10. general equilibrium i

7/25/2019 10. General Equilibrium I

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General Equilibrium IPure Exchange (no production)

Analyzing exchange b/w 2 people; Background material from consumer theory, optimization, tradingfrom endowment

Exchange

Consumers A B

!heir endowments of goods " 2#

o wA = (w1A,w2

A) wB = (w1B,w2

B)

ex. wA = (6,4) wB = (2,2)total $uantities a%ailable #

o w1A+ w1

B=6 + 2 = 8 units of good 1

o w2A+ w2

B= + 2 = 6units of good 2

Edge!orth box"&dgeworth and Bowley's diagram

o to show all possible allocationsof a%ailable $uantities of goods " 2 between 2

consumerso dimensions of the box ( $uantities a%ailable of the goods

height ( units of good 2; width ( units of good "

o Assumptions#

#. )ndi* cur%es con%ex and appropriate to origin2. )ndi* cur%es are smooth$. Both goods essential to consumer. +uantity of the 2 goods consumed are the only %ariables that a*ect economic well

being

%ea&ible 'llocation&

allocations of - units of good " and . units of good 2 are feasible?

ow are all feasible allocations depicted in the &dgeworth box diagram0

Endo!ment allocation# beforetrade allocation 1" feasible allocation

Endo!ment 'llocation

Generally:

ther %ea&ible 'llocation&

(x#'x2

') ( allocation to consumer A

(x#*x2*) ( allocation to consumer B

feasibleonly if

o x#'+ x#

* !#'+ !#

*

o x2'+ x2

* !2'+ !2

*

%ea&ible ,eallocation&

all pnts in box 1with boundary ( feasible allocations of combined endowments

which allocations make consumer better o*0

3hich allocations will be blocked by "/both consumers0

-

.

-idth = w1A+ w1

B=6 +

&ndowment

allocation is!' = (6) !* = (22)

eight=

w2A+w2

B=+ 2= 6


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'dding Pre/erence to the *ox rotated'0& add *

4ize of box 56! ( to 4um of &ndowments

edge!orth0& box (done right)

-el/are Impro1ement& Pareto

7utilitarian8 prescription is to 9nd an allocation that achie%es 7the greatest good for the greatest

number8

any change in allocation is a 7good8 thing if the gain in utility to someone exceeds the sacri9ce by

someone else

operationalcomparingutility across indi%idualso not possible utility ( ordinal1cant be measured

if a change makes e%eryone better of then it should be made

Pareto"isn't necessary to make e%eryone better o*, as long as someone is made better o* and

nobody made !or&eo*

3!hen indi4erence cur1e& are tangent at a point in Edge!orth box that point = Pareto5optima

Pareto5Impro1ement

Pareto5impro1ing allocation" Allocation endowment that impro%es welfare of a consumer w/o

reducing welfare of another

3here are the :aretoimpro%ing allocations0

A better o*, B no worse o* B better o*, A no worse o* Both Better 6*

:areto )mpro%ements

Consumer can re/u&eto trade only possible outcomes from exchange are Pareto-improving

allocations

-hich particular :aretoimpro%ing allocation will be the outcome of trade0


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Pareto ptimal 'llocation&

A change in a 7:areto )mpro%ement8 if the change makes nobody worse o*, and at least one person

better o4 Pareto Ecient7optimalif it is not possible to make any change that results in a :areto

)mpro%ement

:areto6ptimality black dot# optimality# allocation where con%ex

indi*erence cur%es are 7backtoback8 is :areto optimal

allocation is pareto5optimalsince only way one consumer'swelfare can be increasedis to decrease welfare of ther other con&umer

contract cur1e"set of all :aretooptimal allocations 1pass through pts of tangency

Pareto5ptimalit

which of the allocations on the contract cur%e will consumers trade0

o


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core"set of all :areto6ptimal allocations that are welfareimpro%ing for bothconsumers relati%e to

their own endowments rational tradeshould achie%e a core allocation

whichcore allocation0

o


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o! can Equilibrium be 'ttainedA

&xcess demand for commodity 2 p2 will rise

&xcess supply of commodity " p" will fall

4lope of budget constraint ( 5 p#7p2willpiot abo!t the endowment point and become less

steep

A's utility aximization at 5ew :rices B's utility aximization at 5ew :rices

9ol1ing /or Pareto5ptimal 'llocation '&&ociated !ith a 9peciBed Endo!ment

6nly relati%e prices matter#

o &ndowment of goods 1some , some D and prices are :x and :yo 4et of consumption opportunities 1budget set is the same as if prices were

"EE:x and "EE:y; 2E:x and 2E:y F:x and F:y , for any %e number F

multiplying prices by same %e constant lea%es the budget set unchanged

increase in price of what is boughtis compensated by an ( increase in the %alue of what is &old

P = #Px = Px7P = p price 1ariable i& the relati1e price o/ good C

9ol1ing" ;he 9tep&

%ir&t"9nd expressions for +< by each person as a function of endowment amounts and p'

9econd" impose condition of exactly E excess demand in market or D 1if " market clears G E excess

demand G so will the othero 4ol%e that 7E excess demand8 e$uation for p'

at gi%en prices p" p2 #o excess supply of commodity "

o excess demand for commodity 2

neither markets clear at prices p" p" ( no general equilibrium


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;hird# substitute solution %alue of p' into demand functions compute $uantities of each good that

each consumer demands

&xample# ick endowed with HE units , I units D Jeith endowed with 2E units , 2E units D Ktility Lunctions#

5 %ind maret5clearing relati1e price o/ C and amount& con&umed in equilibrium b @ic anDeith

o if utility is form of#

o satis9es budget constraint#

o optimal $uantities consumers are#

" #obb"$o!glas $emands%o Malue of endowments#

o


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4et total +< ( +uantity a%ailable 1sum of +< of B ( sum of endowments of B; sol%e for

e$uilibrium price

Special Case- Identical C !tilities

&$uilibrium price ( common O4 G function of the relati%e sums of

endowments of 2 goods

6ptimal ratio of goods consumers are functions of priceoptimal

ratios of goods consumed are functions of relati%e sums ofendowments of 2 goods

;rade in

following approp redistributi%e transfers

-alra&0 a! and GEE,' Equilibrium

Malue of what a consumer wants to sell must ( %alue of what consumer wants to buy G e%aluated at

any set of priceso !rue for any prices

5obody sells something w/o spending al of the proceeds, and the %alue of what is purchased cannot

exceed the re%enue recei%ed from selling something

-alra&0 a!

)s an identit1statement that is true for an%e prices 1p",p2, whether these are e$uilibrium price

or not

Consumer's preferences are wellbeha%ed for any %e prices each consumer spends all of his

budget

Consumer A#

o p#x#'3 + p2x2

*3 = p#!#' + p2!2

*

consumer B#


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o p#x#'3 + p2x2

*3 = p#!#' + p2!2

*

summing

summed market %alue of excess demands is E for an%e prices p" p2 -alra&0 a!

Implication& o/ -alra&0 a!

suppose market for commodity A is in e$uilibrium

1" implication for a twocommodity exchange economyif " market is in e$uilibrium, then other

market must also be in e$uilibrium

if there is excess +4 of commodity "

12 implication for a twocommodity exchange economyexcess supply in " market implies an excessdemand in the other market

10. general equilibrium i

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