Price Level Volatility: A Simple Model of Money Taxes and Sunspots*

Transcription

1 journal of economic teory 81, (1998) article no. ET Price Level Volatility: A Simple Model of Money Taxes and Sunspots* Joydeep Battacarya Department of Economics, Fronczak Hall, SUNY-Buffalo, Buffalo, New York joydeepbacsu.buffalo.edu Mark G. Guzman Department of Economics, Wellesley College, Wellesley, Massacusetts mguzmanwellesley.edu and Karl Sell - Department of Economics, Cornell University, 402 Uris Hall, Itaca, New York ks22cornell.edu Received May 27, 1997; revised September 20, 1997 We investigate sunspot equilibria in a static, one-commodity model wit taxes and transfers denominated in money units. Volatility in tis economy is purely monetary, since te only uncertainty is about te price level. We construct simple, robust examples of sunspot equilibria tat are not mere randomizations over certainty equilibria. We also identify te source of tese SSEs: Equilibrium in te securities market is determined as if tere were no restricted consumers and te unrestricted consumers face intrinsic uncertainty. Perfect securities markets eliminate allocation uncertainty, but tey exacerbate price-level volatility. Journal of Economic Literature Classification Numbers: D50, D52, D84, E31, E Academic Press * Te autors gratefully acknowledge researc support from National Science Foundation Grant SES to Cornell University, te Torne Fund, and te Center for Analytic Economics. Tom Muenc and Subir Cattopadyay motivated us to find anoter ``convex'' example of an SSE tat is not a randomization over CEs. We tank Jess Benabib and Neil Wallace for teir advice and encouragement, and Cristian Giglino, Aditya Goenka, Todd Keister, Jim Peck and seminar participants at NYU, Princeton, Cornell, Illinois, Cambridge, Essex, Dosisa, Kyoto, Tokyo and Keio for teir comments. - To wom correspondence sould be addressed Copyrigt 1998 by Academic Press All rigts of reproduction in any form reserved.

2 402 BHATTACHARYA, GUZMAN, AND SHELL 1. INTRODUCTION We analyze sunspot equilibria (SSE) in a simple, one-period, generalequilibrium model wit one commodity and a single fiat money. Te fiat money is created by te government troug its fiscal policy, a system of lump-sum taxes and transfers denominated in money units. Uncertainty is purely extrinsic; i.e., te fundamentals of te economy are nonstocastic. We do tree tings in te present paper: (1) We make te logical first step toward integrating te teory of sunspot equilibrium 1 and te teory of money taxes and transfers. 2 (2) We present very simple, numerical examples of SSE tat are not mere randomizations over te corresponding certainty equilibria (CE). 3, 4 Te examples are generated wit te elp of a new and powerful tool: te tax-adjusted Edgewort box. (3) We observe an important difference between economies wit government money and economies in wic all securities are private. 5 Perfect securities markets eliminate volatility in te economy witout any government securities. 6 In te economy wit government money, perfect securities markets do eliminate allocation volatility but tey can also exacerbate price-level volatility. We introduce te certainty economy in Section 2 and te corresponding sunspots economy in Section 3. Te simplicity of our model allows us to identify te basic source and nature of sunspot equilibrium. We do tis in Section 4: Te seed of te stocastic allocation is in te set of restricted consumers, i.e., tose wo are unable to edge against te effects of pricelevel uncertainty. Teir allocations are uncertain and tis typically causes te aggregate allocation to te unrestricted consumers to be uncertain. Te 1 See Sell [18] and especially Cass and Sell [8]. See also Sell [19] and Sell and Smit [21]. 2 See Balasko and Sell [5] and especially Balasko and Sell [6]. See also Balasko and Sell [4] and Sell and Smit [20]. 3 Hence tis paper provides a partial substitute to te necessarily complicated appendix to Cass and Sell [8]. 4 Goenka [12, Example 4.1] provides an example wic is based on a convex, competitive economy wit rationing and Sell and Wrigt [22] provide examples, based on te indivisible good economy, of SSE tat are not randomizations over CE. For an imperfectly competitive economy, Peck and Sell [17] provide examples of SSE tat are not mere randomizations over pure-strategy Nas equilibria. 5 Since our present model is a static one, in every proper monetary equilibrium, aggregate government money must be zero. In tis sense, aggregate outside money will be zero. Te present model is, owever, easily extended to a multiperiod setting in wic aggregate outside money can be nonzero in eac period. See Balasko and Sell [5], te references terein, and Giglino and Sell [11]. Wallace [24 and elsewere] says tat outside money is positive only if te government's net liability is positive. Tis interpretation of ``outside money'' requires te (infinite) overlapping-generations model; see, e.g., Balasko and Sell [4]. 6 See Cass and Sell [8, Propositions 2 and 3].

3 PRICE LEVEL VOLATILITY 403 unrestricted consumers seek to edge against te effects of price-level volatility. Te equilibrium allocations of te unrestricted consumers and te state-contingent price ratio are determined in te tax-adjusted Edgewort box. We sow tat te determination of equilibrium in tis economy reduces to te determination of equilibrium in a smaller economy wit no restrictions on market participation but in wic uncertainty does affect te economic fundamentalsi.e., in wic uncertainty is intrinsic. In te smaller economy, tere is typically economy-wide risk. Hence allocations are usually stocastic and income is typically transferred across states of nature. Te transfer of income across states of nature makes te SSE allocations different from a randomization across CE allocations. We make some concluding remarks in Section 5. In te appendix, we establis te equivalence of contingent-commodity edging and securities (or, contingent-money) edging. 2. THE CERTAINTY ECONOMY We begin wit te underlying certainty economy, a simple pureexcange, monetary economy witout sunspots. 7 Consumer as an endowment of te single commodity, >0. He must pay { units of money in taxes. 8 Te scalar { can be positive, zero, or negative; if it is negative, te absolute value of { is is money transfer. His consumption set X is te set of positive scalars, i.e., we ave X =R ++. Te set of consumers H is finite. Consumer cooses x to maximize u (x ) subject to (2.1) px = p & p m { and x >0, were u is a strictly increasing utility function, p>0 is te price of te commodity, and p m is te price of money. Te price of money must be nonnegative, i.e., we ave p m 0. 7 See Balasko and Sell [6]. In te present paper, tere is only one commodity, i.e., we analyze te special case of l=1. 8 One migt question te realism of nominal taxation. Are not, for example, income taxes typically ``real'' taxes, i.e. automatically adjusted for price level canges? Wile it migt be argued tat 1996 income taxes are partially adjusted for te 1996 price level, te amount of 1996 tax collected in 1997 is unadjusted for te 1997 price level. Hence some parts of actual tax bills are not adjusted for inflation.

