DP2019-03

On the Real Determinacy andIndeterminacy of Stationary
E q u i l i b r i a i n M o n e t a r y M od e l s
Kazuya KAMIYA

Revised March 17, 2021

On the Real Determinacy and Indeterminacy of Stationary
Equilibria in Monetary Models
Kazuya Kamiya∗†
March 15, 2021

Abstract
It is known that stationary equilibria are indeterminate in some monetary models, especially inmoney search models with divisible money. However, most of the indeterminacy results are limited
to the case that money holdings distributions have ﬁnite supports. In the case of inﬁnite supports,
both determinacy and indeterminacy results are known. In this paper, using the Borsuk-Ulamtheorem in Banach Space, I investigate what determines the diﬀerences.
Keywords: Real Determinacy, Real Indeterminacy, Monetary Search Model, All-Pay Auction,
Divisible Money, Inﬁnite Dimensional Space, Borsuk-Ulam Theorem.
Journal of Economic Literature Classiﬁcation Number: C78, D51, D83, E40.

∗ RIEB, Kobe University. (E-mail: kkamiya at rieb.kobe-u.ac.jp)† I thank Hidetoshi Komiya, Mitsunori Noguchi, Yoshinao Sahashi, and Ken Urai for helpful comments. This research

is ﬁnancially supported by Grant-in-Aid for Scientiﬁc Research from JSPS (16K13349). Of course, any remaining errors
are my own.

11

Introduction
It is known that real indeterminacy of stationary equilibria arises in some monetary models, espe-

cially in money search models with divisible money. (See, for example, Green and Zhou (1998, 2002),
Kamiya and Shimizu (2006), Matsui and Shimizu (2005), and Zhou (1999).) However, most of the indeterminacy results are limited to the case that the supports of equilibrium money holdings distributions
are ﬁnite sets. The exceptions are Green and Zhou (1998) and Kamiya and Shimizu (2011). In theformer model, the money holdings distribution has a support {0, p, 2p, . . .} for some p > 0, and in thelatter model, it is a non-discrete money holdings distribution. On the other hand, Kamiya and Shimizu(2013) suggest that a dynamic all-pay auction model with ﬁat money could have a determinate equilibrium, where the support is an interval. The purpose of this paper is to investigate what determines thediﬀerences.
In the case that a money holdings distribution has a ﬁnite support {0, p, . . . , N p} for some p > 0
and some integer N > 0, Kamiya and Shimizu (2006) show that the condition for stationarity ofmoney holdings distribution always has a hidden identity (conservation law) in money search models.
Therefore, the number of equations in stationary condition is one less than the number of variables
due to the identity, and, applying the implicit function theorem, there is a continuum of stationary
equilibria. 1 More precisely, the variables are h0 , h1 , . . . , and hN , and strategies, where hn is a measureof agents with np amount of money. On the other hand, the conditions for stationary equilibria areIO= 0, and the Nash equilibrium condition, where fnO and fnI− fN
f0O − f0I = 0, f1O − f1I = 0, . . ., and fN

are the outﬂow of agents from np and the inﬂow of agents to np, respectively. The number of equations
in Nash equilibrium condition is typically equal to the number of variables in strategies. Thus, from theidentity, the number of equations is one less than the number of variables. That is, the dimension of therange is one less than that of the domain. Thus under some regularity condition the implicit function
theorem can be applied, and the set of stationary equilibria is one-dimensional.
In the case that a support B ⊂ R+ has an inﬁnite number of elements, the variable is a distribution
on B and is an element in an inﬁnite dimensional space. On the other hand, the condition for stationary
distribution cannot be expressed by a ﬁnite number of equations. Thus both the numbers of variables
and equations cannot be counted. In other words, the dimensions of the domain and range cannot
1 Lagos and Wright (2005) present a model with both Walrasian markets and search markets, and there is a uniquestationary equilibrium. In their model, the identity does not hold due to the Walrasian Market.

