CONVEXITY
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Convexity
1
Convex sets
 Ideas of convexity used throughout microeconomics
 Restrict attention to real space Rn
 I.e. sets of vectors (x1, x2, ..., xn)
 Use the concept of convexity to define
• Convex functions
• Concave functions
• Quasiconcave functions
March 2012
Frank Cowell: Convexity
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Overview...
Convexity
Sets
Basic definitions
Functions
Separation
March 2012
Frank Cowell: Convexity
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Convexity in R2
 A set A in R2
 Draw a line between any two points in A
x2

Any point on this
line also belongs
to A

...so A is convex
x1
March 2012
Frank Cowell: Convexity
4
Strict Convexity in R2
 A set A in R2
 A line between any two boundary points of A
x2

Any intermediate
point on this line is
in interior of A

...so A is
strictly convex
x1
March 2012
Frank Cowell: Convexity
Examples of
convex sets in
R3
5
The simplex
x1
x1 + x2 + x3 = const
 The simplex is convex,
but not strictly convex
0
x3
March 2012
Frank Cowell: Convexity
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The ball
x1
Si [xi– ai]2 = const
 A ball centred on the
point (a1,a2,a3) > 0
It is strictly convex
0
x3
March 2012
Frank Cowell: Convexity
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Overview...
Convexity
Sets
For scalars and
vectors
Functions
Separation
March 2012
Frank Cowell: Convexity
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Convex functions
A function f: RR
 Draw A, the set "above" the
function f
y
A := {(x,y): y  f(x)}
 If A is convex, f is a convex
function
y = f(x)
 If A is strictly convex, f is a
strictly convex function
x
March 2012
Frank Cowell: Convexity
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Concave functions (1)
A function f: RR
y
Draw the function –f
y = f(x)
 Draw A, the set "above" the
function –f
 If –f is a convex function, f is a
x concave function
 Equivalently, if the set "below" f
is convex, f is a concave function
If –f is a strictly convex function, f
is a strictly concave function
March 2012
Frank Cowell: Convexity
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Concave functions (2)
y
y = f(x)
0
A function f: R2R
 Draw the set "below" the
function f
x2
 Set "below" f is strictly
convex, so f is a strictly
concave function
March 2012
Frank Cowell: Convexity
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Convex and concave function
y
y = f(x)
An affine function f: RR
 Draw the set "above" the function f
 Draw the set "below" the function f
 The graph in R2 is a straight line
x
 Both "above" and “below" sets
are convex
So f is both concave and convex
 Graph in R3 is a plane
The graph in Rn is a hyperplane
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Frank Cowell: Convexity
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Quasiconcavity
x2
Draw contours of function f: R 2R
Pick contour for some value y0
Draw the "better-than" set for y0
 If the "better-than" set B(y0) is
convex, f is a concavecontoured function
B(y0)
 An equivalent term is a
"quasiconcave" function
If B(y0) is strictly convex, f is a
strictly quasiconcave" function
y0 = f(x)
x1
March 2012
Frank Cowell: Convexity
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Overview...
Convexity
Sets
Fundamental
relations
Functions
Separation
March 2012
Frank Cowell: Convexity
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Convexity and separation
convex
Two convex sets in R2
 Convex and nonconvex sets
 Convex sets can be separated
by a hyperplane...
 ...but nonconvex sets
sometimes can't be separated
non-convex
convex
convex
March 2012
Frank Cowell: Convexity
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A hyperplane in R2
x2
Hyperplane in R2 is a straight line
Parameters p and c determine
the slope and position
{x: Si pixi  c}
Draw in points "above" H
Draw in points "below" H
{x: Si pixi  c}
x1
March 2012
Frank Cowell: Convexity
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A hyperplane separating A and y
x2
A convex set A
H
A point y "outside" A

y
The point x* in A that is closest to y
The separating hyperplane

A
x*
yA
 y lies in the "above-H" set
 x* lies in the "below-H" set
x1
March 2012
Frank Cowell: Convexity
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A hyperplane separating two sets
Convex sets A and B
A and B only have no points in common
The separating hyperplane
H
A
B
March 2012
 All points of A lie strictly above H
 All points of B lie strictly below H
Frank Cowell: Convexity
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Supporting hyperplane
Convex sets A and B
A and B only have boundary
points in common
H
The supporting hyperplane
A

B
 Interior points of A lie strictly
above H
 Interior points of B lie strictly
below H
March 2012
Frank Cowell: Convexity
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