Stochastic multiplier in the (S, s) economy
Naoto Yagi ∗†This Version: December 8, 2015

Abstract

This paper proposes a new multiplier process model with discrete inventory adjustment and input-output structure. The (S, s)inventory policy and a regular network are employed. We deﬁne
the multiplier process as a branching and derive a statistics of themultiplier in the statistical steady state. We show that convergenceand termination of the multiplier process are diﬀerent matters ina statistical model. We prove that the suﬃcient condition for convergence of expected multiplier is the same as the Brauer–Solow
condition, which is a suﬃcient condition for a linear input-output
model. We also prove that the necessary and suﬃcient condition
for termination of a multiplier process is weaker than the Hawkins–
Simon condition, which is a necessary and suﬃcient condition forthe linear model. We examine the relation between the rate of return and the multiplier. We prove that the multiplier is ﬁnite andits probability distribution function decays exponentially if the rateof return is suﬃciently large, whereas the multiplier asymptoticallydiverges and the probability distribution function is asymptoticallya power law if the rate of return is close to zero. This indicates therelation between low proﬁt rate and high output volatility.
JEL Classiﬁcation: C60, E32
Keywords: Multiplier, (S, s) Economy, Inventory Dynamics, Inputoutput Structure, Endogenous Business Cycle

1

Introduction

Inventory adjustment has been assumed to play an important role in multiplierprocesses ever since the concept was introduced by Richard Kahn and John Maynard Keynes in the 1930s. During a multiplier process, the aggregate demandis higher than the aggregate output and the total inventory in the economy
continues to fall. To prevent the total inventory from falling to an undesirably
low level, the ﬁrm must produce some extra output and restore its inventory∗†

Department of Economics and Business Administration, Niigata Sangyo University.
E-mail : yagi@econ.nsu.ac.jp

1

levels during the multiplier process. The ﬁrm’s decision on how and when to
do so would aﬀect the economy’s reaction to disequilibrium and its propagation
mechanism. There has been considerable research on linear and continuous adjustment models. Nevertheless, when it comes to the discrete adjustment model,
it is still unclear as to how the inventory adjustment process moves the economy
to a new equilibrium after the multiplier process.
In inventory theory, the (S, s) policy is a well-known adjustment eﬀective insituations involving non-convex adjustment costs. Arrow, Harris, and Marschak(1951) introduced the (S, s) form of inventory policy, while a general proof ofthe optimality of these policies was provided by Scarf (1960). We describe thecharacteristics of the (S, s) policy below. The ﬁrm establishes a lower stock points and an upper stock point S. If the inventory level is above the lower point, theﬁrm satisﬁes the requirement of inventory and no production takes place. If theinventory level falls to or below the point s, the ﬁrm produces a certain amountof goods in order to restore the inventory level to the upper point S.
This class of economies, called the (S, s) economy, has attracted a large number of researchers concentrating on its aggregate consequences. For a study ofmacroeconomic impact, we need to aggregate the individual policy. However,this is a diﬃcult task because of the nonlinearity and discreteness of the (S, s)
policy due to its non-convex environment. Caplin (1985) considered the aggregate implication of exogenous (S, s) inventory policies across ﬁrms that follow
a time-invariant policy. Using a Markov process model, he concluded that in
the long run, the inventory levels of individual retailers are mutually independent regardless of correlation in sales. Caballero and Engle (1991) provide aframework for analysis of the aggregate dynamics of the (S, s) economy and givethe conditions under which the (S, s) economy achieves a steady state; here, thesteady state is deﬁned as a condition in which the distribution of inventories isinvariant to the distribution of demand.
Bak, Chen, Scheinkman, and Woodford (1993), hereafter BCSW, built a production and inventory economy model with large endogenous ﬂuctuations. They
demonstrated that random shocks to an economy do not average out in the aggregate but might produce signiﬁcant aggregate ﬂuctuations. They incorporate
local interactions through production networks into the (S, s) economy; these
are absent in Caplin (1985) and Caballero and Engle (1991). BCSW used thesandpile model (Bak et al. (1989)) and considered a simple square economy.
They assumed that ﬁrms connect through a hierarchical network involving someamount of inventory. This mechanism demonstrated that random shocks to theeconomy do not average out in the aggregate but might produce signiﬁcant ﬂuctuations. Although the BCSW model is very attractive, it depended heavily ona particular kind of production network, that is, the hierarchy of order ﬂow.
In this paper, we generalize the model and study a more general patternof connections between ﬁrms and the multiplier process with the (S, s) discreteadjustment economic policy. The rest of the paper is organized as follows. The
next section presents the model we use and deﬁnes the multiplier process, including the discrete inventory adjustment process. In section 3, we characterize
the statistical property of the model. Section 4 proves four key theorems, andsection 5 concludes the paper.

