Computing Equilibrium Prices in Exchange Economies with Tax Distortions

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1 Computng Equlbrum Prces n Exchange Economes wth Tax Dstortons Bruno Codenott 1 Lus Rademacher 2 Kastur Varadaraan 3 Abstract We consder the computaton of equlbrum prces n market settngs where purchases of goods are subect to taxaton. Whle ths scenaro s the standard one n appled computatonal work, so far t has not been an obect of study n theoretcal computer scence. Taxes ntroduce sgnfcant dstortons: equlbra are no longer Pareto optmal, suffcent condtons for unqueness do not contnue to guarantee t, exstence tself must be revsted. We analyze the effects of these dstortons on scenaros whch, n the absence of taxes, admt polynomal tme algorthms, and prove a number of results: For Fsher s model wth homothetc preferences: f consumers are subect to a unform tax regme, the model loses the representatve consumer and becomes prone to multple dsconnected equlbra; however we show that the dstorton has a structure whch leads to two (as opposed to one) representatve consumers. We take advantage of ths property to develop polynomal tme algorthms, n spte of the presence of multple dsconnected equlbra. To obtan the result above, we develop a technque to estmate the senstvty of Fsher s equlbrum prces wth respect to the consumers ncome, whch mght fnd wder applcatons n the feld. For the exchange model wth Cobb-Douglas utlty functons: f consumers are subect to a dfferentated tax regme, we show that the model s equvalent to a Cobb-Douglas exchange economy wth an extra good. Ths mples that the polynomal tme computablty s preserved. For the exchange model wth utlty functons satsfyng weak gross substtutablty: f consumers are subect to a dfferentated tax regme, the technques based on convex programmng seem to break down. However we are able to modfy certan combnatoral algorthms and prce adustment methods to obtan polynomal tme approxmaton schemes n two specal cases: an economy wth lnear utltes and an economy where the demands do not change too rapdly as a functon of the prces. 1 IIT-CNR, Psa, Italy. E-mal: bruno.codenott@t.cnr.t. Part of ths work has been done whle vstng the Toyota Technologcal Insttute at Chcago. 2 Department of Mathematcs, MIT. Work done whle vstng Toyota Technologcal Insttute at Chcago, Chcago IL E-mal: lrademac@math.mt.edu. 3 Department of Computer Scence, The Unversty of Iowa, Iowa Cty IA E-mal: kvaradar@cs.uowa.edu. 1

2 1 Introducton The equlbrum problem for a pure exchange economy amounts to fndng a set of prces and allocatons of goods to economc agents such that each agent maxmzes her utlty, subect to her budget constrants, and the market clears. The equlbrum depends only on the agents utlty functons and ntal endowments of goods. If one ams at analyzng equlbrum problems arsng from real world applcatons, the scenaro outlned above has often to be extended. Indeed one needs to take nto account the presence of sutable dstortons, whch mght be, dependng on the specfc applcaton, transacton costs, transportaton costs, tarffs, and/or taxes. In these frameworks, whch are the standard ones for appled computatonal work, one has to deal wth equlbrum condtons nfluenced by addtonal parameters whch often change the mathematcal propertes of the problem. For nstance, n models wth taxes, () the equlbrum allocatons mght lose ther Pareto optmalty; () restrctons whch mply, n the absence of taxes, the unqueness of equlbrum prces mght become compatble wth multple dsconnected equlbra. In ths paper, we consder exchange economes wth ether unform or dfferentated ad valorem taxes (see Secton 2 for the approprate defntons). We explore the effects of such tax dstortons on models whch admt - n the absence of taxes - polynomal tme algorthms. In spte of the loss of certan structural propertes (ncludng unqueness), we are able to obtan polynomal tme algorthms or approxmaton schemes n several nstances where the model wthout taxes admtted them. Background. We now descrbe the model of an exchange economy, and provde some basc defntons. Let us consder m economc agents whch represent traders of n goods. Let R n + denote the subset of R n wth all nonnegatve coordnates. The -th coordnate n R n wll stand for good. Each trader has a concave utlty functon u : R n + R +, whch represents her preferences for the dfferent bundles of goods, and an ntal endowment of goods w = (w 1,...,w n ) R n +. At gven prces π R n +, trader wll demand a bundle of goods x = (x 1,...,x n ) R n + whch maxmzes u (x) subect to the budget constrant π x π w. Let W = w denote the total amount of good n the market. An equlbrum s a vector of prces π = (π 1,...,π n ) R n + at whch, for each trader, there s a bundle x = ( x 1,..., x n ) R n + of goods such that the followng two condtons hold: () for each trader, the vector x maxmzes u (x) subect to the constrants π x π w and x R n +; () for each good, x W. The celebrated result of Arrow and Debreu [2] states that, under qute mld assumptons, such an equlbrum exsts. We now gve the defnton of approxmate equlbrum. We assume that all the utlty functons u() dscussed n ths paper satsfy u(0) = 0. A bundle x R n + s an ε-approxmate demand, for 0 < ε < 1, of trader at prces π f u (x ) (1 ε)u and π x (1 + ε)π w, where u = max{u (x) x R n +,π x π w }. A prce vector π R n + s an ε-approxmate equlbrum f there s a bundle x for each such that (1) for each trader, x s an ε-approxmate demand of trader at prces π, and (2) x (1 + ε) w for each good. An mportant specal case of an exchange economy s the dstrbutonal economy, where the ntal endowments are all collnear,.e., w = δ w, δ > 0, so that the relatve ncomes of the traders are ndependent of the prces. Ths specal case s equvalent to Fsher model, whch s 1

