Moneyspots: Extraneous Attributes and the Coexistence of Money and Interest-Bearing Nominal Bonds

Ricardo Lagos New York University

It is folklore among monetary theorists that, under laissez faire, without ad hoc assumptions that favor money over bonds, there do not exist equilibria in which government-issued fiat money coexists with nominal default-free, interest-bearing government bonds with similar physical characteristics. This proposition is the basis for the strongest version of the rate-of-return-dominance puzzle. In this paper I show that if—as has been the case throughout monetary history—the physical object used as fiat money is heterogeneous in an extraneous attribute, then there exist equilibria in which money coexists with interest-bearing bonds.

What has to be explained is the decision to hold assets in the form of barren money, rather than of interest- or profit-yielding securities. . . . This, as I see it, is really the central issue in the pure theory of money. Either we have to give an explanation of the fact that people do hold money when rates of interest are positive, or we have to evade the difficulty somehow. ð John Hicks 1935, 5Þ I. Introduction At least since Hicks ð1935Þ, the basic observation that in actual economies money coexists with securities that bear a higher financial return I thank David Andolfatto, Huberto Ennis, Daniela Puzzello, Tom Sargent, and Neil Wallace, for their feedback. [ Journal of Political Economy, 2013, vol. 121, no. 1] © 2013 by The University of Chicago. All rights reserved. 0022-3808/2013/12101-0001$10.00

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has been regarded as the central challenge facing monetary theory. Because this observation has proven difficult to rationalize in economic models, it has come to be known as the rate-of-return-dominance puzzle ðor coexistence puzzleÞ. The rate-of-return-dominance puzzle is one of the two fundamental questions that constitute the core of monetary theory. The other, more elementary, question is that of existence of a monetary equilibrium as a way to rationalize the fact that fiat money sells at a positive price for valuable goods and services in actual economies, despite its being intrinsically useless, and a formal claim to nothing, against no one. Much has been learned about the existence question by building models that explicitly incorporate frictions ðe.g., limited commitment and enforcement, decentralized exchange, double-coincidenceof-wants problemsÞ and identify conditions that can make monetary exchange an equilibrium. In contrast, the issue of rate-of-return dominance has received scant attention. One reason why monetary economists may have been disinclined to tackle the rate-of-return-dominance puzzle is that the stronger versions of the puzzle, that is, those in which the security that dominates money in rate of return is very similar to money, appear unsolvable: It is folklore among monetary theorists that, under laissez faire, without ad hoc assumptions that favor money over bonds, there do not exist equilibria in which government-issued fiat money coexists with nominal default-free, interest-bearing, payable-to-the-bearer government bonds with similar physical characteristics. Because there have been instances in which such securities were issued and money remained in circulation, this folk impossibility proposition is the basis for the strongest version of the rate-ofreturn-dominance puzzle. In this paper I show that if the physical object used as fiat money is heterogeneous in some extraneous attribute—a moneyspot such as a serial number—then there exist equilibria in which money coexists with interest-bearing bonds.1 I provide an explicit characterization of a class of equilibria that can rationalize the rate-of-return-dominance puzzle in its strongest form and show that these equilibria can exhibit “liquidity effects” of open-market operations that resemble those found in limitedparticipation models, but without actually assuming that participation in asset markets is limited. I also show that moneyspots can overturn an1 Throughout this paper, “moneyspots” is used to represent extraneous attributes of fiat money, e.g., the serial numbers on notes, the mint years stamped on coins, the markings that identify the region where the issuing mint is located, the degree of wear and tear, etc. The term is intended to be reminiscent of “sunspots” ðCass and Shell 1983Þ. Sunspots, the realization of a publicly observed extrinsic random variable, can sometimes be used to coordinate actions in a way that generates self-fulfilling prophecies and enlarges the set of equilibria. I will show that, by introducing the possibility of symmetry-breaking self-fulfilling prophecies, moneyspots can enlarge the set of monetary equilibria. This observation will be the basis for the coexistence result.

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other folk proposition, that is, that it is impossible for the nominal interest rate on government bonds to ever become negative if private agents have access to a costless technology to store money. The rest of the paper is organized as follows. Section II presents the model. Section III considers the special case of no moneyspots and formulates a version of the folk impossibility proposition. Section IV analyzes the general case with moneyspots and presents the coexistence proposition. Section V discusses additional theoretical implications of moneyspots for the behavior of the nominal interest rate, such as the liquidity effects often associated with open-market sales, and the possibility that nominal interest rates become negative. Section VI contains a discussion of the findings and a review of the related literature. Section VII presents conclusions. Appendix A contains the proofs of the main propositions. Subsidiary proofs are in Appendix B. Appendix C contains supplementary material. II. The Model The model builds on Lagos and Wright ð2005Þ. Time is represented by a sequence of periods indexed by t 5 0; 1; : : : . Each time period is divided into two subperiods in which different activities take place. There is a continuum of infinitely lived agents, each identified with a point in the set I 5 ½0; 1. There are two nonstorable and perfectly divisible consumption goods at each date: general goods and special goods. In each subperiod, every agent is endowed with h
units of time that can be employed as labor services. In the second subperiod, each agent has access to a linear production technology that transforms labor services into general goods. General goods are homogeneous and are consumed by all agents. Special goods instead come in many varieties. In the first subperiod, each agent has access to a linear production technology that transforms his own labor input into a particular variety of the special good that he himself does not consume. This specialization is formalized as follows. Given two agents i and j drawn at random, the probability that i consumes the variety of special good that j produces but not vice versa ða single coincidenceÞ is denoted a, with a ≤ 1=2. The probability that j consumes the special good that i produces but not vice versa is also a. The probability that neither i nor j wants what the other agent can produce is 1 2 2a. In a single-coincidence meeting, the agent who wishes to consume is the buyer and the agent who produces the seller. Let Itb ⊆ I and Its ⊆ I denote the subsets of agents that act as buyers and sellers, respectively, in the first subperiod of period t. In the first subperiod, agents participate in a decentralized market, where trade is bilateral ðeach meeting is a random draw from the set of pairwise meetingsÞ, and the terms of trade are determined by bargaining con-

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sisting of a take-it-or-leave-it offer by the buyer. The specialization of agents over consumption and production of the special good combined with bilateral trade gives rise to a double-coincidence-of-wants problem. In the second subperiod, agents trade in a competitive centralized market. Agents cannot make binding commitments, and histories of actions are private in a way that precludes any borrowing and lending, so all trade— in both centralized and decentralized markets—is quid pro quo. There is a monetary authority or “government” that can issue a financial security called money. Each unit of money is physically represented by a note —a piece of paper that is durable, uncounterfeitable, and intrinsically useless ðthe note itself is not an argument of any utility or production functionÞ. The initial stock of notes outstanding at the beginning of period 0 is represented by the set M0 5 ½0; M 0 , where M 0 ∈ R1 is given. Every note is physically identical to every other note ðe.g., equal in size, color, denomination, and other markingsÞ, except for the fact that each note is uniquely identified by an extraneous attribute or moneyspot : a serial number n ∈ M0 that is printed on it.2 The government can also issue nominal one-period bearer default-free pure discount bonds. Each bond is represented by a piece of paper that is durable, uncounterfeitable, and intrinsically useless. A bond issued in period t entitles the bearer to collect a note in period t 1 1. The stock of bonds outstanding at the beginning of period t is represented by the set Bt 5 ½0; Bt , where Bt ∈ R1. Every bond outstanding in period t is different from every outstanding note and is uniquely identified by a serial number s ∈ Bt.3 The timing of government interventions in a typical period t is as follows. The set of bonds outstanding at time t, Bt , is redeemed after the round of decentralized trade, right before agents trade in the centralized market of period t. The new bond issue, Bt11 , is sold competitively for notes after the round of centralized trade of period t, right before agents enter period t 1 1. The government finances bond redemptions 2 Franc¸ois Velde has pointed out to me that the practice of identifying notes with serial numbers is very old; e.g., the edict creating John Law’s bank in 1716 specifies in its annex the form that the notes to the bearer should take, and they were numbered. The serial number is also one of the most universal features of paper money. It is difficult to find a class of notes that has ever circulated in which each note was not uniquely identified by a serial number ðsome rare cases have been recorded; e.g., the Canadian “shinplasters,” 256 fractional notes issued in 1870 and in 1900, had no serial numberÞ. These considerations make serial numbers a natural leading example of moneyspots. As will become clear below ðe.g., see App. CÞ, the main results of this paper generalize to coarser moneyspots, i.e., to specifications with extraneous attributes that, unlike serial numbers, allow agents to identify only proper subsets of notes rather than each individual note. 3 For example, notes are green while bonds are blue. Whether or not every bond is assumed to be indistinguishable from every other bond is inessential for the main results; the main proposition goes through even if every bond is assumed to be identical to every other bond.

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with notes acquired in previous bond sales, and if these notes were not enough, it makes up the difference by printing new notes.4 Each newly printed note is marked with a unique serial number ðnotes are numbered consecutivelyÞ, but newly printed notes are otherwise identical to 2 denote the set of notes withdrawn from pripreexisting notes. Let Nt11 1 be the set of vate circulation by the bond sale of period t, and let Nt11 notes injected by the corresponding bond redemption in period t 1 1. Let Mt represent the set of notes outstanding at the beginning of pe
t be the set of notes outstanding at the end of the riod t, and let M centralized trading session of period t, before the new bond sale, that is,
t 5 Mt [ N 1 . The law of motion for the postredemption set of outM t standing notes is
tnN 2 Þ [ N 1 :
t11 5 ðM M t11 t11 Let Mt represent the size of the set of notes outstanding at the be
t be the size of the set of notes outstanding ginning of period t, and let M at the end of the centralized trading session of period t, before the new bond sale. Since Bt is the size of the set of bonds outstanding at the be
t represent
t 5 Mt 1 Bt . Intuitively, Bt , Mt , and M ginning of period t, M the period t quantities of bonds, notes outstanding at the beginning of the period, and notes outstanding after that period’s bond redemption, respectively. Formally, consider the measure space ðR1 ; F ðR1 Þ; mÞ, where F ðR1 Þ denotes the Borel j-field on R1 , and m is the Lebesgue
t Þ, and Bt 5 mðBt Þ 5 mðN 1 Þ.5 A
t 5 mðM measure. Then Mt 5 mðMt Þ, M t bond issue Bt11 injects Bt11 notes in period t 1 1 and withdraws from 2 private circulation mðNt11 Þ 5 ∫Bt11 qt ðsÞmðdsÞ ; Q t ðBt11 Þ notes at the end of period t, where qt ðsÞ is interpreted as the quantity of notes needed to purchase a bond with serial number s at time t.6 Formally, let F ðBt Þ 5 0 0 fB ∈ F ðR1 Þ : B ⊆ Bt g; then qt : Bt11 → R1 [ ` is an F ðBt11 Þ-measurable function. The law of motion for the measure of the postredemption set of outstanding notes is

4 The analysis will abstract from fiscal considerations; in particular, I will not assume that the monetary authority has the ability to levy taxes. 5
t , N 1 , and N 2 are eleThroughout, the maintained assumption will be that Mt , M t t ments of F ðR1 Þ. This does not entail any substantial restriction on individual behavior and is inconsequential for the economic results. 6 I will work under the assumption that the government treats notes symmetrically regardless of serial number. One could imagine that the government gives certain serial numbers differential treatment in open-market operations, e.g., by requiring that bond purchases be paid with those specific serial numbers. I will not explore such formulations, but as will become clear below, the main insights would not be affected. In what follows, given an F ðR1 Þ-measurable function f, I will denote the Lebesgue integral ∫f ðnÞmðdnÞ by ∫f ðnÞdm or, when there is no possibility of confusion, by the more concise ∫fdm.

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t 2 Q t ðBt11 Þ 1 Bt11 :
t11 5 M M

The size of the bond issue, Bt11 , expressed relative to the size of the
t , will be denoted by xt . That is, if there postredemption money supply, M
are Mt notes outstanding after the round of centralized trade, at the end
t notes of the period the government auctions off claims to Bt11 5 xt M payable at the beginning of the centralized trading session one period
t11 5 M
t 2 qt Bt11 1 Bt11 , and hence. Note that if qt ðsÞ 5 qt for all s, then M
t11 5 ½1 1 ð1 2 qt Þxt M
t : M

ð1Þ

The motion of xt is assumed to follow a Markov process with transition function F ðx 0 ; xÞ 5 Prðxt11 ≤ x 0 jxt 5 xÞ, where F : R1  R1 → R is continuous. For each fixed x, F ð; xÞ is a distribution function with support X ⊂ R1 . It is assumed that the process defined by F has a stationary distribution w, the unique solution to wðx 0 Þ 5 ∫F ðx ;0 xÞdwðxÞ, and F has the Feller property; that is, for any continuous real-valued function g on X, ∫g ðx 0 ÞdF ðx ;0 xÞ is a continuous function of x. The realization xt is observed by all agents and is common knowledge at the beginning of period t ðbefore the round of decentralized tradeÞ. Aside from these stochastic open-market operations, there are no shocks to the fundamentals of the economy. Let the utility function for special goods, u : R1 → R, be continuously differentiable, increasing, and strictly concave, with uð0Þ 5 0. Suppose
Each agent that there exists c * ∈ ð0; `Þ defined by u 0 ðc * Þ 5 1, with c * ≤ h. ranks consumption and labor supply bundles according to
` E0

 t b ½uðc Þ 2 l 1 y 2 h  ; t t t t o t50

where b ∈ ð0; 1Þ, ct and lt are the quantities of special goods consumed and produced in the decentralized market, yt denotes consumption of general goods, and ht represents the hours worked in the second subperiod;7 Et is the expectations operator conditional on the information available to the agent at time t, defined with respect to the matching probabilities and the probability measure over fxt g induced by F. 7 Many of the assumptions on preferences are made for convenience and can be relaxed without changing anything of substance. All that is needed for analytical tractability is that the period utility function is quasi-linear in yt or ht ðsee Lagos and Wright 2005Þ.

moneyspots III.

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The Impossibility Proposition

In this section I focus on the special case in which notes have no extraneous attributes, that is, there are no serial numbers, so every note is indistinguishable from every other note. In this context, I establish a wellknown impossibility result, namely, that there cannot exist a monetary equilibrium in which money coexists with nominal default-free, interestbearing bonds.8 Begin by describing the individual optimization problems faced by an agent in a typical period, starting at the end of the period. Consider an agent who enters the competitive market for government bonds at the end of period t with m
t ∈ R1 notes. His problem consists of choosing a portfolio a t11 5 ðmt11 ; bt11 Þ, where bt11 is the quantity of bonds he purchases, and mt11 is the quantity of notes he holds after the bond purchase. Formally, in the market for government bonds this agent solves Ut ðm
t Þ 5 max bEtVt11 ða t11 Þ a t11

subject to qt bt11 1 mt11 ≤ m
t ; mt11 ; bt11 ∈ R1 , where qt is the money price of a bond, and Vt11 ða t11 Þ is the maximum expected utility that the agent can attain by entering the decentralized market of period t 1 1 with portfolio a t11 . Let Wt ða t Þ denote the maximum expected discounted utility that an agent can attain when he enters the centralized market holding a portfolio a t in period t. This value satisfies Wt ða t Þ 5 maxfyt 2 ht 1 Ut ðm
t Þg yt ;ht ;m
t

ð2Þ

subject to yt 2 ht 1 ft m
t 5 ft ðmt 1 bt Þ;
where ft is the price of a note in terms of genyt ; ht ; m
t ∈ R1 ; and ht ≤ h, eral goods. On the left side of the budget constraint are the agent’s net consumption of general goods and the real value of the money holdings 8 Versions of this impossibility result in the context of other models can be found, for instance, in Wallace ð1983, 1990Þ, Hellwig ð1993Þ, Aiyagari, Wallace, and Wright ð1996Þ, and Kocherlakota ð2003Þ.

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he is choosing. On the right side is the real value of his postredemption money holdings.9 Consider a bilateral meeting in the decentralized market of period t ^ t ; ^bt Þ, between a buyer and a seller who hold portfolios ðmt ; bt Þ, and ðm respectively. A bargaining outcome is a quantity of special good, ct , that the seller produces for the buyer in exchange for a portfolio of notes and bonds, ðmt0 ; bt0 Þ, that the buyer offers as payment. The gains from trade from such an outcome would be uðct Þ 1 Wt ðmt 2 mt0 ; bt 2 bt0 Þ 2 ^ t 1 mt0 ; ^bt 1 bt0 Þ 2 Wt ðm ^ t ; ^bt Þ for the Wt ðmt ; bt Þ for the buyer and 2ct 1 Wt ðm seller. Hence, the buyer’s take-it-or-leave-it offer solves fuðct Þ 2 ft ðmt0 1 bt0 Þg max 0 0 ct ;mt ;bt

subject to
m 0 ≤ mt ; b 0 ≤ b t : ct ≤ ft ðmt0 1 bt0 Þ; 0 ≤ ct ≤ h; t t The first constraint requires that the trade be individually rational for the seller, and the last two constraints state that the buyer can pay only with the notes and bonds that he owns. The bargaining outcome can be described as follows. If c * ≤ ft ðmt 1 bt Þ, the buyer exchanges some portion ðmt0 ; bt0 Þ of his portfolio with value ft ðmt0 1 bt0 Þ 5 c * for a quantity c * of the special good. Else, the buyer gives the seller his whole portfolio, that is, ðmt0 ; bt0 Þ 5 ðmt ; bt Þ, in exchange for ct 5 ft ðmt 1 bt Þ special goods. Hence, the quantity traded can be written as a function c : R1 → ½0; c *  of the real value of the buyer’s portfolio, ft ðmt 1 bt Þ. Specifically, the quantity traded in the decentralized market of period t between a buyer who holds portfolio ðmt ; bt Þ and a seller is ct 5 cðft ðmt 1 bt ÞÞ, where cðxÞ ; minðx; c * Þ:

ð3Þ

The maximum expected discounted utility attainable by an agent who enters the decentralized market of period t with portfolio a t can be written as Vt ða t Þ 5 L½ft ðmt 1 bt Þ 1 Wt ð0Þ;

ð4Þ

LðxÞ ; afu½cðxÞ 1 x 2 cðxÞg 1 ð1 2 aÞx

ð5Þ

where

9 To simplify the exposition, it is convenient to abstract from the agent’s decision of whether to redeem a bond for a note and simply assume that all bonds are redeemed at maturity. A mechanical way to think about this would be to imagine that the bond automatically turns into a note at maturity. In Sec. VI, I discuss the implications of endogenizing the agents’ decision of whether and when to redeem their bond holdings.

