Capacity Constraints and Asymmetric Cost Behavior

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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.

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Page 2 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_{See, for example, Banker, Byzalov, and Plehn-Dujowich [2014].}

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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:

t t t

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

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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

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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]}

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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]

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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 pc ac

m



is critical in the decision of whether to expand capacity

and the ratio pc dc

m



is critical in the decision of whether to reduce capacity. Assuming that the

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Proof:

. Then dc > pc, and the deactivation cost would exceed the periodic 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 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

1

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Expected Economic Profit14

The expected economic profit depends on whether capacity is expanded, remains the same, or is contracted. 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Q_t_₁



, exceeds the critical value pc ac m



, then capacity should be expanded to

the point, Qt*, where the probability that demand exceeds capacity, 



qtQt*



, equals the

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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

m



is less than 1.00,

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



*



1 1

q Q

   equals the critical value pc ac

m

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Exhibit 1

Example of Proposition 1

Period 1: Qt-1 = 0

0 

q1 Q0



  

Q

1*

pc ac m





q1 Q



  

Q

1

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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



*



2 2

q Q

   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.

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Exhibit 2

Period 2



*



2 1

q Q   

Q

2*

pc ac m





q

2

Q









Q

1*

1

0

Probabilities based on new information

available at the beginning of Period 2.

Probabilities based on information available at the beginning

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Proposition 2

If and only if m



q_tQ_t_₁



< pc− dc, then Qt* < Qt-1 and capacity should be reduced to the

point where



*



t t

m q Q = 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,



qt Qt 1



   _ , is less than the critical value pc dc m



then capacity should be reduced to the point

Qt* where the probability that demand exceeds capacity, 



q_tQ_t*



equals the critical value

pc dc m



. To make this more concrete, in the simulations at the end of this paper, the assumed

value of pc dc

m



is usually 0.25. Proposition 2 states that if the probability of exceeding the

endowed capacity is less than 0.25, then capacity should be reduced. Somewhat counter-intuitively, 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Q_t_₁



≤pc + ac AND m



q_t Q_t_₁



≥pc− dc, then Qt* = Qt-1 and

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Proof: Proposition 3 follows from Proposition 1 and Proposition 2.

Proposition 3 establishes that if the probability that demand exceeds the endowed capacity,



qt Qt 1



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

m



and pc ac

m



, the more likely it is that capacity will

not change. The size of this gap is ac dc

m



, which is increasing in the adjustment costs ac and dc 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

m



and pc dc

m



and therefore, as depicted in

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Exhibit 3

Period 3



*



3 2

q Q   

Q

3*

= Q

2*



pc ac m



q

3

Q









Q

1

0

pc dc

m



Probabilities based on new

information available at the

beginning of Period 3.

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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 m



. Consequently, by Proposition 2, capacity will be reduced to the level Q4* where the

probability that demand exceeds capacity,



*



4 4

q Q

   , equals the critical value pc dc

m

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Exhibit 4

Example of Proposition 2 Period 4

Q

4*

Q

3*



*



4 3

q Q   

pc ac m





q

4

Q









Q

1

0

pc dc

m



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

Probabilities based on new information available at the

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The Relevant Range

carried over from period t-1, conditional on the expected demand ( )E qt . As shown in Appendix

B,



*



1 ( )

t t t

q Q E q

     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

( )_t

of LL then the probability of demand exceeding the endowed capacity,



*



1 ( )

t t t

q Q E q

     , is

less than the critical value pc dc

m

 _{and hence, by Proposition 2, capacity should be reduced.}

The point UL in Exhibit 5 is defined by the value of expected demand ( )E qt that satisfies

the probability of demand exceeding the endowed capacity,



*



1 ( )

t t t

q Q E q

     , will be greater

than the critical value pc ac

m

 _{and hence, by Proposition 1, capacity should be increased.}

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Exhibit 5

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

*

t

Q

* * 1

t t

Q

=

Q

₋

( )

_t

E q



* 1

t

Q

₋

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“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 Q_t*₁ E q( )_t q_t ₁



pc dc

If the realized demand qt-1 is to the left of LL then the probability of demand exceeding the

endowed capacity,



*



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



*



then the probability of demand exceeding the endowed capacity,



*



1 ( ) 1

t t t t

q Q E q q

       , will

be greater than the critical value pc ac

m

 _{and hence, by Proposition 1, capacity should be}

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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)

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

demand will exceed the capacity carried over from the period is closer to pc ac

m

 _or pc dc 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.