4 404 BHATTACHARYA, GUZMAN, AND SHELL Let P m be te commodity price of money. Since tere is only one commodity, we can also define te general price level, P, to be te money price of te commodity. 9 Hence we ave and P m = p m p, P= pp m =1P m, 0P m <+ and tus 0<P+. It will be advantageous to associate allocations in tis money-taxation economy wit counterparts in a no-taxation economy. To do tis we define ~, te tax-adjusted endowment of Mr., by ~ = &P m {. (2.2) Te after-tax income of consumer, w, is given by w = p & p m { = p[ &(p m p) { ]=p ~. (2.3) Tus te scalar ~ is also is effective demand for te commodity and so we ave x = ~ (2.4) for ~ >0. Te price of money is not completely arbitrary. If it is too ig, some taxed individual would be bankrupt. Tat is, is tax-adjusted income by (2.3) would be negative and ence tere would be no solution to te constrained-maximization problem (2.1). From (2.3), it follows tat we ave if { >0 ten P m < {. (2.5) (Only te consumers facing positive money taxes are involved in te restriction (2.5), because only tey face te possibility of bankruptcy.) In competitive equilibrium, aggregate demand and supply must be equal, i.e., : x =:, (2.6) H 9 In te analysis of monetary economies, it is better to use te ``commodity price of money'' rater tan te more familiar ``money price of commodity'' in order to comfortably include te analytically important case of wortless money, i.e., te case of P m =0. H

5 PRICE LEVEL VOLATILITY 405 and te price of te commodity must be positive, i.e., p>0. Substituting (2.2) and (2.4) into te left side of (2.6) yields P m : { =0. (2.7) H From (2.7), it follows tat in a proper monetary equilibrium (i.e., one wit P m >0) we ave : { =0, (2.8) H i.e., taxes are exactly offset by transfers. 10 Note tat if money is wortless (i.e., P m =0), ten autarky is te competitive equilibrium allocation, since x = ~ = >0 for eac. On te oter and, if all taxes are zero (i.e., { =0 for eac ), ten autarky is te unique equilibrium allocation (i.e., x = ~ = for eac ), but ten P m is completely indeterminate. Next we apply tese ideas to a concrete pure-excange, competitive economy wit lump-sum money taxes and transfers. We focus on te nondegenerate cases, namely ones in wic taxes are nontrivial (i.e., not all zero), taxes are balanced (i.e., (2.8) olds), and te price of money is strictly positive. Tere are tree consumers: 2.1. Parameters for te Certainty Economy H=[1, 2, 3]. Te vector of before-tax endowments is given by =( 1, 2, 3 )=(20, 10, 5). Te vector of money lump-sum taxes { is given by {=({ 1, { 2, { 3 )=(5,0,&5). Since { 1 +{ 2 +{ 3 =0, we see tat { is balanced and ence we know tat tere will be some equilibrium in wic te price of money is positive A tax vector {=(..., {,...) is said to be balanced if (2.8) olds. A tax vector { is said to be bonafide if it permits an equilibrium in wic money is not wortless. In finite economies, { is bonafide if and only if it is balanced; see Balasko and Sell [6] for te proof (Corollary 3.7) and a caveat (in Section 5). 11 See Balasko and Sell [6].

6 406 BHATTACHARYA, GUZMAN, AND SHELL 2.2. Competitive Equilibrium in te Certainty Economy Te set of certainty equilibrium (CE) allocations, X CE, is given by [x=(x 1, x 2, x 3 )#R 3 x ++ 1=20&5P m, x 2 =10, x 3 =5+5P m, P m # R + ], (2.9) were P m satisfies inequalities (2.5), wic reduce in tis example to te requirement P m # [0, 4). Tus, te set of certainty equilibrium prices, P m CE, is given by Combining (2.9) and (2.10) yields P m CE =[Pm 0P m <4]. (2.10) X CE =[(x 1, x 2, x 3 ) x 1 =20&5P m, x 2 =10, x 3 =5+5P m,0p m <4]. (2.11) Hence te set X CE of certainty equilibrium allocations is one-dimensional, parameterized by te commodity price of money P m Randomizations over Certainty Equilibria Before analyzing te sunspots economy, it will be useful to make precise te notion of a ``randomization over certainty equilibria,'' or equivalently a ``lottery over certainty equilibria.'' Let s be a random variable wic, for simplicity, we assume to take on one of two possible realizations, : and ;, wit probability?(:)>0 and?(;)=1&?(:)>0 respectively. Let x (s) be te allocation to consumer in state s and x(s)=(x 1 (s), x 2 (s), x 3 (s)) # R Te allocation (x(:), x(;)) # R 6 ++ is said to be a (mere) randomization over certainty equilibria if we ave x(s)#x CE for s=:, ;. Hence X RCE, te set of randomizations over CE allocations, is given by X RCE =X CE _X CE. Let P m (s) be te goods price of money in state s. Te price vector (P m (:), P m (;)) # R 2 + is said to be a randomization over CE prices if P m (s)#p m CE for s=:, ;. Terefore te set of randomizations over CE prices, P m RCE, is given by P m RCE =Pm CE _Pm CE =[0, 4)_[0, 4). (2.12)

7 PRICE LEVEL VOLATILITY 407 Hence te set X RCE of randomizations over certainty equilibria is two-dimensional, parameterized by te vector (P m (:), P m (;)). 3. THE SUNSPOTS ECONOMY We extend te model of Section 2 to allow individuals to face extrinsic uncertainty about te price level. We introduce te extrinsic random variable s, wic, as above, is assumed to satisfy s # [:, ;]. We assume tat consumers sare te same beliefs; 12 ence te probabilities wit wic state : or ; occur, denoted by?(:) and?(;)=(1&?(:)) respectively, are eld in common. By definition, extrinsic uncertainty does not affect te fundamentals of te economy. In te present example, uncertainty is extrinsic because we assume: 1. Endowments: (:)= (;)= (3.1) for # H, were (s) is te endowment of consumer in state s=:, ;. 2. Taxes: { (:)={ (;)={ (3.2) for # H, were { (s) is te money tax on consumer in state s=:, ;, and 3. Preferences: v [x (:), x (;)]=?(:) u (x (:))+?(;) u (x (;)), (3.3) were v [ } ] is te ex ante von NeumannMorgenstern utility function for Mr.. It is obvious tat symmetry assumptions (3.1) and (3.2) are required if uncertainty is to be extrinsic. Symmetry assumption (3.3), along wit (3.1) and (3.2), implies tat uncertainty is extrinsic because v [ } ] is unaffected by merely permuting te labels for te states of nature. 13 Te set H of consumers is partitioned into two classes, G 0 and G 1. Every consumer as access to te spot-market for trading te commodity (and money) after state s is revealed. Consumers in G 0 (possibly ``te older generation'') also ave access to trading on state-contingent markets. Tey 12 Tus we make a strong rational-expectations ypotesis; see Sell [19]. (For a sunspots economy in wic information is asymmetric, see Peck and Sell [17].) 13 For te formalization and te generalization, see Balasko [2].

