Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of people who exchange good through ths Walrasan mechansm. Loong at consumers behavor gven fxed prces s often called partal equlbrum analyss, whle loong at a settng where prces must equlbrate (to match supply wth demand) s called general equlbrum analyss. Snce we wll be loong at multple consumers wth potentally dfferent utlty functons and endowments, we wll ndex them wth the letter, where {,..., M} and M s the number of consumers n the economy. So we have M consumers n the economy and consumer has an endowment e and utlty functon u. An equlbrum n ths settng must satsfy two condtons. Let x be the equlbrum consumpton of consumer and p be the equlbrum prces. Frst, consumers must choose optmally gven prces, that s, for all x argmax x R N + u (x) s.t. p x p e Second, the goods maret must clear for each good, so for all M x = = M = e It turns out that there s a lot of ntuton to be ganed by loong at the case of an economy wth two consumers and two goods. Remember, we are free to normalze the prce of one of the goods to, so let the prce of good
be and the prce of good 2 be p. Now we have utlty functons u and u 2 and endowments e and e 2. The budget constrants are The maret clearng condtons are x + px 2 = e + pe 2 () x 2 + px 2 2 = e 2 + pe 2 2 (2) x + x 2 = e + e 2 (3) x 2 + x 2 2 = e 2 + e 2 2 (4) In addton to the above equatons, we also want the consumer to choose optmally, whch under mld assumptons means the frst order condtons must hold u (x, x 2) = λ u 2(x, x 2) = pλ u 2 (x 2, x 2 2) = λ 2 u 2 2(x 2, x 2 2) = pλ 2 Notce that each consumer has ther own Lagrange multpler. Dvdng, we get u (x, x 2) u 2(x, x 2) }{{} MRS = u2 (x 2, x 2 2) = p (5,6) u 2 2(x 2, x }{{ 2 2) } MRS 2 So we can see that snce consumers face a prce of p and any optmal choce wll set MRS = p, n fact the consumers must have the same MRS n equlbrum. Consder what we have now: there are 6 equatons (numbered above) that must be satsfed and only 5 unnown varables (x,x 2,x 2,x 2 2, and p). Normally, we want to have the same number of equatons as unnowns. One way to reconcle ths s to recall that we started wth p and p 2 but normalzed to p =. Another way s to observe the followng Proposton (Walras s Law). If equatons (), (2), and (3) hold, then equaton (4) must hold as well. 2
Proof. () x = e + pe 2 px 2 (2) x 2 = e 2 + pe 2 2 px 2 2 Pluggng these nto (3) e + pe 2 px 2 + e 2 + pe 2 2 px 2 2 = e + e 2 pe 2 px 2 + pe 2 2 px 2 2 = 0 x 2 + x 2 2 = e 2 + e 2 2 So the budget constrants and maret clearng n one of the goods mples maret clearng n the other good. In fact, ths result s true for the general settng wth N goods. If the maret for N goods clears, then the maret for the N th good must clear as well. As t happens, there exsts a truly excellent method for vsualzng a 2 good, 2 consumer economy called the Edgeworth Box. Essentally, we tae the standard graph of consumer s budget set and supermpose on that the graph of consumer 2 s budget set rotated by 80. Now any pont n ths box specfes a full allocaton. Consumer s s gven by the dstance from the southwest orgn, whle consumer 2 s s gven by the dstance to the northeast orgn. Furthermore, f the sze of the box s (e, e 2 ), then maret clearng wll be satsfed as well. Now we defne a new concept called excess demand. Ths s smply the dfference between what s beng demanded and the total amount of goods n the economy. Ths wll depend on the prce p. z (p) = x (p) + x 2 (p) e e 2 z 2 (p) = x 2(p) + x 2 2(p) e 2 e 2 2 An equlbrum wll satsfy z (p ) = z 2 (p ) = 0. Proposton 2. For any p, z (p) + pz 2 (p) = 0. Proof. The consumer s budget constrants mply x (p) + px 2(p) e pe 2 = 0 x 2 (p) + px 2 2(p) e 2 pe 2 2 = 0 3
