M�t

TUSNBSer

E

THE ACCOUNTING REVIEW American Accounting AssociationVol. 85, No. 6 DOI: 10.2308/accr.2010.85.6.18392010pp. 1839–1860

Informativeness, Incentive Compensation, andthe Choice of Inventory Buffer

Stanley BaimanSerguei Netessine

University of Pennsylvania

Richard SaoumaUniversity of California, Los Angeles

ABSTRACT: Previous research in management accounting and economics has notedthe potential for complementarities between the firm’s performance measurement sys-tem and its other organizational design choices. We add to this literature by studyinghow the informativeness and incentive properties of a performance metric can be influ-enced by one particular organizational design choice—the size of the firm’s inventorybuffers. We model a manufacturing setting in which an agent manages a workstationthat processes intermediate units. As intermediate units arrive, they are stored in aninventory buffer until the agent can process them. The buffer can hold a maximumnumber of intermediate units—its buffer size. The agent is compensated on the basis ofhis workstation’s throughput. We characterize the conditions under which reducing theinventory buffer enhances/degrades the informativeness of the performance metricand, hence, mitigates/exacerbates the agent’s incentive problem.

Keywords: agency theory; inventory buffers; incentive complementarities; organiza-tional design; slack.

I. INTRODUCTIONost of the agency theory work in managerial accounting has studied the relation betweenperformance metric properties �e.g., informativeness, precision, congruity� and optimalincentive compensation, holding fixed the firm’s other organizational design decisions

e.g., hierarchical structure, job assignment, production technology�. However, by holding fixedhese other organizational design choices, one cannot study whether and how they affect the

his paper has benefited from the comments of workshop participants at the University of Wisconsin–Madison, Theniversity of Iowa, University of Houston 2007 Accounting Symposium, 5th Accounting Research Workshop �Fribourg,witzerland�, The University of Chicago, University of Pennsylvania, Tel Aviv University, Hebrew University �Jerusalem�,orwegian School of Economics and Business Administration, and Catholic University of Portugal, especially those of R.alakrishnan, P. Berger, R. Cazier, D. DeJong, J. Gerakos, F. Gjesdal, J. Glover, T. Hemmer, E. M. Matsumura, M. Penno,. Radhakrishnan, K. Sivaramakrishnan, and J. Stecher. We are particularly indebted to Jack Hughes, Cathy Schrand, theditor, and two anonymous reviewers for many helpful suggestions, and to Romanos Malikiosis for his outstandingesearch assistance.

ditor’s note: Accepted by John H. �Harry� Evans III.

Submitted: May 2008Accepted: May 2010

Published Online: November 2010

1839

pmon

nzarwtam

wiibabfo

oicaappam

ic

ecaMsG

1

2

1840 Baiman, Netessine, and Saouma

TA

erformance metric properties. As Hemmer �1998, 321–322� notes, “the value of a performanceeasure is determined not simply by its congruity and precision but by its influence on the optimal

rganizational design … Much of the recent theoretical accounting literature … has largely ig-ored complementarities between performance measures and organizational design.”

In this study, we expand on Hemmer’s �1998� observation by studying how the informative-ess and incentive properties of a performance metric are influenced by one of the firm’s organi-ational design choices—the size of its inventory buffers. The introduction of just-in-time �JIT�nd, more generally, lean manufacturing has led to an increased emphasis on controlling andeducing inventory levels. Managerial accounting has responded to this reduction in inventoryith new costing techniques such as backflush costing �Horngren et al. 2008�. However, we show

hat the choice of inventory buffers has a more subtle effect on the design of the managerialccounting system, in that it affects the informativeness of performance metrics produced by theanagerial accounting system.

We model a manufacturing setting in which an agent expends effort to set up or program aorkstation that processes intermediate units. As intermediate units arrive, they are stored in an

nventory buffer until the agent can process them. The buffer can hold a maximum number ofntermediate units, b—its buffer size. The buffer size represents the workstation’s maximum orderacklog. If an intermediate unit arrives while the buffer is full, then the unit is diverted and thegent loses the opportunity to process it. If the workstation is ready to process another unit but theuffer is empty, then it remains idle until the next intermediate unit enters the buffer. While werame the analysis in terms of a manufacturing setting, the analysis is also applicable to serviceperations such as call centers that can have a maximum number of calls awaiting service.

