Thinking outside the (Edgeworth) Box

Transcription

1 Tinking outsid t (dgwort) ox by Jon G. Rily Dartmnt of conomics UCL 0 Novmbr 008

2 To dvlo an undrstanding of Walrasian quilibrium allocations, conomists tyically start wit t two rson, two-commodity xcang conomy. To nsur as of analysis it is common to assum Cobb-Douglas rfrncs. Wit idntical rfrncs it is an asy mattr to caractriz t Walrasian quilibrium allocations, sinc in tis cas rfrncs can b aggrgatd. Tn t uniqu Walrasian quilibrium ric ratio is t marginal rat of substitution of t rrsntativ individual at t aggrgat ndowmnt. Tus t quilibrium ric ratio is indndnt of t distribution of ndowmnts. Hr w sow tat, for t cntral Cobb-Douglas cas, it is almost as asy to fully caractriz t Walrasian quilibrium rics and allocations wn rfrncs diffr. W tn us a modifid Cobb-Douglas modl to illustrat som of t uzzling faturs of gnral quilibrium. In articular, w rsnt an xaml in wic a gift by on of t individuals lads to a ric cang tat maks t donor bttr off tan bfor making t gift! It is wll undrstood tat tr may b multil quilibria in a gnral quilibrium modl and tat t numbr of quilibria is almost always finit. For our modifid Cobb- Douglas modl t two fr aramtrs ar t ndowmnts of t first consumr. W sow tat, for on articular focal ndowmnt, tr is a continuum of Walrasian quilibria. For lss xtrm initial ndowmnts tr is a uniqu quilibrium. For mor xtrm initial ndowmnts tr ar tr quilibria. Cobb Douglas rfrncs and t focal oint For t Cobb-Douglas cas, individual as a utility function θ θ ( ) = ( ) ( ), 0< θ <, = {, } U x x x H W will rfr to t two individuals as lx and v. W normaliz commoditis so tat t total ndowmnt of ac commodity is. W assum tat lx as a strongr rfrnc for commodity so tat θ > θ. It follows tat, along t diagonal of t dgwort ox, lx s indiffrnc curv as a mor ngativ slo. Tn, as dictd blow, t Parto fficint allocations li blow t diagonal. Suos tat t ndowmnt of commodity j is ω j. Dfin cj = xj / ω j. Tn cj = xj / ω j and U ( x ) = U ( ωc, ω c) = U ( ω) U ( c ) H H

3 slo = slo = f = ( f, f) Considr t oint Fig. dgwort box and t Focal oint wr lx as t ntir ndowmnt. t tis oint MRS U U θ (,) = / = x x θ. T suorting ric lin troug tis oint tus as a slo. Similarly, at t oint wr v as t ntir ndowmnt, r marginal rat of substitution is U U θ MRS (,) = / =. x x θ T suorting ric lin troug is also dictd. Sinc θ > θ, it follows tat f = ( f, f ), to t > and so t two suorting lins must mt at som allocation Sout-Wst of t dgwort ox. W will rfr to tis as t focal oint. From Fig. θ f = = θ f θ f and = = θ f Tus t focal oint, ( f, f ), satisfis t following two linar quations. Solving, θ f+ ( θ ) f = 0 and θ f+ ( θ ) f =. θ θ ( f, f) = (, ). (.) θ θ θ θ

4 3 W ar now rady to stat our ky obsrvation. Proosition : T Cobb-Douglas Focal Point If t two individuals av diffrnt Cobb-Douglas rfrncs, t focal oint lis on t suorting (tangnt) lin for vry Parto fficint allocation Proof: Considr t lin troug t focal oint f and an arbitrary Parto fficint. allocation * x dictd in Fig.. * x f W nd to sow tat tis lin is a suorting lin. Tat is, t indiffrnc curvs for lx and v ar tangntial at * x. ut tis is t cas if and only if x * is also a Walrasian quilibrium allocation for any ndowmnt on t lin troug f and t roosition by sowing tat * x is a Walrasian quilibrium allocation. For Cobb-Douglas rfrncs, t aramtr snt on commodity. Tn x = ( θ )( ω + ω ). Fig. : Suorting lin Consumr trfor as an xcss dmand of θ (, ω ) = x ω = ( θ ) ω θ ω, wr / * x. Tus w rov is t fraction of incom

