Capacity Constraints and Asymmetric Cost Behavior

Capacity Constraints and Asymmetric Cost Behavior
Eric W. Noreen
Professor Emeritus, University of Washington
Accounting Circle Professor of Accounting, Temple University

Abstract:
This paper provides a formal model to help understand and motivate the empirical "sticky" cost
literature. The verbal arguments used to generate hypotheses in the sticky cost literature
emphasize the costs of adding or removing capacity and are often concerned with how managers
are likely to react to various scenarios involving more or less excess capacity. Likewise, the
model in this paper focuses on the decision to expand or contract capacity, which is a constraint
that either limits output or results in a discontinuous increase in marginal cost. Tests run on
simulated data from this model are strikingly similar to the results of tests on real world data and
suggest that the formal model captures important real world phenomenon.

Subject matter keywords: Economics, Cost management, Analytical

Acknowledgements: I would like to thank Alister Hunt, Dave Burgstahler, Rajib Doogar, and Ed
Rice for their comments on an earlier version of this paper.

6/22/2017 Version

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Electronic copy available at: https://ssrn.com/abstract=2991369

INTRODUCTION
Empirical work over the last two decades has documented that cost behavior in the real world is
much more complex than assumed in cost accounting systems and in standard textbook
treatments—in particular, costs appear to respond differently to increases and decreases in
activity. The purpose of this paper is to provide a formal model of the decision to expand or
contract capacity that can be used to help understand these empirical insights about real cost
behavior. The model in this paper differs in important ways from models that have been cited in
the cost behavior literature. The model that is presented here features a constraint that either
limits output or results in a discontinuous increase in marginal cost. In contrast, the cost behavior
literature typically relies for theoretical support on a single period Cobb-Douglas type continuous
production function in which an investment in capital goods affects the marginal productivity of
labor or some other input. 1 There is a fundamental mismatch between these theoretical models
and the verbal arguments used in the empirical literature to motivate hypotheses. Those verbal
arguments emphasize the costs of adding or removing capacity and are often concerned with how
managers are likely to react to various scenarios involving more or less excess capacity. In the

Cobb-Douglas world, no capacity is carried over from the prior period, a constraint does not
exist, and there is no such concept as excess capacity.
BACKGROUND
Cost accounting systems (including activity-based costing systems) typically assume that a cost
is proportional to some measure(s) of activity. 2 In a proportional model, an x% increase
(decrease) in activity results in an x% increase (decrease) in cost and is equivalent to saying that
marginal cost always equals average cost. In perhaps the first study in the accounting literature to
empirically test this proportionality assumption underlying cost accounting systems, Noreen and
Soderstrom [1994] examine cross-sectional data from hospital service departments in
Washington State. They find that “on average across the accounts [i.e., departments], the average
cost per unit of activity overstates marginal costs by about 40% and in some departments by over
100%. (p. 225)” In a subsequent time-series and cross-sectional study of hospital service

1
2

See, for example, Banker, Byzalov, and Plehn-Dujowich [2014].
See Noreen [1991] for the assumptions concerning cost behavior that typically underlie cost systems.

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departments, Noreen and Soderstrom [1997] find that “more accurate predictions of costs are
usually generated by assuming a cost will not change at all (except for inflation) than by
assuming that the cost will change in proportion to changes in activity. (p. 89-114)”
In an aside, Noreen and Soderstrom [1997] run a regression of the form:

 TC t 
 qt 
 qt 
ln 
 = α + β ln 
 + β− Decline t ln 

 TC t −1 
 q t −1 
 q t −1 

(1)

where
TCt = Total cost in period t for a particular hospital department
qt = measure of activity for the hospital department in period t
Declinet = dummy variable which has the value of 1 if the measure of activity declines
in period t and is zero otherwise
If the proportional cost model is correct, then α should be zero, β should be 1, and β− should be
zero. If costs are “sticky” downward (i.e., costs are more responsive to increases than decreases
in activity), then the coefficient β− should be negative. Noreen and Soderstrom [1997] found that
“While very few of the interactive dummy coefficients were significantly different from zero, all
but three were negative. The probability of obtaining 13 out of 16 coefficients with the “right
sign” is about 0.01. This suggests that costs are indeed more difficult to adjust when decreasing
activity…. (p. 103)” Some variation on the above log model has generally been used in
subsequent studies of asymmetric cost behavior. 3
It is important to note that stickiness is a different phenomenon from the existence of
fixed costs. Suppose that a cost has the simple linear form TC = F + vq, where TC is the total
cost for that cost category, F is the fixed cost, v is the variable cost per unit of activity, and q is
the amount of activity. If F>0, then a given percentage increase (decrease) in activity results in a
smaller percentage increase (decrease) in cost 4. Nevertheless, in this standard linear model the

3

Noreen and Soderstrom [1997] included the independent variable ln(pt/pt-1) in their model, where pt is the
estimated cost per unit of activity in period t.. This has not generally been done in subsequent work by others.
4
If the simple linear cost model is correct, the relation between the percentage change in total cost, y%, and the
percentage change in activity, x%, is given by y% = [vq/(F+vq)]x%.

