Slides on general equilibrium mathematics in economics

8/12/2019 Slides on general equilibrium mathematics in economics

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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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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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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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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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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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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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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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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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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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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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- 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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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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- 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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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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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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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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48

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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-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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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!


53/133

53


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#&


54/133

54


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"


55/133

55

"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@


56/133

56


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+


57/133

57


- 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"!,(


58/133

58


- 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


59/133

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


60/133

60


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


61/133

61


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"!,(


62/133

62

(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&"


63/133

63

(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%"= ,%.


64/133

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!&


65/133

65


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!"


66/133

66


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


67/133

67


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#!


68/133

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


69/133

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


70/133

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


71/133

71


%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


72/133

72


%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 +=


74/133

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*!


75/133

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!"


76/133

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,!


77/133

77

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


78/133

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!"


79/133

79


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


80/133

80


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


118/133

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%&!


119/133

119

$istributionOJ

OS

U#

U#>

U#2

U#6

US

US>

US2

US6

A

Any trae in this area is

an improvement overA


120/133

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((


121/133

121

Contract CurveOJ

OS

U#

U#>

U#2

U#6

US

US>

US2

US6

A

Contract curve

E#chan!e with 3nitial


122/133

122


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



123/133

123


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!



124/133

124


Enowments OJ

OS

U#A

USA

A

Suppo"! th%tAr!pr!"!#t"

th! i#iti%, !#&o.*!#t"



125/133

125


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'!"



126/133

126


Enowments OJ

OS

U#A

USA

A

?#,y %,,o%tio#" b!t.!!# M6

%#& M2.i,, b! %!pt%b,! to

both

M1

M2


127/133

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"


128/133

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


129/133

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&"


130/133

130


- 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


131/133

131


- 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


132/133

132


- 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


133/133


- 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#

Slides on general equilibrium mathematics in economics

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