5 4 Summing ovr consumrs, (.) (, ω ) = ( θ ) ω θ ω H H Considr t dgwort box diagram dictd in Fig.. W av argud tat t tangnt lin troug and f is a Walrasian quilibrium ric lin for t Parto fficint allocation. Tus t ric ratio is a Walrasian quilibrium ric ratio for vry ndowmnt on t lin sgmnt f. Similarly, t ric ratio quilibrium ric ratio for vry ndowmnt on t lin troug is a Walrasian and f. T oint f is not an ndowmnt oint in t usual sns, sinc it lis outsid t dgwort box. Howvr w can tink of it as an ndowmnt oint if w allow lx to ntr t markt sort in commodity. Tat is, lx ows units of commodity to v. Similarly v bgins sort in commodity. Tn suos tat t initial allocation is t focal oint f. Sinc f is on bot tangnt lins, it follows tat bot ric ratios ar Walrasian quilibrium ric ratios. Tat is (, f) = (, f) = 0 λ Dfin ( λ) + λ, 0 < λ <. Sinc t xcss dmand function is a linar function of t ric ratio, (, f) = ( λ) (, f) + λ (, f) = 0 λ Tus vry ric ratio btwn and is a Walrasian quilibrium ric ratio. Sinc vry Walrasian quilibrium is Parto fficint, it follows tat vry suorting ric lin must b a lin troug t focal oint. QD 3 Not tatω = ω, =, so w can xrss aggrgat dmand as a function only j j j of lx s ndowmnt. 3 ltrnativly, sinc t aggrgat ndowmnt of ac commodity is w can rwrit quation (.) as follows. (, ω ) = [( θ ) ω + ( θ )( ω )] [ θ ω + θ ( ω )] Substituting from (.) = [( θ ) ( θ θ ) ω ] [ θ ω + ( θ θ ) ω ].

6 5 Givn tis roosition, w av t following rsult. Proosition : Caractrization of t Walrasian wit Cobb-Douglas rfrncs If t two individuals av diffrnt Cobb-Douglas rfrncs and individual as a strongr rfrnc for commodity tn any cang in ndowmnt (in t dgwort box) tat raiss t valu of individual s ndowmnt at currnt quilibrium rics, lads to a nw Walrasian quilibrium in wic t rlativ ric of commodity is igr and consumr is bttr off. Proof: Considr Fig. 3. ω ω * x * x f Fig. 3: Caractrizing quilibrium allocations For t ndowmnt ω, t suorting lin is t lin troug t focal oint. Suos tat t ndowmnt sifts to t oint ω abov tis suorting lin. T nw suorting lin is dictd. Sinc it is str, t nw quilibrium ric ratio = / is igr and individual is strictly bttr off. QD (, ω ) = ( θ θ )[ ( f ω ) ( ω f )]. Tus xcss dmand is zro for a continuum of rics if ω = f.

7 6 Tinking insid t box Tr ar two siml ways to mov t focal oint insid t dgwort box. T first is modify t Cobb-Douglas rfrncs and introduc a minimum consumtion γ. T consumtion st is tn t st C = { x x γ } and rfrncs ovr tis st ar givn by t modifid Cobb-Douglas function θ γ γ θ u ( x) = ( x ) ( x ). (.3) Prfrncs ar tn as dictd blow. x C γ γ x Fig.4a: Modifid Cobb-Douglas rfrncs If t minimum consumtion is sufficintly larg, t focal oint lis in t dgwort box as sown in Fig. 4b. x x f ω ω Fig. 4b: Focal oint insid t box

