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Inter-temporal Dynamics of Cost Asymmetry
Eti Einhorn
Tel Aviv University
einhorn@post.tau.ac.il
Efrat Shust
The Open University of Israel
efratshu@openu.ac.il
Abstract: Despite its large body and fundamental importance, the literature on asymmetric cost behavior
has evolved empirically and lacks theoretical foundation. To address this gap in the literature, we introduce
a novel theoretical framework for studying the dynamics of cost behavior over time in a competitive
environment. Our theoretical analysis delves into the primitives of asymmetric cost behavior and explores
its inter-temporal dynamics. The analysis yields the result that cost behavior is asymmetric by default,
transpiring in the pattern of cost stickiness following resource expansion while taking the form of cost
anti-stickiness subsequent to resource contraction. The analysis further suggests that the magnitude of
cost asymmetry, either stickiness or anti-stickiness, is decreasing in demand uncertainty and competition
intensity, increasing in adjustment costs and time discounting, and greater for small demand shocks than
for large demand shocks.
Keywords: cost behavior; cost asymmetry; cost stickiness; cost anti-stickiness.
1
1. Introduction
The topic of cost behavior lies at the heart of the academic literature in managerial accounting and
attracts wide research interest due to its critical implications for firms’ profitability. Modern cost behavior
research deviates from the traditional view that costs are proportionally adjusted to changes in demand. This
paradigm shift is manifested in the influential study of Anderson, Banker, and Janakiraman (2003), showing
empirical evidence of the tendency of costs to increase more when revenues rise than to decrease when
revenues fall by an equivalent amount. The observed asymmetric cost response to positive versus negative
changes in demand, known as cost stickiness, implies that firms choose different resource levels for the same
demand level depending on whether demand increased or decreased from the prior period. This surprising
finding has triggered an extensive empirical research into the fundamental determinants of the cost stickiness
pattern and the reasons for its varying degrees across different firms.
1
The explanations provided in the
literature for cost stickiness include asymmetric upward versus downward resource adjustment costs (e.g.,
Banker, Byzalov, and Chen 2013), asymmetric persistence of positive versus negative demand shocks
(e.g., Banker and Byzalov 2014), managerial incentives and preferences such as empire building
motivation (e.g., Anderson, Banker, and Janakiraman 2003; Chen, Lu, and Sougiannis 2012), managerial
beliefs and behavioral biases such as optimism or overconfidence (e.g., Qin, Mohan, and Kuang 2015; Chen,
Kama, and Lehavy 2019), and flaws in empirical measurement (e.g., Banker and Byzalov 2014; Riegler and
Weiskirchner-Merten, 2021). The literature also identifies factors that mitigate the pattern of cost stickiness
and even trigger the opposite pattern of cost anti-stickiness, as introduced by Weiss (2010). Among them,
unutilized resources (e.g., Chen, Kama and Lehavy 2019), strong corporate governance (e.g., Chen, Lu, and
Sougiannis 2012), and managerial incentives to meet earnings targets (e.g., Dierynck, Landsman, and Renders
2012; Kama and Weiss 2013). This line of literature, in its vast majority, has evolved empirically, and it
thus lacks a theoretical foundation.
2
Recognizing this gap in the literature, we offer a novel theoretical
framework to analyze the rationale behind cost asymmetry and its inter-temporal dynamics.
Unlike the empirical literature, which suggests both rational and irrational explanations for
asymmetric cost behavior, but faces difficulties in disentangling between them, we establish cost
asymmetry on a purely rational ground. Our analysis is built on a multi-period setting with multiple
identical firms that operate in the same product market, producing and selling the same product. The
firms in our model repeatedly compete à la Cournot over an infinite number of successive periods. The Cournot
competition game between the firms in each period is subject to changing demand over time, and it departs
1
See Banker and Byzalov (2014) for a survey of the cost stickiness literature.
2
See Labro (2015) on the lack of theoretical underpinning of the cost stickiness pattern, which hardens the work of empiricists.
2
from the classical Cournot setting by the inclusion of resource adjustment costs. To capture the intertemporal
dynamics of the demand for the products of the firms, we assume an exogenous random demand that follows
a Markovian stochastic process, where the demand of each period is realized at the beginning of the period
and its realization is drawn from a distribution whose mean is the demand of the previous period. We preclude
from the model any frictions already known in the literature to cause cost stickiness or cost anti-stickiness.
Specifically, the adjustment cost per unit of resource is assumed to be symmetric for upward and downward
adjustments. The exogenous random demand in each period is assumed to be distributed symmetrically,
upward and downward, around the demand of the previous period. Since unutilized resources are known as a
determinant of cost anti-stickiness, we also restrict the analysis to fully utilized resources by assuming the
adjustment cost per unit of resource adjustment is lower than the ongoing cost per unit of retaining resources.
We further assume that the managers of the firms are risk-neutral and rational, devoid of any behavioral biases
and motivated to maximize the economic value of their firm.
As a natural starting point of reference, we consider first the benchmark case where resource
adjustments are costless. In the absence of resource adjustment costs, intertemporal concerns are excluded
from the managers’ strategic considerations when making resource adjustments in response to demand shocks,
and thus the competition game in each period is independent of other periods. The benchmark case
therefore yields the classic Cournot equilibrium in each period. Accordingly, the optimal choice of
resource adjustment of each firm in each period is proportional to the demand shock realization. Hence,
when resource adjustment costs are absent, firms adjust resources upward in response to a positive
demand shock and symmetrically adjust resources downward in response to a negative demand shock of
an identical magnitude. This symmetric cost behavior, however, does not carry over to the case where
resource adjustments are costly.
To demonstrate that the existence of resource adjustment costs generates asymmetric cost behavior,
even when assuming an otherwise frictionless setting, we move on to analyzing the case where resource
adjustments are costly. When expanding production upon a positive demand shock, each additional unit of
production is more expensive since it involves both the resource adjustment cost and the production cost.
