Commodity money equilibrium in a convex trading post economy with transaction costs

Transcription

1 Avalable onlne at Journal of Mathematcal Economcs 44 (008) Commodty money equlbrum n a convex tradng post economy wth transacton costs Ross M. Starr Economcs Department 0508, Unversty of Calforna, San Dego, 9500 Glman Dr., La Jolla, CA , USA Receved 8 June 006; receved n revsed form December 007; accepted 7 December 007 Avalable onlne 7 January 008 Abstract Exstence and effcency of general equlbrum wth commodty money s nvestgated n an economy where N commodtes are traded at N(N )/ commodty-parwse tradng posts. Trade s a resource-usng actvty recoverng transacton costs through the spread between bd (wholesale) and ask (retal) prces. Budget constrants, enforced at each tradng post separately, mply demand for a carrer of value between tradng posts. Exstence of general equlbrum s establshed under conventonal convexty and contnuty condtons whle structurng the prce space to account for dstnct bd and ask prce ratos. Commodty money flows are dentfed as the dfference between gross and net nter-post trades. 008 Elsever B.V. All rghts reserved. JEL classfcaton: C6; D5; E40 Keywords: General equlbrum; Money; Transacton cost; Tradng post; Bd; Ask; Commodty money [An] mportant and dffcult queston...[s] not answered by the approach taken here: the ntegraton of money n the theory of value... Gerard Debreu, Theory of Value (959). Introducton It s well-known that the Arrow Debreu model of Walrasan general equlbrum cannot account for money. Professor Hahn (98) wrtes The most serous challenge that the exstence of money poses to the theorst s ths: the best developed model of the economy cannot fnd room for t. The best developed model s, of course, the Arrow Debreu verson of a Walrasan general equlbrum. A frst, and...dffcult...task s to fnd an alternatve constructon wthout... sacrfcng the clarty and logcal coherence...of Arrow Debreu. Ths paper pursues development of foundatons for a theory of money based on elaboratng the detal structure of an Arrow Debreu model. The elementary frst step s to create a general equlbrum where there s a well-defned demand Tel.: ; fax: E-mal address: rstarr@ucsd.edu /$ see front matter 008 Elsever B.V. All rghts reserved. do:0.06/j.jmateco

2 44 R.M. Starr / Journal of Mathematcal Economcs 44 (008) for a medum of exchange a carrer of value between transactons. Ths s arranged by replacng the sngle budget constrant of the Arrow Debreu model wth the requrement that the typcal household or frm pays for ts purchases drectly at each of many separate transactons. Transactons take place at commodty-parwse tradng posts. Then a well-defned demand for meda of exchange (commodty mones, not necessarly unque) arses endogenously as an outcome of the market equlbrum. The use of meda of exchange s partcularly evdent when the structure of demands s characterzed by an absence of double concdence of wants (Jevons, 875). Meda of exchange are characterzed as the carrer of value between transactons (not fulfllng fnal demands or nput requrements themselves), the dfference between gross and net trades. Related general equlbrum models wth transacton cost nclude Foley (970); Hahn (97, 973); Kurz (974); Starrett (973); Starr (003c). The tradng post model s ntended to provde a parsmonous addton to the Arrow Debreu model suffcent to generate a theory of money. The monetary structure of trade s shown to be a consequence of the prce theory general equlbrum, not a separate assumpton... Structure of the tradng post model In the tradng post model, transactons take place at commodty-parwse tradng posts (Shapley and Shubk, 977; Walras, 874; Wcksell, 936) wth budget constrants (you pay for what you get n commodty terms) enforced at each post. Prces bd (wholesale) and ask (retal) are quoted as commodty rates of exchange. Trade s arranged by frms, typcally buyng at bd prces and sellng at ask prces, ncurrng costs (resources used up n the transacton process) and recoupng them through the bd/ask spread. Market equlbrum occurs when bd and ask prces at each tradng post have adjusted so that all tradng posts clear... Structure of the proof The structure of the proof of exstence of general equlbrum follows the approach of Arrow and Debreu (954); Debreu (959); Starr (997). The usual assumptons of contnuty, convexty (tradtonal but by no means nnocuous n ths context), and no free lunch/rreversblty are used. The prce space at a tradng post for exchange of one good at bd prce for another at ask prce s the unt -smplex, allowng any possble nonnegatve relatve prce rato. The prce space for the economy as a whole then s a Cartesan product of unt -smplces. The attanable set of tradng post transactons s compact. As n Arrow and Debreu (954), the model consders transacton plans of frms and households artfcally bounded n a compact set ncludng the attanable set as a proper subset. Prce adjustment to a fxed pont wth market clearng leads to equlbrum of the artfcally bounded economy. But the artfcal bounds are not a bndng constrant n equlbrum. The equlbrum of the artfcally bounded economy s as well an equlbrum of the orgnal economy..3. Concluson: The medum(a) of exchange The general equlbrum specfes each household and frm s tradng plan. At the concluson of trade, each has acheved a net trade. Gross trades nclude tradng actvty that goes to payng for acqustons and acceptng payment for sales rather than drectly mplementng desred net trades. It s easy to calculate gross trades and net trades at equlbrum. For households, the dfference gross trades mnus net trades represents tradng actvty n carrers of value between trades, meda of exchange (perhaps ncludng some arbtrage). Snce frms perform a market-makng functon wthn tradng posts, dentfcaton of meda of exchange used by frms s not so straghtforward. After nettng out ntra-post trades, the remanng dfference between nter-post gross and net trades represents the frms trade flows of meda of exchange. In some examples (see Starr, 003a,b, 008) the medum of exchange may be a sngle specalzed The present model s an alternatve to the fat money models of overlappng generatons (Wallace, 980) and of search (Kyotak and Wrght, 989). There a unque unbacked fat money of postve value s typcally assumed and presented as a bubble. The models allow, as well, a nonmonetary no-trade equlbrum where the fat money has a value of zero. In the present model, the exstence of meda of exchange and ther values are endogenously determned. It s possble to accommodate n an Arrow Debreu settng an ntrnscally worthless paper money tradng at a postve value and used as a common medum of exchange. The ratonale s that taxes payable n paper money provde backng for a postve value, and low transacton cost ensures use as medum of exchange (Goldberg, 005; Smth, 776; Starr, 003a,b). Consstent wth Ockham s razor.