8 408 BHATTACHARYA, GUZMAN, AND SHELL ave perfect foresigt about te spot-market prices tat will prevail after te state s is revealed. Consumers in G 1 (possibly ``te younger generation'') cannot trade in contingent securities. Our time line (Fig. 3.1) can be used for an overlapping-generations interpretation of te (natural) restrictions on market participation. Let x (s) be consumption of te commodity by Mr. in state s. Let te utility function u be strictly increasing, smoot, and strictly concave. Also assume tat te beavior of indifference curves at te axes is suc tat free commodities are ruled out. Let p(s)>0 be te price of te commodity to be delivered in state s and p m (s)0 be te price of money delivered in state s. Formally, consumer # G 1 cooses x (s)#r ++ to maximize u (x (s)) subject to (3.4) p(s) x (s)=p(s) & p m (s) { for s=:, ;. Define te money price P m (s) and te tax-adjusted endowment ~ (s) by P m (s)=p m (s)p(s) and ~ (s)= &P m (s) { (3.5) for s=:, ;. Ten te budget constraint in (3.4) can be rewritten as Hence for # G 1 we ave p(s) x (s)=p(s) ~ (s). (3.4)$ x (s)= ~ (s) (3.6) for s=:, ;; i.e., ~ (s) is restricted-consumer 's demand for te commodity in state s. Terefore, aggregating over te set of restricted consumers yields for s=:, ;. : x (s)=: ~ (s) (3.7) G 1 G 1 FIG Timeline for te OG interpretation of te restrictions on market participation.

9 PRICE LEVEL VOLATILITY 409 Consumer # G 0 cooses (x (:), x (;)) # R 2 ++ to maximize?(:) u (x (:))+?(;) u (x (;)) subject to (3.8) p(:) x (:)+p(;) x (;)=(p(:)+p(;)) &(p m (:)+p m (;)) {. Using (3.5), te budget constraint in (3.8) can be rewritten as p(:) x (:)+p(;) x (;)=p(:) ~ (:)+p(;) ~ (;). (3.8)$ Te simplest (but least realistic) interpretation of (3.8) is tat consumer # G 0 trades only before state s is revealed. He buys and sells contingent commodities. Taxes valued at contingent money prices are deducted from is income. We sow in te appendix tat (3.8) is equivalent to oter, more interesting market arrangements in wic consumer # G 0 trades on te spot market and edges troug te contingent commodity market andor te contingent money market. A competitive equilibrium for te monetary, sunspots economy is a price vector ( p(:), p(;), p m (:), p m (;)) wit p(s)>0 and p m (s)0 for s=:, ; wit te property tat if consumers beave according to (3.4) and (3.8), ten demand is equal to supply, i.e., we ave : x (s)=: (3.9) for s=:, ;. Summing over te budget constraints (3.8)$ yields H H p(:) : G 0 x (:)+p(;) : G 0 x (;)=p(:) : G 0 ~ (:)+p(;) : G 0 ~ (;). (3.10) Hence te equilibrium beavior of te unrestricted consumers can be described in terms of teir state-specific tax-adjusted endowments in a tax-adjusted Edgewort box of dimensions : ~ (:)_: ~ (;). G 0 G 0 Substituting (3.7) and (3.10) into (3.9) yields [ p m (:)+p m (;)] : { =0. (3.11) H

10 410 BHATTACHARYA, GUZMAN, AND SHELL Hence in a proper monetary equilibrium (one in wic p m (s)>0 for s=:, ;) 14 15, 16 taxes must be balanced, i.e., H { =0. Next we specify te underlying parameters for te sunspots economy from wic we build our numerical examples Parameters for te Sunspots Economy Te sunspots economy is based on te certainty economy described in Example (2.1): H=[1, 2, 3], =( 1, 2, 3 ), and {=({ 1,0,&{ 1 ). In addition, we make te following specifications: and so tat we ave u (x (s))=log x (s) for # H, (3.12)?=(?(:),?(;))=(34, 14), (3.13) v =(34) log x (:)+(14) log x (;) for # G 0. (3.14) Four numerical examples follow Totally Restricted Market Participation We begin wit a sunspots economy example in wic all consumers are restricted from participating in te state-contingent markets. Hence we ave G 0 =< and G 1 =H. Te parameters of te economy are identical to tose described in example (2.1): =( 1, 2, 3 )=(20, 10, 5) and {=({ 1, { 2, { 3 )=(5, 0, &5). Since no trading across states of nature is possible in tis degenerate case, te set 14 A seemingly weaker requirement would be p m (s)>0 for at least one s. 15 Hence bonafidelity of taxes in te monetary sunspots economy requires balanced taxes. It is also easy to sow tat balancedness of taxes implies teir bonafidelity in tis economy. 16 Wit state-specific taxation, equilibrium condition (3.11) generalizes to p m (:) H { (:)+ p m (;) H { (;)=0. Under tis interpretation te government may ave to accept for tax payment inside money issued by a consumer in G 0 and financed by is purcases of money in te oter state. If H { (:) and H { (;) are not zero, ten tey must be of opposite sign. Tus te price ratio p m (:)p m (;) is (uniquely) given by te ratio (& H { (:) H { (;)). Tis observation is reminiscent of Fiser [10], in wic it is sown tat te international excange rate is equal in absolute value to te ratio of current account deficits.

11 PRICE LEVEL VOLATILITY 411 of equilibrium allocations X NP, were NP denotes ``no participation,'' is given by X NP =[((x 1 (:), x 2 (:), x 3 (:)), (x 1 (;), x 2 (;), x 3 (;))) # R 6 ++ x 1 (s)=20&5p m (s), x 2 (s)=10, x 3 (s)=5+5p m (s), and 0P m (s)<4 for s=:, ;]. (3.15) For fixed endowments and taxes, x (s) depends only on P m (s). Tus, te set of SSE allocations X NP and te set of sunspot equilibrium prices P m NP satisfy, respectively, X NP =X RCE =X CE _X CE and (3.16) P m NP =Pm RCE =Pm CE _Pm CE =[0, 4)_[0, 4). Hence te set of SSE allocations is two-dimensional, parameterized by te price vector (P m (:), P m (;)) Partially Restricted Market Participation In te two examples wic follow, we divide te consumers into te following two groups: G 0 =[1, 2] and G 1 =[3]. Mr. 1 and Mr. 2 are unrestricted and can trade in securities or contingent commodities, wereas Mr. 3 cannot edge against te effects of sunspots. Since Mr. 3 cannot trade across states of nature, e consumes (by (3.6)) is tax-adjusted endowments, i.e., we ave x 3 (:)= ~ 3 (:) and x 3 (;)= ~ 3 (;), (3.17) were ~ 3 (:)= ~ 3 (;) only if P m (:)=P m (;). (In te examples wic follow, we always ave P m (:){P m (;).) Any trading on te state-contingent markets must tus occur between Mr. 1 and Mr. 2. We can analyze teir decisions and te resulting equilibrium allocations by means of a tax-adjusted Edgewort box (see Figs. 3.2 and 3.3). Te dimensions, G 0 ~ (:)_ G 0 ~ (;), represent te state-specific tax-adjusted endowments summed over Mr. 1 and Mr. 2. Wit te (identical, log-linear) utility functions specified in (3.14), te contract curve is te minor diagonal (wit slope G 0 ~ (;) G 0 ~ (:)) of te tax-adjusted Edgewort box. (For tese utility functions, an allocation is