2

2

The model

2.1

General framework of the (S, s) economy

In an economy consisting of a large number of ﬁrms, assume that N = {1, 2, · · · , n, · · · , N }is a set of ﬁrms. Each ﬁrm produces diﬀerentiated goods indexed by i ∈ N .
Further, integer xi represents the number of inventories held by ﬁrm i, and a
conﬁguration of inventory holdings x′ = (x1 , · · · , xN ) is a vector of non-negative
integers.
Each ﬁrm i sets a pair of integers (Si , si ) with Si > si + 1 so as to minimize
costs, where Si is the upper target of inventory level and si , the lower point. Two
positive integers, (Si , si ), specify the inventory policy of ﬁrm i. If the inventory
level falls to si or below, that is, xi ≤ si , we say that ﬁrm i is unstable withregard to inventory. Similarly, if si < xi , ﬁrm i is stable. A stable conﬁguration
indicates stability at all ﬁrms. We denote the set of all stable conﬁgurations by
S; that is,
S = {(x1 , · · · , xn ) ∈ Zn+ | Si ≥ xi > si , ∀i ∈ N }.

(1)

We further assume that (Si , si ) is time-invariant for all ﬁrms, so that stable set S
is also time-invariant. All ﬁrms monitor their inventory levels in order to avoid
costly stock-outs each period and maintain the stability of their inventory levels.
When a ﬁrm becomes unstable, it needs to increase its production immediately
and restore its target level.
Similarly, (Si , si ) determines its production lot size, ∆i = Si − si > 1, whichindicates the amount ﬁrm i can produce in a period. Whether or not a ﬁrm
produces goods depends on the level of inventory the ﬁrm has at the beginningof the period and the order for that period. On receiving an order, the ﬁrm
satisﬁes its inventory position. If the ﬁrm is stable after execution of the order,
no need exists for further production. If ﬁrm i becomes unstable following theorder, it must produce ∆i = Si − si units of goods in that period.
In this economy, ﬁrms are interdependent in terms of input. Every ﬁrm needs
to procure certain units of output from other ﬁrms. For its production, ﬁrm
j needs (∆1j , · · · , ∆ij , · · · , ∆N j ) of goods produced by other ﬁrms (notice thatthese inputs include ﬁrm j’s own products, ∆jj ). One production by ﬁrm j meansthat its inventory level increases by ∆j = Si − si , while the inventory levels of all
the associated ﬁrms decrease by ∆ij . Besides the assumption of production lot
size ∆j , we assume that the vectors of inputs (∆1j , · · · , ∆ij , · · · , ∆N j ), ∀j ∈ N
are time-invariant.
We deﬁne the non-negative integer mi as the amount of production taking
place at ﬁrm i during the given period and note that vector m′ = (m1 , · · · , mN ) ∈ZN+ . Let integer yi ≥ 0 denote the order as an exogenous demand on ﬁrm i. Furthermore, let xi and x′i denote respectively ﬁrm i’s ex-ante and ex-post inventory
levels. The following equations represent the inventory dynamics of the economy:∑
x′i = xi − yi + mi ∆i −mj ∆ij , ∀i ∈ N .
(2)
j∈N

The term mi ∆i represents the total amount of products ﬁrm i produces during
the given period. This term has a positive role in inventory accumulation. Term
mj ∆ij represents the total amount of inputs moved from i to j for the production3

of ﬁrm j. This term has a negative, or at least non-positive, role in inventory
accumulation.
We consider the equations of inventory dynamics as the equation systemrepresenting the structure of the (S, s) economy.






m1 ∆1
m2 ∆2
..
.





∆11
  ∆21 
 =  ..
  .

∆12
∆22
..
.

···
···
..
.

∆N 1 ∆N 2 · · ·

mN ∆N



 
  ′

m1
y1
x1 − x1
  m2   y2   x′ − x2 

 
  2

  ..  +  ..  + 
 .
..
 .   .  

