Chap. 16 Convex set and
optimization
A is a convex set 
 (0    1) x1 , x 2  A x1  (1   )x 2  A
convex
x  x1  (1   ) x2
nonconvex
1
Upper contour set
lower contour set
U(y)
y  f ( x1, x2 ,.., xn )
Upper contour set
U ( y )  {( x1, x2 ,..., xn )   | f ( x1, x2 ,..., xn )  y}
n
Lower contour set
D( y )  {( x1, x2 ,..., xn )   | f ( x1, x2 ,..., xn )  y}
n
2
x2
upper contour set
y≤f(x1,x2)
lower contour
set
y≥f(x1,x2)
y=f(x1,x2)
x1
3
y=4-(x1-4)2-(x2-3)2
x2
lower
contour set
y≥f(x1,x2)
upper contour
set
y≤f(x1,x2)
y=f(x1,x2)
x1
4
y=-4+(x1-4)2+(x2-3)2
x2
lower contour
set
y≥f(x1,x2)
upper
contour set
y≤f(x1,x2)
y=f(x1,x2)
x1
5
x2
U(y)
upper contour
set
y≤f(x1,x2)
lower
contour set
y≥f(x1,x2)
f1>0
f2>0
y=f(x1,x2)
x1
upper contour set is convex
=quasi-concave function
6
x2
lower
contour set
y≥f(x1,x2)
U(y)
upper
contour set
y≤f(x1,x2)
f1<0
f2<0
y=f(x1,x2)
x1
upper contour set is convex
=quasi-concave function
7
x2
f1>0
upper
contour set
f2>0
y≤f(x1,x2)
lower
contour set
D(y)
y≥f(x1,x2)
y=f(x1,x2)
x1
lower contour set is convex
=quasi-convex function
8
x2
D(y)
lower
contour set
f1<0
f2<0
y≥f(x1,x2)
upper
contour set
y≤f(x1,x2)
y=f(x1,x2)
x1
lower contour set is convex
=quasi-convex function
9
hypo graph = set of (x,y) which
satisfies y≤f(x)
y
y≤f(x)
hypograph
y=f(x)
x
hypo graph is a convex set
=a concave function
10
epigraphe = set of (x,y) which
satisfies y≥f(x)
y
y=f(x)
epigraphe
y≥f(x)
x
epigraphe is a convex set
=a convex function
11
The semi- concavity function,
the semi- convexity functionn
multivariable function y=f(x)
(x  )
y  f (x) is quasi - concave
 U ( y) is a convex set for any y
y  f (x) is quasi - convex
 D( y) is a convex set for any y
12
The necessary and sufficient
condition of quasi-concavity
y  f (x) is quasi - concave 
f (x1 )  f (x2 )  f (x1  (1   )x2 )  f (x2 )
 (0    1)
x1 , x2 U ( y)  f (x1 )  y, f (x2 )  y
Suppose f (x1 )  f (x2 )  y
f (x1  (1   )x2 )  f (x2 )  y  x U (y)
13
Strictly quasi-concave functions,
strictly quasi-convex functions
y  f (x) is strictly quasi - concave 
f (x1 )  f (x2 )  f (x1  (1   )x2 )  f (x2 )
 (0    1)
y  f (x) is (strictly)quasi - convex 
f (x1 )  f (x2 )  f (x1  (1   )x2 )  f (x2 )
 (0    1)
[]
14
x1
x1  (1   )x 2
x2
15
Strictly quasi-concave functions
and marginal rate of substitution
y  f ( x1 , x2 ) is strictly quasi - concave
f2  0
On the border line of U ( y ) the absolute of
dx2
the slope 
is decreasing
dx1
dx2
f1
By theimplicit function theorem 

dx1 f 2
16
Strictly quasi-concave functions
and marginal rate of substitution
d  dx2  d  f1    f1    f1  dx2
 
 
  
  
 
dx1  dx1  dx1  f 2  x1  f 2  x2  f 2  dx1
f11 f 2  f1 f 21 f12 f 2  f1 f 22  f1 
  


2
2
( f2 )
( f2 )
 f2 
2
2
2
2
f11 f 2  f1 f 21 f 2  f12 f 2 f1  f1 f 22 f11 f 2  f 22 f1  2 f12 f1 f 2


0
3
3
( f2 )
( f2 )
17
Necessary and sufficient conditions
for strictly quasi-concave functions
If f 2  0, f　is strictly quasi - concave
　2 f 21 f1 f 2  f11 f 2  f 22 f1  0
2
2
d  dx2 
  0　If f 2  0, with  
,
dx1  dx1 
f11 f 2  f f  2 f12 f1 f 2
0
3
( f2 )
2
2
22 1
f　is stricly quasi - concave
　2 f 21 f1 f 2  f11 f 2  f 22 f1  0
2
2
18
Strictly quasi-concave functions
f1>0
f1<0
f2<0
f2>0
The slope approaches
0 as x1 goes up.
The slope increases
as x1 goes up.
19
Necessary and sufficient conditions
for strictly quasi- concave, quasiconvex functions
f is strictly quasi - concave
 2 f 21 f1 f 2  f11 f 2  f 22 f1  0
f　is quasi - concave
2
2
　2 f 21 f1 f 2  f11 f 2  f 22 f1  0
f　is (strictly)quasi - convex
2
2
　2 f 21 f1 f 2  f11 f 2  f 22 f1  0
[ ]
2
2
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Quasi- concavity of concave functions
y  f (x) is concave  quasi - concave
f (x )  f (x1  (1   )x2 )  f (x1 )  (1   ) f (x2 )
x1 , x2 ,  (0    1)
x1 , x2 U ( y)  f (x1 )  y　f (x2 )  y
f (x )  f (x1 )  (1   ) f (x2 )  y  (1   ) y  y
 x U (y)
y  f (x) is convex  quasi - convex
21
Quasi-concavity of Cobb-Douglas
function


F ( K , L)  AK L
The iso-quant curve
A, ,   0
 
Q  F ( K , L)  AK L
L  h( K , Q )  K



1
 Q 
 
 A 
1
h( K , Q)
   1  Q  
 RTS12 
 K
 
K

 A
22
Quasi-concavity of Cobb-Douglas
function
1

h( K , Q)
   1  Q  
 RTS12 
 K
 
K

 A

 h( K , Q )   
   1 K
2
K
  
2

 2

1
 Q 
  0
 A
always quasi-concave
FKK FLL  F
2
KL
 { (  1)  (   1)   2  2}A2 K 2 2 L2  2
  (1     ) A K
2
2 2
1      convex
2  2
L
0
23
It mediates between CobbDouglas function.


F ( K , L)  AK L
A, ,   0
always quasi-concave
    1  decreasing return to scale,
concave
    1  constant return toscale
    1  increasing return toscale
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