We assume that the agent is compensated on the basis of his realized throughput; i.e., theutput per unit of time of his workstation.1 We show that the principal’s choice of buffer sizendirectly affects the effectiveness with which throughput reflects the agent’s chosen effort andonsequently affects the optimal compensation weight placed on throughput. The buffer sizeffects both the probability that the workstation’s buffer will be full when an intermediate unitrrives, causing it to lose the opportunity to complete that unit �referred to as blocking�, and therobability that the buffer will be empty when the workstation is free, causing it to forgo furtherroduction until another intermediate unit arrives �referred to as starving�. Both probabilitiesffect the marginal productivity of the agent’s setup effort with respect to throughput �his perfor-ance metric� and, hence, the extent to which that effort is reflected in throughput.

Next we discuss the relevant literature on organizational design and incentives. Section IIntroduces the model and Section III presents the results. Section IV discusses extensions andoncludes.

A number of articles in both the accounting and economics literatures have considered theffect of organizational design choice on the design of optimal incentives. Among the issuesonsidered are: job design �Riordan and Sappington 1987; Holmstrom and Milgrom 1991; Bal-krishnan et al. 1998�; the choice of production systems and technologies �Hemmer 1995, 1998;ilgrom and Roberts 1995�; the hierarchical structuring of work �Melumad et al. 1995�; routing

chemes for product rework �Lu et al. 2009�; and production bottlenecks �Datar and Rajan 1995;ietzmann and Hemmer 2002�.2

Realized throughput is measured as realized total output during the shift that the workstation has been operating dividedby the length of the shift.Also somewhat related is Cremer �1995�, which examines the incentive effects on both the buyer and supplier when thebuyer goes to a zero-inventory system for the supplier’s product. However, the two motivations are different in thatCremer �1995� is based on a double moral hazard model.

he Accounting Review November 2010merican Accounting Association

hibib�pfIb

awmvaptas

ccmduhatbwt

rwu

3

4

5

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1841

T

Similar to our model, Hemmer �1995, 1998� and Gietzmann and Hemmer �2002� examineow different workflow arrangements between agents affect the information available for contract-ng, and the incentives facing agents. Our work is distinct from these models that do not consideruffer size as one of the principal’s choice variables. Nagar et al. �2009� examine the role ofnventory buffers in agency problems, although unlike our model, in their setting buffers are filledy agents in order to signal private information and buffer size is not a choice variable. Alles et al.1995� do focus on the effect that the choice of buffer size can have on the informativeness oferformance metrics. The major difference between our work and theirs is that they do notormally model how buffers affect performance metrics, but instead assume a monotonic relation.n contrast, we formally model the inventory process and derive a non-monotonic relation betweenuffer size and the informativeness of the agent’s performance metric.

II. MODEL AND INITIAL ANALYSISThe model consists of a single-period and a firm comprised of a risk-neutral principal and

gent. Contracting takes place at the start of the period when the agent is hired to set up aorkstation to process incoming intermediate units �e.g., test completed computers or weld auto-otive chassis�. The intermediate units arrive stochastically into the workstation’s incoming in-

entory buffer at a commonly known, mean arrival rate of �. The buffer has capacity b � N,llowing a maximum of b � 1 units to be held in front of the workstation while the workstationrocesses one unit. If an intermediate unit arrives and the buffer is not full, then the unit is addedo the inventory. However, if the buffer is full, then blocking occurs and intermediate units ceaserriving until there is space in the buffer, whereupon the intermediate units begin arriving at theame stochastic rate as before.3