.
′
∆N N
mN
yn
xN − xN
∆1N
∆2N
..
.

Matrix D = (∆ij )i,j∈N is an N × N adjacency matrix of a weighted graphrepresenting the input network between ﬁrms.
Now, the model has been fully described. Following Caplin (1985), Deﬁnition
2.1 summarizes the components of the model below.
Definition 2.1. The (S, s) economy consists of• a set of firms: N = {1, 2, · · · , n, · · · , N }{ }• N pairs of (S, s) policies: ∆i i∈N with ∆i = Si − si > 1• an N × N adjacency matrix of a weighted graph representing the input
network of the economy: D = (∆ij )i,j∈N• an exogenous demand vector: y = (y1 , · · · , yN ) ∈ ZN+
• transition equations of inventory level given by (2).
We deﬁne the input coeﬃcient as
aij =

∆ij
∈ Q+ ,∆i(3)

and the input coeﬃcient matrix as A = (aij )i,j∈N . All ∆i are positive integers
and all ∆ij are non-negative; therefore, input coeﬃcients aij are non-negative
rational numbers. Since both ∆i and ∆ij are time-invariant, the input coeﬃcientsaij are constant. From aij , we obtain


 
  ′

1 − a11 −a12 · · ·
−a1N
m1 ∆1
y1
x1 − x1
 −a21 1 − a22 · · ·

 
  ′

−a2N   m2 ∆2   y2   x2 − x2 
 =  ..  + 
 ,
..
..
..
..
..
..


  .  

.
.
.
.
.
.
−aN 1

−aN 2

···

1 − aN N

mN ∆N

yN

x′N − xN

or its matrix form representation∆[I − A]m = y + x′ − x ,(4)where A, ∆, and y are the input coeﬃcient matrix, the vector of production lot
size, and an exogenous demand vector, respectively.
The problem of the (S, s) economy is to determine m (and hence x′ ) for
given ∆, A, and x when the exogenous demand takes a certain vector y. In
4the linear model, as with the Leontief input-output model, we can obtain thesolution from the inverse matrix [I − A]−1 = I + A + A2 + · · · , which is called
the matrix multiplier. However, the adjustment process of the (S, s) economy isnon-linear and composed of many (Si , si ) policies, which are also non-linear adjustment policies. In addition, individual non-linearity can be ampliﬁed throughinteraction with the input network.

2.2

Multiplier process of the (S, s) economy

To determine m in such a non-linear model, we describe how m can be constructed through the inventory adjustment process. The dynamics is deﬁned asfollows. At ﬁrst, the initial state of inventory holding for each ﬁrm would randomly take a conﬁguration x′ = (x1 , · · · , xN ) from the stable set S. Then, we
have one unit of exogenous demand occurring at a time to a randomly chosen
ﬁrm i. This decreases i’s inventory level by one. If xi → xi − 1 < si , x becomes
unstable at ﬁrm i and the inventory adjustment process starts, but otherwise x
will be stable and the next perturbation occurs.
The inventory adjustment process starts with one single production; we callthis step t = 0 of the process. Let i0 be the index of this ﬁrm, which must update
mi0 = 0 → mi0 = 1. At the same time, i0 needs (∆i0 1 , · · · , ∆i0 N ) of input andderived demand occurs for other ﬁrms. This derived demand decreases ∆ij units
of inventory of each ﬁrm and may render one or more of the ﬁrms unstable. Let{i1 } be the list of unstable ﬁrms at t = 1. Firms in {i1 } have to produce andincrease their inventories within step t = 1 and update mi1 → mi1 +1, ∀i1 ∈ {i1 }.
Similarly, the production in step t − 1 may lead to further production in step t.
(i)
Let Zt = mi,t − mi,t−1 ∈ {0, 1} be the update score of ﬁrm i at step t, which(1)(N )is a random variable in a large economy, and let Zt = (Zt , · · · , Zt ) be thevector of scores. Furthermore, let mt be the vector of the total score until t, mt
is generated by random vector Zt .
Definition 2.2. A multiplier process in the (S, s) economy is defined as
the stochastic process {mt }∞t=0 generated by
mt = mt−1 + Zt

(5)

Unfortunately, we ﬁnd it very diﬃcult to explicitly solve (5) because the (S, s)economy described here is rather general and contains several heterogeneities andirregularities. Inventory policies and input networks are diﬀerent from ﬁrm toﬁrm. Each ﬁrm’s updating therefore evolves diﬀerent (Si , si ) policies and so(i){Zt } are not identically distributed. Furthermore, the timing of updating is(i)dependent on other ﬁrms and so {Zt } is not independent.