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effect is symmetric. A 1% decrease in activity in activity should result in the same percentage
change in cost (but with the opposite sign) as a 1% increase in activity. 5 This is not sticky cost
behavior, which is asymmetric.
Subsequent to Noreen and Soderstrom, a number of published and unpublished papers
have consistently found evidence that is inconsistent with the simple proportional and linear
models of cost behavior in many different organizations across the world using very large as well
as small datasets. 6 For example, Anderson, Banker, and Janakiraman, [2003] report that “for
7,629 firms over 20 years, … selling, general, and administrative (SG&A) costs increase on
average 0.55% per 1% increase in sales but decrease only 0.35% per 1% decrease in sales. (p.
47)” In terms of the natural log model of cost behavior (1), their estimate of β is 0.55 and their

estimate of β− is -0.20. These figures are economically as well as statistically significant. Most
recently, Banker and Byzalov [2014] test five hypotheses concerning asymmetric cost behavior
using all companies in Global Compustat with sufficient time series data and find that
“asymmetric cost behavior is a pervasive global phenomenon. (p.65)”
The essence of the verbal arguments for asymmetric cost behavior that are found in the
literature is that in the presence of a constraint managers are faced with a complex series of
decisions. If the constraint is not currently binding, the manager must trade off the costs that
could be saved by removing excess capacity with the possible loss of sales in the future when the
constraint may become binding. If the constraint is binding, the manager must trade off the costs
of expanding capacity (and taking on additional fixed periodic costs) against the benefits of
being able to increase sales or to do so without incurring abnormally large marginal costs. These

5

However, Balakrishnan, Labro, and Soderstrom [2014] point out that the log version of the empirical model
relating changes in cost to changes in activity is biased toward finding sticky costs if the standard linear model is
valid and fixed costs are present. They document this effect, but find that the magnitude of the bias is small and the
β− coefficient is reliably negative even after controlling for this bias.
6
The published papers include Anderson, Banker, and Janakiraman, [2003]; Balakrishnan, Peterson, and

Soderstrom. [2004]; Calleja, Steliaros, and Thomas [2006]; Balakrishnan and Gruca [2008]; Uy [2011]; Baumgarten
[2012]; Chen, Lu, and Sougiannis [2012]; Banker, Byzalov, and Chen [2013]; Banker, Byzalov, Ciftci, and
Mashruwala [2014]; Cannon [2014]; and Banker and Byzalov [2014]. The latter paper also contains an excellent
review of the literature. A number of published studies have also focused on the implications of asymmetric cost
behavior for forecasting reported expenses and earnings including: Banker and Chen [2006]; Anderson, Banker,
Huang, and Janarkiraman [2007]; Weiss [2010]; Baumgarten, Bonenkamp, and Homburg [2010]; Janakiraman
[2010], and Shust and Weiss [2014].

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decisions are impacted by the amount of excess capacity currently in place, the costs of adjusting
capacity, and expectations about future sales as well as selling prices, variable costs, and periodic
costs of maintaining capacity. The formal model developed in the next section makes these
trade-offs explicit.
THE MODEL
In this model, capacity is carried over from the prior period. Cost is a linear function of demand
and capacity, but if demand exceeds capacity, an additional marginal cost must be incurred. 7
This latter feature is consistent with The Theory of Constraints (TOC) which claims that when
the constraint is an internal bottleneck, there often exist many ways to temporarily relax that
constraint. 8,9 Examples include working overtime on the bottleneck, outsourcing some of the

processing that would be done on the bottleneck, shifting resources from non-bottlenecks to the
bottleneck, and reducing the number of defective units that are processed through the bottleneck
by inspecting units prior to the bottleneck. Some of these actions are free, but some are not.
Those that are free are equivalent to uncovering hidden amounts of capacity. Those that are not
free result in higher marginal costs when realized demand exceeds capacity. In each period,
managers must choose to increase capacity, decrease capacity, or simply carry forward the
capacity from the prior period. Increasing capacity incurs a one-time acquisition cost and a
periodic cost. Decreasing capacity incurs a deactivation cost. 10 Finally, I assume that production
always equals demand. If demand exceeds capacity, demand is satisfied by paying the additional
marginal cost m. If capacity exceeds demand, the excess capacity is not used to build
inventories. 11

7

This model can also accommodate, as a special case, the situation in which capacity is a hard constraint and
demand that exceeds capacity will not be satisfied.
8
Goldratt and Cox [2014]
9
Another strategy when an internal constraint is binding is to ration the use of the constraint by raising prices. In

this paper, I assume that this strategy is not followed and instead the organization incurs additional costs to
temporarily relax the constraint.
10
The situation modeled in this paper is similar in some ways to the Newsvendor problem which focuses on the
problem of how much inventory a retailer should acquire in the face of uncertain demand. However, in the situation
modeled in this paper, capacity (e.g., inventory) can be carried over from one period to the next and capacity can be
exceeded, but at an incremental cost.
11
This assumption is consistent with Just-In-Time production, TOC, and Lean Production. Nevertheless, if output is
non-perishable and can be saved in the form of inventories, the option to build inventories could ameliorate the
incentives to build additional capacity and could moderate asymmetric cost behavior.

