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Chapter 12
GENERAL EQUILIBRIUM AND
WELFARE
Copyright 2005 by SouthW!"t!r#$ % &i'i"io# o( )ho*"o# L!%r#i#g+ A,, right" r!"!r'!&+
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Perfectly Competitive
Price System- W! .i,, %""u*! th%t %,, *%r/!t" %r!
p!r(!t,y o*p!titi'!
1 th!r! i" "o*! ,%rg! #u*b!r o( ho*og!#!ou"goo&" i# th! !o#o*y
- both o#"u*ptio# goo&" %#& (%tor" o(pro&utio#
1 !%h goo& h%" %# !ui,ibriu* pri!1 th!r! %r! #o tr%#"%tio# or tr%#"port%tio#
o"t"
1 i#&i'i&u%," %#& (ir*" h%'! p!r(!t i#(or*%tio#
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Law of One Price- A ho*og!#!ou" goo& tr%&!" %t th!"%*! pri! #o *%tt!r .ho buy" it or
.ho "!,," it
1 i( o#! goo& tr%&!& %t t.o &i((!r!#t pri!"$
&!*%#&!r" .ou,& ru"h to buy th! goo&
.h!r! it .%" h!%p!r %#& (ir*" .ou,& try to
"!,, th!ir output .h!r! th! pri! .%" high!r- th!"! %tio#" .ou,& t!#& to !u%,i3! th! pri!
o( th! goo&
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Assumptions of Perfect
Competition- )h!r! %r! % ,%rg! #u*b!r o( p!op,!
buyi#g %#y o#! goo&
1 !%h p!r"o# t%/!" %,, pri!" %" gi'!# %#&"!!/" to *%4i*i3! uti,ity gi'!# hi" bu&g!t
o#"tr%i#t
- )h!r! %r! % ,%rg! #u*b!r o( (ir*"pro&ui#g !%h goo&
1 !%h (ir* t%/!" %,, pri!" %" gi'!# %#&
%tt!*pt" to *%4i*i3! pro(it"
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General Equilibrium- A""u*! th%t th!r! %r! o#,y t.o goo&"$x
%#& y
- A,, i#&i'i&u%," %r! %""u*!& to h%'!i&!#ti%, pr!(!r!#!"
1 r!pr!"!#t!& by %# i#&i((!r!#! *%p
- )h! pro&utio# po""ibi,ity ur'! %# b!u"!& to "ho. ho. output" %#& i#put" %r!
r!,%t!&
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E!eworth "o# $ia!ram- Co#"trutio# o( th! pro&utio# po""ibi,ity
ur'! (orx%#& y"t%rt" .ith th!
%""u*ptio# th%t th! %*ou#t" o( k%#& l
%r! (i4!&
- A# E&g!.orth bo4 "ho." !'!ry po""ib,!
.%y th! !4i"ti#g k%#& l*ight b! u"!& to
pro&u!x%#& y1 %#y poi#t i# th! bo4 r!pr!"!#t" % (u,,y
!*p,oy!& %,,o%tio# o( th! %'%i,%b,!
r!"our!" tox%#& y
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E!eworth "o# $ia!ram
Ox
Oy
%otal Labor
%
otalCapital
A
Capital
forx
Capitalfory
Labor for yLabor forx Capital
in y
prouction
Capital
inx
prouction
Labor in yprouction
Labor inxprouction
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E!eworth "o# $ia!ram- M%#y o( th! %,,o%tio#" i# th! E&g!.orth
bo4 %r! t!h#i%,,y i#!((ii!#t
1 it i" po""ib,! to pro&u! *or!x%#& *or! yby"hi(ti#g %pit%, %#& ,%bor %rou#&
- W! .i,, %""u*! th%t o*p!titi'! *%r/!t"
.i,, #ot !4hibit i#!((ii!#t i#put hoi!"- W! .%#t to (i#& th! !((ii!#t %,,o%tio#"
1 th!y i,,u"tr%t! th! %tu%, pro&utio# outo*!"
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E!eworth "o# $ia!ram- W! .i,, u"! i"ou%#t *%p" (or th! t.o
goo&"
1 th! i"ou%#t *%p (or goo&xu"!" Ox%" th!origi#
1 th! i"ou%#t *%p (or goo& yu"!" Oy%" th!
origi#- )h! !((ii!#t %,,o%tio#" .i,, our .h!r!
th! i"ou%#t" %r! t%#g!#t to o#! %#oth!r
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E!eworth "o# $ia!ram
Ox
Oy
%otal Labor
%
otalCapital
x2
x1
y1
y2
A
oi#tAi" i#!((ii!#t b!%u"!$ by *o'i#g %,o#g y6$ .! %# i#r!%"!
x(ro*x6tox2.hi,! ho,&i#g yo#"t%#t
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E!eworth "o# $ia!ram
Ox
Oy
%otal Labor
%
otalCapital
x2
x1
y1
y2
A
W! ou,& %,"o i#r!%"! y(ro* y6to y2.hi,! ho,&i#gxo#"t%#t
by *o'i#g %,o#gx6
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E!eworth "o# $ia!ram
Ox
Oy
%otal Labor
%
otalCapital
At !%h !((ii!#t poi#t$ th! RTS7o( k(or l8 i" !u%, i# both
x%#& ypro&utio#
x2x1
x&
x'
y1
y2
y'
y&
p&
p'
p2
p1
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Prouction Possibility (rontier- )h! ,ou" o( !((ii!#t poi#t" "ho." th!