Addng these together yelds [ ] [ ] x (p) + x 2 (p) e e 2 + p x 2 (p) + x 2 2(p) e 2 e 2 2 = 0 z (p) + pz 2 (p) = 0 Notce that the above actually mples Walras s Law, whch can be stated smply as [z (p) = 0] [z 2 (p) = 0] Example. Consder the two consumer, two good case. We ll use Cobb- Douglas utlty for both consumers u (x, x 2) = α log(x ) + ( α ) log(x 2) As we ve seen before, the demand s x (p) = α w x 2(p) = ( α )w p where w = e + pe 2 s the wealth of consumer. Because of Walras s Law, we smply need to fnd some p such that z (p) = 0. 0 = z (p) 0 = x (p) + x 2 (p) e e 2 0 = α w + α 2 w 2 e e 2 0 = α (e + pe 2) + α 2 (e 2 + pe 2 2) e e 2 0 = p(α e 2 + α 2 e 2 2) ( α )e ( α 2 )e 2 Now we solve for p to fnd the equlbrum prce p = ( α )e + ( α 2 )e 2 α e 2 + α 2 e 2 2 4
From here we can fnd optmal consumpton x = α (e + p e 2) ( ) e x 2 = ( α ) + p e 2 Consder the specal case where e = e 2 = e 2 = e 2 2 = /2. Ths yelds prces p and consumpton p = ( α ) + ( α 2 ) α + α 2 + p = α + α 2 x = α α + α 2 and x 2 = α ( α ) + ( α 2 ) So each person consumes n proporton to ther preference parameter. 2 Welfare Up untl now, we have been dealng wth ndvdual utlty functons n solaton. Ultmately, we would le to use the utlty specfcatons to mae statements about the desrablty of partcular allocatons of goods. That s, gven the total amount of goods n an economy, how should we best dstrbute them amongst the agent. Not surprsngly, there s no one rght answer to ths queston, even f we now exactly what people s utlty functons are. We can, however, narrow our focus to a set of allocatons that are consdered to be better than the rest. Frst, we must defne some terms Allocaton: a specfcaton of consumpton for each consumer x = (x,..., x M ). such that M x = = M e = e = 5
Notce that the above s the same as the maret clearng condton from a Walrasan equlbrum. In ths case, we call f feasblty. Now we defne the standard noton of effcency n economcs Pareto Effcent: An allocaton x such that there s no other allocaton ˆx wth u (ˆx ) u (x ) u (ˆx ) > u (x ) for all for some A related term that we may use as well s Pareto Domnated: An allocaton ˆx Pareto domnates x f u (ˆx ) u (x ) u (ˆx ) > u (x ) for all for some So you can see that an allocaton s Pareto effcent f t s not Pareto domnated by any other allocaton. In plan Englsh, an allocaton s Pareto optmal f there s no way to transfer goods so that everyone s mae wealy better off and at least one person s made strctly better off. Pareto effcency does not guarantee that an allocaton has other propertes that are consdered desrable, such as farness. It s Pareto effcent for me to have everythng and you to have nothng. The reverse s also Pareto effcent. In general, there s a set of Pareto effcent allocatons Proposton. Gven β,..., β M wth β = and β > 0, f x maxmzes W (x β) = M β u (x ) = then t s Pareto effcent. Proof. Suppose x maxmzed W (x β) and was not Pareto effcent. there s some ˆx such that Then u (ˆx ) u (x ) u (ˆx ) > u (x ) for all for some 6
