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 u  x   x1 x2
2.3.B.1 u  x  quasiconcave and strictly quasiconcave
2.3.B.2 Example with the Cobb-Douglas utility function u  x   x1 x2
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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
1
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 C 2 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 C 2 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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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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 
L 2
,
, with D a convex subset of
is concave if, for all x, y  D ,3 the utility of a convex combination
z   x  1    y of x and y is no less than the weighted average of their separate
utilities, namely
(2.1)
u  z   u  x  1    y    u  x   1    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)
u  x   u  x  1    y    u  x   1    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
u  x   u  x  1    x    u  x   1    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  .
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).
3
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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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 u  u  x1 , x2  , 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 z   x  1    y is
preferred to both x and y .7 It follows that the convex combination z   x  1    y lies on
a higher indifference curve with the implication, in terms of the utility representation of
preferences, that u  z   u  x  1    y   u  x    u  x   1    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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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Proposition 2.2 Convex preferences admit of a numerical representation if only if the
utility function is (strictly) concave.
2.1.B. Quasiconcavity
Definition 2.2 The real valued function u  x  : D 
D
L

, defined in the convex set
, is quasiconcave if, for all x, y, z   x  1    y  D , we
with values in
have
u  z   u  x  1    y   min u  x  , u  y   for all    0,1
(2.3)
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 
D

with values in
, defined in the convex set
, is quasiconvex if, for all x, y  D , we have
h  x  1    y   max  h  x  , h  y   for all    0,1
(2.4)
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 
D
L

with values in
, defined in the convex set
, 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)
u  x  1    y    min u  x  , u  y  
Noting that the left hand side of (2.5) is, by definition h  x  1    y  and that in the right
hand side  min u  x  1    y    max  h  x  1    y   , we obtain definition (2.4) of
quasiconvexity of a function.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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Consider first the one-commodity case, L  1 . 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
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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
6
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 z   y  1    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
u  z   min 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 
L
,
+
, with D a convex subset of
is
quasiconcave if and only if the upper contour set I   x  is convex.
Proof. To prove the “if” part choose x 0 
L
,
+
 
let I  x 0 be the upper contour set of x 0 and
 
 
take x, y  I  x 0 so that x · x0 and y · x0 . By representation we then have u  x   u x0 ,
 
u  y   u x 0 and by quasiconcavity
(2.6)
 
u  x  1    y   min u  x  , u  y   u x 0
 
 
We then have  x  1    y · x0 , which implies  x  1    y  I x 0 . Hence I  x 0
is
convex.
 
To prove the “only if” part, assume that I  x 0 is convex for all x 0 
L
.
+
Take x, y 
L
+
 
with u  x   u  y  and suppose x  x0 . Hence, by construction x, y  I  x 0 , by convexity
  and
u  x  1    y   u  x  .
 x  1    y  I  x0
finally,
by
representation,
we
obtain
quasiconcavity
0
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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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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 u  x  1    y   min 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 I   x   y 
L


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;
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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Consider now the quasiconvex function g(x). According to Definition 2.3 above, g(x) satisfies
the condition g  x  1    y   max  g  x  , g  y   for all    0,1 . Take x and y in the

 x  y 
same level set I  x   y 
L

respectively I 
L


g  y   g  x 
g  y   g  x  so that the upper and the lower contour sets are

and I   x   y 
L


g  y   g  x  ; then by
definition of quasiconvexity we have that g is convex if g  x  1    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 x int D, while definition 2.5
uses the second order derivative to identify the curvature of the function in the neighborhood
of every x int D.
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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).
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“     ”. Graphically, a function is convex at every x  int D if only if the tangent plane lies
below or at most on the function itself.
Definition 2.5 The twice continuously differentiable function u  x  : int D 
is
concave for every x  int D if only if the second derivative is non positive
(2.8)
d 2u  x 
dx 2
 u  x   0
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 D 
L
.
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 u  x    u1  x  ... ul  x  ... uL  x   to represent the gradient of
T
u  x  , we have the following Definition.
11
Continuity will take care of the boundary points of the domain D.
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.
12
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Definition 2.7. The function u  x  is concave if and only if13,14
(2.9)
for all x 
u  x  z   u  x   u  x   z
L

and all z 
L
, with x  z 
if the inequality (2.9) holds strictly for all x 
L
 .
L

