On the Indeterminacy of Monetary Equilibria
Kazuya Kamiya
Kobe University

2 / 50Indeterminacy of Monetary Equilibria

Money has value as a medium of exchange. (Iwai, Kiyotaki andWright )3 / 50

CAAB

In their models, money is indivisble. More precisely, each agent canhave just one unit of money.

4 / 50Indeterminacy of Monetary Equilibria

There are a finite number of stationary equilibria.

5 / 50Indeterminacy of Monetary Equilibria

I In a special random matching model with divisible money,

Green and Zhou (1998, 2002) show that equilibria areindeterminate, i.e., the set of equilibria is a continuum. (Realindeterminacy!)“It is an open question whether this indeterminacy
reflects a fundamental fact about random matchingmodels.” (Green and Zhou (2002).)6 / 50Indeterminacy of Monetary Equilibria

I Kamiya and Shimizu (2006) show that it is an intrinsic

property of random matching models. That is in any random
matching model with divisible money, stationary equilibria are(generically) indeterminate.

7 / 50Indeterminacy of Monetary Equilibria

In Iwai, Kiyotaki, and Wright’s models, if everyone believes thatmoney has value, it indeed has value in equilibria.
In Green, Zhou, Kamiya, and Shimizu’s models, if everyone
believes that money has big value, it indeed has big value inequilibria, and if everyone believes that money has small value, it
indeed has small value in equilibria.
Note that indeterminacy occurs in matching models withdivisible modey.
Even when money is divisible, indeterminacy does not occur incentralized Walrasian markets with cash-in-advanceconstraints.

8 / 50CAABCAAB

Indeterminacy of Monetary Equilibria

The logic of indeterminacy:

M > 0: The amount of fiat money.
p > 0: a price of good

9 / 50Indeterminacy of Monetary Equilibria

Example 1:{0, p , 2p }: the set of money holdings{A, B , C , D }:the set of agents. (In the next example, I consider
the case of a continuum of agentsj

A is p
I The money holding of B is 2p
I The money holding of C is 0
I The money holding of D is 2p

I The money holding of10 / 50

Indeterminacy of Monetary Equilibria

At time t ,1. (i) A meets B , and (ii) C meets D

2. Then (i) B buys goods from A (ii) D buys goods from C(the price is p .)A: p → 2p
I The money holding of B : 2p → p
I The money holding of C : 0 → p
I The money holding of D : 2p → p
I The money holding of

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Indeterminacy of Monetary Equilibria

On : The set of agents who have np before the trades.
In : The set of agents who have np after the trades.

A ∈ O1 , A ∈ I 2I B ∈ O2 , B ∈ I1I C ∈ O0 , C ∈ I1I D ∈ O2 , D ∈ I1
I

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Indeterminacy of Monetary EquilibriaThen(∗)2X

n =0#On =2X

n =0#In(identityjholds.

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Indeterminacy of Monetary Equilibria

Moreover(i) B buys goods from A →The money holding of A before the trade+
The money holding of B before the trade=
The money holding of A after the trade+
The money holding of B after the trade

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Indeterminacy of Monetary Equilibria(ii) D buys goods from C →The money holding of C before the trade+
The money holding of D before the trade=The money holding of C after the trade+The money holding of D after the trade

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Indeterminacy of Monetary EquilibriaThen(∗∗)

2
X

n =0

np#On =2X

n =0

np#In

(identity)

p#O1 + 2p#O2 = p#I1 + 2p#I2

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Indeterminacy of Monetary Equilibria

In a stationary equilibrium, the money holdings distribution should
be stationary.
The stationarity condition:#I0 = #O0
#I1 = #O1
#I2 = #O2

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Indeterminacy of Monetary Equilibria

By (∗), #I1 = #O1 and #I2 = #O2 imply #I0 = #O0 .
By (∗∗), #I2 = #O2 implies #I1 = #O1 .
Thus #I2 = #O2 implies #I1 = #O1 and #I0 = #O0 .

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Indeterminacy of Monetary Equilibria

Example 2:{0, p , 2p }: The set of money holdings[0, 1]: The set of agents
On : The measure of agents with np before trade.
In : The measure of agents with np after trade.