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Definitions and Assumptions:
A period is the amount of time that must elapse before capacity can be adjusted.
p = the constant selling price per unit of output. [p ≥ 0]
v = the constant variable cost per unit of output [v ≥ 0]
Assume that p –v ≥ 0.
FC = the constant fixed cost per period

qt = realized demand in the current period t [qt > 0]
(qt ) = probability of realized demand qt. This probability function can change from period to
period, but a priori the best estimate of next period’s probability function is this period’s
probability function. 12 Technically, this probability function should be written as (qt | t )
where t is an n-dimensional vector of signals that represents the state of relevant
knowledge as of the beginning of period t. t includes the realized demand from the
previous period, but could include many other signals as well. For the sake of brevity, (qt )
is understood to stand for (qt | t ) throughout this paper.
Qt-1 = endowed capacity. The amount of the constrained resource brought forward from the
previous period. Each standard unit of the constrained resource can be used to produce one
unit of the output. [Qt-1 ≥ 0]
Qt = amount of capacity (i.e., the constrained resource) chosen by management for the current
period [Qt ≥ 0]
pc = the periodic avoidable cost of a standard unit of the constrained resource. This cost could
consist of rent, real periodic depreciation (i.e., decline in resale value through time), periodic
maintenance costs, or other costs that are required to maintain a unit of capacity but that can
be avoided by removing that capacity. I assume pc is a constant. [p –v ≥ pc ≥ 0]

12

I hope that this assumption about probabilities is sufficient to decompose what is a multi-period dynamic
programming problem into a sequence of single period decisions. If it isn’t, I am not too concerned. My purpose is
to build a model that is good enough rather than one that is perfect.

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ac = the periodic amortized cost to acquire and activate a standard unit of the constraint. This is a
sunk cost once the capacity is acquired and activated. This cost is relevant when deciding
whether to add capacity, but is irrelevant thereafter. However, this cost will continue to be
included in reported expenses until the capacity is deactivated, at which point any
unamortized cost would be written off. [ac ≥ 0]
dc = one-time cost of deactivating a standard unit of the constrained resource. This cost includes
any out of pocket cost as well as the opportunity cost that arises from surrendering the
valuable option to use this marginal unit of capacity in the future without incurring an
activation cost. This latter opportunity cost would not appear in the accounting records. It is
likely that this cost is not a constant and depends on (qt ) as well as all of the other
parameters in the model and ideally should be endogenously determined. Nevertheless, for
tractability, it will be treated as a constant in this model. [dc ≥ 0]
m = the additional marginal cost that must be incurred to produce a unit if production exceeds
capacity. The parameter m is bounded above by the contribution margin because if m exceeds
the contribution margin, the optimal choice would be to not satisfy demand completely and
forego the contribution margin. In the special case where capacity cannot be exceeded, m
equals the opportunity cost of lost sales, which is the contribution margin. [p –v ≥ m > 0]

Critical values:
As we shall see below, the ratio
and the ratio

pc  ac

m

is critical in the decision of whether to expand capacity

pc  dc
is critical in the decision of whether to reduce capacity. Assuming that the
m

marginal cost of exceeding capacity m is strictly positive, some relations are immediately
apparent without adding any more structure to the model. If it ever makes sense economically to
expand or contract capacity and capacity adjustment costs are positive, it must be the case that
0

pc  dc pc  ac

 1 . In other words, these two critical values, like probabilities, lie
m
m

between zero and one.

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Proof:
° If the capacity adjustment costs are positive, then ac + dc > 0 and ac > −dc. And if m > 0,
pc  dc pc  ac
.

m
m

° Suppose

pc  dc
 0 . Then dc > pc, and the deactivation cost would exceed the periodic
m

avoidable cost of maintaining capacity and it would never make sense to deactivate capacity.
Therefore, if it ever makes sense to deactivate capacity, it must be that case that dc < pc and
hence

pc  dc
 0 . 13
m

° Suppose

pc  ac

m

 1 . Then pc + ac > m. In words, the periodic cost of a unit of capacity would

exceed the marginal cost of exceeding capacity. In that case it would never make sense to expand
capacity. Therefore, if it ever makes sense to expand capacity, it must be the case that
pc  ac

m

 1.

13
This brings up a nuance concerning the deactivation cost dc. If the marginal unit of capacity has only one period
of useful life remaining, the comparison between the periodic avoidable cost pc and dc is straight-forward. If dc is
greater than pc, it would never make sense to deactivate capacity. However, suppose that the marginal unit of
capacity has a useful remaining life of two periods. Then it might make sense to deactivate the marginal unit of
capacity even though its gross deactivation cost exceeds the periodic avoidable cost because that avoidable cost
would then disappear for two periods. The deactivation cost dc is best interpreted as a sort of normalized
deactivation cost where the gross cost is spread across the remaining useful life of the marginal unit of capacity.
Roughly speaking, dc is the gross deactivation cost divided by the remaining useful life of the marginal unit of
capacity.

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Expected Economic Profit 14
The expected economic profit depends on whether capacity is expanded, remains the same, or is
contracted.
For Qt ≥ Qt-1, an expansion in capacity:

E  profit ( Q=
t ) | Qt ≥ Qt −1 


∞

∫

( p − v)qtφ ( qt )dqt −

qt =1

∞

∫

m × (qt − Qt )φ ( qt )dqt

qt =Qt

−ac × ( Qt − Qt −1 ) − pc × Qt – FC
For Qt ≤ Qt-1, a contraction in capacity:

E  profit ( Q=
t ) | Qt ≤ Qt −1 


∞

∫

( p − v)qtφ ( qt )dqt −

qt =1

∞

∫

m × (qt − Qt )φ ( qt )dqt

qt =Q

−dc × ( Qt −1 − Qt ) − pc × Qt – FC

Proposition 1
If and only if m qt  Qt1  > pc + ac, then Qt* > Qt-1 and capacity should be expanded to the
point where m qt  Qt *  = pc + ac. [In words, if the expected additional marginal cost of
producing one more unit beyond the current capacity exceeds the cost of acquiring and
maintaining one more unit of capacity, then capacity should be expanded.]
Proof: See Appendix A.
Proposition 1 establishes that if the probability that demand exceeds the endowed
capacity,  qt  Qt1  , exceeds the critical value

pc  ac
, then capacity should be expanded to
m

the point, Qt*, where the probability that demand exceeds capacity,  qt  Qt *  , equals the
critical value

pc  ac
. To give this some concreteness, in the simulations at the end of this
m

14

Economic profit ignores activation costs once they are sunk and takes into account the opportunity cost of giving
up capacity when deciding to deactivate capacity.