2

be directly comparable. In this paper, using the Borsuk-Ulam theorem in Banach space, I present a
method to compare two Banach spaces, and to apply the implicit function theorem in Banach spaces tomonetary models.
In order to compare the topological properties of two Euclidean spaces, the Borsuk-Ulam theorem
in a ﬁnite dimensional space can be applied. Let Rn and S n be an n-dimensional Euclidean space and an
n-dimensional unit sphere. The theorem says that, for a continuous function f : S n ⊂ Rn+1 → Rn , there
exists an x ∈ S n such that f (x) = f (−x). This immediately implies that Rn+1 is not homeomorphic to
Rn .2 Similarly, if an inﬁnite dimensional version of the Borsuk-Ulam theorem can be applied to inﬁnite
dimensional spaces X and Y and a function f : X → Y , then X and Y are topologically diﬀerent
spaces, and the dimension of X is in some sense larger than that of Y . Therefore, the implicit function
theorem in inﬁnite dimensional spaces could be applied to X, Y and f . Conversely, if the condition for
the Borsuk-Ulam theorem in inﬁnite dimensional space is not satisﬁed, then the stationary equilibria
could be determinate. Indeed, in the all-pay auction in Kamiya and Shimizu (2013), the condition for
the Borsuk-Ulam theorem is not satisﬁed, and the stationary equilibrium is determinate.
The plan of this paper is as follows. In Section 2, some mathematical tools are presented. Section 3
is the main part of this paper and is devoted to a general monetary model and a condition for the implicit
function theorem in Banach space. In Section 4, a model with a determinate stationary equilibrium ispresented, where the condition for the implicit function theorem is not satisﬁed. Section 5 concludes
the paper.

2

Implicit Function Theorem and Borsuk-Ulam Theorem in
Banach SpaceLet X and Y be Banach spaces, and L(X, Y ) be the set of continuous linear function from X to Y .

For C ∈ L(X, Y ), if there exists a D ∈ L(Y, X) such that CD = IY , where IY is the identity function
on Y , then D is called a right inverse of C.
Theorem 1. : The image of the right inverse of Y , denoted by D(Y ), is a closed linear subspace of X
and X = Ker C + D(Y ), where + denotes the direct sum of two linear spaces and Ker C is the kernel
2 If there exists a homeomorphism f : Rn+1 → Rn , then for all x, y ∈ S n such that x ̸=y, f (x) ̸= f (y) holds. However,there exists an x ∈ S n such that f (x) = f (−x). This is a contradiction.

3

of C.
Proof. 3 For x ∈ X, let x = (x − DC(x)) + DC(x). Then x − DC(x) ∈ Ker C. Let K = IX − DC. Then
K = L(X, X) and Ker K = D(Y ). Clearly, DC(X) ⊂ D(Y ). Moreover, for y ∈ Y , D(y) = DCD(y) andthus D(Y ) ⊂ DC(X). Thus D(Y ) is a closed linear space and X = K(X) + D(Y ) = Ker C + D(Y ).
Next, I present the implicit function theorem in Banach space.
Theorem 2. (Theorem 5.9 in Chapter 1 in Lang (1999)): Let X, Y , and Z be Banach spaces andf : X × Y → Z be a Fréchet diﬀerentiable function. Suppose f (x0 , y0 ) = 0 at (x0 , y0 ) ∈ X × Y
and y → df(x0 ,y0 ) (0, y) is isomorphism between Banach spaces Y and Z, where df(x0 ,y0 ) is the Fréchet
diﬀerential at (x0 , y0 ). Then there exist neighborhoods U of x0 and V of y0 , and a Fréchet diﬀerentiable
function g : U → V such thatg(x0 ) = y0 and ∀x ∈ U, f (x, g(x)) = 0.
Next, I present an inﬁnite dimensional version of the Borsuk-Ulam Theorem.
Deﬁnition 1. : f : X → Y is said to be an odd function if f (−x) = −f (x).
Deﬁnition 2. : f : X → Y is said to be a completely continuous function if for a weak convergent
sequence {xq }, {f (xq )} is norm convergent.
Theorem 3. (Gel′ man (2002)): Let α : X → Y be a surjective continuous linear function, and f :Sr (0) → Y be an odd completely continuous function, where Sr (0) is a sphere with radius r > 0 in X.
If dim(Ker α) ≥ 1, then the equation
α(x) = f (x)has a nonempty solution set N (α, f ) and the dimension of the set is larger than or equal to dim(Ker α)−1.
This theorem is a generalized version of the Borsuk-Ulam theorem. Indeed, for a function g :Sr (0) → Y , let f (x) = g(x) − g(−x). Then f is an odd function, and applying the above theorem there
exists an xλ ∈ Sr (0) such that λα(xλ ) = f (xλ ), where λ > 0. Then, if Sr (0) is sequentially compact,