3 R.M. Starr / Journal of Mathematcal Economcs 44 (008) commodty (the common medum of exchange). The approach of the present model s ntended to provde a Walrasan general equlbrum theory of (commodty) money as a medum of exchange. It s suffcently general to nclude both a sngle common medum of exchange and many goods smultaneously actng as meda of exchange. When wll meda of exchange actually be used n the tradng post economy? Two condtons seem to be suffcent: desrablty of trade, net of transacton costs; absence of double concdence of wants. The logc s smple. If trade s desrable at prevalng equlbrum prces (net of transacton costs ncludng the transacton cost of meda of exchange) and there s no double concdence of wants, then n order for trade to proceed fulfllng the budget constrant at each tradng post separately, meda of exchange wll be used as carrers of value between tradng posts. However, the absence of double concdence of wants depends on prevalng prces as well as endowments and technology. It s problematc to characterze necessary and suffcent ntal condtons so that absence of double concdence s fulflled. Hence, the relance on smple llustratve examples below. Nevertheless, the examples are ntended to be robust. The parametrc examples should be contaned n an open subset of parameter space where the results of the example reman vald. Conversely, there are two cases where tradng post equlbra wll have no use of meda of exchange: full double concdence of wants (subject to drect trade experencng no hgher transacton costs than ndrect trade); and a no-trade equlbrum. Agan, necessary and suffcent condtons, a pror, to fulfll these characterstcs are not mmedately evdent.. Tradng posts There are N tradeable goods denoted,,...,n. They are traded for one another parwse at tradng posts. {, j} (or equvalently {j, }) denotes the tradng post where goods and j are traded for one another. There are N(N )/ dstnct tradng posts. 3. Prces Goods are traded drectly for one another wthout dstngushng any sngle good as money. Let Δ represent the unt -smplex. At tradng post {, j}, the (relatve) ask prce of good and (relatve) bd prce of good j are represented as p {,j} (a {,j} j ) Δ. In a (mnor) abuse of notaton, the orderng of and j n the superscrpt on p wll matter. The relatve ask prce of good j and bd prce of are represented as p {j,} (a {,j} j,b {,j} ) Δ. Thus, there are two operatve prce -smplces at each tradng post. The full prce space then s Δ N(N ), the N(N )- fold Cartesan product of Δ wth tself; ts typcal element s p Δ N(N ). Then the ask prce of at {, j} n unts of j s a {,j} /b {,j} j and the bd prce of s b {,j} /a {,j} j. Prces can then be read as rates of exchange between goods, dstngushng between bd (sellng or wholesale) prces and ask (buyng or retal) prces. Thus the ask prce of a hamburger mght be 5.0 chocolate bars and the bd prce 3.0 chocolate bars. Note that the ask prce of a chocolate bar then s the nverse of the bd prce of a hamburger. That s, the ask prce of a chocolate bar s hamburger and the bd prce of a chocolate bar s 0. hamburger. 4. Budget constrants and tradng opportuntes The budget constrant s smply that at each parwse tradng post, at prevalng prces, n each transacton, payment s gven for goods receved. That s, at tradng post {, j}, an ask/bd prce par s quoted p {,j} (a {,j} j ) Δ expressng the ask prce of n terms of j and a bd prce of j n terms of. A frm or household s tradng plan (y, x) R N(N ) specfes the followng transactons at tradng post {, j}: y {,j} (at ask prces, retal) n, y {,j} j (at ask prces, retal) n j, x {,j} (at bd prces, wholesale) n, x {,j} j (at bd prces, wholesale) n j. Postve values of these transactons are purchases. Negatve values are sales. At each tradng post (of two goods) there are four quanttes to specfy n a tradng plan. Then the budget constrant facng frms and households at each tradng post s that value delvered must equal value receved. That s 0 = (a {,j} j ) (y {,j},x {,j} j ), 0 = (a {,j} j,b {,j} ) (y {,j} j,x {,j} ) (B) (B) says that purchases of at the bd prce are repad by sales of j at the ask prce, purchases of at the ask prce are repad by sales of j at the bd prce.

4 46 R.M. Starr / Journal of Mathematcal Economcs 44 (008) Gven a prce vector p Δ N(N ) the array of trades fulfllng (B) s the set of trades fulfllng the N(N ) local budget constrants at the tradng posts. Denote ths set M(p) {(y, x) R N(N ) (y, x) fulflls (B) at p for all, j =,...,N,= j} 5. Frms The heavy lftng n ths model s done by frms. They perform the market-makng functon, ncurrng transacton costs. The populaton of frms s a fnte set denoted F, wth typcal element f F. Thus, frm f s technology set may specfy that f s purchase of labor (retal) n exchange for on the {, labor} market and purchase of and j wholesale on the {, j} market allows f to sell and j (retal) on the {, j} market. That s how f can become a market-maker. If there s a suffcent dfference between bd and ask prces so that f can cover the cost of ts nputs wth a surplus left over, that surplus becomes f s profts, to be rebated to f s shareholders. 5.. Transacton and producton technology Frm f s technology set s Y f. We assume P.0 Y f R N(N ) The typcal element of Y f s (y f,x f ), a par of N(N )-dmensonal vectors. The N(N )-dmensonal vector y f represents f s transactons at ask (retal) prces; the N(N )-dmensonal vector x f represents f s transactons at bd (wholesale) prces. The -dmensonal vector y f {,j} represents f s transactons at ask (retal) prces at tradng post {, j}; the -dmensonal vector x f {,j} represents f s transactons at bd (wholesale) prces at tradng post {, j}. The typcal co-ordnates y f {,j},x f {,j} are f s acton wth respect to good at the {, j} tradng post. Snce f may act as a wholesaler/retaler/market-maker, entres anywhere n (y f {,j},x f {,j} ) may be postve or negatve subject of course to constrants of technology Y f and prces M(p). Ths dstngushes the frm from the typcal household. The typcal household can only sell at bd prces and buy at ask prces. The entry y f {,j}, represents f s actons at ask prces wth regard to good at tradng post {, j}. y f {,j} > 0 represents a purchase of at the {, j} tradng post (at the ask prce). y f {,j} < 0 represents a sale of at the ask prce. The entry x f {,j}, represents f s actons at bd prces wth regard to good at tradng post {, j}. x f {,j} > 0 represents a purchase of at the tradng post (at the bd prce). x f {,j} < 0 represents a sale of at the bd prce. A frm that s an actve market-maker at {, j} wll typcally buy at the bd prce and sell at the ask prce. A frm that s not a market-maker may have to pay retal lke the rest of us sellng at the bd prce and buyng at the ask prce. In addton to ndcatng the transacton possbltes, Y f ncludes the usual producton possbltes. The usual assumptons on producton technology apply. For each f F, assume P.I Y f s convex. P.II 0 Y f, where 0 ndcates the zero vector n R N(N ). P.III Y f s closed. The aggregate technology set s the sum of ndvdual frm technology sets. Y f F Y f. It fulflls the famlar no free lunch and rreversblty condtons. P.IV [(a)] f (y, x) Y and (y, x) /= 0, then y {,j} + x {,j} > 0 for some, j. [(b)] f (y, x) Y and (y, x) /= 0, then (y, x) / Y. Denote the ntal resource endowment of the economy as r R N +. Then we defne the attanable producton plans of the economy as Ŷ {(y, x) Y r j (y {,j} + x {,j} ) all =,,...,N}