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paper the critical value is assumed to be 0.60. Proposition 1 states that if the probability that
demand will exceed capacity is greater than 0.60, then capacity should be expanded. The critical
value itself is determined by the parameters pc, ac, and m. Ceteris paribus, the higher the
periodic cost of maintaining capacity, pc, the larger the critical value and the greater the
tolerance for exceeding capacity. Likewise, ceteris paribus, the higher the amortized cost of
activating capacity, ac, the larger the critical value and the greater the tolerance for exceeding
capacity. In contrast, ceteris paribus, the higher the marginal cost of exceeding capacity, m, the
smaller the critical value and the less the tolerance for exceeding capacity. All of this should be
intuitively appealing.
Exhibit 1 provides an example of the application of Proposition 1. Exhibit 1 is concerned
with Period 1, which begins with no endowed capacity (i.e., Qt-1 = 0). This exhibit shows a
hypothetical plot of  q1  Q  as a function of Q. Because Qt-1 = 0 and qt  0 , the probability
that demand will exceed capacity is 1.00. Because the critical value

pc  ac
is less than 1.00,
m

then by Proposition 1, capacity should be expanded beyond zero. As Q increases,  q1  Q 
decreases. As depicted in Exhibit 1, the optimal level of capacity, Q1*, occurs where the
probability  q1  Q1*  equals the critical value

pc  ac
.
m

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Exhibit 1
Example of Proposition 1
Period 1: Qt-1 = 0

 q1  Q
 q1  Q0  1

pc  ac
m

0

Q1*

Qt-1

Page 11

Q

We assume in Period 2 that as a result of new information, the probability that realized
demand exceeds any given level Q is greater than it was in Period 1. This could have happened
for a variety of reasons including realized demand in Period 1 that was higher than expected. The
capacity carried over from Period 1 is the endowed capacity in Period 2. Because the probability
function has shifted up, the probability that demand in Period 2 will exceed the endowed
capacity from Period 1,  q2  Q1*  , exceeds the critical value

pc  ac

m

. Consequently, by

Proposition 1 and as depicted in Exhibit 2, capacity will be expanded to the point where

 q2  Q2*  equals the critical value

pc  ac

m

. This response to an increase in expected demand

should be intuitive. If management had already optimally set the capacity level based on
expected demand but expected demand increases, then an investment in more capacity is
warranted.
Proposition 1 establishes the conditions under which it would make sense to increase
capacity. Now let’s consider when it would make sense to reduce capacity.

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Exhibit 2
Example of Proposition 1
Period 2

 q2  Q
1

Probabilities based on
new information
available at the
beginning of Period 2.

 q2  Q1* 
pc  ac
m

Probabilities
based on
information
available at
the beginning
of Period 1.

0

Q1* Q2*

Page 13

Q

Proposition 2
If and only if m qt  Qt1  < pc − dc, then Qt* < Qt-1 and capacity should be reduced to the
point where m qt  Qt *  = pc – dc. [In words, if the expected additional marginal cost of
producing one more unit beyond the current capacity is less than cost that can be saved by
contracting capacity, then capacity should be reduced.]
Proof: See Appendix A.
Proposition 2 establishes that if the probability that demand exceeds the endowed capacity,

 qt  Qt1  , is less than the critical value

pc  dc
then capacity should be reduced to the point
m

Qt* where the probability that demand exceeds capacity,  qt  Qt *  equals the critical value
pc  dc
. To make this more concrete, in the simulations at the end of this paper, the assumed
m

value of

pc  dc
is usually 0.25. Proposition 2 states that if the probability of exceeding the
m

endowed capacity is less than 0.25, then capacity should be reduced. Somewhat counterintuitively, the higher the critical value, the more likely it is that capacity will be reduced. To
take the extreme, suppose the critical value is 1.00. Then the probability that demand exceeds
any given level of capacity will always be less than this critical value and, by Proposition 2,
capacity should always be reduced. The critical value itself is determined by the parameters pc,
dc, and m. Ceteris paribus, the higher the periodic cost of maintaining capacity, pc, the lower the
tolerance for maintaining excess capacity. And, ceteris paribus, the higher the deactivation cost,
dc, the greater the tolerance for maintaining excess capacity. What is less obvious is that the
higher the marginal cost of exceeding capacity, m, the smaller the critical value and hence the
less likely that capacity will be reduced.

Proposition 3
If and only if m qt  Qt1  ≤ pc + ac AND m qt  Qt1  ≥ pc − dc, then Qt* = Qt-1 and
capacity will remain the same.

Page 14

Proof: Proposition 3 follows from Proposition 1 and Proposition 2.
Proposition 3 establishes that if the probability that demand exceeds the endowed capacity,

 qt  Qt1  , lies between the critical values

pc  dc
pc  ac
and
, then there will be no
m
m

change in capacity during the current period. In the simulations later in the paper the assumed
critical values are usually 0.25 and 0.60. If the probability of demand exceeding the endowed
capacity is between 0.25 and 0.60, no change in capacity will take place. In general, the wider
the gap between the critical values

pc  dc
pc  ac
and
, the more likely it is that capacity will
m
m

not change. The size of this gap is

ac  dc
, which is increasing in the adjustment costs ac and dc
m

and decreasing in marginal cost of exceeding capacity m. Intuitively, the higher the costs of
adjusting capacity, the less likely that capacity will be adjusted and the higher the cost of
exceeding capacity, the more likely that capacity will be adjusted.