3 This proof is based on Omori (1978).

4

then there exists a converging subsequence of {xλ }, and from the complete continuity the limit pointx∗ ∈ Sr (0) satisﬁes 0 = f (x∗ ) = g(x∗ ) − g(−x∗ ).
A linear function is clearly an odd function. Then the following theorem can be proved.
Theorem 4. (Gel′ man (2002)): Let α : X → Y be a surjective linear continuous function, and k :X → Y be a completely continuous linear function. If dim(Ker α) ≥ 1, then the function α + k has a
non-trivial kernel and dim(Ker α + k) ≥ dim(Ker α).
The following Theorem will be used in the following section.
Theorem 5. (Theorem 8.6.2. in Bogachev (2007)): Let (G, ρ) be a separable and complete metricspace, where ρ is a metric on a set G. Let Ω be the set of signed Borel measures on (G, ρ) with weak
convergence topology. Then the following two statements are equivalent.
1. Ω is sequentially compact.
2. Ω is uniformly tight and uniformly bounded.

3

A General Monetary Model and a Condition for Indeterminacy

3.1

A General Model
The set of agents is represented by the interval [0, 1]. The support of real money holdings distri-butions is a closed interval [0, m̄], where m̄ > 0. A real money holdings distribution is expressed as a∫
Borel probability measure φ on [0, m̄] satisfying dφ = 1. For a given money supply M > 0, nominal
money holding distribution is obtained by adjusting the support using p = ∫ M. For example, for a real
ηdφmoney holdings distribution φ expressed by (h0 , h1 , . . . , hN ), where hi is a measure of {ηi } ⊂ [0, m̄], thecorresponding nominal money holdings distribution is (h0 , h1 , . . . , hN ), where hi is a measure of {pηi }with p = ∫ M= ∑N Mη h .
ηdφ
i=0

i

i

Time is discrete and inﬁnite horizon, denoted by t = 1, 2, . . .. In each period, agent i ∈ [0, 1] choosesai ∈ A(ηi ), where A(ηi ) is the set of feasible strategies and ηi ∈ [0, m̄] is her money holding. The
temporal utility of agent i is g(ai , a−i , φ), where a = {ai }i∈[0,1] and ai ∈ A(ηi ). Let f O (a, φ) andf I (a, φ) are an outﬂow and an inﬂow of agents, respectively. For example, f O (a, φ)(C) (f I (a, φ)(C))
5

is the outﬂow (inﬂow) from (to) a Borel set C ⊂ [0, m̄]. Each agent maximizes the discounted sum of
temporal utilities with a discount factor β ∈ (0, 1).
Since an agent in an outﬂow at η ∈ [0, m̄] must be in an inﬂow at some η ′ ∈ [0, m̄],∫d(f O (a, φ) −f I (a, φ)) = 0 holds. Moreover, as shown in Kamiya and Shimizu (2006), if the total money holding∫
before trades is equal to that after trades even out of equilibria, then ηd(f O (a, φ) − f I (a, φ)) = 0
always holds, i.e., an identity. This identity typically holds in money search models and some dynamic
auction models with ﬁat money. Therefore, I make the following assumption.
Assumption 1. For any strategy a and any Borel probability measure φ,∫and ηd(f O (a, φ) − f I (a, φ)) = 0 hold.

∫d(f O (a, φ) − f I (a, φ)) = 0

Let E be the set of signed Borel measures on [0, m̄]. The domain of f O and f I can be extended
to E as follows. For a signed Borel measure φ, let |φ| = φ+ + φ− be the total variation of φ, whereφ+ and φ− are the positive and negative parts of φ. Let ∥φ∥ be the total variation norm of φ. First,
f O and f I can be extended to f˜O and f˜I on the space of signed Borel measures φ with total variation
norm one, i.e., ∥φ∥ = 1, as follows:f˜O (a, φ) = f O (a, |φ|) and f˜I (a, φ) = f I (a, |φ|).
f˜O and f˜I can be extended to fˆO and fˆI on the space of signed Borel measure E as follows:
(
)
(
)
φ
φ
fˆO (a, φ) = ∥φ∥f˜O a,
and fˆI (a.φ) = ∥φ∥f˜I a,
.
∥φ∥
∥φ∥
Deﬁne fˆO (a, 0) = fˆI (a, 0) = 0. Below, I use f O and f I instead of fˆO and fˆI . Note that, from theconstruction, Assumption 1 holds for all signed Borel measures.