3 a market of n goods desred by m utlty maxmzng buyers wth fxed ncomes. For any prce vector π, the vector x (π) that maxmzes u (x) subect to the constrants π x π w and x R n + s called the demand of the -th trader. By addng up the traders demands, one gets the market demand. The utlty functon of an ndvdual trader s sad to satsfy weak gross substtutablty f ncreasng the prces of some of the goods whle keepng some others and her ncome fxed cannot cause a decrease n demand for the goods whose prce s fxed. It s well known and easy to see that the market demand satsfes weak gross substtutablty f the utlty functon of each ndvdual trader does. A utlty functon u( ) s homogeneous of degree one f t satsfes u(αx) = αu(x), for all α > 0, whle t s log-homogeneous f t satsfes u(αx) = u(x) + log α, for all α > 0. Both homogeneous and log-homogeneous utlty functons represent consumers wth homothetc preferences,.e., a bundle x s preferred to a bundle y f and only f the bundle αx s preferred to the bundle αy, for all α > 0. A lnear utlty functon has the form u (x) = a x. A CES (constant elastcty of substtuton) utlty functon has the form u(x ) = ( (a x ) ρ ) 1/ρ, where < ρ < 1, ρ 0. The Cobb-Douglas utlty functon has the form u (x) = (x ) a, where a 0 and a = 1. Related Work. Substantal work has been done on extendng equlbrum models to handle scenaros where good purchases are subect to taxaton [19, 20, 25, 26, 27]. Such efforts have provded exstental results [25, 26], evdence of tax-nduced multplcty [27] and of the loss of Pareto-optmalty of equlbra [19, 20]. Buldng upon ths body of results, appled models have been desgned to explctly take nto account tax dstortons (see for nstance the popular GAMS-MPSGE programmng envronment). Prevous work wthn theoretcal computer scence, whch was ntated by [9], has been focussed on restrctons under whch the market equlbrum problem, n ts verson wthout taxes, can be solved n polynomal tme. These restrctons nclude () dstrbutonal economes (the Fsher settng) where the traders have homogeneous utlty functons [10, 18, 8, 13], () exchange economes whch satsfy weak gross substtutablty [16, 15, 7, 4], and () exchange economes wth some famles of CES and nested CES utlty functons [3, 17]. For a more comprehensve lst of results, see [6]. Our Results. In ths paper we analyze the effects of tax dstortons on scenaros whch, n the absence of taxes, admt polynomal tme algorthms. Frst of all, an exchange economy wth unform ad valorem taxes can be effcently transformed nto an equvalent exchange economy wthout taxes, wth the same utlty functons and dfferent ntal endowments, obtaned by redstrbutng the orgnal ones (Secton 2.1). Therefore algorthms for the exchange model that do not depend on the dstrbuton of endowments carry through the model wth unform taxes. We prove that a dstrbutonal economy (whch s equvalent to Fsher s model) wth unform ad valorem taxes and homothetc consumers can be effcently transformed nto an equvalent two-trader exchange economy wthout taxes (Secton 3.1). We then develop a polynomal-tme algorthm for approxmatng the equlbrum (Secton 3.2). To analyze some parameters related to the accuracy of the algorthm, we use the tool of mplct dfferentaton, whch we beleve wll fnd more general applcatons (Secton 3.4). Note that a two-trader exchange economy wth homothetc consumers admts multple dsconnected equlbra [14]. The example n [14] can be modfed to model an exchange economy that s equvalent to a dstrbutonal economy wth unform ad valorem taxes. Therefore our algorthm provdes the frst sgnfcant example 2

4 of polynomal tme computaton of equlbra n a settng wth multple dsconnected equlbra. We then show that an n-good exchange economy wth dfferentated ad valorem taxes and m Cobb-Douglas consumers can be effcently transformed nto an equvalent (n+1)-good exchange economy wthout taxes, wth m Cobb-Douglas consumers (Secton 2.2). Snce the equlbrum for a Cobb-Douglas exchange economy can be computed n polynomal tme [11], our reducton shows that the same s possble for the model wth non-unform taxaton. We beleve that ths reducton s not possble for more general utlty functons, ncludng some of the commonly studed ones. Consequently, we do not at present have polynomal tme algorthms or approxmaton schemes for the model wth non-unform taxes when the traders have CES functons wth 1 ρ < 0, even though there are explct convex programs and polynomal tme algorthms for the correspondng pure exchange model [3]. The same happens for certan nested CES functons [17]. The stuaton s notably better when the utlty functons satsfy weak gross substtutablty. Here too, methods based on explct or mplct convex programs seem to be unavalable, n contrast wth the pure exchange model. Ths s true even for the case of lnear utlty functons, where explct convex programs are known for the pure exchange model [23, 15]. Interestngly, n spte of ths, we are able to show that the combnatoral algorthm of Garg and Kapoor [13] for lnear utltes can be adapted to handle the dfferentated tax scenaro (Secton 4). We also consder the case of economes wth weak gross substtutablty and demand functons that do not change too rapdly wth prces. In ths case, t s possble to obtan a poly-tme approxmaton scheme va a smple prce adustment (tâtonnement) scheme. The man tool s Lemma 7 from [1], whch guarantees convergence of a normalzed verson of tâtonnement, where the prce of one good (called the numérare) s kept fxed. By expressng the role of taxaton n terms of a fcttous good (the tax good of Secton 2.2) whch acts as the numérare, we can then analyze the convergence of the normalzed process. The lemma above becomes thus relevant because t does not requre gross substtutablty wth respect to the good whose prce s not part of the dynamcs. For lack of space, ths last result s omtted from the extended abstract. 2 Exchange Economes wth Taxes on Consumpton We descrbe the model of an exchange economy wth ad valorem taxes as presented by Kehoe ([19], pp ), and dstngush between the unform case, where the tax rate s unform across consumers, and the non-unform case, where the tax rate s dfferentated among consumers. 2.1 Unform Ad Valorem Taxes Consder a trader wth utlty functon u (x ) and ntal endowment w. Let τ 0 be the unform ad valorem tax assocated wth the consumpton of good. Ths means that f consumer purchases x unts of good at prce π, she wll spend on ths good the amount π x + τ π x. We postulate the presence of a specal actor, the government, whch wll rebate the tax revenues to consumers. Let θ 0, wth θ = 1, be the share of total tax revenues rebated to consumer as a lump sum. Then the classcal consumer s maxmzaton problem gets modfed as follows: 3

5 max u (x ) (1) s.t. π (1 + τ )x π w + θ R, (2) where x R n +, and R s the total amount of revenues dstrbuted by the government. In ths context, the market equlbrum problem conssts of fndng ( π, x, R) such that at prces π, x solves (1) (2), (optmalty and budget constrant are satsfed for all consumers); x = w, (the market clears all the goods); R = π τ x (the amount of taxes dstrbuted s equal to the amount of taxes collected). We now show that the equlbra for such an economy are n a one-to-one correspondence wth the equlbra of an exchange economy wthout taxes, where the traders have a dfferent set of ntal endowments, obtaned by a sutable redstrbuton of the orgnal ones. Ths correspondence has been establshed n [5], where models of taxaton were one of the targets of some expermental work. Whenever the equlbra of two economes are n a one-to-one correspondence, and can be mmedately computed one from the other, we say that the two economes are equvalent. Proposton 1 [5] Let E = E(u ( ),w,τ,θ) be an exchange economy wth unform ad valorem taxes τ = (τ 1,...,τ n ), and tax shares θ = (θ 1,...,θ m ). E s equvalent to an exchange economy wthout taxes E = E (u ( ),w ) where w = w τ 1+τ + θ 1+τ w. Proof : The proof s n Appendx A. 2.2 Dfferentated Taxes We now consder a more general model n whch the taxes on purchases are dfferent for each trader. We call ths scheme specfc taxaton or dfferentated ad valorem taxaton. In an exchange economy wth specfc taxes, τ 0 s the tax rate mputed to trader on purchase of good. The settng for an exchange economy wth specfc taxes dffers from that wth unform taxes n the budget constrant, whch s now gven by π (1 + τ )x π w + θ R. (3) At equlbrum, we must have R = π τ x. The lack of unformty of ths model, whch dfferentates between consumers, prevents the possblty of a drect reducton to a pure exchange economy, obtaned by redstrbutng the ndvdual endowments, as n Proposton 1. Nevertheless, ths model can be made smlar to a pure exchange economy wth an extra good. Indeed, note that the budget constrant can be rewrtten as π x + Rx,n+1 π w + Rθ (4) 4