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is the expected utility that an agent obtains over both subperiods of any given period, if he enters the first subperiod with a portfolio of money and nominal bonds of real value x ðexpressed in terms of current-period general goodsÞ.10 To interpret LðxÞ, note that with probability a, the agent acts as a buyer in the first subperiod: the portfolio allows him to buy cðxÞ special goods ðwhich give him utility u½cðxÞÞ, and the real value of the unspent portfolio, x 2 cðxÞ, can be sold for x 2 cðxÞ general goods in the following centralized market. With probability 1 2 a, the agent has no opportunity to use his portfolio in a bilateral exchange, and he carries it into the centralized market, where he can sell it for x general goods. Equivalently, since x is the real resale value of the portfolio that the agent carries into the period and given that the expected gain from trade in the decentralized market, afu½cðxÞ 2 cðxÞg, can be interpreted as an expected “liquidity dividend” from holding the portfolio, one can write LðxÞ 5 x 1 afu½cðxÞ 2 cðxÞg and interpret LðxÞ as the expected cum ðliquidityÞ dividend value ðin terms of current-period general goodsÞ of the portfolio of money and bonds.11 Definition 1. An equilibrium of the economy with no extraneous ` Þi ∈I , bargaining outattributes is an allocation ðfyit ; hit ; m
it ; mit11 ; bit11 gt50 ` ` comes fðcijt Þi; j ∈I gt50 , and prices fft ; qt gt50 such that ðiÞ given prices and ` solves agent i’s optithe bargaining protocol, fyit ; hit ; m
it ; mit11 ; bit11 gt50 mization problem in the competitive markets for all i ∈ I ; ðiiÞ the bilateral terms of trade are determined by Nash bargaining; that is, if agent i is the buyer and agent j the seller in a bilateral meeting at time t, then j produces cijt 5 min½ft ðmit 1 bit Þ; c *  special goods for i ðcijt 5 0 if i and j are not in a single-coincidence meeting at tÞ; and ðiiiÞ the centralized markets clear for all t. An equilibrium is said to be “monetary” if ft > 0 for all t, and in this case the allocation must also satisfy ða Þ the money
t , and ðbÞ the bond market– market–clearing condition, ∫I m
it dm 5 M clearing condition, ∫I bit11 dm 5 Bt11 , for all t ≥ 0. In a monetary equilibrium, money is said to coexist with bonds in period t if Bt > 0 and ∫I ft mit dm > 0. Proposition 1. Let it11 5 1=qt 2 1 denote the nominal interest rate on nominal bonds between period t and period t 1 1. The set of monetary equilibria of the economy with no extraneous attributes is nonempty if u 0 ð0Þ > 1 1 ½ð1 2 bÞ=ab. In any monetary equilibrium of the economy with no extraneous attributes, money coexists with bonds in period t 1 1 only if it11 5 0. Each bond issued at the end of period t becomes a note with certainty after the round of decentralized trade of period t 1 1. Agents anticipate 10 The symbol 0 will be used to denote the origin in Rk as well as the real number zero; no confusion will result. 0 11 Notice that LðxÞ is twice differentiable everywhere, with L ðxÞ 5 1 1 a½u 0 ðxÞ 2 1Ifx
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this fact in the decentralized market of period t 1 1 and therefore value a bond and a note identically at that point. When agents choose their money and bonds portfolio in the bond market of period t, they anticipate that a note and a bond will be treated as the same asset by everyone in the following period, so they choose to hold no money if qt < 1 and are indifferent between holding money and bonds if qt 5 1. ðBond market clearing is inconsistent with qt > 1.Þ A monetary equilibrium may exhibit periods with qt 5 1 and periods with qt < 1. Money will circulate ðalongside bondsÞ, or, equivalently, money and bonds will coexist in the portfolios that agents choose to carry into the decentralized market, only in periods when the nominal interest rate is zero. In any period t with 1=qt 2 1 > 0, agents use all their notes to purchase government bonds and enter period t 1 1 holding no money. Hence, in such a period t 1 1, money does not circulate: only bonds are used as medium of exchange in decentralized trades. Proposition 1 is a version of a well-known result: Money cannot coexist with interest-bearing default-free nominal bonds in the portfolio of assets held by the private sector.12 IV.

Coexistence of Money and Interest-Bearing Nominal Bonds

In this section I consider the general setup with nominal bonds and notes with extraneous attributes. In this context, I establish the existence of monetary equilibria in which money coexists with interest-bearing nominal bonds. The set of postredemption notes outstanding during the central
t Þ 5 fM ∈ F ðR1 Þ :
t ⊂ R1 . Let F ðM ized trading session of period t is M

M ⊆ Mt g, and use finite measures on F ðMt Þ to represent portfolios
t denote a finite meaof time t postredemption notes. Specifically, let m

sure on F ðMt Þ, and let Mt denote the collection of all such measures;
t is the portfolio of postredemption notes held by agent i
it ∈ M then m at time t. Similarly, let Mt denote the collection of all finite measures on F ðMt Þ 5 fM ∈ F ðR1 Þ : M ⊆ Mt g, and let mit be a typical element, interpreted as the beginning of period t portfolio of notes held by agent i. Also, let Bt be the set of all finite measures on F ðBt Þ, and use bit11 ∈ Bt11

12 This is so, of course, as long as the environment does not give money an advantage relative to bonds in the exchange process. For example, if one exogenously imposes that agents can use only money as a medium of exchange, then it is easy to construct a monetary equilibrium in which money circulates in periods when bonds pay interest. Such exogenous restrictions on the use of bonds as media of exchange seem to be the standard assumption underlying cash-in-advance models, as well as an assumption used in some versions of monetary search models with multiple assets. As another example, Zhu and Wallace ð2007Þ obtain coexistence of money and interest-bearing nominal bonds by assuming a bilateral trading protocol that effectively confers larger gains from trade to buyers who hold a larger proportion of money in their portfolios.

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to represent the portfolio of bonds held by agent i at the end of period t. In general, given a set of assets S with associated Borel j-algebra F ðSÞ, a real-valued F ðSÞ-measurable price function p on S, and a portfolio f, which is an element of the collection of all finite measures on F ðSÞ, the Walrasian value of the portfolio f is computed in the usual way, that is, p  f 5 ∫S pðnÞfðdnÞ.13 I proceed by describing the individual optimization problems faced by an agent in a typical period, starting at the end of the period. Consider
t when he
it ∈ M an agent i ∈ I who is holding a portfolio of notes m enters the competitive market for government bonds at the end of period t. This agent’s problem consists of choosing measures n2it11 and ait11 5 ðmit11 ; bit11 Þ, where bit11 is the portfolio of bonds that he purchases,
tÞ n2it11 is the portfolio of notes that he trades in the bond market ðn2it11 ðM is the quantity of notes that he uses to pay for the bond purchaseÞ, and mit11 is the portfolio of notes that he holds after the bond purchase. The value of this agent’s problem in the bond market is
it Þ 5 max bEtVt11 ðait11 Þ U t ðm 2

ð6Þ


t Þ;
it ðMÞ for all M ∈ F ðM n2it11 ðMÞ 1 mit11 ðMÞ ≤ m

ð7Þ


t Þ; qt  bit11 ≤ n2it11 ðM

ð8Þ


t ; bit11 ∈ Bt11 : mit11 ; n2it11 ∈ M

ð9Þ

ait11 ;nit11

subject to


t and bt11 ∈ Bt11 , Vt11 ðat11 Þ denotes the maxFor any ait11 with mit11 ∈ M imum expected discounted utility that the agent can attain when he enters the decentralized market of period t 1 1 holding portfolio at11 ðafter the size of the open-market operation of period t 1 1 has been announced but before the realization of the bilateral matchingÞ. The first constraint states that the notes that the agent uses to purchase bonds and the notes that he carries into the next round of decentralized trade must belong to the portfolio of notes that the agent was holding before the open-market operation. The second constraint states that given the price function for bonds, qt , the agent has enough notes to finance ait11 . 13 This formulation is often used in the theory of general Walrasian equilibrium with infinitely many commodities ðe.g., Jones 1983Þ, and it encompasses the following special cases. If0 f has a Radon-Nikodym derivative f with respect to m, then p  f 5 ∫S pðnÞf ðnÞmðdnÞ, 0 of point masses of size fi at k points and fðS Þ 5 ∫S 0 f ðnÞmðdnÞ for all S ∈ F ðSÞ. If f consists 0 k k n1 ; : : : ; nk ∈ S, then p  f 5 oi51 pðni Þfi and fðS Þ 5 oi51 Ifni ∈S 0 g fi . For any two vectors f 5 1 k ðf ; : : : ; f Þ and m 5 ðm 1 ; : : : ; m k Þ, the expression fm denotes the scalar product oks51 fs m s .

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ðImplicit in this constraint is the maintained assumption that the government ignores extraneous attributes.Þ As stated in Section II, each bond represents a sure claim to a note. However, since notes are differentiated by serial numbers, in order to price a bond, it may not be enough for an agent to know that the bond will be redeemed with a note: the agent may also wish to predict the serial number of the note that will be used to redeem a particular bond. For this reason, a government redemption lottery is specified for each period t. Intuitively, a redemption lottery is a probability measure over assignments of the notes that the government uses for redemption to bonds, specifying the probability that a collection of bonds with serial 0 numbers in a ðBorel measurableÞ set B ⊆ Bt is redeemed with a set of 0 notes with serial numbers in the ðBorel measurableÞ set N ⊆ Nt .1 Let Qt 1 5 fq : Bt → Nt jqð∅Þ 5 ∅, q is bijective, m-measure preserving, and Borel measurableg. A time t assignment ðof notes to bondsÞ is an element q ∈ Qt ; that is, an assignment maps measurable sets of outstanding bonds onto measurable sets of notes ðof equal sizeÞ being used for redemption. 0 0 That is, each function q ∈ Qt assigns a set N ⊆ Nt 1 to each B ⊆ Bt via 0 0 0 0 1 N 5 fn ∈ Nt jn 5 qðsÞ for s ∈ B g or, more compactly, via N 5 qðB Þ. Let F Qt denote the j-field generated by Qt, and let nt be a probability measure on F Qt . The probability space ðQt ; F Qt ; nt Þ represents the government redemption lottery at time t. The period t assignment ði.e., the realization q ∈ Qt Þ becomes known after the round of decentralized trade, right before the bonds are redeemed.
t → R1 [ ` be an F ðM
t Þ-measurable function. Intuitively, Let ft : M ft is the price function for notes at time t; that is, ft ðnÞ is interpreted as the price in terms of general goods of a note with serial number n.
t21 and bit ∈ Bt , let Wt ðait Þ denote the For a given ait 5 ðmit ; bit Þ with mit ∈ M maximum expected discounted utility that an agent can attain when he enters the time t centralized market ðafter the realization of the redemption lottery has been observedÞ with ait . This value satisfies
it Þg Wt ðait Þ 5 max fyit 2 hit 1 Ut ðm

ð10Þ


it 5 ft  mit 1 ft  n1it ; yit 2 hit 1 ft  m

ð11Þ


it yit ;hit ;m

subject to


t , and hit ≤ h,
where ft  mit 5 ∫M ft ðnÞmit ðdnÞ, and
it ∈ M yit , hit ∈ R1 , m t ft  n1it ; ∫N 1t ft ðnÞn1it ðdnÞ. The measure n1it represents the portfolio of notes that the agent receives from the government upon redeeming his bond holdings, bit . Formally, let F ðNt 1 Þ denote the Borel j-field on Nt 1; then n1it is the finite measure defined by n1it ðN Þ 5 bit ðq21 ðN ÞÞ for each

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1

N ∈ F ðNt Þ.14 Hence, the second integral on the right side of the budget constraint can be written as ft  n1it 5

E

Nt

ft ðnÞbit ðq21 ðdnÞÞ ; ft  ðbit q21 Þ 1

and represents the real value ðin terms of general goodsÞ of the portfolio of notes that the agent receives from the government upon redeeming his bond holdings, bit . The first integral on the right side of the budget constraint represents the real value ðin terms of general goodsÞ of the portfolio of notes that the agent brought into the decentralized market. On the left side of the budget constraint, the difference yit 2 hit represents the net quantity of general goods consumed ðor produced, if negativeÞ, and the integral is the real value of the set of notes that the agent is choosing. Consider a bilateral meeting in the decentralized market of period t between a buyer i ∈ I and a seller j ∈ I who are holding portfolios ðmit ; bit Þ and ðmjt ; bjt Þ, respectively, where mit , mjt ∈ Mt and bit , bjt ∈ Bt . A bargaining outcome is a quantity of special good, cijt , that the seller produces for the buyer in exchange for a portfolio ðmt0 ; bt0 Þ, with mt0 ∈ Mt and bt0 ∈ Bt , that the buyer offers as payment. Define the measures 0 mjt  m0, mit  m0, bjt  b0, and bit  b0 by mjt  m ðMÞ 5 mjt ðMÞ 1 m0 ðMÞ and mit  m0 ðMÞ 5 mit ðMÞ 2 m0 ðMÞ for each M ∈ F ðMt Þ and by bjt  b0 ðBÞ 5 bjt ðBÞ 1 b0 ðBÞ and bit  b0 ðBÞ 5 bit ðBÞ 2 b0 ðBÞ for each B ∈ F ðBt Þ. Note that mjt  m0 ∈ Mt and bjt  b0 ∈ Bt , while mit  m0 and bit  b0 are finite signed measures on F ðMt Þ and F ðBt Þ, respectively. The gains from trade from a bargaining outcome ðcijt ; mt0 ; bt0 Þ would be uðcijt Þ 1 Et ½Wt ðmit  m 0t ; bit  bt0 Þ 2 Et ½Wt ðmit ; bit Þ for the buyer and 2cijt 1 Et ½Wt ðmjt  m0t ; bjt  bt0 Þ 2 Et ½Wt ðmjt ; bjt Þ for the seller, where the expectation is with respect to the measure nt , over assignment realizations of the redemption lottery. The buyer’s takeit-or-leave-it offer solves uðct Þ 2 Lt ðm0t ; bt0 Þ max 0 0 ct ;mt ;bt

subject to
mit  m0t ∈ Mt ; bit  bt0 ∈ Bt ; ct ≤ Lt ðm0t ; bt0 Þ; 0 ≤ ct ≤ h; 14

1

Note that q21 ðN Þ ∈ F ðBt Þ for any N ∈ F ðNt Þ because q is Borel measurable.

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where15 Lt ðm0t ; b0t Þ ; ft  m0t 1

E

ft  ðb0t q21 Þnt ðdqÞ:

Qt

The first constraint requires that the trade be individually rational for the seller. The last two constraints indicate that the buyer can pay only with the notes and bonds that he owns. The bargaining outcome is as follows. If c * ≤ Lt ðmit ; bit Þ, the buyer exchanges a part ðmt0 ; bt0 Þ of his asset portfolio with value Lt ðmt0 ; bt0 Þ 5 c * for a quantity c * of the special good. Else he gives the seller all his asset holdings, that is, ðmt0 ; bt0 Þ 5 ðmit ; bit Þ, in exchange for cijt 5 Lt ðmit ; bit Þ special goods. Hence, the quantity traded in a single-coincidence meeting is cðLt ðmit ; bit ÞÞ. With this bargaining solution, ð10Þ, and ð11Þ, the maximum expected discounted utility attainable by an agent who enters the decentralized market of period t holding ait 5
t21 and bit ∈ Bt , can be written as ðmit ; bit Þ, with mit ∈ M Vt ðait Þ 5 L½Lt ðait Þ 1 Kt ; ` satisfies Kt 5 Dt 1 bEt Kt11 , and where fKt gt50

Dt ; max 2ft  mit 1 max bEt L½Lt11 ðait11 Þ ; 2
t
it ∈ M m

ait11 ;nit11

ð12Þ

ð13Þ

subject to ð7Þ, ð8Þ, and ð9Þ. ðSee lemma 1 and lemma 2 in App. B for details.Þ
it ; mit11 ; Definition 2. An equilibrium is an allocation ðfyit ; hit ; m ` ` Þi ∈I , bargaining outcomes fðcijt Þi; j ∈I gt50 , and pricing functions bit11 gt50 ` such that ðiÞ given prices and the bargaining fðft ðnÞÞn ∈ M
t ; ðqt ðsÞÞs ∈Bt11 gt50 `
it ; mit11 ; bit11 gt50 protocol, fyit ; hit ; m solves agent i’s optimization problem in the competitive markets for almost every i ∈ I ; ðiiÞ for almost every i, j ∈ I in a bilateral meeting, the bilateral terms of trade are determined by Nash bargaining; that is, if agent i is the buyer and agent j the seller in a bilateral meeting at time t, then j produces cijt 5 min½Lt ðmit ; bit Þ; c *  special goods for i ðcijt 5 0 if i and j are not in a singlecoincidence meeting at tÞ; and ðiiiÞ the centralized market clears for all t.
t : ft ðnÞ > 0g and F ðM
* Þ 5 fM ∈ F ðM
tÞ : M ⊆ M
* g;
* 5 fn ∈ M Let M t t t
* ≠ ∅ for all t, and in this case, an equilibrium is said to be “monetary” if M t the allocation must also satisfy, for all t, ðaÞ the money market–clearing
* Þ, and ðbÞ the bond
it ðdnÞmðdiÞ 5 mðMÞ for all M ∈ F ðM condition, ∫I ∫M m t 15

In the proof of proposition 2, I will show that ft  ðbt0 q21 Þ is F Qt -measurable and that

∫Qt ft  ðbt0 q21 Þnt ðdqÞ is well defined in the class of equilibria of interest.

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market–clearing condition, ∫I ∫B bit11 ðdsÞmðdiÞ 5 mðBÞ for all B ∈ F ðBt11 Þ. In a monetary equilibrium, money is said to coexist with bonds in period t ≥ 1 if Bt > 0 and ∫I ðft  mit ÞmðdiÞ > 0. A monetary equilibrium is said to
t with ft ðnÞ ≠ ft ðn 0 Þ be a “moneyspot equilibrium” if there exist n, n 0 ∈ M for some t. Proposition 2. Let it11 ðsÞ 5 1=qt ðsÞ 2 1 denote the nominal interest rate on a nominal bond with serial number s ∈ Bt11 between period t and 0 period t 1 1. If u ð0Þ > 1 1 ½ð1 2 bÞ=ab and supX < ` , the economy with extraneous attributes admits a continuum of monetary equilibria in which money coexists with bonds for all 1 ≤ t < ` , even if it11 ðsÞ > 0 for all s ∈ Bt11 . The proof of proposition 2 consists of constructing a family of moneyspot equilibria and showing that money coexists with bonds in any equilibrium that belongs to the family. The basic logic is as follows. If notes are distinguishable, for example, by a serial number, then they need not be treated symmetrically by agents’ beliefs, and different subsets of the set of outstanding notes may be valued differently in equilibrium. The differences in valuations of extraneous attributes supported by these symmetry-breaking self-fulfilling beliefs can be used to construct rational expectations monetary equilibria with the property that fiat money coexists with interest-bearing nominal bonds. To further illustrate the argument formalized in the proof of proposition 2, consider the following streamlined version. Suppose that in every period t, agents expect other agents to be willing to give up f0t goods for notes whose serial numbers lie in some set
0 and f1 goods for notes whose serial numbers lie in another set M t t
0; M
1 i is a partition of the set of time t postredemp
1 , where hM M t t t
t , and 0 < f1 < f0 . In addition, suppose that agents expect tion notes, M t t other agents to assign all notes injected by the government in period
1 , and to value newly injected notes accordingly, at t 1 s to the set M t1s f1t1s ∈ ð0; f0t1s Þ. Given these price paths for notes, agents are always will
1 in the bond market of period t. ing to purchase bonds with notes in M t
0 unless qt , the However, they will not purchase bonds with notes in M t nominal bond price in period t, is lower than f1t11 =f0t11 , the relative price
0 Þ.
1 in period t 1 1 ðin terms of notes in M of notes in M t11 t11 Given a spread f0t11 =f1t11 implied by the price paths for notes, it is possible for an open-market operation to be large enough to require bonds to be priced at a discount but for this discount to be too small ðrelative
0 to purchase to the spread f0t11 =f1t11 Þ to induce agents to use notes in M t bonds. In turn, for a given open-market operation, the size of the spread f0t11 =f1t11 is supported by the self-fulfilling expectation that bonds are
1 : Since all newly injected notes are exredeemed with notes in M t11
1 and in equilibrium they are treated pected to be treated as notes in M t11 1 1
bear a higher endogenous inflation rate
, notes in M as notes in M t11 t

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, which rationalizes the belief that notes in M
1 will be than notes in M t11 0
. Notice that agents’ expectations are all less valuable than notes in M t11
1 ig ` , and the se
0; M that determine the sequence of partitions, fhM t t t50 quence of price differentials between notes that support the equilibrium bond prices and the ensuing portfolio choices. The proof of proposition 2 shows that it is possible to construct equilibria in which all these expectations are mutually consistent and become self-fulfilling. In the equilibria constructed in proposition 2, the nominal rate can be positive or zero depending on the size of the open-market sale, xt .
1 Þ=M
t denote the proportion of the postSpecifically, let 12 vt ; mðM t redemption time t money supply that agents are willing to use for bond purchases: If xt ≤ 1 2 vt , then it11 5 0 and positive real money balances
1 Þ. If
0 and in M are held in private portfolios ðwhich include notes in M t t xt > 1 2 vt , then positive real money balances are held in private portfolios even though it11 > 0 ðin this case private portfolios contain notes
0 onlyÞ. Therefore, for large enough open-market sales, an outside in M t analyst who is unmindful of moneyspots would be puzzled by the observation that notes coexist in private portfolios with nominal interestbearing bonds that are default free in the usual sense that the government redeems each bond with a note, with probability one at maturity. Specifically, in periods with it11 > 0, this analyst would observe that money is held by private agents even though its measured real rate of return is negative, while the nominal default-free government bonds, which are just as liquid as money, yield a positive real rate of return.16 0 t

V.