Consequently, production expands in response to an increase in demand, but reaches a lower level compared
to the benchmark case (under-production). Conversely, when reducing production in response to a negative
demand shock, the removal of each unit entails a smaller cost saving since the firm avoids the production cost
but incurs the resource adjustment cost. As a result, the production contracts in response to a decrease in
demand, but to a higher level relative to the benchmark case (over-production). Hence, adjustment costs
restrain both upward and downward resource adjustments relative to the benchmark case. So, the equilibrium
3
resource adjustment in the presence of resource adjustment costs is no longer proportional to the demand
shock. The analysis reveals that firms are more likely to adjust their resources and adjust them more
significantly in response to a demand shock that follows the course of the previous shock (positive or negative)
than they do in response to a demand shock reversed to the previous shock. Subsequent to an upward resource
adjustment, a consecutive expansion shifts the production upward from under production in the past to a similar
under production in the present, thereby generating full adjustment (equal to the benchmark). Resource
contraction following prior resource expansion, however, shifts the production downward from under
production in the past to over production in the present, thus triggering a smaller adjustment. Therefore, a prior
resource expansion stimulates cost stickiness. Similarly, following a downward resource adjustment, a
consecutive negative demand shock shifts the production downward from over production in the past to a
similar over production in the present, while resource expansion causes a smaller adjustment that shifts the
production upward from over production in the past to under production in the present. Hence, a prior resource
contraction induces cost anti-stickiness.
We therefore show that resource adjustment costs generate asymmetric cost behavior, even when
assuming otherwise frictionless setting. Both cost stickiness and cost anti-stickiness are expected to rationally
and naturally appear in equilibrium. Our analysis further explores the dynamic over time of cost behavior,
suggesting that the direction of cost asymmetry is determined by the direction of the prior resource adjustment,
and showing that cost asymmetry emerges in the form of cost stickiness following resource expansion and
transpires in the form of cost anti-stickiness after resource contraction. In addition to exploring the determinants
of the direction of cost asymmetry, the analysis also sheds light on the economic forces that affect the
magnitude of cost asymmetry. We show that the magnitude of cost asymmetry, either stickiness or anti-
stickiness, increases in the resource adjustment cost, increases in the time discounting, decreases in the number
of competing firms, and decreases in the demand uncertainty. A higher resource adjustment cost has a more
significant restraining impact on both upward and downward resource adjustments, and accordingly the
resulting cost asymmetry is increasing in the resource adjustment cost. An increase in the number of competing
firms has the opposite effect of decreasing the cost asymmetry because the response of firms to demand shocks
is less significant under enhanced competition, and so is the proportional restraint in resource adjustments due
to adjustment costs. A decrease in the time discounting also has a mitigating effect on the restraint in resource
adjustment and on the resulting cost asymmetry, because resource adjustment in response to the demand shock
in each period is likely to continue serving the firm in the future, so managers are more willing to adjust
resources when the time discounting is lower. The effect of demand uncertainty on cost asymmetry is subtler.
As uncertainty increases, the potential demand shocks become larger, in absolute terms, and so does the size
4
of the corresponding resource adjustments. Conversely, the magnitude of the restraint in resource adjustment
remains constant, regardless of the demand shock. Therefore, it becomes smaller relative to the adjustment
size as demand uncertainty increases, mitigating the cost asymmetry.
The paper proceeds as follows. In Section 2, we present the model underlying the analysis, which is
by design devoid of any friction already known in the literature to cause cost asymmetry. In Section 3, we
derive the equilibrium outcomes that the model yields and analyze their economic properties and implications.
Section 4 summarizes and offers concluding remarks. The proofs of all the results appear in the appendix.
2. Model
We consider a multi-period setting with identical firms. The firms operate in the product market,
producing and selling the same product, for an infinite number of periods. The special case of captures
a monopoly market, whereas the case of depicts an oligopoly, ranging from a duopoly when to
a perfect competitive market when approaches infinity. In each period, the firms compete à la Cournot,
choosing simultaneously and independently the quantity they produce and sell in response to an exogenous
demand shock. As our focus is the analysis of the outcomes of the competition in a representing period , we
assume an infinite number of past periods and an infinite number of future periods, and thus allow the period
index to be any integer number.
The number of product units that firm chooses to produce and sell in period is
denoted . For simplicity, and without loss of generality, we assume that the production of each unit of
product requires one unit of resources, and thus represents the resource quantity of firm in period .
Accordingly, represents the resource adjustment of firm in period . Assuming a standard linear
demand function for the product, we represent the retail price for each unit of product in period by
, where is the demand for the product in period . The uncertain demand for the product in period
is represented by the random variable , whose realization becomes publicly known at the beginning of
period . We assume that the dynamics over time of the demand follows a Markovian stochastic process.
Specifically, the random variable is assumed to be uniformly distributed around a mean that equals to the
demand of the previous period and with a variance of . Hence, .
3
3
By the properties of the uniform distribution, the mean of the random variable equals
, and its variance equals . The
uniform distribution of the demand simplifies the analysis significantly, but it might generate over time a negative demand
realization. To approximately preclude scenarios where the demand realization gets negative, we assume that the demand
mean exceeds the demand variance by a large amount, so that the probability of a negative demand is negligible.
5
The marginal cost of operating one unit of resources, per one period, is given by the positive scalar .
Consistent with prior studies (e.g., Banker and Hughes, 1994; Göx 2002), resource adjustment is assumed
to be costly. Our assumption is that any resource adjustment of one unit, upwards or downwards, entails
a positive one-time adjustment cost of . The parameter reflects, for example, the one-time cost
per unit of hiring (firing) labor or installing (liquidating) machinery, which is associated with an upward
(downward) resource adjustment. We assume that , implying that it is always worthwhile for firms
to avoid unutilized resources.
We denote the profit of firm in period by , and assume that the profit is distributed as a
dividend to the shareholders at the end of period . The profit of firm in period is given by
or, equivalently, can be also represented as
. We assume that the managers of the firms are rational and risk neutral, and they all aim at
maximizing the expected economic value of their firm. Hence, in making the production quantity decision in
each period , the manager of each firm strives to maximize the present value
of the
accumulated profits of firm in period and in the successive periods discounted by a discount rate of .
