AAE 635 Discussions
10/09/2006
Bordered Hessian, Quasi-concave and Quasi-convex functions
After talking about select problems from Homework2, we now discuss quasi-concave and quasiconvex functions.
Quasi-concave and quasi-convex functions
A function f: X  R is (strictly) quasi-concave on X if and only if for all x1, x2 X and 0   
1, it is the case that f( x1 + (1-) x2) (>)  min{f(x1), f(x2)}
Sometimes, it may prove easier to check quasi-concavity by the following alternative definition:
A function f(x), where x is a vector of variables, is (strictly) quasi-concave if and only if, for any
constant z, the set A = {x  X: f(x)  z} is a (strict) convex set.
And,
A function f(x), where x is a vector of variables, is (strictly) quasi-convex if and only if, for any
constant z, the set B = {x  X: f(x) ≤ z} is a (strict) convex set.
An Examples of Quasi-concave Functions:
Exercise1: Are the following functions concave, convex, quasi-concave or quasi-convex?
(1) f(x) = -(x-2)2 + 25
30
20
f(x)
10
0
-5
-10
0
5
10
-20
x
1
f(x)
(2) f(x) = ln x; x>0
3
2
1
0
-1 0
-2
-3
-4
-5
2
4
6
8
10
12
x
(3)
f(x)
x
(4) f(x) = ex.
25
20
f(x)
15
10
5
0
-6
-4
-2
0
2
4
x
2
(5) f(x1, x2) =x10.5x20.5.
5
4.5
4
3.5
3
f(x1,x2)
2.5
2
1.5
1
5
4
0.5
3
0
x2
2
0
0.5
1
1
1.5
2
2.5
3
x1
3.5
0
4
4.5
5
(6) f(x1,x2)=x11.5x21.5
140
120
100
80
f(x1,x2)
60
40
5
20
4
3
0
2
0
0. 5
1
x2
1
1. 5
2
x1
2. 5
3
3. 5
4
0
4. 5
5
3
(7) f(x1,x2) = - x1x2.
0
-0.5
-1
-1.5
-2
f(x1,x2) -2.5
-3
-3.5
-4
4.5
-4.5
3
x2
-5
0
1.5
0.5
1
1.5
x1
2
2.5
3
3.5
4
4.5
0
5
Some facts:
(All these facts can be easily extended to quasi-convexity.)
(1) If f(x) is (strictly) quasi-concave, then –f(x) is (strictly) quasi-convex.
(2) Any concave (convex) function is quasi-concave (quasi-convex).
Proof: Suppose f is concave. Then, for all x1, x2  X, and   (0,1), we have
f( x1 + (1-) x2)  f(x1) + (1-)f(x2)
 min{f(x1), f(x2)} + (1-) min{f(x1), f(x2)}
= min {f(x1), f(x2)}. So f is also quasi-concave.
In graph, call the set of all points under the function A. Now, if A is convex, (i.e. the function is
concave), then by inspection, each of its upper contour sets, formed by taking a horizontal cross
section of A, must be convex also. This argument is demonstrated in the figure below.
Y
A
X
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(3) A linear function is quasi-concave and quasi-convex.
(4) If f(x) is quasi-concave and F (f) is an increasing function, then F (f(x)) is also quasi-concave.
(5) If f(x) is continuously differentiable, the function f(x) is quasi-concave if and only if f(x2) (>)
 f(x1) implies that f'(x1) (x2 - x1) (>)  0.
(6) To determine whether a twice-differentiable function on a convex set S is quasiconcave or
quasiconvex, we can examine the determinants of the bordered Hessians of the function, defined
as follows:
0
f1
fk
D 
f1
f11
f1k
fk
f k1
f kk
0
D1 
f1
f1
,
f11
0
f1
f2
D2  f1
f2
f11
f 21
f12 , …
f 22
Dk 
0
f1
fk
f1
f11
f1k
fk
f k1
f kk
If D1(x) < 0, D2(x) > 0,…, Dn(x) < 0 if n is odd and Dn(x) > 0 if n is even for all x in S then f is
quasi-concave. If f is quasi-concave then D1(x)  0, D2(x)  0, ..., Dn(x)  0 if n is odd and
Dn(x)  0 if n is even, for all x in S.
If Dk(x) < 0 for all k, for all x in S then f is quasi-convex. If f is quasi-convex, then Dk(x)  0 for
all k,for all x in S.
Exercise2: Show that f ( x1 , x2 )  x1a x2b ( x1 , x2  0;0  a, b  1) is quasi-concave.
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