Propositions 2 and 3 are illustrated in Exhibits 3 and 4. Exhibit 3 illustrates Proposition 3.
In that exhibit, we assume that due to new information the probability of demand exceeding any
given level of capacity has decreased from the beginning of Period 2 to the beginning of Period
3, but that decrease in expected demand is not enough to trigger a reduction in capacity. In this
example, the probability of demand in Period 3 exceeding the endowed capacity carried over
from Period 2 falls within the gap between

pc  ac
pc  dc
and
and therefore, as depicted in
m
m

Exhibit 3, there is no change in capacity.

Page 15

Exhibit 3
Example of Proposition 3
Period 3

 q3  Q
1

Probabilities based on
information available at
the beginning of Period 2.

pc  ac
m

 q3  Q2* 
pc  dc
m

Probabilities
based on new
information
available at the
beginning of
Period 3.

0

Q3*= Q2*

Page 16

Q

In Period 4 we assume a further decrease in the probability that the realized demand
exceeds any given level Q and that this shift is large enough to trigger a decrease in capacity.
This is illustrated in Exhibit 4. Because the probability function has shifted down sufficiently, the
probability that demand will exceed endowed capacity  q4  Q3*  is less than the critical value
pc  dc
. Consequently, by Proposition 2, capacity will be reduced to the level Q4* where the
m

probability that demand exceeds capacity,  q4  Q4*  , equals the critical value

Page 17

pc  dc
.
m

Exhibit 4
Example of Proposition 2
Period 4

 q4  Q

1

Probabilities based on
information available at
the beginning of Period 3.

pc  ac
m

pc  dc
m

 q4  Q3* 
Probabilities
based on new
information
available at the
beginning of
Period 4.

0

Q4* Q3*

Page 18

Q

The Relevant Range
Let  qt  Qt*1 E (qt ) be the probability that demand in period t will exceed the capacity
carried over from period t-1, conditional on the expected demand E (qt ) . As shown in Appendix
B,  qt  Qt*1 E (qt ) is increasing in E (qt ) ; that is, the higher the expected demand, the higher
the probability that demand in current period will exceed the capacity carried over from the prior
period.
Exhibit 5 illustrates how the choice of optimal capacity in the current period is impacted
by expected demand. The point LL in Exhibit 5 is defined by the value of the expected demand
E (qt ) that satisfies the condition  qt  Qt*1 E (qt )  pc  dc . If expected demand is to the left
m

of LL then the probability of demand exceeding the endowed capacity,  qt  Qt*1 E (qt ) , is
less than the critical value pc  dc and hence, by Proposition 2, capacity should be reduced.
m

The point UL in Exhibit 5 is defined by the value of expected demand E (qt ) that satisfies
the condition  qt  Qt*1 E (qt )  pc  ac . If expected demand is to the right of this point, then
m

the probability of demand exceeding the endowed capacity,  qt  Qt*1 E (qt ) , will be greater
than the critical value pc  ac and hence, by Proposition 1, capacity should be increased.
m

If expected demand falls between LL and UL there should be no change in capacity.
If this model is correct, then there are two relevant ranges—one for the capacity decision,
which is based on expected activity, and one for actual activity. If expected activity falls within
the interval (LL, UL), then capacity will not be adjusted and one source of nonlinear cost
behavior is eliminated. And if actual activity falls within the interval (0, Q*t-1), then the other
source of nonlinear cost behavior—the additional marginal costs that are incurred when activity
exceeds capacity—is eliminated.

Page 19

Exhibit 5
The Optimal Level of Capacity in Period t
as a Function of the Expected Demand in Period t
*

Qt

Qt* = Qt*−1

LL

Qt*−1

Page 20

UL

E ( qt )

“Sticky” and “Anti-Sticky” Costs
If we further assume that realized demand from period t-1 becomes the expected demand
for period t, then Exhibit 6 illustrates how the choice of optimal capacity in the current period is
impacted by the prior period’s realized demand. 15 The point LL in Exhibit 6 is defined by the
value of the realized demand qt-1 that satisfies the condition  qt  Qt*1 E (qt )  qt1   pc  dc .
m

If the realized demand qt-1 is to the left of LL then the probability of demand exceeding the
endowed capacity,  qt  Qt*1 E (qt )  qt1  , is less than the critical value pc  dc and hence,
m

by Proposition 2, capacity should be reduced.
The point UL in Exhibit 6 is defined by the value of the realized demand qt-1 that satisfies
the condition  qt  Qt*1 E (qt )  qt1   pc  ac . If realized demand is to the right of this point,
m

then the probability of demand exceeding the endowed capacity,  qt  Qt*1 E (qt )  qt1  , will
be greater than the critical value pc  ac and hence, by Proposition 1, capacity should be
m

increased.

15

Exhibit 6 is basically the same as Panel B of Figure 2 in Banker and Byzalov [2014], except that in their figure
the capacity depends on contemporaneous demand rather than the realized demand from the prior period.