Moneyspots and the Nominal Interest Rate

An expectations-driven theory of nominal interest rate determination underlies the construction of moneyspot equilibria in the proof of proposition 2. In this section, I discuss some implications of this theory for the behavior of the nominal interest rate. A.

Liquidity Effects of Open-Market Operations

There are some similarities between the behavior of the nominal interest rate in moneyspot equilibria and that in models with limited participation ðe.g., Grossman and Weiss 1983; Rotemberg 1984; Lucas 1990Þ. Limited-participation models provide a theory of nominal interest rate 16 In corollary 1 ðApp. AÞ, I derive expressions for the real returns on bonds and on each type of money as they are calculated by agents living in any of the equilibria constructed in the proof of proposition 2. I also provide expressions for the real returns on bonds and on money as they would be calculated by a financial analyst who is unmindful of moneyspots in that he uses a standard price index ðe.g., the GDP deflatorÞ to deflate nominal bond returns and to calculate the real return on money.

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determination with some attractive properties, such as the so-called liquidity effects of open-market operations. Those models rest on two fundamental assumptions that are critical for their ability to finesse the rate-of-return-dominance puzzle and to generate these liquidity effects. First, by construction, they do not have to confront the rate-of-returndominance puzzle: Those models are in the cash-in-advance tradition, in that money is assumed to be the only asset with which goods ðand securitiesÞ can be bought. Second, the liquidity effects rely on specific timing assumptions as well as on particular restrictions on portfolio reallocations. In Lucas’s version, for instance, there is a representative household composed of three members, each of whom carries out his own activity during a period, with the three regrouping at the end of the day to pool goods, assets, and information. One member collects the period endowment and sells it to other households on a cash-in-advance basis.17 A second member takes Nt 2 Zt ≥ 0 of the household’s initial cash balances, Nt , and uses it to purchase goods from other households on a cash-inadvance basis. A third member of the household takes the remaining cash balances, Zt , and engages in bond trading. A critical assumption is that the household commits itself to a division of cash among the three members before the size of the current-period open-market operation is announced. Hence, there may sometimes be too much and sometimes too little cash to be traded for bonds, which is the reason for the “liquidity effects.” For example, the nominal rate will be low when the size of the open-market sale is large relative to the quantity of money that was committed in advance to the household member who participates in the bond market. In Lucas’s simplest formulation ðe.g., Lucas 1990, 242Þ, the equilibrium price of bonds is minðzt* =xt ; 1Þ, where zt* is the proportion of the money supply that the representative household allocates to bond trading, and xt is, as in this paper, the size of the open-market operation relative to the outstanding money supply. In each of the equilibria that I construct in the proof of proposition 2, the price of bonds is
1 Þ=M
t is the proportion of the minðð1 2 vt Þ=xt ; 1Þ, where 1 2 vt ; mðM t money supply that agents are willing to use for bond purchases. The expression is identical to Lucas’s, with 1 2 vt replacing zt* , but the underlying mechanism is rather different: It is entirely driven by expectations and moneyspots rather than by the specific timing and trading restrictions that the limited-participation formulation relies on.18 17 By assumption, a household cannot consume any of its own endowment or spend cash receipts from date t sales before date t 1 1. 18 The critical timing assumption is that money balances are allocated to the shopper and the financial trader before the size of the open-market operation shock is announced. The critical trading restriction is that money cannot be reallocated between the shopper

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B. Negative Nominal Interest Rates On occasion, nominal interest rates on government securities have become negative. For example, the transaction prices of Treasury bills in the United States sometimes exceeded par during the Great Depression. The same happened with short-term Japanese government bills late in 1998. The nominal interest rate on 4-week US Treasury bills became negative twice in 2008.19 These instances are puzzling because, according to standard monetary theory, negative nominal interest rates are impossible given that agents have access to a costless technology to store money—an assumption that is implicitly adopted in most monetary models. In this subsection I show that the nominal interest rate on government bonds can become negative in a moneyspot equilibrium, even if agents can costlessly store notes. To this end, I consider a special case of the economy of Section IV with a single open-market sale at t 5 0.
0 > 0, where x0 ∈ X is given at Proposition 3. Suppose that B 1 5 x 0 M 0 the beginning of period 0, and Bt11 5 B0 5 0 for all t ≥ 1. If u ð0Þ > 1 1 ½ð1 2 bÞ=ab and x 0 < 1, the economy with extraneous attributes admits a continuum of monetary equilibria in which the nominal interest rate on government bonds is negative. The proof of proposition 3 consists of constructing a family of moneyspot equilibria and showing that the nominal interest rate between period 0 and period 1 is negative in any equilibrium that belongs to the family. The basic logic is as follows. Suppose that agents’ expectations
s and all t, for an expected sequence of are that ft ðnÞ 5 fst for n ∈ M t ` 0 1


15M
1 partitions fhMt ; Mt igt50 with the property that for all t ≥ 1, M t 1 1 1 0 1



5 M0 [ N 1 , where hM0 ; M0 i is a partition of M0 . In words, agents’ expectations partition the set of outstanding notes at each date into notes of “type 0” and notes of “type 1.” All the notes that are of type 1 in period 0 and all the notes that the government uses for bond redemptions in period 1 are regarded as notes of type 1 for all t ≥ 1. Notice that according to these expectations, notes that are of type 0 in period 0 can become notes of type 1 in period 1 if the private sector uses them to purchase bonds in period 0 and the government uses them to redeem and the financial trader once the size of the open-market operation shock has been observed, e.g., in time to take advantage of the low bond prices if the open-market sale was large. There is no counterpart to either of these restrictions in this paper. The indeterminacy of the nominal interest rate in the equilibrium constructed in proposition 2 ðwhich is indexed by vt Þ is reminiscent of the exchange rate indeterminacy discovered by Kareken and Wallace ð1981Þ. 19 The nominal rate on 4-week US Treasury bills was negative on December 11 and December 19, 2008; see the Daily Treasury Bill Rates table published by the US Treasury ðhttp://www.ustreas.gov/offices/domestic-finance/debt-management/interest-rate/daily _ treas _ bill _ rates _ historical _ 2008.shtmlÞ. Cecchetti ð1988Þ offers an account of negative nominal yields on US Treasury bonds during the 1930s and early 1940s. WuDunn ð1998Þ reports on Japan’s negative rates in 1998.

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those bonds in period 1. ðNotes that are regarded as type 0 in period 0 but are not used to purchase bonds in period 0 remain of type 0 forever.Þ If, in addition, agents expect f1t 5 f11 > f0t 5 0 for all t ≥ 1, that is, that notes of type 1 will remain in circulation while notes of type 0 will have no value from date 1 onward, then at date 0 agents will be willing to use notes of type 0 to buy bonds at a premium. The proof of proposition 3 shows that, for relatively small open-market sales at date 0, there exist a continuum of moneyspot equilibria in which these expected partitions and prices are validated, and the nominal interest rate is negative. VI.

Discussion

Abstract though it may seem, the rate-of-return-dominance puzzle has important implications for applied work. Impossibility results like proposition 1 potentially subvert most of existing applied research in monetary economics, which by introducing outside money through shortcuts or reduced-form assumptions takes the coexistence of non-interestbearing money and higher-return assets as given.20 In models used to provide monetary policy advice, government-issued nominal bonds and fiat money are often assumed to coexist, with bonds trading at a discount. The implied nominal interest rate is analyzed, compared to data, used as a policy target, and so on. But given the impossibility propositions, how is it possible that the nominal rate is not identically zero at every date in those models? In models that assume that money is an argument of the utility function, the answer is that it is assumed that outside money yields utility but bonds and inside money do not. In cashin-advance models, the answer is that it is assumed that outside money satisfies the cash-in-advance constraint but bonds and inside money do not. In both cases the “answer” is a reduced-form modeling assumption. In contrast, as shown in Section IV, the nominal interest rate can be positive in a moneyspot equilibrium, even without assuming that money is intrinsically valuable or that it can facilitate exchange in ways that bonds cannot. At first, the idea that payoff-irrelevant attributes of money may be priced could seem odd.21 However, this is theoretically no different than 20 Neil Wallace has been making this point for many years. See Wallace ð1990, 1998Þ and also Hellwig ð1993Þ. 21 I have been asked, “Do you really believe that moneyspots explain why money coexists with nominal bonds in actual economies?” The prudent answer is that this paper is a contribution to the pure theory of money. Specifically, proposition 2 shows that if we start with a standard, abstract model of money ðone that abstracts from the many institutional, technological, legal, and informational complexities present in actual economiesÞ and incorporate a seemingly trivial ingredient ðextraneous attributes such as serial numbersÞ, then the set of monetary equilibria is enlarged dramatically, and interestingly, it is expanded in a way that can offer a coherent theoretical rationalization for one of the oldest and central challenges in monetary theory. Friedman ð1994Þ recounts a historical episode

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having an intrinsically useless object being assigned a positive price, as is the case in any monetary equilibrium. In fact, whenever there is heterogeneity in the extraneous attributes of notes, the existence of equilibria in which moneyspots are priced would appear to be unavoidable in any good model of fiat money.22 An explicit model of extraneous monetary attributes requires making some specific modeling choices that have no counterpart in an ordinary model in which the object that acts as money is assumed to be homogeneous. Let me enumerate what these modeling choices are, how they were made, and the ways in which they influence the main results. First, in the formal modeling I have used serial numbers as the leading example of moneyspots, mainly because they have been one of the most universal features of paper money since its inception. As it was modeled above, this heterogeneity is very refined in that each note is effectively distinguishable from every other note. It should be evident from the construction in the proof of proposition 2 that a version of the proposition can be established for a coarser specification of the heterogeneity in the extraneous attribute.

that can be interpreted as a real-world instance in which moneyspots indeed coordinated people’s expectations, which were in turn validated by the monetary authority. After the Russian Revolution of 1917, the new Soviet government issued a new ruble intended to replace the old imperial rubles that had czarist iconography printed on them. By 1924 inflation had been so rampant that the Soviet government introduced another new ruble, each of which traded for 50 billion old ðSovietÞ rubles. Interestingly, throughout this whole period the original czarist rubles remained in circulation and maintained their purchasing power despite the unlikely prospect that the czarist regime would be reinstated. Friedman’s explanation is that the czarist rubles retained their value precisely because no new imperial rubles with czarist iconography would be created, and hence the quantity available for circulation was fixed, sheltered from inflation. Effectively, what is anchoring expectations in Friedman’s account is the czarist iconography on the notes—a moneyspot. Similarly, in the equilibria constructed in proposition 2, agents believe ðcorrectly, in equilibriumÞ that every newly printed note will have a serial number higher than the threshold, so they correctly believe that the stock of notes with serial numbers below the threshold will not be inflated by money creation. The low serial numbers on the notes in the proof of proposition 2 are playing a role that is analogous to the czarist iconography printed on the old imperial Russian rubles. 22 At some level, this point is no different from the “tenuousness” feature of monetary equilibria that Wallace ð1977Þ deemed unavoidable in all good models of fiat money. This tenuousness alludes to the fact that a monetary equilibrium will typically coexist with a nonmonetary equilibrium. Wallace stresses that rather than being regarded as a defect, this tenuousness is natural and can be helpful in interpreting some properties of monetary systems. Analogously, in the presence of extraneous ðand, arguably, realisticÞ characteristics of the object that will play the role of fiat money, monetary equilibria become even more tenuous, in the sense that the extraneous characteristics can coordinate expectations giving rise to a richer set of monetary equilibria. This higher degree of tenuousness is a natural feature of a model in which expectations are capable of rendering an intrinsically useless object valuable, and moreover, it sheds new light on some central issues in monetary theory. Also, the indeterminacy of the nominal interest rate in the equilibrium constructed in proposition 2 is an unavoidable feature of the economy with moneyspots under laissez faire, much as the indeterminacy of the equilibrium exchange rate was an unavoidable feature in Kareken and Wallace ð1981Þ.

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Second, to simplify the exposition, I have abstracted from the agents’ decisions of whether to redeem a bond for a note. The working assumption has been that all bonds turn into notes at maturity ðrecall n. 9Þ. One could extend the model to explicitly incorporate an individual agent’s redemption decision. I can see two natural ways to do this, and neither affects the main result. One possibility is to assume that the government does not honor bonds that have exceeded their maturity and does not accept matured bonds as payment for newly issued bonds. In this case there will be a monetary equilibrium in which matured bonds are not valued, and in such an equilibrium, all agents would be willing to redeem their bond holdings at maturity. The other possibility is to assume that the government stands ready to redeem any matured note. Then, in the class of equilibria constructed in the proof of proposition 2, a matured bond would be priced just as a note of type 1; since agents would be indifferent between redeeming and not, it would be possible that matured bonds circulate alongside notes of type 0, notes of type 1, and unmatured bonds ðwith all bonds, and notes of type 1, trading at a discount relative to notes of type 0Þ. Third, I have assumed that the government ignores extraneous attributes; for example, in the open-market operation of time t, the government accepts any qt notes as payment for a newly issued bond ðrecall n. 6 and constraint ½8Þ. This symmetric government treatment of notes makes it more difficult to support equilibria in which notes are treated asymmetrically, which biases the theoretical model against the relevance of moneyspots. An explanation of rate-of-return dominance based on an exogenous government policy that favors some notes over others would have been much easier to formulate but also more contrived and less revealing. There is another aspect of government behavior that has been implicitly assumed, namely, that it does not operate a “window” under a set of rules in which agents can swap notes for notes. Can some arrangement of this type rule out the moneyspot equilibria of proposition 2? After all, it is some well-run window of this sort that presumably keeps the exchange rate between $1 notes and $10 notes fixed at 10-for-1.23 The answer is that even if the government in the model were ready to swap a note for another on demand, the moneyspot equilibria of proposition 2 would still exist. The reason is that beliefs can be specified as in the proof of the proposition, so that any newly injected note ðthrough

23 Without a well-functioning window, not even the fixed-exchange-rate regime among different denominations of a single currency can be taken for granted in monetary economies. For example, over the last couple of years in Argentina, coins have been trading privately at prices that exceed their face values against higher-denomination notes. The phenomenon appears to be generalized in some areas of the country and has received substantial press coverage ðe.g., Galva´n 2008; Surowiecki 2009Þ.

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window swaps or bond redemptionsÞ would be deemed of “type 1,” and in any equilibrium in which these expectations are validated, the window would merely be swapping a note whose serial number belongs to the set of serial numbers of type 1 for another note with a serial number in the same set.24 Related literature.—There are several accounts of instances in which money remained in circulation along with interest-bearing, smalldenomination, payable-to-the-bearer bonds issued by a government with the authority to print or tax the money needed to redeem those bonds.25 In light of impossibility results similar to proposition 1, these instances have been regarded as puzzling: they embody the strongest version of the rate-of-return-dominance puzzle.26 The observation goes back to Hicks’s ð1935Þ “Suggestion,” and since then, the issue has periodically caught the interest of monetary theorists. Governments seldom issue bonds that literally fit the description in the impossibility propositions.27 In fact, an early argument for why fiat money consistently circulates even though there exist governmentissued nominal securities that typically bear interest is that the physical ways in which governments choose to materialize their nominal promises can cause marketability problems, for example, due to large denomina-

24 This argument uses the fact that the heterogeneity in moneyspots is fine enough for each note to be distinguished from every other note. If the set of extraneous attributes consisted of just two elements, e.g., if all notes were alike except that some were blue and the rest red, then the window could get rid of moneyspot equilibria in the same way that the Federal Reserve pegs exchange rates between notes of different denominations. 25 Makinen and Woodward ð1986Þ recount that small-denomination bearer bons issued by the government of France during 1915–27 seemed to circulate, at least to some extent, and exchanged only at a discount. Gherity ð1993Þ provides examples of interest-bearing notes that circulated at a discount alongside non-interest-bearing notes ðmoneyÞ in the US North during the Civil War ðe.g., the certificates of indebtedness of 1862, the notes of 1863, the compound-interest notes of 1863, the seven-thirties of 1864 and 1865Þ. Makinen and Woodward ð1999Þ find evidence of the circulation of small-denomination bearer notes issued by the Confederate states during the Civil War. Wallace ð1983, n. 3Þ mentions the liberty bonds, which were issued by the United States during World War I, as smalldenomination bearer securities that seem to have circulated as currency from time to time. More recently, several small-denomination bearer securities circulated extensively as currency alongside fiat money in Argentina between 2001 and 2003 ðe.g., LECOPs, patacones, etc.Þ. 26 I call this the “strongest version” of the puzzle because an explanation is quite demanding in that it calls for a theory that is able to rationalize why money is not driven out by bonds that pay interest and have exactly the same payoff-relevant physical characteristics as money, e.g., issued in the same denominations, nominally default free, just as easy to exchange, just as costly to counterfeit, and so on. The observation that money is not driven out by interest-bearing default-free bonds with different physical characteristics than money is in this sense a weaker version of the coexistence puzzle. 27 In the United States, for instance, the Treasury issues bills that are payable to the bearer and nominally default free, but unlike US Federal Reserve notes, they exist only as a book entry in a ledger at the Treasury. In the past, these bills existed in printed form, but the minimum denomination was rather large ð$1,000Þ.

