1

Chap. 16 Convex set and

optimization

AA

A

2121 )1(,)10(

setconvex a is

xxxx 　

21 )1( xxx

convexnon-

convex

2

Upper contour set

lower contour set

}),...,,(|),...,,{()( 2121 yxxxfxxxyU n

n

n

Upper contour set

Lower contour set

),..,,( 21 nxxxfy

}),...,,(|),...,,{()( 2121 yxxxfxxxyD n

n

n

U(y)

3

upper contour set

lower contour

set

x1

x2

y≤f(x1,x2)

y≥f(x1,x2) y=f(x1,x2)

4

upper contour

set

lower

contour set

y=4-(x1-4)2-(x2-3)2

y≤f(x1,x2)

y=f(x1,x2)y≥f(x1,x2)

x1

x2

5

upper

contour set

lower contour

set

x2

x1

y≤f(x1,x2)

y≥f(x1,x2)

y=f(x1,x2)

y=-4+(x1-4)2+(x2-3)2

6

upper contour

set

lower

contour set

f1>0

f2>0

upper contour set is convex

=quasi-concave function

U(y)

y=f(x1,x2)y≥f(x1,x2)

y≤f(x1,x2)

x1

x2

7

upper

contour set

lower

contour set

upper contour set is convex

=quasi-concave function

U(y)

x2

x1

y≤f(x1,x2)

y≥f(x1,x2)

f1<0

f2<0

y=f(x1,x2)

8

D(y)

upper

contour set

y≤f(x1,x2)

f1>0

f2>0

y≥f(x1,x2)

lower

contour set

lower contour set is convex

=quasi-convex function

y=f(x1,x2)

x1

x2

9

D(y)

upper

contour set

y≤f(x1,x2)

y≥f(x1,x2)

lower

contour set

f1<0

f2<0

y=f(x1,x2)

x1

x2

lower contour set is convex

=quasi-convex function

10

y=f(x)

hypograph

x

y

hypo graph is a convex set

=a concave function

hypo graph = set of (x,y) which

satisfies y≤f(x)

y≤f(x)

11

y=f(x)

y≥f(x)

epigraphe

x

y

epigraphe is a convex set

=a convex function

epigraphe = set of (x,y) which

satisfies y≥f(x)

12

The semi- concavity function,

the semi- convexity functionmultivariable function y=f(x)

concave-quasi is )(xfy

yyU any for set convex a is )(

)( nx

convex-quasi is )(xfy

yyD any for set convex a is )(

13

The necessary and sufficient

condition of quasi-concavity

)10(

concave-quasi is )(xfy

)())1(()()( 22121 xxxxx ffff

yfyfyU )( ,)()(, 2121 xxxx

yff )())1(( 221 xxx )(yU x

yff )()( Suppose 21 xx

14

Strictly quasi-concave functions,

strictly quasi-convex functions

][

convex-quasi (strictly) is )(xfy

)())1(()()( 22121 xxxxx ffff

concave-quasistrictly is )(xfy

)())1(()()( 22121 xxxxx ffff

)10(

)10(

15

21 )1( xx 1x

2x

16

Strictly quasi-concave functions

and marginal rate of substitution

2

1

1

2heoremfunction timplicit By thef

f

dx

dx

0

concave-quasistrictly is ),(

2

21

f

xxfy

decreasing is slope the

of absolute the)( of lineborder On the

1

2

dx

dx

yU

17

1

2

2

1

22

1

12

1

11

2

1 dx

dx

f

f

xf

f

xf

f

dx

d

dx

dx

dx

d

2

1

2

2

221212

2

2

211211

)()( f

f

f

ffff

f

ffff

0)(

2

)( 3

2

2112

2

122

2

211

3

2

22

2

112122211

2

211

f

fffffff

f

ffffffffff

Strictly quasi-concave functions

and marginal rate of substitution

18

02

concave-quasistrictly is ,0 If

2

122

2

2112121

2

fffffff

ff

　

　

,0 with ,0 If1

2

1

2 　　

dx

dx

dx

df

02

concave-quasistricly is

2

122

2

2112121 fffffff

f

　

　

0)(

23

2

2112

2

122

2

211

f

fffffff

Necessary and sufficient conditions

for strictly quasi-concave functions

19

Strictly quasi-concave functions

The slope approaches

0 as x1 goes up.

The slope increases

as x1 goes up.

f1<0

f2<0f1>0

f2>0

20

Necessary and sufficient conditions

for strictly quasi- concave, quasi-

convex functions

02

concave-quasistrictly is

2

122

2

2112121 fffffff

f

02

concave-quasi is

2

122

2

2112121 fffffff

f

　

　

02

convex-quasi (strictly) is

2

122

2

2112121 fffffff

f

　

　

][

21

Quasi- concavity of concave functions

)10(,, 21 xx

concave-quasiconcave is )( xfy

)()1()())1(()( 2121 xxxxx ffff

yfyfyU )()()(, 2121 xxxx 　

yyyfff )1()()1()()( 21 xxx

)(yU x

convex-quasiconvex is )( xfy

22

Quasi-concavity of Cobb-Douglas

functionLAKLKF ),( 0,, A

LAKLKFQ ),(

1

),(

A

QKQKhL

1

1

12

),(

A

QK

K

QKhRTS

The iso-quant curve

23

1

1

12

),(

A

QK

K

QKhRTS

01),(

1

2

2

2

A

QK

K

QKh

always quasi-concave

0)1( 22222 LKA

22222222})1()1({ LKAFFF KLLLKK

convex1

Quasi-concavity of Cobb-Douglas

function

24

It mediates between Cobb-

Douglas function.

LAKLKF ),( 0,, A

always quasi-concave

scale return to increasing1

scale return toconstant 1

concave

scale, return to decreasing1