Stationary Monetary Equilibria with Strictly Increasing Value Functions and Non-Discrete Money Holdings Distributions: An Indeterminacy Result

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1 CIRJE-F-615 Stationary Monetary Equilibria with Strictly Increasing Value Functions and Non-Discrete Money Holdings Distributions: An Indeterminacy Result Kazuya Kamiya University of Toyo Taashi Shimizu Kansai University March 2009 CIRJE Discussion Paers can be downloaded without charge from: htt:// Discussion Paers are a series of manuscrits in their draft form. They are not intended for circulation or distribution excet as indicated by the author. For that reason Discussion Paers may not be reroduced or distributed without the written consent of the author.

2 Stationary Monetary Equilibria with Strictly Increasing Value Functions and Non-Discrete Money Holdings Distributions: An Indeterminacy Result Kazuya Kamiya and Taashi Shimizu February 2009 Abstract In this aer, we resent a search model with divisible money in which there exists a continuum of monetary equilibria with strictly increasing continuous value functions and with non-discrete money holdings distributions. Keywords: Real Indeterminacy, Random Matching, Money Journal of Economic Literature Classification Number: C72, C78, D44, D51, D83, E40 1 Introduction Recently, real indeterminacy of stationary equilibria has been found in both secific and general search models with divisible money. (See, for examle, Green and Zhou [1] [2], Kamiya and Shimizu [3], Matsui and Shimizu [4], and Zhou [7].) However, all the indeterminacy results found so far are limited to the case that value functions are ste functions and money holdings distributions are discrete ones. In this aer, we show that real indeterminacy can occur even in the case of strictly increasing continuous value functions and non-discrete money holdings distributions. In his introductory aer of symosium volume of Journal of Economic Theory, Wallace [5] resents a conjecture that the indeterminacy result is not robust in the following sense: This research is financially suorted by Grant-in-Aid for Scientific Research from JSPS and MEXT. Of course, any remaining error is our own. Faculty of Economics, University of Toyo, Bunyo-u, Toyo JAPAN ( amiya@e.utoyo.ac.j) Faculty of Economics, Kansai University, Yamate-cho, Suita-shi, Osaa JAPAN ( tshimizu@icu.ansai-u.ac.j) 1

3 The multilicity is almost certainly not robust to dearting from the assumtion that the money is a fiat object. That, is if nominal holdings of the fiat object give utility (can be as aer weights or decoration or burned as fuel), then the ind of multilicity that has eole treating x units of a fiat asset as a new fiat object disaears. ([5].225) Following this conjecture, Wallace and Zhu [6] introduce the concet of commodity-money refinement. To ut it shortly, a stationary equilibrium satisfies the commodity-money refinement if it is also a limit of stationary equilibria as the consumtion utility of money goes to zero. Wallace and Zhu aly the concet to two secific models. In one model, the commodity-money refinement eliminates stationary equilibria with discrete money holdings distributions, and in the other model the commodity-money refinement eliminates stationary equilibria with value functions that are not strictly increasing, such as ste-functions. 1 One might thin that Wallace and Zhu s result verifies the above conjecture, for all the revious results of real indeterminacy in money search models are limited to the case of discrete money holdings distributions and ste value functions. 2 The urose of this note is to resent a money search model in which there is a continuum of stationary equilibria with non-discrete money holdings distribution and strictly increasing continuous value functions. Moreover, they satisfy the commodity-money refinement. In Section 2, we resent the model, and then in Section 3 we resent the indeterminacy result and show that the stationary equilibria indeed satisfies the commodity-money refinement in the sense of Wallace and Zhu. 2 The Model There is a continuum of agents with a mass of measure one. There are 3 tyes of agents with equal fractions and the same number of tyes of erfectly divisible goods. A tye i 1 agent can roduce tye i good. (We assume that a tye agent roduces tye 1 good.) The roduction technology of each agent is characterized by a fixed cost c > 0, zero 1 Note that their results also deend uon their definition of refinement. Zhou [8] defines the commoditymoney refinement in another way and shows that there is a continuum of robust stationary equilibria with discrete money holdings distributions and ste value functions. 2 Green and Zhou [2] construct non-stationary equilibria with non-discrete money holdings distributions, but the stationary equilibria in their model have discrete money holdings distributions. 2