6 P where x,n+1 = π τ x R. If we nterpret R as the prce of an addtonal (fcttous) good, that we call the tax good, and θ and x,n+1 as the -th trader s ntal endowment and demand of such good, then nequalty (4) corresponds to the budget constrant of a consumer n a pure exchange economy wth an extra good. To get a reducton to an exchange economy wthout taxes, one would now need to exhbt a utlty functon, defned on n + 1 goods, whch, combned wth the budget constrant (4), gves P π τ x R a demand of x (π,r) for the frst n goods, and x,n+1 (π,r) = for the tax good. We do not know f such a reducton s possble n general. However we show below (Proposton 2) that t can be done n the case of exchange economes where the traders have Cobb- Douglas utlty functons. Proposton 2 Let n and m be the number of goods and the number of traders, respectvely. Let u ( ), = 1,...,m, be Cobb-Douglas utlty functons. We denote by E n,m = E n,m (u ( ),w,τ,θ) a Cobb-Douglas exchange economy wth dfferentated ad valorem taxes τ = (τ 1,...,τ n ), and tax shares θ = (θ 1,...,θ m ). E n,m s equvalent to an exchange economy wthout taxes E n+1,m = E n+1,m (v ( ),w ) where the v ( ) s are Cobb-Douglas utlty functons, and w = (w 1,...,w n,θ ). Proof : The proof s n Appendx B. 3 Collnear endowments dstorted by unform taxaton The general reducton of Proposton 1 shows that unform taxaton does not affect algorthms whch compute equlbrum prces for exchange economes wthout explotng any partcular property of the ntal endowments. Therefore, several results for pure exchange economes extend to the model wth unform taxaton. One nterestng case where the redstrbuton of ntal endowments potentally carres negatve computatonal consequences s that of exchange economes wth collnear endowments and homothetc preferences. In ths settng an equlbrum (wthout taxes) can be computed n polynomal tme by convex programmng [8], based on certan aggregaton propertes of the economy whch mply the exstence of a representatve consumer [12]. The redstrbuton of endowments assocated wth taxaton clearly destroys the collnearty of endowments, and thus the collapse to a sngle consumer s problem. We show that n an exchange economy wth collnear endowments and homothetc consumers, the model wth unform taxaton s equvalent to an exchange economy wth two representatve consumers. Buldng upon ths property, we then show how to compute an approxmate equlbrum n polynomal tme for a wde famly of problems. 3.1 Reducton to two representatve consumers Recall that a dstrbutonal economy s an exchange economy where the ntal endowments of the traders are collnear. In other words, the k-th trader has endowments of the form w k = γ k w, for k = 1,...,m, where w = (w 1,...,w n ) descrbes the overall amount of each good n the market, and γ k s a postve constant less then one. We have k γ k = 1. In ths scenaro, the relatve ncomes of the traders are constants, so that the model s equvalent to Fsher s model. 5

7 If we specalze the model of Secton 2.1 to an economy wth collnear ntal endowments, then the redstrbuton descrbed by Proposton 1, gves w = γ w τ 1+τ + θ 1+τ w. Notce that the matrx whose columns represent the new ntal endowments of the traders has rank at most two. Indeed all the columns are lnear combnatons of the vectors z and s, whose -th entres are z = w 1+τ, and s = τ 1+τ w, respectvely. Note that w = γ z + θ s. (Note also that w = z + s and verfy that w = w.) Thus the effect of unform taxaton on economes wth proportonal endowments amounts to an ncrease of the rank of the endowment matrx from one to two. Whenever the consumers are homothetc, exchange economes wth a rank two endowment matrx can be reduced to a two-trader economy, accordng to the followng scheme: 1. Let z be the n-vector whose -th component s s wτ 1+τ. w 1+τ, and s be the n-vector whose -th component 2. For all k, splt the k-th trader nto two traders, whch have the same utlty functon of the orgnal trader and ntal endowments γ k z, and θ k s, respectvely. Ths procedure produces two groups of m traders each, where the traders n each group have proportonal endowments. 3. Based on the propertes n [12], aggregate all the consumers from each group nto one representatve consumer, wth endowment gven by the sum of ther endowments and utlty functon obtaned by aggregatng the utlty functons as n [12]. Ths gves a two-trader economy. These arguments lead to the followng result. Proposton 3 Let u ( ), = 1,...,m, be log-homogeneous utlty functons. Let E m = E(u ( ),w,τ,θ,γ) be an m-trader dstrbutonal economy wth unform ad valorem taxes τ = (τ 1,...,τ n ), tax shares θ = (θ 1,...,θ m ), and ncome shares γ = (γ 1,...,γ m ). E m s equvalent to a twotraders exchange economy wthout taxes E 2 = E 2 (v 1( ),v 2 ( ),z,s), where v 1 and v 2 are the log-homogeneous utlty functons of the two consumers, and z and s are ther ntal endowment vectors. Here, v 1 (x) (resp. v 2 (x)) s defned to be the maxmum of γ u (x ) (resp. θ u (x )) over all x 1,...,x m R n + such that x = x. 3.2 The algorthm The reducton summarzed n Proposton 3, combned wth some results of Mantel [21] on two-trader economes, suggest the followng algorthm, whch we call RSR (Reduce-Solve- Reconstruct), for the computaton of an approxmate equlbrum. For the analyss, we assume that τ > 0 for each. Let τ mn = mn τ and τ max = max τ. Algorthm RSR 1. The nput s gven n terms of m log-homogeneous utlty functons u, = 1,...,m, and vectors w, γ, θ, τ. 2. Apply the transformaton of Proposton 3, whch returns an economy wth two homothetc consumers (wth utlty functons v 1 and v 2, and ntal endowments z and s) and n goods. 3. Consder the followng constraned maxmzaton problem: max αv 1 (x 1 ) + (1 α)v 2 (x 2 ) s.t. x 1 + x 2 = w x 1, x 2 0 6