The function u  x  is strictly concave
and all z  0 .
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 x 
L

, 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 x   x1 , x2  . In analytical terms, for all y 
2

let z  y  x be the
vector representing deviations from the chosen point x. The slope of the line through the
y x
z
points y and x is 2 2  2 , while the slope of the indifference curve is equal to the
y1  x1 z1
marginal rate of substitution: MRS1,2  x  
u1  x 
. Equating these two slopes we obtain
u2  x 
u1  x  y1  x1   u2  x  y2  x2   u  x    y  x   u  x   z  0 . The last equality, which is
the standard general definition of the tangent hyperplane at point x 
L

, will be later used
in the determination of the property of quasiconcavity.
For completeness, the function h  x  is convex for all x 
L

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.
L
y  x   yl xl
l 1
14
Condition (2.9) is derived utilizing again the concavity of the auxiliary function g  t   u  x  tz  with z 
L
and
x  tz  L . See JR, Theorem A2.4, p. 467.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
12
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 x 
L

and for all z 
L
if and
only if the quadratic form z  H  x  z is negative semidefinite,15 i.e. if
(2.10)
z  H  x z  0
The  L  L  matrix of second order partial derivatives
(2.11)
 u11 u12
u
u22
H  x    21
 ... ...

 u L1 u L 2
... u1L 
... u2 L 
... ... 

... uLL 
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 t  0 coincides with the quadratic form z  H  x  z ; (2.10) then follows from g   t  0   0 . See JR, pp.
467-469. MWG (p. 933) offer a different approach considering a second order Taylor expansion around t  0 .
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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If (2.10) holds with the strict inequality sign for all z  0 , 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 - ulm  uml - 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 zl  0 and zk l  0 ,
since in this case z  H  x  z  ull  zl  which is non positive only if ull  0 . Again, the
2
function h  x  is convex at every x 
L

if and only if the inequality sign in (2.10) is
reversed.
2.2.B. Quasiconcavity
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 
D
(2.12)

with values in
, defined in the convex set
, is quasiconcave if and only if, for all x, y  D ,
u   x  y  x   0
whenever
u  y  u  x
If u   x  y  x   0 whenever u  y   u  x  , then u  x  is strictly quasiconcave.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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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 Z  z 
L


u  x   z  0 .
The relevant matrix is now the Bordered Hessian H B  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  L  1   L  1 matrix. There are two equivalent
ways to write, in the compact block notation, the bordered Hessian matrix
(2.13)
 H  x
H B  x  
T
u  x 
u  x  

0 
(2.14)
T
 0
u  x  
H B  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  L  1 column vector u  x  is the vector of first derivatives and the
vector u  x 
T
1 L 
row
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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
15
Definition 2.10. u  x  is quasiconcave if and only if the quadratic form z  H  x  z is

negative semidefinite in the subspace Z  z 
L


u  x   z  0 ; if the quadratic
form z  H  x  z is negative definite in the subspace Z 
L
 ,
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 Z 
Z
L

L
 ;
if the quadratic form z  H  x  z is positive definite in the subspace
, 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 H B  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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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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 x 
negative definite for every x 
L

