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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100X = [0,1] i.e., X compact ; f discontinuous

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100X = (0,1] i.e., X not compact; f continuous

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

50

100X = [0,1] i.e., X compact; f continuous

Figure 1. Compactness plus continuity implies boundedness

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(

(

X

Y

W

X

Y

[ )

[

[xx xxxn xn

) f(x)

n)

f(xn)

S

f1(S)

O

f1(O)

if part of the proof (contra-positive)

f continuous, f1(S) not open impliesSnot open

only if part of the proof (contra-positive)

fdiscontinuous,O open,f1(O) not open

Figure 1. fis continuous iffSopen = f1(S) open

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3/14

Continuity & Hemi-continuity defns: continuity

1(a) Inverse image formulation of continuity.

A functionf : X Y is calledcontinuousif for every open setO Y,f1(O)is an open set ofX.

1(b) Neighborhood formulation of continuity.

A functionf : X Y is calledcontinuousif for everyx Xand everyneighborhoodUoff(x)there is a neighborhoodV ofxsuch thatf(x) U foreveryx V.

1(c) Sequential formulation of continuity.

A functionf : XRn is calledcontinuousatx0 Xif whenever {xm}m=1convergesx0 then {f(xm)} converges tof(x0); the functionfis continuous if itis continuous atxfor everyx X.

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Continuity & Hemi-continuity defns: upper hemi-continuity

2(a) Inverse image formulation of upper hemicontinuity.A correspondence : S T is calledupper hemicontinuousif for every open

setO T, the upper inverse image ofOunder,1

(O) S, is an open set.Explosions allowed;implosions are not

2(b) Neighborhood formulation of upper hemicontinuity.A correspondence : S T is calledupper hemicontinuousif for everys Sand every neighborhoodUof(s)there is a neighborhoodV ofssuch that(s) Ufor everys V.

2(c) Sequential formulation of upper hemicontinuity.

A compact-valued correspondence : S T is calledupper hemicontinuousif for everys S, every sequence {sn} converging tosand every sequence{tn} withtn (sn), there is a convergent subsequence {tnk} of {tn} such thatlim

ntnk=

t (

s)

.

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Continuity & Hemi-continuity defns: lower hemi-continuity

3(a) Inverse image formulation of lower hemicontinuity.A correspondence : S T is calledlower hemicontinuousif for every opensetO T, the lower inverse image ofOunder,1(O) S, is an open set.Implosions allowed;explosions are not

3(b) Neighborhood formulation of lower hemicontinuity.

A correspondence : S T is calledlower hemicontinuousif for everys S,and every open setU T withU

(s) = /0, there exists a neighborhoodV of

ssuch thatU(z) = /0, for everyz V.

3(c) Sequential formulation of lower hemicontinuity.

A correspondence : S T is calledlower hemicontinuousif for everys S,anyt (s), and any sequence {sn} converging tos, there exists a sequence{tn} such thattn (sn)and lim

ntn= t.

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Explosions, Implosions and Hemi-continuity

Thegraphof a correspondence:

Graph() = {(s, t) ST :t (s)}

Theouter accumulationof a correspondence.

OAc

(s) = {t T : (s,

t)is an accumulation point of Graph()}Theinner accumulationof a correspondence.

IAc(s) = {t T : {sn} s,{tn} s.t.n, tn (sn)and {tn} t}

Definition: explodesat sif(s) IAc(s).

Definition: implodesat sifOAc(s) (s).

Theorem: If:S T,Tis compact and is compact-valued, then is u.h.c atsiff does not implode ats

Theorem: is l.h.c atsiff does not explode at s() August 18, 2013 6 / 14
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O)

(

( )( )1(O)

1(O)

Figure 3. Upper and Lower inverse images of

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(1,

2)

(0,0)

x

u

is

uhsb

ut

not

lhc

u1((1, 2)) = {0} closed

u1

((1, 2)) =

(1

,1)

(0,0)

is

lhc

but

not

uhc

1

((1, 1)) = {0} closed

1((1, 1)) = R

Figure 4. u and

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0 1

U

O

yz w

(

(

)

)

x

Figure 5. : [0, 1] R() August 18, 2013 9 / 14
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0 1

U

O

yz w

(

(

)

)

v x

Figure 6. : [0, 1] R() August 18, 2013 10 / 14
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Berges Theorem

Berges Theorem of the Maximum: If :X Yis a continuous

correspondence with nonempty and compact values and:YR is acontinuous function, theny :X Ydefined byy(x) =argmaxy(x)(y)isu.h.c. and :XR defined by(x) =maxy(x)(y)is a continuousfunction.

For our purposes, think of:Xas a space of price vectors

Yas a space of commodity vectors

as a budget correspondence, continuous, compact-valued

as a utility function, continuousy as a demand correspondence. Result: its u.h.c.

as an indirect utility function. Result: its continuous

(Ill adduto the pics, its a direct utility function)

() August 18, 2013 11 / 14
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Berges theorem: Introduction

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 01 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 01 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 11 1 1 1 10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 2.5

* **

y1y1y1

y2y2y2

y(1)

y(2)

y(0.5) (1) (2)(0.5)

P = x1x2

Figure 7. The demand correspondence is u.h.c

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Berges theorem: Role of upper-hemi-continuity

(

(

}(y) = (x)y O (nbd of(x))inf(O)

U =1(O)

(x)

(x)

x

x

V = 1(U)

RX Y

Figure 1. Lower hemi-continuity of implies that (x)>inf(O).

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Berges theorem: Role of upper-hemi-continuity

Y

(

(

(

(

}

}(y) = (x) O (nbd of(x))

Uh =1(O)

O (nbd of((x))

O

=O O

sup(O)

(x)

x

Vh =1(Uh)

RX

Figure 2. Upper hemi-continuity of implies that (x)< sup(O).

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