Inventory Adjustment and Aggregate Fluctuations
Naoto Yagi ∗†October 16, 2015 ‡Abstract

In this paper, we investigate the relationship between lumpy inventory adjustment and aggregate ﬂuctuations. We present a general equilibrium model composed of many ﬁrms interacting in the
input–output network. Each ﬁrm follows a threshold-type inventory
policy that generates a lumpy production pattern. First, we provide
a tractable method to compute the competitive equilibrium when
the network structure is symmetric and all ﬁrms are homogeneous.
The branching process is employed to estimate the key statistics
of complex adjustment process, and we derive the stationary distribution in the long-run equilibrium. Second, we prove that theaggregate ﬂuctuations almost surely converge to zero and the law oflarge numbers is satisﬁed in our framework. We also show that theprobability distribution function of impulse response decays exponentially if the labor share is positive. We conclude that the positive
labor share reduces the high inventory ﬂuctuations and moderates
the business cycle.
JEL Classification: C60, E32.
Keywords: (S, s) inventory policy, lumpy adjustment, input-output
structure, law of large numbers, aggregate ﬂuctuation.

1

Introduction

Business cycles have always been an important issue in economics. Innumerable
studies have tried to explain how and why the economy ﬂuctuates over time.
The various theories are roughly divided into two types on the basis of whether
or not microeconomic characteristics aﬀect the aggregate economic activities.
Mainstream theories emphasize the importance of macroeconomic shocks suchas monetary and ﬁscal policy shocks, which aﬀect all sectors simultaneously andtrigger aggregate ﬂuctuations. New theories, however, suggest the existence ofchannels that play an amplifying role in the propagation of micro-level shocks
behind the economy-wide ﬂuctuations.
∗

Department of Economics and Business Administration, Niigata Sangyo University.
E-mail : yagi@econ.nsu.ac.jp‡
First Version: Septamber 13, 2014.
†1

Inventory Adjustment and Aggregate Fluctuations

2

In the macroeconomic literature, leading theories have paid less attention to
the role of micro-level shocks. Lucas (1977) argues that independent micro-level
shocks would average out as the number of units approaches inﬁnity. Lucas’s
argument, which supports the mainstream view, is a direct application of the lawof large numbers to economics. According to this view, macroeconomic shocks
are more important than micro shocks for the aggregate ﬂuctuations. On theother hand, recent researchers tend not to think that macro shocks can oﬀer afull explanation of all the ﬂuctuations. The new view, which has been attracting
much attention in the recent years, emphasizes the eﬀects of interactions between
micro units, especially input–output linkages. Indeed, in the economy composedof many independent units, a shock aﬀecting a single unit might have little
impact on the aggregate level. However, micro shocks to individual units areunlikely to be independent. If the micro units are linked, a positive shock to one
unit produces a positive response among other units through this linkage. Then,small shocks are ampliﬁed and propagated in the economy through linkages tocause signiﬁcant aggregate ﬂuctuations. In this situation, the sum of these small
shocks may fail to produce the kind of cancellation required by Lucas’s argument.
Bak et al. (1993) is one of earlier studies following the new view, where
the inventory adjustment has been considered to play an important role behindthe business cycle. They focus on the fact that non-convex adjustment costs
make inventory adjustment lumpy at the micro-level. They employ the thresholdinventory policy through the so-called “(S, s) rule” and assume that the ﬁrms
connect through a hierarchical network with a certain amount of inventories.
The lumpy inventory adjustment generates micro-level perturbation, and theexistence of production linkages provides a possible ampliﬁcation mechanism.
Bak et al. (1993) describe the large aggregate ﬂuctuation as critical phenomena; we refer to this line of approaches as criticality hypothesis. Criticalphenomena are emergent properties associated with critical points at which thecorrelation length diverges to inﬁnity. At the critical point, aggregate quantities
follow power laws, whose properties include scale invariance and the lack of awell-deﬁned average value. Then, micro-level shocks fail to cancel out at theaggregate level, and hence, large economic ﬂuctuations can occur even without
macro-level shocks. Their model displays self-organized criticality, and the effects of small perturbations from independent inventory adjustment fail to cancel
out. The large ﬂuctuations in aggregate productions follow a power law distribution. Nirei (2006) proposes a more general model along this line, where many
individuals follow (S, s) rules and aggregate ﬂuctuations arise from interaction
with a positive feedback.
The importance of multi-sectoral linkage has also been explored in the realbusiness cycle literature, both theoretically and empirically, after Long andPlosser (1983). More recently, Acemoglu et al. (2012) propose the network
hypothesis, which is receiving considerable attention. They interpret the input–
output structure as a weighted network, where the nodes correspond to theindustries and the links to the input–output trade ﬂows. They focus on theinequality across ﬁrms or sectors in terms of importance in an input–output network. They also derive conditions about the structure of networks to deliver
low convergence rates. One of the necessary conditions is that the number ofcustomer links follows a power law distribution with a tail exponent lower than
2. They also provide some empirical evidence that idiosyncratic shocks to theInventory Adjustment and Aggregate Fluctuations3

top 100 ﬁrms explain a large fraction of aggregate volatility in the United States.
Recent literature has also pointed out that the heterogeneity of ﬁrm sizes isimportant to understand the business cycle. Gabaix (2011) propose the granularhypothesis of economic ﬂuctuations based on the fact that the distribution ofﬁrm sizes typically follows the power law. When large ﬁrms have a large share of
the economy, aggregate ﬂuctuations may arise from idiosyncratic shocks to theseﬁrms. The power law distribution makes the law of large numbers break down,
and idiosyncratic shocks to large ﬁrms aﬀect aggregate outputs.
All of these hypotheses relate to power law. Hence, in a broad sense, the large
ﬂuctuations arise because of critical phenomena in all cases. However, speciﬁcally, we interpret the criticality hypothesis to be mutually exclusive with theother two hypotheses. The criticality hypothesis claims that the power law distributions of aggregate quantities can be generated only by the system criticality,
without any network or granular eﬀect. From this viewpoint, Bak et al. (1993)
provide an interesting perspective on inventory adjustment process as the sourceof large aggregate ﬂuctuations. The essential point of their model is that the system criticality solely arises from the combination of lumpy inventory adjustment
and local interaction through input–output linkage. In a multi-sector economy
with linkage, this implies that the lumpiness attributed to non-convexity can be
the source of large aggregate ﬂuctuations by itself, even in the absence of network
asymmetry or granularity of ﬁrms.
Although Bak et al.’s (1993) argument is very attractive, their results relyon the speciﬁc structure of network and production technologies. The conditions
for the emergence of criticality also seem rather unique. In addition to that, themodel generates wider ﬂuctuations than those observed in reality.
The purpose of this paper is twofold. Firstly, we propose a multi-sector
general equilibrium model with a non-convex cost environment as an extensionof Bak et al. (1993). Then, we provide a tractable method to compute thecompetitive equilibrium. Along the lines of Acemoglu et al. (2012), we consider
a static version of Long and Plosser (1983) but modify the model for it to contain(S, s) inventory policies that generate lumpy inventory adjustments. Owing to
the complexity of the propagation mechanism through linkage, it is hard tofollow the details of the micro-level lumpy adjustment process in a large economy.
Instead, we employ the stochastic approach to estimate the macro-level statisticsof the inventory adjustment process. Secondly, we assess the relevance of thecriticality hypothesis for the study of inventory ﬂuctuations in comparison withthe other two hypotheses. We verify whether the law of large numbers applies tothe model with non-convex costs, and we examine the behavior of the responseto the uncorrelated micro-level shocks and how quickly their impact on aggregate
volatility decays in the equilibrium. For the purpose of comparison, we get ridof the asymmetry of network structure and the heterogeneity across ﬁrms. The
non-convex cost environment is the only diﬀerence between our model and theordinary multi-sector model.
The rest of the paper is organized as follows. The next section presents a
structural model with inventory adjustment and explicit production linkages. We
deﬁne the competitive equilibrium and derive the aggregate relation. In section
3, we construct the representation of inventory adjustment process and provide a
tractable method to estimate the key statistics. In section 4, we prove the main
theorems of this study. Section 5 concludes the paper.