Ths mples that M β u (ˆx ) > = M β u (x ) = or W (ˆx β) > W (x β). So x does not maxmze W ( β), contradctng our ntal assumpton that t dd, so x must be Pareto effcent. A partal converse to ths statement, whch we wll not prove, s Proposton 2. For an Pareto effcent allocaton x, there s some β such that x maxmzes W ( β). So by loong over all β values, we could concevable map out the set of Pareto effcent allocatons. We can vsualze Pareto effcent allocatons usng the Edgeworth box. Here, the set of Pareto effcent allocatons wll le on a lne extendng form consumer s orgn to consumer 2 s orgn. Ths lne s called the contract curve. Let s wor out the Pareto problem n the 2 consumer, 2 good case. Here our welfare functon s And we wsh to solve W (x β) = βu (x ) + ( β)u ( x 2 ) max x R N + βu (x, x 2) + ( β)u 2 (x 2, x 2 2) s.t. x + x 2 = e + e 2 = e x 2 + x 2 2 = e 2 + e 2 2 = e 2 Thus we wll have two Lagrange multplers, one for each good, and the Lagrangan functon s L = βu (x, x 2) + ( β)u 2 (x 2, x 2 2) + λ (e + e 2 x x 2 ) + λ 2 (e 2 + e 2 2 x 2 x 2 2) Tang the FOC s βu (x, x 2) = λ ( β)u 2 (x 2, x 2 2) = λ βu 2(x, x 2) = λ 2 ( β)u 2 2(x 2, x 2 2) = λ 2 7
Dvng to cancel the λ s, we fnd Cross multplyng yelds u (x, x 2) u 2 (x 2, x 2 2) = β β = u 2(x, x 2) u 2 2(x 2, x 2 2) u (x, x 2) u 2(x, x 2) = u2 (x 2, x 2 2) u 2 2(x 2, x 2 2) whch s smply MRS = MRS 2. So the margnal rates of substtuton are equal at a Pareto optmal allocaton. Furthermore, f you an allocaton where the MRS s are equal, that must be Pareto optmal. Example 2. Agan we ll use Cobb-Douglas utlty for both agents u (x, x 2) = α log(x ) + ( α ) log(x 2) The MRS n ths case s MRS = u (x,, x 2) u 2(x, x 2) = α /x ( α )/x 2 = α α l x 2 x Equatng these two, we get ( α α ) x 2 x ( ) α 2 x 2 = 2 α 2 x 2 Usng the feasblty condtons, ths becomes ( α α ) x 2 x ( ) α 2 e2 x 2 = α 2 e x Now we want to solve for x 2 n terms of x. Ths wll lead us to the contract curve. α ( α 2 )(e x )x 2 = α 2 ( α )(e 2 x 2)x [ α ( α 2 )e + (α 2 α )x ] x 2 = α 2 ( α )e 2 x x 2(x ) = α 2 ( α )e 2 x α ( α 2 )e + (α 2 α )x Notce that x 2(0) = 0 and x 2(e ) = e 2. 8
3 Effcency of Equlbrum Now we move on to the queston of whether a Walrasan equlbrum s effcent. It turns out that t s. Remember that any allocaton where MRS = MRS 2 s effcent. Also recall that we proved that any equlbrum satsfes MRS = MRS 2 = p, so an equlbrum s effcent. Ths of course requres that utlty s ncreasng and concave. However, we can prove t wth weaer assumptons. Theorem (Frst Basc Welfare Theorem). When utlty s ncreasng, any Walrasan equlbrum s effcent. Proof. Suppose we have an equlbrum prce vector p wth allocaton x demanded, and that ths allocaton s not Pareto effcent. Ths means that there s some allocaton ˆx that Pareto domnates x, that s u (ˆx ) u (x ) u (ˆx ) > u (x ) for all for some Recall that snce utlty s ncreasng, any optmal choce wll le on the budget lne, otherwse the consumer could consume a lttle bt more of each good and be better off, so p x = p e Step : For any agent, t must be that p ˆx p e If ths were not the case, then we would have p ˆx < p e Then there would be some pont x wth p x < p e 9
and x > ˆx for all that s also n the budget set. Snce u s ncreasng, ths wll satsfy u ( x ) > u (ˆx ) u (x ). However, ths cannot be, snce we assumed x was chosen optmally, but here we have an affordable pont that gves hgher utlty. Step 2: For, we must have p ˆx If ths were not the case, then p ˆx > p e p e But here we have the same contradcton to the fact that x was chosen optmally, but we have a pont ˆx that s affordable wth u (ˆx ) > u (x ). So now we now that p ˆx p ˆx > p e p e for all Addng these nequaltes together for all, ncludng p ˆx > p ˆx > p ˆx > p ˆx [ p ˆx x p x p x p x > 0 x ] > 0 0
However, we now that ˆx s a feasble allocaton, meang ˆx = e for all Pluggng ths nto the above, we get p 0 > 0 0 > 0 Obvously, ths cannot be, so have reached a contradcton. It must be that x was Pareto effcent after all!