4 marginal cost, and a caacity constraint q > 0, where c and q are common to all agents. More recisely, the cost function of each agent is exressed as: 0, if q = 0, C(q) = c, if 0 < q q,, if q < q. A tye i agent obtains utility only when she consumes tye i good and her utility function is exressed by a linear function, u(q) = aq, where a > 0 is given and q is the amount of good i. Note that a is common to all agents. Time is discrete. In each eriod, each agent first chooses either to be a consumer or a seller. Then airwise random matchings tae lace. Note that if an agent of tye i chooses to be a seller, then she cannot buy tye i good even when she meet a tye i 1 agent. If a tye i seller meets a tye i + 1 buyer, then the seller maes a tae-it-or-leave-it offer of (q s, s ) without nowing the artner s money holding, where q s is the maximum amount of tye i + 1 good she can sell and s is the rice of the good. Note that the seller nows the money holdings distribution of the economy. Finally, when (q s, s ) is offered, the tye i buyer chooses the amount of good q b q s he wants to consume. Let m 0 [0, 1] be the measure of agents without money and f : (0, ) R + be a density function of money holdings on (0, ). Of course, = (0, ) fdη must hold. Let M > 0 be the nominal stoc of fiat money and β (0, 1) be the discount factor. The conditions for a stationary equilibrium are (i) each agent maximizes the exected value of utility-streams, i.e., the Bellman equation is satisfied, (ii) the money holdings distribution of the economy is stationary, i.e., time-invariant, and (iii) the total amount of money the agents have is equal to M. We focus on stationary equilibria in which all agents with identical characteristics act similar and in which all of the tyes are symmetric. 3 The Results The following theorem is the main result of the resent aer. Theorem 1 Let d = a q c. Suose 3 2 < d 3. Then there exists a β (0, 1) such that, for any given β (β, 1), there exists a continuum of stationary equilibria in which (i) 3

5 the value functions are continuous, strictly increasing, and concave, and (ii) the money holdings distributions have a full suort in some closed interval with a nonemty interior. Proof: (I) We focus on the strategy satisfying the following: for some > 0, an agent without money always chooses to be a seller and an agent with money holding η > 0 always chooses to be a buyer, a seller always offers (, q), a buyer with money holding η > 0 consumes the following amount of her consumtion good: there exists a (η) such that, for given ( s, q s ), q b (η, s, q s ) = { min{η/ s, q s } if s (η), 0 if s > (η), (1) for some λ and σ, f is exressed by { 2λη + σ, for η (0, q], f(η) = 0, for η ( q, ]. (2) Note that, (η), λ, and σ will be determined as functions of m 0 later. (II) Next, we obtain a candidate for a value function V : R + R consistent with the above strategy. From the above strategy, V (η) for η (0, ) can be written as a function of V (0) as follows. First, for η (0, q], V (η) = m 0 (a η ) ( + βv (0) + ) βv (η) holds. Thus where A(m 0 ) = written as: V (η) = A(m 0 ) (a η ) + βv (0), for η (0, q], (3) m 0 ( m 0 )β. Note that A(m 0) < 1. Similarly, V (η) for η ( q, ) is V (η) = A(m 0 ) (a q + βv (η q)), for η ( q, ). (4) 4

6 Next, since an agent without money always chooses to be a seller, then V (0) is determined by V (0) = [ c + (0, q] ] βv (η) f(η) ( dη m ) 0 βv (0). (5) (III) Below, we focus on equilibria with V (0) = 0 and obtain (, λ, σ) as functions of m 0. 3 First, we decomose η 0 into an multile of q and a residual; that is, η = n q +ι, where n is a nonnegative integer and ι is a nonnegative real number less than q. Then, by (3) and (4), V (n q + ι) = aa(m [ 0) { q (βa(m 0 )) n q (1 βa(m 0 )) ι ]} 1 βa(m 0 ) (6) holds. On the other hand, by (2) and (5), ( ) c = aβa(m ( 0) 2 ηf(η)dη = aβa(m 0 ) 3 λ2 q ) 2 σ q2 (0, q] (7) holds. Below, we obtain (, λ, σ) as functions of m 0. First, = (0, ) fdη can be written as follows: = (0, q] fdη = λ 2 q 2 + σ q. (8) Since the total amount of money the agents have is equal to M, the following equation must be satisfied: M = (0, q] By (7), (8), (9), and d = a q c, we obtain ηfdη = 2 3 λ3 q σ2 q 2. (9) = MaβA(m 0) ( ) c, (10) λ = 3() 3 (2 βda(m 0 )) M 2 β 3 d 3 (A(m 0 )) 3, (11) σ = 2() 2 ( 3 + 2βdA(m 0 )) Mβ 2 d 2 (A(m 0 )) 2. (12) 3 If V (0) > 0, then an agent with a small amount of money does not choose to be a buyer. Indeed, by (3), lim η 0 V (η) = βv (0) < V (0) when V (0) > 0. 5