The existence of resource adjustment costs generates a linkage between the profit of the firms in the current
period to their production quantity in the previous period, and it is thus the source of the intertemporal dynamics
of the firms’ strategic behavior.
We preclude from the model any frictions already known in the literature to cause cost stickiness or
cost anti-stickiness. Specifically, the cost per unit of resource adjustment is assumed to be symmetric for
upward and downward adjustments. The exogenous demand shock in each period is depicted in the model
by the realization of the random variable , which is assumed to be symmetrically
distributed around zero. Since unutilized resources are known as a determinant of cost anti-stickiness, we also
restrict the analysis to fully utilized resources by assuming the adjustment cost per unit of resource
adjustment is lower than the marginal production cost . We further assume that the managers of the firms are
all rational, devoid of any behavioral biases, and motivated to maximize the economic value of their firm.
Equilibrium in the competing game between the firms in each period consists of the production
quantities of the firms in period . Accordingly, equilibrium in the competition game between the firms in
each period is formally defined as a vector of their production quantities. We look for
Nash equilibrium, in which the managers of all firms make optimal decisions that maximize the economic
value of their firms given all their available information, as well as their rational expectations regarding the
6
strategic behavior of their rivals, utilizing Bayes’ rule to make inferences and update their beliefs. In
equilibrium, for any , is the optimal production quantity that maximizes the expected
economic value of firm in period , given the production quantity of firm in the previous period
, conditional on the realization of the demand shock in period , and based on the rational expectations
about the production quantity of any other firm in period .
Cost stickiness (anti-stickiness) transpires when costs tend, on average, to increase more (less)
when the demand rises than to decrease when the demand falls by an equivalent amount.
4
To analyze the
direction and magnitude of the cost asymmetry that the model yields in equilibrium, we formulate a measure
of cost asymmetry of firm in period and denote it . In defining the cost asymmetry
measure, we use the notation to denote the last period of conducting resource adjustment prior to period
, and we similarly use the notation to denote the last period of conducting resource adjustment prior
to period . Utilizing this notation, the cost asymmetry measure is defined as
.
5
The measure is the difference between two
terms. The former is the ratio of the average resource adjustment of firm in period to the corresponding
average demand shock, conditional on a positive demand shock. The latter is the ratio of the average
resource adjustment of firm in period to the corresponding average demand shock, conditional on a
negative demand shock. A positive (negative) value of this measure reflects cost stickiness (anti-stickiness),
because it indicates that the upward resource adjustment upon a given demand increase is, on average, greater
(smaller) than the downward resource adjustment upon an equivalent demand decrease. A value of zero stands
for symmetric cost behavior. While the sign of the cost asymmetry measure indicates whether the cost
behavior is sticky or anti-sticky, its absolute value captures the magnitude of cost asymmetry.
6
4
In the extant literature revenues and demand are used interchangeably when discussing cost stickiness. We note that demand
is an exogenous variable, whereas revenues are endogenously determined based on current and expected demand, available
resources and adjustment costs. Therefore, while revenues serve in empirical studies as a proxy for demand, on a theoretical
ground the driving force behind resource adjustment decisions of managers is change in demand.
5
As we show below, in equilibrium, firms may not adjust resources if the demand shock is sufficiently small and the
adjustment is costly. Therefore, the amount of resource in period is the consequent of demand in period , which
is the most recent period in which resource adjustment took place.
6
The measure of cost asymmetry can be alternatively defined in terms of the change in costs, instead of the
change in quantities, , but this alternative measure equals and thus it is proportional to , so that the
two alternative definitions are equivalent. Empirical measures of cost asymmetry are defined in terms of costs, since data
on quantities is not available to researchers. They usually rely on the total reported costs, although only the change in the
ongoing production costs is relevant for depicting resource adjustments, due to the difficulty in separating between ongoing
production costs and one-time adjustment costs. Thus, the commonly used empirical measure of changes in costs as a proxy
for resource adjustments ascribe a cost of per unit of upward adjustment and a lower cost of per unit of
downward adjustment, generating a systematic empirical bias toward cost stickiness in estimating cost asymmetry.
7
3. Equilibrium Analysis
3.1. Benchmark
As a natural starting point of reference, it is convenient to consider first the benchmark case of
where resource adjustments are costless. The assumption excludes intertemporal concerns from the
firms’ strategic considerations and thereby eliminates the linkage between their strategic decisions over time.
In the absence of resource adjustment costs, the choice of the production quantity of each firm in each
period is independent of the production quantity in the previous period and has no effect on future
profits. The benchmark case of therefore yields the classic Cournot equilibrium in each period .
This argument is formally stated in Observation 1, using the superscript in presenting the
equilibrium outcomes of the benchmark case.
OBSERVATION 1. In the benchmark case of , there exists a unique equilibrium
in each period , which is given by
. In
equilibrium, the resource adjustment of each firm in each period is
. The cost
asymmetry measure satisfies
independently of and .
As shown in Observation 1, in the benchmark equilibrium, the production quantity of each firm in
each period upon the demand realization is
, as in the classical equilibrium that a one-
shot Cournot competition game yields. Hence, in the absence of resource adjustment costs, the
competition game in each period is independent of the competition games in other periods. Accordingly,
the optimal choice of resource adjustment
of each firm in each period is given by
and is proportional to the demand shock realization . Hence, when resource
adjustment costs are absent, firms adjust resources upward in response to a positive demand shock and
symmetrically adjust resources downward in response to a negative demand shock of an identical
magnitude. Figure 1 graphically illustrates this symmetric cost behavior. The blue diagonal line describes
the resource quantity
of each firm in period as a function of the demand in period .
