Spring 2004
John Rust
Economics 425
University of Maryland
Properties of Concave and Convex Functions
Definition 1:
A function f
:
R
n
→
R
n
is
concave
if and only if for every x
,
y
∈
R
n
and for every
θ
∈
[
0
,
1
]
f
((
1

θ
)
x
+
θ
y
)
≥
(
1

θ
)
f
(
x
)+
θ
f
(
y
)
(1)
Definition 2:
A function f
:
R
n
→
R
n
is
convex
if and only if for every x
,
y
∈
R
n
and for every
θ
∈
[
0
,
1
]
f
((
1

θ
)
x
+
θ
y
)
≤
(
1

θ
)
f
(
x
)+
θ
f
(
y
)
(2)
Comment 0:
Note that the
(
1

θ
)
multiplying the
x
and
f
(
x
)
and the
θ
multiplying the
y
and
f
(
y
)
in
the definitions above is just an arbitrary convention. We can switch this, and get an equivalent definition
of concavity:
f is concave if and only if for each x and y and every
θ
∈
[
0
,
1
]
we have
f
(
θ
x
+(
1

θ
)
y
)
≥
θ
f
(
x
)+(
1

θ
)
f
(
y
)
(3)
Lemma 0:
A function f is concave if and only if its negative,

f, is convex.
Proof:
If
f
is concave then we have from Definition 1,
f
((
1

θ
)
x
+
θ
y
)
≥
(
1

θ
)
f
(
x
)+
θ
f
(
y
)
(4)
Multiplying both sides of this inequality by

1 we get

f
((
1

θ
)
x
+
θ
y
)
≤
(
1

θ
)[

f
(
x
)]+
θ
[

f
(
y
)]
(5)
(Notice that multiplying an inequality by

1 flips the direction of the inequality). But this latter in
equality, (5), satisfies the second definition for convexity, i.e.

f
is a convex function.
Comment 1:
The simple geometric intuition is that a concave function
f
has the curvature of an over
turned bowl.
But taking the negative,

f
, is just like flipping the bowl up, so an upturned bowl is
convex.
Figure 1, below, illustrates the definition of concavity for a particular concave function, log
(
x
)
(where log denotes the natural logarithm, i.e. the inverse of the exponential function, log
(
exp
{
x
}
) =
exp
{
log
(
x
)
}
=
x
). The figure shows two points
x
and
y
and a point
(
1

θ
)
x
+
θ
y
lying on the line seg
ment (on the xaxis) between
x
and
y
. According to the definition of concavity, log
(
x
)
is concave if and
only if for any
x
and
y
and any
θ
∈
[
0
,
1
]
we have log
((
1

θ
)
x
+
θ
y
)
≥
(
1

θ
)
log
(
x
) +
θ
log
(
y
)
. This
latter point,
(
1

θ
)
log
(
x
) +
θ
log
(
y
)
, is just a point on the line joining log
(
x
)
and log
(
y
)
(to see this,
note that the function
l
(
θ
)
≡
(
1

θ
)
log
(
x
)+
θ
log
(
y
) =
log
(
x
)+
θ
[
log
(
y
)

log
(
x
)]
is linear with slope
log
(
y
)

log
(
x
)
and goes through the two points
l
(
0
) =
log
(
x
)
and
l
(
1
) =
log
(
y
)
).
Thus,
the geometric intuition behind concavity is that a function is concave if the line seqment
joining the function at any two points x and y lies below the function.
In the case of the function
f
(
x
) =
log
(
x
)
we can see visually that this is the case, i.e.
(
1

θ
)
log
(
x
)+
θ
log
(
y
)
≤
log
((
1

θ
)
x
+
θ
y
)
.
1
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Example of Concavity Definition: log(x)
x
y
(1
−
θ
)x+
θ
y
log(y)
log(x)
log((1
−
θ
)x+
θ
y)
(1
−
θ
)log(x)+
θ
log(y)
However it seems that the general definition of concavity is a hard one to verify.
How can we
determine in general whether for any particular function
f
whether for any two points
x
and
y
the
function lies above the line segment joining
f
(
x
)
and
f
(
y
)
? For example take the function
f
(
x
) =
√
x
.
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 Spring '06
 cramton
 Economics, Derivative, lim, Rn → Rn

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