x z is negative definite in the subspace LZ , then u x is strictly

quasiconcave.

The function h x is quasiconvex if the quadratic form z H x z is positive semidefinite in

the subspace LZ ; if the quadratic form z H x z is positive definite in the subspace

LZ , then h x is strictly quasiconvex.

2.3 Determinant rules for (strict) concavity and (strict) quasiconcavity of the utility

function

As stated in Definitions (2.9) and (2.10), the nature of the function utility u x - (strictly)

concave (convex) or (strictly) quasiconcave (quasiconvex) – depends on the properties of a quadratic

form involving the matrix of second order derivatives of the function. Necessary and sufficient

conditions for the quadratic form z H x z to be negative (positive) definite (semidefinite)

were established by Debreu (1952). These conditions are expressed in terms of the properties

of the Hessian matrix H x and of the Bordered Hessian BH x , which are directly referred

to as being respectively negative (positive) definite and negative (positive) semidefinite. We

will subsequently concentrate our attention on the conditions for negative definiteness and

semidefiniteness. The conditions for positive definiteness and semidefiniteness are stated at

the end of each subsection.

2.3.A.1 u x concave and strictly concave

As stated in Definition 2.8, u x is concave if and only if the quadratic form z H x z is

negative semidefinite and strictly concave only if the quadratic form z H x z is negative

definite. The connection between concavity of u x and the property that the Hessian matrix

H x is negative semidefinite is stated in the following proposition, a proof of which - for

the two commodity case - is presented after the complete indication of the determinant rules

for a matrix to be negative semidefinite.


Proposition 2.5. The twice continuously differentiable function u x is concave if and

only if the Hessian matrix H x is negative semidefinite for every Lx . If H x is

negative definite for every Lx , then the function u x is strictly concave.

17

For convexity of the function h x an analogous proposition holds replacing the word

“negative” with “positive”.

The rules for ascertaining that a matrix is negative semidefinite and negative definite are

based on the sequence of signs of the determinants of particular submatrices of H x . It is

convenient to start with the rules concerning a negative definite matrix.

Definition 2.11. The Hessian matrix H x is negative definite if the leading principle

minors of the matrix are of alternating sign starting with a minus sign, i.e. if:

(2.15) 1 0r

rrH x with 1,...,r L

where r rH x is the leading principle minor of order r . In words, the definition of

negative definiteness requires that the minors r rH x be negative, when the index r is

odd, and positive when r is even.

The leading principle minors r rH x are the determinants of the matrices resulting when

only the first r rows and columns of the Hessian H x are retained, alternatively when the

last L r rows and columns are deleted. They are called principal minors because they are

the determinants of the submatrices obtained moving down the principal diagonal of the

matrix H x . Suppose that H x is a 3x3 matrix. There are three leading principle minors:

- the determinant 1 1H x of the 1x1 submatrix 11u ;

- the determinant 2 2H x of the 2x2 submatrix 11 12

21 22

u u

u u

;

- the determinant 3 3H x of the 3x3 Hessian matrix

11 12 13

21 22 23

31 32 33

u u u

u u u

u u u

.

17

Note that second part of Proposition 2.5 is in the form of a sufficient, not a necessary condition. In fact, a function

may be strictly concave while the Hessian may fail to be negative definite.


The sign rule requires that the signs of these principle minors alternate starting from the first,

which must be negative; hence 11det 0u ,

11 12

11 22 12 21

21 22

det 0u u

u u u uu u

and

det 0H x . Note that 11det 0u implies

110u ; the marginal utility of commodity 1

must, therefore, be decreasing.

The rules for ascertaining that a matrix is negative semidefinite are stated in the following

definition.

Definition 2.12. The Hessian matrix H x is negative semidefinite if the leading

principle minors of the matrices obtained by all possible permutations of rows and

columns of H x alternate in sign starting with a minus sign

(2.16) 1 0r

r rH x

where H x indicates the 1,..., L permutation of the Hessian and r rH x the

leading principal minor of order r of the permuted matrix H x .