8 For a gven 0 α 1, let x 1 (α) and x 2 (α) be maxmzng allocatons, and let π(α) be the vector of shadow prces (Lagrange multplers). It can be shown that π(α) x 1 (α) = α, π(α) x 2 (α) = 1 α, and thus π(α) w = 1. Moreover, x 1 (α) and x 2 (α) have the rght shape they are proportonal to the optmal bundles demanded by the two traders at the prce π(α). Let B 1 (α) = π(α) (z x 1 (α)) and B 2 (α) = π(α) (s x 2 (α)) be the functons expressng the (postve or negatve) savngs of consumer 1 and 2, respectvely. Note that B 1 (α)+b 2 (α) = 0, snce z+s = w. Thus, f B 1 (α) = 0, then we have π(α) x 1 (α) = π(α) z, and π(α) x 2 (α) = π(α) s. Thus, x 1 (α) and x 2 (α) are not merely proportonal to the optmal bundles, but they are the optmal bundles of the two traders at prce π(α). Thus π(α) s an equlbrum for the two trader economy [22]. We therefore fnd an approxmate equlbrum for the two-trader economy by fndng a value of α such that B 1 (α) and B 2 (α) are suffcently close to zero. In such a case, π(α), x 1 (α) and x 2 (α) form an approxmate equlbrum, provded that the functons B (α) are smooth enough (see the extensve dscusson below). The search for an approprate value of α can be done by the bsecton method,.e., bnary search, guded by the value of B (α), computed from the values x (α) and π(α) returned by the soluton of the maxmzaton problem above. The applcablty of bsecton method bulds upon some results by Mantel [21] on the global convergence of the welfare adustment process when appled to two-trader economes. 4. From the soluton to the two-trader problem, reconstruct the soluton to the 2m-trader problem,.e., the correspondng allocatons, and then to the m-trader problem wthout taxes. 5. Compute approxmate equlbrum prces π, = 1,..., for the orgnal economy wth taxes, by scalng prces π (α),.e., π = π(α) 1+τ. 3.3 Analyss of algorthm RSR For any prce vector π, t s easy to see that π z π w les n the nterval [α 1 mn = 1+τ max,α max = 1 1+τ mn ]. Thus f B 1 (ᾱ) = 0, then ᾱ = π(ᾱ) x 1(ᾱ) π(ᾱ) w = π(ᾱ) z π(ᾱ) w les n the range [α mn,α max ]. It s also easy to verfy that B 1 (α mn ) 0 and B 1 (α max ) 0. So we perform our bnary search n the nterval [α mn,α max ]. The bnary search s descrbed and analyzed n Appendx C, where we show that Algorthm RSR computes an ε-approxmate equlbrum n tme of the order of T(n)(log M + log 1 ε log α mn + log (1 α max) ), where T(n) s the polynomal bound on the tme requred to solve the convex program, and M s an upper bound on the absolute value of the dervatve of B 1 (α) n the nterval [α mn,α max ]. The next secton takes a close look at the parameter M that nfluences the runnng tme. 3.4 Senstvty of the welfare maxmzaton problem The runnng tme of algorthm RSR depends on the logarthm of M, where M s an upper bound on B 1 (α) for α n [α mn,α max ]. We now show how to estmate M for a wde famly of utlty functons. The two-trader maxmzaton problem occurrng n step 3 of Algorthm RSR, when wrtten n ts expanded form, becomes the 2m-trader maxmzaton problem: max α γ u 1 (x 1 ) + (1 α) θ u 2 (x 2 ) s.t. x x1 m + x x2 m = w x 1, x2 0 7

9 where u 1 () = u 2 () = u (). Let x l (α) denote the soluton of ths problem, and π(α) the correspondng shadow prces. We wll smply denote x l (α) by xl and π(α) by p. Snce B 1(α) = p z α, we can upper bound B 1 p (α) by boundng the elements of the vector α. Let us consder the frst order optmalty condtons for the problem above, whch we wll denote by y {}}{ H( x,p,α) = 0. These equatons can be explctly wrtten as: αγ u 1 (x 1 ) p = 0; (1 α)θ u 2 (x 2 ) p = 0; x x1 m + x x2 m w = 0. By Implct Dfferentaton, from the frst order condtons we obtan the followng equaton: y H(x,p,α) α y(α) + H(x,p,α) = 0. (5) α Let A and B denote the Hessan αγ 2 u 1 (x 1 ), and (1 α)θ 2 u 2 (x 2 ), respectvely. Equaton 5 takes the form: A 1 I A m B 1 I I B m I I... I I... I x 1 1 α... x 1 m α x 2 1 α... x 2 m α p α γ 1 u 11(x 1 1). γ m u 1m(x 1 m) θ 1 u 21(x 2 1). θ m u 2m(x 2 m) = Assume that the A s and B s be nonsngular,.e., that they are negatve defnte. In ths case the matrx of the lnear system above has an nverse, whch s 2 A 1 1 A 1 1 KA 1 1 A 1 1 KA 1 2 A 1 1 KB 1 m A 1 1 K , 7 4 Bm 1 KA Bm 1 Bm 1 KBm 1 Bm 1 K 5 KA KBm 1 K ) 1. where K = ( A A 1 m + B Bm 1 Let now d 1 = γ u 1 (x 1 ), and d 2 = θ u 2 (x 2 ). We obtan the expresson p α = K( A 1 d B 1 d 2 ), (6) from whch we can upper bound the absolute value of any element of p α n terms of A 1, B 1, K, and u l (x l ), where denotes the spectral norm of a matrx, whch, n the case of sem-defnte matrces, concdes wth the spectral radus. In order to bound the spectral norm of K n terms of those of A and B, we now use a classcal result from lnear algebra. The theorem we need (see [24], p.192) states that f C 1 and 8