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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 pc dc 0.25

m



 and pc ac 0.60

m



 . From these assumptions, the realized

16_{In Period 1, demand (}

0

q ) 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}

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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



*



1 1 0.60

pc ac

q Q

m

      . Under

the assumption that the demand is lognormally distributed with the mean of 1,000,



q1 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.

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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 q _t, the periodic avoidable and amortized costs of the installed capacity, (pcac)Q_t*, fixed costs, FC, and the costs of exceeding capacity, m



q_tQ_t*

 

 q_tQ_t*



, where 



q_t Q_t*



=1 when *

t t

q Q 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



 



* * *

1 2 3

t t t t t t t

TC       q  Q    q Q  q Q

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.

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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.

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Exhibit 7

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

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Exhibit 7

Panel B: Simulation of 21 Periods Costs

  is 1 when realized demand exceeds capacity and is zero otherwise.

# ₍ ₎ *



*

 

*



t t t t t t t

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Exhibit 7

Panel C: Statistics Concerning the Simulation of 21 Periods

Percentage of periods in which:

Demand exceeds capacity 38%

There is excess capacity 62%

100%

Capacity increases 25%

Capacity does not change 35%

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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 R}2_{s 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.

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Exhibit 8

Summary Statistics Concerning 1,000 Simulations of 21 Periods (The deactivation cost dc is NOT included in reported total cost)

Demand exceeds capacity 43.0%

There is excess capacity 57.0%

100.0%

Capacity increases 32.5%

Capacity does not change 33.6%

Capacity decreases 33.9%

Averages across all 1,000 simulations

adj R2 0.971

Coefficient t Stat % Negative

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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

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Exhibit 9

Summary Statistics Concerning 1,000 Simulations of 21 Periods (The deactivation cost dc is included in reported total cost)

100.0%

adj R2 0.973

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Exhibit 10

Summary Statistics Concerning 1,000 Simulations of 21 Periods

(The probability distribution of qt never changes; hence capacity never changes.)

100.0%

Capacity increases 0%

Capacity does not change 100%

Capacity decreases 0%

adj R2 0.986

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Another variation on the simulations would be to ask the question: “What happens if the costs of deactivating capacity are prohibitive?” This question can be addressed using exactly the same simulation process as described in Exhibit 8 except that now we assume that dc equals pc. With this assumption, it never pays to deactivate capacity and capacity is subject to a ratcheting effect—it can increase but not decrease. The results of making this assumption are displayed in Exhibit 11. In this case, the phenomena observed in empirical work does not disappear. On average, the β_ coefficient is negative, and while the average t-statistic is not statistically

significant, 73% of the β_ coefficients are negative and a substantial proportion of the individual t-statistics would be considered significant.

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Exhibit 11

Summary Statistics Concerning 1,000 Simulations of 21 Periods (The parameter dc is set equal to pc; hence capacity never decreases.)

100.0%

adj R2 0.977

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Exhibit 12

Summary Statistics Concerning 1,000 Simulations of 21 Periods

(Assume m = (p – v) + ε; hence capacity is never exceeded. Capacity is a hard constraint.)

100.0%

adj R2 0.944

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An interesting question to address with the simulations is: “What happens if the expected demand in the current period is not equal to the demand in the prior period?” 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 mean expected demand is a random walk generated by the

following process:

1

( )

_t

(

_t

) (1

)

E q





E q



_

 

z



Where

z



is a normally distributed random variable with mean 0 and standard deviation 0.09.21 We use exactly the same process used in Exhibit 8 to generate demand and to determine optimal capacity; except that the probability distribution of demand is now decoupled from the prior period’s realized demand. The results of making this assumption are displayed in Exhibit 13. In this case, the phenomena observed in empirical work disappears completely. On average, the β_ coefficient is zero and is rarely statistically significant. This suggests that the prior period realized demand has a powerful influence on the current period’s expected demand.

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Exhibit 13

Summary Statistics Concerning 1,000 Simulations of 21 Periods

(The expected value of demand follows the random walk process:

E q

( )



_t



E q

(



_t_₁

) (1

 



z

)

, where



z

is a normally distributed random variable with mean 0 and standard deviation 0.09.)