5 R.M. Starr / Journal of Mathematcal Economcs 44 (008) Attanable producton plans for frm f can then be descrbed as Ŷ f { (y f,x f ) Y f there s (y k,x k ) Y k for each k F, k /= f, so that (y k,x k ) + (y f,x f. ) Ŷ k F,k /= f Lemma 5.. Assume P.0 P.IV. Then Ŷ and Ŷ f are closed, convex, and bounded. Proof. Starr (997), Theorems 8. and Frm maxmand and transactons functon The frm formulates a producton plan and a tradng plan. The frm s opportunty set for net yelds after transactons fulfllng budget s E f (p) [M(p) Y f ] R N(N ) +. That s, consder the frm s producton, purchase, and sale possbltes, net after payng for them, and what s left s the net yeld. Usng the sgn conventons we have adopted purchases are postve co-ordnates, sales are negatve co-ordnates the net yeld s then the negatve co-ordnates (supples) n a tradng plan that are not absorbed by payments due and the net purchases not requred as nputs to the frm. The supples are subtracted out, so the surpluses enter E f (p) as postve co-ordnates. A typcal element of these surplus supples s denoted (y,x ) E f (p). In ths notaton y and x are dummes, not actual marketed supples and demands. Now consder (y,x ) E f (p). In each good, the net surplus avalable n good s w f N j= (y {,j} + x {,j} ) and frm f s surplus s the vector w f of these co-ordnates. To gve ths noton a functonal notaton, let W(y,x ) w f descrbed here. There are N tradng posts where each good s traded, at N rates of exchange. The noton of proft s not well defned. In the absence of a sngle famly of well-defned prces, t s dffcult to characterze optmzng behavor for the frm. Fautes de meux we ll gve the frm a scalar maxmand wth argument p, y,x. Frm f s assumed to have a real-valued, contnuous maxmand v f (p; y,x ). We take v f to be strctly monotone and concave n (y,x ). Ths descrpton of v f ncludes as a specal case the usual frm proft functon (when p s suffcently unform across tradng posts that the usual noton of proft s well defned). The frm s optmzng choce (whch may not be well defned) then s G f (p) {argmaxv f (p; y,x ) E f (p)}. Ths results n the frm s market behavor (wthout any constrant requrng actons to stay n a bounded range) descrbed by H f (p) {(y, x) M(p) [(y, x) + (y,x )] Y f, (y,x ) G f (p)}. Ths marketed plan then results n the market and dvdend plan S f (p) {(y, x; w) (y, x) H f (p), [(y, x) + (y,x )] Y f, (y,x ) G f (p); w = W(y,x )} The logc of ths defnton s that (y,x ) 0 s the surplus left over after the frm f has performed accordng to ts technology and subject to prevalng prces. It s possble that S f (p) s not well defned, snce the opportunty set may be unbounded. In the lght of Lemma 5., there s a constant c>0 suffcently large so that for all f F, Ŷ f s strctly contaned n a closed ball, denoted B c of radus c centered at the orgn of R N(N ). Followng the technque of Arrow and Debreu (954), constraned market behavor for the frm wll consst of lmtng ts producton choces to Y f B c. Ths leads to the constraned surplus Ẽ f (p) [[M(p) B c ] [Y f B c ]] R N(N ) +. G f (p) {argmaxv f (p; y,x ) Ẽ f (p)}. H f (p) {(y, x) M(p) [(y, x) + (y,x )] Y f B c, (y,x ) G f (p)}.

6 48 R.M. Starr / Journal of Mathematcal Economcs 44 (008) The frm s constraned (to B c ) market behavor then s defned as S f (p) {(y, x; w) (y, x) H f (p), [(y, x) + (y,x )] Y f B c, (y,x ) G f (p); w = W(y,x )}. Lemma 5.. Assume P.0 P.IV. Then Ẽ f (p) s convex-valued, nonempty, upper and lower hem-contnuous. Proof. (Note to the reader: The notaton x o appears n two dstnct unrelated forms n ths proof. Usually as part of frm f s planned transactons, but later completely dstnctly n the quotaton from Green and Heller (98).) Upper hem-contnuty and convexty follow from closedness and convexty of the underlyng sets. 0 Ẽ f (p) always, so nonemptness s fulflled. Lower hem-contnuty requres some work. Let p ν p o,(y o,x o ) Ẽ f (p o ). We seek (y ν,x ν ) Ẽ f (p ν ) so that (y ν,x ν ) (y o,x o ). If (y o,x o ) = 0, exstence of (y ν,x ν ) (y o,x o ) s trvally satsfed. Suppose nstead (y o,x o ) 0 (the nequalty apples co-ordnatewse). Then n an ɛ-neghborhood of (y o,x o ), for ν suffcently large, we seek to show that there s (y ν,x ν ) Ẽ(p ν ). (y ν,x ν )sthe requred sequence. To demonstrate ths, note that Ẽ(p ν ) s defned as the ntersecton of a convex-valued correspondence lower hem-contnuous n p wth a constant convex set. When (y o,x o ) 0 and (y o,x o ) Ẽ f (p o ) t follows that the relatve nteror of Ẽ f (p o ) s nonempty. It s suffcent then to apply Green and Heller (98), p. 48, (8, lower), If γ,=,, are two l.h.c. convex-valued correspondences such that ntγ (x o ) ntγ (x o ) /=, then γ γ s l.h.c. at x o. Lemma 5.3. Assume P.0 P.IV. Then G f (p), H f (p), S f (p) are well defned, nonempty, upper hem-contnuous, and convex-valued for all p Δ N(N ). Proof. Note compactness of B c. Apply Theorem of the Maxmum, contnuty and concavty of v f. Lemma 5.4. Assume P.0 - P.IV. Let [ G f (p) + H f (p)] Ŷ f /=. Then [ G f (p) + H f (p)] [G f (p) + H f (p)]. Proof. Recall that B c strctly ncludes Ŷ f. Then the result follows from convexty of Y f and Ŷ f and concavty of v f (p; y,x ). The proof follows the model of Starr (997), Theorem 8.3. Let (y,x ) G f (p), (y,x ) H f (p), [(y,x ) + (y,x )] Ŷ f B c. Use a proof by contradcton. Suppose not. Then there s (y, x) Y f so that (y, x) (y o,x o ) = (y,x ), where v f (p; y,x ) >v f (p; y,x ), (y,x ) E f (p), and (y o,x o ) M(p). But convexty of Y f and concavty of v f mply that on the chord between (y,x ) and (y, x) there s [α(y,x ) + ( α)(y, x)] B c for α>0 where v f (p;[α(y,x ) + ( α)(y,x )]) > v f (p; y,x ). Ths s a contradcton Incluson of constraned supply n unconstraned supply (y, x; w) S f (p) mples (y, x) B c, a bounded set. w R N + s f s profts. By constructon there s K>0 so that w s contaned n the nonnegatve quadrant of a ball of radus K centered at the orgn, denoted B K R N +. Lemma 5.5. Let p Δ N(N ) such that S f (p) [Ŷ f B K ] /=. Then S f (p) s well defned and nonempty. Further S f (p) S f (p). Proof. Lemma Households There s a fnte set of households, H, wth typcal element h. 6.. Endowment and consumpton set h H has a possble consumpton set, taken for smplcty to be the nonnegatve quadrant of R N, R N +. h H s endowed wth r h >> 0 assumed to be strctly postve to avod boundary problems. h H has a share α hf 0 of frm f, so that h H αhf =.