10 C 2 are real symmetrc matrces wth egenvalues λ 1 λ 2... λ n and µ 1 µ 2... µ n, respectvely, then the egenvalues s 1 s 2... s n of C 1 + C 2 satsfy λ k + µ n s k λ k + µ 1. For any matrx Q, let λ n (Q) denote ts smallest egenvalue. Then, f Q s postve defnte, 1 the spectral norm of ts nverse s gven by λ n(q). We apply the theorem to the postve defnte matrx K, and obtan (the calculaton s shown for m = 1, for the sake of clarty) ( A 1 1 B 1 1 ) 1 = λmax ( A 1 1 B 1 1 ) 1 = 1 λ mn ( A 1 1 B 1 1 ) 1 λ mn ( A 1 1 ) + λ mn( B1 1 ) = 1 1 A + 1 mn{ A 1, B 1 }. 1 B 1 Let now A m = mn A, B m = mn B, A 1 M = max A 1, and B 1 M = max B 1. From Equaton 6, we then get the crude upper bound p α 2mdmn{ A m, B m }max{ A 1 M, B 1 M }, = 1,...,m, where d s the maxmum of the d l s. We have shown how to bound B 1 (α) n terms of the Hessans of the utlty functons u l evaluated at x l. For utlty functons wth negatve defnte Hessans, ths gves an approach for boundng B 1 (α). We show how we can proceed even further for an nterestng class of such functons. (Incdentally, recall that negatve defnteness of the Hessan s a suffcent condton for strct concavty of the utlty functon, but t s not necessary.) Let f(x 1,...,x n ) be a concave log-homogeneous utlty functon. Its Hessan F(x 1,...,x n ) s negatve sem-defnte. Consder the functon g(x 1,...,x n ) = 1 n (log x log x n ). Its Hessan G(x 1,...,x n ) s the dagonal matrx whose (,)-th entry s 1, so that the maxmum nx 2 1 egenvalue of G s at most, where c s the mnmum consumpton of any good n x, and nc 2 1 ts mnmum egenvalue s at least, where C s the maxmum consumpton. nc 2 Let now consder the functon h(x 1,...,x n ) = (1 σ)f(x 1,...,x n ) + σg(x 1,...,x n ), for a small postve constant σ less than one. Ths functon s log-homogeneous, and ts Hessan H(x 1,...,x n ) s negatve defnte (agan by applyng the result n [24]). Let f max = max 2 f. x x We obtan: H (1 σ)nf max + σ nc 2, H 1 1 (1 σ)λ mn ( F) + σλ mn ( G) nc2 σ. (7) Therefore the functon h (whch s ntended as an approxmaton of the concave log-homogeneous functon f) s such that: () the Hessan H s a negatve defnte matrx (f there s postve consumpton of each of the goods);() the spectral norm of H can be bounded n terms of the mnmum consumpton, and the absolute value of the maxmum entry of F, whle the spectral norm of ts nverse can be bounded n terms of the maxmum consumpton. In order to lower bound the mnmum consumpton c we can proceed as follows. Let us consder the maxmzaton problem of a consumer wth a utlty functon of the form h(x) = (1 σ)f(x) + σg(x), where f and g are as above, and budget constrant π x = I. The frst order condtons for ths problem are of the form (1 σ) f(x) x k + σ 1 nx k = λπ k. 9

11 If we multply both sdes of ths equaton by x k, and then sum over k, we obtan that λ = 1 I. Snce f(x) x k 0, we get x k I σ nπ k. Snce α [α mn,α max ], the mnmum expendture I mn over all traders at α s bounded below by mn mn{γ α mn,θ (1 α max )}. Snce the total expendture of the 2m traders n the welfare maxmzaton problem above s one, then p k w k 1, from whch we must have x k I σw k n. Therefore, the mnmum consumpton s lower bounded by σi mnw mn where w mn s the quantty of the most scarce good. The maxmum consumpton C can be upper bounded by observng that π x 1. Now notce that at least one of the 2m traders has an expendture of at least 1/2m. Ths trader wll σ 2π nm n, σ demand at least unts of good. If π s to be an equlbrum, we must have 2π nm w. Therefore we obtan C 2nmwmax σ, where w max the quantty of the most abundant good. Fnally, for a wde famly of utlty functons, we can upper bound f max n terms of mnmum and maxmum consumptons. The reader s nvted to check ths when f s the log of a CES functon. 4 Lnear Exchange Economes wth Dfferentated Taxes We consder an economy wth m traders and n goods where each trader has a lnear utlty functon. Let u = a x denote the utlty functon of the -th trader. Each trader has the ntal endowment w R n +. Let τ 0 denote the tax rate of the -th trader for the consumpton of the -th good. And let θ denote the share of the -th trader n the overall tax collected. We have θ = 1. An equlbrum s a prce vector π = (π 1,...,π n ) and a number R 0 at whch there are bundles x R n + for each trader such that (1) x maxmzes u (x) over all x R n + such that the cost π (1 + τ )x of bundle x s at most the ncome θ R + π w ; (2) x w for each good ; and (3) T (x,π) = R, where T (x,π) s defned to be τ π x, the tax that has to pay to consume x at prce π. For ε > 0, we defne an ε-approxmate equlbrum to be a prce vector π = (π 1,...,π n ) and a number R 0 at whch there are bundles x R n + for each trader such that (1) π x + T(x,π) (1 + ε)(θ R + π w ), and u (x ) (1 ε)v (π,r), where v (π,r) s the maxmum value of u (x) over all x R n + such that π (1 + τ )x θ R + π w ; (2) x (1 + ε) w for each good ; and (3) (1 ε)r T (x,π) (1 + ε)r. We now descrbe our algorthm, an adaptaton of the aucton based algorthm of Garg and Kapoor [13], for computng an approxmate equlbrum of the model. The analyss of the algorthm assumes that a > 0 for each and. Let τ max denote max, τ. For smplfyng some expressons, we also assume, wthout loss of generalty, that w = 1 for each. Let w mn = mn, w. Our analyss also assumes that w mn > 0. Let κ = 1/w mn. ε Let δ = 90n(1+τ. The algorthm has varables π max)κ for the prces, and a varable R that stands for the tax money that s dstrbuted to the buyers. The algorthm starts wth all prces set to 1. From tme to tme, t ncreases the prce of some good by a multplcatve factor of 1 + δ. It has varables y and h correspondng to the amounts of good allocated to at the current prce π and the prevous prce π /(1 + δ), respectvely. Let x = y + h. Let T (y,h,π) = τ (π y + π 1+δ h ), the tax that pays n consumng y and h. Let a D (π) = { (1 + τ )π a k (1 + τ k )π k for 1 k n}. Intalze. Let π = 1 for 1 n, R = 1, y = 0 and h = 0 for each and. Phase 1: 10