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Indeterminacy of Monetary EquilibriaThen(∗)

2X

n =0(∗∗)

2X

n =0On =npOn =2X

n =0In (identity)

2Xn =0npIn (identity)

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Indeterminacy of Monetary Equilibria

I0 = O0
I1 = O1
I2 = O2
By (∗), I1 = O1 and I2 = O2 imply I0 = O0 .
By (∗∗), I2 = O2 implies I1 = O1 .
Then I2 = O2 implies I1 = O1 and I0 = O0 .

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Indeterminacy of Monetary Equilibria

hn : The measure of agents with np
h = (h0 , h1 , h2 )Suppose On and In are functions of h .
The stationarity condition:

I0 = O0
I1 = O1
I2 = O2h0 + h1 + h2 = 1

The number of independent equations=2
The number of variables=3
The degree of freedom =122 / 50

Indeterminacy of Monetary Equilibria
The condition for a stationary equilibrium:

I0 = O0
I1 = O1
I2 = O2h0 + h1 + h2 = 12X

n =0pnhn = M(The other equations, e.g. Bellman equations)
In the other equations, the number of equations is typically equal
to the number of variables.
→ the degree of freedom is equal to one.
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Indeterminacy of Monetary Equilibria

How to understand the indeterminacy?
The general consensus of indeterminacy is either
1. due to the absence of some important equation, or
2. equilibria in the real world economy are intrinsically fragile.
Thus we need a policy which induces a determinate andefficient equilibrium.
3. A specific equilibrium is selected as a focal point.
(Experiment)

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Indeterminacy of Monetary Equilibria

The first approach: Lagos and Wright (2005)
Some goods are traded in centralized Walrasian markets.
The second approach: Kamiya and Shimizu (2007)
A certain tax-subsidy scheme induces a determinate and efficientequilibrium.

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Literature

I Duffy and Ochs (1999, 2002): the 1st gereneration model,i.e., money is indivisible
I Duffy and Puzzello (2014): Lagos and Wright model )degenerate money holdings distributions
I This paper: a continuum of non-degenerate money holdings

distributions

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Results

I In some treatment, there is a tendency to converge to themost efficient stationary equilibrium
I However, as a whole, there is some degree of indeterminacy
I There are systematic deviations from our target equilibria
IIMany subjects avoid spending all their money holdings
Some subjects become inactive when they are sellers as thesession lasts longer

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Environment

I Baseline model = a variant of Zhou (1999)’s model
I Time:t = 1, 2, . . .

I Each agent can produce one unit of goods in each period
I She cannot consume the good she produces by herself and canconsume the goods the other agents produce
I Goods: indivisible and perishable

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Money and Matching

I Money is divisible and each agent can hold any amount ofmoney
I Each agent can observe the current money holdingsdistribution in the beginning of each period
I Pairwise random matchings take place in each period
I In each matching, one agent becomes a seller and the otherbecomes a buyer (random assignment)
I The bargaining protocol: the seller’s take-it-or-leave-it offer
I The seller cannot observe the buyer’s money holding

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Parametersu : the utility of consumption
I c : the cost of production
I u > c > 0I

I In the end of each period, the economy ends with probability

1 − δ, while it goes to the next period with probability δI Agents do not discount future payoffs

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Single Price Equilibrium

We focus on single price equilibria (SPE) as follows:

I : total number of agents
I In : number of agents holding np amount of money
I (I0 /I , I1 /I ): stationary money holdings distribution
I p is determined by M = pI1 where M is the nominal stock ofImoney

p if her current money holding is η < p,and otherwise she offers a price that cannot be accepted byany buyer

I A seller offers a priceI A buyer accept a price offerη≥pp whenever her money holding is

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Single Price Equilibrium (cont’d)

In a nutshell, on the equilibrium path of SPE with p ,I transaction only occurs between a seller with 0 and a buyer

with p ,I # of potential buyers = # of money holders, andI an individual money holding stochastically alternates between

0 and p32 / 50

Multiple SPEs

II =6I δ = 0.9

c = 10I M = 600
I

1. u = 14⇒ SPE: (p , # of money holders) = (200,3), (300,2), (600,1)
2. u = 20⇒ SPE: (p , # of money holders) = (150,4), (200,3)

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Variation: SPE with Residuals