100.0%

adj R2 0.559

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Finally, the simulations can be used to address the question “what happens if the deactivation cost dc represents the option value of having capacity in place that can be used in the next period without incurring the activation cost ac?” In this case, dc is endogenous. The value of dc helps determine the optimal level of capacity, but you have to know the optimal level of capacity—the amount of capacity that will be carried over to the next period—to determine dc. Ideally, dc would be determined simultaneously with the optimal level of capacity, which makes it an endogenous variable. Analytically, this would be challenging. In the context of the simulations, however, the value of dc can be approximated via an iterative process. The variable dc is the expected savings in the next period from carrying over capacity from this period and thereby possibly avoiding the capacity activation cost ac in the next period. Specifically, I

assume that



_| *



t t

dcac q x Q , where x is the value of the realized demand in period t that would induce a marginal increase in capacity in period t+1, given the capacity that is selected in period t. The iterative process unfolds as follows. In first step, the value of x is determined by solving for the realized demand in period t that would induce a marginal increase in capacity in period t+1 assuming that there is no change in capacity in period t. The parameter dc is set equal to the product of ac and the probability that demand in period t will exceed this value of x. From this tentative value of dc, a tentative value for the optimal capacity *

t

Q is derived using the analytical model as before. In the second step, this tentative value of *

t

Q is used to compute an updated value for x, which in turns determines an updated value of dc, which is then used to derive an updated value of *

t

Q . This second step is repeated until * t

Q converges to a stable value. At that point, the value of *

t

Q that is used to compute dc is the same as the optimal capacity that is computed using that value of dc and the selection of has been endogenized. Experimentation revealed that five iterations was sufficient to attain stability in the sense that the optimal value of

* t

Q changed by less than one unit.

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Exhibit 14

Summary Statistics Concerning 1,000 Simulations of 21 Periods (The deactivation cost dc is dynamic and is defined by dc ac 



q_tx



,

where x is the minimum realization of demand in period t that would induce a marginal increase in capacity in period t+1 beyond the capacity carried over from period t.

dc is NOT included in reported total cost)

100.0%

adj R2 0.973

α -0.010 -2.473 100%

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CONCLUSION

This paper develops a formal model that is highly consistent with both the empirical results from the sticky cost literature and with the verbal theorizing that has motivated the main hypotheses in this literature. Where formal models have been referred to in this literature in the past, those models have fundamentally differed from the economic setting assumed in the verbal theorizing. The formal model in this paper reinforces the verbal theorizing and fleshes out that theory by helping to identify the salient variables and how their impacts on cost should be modeled in empirical work.

In this paper, costs are generated by a two-stage model. In the first stage, the level of capacity is determined based on the endowed capacity, the probability of demand exceeding capacity, and the costs of acquiring and maintaining capacity, the cost of exceeding capacity, and the cost of deactivating capacity. In the second stage, costs are computed conditional on the level of capacity that was chosen in the first stage. The results of simulations indicate that the

characteristic phenomenon observed in the cost behavior literature—a negative sign for the coefficient for a decline in activity—is strongest when all of the elements in this model are present. Moreover, it appears that in the real world expectations about demand are strongly influenced by the prior period’s realized demand.

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REFERENCES

Anderson, Mark C., Rajiv D. Banker, and Surya N. Janakiraman, 2003, Are Selling, General, and Administrative Costs “Sticky”? Journal of Accounting Research 41 (1): 47-63

Anderson, Mark, Rajiv Banker, Rong Huang, and Surya Janakiraman. 2007. Cost Behavior and Fundamental Analysis of SG&A Costs. Journal of Accounting, Auditing and Finance 22 (1): 1-28.

Balakrishnan, Ramji, and Thomas S. Gruca. 2008. Cost Stickiness and Core Competency: A Note. Contemporary Accounting Research 25 (4): 993-1006.

Balakrishnan, Ramji, Eva Labro, and Naomi Soderstrom. 2014. Cost Structure and Sticky Costs. Journal of Management Accounting Research 26(2), 91-116

Balakrishnan, Ramji, Michael J. Peterson, and Naomi S. Soderstrom. 2004. Does Capacity Utilization Affect the 'Stickiness' of Costs? Journal of Accounting, Auditing and Finance 19: 283-299.

Banker, Rajiv D. and Dmitri Byzalov, 2014. Asymmetric Cost Behavior, Journal of Management Accounting Research 26(2), 43-79

Banker, Rajiv D., Dmitri Byzalov, and Lei Chen. 2013. Employment Protection Legislation, Adjustment Costs and Cross-Country Differences in Cost Behavior. Journal of Accounting and Economics 55 (1), 111-127.