12 1. We make a call to the procedure Allocatemore(), descrbed below. 2. If T (y, h, π) = R, let R R(1 + δ) and go to Step 1 of Phase If for each, we have (π y + π 1+δ h ) + T (y, h, π) = θ R + π w, the algorthm ends. 4. If for some, we have (π y + π 1+δ h )+T (y, h, π) < θ R+ π w, consder any D (π). An nspecton of Allocatemore() tells us that we must have h = 0 for every trader and y = 1. We call Raseprce(). If π k > 0 for every good k, we ump to Step 1 of Phase 2. Otherwse, return to Step 1 of Phase 1. Phase 2: 1. If R δ π w for each, let R T (y, h, π); the algorthm ends. 2. Make a call to Allocatemore(). 3. If T (y, h, π) = R, the algorthm ends. 4. If for each, we have (π y + π 1+δ h ) + T (y, h, π) = θ R + π w, the algorthm ends. 5. If for some, we have (π y + π 1+δ h )+T (y, h, π) < θ R+ π w, consder any D (π). We must have h = 0 for every trader and y = 1. We call Raseprce() and return to Step 1 of Phase 2. To complete the descrpton of the algorthm, we need to specfy the two procedures Allocatemore and Raseprce. The procedure Allocatemore. In ths procedure, we frst solve a lnear program, whch uses as data the current values of the varables π, y, h, and R. The lnear program has varables y and h for each trader and good. The lnear program s: X Maxmze X, y Subect to X (y + h ) 1 for each X X (y + h ) = 1 for each such that π > 1 X T (y, h, π) R π y + π 1 + δ h + T (y, h, π) θ R + X π w for each y = 0 for each and D (π). h = 0 for each and such that h = 0. y 0 for each, h 0 for each, As we argue below, ths lnear program wll always be feasble. Thus the maxmzaton s well defned. After solvng the lnear program, we set y = y and h = h for each and. Ths completes the descrpton of the procedure Allocatemore. The procedure Raseprce(). In ths procedure, we set π π (1 + δ), h y for each, and y 0 for each. Ths completes the descrpton of Raseprce. 11

13 Phases 1 and 2 of our algorthm are smlar to the basc algorthm of Garg and Kapoor [13] - a call to Allocatemore() replaces the sequence of steps n ther algorthm occurrng between two prce rases. Our algorthm needs to track the relaton of T (y,h,π) the tax pad as a consequence of consumpton to R, the tax that s dstrbuted as ncome. The reason we have Phase 2 s that at the end of Phase 1, T (y,h,π) can be sgnfcantly smaller than R. Theorem 4 For any ε > 0, our algorthm computes an ε-approxmate equlbrum for the model n tme that s polynomal n the nput sze, 1/ε, and κ. For lack of space, the proof of ths theorem s gven n Appendx D. In the process, we state some nvarants that help n understandng the algorthm. References [1] K. Arrow; H. Block; L. Hurwcz, On the Stablty of the Compettve Equlbrum, II, Econometrca, Vol. 27, No. 1. (Jan., 1959), pp [2] K.J. Arrow and G. Debreu, Exstence of an Equlbrum for a Compettve Economy, Econometrca 22 (3), pp (1954). [3] B. Codenott, B. McCune, S. Penumatcha, and K. Varadaraan, Exstence, Multplcty and Computaton of Equlbra for CES Exchange Economes, FSTTCS [4] B.Codenott, B. McCune and K. Varadaraan, Market Equlbrum va the Excess Demand Functon. Proc. STOC [5] B.Codenott, B. McCune and K. Varadaraan, Computng Equlbrum Prces: Does Theory Meet Practce? Proc. ESA [6] B.Codenott, S. Pemmarau and K. Varadaraan, Algorthms Column: The Computaton of Market Equlbra, ACM SIGACT News 35(4) December [7] B. Codenott, S. Pemmarau, K. Varadaraan, On the Polynomal Tme Computaton of Equlbra for Certan Exchange Economes. Proc. SODA [8] B. Codenott and K. Varadaraan, Effcent Computaton of Equlbrum Prces for Markets wth Leontef Utltes. ICALP [9] X. Deng, C. Papadmtrou, and S. Safra. On the complexty of equlbra. Proc. STOC [10] N. R. Devanur, C. H. Papadmtrou, A. Saber, V. V. Vazran, Market Equlbrum va a Prmal-Dual-Type Algorthm. FOCS 2002, pp [11] B. C. Eaves, Fnte Soluton of Pure Trade Markets wth Cobb-Douglas Utltes, Mathematcal Programmng Study 23, pp (1985). [12] E. Esenberg, Aggregaton of Utlty Functons. Management Scences, Vol. 7 (4), (1961). [13] R. Garg and S. Kapoor, Aucton Algorthms for Market Equlbrum. In Proc. STOC, [14] S. Gerstad, Multple Equlbra n Exchange Economes wth Homothetc, Nearly Identcal Preference, Unversty of Mnnesota, Center for Economc Research, Dscusson Paper 288, [15] K. Jan, A polynomal tme algorthm for computng the Arrow-Debreu market equlbrum for lnear utltes. FOCS

14 [16] K. Jan, M. Mahdan, and A. Saber, Approxmatng Market Equlbra, Proc. APPROX [17] K. Jan and K. Varadaraan. Equlbra for Economes wth Producton: Constant-Returns Technologes and Producton Plannng Constrants. To appear n SODA 06. [18] K. Jan, V. V. Vazran and Y. Ye, Market Equlbra for Homothetc, Quas-Concave Utltes and Economes of Scale n Producton, SODA [19] T.J. Kehoe, Computaton and Multplcty of Equlbra, n Handbook of Mathematcal Economcs Vol IV, pp , Noth Holland (1991). [20] T.J. Kehoe, The Comparatve Statcs Propertes of Tax Models, The Canadan Journal of Economcs 18(2), (1985). [21] R. R. Mantel, The welfare adustment process: ts stablty propertes. Internatonal Economc Revew 12, (1971). [22] T. Negsh, Welfare Economcs and Exstence of an Equlbrum for a Compettve Economy, Metroeconomca 12, (1960). [23] E. I. Nenakov and M. E. Prmak. One algorthm for fndng solutons of the Arrow-Debreu model, Kbernetca 3, (1983). [24] B.N. Parlett, The Symmetrc Egenvalue Problem, Prentce Hall (1980). [25] J.B. Shoven, J. Whalley, Appled General Equlbrum Models of Taxaton and Internatonal Trade, Journal of Economc Lterature 22, (1984). [26] J.B. Shoven, J. Whalley, Applyng General Equlbrum, Cambrdge Unversty Press (1992). [27] J. Whalley, S. Zhang, Tax nduced multple equlbra. Workng paper, Unversty of Western Ontaro (2002). 13