I Suppose (200, 200, 200, 0, 0, 0) andp = 200 constitute anSPE

p = 180 also consitute an
SPE as long as the discount factor is not so large

I (190, 190, 190, 10, 10, 10) andI Residual=10: a small portion of money that is not used intransaction and has no value by itself

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Experimental Design

I Each session consists of several sequences
I Each sequence consists of an indefinite number of periods
I The experiments were conducted at Kansai University on

January 2015–July 2016
I In each session, 24 or 18 subjects interacted through z-Treesoftware (Fischbacher 2007)
I Total points = 300 (showup fee) + points acquired in thesession
I 1 point = 10 JPY

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cycle0.

cycle.

cycle.

one hour

cycle

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end

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Summary of Experimental Sessions

Session No., Treatment
1: u=14, dis200, ShowPrice=02: u=14, dis200, ShowPrice=03: u=14, dis100, ShowPrice=04: u=14, dis100, ShowPrice=0
5: u=14, dis300, ShowPrice=0
6: u=14, dis300, ShowPrice=0
7: u=14, dis100, ShowPrice=08: u=14, dis200, ShowPrice=0
9: u=14, dis300, ShowPrice=0
10: u=14, dis200, ShowPrice=1
11: u=14, dis300, ShowPrice=1
12: u=14, dis300, ShowPrice=1
13: u=14, dis100, ShowPrice=1
14: u=14, dis100, ShowPrice=1
15: u=14, dis200, ShowPrice=1
16: u=20, dis200, ShowPrice=1
17: u=20, dis200, ShowPrice=1
18: u=20, dis100, ShowPrice=1
19: u=20, dis100, ShowPrice=1
20: u=20, dis150, ShowPrice=1
21: u=20, dis150, ShowPrice=1

Subjects
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×4
6×3
6×4
6×3
6×4
6×4
6×4

Total Periods
447645
4850
55
55
5355
4955
5453
57
49
4464464848
66Sequences
24981078727107466482467Ave. Duration
22195657.9
6.9
7.6
27.5
75.5
7.7
13.25
9.5
8.2
11823
1289.436 / 50

Convergence of Offer Price

I Samples: SequenceLength ≥ 21
I Data: PeriodInSequence ≥ 6IPt :average serious offer price within the group
Serious offer=offering a price that can be accepted by somemember
I No-sale offer=offering a price that cannot be accepted by any
member
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Convergence of Offer Price (cont’d)

Pt − Pt −1 = α + βPt −1 + εt
I Dicky-Fuller test: H0 : β = 0I

I Non-convergence: groups in which the null hypothesis is notrejected with significance level of 10%
I LimitPrice: estimated value of − αβ
I LB: lower bound of 95% confidence interval of LimitPrice+ cautious proxy of limit price
I Rich: average number of subjects whose money holdings areabove LB within the last 5 periods+ number of hypothetical buyers

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Convergence of Offer Price (cont’d)

I Residual: {M–LimitPrice*Rich } / 6
I Inconsistent groups: Residual ≥ LimitPriceI Class-N: convergent and consistent groups whose Rich is theclosest to N
+ groups converging to SPE with N buyers

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Result 2: Residuals

All convergent and consistent groups have a significant amount ofresiduals
MeanMinMax

Residual
46.86
15.57
77.15

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Random Effects, Probit Analysis on Buyer’s Acceptance

PeriodOffer Price
Money Holding

(1)

(2)

(3)

(4)

0.0398***(0.00242)-0.0155***(0.000668)
0.00776***(0.000544)

0.0365***(0.00228)

0.0385***(0.00310)-0.0155***(0.000867)
0.00806***(0.000704)

0.0354***(0.00295)

Remaining Amount-0.00120(0.000813)
3.076***(0.197)

Constant

0.487***(0.0802)-0.820***(0.0825)-0.0598*(0.0336)0.737***(0.239)-0.000451(0.00106)
2.837***(0.251)-0.0685**(0.0343)-0.415*(0.248)Observations
Standard errors in parentheses*** p < 0.01, ** p < 0.05, * p < 0.1

4,349

4,322

2,523

2,511

Remaining ProportionHolt-Laury Score

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. Number of Hypothetical Buyers: u = 14.