*%4i*u* output o( yth%t %# b!
pro&u!& (or %#y ,!'!, o(x1 .! %# u"! thi" i#(or*%tio# to o#"trut %
pro&utio# po""ibi,ity (ro#ti!r
- "ho." th! %,t!r#%ti'! output" o(x%#& yth%t
%# b! pro&u!& .ith th! (i4!& %pit%, %#&
,%bor i#put" th%t %r! !*p,oy!& !((ii!#t,y
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Prouction Possibility (rontier
Qu%#tity o(x
Qu%#tity o( y
p&
p'
p2
p1
y1
y2
y'
y&
x1 x2 x' x&
Ox
Oy
E%h !((ii!#t poi#t o( pro&utio#
b!o*!" % poi#t o# th! pro&utio#
po""ibi,ity (ro#ti!r
)h! #!g%ti'! o( th! ",op! o(
th! pro&utio# po""ibi,ity
(ro#ti!r i" th! r%t! o( pro&ut
tr%#"(or*%tio# 7RPT8
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)ate of Prouct %ransformation- )h! r%t! o( pro&ut tr%#"(or*%tio# 7RPT8
b!t.!!# t.o output" i" th! #!g%ti'! o(
th! ",op! o( th! pro&utio# po""ibi,ity(ro#ti!r
(ro#ti!rypo""ibi,it
pro&utio#o(",op!8(or7o( =yxRPT
87%,o#g8(or7o( yxOOdx
dyyxRPT =
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)ate of Prouct %ransformation- )h! r%t! o( pro&ut tr%#"(or*%tio# "ho."
ho.x%# b! t!h#i%,,y tr%&!& (or y
.hi,! o#ti#ui#g to /!!p th! %'%i,%b,!pro&uti'! i#put" !((ii!#t,y !*p,oy!&
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Shape of the Prouction
Possibility (rontier
- )h! pro&utio# po""ibi,ity (ro#ti!r "ho.#
!%r,i!r !4hibit!& %# i#r!%"i#g RPT1 thi" o#%'! "h%p! .i,, h%r%t!ri3! *o"t
pro&utio# "itu%tio#"
-RPTi" !u%, to th! r%tio o( MC
xto MC
y
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Shape of the Prouction
Possibility (rontier- Suppo"! th%t th! o"t" o( %#y output
o*bi#%tio# %r! C7x$y8
1 %,o#g th! pro&utio# po""ibi,ity (ro#ti!r$
C7x$y8 i" o#"t%#t
- W! %# .rit! th! tot%, &i((!r!#ti%, o( th!
o"t (u#tio# %"
0=
+
= dy
y
Cdx
x
CdC
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Shape of the Prouction
Possibility (rontier- R!.riti#g$ .! g!t
y
xyx
MCMC
yCxCOO
dxdyRPT =
==
9987%,o#g
- )h! RPTi" % *!%"ur! o( th! r!,%ti'!
*%rgi#%, o"t" o( th! t.o goo&"
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Shape of the Prouction
Possibility (rontier- A" pro&utio# o(xri"!" %#& pro&utio#
o( y(%,,"$ th! r%tio o( MCxto MC
yri"!"
1 thi" our" i( both goo&" %r! pro&u!&
u#&!r &i*i#i"hi#g r!tur#"
- i#r!%"i#g th! pro&utio# o(xr%i"!" MCx$ .hi,!
r!&ui#g th! pro&utio# o( y,o.!r" MCy
1 thi" ou,& %,"o our i( "o*! i#put" .!r!
*or! "uit!& (orxpro&utio# th%# (or y
pro&utio#
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Shape of the Prouction
Possibility (rontier- But .! h%'! %""u*!& th%t i#put" %r!
ho*og!#!ou"
- W! #!!& %# !4p,%#%tio# th%t %,,o."
ho*og!#!ou" i#put" %#& o#"t%#t
r!tur#" to "%,!
- )h! pro&utio# po""ibi,ity (ro#ti!r .i,, b!
o#%'! i( goo&"x%#& yu"! i#put" i#
&i((!r!#t proportio#"
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Opportunity Cost
- )h! pro&utio# po""ibi,ity (ro#ti!r
&!*o#"tr%t!" th%t th!r! %r! *%#y
po""ib,! !((ii!#t o*bi#%tio#" o( t.ogoo&"
- ro&ui#g *or! o( o#! goo&
#!!""it%t!" ,o.!ri#g th! pro&utio# o(th! oth!r goo&
1 thi" i" .h%t !o#o*i"t" *!%# by opportu#ity
o"t
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Opportunity Cost
- )h! opportu#ity o"t o( o#! *or! u#it o(
xi" th! r!&utio# i# yth%t thi" !#t%i,"
- )hu"$ th! opportu#ity o"t i" b!"t*!%"ur!& %" th! RPT7o(x(or y8 %t th!
pr!'%i,i#g poi#t o# th! pro&utio#
po""ibi,ity (ro#ti!r1 thi" opportu#ity o"t ri"!" %" *or!xi"
pro&u!&
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Concavity of the Prouction
Possibility (rontier- Suppo"! th%t th! pro&utio# o(x%#& y
&!p!#&" o#,y o# ,%bor %#& th! pro&utio#
(u#tio#" %r!5+087 xxfx ll ==
5+087 yyfy ll ==
-I( ,%bor "upp,y i" (i4!& %t 600$ th!#
lx: ly; 600
- )h! pro&utio# po""ibi,ity (ro#ti!r i"
x2: y2; 600 (orx$y0
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Concavity of the Prouction
Possibility (rontier- )h! RPT%# b! %,u,%t!& by t%/i#g th!
tot%, &i((!r!#ti%,
y
x
p
p= ",op!