15 Appendx A: Proof of Proposton 1 From the market clearance condton, t follows that the tax revenue at equlbrum must be R = π τ x = π τ w. We can therefore elmnate the equlbrum requrement R = π τ x by substtutng π τ w for R n the rght hand sde of nequalty (2), whch thus becomes π (w + θ τ w ). After dvdng and multplyng each term of ths expresson by 1 + τ, the budget constrant can be rewrtten as π (1 + τ )x π (1 + τ )w, where w = w τ + θ w. 1 + τ 1 + τ We have therefore reduced the equlbrum n the orgnal economy to an equlbrum n a pure exchange economy, where the -th trader now has an ntal endowment w. It s easy to check that w = w for each, so that the two economes have the same quantty of each good. Fnally we have that (π 1,...,π n ) s an equlbrum for the orgnal economy E f and only f ((1 + τ 1 )π 1,...,(1 + τ n )π n ) s an equlbrum for the pure exchange economy E. Appendx B: Proof of Proposton 2 Suppose that consumer has a Cobb-Douglas utlty functon of the form n u (x ) = where α 0, n =1 α = 1. It s known (and easy to see) that the resultng demand s such that consumer wll spend on good an α fracton of her ncome. In vew of the budget constrant (3), the -th consumer ncome s π w + θ R, wth apparent prces π (1 + τ ). Thus, α x (π,r) = π (1 + τ ) (π w + θ R), = 1,...,n. We want to derve a consumer s problem that looks lke the consumer s problem of an economy wthout taxes, and that nduces the same demand as the orgnal one. The new problem wll have an addtonal good, the tax good, a utlty functon v( ) defned on n + 1 goods, and a budget constrant as n (4). The demand for the tax good has to be π τ x (π,r) R =1 x α α τ 1+τ x r (π,r) = = (π w + θ R). R Observe that ths demand looks exactly lke the demand generated by a Cobb-Douglas utlty functon: a constant dvded by the prce and multpled by the ncome. Thus, f we set α r = α τ 1+τ, α = α /(1 + τ ), the utlty functon v that we want s the Cobb-Douglas functon v(x 1,...,x n,x r ) = x α r x α r. 14

16 Appendx C: Analyss of algorthm RSR The correctness of the aggregaton and dsaggregaton processes executed n steps 2 and 4 of Algorthm RSR readly follows from Esenberg s results [12]. Consder now the two-trader problem of step 3, where we approxmately compute a zero of B 1 by means of the bsecton method, startng from the nterval [α mn,α max ]. Each bsecton step depends on the sgn of B 1 and s thus guded by the soluton to an nstance of the constraned maxmzaton problem. The next Lemma gves a stoppng crteron whch guarantees an ε-equlbrum: Lemma 5 If for some α we have 0 B 1 (α) α(1 α)ε then the prces π(α) and the allocatons x 1 (α) and (1 ε)x 2 (α) are an ε-equlbrum. Proof : The hypothess of the Lemma mples that B 1 (α) εmn{α,1 α}. The ε-optmalty of the frst consumer s guaranteed when B 1 (α) εα: snce α > 0, we have that 0 B 1 (α) εα s equvalent to 0 π(α) z π(α) x 1 (α) αε and, usng that α = π(α) x 1 (α), t s equvalent to ε π(α) z π(α) x 1(α) π(α) z. (8) On the other hand, f we want a smlar nequalty for the other trader, we observe that s equvalent to 0 B 2 (α) ε(1 α) (9) π(α) s π(α) x 2 (α) 1 π(α) s. (10) 1 ε Thus the bundles x 1 (α) and x 2 (α) nearly exhaust but do not sgnfcantly exceed the respectve budgets π(α) z and π(α) s. Snce they have the rght shape, we are at an O(ε)- equlbrum. Corollary 6 Suppose that B 1 ( ) s dfferentable n (0,1) and that there exsts M > 0 such that for all α [α mn,α max ] we have d dα B 1(α) M. If the bsecton method appled to B 1 n [α mn,α max ] stops when t determnes an nterval [α,β] such that β α α(1 α)ε M and B 1 (α) 0, B 1 (β) 0, then the concluson of Lemma 5 follows. Proof : Under the hypotheses, B 1 (α) B 1 (β) α(1 α)ε. In partcular, B 1 (α) α(1 α)ε and the concluson of Lemma 5 follows. Now we need to argue that from an ε-approxmate equlbrum to the two trader economy we can recover an ε-approxmate equlbrum to the orgnal economy wth taxes. We wll consder the mplcatons of undong the aggregatons and dsaggregatons nto equvalent economes one by one, n reverse order. 15

17 Suppose that the algorthm reaches the concluson of Lemma 5 at α = α 1. Let π = π(α 1 ), x 1 = x 1 (α 1 ), x 2 = x 2 (α 1 ), and α 2 = 1 α 1. Wrte the maxmzaton problem n Step 5 of the algorthm n ts extensve form: max s.t. α 1 ( γ u (x )) + (1 α 1)( x = x 1 x = x 2 x 1 + x 2 = w x, x 0 θ u (x )) Let x = x (α 1) and x = x (α 1). Obvously, we have x = x 1 and x = x 2. It can be shown, usng the log-homogenety of the utlty functons, that π x = γ α 1 and π x = θ α 2. For the dsaggregaton nto 2m traders, we further note that x and x have the rght shape they are proportonal to the bundles demanded by the two nstances of the -th trader. Snce α 1 = π x 1 roughly equals π z (recall the proof of Lemma 5), t follows that π x = γ α 1 roughly equals γ π z. Smlarly, π x = θ α 2 roughly equals θ π s. So we are at an approxmate equlbrum of the 2m trader economy. For the reaggregaton of pars of traders, t s mmedate that π s an approxmate equlbrum of the m trader pure exchange economy. The approxmate utlty maxmzng bundles are x = x + x. The natural canddate equlbrum prces for the orgnal economy wth taxes are π = π /(1 + τ ). We set R = π τ w. Snce w = x, R = π τ x. Obvously, we have x = w. We can easly verfy that the x are approxmate utlty maxmzng bundles for each trader at prces π and R. The runnng tme of Algorthm RSR s domnated by step (3), whch can be executed n tme proportonal to T(n)(log M +log 1 ε +log 1 1 α mn +log ), where T(n) s the polynomal bound on the tme requred to solve the convex program. (1 α max) Appendx D: Proof of Theorem 4 Invarants. We begn by statng some mportant nvarants mantaned by the algorthm. Proposton 7 At any stage n the algorthm, we have (1) y > 0 = D (π); (2) h > 0 = D (( π 1+δ,π )), where (r,π ) s defned to be the vector that dffers from π only n ts -th component, whch s r; (3) π > 1 = x = 1; (4) The lnear program n Allocatemore() s always feasble. Proof : The varables y, h, and π, are only changed by calls to Allocatemore() or Raseprce(). Note that the lnear program n Allocatemore() s feasble f (1) and (3) hold: y = y and h = h yelds a feasble soluton. On the other hand, t s evdent that Allocatemore() preserves (1), (2), and (3). 16