8 7 Rdistribution W nxt considr t ffct on t Walrasian allocation and rics if ndowmnts ar rdistributd. Lt ω b lx s initial ndowmnt and lt b t quilibrium ric ratio. W fix t ric of commodity and considr cangs in t ric of commodity. Suos tat lx s nw ndowmnt, ω, as a lowr valu wn calculatd at t initial quilibrium rics. Sinc v as a strongr rfrnc for commodity, s as a igr marginal ronsity to consum commodity as r walt incrass. Tus, wit t walt transfr, v s dmand for commodity riss mor tan lx s dmand falls. Trfor, at t old quilibrium ric of commodity, tr is xcss dmand for commodity. T old and nw xcss dmand curvs ar dictd blow (, ) ω (, ) ω Fig. 5: xcss dmand for commodity Sinc (, ω ) > (, ω ) = 0, it follows tat > and so If lx as a larg ndowmnt of commodity is a nt sulir of tis commodity. Tn, as t rlativ ric of commodity falls, lx is vn wors off tan bfor t rlativ ric cang. W now sow tat tis argumnt is not ncssarily corrct. Considr Fig. 4b onc mor. Suos lx as an ndowmnt in t sadd rgion to t Sout-ast of t focal oint. From Proosition w know tat t uniqu Walrasian quilibrium allocation is on t lin troug t ndowmnt oint and t focal oint. Suos tat < lx as an initial ndowmnt ω but gnrously dcids to donat art of is ndowmnt to v. His rsulting ndowmnt is ω wic lis to t Sout-Wst of t

9 8 lin joining ω and f. It follows tat t nw Walrasian quilibrium ric ratio = / riss and so t nw Walrasian quilibrium lis to t Nort-ast of t old quilibrium allocation. Tn t gift maks lx bttr off! W summariz tis aradoxical rsult blow. Proosition 3: To tos wo giv, mor sall b givn For any ndowmnt in t sadd rgion to t Sout-ast of t focal oint, if on individual donats art of is ndowmnt to t otr individual, it is t donor wo is strictly bttr off in t nw Walrasian quilibrium. Givn tis rsult tr is somting vry wrong wit our informal argumnt tat lx must b wors off. s w sall s, t flaw in t argumnt is t assumtion tat t xcss dmand curv must av a ngativ slo. In fact t slo must b ositiv as dictd blow (, ) ω (, ) ω Fig. 6: Uward sloing xcss dmand curvs rguing as bfor, aftr lx as mad is donation to v, tr is xcss dmand for commodity at t old quilibrium rics. ut wit an uward sloing xcss dmand function, t nw quilibrium ric of commodity is lowr. Tn if lx is a nt sulir of commodity, t ric ffct is favorabl sinc t rlativ ric of commodity riss. In fact, as Proosition 3 stabliss, for sufficintly xtrm ndowmnts t ric ffct dominats and so lx is bttr off in t nw quilibrium. W now sow formally tat t xcss dmand curv is ositivly slod for all ndowmnts in t kit-sad sadd ara of Fig. 4b.

10 9 Proosition 4: Positivly slod xcss dmand curv For any ndowmnt in t sadd ara to t Sout-ast of t focal oint t xcss dmand function for commodity, (, ) ω, is a strictly incrasing function of. Considr Fig. 4b. Lt b t quilibrium ric ratio wn lx as an ndowmnt ω. Sinc x is t Walrasian quilibrium allocation, (, ω ) = (, f) = (, x ) = 0. Considr t xcss dmand function (, ) x. t ac individual as a zro nt dmand. For all lowr rics of commodity, ac individual s dmand for commodity riss and so tr is xcss aggrgat xcss dmand. Tus t aggrgat xcss dmand function (, ) x is a strictly dcrasing function of. Now considr an ndowmnt on t lin troug x and t focal oint f. Tat is λ ω = ( λ) x + λf. For valus of λ btwn 0 and tis is on t lin sgmnt conncting λ < 0 t ndowmnt is to t Nort-Wst of x and f. If x. Finally, if λ > t ndowmnt lis in t sadd kit-sad ara to t Sout-ast of t focal oint. Sinc t aggrgat xcss dmand function is linar in t ndowmnt vctor, it follows tat (, ω ) = ( λ) (, x ) + λ (, f). λ ut w av sn tat (, f ) = 0. Tn (, ω ) = ( λ) (, x ). λ W av just argud tat (, ) x is a dcrasing function of. Tn (, λ ) ω is a dcrasing function of for λ < and an incrasing function for λ >. QD Multil quilibria Tr is a scond way of lacing t focal oint insid t dgwort box. s bfor, w sift t origin of t Cobb-Douglas utility function so tat utility is zro at