7 6.. Trades and payment constrant R.M. Starr / Journal of Mathematcal Economcs 44 (008) h H chooses (y h,x h ) R N(N ) subject to the followng restrctons. A household always balances ts budget, sells wholesale and buys retal: () 0 x h{,j} for all, j. () y h{,j} 0 for all, j. () (y h,x h ) M(p) 6.3. Maxmand and demand Household h s share of profts from frm f s part of h s endowment and enters drectly nto consumpton. When the profts of all frms f F, w f n (y f,x f ; w f ), are well defned, f dstrbutes to shareholders w f, and h s consumpton of good s [ ] (v) c h r h + f F αhf w f + N j= x h{,j} + N j= y h{,j} However, prces p may be such that S f (p) s not well defned for some f. Then we may wsh to dscuss the constraned verson of (v), where w f comes from (y f,x f ; w f ) S f (p). (v ) c h r h + [ f F αhf w f ] + N j= x h{,j} + N j= y h{,j} In addton, h s consumpton must be nonnegatve. (v) c h 0. The nequalty apples co-ordnatewse. C.I For all h H, h s maxmand s the contnuous, quas-concave, real-valued, strctly monotone, utlty functon u h (c h ). u h : R N + R. h s planned transactons functon s defned as D h : Δ N(N ) R N#F R N(N ). Let w denote (w,w,w 3,...,w f,...,w #F ). D h (p, w) {(y h,x h ) R N(N ) (y h,x h ) maxmzes u h (c h ), subject to (), (), (), (v) and (v)}. However, D h (p, w) may not be well defned when opportunty sets are unbounded (when ask prces of some goods are zero) and w may not be well defned for p such that S f (p) s not well defned for some f. To treat ths ssue, let B K #F be the #F-fold Cartesan product of B K, and defne D h : Δ N(N ) B K #F B c. D h (p, w) {(y h,x h ) (y h,x h )maxmzes u h (c h ), subject to (), (), (), (v ), (v), and (y h,x h ) B c }. The restrcton to B c n ths defnton assures that D h (p) represents the result of optmzaton on a bounded set, and s well defned. Lemma 6.. Assume P.0 P.IV, C.I. Then D h (p, w) s nonempty, upper hem-contnuous and convex-valued, for all p Δ N(N ), w B K #F. The range of D h (p, w) s compact. For (p, w) such that (y h,x h ) <c for (some) (y h,x h ) D h (p, w), t follows that D h (p, w) D h (p, w). Proof. (Note to the reader: Ths proof ncludes an unfortunate confuson of notaton. c wthout superscrpt denotes a large real number ndcatng the radus of B c, a ball strctly contanng all attanable transactons of the typcal frm. c h and c (wth superscrpt) denote consumpton vectors.) Apply Theorem of the Maxmum, notng contnuty and quas-concavty of u h, convexty of constrant sets defned by () (v) or by (), (), (), (v ), (v). Incluson of D h (p, w) nd h (p, w) follows the pattern of Starr (997), Theorem 9.(b). Proof by contradcton. Suppose not. Then there s (y,x ) D h (p, w) wth assocated c so that u h (c ) >u h (c h ). But recall (y h,x h ) <c. On the chord between (y h,x h ) and (y,x ) there s [α(y,x ) + ( α)(y h,x h )], >α>0, fulfllng (), (), (), (v ), (v), and [α(y,x ) + ( α)(y h,x h )] =c so that u(αc + ( α)c h ) >u(c h ). Ths s a contradcton.

8 40 R.M. Starr / Journal of Mathematcal Economcs 44 (008) Excess demand Let (p, w ) Δ N(N ) B K #F. Constraned excess demand and dvdends at (p, w ) s defned as Z : Δ N(N ) B K #F RN(N ) B K #F. Z(p, w ) {( (y f,x f ) + D h (p, w ),w,w,...,w f,...,w #F ) (y f,x f,w f ) S f (p)}. f F h H Lemma 7.. Assume P.0 P.IV, and C.I. The range of Z s bounded. Z s upper hem-contnuous and convex-valued for all (p, w ) Δ N(N ) B K #F. Lemma 7.. (Walras Law): Let (p, w ) Δ N(N ) B K #F. Let (y, x, w) Z(p, w ). Then for each, j =,..., N, /= j, we have 0 = (a {,j} j ) (y {,j},x {,j} j ), 0 = (a {,j} j,b {,j} ) (y {,j} j,x {,j} ) (W) Proof. The element (y, x)of(y, x, w) Z(p, w ) s the sum of elements (y f,x f )of S f (p) and (y h,x h )of D h (p, w ) each of whch s subject to (B). 8. Equlbrum Let Ξ denote a compact convex subset of R N(N ) so that Ξ B K #F ((y {,},x {,} ),...,(y {,j},x {,j} j ),...,(y {N,N} N,x {N,N} N )). Defne ρ : Ξ Δ N(N ) ρ(z) {p o Δ N(N ) for each, j =,,...,N, /= j, p o{,j} Δ maxmzes p {,j} (y {,j},x {,j} j ) subject to p {,j} Δ}. Lemma 8.. ρ s upper hem-contnuous and convex-valued for all z Ξ. Defne Γ : Δ N(N ) Ξ B K #F ΔN(N ) Ξ B K #F. Γ (p, z, w ) ρ(z) Z(p, w ). ncludes the range of Z. Let z Ξ, z Lemma 8.. Assume P.0 P.IV, and C.I. Then Γ s upper hem-contnuous and convex-valued on Δ N(N ) Ξ B #F K. Γ has a fxed pont (p,z,w ) and 0 = z. Proof. Upper hem-contnuty and convexty are establshed n Lemmas 7. and 8.. Exstence of the fxed pont (p,z ) then follows from the Kakutan fxed pont theorem. To demonstrate that z = 0, note Lemma 7. and strct monotoncty of u h and v j. Defnton: (p,w ) Δ N(N ) B K #F s sad to be an equlbrum f (0,w ) {( f F (yf,x f ) + h H Dh (p,w ),w,w,...,w f,...,w #F ) (y f,x f,w f ) S f (p )} where 0 s the orgn n R N(N ). Theorem 8.. Assume P.0 P.IV, C.I. Then there s an equlbrum (p,w ) Δ N(N ) B K #F. Proof. Apply Lemmas 5.5, 6. and 8.. Lemmas 8. provdes (p,z,w ) Δ N(N ) Ξ B K #F so that 0 = z, where (z,w ) {( f F (yf,x f ) + h H D h (p,w ),w,w,...,w f,...,w #F ) (y f,x f,w f ) S f (p )}. Then S f (p ) [Ŷ f B K ] /=, so by Lemma 5.5, S f (p ) S f (p ). 0 = z, mples that (y h,x h ) <c, so by Lemma 6., D h (p,w ) D h (p,w ). But then (0,w ) {( f F (yf,x f ) + h H Dh (p,w ),w,w,...,w f,...,w #F ) (y f,x f,w f ) S f (p )}. Then (p,w ) s an equlbrum.