18 It s also easy to check that a call to Raseprce() preserves (1), (2), and (3). Arguments are smlar to those n [13]. Proposton 8 At any stage of the algorthm, we have: (1) x 1 for each ; (2) T (y,h,π) R. Proof : Easly follows by nspectng Allocatemore() and Raseprce(). Proposton 9 At any stage of the algorthm, except possbly after the executon of the (termnal) assgnment statement n Step 1 of Phase 2, we have, for each, that π y + π 1+δ h + T (y,h,π) θ R + π w. Proposton 10 Let π max denote the largest component of π and π mn the smallest component of π. At any stage of the algorthm, we have (1) πmax π mn 2amax(1+τmax) a mn ; (2) R max{1,2nτ max π max }. Proof : To prove (1), t s suffcent to consder the case when π max > 1. Suppose π max = π > 1. Then by statement (3) of Proposton 7, we must have x > 0 for some. By Proposton 7, we ether have D (π) or D (( π 1+δ,π )). In ether case, t s true that D (( π 1+δ,π )). Thus a (1+δ) (1+τ )π whch completes the proof of (1). a k (1+τ k )π k for every k. Rearrangng, we have π π k a (1+τ k )(1+δ) a k (1+τ ), For (2), observe that f at some stage R > 1, then R = (1+δ)R, where R = T (ŷ,ĥ, ˆπ), where ŷ,ĥ, and ˆπ are the values at some prevous stage of y, h, and π. Usng statement (1) of Proposton 8, we see that T (ŷ,ĥ, ˆπ) T (ˆx, ˆπ) = τ ˆx ˆπ τ maxˆx ˆπ = τ max ˆπ τ max ˆπ. Thus R (1 + δ)τ max ˆπ (1 + δ)nτ max π max, snce π max π ˆπ for every (the algorthm never decreases a prce). Ths completes the proof. Runnng Tme. For each phase, we bound the number of tmes we call Raseprce and the number of tmes we rase the value of R. Frst observe that the moment π mn exceeds 1 n Phase 1, we ump to Phase 2 va Step 4 of phase 1. Thus π mn (1 + δ) 2 n Phase 1, and by statement (1) n Proposton 10, we have π max 4amax(1+τmax) a mn n Phase 1. Thus, by statement (2) n Proposton 10, we have R max{1, 8nτmax(1+τmax)amax a mn } 8n(1+τmax)2 a max a mn n Phase 1. Snce R s never rased n Phase 2, we have that ˆx R α = 8n(1 + τ max) 2 a max a mn (11) 17

19 throughout the algorthm. We now use ths bound to bound π max n Phase 2. Phase 2 stops when R δ π w for each, due to the stoppng rule n Step 1 of Phase 2. To have R > δ π w for some, we must have R > δπ mn w mn, whch mples that π mn < R δw mn. Thus we must have π mn < (1+δ)R δw mn 2R δw mn throughout Phase 2. Usng statement (1) n Proposton 10, and Equaton 11, we have π max 16n(1+τmax)2 a max a mn δw mn. Snce ths bound s larger than the bound we establshed for π max n Phase 1, we conclude that π max β = 16n(1 + τ max) 2 a max a mn δw mn (12) It follows that the number of calls to Raseprce plus the number of tmes R s rased s bounded by n log 1+δ α+log 1+δ β, whch s a polynomal n the nput sze, 1/ε, and κ. It follows that the runnng tme of the algorthm s bounded by a polynomal n the nput sze, 1/ε, and κ. Correctness. We now argue that the algorthm termnates at an ε-approxmate equlbrum. Suppose that the algorthm termnates ether va Step 3 of Phase 1 or Step 4 of Phase 2. Then, we have for each, T (y,h,π) + π x T (y,h,π) + (π y + π 1+δ h ) = θ R + π w. Addng over all, we get T (y,h,π) + π x R + π w. (13) Multplyng the -th nequalty n statement (1) of Proposton 8 by π, addng these nequaltes and the nequalty n statement (2) of Proposton 8, we get T (y,h,π) + π x R + π w. (14) Thus for Equaton 13 to hold, the nequaltes n Proposton 8 must be equaltes. So we have for each that x = w. Also, we have T (x,π) (1 + δ)t (y,h,π) = (1 + δ)r, whereas T (x,π) T (y,h,π) = R, so we have R T (x,π) (1 + δ)r. For each trader, we have T (x,π) + π x T (y,h,π) + π x θ R + π w. Thus the bundle x costs at least her ncome. But by Proposton 7, we have that f x > 0 then ether D (π) or D (( π 1+δ,π )). Thus only spends money on goods that nearly optmze her bang for the buck. It follows va standard arguments that u (x ) v (π,r) 1+δ. On the other hand, T (x,π) + π x (1 + δ)(t (y,h,π) + (π y + π 1+δ h )) = (1 + δ)(θ R + π w ), so the bundle x does not cost more than 1 + δ tmes ts ncome. Thus we are at a δ-approxmate equlbrum. Now suppose that the algorthm termnates at ether Step 1 of Phase 2 or Step 3 of Phase 2. Note that we enter Phase 2 when π k > 1 for every good k, so by statement (3) of Proposton 7 we have x k = 1 for every good k. Moreover, termnaton at ether of these steps mples that T (y,h,π ) = R, whch mples that R T(x,π) (1 + δ)r. We now reason about the budget constrants of the traders. By Proposton 9, we have, for each T (x,π) + π x (1 + δ)(t (y,h,π) + π y + π 1 + δ h ) (1 + δ)(θ R + π w ), 18

20 f we termnate va Step 3 of Phase 2. If we termnate after makng the assgnment to R n Step 1 of Phase 2, then we have R δ π w for each ust before the assgnment, whch mples that θ R + π w (1 + δ) π w ust before the assgnment. So the assgnment decreases the ncome of each trader by at most a multplcatve factor of (1+δ). Snce the nequalty n Proposton 9 holds ust pror to the assgnment, we have that for each, T (y,h,π) + π y + π 1 + δ h (1 + δ)(θ R + π w ) after the assgnment. It follows that on termnaton, we have for each T (x,π) + π x (1 + δ) 2 (θ R + π w ) (1 + 3δ)(θ R + π w ). (15) Note that ths nequalty also holds f we termnate va Step 3 of Phase 2. Havng argued that no trader exceeds hs budget by much, we now argue that each trader almost exhausts hs budget. We have (θ R+ π w π x T (x,π)) = π ( w x )+R T (x,π) 0. (16) It follows that for any trader, we have θ R + π w π x T (x,π) max{0, π x + T (x,π) θ R π w }. Usng Equaton 15, and Proposton 10, we have θ R + π w π x T (x,π) 3δ(θ R + π w ) = 3δ(R + π ) 3δ(2nπ max + 2nπ max τ max + π ) 3δ3nπ max (1 + τ max ) ε 10 π kw k ε 10 ( π w + θ R). Here k s the good for whch π max = π k. We have argued that each trader spends at least a fracton 1 1+ε of her budget on bundle x. But by Proposton 7, we have that f x > 0 then ether D (π) or D (( π 1+δ,π )). Thus only spends money on goods that nearly optmze her bang for the buck. It follows that u (x ) (1 cε)v (π,r), where c 1 s some constant. Thus we are at a cε-approxmate equlbrum. 19