Inventory Adjustment and Aggregate Fluctuations

2

Model

2.1

Environment

2.1.1

Firms

4

There are a large number of ﬁrms producing diﬀerentiated goods indexed byj ∈ {1, 2, · · · , n}. The produced goods of any given ﬁrm are not only purchasedby consumers but also used by other ﬁrms as inputs (intermediate goods) for
their own production. Each ﬁrm produces its ﬁnal goods through labor and theintermediate goods purchased from other ﬁrms. The technology that produces
the ﬁnal goods xj is given by the Cobb–Douglas function:(1−γj )

xj = ljn∏axijij ,

(1)i=1where lj and xij denote the amount of labor hired by ﬁrm j and intermediate
goods i used in the production of good j. Constant returns to scale is assumed
to hold and is simply expressed by
γj =

n
∑

aij

(2)

i=1

for all j ∈ {1, 2, · · · , n}.
We assume perfectly competitive markets and price-taking ﬁrms. The ﬁrm’s
has two problems: a cost minimization problem and a proﬁt maximization problem. At ﬁrst, we consider a ﬁrm that chooses to minimize cost subject to producing xj . The optimal input use problem is given by


n


∑
c(xj ) = minwlj +pi xij ,{lj ,xij } 
j=1

subject to (1) and (2). From ﬁrst order conditions, cost shares of labor input
and intermediate inputs are obtained, respectively, as
wlj = (1 − γj )(pj xj ) ,(3)pi xij = aij (pj xj ) .

(4)

The assumption of constant returns to scale leads us to the linear cost functions,c(xj ) = pj xj(5)

for all j, where pj is (constant) marginal cost to produce xj . If there are noquantity discounts and the costs of holding inventories, then each ﬁrm has no
incentive to store the inventory and the production takes place at the same timethat the order is received.
2.1.2

Inventory Policy

Now, we introduce a substantial ﬁxed cost to conduct production. For example,
it could be the cost to activate the manufacturing device, or the cost for material

Inventory Adjustment and Aggregate Fluctuations

5

transportation, and so on. In this case, the cost function is non-convex, that is,{ψj + pj xj if xj > 0ψ(xj ) =,(6)0
if xj = 0
where ψj is the ﬁxed cost to produce xj units of goods. Notice that all costs are
incurred only when xj is positive.
This type of non-convex cost function leads to the so-called “(S, s) policy”
of inventory behavior. The striking features of the (S, s) policy are the ﬁxed
lot size and the lumpy pattern of productions. In the (S, s) policy, the ﬁrm
constantly monitors inventories and chooses a lower level, s, below which it does
not let inventories fall, and an upper level, S. Inventories are allowed to vary
between two target levels, S and s. When inventory stocks reach s, the ﬁrm
immediately produces to increase its inventory stocks to S. The quantity (S − s)is referred to as the optimal lot size, allowing the ﬁrms to separate the timingof orders and productions. If the ﬁxed cost is relatively large, the ﬁrm produces
infrequently and the optimal lot size would be large. It is well known that the(S, s) inventory policy is eﬀective in variety situations where non-convex costs
exist. Arrow, Harris, and Marschak (1951) introduce the ﬁrst mathematicalformulation of the (S, s) policy, and Scarf (1960) provide a general proof of its
optimality.
We suppose that each ﬁrm follows its (Sj , sj ) policy and has its own lot
size (Sj − sj ) associated with (6). The proﬁt maximization generates a statedependent behavior based on the inventory level at the beginning of a period:{
Sj − sj > 1 if hj ≤ sj
Qj =
(7)
0
otherwise
where hj denotes the inventory level for ﬁrm j.
We refer to the ﬁrm as unstable if its inventory stocks fall to its lower level or
below, that is, hj ≤ sj . Similarly, we refer to the ﬁrm as stable if sj < hj ≤ Sj .
The non-negative vector h = (h1 , · · · , hn ) represents a conﬁguration of inventory
stock levels. A stable conﬁguration is the one where all ﬁrms are stable. The setof all stable conﬁgurations are denoted as below.
{}
S = (h1 , · · · , hn ) ∈ Zn+ | sj < hj ≤ Sj , ∀j ∈ {1, · · · , n} .
Because all (Sj , sj ) are time-invariant, the stable set S is also time-invariant.
The total output of ﬁrm j takes a discrete value,
xj = mj Qj ,mj ∈ N = {0, 1, 2, · · · }(8)where mj is the number of productions conducted by j. The non-negative integer
vector m = (m1 , · · · , mn ) ∈ Zn≥0 summarizes the number of productions.
Let lj∗ denote the amount of labor used to produce the optimal lot size (Sj −sj ). The labor income from one lot size is given by wlj∗ = (1−γj )[pj (Sj −sj )−ψj ].
Henceforth, we suppose that the ﬁxed cost ψj is relatively small enough to beignored at the aggregate level. Then, let us consider
Y ≡w

n
∑
j=1

lj∗ mj =n∑(1 − γj )pj (Sj − sj )mj
j=1

(9)

Inventory Adjustment and Aggregate Fluctuations

6

as an aggregate income and aggregate output.
In a suﬃciently large economy, the aggregate income Y can be regarded asa stochastic variable. Each ﬁrm produces a large amount of goods at a time sothat the production pattern becomes lumpy and ﬂuctuates more than external
demands. This lumpiness and the ﬂuctuations propagate through the network ofintermediate inputs. Any ﬁrm needs to produce some units of output from otherﬁrms. Once a production takes place, it must reduce the inventory levels of othergoods. They may generate inventory stock-out of other ﬁrms and trigger otherproductions. The chain of productions may occur and continue until all ﬁrms’
inventory levels are recovered. The total number of productions is sensitively
dependent on the state of initial inventory conﬁguration. It is possible to createa big chain of productions when there exists a big cluster of low-level inventory
stocks in the initial state.
2.1.3

Consumer

The representative consumer has Cobb–Douglas preferences over n distinct goods,u(c1 , c2 , · · · , cn ) = Θnn∏cθi i ,

(10)i=1

where ci is the consumption of good i and Θn is a normalization constant discussed below. The preference weights {θi } become known before making a decision, and are given by1
exp(εi ) ,n(11)

pi ci = E[Y ]

(12)

θi =

∑
where {εi } are bounded by −∞ < ε ≤ εi ≤ ε < ∞. We suppose i exp(εi ) = n.
The representative consumer is endowed with one unit of labor, which issupplied inelastically. She supplies 1/n units of labor to the ﬁrm j and receives
labor income wmj /n from ﬁrm j in proportion
to the number of its productions.
∑
The total labor income is Y = (w/n) nj=1 mj . The representative consumer isalso considered to be a price taker, and maximizes her utility (10) by choosing a
consumption bundle (c1 , c2 , · · · , cn ) subject to her budget constraint,n∑i=1

where E[ Y ] is the expectation of aggregate income. The ﬁrst-order necessary
conditions imply that consumer expenditures on individual goods are proportional to their respective preference weights:
pj c jpi ci=, ∀i, j ∈ {1, 2, · · · , n} .
θiθjSubstituting this into the consumer budget constraint (12), we have demand
functions for goods,
pi ci = θi E[Y ]for all i ∈ {1, 2, · · · , n}.