∫
Let F be the product space of two sets, the set of signed Borel measure φ satisfying dφ = 0∫
and ηdφ = 0, and the set of real numbers R. That is, F is the set of values of f O (a, φ) − f I (a, φ)∫and 1 − dφ. Note that E is a Banach space with total variation norm, and so is F with the product∫topology of total variation norm and Euclidean norm on R. Let f (a, φ) = (f O (a, φ)−f I (a, φ), 1− dφ).
For a given a, a stationary distribution is a solution to f (a, φ) = (0, 0), where the ﬁrst 0 is the zeromeasure and the second zero is the real number 0.
The stationary equilibrium is deﬁned as follows.
Deﬁnition 3. Let ai : [0, m̄] → A, i ∈ [0, 1], and a = {ai }i∈[0,1] . A pair (a, φ) is said to be a stationary
equilibrium, where φ is a Borel probability measure on [0, m̄], if6

1. ∀i ∈ [0, 1], ai maximizes discounted sum of expected utilities for given a−i , φ, and her money
holding,
2. f O (a, φ) − f I (a, φ) is a zero measure, i.e., (f O (a, φ) − f I (a, φ))(C) = 0 for all Borel set C.
In a stationary equilibrium, if the discounted sum is not constant in money holdings, then it iscalled a monetary equilibrium. Below, I suppose that an equilibrium strategy a∗ is ﬁxed, and thus f O
and f I are functions of only a measure. I redeﬁne f (φ) = f (a∗ , φ).

3.2

A Condition for Implicit Function Theorem
In order to apply the Borsuk-Ulam theorem in Banach space, it is shown that there exists a surjective

linear continuous function α : E → F .
Lemma 1. There exists a surjective linear continuous function α : E → F . Moreover, the dimensionof the kernel of α is larger than or equal to one.
From the construction, F = E1 × R, where E1 is the linear subspace in E which consists of∫∫ν ∈ E satisfying dν = 0 and ηdν = 0.
Proof.

Let φ̄ be a measure with a support {0} and a mass h0 > 0 on {0}. Consider the product space ofE1 and the space spanned by the measure φ̄. Denote the product space by E2 . φ ∈ E can be expressedas φ = φ1 + cφ̄ + φ−2 , where φ1 ∈ E1 , c is a scalar, and φ−2 ∈ E \ E2 . Let α(φ) = (φ1 , c) ∈ F . Then α
is a surjective and continuous linear function.
Below, I show that E \ E2 is at least one dimensional. Let φ′ be a measure with a support {q}and a mass hq > 0 on {q}, where q > 0 is in [0, m̄]. Suppose φ′ is in E2 . Then φ′ = φ′1 + c′ φ̄ forsome φ′1 ∈ E1 and c′ . Then φ′1 has masses −c′ h0 and hq on {0} and {q}, respectively, and the support∫
is {0, q}. However, ηdφ′ = −0c′ h0 + qhq = qhq > 0 contradicts the deﬁnition of E1 and thereforeφ′ ∈ E \ E2 . Thus E \ E2 is at least one dimensional and the dimension of kernel of α is at least one.
Below, I show that the kernel of dfφ∗ , denoted by Ker dfφ∗ , is at least one-dimensional.
Lemma 2. Suppose dfφ∗ is a completely continuous function.
dimensional.