Suppose again, for example, that H x is a 3x3 matrix. There are six possible permutations

of rows and columns: starting from the natural order 1, 2,3 of rows and columns, the

other five permutations are 1,3,2 , 2,1,3 , 2,3,1 , 3,1,2 , 3,2,1 . The permutation,

for instance, 2,3,1 is therefore represented by the following matrix18

(2.17)

22 23 21

2,3,1

32 33 31

12 13 11

u u u

H x u u u

u u u

According to the definition, the alternating sign rule starting with a minus sign applies to the

leading principle minors of all the possible permutations. This means that one would have to

control the sign of the determinant of 10 matrices: 3 of order one, since there are three

1 1H x minors of the three possible 1x1 submatrices 11u , 22

u e 33u ; 6 of order two and

only one of order three, since being all permutations at the same time of one row and one

column, the determinant remains unchanged. Note that, in order to satisfy the sign rule for

negative semidefiniteness, we must have 11 22 33

, , 0u u u ; this means, in economic terms, that

the marginal utilities of all commodities should not be increasing.

18

The permutation (2,3,1) is obtained by first permuting column 2 in column 1 and column 3 in column2 and column 1

in column 3 and, subsequently, permuting row 2 in row 1, row 3 in row 2 and row 1 in row 3. Alternatively, one can

permute the rows first and then the columns. The result is the same, as can be easily checked.


The condition that the Hessian matrix of second order derivative of the utility function be

negative semidefinite is, as already mentioned, a generalization to matrices of the notion of a

nonpositive number. Thus the condition that the matrix of second order derivatives of a

function of several variable is negative semidefinite takes, in the definition of a concave

function and in the formulation of the second order necessary condition for a maximum, the

role of the condition that the second order derivative of a single variable function is

nonpositive.

We are now ready to turn, as previously anticipated, to a proof of Proposition 2.5. The

technique used in the proof will be used in Lecture Note 3 to establish the second order

condition for a maximum in a problem of utility maximization under the wealth constraint.

For this reason the analytical operations of taking first and second order derivatives of a

vector valued function are carefully specified.

Proof of proposition 2.5 Let 1 2,u x u x x be a twice continuously differentiable utility

function. Fix 2intx ; let t R and define the following function of the single variable t

(2.18) 1 1 2 2,g t u x tz x tz

which describes movements away from the point x in any direction z. The first order

derivative of g t is

(2.19) g t u x tz x tz z u x tz x tz z , ,1 1 1 2 2 1 2 1 1 2 2 2

and, by differentiating (2.19), we can obtain the second order derivative of g t

(2.20) g t u z u z z u z z u z z H x z , , , ,2 211 1 12 2 1 21 1 2 22 2

where lmu , , with , 1,2l m , are the second order partial derivatives of the utility function

u x .

Note that, if u x is concave, so is g t . This implies that, if 0g t , we have, in view of

(2.20), 0z H x z ; this in turn means that the quadratic form z H x z is negative

semidefinite as well as the Hessian matrix H x . On the other hand, if H x is negative

semidefinite, then g t is concave and so is the utility function u x .

A similar test is available for positive definite and positive semidefinite matrices.

Definition 2.13. The Hessian matrix H x is positive definite if the leading principle

minors of the matrix are all positive, i.e. if:


(2.21) 0r rH x with 1,...,r L

Definition 2.14. The Hessian matrix H x is positive semidefinite if the leading

principle minors of the matrices obtained by all possible permutations of rows and

columns are of H x all nonnegative

(2.22) 0r rH x x

2.3.A.2 Example with the Cobb-Douglas utility function 1 2u x x x , 0

The first order derivatives of the function are

(2.23) 1 11 2 1 2

1 2

T

u x x x x x u x u xx x

.

Deriving the last expression of the first order derivatives, the Hessian matrix is

(2.24)

21 21

21 2 2

1

1

u x u xx xx

H x

u x u xx x x

and its permutation of rows and columns is

(2.25)

21 22,1 2

21 2 1

1

1

u x u xx xx

H x

u x u xx x x

Note that the Hessian matrix with rows and columns arranged in the natural order is indicated

as H x while the permuted matrix is indicated as 2,1

H x . According to Definition 2.12,

the function 1 2u x x x is concave if 1 0r

r rH x . This requires that the principal

minors of order 1 – i.e. with the index 1r in the Definition (2.12) - of the natural and of the

permuted matrix be negative. This obtains if and as well as 1 and 1 are

positive; in economic terms, if the marginal utilities are both positive and decreasing. Note

next that the principal minor of order 2 is the determinant of the whole matrix. The function is

concave if


(2.26)

22 2

2 2

1 2 1 2

2

2

1 2

det 1 1

1 0

H x u x u xx x x x

u xx x

This occurs only if 1 . The function is strictly concave if 1

2.3.B.1 u x quasiconcave and strictly quasiconcave

As stated in Definition 2.10, u x is quasiconcave if and only if the quadratic form z H x z is

negative semidefinite in the subspace 0LZ z u x z ; if the quadratic form

z H x z is negative definite in the subspace LZ , then u x is strictly quasiconcave.