9 8.. No-arbtrage condton n tradng post equlbrum R.M. Starr / Journal of Mathematcal Economcs 44 (008) At tradng post equlbrum proftable arbtrage by households should not be possble at prevalng equlbrum prces. Otherwse, arbtrarly large tradng profts would seem possble to the household. For smplcty, consder arbtrage among only two commodtes, wthout loss of generalty denoted and. There s only one tradng post {, } under consderaton so the superscrpt desgnatng the tradng post can be omtted to smplfy notaton. The prce vector s ((a,b ), (a,b )) Δ Δ where Δ s the unt -smplex. Recall that households sell at bd prces, b, b and buy at ask prces a, a. Then from the household sde the no-arbtrage condton can be stated as b a. a b Ths s demonstrated n the followng way. Consder a sngle household, omttng the household superscrpt for smplcty. We have the followng relatons from the structure of the model: x 0,x 0,y 0,y 0 b x = a y, b x = a y,x = a y,y = b x,x = a y,y = b x. b a b a Consder household arbtrage n good, to accumulate large profts n good. Set x = y = ξ>0. Then x = (a /b )ξ and y = (b /a )( ξ)ory + x = ξ[b /a a /b ] = arbtrage proft. Hence, the suffcent condton for arbtrage proft to be nonpostve s b /a a /b. Smlarly consder household arbtrage n good to accumulate large profts n good. Set x = y = ξ>0. Then x = (a /b )ξ and y = (b /a )( ξ) ory + x = ξ[b /a a /b ] = arbtrage proft. Hence a suffcent condton for arbtrage proft to be nonpostve s b /a a /b or equvalently b a. a b 9. Meda of exchange, commodty mones Let (y h,x h ) D h (p, w ) be household h s N(N )-dmensonal transacton vector. The x co-ordnates are typcally sales (negatve sgn) at bd prces; the y co-ordnates are typcally purchases (postve sgn) at ask prces. Then we can characterze h s gross transactons n good as y h{,j} x h{,j} γ h. j j Further, the absolute value of h s net transactons n good, s j j y h{,j} + j x h{,j} ν h. The N-dmensonal vector γ h wth typcal element γ h s h s gross trade. The N-dmensonal vector ν h wth typcal element ν h s h s net trade vector (n absolute value). μ h γ h ν h s h s flow of goods as meda of exchange, gross trades mnus net trades. Snce frms perform a market-makng functon, buyng and sellng the same good at a sngle tradng post, a more complex vew of ther transactons s needed to sort out tradng flows used as meda of exchange. In partcular, for frms, we should net out offsettng transactons wthn a sngle tradng post. Thus, for f F, f s gross transactons n, nettng out ntra-post transactons s [y f {,j} + x f {,j} ] γ f.

10 4 R.M. Starr / Journal of Mathematcal Economcs 44 (008) The correspondng net transacton s j [y f {,j} + x f {,j} ] ν f. The N-dmensonal vector γ f wth typcal element γ f s f s gross nter-post trade. The N-dmensonal vector ν f wth typcal element ν f s h s net nter-post trade vector (n absolute value). μ f γ f ν f s f s flow of goods as meda of exchange, gross (nter-post) trades mnus net trades. The total (N-dmensonal vector) flow of meda of exchange among households and frms s then h H μh + f F μf. Ths expresson, h H μh + f F μf, s the flow of commodty mones. Thus, the tradng post equlbrum establshes a well-defned demand for meda of exchange as an outcome of the market equlbrum. Meda of exchange (commodty mones) are characterzed as goods flows actng as the carrer of value between transactons (not fulfllng fnal demands or nput requrements themselves), the dfference between gross and net trades. 0. Walrasan equlbrum, tradng post equlbrum, and demand for meda of exchange 0.. Transacton costs, essental and nessental sequence economes The ssues of general equlbrum wth transacton cost, effcency of allocaton and the mplcatons for the role of money appear n Foley (970); Hahn (97, 973); Starrett (973). Foley (970) consders a statc equlbrum wth (consstent wth the Arrow Debreu treatment) a sngle market meetng. All of the formal structure of the Arrow Debreu economy s mantaned whle the transacton process s treated as a producton actvty. Each of N goods has a bd and ask (wholesale and retal) prce wth the resultng dmensonalty of the prce space at N. As n Debreu (959) the count N ncludes futures markets for all of the relevant goods. Foley (970) s dstnctve powerful nsght s that ths structure s mathematcally equvalent to the Arrow Debreu model. Assumng the usual contnuty and convexty assumptons, a compettve equlbrum exsts n the convex transacton cost economy, and the resultng allocaton s Pareto effcent. The noton of Pareto effcency here needs to take account of transacton costs: movng ownershp from one frm or household to another s a resource-usng actvty. Effcency conssts of effcent allocaton net of the necessary resource cost of reassgnng ownershp. Hahn (973) treats the reopenng of markets over tme n a sequence economy, dstngushng between essental and nessental sequence economes. The ssue treated s whether two otherwse dentcal economes have sgnfcantly dfferent equlbrum prces and resource allocaton dependng on the character of the budget constrant: a sngle Arrow Debreu budget for each household versus a tme-dated sequence of budget constrants n a sequence economy. In ths comparson t s necessary to take account of transacton costs, so the reference pont s not the conventonal Arrow Debreu equlbrum wthout transacton costs (Debreu, 959). Rather, t s the allocaton n an Arrow Debreu economy wth transacton costs (Foley, 970). Ths paper adopts the same usage. The effcency concept s subject to techncally necessary transacton costs. A tradng post equlbrum s nessental f the resultng allocaton s Walrasan, the same as n an Arrow Debreu (Foley) economy wth transacton costs. The equlbrum s nessental f the mult-faceted structure of the tradng post budget constrant has no effect n tself on the resultng allocaton of resources. Conversely, the tradng post equlbrum wll be descrbed as essental f the equlbrum resource allocaton s non-walrasan, dfferng because of the structure of budget constrants. Then the resource allocaton n an nessental tradng post economy s a Walrasan equlbrum allocaton and t s Pareto effcent by the Frst Fundamental Theorem of Welfare Economcs. Conversely, a tradng post economy s essental when the mult-faceted structure of budget constrants renders the equlbrum allocaton of resources dfferent from an Arrow Debreu equlbrum (takng full account of the effect of transacton costs, wth a complete array of futures markets). Then the equlbrum allocaton wll not be a Walrasan equlbrum and may be Pareto neffcent. The neffcency arses n ether of two ways: addtonal resources may be expended n fulfllment the multplcty of budget constrants, or the allocaton may be shfted (relatve to Walrasan equlbrum) to fulfll the addtonal constrants. Snce these crcumstances represent real resource allocatons to fulfll a purely admnstratve constrant, the reallocaton s regarded as Pareto neffcent. Ths treatment s smlar to Hahn (973) s treatment of sequence economes. A full