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tions with indivisibilities. In a series of well-known papers, Bryant and Wallace ð1980Þ and Wallace ð1983, 1990Þ advanced the hypothesis that fiat money is dominated in rate of return by government bonds in actual economies because of legal restrictions that prevent financial companies from intermediating the large-denomination bonds into securities or claims that are similar to money. There are a number of differences between moneyspots and legal restrictions as potential explanations of the coexistence puzzle. The main one is that according to the latter, without legal and regulatory obstacles, interest-bearing bonds would be used in exchange, and there could not exist an equilibrium with noninterest-bearing notes. In contrast, moneyspot equilibria can rationalize coexistence even in the absence of legal restrictions.28 Researchers who are mindful of the foundations of monetary exchange, a subset of whom tend to work on search models of money that satisfy the Wallace dictum ðe.g., Wallace 1998Þ, are well aware of the rateof-return-dominance puzzle. In order to avoid internal inconsistencies and ad hoc restrictions on portfolios or transaction patterns, they typically refrain from explicitly introducing nominal government bonds in their models. In order to introduce government-issued nominal bonds explicitly, some of these researchers impose explicit marketability restrictions that can be interpreted as versions of Wallace’s legal restrictions ðe.g., Aiyagari et al. 1996; Andolfatto 2005; Berentsen and Waller 2008; Marchesiani and Senesi 2009Þ. All in all, for the most part the search literature has not attempted to tackle the rate-of-return-dominance puzzle. There are two notable exceptions to this statement: Aiyagari et al. ð1996Þ and Zhu and Wallace ð2007Þ. Aiyagari et al. ð1996Þ study a version of Trejos and Wright ð1995Þ with two assets: fiat money and a pure discount nominal bond, which can be distinguished from money but has all the same relevant physical characteristics as money. They introduce a positive measure of “government agents” whose role is to issue and redeem bonds. Government agents are just like private agents in every respect, except for the fact that their trading behavior is specified exogenously in three instances. First, when 28 For instance, the examples in n. 25, which run counter to the legal restrictions hypothesis, can be rationalized as a moneyspot equilibrium. The legal restrictions hypothesis as it is usually stated does not seem compelling for the present-day United States. Since April 7, 2008, the minimum amount that can be purchased of any given US Treasury bill, note, bond, or Treasury inflation-protected securities is $100. A loose argument is often made suggesting that contemporary bonds could not possibly be used as means of payment because they exist only as electronic book entries in a central electronic ledger. But whether ownership of the bond is manifested by a printed title or by a computer entry surely cannot matter. In fact, with the electronic trading systems that currently exist, e.g., the Commercial Book-Entry System, these book entries can be transferred pretty much instantaneously at a negligible technological cost, and the transfer could in principle be readily verified by the payer and the payee.

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in a single-coincidence meeting with a private seller, a government agent holding a unit of money is programmed to pay with the unit of money with some exogenous probability or to issue a security and use the newly issued security as a means of payment with complementary probability. ðIn this context, “issuing a security” effectively means physically transforming the unit of money into a bond.Þ Second, when in a singlecoincidence meeting with a private buyer, the government agent refuses to trade with some exogenous positive probability if the buyer carries an unmatured bond. Third, any government agent who meets a private agent who is holding a matured bond turns the matured bond into a unit of money. This model provides two explanations for the coexistence of money and interest-bearing securities, in the sense that the model has two types of steady states in which such coexistence can occur. The first rationalization is based on a steady state ðwhich exists for all parameterizationsÞ in which money and matured bonds trade at par but unmatured bonds are traded at a discount because—by assumption— government agents refuse to accept unmatured bonds in exchange for goods, with positive probability. The second rationalization is based on a steady state ðwhich exists only for some parameterizationsÞ in which matured bonds trade at a discount. ðSince both assets are fiat objects, it is not surprising that there exist steady states with different relative values between them.Þ In this case, given that private agents regard money as being more valuable than a matured bond, both matured and unmatured bonds will trade at a discount. This discount originates from the fact that in an environment in which it takes an exogenous, random period of time to be able to find a government agent who will redeem a bond, a private agent may end up locked into a matured bond for some time and hence forced to hold an asset that is less valuable than money. In the first rationalization, the discount on bonds is possible by the assumption that government agents discriminate against unmatured bonds. In the second rationalization, it is private agents who endogenously “discriminate” against matured bonds, and this brings about a discount on all bonds. Clearly, for this result to follow, it is necessary to assume that the redemption of matured bonds involves search-type frictions or other imperfections that may prevent a bondholder from redeeming the bond at maturity. For the last 15 years, the study by Aiyagari et al. ð1996Þ has been on the frontier of what search-theoretic models of money have to offer in way of explanation of the rate-of-return-dominance puzzle. However, because the authors were working with a pure random-search model, they were forced to adopt a rather contrived bond sales and redemption mechanism. In their formulation, bonds can be issued as payment for goods only in bilateral exchanges ðso there is nothing that approximates a real-

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world open-market auction of government securitiesÞ, and matured bonds can get redeemed only at random times, whenever the bondholder happens to find himself in a bilateral meeting with a government agent. There is a sense in which there is “too much random matching” in the model. This is not just an aesthetic concern; the fact that bondholders are forced to go through a time-consuming period of search for a government agent in order to redeem their bonds is critical for their results. Specifically, if no private agent is locked in with matured bond holdings that he would like to redeem, then coexistence of money and interest-bearing securities can be the result of only a deliberate government policy to discriminate against its own government bonds by arbitrarily instructing government agents to refuse unmatured bonds in bilateral trades. The moneyspots model I have developed above has several advantages relative to Aiyagari et al.’s model. The most obvious is that all assets are divisible and there are no restrictions on individual asset holdings, so the model lends itself to standard policy analysis. Also, notice that the coexistence results in Section IV do not rely on bond redemptions being subject to random-matching frictions, since all agents can ðand in fact doÞ redeem their bond holdings at maturity.29 Zhu and Wallace ð2007Þ ðsee also the related discussion piece, Wallace ½2003Þ obtain coexistence of money and interest-bearing nominal bonds as an equilibrium by assuming a bilateral trading protocol that confers larger gains from trade to buyers who hold a larger proportion of money in their portfolios. Specifically, their trading protocol is modeled as a twostep maximization problem. In the first step, buyer utility is maximized subject to a cash-in-advance constraint and to the utility of no trade as a lower bound on seller utility. In the second step, seller utility is maximized subject to the maximum value of utility of the buyer in the first step as a lower bound on buyer utility. There are many allocations ða continuumÞ in the pairwise core implied by the portfolios of the buyer and seller in the bilateral meeting, which range from giving all the gains from trade to the buyer to giving all the gains from trade to the seller. This feature gives Zhu and Wallace freedom to make a selection from the pairwise core that depends on the composition of the portfolios brought into the meeting ðalthough the compositions of those portfolios are payoff irrelevantÞ. Effectively, the two-step maximization problem selects 29 Without assuming frictions that prevent agents from redeeming bonds when they so desire and without moneyspots, having nominal bonds and money circulate at different prices is not as easy as constructing an equilibrium in which “blue money” and “red money” circulate at different prices. The reason is that, by standing ready to redeem matured bonds with notes, the government imposes a boundary condition that fixes the relative price of bonds and notes at unity. No such boundary condition exists in the case with blue and red money.

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an allocation from the pairwise core in such a way as to give potential buyers an incentive to bring money to the meetings. The outcome that results from the two-step maximization problem, and hence the ensuing coexistence theory, could be formulated by replacing the maximization problem with a bargaining procedure in which the buyer’s bargaining power is an increasing function of his money holdings. In contrast, the protocol used to divide the gains from trade between buyers and sellers in the moneyspot model of Section IV does not favor money over bonds: buyers get all the gains from trade in every meeting, regardless of the composition of the portfolio that they bring into the meeting. In terms of similarities, the multiplicity of moneyspot equilibria has a parallel in the multiplicity of core allocations in Zhu and Wallace. VII.

Conclusion

I have shown that if the physical object that society uses as fiat money is heterogeneous in some extraneous attribute—a moneyspot—then there exist equilibria that induce outcomes that have traditionally proven difficult to rationalize, such as the coexistence of money and nominal default-free interest-bearing government bonds, the liquidity effects of open-market operations, and the possibility of negative nominal interest rates. The formulation also articulates an expectations-driven theory of the nominal interest rate and illustrates the fact that, contrary to established wisdom, a positive nominal interest rate need not indicate a marketability advantage of money relative to bonds. From the standpoint of the pure theory of money, all this suggests that moneyspots can matter and that they are worth studying. In modern economies, money ðe.g., Federal Reserve notesÞ is held despite being dominated in rate of return by default-free claims to money ðe.g., US Treasury billsÞ. This is perhaps the most obvious instance of an asset pricing anomaly. This puzzle is central to monetary economics because a resolution requires a theory that explains the role that money is uniquely suited to play in actual economies, a role seemingly so specific that justifies forgoing the rate of return on assets that, through the lenses of our standard theories, ought to be regarded as equivalent to money. It seems to me that with the modern degree of financial and technological sophistication, the research agenda that consists of trying to uncover the “frictions” that explain why money is held despite being dominated in rate of return by default-free nominal bonds is as challenging today as it was when Hicks wrote his “Suggestion.” I do not see a way of developing this agenda that does not begin by studying the set of equilibrium outcomes that can be implemented in the absence of such frictions, which is what I have done in this paper.

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Appendix A Proofs Proof of Proposition 1 The proof is organized in two parts. The first part shows that money cannot coexist with interest-bearing nominal bonds. The second part shows that the set of monetary equilibria is nonempty. Part 1: The agent’s problem in the bond market is Ut ðm
t Þ 5

max bEt Vt11 ðmt11 ; bt11 Þ

0≤qt bt11 ≤m
t

subject to mt11 5 m
t 2 qt bt11 , and his problem in the general goods market is Wt ðmt ; bt Þ 5 ft ðmt 1 bt Þ 1 maxf2ft m
t 1Ut ðm
t Þg: 0≤ m
t

Combine these expressions to arrive at
 Wt ðmt ; bt Þ 5 ft ðmt 1 bt Þ 1 max 2ft m
t 1 max bEt Vt11 ðmt11 ; bt11 Þ ; 0 ≤ m
t

0 ≤qt bt11 ≤m
t

where mt11 5 m
t 2 qt bt11 . Notice that Wt ðmt ; bt Þ 5 ft ðmt 1 bt Þ 1 kt , where
 kt ; max 2ft m
t 1 max bEt Vt11 ðm
t 2 qt bt11 ; bt11 Þ : 0 ≤ m
t

0 ≤qt bt11 ≤m
t

From ð4Þ, Vt11 ðm
t 2 qt bt11 ; bt11 Þ 5 Lðft11 ½m
t 1 ð1 2 qt Þbt11 Þ 1 kt11 : Thus, the agent’s optimization problems in the centralized market and in the bond market are
 max 2ft m
t 1 max bEt Lðft11 ½m
t 1 ð1 2 qt Þbt11 Þ : 0 ≤ m
t

0 ≤ qt bt11 ≤m
t

The solution to the inner maximization is 8 if 1 < qt < 50 if qt 5 1 qt bt11 ∈ ½0; m
t  : 5 m
t if qt < 1:

ðA1Þ

Notice that qt > 1 cannot be part of a monetary equilibrium, as it implies ∫I bit11 dm 5 0 < Bt11 , and the bond market would not clear. Hence, qt ≤ 1 for all t in any monetary equilibrium. Suppose that qt < 1 at time t in a monetary equilibrium. Then, from ðA1Þ, every agent i chooses mit11 5 0 in the bond market at t, and therefore ∫I mit11 dm 5 0, so money does not coexist with bonds in period t 1 1. Part 2: In order to establish that the set of monetary equilibria of the economy with no extraneous attributes is nonempty, I construct a recursive monetary equilibrium for this economy.

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Definition 3. Let s t 5 ðxt ; M
t Þ denote the aggregate state at time t. A recursive equilibrium of the economy with no extraneous attributes is a collection
t Þ, mt11 5 mðs t Þ, bt11 5 bðs t Þ, together with barof individual decision rules m
t 5 mðs gaining outcomes ct 5 cðs t Þ, and pricing functions ft 5 fðs t Þ and qt 5 qðs t Þ, such that ðiÞ given prices and the bargaining protocol, the decision rules mðÞ,
mðÞ, and bðÞ solve the agent’s problem in the centralized markets; ðiiÞ the bilateral terms of trade are determined by Nash bargaining, that is, cðs t Þ 5 min½fðs t Þmðs
t Þ; c * ; and ðiiiÞ prices are such that centralized markets clear. The equilibrium is “monetary” if fðs t Þ > 0 for all s t , and in this case the money market–clearing condition, mðs
tÞ 5 M
t , and the bond market–clearing condition, bðs t Þ 5 xt M
t , must hold. Given ðA1Þ, the agent’s problem in the centralized market before the bond issue is maxf2ft m
t 1 bEt Lðft11 ½m
t 1 ð1 2 qt Þgt ðm
t ÞÞg; 0 ≤ m
t

where 8 50 > >  > > m
t < ∈ 0; gt ðm
t Þ qt > > m
t > > :5 qt

if 1 < qt if qt 5 1 if qt < 1:

Since 

 1 ; 1 m
t ; m
t 1 ð1 2 qt Þgt ðm
t Þ 5 max qt

ðA2Þ

this problem can be written as
    1 max 2ft m
t 1 bEt L max ; 1 ft11 m
t ; 0 ≤ m
t qt and the corresponding first-order condition is
       1 1 0 2ft 1 bEt L max ; 1 ft11 m
t max ; 1 ft11 ≤ 0; qt qt

ðA3Þ

5 if m
t > 0: In a monetary equilibrium, m
t 5 M
t > 0 ðmoney market clearingÞ, and gt ðM
t Þ 5 Bt11 ðbond market clearingÞ, so ðA2Þ implies   1 ; 1 M
t : ðA4Þ M
t11 5 max qt Thus in a monetary equilibrium, ðA3Þ together with ðA4Þ give the Euler equation 0 ft M
t 5 bEt ½L ðft11 M
t11 Þft11 M
t11 :

ðA5Þ

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155

The equilibrium conditions m
t 5 M
t and bt11 5 Bt11 and ðA1Þ imply qt 5 minðM
t =Bt11 ; 1Þ. Since Bt11 5 xt M
t , the equilibrium price of bonds is qt 5 minðxt21; 1Þ

ðA6Þ

and the nominal interest rate is it11 5 maxðxt 2 1; 0Þ: Given ðA4Þ and ðA6Þ, the postredemption stock of notes evolves according to M
t11 5 maxðxt ; 1ÞM
t

ðA7Þ

in a monetary equilibrium. The quantity of money that agents carry out of the bond market into the search market is Mt11 5 maxð1 2 xt ; 0ÞM
t : Define real money balances, zt ; ft M
t , and look for a recursive equilibrium in which zt 5 zðxt Þ ; fðs t ÞM
t . Then the Euler equation ðA5Þ becomes

E

0

zðxÞ 5 b L ½zðx 0 Þzðx 0 ÞdF ðx 0 ; xÞ:

ðA8Þ

Given a positive zðÞ that solves ðA8Þ, a recursive monetary equilibrium is then an allocation mðs
t Þ 5 M
t , bðs t Þ 5 xt M
t , mðs t Þ 5 max½ð1 2 xt ÞM
t ; 0, bargaining outcomes cðxt Þ 5 min½zðxt Þ; c * , together with the pricing function for notes, fðs t Þ 5 zðxt Þ=M
t and the pricing function for bonds, qðxt Þ 5 minðxt21 ; 1Þ. Notice that a solution to ðA8Þ is zðxÞ 5 z for all x, where z ∈ ð0; c * Þ is the unique solution to 0 bL ðzÞ 5 1. QED Proof of Proposition 2 The proof is organized in three parts. Part 1 derives the set of equilibrium conditions that characterize a monetary equilibrium that belongs to a certain class. Part 2 constructs a particular allocation and price system and establishes that they constitute a family ða continuumÞ of monetary equilibria that belong to the class described in part 1. Part 3 establishes that in the family of monetary equilibria described in part 2, money coexists with bonds for all t, even if qt ðsÞ < 1 for all s ∈ Bt11 . Part 1: Focus on a class of monetary equilibria in which agents’ beliefs endogenously partition the set of outstanding notes into two subsets such that any two notes are treated identically if and only if their serial numbers belong to the same subset. Let hM0t ; M1t i be a partition of the set of beginning of period t outstanding notes, Mt , and let hNt 01 ; Nt 11 i be a partition of the set of notes injected in period t, Nt 1 . Given hM0t ; M1t i and hNt 01 ; Nt 11 i, posit that the post-

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s 5 Ms [ N s1 . ðHereafter, I will redemption set of notes of type s 5 0, 1 is M t t t s
s, as notes of type s.Þ At the refer to notes whose serial numbers lie in Mt , Nt s1 , or M t end of period t, the government issues a set of bonds Bt11 of size Bt11 5 mðBt11 Þ, 2 thereby withdrawing a set of notes Nt11 . The set of notes withdrawn in period t, 2 Nt11 , is determined by the equilibrium trades in the bond market along with the 02 12 s2 2
s represents the set of notes of type s partitionhNt11 ; Nt11 i, where Nt11 5 Nt11 \M t withdrawn by the bond issue of period t. Next, I obtain the set of equilibrium conditions that characterize a monetary equilibrium within a class with certain properties. Property 1. At t 5 0, agents’ beliefs are that M00 5 ½0; v0 M 0 

and M10 5 ðv0 M 0 ; M 0 ;

ðA9Þ

where v0 is an arbitrary number in the interval ð0, 1Þ. For t ≥ 0, agents’ beliefs specify Nt 01 5 ∅

ðA10Þ


s nN s2 Þ [ N s1
s 5 ðM M t11 t11 t11 t

ðA11Þ

for newly injected notes and

for the law of motion of the postredemption set of outstanding notes of type s.
s. Property 2. For s 5 0, 1 and every t, ft ðnÞ 5 fst > 0 for all n ∈ M t
* 5M
t for every t. With regard to An implication of property 2 is that M t
0 5 M0 and that all bonds have the same property 1, notice that ðA10Þ implies M 0 0 probability of being redeemed by a note of type s ðthis probability is one for s 5 1 and zero for s 5 0Þ, which makes each bond effectively identical to every other bond. Since every bond is worth the same to private agents regardless of the serial number printed on it, competitive trade at the time of issue implies qt ðsÞ 5 qt for all s ∈ Bt11 . With this, I can write
N

11 t11

1 t11

5N

5

2 [ ðM
t ; ½1 1 ð1 2 qt Þxt M
t  Nt11 2 \ ½0; ht ðxt M
t Þ Nt11

if qt ≤ 1 if qt > 1


t jmðN 2 \ ½0; nÞ 5 zg. for t ≥ 0, where ht ðzÞ ; inffn ∈ M t11 s2 s s s s s

Let Mt 5 mðMt Þ, Mt 5 mðMt Þ, and for qt > 0, let Bt11 5 mðNt11 Þ=qt . Intuitively, s Mt represents the quantity of notes of type s outstanding at the beginning of period t, M
ts represents the quantity of notes of type s outstanding after the s period t bond redemption, and Bt11 is the measure of bonds issued in period t that are purchased with notes of type s. The law of motion ðA11Þ implies s1 s s M
t11 5 M
ts 2 qt Bt11 1 mðNt11 Þ

for s 5 0, 1, and together, the two conditions in ðA12Þ imply ð1Þ.