C
C
)h! pri! o(x.i,, ri"! %#&th! pri! o( y.i,, (%,,
x1*
y1*
)h!r! i" !4!"" &!*%#& (orx%#&
!4!"" "upp,y o( y
e#cess
supply
e#cess eman
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$etermination of
Equilibrium Prices
Qu%#tity o(x
Qu%#tity o( y
y1
x1
U6
U2
U>
y
x
p
p= ",op!
C
C
x1*
y1*
)h! !ui,ibriu* output .i,,
b!x6@ %#& y6@y1+
x1+
)h! !ui,ibriu* pri!" .i,,
b!px@ %#&py@
C+
C+
@@",op!
y
x
pp=
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Comparative Statics Analysis
- )h! !ui,ibriu* pri! r%tio .i,, t!#& to
p!r"i"t u#ti, !ith!r pr!(!r!#!" or
pro&utio# t!h#o,ogi!" h%#g!- I( pr!(!r!#!" .!r! to "hi(t to.%r& goo&
x$px9py.ou,& ri"! %#& *or!x%#& ,!""
y.ou,& b! pro&u!&1 .! .ou,& *o'! i# % ,o/.i"! &ir!tio#
%,o#g th! pro&utio# po""ibi,ity (ro#ti!r
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Comparative Statics Analysis
- )!h#i%, progr!"" i# th! pro&utio# o(
goo&x.i,, "hi(t th! pro&utio#
po""ibi,ity ur'! out.%r&1 thi" .i,, ,o.!r th! r!,%ti'! pri! o(x
1 *or!x.i,, b! o#"u*!&
-i(xi" % #or*%, goo&
1 th! !((!t o# yi" %*biguou"
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%echnical Pro!ress in the
Prouction of #
Qu%#tity o(x
Qu%#tity o( y
U6
U2
U>
x1+
)h! r!,%ti'! pri! o(x.i,, (%,,
Mor!x.i,, b! o#"u*!&
x2+
)!h#i%, progr!"" i# th! pro&utio#
o(x.i,, "hi(t th! pro&utio# po""ibi,ity
ur'! out
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General Equilibrium Pricin!- Suppo"! th%t th! pro&utio# po""ibi,ity
(ro#ti!r %# b! r!pr!"!#t!& by
x2: y2; 600
- Suppo"! %,"o th%t th! o**u#ity="
pr!(!r!#!" %# b! r!pr!"!#t!& by
U7x$y8 ;x0+5y0+5
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General Equilibrium Pricin!- ro(it*%4i*i3i#g (ir*" .i,, !u%t! RPT%#& th! r%tio o(px9py
y
x
pp
yxRPT ==
- Uti,ity *%4i*i3%tio# r!uir!" th%t
y
x
p
p
x
yMRS ==
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General Equilibrium Pricin!- Eui,ibriu* r!uir!" th%t (ir*" %#&i#&i'i&u%," (%! th! "%*! pri! r%tio
MRSxy
pp
yxRPT
y
x ====
or
x@ ; y@
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%he Corn Laws $ebate- igh t%ri((" o# gr%i# i*port" .!r!
i*po"!& by th! Briti"h go'!r#*!#t %(t!r
th! N%po,!o#i .%r"
- Eo#o*i"t" &!b%t!& th! !((!t" o( th!"!
or# ,%." b!t.!!# 62 %#& 65
1 .h%t !((!t .ou,& th! !,i*i#%tio# o( th!"!
t%ri((" h%'! o# (%tor pri!"
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%he Corn Laws $ebate
Qu%#tity o( Gr%i# 7x8
Qu%#tity o(*%#u(%tur!&
goo&" 7y8
U6
U2
x,
I( th! or# ,%." o*p,!t!,y pr!'!#t!&
tr%&!$ output .ou,& b!x0%#& y0
y,
)h! !ui,ibriu* pri!" .i,, b!
px@ %#&py@
@
@",op!
y
x
p
p=
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%he Corn Laws $ebate
Qu%#tity o( Gr%i# 7x8
Qu%#tity o(*%#u(%tur!&
goo&" 7y8
x,
U6
U2
y,
R!*o'%, o( th! or# ,%." .i,, h%#g!
th! pri!" topx= %#&py=
H
H",op!
y
x
p
p=
?utput .i,, b!x6= %#& y6=
x1*
y1*
y1
x1
I#&i'i&u%," .i,, &!*%#&x6%#& y6
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%he Corn Laws $ebate
Qu%#tity o( Gr%i# 7x8
Qu%#tity o(*%#u(%tur!&
goo&" 7y8
y1
x,
U6
U2
x1
y,
H
H",op!
y
x
p
p=
x1*
y1*
Gr%i# i*port" .i,, b!x61x6=
imports of !rain
)h!"! i*port" .i,, b! (i#%#!& by
th! !4port o( *%#u(%tur!& goo&"
!u%, to y6= 1 y6e#portsof
!oos
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%he Corn Laws $ebate
- W! %# u"! %# E&g!.orth bo4 &i%gr%*
to "!! th! !((!t" o( t%ri(( r!&utio# o#
th! u"! o( ,%bor %#& %pit%,- I( th! or# ,%." .!r! r!p!%,!&$ th!r!
