ECO 70, Problem Set en Cbrer. Suose two consumers hve lecogrhc references where erson rnks bundles b the level of commodt nd onl consders the level of commodt two f bundles hve the sme mount of commodt. Person does the reverse lookng frst t the level of commodt. Fnd the set of Preto effcent lloctons n n Edgeworth bo. Fnd the comettve equlbrum rce rto nd the comettve equlbrum qunttes for n rbtrr dstrbuton of ntl endowments. Does the Second Welfre Theorem hold n ths econom? Put consumer n the lower left, corner. We rell cn't drw ndfference curves, but relze tht consumer s best off movng to the rght frst.e., more of good, then consders movng u.e., more of good f he cn't move rght. Consumer goes n the uer, rght corner. Consumer refers good so he s best off movng down frst.e., more of good. We don't hve ndfference curves to check tngenc onts. To fnd Preto otml onts, we cn strtegcll lce few onts n the Edgeworth bo nd determne f those onts re Preto otml.e., we cn't mke one erson better off wthout mkng the other worse off. It's enough to check ont on ech edge nd one n the nteror to determne where ll the Preto otml onts re. To kee from clutterng the cture more thn t lred s, red rrows show the references for consumer nd blue rrows for consumer. The rrows re onl ncluded f t s fesble for the consumer ncrese consumton n tht drecton. A blck rrow t ont shows w to chnge the llocton so tht both consumers re better off. Therefore, n ont wth blck rrow s not Preto otml. The end result s tht ll onts on the bottom nd rght edges of the Edgeworth bo re Preto otml. Tht s, ll onts where consumer hs ll of good or consumer hs ll of good. Consumer wnts to go down frst, then left Consumer Consumer wnts to go rght frst, then u Consumer PO lloctons re onts on the bottom nd rght edges. A comettve equlbrum n ths ure echnge emle hs two condtons: both consumers mmze ther utltes subject to ther budget constrnts.e., bsed on the current rce rto nd endowments nd b mrket of 6
clerng the totl mount of ech commodt tht s consumed s equl to the totl mount of the endowment for tht commodt. Another w to look t comettve equlbrum s to s tht the ont tht ech consumer wnts to be t for gven rce rto s the sme s the other consumer's ont nd s wthn the Edgeworth bo.e., s fesble. There re three dfferent tes of ntl endowment to consder: n nteror ont, on the bottom edge, on the rght edge. Note: onts on the left nd to edge of the Edgeworth bo wll hve the sme result s n nteror ont. Interor Pont - In order for consumer to to mmze hs utlt, he wnts to trde hs entre endowment of good to get s much of good s the rce rto llows; dtto for consumer wth good. There re three dfferent rce rtos to consder shown t rght. Frst consder rce rto tht llows consumer to trde ll of hs endowment of good for more good thn s vlble to socet. Nether consumer knows wht the lmts re. In ths cse, consumer wnts to be t the red ont whch s outsde the Edgeworth bo. As consumer tres to get more of good, he'll end u drvng u the rce of good reltve to good.e., the budget lne wll get steeer. Therefore, ths rce rto wll not result n comettve equlbrum. Net, consder rce rto tht llows consumer to trde ll of hs endowment of good for more good thn s vlble to socet. In ths cse, consumer wll wnt to be t the blue ont whch s outsde the Edgeworth bo. As n the revous roblem, consumer 's ttemt to ttn consumton tht s not fesble wll drve u the rce for good mkng the budget lne more shllow. Therefore, ths wll not be comettve equlbrum. Fnll, consder rce rto tht connects the endowment ont to the bottom, rght corner of the Edgeworth bo. In ths cse, Consumer Consumer Consumer Consumer Consumer Consumer consumer trdes hs entre endowment of good to gn the remnng mount of good n socet. Smlrl, consumer trdes hs entre endowment of good for the remnng mount of good n socet. The ont tht both consumers wnt to be t s the sme nd s fesble. Therefore, the bottom, rght corner of the Edgeworth bo s the onl comettve equlbrum for n endowment n the nteror or left nd to edge. Summr - CE s the bottom, rght corner of the Edgeworth bo nd the rce rto s lne between the endowment ont nd the CE ont. Bottom Edge - At n endowment ont on the bottom edge, consumer wnts to consume more of good, but he hs no good to trde for t so no rce rto wll ncrese hs utlt. Consumer, however, wll hve some of good to trde w for more good bsed on the rce rto. Relze ths would be the sme s strtng t the red ont n the mddle grh bove. An rce rto ecet one wth the rce of good equl to zero would hve consumer of 6