L

. If H  x  is
, 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:
 1r r H r  x   0
(2.15)
where
r Hr
 x
with r  1,..., L
is the leading principle minor of order r . In words, the definition of
negative definiteness requires that the minors r H r  x  be negative, when the index r is
odd, and positive when r is even.
The leading principle minors
r Hr
 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 H1  x  of the 1x1 submatrix  u11  ;
 u11 u12 
- the determinant 2 H 2  x  of the 2x2 submatrix 
;
u21 u22 
 u11 u12
- the determinant 3 H 3  x  of the 3x3 Hessian matrix u21 u22
u31 u32
u13 
u23  .
u33 
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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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The sign rule requires that the signs of these principle minors alternate starting from the first,
which must be negative; hence
det u11   0 ,
u u 
det  11 12   u11u22  u12u21  0
u21 u22 
and
det H  x   0 . Note that det u11   0 implies u11  0 ; 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
 1r r H   x r  0
(2.16)
where H   x  indicates the   1,..., L permutation of the Hessian and r H   x r 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)
H  x
 2,3,1
u22
 u32
 u12
u23 u21 
u33 u31 
u13 u11 
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 H1
 x
minors of the three possible 1x1 submatrices  u11  ,  u 22  e u33  ; 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 u11 , u22 , u33  0 ; 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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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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 u  x   u  x1 , x2  be a twice continuously differentiable utility
function. Fix x  int
(2.18)
2
;
let t  R and define the following function of the single variable t
g  t   u  x1  tz1 , x2  tz2 
which describes movements away from the point x in any direction z. The first order
derivative of g  t  is
(2.19)
g t   u1  x1  tz1 , x2  tz2  z1  u2  x1  tz1 , x2  tz2  z2
and, by differentiating (2.19), we can obtain the second order derivative of g  t 
(2.20)
g   t   u11   , z12  u12  , z2 z1  u21  ,  z1 z2  u22  ,  z22  z  H  x  z
where ulm  , , with l , m  1, 2 , 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 g   t   0 , we have, in view of
(2.20), z  H  x  z  0 ; 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:
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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(2.21)
r Hr
 x  0
with r  1,..., 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)
rH

 x r  x   0
2.3.A.2 Example with the Cobb-Douglas utility function u  x   x1 x2  ,   0
The first order derivatives of the function are
(2.23)
u  x     x1 1 x2
T

u  x
 x1
 x1 x2 1   

u  x  .
x2


Deriving the last expression of the first order derivatives, the Hessian matrix is
(2.24)




u  x
  1    x 2 u  x 

x1 x2
1

H  x  




u  x
 1    2 u  x  

x1 x2
x2


and its permutation of rows and columns is
(2.25)




u  x
  1    x 2 u  x 

x1 x2
2
 2,1


H
 x 




u  x
 1    2 u  x  

x1 x2
x1


Note that the Hessian matrix with rows and columns arranged in the natural order is indicated
2,1
as H  x  while the permuted matrix is indicated as H    x  . According to Definition 2.12,
the function u  x   x1 x2 is concave if  1 r H   x r  0 . This requires that the principal
r
minors of order 1 – i.e. with the index r  1 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
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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2
 

2
det H  x   1   1   
u x 
u x 
2  
2  
 x1x2 
 x1x2 

(2.26)
 1      

 x1 x2 
2
u  x  0
2
2
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 Z  z 
L
z  H  x  z is negative definite in the subspace Z 


u  x   z  0 ; if the quadratic form
L
 ,
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


definite in the subspace Z   z 
Z  z
L

u  x   z  0 for every x in the convex domain D. If H  x  is negative
L


u  x   z  0 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
u  x   z  0 .
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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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Fig. 2.7 - H  x  is negative semidefinite along the tangent line u  x   z  0
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 Z 
L

are again based on the sequence of signs of the determinants of particular
submatrices of the bordered Hessian matrix H B  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 Z 
L

of dimension S if
the determinants of the leading principle minors of H B  x  of order r  S  1,..., L are
of alternating sign. In compact notation, if
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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 1r r H B r  x   0
(2.27)
where
B
rH r
 x
is the principle leading minor of order r of the Bordered Hessian
(2.13) here repeated for a more convenient reference
(2.13)
 H  x
H B  x  
T
u  x 
u  x  

0 
Suppose that H  x  is a 2x2 matrix. Remembering then that S  1 , since we have only one
constraining linear subspace, i.e. the subspace u  x   z  0 , H B  x  is the 3x3 matrix
(2.28)
 u11 u12
H  x   u21 u22
 u1 u2
B
u1 
u2 
0 
The leading principle minor of order r  2 is the Hessian itself. We have therefore only one
leading principle minor, namely
(2.29)
B
2H 2
 u11 u12
 det u21 u22
 u1 u2
u1 
u2 
0 
to take into consideration and conclude that the quadratic form z  H  x  z is negative definite
in the linear space Z 
(2.30)
L

if
 122 H B 2  0
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 S  1 , H B  x  is the 4x4 matrix
(2.31)
 u11 u12
u
u22
B
H  x    21
u31 u32

 u1 u2
u13
u23
u33
u3
u1 
u2 
u3 

0
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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We would have in this case to check the signs of the two leading principle minors 2 H B 2 and
B
3H 3.
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
Z
L

if the leading principle minors of the matrices obtained by all possible
permutations of rows and columns of H B  x  alternate in sign starting with a minus
sign
 1r  r H B r  x  
(2.32)