12 412 BHATTACHARYA, GUZMAN, AND SHELL Pareto efficient for te community of unrestricted consumers if and only if te ratio of :-consumption to ;-consumption is te same for eac of tese consumers and tere are no unallocated consumption goods.) Tere are two important points to be made regarding te contract curve: (1) Since te tax-adjusted Edgewort box is now rectangular (given P m (:){P m (;)), and not square like te pre-tax Edgewort box, te equilibrium allocation must be sunspot dependent for at least one consumer in G 0. (2) As long as te tax-adjusted endowments do not lie on te minor diagonal (and tey do not in our examples), tere must be trading across te states of nature. Given te utility functions defined in (3.14), te demand functions are x (s)=?(s) p(s) w for s=:, ; and =1, 2, were te income of consumer, w, is defined by w =(p(:)+p(;)) &(p(:) P m (:)+p(;) P m (;)) { for =1, 2. (3.18) Equating supply and demand for te state-s commodity witin te taxadjusted Edgewort box yields te equilibrium condition x 1 (s)+x 2 (s)= ~ 1 (s)+ ~ 2 (s) for s=:, ;. (3.19) Substituting te demand functions into (3.19) yields te market clearing equilibrium condition p(:) p(;) = \?(;)+\?(:) ~ 1(;)+ ~ 2 (;) ~ 1 (:)+ ~ 2 (:)+. (3.20) More generally, te equilibrium condition (3.20) is p(:) p(;) = \?(;)+\?(:) G 0 ~ (;) G 0 ~ (:)+. (3.21) Equation (3.21) states tat te contingent-commodity price ratio must equal te after-tax scarcity ratio multiplied by te likeliood ratio. Te commodity prices of money (P m (:), P m (;)) are restricted by te requirements tat (x 1 (:), x 1 (;)) # R 2 and (x ++ 2(:), x 2 (;)) # R 2, wic ++ reduce to te requirement tat 0<x 1 (s)< ~ 1 (s)+ ~ 2 (s)

13 PRICE LEVEL VOLATILITY 413 for s=:, ;. Te set of SSE prices P m RP, were RP denotes restricted participation, is te set of (P m (:), P m (;)) # R 2 + tat satisfy and P m 1 ( ) (:)# _0, (3.22) { 1 ( 1 +?(:) 2 )+ P m (;)# _0, 1( )&{ 1 ( 1 +?(:) 2 ) P m (:) { 1 ( 1 +?(;) 2 &{ 1 P m (:)) +. (3.23) Te set of SSE allocations X RP is given by X RP =[(x 1 (:), x 2 (:), x 3 (:)), (x 1 (;), x 2 (;), x 3 (;)) # R 6 ++ x 1 (:)=34[20&5P m (:)+(13_)(20&5P m (;))], x 2 (:)=304[1+(13_)], x 3 (:)=5+5P m (:), x 1 (;)=14[3_(20&5P m (:))+20&5P m (;)], x 2 (;)=104[3_+1], and x 3 (;)=5+5P m (;)], were _=(30&5P m (;))(30&5P m (:)) is te slope of te contract curve in te tax-adjusted Edgewort box, and were, as in te previous section, =( 1, 2, 3 )=(20, 10, 5) and {=({ 1, { 2, { 3 )=(5, 0, &5). Given and {, te allocation of te restricted consumer, x 3 (s) depends solely on te commodity price of money, P m (s), for s=:, ;. (Consequently, te taxadjusted Edgewort box depends on P m (:) and P m (;) and tus in general te equilibrium allocations x (s) for =1, 2 also depend on te state contingent commodity prices of money.) Hence te set X RP is two-dimensional, parameterized te price vector (P m (:), P m (;)) Price Level Volatility Example For te remaining tree examples, we continue wit te economy tat satifies =( 1, 2, 3 )=(20, 10, 5) and {=({ 1, { 2, { 3 )=(5, 0, &5). In addition, we set P m (:)=1 and P m (;)=2. (3.24)

14 414 BHATTACHARYA, GUZMAN, AND SHELL Relative to state :, state ; is ``deflationary.'' Given te values for { and, (3.22) and (3.23) respectively yield P m RP =[(Pm (:), P m (;)) # R Pm (:)+[9&2P m (:)] P m (;)<48 and 0P m (:)<4811]. Hence te values cosen for te prices in te two states are consistent wit equilibrium, i.e., we ave (1, 2) # P m RP. Substituting te prices from (3.24) in (3.5) yields te tax-adjusted endowments and ( ~ 1 (:), ~ 1 (;))=(15, 10), ( ~ 2 (:), ~ 2 (;))=(10, 10), ( ~ 3 (:), ~ 3 (;))=(10, 15). Since Mr. 3 cannot trade across states of nature, e consumes (by (3.17)) is tax adjusted endowments x 3 (:)= ~ 3 (:)=10 and x 3 (;)= ~ 3 (;)=15. (3.25) Only Mr. 1 and Mr. 2 trade on te state-contingent markets. Figure 3.2 denotes te tax-adjusted Edgewort box for tis example, wic is of FIG Te tax-adjusted Edgewort box.