11 R.M. Starr / Journal of Mathematcal Economcs 44 (008) development of effcency condtons and detaled characterzaton of (n)essentalty s a sgnfcant topc, beyond the scope of ths paper. The array of economes subject to general equlbrum modelng ncludes essental and nessental tradng post economes wth resultant Walrasan and non-walrasan allocatons. Snce the desgnaton essental or nessental s based on the character of endogenous equlbrum prcng, t seems problematc to dstngush essental from nessental tradng post economes a pror. The alternatve s to revew examples, several of whch are presented below. 0.. Economes actvely usng meda of exchange The examples of Sectons 0.3. and 0.4. below llustrate the noton of tradng post economes usng meda of exchange n equlbrum. They are characterzed by economes where trade s mutually advantageous but drect trade between supplers and fnal demanders at tradng posts may be more costly n resources than ndrect trade through a lower transacton cost nstrument. Ths typcally reflects two elements of the example: drect exchange s not fully mutually satsfactory because of absence of double concdence of wants; transacton costs n some commodty may be lower than others, favorng ts use as a carrer of value n exchange. For a partcularly smple example, see Starr (008). It s dffcult fully to characterze the attrbutes of an economy, a pror, that wll lead to these condtons, hence the relance on examples. Nevertheless, the examples are ntended to be robust. The parameters of the examples are ntended to be elements of an open subset of parameter space where smlar results hold Pareto effcency of tradng post equlbrum wth transacton costless meda of exchange When there s a generally avalable zero-transacton cost medum of exchange, the tradng post equlbrum wll be nessental and the resultng allocaton of resources Pareto effcent (takng nto account transacton costs). The allocaton wll be a Walrasan equlbrum. Supposng that the transacton costs of meda of exchange n advanced monetary economes are low (f not nl), the zero-cost case should be a sgnfcant lmtng case. However mportant, the result s not deep. The presence of a costless medum of exchange means that prce ratos n a tradng post economy wll be the same as those of the correspondng Arrow Debreu economy. The example of Secton 0.3. below llustrates the effcency. The pont of comparson s an economy wth transacton costs, complete markets, effcent allocaton n general equlbrum, a sngle budget constrant for each household and well-defned proft maxmand for each frm, as n Foley (970). Then apply the Frst Fundamental Theorem of Welfare Economcs Example: A natural money absent double concdence of wants; Pareto effcent allocaton n tradng post equlbrum Let H {h =,,...,N} where rh h = and where uh (c h ) = 0ch+ h + N n/= h+,n= cn h for h =,...,99, and for h = N, u h (c h ) = 0c h + N n/=,n= cn h. There are N households named h =,,...,N; each endowed wth unt of good h and strongly preferrng good h + (mod N) to all others. There are N(N )/ frms denoted {, j},j>,,j =,,...,N. The transacton technology of {, j}, /= s Y {,j} {(y, x) for k =, j, 0 y k 0.8x k ; for k/=, j, y k = x k = 0}. For{, j},=,y {,j} {(y, x) for k =,y = x, for j/=, 0 y j 0.8x j ; for k/=, j, y k = x k = 0}. That s, for each par of goods there s a dstnct tradng post frm {, j} and there s no arbtrage by frms between posts. Trade n all goods except good experences a 0% loss n the tradng process. The resultng equlbrum prces, for, j /= are (a {,j} j ) = (5/8, 3/8). For =,j /= we have, (a {,j},b {,j} j ) = (/, /), (a {,j} j,b {,j} ) = (5/9, 4/9). For {, } we have (a {,},b {,} ) = (/, /), (a {,},b {,} ) = (5/9, 4/9). The trade flows for h =, 3,...,N, are (x h{h,} h,y h{h,} ) = (, ), (x h{,h+},y h{,h+} h+ ) = (, 0.8). For h = N, (x N{,N} N,y N{,N} ) = (, 0.8). For h =, (x {,},y {,} ) = (, 0.8). That s, drect trade of most goods for j s prohbtvely expensve, losng 40% of the goods n the transacton process. Indrect trade, through good, s more attractve snce good tself s transacton costless. The typcal pattern of trade then s that household h sells endowment, good h, for good, then sells good for the desred good, h +. In the process, only 0% of goods are