11 0 γ. Tis is dictd blow. Howvr w no longr intrrt t sadd ara C as t st of fasibl consumtion vctors. Instad it is t rgion ovr wic rfrncs ar modifid Cobb-Douglas rfrncs. Not tat t contour st troug ( γ, γ ) dictd in Fig. 4a is L- sad. W can trfor xtnd t st ovr wic rfrncs ar dfind by introducing L-sad indiffrnc curvs for all allocations not in Formally, Min{ c, c }, c C u ( c ) = γ γ ( c ) ( c ), c C θ θ γ γ Tis utility function is continuous, quasi-concav and strictly incrasing. Considr t dgwort box dictd in Fig. 7. C. 3 f ω Fig. 7: Tr quilibria For all ndowmnts in t sadd rgion w know tat t Walrasian quilibrium is obtaind by conncting t lin joining t ndowmnt and t focal oint. Considr nxt all P allocations to t Nort-ast of t sadd ara. v s rfrncs ar L- sad. Sinc t aggrgat ndowmnt of ac commodity is t sam, t slo of

12 lx s indiffrnc curv is -. Tus t slo of all t suorting ric lins is. Tus for any ndowmnt in tis unsadd rgion t uniqu Walrasian quilibrium ric ratio is. rguing symmtrically, for any ndowmnt in t unsadd rgion to t Sout-Wst of t sadd rgion, t uniqu Walrasian quilibrium ric ratio is. Now considr an ndowmnt ω to t Sout-ast of t focal oint. y t abov argumnts tr ar tr suorting lins troug tis ndowmnt and tus tr ar tr Walrasian quilibria. From Proosition 4 w know tat t slo of t aggrgat xcss dmand function is ositiv for t middl quilibrium. Tus t xcss dmand function must b as dictd blow. 0 (, ) ω Fig. 8: Tr quilibria s our final xloration, considr ndowmnts on t lin troug ω and f in Fig. 7. Tat is μ ω = ( μω ) + μf Suos tat t ndowmnt lis btwn ω and f, so tat 0 < μ <. From Proosition t middl quilibrium ric ratio is t sam for all suc ndowmnts. From our argumnt abov, t xtrm quilibrium ric ratios ar and. Tus non of t

13 tr quilibrium rics is affctd by t cang in ndowmnt. 4 Howvr, as μ aroacs, t xcss dmand function lis closr and closr to t vrtical axis for ric ratios btwn and. t μ = t ndowmnt oint is t focal oint so aggrgat xcss dmand is zro at all ric ratios on t intrval [, ]. For all μ > t ndowmnt oint is no longr in t kit-sad rgion so is t only quilibrium ric. Tis is illustratd blow. In t first diagram μ = δ wr δ is small. In t scond μ =. In t tird μ = + δ. μ (, ω ) μ (, ω ) μ (, ω ) (a) μ = δ (b) μ = (c) μ = + δ Fig.9: xcss dmand for commodity as μ incrass Concluding Rmark W av sown tat for t cntral Cobb-Douglas modl wit trognous rfrncs tr is a critical focal ndowmnt outsid t dgwort box. ll Walrasian quilibrium ric lins ass troug tis focal ndowmnt. T focal ndowmnt satisfis two siml linar quations so it is a siml mattr to solv for t Walrasian quilibrium ric ratio for any ndowmnt. Mor imortant, using t focal oint, it is ossibl to 4 Not tat it is t xtrm quilibria tat ar locally stabl undr t Walrasian tatonnmnt rocss.

14 3 rovid a strong caractrization of t ffcts of an ndowmnt rdistribution on rics and utility. Using modifid Cobb-Douglas rfrncs w also offr usful insigts for Gnral quilibrium tory. Wil it is wll known tat G dmand curvs can av all sorts of unaaling rortis, tr ar fw good xamls and ts ar tyically muc mor comlicatd tan t modl rsntd r. Pras most intriguingly w sow tat a donor can mak imslf bttr off by giving away art of is ndowmnt. Tis aradoxical rsult is xlaind by sowing tat t undrlying gnral quilibrium aggrgat dmand function must b uward sloing. Finally w not tat wil w work wit a rson commodity modl, t rsults old gnrally for Cobb-Douglas rfrncs. If tr ar n consumrs and m commoditis and nc t sac of ndowmnts is of dimnsion mn, t focal ndowmnt bcoms a subsac of dimnsion mn n m. Rfrncs dgwort, Francis Ysidro Matmatical Psycics: n ssay on t alication of matmatics to t moral scincs, 88. Parto, Vilfrdo Manual of Political conomy 906