.ou,& b! %# i#r!%"! i# th! pro&utio#
o( *%#u(%tur!& goo&" %#& % &!,i#! i#th! pro&utio# o( gr%i#
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%he Corn Laws $ebate
Ox
Oy
%otal Labor
%otalCapital
A r!p!%, o( th! or# ,%." .ou,& r!"u,t i# % *o'!*!#t (ro*p>to
p6.h!r! *or! y%#& ,!""xi" pro&u!&
x2x1
x&
x'
y1
y2
y'
y&
p&
p'
p2
p1
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%he Corn Laws $ebate
- I( .! %""u*! th%t gr%i# pro&utio# i"
r!,%ti'!,y %pit%, i#t!#"i'!$ th!
*o'!*!#t (ro*p>top6%u"!" th! r%tioo( kto lto ri"! i# both i#&u"tri!"
1 th! r!,%ti'! pri! o( %pit%, .i,, (%,,
1 th! r!,%ti'! pri! o( ,%bor .i,, ri"!- )h! r!p!%, o( th! or# ,%." .i,, b!
h%r*(u, to %pit%, o.#!r" %#& h!,p(u, to
,%bor!r"
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Political Support for
%rae Policies- )r%&! po,ii!" *%y %((!t th! r!,%ti'!
i#o*!" o( '%riou" (%tor" o( pro&utio#
- I# th! U#it!& St%t!"$ !4port" t!#& to b!
i#t!#"i'! i# th!ir u"! o( "/i,,!& ,%bor
.h!r!%" i*port" t!#& to b! i#t!#"i'! i#
th!ir u"! o( u#"/i,,!& ,%bor1 (r!! tr%&! po,ii!" .i,, r!"u,t i# ri"i#g r!,%ti'!
.%g!" (or "/i,,!& .or/!r" %#& i# (%,,i#g
r!,%ti'! .%g!" (or u#"/i,,!& .or/!r"
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E#istence of General
Equilibrium Prices- B!gi##i#g .ith 6th !#tury i#'!"tig%tio#"
by L!o# W%,r%"$ !o#o*i"t" h%'!
!4%*i#!& .h!th!r th!r! !4i"t" % "!t o(pri!" th%t !ui,ibr%t!" %,, *%r/!t"
"i*u,t%#!ou",y
1 i( thi" "!t o( pri!" !4i"t"$ ho. %# it b!(ou#&
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E#istence of General
Equilibrium Prices- Suppo"! th%t th!r! %r! ngoo&" i# (i4!&
"upp,y i# thi" !o#o*y
1 ,!t Si7i;6$$n8 b! th! tot%, "upp,y o( goo& i
%'%i,%b,!
1 ,!tpi7i;6$n8 b! th! pri! o( goo& i
- )h! tot%, &!*%#& (or goo& i&!p!#&" o#%,, pri!"
Di 7p6$$pn8 (or i;6$$n
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E#istence of General
Equilibrium Prices- W! .i,, .rit! thi" &!*%#& (u#tio# %"
&!p!#&!#t o# th! .ho,! "!t o( pri!" 7P8
Di 7P8
- W%,r%"= prob,!*< Do!" th!r! !4i"t %#
!ui,ibriu* "!t o( pri!" "uh th%t
Di 7P*8 ; Si
(or %,, '%,u!" o( i
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47
E#cess $eman (unctions
- )h! !4!"" &!*%#& (u#tio# (or %#y
goo& i%t %#y "!t o( pri!" 7P8 i" &!(i#!&
to b!EDi 7P8 ; Di 7P8 1 Si
- )hi" *!%#" th%t th! !ui,ibriu*
o#&itio# %# b! r!.ritt!# %"
EDi 7P@8 ; Di 7P@8 1 Si; 0
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E#cess $eman (unctions
- D!*%#& (u#tio#" %r! ho*og!#!ou" o(
&!gr!! 3!ro
1 thi" i*p,i!" th%t .! %# o#,y !"t%b,i"h!ui,ibriu* r!,%ti'! pri!" i# % W%,r%"i%#
typ! *o&!,
- W%,r%" %,"o %""u*!& th%t &!*%#&
(u#tio#" %r! o#ti#uou"
1 "*%,, h%#g!" i# pri! ,!%& to "*%,, h%#g!"
i# u%#tity &!*%#&!&
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-alras* Law
- A (i#%, ob"!r'%tio# th%t W%,r%" *%&!
.%" th%t th! n!4!"" &!*%#& !u%tio#"
%r! #ot i#&!p!#&!#t o( o#! %#oth!r- W%,r%"= ,%. "ho." th%t th! tot%, '%,u! o(
!4!"" &!*%#& i" 3!ro %t %#y "!t o(
pri!"
=
=n
i
ii PEDP6
087
-
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50
-alras* Law
- W%,r%"= ,%. ho,&" (or %#y "!t o( pri!"
