Post on 16-May-2020
transcript
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)
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upper contour set
lower contour
set
x1
x2
y≤f(x1,x2)
y≥f(x1,x2) y=f(x1,x2)
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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
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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
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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
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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)
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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
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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
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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)
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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)
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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 )(
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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
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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
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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
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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
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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
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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
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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
][
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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
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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
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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