15 PRICE LEVEL VOLATILITY 415 dimensions 25_20. Te contract curve is given by te minor diagonal and as slope From Eq. (3.20), te price ratio consistent wit market clearing in tis example is p(:) p(;) = \ \ =12 5. (3.26) Te resulting consumption allocations for Mr. 1 and Mr. 2 are given by x 1 (:)=14 3 8, x 1(;)=11 1 2, x 2(:)=10 5 8, and x 2(;)= (3.27) Te next proposition follows directly from our example (3.3.1). It establises tat tere is an SSE allocation wic is not contained in te set X RCE. Proposition 3.1. Te SSE allocation [(x 1 (:), x 1 (;)), (x 2 (:), x 2 (;)), (x 3 (:), x 3 (;))], described by (3.25) and (3.27), is not a randomization over certainty equilibria. Proof. Te allocation described by (3.25) and (3.27) is a SSE allocation for te economy defined in example (3.3.1). Tis economy is based on te certainty economy defined in example (2.1). Te set of CE allocations is given by (2.11). Because Mr. 2 is untaxed, is CE allocation must be x 2 =10, and consequently, in any randomization over CE Mr. 2's allocation will also be x 2 (s)=10 for s=:, ;. But we ave from (3.27) tat is allocation in te corresponding sunspots economy is x 2 (:)= >10 and x 2(;)=8 1 2 <10. Hence we ave sown tat tis SSE allocation is not a mere randomization over any CE. Tat is, we ave found an allocation in X RP tat is not in X RCE. K In te next example, we sow tat te SSE price levels are not necessarily randomizations over CE price levels Hig Price-Level Volatility Example: We alter te prices given in (3.24) for example (3.3.1) to be P m (:)=1 and P m (;)=5, (3.28) and ence state ; is even more ``deflationary'' tan before. Notice tat in te certainty economy, randomizations over te certainty economy, and te sunspots economy wit totally restricted market participation, te respective equilibrium prices of money, P m and P m (s) for s=:, ;, lie in te

16 416 BHATTACHARYA, GUZMAN, AND SHELL interval [0, 4) by (2.10), (2.12) and (3.16). Hence if (3.28) is consistent wit SSE, ten we ave sown tat for one state te goods price of money is greater tan could be acieved in te certainty economy or any randomization over te certainty economy. Te tax-adjusted endowments obtained by using (3.28) and (3.5) are given by ( ~ 1 (:), ~ 1 (;))=(15, &5), ( ~ 2 (:), ~ 2 (;))=(10, 10), and (3.29) ( ~ 3 (:), ~ 3 (;))=(10, 30). Since Mr. 3 cannot trade in te contingent markets, is allocation is identical to is tax-adjusted endowment x 3 (:)= ~ 3 (:)=10 and x 3 (;)= ~ 3 (;)=30. (3.30) In tis economy, as in example (3.3.1), only Mr. 1 and Mr. 2 trade in te commodity spot-market. Te tax-adjusted Edgewort box for tis example is given in Fig. 3.3 and is of dimensions 25_5. Notice, owever, tat Mr. 1's tax-adjusted endowment is outside te tax-adjusted Edgewort box, and ence it is outside is consumption set (in state ;), since all allocations must lie in te strictly positive ortant. FIG ``Endowment'' outside te tax-adjusted Edgewort box.

17 PRICE LEVEL VOLATILITY 417 Te contract curve is te minor diagonal (wit slope 15); te ratio of prices consistent wit market clearing in te goods market and ence wit competitive equilibrium is given by p(:) p(;) = \ 25+\ =3 5, (3.31) from (3.20). Te sunspot equilibrium allocations for Mr. 1 and Mr. 2 in tis economy are given by x 1 (:)=5, x 1 (;)=1, x 2 (:)=20, and x 2 (;)=4. (3.32) As in te previous example, we ave constructed a SSE wic is not a randomization over certainty equilibrium, as evidenced by te SSE allocations associated wit Mr. 2. In addition, tis economy can exibit greater price volatility tan is possible in te CE economy or te sunspots economy wit totally restricted market participation since P m (;)=5 P m CE =[0, 4).17 As a result Mr. 1's tax-adjusted endowment is negative in state ;, but because e as positive income, w, e can afford a strictly positive bundle. (Mr. 1 can afford a strictly positive bundle if and only if?(:)>58. Tat is, Mr. 1 is viable if and only if te probability of te bad state, ; for im, is less tan 38.) In our next example, we sow tat even wit full market participation, te price level can be volatile in te sunspots economy and in fact, tat tere is room for wider price fluctuations in te economy wit full edging markets tan in te corresponding economy wit no scope for edging Unrestricted Market Participation Before we use te tax-adjusted Edgewort box to generate our last example of a SSE, we recall tat if markets are perfect ten te equilibrium allocations in te economy are immune to te effects of sunspots. We also register a caveat: Even wit perfect markets tere is price-level volatility in te economy wit money taxes and transfers. Furtermore, tere is room for greater price-level volatility wit perfect edging markets tan in te cases wit imperfect edging markets. Proposition 3.2. Consider te sunspots economy wit money taxes and transfers and unrestricted market participation, i.e., te economy in wic 17 Te relationsip between te magnitude of te price-level volatility for te certainty economy and te sunspots economy wit restricted market participants will be more fully explored in Section

18 418 BHATTACHARYA, GUZMAN, AND SHELL G 0 =H (and ence G 1 =<). If (..., x (:), x (;),...) is a competitive equilibrium allocation in tis economy, ten we ave x (:)=x (;), (3.33) for # H. Tat is, te allocations are te same in eac state of nature. Proof. Consider te economy represented by te related no-taxation economy wit endowments ~ (s), s=:, ;, given by te (3.5). Since : ~ (s)=: (3.34) H for s=:, ;, tere is no aggregate uncertainty in te tax-adjusted economy, i.e., te tax-adjusted Edgewort diagram is a cube. Using (3.34) allows us to adopt te standard proof from te sunspots literature. 18 Let (x (:), x (;)) be te equilibrium allocation of consumer. Assume tat for some, x (:){x (;). Let x be defined by x =?(:) x (:)+?(;) x (;) for # H. Ten we ave H x = H and v (x, x )ev (x (:), x (;)) for eac # H and wit strict inequality for at least one # H. Hence we ave found a competitive equilibrium tat is not Pareto optimal, contradicting te first teorem of welfare economics. Terefore, (nonsunspots) condition (3.33) is establised. K In te context of te economy witout outside money, Cass and Sell use te condition in (3.33) to define te case ``were sunspots do not matter.'' In teir model, if te allocations are symmetric ten so are te prices. However, in te economy wit outside money, tere can be price-level volatility even wen te allocations are symmetric, i.e., tey satisfy (3.33). Hence te CassSell definition of ``sunspots do not matter'' is incomplete for te economy wit money taxes and transfers. 19 Te set of SSE allocations X FP were FP denotes ``full participation,'' is defined by X FP =[((x 1 (:), x 2 (:), x 3 (:)), (x 1 (;), x 2 (;), x 3 (;))) # R 6 ++ H x 1 (:)=x 1 (;)=[(34)(20&5P m (:))+(14)(20&5P m (;))], x 2 (:)=x 2 (;)=10, x 3 (:)=x 3 (;)=[(34)(5+5P m (:))+(14)(5+5P m (;))]], 18 See Malinvaud [16]. Also see Cass and Sell [8, Proposition 3] and Goenka and Sell [13, 14]. 19 Te extent of te possible price-level volatility wit perfect edging markets will be made clear in numerical Example