(13)

Inventory Adjustment and Aggregate Fluctuations

7

The indirect utility function is obtained by substituting demand functions
into the utility function:)n (n ( )θi∏∏
θi
θi E[Y ] θi
= Θn E[Y ].
U = Θn
pi
pi
i=1

i=1

By setting the utility shift parameter as Θn ≡is given by
P ≡

∏n

−θi
, the ideal price index
i=1 (θi )n∏pθi i .

(14)

i=1

Then, it turns out that real planned aggregate expenditure,y ∗ ≡ E[Y ]/P(15)

provides the welfare criteria and we refer to this as an eﬀective demand.

2.2

Network and Assumptions

The structure of the input–output network can be represented by a weighted,
directed graph G = (V, E, A). In the graph G, each vertex, vj , corresponds tothe ﬁrm j in this economy so that the vertex set is V = {1, 2, · · · , n}. A directededge (or arc) (i, j) ∈ E is an ordered pair of vertices, and the weight aij ≥ 0
assigned to its directed edge represents an input ﬂow from vertex i to vertex
j. The non-negative matrix An ∈ Rn×n
≥0 with entry aij is a weighted adjacency
matrix of the graph G. As we shall see later, under Cobb–Douglas technologies
and competitive factor markets, aij also corresponds to the entries of input–
output tables, measuring the value of spending on input i per value of produced
good j. The entry takes a positive value, aij > 0, if the share of ﬁrm i’s output
in ﬁrm j’s production is positive, and aij = 0 otherwise. By deﬁnition of themodel, self-loops (j, j) ∈ E could be allowed whenever ajj > 0.
We also introduce the notions of in-degree and out-degree to weighted digraphs. In a weighted digraph G, the in-degree of vj refers to the number ofweighted arcs incident to (i.e., directed towards) vj . Similarly, the out-degree of
vj refers to the number of weighted arcs incident from (i.e., directed away from)vj . The cost-shares γj for ﬁrm j coincide with its in-degree.
We have already made suitable specializations of labor supply and production technologies. These assumptions are widely acceptable and make our model
exactly similar to that of Acemoglu et al. (2012). Now, we impose other assumptions on the network structure. In order to verify the criticality hypothesis
to hold without network eﬀect or granular eﬀect, we get rid of the asymmetry of
the network structure and the heterogeneity of ﬁrms.
Assumption 2.1. We assume the following:(i) The cost shares of labor are equal for all ﬁrms: γj = γ.
(ii) All input sizes are equal to one if it is used in production:{
1 if (i, j) ∈ E,xij =0 otherwise.

Inventory Adjustment and Aggregate Fluctuations

8(iii) The optimal inventory policies are homogeneous, and identical lot sizes takepositive integer value: (S − s) ∈ Z≥0 .
Assumptions 2.1 (i) (ii) are also imposed by Acemoglu et al. (2012). Assumption 2.1 (i), which means in-degree equivalence, can be regard as a condition foruniform rate of proﬁt. This would be natural when the economy is in the longrun equilibrium. Assumption 2.1 (ii) is additionally imposed to make the model
more tractable. In addition to these assumptions, we make 2.1 (iii), that is, thehomogeneity of inventory policies. Assumption 2.1 states that ﬁrms are homogeneous in all ways except through the input–output structure. The structure of
the input–output network has no inequality without network formation between
ﬁrms.
Under Assumption 2.1, each∏ price and the price index are given by p =w/(1 − γ)(S − s)n and P ≡ ni=1 pθi i = p. The real total income and the real
total output are expressed by
∑
w ∑
Yn
=
yn ≡
mj = (1 − γ)(S − s)
mj ,
P
np
n

n

j=1

j=1

(16)

which we refer to as an actual output.

2.3

Competitive Equilibrium

In this section, we deﬁne the competitive equilibrium with inventory and derive
the equilibrium quantities under Assumption 2.1. The competitive equilibriumis deﬁned as follows.
Definition 2.1. A competitive equilibrium consists of prices (p1 , p2 , · · · , pn ),wage w, consumption bundle (c1 , c2 , · · · , cn ) and quantities (lj , xj , (xij )) suchthat(i) {ci }ni=1 solves the utility maximization problem of a representative consumer,
taking {pi }ni=1 and w as given.
(ii) {xij }ni=1 and lj solve the proﬁt maximization problem of ﬁrm j for all j ∈{1, 2, · · · , n}; taking {pj }nj=1 and w as given.
(iii) labor and commodity markets are clear, that is,n∑li = 1 ,

(17)

i=1

xi =

n
∑

xij + ci + (hi − h̄i ),(i = 1, 2, · · · , n)

(18)

j=1

where h̄i denotes initial inventories for ﬁrm i.
(iv) All ﬁrms are stable in terms of inventory level, that is,
h = (h1 , h2 , · · · , hn ) ∈ S .

(19)

Inventory Adjustment and Aggregate Fluctuations

9

Notice that Deﬁnition 2.1 (iv) implies the terminal condition for inventory
adjustment process. If there are any ﬁrms out of inventory, then the productions
must take place to recover their stock levels according to the (S, s) policies.
Those productions vary inventory conﬁguration and may cause subsequent stockouts, so that the existence of stock-outs is inconsistent with equilibrium. Thisadditional condition for competitive equilibrium gives a distinction between themodel and the ordinary multi-sector model.
We express the general equilibrium by a closed system of equations. Multiplying both sides of the market clearing condition (18) for good i by its price pi ,and plugging in demand function (13) and factor demand function (4) for goodi, we obtainpi xi =n∑aij (pj xj ) + θi E[Y ] + pi (hi − h̄i ) .

j=1

Let xj = (Sj − sj )mj denote the amount of products for j, and we have thefollowing closed system of equations,



  

1 − a11 −a12 · · ·−a1np1 (S1 − s1 )m1
θ1
p1 (h1 − h̄1 ) −a21 1 − a22 · · · θ2   p1 (h2 − h̄2 ) −a2n   p2 (S2 − s2 )m2 
  

=
E[Y
]
 ..



 ..  + 
 .
..
..
..
 .







.
.
.
.
−an1

−an2

···

1 − ann

pn (Sn − sn )mnθnp1 (hn − h̄n )Hereafter, we suppose the assumption 2.1. Summing up the rows of theclosed system and dividing both sides by the price index P = p, we obtain theaggregate relation in the competitive equilibrium as follows:∗yn = y +n∑(hi − h̄i ).