12 44 R.M. Starr / Journal of Mathematcal Economcs 44 (008) lost to transacton costs. In ths s example all trade goes through good, and for N ofn traders good s a medum of exchange. The allocaton s Pareto effcent. Is the tradng post equlbrum a Walrasan equlbrum? Indvdual agent tradng behavor n the tradng post model dffers from Walrasan behavor (e.g. n Foley, 970) snce t ncludes actve use of a medum of exchange, good. But those trades are costless and net out to zero. The resultng resource allocaton s fully consstent wth Walrasan equlbrum and n a Foley (970) economy (Arrow Debreu wth transacton costs) the allocaton could be supported by Walrasan equlbrum prces. The allocaton s Pareto effcent. Ths tradng post economy s nessental Pareto neffcency of tradng post equlbrum wth costly meda of exchange; An essental tradng post economy As n Hahn (973); Starrett (973) s analyss of a sequence economy, when the mult-faceted structure of the budget constrant n the tradng post economy sgnfcantly affects the real allocaton of resources, the resultng allocaton s Pareto neffcent. Ths occurs because real resources spent or reallocated n fulfllment of the admnstratve requrement of budget constrants represent a waste. The expendture or reallocaton s admnstratvely requred but techncally unnecessary Example: An essental tradng post economy; Pareto neffcent allocaton n tradng post equlbrum The followng example smply follows the format of the prevous example, except that there s no costless medum of exchange. The result s a non-walrasan Pareto neffcent allocaton. The mechansm of neffcency s transparent. Transactons wll use the medum of exchange and ncur the cost of dong so. The cost s a wasted resource; t s admnstratvely requred but fulflls no techncal functon. Let the populaton H and H s endowments and preferences be as descrbed n Secton There are N(N )/ frms denoted {, j},j>,,j =,,...,N. The transacton technology of {, j}, /= sy {,j} {(y, x) for k =, j, 0 y k 0.8x k ; for k/=, j, y k = x k = 0}. For {, j},=,y {,j} {(y, x) for k =,y = x, for j/=, 0 y + y j 0.9x 0.8x j ; for k/=, j, y k = x k = 0}. That s, for each par of goods there s a dstnct tradng post frm {, j} and there s no arbtrage by frms between posts. Trade n all goods except good experences a 0% loss of each good n the tradng process; tradng two goods ncurs two 0% losses, 0% of each. Trade n good wth any other good j experences a 30% loss n good j (a 0% savng compared to usng any good other than as medum of exchange, hence the desrablty of tradng through good f a medum of exchange s to be used). The resultng equlbrum prces, for, j /= are (a {,j} j ) = (5/8, 3/8). For =,j /= we have, (a {,j},b {,j} j ) = (/, /), (a {,j} j,b {,j} ) = (0/7, 7/7). For {, } we have (a {,} ) = (/, /), (a {,},b {,j} ) = (0/7, 7/7).,b {,j} The trade flows for h =, 3,...,N, are (x h{h,} h,y h{h,} ) = (, ), (x h{,h+},y h{,h+} h+ ) = (, 0.7). For h = N, (x N{,N} N,y N{,N} ) = (, 0.7). For h =, (x {,},y {,} ) = (, 0.7). That s, drect trade of most goods for j s prohbtvely expensve, losng 40% of the goods n the transacton process. Ths reflects the absence of double concdence of wants. A typcal household drectly tradng good h for good h + necessarly ncurs transacton costs on both sdes of the bargan. Indrect trade, through good, s more attractve snce good tself carres lower transacton costs. The typcal pattern of trade then s that household h sells endowment, good h, for good, then sells good for the desred good, h +. In the process, only 30% of good h + s lost to transacton costs. In ths example all trade goes through good, and for N out of N traders good s a medum of exchange. The allocaton s not however Pareto effcent. Some of the resources used n the transacton process, 0% of gross endowment, are techncally necessary to the reallocaton. It s not wasted. But the transacton costs assocated merely wth fulfllng the parwse tradng post budget constrant, 0% of total endowment, s admnstratvely necessary but not techncally necessary. It s a waste. The equlbrum allocaton represents the outcome n an essental tradng post economy. It s not Pareto effcent. Is the tradng post equlbrum a Walrasan equlbrum? Indvdual agent tradng behavor n the tradng post model dffers from Walrasan behavor (e.g. n Foley, 970) snce t ncludes actve use of a medum of exchange, good. Those trades net out to a loss. The resultng resource allocaton s nconsstent wth Walrasan equlbrum. In a Foley

13 R.M. Starr / Journal of Mathematcal Economcs 44 (008) (970) economy (Arrow Debreu wth transacton costs) the allocaton cannot be supported by Walrasan equlbrum prces and t s Pareto neffcent. Ths tradng post economy equlbrum s essental Economes not usng meda of exchange: Double concdence of wants and nactve trade Economes wth full double concdence of wants wll typcally not use meda of exchange n tradng post equlbrum. Supples are drectly exchanged for demands 3. Alternatvely, the economy may not use meda of exchange smply because trade s unattractve. There are two obvous cases: a Pareto effcent endowment and prohbtve transacton costs Full double concdence of wants wth lnear transacton costs Consder the followng economy wth full double concdence of wants. Let N be an even nteger. Let H {h =,,...,N} where rh h = and where for h odd uh (c h ) = 0ch+ h + N n/= h+,n= cn h, and for h even, uh (c h ) = 0ch h + N n/= h,n= cn h. There are N households named h =,,...,N; each endowed wth unt of good h and the odd numbered households strongly preferrng good h +, the even numbered households strongly preferrng good h. Drect trade wth the neghbor s the obvous polcy. Ths wll be true even f there s a low transacton cost nstrument avalable, so long as drect trade s no more costly than ndrect trade through the low transacton cost nstrument. Assume a populaton of frms and transacton technologes the same as n Secton The resultng equlbrum prces, for, j /= are (a {,j} j ) = ((5/9), (4/9)). For {, } we have (a {,} ((0/7), (7/7)), (a {,},b {,} ) = ((/), (/)). The trade flows for h odd, h/=, are (x h{h,h+} h For h = even, (x h{h,h } h,y h{h,h } h ) = (,.8), (x h{h,h } (, 0.7), (x {,},b {,} ) =,y h{h,h+} h+ ) = (,.8), (x h{h,h+} h+,y h{h,h+} h ) = (0, 0). h,y h{h,h } h ) = (0, 0). For h =,, (x {,},y {,} ) =,y {,} ) = (0, 0), (x {,},y {,} ) = (0, 0), (y {,},x {,} ) = (, ). All of the trade flows n ths allocaton are drect trade. There s no trade n meda of exchange. Ths reflects the endowment, demand, and transacton cost structure: there s a double concdence of wants, so there s lttle ncentve to trade ndrectly, and no transacton cost advantage to ndrect trade. Thus, the example generates a tradng post equlbrum wthout use of a medum of exchange. The tradng structure and resultng allocaton are Pareto effcent, and consttute a Walrasan equlbrum (allowng for transacton costs). The tradng post economy s nessental. That s, the trade flows and resultng allocatons would be the same allowng for smlar transacton technology n a unfed (Foley (970)) tradng settng Inactve trade: Pareto effcent endowment In an economy where there s no need for trade, there s no use for meda of exchange. If the endowment s Pareto effcent, there wll be no use of meda of exchange n a tradng post equlbrum Inactve trade: Prohbtve transacton costs A far more nterestng reason for a nl demand for meda of exchange s overwhelmng transacton costs. Costs hgh enough to dscourage all trade wll elmnate the demand for meda of exchange as well. Assume household populaton, tastes and endowments, the same as n Secton There are N(N )/ frms denoted {, j},j>,,j =,,...,N. The transacton technology of {, j}, all, j, s Y {,j} {(y, x) for k =, j, 0 y k 0.x k ; for k/=, j, y k = x k = 0}. That s, for each par of goods there s a dstnct tradng post frm {, j} and there s no arbtrage by frms between posts. Trade n all goods experences a 90% loss n the tradng process. Two sdes to the trade compounds the loss: 99% loss n two trades. The resultng equlbrum prces, for, j are (a {,j} j ) = ((99/00), (/00)). The endowment s the equlbrum allocaton. No household wshes to trade at a dscount of 99% but ths s just break-even for the frms consderng 3 Exceptons to ths generalzaton occur where multple trades through a medum of exchange ncur lower cost than a sngle drect trade. That reflects some cost assocated wth the nteracton between the goods traded drectly (e.g. gasolne and matches) or economes of scale n a hgh volume market wth a common medum of exchange (Starr, 003b).

