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tion operations.
Theorem 2.9.1 (Fundamental Theorem of Calculus) The integration and dif-
ferentiation operators are inverses in the following senses:
1.
b
d
f(x)dx = f(b)
db a
2.
b
f (x)dx = f(b) - f(a)
a
This can be illustrated graphically using a picture illustrating the use of integration
to compute the area under a curve.
2
Should this not be the closed interval?
Revised: December 2, 1998
CHAPTER 3. CONVEXITY AND OPTIMISATION 27
Chapter 3
CONVEXITY AND
OPTIMISATION
3.1 Introduction
[To be written.]
3.2 Convexity and Concavity
3.2.1 Definitions
Definition 3.2.1 A subset X of a vector space is a convex set
�!�!
"x, x " X, � " [0, 1], �x + (1 - �)x " X.
Theorem 3.2.1 A sum of convex sets, such as
X + Y a" {x + y : x " X, y " Y } ,
is also a convex set.
Proof The proof of this result is left as an exercise.
Q.E.D.
Definition 3.2.2 Let f : X �! Y where X is a convex subset of a real vector
space and Y �" . Then
Revised: December 2, 1998
28 3.2. CONVEXITY AND CONCAVITY
1. f is a convex function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x ) d" �f(x) + (1 - �)f(x ). (3.2.1)
(This just says that a function of several variables is convex if its restriction
to every line segment in its domain is a convex function of one variable in
the familiar sense.)
2. f is a concave function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x ) e" �f(x) + (1 - �)f(x ). (3.2.2)
3. f is affine
�!�!
f is both convex and concave.1
Note that the conditions (3.2.1) and (3.2.2) could also have been required to hold
(equivalently)
"x, x " X, � " [0, 1]
since they are satisfied as equalities "f when x = x , when � = 0 and when
� = 1.
Note that f is convex �!�! -f is concave.
Definition 3.2.3 Again let f : X �! Y where X is a convex subset of a real vector
space and Y �" . Then
1. f is a strictly convex function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x )
1
A linear function is an affine function which also satisfies f(0) = 0.
Revised: December 2, 1998
CHAPTER 3. CONVEXITY AND OPTIMISATION 29
2. f is a strictly concave function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x ) > �f(x) + (1 - �)f(x ).
Note that there is no longer any flexibility as regards allowing x = x or � = 0 or
� = 1 in these definitions.
3.2.2 Properties of concave functions
Note the connection between convexity of a function of several variables and con-
vexity of the restrictions of that function to any line in its domain: the former is
convex if and only if all the latter are.
Note that a function on a multidimensional vector space, X, is convex if and only
if the restriction of the function to the line L is convex for every line L in X, and
similarly for concave, strictly convex, and strictly concave functions.
Since every convex function is the mirror image of a concave function, and vice
versa, every result derived for one has an obvious corollary for the other. In gen-
eral, we will consider only concave functions, and leave the derivation of the
corollaries for convex functions as exercises.
Let f : X �! and g : X �! be concave functions. Then
1. If a, b > 0, then af + bg is concave.
2. If a
3. min{f, g} is concave
The proofs of the above properties are left as exercises.
Definition 3.2.4 Consider the real-valued function f: X �! Y where Y �" .
1. The upper contour sets of f are the sets {x " X : f(x) e" �} (� " ).
2. The level sets or indifference curves of f are the sets {x " X : f(x) = �}
(� " ).
3. The lower contour sets of f are the sets {x " X : f(x) d" �} (� " ).
In Definition 3.2.4, X does not have to be a (real) vector space.
Revised: December 2, 1998
30 3.2. CONVEXITY AND CONCAVITY
Theorem 3.2.2 The upper contour sets {x " X : f(x) e" �} of a concave
function are convex.
Proof This proof is probably in a problem set somewhere.
Q.E.D.
Consider as an aside the two-good consumer problem. Note in particular the im-
plications of Theorem 3.2.2 for the shape of the indifference curves corresponding
to a concave utility function. Concave u is a sufficient but not a necessary condi-
tion for convex upper contour sets.
3.2.3 Convexity and differentiability
In this section, we show that there are a total of three ways of characterising
concave functions, namely the definition above, a theorem in terms of the first
derivative (Theorem 3.2.3) and a theorem in terms of the second derivative or
Hessian (Theorem 3.2.4).
Theorem 3.2.3 [Convexity criterion for differentiable functions.] Let f : X �!
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tion operations.