ðA12Þ

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157

Next, I derive two implications of properties 1 and 2. Agent i’s problem in the bond market at time t ðthe inner maximization in ½13Þ depends only on the portfolio of assets being chosen, ait11 , through Lt11 ðait11 Þ. Given properties 1 and 2, Lt11 ðait11 Þ 5 ft11 m it11 1 f1t11 bit11 ; 0 1 s
s Þ, and bit11 ; where ft11 ; ðf0t11 ; f1t11 Þ, m it11 ; ðmit11 ; mit11 Þ with mit11 ; mit11 ðM t 1 bit11 ðBt11 Þ 5 n1it11 ðNt11 Þ. This leads to the first implication: the agent’s problem in the bond market consists of choosing the quantity of bonds to buy from the government and the quantities of notes of each type to use as payment. The relevant notation is as follows. The measure of notes of type s that agent i exs2
s Þ, while n 2 ; n2 ðM
t Þ 5 n 02 1 changes for bonds at time t is nit11 ; n2it11 ðM t it11 it11 it11 12 s nit11 is the total quantity of notes that i uses to buy bonds at t. For qt > 0, bit11 s2 ; nit11 =qt denotes the measure of bonds that agent i chooses to buy with notes of 0 1 0 1 type s. It is also convenient to define bit11 ; ðbit11 ; bit11 Þ, where bit11 1 bit11 is the total 0 1 quantity of bonds purchased by agent i, that is, bit11 1 bit11 5 bit11 . The second useful implication of properties 1 and 2 is that the objective that agent i seeks to maximize in the centralized market at time t ðthe outer maximization in ½13Þ de
it only through ft  m
it and that ft  m
it 5 ft m
it , where m
it ; ðm
it0 ; m
it1 Þ, pends on m
s Þ.
it ðM with m
its ; m t Lemma 2 ðsee App. BÞ uses the two previous implications of properties 1 and
0 ; M
1 ig ` and 2 to establish that in a monetary equilibrium in which fhM t11 t11 t50 ` fft gt50 satisfy properties 1 and 2, a solution to agent i’s time t optimization prob
t, ð7Þ lem, ð13Þ, is a collection of measures ðm
it ; mit11 ; bit11 ; n2it11 Þ that satisfy m
it ∈ M s s s2 with equality, and ð9Þ such that the associated quantities ðm
its ; mit11 ; bit11 ; nit11 Þs50;1 achieve
 Dt 5 max 2ft m
it 1 max bEt L½lt11 ðm
it ; bit11 Þ ; ðA13Þ m
it ∈R1 R1

0 ≤ qt bit11 ≤m
it

with s s mit11 5 m
its 2 qt bit11

ðA14Þ

s2 s 5 qt bit11 for s 5 0, 1 and with lt11 : R4 → R defined by and nit11

lt11 ðm
it ; bit11 Þ 5

1

o ½f

s t11

s m
its 1 ðq
ts 2 qt Þfst11 bit11 ;

s50

where q
ts ; f1t11 =fst11 . ` Property 3. The sequence ff0t ; f1t gt50 is such that fst11 is known at time t. If property 3 holds, the solution to the maximization problem of an agent i who enters the market for bonds at time t carrying a vector m
it of notes ðthe ins ner maximization on the right side of ½A13Þ is bit11 5 gts ðm
its Þ, where 8 50 if q
ts < qt > >  > s > m
< if qt 5 q
ts ∈ 0; it gts ðm
its Þ ðA15Þ qt > s > m
> it s > if qt < q
t : :5 qt

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journal of political economy

With ðA15Þ, agent i’s problem in the centralized market of period t ðthe outer maximization on the right side of ½A13Þ becomes max f2ft m
it 1 bEt L½l*t11 ðm
it Þg;

m
it ∈R1 R1

where l*t11 ðm
it Þ 5

1

o ½f

m
its 1 ðq
ts 2 qt Þfst11 gts ðm
its Þ  s  1 q
5 o max t ; 1 fst11 m
its : qt s50 s t11

ðA16Þ

s50

ðA17Þ

With property 3, it is easy to check that the objective, 2ft m
it 1 bEt L½l*t11 ðm
it Þ, is a concave function of m
it , so the following first-order conditions are necessary and sufficient for an optimum of the agent’s problem in the centralized market of period t :
  s  q
0 2fst 1 bEt L ½l*t11 ðm
it Þmax t ; 1 fst11 ≤ 0; qt ðA18Þ s 5 if m
it > 0; for s 5 0; 1: Property 4. m
it 5 m
t for almost every ða.e.Þ i ∈ I . The money market–clearing condition is ∫I ∫M m
it ðdnÞmðdiÞ 5 mðMÞ for all
t Þ. Evaluate it at M 5 M
s to get ∫I m
s mðdiÞ 5 M
s , which combined with M ∈ F ðM t t it 0 1 property 4 implies m
t 5 ðM
t ; M
t Þ ; M
t . Together with ðA15Þ, this implies gts ðm
its Þ s 0 1 5 gts ðM
ts Þ, so from ðA14Þ, mit11 5 M
ts 2 qt gts ðM
ts Þ, so m it11 5 ðMt11 ; Mt11 Þ ; M t11 . s2 s s
s s s
s
Therefore, qt gt ðMt Þ 5 Mt 2 Mt11 , which implies that qt gt ðMt Þ 5 mðNt11 Þ; but since s2 s s qt Bt11 ; mðNt11 Þ, it follows that gts ðM
ts Þ 5 Bt11 for s 5 0, 1. In summary, property 4 s s s s s s implies that, in equilibrium, m
it 5 M
t , mit11 5 Mt11 , and bit11 5 Bt11 for s 5 0, 1. Since ðA9Þ, ðA10Þ, ðA11Þ, ðA12Þ, and ðA16Þ imply
t Þ 5 ft11 M
t11 ; zt11 ; l*t11 ðM in a monetary equilibrium in which both types of notes are held, ðA18Þ becomes    s  q
0 for s 5 0; 1: fst 5 bEt L ðzt11 Þmax t ; 1 fst11 qt

ðA19Þ

Also, ðA9Þ, ðA10Þ, ðA11Þ, ðA12Þ, and ðA15Þ yield   g 0 ðM
0 Þ 0 M
t11 5 1 2 t
0 t qt M
t0 ; Mt

ðA20Þ

    1 g 0 ðM
0 Þ 1 5 max ; 1 1 t
1 t M
t1 ; M
t11 qt Mt

ðA21Þ

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159

so with z ; f M
for s 5 0, 1, ðA19Þ can be written as s t

s t

s t

maxðq
t 0 =qt ; 1Þ 0 0 bEt ½L ðzt11 Þzt11 ; 1 2 ½g 0 ðM
0 Þ=M
0 qt

ðA22Þ

1 0 1 : bEt ½L ðzt11 Þzt11 0
1 1 ½g ðMt Þ=maxð1=qt ; 1ÞM
t1 

ðA23Þ

zt0 5

t

zt1 5

t

t

0 t

The bond market–clearing condition requires 0 1 1 Bt11 5 Bt11 : Bt11

ðA24Þ

s The supply of bonds at time t is Bt11 5 xt M
t . With gts ðM
ts Þ 5 Bt11 and ðA15Þ, the 1 s
s
aggregate demand for bonds, Gt ðM t Þ ; os50 gt ðMt Þ, is

8 > > > > > > > > > > > > > > > <
Gt ðM t Þ > > > > > > > > > > > > > > > :

50   Ifq
1 ≤q
0 g M
t0 1 Ifq
t 0 ≤q
t1 g M
t1 ∈ 0; t t maxðq
t0 ; q
t1 Þ Ifq
t1
if maxðq
t 0 ; q
t 1 Þ < qt if qt 5 maxðq
t 0 ; q
t1 Þ if minðq
t 0 ; q
t1 Þ < qt < maxðq
t 0 ; q
t1 Þ

if qt 5 minðq
t 0 ; q
t 1 Þ if qt < minðq
t 0 ; q
t 1 Þ;

where Ifq
t 1 ≤q
t 0 g is an indicator function that takes the value one if q
t1 ≤ q
t 0 . Hence
t Þ 5 xt M
t , which implies that the equilibrium price of ðA24Þ can be written as Gt ðM bonds satisfies

qt 5

8 > > > maxðq
t 0 ; q
t 1 Þ > > > > > > Ifq
t 1 > > > > xt > > > > > > < > > > minðq
t 0 ; q
t 1 Þ > > > > > > > > > > > > > > > 1 > > : xt

Ifq
t 1 ≤q
t 0 g vt 1 Ifq
t 0 ≤q
t 1 g ð1 2 vt Þ maxðq
t 0 ; q
t 1 Þ Ifq
t 1 ≤q
t 0 g vt 1 Ifq
t 0 ≤q
t 1 g ð1 2 vt Þ < xt if maxðq
t 0 ; q
t 1 Þ Ifq
1
where vt ; M
t0 =M
t . Finally, the bond market constraint ðA14Þ implies

ðA25Þ

160

journal of political economy
t 2 qt B t11 ; M t11 5 M

ðA26Þ

0 1 ; Bt11 Þ. where B t11 5 ðBt11 In summary, a monetary equilibrium that satisfies properties 1–4 consists `
it ; mit11 ; bit11 ; n2it11 Þi ∈I gt50 of an allocation fðyit ; hit ; m , prices fðft ðnÞÞn ∈ M
t ; ` ` ðqt ðsÞÞs ∈Bt11 gt50 , and bargaining outcomes, fðcijt Þi;j ∈I gt50 , such that

ðE1Þ cijt 5 minðzt ; c * ÞIfi ∈I bt g ðiÞ for a.e. i, j ∈ I , where Ifi ∈I bt g ðiÞ is an indicator function that takes the value one if i ∈ I bt ; ðE2Þ yit and hit satisfy

hit 5 yit 1 ft M
t Ifi ∈I bt g ðiÞ 2 ft M
t Ifi ∈I st g ðiÞ;

ðE3Þ ðE4Þ ðE5Þ

ðE6Þ ðE7Þ ðE8Þ ðE9Þ

with 0 ≤ yit and 0 ≤ hit ≤ h
for a.e. i ∈ I ; qt ðsÞ 5 qt for all s ∈ Bt11 , with qt given by ðA25Þ; ` ` ft ðnÞ 5 f0t Ifn ∈ M
0t g ðnÞ 1 f1t Ifn ∈ M
1t g ðnÞ, with fðfst Þs50;1 gt50 5 fðzts =M
ts Þs50;1 gt50 , ` where fðzts Þs50;1 gt50 satisfies ðA22Þ and ðA23Þ, and zt 5 zt0 1 zt1 ;
0 is given by ðA20Þ with M
0 5 v0 M 0 , M
1 is given by ðA21Þ with M t11 0 t11 M
01 5 ð1 2 v0 ÞM 0 1 B 0 , and ðA26Þ is satisfied ðthe first two conditions ` imply that fM
t11 gt50 follows ½1 with M
0 5 M 0 1 B 0 given and Mt11 5 M
t 2 qt Bt11 for all t ≥ 0Þ;
t , m
it 5 M
t for a.e. i ∈ I , and ∫I ∫M m
it ∈ M
it ðdsÞmðdiÞ 5 mðMÞ for all m
t Þ; M ∈ F ðM bit11 ∈ Bt11 , bit11 5 Bt11 for a.e. i ∈ I , and ∫I ∫B bit11 ðdsÞmðdiÞ 5 mðBÞ for all B ∈ F ðBt11 Þ;
t , and mit11 5 Mt11 for a.e. i ∈ I ; mit11 ∈ M
t , n s2 =qt 5 b s satisfies ðA15Þ, n 2 5 qt Bt11 , and for a.e. i ∈ I , n2it11 ∈ M it11 it11 it11 2
t Þ.
it ðMÞ for all M ∈ F ðM nit11 ðMÞ 1 mit11 ðMÞ 5 m

Part 2: Let agents’ beliefs be as described in property 1. Let v0 ∈ ð0, 1Þ be the arbitrary number introduced in part 1, and let   1 r∈ ; 1 ðA27Þ 1 1 D0 be an arbitrary number, where D0 ;

  1 2 v0 1 2 v0 min ; 1 : v0 supX

Next, I construct a monetary equilibrium indexed by v0. For all t, and a.e. i, j ∈ I , set cijt 5 zIfi ∈I bt g ðiÞ;

yit 5 zIfi ∈I st g ðiÞ;

where z is the unique solution to

and

hit 5 zIfi ∈I bt g ðiÞ;

ðA28Þ

moneyspots

161 0

bL ðzÞ 5 1:

ðA29Þ

M
t0 5 v0 M 0 ;

ðA30Þ

1 M
t11 5 maxðxt M
t ; M
t1 Þ;

ðA31Þ

M
01 5 ð1 2 v0 ÞM 0 1 B 0 ;

ðA32Þ

For all t, let

where M
t ; M
t0 1 M
t1 , with

and M
0 5 M 0 1 B 0 given by the initial conditions M 0 and B 0 . For all t ≥ 0, let Mt0 5 M
00

1 and Mt11 5 maxð1 2 vt 2 xt ; 0ÞM
t ;

ðA33Þ

with M 01 5 ð1 2 v0 ÞM 0 and vt ; M
t0 =M
t . Then for all t, set
ft ðnÞ 5

for n ∈ ½0; M
00  for n ∈ ðM
00 ; M
t ;

f00 f1t

ðA34Þ

with rz f00 5
0 M0

and

f1t 5

ð1 2 rÞz : M
1

ðA35Þ

t

For every t and all s ∈ Bt11 , set 

 1 2 vt qt ðsÞ 5 min ; 1 ; qt : xt

ðA36Þ

s bonds with notes of type s in period t ≥ 0, Posit that each agent purchases Bt11 as follows: 0 Bt11 5 0;

ðA37Þ

1 Bt11 5 xt M
t :

ðA38Þ

For all i ∈ I and t ≥ 0, let individual portfolios be 1
it ðMÞ 5 Ifi M
00 ∈Mg ðiÞM
00 1 IfM
00 1iMt11 1 ∈Mg ðiÞM m t11 1 1iq B 1 IfM
00 1Mt11 ðiÞqt Bt11 ; t t11 ∈Mg

1 1 ∈Mg ðiÞM mit11 ðMÞ 5 fIfiM00 ∈Mg ðiÞM00 1 IfM 00 1iMt11 t11 gIfM∈F ðMt11 Þg ;


it ðMÞ 2 mit11 ðMÞ n2it11 ðMÞ 5 m

ðA39Þ

ðA40Þ ðA41Þ

162

journal of political economy


t Þ, and for all B ∈ F ðBt11 Þ, for all M ∈ F ðM bit11 ðBÞ 5 IfiBt11 ∈Bg ðiÞBt11 :

ðA42Þ

The proposed paths ðA34Þ and ðA35Þ satisfy properties 2 and 3. From ðA29Þ
t 5 z ∈ ð0; c * Þ, so the cijt , yit , and hit proposed in ðA28Þ satisfy ðE1Þ and ðA35Þ, ft M and ðE2Þ. With ðA35Þ, q
t 0 ; f1t11 =f0t11 becomes q
t 0 5

1 2 r M
00 : 1 r M
t11

ðA43Þ

Since ðA31Þ and ðA36Þ imply

  1 1 5 max ; 1 M
t1 ; M
t11 qt

ðA44Þ

I have 0 q
t11 5 minðqt11 ; 1Þq
t 0 ≤ q
t 0

for all t:

ðA45Þ

Notice that q
00 5

1 2 r v0 1 2 r v0 minðq0 ; 1Þ ≤ r 1 2 v0 r 1 2 v0     1 2 v0 1 2 vt < min ; 1 5 qt ; ; 1 ≤ min supX xt

ðA46Þ

where the strict inequality follows from ðA27Þ, and the last inequality uses the fact that vt11 ≤ vt for all t. Therefore, combining ðA45Þ and ðA46Þ, q
t 0 ≤ q
00 < qt ≤ 1 5 q
t 1

for all t:

ðA47Þ

Given ðA43Þ and ðA47Þ, ðA25Þ reduces to

qt 5

8 1 > > > > 1 2 vt > > > > < xt q
t 0 > > > > > > 1 > > : xt

if xt ≤ 1 2 vt

1 2 vt q
t 0 1 2 vt 1 if ≤ xt ≤ 0 q
t q
t 0 1 if 0 < xt : q
t if 1 2 vt < xt <

ðA48Þ

For all t, ðA45Þ, ðA46Þ, and vt11 ≤ vt imply   1 2 r v0 1 2 v0 1 2 v0 1 2 vt < q
t ≤ q
≤ min ; ; 1 ≤ ≤ r 1 2 v0 supX supX xt 0

0 0

where the strict inequality follows from ðA27Þ. Thus ðA48Þ ðand therefore ½A25Þ reduces to ðA36Þ, so ðE3Þ is satisfied.

moneyspots

163

With ðA37Þ, notice that for all t, ðA20Þ and ðA21Þ reduce to ðA30Þ and ðA44Þ, respectively, with initial conditions M
00 5 v0 M 0 and M
01 5 ð1 2 v0 ÞM 0 1 B 0 given by ðA30Þ and ðA32Þ. Moreover, ðA30Þ and ðA44Þ imply M
t11 5 maxðxt 1 vt ; 1ÞM
t ; which is the same as ð1Þ with qt given by ðA36Þ. Also, with ðA36Þ, the conditions in ðA26Þ reduce to the conditions in ðA33Þ. Hence ðE5Þ is satisfied.
t Þ 5 M
0 1 M 1 1
t , and m
it ; m
it satisfies m
it ∈ M
it ðM As defined in ðA39Þ, m 0 t11 qt Bt11 5 M
t ðthe last equality follows from ½A30, ½A33, ½A36, ½A37, and ½A38Þ.
t Þ, Also, for any M ∈ F ðM

EE I


it ðdsÞmðdiÞ 5 m

M

E E


it ðMÞmðdiÞ m

I

5

I

1

Ifi M
00 ∈Mg ðiÞM
00 mðdiÞ 1

E

E

1 1 ∈Mg ðiÞM IfM
00 1iMt11 t11 mðdiÞ

I

1 1iq B IfM
00 1Mt11 ðiÞqt Bt11 mðdiÞ t t11 ∈Mg

I

1 1 5 mðfi M
00 ∈ MgÞM
00 1 mðfM
00 1 iMt11 ∈ MgÞMt11 1 1 mðfM
00 1 Mt11 1 iqt Bt11 ∈ MgÞqt Bt11

5 mðMÞ; where the last equality follows from the relative dilation invariance of the Lebesgue measure ðtheorem D in Halmos ½1974, 64Þ. Hence property 4 and ðE6Þ are satisfied. As defined in ðA42Þ, bit11 ∈ Bt11 , and bit11 ; bit11 ðBt11 Þ 5 IfiBt11 ∈Bt11 g ðiÞBt11 5 Bt11 . Also, for all B ∈ F ðBt11 Þ,