The resulting resource adjustment of each firm in period is illustrated in the figure under two scenarios
with respect to the demand shock in period . The first scenario (illustrated in green) pertains to a positive
demand shock in period , which causes an upward resource adjustment of
units,
marked by the green arrow, from the prior quantity of
to the current higher quantity of
8
. The second scenario (illustrated in orange) pertains to a negative demand shock of
the same amount in period , which causes a downward resource adjustment of
units, marked by the
orange arrow, from the prior quantity of
to the current lower quantity of
. This
symmetric resource adjustment in response to positive and negative demand shocks of the same amount
is reflected in the cost asymmetry measure
, which is always zero, as indicated by Observation 1.
The symmetric cost behavior, however, does not carry over to the case where resource adjustments are
costly, as will be established in the next sections.
[Figure 1]
3.2. Equilibrium under extreme time discounting
To demonstrate our argument that the existence of resource adjustment costs generates both cost
stickiness and cost anti-stickiness, even when assuming an otherwise frictionless setting, we move on to
analyzing the case of where resource adjustments are costly. To facilitate the exposition, we build and
present the analysis in two steps. We first consider in this section the case where the discount rate approaches
infinity. In the next section, we relax this assumption and analyze the unrestricted model. The case where
the discount rate approaches infinity captures situations of an extreme time discounting, where decision
making aims only at maximizing the profit of the current period while ignoring the expected profits in
future periods. The analysis of this case highlights that resource adjustment costs trigger an intertemporal
linkage between the managers’ strategic decisions even when they consider only the current period.
Proposition 2 explores the equilibrium outcomes under such extreme time discounting, using the superscript
in presenting them.
PROPOSITION 2. In the case of , there exists a unique equilibrium
in each
period . The equilibrium in period takes the following form:
If , then
If , then
9
Proposition 2 implies that the presence of resource adjustment costs results in under-production in
response to positive demand shocks and over-production in response to negative demand shocks, as compared
to the benchmark case. Hence, resource adjustment costs work to restrain both upward and downward resource
adjustments relative to the benchmark case. This restraint stems from the effect of adjustment costs on the
marginal production cost. Specifically, when expanding production, each additional unit of product is more
expensive to produce since it entails both the adjustment cost and the operating cost per one unit of
resource, so the cumulative marginal cost is . Consequently, a positive demand shock increases
production, but to a level lower by the amount of
relative to the benchmark case (under-production).
Conversely, when scaling down production, the elimination of each unit of product entails a smaller cost saving
since the firm avoids the production cost but incurs the adjustment cost , hence costs decline by per
unit. As a result, the production contracts upon a negative demand shock, but to a level higher by the amount
of
relative to the benchmark case (over-production).
Note that the resource level of each firm in each period depends not only on the latest demand shock
but also on the sign of the previous demand shock that triggered a resource
adjustment. The sign of the previous demand shock determines whether the firm begins the current period in
a state of under-production or over-production, thus influences the response to the current shock. Moreover,
unlike the benchmark case, where all demand shocks trigger resource adjustment, in the presence of resource
adjustment costs firms sometimes strategically choose to refrain from adjusting resources in response to
demand shocks. Such is the case for a sufficiently small negative shock following prior expansion. The
rationale is that ensuing the prior positive shock each firm operates in under-production anyhow. Therefore, a
small decrease in demand does not merit further resource cut down considering the attached adjustment cost.
The same applies for a sufficiently small positive shock following a prior contraction. In this case each firm
already operates in over-production, so a small increase in demand does not trigger a costly resource addition.
[Figure 2]
These results of Proposition 2 are graphically illustrated in Figure 2. The green and the orange
curves describe the resource quantity
of each firm in period as a function of the demand in
period , under two scenarios with respect to the last prior demand shock that triggered
a resource adjustment. The green curve pertains to the scenario where the prior demand shock was
positive, and the orange curve pertains to the scenario where it was negative. For comparison, the blue
diagonal dashed line describes the benchmark resource quantity as a function of the demand . The
10
green curve demonstrates that, following a prior resource expansion in period , each firm keeps resource
quantity at its previous level for sufficiently small negative demand shocks in the range in period .
Larger negative shocks decrease resource quantity to the level of
, which reflects over-production
relative to the lower benchmark quantity of
. Positive shocks, however, always trigger an upwards
adjustment to the level of
, which reflects under-production relative to the benchmark quantity.
Therefore, following an upward resource adjustment, a successive upward adjustment is more likely to occur
than a subsequent downward adjustment. The orange curve depicts a mirror image. Following a resource
contraction in period , each firm keeps resource quantity at its level in period for sufficiently small
positive demand shocks in the range in period . Upon larger positive demand shocks, the resource
quantity increases to the level of
, which reflects under-production relative to the benchmark quantity.
Nevertheless, all negative shocks trigger a downward adjustment to the level of
, reflecting over-
production relative to the benchmark. Thus, following a downward resource adjustment, a successive
downward adjustment is more likely to occur than a subsequent upward adjustment. The green and the orange
curves coincide into the same curve for sufficiently large demand shocks, but they diverge from each other for
small demand shocks that belong to the range . This implies that firms may choose in equilibrium
different resource levels in response to the same demand level and to the same demand shock ,
depending on whether the previous demand shock was positive or negative. While
Proposition 2 and Figure 2 present the equilibrium outcomes in terms of the relationship between resource
quantity and the corresponding demand shock, the following Corollary recasts the equilibrium results in terms
of the relationship between resource adjustment (the amount of resources added or removed in the current
period) and the corresponding demand shock.
COROLLARY TO PROPOSITION 2. In the case of , the equilibrium in period satisfies for any
:
If , then
If , then
11
The corollary indicates that subsequent to a positive demand shock that caused a prior
upward resource adjustment in period , a successive positive demand shock in period generates an
upward resource adjustment which is greater than the downward adjustment that an equivalent negative
demand shock generates. This implies that a prior upward resource adjustment stimulates cost stickiness. The
opposite occurs following a negative demand shock that caused a prior downward resource adjustment in
period . Hence, a prior downward resource adjustment induces cost anti-stickiness. These results are
graphically illustrated in Figure 3, where Figure 3a pertains to the case of and Figure 3b
relates to the case of .