2.3

The n-regular (S, s) economy

We introduce regularities into the economy to make the model more tractablefrom this point onward.
Assumption 2.1. We assume two regularities in the (S, s) economy.
1. Regularity of (S, s) policy: ∆i = ∆ = S − s > 1, ∀i ∈ N

5

2. Regularity of input matrix:(a) ∆ij ∈ {1, 0}∑(b)j ∆ij = n, ∀i ∈ N and 0 < n << N .
Assumption 2.1 (1) is about the regularity of (S, s) policy. We assume that
all ﬁrms have equal (identical) (S, s) policies. This assumption means that every
ﬁrm has the same lot size and produces the same amount in a single production.
Assumptions 2.1 (2.a) and (2.b) are about the regularity of inputs. We
assume that if there is an input relation from i to j, then element ∆ij = 1,otherwise ∆ij = 0. We also assume that the number of inputs the ﬁrm needs ina single production of ∆ is equal (identical) for all ﬁrms. For a given ﬁrm, thelist of ﬁrms from which it gets inputs is ﬁxed. These two assumptions on input
regularity suggest that we restrict the graph to a regular graph representing an
input network between ﬁrms under assumption 2.1. When the number of inputs
is equal to n, the graph is referred to as an n-regular graph.
Now, under assumption 2.1, we can summarize an n-regular (S, s) economy
by deﬁnition 2.3.
Definition 2.3. An n-regular (S, s) economy consists of• a set of firms: N = {1, 2, · · · , n, · · · , N }• identical (S, s) policies for all firms, given by assumption 2.1 (1)• an N × N adjacency matrix of n-regular graph representing the input
network of economy: D = (∆ij )i,j∈N , given by assumptions 2.1 (2.a) and(2.b)• an exogenous demand vector: y = (y1 , · · · , yN ) ∈ ZN+
• transition equations of inventory level given by (2).

From the perspective of analytic solvability, we beneﬁt from these assumptions. The input coeﬃcients of the n-regular (S, s) economy become identical{ ( 1)
a = ∆ if ﬁrm j needs input from ﬁrm i
.
aij =
(6)
0
otherwize,
and∑aij = na, ∀i ∈ N .

(7)

jConsidering the regularities of ∆, we deﬁne the total number of productions instep t, Zt and that until step t, Mt , as
∑ (i)
Zt ≡
Zt
(8)
i∈N

∑

Mt ≡

i∈N

6

mi

(9)

Figure 1: Description of inventory adjustment process in a 2-regular(S, s) economy. Arrows −→ represent the direction of inputs. Points• represent ﬁrms with stock-out and production in process, and points◦ represent ﬁrms with suﬃcient inventories to terminate the chain of
production.

Figure 2: Multiplier process in a 2-regular (S, s) economy. Z0 ≡ 1, Z1 =2, Z2 = 3, Z3 = 1, Z4 = 1, Z5 = 0. M5 = Z0 + Z1 + Z2 + Z4 + Z5 = 8.

7

With the regularity of number of inputs, n, the random variable Zt can be
represented as follows. Let K be the number of ﬁrms that start production of
new inventories from the derived demand of a single ﬁrm. Whether a ﬁrm starts
production depends on its inventory level. More precisely, K is decided by theratio of ﬁrms whose inventory level becomes xi → xi − 1 < si by one order. Let
π be the ratio of ﬁrms whose inventory level is equal to 1 per N .
π≡# of ﬁrms whose inventory level is equal to 1
.
N

(10)

The probability that a ﬁrm that produced once will produce again in the next
steps of the same process is of the order of 1/N . If N is suﬃciently large, we caneliminate the possibility that the chain of production forms loops. A binomial
distribution gives the probability of k times of production triggered by a single
production,()
npk = P (K = k) =(11)
π k (1 − π)n−k , (k = 0, 1, · · · n) .
k
From this nature of K, we can construct the stochastic process {Zt }∞t=0 asfollows. We start with Z0 = 1. Z1 is the number of ﬁrms that produced in stept = 1. Obviously, Z1 = K. Since Zt is the number of ﬁrms that produced in stept, Zt is obtained from the recursive equation
∑