Banker, Rajiv D., Dmitri Byzalov, Mustafa Ciftci, and Raj Mashruwala. 2014. The Moderating Effect of Prior Sales Changes on Asymmetric Cost Behavior. Journal of Management Accounting Research 26(2): 221-242

Banker, Rajiv D., Dmitri Byzalov, and Jose M. Plehn-Dujowich 2014. Demand Uncertainty and Cost Behavior, The Accounting Review 89(3), 839-865

Banker, Rajiv D., and Lei (Tony) Chen. 2006. Predicting earnings using a model based on cost variability and cost stickiness. The Accounting Review 81: 285-307.

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Baumgarten, D., 2012. The Cost Stickiness Phenomenon: Causes, Characteristics, and Implications for Fundamental Analysis and Analysts’ Forecasts, Springer Gabler Calleja, K. M., M. Steliaros, and D. Thomas. 2006. A Note on Cost Stickiness: Some

International Comparisons. Management Accounting Research 17: 127-140.

Cannon, James N. 2014. Determinants of “Sticky Costs”: An Analysis of Cost Behavior Using United States Air Transportation Industry Data. The Accounting Review 89(5): 1645-1672 Chen, Clara Xiaoling, Hai Lu, and Theodore Sougiannis. 2012. The Agency Problem, Corporate

Governance, and the Asymmetrical Behavior of Selling, General, and Administrative Costs. Contemporary Accounting Research 29 (1): 252-282.

Goldratt, Eliyahu M. and Jeff Cox, 2014, The Goal: A Process of Ongoing Improvement, Third Revised Edition, , North River Press

Janakiraman, Surya, 2010. Discussion of The Information Content of the SG&A Ratio, Journal of Management Accounting Research 22: 23-30

Noreen, Eric, 1991, Conditions under Which Activity-Based Costing Systems Provide Relevant Costs, Journal of Management Accounting Research 3: 159-168

Noreen, Eric and Naomi Soderstrom, 1994, Are Overhead Costs Strictly Proportional to Activity? Journal of Accounting and Economics 17 (1-2): 255-278

Noreen, Eric and Naomi Soderstrom, 1997, The Accuracy of Proportional Cost Models: Evidence from Hospital Service Departments, Review of Accounting Studies 2 (1): 89-114 Shust, Efrat and Dan Weiss, 2014, Discussion of Asymmetric Cost Behavior—Sticky Costs:

Expenses versus Cash Flows, Journal of Management Accounting Research 26(2), 81-90 Weiss, Dan, 2010, Cost Behavior and Analysts’ Earnings Forecasts, The Accounting Review 85

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APPENDIX A

PROOFS OF PROPOSITIONS

The expected economic profit depends on whether capacity is expanded, remains the same, or is contracted. We take the expansion case first.

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Moreover, Qt* exists because as Qt increases, 



qt Qt



decreases.

expected additional marginal cost of producing one more unit beyond the current capacity shrinks as well. Equilibrium is reached when the expected additional marginal cost of producing one more unit beyond the current capacity equals the cost of acquiring and maintaining

additional capacity.

decreases as capacity is expanded from Qt-1.

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To see this, note that





 

the expected additional marginal cost of producing one more unit beyond the current capacity increases as well. Equilibrium is reached when the expected additional marginal cost of producing one more unit beyond the current capacity equals the savings from deactivating one unit of capacity.

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Proof of Proposition 3:

First, verify that the conditions m



qt Qt₁



≤pc + ac and m



qt Qt1



≥pc – dc are compatible. Together, they imply pc + ac≥ m



qt Qt₁



≥pc – dc which implies pc + ac≥ pc

– dc which implies ac≥ -dc which implies ac+ dc ≥ 0. If this were not true, the company could activate capacity and then immediately deactivate it for a profit.

⇒Suppose m



q_tQ_t_₁



≤pc + ac AND m



q_tQ_t_₁



≥pc – dc. Proposition 1 establishes

that Qt* > Qt-1 only if m



qtQt₁



> pc + ac. Therefore, Qt* ≤ Qt-1. Proposition 2 establishes

that Qt* < Qt-1 only if m



qtQt₁



< pc – dc. Therefore, Qt* ≥ Qt-1. Consequently, Qt* = Qt-1.

 Suppose Qt* = Qt-1. Proposition 1 establishes in that case that m



qtQt₁



≤pc + ac.

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APPENDIX B

THE FIRST DERIVATIVE OF A CDF WITH RESPECT TO THE MEAN It is intuitive that, ceteris paribus, as the mean of a random variable increases, the probability that the random variable will exceed any given level should increase as well. Let x be a random variable with a probability density function f x( ) defined on the interval [0,∞). Given z,

   before the mean shift is equivalent to the probability





   after the mean shift. The difference between these two probabilities is

1