Once parties agree to the contract, but before the workstation begins processing, the agenthooses his personally costly setup effort r, which affects the average rate at which his workstationan process the intermediate units. That effort is unobserved by the principal and is subject tooral hazard. The arriving intermediate units are sufficiently heterogeneous that the agent’s effort

oes not perfectly control the rate at which the workstation processes the incoming intermediatenits. Rather, the intermediate units are processed at a stochastic rate with mean r�1 � h�, where� ��1,�� represents the productivity-enhancing resources the principal has allocated to the

gent. Examples of the latter include: machine and tool upgrades, improved working conditions,raining, etc. As the workstation finishes processing an intermediate unit, the agent reaches into theuffer for the next unit on which to work. If the buffer is empty, then starving occurs and theorkstation is idle until the next intermediate unit arrives. Figure 1 provides a graphical descrip-

ion of the process.The principal chooses both the agent’s compensation scheme and the buffer size, b. We

estrict our attention to linear compensation schedules for the agent consisting of a fixed payment,

0, plus an incentive wage, w, based on realized throughput—the number of units processed pernit of time.4,5

Alternatively, one can interpret the model as one in which the intermediate units continue arriving while the inventoryis blocked but are diverted elsewhere. What is important is that in both cases, blocking imposes an opportunity cost onthe agent because his workstation loses the chance to process the intermediate units that might have been added to thebuffer had it not been full.Given that the basic properties of queuing systems are stated in terms of rates �i.e., arrivals per unit of time, number ofunits processed per unit of time� we assume that the principal compensates the agent on throughput per unit of timerather than total throughput. Linear compensation schemes are frequently used in practice �Berg and Fast 1975; Hall etal. 2000� and are extensively used in the analytical literature �Feltham and Xie 1994; Alles et al. 1995; Hemmer 1995,1998; Gietzmann and Hemmer 2002�.In Section IV, we discuss the possibility of contracting on inventory levels.

he Accounting Review November 2010American Accounting Association

tr

6

Iiuae

1842 Baiman, Netessine, and Saouma

TA

Last, we assume that the marginal cost associated with each unit of buffer capacity is c � 0,he agent’s marginal cost of effort is a � 0, and we normalize the marginal cost of the additionalesources provided to the agent by the principal, h, at one dollar per unit of resource.

All notation and assumptions are as follows:

� � The mean arrival rate of the intermediate units. The arrival rate is Poissondistributed, implying that the time between arrivals is distributed according to anexponential distribution with mean of �−1.6

r � The level of effort the agent exerts at private cost, ar.h � The amount of productivity-enhancing resources allocated to the agent. The

principal’s out-of-pocket cost is h.r�1 � h� � The mean rate at which the agent’s workstation processes the intermediate units.

The actual processing time is distributed according to an exponential distribution.b � The maximum number of units the station can simultaneously store �buffer size�.

The principal incurs the associated capacity cost, cb, each period.

For an exponential distribution, the same parameter that represents the mean also represents the variance. For simplicitywe refer to the choice of this parameter as the choice of the mean arrival rate.

FIGURE 1The Effect of Inventory on the Production Process

No incomingunits currentlyarriving atworkstation

Emptybuffer

Workstation idle (starving)

Incoming unitsblocked(blocking)

Fullbuffer

Workstation processing a unit

Workstation processing a unitNon-full,non-emptybuffer

Incoming unitssent to buffer

n the top panel, the buffer is not full, incoming units are added to the buffer, and the workstation starts process-ng a unit from the buffer once it completes the current unit. In the middle panel, the buffer is full, incomingnits are blocked from entering the buffer (blocking) until the workstation finishes processing its current unitnd the agent withdraws the next intermediate unit from the buffer. In the bottom-most panel, the buffer ismpty, implying that the workstation is idle (starved) until the next unit arrives.

he Accounting Review November 2010merican Accounting Association

itsTdefhr

mmpTaoqctsoimtedtcimdAaCc

7

8

9

1

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1843

T

K � The steady-state mean throughput per period, with K̃ denoting the realizedthroughput.

w, w0 � The incentive payment made to the agent per unit completed per period, and thefixed salary payment, respectively.

