10. General Equilibrium I
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Transcript of 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