0
where   1,..., L indicates the permutation of the leading principal minor of order r of
the permuted Bordered Hessian H B  x  and
rH

 x r
H   x  and
r u
 x 
the
permutation of the rows of the column vector u  x  .
Suppose again L  2 so that H  x  is a 2x2 matrix. Then H B  x  is the 3x3 matrix
(2.33)
 u11 u12
H  x   u21 u22
 u1 u2
B
u1 
u2 
0 
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)
 H
B
 x 
 2,1
u22
  u11
 u2
u21 u2 
u12 u1 
u1 0 
Since the determinants of H B  x  and  H B  x  
 2,1
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 r  2 , 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  .
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
24
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 Z  z 
L


h  x   z  0 are

of dimension S if
the following.
Definition 2.17 H  x  is positive definite in the subspace Z 
L
the determinants of the leading principle minors of H B  x  of order r  S  1,..., L are
positive if S is even, negative if S is odd. In compact notation, if
 1S r H Br  x   0
(2.35)
Definition 2.18 The Hessian matrix H  w  is positive semidefinite in the linear space
Z
L

if the leading principle minors of the matrices obtained by all possible
permutations of rows and columns of H B  x  are non negative if S is even, non positive
if S is odd. In compact notation, if
(2.36)
 1S  r H B r  x 

0
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 x 0 ,
can be taken into consideration.
2.3.B.2. Example with the Cobb-Douglas utility function u  x   x1 x2
Taking into account the definitions (2.23) and (2.24) of the first and second order derivatives
of the Cobb-Douglas function u  x   x1 x2 , the Bordered Hessian is
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
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


u  x
  1    2 u  x 
x1 x2
x1




H B  x  
u  x
 1    2 u  x 
x1 x2
x2




u  x
u  x

x1
x2

(2.37)

u  x 
x1



u  x 
x2


0 


and its permutation of rows and columns is



u  x
  1    2 u  x 
x
x
x
1
2
2


2,1




 H B  x 

u
x

1


u  x







x1 x2
x12




u  x
u  x

x2
x1

(2.38)

u  x 
x2



u  x
x1


0 


According to Definition 2.16, the function u  x   x1 x2 is quasiconcave if the leading
principle minors of H B  x  of order r  S  1,..., L of all permutations are alternating in sign
starting with a nonnegative sign. Since in the two-commodity case the determinants of
 H B  x 


 2,1
is the same as the determinant of H B  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)
det  H B  x  
 2,1


 x1x2 
2
    u  x 3   2    2  x13 2 x23 2
which is greater than zero for all positive  and  . We conclude that, while u  x   x1 x2 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.
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
26
Quasiconcavity
Concavity
u  x  is concave if and only if
 1 r H r  x   0
r

where r H r

 x
with r  1,..., L
u  x  is quasiconcave if and only if
 1r  r H B r  x  

0
r  S  1, 2,..., L
are the leading principal

 x r
minor of order r of the permuted matrix
where
H   x  ,   1,..., L
minor of order r of the Hessian H  x 
rH
is the principle leading
relative to permutation  .
Strict Concavity
Strict Quasiconcavity
If
If
 1r r H B r  x   0
 1r r H r  x   0
where
r Hr
 x
with r  1,..., L
are the leading principal
r  S  1, 2,..., L
where
rH
 x r
is the principle leading
minor of order r of the Hessian matrix
minor of order r of the Hessian matrix
H  x ,
H  x ,
then u  x  is strictly concave
then u  x  is strictly quasiconcave
D. Tosato – Appunti di Microeconomica – Lecture Notes of Microeconomics – a.y. 2016-17
27
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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