Zt−1

Zt =

Kj ,

(12)

j=1

where Ki corresponds to the number of ﬁrms that produced in step t from thederived demand of ﬁrm i, which produced in step (t − 1).
From this recursion of Zt , we deﬁne the multiplier process in the n-regular(S, s) economy as follows:
Definition 2.4. A multiplier process in the n-regular (S, s) economy isdefined as a stochastic process {Mt }∞t=0 , that is,
Mt = Mt−1 + Zt

(13)

3

Stochastic properties of multiplier process

3.1

Probability generating function

Now, let ϕ be the probability generating function of K as
ϕ(θ) = ϕK (θ) = E(θK ) =∞∑pk θ k ,|θ| ≤ 1.

(14)

k=0

We also deﬁne the probability generating function of Zt as
ϕZt (θ) = E(θZt ) =∞∑P (Zt = k)θk ,k=08|θ| ≤ 1.

(15)

Since K1 , · · · , Kk are independent and ϕ(θ) = ϕZ1 (θ) = ϕK (θ) because Z1 = K,
we haveϕZt (θ) = E(θZt )
∞
∑
[
]
=
E θZt | Zt−1 = k · P (Zt−1 = k)
k=0
∞
∑
[
]
=
E θK1 +···+Kk · P (Zt−1 = k)
k=0
∞
∑

=

E(θK )k · P (Zt−1 = k)

k=0
∞
∑

=

ϕ(θ)k P (Zt−1 = k) .

(16)

k=0

From the deﬁnition of ϕ,
ϕZt−1 (ϕ(θ)) =∞∑ϕ(θ)k P (Zt−1 = k) ,

(17)

k=0

we haveϕZt (θ) = ϕZt−1 (ϕ(θ)) .

(18)

ϕZt (θ) = ϕt (θ)

(19)

Thus,

3.2

Expected values

Following the binomial theorem, the generating function becomes
)
n (
∑
n
ϕ(θ) =
π k (1 − π)n−k θk = (πθ + (1 − π))n .
k

(20)

k=0

Thus, we have the following lemmas.
Lemma 3.1.

E(K) = nπ < ∞.

Proof. By diﬀerentiating (20), we have′ϕ (θ) =n∑kpk θk−1 = n[πθ + (1 − π)]n−1 · π

k=0Taking the limit z → 1 − 0, and recalling that 0 < n ≤ ∆ and 0 ≤ π ≤ 1, we
have
E(K) = lim ϕ′ (θ) = nπ < ∞.
θ→1−0

Lemma 3.2.

E(Zt ) = (nπ)t < ∞.
9Proof. Diﬀerentiating (18), we have
ϕ′Zt (θ) = ϕ′Zt−1 (ϕ(θ)) · ϕ′ (θ).

(21)

From limit z → 1 − 0 and lemma 3.1, we havelim ϕ′Zt (θ)

E(Zt ) =θ→1−0

lim ϕ′Zt−1 (ϕ(θ)) · lim ϕ′ (θ)

=

θ→1−0

θ→1−0

= E(Zt−1 ) · E(K)= E(K)t= (nπ)t .

The expected number of productions conditioned to the value of previous
productions, E(Zt+1 | Zzt ), can be obtained, since Zt = zt is ﬁxed, from the∑t z=texpectation of Zt+1 = i=1 Ki ; that is,
(z
)
t
∑
(22)
E(Zt+1 | Zt = zt ) = E
Ki = zt · E(K) .
i=1

4

Multiplier in the statistical equilibrium

4.1

Definition of stochastic multiplier

Before discussing the stochastic multiplier, let us deﬁne the statistical steadystate of the economy, the so-called statistical equilibrium. Following repeated
perturbations of exogenous demand, the economy reaches a statistical equilibrium state, when the aggregate inﬂow of inventories into the economy must be
equal to the aggregate outﬂow of inventories from the economy. Let Xin and
Xout denote the total inﬂow and outﬂow of inventories, respectively, following
the multiplier process triggered by one exogenous demand; E[Xin ] and E[Xout ]are respectively the expected values.
Definition 4.1 (statistical equilibrium). The statistical equilibrium of the nregular (S, s) economy is defined as a state in which the expected aggregate inflow
equals the expected aggregate outflow; that is,
E[Xin ] = E[Xout ].