The connection between quasiconcavity of u x and the Hessian matrix H x is stated,

without proof, in the following proposition.

Proposition 2.6. The twice continuously differentiable function u x is quasiconcave

if and only if the Hessian matrix H x is negative semidefinite in the subspace

0LZ z u x z for every x in the convex domain D. If H x is negative

definite in the subspace 0LZ z u x z for every x D , then the function

u x is strictly quasiconcave.19

To help getting an intuitive grasp of the meaning of the analytical condition of quasiconcavity

of a function, we should first of all remember that the condition that the Hessian matrix H x

is negative semidefinite is the analogue for a function of several variables of the condition of

nonpositive second order derivative for a function of a single variable. This condition

excludes, therefore, that a displacement from a given point x may lead to an increase in value

of the function u x . This negative – more generally, nonpositive – effect on the value of the

function must however be ascertained in a particular direction, that defined by the linear space

0u x z .

19

Note that second part of Proposition 2.3 is in the form of a sufficient, not a necessary condition. In fact, a function

may be strictly concave while the Hessian may fail to be negative definite.


Fig. 2.7 - H x is negative semidefinite along the tangent line 0u x z

We depict to this end in Fig. 2.7, for the usual case that the function u x has only two

arguments, the level set I x of u x for a given x and the tangent line AB to the level set at

x. Think of the function u x as being represented by a family of non intersecting convex

level curves and increasing in the north-east direction. Only a second level curve is

represented in Fig. 2.7, the level curve I y with u x u y . The function u x is

quasiconcave if, for small movements in the neighborhood of x along the tangent line AB,

u x decreases to the lower level u y , as in the case of strict convexity of the level curves

shown in Fig. 2.7, or does not increase if the point x is located on a flat part of a level curve.

For convexity of the function h w an analogous proposition holds replacing the word

“negative” with “positive”.

The rules for ascertaining that a matrix is negative semidefinite and negative definite in the

subspace LZ are again based on the sequence of signs of the determinants of particular

submatrices of the bordered Hessian matrix BH x . It is convenient to start with the rules

concerning a negative definite matrix.

Definition 2.15. H x is negative definite in the subspace LZ of dimension S if

the determinants of the leading principle minors of BH x of order 1,...,r S L are

of alternating sign. In compact notation, if


(2.27) 1 0r B

rrH x

where Br rH x is the principle leading minor of order r of the Bordered Hessian

(2.13) here repeated for a more convenient reference

(2.13)

0

B

T

H x u xH x

u x

Suppose that H x is a 2x2 matrix. Remembering then that 1S , since we have only one

constraining linear subspace, i.e. the subspace 0u x z , BH x is the 3x3 matrix

(2.28) 11 12 1

21 22 2

1 20

B

u u u

H x u u u

u u

The leading principle minor of order 2r is the Hessian itself. We have therefore only one

leading principle minor, namely

(2.29)

11 12 1

2 2 21 22 2

1 2

det

0

B

u u u

H u u u

u u

to take into consideration and conclude that the quadratic form z H x z is negative definite

in the linear space LZ if

(2.30) 2

221 0BH

In the two commodity case one has, therefore, to check only the sign of the determinant of the

bordered Hessian.

Suppose that H x is a 3x3 matrix. With 1S , BH x is the 4x4 matrix

(2.31)

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

0

B

u u u u

u u u uH x

u u u u

u u u


We would have in this case to check the signs of the two leading principle minors 2 2BH and

3 3BH . The first coincides with the expression in (2.29), the second with the determinant of

the Bordered Hessian (2.31).

The rules for ascertaining that a matrix is negative semidefinite are stated in the following

definition.

Definition 2.16 The Hessian matrix H x is negative semidefinite in the linear space

LZ if the leading principle minors of the matrices obtained by all possible

permutations of rows and columns of BH x alternate in sign starting with a minus

sign

(2.32) 1 0r B

r rH x

where 1,..., L indicates the permutation of the leading principal minor of order r of

the permuted Bordered Hessian BH x and r rH x H x and r u x

the

permutation of the rows of the column vector u x .