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Kazuya Kamiya, Hajime Kobayashi, Tatsuhiro Shichijo, and Takashi
Equilibrium
Shimizu Selection in Monetary Search Models: An ExperimeResult 4: Most Efficient Equlibrium Is Most Likely in u = 14

In the case of u = 14, the sequences of offer prices are most likely
to converge to the most efficient stationary equilibrium. most likely
to converge to43 / 50

. Number of Hypothetical Buyers: u = 20.

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Kazuya Kamiya, Hajime Kobayashi, Tatsuhiro Shichijo, and TakashiEquilibrium
Shimizu Selection in Monetary Search Models: An ExperimeResult 5: More Likely to Converge in u = 20

The frequency of non-convergent groups is much less in the case ofu = 20 than in the case of u = 14

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Interpretation of Result 5:Higher Buyer’s Acceptance Rate in u = 20

u=14
u=20

Acceptance Rate
0.34
0.41

I Wilcoxon-Mann-Whitney test: the null hypothesis is rejected(P=0.0000)
I Conjecture: More acceptance rate ⇒ more likely to converge

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Result 6: Fewer Buyers in u = 20

In the case of u = 20, the number of hypothetical buyers (Rich)is less than that in SPEs

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Interpretation of Result 6: Inactive Sellers

I InactiveSellerRate: the rate of no-sale offers among all offers

when seller’s current money holdings are less than LB within
the last 10 periods
I InactiveSellerRate is above 0.4 among 5 out of 7 samples

with fewer buyers than the theory predicts

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Random Effects, Probit Analysis on Seller’s No-Sale Offer

I Samples: sellers whose money holdings are the least within

the groups they are belonging to(1)

(2)

(3)

0.0470***(0.00268)

0.0465***(0.00568)
0.00226(0.00353)

0.0494***(0.00376)

Constant-3.441***(0.182)-3.762***(1.142)0.00458(0.103)-3.983***(0.728)Observations
Standard errors in parentheses*** p < 0.01, ** p < 0.05, * p < 0.1

4,073

1,386

2,102

PeriodAcquiredPoints
Holt-Laury score

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Conclusion

I We investigate the convergence of sellers’ offer prices in amoentary serach model in which there are multiple stationary
monetary equilibria
I In some treatment, there is a tendency to converge to themost efficient equilibrium
I However, as a whole, there is some degree of indeterminacy
I There are systematic deviations from typical SPEs

Many subjects avoid spending all their money holdings andkeep some residuals with them
I Some subjects become inactive when they are assigned theroles of sellers as the session lasts longerI50 / 50

Introduction
The Model
Theoretical Result
Experimental Design
Experiment Results. ReferenceJohn Duﬀy and Jack Ochs (1999) “Emergence of Money as aMedium of Exchange: An Experimental Study” AmericanEconomic Review 89(4): 847-877
John Duﬀy and Jack Ochs (1999) “Intrinsically Worthless
Objects as Media of Exchange: Experimental Evidence”
International Economic Review 43(3): 637-673
John Duﬀy and Daniela Puzzello (2014) “Gift Exchange
versus Monetary Exchange: Theory and Evidence” American
Economic Review 104(6): 1735-1776
Akihiko Matsui and Takashi Shimizu (2005) “A Theory ofMoney and Marketplaces” International Economic
Review 46(1): 35-59
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Kazuya Kamiya, Hajime Kobayashi, Tastuhiro Shichijo, and Takashi
Equilibrium
Shimizu Selection in Monetary Search Models: An ExperimeIntroduction
The Model
Theoretical Result
Experimental Design
Experiment Results. Reference (cont’d)
Kazuya Kamiya and Takashi Shimizu (2006) “RealIndeterminacy of Stationary Equilibria in Matching Modelswith Divisible Money” Journal of Mathematical Economics
42(4-5): 594-617
Kazuya Kamiya and Takashi Shimizu (2007) “On the Role ofTax-Subsidy Scheme in Money Search Models” International
Economic Review 48(2): 575-606
Ricardo Lagos and Randall Wright (2005) “A Unified
Framework for Monetary Theory and Policy Analysis” Journalof Political Economy 113(3): 463-484
Zhou (1999) “Individual and Aggregate Real Balances in aRandom-Matching Model” International Economic Review
40(4) 1009-1038
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Kazuya Kamiya, Hajime Kobayashi, Tastuhiro Shichijo, and Takashi
Equilibrium
Shimizu Selection in Monetary Search Models: An Experime