(20)

i=1

Equation (20) shows the ex-post identity. The left-hand side is the actual output
given by (16) and the ﬁrst term of the right-hand side denotes the eﬀective
demands given by (15). IF the second term, the unplanned inventory investment,is zero, then the actual output is equal to aggregate demands. Therefore, theaggregate ﬂuctuations in this model have the same meaning as the unplanned
inventory ﬂuctuations.
We consider an economy in which there are a large number of homogeneousﬁrms that face a given demand distribution. All ﬁrms select the same S ands points. However, during any particular period, each ﬁrm inherits a diﬀerent
initial inventory stock, h̄j , and receives a diﬀerent order, not only as ﬁnal goods
cj but also as intermediate goods xij . The distribution of the initial inventory
holdings results in a little diﬀerence in the micro-economic state.
In order to solve equilibrium quantities, we need to know the vector m =(m1 , m2 , · · · , mn ) as a function of underlying parameters An = [aij ] and h̄ =(h1 , · · · , hn ), when the demand shares take a certain vector θ = (θ1 , · · · , θn ). If it
were the linear model, we could obtain the solution by inverse matrix [ I−A ]−1 =I + A + A2 + · · · , which is called the matrix multiplier. However, this method,owing to the complexity of lumpiness of adjustment, cannot be applied to themodel where the adjustment process is governed by (S, s)-type inventory policies.

Inventory Adjustment and Aggregate Fluctuations

3

Inventory Adjustment Process

3.1

Operator and Dynamics

10

In counting m, we need to pay attention to the action of individual productions.
Whether or not a ﬁrm produces goods depends on the level of inventory. Recall
that the ﬁrms are divided into two types on the basis of the inventory: stableﬁrms and unstable ﬁrms. Each production takes place only by an unstable ﬁrm.
A stable ﬁrm never produce goods at that moment. All ﬁrms monitor their own
inventory levels to avoid stock-outs at each period and want to maintain thestability of their inventory level. When a ﬁrm becomes unstable, the production
takes place immediately to restore to its target level. In producing goods, theﬁrm needs certain units of goods from other ﬁrms. A single production conductedby ﬁrm j increases its inventory level by (S − s), while it decreases the inventory
level at all associated ﬁrms by (S − s)aij . This inter-dependency of inputs is
captured by the matrix [In − An ].
Here, we deﬁne the production operator ϕj , which describes the action of
inventory adjustment caused by single production of ﬁrm j.
Definition 3.1. The production operator is a mapping ϕj : Zn+ → Zn+ , that is,{
h + (S − s) [ In − An ] ej (hjt < sj )ϕj (h) =(21)h(sj ≤ hjt )where ej = (0, · · · , 1, · · · , 0) is the unit vector in the direction of the j-axis in a
Cartesian coordinate system.
Although the operator ϕj is expressed in terms of the Leontief matrix [In −An ], ϕj is not linear itself. The operator ϕj consists of two cases. If the ﬁrm
j is unstable, then mapping ϕj means the addition of the j-th column of (S −s)[In − An ] to the given inventory conﬁguration h. If the ﬁrm j is stable, thenϕj is ineﬀective. In the former case, we say that the operator ϕj is eﬀective, andin the latter case, we say that ϕj is ineﬀective.
Using these operators, the dynamics of inventory adjustment process can be
represented by
ht =

n
∏

ϕj (ht−1 ) = ht−1 + (S − s)[ In − An ] mt ,

(22)

j=1

where mt denotes the vector of mj obtained by composite applications of ϕ1 ϕ2 · · · ϕnin t. If the corresponding operator ϕj is eﬀective, then mj = 1; otherwise,
mj = 0.
The next example shows us how the adjustment process is carried out andhow m is constructed by (22).
Example 3.1. Suppose that (S, s) = (3, 0) and the adjacency matrix
0 1 0 0 0 1
1 0 1 0 0 01
0 1 0 1 0 0 .

A6 = 
3 0 0 1 0 1 0
0 0 0 1 0 1
1 0 0 0 1 0

Inventory Adjustment and Aggregate Fluctuations

11∑
The optimal lot-size is given by (S−s) = 3. Since in-degree is γ = 6i=1 aij = 2/3, the number of input nodes is given by γ(S − s) = 2. Further, suppose that theinitial inventory conﬁguration is given by h0 = {1, 1, 0, 1, 1, 2}. For simplicity, let us focus on eﬀective operations. Then, the time evolution of inventory
conﬁguration is described as follows:h0 =t (1, 1, 0, 1, 1, 2)

ϕ3 h0 =t (1, 0, 3, 0, 1, 2)⇔ m1 = t (0, 0, 1, 0, 0, 0) ̸= 0

h2 = ϕ2 ϕ4 h1 =t (0, 3, 1, 3, 0, 2)⇔ m2 = t (0, 1, 0, 1, 0, 0) ̸= 0

h3 = ϕ1 ϕ5 h2 =t (3, 2, 1, 2, 3, 0)⇔ m3 = t (1, 0, 0, 0, 1, 0) ̸= 0h4 =ϕ6 h3 =t (2, 2, 1, 2, 2, 3)⇔ m4 = t (0, 0, 0, 0, 0, 1) ̸= 0

h5 =

h4 =

t (2, 2, 1, 2, 2, 3)⇔ m5 = t (0, 0, 0, 0, 0, 0) = 0
..
.

h1 =

..
.

In the beginning of the process, ﬁrm 3 is unstable so that only ϕ3 is the eﬀective
operator to h0 . At t = 1, ϕ3 h0 yields another unstable conﬁguration h1 with
two unstable ﬁrms in position 2 and 4. At t = 2, both ϕ2 and ϕ4 are eﬀective,
but h2 = ϕ2 ϕ4 h1 is still unstable. At t = 3, both ϕ1 and ϕ5 are eﬀective, but
h3 = ϕ1 ϕ5 h1 is still unstable. At t = 4, eﬀective application of ϕ6 gives stable
conﬁguration h4 = ϕ6 h3 . For t ≥ 5, there is no eﬀective operation, whichmeans that inventory conﬁgurations are invariant h4 = h5 = · · · = h∞ andm5 = · · · = m∞ = 0.

3.2

Branching Process

In Example 3.1, {mt }∞t=0 can be solved by hand because the number of ﬁrms, n,
is very small. In general, however, it is hard to get the exact solution of {mt }∞t=0when n is very large. Therefore, we substitute to estimate the aggregation ofvector mt to count its exact components.
In particular, we consider the case where the adjustment process starts with
one unstable ﬁrm. Now, let
Mt ≡

n
∑

mjt .

(23)

j=1

denote the sum of all components of vector mt . The sequence {1, M1 , M2 , · · · }represents the aggregate quantities of inventory adjustment process. A simple
way to account for the statistics of Mt is through the branching process. The
branching process is constructed in the following manner. Let ξ denote thenumber of unstable ﬁrms generated by ∑eﬀective operation of ϕ. From Assumption
2.1, each ﬁrm has the same in-degree, i xij = γ(S − s), and then, ξ is identicalfor all ϕ. By use of ξ, we can construct the sequence {Mt }∞t=1 as follows. At
t = 0, by deﬁnition, the number of unstable ﬁrms is one: M0 = 1. Thus, single
production takes place at t = 1 and generates ξ unstable ﬁrms. Then, M1 = ξ1 .
M1 times of production take place at t = 2, and each single production generates
ξj unstable ﬁrms. Then, M2 = ξ1 + · · · + ξM1 . Similarly, M3 = ξ1 + · · · + ξM2
at t = 3. In the same manner, the sequence {Mt }∞t=1 can be obtained from theInventory Adjustment and Aggregate Fluctuations

12

recursive equation:∑

Mt−1

M0 = 1 and

Mt =

ξj .