Suppose again 2L so that H x is a 2x2 matrix. Then BH x is the 3x3 matrix

(2.33) 11 12 1

21 22 2

1 20

B

u u u

H x u u u

u u

Note that permutations apply to rows and columns of the Hessian matrix H x and not to the

bordering row and column. Since 2,1 is the only possible permutation of rows and

columns one and two,20

we obtain

(2.34)

22 21 22,1

11 12 1

2 10

B

u u u

H x u u u

u u

Since the determinants of BH x and 2,1

BH x are the same due to the symmetry of the

Bordered Hessian, there is only one determinant to compute. Hence u x is quasiconcave,

according to Definition 2.16 and remembering that we have 2r , if the determinant of the matrix

(2.34) is non negative.

20

The natural order 1,2 is not explicitly indicated since its is implied in the definition of H x .


The two commodity case makes the verification of quasiconcavity of the utility function

particularly simple. The three commodity case is definitely more complex due to the fact there

are now six possible permutation.

For a quasiconvex function h x , the matrix rules needed to ascertain that the Hessian is

positive definite and positive semidefinite in the subspace 0LZ z h x z are

the following.

Definition 2.17 H x is positive definite in the subspace LZ of dimension S if

the determinants of the leading principle minors of BH x of order 1,...,r S L are

positive if S is even, negative if S is odd. In compact notation, if

(2.35) 1 0S B

rrH x

Definition 2.18 The Hessian matrix H w is positive semidefinite in the linear space

LZ if the leading principle minors of the matrices obtained by all possible

permutations of rows and columns of BH x are non negative if S is even, non positive

if S is odd. In compact notation, if

(2.36) 1 0S B

r rH x

A final word of caution. Conditions have been formulated for the Hessian matrix of the

second order derivatives to be negative semidefinite in connection with the consideration of a

quasiconcave utility function. The notion of a semidefinite matrix is more general. Quasi

concavity requires studying the properties of the Hessian restricted to a linear subspace.

Actually, several restrictions, possibly non linear, but to be evaluated at a specific point 0x ,

can be taken into consideration.

2.3.B.2. Example with the Cobb-Douglas utility function 1 2u x x x

Taking into account the definitions (2.23) and (2.24) of the first and second order derivatives

of the Cobb-Douglas function 1 2u x x x , the Bordered Hessian is


(2.37)

21 2 11

21 2 22

1 2

1

1

0

B

u x u x u xx x xx

H x u x u x u xx x xx

u x u xx x

and its permutation of rows and columns is

(2.38)

21 2 22

2,1

21 2 11

2 1

1

1

0

B

u x u x u xx x xx

H x u x u x u xx x xx

u x u xx x

According to Definition 2.16, the function 1 2u x x x is quasiconcave if the leading

principle minors of BH x of order 1,...,r S L of all permutations are alternating in sign

starting with a nonnegative sign. Since in the two-commodity case the determinants of

2,1

BH x is the same as the determinant of BH x due to the symmetry of the bordered

Hessian, there is only one determinant to compute. Hence u x is quasiconcave if the

determinant of the matrix (2.38) is non negative. We have

(2.39)

2,1 3 2 2 3 2 3 21 22

1 2

det BH x u x x xx x

which is greater than zero for all positive and . We conclude that, while 1 2u x x x is

concave only if 1 , it is quasiconcave for all positive values of the parameters and

.

The following table offers a summing up of the determinant rules for concavity and

quasiconcavity.


Concavity

Quasiconcavity

u x is concave if and only if

1 0r


where r rH x are the leading principal

minor of order r of the permuted matrix

H x , 1,..., L

u x is quasiconcave if and only if

1 0r B

r rH x

1,2,...,r S L

where r rH x is the principle leading

minor of order r of the Hessian H x

relative to permutation .

Strict Concavity Strict Quasiconcavity

If

1 0r


where r rH x are the leading principal

minor of order r of the Hessian matrix

H x ,

then u x is strictly concave

If

1 0r B

rrH x

1,2,...,r S L

where r rH x is the principle leading

minor of order r of the Hessian matrix

H x ,

then u x is strictly quasiconcave


References

Debreu, G. (1952), “Definite and semi-definite quadratic forms”, Econometrica, vol. 20,

pp. 295-300.

Jehle, J.A. and P.J. Reny, Advanced Microeconomic Theory, Boston, Addison Wesley,

2001, 2nd

ed.

Mas Colell, A. Whinston, M.D. and J.R. Green, Microeconomic Theory, Oxford

University Press, 1995, Mathematical Appendix M.C and M.D.

Simon, C.P. and L.E. Blume, Mathematics for Economists, W.W. Norton & Company,

1993

Sundaran, R.K.. A First Course in Optimization Theory, Cambridge University Press,

1996

Varian, H.R. (1992), Microeconomic Analysis, New York, W.W. Norton & Company, 3rd

ed.

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