19 PRICE LEVEL VOLATILITY 419 were =( 1, 2, 3 )=(20, 10, 5) and {=({ 1, { 2, { 3 )=(5, 0, &5). Notice tat x (:)=x (;) depends on P m (:) and P m (;) solely troug te sum?(:) P m (:)+?(;) P m (;)=(34) P m (:)+(14) P m (;). Te set of full-participation SSE money prices P m FP is defined by P m FP =[(Pm (:), P m (;)) # R Pm (:)+P m (;)<16]. (3.35) Te equilibrium allocations x (s) for # H and s=:, ;, are functions of te tax-adjusted endowments, and tus, given and { depend only on te commodity prices of money, P m (:) and P m (;), solely troug te weigted average of prices?(:) P m (:)+?(;) P m (;). (Tis results from Mr. 3 no longer being restricted from trading on te securities markets. Consequently, te ratio of commodities prices from (3.20) reduces to p(:)p(;)=?(:)?(;)=3.) Hence te set X FP is one dimensional, parameterized by te scalar?(:) P m (:)+?(;) P m (;)=(34) P m (:)+(14) P m (;). Tis indeterminacy is te same as one finds in general-equilibrium models wit incomplete markets (GEI) and wit nominal financial instruments; see, e.g., Cass [7] and Werner [25]. In tat literature, te financial assets are available in zero net supply. Our financial asset is money; see Appendix A.1 for te best interpretation. Te nominal (coupon) return on tis money is zero in eac state. (Oter coupon rates are easily accomodated.) In our model Mr. 's endowment of te financial asset, &{, is not in general zero, but te balancedness condition (2.8) corresponds to an aggregate zero supply condition. In our model, markets are complete but edging on te state-contingent money market is restricted to consumers in G 0. For an exposition of te monetary tax-transfer model in wic &{ is explicitly treated as Mr. 's endowment of money, see Sell [18]. For a recasting of te monetary tax-transfer model in terms of te GEI literature, see Vilanacci [23]. It sould be remarked tat not all financial markets models ave te same indeterminacy properties as tose based on Arrow securities. For a model wit different financial securities and different equilibrium properties, see Antinolfi and Keister [1] Hig Price-Level Volatility Example Building on example (3.3.2), we replace te restricted participation assumption wit te assumption tat participation on te securities market is unrestricted, i.e., we ave G 0 =H (and ence G 1 =<). Te prices are te same as in (3.28) and te resulting tax-adjusted endowments are te

20 420 BHATTACHARYA, GUZMAN, AND SHELL same as in (3.29). From (3.2.1), te ratio of prices necessary for te commodity market to clear is given by p(:) p(;) = \ 30+\ 34 =3. (3.36) 14+ Since all individuals ave access to contingent markets and tus can insure against te possibility of sunspots, allocations are invariant across states of nature; more specifically we ave x 1 (:)=x 1 (;)=10, x 2 (:)=x 2 (;)=10 and (3.37) x 3 (:)=x 3 (;)=15. However, as in te previous example, (3.3.2), we obtain a price level not sustainable as a CE or as a randomization over CE; in particular, P m (;)=5 is consistent wit competitive equilibrium in tis example. In fact in tis example, if we fix P m (:)=1, ten we ave tat equilibrium exists for every P m (;) satisfying P m (;)#[0,13). Tus substantial price level volatility can exist. Also note tat from (3.35) we ave tat for one state te SSE can exibit greater deflation tan is possible in te certainty economy or in te sunspots economy witout securities markets. 20 Price-Level Volatility 21. As we ave suggested in examples (3.3.2) and (3.4.1), potential price-level ``volatility'' as grown wit eac succeeding example, i.e., it increases as te restrictions on market participation are removed. We make tis idea concrete by calculating te ``maximum'' variance in money prices for (1) te sunspots economy wit totally restricted market participation (3.2), (2) te sunspots economy wit partially 20 Fix P m (:)=1 and allow?(:) # (0, 1) to vary; ten for every (large) number N, tere is a?(:) sufficiently close to unity, suc tat te interval [0, N] is included in te set of equilibrium values for P m (;). 21 Te analysis of ``general-price level volatility'' suggests examination of te random variable P(s). Witout losing tis spirit, we work instead wit te random variable P m (s), te inverse of te general price level, and tereby avoid difficulties caused by te fact tat P(s) can ave te realization +.

21 PRICE LEVEL VOLATILITY 421 restricted market participation (3.3), and (3) te sunspots economy in wic all market participants are unrestricted (3.4). Te variance of te price is given by var(p m (s))= 3 16 [Pm (:)&P m (;)] 2, (3.38) for eac of te examples. Hence we ave sup var(p m (s))=3<supvar(p m (s)) = 16 < sup P m P m 3 P m NP RP FP var(p m (s)) = 48 from (3.38). Tus, if price-level volatility is measured in terms of te ``maximum'' potential variance in money prices, we see tat tere is greater potential price-level volatility in te sunspots economy wit partially restricted market participation tan in te sunspots economy wit totally restricted market participation. Similarly, te sunspots economy in wic all market participants are unrestricted as even greater potential pricelevel volatility tan eiter of te oter two economies. Tis suggests tat in an economy wit outside money, nonsunspot equilibria migt not be robust in te face of even small perturbations of te restrictions on market participation or oter data defining te economy. 4. THE SOURCE OF SUNSPOT EQUILIBRIA We ave analyzed sunspot equilibrium in a very simple monetary model. Te simplicity makes te nature and te source of te sunspot equilibria apparent. Te existence of restricted consumers makes a stocastic equilibrium possible. Te interaction of te restricted and unrestricted consumers is essential for producing a SSE wic is not a randomization over CEs. Te restricted consumerstose in G 1, wo cannot trade on sunspot contingent marketstypically face uncertain tax-adjusted endowments and ence ave stocastic consumptions. Typically, te aggregate consumption of te consumers in G 1 is stocastic and ence te aggregate after-tax endowment of te consumers in G 0 will be stocastic. Te unrestricted consumerstose in G 0, wo can trade on sunspot contingent markets attempt to re-allocate te sunspot risks among temselves, but tey do not completely rid temselves of uncertainty: te unrestricted consumers will typically ave uncertain consumptions. Because of te risk-saring among te unrestricted consumers, te resulting SSE allocation will typically differ from a mere randomization over CE allocations. Unrestricted consumers transfer income across states of nature; teir beavior is typically not maximal on a state-by-state basis.