We start by considering the illustration in Figure 3a, which pertains to the case where the most recent
resource adjustment was an upward adjustment triggered by a positive demand shock .
Following the upward adjustment in period , production level was set to
, lower than the benchmark production for a similar demand by
. A successive positive
demand shock triggers an upward adjustment to a level of
, which is also smaller
by
than the benchmark quantity. Since each firm shifts from under-production to under-production and
the shortfall is constant, the resource adjustment is equal to
, similar to the
adjustment that would occur following a similar sequence of positive demand shocks in the benchmark case.
This adjustment is as illustrated in Figure 3a by the green arrow. However, a negative demand shock following
a prior expansion triggers a downward resource adjustment only if it is sufficiently large. In such case, each
firm reduces resources to a quantity of
, which is greater by
than the benchmark quantity.
Here, the firm starts with under-production of
and decreases resources to a level that reflects over-
production of
relative to the benchmark. Therefore, the downward resource adjustment equals to
, as illustrated in Figure 3a by the orange arrow, and its absolute value
is smaller by
compared to the adjustment that would have been made upon a similar sequence of shocks
in the benchmark case. Thus, subsequent to an upward resource adjustment, a successive positive demand
shock generates greater resource adjustment than an equivalent negative demand shock, and accordingly cost
stickiness arises.
The opposite pattern emerges when the most recent resource adjustment was a downward adjustment
triggered by a negative demand shock , as Figure 3b demonstrates. In this case, production
12
level was set in period to
, higher than the benchmark production for a
similar demand by
. A successive negative demand shock triggers a downward
adjustment to a level of
, which is also greater by
than the benchmark quantity. Since
each firm shifts from over-production to over-production and the surplus is constant, the resource adjustment
is equal to
, marked in Figure 3b by the orange arrow. It is equal to the adjustment
that would occur for similar consecutive negative demand shocks in the benchmark case. However, a positive
demand shock following a prior contraction triggers an upward resource adjustment only if it is sufficiently
large. In such case, each firm increases resources to a quantity of
, which is smaller by
than the benchmark quantity. Here, the firm starts with over-production of
and increases resources to a
level that reflects under-production of
relative to the benchmark. Therefore, the upward resource
adjustment equals to
, as illustrated in Figure 3b by the green arrow, and it is
smaller by
compared to the adjustment that would have been made upon a similar sequence of shocks in
the benchmark case. Thus, following a downward resource adjustment, a successive positive demand shock
generates smaller resource adjustment than an equivalent negative demand shock, and accordingly cost anti-
stickiness appears.
[Figure 3]
Cumulatively, the Corollary to Proposition 2 implies that the presence of resource adjustment cost
triggers asymmetric cost behavior, which can take the form of cost stickiness or cost anti-stickiness, depending
on the demand intertemporal dynamics. The asymmetric cost behavior stems from the dissimilar response to
different sequences of demand shocks when resource adjustments are costly. Firms are more likely to adjust
their resources, and adjust them more significantly, in response to a demand shock that follows the course of
the last shock (positive or negative) than they do in response to a shock that reverses the last shock. Hence,
cost asymmetry takes the shape of cost stickiness following resource expansion and transpires in the form of
cost anti-stickiness after resource contraction. Proposition 3 formally establishes this argument in terms of the
cost asymmetry measure
.
13
PROPOSITION 3. In the case of , the cost asymmetry measure
is independent of and . It is
given by
where
and
.
Proposition 3 indicates that, in the presence of adjustment costs, the cost asymmetry measure always
deviates from the benchmark of zero. This implies that the existence of resource adjustment costs is sufficient
to generate asymmetric cost behavior, even when assuming otherwise frictionless setting. Proposition 3 further
indicates that the sign of the cost asymmetry measure is determined by the direction of the last resource
adjustment, which took place in period . If resources were adjusted upward (downward) in period
in response to a positive (negative) demand shock , then the cost asymmetry measure in period
takes a positive (negative) value. Thus, both cost stickiness and cost anti-stickiness are expected to rationally
and naturally appear in equilibrium. Cost stickiness emerges following resource expansion, and cost anti-
stickiness transpires after resource contraction.
7
While cost behavior is asymmetric by default, its magnitude
depends on the characteristics of the economy and thus varies across industries. Proposition 4 presents the
sensitivity of the magnitude of the cost asymmetry measure, as captured by
, to the modeling
parameters.
PROPOSITION 4. In the case of , as long as , the magnitude of cost asymmetry, as
captured by the measure , is increasing in the adjustment cost , decreasing in the number of
competing firms, and decreasing in the demand uncertainty . For , the measure is
decreasing in , and independent of and .
Proposition 4 shows that, as long as , the magnitude of cost asymmetry, either stickiness or
anti-stickiness, increases in the resource adjustment cost, decreases in the number of competing firms, and
decreases in the demand uncertainty. As detailed above, resource adjustment costs generate cost asymmetry
due to their restraining effect on resource adjustments. As resource adjustment costs increase, their restraining
impact on both upward and downward resource adjustments becomes more significant, and accordingly the
resulting cost asymmetry increases. An increase in the number of competing firms has the opposite effect of
decreasing the cost asymmetry. This is because enhanced competition decreases the response of firms to
7
If demand shocks are uncorrelated between industries, we would expect the entire economy to exhibit, on average, cost
symmetry because some industries demonstrate cost stickiness while others demonstrate cost anti-stickiness. However,
since demand in most industries is positively correlated with macro-economic activity, we expect to observe a similar
pattern of cost asymmetry across industries.