Theorem 2.9.1 (Fundamental Theorem of Calculus) The integration and dif-
ferentiation operators are inverses in the following senses:
1.
b
d
f(x)dx = f(b)
db a
2.
b
f (x)dx = f(b) - f(a)
a
This can be illustrated graphically using a picture illustrating the use of integration
to compute the area under a curve.
2
Should this not be the closed interval?
Revised: December 2, 1998
CHAPTER 3. CONVEXITY AND OPTIMISATION 27
Chapter 3
CONVEXITY AND
OPTIMISATION
3.1 Introduction
[To be written.]
3.2 Convexity and Concavity
3.2.1 Definitions
Definition 3.2.1 A subset X of a vector space is a convex set
�!�!
"x, x " X, � " [0, 1], �x + (1 - �)x " X.
Theorem 3.2.1 A sum of convex sets, such as
X + Y a" {x + y : x " X, y " Y } ,
is also a convex set.
Proof The proof of this result is left as an exercise.
Q.E.D.
Definition 3.2.2 Let f : X �! Y where X is a convex subset of a real vector
space and Y �" . Then
Revised: December 2, 1998
28 3.2. CONVEXITY AND CONCAVITY
1. f is a convex function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x ) d" �f(x) + (1 - �)f(x ). (3.2.1)
(This just says that a function of several variables is convex if its restriction
to every line segment in its domain is a convex function of one variable in
the familiar sense.)
2. f is a concave function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x ) e" �f(x) + (1 - �)f(x ). (3.2.2)
3. f is affine
�!�!
f is both convex and concave.1
Note that the conditions (3.2.1) and (3.2.2) could also have been required to hold
(equivalently)
"x, x " X, � " [0, 1]
since they are satisfied as equalities "f when x = x , when � = 0 and when
� = 1.
Note that f is convex �!�! -f is concave.
Definition 3.2.3 Again let f : X �! Y where X is a convex subset of a real vector
space and Y �" . Then
1. f is a strictly convex function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x )
1
A linear function is an affine function which also satisfies f(0) = 0.
Revised: December 2, 1998
CHAPTER 3. CONVEXITY AND OPTIMISATION 29
2. f is a strictly concave function
�!�!
"x = x " X, � " (0, 1)
f (�x + (1 - �)x ) > �f(x) + (1 - �)f(x ).
Note that there is no longer any flexibility as regards allowing x = x or � = 0 or
� = 1 in these definitions.
3.2.2 Properties of concave functions
Note the connection between convexity of a function of several variables and con-
vexity of the restrictions of that function to any line in its domain: the former is
convex if and only if all the latter are.
Note that a function on a multidimensional vector space, X, is convex if and only
if the restriction of the function to the line L is convex for every line L in X, and
similarly for concave, strictly convex, and strictly concave functions.
Since every convex function is the mirror image of a concave function, and vice
versa, every result derived for one has an obvious corollary for the other. In gen-
eral, we will consider only concave functions, and leave the derivation of the
corollaries for convex functions as exercises.
Let f : X �! and g : X �! be concave functions. Then
1. If a, b > 0, then af + bg is concave.
2. If a
3. min{f, g} is concave
The proofs of the above properties are left as exercises.
Definition 3.2.4 Consider the real-valued function f: X �! Y where Y �" .
1. The upper contour sets of f are the sets {x " X : f(x) e" �} (� " ).
2. The level sets or indifference curves of f are the sets {x " X : f(x) = �}
(� " ).
3. The lower contour sets of f are the sets {x " X : f(x) d" �} (� " ).
In Definition 3.2.4, X does not have to be a (real) vector space.
Revised: December 2, 1998
30 3.2. CONVEXITY AND CONCAVITY
Theorem 3.2.2 The upper contour sets {x " X : f(x) e" �} of a concave
function are convex.
Proof This proof is probably in a problem set somewhere.
Q.E.D.
Consider as an aside the two-good consumer problem. Note in particular the im-
plications of Theorem 3.2.2 for the shape of the indifference curves corresponding
to a concave utility function. Concave u is a sufficient but not a necessary condi-
tion for convex upper contour sets.
3.2.3 Convexity and differentiability
In this section, we show that there are a total of three ways of characterising
concave functions, namely the definition above, a theorem in terms of the first
derivative (Theorem 3.2.3) and a theorem in terms of the second derivative or
Hessian (Theorem 3.2.4).
Theorem 3.2.3 [Convexity criterion for differentiable functions.] Let f : X �!
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