EE I

bit11 ðdsÞmðdiÞ 5

B

E E

bit11 ðBÞmðdiÞ

I

5

IfiBt11 ∈Bg ðiÞBt11 mðdiÞ

I

5 mðfiBt11 ∈ BgÞBt11 5 mðBÞ:
t , and Hence ðE7Þ is satisfied. As defined in ðA40Þ, mit11 satisfies mit11 ∈ M 0 1 0 0 mit11 ðMt11 Þ5M0 1Mt115Mt11 ðsince by ½A33Mt11 5M0 for all tÞ, so ðE8Þ is satisfied. From ðA47Þ, q
t 0 < qt for all t, so ðA15Þ for s 5 0 reduces to ðA37Þ. Also, from ðA36Þ, qt ≤ 1 5 q
t1 for all t, so ðA15Þ for s 5 1 reduces to 8 < ∈ ½0; M
t1  if qt 5 1 1 M
1 Bt11 if qt < 1: :5 t qt

ðA49Þ

164

journal of political economy

With ðA36Þ, it is clear that ðA38Þ satisfies ðA49Þ, so I can conclude that ðA37Þ and
t , and evaluating ðA41Þ at M 5 ðA38Þ satisfy ðA15Þ. As defined in ðA41Þ, n2it11 ∈ M s s s2 s s

Mt implies nit11 5 Mt 2 Mt11 5 qt Bt11 , where the last equality follows from ðA30Þ,
t implies ðA33Þ, ðA36Þ, ðA37Þ, and ðA38Þ. Similarly, evaluating ðA41Þ at M 5 M 2 nit11 5 M
t 2 Mt11 5 qt Bt11 . Hence ðE9Þ is satisfied. With ðA36Þ, ðA37Þ, and ðA47Þ, the Euler equations ðA22Þ and ðA23Þ reduce to 0

s zts 5 bEt ½L ðzt11 Þzt11  for s 5 0; 1;

ðA50Þ

which combined, and using zt0 1 zt1 5 zt , imply 0

zt 5 bEt ½L ðzt11 Þzt11 :

ðA51Þ

` ` 5 frz; ð1 2 rÞz; zgt50 , with z But then notice that the constant path fzt0 ; zt1 ; zt gt50 defined by ðA29Þ, satisfies ðA50Þ and ðA51Þ. Thus, ðA34Þ and ðA35Þ satisfy ðE4Þ. In summary, for a given v0, the proposed allocations and prices constitute a monetary equilibrium. Since v0 is arbitrary, the construction characterizes a continuum of monetary equilibria, one for each v0 ∈ ð0, 1Þ. Part 3: In the equilibrium described in part 2, Bt11 5 xt M
t > 0, and bonds are always held. The price of bonds satisfies 0 < qt ≤ 1 for all t ðwith qt < 1 if 1 2 vt < xt Þ, and   xt it11 5 max 2 1; 0 : ðA52Þ 1 2 vt

The real value of the money holdings brought into the decentralized market of period t 1 1 is

E

 ðft11  mit11 ÞmðdiÞ 5 rz 1 max 1 2

I

 xt ; 0 ð1 2 rÞz > 0: 1 2 vt

Hence ∫I ðft11  mit11 ÞmðdiÞ > 0 for all t, even if it > 0, as is the case whenever the realization of the relative size of the open-market operation, xt , lies in the interval ð1 2 vt ; supX. QED s Corollary 1. Let pst11 ; pt11 =pts 2 1, with pts ; 1=fst , where f0t 5 f00 and f1t are given by ðA35Þ. Let p
t11 ; p
t11 =p
t 2 1, with p
t ; M
t =z, where z is given by ` ðA29Þ, and M
t 5 M
t0 1 M
t1 , where M
t0 is given by ðA30Þ, and fM
t1 gt51 satisfies ðA31Þ
and ðA32Þ given the initial condition M0 . Let it11 be as in ðA52Þ. In any of the equilibria characterized in the proof of proposition 2,

i. p0t11 5 0 is the inflation rate between t and t 1 1 when the price level is measured in terms of notes of type 0;

ii. p1t11 5 it11 is the inflation rate between t and t 1 1 when the price level is measured in terms of notes of type 1; and

iii. p
t11 5 ð1 2 vt Þit11 is the inflation rate between t and t 1 1 when the price level is measured using the GDP deflator, p
t .

moneyspots

165

m Let 1 1 r ; 1=ð1 1 p Þ for s 5 0, 1, 1 1 r ; ð1 1 it11 Þ=ð1 1 p Þ, 1 1 r
t11 ; 1=ð1 1 p
t11 Þ, and 1 1 r
; ð1 1 it11 Þ=ð1 1 p
t11 Þ. In any of the equilibria characterized in the proof of proposition 2, s t11

s t11 b t11

b t11

1 t11

s iv. rt11 5 0 is the real return from holding a note of type s between t and t 1 1

ðnotes of type 1 are held only if xt < 1 2 vt Þ;

b v. rt11 5 0 is the real return from investing a note of type 1 in nominal bonds

at t ;

vi. r
mt11 5 minðð1 2 vt 2 xt Þ=ðvt 1 xt Þ; 0Þ is the measured real return from holding money between t and t 1 1 ðbased on the GDP deflatorÞ;

vii.


b r
t11 5 max

 vt ½xt 2 ð1 2 vt Þ ; 0 ð1 2 vt Þðvt 1 xt Þ

is the measured real return from holding nominal bonds between t and t 1 1 ðbased on the GDP deflatorÞ; and b m viii. ð1 1 r
t11 Þ=ð1 1 r
t11 Þ 5 1 1 it11 ≥ 1, with strict inequality for xt > 1 2 vt ; that is, the measured real return on bonds exceeds the measured real return on money. Proof of corollary 1. In any equilibrium of proposition 2, in a bilateral trade the
t general seller exchanges z special goods for a portfolio ðM t ; Bt Þ that is worth ft M goods. Hence, given ðA35Þ, the relative price of any special good in terms of the general good is
t ft M rz 1 ð1 2 rÞz 5 1: 5 z z Then real GDP in period t ðexpressed in terms of the general or any of the special goodsÞ is Yt 5

E E

ðcijt 1 yit ÞmðdiÞ

 5

Ifi ∈I bt g ðiÞmðdiÞ 1

E

 Ifi ∈I st g ðiÞmðdiÞ z

5 2az: Notice that pts is the time t price of goods in terms of notes of type s, so pst11 is the corresponding inflation rate. Part i: The fact that p0t11 5 0 is immediate from ðA34Þ. Part ii: Notice that ðA31Þ, ðA35Þ, and ðA36Þ imply 1 1 p1t11 5

  1 xt 1 M
t11 5 max ; 1 5 5 1 1 it11 : qt 1 2 vt M
t1

ðA53Þ

Part iii: Nominal expenditure in the decentralized market is equal to aðM
t0 1 M
t1 Þ, which is also the amount of nominal expenditure in the centralized market. The GDP deflator, that is, the ratio of nominal GDP to real GDP, in this economy is

166

journal of political economy p
t 5

2aM
t M
t 5 ; Yt z

so the gross inflation rate between t and t 1 1 when the price level is measured using the GDP deflator, denoted 1 1 p
t11 , is equal to 0 1 M
t11 M
t11 M
t11 5 v 1 ð1 2 v t tÞ 0 M
t M
M
1 t

t

1 5 vt 1 ð1 2 vt Þð1 1 pt11 Þ

ðA54Þ

5 vt 1 ð1 2 vt Þð1 1 it11 Þ; where the second equality follows from ðA30Þ and the first equality in ðA53Þ, and the third equality follows from the last equality in ðA53Þ. Hence, p
t11 5 ð1 2 vt Þit11 . Part iv: The real gross return from holding a note of type 0 between t and t 1 1 0 is 1 1 rt11 5 1 by part i. According to ðA33Þ, notes of type 1 are held between periods t and t 1 1 only if xt < 1 2 vt , and in this case the real gross return from holding a 1 note of type 1 between t and t 1 1 is 1 1 rt11 5 1 by ðA53Þ. Part v: The real gross return on bonds ðwhich are bought only with notes of b type 1 in equilibriumÞ between t and t 1 1, denoted 1 1 r t11 , is equal to f1t11 qt21 1 1 it11 5 5 5 1; 1 qt ft 1 1 p1t11 1 1 p1t11 where the last equality follows from ðA53Þ. Part vi: The measured real gross return from holding money between t and m t 1 1 ðbased on the GDP deflatorÞ, denoted 1 1 r
t11 , is equal to 1 1 5 1 1 1 p
t11 vt 1 ð1 2 vt Þð1 1 pt11 Þ 1 5 vt 1 ð1 2 vt Þmaxðxt =ð1 2 vt Þ; 1Þ   1 ; 1 ; 5 min vt 1 xt where the first equality follows from ðA54Þ and the second equality follows from ðA53Þ. Part vii: The measured real gross return from holding nominal bonds between t b and t 1 1 ðbased on the GDP deflatorÞ, denoted 1 1 r
t11 , is equal to 1=p
t11 qt21 1 1 it11 1 1 it11 5 5 5 qt ð1=p
t Þ 1 1 p
t11 1 1 p
t11 vt 1 ð1 2 vt Þð1 1 it11 Þ   xt 5 max ; 1 ; ð1 2 vt Þðvt 1 xt Þ

ðA55Þ

where the third equality follows from ðA54Þ and the fourth equality follows from ðA53Þ.

moneyspots

167

Part viii: Since 1 1 r
follows that

m t11

; 1=ð1 1 p
t11 Þ, from the second equality in ðA55Þ, it

b 5 1 1 r
t11

1 1 it11 m 5 ð1 1 it11 Þð1 1 r
t11 Þ; 1 1 p
t11

and the result is immediate from ðA52Þ. QED Proof of Proposition 3 Let the notation be as in the proof of proposition 2. Also as in that proof, let the
0; M
1 ig ` be given by ðA9Þ–ðA11Þ, and M s 5 M
s 2 q0 B s sequence of partitionsfhM t t 0 t50 1 1 0 1 0

for s 5 0, 1, M1 5 M1 , and M1 5 M11 1 B1 . In addition, the current assumptions on ` fBt gt50 imply M 00 5 M
00 5 v0 M 0, M 01 5 M
01 5 ð1 2 v0 ÞM 0 , and M
ts 5 Mts 5 M
1s for all t ≥ 2. To illustrate the possibility of negative nominal rates, I construct an equilibrium as follows. Begin with the optimization problem of an agent in the centralized market of period t ≥ 1: max½2ft m
t11 1 bLðft11 m
t11 Þ:

0≤m
t11

This is the special case of ðA13Þ that obtains by replacing the function lt11 ðm
t11 ; bt11 Þ with ft11 m
t11 . ðSince Bt11 5 0 for t ≥ 1, this is a purely monetary economy from the centralized market of date 1 onward.Þ The Euler equation for money of type s is 0

2fst 1 bL ðft11 m
t11 Þfst11 ≤ 0;

5

s > 0; for all t ≥ 1: if m
t11

Focus on an equilibrium with ft ðnÞ 5 os50 fst Ifn ∈ M
st g ðnÞ, where f0t 5 0 and f1t 5 f11
1 5 M
1 for all t ≥ 1, real balances are constant from pe> 0, for all t ≥ 1. Since M t 1 riod 1 onward in the equilibrium under consideration; that is, f1t M
t1 5 f11 M
11 ; z for all t ≥ 1, where z is the unique solution to ðA29Þ. Hence given z ðwhich is guaranteed to be strictly positiveÞ, 1

z f1t 5 f11 5
1 M1

for all t ≥ 1:

ðA56Þ

At the end of period 0, an agent who enters the bond market with m
0 5 ðm
00 ; m
01 Þ solves
m
0 ; b1 Þ; max bL½lð

0≤q0 b1 ≤m
0

where
m
0 ; b1 Þ ; f1 m
1 1 f1 b 0 1 ð1 2 q0 Þf1 b 1 : lð 1 0 1 1 1 1 This is just a special case of the inner maximization in ðA13Þ, for t 5 0, with f01 5 0, and no uncertainty. The implied individual bond demands are b10 5 m
00 =q0 and b11 5 g01 ðm
01 Þ, where g01 ðÞ is as in ðA15Þ. In the centralized market of period 0, the agent solves

168

journal of political economy maxf2f0 m
0 1 bL½l
* ðm
0 Þg; 0 ≤ m
0

where     m
0 f1 m
0 1 ; 1 f11 m
01 : l
* ðm
0 Þ ; l
m
0 ; 0 ; g01 ðm
01 Þ 5 1 0 1 max q0 q0 q0 Focus on an equilibrium in which both types of moneys are held at t 5 0. Then the Euler equations corresponding to this problem are 1 0 f00 5 bL ½l
* ðm
0 Þ f11 q0 and 0 f10 5 bL ½l
* ðm
0 Þmax



 1 ; 1 f11 : q0


0 and l
* ðM
0 Þ 5 z, so these conditions become In equilibrium, m
0 5 M f00 5

1 1 f; q0 1

 f10 5 max

ðA57Þ

 1 ; 1 f11 : q0

ðA58Þ


10 Þ denote the aggregate demand for bonds at Let G 0 ðM
0 Þ ; M
00 =q 0 1 g 01 ðM
0 Þ 5 B1 , yields t 5 0. The bond market equilibrium condition at t 5 0, G 0 ðM 8 v0 > > if x 0 < v0 > < x0 q0 5 1 ðA59Þ if v0 ≤ x 0 ≤ 1 > > 1 > : if 1 < x 0 : x0 The individual bond demands, the constraint q0 B1s 1 M1s 5 M
0s , and ðA59Þ imply M10 5 0

and q0 B10 5 v0 M 0 ;

ðA60Þ

and 8 < ð1 2 v0 ÞM 0 M11 5 ð1 2 x 0 ÞM 0 : 0 8 < 0 q0 B11 5 ðx 0 2 v0 ÞM 0 : ð1 2 v0 ÞM0

if x 0 < v0 if v0 ≤ x 0 ≤ 1 if 1 < x 0 ; if x 0 < v0 if v0 ≤ x 0 ≤ 1 if 1 < x 0 :

ðA61Þ

moneyspots

169

Since M
11 5 M
01 2 q0 B11 1 B1 5 ð1 2 v0 1 x 0 ÞM 0 2 q 0 B11 ; ðA61Þ implies 8 < ð1 2 v0 1 x 0 ÞM 0 M
11 5 M 0 : x 0M 0

if x 0 < v0 if v0 ≤ x 0 ≤ 1 if 1 < x 0 :

ðA62Þ

Combined, ðA56Þ and ðA62Þ imply 8 z > > > ð1 2 v 1 x 0 ÞM 0 > 0 < z f11 5 > M0 > > z > : x 0M 0

if x 0 < v0 if v0 ≤ x 0 ≤ 1

ðA63Þ

if 1 < x 0 :

Finally, with ðA59Þ and ðA63Þ, ðA57Þ and ðA58Þ become 8 x 0z > if x 0 < v0 < v0 ð1 2 v0 1 x0 ÞM0 0 f0 5 z > : if v0 ≤ x 0 ; M0 8 z > if x 0 < v0 < ð1 2 v0 1 x 0 ÞM 0 1 f0 5 z > : if v 0 ≤ x 0 : M0 At this point a family of monetary equilibria with moneyspots, indexed by the arbitrary number v0 ∈ ð0, 1Þ has been constructed. Given a value for v0, the equilibrium can be summarized as follows. The prices of bonds and the two types of ` money are q 0 and fðfst Þs50;1 gt50 , where q 0 is given by ðA59Þ, f00 and f10 are given by 0 1 ðA64Þ, ft 5 0 for t ≥ 1, and ft 5 f11 for t ≥ 1, with f11 given by ðA63Þ. Each agent carries M1s units of money of type s into period 1 and uses q0 B1s units of money of type s to purchase bonds in period 0, with the corresponding expressions given by ðA60Þ and ðA61Þ, for s 5 0 and s 5 1, respectively. Along the equilibrium path, M 00 5 M
00 5 v0 M 0 , M 01 5 M
01 5 ð1 2 v0 ÞM 0 , M 10 5 M
10 5 0, M 11 is as in ðA61Þ, M
11 as in ðA62Þ, and ` ` ` fðMts Þs50;1 gt52 5 fðM
ts Þs50;1 gt52 5 fðM
1s Þs50;1 gt52 :

The desired result is immediate from ðA59Þ: the equilibrium nominal interest rate on government bonds between period 0 and period 1, that is, i1 5 1=q0 21, is 8 x0 > if x 0 < v0 > > : x0 2 1 if 1 < x 0 ;

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which is negative if the size of the open-market operation is small enough, that is, x 0 < v0 . Equivalently, for any x 0 < 1, there exists a continuum of moneyspot equilibria with i1 < 0, indexed by the arbitrary number v0 ∈ ðx 0 ; 1Þ. QED Appendix B Subsidiary Proofs The following lemma provides closed-form expressions for the value functions that will summarize the maximum expected utility attainable by an agent i ∈ I in any equilibrium. Lemma 1. Agent i’s values are Wt ðait Þ 5 ft  mit 1 ft  ðbit q21 Þ 1 Kt ;

ðB1Þ

Vt ðait Þ 5 L½Lt ðait Þ 1 Kt ;

ðB2Þ


it Þ 5 max Ut ðm bEt fL½Lt11 ðait11 Þ 1 Kt11 g; 2

ðB3Þ

ait11 ;nit11

where fKt g satisfies Kt 5 Dt 1 bEt Kt11 ;

ðB4Þ



it 1 max Dt ; max 2ft  m bE L½L ða Þ t t11 it11 2

ðB5Þ


t Þ;
it ðMÞ for all M ∈ F ðM n2it11 ðMÞ 1 mit11 ðMÞ 5 m

ðB6Þ


t Þ; qt  bit11 5 n2it11 ðM

ðB7Þ

Dt is given by


t m
it ∈ M

ait11 ;nit11

subject to


t ; bit11 ∈ Bt11 , and the maximization on the right side of ðB3Þ is mit11 ; n2it11 ∈ M subject to ðB6Þ and ðB7Þ. Proof of lemma 1. Agent i’s value in the centralized market is given by ð10Þ. Substitute ð11Þ into the objective function on the right side of ð10Þ to get ðB1Þ, where
it 1 Ut ð
Kt 5 max f2ft  m mit Þg:
t m
it ∈ M

ðB8Þ

Let hct ðait ; ajt Þ; ðm0t ðait ; ajt Þ; b0t ðait ; ajt ÞÞi denote the bargaining outcome in a bilateral trade between agent i, who is acting as buyer and holding portfolio ait 5 ðmit ; bit Þ, and agent j, who is acting as seller and holding portfolio ajt 5 ðmjt ; bjt Þ, at time t. Upon entering the decentralized market of period t holding ait 5
t21 and bit ∈ Bt , agent i’s value is ðmit ; bit Þ, with mit ∈ M

moneyspots

E

171

Vt ðait Þ 5 a fu½ct ðait ; ajt Þ 1 Et Wt ½mit  m0t ðait ; ajt Þ; bit  b0t ðait ; ajt ÞgϜ ðdajt Þ