22 422 BHATTACHARYA, GUZMAN, AND SHELL Tis is also made clear in te tax-adjusted Edgewort box, in wic te contingent-claims price ratio and te final allocations to te individual unrestricted consumers are determined. Te after-tax endowments of eac consumer, and terefore te aggregate endowment for te community of unrestricted consumers, are typically stocastic. Consequently, as a result of te stocastic nature of aggregate endowments, te dimensions of te sides of te Edgewort box are typically unequal. Hence te analysis is as if tis were an economy in wic all consumers are unrestricted and all uncertainty is intrinsic. Competitive-risk saring leads to final allocations wic are typically stocastic and different from te (tax-adjusted) endowments. To summarize: te seed of te sunspot allocation is in te set of restricted consumers. Teir net tax payments are uncertain because te price level is uncertain. Hence te net tax receipts of te unrestricted consumers are also uncertain. Te stocastic tax-adjusted endowments of te unrestricted consumers flower into a proper sunspot equilibrium allocation troug te edging against price level uncertainty by te unrestricted consumers. In finite, ``convex,'' competitive economies wit perfect markets for edging against te effects of sunspots, te equilibrium allocations are not uncertain. Noneteless in te monetary economy, edging marketseven perfect edging marketsincrease te possible range of price-level volatility. In nonmonetary economies, small market imperfections do not induce sunspots. 22 However, for monetary economies, because of price-level volatility, it would seem tat te sligtest imperfection in markets would leave te door open to sunspot effects on real allocations. Tis remains to be investigated torougly. Te case in wic te government possesses a (possibly unlimited) stock of commodity is probably not of muc direct economic interest, but it is instructive. Wit a positive government stockpile, balancedness of te tax policy across private agents is not necessary for bonafidelity. Te model ten also allows for government aggregate real transfers to vary across states. Hence tere can ten be proper SSE allocations even wit complete markets and unrestricted participation. Tis is because after-tax Edgewort box would typically not be ``square''. In tis caseeven if markets are perfectequilibrium allocation will typically be stocastic and not mere randomizations over CE allocations. 5. CONCLUDING REMARKS In constructing examples, we employ a new tool: te tax-adjusted Edgewort box. We ope tat tis tool will be useful in te systematic 22 See Balasko, Cass, and Sell [3].

23 PRICE LEVEL VOLATILITY 423 analysis of te structure of te set of competitive equilibria (and te related comparative statics) for tis and more general monetary models. We do not envision difficulty in moving to general von Neumann Morgenstern utility functions. Moving from one commodity to several, owever, will require a more sensitive use of te tax-adjusted Edgewort box. In tis case, intra-state commodity price ratios are jointly determined by te restricted and unrestricted consumers. Tis is only a first attempt at integrating monetary equilibrium teory and sunspot equilibrium teory. Extension of te analysis of te present paper to dynamic economies is essential; static monetary analysis is clearly very incomplete. In te perfect-foresigt overlapping-generations economy, it is neiter necessary nor sufficient for proper monetary equilibrium tat te public debt be retired. 23 Is tis result substantially strengtened wen we allow for te possibility of sunspot equilibria? In particular, are te existing caracterizations of tose fiscal policies wic are consistent wit positively priced money 24 significantly altered wen moving from perfectforesigt equilibrium to te more general case of sunspot equilibrium? Te examples of ``volatility'' given in te present paper are meant to be suggestive. We give no defense of te particular volatility measure (maximum potential variance) tat we use. Peraps our calculations migt prove to be provocative for te development of a teory in wic government policies can be judged in terms of teir implied efficiency, equity, and stability (te ``inverse'' of volatility). 25 A. APPENDIX Here we provide microeconomic justification for te budget constraints in (3.4) and (3.8) for te monetary, sunspots economy of Section 3. We explicitly adopt te overlapping-generations economy; te timing is given by Fig Every consumer in H can trade for money and commodity in te spotmarket (te market wic meets after stated s is revealed). Tese are te only trades tat te restricted consumers (tose in G 1 ) can make (because tey are born after s is revealed). Te unrestricted consumers (tose in G 0 ) can also trade in eiter state-contingent commodity or state-contingent money. Consumers in G 0 ave perfect foresigt about spot-market prices. 23 See Balasko and Sell [5]. 24 See, e.g., Balasko and Sell [4, Sections 5 and 6, and especially Proposition 5.5] and Esteban, Mitra, and Ray [9]. 25 See Keister [15] for recent work in tis direction based in part on te ideas in our present paper.

24 424 BHATTACHARYA, GUZMAN, AND SHELL Te notation is necessarily elaborate. We must employ a precise generalequilibrium-style approac. 26 Te superscript 1 denotes prices and transactions on te spot-market. Te superscript 0 denotes prices and transactions in state-contingent commodity or state-contingent money made before te state of nature is revealed. Let x 1 1 (s) and xm, (s) be respectively te amount of commodity and money purcased by Mr. in te spot-market after state s as been revealed. Let p 1 (s)>0 and p m,1 (s)0 be te corresponding spot-market prices. Let x 0 0 (s) and xm, (s) be respectively te amount of commodity to be delivered if state s occurs and te amount of money to be delivered if state s occurs. Tese contingent commodities are purcased on te market tat meets before te state s is revealed. Let p 0 (s) and p m,0 (s) be te corresponding prices of tese contingent goods. If consumer pays is money taxes in eac state, ten we ave x m,0 (s)+x m,1 (s)={ (A.1) for s=:, ; and # H, but, of course, for # G 1 we ave x m,0 (s)=0. Te consumption of consumer in state s, x (s) is given by x (s)=x 0 (s)+x1 (s) (A.2) but, of course, x 0 (s)=0 for # G 1. We treat separately two cases: (1) Consumers in G 0 do all teir edging troug purcases and sales of state-contingent money. (2) Consumers in G 0 do teir edging troug purcases and sales of state-contingent commodity. We sow tat te model based on assumption (1) is equivalent to te model based on assumption (2) and tat tey are equivalent to te model of Section 3, for wic it is implicitly assumed tat tere are contingent-money markets and contingent-commodity markets available to te consumers in G 0 for edging against te effects of sunspots. A.1. Hedging troug State-Contingent Money Assume, for te moment, tat tere is at least one consumer in G 0, i.e., we ave G 0 {<. Consumer # G 0 cooses (x m,0 (:), x m,0 (;), x m,1 (:), (;), x 1 (:), x1(;)) # R6 to x m,1 maximize?(:) u (x (:))+?(;) u (x (;)) (A.1.1) 26 See Balasko and Sell [4, especially Section 2, Eq. 2.1].