14
demand shocks and thus decreases proportionally the restraint in resource adjustments due to adjustment costs
and the consequent cost asymmetry. The effect of demand uncertainty is subtler. As uncertainty increases, the
potential demand shocks become larger, in absolute terms, and so does the extent of the consequent resource
adjustments. However, the distortion caused by adjustment costs remains constant. Therefore, even though the
absolute size of the restraint in resource adjustments due to adjustment costs is independent of demand
uncertainty, it becomes smaller relative to the adjustment size when demand uncertainty increases, making the
asymmetry between upward and downward adjustments less significant. In the rare circumstances where the
adjustment cost per one unit of resource adjustment is extremely large such that twofold the adjustment cost
is at least the maximum possible demand shock (i.e., ), the magnitude of cost asymmetry reaches
its peak level of
and it no longer depends on the parameters and . In such cases the adjustment cost
effectively prevents any resource adjustment upon a demand shock reversing the previous shock, while a
demand shock prolonging the previous shock triggers a full adjustment (equal to the benchmark), hence the
magnitude of cost asymmetry is the greatest possible.
3.3. Equilibrium in the unrestricted model
The unrestricted model captures situations where the managers of the firms care not only about the
profit of the current period but also about the expected profits in future periods. Thus, when contemplating
resource adjustment in response to a demand shock, the managers consider both its current and future
consequences. The future implication of resource adjustment in the current period depends on whether the
current demand shock persists in the next period or reverts. The Markovian stochastic process of the demand
implies that a positive (negative) demand shock in the current period also increases (decreases) the expected
demand in the future. So, resource adjustment in response to the current demand shock is likely to continue
serving the firm in the future. Therefore, managers are more willing to adjust resource in the unrestricted case
relative to the case of , where the managers care only about the profit of the current period. This
equilibrium outcome is formally presented in Proposition 5.
PROPOSITION 5. In the unrestricted model, there exists a unique equilibrium in each
period of the following form, where is a scalar:
If , then
15
If , then
In this unique equilibrium, the scalar is given by
The scalar
satisfies , and it is decreasing in the adjustment cost , increasing in the demand uncertainty
, and increasing in the discount rate , converging to when approaches .
Proposition 5 shows that the equilibrium for the case where can be generalized to the
unrestricted model. Indeed, when the time discounting rate approaches , converges to , and the
equilibrium presented in Proposition 5 converges to the equilibrium of Proposition 2. However, while
establishing the existence and uniqueness of equilibrium in the unrestricted model that takes this generalized
form, we do not eliminate the potential existence of equilibria that have another structure due to tractability
constraints. Proposition 5 indicates that in the unrestricted model each firm under-produces in response to
positive demand shocks and over-produces in response to negative demand shocks, as compared to the
benchmark case of where resource adjustments are costless. This restrained resource adjustment
strategy is similar to the equilibrium strategy obtained in the case of , where the managers care only
about the profit of the current period in making their resource adjustment decision. However, the magnitude
of the restrain in adjusting resources in the unrestricted model is smaller by a factor relative to the
case of . Intuitively, while the benefit from adjusting resources in response to the current demand shock
is limited to the current period in the case of , the forward-looking managers in the unrestricted model
get an incremental future benefit from the current resource adjustment, which enhances their motivation to
adjust resources. Their incremental future benefit from the current resource adjustment is increasing in the
adjustment cost and decreasing in both the discount rate and the demand uncertainty . Accordingly, the
restraint factor is decreasing in and increasing in both and .
[Figure 4]
The results of Proposition 5 are graphically illustrated in Figure 4. The illustration in Figure 4 is
similar to that of Figure 2, demonstrating the restrain in the resource adjustment relative to the
benchmark. This restrain in adjusting resources is however smaller by the factor relative to the case of
as illustrated in Figure 2. The green and the orange curves describe the resource quantity of
each firm in period as a function of the demand in period , for the case of
16
and the case of , respectively. For comparison, the corresponding functions in the
case of are illustrated by the dashed green and orange curves, whereas the blue diagonal dashed
line describes the benchmark resource quantity
as a function of the demand . Unlike the
benchmark strategy, where all demand shocks trigger resource adjustment, in the unrestricted model firms
refrain from adjusting resources when the prior demand shock is followed by an opposite and sufficiently small
demand shock (whose magnitude is less than ), as illustrated by the horizontal sections in the green and
orange curves. The range of demand shocks that do not trigger resource adjustment is however narrower in the
unrestricted model as compared to the case of . Upon a sufficiently large positive (negative) demand
shock, the firm adjusts resources upward (downward) to a lesser extent than it would in the benchmark case
of but to a greater extent than it would in the case of . The green and the orange curves coincide
into the same curve for sufficiently large demand shocks, but unlike the benchmark case the two curves diverge
from each other for small demand shocks that belong to the range . This range is however
narrower than the corresponding divergent range in the case of . Hence, unlike the
benchmark case, the equilibrium resource adjustments in the unrestricted model are not proportional to the
corresponding demand shocks. But, as formally stated in the following corollary to Proposition 5, this
disproportion is of a lower magnitude relative to the case of extreme time discounting.
COROLLARY TO PROPOSITION 5. In the unrestricted model, using the notation
, the
equilibrium in period satisfies for any :
If , then
If , then
As in the case of extreme time discounting, cost asymmetry emerges in the unrestricted model because
of the dissimilar response to a demand shock that continues the course of the last shock versus a demand shock
that reverses it. Cost asymmetry takes the shape of cost stickiness subsequent to resource expansion while
transpiring in the form of cost anti-stickiness following resource contraction. This is the same cost asymmetry
pattern that emerges in the case of but its magnitude is smaller by a factor of .
17
[Figure 5]
The results of the corollary are graphically illustrated in Figure 5, where Figure 5a pertains to the case
of and Figure 5b relates to the case of . Figure 5 demonstrates how
the presence of resource adjustment costs triggers asymmetric cost behavior, which takes the form of cost
stickiness following past resource expansions (the case of ) and the opposite form of cost
anti-stickiness subsequent to past resource contractions (the case of ). The asymmetric
cost behavior follows the same pattern as illustrated in Figure 3 for the case of , but its magnitude is
smaller. This is reflected in the cost asymmetry measure that the unrestricted model yields in equilibrium,
as presented in Proposition 6.
PROPOSITION 6. In the unrestricted model, the cost asymmetry measure is independent of and . It
is given by
where
and
.