(23)

Statistical equilibrium can be characterized by a single parameter.
Proposition 4.1. In statistical equilibrium, π = a.
Proof. The inﬂow at step t can be given by Zt ∆. From lemma 3.2, the expectedvalue of Xin isE(Xin ) =∞∑E(Zt )∆ = ∆

t=0

∞
∑
t=0

10

(nπ)t .

(24)

Note that the outﬂow consists of two parts. An exogenous demand of 1/π isrequired before an adjustment process starts. Once the process starts, the outﬂow at step t can be expressed by Zt−1 · n. Then, the expected number of Xout
becomes
∞
∞
∞
1 ∑
1 ∑
1∑
t−1
(nπ)t .
E(Xout ) = +E(Zt−1 )n = +(nπ) n =
(25)
π
π
π
t=1

t=1

t=0

In the equilibrium E(Xin ) = E(Xout ), we get
π=

1
=a
∆

(26)

Assume that M denotes the limit of multiplier process {Mt },M = lim Mt .
t→∞

(27)

Definition 4.2 (stochastic multiplier). In statistical equilibrium, a stochastic
multiplier of the n-regular (S, s) economy is defined as the expectation of the limitof stochastic process {Mt } with probability π = a; that is,()(28)
E(M ) = E lim Mt
with π = a .
t→∞

4.2

Convergence of stochastic multiplier

Theorem 4.1 (convergence). In statistical equilibrium, the multiplier processof the n-regular (S, s) economy converges and can be expressed exactly asE(M ) =

1
= 1 + na + (na)2 + (na)3 + · · · ,1 − na(29)

if the Brauer–Solow suﬃcient condition holds, that is,N∑

aij < 1,∀i ∈ N .

(30)

j=1

Proof. For the interchange of limits and expectations and the linearity of expectations, the RHS of (28) can be()
E lim Mt
= lim E(Mt )
t→∞
t→∞
[ t
]
∑
= lim E
Zτ
t→∞

=

lim

t→∞

τ =0
t
∑

E(Zτ )

τ =0= 1 + E(Z1 ) + E(Z2 ) + E(Z3 ) · · ·= 1 + na + (na)2 + (na)3 + · · ·1=(|na| < 1) .
1 − na∑N
If (30) holds, 1 >i=1 aij = na > 0 for all i ∈ N . This satisﬁes theconvergence radius of (29).
11

4.3

Termination of multiplier process

Next, we examine the probability that the multiplier process terminates in ﬁnite
steps, or, in another words, the multiplier process contains ﬁnite mass; that is,()
t∑Zτ < ∞ = 1 .
(31)
P (M < ∞) = P lim
t→∞

τ =0

Note that this problem is diﬀerent from the convergence of multiplier, which we
proved in Theorem 4.1. Theorem 4.1 gives the condition for convergence of the“expectation value” of the multiplier. The following theorem gives the condition
for convergence of the multiplier “process” itself.
Let ωt be the probability that the multiplier process terminates by the t-thstep,
ωt = P (Zt = 0) .

(32)

Further, let ω be the limit, if one exists,lim ωt = ω.

t→∞

(33)

If the multiplier process terminates, ω = 1, otherwise ω < 1.
Theorem 4.2 (termination probability). In statistical equilibrium, the multiplierprocess of the n-regular (S, s) economy terminates in finite steps if and only if
na ≤ 1; that is,⇐⇒

ω=1na ≤ 1.

(34)

Proof. Since ϕZt (θ) = ϕzt−1 (ϕ(θ)) = ϕt (θ) = ϕ(ϕZt−1 (θ)) from (18) and ϕZt (1) =P (Zt = 0) = ωt , we haveωt = ϕ(ωt−1 ).

(35)

For the initial value ω0 = 0, {ωt }tt=0 satisﬁes
ω1 = ϕ(ω0 ) = ϕ(0) = p0 ≤ ϕ(1) = 1(36)

ω2 = ϕ(ω1 ) ≤ ϕ(1) = 1
..
..
.
.

(37)

ωt = ϕ(ωt−1 ) ≤ ϕ(1) = 1.

(39)

(38)

Therefore, {wt }∞t=0 is a bounded monotonic sequence and converges to the limit
lim ωt = ω ≤ 1 .

t→∞

(40)

In the limit, ω is a ﬁxed point of ϕ in [0, 1]; that is, ω = ϕ(ω).
We assume that p0 > 0 to avoid a trivial case. Map ϕ is continuous forz ∈ [0, 1]. Further, ϕ is non-decreasing and convex becauseϕ′ (θ) =∞∑kpk θk−1 ≥ 0 ,(41)

k(k − 1)pk θk−2 ≥ 0 .