(24)

j=1

The next example describes the method to construct the sequence {Mt }∞t=1 .
Example 3.2. Consider the same situation as in Example 3.1. Since the numberof input nodes is 2, then ξ ∈ {0, 1, 2}. The branching process representation of
the inventory adjustment process is obtained as follows:h0 =t (1, 1, 0, 1, 1, 2)

⇔ M0

=1

ϕ3 h 0 =t (1, 0, 3, 0, 1, 2)⇔ M1 = ξ1=2h2 = ϕ2 ϕ4 h1 =t (0, 3, 1, 3, 0, 2)⇔ M2 = ξ1 + ξ2 = 1 + 1 = 2

h3 = ϕ1 ϕ5 h2 =t (3, 2, 1, 2, 3, 0)⇔ M3 = ξ1 + ξ2 = 0 + 1 = 1h4 =ϕ6 h 3 =t (2, 2, 1, 2, 2, 3)⇔ M4 = ξ1

=0

h4 =

t (2, 2, 1, 2, 2, 3)

⇔ M5
..
.

=0

h1 =

h5 =

..
.

Whether or not a ﬁrm would become unstable entirely depends on the respective ﬁrms’ inventory levels. In addition to the division of ﬁrms into stableand unstable ones, we divide the stable ﬁrms into two types: critical and notcritical ﬁrms. We refer to the stable ﬁrm as critical if its inventory is just at thecritical level, that is, sj = hj . Let π be the ratio of critical ﬁrms,
π≡# of critical ﬁrms.
n

The probability that a certain ﬁrm becomes unstable twice in the same adjustment process is of the order of 1/n. If n is suﬃciently large, we can eliminate
the possibility that the chain of productions forms loops.
In the large economy, ξ can be regarded as a random variable according to π.
The characteristics of the stochastic process {Mt }∞t=1 depend on the probability
distribution of ξ. Under Assumption
2.1,
thenumberof ﬁrms who supply to ﬁrm∑
j is equal to its in-degree, ni=1 xij = γ(S − s). Since all ﬁrms are homogeneousand ξ are i.i.d., then the probability distribution of ξ is given by the binomial
distribution,()γ(S − s)pk = P (ξ = k) =π k (1 − π)γ(S−s)−k ,(25)
kwhere k ∈ {0, 1, · · · , γ(S − s)} denotes the number of unstable ﬁrms generated
by a single production. Using the binomial theorem, the probability generating
function of ξ is given by
f (z) =∞∑

pk z k = (πz + (1 − π))γ(S−s) .

k=0

The mean of ξ is ﬁnite and given by E[ ξ ] = f ′ (1) = πγ(S − s).

(26)

Inventory Adjustment and Aggregate Fluctuations

13

Next, let
M =1+

∞
∑

Mt

(27)

t=1

denote the total size of productions taken during the adjustment process. We
consider the probability that the inventory adjustment process terminates after
a ﬁnite number of steps, that is, P (M < ∞).
Lemma 3.1. If πγ(S − s) ≤ 1, then P (M < ∞) = 1.
Proof. See Appendix A.1.
Lemma 3.1 can be considered as the suﬃcient condition for the existence ofcompetitive equilibrium satisfying Deﬁnition 2.1. If πγ(S − s) ≤ 1, the adjustment process almost surely terminates in ﬁnitely many steps. Otherwise, thereis the possibility that the adjustment process never terminates. To be precise,even in the case πγ(S − s) > 1, the adjustment process may terminate, but thisis not certain. This includes the case that contradicts Deﬁnition 2.1 (iv).
The sequence {Mt }∞t=1 is stochastic so that M gives a probability distribution
on N = {0, 1, 2, · · · }. We deﬁne the probability generating function of p(m) =P (M = m) as follows:g(z) = E[z M ] =∞∑p(m)z m(28)

m=0

The next lemma provides a key role in evaluating statistical properties of M .
Lemma 3.2. For πγ(S − s) ≤ 1, the generating function g(z) of M satisﬁes thefollowing equation:g(z) = zf (g(z)) .

(29)

Proof. See Appendix A.2.
By using lemma 3.1, we are able to derive the mean of M .
Lemma 3.3. For πγ(S − s) < 1, the mean of the total size M is ﬁnite and givenbyE[M ] =

1.
1 − πγ(S − s)(30)

Proof of Lemma 3.3. Diﬀerentiating (29), we obtain g ′ (z) = f (g(z))+zf ′ (g(z))g ′ (z).
Rearranging and considering z = 1 leads to E[M ] = g ′ (1) = 1/(1 − f ′ (1)) =1/(1 − πγ(S − s)), which has a ﬁnite value when πγ(S − s) < 1.

4

Properties of the equilibrium

4.1

Stationary Distribution

In this section, we examine the properties of the equilibrium and the aggregate
ﬂuctuations. First, we investigate the stationary distribution in the long-run

Inventory Adjustment and Aggregate Fluctuations

14

equilibrium, as it is called a statistical equilibrium. Let π denote the state probability that the stable ﬁrm is at the critical level of inventory. We notice that
there is no unstable ﬁrm in the equilibrium, and hence, the probability that thestable ﬁrm is not at the critical level can be denoted by 1 − π. In the short-run
equilibrium, the actual output, yn , is obtained under given π. In the long run,however, the probability distribution (π, 1 − π) would vary according to the average of inventory levels changes. The state probability π would increase when
the average level of inventories decreases. In contrast, π would decrease when
the average level increases.
The average level of inventory is aﬀected by exogenous demands and derived
demands caused by inventory adjustments. One unit of exogenous demand makes
the ﬁrm become unstable with probability π. Once a ﬁrm becomes unstable andthe inventory adjustment process starts, it is expected that the total amount of
inﬂows of inventory during the process is (S − s)E[M ] and the total amount ofoutﬂows is γ(S − s)E[M ]. Then, the dynamics of π can be written by
[
{
}
]
dπ
= −β − 1 + π (S − s) − γ(S − s) E[M ]
dt

(31)

where the parameter β represents the speed of adjustment. The statistical equilibrium (π ∗ , 1 − π ∗ ) corresponds to the case dπ/dt = 0.
Proposition 4.1. If the labor share is positive, γ < 1, then there is a uniquestationary distribution given by
π∗ =

1
.
S−s

(32)

Proof of Proposition 4.1. For πγ(S − s) < 1, by combining (30) with (31), we
have
π⪌

1
S−s

⇔

dπ
⪋0.
dt

This implies that there is a unique and stable stationary distribution π ∗ = 1/(S −s). In this case, 1 − π ∗ γ(S − s) = 1 − γ > 0.
Notice that, if γ = 1, the expected inﬂows and outﬂows are equal, and thus,
the external perturbation cannot be absorbed by inventory adjustment process.
Following is a straightforward corollary of Proposition 4.1 and Lemma 3.1.
Corollary 4.1. If the labor share is positive, γ < 1, then there exists an competitive equilibrium satisfying statistical equilibrium almost surely.
Proposition 4.1 also allows us to use the Poisson approximation, that is, whenπγ(S − s) = γ (> 0),p(k) = P (ξ = k) =γ k e−γ,k!