7

Then Ker dfφ∗ is at least one-

Proof. Let S be the set of signed Borel measures of which total variation norms are one. Since dfφ∗ isa completely continuous function, then, from Lemma ??, Theorem 4 can be applied and there exists a
nontrivial solution to(α + dfφ∗ )(φ) = 0.
Since λα, λ > 0, is a surjective linear continuous function, then there exists a nontrivial solution to(λα + dfφ∗ )(φ) = 0.
Since the solution is nontrivial, then from the linearity of λα+dfφ∗ there exists a solution φλ ∈ S. In ourenvironment, the support of measure is a compact set [0, m̄]. Thus, from Theorem 5, S is sequentiallycompact. Indeed, the set of signed Borel measures on [0, m̄] is uniformly tight and uniformly bounded.4
Let λ → 0. Then, from sequential compactness there exists a subsequence {φλq } weakly converging tosome φ̂ ∈ S. Then, from complete continuity and λq → 0, dfφ∗ (φ̂) = 0. Since φ̂ ̸= 0, the linear spacespanned by φ̂ is one dimensional and Ker dfφ∗ is at least one-dimensional.
If dfφ∗ is completely continuous and has a right inverse, then from Theorems 1 and 2 the set of
stationary equilibria is at least one-dimensional.
Theorem 6. Let U ⊂ E be an open set. Suppose f is a Fréchet diﬀerentiable function from U to F
and f (φ∗ ) = 0. Suppose dfφ∗ ∈ L(E, F ) is a completely continuous function and has a right inverse D.
Then E = Ker dfφ∗ + D(F ) and φ∗ = (x0 , y0 ), where x0 ∈ Ker dfφ∗ and y0 ∈ D(F ).5 Moreover, there
exist neighborhoods U1 of Ker dfφ∗ , U2 of D(F ), and V of F such that U1 + U2 ⊂ U and there exists a
unique Fréchet diﬀerentiable function g : Ker dfφ∗ → F such that(*) g(x0 ) = y0 , (x0 , g(x0 )) ∈ U , and f (x, g(x)) = 0 for all x ∈ U1 .
Proof. From Theorem 1, E = Ker dfφ∗ + D(F ) and φ∗ = (x0 , y0 ), where x0 ∈ Ker dfφ∗ and y0 ∈ D(F ).
First, I show that both Ker dfφ∗ and D(F ) are Banach spaces and the implicit function theorem canbe applied. Beside the completeness of the space, all the properties of Banach space are satisﬁed.
4 The set of signed Borel measures on [0, m̄] is called uniformly tight if for all ε > 0 there exists a compact set K suchεthat |φ|([0, m̄] \ Kε ) < ε for all signed Borel measure φ in the set, where |φ| is the total variation of φ. Clearly, Kε = [0, m̄]satisﬁes the condition.
5 The product and direct sum of ﬁnite number of linear spaces are the same. Therefore, I will use x + y and (x, y) for
the same object.

8

Consider a Cauchy sequence aq ∈ Ker dfφ∗ , q = 1, 2, . . . , where aq norm converges to some a ∈ E. Since
norm convergence implies weak convergence, then from the complete continuity of dfφ∗ , dfφ∗ (aq ) norm
converges to dfφ∗ (a). That is,
0 = lim ∥dfφ∗ (aq ) − dfφ∗ (a)∥ = ∥ − dfφ∗ (a)∥q→∞Therefore, a ∈ Ker dfφ∗ and thus Ker dfφ∗ is complete. Since E = Ker dfφ∗ + D(F ) is complete andD(F ) is closed from Theorem 1, then D(F ) is complete.
Next, D is clearly an isomorphism between F and D(F ) so that y → dfφ∗ (0, y) = df(x0 ,y0 ) (0, y) is also
an isomorphism between F and D(F ). Thus, applying Theorem 2, (*) holds.
From Lemma ??, Ker dfφ∗ is at least one-dimensional. Therefore, from Theorem ??, the set of
stationary equilibria is at least one-dimensional.
Theorem 7. Let U ⊂ E be an open set. Suppose f is a Fréchet diﬀerentiable function from U to F
and f (φ∗ ) = 0. Suppose dfφ∗ ∈ L(E, F ) is a completely continuous function and has a right inverse.
Then the set of stationary equilibria is at least one-dimensional.