Page 21

Exhibit 6
The Optimal Level of Capacity in Period t
as a Function of the Realized Demand in Period t-1
(This assumes that the expected demand in period t
equals the realized demand from period t-1)
*

Qt

Qt* = Qt*−1

LL

Qt*−1

UL

qt −1

In the literature, “sticky’ costs are characterized by costs that decrease less when activity
declines than they increase when activity increases. “Anti-sticky” costs are characterized by
costs that decrease more when activity declines than they increase when activity increases. When
activity fluctuates around the upper limit UL in Exhibit 6, costs will be sticky. When activity
fluctuates around the lower limit LL in Exhibit 6, costs will be anti-sticky. A firm may exhibit
sticky or anti-sticky cost behavior in any given period, depending on whether the probability that
pc  dc
demand will exceed the capacity carried over from the period is closer to pc  ac or
.
m
m

Within this framework, there is no such thing as a “sticky cost” firm or an “anti-sticky” cost
firm. Any one firm can exhibit either type of behavior depending on the circumstances.

Page 22

Sources of Asymmetric Cost Behavior
In this model there are two sources of asymmetric cost behavior where changes in cost are not
proportional to changes in contemporaneous activity. First, when activity exceeds capacity, an
additional marginal cost is incurred. Second, depending on expectations about demand, capacity
may be expanded, reduced, or kept the same. A simple model in which changes in cost are
proportional to changes in contemporaneous activity will only be valid in situations where (1)
demand never exceeds capacity and (2) capacity is always being increased, capacity is always
being decreased, or capacity never changes.
In the real world, many other sources of non-linear, asymmetric cost behavior certainly
exist. However, an interesting question is whether the relatively simple model developed in this
paper would by itself generate the sort of cost behavior we actually observe using real world
data.
NUMERICAL SIMULATIONS AND EMPIRICAL IMPLICATIONS
In this section the model is used to generate data to simulate empirical tests of cost behavior. The
generating process can be briefly described as follows. In Period 1 when endowed capacity is
zero, it is assumed that demand is lognormally distributed with the mean 1,000. 16 In all
subsequent periods, demand qt is assumed to be lognormally distributed with a mean equal to
the prior period’s realized demand. The cost parameters are arbitrarily set as follows: m = $100,
pc = $35, ac = $25, dc = $10, v = $200, and FC = $100,000. 17 Given these parameter values, the
critical values are

16

pc  ac
pc  dc
 0.25 and
 0.60 . From these assumptions, the realized
m
m

In Period 1, demand ( q0 ) is assumed to be lognormally distributed with parameters σ = ln(1.1) and μ = ln(1000) –

½σ2. Given these parameters, the expected value of demand is 1000. In subsequent periods, demand in Period t is
assumed to be lognormally distributed with parameters σ = ln(1.1) and μ = ln(qt-1) – ½σ2. With these parameters, the
expected value of demand in Period t is the realized demand in the prior period. The parameter value σ = ln(1.1) was
arbitrarily chosen so as to generate a reasonable looking variation in demand and this value was used in all of the
simulations. For example, with the lognormal distribution parameters σ and μ set as described above, the 90%
confidence interval in Period 1 is [851, 1164].
17
These parameter values were selected to generate a substantial number of periods in each of the categories of an
increase in capacity, a decrease in capacity, and no change in capacity. The parameter values were selected before
the regression model from the cost behavior literature was applied to the simulated data.

Page 23

demand for each period can be randomly generated and the optimal level of capacity and costs
can be computed by using Propositions 1, 2, and 3.
The results of following this process for one simulation of 21 periods, a representative
sample size found in the empirical literature, is displayed in Panels A and B of Exhibit 7. In
Period 1, the optimal level of capacity is selected to satisfy  q1  Q1*  

pc  ac
 0.60 . Under
m

the assumption that the demand is lognormally distributed with the mean of 1,000,
 q1  972  0.60 , so the level of capacity in Period 1 is set at 972. A random draw from the

lognormal distribution with mean 1,000 resulted in a realized demand of 1,033 units. Therefore,
demand exceeded capacity by 61 units (= 1,033 units – 972 units).
In Period 2, the endowed capacity is 972 units. By construction, the mean of the demand
for Period 2 is 972 units—the realized demand in Period 1. Because the mean has changed, the
probability that demand will exceed the endowed capacity of 972 units is now 0.722. Because
this exceeds the upper limit of 0.60, the capacity is increased to 1,003 units. The random realized
demand in Period 2 is 1,019 units.
In Period 3, the endowed capacity is 1,003 units. Once again, the mean of the distribution
of demand has shifted—this time to 1,019 units—the realized demand in Period 2. The
probability that demand will exceed the endowed capacity of 1,003 units is 0.544. This value is
between the lower limit of 0.25 and the upper limit of 0.60, so capacity is not changed. Demand
falls to 762 units and there is excess capacity.
In Period 4, the endowed capacity is once again 1,003 units. However, the mean of the
distribution of demand has now shifted down to 762 units—the realized demand in Period 3.
Given this new mean, the probability of exceeding the endowed capacity is now only 0.002,
which triggers a reduction in capacity to 809 units. The process continues on in this fashion
through Period 21.
Costs are computed in Panel B of Exhibit 7.These calculations involve some subtleties.
The propositions rely on the calculation of expected economic profits, which in turn rely on the
calculation of economic costs. However, empirical work, which is the focus of this section,
necessarily must deal with reported costs. Reported costs do not, for example, include