7#ot Ju"t !ui,ibriu* pri!"8
- )h!r! %# b! #!ith!r !4!"" &!*%#& (or%,, goo&" tog!th!r #or !4!"" "upp,y
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-alras* Proof of the E#istence
of Equilibrium Prices- )h! *%r/!t !ui,ibriu* o#&itio#"
pro'i&! 7n68 i#&!p!#&!#t !u%tio#" i#
7n68 u#/#o.# r!,%ti'! pri!"1 %# .! "o,'! th! "y"t!* (or %# !ui,ibriu*
o#&itio#
- th! !u%tio#" %r! #ot #!!""%ri,y ,i#!%r- %,, pri!" *u"t b! #o##!g%ti'!
- )o %tt%/ th!"! &i((iu,ti!"$ W%,r%" "!t up
% o*p,i%t!& proo(
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-alras* Proof of the E#istence
of Equilibrium Prices- St%rt .ith %# %rbitr%ry "!t o( pri!"
- o,&i#g th! oth!r n6 pri!" o#"t%#t$
(i#& th! !ui,ibriu* pri! (or goo& 6 7p6=8
- o,&i#gp6= %#& th! oth!r n2 pri!"
o#"t%#t$ "o,'! (or th! !ui,ibriu* pri!
o( goo& 2 7p2=8
1 i# h%#gi#gp2(ro* it" i#iti%, po"itio# top2=$
th! pri! %,u,%t!& (or goo& 6 &o!" #ot
#!!& to r!*%i# %# !ui,ibriu* pri!
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-alras* Proof of the E#istence
of Equilibrium Prices- U"i#g th! pro'i"io#%, pri!"p6= %#&p2=$
"o,'! (orp>=
1 pro!!& i# thi" .%y u#ti, %# !#tir! "!t o(
pro'i"io#%, r!,%ti'! pri!" h%" b!!# (ou#&
- I# th! 2#&it!r%tio# o( W%,r%"= proo($p2=$
$pn= %r! h!,& o#"t%#t .hi,! % #!.
!ui,ibriu* pri! i" %,u,%t!& (or goo& 6
1 pro!!& i# thi" .%y u#ti, %# !#tir! #!. "!t o(
pri!" i" (ou#&
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-alras* Proof of the E#istence
of Equilibrium Prices- )h! i*port%#! o( W%,r%"= proo( i" it"
%bi,ity to &!*o#"tr%t! th! "i*u,t%#!ou"
#%tur! o( th! prob,!* o( (i#&i#g!ui,ibriu* pri!"
- B!%u"! it i" u*b!r"o*!$ it i" #ot
g!#!r%,,y u"!& to&%y- Mor! r!!#t .or/ u"!" "o*! r!,%ti'!,y
"i*p,! too," (ro* %&'%#!& *%th!*%ti"
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"rouwer*s (i#e.Point %heorem
- A#y o#ti#uou" *%ppi#g KF7X8 o( %
,o"!&$ bou#&!&$ o#'!4 "!t i#to it"!,(
h%" %t ,!%"t o#! (i4!& poi#t 7X@8 "uh
th%t F7X@8 ;X@
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"rouwer*s (i#e.Point %heorem
x
f7X8
1
1
,
&/
A#y o#ti#uou" (u#tio# *u"tro"" th! 5,i#!
Suppo"! th%t f7X8 i" % o#ti#uou" (u#tio# &!(i#!&
o# th! i#t!r'%, K0$6 %#& th%t f7X8 t%/!" o# th!
'%,u!" %,"o o# th! i#t!r'%, K0$6
)hi" poi#t o( ro""i#g i" %
(i4!& poi#t b!%u"! f*%p"
thi" poi#t 7X@8 i#to it"!,(
X+
f0X+
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57
"rouwer*s (i#e.Point %heorem
- A *%ppi#g i" % ru,! th%t %""oi%t!" th!
poi#t" i# o#! "!t .ith poi#t" i# %#oth!r "!t
1 ,!tXb! % poi#t (or .hih % *%ppi#g 7F8 i"&!(i#!&
- th! *%ppi#g %""oi%t!"X.ith "o*! poi#t Y ; F7X8
1 i( % *%ppi#g i" &!(i#!& o'!r % "ub"!t o( n
&i*!#"io#%, "p%! 7S8$ %#& i( !'!ry poi#t i# Si" %""oi%t!& 7by th! ru,! F8 .ith "o*! oth!r
poi#t i# S$ th! *%ppi#g i" "%i& to *%p Si#to
it"!,(
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58
"rouwer*s (i#e.Point %heorem
- A *%ppi#g i" o#ti#uou" i( poi#t" th%t %r!
,o"! to !%h oth!r %r! *%pp!& i#to
oth!r poi#t" th%t %r! ,o"! to !%h oth!r- )h! Brou.!r (i4!&poi#t th!or!* o#"i&!r"
*%ppi#g" &!(i#!& o# !rt%i# /i#&" o( "!t"
1,o"!& 7th!y o#t%i# th!ir bou#&%ri!"81 bou#&!& 7#o#! o( th!ir &i*!#"io#" i" i#(i#it!,y
,%rg!8
1 o#'!4 7th!y h%'! #o ho,!" i# th!*8
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59
Proof of the E#istence of
Equilibrium Prices- B!%u"! o#,y r!,%ti'! pri!" *%tt!r$ it i"
o#'!#i!#t to %""u*! th%t pri!" h%'!
