Lecture2

ur u r 2 2

lecture 2: General Equilibrium

Mauricio Romero

Lecture 2: Genera l Equilibrium

Cobb-Douglas

Cobb-Douglas

Fo.-graphsuppose

Cobb-Douglas

15

~}: 0

UA(x . y) = x" y 1-" us(x , y) = x13y1- /J

a = 0.7

fJ= 0.3

WA= (l , l)

W B = (1.1)

0 0.5 1 ~ X

04 Cobb-Douglas

~ Indifference curves must be tangent (formalize th is later)

~ Thus, the MRS must be equalized across the two consumers

MRS:Y = ~ = 1 : 0 x::;~~0 = 1 : 0 ~ t::'::I\ ~ f ,_ , , _, ,(71 0 =$=N~~

r~~~~~ J Cobb-Douglas

But we haven't used the fact that -------

Cobb-Douglas But we haven't used the fact that

1-o

Cobb-Douglas But we haven 't used the fact that

Cobb-Douglas But we haven't used the fact that

Cobb-Douglas But we haven't used the fact tha t

YA = XA. ~. _I!_ ( i;.,"y - YA) O 1- (:J w,, -xA

Y l +- -~ -~x _ A 1- o (3 w , ( 1-o p , ) o l -/3 Wx - xA - x · ----;-· N · w".:_"A

Cobb-Douglas But we haven't used the fact that

l:o~= l~(:J ::=:: YA = XA· ~. _I!_ (1.-'y -yA)

0 1 -{:J Wx - xA

rA(1 + ~ -_1!_ -~ ) - xA 1-o f3 w

Theo o t p"::~;:,~~1:::j p ~ Cobb-Douglas

~ ~0.5:::::::::=:±:l :___!.5~ 2 2 X

o( y" W v• /' R,/\ A A 4 " A A 4 A 'f.. -a

ua(xB,yB) 2'.: Y.s

XB +xA

Intuition

1S~ ocationwhere ~anwemakebothconsumersbetter off7

Intu ition

Suppose that we are at an alloca tion where MRS:1 = 2 > MRS!,1 = L Can we make both consumers better off7

~ A gives up 1 un it of y to person 8 in exchange for uni t of x

~ B is indifferent since his MRS!,1 = 1.

~ A receives a unit of x and only needs to give one unitofy (he was willing to give two)

~ WehavereallocatedgoodstomakeAstrictly better off without hurting 8

m" ((~,', ,x/J . .. (>

General case

Theorem Suppose that all utility functions are strictly increasing and quasi-concave. Suppose also that ((Xf , , Xf) , , (X/ , , XL)) is a feasibleinteri(){"al/ocation . Then ((Xf, .. Xf), .. . (X/, .. Xt)) is Pareto efficient ifandonlyif((Xf, .. , Xl). .. A 1, .. , Xl)) exhausts a/I resources and forallpairsofg, (,€' .

MRSf,,, (xl, ... Xf) = = MRSl,,, (xf, .. , xl)

~ Util ity functions must be strictly increasing, quasi-concave, anddiffe rentiablel

Lecture 2: Genera l Equil ibrium

Perfect substitutes

Perfect substitutes

Suppose that

~ l)A'. L;

~{c.1(lk : Jf

/

(' uA(xA ,yA) = 2xA+ yA

Gil2.:: :a-,•w---

Perfect substitutes

l0 r"0 15

~ l

0.5

W A = (1.1) w8 = (1, l)

o.s oi 0.5

0 11VI?

fl o 05 ! ~\oY (J-,o') Perfect substitutes

2 15 2

15

~ l

Perfect substitutes

2 1.5 0.5 0 2------~o

15 0.5

~ l

0.5 15

0 - 2 0 0.5 1 1.5 2

X

Perfect substitutes

0 0 0.5 1 15 2

X

Perfect substitutes

15

2

y '3=- 1./i3 - -/3 c-'!-=- u-s- -z,,-'K.~) vrs-f' = 11t

rN~~lO..S, 1./rJlvtv T .J\"tcJ.c>S,

< '- ror,< t-to VMz&to

I

l ,_

../

vs I \

" ----', Vi:s

' '

I~-

2 1.5 , ______ _. ~I> C.1ti.v~ C ~-,.,. illrtc) \ 0.5 15 .5

~ l

0.5

X

Lecture 2: Genera l Equil ibrium

Perfect complements

c r}

b

I

Perfect complements

Suppose that

uA(xA,yA) = min(xA,2yA) UB(x 8 , y 8 ) = min(2x 8 ,y8 )

:::o~ (A ~---_=i \by

,llb:v,1 ~J=-1.. ~ '1-,,.y t 11

1 0.5

0 I 2 "

·1 1 I ... ~ u;

0 '

·~ 0 ,

[2 0 '

VA, (\A+ YA Ur~:~'i.~~ 1~

"

c c)

Make A as well as we can without making B worse off

Make A as well as we can without making B worse off

Make A as well as we can without making B worse off

'ti ug : : : -------- -- -- ~ 0 '

Make A as well as we can without making 8 worse off

+~ 0 , Make A as well as we can without making 8 worse off

Make A as well as we can without making 8 worse off

E~ 0 '

' ~

0 '

'~

0 '

'~

0 '

,ar

to Try it at home !

Recap

to Weexpectallexchan (hence its name) ges to happen on the contract curve

to Weexpectall vo lunt ary exchangestobeintheorangebox

otw1thoutprices to Can we sa~ more? N .