Page 24

opportunity costs, which could be a large component of the deactivation cost dc. Initially we
assume that dc consists entirely of non-reported opportunity costs so that the reported total
operating cost is the sum of the variable cost of production, v  qt , the periodic avoidable and
amortized costs of the installed capacity, ( pc  ac)Qt * , fixed costs, FC , and the costs of
exceeding capacity, m qt  Qt* qt  Qt*  , where  qt  Qt*  =1 when qt  Qt* and 0
otherwise. We assume that when capacity is deactivated any unamortized costs of acquiring
capacity are written off and not included in total operating cost.
Panel C of Exhibit 7 provides statistics concerning this particular simulation of 21
periods. In 38% of the periods demand exceeded capacity and, in the remaining 62%, there was
excess demand. This should make sense because in any given period, depending on whether
capacity was increased, decreased, or remained the same, the probability of demand exceeding
capacity could lie anywhere within the range (0.25, 0.60). Note also that in this particular
simulation, capacity increased in 25% of the periods, decreased in 40% of the periods, and
remained the same in 25% of the periods.
By construction, a regression of the form

TCt    1  qt  2 Qt *  3  qt  Qt* qt  Qt* 
would yield an R2 of 1.000 and coefficient estimates of   FC , 1  v , 2  ( pc  ac) , and

3  m with infinite t-statistics. However, to use this model in empirical work, one would have
to know the total reported cost in period t, the realized demand in period t, and the capacity in
period t. Unfortunately, the latter datum—the capacity in period t—is likely to be particularly
elusive in most settings.
Moreover, rather than a linear model based on levels as developed in this paper, recent
empirical work on cost behavior has usually used some variation on the regression model (1) of
Noreen and Soderstrom [1997] that focuses on changes:

 TC t 
 qt 
 qt 
ln 
 = α + β ln 
 + β− Decline t ln 

 TC t −1 
 q t −1 
 q t −1 

Page 25

(1)

This regression equation is clearly misspecified if the data are generated following the process as
described in this paper. Nevertheless it is interesting to see what happens when the
“misspecified” regression model (1) is applied to the data generated by the simulation. The
results of running this test are reported in the bottom of Panel C of Exhibit 7. Those who are
familiar with the recent empirical work on cost behavior should be struck by the similarity
between the results in Panel C and the results typically reported in the literature for real world
data. 18 In particular, the β coefficient is positive and less than one and the β− coefficient is
negative, but much smaller in magnitude than the β coefficient and much less significant.

18

Given the strong resemblance of the regression results in Exhibit 5 to real world regressions of actual cost and
activity data, the reader may naturally be skeptical. However, I did not tweak any of the parameters to get this
striking result. Before I ran any regressions, I simply ran the simulation repeatedly until I got a result in which
Period 2 involved an increase in capacity, Period 3 involved no change in capacity, and Period 4 involved a decrease
in capacity. The only reason I had for doing this was to parallel the discussion of Exhibits 1 through 4 which also
involved an increase in capacity in Period 2, no change in capacity in Period 3, and a decrease in capacity in Period
4.

Page 26

Exhibit 7
Panel A: Simulation of 21 Periods
Optimal Capacity and Realized Demand

Period t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

Qt*1
0
972
1,003
1,003
809
798
798
798
798
729
717
656
670
734
757
670
670
670
616
576
589

pc  dc
m
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25

 qt  Qt*1 
1.000
0.722
0.544
0.002
0.206
0.526
0.520
0.388
0.052
0.197
0.054
0.682
0.889
0.716
0.025
0.254
0.339
0.060
0.084
0.687
0.328

pc  ac
m
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60
0.60

Change in
Q*
Increase
Increase
No Change
Decrease
Decrease
No Change
No Change
No Change
Decrease
Decrease
Decrease
Increase
Increase
Increase
Decrease
No Change
No Change
Decrease
Decrease
Increase
No Change

Page 27

Qt*

 qt  Qt* 

972
1,003
1,003
809
798
798
798
798
729
717
656
670
734
757
670
670
670
616
576
589
589

0.600
0.600
0.544
0.250
0.250
0.526
0.520
0.388
0.250
0.250
0.250
0.600
0.600
0.600
0.250
0.254
0.339
0.250
0.250
0.600
0.328

Realized
qt
1,033
1,019
762
752
807
806
780
687
675
618
689
756
779
631
632
647
580
542
606
567
566

Demand in
excess of
capacity
61
15
0
0
9
8
0
0
0
0
33
86
45
0
0
0
0
0
30
0
0

Exhibit 7
Panel B: Simulation of 21 Periods
Costs

Period t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

v×qt
$206,502
$203,726
$152,469
$150,344
$161,326
$161,110
$156,020
$137,339
$135,010
$123,533
$137,826
$151,136
$155,764
$126,172
$126,321
$129,330
$116,021
$108,467
$121,155
$113,360
$113,191

(pc+ac)
×Qt*
$58,303
$60,199
$60,199
$48,557
$47,880
$47,880
$47,880
$47,880
$43,738
$42,997
$39,342
$40,179
$44,058
$45,408
$40,182
$40,182
$40,182
$36,949
$34,544
$35,318
$35,318

m×
 qt  Qt*  ×

qt  Qt*  §
$6,079
$1,532
$0
$0
$863
$755
$0
$0
$0
$0
$3,344
$8,604
$4,451
$0
$0
$0
$0
$0
$3,005
$0
$0

FC
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000
$100,000

TCt #
$370,884
$365,457
$312,668
$298,901
$310,068
$309,745
$303,900
$285,219
$278,748
$266,530
$280,512
$299,918
$304,274
$271,579
$266,503
$269,512
$256,203
$245,417
$258,703
$248,678
$248,509

§   X  is a function whose value is 1 when X is true and 0 otherwise. So in this case,
 qt  Qt*  is 1 when realized demand exceeds capacity and is zero otherwise.