(k = 0, 1, 2, · · · ).

(33)

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to inﬁnity and the expected number ofsuccesses remains ﬁxed. Therefore, it can be used as an approximation of theInventory Adjustment and Aggregate Fluctuations

15

binomial distribution if γ(S − s) is suﬃciently large and π is suﬃciently small.
Under the Poisson approximation (33), the generation function of ξ is given byξf (z) = E[z ] =∞ ( k −γ )
∑
γ e
k=0

k!

z k = e−γ(1−z) .

(34)

It yields both E[ξ] = f ′ (1) = γ and V ar[ξ] = f ′′ (1) + f ′ (1)(1 − f ′ (1)) = γ. The
mean and variance of M can be calculated as below.
Proposition 4.2. For γ < 1, both mean and variance of M are ﬁnite and given,
respectively, by
E[M ] =

1
1−γ

and

V ar[M ] =γ.
(1 − γ)3

Proof of Proposition 4.2. From Lemma 3.2 and (34), E[M ] = g ′ (1) = 1/(1 −f ′ (1)) = 1/(1 − γ). Similarly, V ar[M ] = g ′′ (1) + g ′ (1)(1 − g ′ (1)) = (f ′′ (1) +f ′ (1) − f ′ (1)2 )/(1 − f ′ (1))3 = γ/(1 − γ)3 .

4.2

Aggregate Fluctuation

The main purpose of this paper is to show how the aggregate ﬂuctuations behave in the statistical equilibrium. In particular, we are interested in testingwhether or not the output gap disappears if there is a large number of ﬁrms. We
ﬁrstly express the actual output in terms of M . Let m(θ) be the vector that
represent the number of productions corresponding to the demand share vectorθ = (θ1 , θ2 , · · · , θn ). When n is suﬃciently large, we can eliminate the possibility
that the chain of productions form loops. Then, it seems natural to assume thatm(θ) = θ1 m(e1 ) + θ2 m(e2 ) + · · · + θn m(e∑ n ), where ej is the j-th unit vector.
For any given distribution (π, 1 − π) , nj=1 mj (ei ) = πM + (1 − π)(0) is heldin the short-run equilibrium. Then the actual output can be expressed by theweighted sum of Mi ,yn = (1 − γ)(S − s)n∑mi (y ∗ θ) = π(1 − γ)(S − s)y ∗

i=1

n
∑

θi Mi .

i=1

Consequently, the output gap in the short-run equilibrium can be represented as
(
)
( )
n
∑
yn
log
= log π(1 − γ)(S − s )θi Mi .
(35)
y∗
i=1

The following theorem is the ﬁrst main result of this paper.
Theorem 4.1. We suppose that the labor share is positive, γ < 1, and the economy is in the statistical equilibrium. For arbitrary demand share, (θ1 , θ2 , · · · , θn ),a sequence of output gap converges to zero almost surely, that is,
( )
yn
a.s.
log
−−→ 0 .
∗y
Next, we use two lemmas in proving Theorem 4.1.

Inventory Adjustment and Aggregate Fluctuations

16

Lemma 4.1 (Kolmogorov’s strong law of large numbers). Let {Xk }nk=1 be a sequence of independent random variables, with ﬁnite expectations. If the varianceof Xk is ﬁnite for each k, and
∞
∑
1
k=1

then

(
P

k2

V ar[Xk ] < ∞ ,1∑(Xk − E[Xk ]) = 0
lim
n→∞ n
n

(36)

)
=1.

(37)

i=1

Lemma 4.2 (Limit comparison test). Suppose that we have two series Σk ak andΣk bk with ak , bk ≥ 0 for all k. If limk→∞ (ak /bk ) = c with 0 < c < ∞, then both
series either converge or diverge.
Proof of Theorem 4.1. Because {θi } are given before making a decision, each θiand Mi are independent. From Proposition 4.2, both mean and variance of θi Mi
are ﬁnite.
Recall that exp(εi ) = nθi , and hence,n∑1∑
exp(εi )Mi .
nn

θi Mi =

i=1

(38)

i=1

We set ak = exp(εk )2 /k 2 > 0 and bk = 1/k 2 > 0. Without loss of generality, werearrange {εk }nk=1 in descending order, that is, n > ε1 ≥ ε2 ≥ · · · ≥ εn ≥ ε > 0.
Since {exp(εk )2 } is decreasing and bounded by its inﬁmum exp(ε)2 > 0, thesequence {ak /bk } is convergent and the limit is given as follows:exp(εk )2 /k 2ak
= lim
= lim exp(εk )2 = exp(ε)2 > 0.
n→∞k→∞k→∞ bk1/k 2∑
On the other hand, series k bk has the limit
lim

lim

n→∞

∑

bk = limn→∞k

By Lemma 4.2, the series

(
)
∫ n
1
< lim 1 +dx = 2.
2
k 2 n→∞
1 x

n
∑
1
k=1

∑

k ak =∑

2
2
k exp(εk ) /k converges and we have

∞
∑
1

∞

k=1

k=1

∑ exp(εk )2γVar[exp(ε
)M
]
=
<∞.
i
i
k2
(1 − γ)3k2

Now, applying Lemma 4.1 to (38), we have
(
)
n
)
1 ∑(
P lim
exp(εi )Mi − E[exp(εi )Mi ] = 0 = 1.
n→∞ n
i=1

Since exp(εi ) = nθi and Mi are independent, we have
(
)
n
∑
P lim
θi Mi = E[M ] = 1.
n→∞

i=1

(39)

(40)

Inventory Adjustment and Aggregate Fluctuations

Using (32) and Proposition 4.2, we conclude that
(
( )
)
yn
P lim log
= log(1) = 1,
n→∞
y∗

17

(41)

which states that the output gap converges to zero with probability 1.
Theorem 4.1 states that the aggregate ﬂuctuations disappear in the limitn → ∞. This result is in complete contrast to that of Bak et al. (1993), whodemonstrate that small shocks can create large ﬂuctuations in aggregate level.
The diﬀerence between the two arise from the role of labor share. In Bak et al.
(1993), the results rely on the speciﬁc structure of network and the production
technology. It is possible to consider their model as a special case of our modelin which the labor share equals to zero, γ = 1. In fact, even in the case γ = 1,
the suﬃcient condition for the existence of (short-run) competitive equilibrium
is satisﬁed, so that a competitive equilibrium exists. However, γ = 1 violates
the condition for (long-run) statistical equilibrium stated in Corollary 4.1. Bothmean and variance diverge to inﬁnity. The expected inﬂows and outﬂows ofinventory are equal so that the external perturbation caused by demands cannot
be absorbed by the inventory adjustment process. The energy of adjustment
is conserved in the very long time periods, generating the large ﬂuctuations of
aggregate output. The assumption of zero labor share seems to be special. As wehave already seen, for γ < 1, which seems more realistic, aggregate ﬂuctuations
disappear.