wntng to get more good whch s not fesble. At rce of zero for good, however, we end u wth Arrow's ecetonl cse. Both consumers hve unbounded utlt mmzton roblems so nether cn ctull mmze utlt. Therefore, t the bottom edge, there s no equlbrum ecet the rght corner dscussed n the revous secton. Rght Edge - The rght edge follows the sme rgument s the bottom edge, ecet we're delng wth consumer hvng no endowment of good to trde nd we'll end u wth the rce of good equl to zero. Agn, we fce Arrow's ecetonl cse wth ech consumer hvng unbounded utlt mmzton roblems. Therefore, there s no equlbrum on the rght edge ecet the bottom corner dscussed n the frst secton.. The second welfre theorem sttes tht gven n Preto otml llocton, there ests rce rto nd n ntl endowment such tht comettve equlbrum results n the sme llocton. As rt just covered, the onl comettve equlbrum n ths scenro s the bottom, rght corner of the Edgeworth bo. Snce there re other onts tht re Preto otml, t would er the second welfre theorem does not hold. Recll, however, tht the ssumtons we mde to rove the second welfre theorem nvolved references beng comlete, conve, trnstve, locll nonstted, nd contnuous. ecogrhc references n ths roblem volte contnut. b. Now ssume lecogrhc references where both consumers cre frst bout the level of commodt nd onl for bundles wth the sme mount of commodt one do the look t the level of commodt. Fnd the set of Preto otml lloctons n n Edgeworth bo. Are there n dstrbutons of ntl endowments t whch comettve equlbrum ests? Consumer wnts to go left frst, then down Al the sme strteg s rt.e., ut onts on ech Consumer edge nd the nteror. In ths cse, both consumers' wnts to go rght frst, references re ectl ooste so one cn never be then u mde better wthout hurtng the other. Tht mens ever ont s Preto otml. Gven endowment ω nd n rce rto, consumer wll wnt ω to trde to get s much good s ossble. So wll consumer. Note n the lower cture tht consumer Consumer mmzes utlt t the red ont nd consumer does so t the blue ont. These onts re not the sme nd won't be regrdless of the rce rto so comettve equlbrum does not est. Consumer of 6
4 of 6. Consder two-erson, two-commodt ure echnge comettve econom,. The consumers' utlt functons re: u nd u Suerscrts refer to consumers nd subscrts to goods. Consumer hs n endowment, nd consumer hs n endowment 0,0 Determne the ecess demnd functons for the two consumers. Fnd the equlbrum rce rto for ths econom. Consumer :, m u s.t.. 0. 0. 0 Solve. nd. for nd set them equl to ech other: Now solve tht for n terms of : Substtute ths nto.: 0 4 Now solve tht for : Plug tht bck nto the equton: Consumer :, m u s.t. 0 0 0 0
of 6. 0. 0. 0 0 0 Solve. nd. for nd set them equl to ech other: Now solve tht for n terms of : Substtute ths nto.: 0 0 0 Now solve tht for : Plug tht bck nto the equton: Ecess demnds re: 0 Good : z 0 Good : z Solve for the rce rto b settng ecess demnds to zero Wlrs' w... onl need one of these, but dd both to check m work: 0 0
. Consder n echnge econom wth two dentcl consumers. Ech hs the utlt functon u for 0 < <. Socet hs 0 unts of good nd 0 unts of good. Fnd set of endowments nd wth nd Wlrsn equlbrum rce rto whch wll suort equlbrum consumton bundles of, for ech consumer. Ech consumer must mmze hs utlt: m u s.t., We know the rce rto wll be equl to the mrgnl rte of substtuton: u / u / At the ont,, the rce rto becomes: At, both ndfference curves nd the budget lne re tngent. In order to ttn, s comettve equlbrum, ll we hve to do s set the endowment ont on the budget lne. Therefore, the set of endowments re ll those onts on the lne through ont, wth sloe -/ -... lmted to the totl mount of ech good beng 0.e., 0,0. Equton of lne: b Gven ont, solve for b: b b So the set of endowments s gven b: 0 0 Documentton. I went over ll three roblems wth Prof Slutsk. He told me m method nd nswer for ws correct. He told me the bottom, rght corner ws the comettve equlbrum n nd elned wh. He elned wh comettve equlbrum doesn't est n b. He told me I onl needed one equton nd one unknown rce rto n. He onted out tht sked for set of endowments, not secfc ont. Chrstne cught errors n m work. One n roblem where I ddn t dvde the b when I solved for. The other ws on roblem where I ddn't cncel correctl to get /. Gulle cught more errors: confused left nd rght n nd forgot the negtve sgn on the sloe n roblem. 6 of 6