# TCt  v  qt  ( pc  ac)Qt *  m qt  Qt* qt  Qt*   FC

Page 28

Exhibit 7
Panel C: Statistics Concerning the Simulation of 21 Periods

Percentage of periods in which:
Demand exceeds capacity
There is excess capacity

38%
62%
100%

Capacity increases
Capacity does not change
Capacity decreases

25%
35%
40%
100%

Cost generating model: TCt  v  qt  ( pc  ac)Qt *  m qt  Qt* qt  Qt*   FC
 TC 

 q 

 q 

t 
 t 
 t 
Empirical model: ln 

 TC      ln  q     qt  qt1  ln  q 
t 1
t 1
t 1

adj R2
α
β
β−

0.963
Coefficient
-0.008
0.626
-0.138

t Stat
-2.33
9.25
-1.62

Page 29

To be sure, this is just one set of simulated data for 21 periods. The results might be quite
different if another 21 periods were simulated using the same assumptions—and indeed it turns
out that they often are. Exhibit 8 summarizes the results from generating 1000 sets of simulated
data for 21 periods following exactly the process described above. In all cases, the mean demand
for Period 1 is assumed to be 1,000 units and exactly the same parameter values were used as
before. Each of the 1000 simulations of 21 periods is unique as a consequence of the random
realizations of demand in Period 1 and all subsequent periods. Once again, those who are
familiar with the results of running the “misspecified” regression model on real world data will
be struck in Exhibit 8 by how closely the results from these simulations resemble what has been
reported in the literature—except perhaps for the high R2s and high t-statistics. 19 Most
importantly, the average β coefficient is positive and less than one and the average β− coefficient
is negative, but much smaller in magnitude than the β coefficient and much less significant. 20

19

The R2s and the t-statistics could be driven down to any desired level simply by adding a random term to the cost
to reflect the fact that in the real world costs are influenced by factors not found in this model.
20
Once again, a skeptical reader might wonder whether I tweaked the parameters of the simulations to achieve these
remarkable results. I will make the Excel © workbook that I used to generate these simulations available to anyone
who requests it.

Page 30

Exhibit 8
Summary Statistics Concerning 1,000 Simulations of 21 Periods
(The deactivation cost dc is NOT included in reported total cost)
Percentage of periods in which:
Demand exceeds capacity
There is excess capacity

Capacity increases
Capacity does not change
Capacity decreases

43.0%
57.0%
100.0%
32.5%
33.6%
33.9%
100.0%

Cost generating model: TCt  v  qt  ( pc  ac)Qt *  m qt  Qt* qt  Qt*   FC
 TC 

 q 

 q 

t 
 t 
 t 
Empirical model: ln 

 TC      ln  q     qt  qt1  ln  q 
t 1
t 1
t 1

Averages across all 1,000 simulations
adj R2
0.971
Coefficient
t Stat
α
-0.009
-2.304
β
0.775
15.330
β--0.236
-2.596

Percentiles
0.10
0.25
0.50
0.75
0.90

Adj R2
0.952
0.964
0.974
0.981
0.986

α
-0.014
-0.012
-0.009
-0.007
-0.005

t-Stat
-3.526
-2.856
-2.237
-1.682
-1.162

% Negative
100%
0%
97%
β
0.634
0.713
0.788
0.857
0.900

t-Stat
8.563
11.262
14.851
18.596
23.112

Page 31

β-0.370
-0.308
-0.242
-0.170
-0.102

t-Stat
-4.318
-3.484
-2.547
-1.748
-1.053

Sensitivity Analysis
These simulations could be tweaked in many different ways. One way is to assume that
the deactivation cost dc represents out of pocket costs rather than opportunity costs and is
included in reported total operating cost. The results of generating 1000 sets of new simulations
of 21 periods using exactly the same generating process as before except for the inclusion of the
deactivation costs in reported total cost are displayed in Exhibit 9. A quick comparison of
Exhibits 8 and 9 indicates that this tweak had little impact on the tenor of the results—which
continue to be strikingly similar to regression results using real world data.
Another variation would be to ask the question: “Are these results just an artifact of the
additional marginal cost of exceeding capacity? Do these results have nothing to do with the
transaction costs involved in expanding and contracting capacity?” This question can be
addressed using exactly the same simulation process as described in Exhibits 7 and 8 except that
now we assume that the probability density function for realized demand qt never changes. In
this case there is no reason to change the optimal level of capacity once it is set in the first
period. (As always in these simulations, we assume that capacity is set based on the probability
density function that actually generates the realized demand. In other words, we assume that
while demand is uncertain, the decision maker knows the true probability density function.) The
results of assuming that the optimal level of capacity never changes and hence the only source of
asymmetric cost behavior is the additional marginal cost of exceeding capacity are displayed in
Exhibit 10. In this case, the phenomenon observed in empirical work completely disappears. On
average, the β_ coefficient is zero and statistically insignificant; there is no significant difference
between the responsiveness of cost to increases and decreases in activity. This suggests that the
capacity adjustment costs play a critical role in the asymmetric cost behavior documented in the
literature.

Page 32

Exhibit 9
Summary Statistics Concerning 1,000 Simulations of 21 Periods
(The deactivation cost dc is included in reported total cost)
Percentage of periods in which:
Demand exceeds capacity
There is excess capacity

Capacity increases
Capacity does not change
Capacity decreases

42.9%
57.1%
100.0%
32.1%
33.5%
34.4%
100.0%

Cost generating model:
TCt  v  qt  ( pc  ac)Qt *  m qt  Qt*