14 46 R.M. Starr / Journal of Mathematcal Economcs 44 (008) the oppressve transacton technology. The allocaton s non-walrasan and s far from Pareto effcent one-step rearrangements for each good would be a grand Pareto mprovement, even ncurrng 90% transacton costs. But that calculaton gnores the 90% transacton cost on payment of qud pro quo, necessarly ncurred n a tradng post equlbrum. Ths calculaton reflects the dual problems of transacton costs and absence of double concdence of wants f there were a better match of supplers wth demanders even 90% transacton costs could be borne and mutually benefcal trades undertaken. But the absence of double concdence of wants means that each trade undertaken benefts drectly only one sde. Two trades and two sets of transacton costs must be ncurred n the tradng post economy, and transacton costs then swamp the gans from trade.. Concluson Ths essay creates a parsmonous model where a medum of exchange (commodty money) s an outcome of the (slghtly augmented) Arrow Debreu general equlbrum. The monetary structure of trade s a result of the prce theory general equlbrum. Monetary trade s not a separate assumpton; monetary exchange s an outcome, a drect mplcaton of the general equlbrum when there are multple dstnct budget constrants facng each agent. The trades of frms and households n a tradng post economy may be characterzed by many separate transactons, each fulfllng a separate budget constrant. In an economy of N commodtes there are N(N )/ tradng posts, one for each par of goods. The tradng post model reformulates the budget so that each of many separate transactons fulflls ts own budget constrant. Ths treatment generates a demand for carrers of value (meda of exchange) movng among successve trades (Starr, 003a,b). Vrtually the same axomatc structure (Arrow and Debreu, 954) that ensures the exstence of general equlbrum n the model of a unfed market wthout transacton costs yelds exstence of equlbrum and a well-defned demand for meda of exchange n ths dsaggregated settng. Tradng post equlbra are Pareto effcent when they are smply the elaboraton of an underlyng Walrasan equlbrum, an nessental tradng post economy; see also Hahn (973). However, the multplcty of separate budget constrants and the addtonal transacton costs ncurred or avoded may skew the allocaton and prcng (an essental tradng post equlbrum). Then the equlbrum cannot be supported by a Walrasan prce structure and the allocaton wll be Pareto neffcent; see also Starrett (973). The prce system s nformatve not only about scarcty and desrablty. It also prces lqudty. Transacton costs generate a spread between bd and ask prces at each tradng post. The bd ask spread tells frms and households whch goods are lqud, easly traded wthout sgnfcant loss of value, and whch are llqud, unsutable as carrers of value between trades, Menger (89). The multplcty of budget constrants creates the demand for lqudty; the bd ask spreads sgnal ts supply. When lqudty s too expensve (example 0.5.), meda of exchange wll not be used. When lqudty s nexpensve and helpful n achevng a Pareto mprovng allocaton (example 0.3.), meda of exchange wll be actvely traded n equlbrum. The tradng post model endogenously generates a desgnaton and a flow of commodty money(es). The exstence of (commodty) money and the monetary structure of trade s an outcome of the general economc equlbrum. Money s not a separate assumpton; t s a result of the equlbrum allocaton. Acknowledgement I am ndebted to ths journal s edtor, guest edtor, and referee, to partcpants n the Legacy of Gerard Debreu conference at Berkeley n October 005, to Omar Lcandro, Garey Ramey, and Joel Sobel for helpful advce. Mstakes are the author s. References Arrow, K.J., Debreu, G., 954. Exstence of equlbrum for a compettve economy. Econometrca, Debreu, G., 959. Theory of Value. Yale Unversty Press, New Haven. Foley, D.K., 970. Economc equlbrum wth costly marketng. Journal of Economc Theory (3), Goldberg, D., 005. The tax-foundaton theory of money, unpublshed, Texas A&M Unversty, College Staton, Texas. Green, J., Heller, W.P., 98. Mathematcal analyss and convexty wth applcatons to economcs. In: Arrow, K.J., Intrlgator, M.D. (Eds.), Handbook of Mathematcal Economcs, vol.. North Holland, New York. Hahn, F.H., 97. Equlbrum wth transacton costs. Econometrca 39 (3), Hahn, F.H., 973. On transacton costs, nessental sequence economes and money. Revew of Economc Studes XL (October (4)),

15 R.M. Starr / Journal of Mathematcal Economcs 44 (008) Hahn, F.H., 98. Money and Inflaton. Basl Blackwell, Oxford. Jevons, W.S., 875. Money and the Mechansm of Exchange. C. Kegan Paul, London. Kyotak, N., Wrght, R., 989. On money as a medum of exchange. Journal of Poltcal Economy 97, Kurz, M., 974. Equlbrum n a fnte sequence of markets wth transacton cost. Econometrca 4 (), 0. Menger, C., 89. On the orgn of money. Economc Journal II, (translated by Carolne A. Foley). Shapley, L.S., Shubk, M., 977. Trade usng one commodty as means of payment. Journal of Poltcal Economy 85 (October (5)), Smth, A., 776/966. An Inqury nto the Nature and Causes of the Wealth of Natons, vol. I. W. Strahan and T. Cadell/A.M. Kelley, London/New York (Book II, Chapter II). Starr, R., 997. General Equlbrum Theory: An Introducton. Cambrdge Unversty Press, New York. Starr, R., 003a. Why s there money? Endogenous dervaton of money as the most lqud asset: a class of examples. Economc Theory (March ( 3)), Starr, R., 003b. Exstence and unqueness of money n general equlbrum: natural monopoly n the most lqud asset. In: Alprants, C.D., Arrow, K.J., Hammond, P., Kubler, F., Wu, H.-M., Yannels, N.C. (Eds.), Assets, Belefs, and Equlbra n Economc Dynamcs. BertelsmanSprnger, Hedelberg. Starr, R., 003c. Monetary general equlbrum wth transacton costs. Journal of Mathematcal Economcs 39 (June (3 4)), Starr, R., 008. Mengeran Saleableness and commodty money n a Walrasan tradng post example, Econ. Lett. Starrett, D.A., 973. Ineffcency and the demand for Money n a sequence economy. Revew of Economc Studes XL (4), Wallace, N., 980. The overlappng generatons model of fat money. In: Kareken, J., Wallace, N. (Eds.), Models of Monetary Economes. Federal Reserve Bank of Mnneapols, Mnneapols. Walras, L., 874. Elements of Pure Economcs, Jaffe translaton (954). Irwn, Homewood, IL. Wcksell, K., 936, Interest and prces [Geldzns und Güterprese]: a study of the causes regulatng the value of money translated by R.F. Kahn. Macmllan, London [reprnt A.M. Kelley, New York, 96].