Proposition 6 indicates that the cost asymmetry measure always deviates from the benchmark of zero,
implying that the existence of resource adjustment costs is sufficient to generate asymmetric cost behavior,
even when assuming otherwise frictionless setting. This result stands in contrast to the conventional perception
in the existing literature that some friction is necessary in order to shift cost behavior from its perceived
fundamental symmetric pattern. Proposition 6 suggests that cost behavior is asymmetric by default, and further
points to the direction of the most recent resource adjustment as the determinant of the sign of the cost
asymmetry measure. Our result is consistent with the empirical evidence of Banker, Byzalov, Ciftci and
Mashruwala (2014), who report that the direction of cost asymmetry depends on prior change in sales,
but explain it by the effects of past change in sales on managers’ expectations for future change in sales
and on the amount of resource slack carried from previous period. Our theoretical analysis reveals that cost
asymmetry is more fundamental, as it exists independently of these effects. In our model past changes in
demand do not affect managers’ expectations for the direction of future demand shock (either rational or
irrational), nor does the model encompass resource slack. While the sign of the cost asymmetry measure is the
same as in the case of , its absolute value is lower, as reflected by the inequality presented
in Proposition 6. Proposition 7 establishes the sensitivity of the magnitude of the cost asymmetry measure, as
captured by , to the modeling parameters.
18
PROPOSITION 7. In the unrestricted model, as long as
, the magnitude of cost asymmetry,
as captured by the measure , is increasing in the adjustment cost , decreasing in the number of
competing firms, decreasing in the demand uncertainty , and increasing in the discount rate . For
, the measure is decreasing in , and independent of ,
and .
Proposition 7 indicates that cost asymmetry in the unrestricted model is affected by the adjustment cost
, the number of competing firms and demand uncertainty in the same manner as in the case of .
Additionally, in the unrestricted model where the managers care about future profits, the discount rate is also
a determinant of the extent of cost asymmetry. Resource adjustment in response to the current demand shock
not only benefits the firm in the current period, it is also likely to continue serving the firm in the future because
of the intertemporal dynamics of the demand. For this reason, forward looking managers care less for the
adjustment costs, and they distort less their adjustment decision, thus mitigating cost asymmetry. The value of
future benefits stemming from current resource adjustment is decreasing in . Therefore, as increases, cost
asymmetry is enhanced, reaching its peak in the case of .
The results of Proposition 7 offer useful guidance to the empirical literature by providing new empirical
predictions, as well as alternative explanations to existing empirical observations. Consistent with Proposition
7, empirical studies indeed document that the degree of cost stickiness is increasing in the magnitude of
resource adjustment costs (e.g., Anderson, Banker and Janakiraman, 2003; Banker, Byzalov, and Chen, 2013;
Banker and Byzalov, 2014). The literature, however, attributes this empirical finding to asymmetry, upward
and downward, in either adjustment costs or persistence of demand shocks. We show that the positive relation
between cost asymmetry and adjustment costs applies not only to cost stickiness but also to cost anti-stickiness,
further indicating that this is true even in a setting with symmetric upward and downward adjustment costs
and under symmetric distribution of the demand shocks around zero. The empirical findings on other
determinants of cost asymmetry are limited and inconclusive. Addressing the effect of competition, Li and
Zheng (2017) find that cost stickiness is increasing in product market competition, but Li and Lou (2021)
report that product market competition reduces cost stickiness in emerging markets. The effect of demand
uncertainty on cost asymmetry was hardly explored, except from a handful of empirical papers that address
political uncertainty and yield mixed results. Lee, Pittman and Saffar (2020) show that cost stickiness rises
during election periods, while Jin and Wu (2021) find that cost stickiness decreases with economic policy
uncertainty. The mixed evidence is consistent with our theoretical result that these determinants can enhance
either cost stickiness or cost anti-stickiness, depending on the context. Lastly, the literature insofar does not
examine the effect of the interest rate on cost asymmetry. To complete the analysis, we examine how the
19
magnitude of the demand shock affects cost asymmetry. Proposition 8 shows the difference in cost asymmetry
for small versus large demand shocks.
PROPOSITION 8. In the unrestricted model, when
,
for any
and , where
and
.
Proposition 8 splits our measure of cost asymmetry into two separate measures: the measure
that applies to small demand shocks in the range , and the measure
that applies to large
demand shocks whose absolute value is greater than .
8
Using these two separate measures, Proposition 8
suggests that cost asymmetry arises for both small and large demand shocks, but it is of a greater magnitude
for small demand shocks than for large demand shocks, as capture by the inequality
. This
is because small demand shocks trigger a full resource adjustment in case of a prolonging shock but no
adjustment in case of a reversing shock, whereas large demand shocks always trigger a resource adjustment
even in response to a reversing shock, albeit of a relatively lower magnitude as compared to the resource
adjustment in response to a prolonging shock. Our result that the magnitude of cost asymmetry is inversely
correlated with the size of the demand shock is consistent with the empirical evidence in Ciftci and Zoubi
(2019). Their empirical findings suggest that cost asymmetry is greater for small changes in sales than for large
changes in sales. As a reasoning for their empirical observation, Ciftci and Zoubi (2019) argue that following
a prior increase in sales, managers are likely to consider small decreases in sales as temporary and large
decreases in sales as permanent. In addition, following a prior decrease in sales, slack resources retained from
the prior period have a greater impact on cost behavior for small changes in sales than for large ones.
Notwithstanding these arguments, our analysis demonstrates that stronger cost asymmetry for modest demand
changes is expected to occur irrespective of managerial expectations or slack resources.
4. Summary and Conclusions
Challenged by the empirical finding of the cost stickiness pattern by Anderson, Banker, and
Janakiraman (2003), extensive empirical research has been conducted to explore the fundamental
determinants of the cost stickiness phenomenon and to explain its observed varying degrees in different
8
Large demand shocks whose absolute value is greater than belong to the intervals and Hence,
the measure
is well defined only under the assumption , which is equivalent to
.