25 PRICE LEVEL VOLATILITY 425 subject to and p 1 (s) x 1 (s)+pm, 1 (s) x m,1 (s)=p 1 (s), for s=:, ;, (A.1.2) p m,0 (:) x m,0 (:)+p m,0 (;) x m,0 (;)=0, (A.1.3) x (s)=x 1 (s)>0 and x m,0 (s)+x m,1 (s)={ for s=:, ;. (A.1.4) Equation (A.1.2) says tat te spot-market purcases of commodity and net accretions to money oldings are financed by te sale of te endowment of commodity. Equation (A.1.3) says tat purcases of money deliverable in one state are financed by te sale of money deliverable in te oter state. Te first equation in (A.1.4) says tat Mr. consumes te commodities wic e as purcased. Te second equation in (A.1.4) says tat te sum of Mr. 's money accretions must equal is money tax obligation. Substituting (A.1.4) and (A.1.3) in (A.1.2) gives and p 1 (:)(x (:)& )+p m,1 (:) { =x m,0 p m,1 (:) (A.1.5) (:) \ pm, 0 (;) p (:)+\ p1 (;)(x (;)& )+p m,1 (;) { m,0 p m,1 (;) + =&xm, 0 (:). (A.1.6) Given prices, is endowment, and is tax-obligation, wen Mr. selects x (:) te left side of (A.1.5) is determined. Similarly, x (;) determines te left side of (A.1.6). Hence (A.1.5) and (A.1.6) can be combined into a single budget constraint witout altering te opportunities of Mr. over final consumption bundles (x 1(:), x1 (;)). Adding (A.1.5) and (A.1.6) results in te single budget constraint for # G 0, \ p1 (:) p m,0 (:) p m,1 (:) + (x (:)& )+ \ p1 (;) p m,0 (;) p m,1 (;) + (x (;)& ) =&(p m,0 (:)+p m,0 (;)) {. (A.1.7) Te (single) budget constraint (A.1.7) reduces to te (single) budget constraint in (3.8) if we ave p(s)=p 1 (s) p m,0 (s)p m,1 (s).

26 426 BHATTACHARYA, GUZMAN, AND SHELL and p m (s)=p m,0 (s). (A.1.8) Substituting te second of te two equations of (A.1.8) into te first yields p 1 (s)p m,1 (s)=p(s)p m (s). (A.1.9) for s=:, ;. We next turn to consumer # G 1 (if tere is one). He cooses (x m,1 (:), (;), x 1 (:), x1(;)) # R4 to x m,1 maximize u (x (s)) (A.1.10) subject to p 1 (s) x 1 (s)+pm, 1 (s) x m,1 (s)=p 1 (s) for s=:, ; (A.1.11) and x (s)=x 1 1 (s)>0 and xm, (s)={ for s=:, ;. (A.1.12) Substituting from (A.1.12) in (A.1.11) yields te single budget constraint for eac state p 1 (s) x 1 (s)=p1 (s) & p m,1 (s) { (A.1.13) for s=:, ;. Te budget constraints (A.1.13) are equivalent to te budget constraints in (3.4) if we ave p(s)p m (s)=p 1 (s)p m,1 (s). for s=:, ;. But by (A.1.9) we ave already cosen te prices p(s) and p m (s) in a way tat assures tis equality. If G 0 is empty, (A.1.9) is te only restriction tat needs to be imposed on ( p(s), p m (s)) for s=:, ;. We ave sown tat if an allocation is an equilibrium to te economy of Subsection A.1, ten it is also an equilibrium for te corresponding economy in Section 3. Te budget constraint in (3.8) is equivalent to (A.1.2)(A.1.4) if (A.1.8) olds. Te budget constraint in (3.8) is equivalent to (A.1.11)(A.1.12) if (A.1.9) olds. Hence an allocation tat is an equilibrium for te economy of Section 3 is also an equilibrium for te economy of Subsection A.1. K

27 PRICE LEVEL VOLATILITY 427 A.2. Hedging troug State-Contingent Commodity Consumer # G 0 cooses (x m,0 (:), x m,0 (;), x m,1 (:), x m,1 (;), x 1 (:), x 1 (;)) # R6 to maximize?(:) u (x (:))+?(;) u (x (;)) (A.2.1) subject to p 1 (s) x 1 (s)+pm, 1 (s) x m,1 (s)=p 1 (s) for s=:, ;, (A.2.2) p 0 (:) x 0 (:)+p0 (;) x 0 (;)=0, (A.2.3) and x (s)=x 0 (s)+x1 1 (s) and xm, (s)={ for s=:, ;. (A.2.4) Equation (A.2.2) says tat in te spot-market purcases of commodity and money are financed by te sale of commodity endowment. Equation (A.2.3) says tat purcases of te contingent commodity to be delivered in one of te states are financed by sales of te commodity to be delivered in te oter state. Te first equation in (A.2.4) says tat consumption of commodity in a given state is equal to te sum of te spot-market purcases of commodity and te purcases of commodity contingent on tat state. Te second equation in (A.2.4) says tat te money tax obligation must be met from spot-market purcases. Substituting from (A.2.4) and (A.2.3) in (A.2.2) gives and p 1 (:)(x (:)& )+p m,1 (:) { =x 0 p 1 (:) (A.2.5) (:) \ p0 (;) p (:)+\ p1 (;)(x (;)& )+p m,1 (;) { 0 p 1 (;) + =&x0 (:). (A.2.6) Adding (A.2.5) and (A.2.6) results in te single budget constraint for consumer # G 0, p 0 (:)(x (:)& )+p 0 (;)(x (;)& ) = \ pm, 1 (:) p 0 (:) + pm, 1 (;) p 0 (;) p 1 (:) p 1 (;) + {. (A.2.7)

28 428 BHATTACHARYA, GUZMAN, AND SHELL Te budget constraint (A.2.7) reduces to te budget constraint in (3.8) if we ave p(s)=p 0 (s) and (A.2.8) p m (s)= pm, 1 (s) p 0 (s) p 1 (s) for s=:, ;. Substituting p(s) for p 0 (s) in te second equation in (A.2.8) yields p m (s) p(s) = pm, 1 (s) p 1 (s) (A.2.9) To complete te argument, consider consumer # G 1. He cooses (x (s), x m (s)) # R2 to maximize u (x (s)) (A.2.10) subject to p 1 (s) x 1 (s)+pm, 1 (s) x m,1 (s)=p 1 (s) for s=:, ; (A.2.11) and x (s)=x$ (s)>0 and x m,1 (s)={ for s=:, ;. (A.2.12) Substituting from (A.2.12) in (A.2.11) yields one budget constraint per state, i.e. p 1 (s)(x (s)& )=&p m (s) { for s=:, ;. (A.2.13) Because of (A.2.9), we know tat we ave already cosen prices so tat (A.2.13) is equivalent to te budget constraint in (3.4). Hence we ave sown tat an allocation tat is an equilibrium for te economy of Subsection A.2 is also an equilibrium for te corresponding economy of Section 3. Te budget in (3.8) is equivalent to (A.2.2)(A.2.4) if (A.2.8) olds. Te budget constraint in (3.4) is equivalent to (A.2.11)(A.2.12) if (A.2.9) olds. Hence an allocation tat is an equilibrium for te economy of Section 3 is also an equilibrium for te economy of Subsection A.2. K