b!!# &!(i#!& "o th%t th! "u* o( %,, pri!"i" !u%, to 6
- )hu"$ (or %#y %rbitr%ry "!t o( pri!" 7p6$
$pn8$ .! %# u"! #or*%,i3!& pri!" o(th! (or*
=
=n
i
i
ii
p
pp
6
H
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60
Proof of the E#istence of
Equilibrium Prices- )h!"! #!. pri!" .i,, r!t%i# th!ir origi#%,
r!,%ti'! '%,u!" %#& .i,, "u* to 6
6H6
==
n
i
ip
j
i
j
i
p
p
p
p=
H
H
- )h!"! #!. pri!" .i,, "u* to 6
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61
Proof of the E#istence of
Equilibrium Prices- W! .i,, %""u*! th%t th! (!%"ib,! "!t o(
pri!" 7S8 i" o*po"!& o( %,,
#o##!g%ti'! #u*b!r" th%t "u* to 61 Si" th! "!t to .hih .! .i,, %pp,y Brou.!r="
th!or!*
1 Si" ,o"!&$ bou#&!&$ %#& o#'!41 .! .i,, #!!& to &!(i#! % o#ti#uou" *%ppi#g
o( Si#to it"!,(
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(ree Goos
- Eui,ibriu* &o!" #ot r!%,,y r!uir! th%t
!4!"" &!*%#& b! 3!ro (or !'!ry *%r/!t
- Goo&" *%y !4i"t (or .hih th! *%r/!t"%r! i# !ui,ibriu* .h!r! "upp,y !4!!&"
&!*%#& 7#!g%ti'! !4!"" &!*%#&8
1 it i" #!!""%ry (or th! pri!" o( th!"! goo&"
to b! !u%, to 3!ro
1 (r!! goo&"
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(ree Goos
- )h! !ui,ibriu* o#&itio#" %r!
EDi 7P@8 ; 0 (orpi@ 0
EDi 7P@8 0 (orpi@ ; 0
- Not! th%t thi" "!t o( !ui,ibriu* pri!"
o#ti#u!" to ob!y W%,r%"= ,%.
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64
appin! the Set of Prices
3nto 3tself- I# or&!r to %hi!'! !ui,ibriu*$ pri!" o(
goo&" i# !4!"" &!*%#& "hou,& b!
r%i"!&$ .h!r!%" tho"! i# !4!"" "upp,y
"hou,& h%'! th!ir pri!" ,o.!r!&
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appin! the Set of Prices
3nto 3tself- W! &!(i#! th! *%ppi#g F7P8 (or %#y
#or*%,i3!& "!t o( pri!" 7P8$ "uh th%t
th! ith o*po#!#t o( F7P8 i" gi'!# by
Fi7P8 ;pi: EDi 7P8
- )h! *%ppi#g p!r(or*" th! #!!""%ryt%"/ o( %ppropri%t!,y r%i"i#g or ,o.!ri#g
pri!"
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appin! the Set of Prices
3nto 3tself- ).o prob,!*" !4i"t .ith thi" *%ppi#g
- Fir"t$ #othi#g !#"ur!" th%t th! pri!" .i,,
b! #o##!g%ti'!1 th! *%ppi#g *u"t b! r!&!(i#!& to b!
Fi7P8 ; M%4 Kpi
+ EDi
7P8$0
1 th! #!. pri!" &!(i#!& by th! *%ppi#g *u"t
b! po"iti'! or 3!ro
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67
appin! the Set of Prices
3nto 3tself- S!o#&$ th! r!%,u,%t!& pri!" %r! #ot
#!!""%ri,y #or*%,i3!&
1 th!y .i,, #ot "u* to 6
1 it .i,, b! "i*p,! to #or*%,i3! "uh th%t
==
n
i
i PF6
687
1 .! .i,, %""u*! th%t thi" #or*%,i3%tio# h%"
b!!# &o#!
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68
Application of "rouwer*s
%heorem- )hu"$ F"%ti"(i!" th! o#&itio#" o( th!
Brou.!r (i4!&poi#t th!or!*
1 it i" % o#ti#uou" *%ppi#g o( th! "!t Si#to
it"!,(
- )h!r! !4i"t" % poi#t 7P@8 th%t i" *%pp!&
i#to it"!,(- For thi" poi#t$
pi@ ; M%4 Kpi@ : EDi 7P@8$0 (or %,, i
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69
Application of "rouwer*s
%heorem- )hi" "%y" th%t P@ i" %# !ui,ibriu* "!t o(
pri!"
1 (orpi@ 0$pi@ ;pi@ : EDi 7P@8
EDi 7P@8 ; 0
1 Forpi@ ; 0$pi@ : EDi 7P@8 0
EDi 7P@8 0
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70
A General Equilibrium with
%hree Goos- )h! !o#o*y o( ?3 i" o*po"!& o#,y o(
thr!! pr!iou" *!t%,"< 768 "i,'!r$ 728
go,&$ %#& 7>8 p,%ti#u*1 th!r! %r! 60 7thou"%#&8 ou#!" o( !%h
*!t%, %'%i,%b,!
- )h! &!*%#&" (or go,& %#& p,%ti#u* %r!