20
firms. The literature also indicates circumstances that work to counterbalance the pattern of cost
stickiness and may even lead to the opposite pattern of cost anti-stickiness. The research on the
asymmetric cost behavior, despite its large body and fundamental importance, has evolved empirically
and lacks theoretical guidance. Our study addresses this gap in the literature by offering a theoretical
framework for investigating various aspects of the phenomenon of cost asymmetry, which has been so
far studied in the literature almost solely on the basis of empirical observations.
Our theoretical analysis delves into the primitives of asymmetric cost behavior and explores its
inter-temporal dynamics in a competitive environment. The analysis shows that cost behavior is
asymmetric by its fundamental nature in the sense that both cost stickiness and cost anti-stickiness are
expected to rationally and naturally emerge in equilibrium even in a frictionless setting, which is devoid
of any frictions known in the literature to trigger cost asymmetry. We further explore the dynamic over
time of cost behavior, showing that cost asymmetry transpires in the pattern of cost stickiness following
resource expansion while taking the form of cost anti-stickiness subsequent to resource contraction.
While the direction of cost asymmetry is primarily determined by the direction of the prior resource adjustment,
the magnitude of cost asymmetry, either stickiness or anti-stickiness, is shown to be decreasing in demand
uncertainty and competition intensity, increasing in adjustment costs and time discounting, and it is greater in
response to smaller demand shocks. These insights offer useful guidance to the empirical literature by
providing new empirical predictions, as well as alternative explanations to existing empirical observations.
21
Appendix – Proofs
Prof of Observation 1. In the benchmark case of , the profit of firm in period is given by
or, equivalently, can be also represented as
.
Hence, the production quantity of firm in period affects only the profit of firm in period and has no
effect on the profits of firm in the successive periods. Therefore, the optimal production quantity
that
maximizes the expected economic value
of firm in period is the one that maximizes the profit
of firm in period .
The first order condition is as follows:
(1.1)
The second order condition is as follows:
(1.2)
It follows from equation (1.1) that
, implying
.
Therefore, equation (1.1) is equivalent to
, and thus
. We conclude
that, in the case of , the equilibrium
in the competition game between the
firms in each period is given by
. So, for each firm in each period ,
the resource adjustment is
.
For each firm in each period , the expected positive resource adjustment
conditional on
a positive demand shock is
and
after algebraic rearrangements it can be rewritten as follows:
(1.3)
For each firm in each period , the expected negative resource adjustment
conditional on
a positive demand shock is
and
after algebraic rearrangements it can be rewritten as follows:
(1.4)
22
For each firm in each period , the expected demand shock conditional on a positive demand
shock is
and after algebraic
rearrangements it can be rewritten as follows:
(1.5)
For each firm in each period , the expected demand shock conditional on a negative demand
shock is
and after algebraic
rearrangements it can be rewritten as follows:
(1.6)
By equations (1.3)-(1.6),
independently of and . □
Prof of Proposition 2 and its corollary. In the case of , the economic value
of firm in
period equals the profit
of firm in period Therefore, the optimal production quantity
that
maximizes the economic value
of firm in period is the one that maximizes the profit
of firm in period .
We first look for a local maximum of the function
in the interval
. For any
, the function
becomes
.
The first order condition is as follows:
(2.1)
The second order condition is as follows:
(2.2)
It follows from equation (2.1) that
, implying
. Therefore, equation (2.1) is equivalent to
, and thus
.
The solution
belongs to the interval
if and only if
or
equivalently
.
We next look for a local maximum of the function
in the interval
. For any
, the function
becomes
.
23
The first order condition is as follows:
(2.3)
The second order condition is as follows:
(2.4)
It follows from equation (2.3) that
, implying
. Therefore, equation (2.3) is equivalent to
, and thus
.
The solution
belongs to the interval
if and only if
or
equivalently
.
So, the maximum of the function
is obtained at
when
,
and it is obtained at
when
.
We now look for the maximum of the function
under the assumption
. If
, then
, and thus
. Using the assumption
,
it follows that
. If
, then
, and thus
. Using the
assumption
, it follows that
. Thus,
when
, the optimal solution is
.
Therefore, in the case of, the equilibrium in each period is given by
. This implies that
can
be either or
or
.
24
If
, then the inequity
is equivalent to
, and the inequity
is equivalent to . So,
and
If
, then the inequity
is equivalent to
, and the inequity
is equivalent to . So,
and
Since resource adjustment has occurred in period , we get that
is equivalent
to , and
is equivalent to . □
Prof of Proposition 3. For firm in period , equals
, and
after algebraic rearrangements it can be rewritten as follows:
(3.1)
For firm in period , equals
, and after algebraic
rearrangements it can be rewritten as follows:
(3.2)
We consider separately two cases: the case of , and the case of
Case 1:
By Proposition 2, for ,
For firm in period ,
equals
, and after algebraic
rearrangements it can be rewritten as follows:
(3.3)
25
For firm in period ,
equals
when
and zero otherwise, and after algebraic rearrangements it can be rewritten as follows:
(3.4)
By equations (3.1)-(3.4),
Case 2:
By Proposition 2, for ,
.
For firm in period ,
equals
when
and zero otherwise, and after algebraic rearrangements it can be rewritten as follows:
(3.5)
For firm in period ,
equals
, and after algebraic
rearrangements it can be rewritten as follows:
(3.6)
By eq. (3.1),(3.2),(3.5),(3.6),
.
It follows from that , and thus . □
Prof of Proposition 4. For , the derivatives of with respect to the parameters , and are:
,
,
. For , the
derivatives are:
,
. □
26
Prof of Proposition 5 and its corollary. We show that there exists a scalar , such that the following
production quantities constitute equilibrium in the unrestricted model:
, where
and
.
Note that implies
, and
implies
. Therefore, the corresponding profits are as follows:
, where
and
.
We next consider three cases separately: the case of , the case of
, and the case of .
Case 1:
In this case, conditional on the demand realization , the production quantity of each firm in period
is
and the profit is
. Thus, the portion of
that depends upon is given by the following function:
(5.1)
Also,
.
The portion of
that depends upon is given by the following function:
.
Equivalently,