(42)

k=1

ϕ′′ (θ) =

∞
∑
k=2

12

It follows that ω = ϕ(ω) has either one or two ﬁxed points in [0, 1]. Obviously,ω = 1 is one of the ﬁxed points,
ϕ(1) =∞∑

pk = 1.

(43)

k=0

In statistical equilibrium, no other ﬁxed point exists in [0, 1] if ϕ′ (1) = na ≤ 1.
Note that this condition for termination of the multiplier process does notsatisfy the Brauer–Solow suﬃcient conditions as well as the Hawkins–Simon conditions in a linear model. We consider a random case where the distribution
shows a strictly positive variance. If na = 1, the probability is that the multiplier process terminates in ﬁnite steps. Nevertheless, the stochastic multiplier
diverges in this case.
(∞
)
∞
∑
∑
(44)
E (M ) = E
Zt =1t = ∞.
t=0

t=0

Technically, if the variance is zero, we would have limt→∞ P (Zt = 0) = 0 in thiscase. However, this fact indicates that the termination of the multiplier process
and convergence of the multiplier are totally diﬀerent matters. The condition forconvergence in the (S, s) economy model could be weaker than that in a linear
model.

4.4

Criticality and rate of return

Let qi be the price of goods i. From the homogeneity of aij , qi = qj , ∀i, j ∈ N ,we deﬁne the uniform rate of return, r, as∑
qi aijqj − N
1 − na=.
r = ∑N i=1na
i=1 qi aij(45)

Theorem 4.3 (asymptotic divergence). The stochastic multiplier of the nregular (S, s) economy is asymptotically divergent when the uniform rate of returnis very close, but not equal, to 0.
Proof. From (45) na = 1/(1 + r), we have
1

E(M ) =
1−

1
(1 + r)

=

1
+1.
r

(46)

For the limit r → 0,{
lim E(M ) = lim

r→0

r→0

1
+1
r

13

}
=∞.

(47)

Figure 3: Asymptotic divergence of the n-regular (S, s) economy.

This result tells us that r = 0, that is na = 1, is the critical point of thiseconomy. This corresponds precisely to Bak et al.’s (1993) case, which saysthat the system evolves to the state of self-organized criticality wherein thetriggered chain reaction is much larger than the original shock and any size ofthe chain reaction can occur. Their model displays power-law ﬂuctuation andE(M ) diverges to inﬁnity. However, in the usual case where r > 0, that is,na < 1, the economy is said to be subcritical and E(M ) is a ﬁnite size.
Now, we examine the ﬂuctuation of M . The multiplier process is represented
as the branching process of a tree, which is a connected graph with no loops. We
take advantage of this nature and calculate the size distribution of the stochastic
multiplier.
If M = m, it contains m production nodes in the tree constructing themultiplier process, depicted as • in Figure 2. The tree has m × n branches,and m − 1 out of m × n are triggered to produce while m × n − (m − 1) are
not triggered. With regard to Figure 2, where n = 2, the tree has 8 × 2 = 16branches and m × n − (m − 1) = 16 − 7 = 9, the probability of m size of tree
can be expressed by π m (1 − π)m×n−(m−1) , where π is the ratio of inventory level
equal to one.
In general, the size distribution of the stochastic multiplier, M , is given by
(
)
1
m×n
π m−1 (1 − π)m×n−(m−1) with m = 1, 2, 3, · · ·(48)P (M = m) =m
m+1
In order to count the number of trees with m nodes, for the case n = 2, theCatalan number Cm is well known,
(
)
1
(2m)!
2m
Cm =
=
.
(49)
m
m+1
m! (m + 1)!
For n = 2, with this number, we give the size distribution of the stochastic
multiplier, M , by
P (M = m) = Cm π m−1 (1 − π)m+1
14

with

m = 1, 2, 3, · · ·(50)Theorem 4.4 (size distribution). In statistical equilibrium, the distribution function of multiplier M is analytically given as
(
)
1−a
m
− 23
P (M = m) ∼ √ · m · exp −with m = 1, 2, 3, · · · ,
(51)
ξ(a)
a π
where the correlation length ξ(a) is(ξ(a) ∼

1
1 − 2a)2.