7 (IV) Next, we chec the otimality of the secified strategy. (i) The otimality of the strategy of an agent with money holding η > 0: First, we show that there exists a (η) in (1). If η (0, q], then by (6), holds. Thus if ( aq + βv (η s q) = aq 1 βa(m 0 ) ) s + aβa(m 0 ) η 1 βa(m 0 ) s 0 holds, then she clearly chooses the maximum amount she can buy, and otherwise she chooses q b = 0. Note that 1 βa(m 0 ) 0 holds, since βa(m 0 ) < 1. Let (η) =, for η (0, q]. (13) βa(m 0 ) Then, (1) is otimal for η (0, q]. Moreover, (η) clearly holds. Similar arguments aly to the case of η ( q, ). Next, we chec an incentive for an agent with η > 0 to become a buyer instead of becoming a seller and offering (, q ). By (1) and (13), for any > βa(m 0 ), no buyer accets such an offer on the equilibrium, and then the value is the same as that of an offer (, 0), where βa(m 0 ). Therefore, we can restrict our attention to (, q ) such that [ ] 0, and q [0, q]. By (1), the value of becoming a seller and offering (, q ) is βa(m 0 ) [ c + (0, q] ] ( βṽ (η, η ) f(η ) dη m ) 0 βv (η), where Ṽ (η, η ) = { V (η + η ), if η q, V (η + q ), if η > q. (14) On the other hand, when she becomes a buyer, the value is V (η). Thus the difference is [ c + β ( V (η, η ) V (η) ) ] f(η ) dη (1 β)v (η). (15) (0, q] 6

8 Below, we show V (η + η ) V (η) aa(m 0 ) η, for η (0, q]. (16) First, there exits a unique nonnegative integer n such that n q η < (n + 1) q. There are two cases: (a) η + η < (n + 1) q and (b) η + η (n + 1) q. In case (a), by (6) and βa(m 0 ) < 1, V (η + η ) V (η) = aa(m 0) 1 βa(m 0 ) aa(m 0) 1 βa(m 0 ) aa(m 0 ) (βa(m 0 )) n η aa(m 0 ) η. [ { q (βa(m 0 )) n q (1 βa(m 0 )) η + ]} η n q [ { q (βa(m 0 )) n q (1 βa(m 0 )) η n q ]} In case (b), by (6) and βa(m 0 ) < 1, V (η + η ) V (η) = aa(m [ 0) { q (βa(m 0 )) n+1 q (1 βa(m 0 )) η + ]} η (n + 1) q 1 βa(m 0 ) aa(m [ 0) { q (βa(m 0 )) n q (1 βa(m 0 )) η n q ]} 1 βa(m 0 ) [ = aa(m 0 ) (βa(m 0 )) n (1 βa(m 0 )) (n + 1) q + βa(m 0 ) η (1 βa(m 0)) η ] aa(m 0 ) (βa(m 0 )) n η aa(m 0 ) η. The fourth line is obtained by η (n + 1) q η. This comletes the roof of (16). (6), (14), and (16) imly Ṽ (η, η ) V (η) aa(m 0 ) η, for η (0, q]. Then, the first term of (15) is less than or equal to [ ] c + aβa(m 0 ) η f(η ) dη. (0, q] This is equal to zero by the first equality of (7), and thus (15) is non-ositive and she becomes a buyer. 7