6626
>
6
22 ++=
p
p
p
pD 6D2
6
>
6
2> +=
p
p
p
pD
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71
A General Equilibrium with
%hree Goos- Eui,ibriu* i# th! go,& %#& p,%ti#u*
*%r/!t" r!uir!" th%t &!*%#& !u%,
"upp,y i# both *%r/!t" "i*u,t%#!ou",y
606626
>
6
2 =++p
p
p
p
606D26
>
6
2 =+p
p
p
p
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72
A General Equilibrium with
%hree Goos- )hi" "y"t!* o( "i*u,t%#!ou" !u%tio#"
%# b! "o,'!& %"
p29p6; 2 p>9p6; >
- I# !ui,ibriu*ED>
- Sub"tituti#g th! !4!"" &!*%#& (u#tio#"
(or go,& %#& "i,'!r %#& "ub"tituti#g$ .! g!t
>
6
2
>
6
>22
6
>2
6
2
266 D22 pp
p
p
pppp
pp
p
pEDp ++=
6
>
6
2
2
6
2>
2
6
22
6 D22
p
p
p
p
p
p
p
pED +=
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74
Smith*s 3nvisible 4an
4ypothesis- A&%* S*ith b!,i!'!& th%t th!
o*p!titi'! *%r/!t "y"t!* pro'i&!& %
po.!r(u, i#'i"ib,! h%#& th%t !#"ur!&r!"our!" .ou,& (i#& th!ir .%y to .h!r!
th!y .!r! *o"t '%,u!&
- R!,i%#! o# th! !o#o*i "!,(i#t!r!"to( i#&i'i&u%," %#& (ir*" .ou,& r!"u,t i# %
&!"ir%b,! "oi%, outo*!
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75
Smith*s 3nvisible 4an
4ypothesis- S*ith=" i#"ight" g%'! ri"! to *o&!r#
.!,(%r! !o#o*i"
- )h! Fir"t )h!or!* o( W!,(%r!
Eo#o*i" "ugg!"t" th%t th!r! i" %#
!4%t orr!"po#&!#! b!t.!!# th!
!((ii!#t %,,o%tio# o( r!"our!" %#& th!o*p!titi'! prii#g o( th!"! r!"our!"
-
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76
Pareto Efficiency
- A# %,,o%tio# o( r!"our!" i" %r!to
!((ii!#t i( it i" #ot po""ib,! 7through
(urth!r r!%,,o%tio#"8 to *%/! o#! p!r"o#b!tt!r o(( .ithout *%/i#g "o*!o#! !,"!
.or"! o((
-)h! %r!to &!(i#itio# i&!#ti(i!" %,,o%tio#"%" b!i#g i#!((ii!#t i( u#%*biguou"
i*pro'!*!#t" %r! po""ib,!
-
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Efficiency in Prouction
- A# %,,o%tio# o( r!"our!" i" !((ii!#t i#
pro&utio# 7or t!h#i%,,y !((ii!#t8 i( #o
(urth!r r!%,,o%tio# .ou,& p!r*it *or! o(
o#! goo& to b! pro&u!& .ithout
#!!""%ri,y r!&ui#g th! output o( "o*!
oth!r goo&
- )!h#i%, !((ii!#y i" % pr!o#&itio# (or
%r!to !((ii!#y but &o!" #ot gu%r%#t!!
%r!to !((ii!#y
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78
Efficient Choice of 3nputs for a
Sin!le (irm- A "i#g,! (ir* .ith (i4!& i#put" o( ,%bor
%#& %pit%, .i,, h%'! %,,o%t!& th!"!
r!"our!" !((ii!#t,y i( th!y %r! (u,,y!*p,oy!& %#& i( th! RTSb!t.!!#
%pit%, %#& ,%bor i" th! "%*! (or !'!ry
output th! (ir* pro&u!"
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79
Efficient Choice of 3nputs for a
Sin!le (irm- A""u*! th%t th! (ir* pro&u!" t.o
goo&" 7x%#& y8 %#& th%t th! %'%i,%b,!
,!'!," o( %pit%, %#& ,%bor %r! k%#& l- )h! pro&utio# (u#tio# (orxi" gi'!# by
x; f7kx$lx8
- I( .! %""u*! (u,, !*p,oy*!#t$ th!
pro&utio# (u#tio# (or yi"
y; g7ky$ly8 ; g7kkx$l
lx8
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80
Efficient Choice of 3nputs for a
Sin!le (irm- )!h#i%, !((ii!#y r!uir!" th%tx
output b! %" ,%rg! %" po""ib,! (or %#y
'%,u! o( y7y8- S!tti#g up th! L%gr%#gi%# %#& "o,'i#g (or
th! (ir"tor&!r o#&itio#"
U#2
U#6
US
US>
US2
US6
-
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118
$istribution
- A#y poi#t .ithi# th! E&g!.orth bo4 i#
.hih th! MRS(or S*ith i" u#!u%, to
th%t (or o#!" o((!r" %# opportu#ity (or
%r!to i*pro'!*!#t"
1 both %# *o'! to high!r ,!'!," o( uti,ity
through tr%&!
-
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119
$istributionOJ
OS
U#
U#>
U#2
U#6
US
US>
US2
US6
A
Any trae in this area is
an improvement overA
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120
Contract Curve
- I# %# !4h%#g! !o#o*y$ %,, !((ii!#t
%,,o%tio#" ,i! %,o#g % o#tr%t ur'!
1 poi#t" o(( th! ur'! %r! #!!""%ri,y
i#!((ii!#t- i#&i'i&u%," %# b! *%&! b!tt!r o(( by *o'i#g to
th! ur'!
