Indeterminacy in Search Theory of Money: Bilateral vs. Multilateral Trades So Kubota Princeton August 16, 2014
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Introduction I
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Topic: Real Indeterminacy of Stationary Equilibrium in money search I Real: consumption, production, etc are also indeterminate. I Stationary: There is a continuum of steady states. Found by Green and Zhou(1998) and others I Divisible money. I No trick to remove heterogeneity in money holding such as Lagos-Wright(2005). I Analytical result I Indeterminacy in discrete m.h.d 2 / 21
Reason? I
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Question: Why do these model have the indeterminacy? Existing conjectures: I I I I
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nominal nature of money? indivisible good assumption in GZ? discrete money holding distribution in GZ? Kamiya and Shimizu(2013), game-theortical concepts?
These conjectures have a kind of counter examples.
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Literature I
Basic models: Green Zhou(1998, 2002), Zhou(1999), Kamiya Shimizu(2006,2007)
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Two prices: Kamiya Sato(2004)
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Commodity Money: Zhou(2003)
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Divisible Good: Ishihara(2010)
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Continuous distribution: Kamiya Shimizu(2011)
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Applications: Matsui Shimizu(2005), Sugaya(2008), Kamiya Shimizu(2013)
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LW: Jean Rabinovich Wright(2010), Dutu Huangfu Julien(2011) 4 / 21
What I do I
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This paper proposes a new conjecture: one-to-one matching is the reason. I make a tractable price-posting model with divisible money. Two matching assumptions 1. One-to-one random matching ⇒ indeterminacy 2. Multilateral matching ⇒ uniqueness
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Model
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a continuum of agents with index i ∈ [0, 1]. Time is discrete and infinite. two types of goods I non-storable divisible consumption good I divisible money with fixed supply M storable and no intrinsic value
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Preference I
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An agent can be either a buyer or a seller each period. discount factor β Buyer: linear utility function: uq Seller: cost function with a production upper bound 1.
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Market I
Each period is divided into four stages
1. Each agent chooses to be a buyer or a seller. 2. Each seller posts a unit price of consumption good p. Price p offerers form Sub-market p. 3. Given offered price distribution, each buyer chooses a sub-market. 4. Matching and trade in each sub-market. I Random search with one-to-one matching I Multilateral matching
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Case 1: One-to-one matching I
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Price p sub-market: set of sellers Sp and buyers Bp measures: h(Sp ) and h(Bp ) respectively random matching with min{h(Sp ), h(Bp )}. After a successful matching, the buyer feely decides the quantity so as to maximize her discounted utility. q ∗ = arg max uq+βV (m−pq) s.t.pq ≤ m, q ≤ 1 q∈R+
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Equilibrium Concept I I
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Stationary monetary equilibrium Strategy for four stages: x(m) = (x1 (m), . . . , x4 (m)). dist of stra. Ω(x), dist of money λ(m), V (m) 1. λ is stationary under Ω 2. V is obtained by Ω 3. Given λ and V , Ω consists subgame perfect equilibrium in Stage 1, . . . , 4, and Ω is stationary 4. V (m) is increasing.
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Equilibrium with one-to-one matching 1. Buyers spend money as much as they can I
Today’s MU u > Tomorrow’s MU βu
2. Single price equilibrium p ∗ I
Bertrand competition among sellers lower price → higher probablity to meet
3. Stationary money holding distribution is degenerated at 0 and p ∗ I
p ∗ → spend all → 0 → sell → p ∗
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Equilibrium with one-to-one matching I
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Stationary money holding distribution is degenerated at two points: 0 and M/H1 . H1 buyers and H1 in H0 sellers meet and trade. H0 and H1 are indeterminate! amount of trade differs ⇒ real indeterminacy
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Case 2: Multilateral matching I
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All sellers and buyers are allowed to trade given posted unit price p. h(Sp ) sellers and h(Bp ) buyers Each buyer i ∈ Bp offers amount of quantity qi A market maker calculates∫two numbers I Total demand: Q p = Bp qi di. I Total supply: h(S ). p Each seller can produce at most 1 unit
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Case 2: Multilateral matching I I I
All agents can trade. Trade volume is adjusted. Excess supply, h(Sp ) > Qp . Seller Buyer i Qp qi h(Sp )
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Excess demand, h(Sp ) < Qp . Seller Buyer i h(Sp ) 1 qi Qp 14 / 21
Equilibrium with multilateral matching
1. Buyers spend money as much as they can 2. Single price equilibrium p ∗ I
Bertrand competition among sellers zero marginal cost! strong incentive to sell more lower price → sell more good
3. stationary money holding distribution is degenerated at 0 and p ∗
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Equilibrium with multilateral matching I I I I
Same two states distribution. H1 buyers and H0 sellers meet and trade. Sellers receive M money and distribute equally. If H1 6= H0 , the distribution is not stationary!
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Equilibrium with multilateral matching I
The unique steady state equilibrium is H1 = H0 = 1/2
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Intuition I
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Suppose money holding distribution is degenerated at N points. (N = 2 in my model) Need to determine N − 1 populations at each state N − 1 equations for stationarity Inflow(i) = Outflow(i) i = 1, . . . , N − 1
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One-to-one matching introduces additional structure Flow(i → j) = Flow(j → i) i, j = 1, . . . , N
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It makes additional identity in the system of equations. 18 / 21
Conclusion I
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This paper proposes a new conjecture: if one-to-one matching assumption is violated, indeterminacy vanish. One-to-one matching makes an identity in the system of equations to determine money holding distribution. The conjecture seems right even in Walrasian market. I make a cash-in-advance model with a kind of one-to-one matching restriction, then it shows the similar indeterminacy. More general results remain for future research. 19 / 21
Multilateral matching I
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Each buyer has incentive to overclaim in excess demand case. Qp qi · < qi . h(Sp ) The buyer can get more if she offers q˜i > qi . Equilibrium with excess demand does not exist because ∀i, qi → ∞, which do not consista Nash equilibirum. Excess demand is allowed off path. I assume a special rule to distribute good. back Buyer’s optimal behavior is to choose a lowest price market and overclaim ⇒ All sellers choose the single price.
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Cyclical equilibrium? I
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Yes, it exists. But H0 = H1 = 1/2 is the only robust distribution to a small perturbation. Stage 0: ε agents are randomly selected. They must stay autarky. MHD: H0 and H1 . H0 ⇒ εH0 + (1 − ε)H1 ⇒ ε[εH0 +(1−ε)H1 ]+(1−ε)[εH1 +(1−ε)H0 ] ⇒ · · ·
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It converges to H0 = 1/2. The indeterminacy with one-to-one matching is robust to the perturbation. 21 / 21