E

1 a f2ct ðajt ; ait Þ 1 Et Wt ½mit  m0t ðajt ; ait Þ; bit  b0t ðajt ; ait ÞgϜ ðdajt Þ 1 ð1 2 aÞEt Wt ðmit ; bit Þ; where the expectation operator Et is with respect to the measure nt over assignment realizations of the redemption lottery, and Ϝ is the probability measure over portfolios in the population. Use ðB1Þ and the fact that Et Wt ðmit  mt0 ; bit  bt0 Þ 2 Et Wt ðmit ; bit Þ 5 Et Wt ðmit ; bit Þ 2 Et Wt ðmit  mt0 ; bit  bt0 Þ 5 Lt ðmt0 ; bt0 Þ; to arrive at

E

Vt ðait Þ 5 a fu½ct ðait ; ajt Þ 2 Lt ðm0t ðait ; ajt Þ; b0t ðait ; ajt ÞÞgϜ ðdajt Þ

E

1 a f2ct ðajt ; ait Þ 1 Lt ðm0t ðajt ; ait Þ; b0t ðajt ; ait ÞÞgϜ ðdajt Þ 1 Lt ðmit ; bit Þ 1 Kt : The bargaining outcome implies Lt ðm0t ðait ; ajt Þ; b0t ðait ; ajt ÞÞ 5 ct ðait ; ajt Þ 5 cðLt ðmit ; bit ÞÞ; so Vt ðait Þ 5 afu½cðLt ðmit ; bit ÞÞ 2 cðLt ðmit ; bit ÞÞg 1 Lt ðmit ; bit Þ 1 Kt ; which, given the notation introduced in ð5Þ, is the same as ðB2Þ. Substitute ðB2Þ into ð6Þ to obtain ðB3Þ. Finally, substitute ðB3Þ into ðB8Þ to arrive at ðB4Þ, with Dt given by ðB5Þ. The constraints ðB6Þ and ðB7Þ are the constraints to the maximization in ð6Þ. QED According to lemma 1, Wt ðat Þ and Vt ðat Þ are separable functions of the agent’s asset holdings; for example, Wt ðat Þ is composed of the real value of the current postredemption set of assets, ft  mit 1 ft  ðbit q21 Þ, plus the continuation utility, Kt , which is independent of the current asset position. The term Dt in ðB5Þ is the “flow utility” associated with K t , which summarizes the flow value of the agent’s optimization problems in the competitive markets of period t. The following lemma provides a sharper characterization of the solution to the optimization problems that an agent faces in the competitive markets of period t in a monetary equilibrium that satisfies properties 1 and 2. Lemma 2. Let lt11 : R4 → R be defined by lt11 ðm
it ; bit11 Þ 5 ft11 ðm
it 2 qt bit11 Þ 1 f1t11 bit11 :

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; M
ig and fft g satisfy properties 1 and 2. A solution Suppose that fhM
it ; to agent i’s time t optimization problem, ð13Þ, is a collection of measures ðm
t, ð7Þ with equality, and ð9Þ, with associated
it ∈ M mit11 ; bit11 ; n2it11 Þ that satisfy m s s s2 quantities ðm
its ; mit11 ; bit11 ; nit11 Þs50;1 that achieve 0 t11

Dt 5

1 t11

` t50

` t50


2ft m max
it 1

m
it ∈R1 R1

 max bEt L½lt11 ðm
it ; bit11 Þ ;

0≤qt bit11 ≤m
it

ðB9Þ

s s s2 s 0 1 with mit11 5 m
its 2 qt bit11 and nit11 5 qt bit11 for s 5 0, 1, and bit11 5 bit11 1 bit11 .
it 5 ft m
it . Property 1 and property 2 Proof of lemma 2. Given property 2, ft  m imply Lt11 ðait11 Þ 5 ft11 m it11 1 f1t11 bit11. Hence, the value Dt in ð13Þ can be written as
 1 max 2ft m
it 1 max bE Lðf m 1 f b Þ ; ðB10Þ t it11 it11 t11 t11 2
t m
it ∈ M

ait11 ;nit11

subject to
t Þ;
it ðMÞ for all M ∈ F ðM n2it11 ðMÞ 1 mit11 ðMÞ 5 m

ðB11Þ

2 qt bit11 5 nit11 ;

ðB12Þ

and ð9Þ. The constraints ðB11Þ and ðB12Þ are obtained from ð7Þ and ð8Þ at equality 2
t Þ, ðthe objective is increasing in the controlsÞ. In ðB12Þ I have used nit11 ; n2it11 ðM and the fact that property 1 implies qt ðsÞ 5 qt for all s, and hence qt  bit11 5 qt bit11 .
it ; mit11 ; Notice that a solution to ðB10Þ is given by any collection measures ðm
t, ðB11Þ, and ð9Þ, as long as the quantities ðm bit11 ; n2it11 Þ that satisfy m
it ∈ M
it ; m it11 ; 2 bit11 ; nit11 Þ solve
 max 2ft m
it 1 max bEt Lðft11 m it11 1 f1t11 bit11 Þ m
it

m it11 ;bit11

s 2 , bit11 ∈ R1 for s 5 0, 1, and nit11 5 qt bit11 . Since ðB11Þ must hold subject to m
its , mit11
t Þ, it holds for M 5 M
s, which implies n2 ðM
s Þ 1 m s 5 m
s for for all M ∈ F ðM t t it11 it11 it s s2 2 s s2
s 5 0, 1. With nit11 ; nit11 ðMt Þ and qt bit11 ; nit11 , these constraints can be written as s s 1 mit11 5 m
its qt bit11

for s 5 0; 1:

ðB13Þ


it ; mit11 ; bit11 ; n2it11 Þ Thus, a solution to ðB10Þ is given by any collection measures ðm s

it ∈ Mt , ðB11Þ, and ð9Þ such that the associated quantities ðm
its ; mit11 that satisfy m ; s s2 ; nit11 Þs50;1 solve bit11
 max 2ft m
it 1 max bEt Lðft11 m it11 1 f1t11 bit11 Þ ; m
it

m it11 ;bit11

s s2 s subject to ðB13Þ and m
its , mit11 , bit11 ∈ R1 for s 5 0, 1, with nt11 5 qt bt11 and bit11 0 1 5 bit11 1 bit11 . Use ðB13Þ to substitute m t11 from the objective. Then a solution to
it ðB10Þ is given by any collection of measures ð
mit ; mit11 ; bit11 ; n2it11 Þ that satisfy m

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t , ð9Þ, and ðB11Þ such that the associated quantities ðm
; m ; b ∈M achieve
 2ft m
it 1 max bEt L½lt11 ðm
it ; bit11 Þ ; Dt 5 max s it

m
it ∈R1 R1

s it11

s it11

; n

Þ

s2 it11 s50;1

0≤qt bit11 ≤m
it

s s s2 s 0 1 5 m
its 2 qt bit11 and nit11 5 qt bit11 for s 5 0, 1 and bit11 5 bit11 1 bit11 , as with mit11 stated in the lemma. QED

Appendix C Supplementary Material: Countable Moneyspots In this appendix I present an example in which the set of serial numbers is countable. The environment is as in Section II, except for the following modifications. The initial set of notes outstanding at the beginning of period 0 is ½0; M 0 , with M 0 ∈ R1 . Every note has an extraneous attribute or moneyspot : a serial number s ∈ Z1 printed on it ðZ1 denotes the set of nonnegative integersÞ. Each serial number, s, is shared by a quantity ðformally, a measureÞ k > 0 of notes. A bond issued in period t entitles the bearer to collect a note in period t 1 1. For simplicity, every bond is assumed to be indistinguishable from every other bond ði.e., bonds do not have serial numbersÞ. The timing of government interventions in a typical period t is as before. The quantity of bonds outstanding at time t, Bt ∈ R1 , is redeemed after the round of decentralized trade, right before agents trade in the centralized market of period t. A new bond issue, of size Bt11 , is sold competitively for notes after the round of centralized trade of period t, right before agents enter period t 1 1. The government finances bond redemptions with notes acquired in previous bond sales, and if these notes were not enough, it makes up the difference by creating new notes. Whenever notes are created, they are numbered consecutively; that is, each of the first k notes ever printed has serial number “0,” each note printed in the second batch of size k carries serial number “1,” and so on. Thus, the quantity of notes with serial number s outstanding at the beginning of time 0 is M0 ðsÞ 5 min ðmaxð0; M0 2 skÞ; kÞ:

ðC1Þ

2 Let Nt11 ðsÞ denote the quantity of notes with serial number s that are with1 drawn from private circulation by the bond sale of period t, and let Nt11 ðsÞ be the quantity of notes with serial number s that are injected by the corresponding bond redemption in period t 1 1. Let Mt ðsÞ represent the quantity of notes with serial number s, outstanding at the beginning of period t, and let M
t ðsÞ be the quantity of notes with serial number s, outstanding at the end of the centralized trading session of period t, before the new bond sale, that is, M
t ðsÞ 5 Mt ðsÞ 1 Nt1 ðsÞ. The law of motion for the postredemption quantity of notes with serial number s is 2 1 M
t11 ðsÞ 5 M
t ðsÞ 2 Nt11 ðsÞ 1 Nt11 ðsÞ:

ðC2Þ

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2 Given the initial condition ðC1Þ and the endogenous path ffNt11 ðsÞ; ` 1 Nt11 ðsÞgs ∈Z1 gt50 , ðC2Þ gives the path of the postredemption quantity of notes, ` fM
t11 ðsÞgt50 , for each type s ∈ Z1 . Let Mt 5 os ∈Z1 Mt ðsÞ and M
t 5 os ∈Z1 M
t ðsÞ. Intuitively, Mt represents the total quantity of notes outstanding at the beginning of period t, and M
t represents the period t quantity of notes outstanding after that period’s bond redemption ðbut before the new bond saleÞ. Since Bt is the quantity of bonds outstanding at the beginning of period t, M
t 5 Mt 1 Bt . The time t bond issue injects Bt11 1 5 os ∈Z1 Nt11 ðsÞ notes in period t 1 1 and withdraws from private circulation qt Bt11 notes at the end of period t, where qt is the time t price of a bond in terms of notes. As before, Bt11 5 xt M
t , so the law of motion for the postredemption money supply is given by ð1Þ.

A. Equilibrium
t 5 fs ∈ Z1 : M
t ðsÞ > 0g. Intuitively, M
t is the set of serial For each t, define M numbers outstanding at the end of the centralized trading session of period t, before the new bond sale. Let m
it 5 fm
it ðsÞgs ∈ M
t denote the portfolio of postredemption notes held by agent i at time t. That is, m
it ðsÞ ∈ R1 represents the quantity of notes with serial number s that agent i is holding at the end of the time t trading centralized session, before the new bond sale. For each t ≥ 0, define Mt 5 fs ∈ Z1 : Mt ðsÞ > 0g. Intuitively, Mt is the set of serial numbers outstanding at the beginning of period t . Let m it 5 fmit ðsÞgs ∈Mt denote the beginning of period t portfolio of notes held by agent i. Consider an agent i ∈ I who is holding a portfolio of notes m
it when he enters the competitive market for government bonds at the end of period t. This agent’s problem consists of choosing vectors a it11 5 ðm it11 ; bit11 Þ and n 2it11 , where the real 2 number bit11 is the quantity of bonds that he purchases, n 2it11 5 fnit11 ðsÞgs ∈ M
t is the 2 portfolio of notes that he trades away in the bond market ðnit11 ðsÞ is the quantity of notes with serial number s that he exchanges for bondsÞ, and m it11 5 fmit11 ðsÞgs ∈ M
t is the portfolio of notes that he chooses to hold after the bond purchase. The value of this agent’s problem in the bond market is Ut ðm
it Þ 5 max bEt Vt11 ða it11 Þ 2 a it11 ;n it11

ðC3Þ

subject to 2 nit11 ðsÞ 1 mit11 ðsÞ ≤ m
it ðsÞ

qt bit11 ≤

on


t s ∈M

2 it11


t; for all s ∈ M

ðC4Þ

ðsÞ;

ðC5Þ

0 ≤ m it11 ; 0 ≤ n 2it11 ; 0 ≤ bit11 :

ðC6Þ

For any nonnegative vector a it11, the real-valued function Vt11 ða it11 Þ denotes the maximum expected discounted utility that the agent can attain when he enters the

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175

decentralized market of period t 1 1 holding portfolio a it11 ðafter the size of the open-market operation of period t 1 1 has been announced but before the realization of the bilateral matchingÞ. The first constraint states that the notes that the agent uses to purchase bonds and the notes that he carries into the next round of decentralized trade must belong to the portfolio of notes that the agent was holding before the open-market operation. The second constraint states that given the price for bonds, qt , the agent has enough notes to finance the bond purchase. ðImplicit in this constraint is the maintained assumption that the government ignores extraneous attributes.Þ When the government redeems bonds, it may be doing so with notes bearing different serial numbers. For this reason, as explained in Section IV, in order to price a bond, the agent needs to be able to form expectations over the serial numbers of the notes that may be used to redeem his bond holding. To this end, in Section IV, I introduced a general redemption lottery. To ease the exposition, here I assume that each agent i’s time t bond holding, bit , is redeemed by a portfolio of notes n 1it 5 fnit1 ðsÞgs ∈N 1t , with nit1 ðsÞ ; pt ðsÞbit , where pt ðsÞ ; Nt1 ðsÞ=os ∈N 1t Nt1 ðsÞ, 1 and N t 5 fs ∈ Z1 : Nt1 ðsÞ > 0g is the set of serial numbers printed on the notes injected by the government during the period t bond redemption.30
t , let ft ðsÞ ∈ R1 [ ` be the real price ðin terms of general goodsÞ For each s ∈ M of a note with serial number s, and let ft 5 fft ðsÞgs ∈ M
t denote the vector of real prices of all serial numbers outstanding at time t. For a given a it 5 ðm it ; bit Þ, let the real-valued function Wt ða it Þ denote the maximum expected discounted utility that an agent can attain when he enters the time t centralized market with portfolio a it . This value satisfies Wt ða it Þ 5 max fyit 2 hit 1 Ut ðm
it Þg

ðC7Þ

yit 2 hit 1 ft m
it 5 ft m it 1 ft n 1it ;

ðC8Þ

yit ;hit ;m
it

subject to


Since nit1 ðsÞ ; pt ðsÞbit , the last term on the right side yit ≥ 0, m
it ≥ 0, and 0 ≤ hit ≤ h. of ðC8Þ is equal to f
t bit , where f
t ; os ∈N 1t pt ðsÞft ðsÞ. Consider a bilateral meeting in the decentralized market of period t between a buyer i ∈ I and a seller j ∈ I who are holding portfolios ðm it ; bit Þ and ðm jt ; bjt Þ, respectively. A bargaining outcome is a quantity of special good, cijt , that the seller produces for the buyer in exchange for a portfolio ðm 0t ; bt0 Þ that the buyer offers as payment. The gains from trade corresponding to a bargaining outcome ðcijt ; m 0t ; bt0 Þ would be uðcijt Þ 1 Wt ðm it 2 m 0t ; bit 2 bt0 Þ 2 Wt ðm it ; bit Þ for the buyer, and 2cijt 1 Wt ðm jt 1 m 0t ; bjt 1 bt0 Þ 2 Wt ðm jt ; bjt Þ for the seller. The buyer’s take-it-or-leave-it offer solves uðct Þ 2 Lt ðm 0t ; bt0 Þ max 0 0 ct ;m t ;b t

30 In terms of the general formulation of Sec. IV, this particular specification can be interpreted as a sort of uniform redemption lottery.

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subject to
m 0 ≤ m it ; b 0 ≤ bit ; ct ≤ Lt ðm 0t ; bt0 Þ; 0 ≤ ct ≤ h; t t where Lt ðm 0t ; bt0 Þ ; ft m 0t 1 f
t bt0 . As before, the quantity traded in a singlecoincidence meeting is cðLt ðm it ; bit ÞÞ. With this bargaining solution, ðC7Þ, and ðC8Þ, the maximum expected discounted utility attainable by an agent who enters the decentralized market of period t holding a it 5 ðm it ; bit Þ can be written as Vt ða it Þ 5 L½Lt ða it Þ 1 Kt ; ` where fKt gt50 satisfies Kt 5 Dt 1 bEt Kt11 , and
 bE L½L ða Þ ; Dt ; max 2ft m
it 1 max t t11 it11 2 0≤m
it

a it11 ;n it11

ðC9Þ

ðC10Þ

subject to ðC4Þ, ðC5Þ, and ðC6Þ. Definition 4. An equilibrium is an allocation ` ðfyit ; hit ; fm
it ðsÞ; mit11 ðsÞgs ∈ M
t ; bit11 gt50 Þi ∈I ; ` ` , and pricing functions ffft ðsÞgs ∈ M
t ; qt gt50 such bargaining outcomes fðcijt Þi;j ∈I gt50 that ðiÞ given prices and the bargaining protocol, fyit ; hit ; fm
it ðsÞ; mit11 ðsÞgs ∈ M
t ; ` bit11 gt50 solves agent i’s optimization problem in the competitive markets; ðiiÞ for i, j ∈ I in a bilateral meeting, the bilateral terms of trade are determined by Nash bargaining; that is, if agent i is the buyer and agent j the seller in a bilateral meeting at time t, then j produces cijt 5 min½Lt ðm it ; bit Þ; c *  special goods for i ðcijt 5 0 if i and j are not in a single-coincidence meeting at tÞ; and ðiiiÞ the cen
* 5 fs ∈ M
t : ft ðsÞ > 0g; an equilibrium is tralized market clears for all t. Let M t
* ≠ ∅ for all t, and in this case, the allocation must said to be “monetary” if M t also satisfy, for all t, ðaÞ the money market–clearing condition, m
it ðsÞ 5 M
t ðsÞ for
* , and ðbÞ thebond market–clearingcondition, bit11 5 Bt11 . In a monetary all s ∈ M t equilibrium, money is said to coexist with bonds in period t ≥ 1 if Bt > 0 and ∫I ½os ∈Mt ft ðsÞmit ðsÞmðdiÞ > 0. A monetary equilibrium is said to be a “moneyspot
t with ft ðsÞ ≠ ft ðs 0 Þ for some t. equilibrium” if there exist s, s 0 ∈ M

B. The Coexistence Proposition Proposition 4. Let it11 5 1=qt 2 1 denote the nominal interest rate between period t and period t 1 1. If u 0 ð0Þ > 1 1 ½ð1 2 bÞ=ab, supX < `, and M0 > kK0 for some K0 ≥ 1, the economy with extraneous attributes admits a continuum of monetary equilibria in which money coexists with bonds for all 1 ≤ t < `, even if it11 > 0. Proof of proposition 4. The proof is organized in three parts. Part 1 derives the set of equilibrium conditions that characterize a monetary equilibrium that belongs to a certain class. Part 2 constructs a particular allocation and price system and establishes that they constitute a family ða continuumÞ of monetary equilibria that belong to the class described in part 1. Part 3 establishes that in the family of monetary equilibria described in part 2, money coexists with bonds for all t, even if qt < 1.