- A,o#g th! o#tr%t ur'!$ i#&i'i&u%,"=pr!(!r!#!" %r! ri'%,"
1 o#! *%y b! *%&! b!tt!r o(( o#,y by *%/i#g
th! oth!r .or"! o((
-
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121
Contract CurveOJ
OS
U#
U#>
U#2
U#6
US
US>
US2
US6
A
Contract curve
E#chan!e with 3nitial
-
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122
E#chan!e with 3nitial
Enowments- Suppo"! th%t th! t.o i#&i'i&u%,"
po""!"" &i((!r!#t u%#titi!" o( th! t.o
goo&" %t th! "t%rt1 it i" po""ib,! th%t th! t.o i#&i'i&u%," ou,&
both b!#!(it (ro* tr%&! i( th! i#iti%,
%,,o%tio#" .!r! i#!((ii!#t
E#chan!e with 3nitial
-
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123
E#chan!e with 3nitial
Enowments- N!ith!r p!r"o# .ou,& !#g%g! i# % tr%&!
th%t .ou,& ,!%'! hi* .or"! o((
- ?#,y % portio# o( th! o#tr%t ur'!"ho." %,,o%tio#" th%t *%y r!"u,t (ro*
'o,u#t%ry !4h%#g!
E#chan!e with 3nitial
-
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124
E#chan!e with 3nitial
Enowments OJ
OS
U#A
USA
A
Suppo"! th%tAr!pr!"!#t"
th! i#iti%, !#&o.*!#t"
E#chan!e with 3nitial
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125
E#chan!e with 3nitial
Enowments OJ
OS
U#A
USA
A
N!ith!r i#&i'i&u%, .ou,& b!
.i,,i#g to %!pt % ,o.!r ,!'!,
o( uti,ity th%#A gi'!"
E#chan!e with 3nitial
-
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126
E#chan!e with 3nitial
Enowments OJ
OS
U#A
USA
A
?#,y %,,o%tio#" b!t.!!# M6
%#& M2.i,, b! %!pt%b,! to
both
M1
M2
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127
%he $istributional $ilemma
- I( th! i#iti%, !#&o.*!#t" %r! "/!.!& i#
(%'or o( "o*! !o#o*i %tor"$ th!
%r!to !((ii!#t %,,o%tio#" pro*i"!& by
th! o*p!titi'! pri! "y"t!* .i,, %,"o
t!#& to (%'or tho"! %tor"
1 'o,u#t%ry tr%#"%tio#" %##ot o'!ro*!
,%rg! &i((!r!#!" i# i#iti%, !#&o.*!#t"
1 "o*! "ort o( tr%#"(!r" .i,, b! #!!&!& to
%tt%i# *or! !u%, r!"u,t"
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128
%he $istributional $ilemma
- )h!"! thought" ,!%& to th! S!o#&
)h!or!* o( W!,(%r! Eo#o*i"
1 %#y &!"ir!& &i"tributio# o( .!,(%r! %*o#g
i#&i'i&u%," i# %# !o#o*y %# b! %hi!'!&
i# %# !((ii!#t *%##!r through o*p!titi'!
prii#g i( i#iti%, !#&o.*!#t" %r! %&Ju"t!&
%ppropri%t!,y
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129
3mportant Points to 6ote7
- r!(!r!#!" %#& pro&utio#
t!h#o,ogi!" pro'i&! th! bui,&i#g
b,o/" upo# .hih %,, g!#!r%,
!ui,ibriu* *o&!," %r! b%"!&
1 o#! p%rtiu,%r,y "i*p,! '!r"io# o( "uh %
*o&!, u"!" i#&i'i&u%, pr!(!r!#!" (or t.o
goo&" tog!th!r .ith % o#%'! pro&utio#po""ibi,ity (ro#ti!r (or tho"! t.o goo&"
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130
3mportant Points to 6ote7
- Co*p!titi'! *%r/!t" %# !"t%b,i"h
!ui,ibriu* pri!" by *%/i#g *%rgi#%,
%&Ju"t*!#t" i# pri!" i# r!"po#"! to
i#(or*%tio# %bout th! &!*%#& %#&
"upp,y (or i#&i'i&u%, goo&"
1 W%,r%"= ,%. ti!" *%r/!t" tog!th!r "o th%t
"uh % "o,utio# i" %""ur!& 7i# *o"t %"!"8
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131
3mportant Points to 6ote7
- Co*p!titi'! pri!" .i,, r!"u,t i# %
%r!to!((ii!#t %,,o%tio# o( r!"our!"
1 thi" i" th! Fir"t )h!or!* o( W!,(%r!
Eo#o*i"
3 P i 6
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132
3mportant Points to 6ote7
- F%tor" th%t .i,, i#t!r(!r! .ith
o*p!titi'! *%r/!t"= %bi,iti!" to
%hi!'! !((ii!#y i#,u&!
1 *%r/!t po.!r
1 !4t!r#%,iti!"
1 !4i"t!#! o( pub,i goo&"
1 i*p!r(!t i#(or*%tio#
3 t t P i t t 6 t
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3mportant Points to 6ote7
- Co*p!titi'! *%r/!t" #!!& #ot yi!,&
!uit%b,! &i"tributio#" o( r!"our!"$
!"p!i%,,y .h!# i#iti%, !#&o.*!#t" %r!
'!ry "/!.!&
1 i# th!ory %#y &!"ir!& &i"tributio# %# b!
%tt%i#!& through o*p!titi'! *%r/!t"
%o*p%#i!& by ,u*p"u* tr%#"(!r"- th!r! %r! *%#y pr%ti%, prob,!*" i#