9 (ii) The otimality of the strategy of an agent without money: By the construction, an agent without money is indifferent between a buyer and a seller. Thus she has an incentive to be a seller. attention to offers (, q ) such that value of offering (, q ) is [ c + [ 0, (0, q] As in the latter art of (i), we restrict our ] and q [0, q]. By (1) and (6), the βa(m 0 ) ] βṽ (0, η ) f(η ) dη, (17) where Ṽ is defined in (14). If q q, Ṽ (0, η ) = V (η ) for any η (0, q]. Then, (17) is the same for all q q, and therefore the offer (, q) is otimal. If q q, Ṽ (0, η )f(η )dη = V (η )f(η )dη + V ( q)f(η )dη (0, q] = (0, q ] (0, q ] (0, q] V (η )f(η )dη + Ṽ (0, q)f(η )dη, ( q, q] ( q, q] V (η )f(η )dη where the inequality is obtained by (6) and (14). Then, the offer (, q) is otimal. This comletes the roof of (IV). (V) Finally, we chec f(η) 0 for all η (0, q]. Since f is linear, it suffices to show f(0) 0 and f( q) 0. By (10), (11), and (12), and f(0) = σ = 2() 2 ( 3 + 2βdA(m 0 )) Mβ 2 d 2 (A(m 0 )) 2 f( q) = 2λ q + σ = 2() 2 (3 βda(m 0 )) Mβ 2 d 2 (A(m 0 )) 2 hold. A sufficient condition for f(0) 0 and f( q) 0 is clearly 3 2 βda(m 0) 3. By the assumtion d 3, βda(m 0 ) 3 is always satisfied. 3 2 βda(m 0) is equivalent to m 0 It is easily verified that 3(1 β) β(2d 3). (18) 8

10 3 Setting β = 3( 1)+2d, we can show that for any β (β, 1) there exists a continuum of m 0 satisfying (18) and m 0 (0, 1). Indeed, β < 1 follows from the assumtion d > 3 2, and 1 > 3(1 β) β(2d 3) follows from β > β. Note that the stationarity of money holdings distribution clearly holds. This concludes the roof. We show that any equilibrium in Theorem 1 satisfies the commodity-money refinement defined by Wallace and Zhu [6]. Wallace and Zhu assume that agents receive utility εγ(η) by consuming η amount of money, where ε 0 and γ : R + R + is differentiable, bounded, strictly increasing, and concave, and satisfies γ(0) = 0 and γ (0) finite. A stationary equilibrium in the case of fiat money (i.e., ε = 0) satisfies the commodity-money refinement if and only if it is a limit of some sequence of stationary equilibria as ε 0, i.e., a limit of commodity money equilibria. By (6), in any stationary equilibrium described in Theorem 1, the value function is ositively linear with resect to money holding η [0, q]. This imlies that, even in the case of commodity money, if ε is sufficiently small, then agents never consume the money. Therefore, the set of equilibria in the case of small ε > 0 is the same as that in the case of ε = 0. Corollary 1 Any equilibrium in Theorem 1 satisfies the commodity-money refinement. This imlies that there is a continuum of stationary equilibria that satisfies the commoditymoney refinement. We define the welfare as the average of values. Then by (6), (7), (8), and (9), we obtain W = m 0 V (0) + V (η)f(η)dη = c β (). In other words, the smaller m 0 is, the higher the welfare level is, as long as V is an equilibrium value function. Then m 0 = 3(1 β) β(2d 3) yields the highest welfare level among this class of equilibria. This imlies the above indeterminacy result is real; there is a continuum of Pareto-ranable stationary equilibria. (0, q] 9

11 References [1] Edward J. Green and Ruilin Zhou. A rudimentary random-matching model with divisible money and rices. Journal of Economic Theory, 81(2): , [2] Edward J. Green and Ruilin Zhou. Dynamic monetary equilibrium in a random matching economy. Econometrica, 70(3): , [3] Kazuya Kamiya and Taashi Shimizu. Real indeterminacy of stationary equilibria in matching models with divisible money. Journal of Mathematical Economics, 42: , [4] Aihio Matsui and Taashi Shimizu. A theory of money and maretlaces. International Economic Review, 46(1):35 59, [5] Neil Wallace. Introduction to modeling money and studying monetary olicy. Journal of Economic Theory, 81: , [6] Neil Wallace and Tao Zhu. A commodity-money refinement in matching models. Journal of Economic Theory, 117: , [7] Ruilin Zhou. Individual and aggregate real balances in a random-matching model. International Economic Review, 40(4): , [8] Ruilin Zhou. Does commodity money eliminate the indeterminacy of equilibrium. Journal of Economic Theory, 110(1): ,