4.3

Impulse Response

In order to give a closer look to this point, we consider the stochastic impulse
response to exogenous perturbations. We suppose that the expenditure share forgood i increases by dθi . We notice that the decrease in each θj to compensate
for the increase in θi can be ignored in the large economy. Diﬀerentiating (35)with respect to θi in the statistical equilibrium, we haved log(yn /y ∗ )= (1 − γ)M ∝ M .
dθi

(42)

The total size of response to the exogenous perturbation is proportional to M .
We refer to this as the stochastic impulse response to the demand shock for goods
i. Therefore, the probability density function of impulse response is proportional
to the probability density function of M ,()
d log(yn /y ∗ )p(g) = P= g ∼ P (M = m)(43)dθi
The total size M is a random variable. Then, the next proposition provides theprobability distribution p(m) = P (M = m) for the case γ ≤ 1, which was ﬁrst
studied by Otter (1949).
Proposition 4.3. We suppose that the labor share is non-negative, γ ≤ 1. The
probability distribution of total size M satisﬁes Borel distribution,
P (M = m) =1 (mγ)m−1 −(mγ)·
em (m − 1)!

(m = 1, 2, 3, · · · ).

(44)

Inventory Adjustment and Aggregate Fluctuations

18

Proof of Proposition 4.3. Diﬀerentiating (29), we have f (g(z)) = g ′ (z)[1−zf ′ (g(z))].
Plugging this into (29) yields
g(z) = zf (g(z)) =∞∑m · p(m)[1 − zf ′ (g(z))]z m .

(45)

m=0

Comparing (28) and (45), we obtain 1 = m[1 − zf ′ (g(z))].
The inverse Z-transform of (28) and the Cauchy integral theorem give p(m)as follows:
I
I ∑
∞
1
1
−m−1
g(z)z
dz =
p(k)z k−m−1 dz
2πi C
2πi C
k=0
I
∞
∑
1
=
p(k)
z k−m−1 dz = p(m)
2πi C
k=0
H
where C denotes the contour integration along a circle surrounding the origin.
Here, we set ω = g(z), and hence,I1ωp(m) =dz
2πi C (ω/f (ω))m+1I1f (ω)m=f (ω)dz
2πi C ω m
I1
f (ω)m=[1 − zf ′ (ω)]dω
2πi C ω m
I1
1
[f (ω)]m
·
=
dω .
(46)
m 2πi C ω m
In the case of Poisson distribution of ξ, from (34), we have
)
∞ (
k
∑
m
−γm(1−z)−(mγ) (mγ)
zk .
f (z) = e
=
e
k!

(47)

k=0

Substituting (47) for (46) and using the Cauchy integral theorem, we obtain
) k
I ∑
∞ (
k
1
1
ω
−(mγ) (mγ)ep(m) =·dωm 2πi C
k!
ωm
k=0
)I
∞ (
1
1 ∑ −(mγ) (mγ)k=·e
ω k−m dω
m 2πi
k!
C
=

k=0
(mγ)m−11·e−(mγ) .
m (m − 1)!

(48)

Then, we conclude the proposition.
The probability density function p(g) is proportional to p(m). The following
theorem is the second main result in this paper.
Theorem 4.2. In the competitive equilibrium, the probability density function of
impulse response to the demand shock for goods i is analytically given as()
g − 23 · exp −g
if γ < 1
ζ(γ)p(g) ∼(49)
 − 23g
if γ → 1

Inventory Adjustment and Aggregate Fluctuations

19

where correlation length is given by
ζ(γ) =

1.
γ − 1 − ln γ

(50)Proof of Theorem 4.2. From Lemma 4.3,
1 (gγ)g−1 −(gγ) (gγ)g−1 −(gγ)
·
e
=
e
.
g (g − 1)!
g!
√
By Stirling’s formula n! ∼ 2πn (n/e)n , we obtain( (1−γ) )g
γe
3
3
γ −1
p(g) ∼ √· g − 2 = √ · g − 2 · e(1−γ+ln γ)g2πγ2πp(g) ∼(51)

which satisﬁes the conclusion of the theorem.
We notice that the correlation length (50) is related to the labor share. The
probability density function (49) is simply divided into two classes. In the casethat the labor share is positive, γ < 1, the correlation length is ﬁnite. For g
smaller than ζ(γ), p(g) is well approximated by the power law with exponent−3/2. However, for large g, the exponential decay dominates the power law decay. The resulting distribution has an exponential tail, with a characteristic scale
given by the correlation length. The impulse response ﬁnitely converges. The
probability of having the propagation size larger than ζ(γ) is extremely small.
This means that the eﬀect of idiosyncratic shocks get extinct early, and could notpropagate larger than the scale given by (50). The inventory adjustment process
in no way can propagate the micro-level lumpiness larger than the certain scale.
However, precisely in the case that the labor share is zero, γ = 1, the correlation length (50) diverges to inﬁnity. This means that the critical point of thiseconomy is the case where the labor share is zero. Then, the exponential part
disappears and the distribution exhibits a power law decay even for suﬃciently
large g. The size of the impulse response diverges so that the eﬀect of shocks cannot be absorbed and conserved in the economic system. Since the characteristic
scale is absent at the critical point, huge propagations are possible. However,as we have already mentioned, it seems unreasonable to assume that the labor
share is zero.

5

Conclusion

This paper investigates the relationship between lumpy inventory adjustment
and aggregate ﬂuctuations. We present a general equilibrium model composedof many ﬁrms interacting within the input–output network. The main feature of
the model is that each ﬁrm follows the (S, s) inventory policy and the production
processes in any ﬁrm make use of input from any other ﬁrms in the economy.
The (S, s) inventory policy leads the ﬁrm to lumpy production pattern and theinteraction between ﬁrms can create the chain of productions. It is possible that
ﬂuctuations caused by micro-level lumpiness propagate over the input–output
linkages across diﬀerent ﬁrms within the economy.
The contributions of this paper are as follows. First, we provide a tractable
method to compute the aggregate output in a general-equilibrium framework

Inventory Adjustment and Aggregate Fluctuations

20

with (S, s) inventory policies. We characterize the competitive equilibrium andgive the closed form solution. Because of the complexity of the lumpy adjustment process, it is hard to determine the exact amount of micro-level quantities.
Then, we employ the branching process to estimate the key statistics of inventory adjustment process. We deﬁne the operator that represents the inventory
adjustment caused by single production and describe the chain of productions by
sequentially applying these operators. We use the statistics of this sequence instead of the exact calculation of the adjustment process. Using these estimates,
we derive the stationary distribution in the statistical equilibrium. The competitive equilibrium restricts the short-run equilibrium when the distributionof inventory level across ﬁrms is given. In the long run, the economy reaches
the stationary state in distribution. We prove that the stationary distributionof this economy can be determined by the (S, s) band under the homogeneity
assumption.
Second, we show that the strong law of large numbers holds in our framework. We prove that when the network structure is symmetric and all ﬁrms arehomogeneous, the output gap converges to zero almost surely in the statisticalequilibrium. In other words, if the economy is suﬃciently large, the aggregate inventory ﬂuctuations do not solely arise from a combination of micro-level lumpiness and economic interactions among ﬁrms. This implies that the micro-level
lumpiness has negligible impact on the macro-level activity in this framework.
We also show that the positive labor share reduces the high ﬂuctuations
in aggregate inventory level and moderates the business cycle. This can beunderstood from the viewpoint of impulse response to demand shock. If the labor
share is positive, the probability density function of size of impulse response has
an exponential tail. The inventory adjustment process cannot propagate themicro-level lumpiness larger than the certain scale. Further, only in the limit
case where the labor share is zero, the exponential part disappears and the
distribution exhibits a power law decay.
From the results of this paper, we can conclude that the criticality hypothesis
does not hold by itself in our framework. In particular, if labor share is positive, system criticality does not solely arise from a combination of lumpiness,
attributed to non-convexity, and ﬁrms’ local interactions, without network or
granular eﬀects. Both network and granular eﬀects are essential sources of large
ﬂuctuations. Nevertheless, this does not imply that non-convexity is a meaningless factor in aggregate ﬂuctuations. If a network structure is asymmetricor ﬁrm sizes are heterogeneous, the eﬀect of lumpiness from non-convexity may
not be negligible in macro-level activities. In such situations, it seems possiblethat the ampliﬁcation mechanism of both network and granular eﬀects can bestrengthened by micro-level lumpiness in the inventory adjustment process.