4

A Model with a Determinate Equilibrium
Kamiya and Shimizu (2013) suggest that a dynamic all-pay auction model with ﬁat money has adeterminate stationary equilibrium. In the model, all bidders must pay regardless of whether they win
the prize. Although the identity exists, the stationary equilibrium is determinate. This is due to thefact that the support of money holdings distribution in the all-pay auction markets cannot be conﬁned
to a ﬁnite set and the outﬂow minus outﬂow is NOT a completely continuous function.
Below, I precisely explain a dynamic all-pay auction model with ﬁat money. In the model, there
is only one type of indivisible good. At the beginning of each period, an agent chooses either to be abuyer or a seller. If she becomes a seller, then she can produce one unit of good with cost c ≥ 0. Whenshe becomes a buyer, without knowing the other agents’ choices, she chooses a nonnegative bid pricewhich cannot exceeds her money holding. If she buys a good, then she obtains utility u > 0, where∫
βu > c. Let the measure of sellers be ms and the distribution of bid price be φb , where ms + dφb = 1.
∫
Suppose ms > dφb . Then all the buyers obtain the goods, all the sellers equally share the total bids
and randomly chosen sellers with measure 1 − ms produces goods, and the transaction price is the9lowest bid. Note that if the lowest bid does not exist, then the inﬁmum of the bids is the transaction∫
price. Suppose ms ≤ dφb . Then all the sellers produce goods and the transaction price is determined∫∞as the highest p satisfying ms ≤ p dφb . Note that if there is no mass of φb at the highest p, then∫∞ms = p dφ. Then the buyers with a bid price larger than or equal to p obtain goods. If there is amass at the highest p, then the buyers who bid larger than p obtain goods and the buyers who bid p are
randomly chosen and obtain goods. Note that, in the all-pay auction, the total money holdings before
trades is always equal to that after trades, i.e., the identity in the previous section holds.
As in the general model in the previous section, a stationary equilibrium is a pair of agents’ strategies
and a money holdings distribution satisfying that (i) each agent maximizes discounted sum of expectedutilities given the other agents’ strategies, her money holding, and a money holdings distribution, and(ii) the distribution is stationary. Below, I focus on monetary equilibria.
Suppose the strategy is symmetric across agents. Then, from the stationarity of distribution,
the transaction price p is uniquely determined, since, for a given strategy, it only depend on money
holdings distribution. Thus, in a stationary equilibrium, the support of money holdings distribution is{0, p, 2p, . . .} or some distribution which has essentially the same support, e.g., {ε, p + ε, 2p + ε, . . .},where ε has no value. For simplicity, I only consider {0, p, 2p, . . .}. Since each agent can choose either
to be a buyer or a seller, then an agent with η ≥ p always chooses to be a buyer. This is because if
she becomes a seller, she postpones the opportunity of consumption. Thus the support of stationary
equilibrium distribution is {0, p}. Let (h0 , 1 − h0 ) be the money holdings distribution, where h0 is ameasure of agents without money and 1 − h0 is a measure of agents with p amount of money. Suppose
all agents without money choose to be sellers and all agents with p choose to be buyers. Below, it will bechecked that the above choices are optimal strategies. From the deﬁnition of all-pay auction, the sellers’0)0)money holding become p(1−hafter trades. From stationarity, p(1−h= p holds and thus h0 = 12 .
h0h0Note that, from p(1 − h0 ) = M , p = 2M holds.
Below, the optimality of the above equilibrium strategy is rigorously checked. Let v(np), n =0, 1, 2, . . . be the value of np. Then the Bellman equation is as follows.
v(0)=max{−c + βv(p), βv(0)}

v(np)

=

max{u + βv((n − 1)p), −c + βv((n + 1)p), βv(np)}, n = 1, 2 . . . .

(1)

The strategy is (i) an agent without money chooses to be a seller and (ii) an agent with np, n ≥ 1, amountu−βcof money chooses to be a buyer and bids p. From the strategy, v(0) = βu−c1−β 2 , v(p) = 1−β 2 , . . . , v(np) =

10

∑n−1

k=1 β

k−1

u + β n−1 u−βc
1−β 2 , . . . hold. It can be easily checked that the strategy is optimal.

Below, it is shown that the outﬂow minus inﬂow is NOT completely continuous. To see this,M
consider a sequence of probability measure {φq }, where φq has masses 21 − 21q at 0 and 12 + 21q at 1 +
1 ,
q
2

2

respectively. Clearly, {φq } weakly converges to the measure of which masses are 12 at 0 and 12 at 2M .
On the other hand, the sequence of outﬂow minus inﬂow has masses 12 − 21q − ( 12 + 21q ) = − 22q at 0, 12 + 21q
M1
1
M
at 1 +, respectively. As q → ∞, the sequence of the outﬂow minus inﬂow
1 , and −( 2 − 2q ) at 1
− 1q
q
2

2

2

2

clearly converges weakly to zero measure, but it is not norm convergent. Indeed, for all q, one can takean open neighborhood of p = 2M such that the total variation measure is one. Thus the outﬂow minus
inﬂow is not completely continuous and the argument in the previous section cannot be applied.

5

Conclusion
In this paper, I investigate monetary models with ﬁat money and present a condition for indeter-

minacy of stationary equilibria. More precisely, the conditions for implicit function theorem in Banach
space are the complete continuity of Fréchet diﬀerential and the existence of the right inverse. Moreover, if the conditions are not satisﬁed, then the stationary equilibrium could be determinate. Indeed,in Section 4, I present a dynamic all-pay auction model with a determinate equilibrium.
References
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Money and Prices”, Journal of Economic Theory 81, pp. 251-271.
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Economy”, Econometrica 70, pp.929-969.
Kazuya Kamiya and Takashi Shimizu (2006), “Real Indeterminacy of Stationary Equilibria in Matching
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