(52)

Proof. From Stirling’s formula, where m is large enough to replace m + 1 ∼ m,
the Catalan number becomes
(
)
1
(2m)!
1 (2m)2m
1
1
2m
√
=
(53)
Cm =
∼
m
m+1
m + 1 m!(m + 1)!
m + 1 πm m2m
3
1
4m
4m
√
∼
∼ √ · m− 2 .
(54)
m + 1 πmπIn statistical equilibrium, if π = a, then we have
P (M = m) ∼3
1−a
√ · m− 2 · [4a(1 − a)]m .
a π

(55)

Because a < 1/2, a(1 − a) is less than 1/4, and so [4a(1 − a)]m decreases.
Therefore, we have [4a(1 − a)]m = em ln[4a(1−a)] .
From Theorem 4.3, the critical point of this economy is ac = 1/n = 1/2.
Introducing a deviation from the critical point, D ≡ a − ac = a − 1/2, and so we
can write
1a(1 − a) = − D2 .
4
We conclude by using the Taylor expansion and plugging the result into (55).
Figure 4 are graphs of (51), the distribution mass function of multiplier sizes.
The (S, s) economy evolves or self-organizes into a statistically stationary state,
where the distribution of the multiplier sizes can be given by)( m− 32 · exp − m
if a < 12
ξ(a)P (M = m) ∼.
(56)

 − 32
1
if a → 2m
If m is smaller than ξ(a), the probability mass function is well approximatedby power law with exponent 3/2. However, for a large m, the exponential decay
dominates. In the critical case, a = 1/n (= 1/2), the exponential disappears andthe distribution is a pure power law.
Parameter a is an input coeﬃcient that is so regular that it can be interpreted
as a function of the uniform rate of return r, as seen in Theorem 4.3. If a <1/n (= 1/2), then r > 0 and (1 − na) > 0; that is, the proﬁt is positive andthe system is subcritical. An external perturbation of demand is absorbed by acertain proﬁt rate. That is, energy is not conserved in this case. Therefore, thecorrelation length ξ in ﬁnite perturbation cannot propagate a long distance.
IF a = 1/n (= 1/2) then r = 0, that is, zero-proﬁt and critical point. In
this case, external perturbation cannot be absorbed, energy is conserved, andthe system is critical since the distribution function (56) shows a power-law
distribution.
15

Figure 4: Size probability mass function in n-regular (S, s) economy in
the case n = 2.

5

Conclusions

In this paper, we proposed a general quantity adjustment behavior and inputoutput structure model. We employ the (S, s) inventory policy as an inventory
adjustment behavior with a non-linear nature. This nature plays an importantrole in aggregate dynamics. A threshold adjustment allows each ﬁrm to store
the adjustment energy and productions have lumpy perturbations. Once a production takes place, the ﬁrm releases a large amount of energy that propagates
through the economy in a multiplier process.
After deﬁning the general framework, we restrict the input-network structure
to a regular graph. We deﬁne the multiplier process as a branching and derive
the expectation value of the multiplier in the statistical steady state. We give
key theorems in Section 4; one is the suﬃcient condition for the convergence ofthe multiplier in terms of expectations.
This condition is similar to the well∑known Brauer–Solow condition,
aij < 1; it takes the form na < 1 in ourmodel. Another is the necessary and suﬃcient condition for termination of themultiplier process. We prove that the condition is na ≤ 1. However, for the caseof na = 1 in particular, this violates Solow’s condition as well as the Hawkins–
Simon condition. We conclude that the necessary and suﬃcient condition for
termination of the multiplier process is weaker than that in the linear model.
We also prove that the multiplier is ﬁnite and that its probability distributionfunction decays exponentially if the rate of return is positive. Rate of return is
16

associated with the input coeﬃcient a in our model; positive returns mean thatna < 1. On the other hand, if the rate of return has a zero limit ( na = 1),
the multiplier asymptotically diverges and the probability distribution function
is asymptotically a power law. This means that a positive rate of return holds
down the high ﬂuctuation of output and moderate business cycles.

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Mathematical Method in the Social Science, 59, 1960.
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[14] Zapperi, S., Lauritsen, K. B. and Stanley, H.E., ”Self-organized branchingprocesses: mean-ﬁeld theory for avalanches,” Phys. Rev. Lett., 75, 4071,
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