A

Appendix

A.1

Proof of Lemma 3.1

Proof. The inventory adjustment process terminates when Mt = 0. Then, all
the subsequent Mt are also zero. The probability that the adjustment process

Inventory Adjustment and Aggregate Fluctuations

21

terminate until step T is given by(T)∪
q (T ) = P{Mt = 0} ∈ [0, 1] .

(52)

t=1

Note that {Mt = 0} ⊆ {Mt+1 = 0} for all t = {1, 2, · · · }. The probability thatq (T ) is increasing in T and is bounded above by 1, and then, q = limT →∞ q (T )exists.
Under Assumption 2.1, we can consider Mt+1 as the sum of independent
copies of Mt . We suppose that M1 = k, and then, we have(1)

E[z Mt+1 | M1 = k] = E[z Mt

(2)

(k)

+Mt +···Mt] = (E[z Mt ])k(53)

Let ft (z) = E[z Mt ] denote the probability generating function of Mt . Plugging(53) into ft+1 (z), we get the following recursion:
ft+1 (z) = E[z

Mt+1

]=

∞
∑

E[z Mt+1 | M1 = k]P (M1 = k)

k=0∞∑(ft (z))k pk=
k=0= f (ft (z))(54)

Since ft (0) = P (Mt = 0) = q (t) , then q (t+1) = f (q (t) ). We obtain q = f (q) by
taking t → ∞. Therefore, q is the attractive ﬁxed point of the map z 7→ f (z).
From deﬁnition, f (1) = 1. In this model, 0 < pk < 1 for all k ≥ 0, andthen, f (0) = p0 > 0. By diﬀerentiating f (z) repeatedly, we have f ′ (z) > 0 andf ′′ (z) > 0, which implies that f (z) is increasing and strictly convex for z ∈ [0, 1].
If f ′ (1) = πγ(S − s) ≤ 1, the map z → f (z) has a unique ﬁxed point z ∗ = 1. If
f ′ (1) = πγ(S − s) > 1, the map z → f (z) has two ﬁxed points and the smallest
one is attractive: z ∗ = f (z ∗ ) with z ∗ < 1. These results are summarized by{
1 if πγ(S − s) ≤ 1
q=(55)
q ∗ if πγ(S − s) > 1
with q ∗ < 1.

A.2

Proof of Lemma 3.2

Proof. (See also Harris(1963) Theorem 8.2.) Suppose the number of productionsin t = 1 is given by k, then the total number M can be decomposed into M =1 + M (1) + M (2) + · · · + M (k) . The M (j) are independent and have the same
distribution as M . Thus, we haveE(z M | M1 = k) = E(z 1+M(1) +M (2) +···+M (k))= zE(z )E(z ) · · · E(z )M= zg(z)k .

M

M

(56)

Inventory Adjustment and Aggregate Fluctuations

22

From this,g(z) = E(z M ) =∞∑E(z M | M1 = k)P (M1 = k)

k=0
∞
∑

=z

pk [g(z)]kk=0= zf (g(z)).

(57)

References[1] Daron Acemoglu, Vasco M Carvalho, Asuman Ozdaglar, and Alireza
Tahbaz-Salehi. The network origins of aggregate ﬂuctuations. Econometrica, 80(5):1977–2016, 2012.
[2] Kenneth J Arrow, Theodore Harris, and Jacob Marschak. Optimal inventory policy. Econometrica, 19:250–272, 1951.
[3] Per Bak, Kan Chen, José Scheinkman, and Michael Woodford. Aggregate
ﬂuctuations from independent sectoral shocks: self-organized criticality ina model of production and inventory dynamics. Ricerche Economiche,
47(1):3–30, 1993.
[4] Alan S Blinder and Louis J Maccini. Taking stock: a critical assessmentof recent research on inventories. The journal of economic perspectives,
5(1):73–96, 1991.
[5] Ricardo J Caballero and Eduardo MRA Engel. Dynamic (s, s) economies.
Econometrica, 59(6):1659–1686, 1991.
[6] Andrew S Caplin. The variability of aggregate demand with (s, s) inventory
policies. Econometrica, 53(6):1395–1409, 1985.
[7] Vasco M Carvalho. From micro to macro via production networks. Journalof Economic Perspectives, 28(4):23–48, 2014.
[8] Deepak Dhar. Self-organized critical state of sandpile automation models.
Physical Review Letters, 64(23):2837, 1990.
[9] Bill Dupor. Aggregation and irrelevance in multi-sector models. Journalof Monetary Economics, 43(2):391–409, 1999.
[10] Xavier Gabaix. The granular origins of aggregate ﬂuctuations. Econometrica, 79(3):733–772, 2011.
[11] Theodore Edward Harris. The theory of branching process, 1963.
[12] Michael Horvath. Cyclicality and sectoral linkages: Aggregate ﬂuctuations from independent sectoral shocks. Review of Economic Dynamics,
1(4):781–808, 1998.

Inventory Adjustment and Aggregate Fluctuations

23[13] Jr. Long, John B. and Charles I. Plosser. Real business cycles. Journal of
Political Economy, 91(1):39–69, 1983.
[14] Robert E Lucas. Understanding business cycles. In Carnegie-Rochester
conference series on public policy, 5:7–29. Elsevier, 1977.
[15] Makoto Nirei. (s, s) inventory dynamics and self-organized criticality. Research in Economics, 54(4):375–383, 2000.
[16] Makoto Nirei. Threshold behavior and aggregate ﬂuctuation. Journal of
Economic Theory, 127(1):309–322, 2006.
[17] Richard Otter. The multiplicative process. The Annals of Mathematical
Statistics, 206–224, 1949.
[18] Valerie A Ramey and Kenneth D West. Inventories. Handbook of macroeconomics, 1:863–923, 1999.
[19] Herbert Scarf. The optimality of (s,s) policies in the dynamic inventory
problem. mathematical methods in the social science, KJ Arrow, S Karlin,
P Suppes, eds, 1960.
[20] Jose A Scheinkman and Michael Woodford. Self-organized criticality andeconomic ﬂuctuations. The American Economic Review, 417–421, 1994.