moneyspots

177

Part 1: Let K be an arbitrary integer that satisfies 0 ≤ K ≤ ⌊K 0 ⌋ 2 1, and let
g; M
bi hMgt ; Mbt i be a partition of Mt , with Mgt 5 f0; : : : ; K g \ Mt . Also let hM t t g


be a partition of Mt , with Mt 5 f0; : : : ; K g \ Mt . ðHereafter, I will refer to
k, as notes of type k, for k 5 g, b.Þ Nonotes whose serial numbers lie in Mkt or M t tice that M0 5 f0; : : : ; ⌈M 0 =k⌉g, so Mg0 5 f0; : : : ; K g. Notice that the quantity of notes with serial numbers in the set Mg0 that are outstanding at the beginning of period 0 is M 0g 5 v0 M 0 , and the quantity of notes with serial numbers in the set Mb0 that are outstanding at the beginning of period 0 is M 0b 5 ð1 2 v0 ÞM 0 , where

o

K

v0 ;

M 0 ðsÞ kK 5 : M0 M0

ðC11Þ

s50

More generally, define Mtk ; os ∈Mkt Mt ðsÞ and M
tk ; os ∈ M
kt M
t ðsÞ for k 5 g, b, the preredemption and postredemption quantities of notes of type k, respectively. In the remainder I will focus on a class of equilibrium that satisfies the following properties.
k , for k 5 g, b. Property 1. For every t, ft ðsÞ 5 fkt > 0 if s ∈ M t Property 2. For every t, fbt11 < qt fgt11 , and agents expect that for all t, M
tg 5 M
0g ;

ðC12Þ

b M
t11 5 M
tb 1 ð1 2 qt ÞBt11 :

ðC13Þ

According to ðC12Þ, agents expect notes of type g never to be used to purchase bonds from the government ðand consequently its stock never to be augmented by government injectionsÞ. According to ðC13Þ, agents expect bonds to be purchased entirely with money of type b and all money injections to augment the stock of money of type b. Since M
tg 1 M
tb 5 M
t for all t, ðC12Þ and ðC13Þ imply ð1Þ. ` Property 3. The sequence ffgt ; fbt gt50 is such that ðfgt11 ; fbt11 Þ is known at time t. Next, I derive the conditions that characterize a monetary equilibrium that k satisfies properties 1–3. Let m
itk ; os ∈ M
kt m
it ðsÞ, mit11 ; os ∈ M
kt mit11 ðsÞ, and k2 2 k k2 nit11 ; os ∈ M
kt nit11 ðsÞ, for k 5 g, b. Define bit11 ; nit11 =qt , and notice that ðC5Þ g b ðwhich must hold with equality at the optimumÞ implies bit11 5 bit11 1 bit11 . Also notice that ðC4Þ ðwhich must also hold with equality at the optimumÞ implies k k mit11 5 m
itk 2 qt bit11 :

Property 1 implies that ðC10Þ simplifies to
2 o fkt m
itk Dt 5 max k fm
it gk ∈ fg ;bg

1

k ∈fg ;bg

 max bE L t k

fbit11 gk ∈ fg ;bg

o

k ∈fg ;bg

k fkt11 ½m
itk 1 ðq
tk 2 qt Þbit11 

ðC14Þ

 ;

with q
kt ; f
t11 =fkt11 , and where the outer maximization is subject to 0 ≤ m
itk , and k ≤ m
itk for k 5 g, b. Property 2 implies the inner maximization is subject to 0 ≤ qt bit11 f
t11 5 fbt11 for all t, so q
kt 5 fbt11 =fkt11 . The solution to this maximization problem

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gives the agents’ optimal portfolio choice in the centralized market and in the bond market of period t. If property 3 holds, the solution to the maximization problem of an agent i who enters the market for bonds at time t carrying a portfolio of notes ðm
itg ; m
itb Þ k ðthe inner maximization in the expression for Dt Þ is bit11 5 gtk ðm
itk Þ, where 8 50 if q
tk < qt >  > > k > m
< if qt 5 q
tk ∈ 0; it gtk ðm
itk Þ ðC15Þ qt > > m
itk > k > if qt < q
t : :5 qt With ðC15Þ, agent i’s problem in the centralized market of period t ðthe outer maximization in the expression for Dt Þ becomes
 k k g b * max 2 f m
1 bE L½l ð m
; m
Þ ; ðC16Þ t t11 it it o t it g 2 ðm
it ;m
itb Þ∈R1

k ∈fg ;bg

where l*t11 ðm
itg ; m
itb Þ 5 5

o

½fkt11 m
itk 1 ðq
tk 2 qt Þfkt11 gtk ðm
itk Þ

o

max

k ∈fg ;bg



k ∈fg ;bg

 q
tk ; 1 fkt11 m
itk : qt

ðC17Þ

It is easy to check that the objective function in ðC16Þ is a concave function of ðm
itg ; m
itb Þ, so the following first-order conditions are necessary and sufficient for an optimum of the agent’s problem in the centralized market of period t :
  k  q
0 2fkt 1 bEt L ½l*t11 ðm
itg ; m
itb Þmax t ; 1 fkt11 ≤ 0 for k 5 g ; b; ðC18Þ qt with 5 if m
itk > 0. In a monetary equilibrium in which both types of notes are k held, m
itk 5 M
tk . Let Bt11 ; gtk ðM
tk Þ denote the aggregate quantity of bonds purg b chased with notes of type k. Then given property 2, we have Bt11 5 0 and Bt11 5 Bt11 ðprovided qt ≤ 1Þ. Hence ðC12Þ, ðC13Þ, and ðC17Þ imply g b l*t11 ðM
tg ; M
tb Þ 5 fgt11 M
t11 1 fbt11 M
t11 ; zt11 ;

and ðC18Þ becomes    k  q
fkt 5 bEt L0 ðzt11 Þmax t ; 1 fkt11 for k 5 g ; b: qt

ðC19Þ

With ðC15Þ, ðC13Þ can be written as   1 b M
t11 5 max ; 1 M
tb : qt

ðC20Þ

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179

Then ðC12Þ and ðC20Þ can be used to rewrite ðC19Þ as k ztk 5 bEt ½L0 ðzt11 Þzt11  for k 5 g ; b;

ðC21Þ

where ztk ; fkt M
tk . Thus, in an equilibrium that satisfies properties 1–3, the se` quence of real money holdings of type k, fztk gt50 , must satisfy ðC21Þ. The bond market–clearing condition requires 0 1 Bt11 1 Bt11 5 Bt11 :

ðC22Þ

k The supply of bonds at time t is Bt11 5 xt M
t . With gtk ðM
tk Þ 5 Bt11 and ðC15Þ, the aggregate demand for bonds, Gt ðM
tg ; M
tb Þ ; ok ∈fg ;bg gtk ðM
tk Þ, is 8 50 if maxðq
t g ; q
tb Þ < qt > > >  I b g M
g 1 I g b M
b  > fq
≤q
g fq
≤q
g > > ∈ 0; t t t g t b t t if qt 5 maxðq
tg ; q
tb Þ > > > maxðq
t ; q
t Þ > > > > Ifq
b > if minðq
tg ; q
tb Þ < qt <5 t t qt g b

Gt ðMt ; Mt Þ < maxðq
tg ; q
tb Þ > >  >  g b >

Ifq
t b M
t > > ; if qt 5 minðq
tg ; q
tb Þ ∈ > g b > minð q
; q
Þ minð q
tg ; q
tb Þ > t t > > > M
> > : 5 t if qt < minðq
tg ; q
tb Þ; qt

where Ifq
tb ≤q
tg g is an indicator function that takes the value one if q
tb ≤ q
tg . Hence ðC22Þ can be written as Gt ðM
tg ; M
tb Þ 5 xt M
t , which implies that the equilibrium price of bonds satisfies 8 Ifq
b ≤q
g g vt 1 Ifq
tg ≤q
tb g ð1 2 vt Þ > g b > if xt ≤ t t > maxðq
t ; q
t Þ > maxðq
tg ; q
tb Þ > > > > g g g b b b Ifq
≤q
g vt 1 Ifq
tg ≤q
tb g ð1 2 vt Þ > Ifq
t < xt > if t t > > xt maxðq
tg ; q
tb Þ > > > > Ifq
b > < t t < minðq
tg ; q
tb Þ qt 5 ðC23Þ > Ifq
tb g b > ; q
Þ if ≤ x minð q
> t g t t > > minðq
t ; q
tb Þ > > > 1 > > > ≤ g > > minð q

tb Þ t ; q > > > > 1 1 > : < xt ; if xt minðq
tg ; q
tb Þ where vt ; M
tg =M
t . Finally, in equilibrium, ðC14Þ implies k k Mt11 5 M
tk 2 qt Bt11

for k 5 g ; b:

ðC24Þ

In summary, a monetary equilibrium that satisfies properties 1–3 consists of ` ` an allocation ðfyit ; hit ; fm
it ðsÞ; mit11 ðsÞgs ∈ M
t ; bit11 gt50 Þi ∈I , prices ffft ðsÞgs ∈ M
t ; qt gt50 , ` and bargaining outcomes, fðcijt Þi;j ∈I gt50 , such that

180

journal of political economy

ðE1Þ cijt 5 minðzt ; c * ÞIfi ∈I bt g ðiÞ for i, j ∈ I , where Ifi ∈I bt g ðiÞ is an indicator function that takes the value one if i ∈ I bt ; ðE2Þ yit and hit satisfy

hit 5 yit 1 ½Ifi ∈I bt g ðiÞ 2 Ifi ∈I st g ðiÞ

ðE3Þ ðE4Þ ðE5Þ

ðE6Þ ðE7Þ ðE8Þ ðE9Þ

o

fkt M
tk ;

k ∈fg ;bg


for all i ∈ I ; with 0 ≤ yit and 0 ≤ hit ≤ h, qt is given by ðC23Þ; ` ` ft ðsÞ 5 fgt Ifs ∈ M
gt g ðsÞ 1 fbt Ifs ∈ M
bt g ðsÞ, with fðfkt Þk ∈fg ;bg gt50 5 fðztk =M
tk Þk ∈fg ;bg gt50 , ` where fðztk Þk ∈fg ;bg gt50 satisfies ðC21Þ, and zt 5 ztg 1 ztb ; g b is given by ðC12Þ with M
0g 5 v0 M 0 , M
t11 is given by ðC20Þ with M
0b 5 M
t11 ð1 2 v0 ÞM 0 1 B0 , and ðC24Þ is satisfied ðthe first two conditions imply that ` fM
t11 gt50 follows ½1 with M
0 5 M 0 1 B0 given and Mt11 5 M
t 2 qt Bt11 for all t ≥ 0Þ;
t; m
it ðsÞ 5 M
t ðsÞ for all s ∈ M bit11 5 Bt11 ; mit11 5 Mt11 ; s2 s 2 2 nit11 =qt 5 bit11 satisfies ðC15Þ, nit11 5 qt Bt11 , and nit11 ðsÞ 1 mit11 ðsÞ 5 m
it ðsÞ
t. for all s ∈ M

Part 2: Let agents’ beliefs be as described in properties 1 and 2. Recall that the integer K introduced in part 1 was assumed to satisfy 0 ≤ K ≤ ⌊K 0 ⌋ 2 1 but is otherwise arbitrary. Hence the real number v0 defined in ðC11Þ could take on ⌊K 0 ⌋ values depending on the choice of K, but in every case, v0 ∈ ð0, 1Þ. Let   1 r∈ ; 1 ðC25Þ 1 1 D0 be an arbitrary number, where D0 ;

  1 2 v0 1 2 v0 min ; 1 : v0 supX

Next, I construct a monetary equilibrium indexed by r. For all t and i, j ∈ I , set cijt 5 zIfi ∈I bt g ðiÞ;

yit 5 zIfi ∈I st g ðiÞ;

and

hit 5 zIfi ∈I bt g ðiÞ;

ðC26Þ

where z is the unique solution to bL0 ðzÞ 5 1:

ðC27Þ

M
tg 5 v0 M0 ;

ðC28Þ

b 5 maxðxt M
t ; M
tb Þ; M
t11

ðC29Þ

For all t, let

moneyspots

181

where M
t ; M
1 M
, with g t

b t

M
0b 5 ð1 2 v0 ÞM 0 1 B 0 ;

ðC30Þ

and M
0 5 M 0 1 B 0 given by the initial conditions M 0 and B 0 . For all t ≥ 0, let Mtg 5 M
0g

b and Mt11 5 maxð1 2 vt 2 xt ; 0ÞM
t ;

with M0b 5 ð1 2 v0 ÞM0 and vt ; M
tg =M
t . Then for all t, set
g for s ∈ f0; : : : ; K g f0 ft ðsÞ 5
t nf0; : : : ; K g fbt for s ∈ M

ðC31Þ

ðC32Þ

with rz fg0 5
g M0

and fbt 5

ð1 2 rÞz : M
tb

ðC33Þ

For every t, set  qt 5 min

 1 2 vt ; 1 : xt

ðC34Þ

k bonds with notes of type k in period t ≥ 0, Posit that each agent purchases Bt11 where g Bt11 5 0;

ðC35Þ

b Bt11 5 xt M
t :

ðC36Þ

For all i ∈ I and t ≥ 0, let individual portfolios be as follows: m
it ðsÞ 5 M
t ðsÞ;

ðC37Þ

mit11 ðsÞ 5 Mt11 ðsÞ;

ðC38Þ

2 nit11 ðsÞ 5 M
t ðsÞ 2 Mt11 ðsÞ;

ðC39Þ

bit11 5 Bt11 ;

ðC40Þ


t , where for all s ∈ M M
t ðsÞ 5 min ðmaxð0; M
t 2 skÞ; kÞ; Mt ðsÞ 5 min ðmaxð0; Mt 2 skÞ; kÞ for all t. The proposed paths ðC32Þ and ðC33Þ satisfy properties 1 and 3. From ðC27Þ and ðC33Þ, fgt M
tg 1 fbt M
tb 5 z ∈ ð0; c * Þ, so the cijt , yit , and hit proposed in ðC26Þ satisfy ðE1Þ and ðE2Þ. With ðC33Þ, q
tg 5 fbt11 =fgt11 becomes q
tg 5

12r r

M
0g : b M
t11

ðC41Þ

182

journal of political economy

Since ðC29Þ and ðC34Þ imply

  1 b 5 max ; 1 M
tb ; M
t11 qt

ðC42Þ

I have g 5 minðqt11 ; 1Þq
tg ≤ q
tg q
t11

for all t:

ðC43Þ

Notice that 1 2 r v0 1 2 r v0 minðq0 ; 1Þ ≤ r 1 2 v0 r 1 2 v0     1 2 v0 1 2 vt < min ; 1 5 qt ; ; 1 ≤ min supX xt

q
0g 5

ðC44Þ

where the strict inequality follows from ðC25Þ, and the last inequality uses the fact that vt11 ≤ vt for all t. Therefore, combining ðC43Þ and ðC44Þ, q
tg ≤ q
0g < qt ≤ 1 5 q
tb

for all t:

ðC45Þ

ðThe strict inequality means that the equilibrium satisfies property 2.Þ Given ðC41Þ and ðC45Þ, ðC23Þ reduces to

qt 5

8 1 > > > 1 2 vt > > > > > < xt g >q
t > > > > >1 > > : xt

if xt ≤ 1 2 vt

1 2 vt q
tg 1 2 vt 1 if ≤ xt ≤ g q
t q
tg 1 if g < xt : q
t if 1 2 vt < xt <

ðC46Þ

For all t, ðC43Þ, ðC44Þ, and vt11 ≤ vt imply q
tg ≤ q
0g ≤

  1 2 r v0 1 2 v0 1 2 v0 1 2 vt < min ; ; 1 ≤ ≤ r 1 2 v0 supX supX xt

where the strict inequality follows from ðC25Þ. Thus ðC46Þ ðand therefore ½C23Þ reduces to ðC34Þ, so ðE3Þ is satisfied. Notice that ðC28Þ is the same as ðC12Þ, and recall that ðC29Þ and ðC34Þ imply ðC42Þ, which is the same as ðC20Þ, with initial condition M
0b 5 ð1 2 v0 ÞM 0 1 B 0 given by ðC30Þ. Conditions ðC28Þ and ðC42Þ imply M
t11 5 maxðxt 1 vt ; 1ÞM
t ; which is the same as ð1Þ with qt given by ðC34Þ. Also, with ðC34Þ, the conditions in ðC24Þ reduce to the conditions in ðC31Þ. Hence ðE5Þ is satisfied. With m
it ðsÞ, mit11 ðsÞ, and bit11 defined as in ðC37Þ, ðC38Þ, and ðC40Þ, conditions ðE6Þ, ðE7Þ, and ðE8Þ are satisfied. From ðC45Þ, q
tg < qt for all t, so ðC15Þ with k 5 g reduces to ðC35Þ. Also, from ðC34Þ, qt ≤ 1 5 q
tb for all t, so ðC15Þ with k 5 b reduces to

moneyspots

8 b < ∈ ½0; M
t  b M
tb Bt11 :5 qt

183 if qt 5 1 if qt < 1:

ðC47Þ

With ðC34Þ, it is clear that ðC36Þ satisfies ðC47Þ, so I can conclude that ðC35Þ and 2
t , and ðC36Þ satisfy ðC15Þ. As defined in ðC39Þ, nit11 ðsÞ ≥ 0 for all s ∈ M

on

k2 nit11 5


k s ∈M t

2 it11

k k ðsÞ 5 M
tk 2 Mt11 5 qt Bt11 ;

where the last equality follows from ðC28Þ, ðC31Þ, ðC34Þ, ðC35Þ, and ðC36Þ. Similarly, ðC39Þ implies 2 nit11 5

on


t s ∈M

2 it11

ðsÞ 5 M
t 2 Mt11 5 qt Bt11 :

Hence ðE9Þ is satisfied. Given that ztg 1 ztb 5 z, the Euler equations ðC21Þ imply 0

zt 5 bEt ½L ðzt11 Þzt11 :

ðC48Þ

` ` 5 frz; ð1 2 rÞz; zgt50 , with z But then notice that the constant path fztg; ztb; zt gt50 defined by ðC27Þ, satisfies ðC21Þ and ðC48Þ. Thus, ðC32Þ and ðC33Þ satisfy ðE4Þ. In summary, for a given v0, the proposed allocations and prices constitute a monetary equilibrium. Since r is arbitrary, the construction characterizes a continuum of monetary equilibria, one for each r ∈ ðð1 1 D0 Þ21 ; 1Þ. Part 3: In the equilibrium described in part 2, Bt11 5 xt M
t > 0, and bonds are always held. The price of bonds satisfies 0 < qt ≤ 1 for all t ðwith qt < 1 if 1 2 vt < xt Þ, and

 it11 5 max

 xt 2 1; 0 : 1 2 vt

ðC49Þ

The real value of the money holdings brought into the decentralized market of period t 1 1 is

o

s ∈Mt11

g b ft11 ðsÞMt11 ðsÞ 5 fgt11 Mt11 1 fbt11 Mt11

 5 rz 1 max 1 2

 xt ; 0 ð1 2 rÞz > 0: 1 2 vt

Hence

Eo

ft11 ðsÞMt11 ðsÞmðdiÞ > 0

I s ∈Mt11

for all t, even if it > 0, as is the case whenever the realization of xt lies in the interval ð1 2 vt ; supX. QED

184

journal of political economy

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