Note that in our model there are two sources of noise that make it difficult for the principal tonfer the agent’s effort from throughput. As is common in agency problems, the relation betweenhe agent’s effort and the time required for the workstation to process an intermediate unit istochastic. In addition, the rate at which units become available to be processed is also stochastic.hus, even if the relation between the agent’s effort and the time required to process an interme-iate unit were deterministic, instantaneous throughput would still be stochastic in the agent’sffort because of the stochastic arrival rate of the intermediate units. Placing an inventory buffer inront of the agent’s workstation is one way of dampening out the effect of the variation in arrivals;owever, as we will show, such dampening can also reduce the informativeness of throughput withespect to the agent’s effort.

Our assumptions regarding arrival and processing rates follow the standard M/M/1/b queuingodel with finite buffers used in the operations management literature.7 Note that while we areodeling a single-period in that the agent chooses his setup effort only once at the start of the

eriod, we analyze the stochastic evolution of the workstation’s throughput over the entire period.he transient behavior of throughput in queuing systems with finite buffers cannot be describednalytically, although the steady-state behavior can be described in closed form.8 As a result, theperations management literature typically uses simulations to analyze the transient behavior ofueuing models with finite buffers and closed-form analysis to examine steady-state behavior. Inontrast, agency models have emphasized rewarding transient behavior, but simplify the produc-ion process �e.g., using the LEN model� to achieve closed-form solutions.9 In order to incorporatetochastic intermediate unit arrival rates, stochastic processing rates, and finite inventory buffers inur analysis, we focus on the problem’s steady-state. That is, we assume that the principal isnterested in maximizing her steady-state expected profit and that the agent selects his effort toaximize his steady-state expected utility.10 This is a reasonable approximation if, as we assume,

he production process reaches steady-state fairly quickly. Our assumption that the agent’s effort isxpended in setting up the workstation before production begins, rather than managing the pro-uction process as it evolves, is also consistent with our emphasis on the steady-state behavior ofhe workstation. An example of our modeled setting is a robotic line that welds automotivehassis. The agent’s main task is to program the spot-welding robots. This task involves identify-ng the optimal position of the chassis, identifying the optimal position of the robotic arms, and

aking sure that robot paths do not cross each other or the chassis itself �so that nothing isamaged in the process� while ensuring that welding is done in the least possible number of steps.ll of this planning and programming setup is done once by the agent at the beginning of the shift

nd thereafter, as long as the workstation is in control, the agent does not need to intervene.learly, the actions of the agent in programming the robots affect the time it takes to weld thehassis �on average�, but the agent does not reprogram the robot once production has begun.

The M/M/1/b model is the most widely accepted modeling approach for production systems with finite buffers �Berkley1992; Groenevelt 1993; Balsamo et al. 2001�.The process attains steady-state when the probability distributions over inventory levels and throughput are no longerfunctions of the starting conditions, and no longer vary over time. That is, while the realized inventory level andthroughput at any moment are still uncertain, the probability distributions that describe them do not vary over time.Queuing models similar to ours have been used to analyze the role of cost allocation in resolving congestion problems�Radhakrishnan and Balachandran 1995, 2004; Balachandran and Radhakrishnan 1996�.

0 Lu et al. �2009� restrict their analyses to steady-state for the same reason.

he Accounting Review November 2010American Accounting Association

m

wbaapotl

mtawiapwt�

os

GfpeaKapt

cwrr

1

1844 Baiman, Netessine, and Saouma

TA

The workstation’s realized output per unit of time or throughput, K̃, is stochastic, though theean steady-state throughput, K, can be represented as �Hopp and Spearman 2001�:

K�b,r� = ��1 − pb� = �1 − �r�1 + h�/��−b

1 − �r�1 + h�/��−�b+1� , �1�

here pb is the steady-state probability of the buffer being full; i.e., the steady-state probability oflocking. The workstation’s steady-state mean throughput is thus equal to the unblocked meanrrival rate of intermediate units, �, times the steady-state probability that the buffer can accept andditional unit when it arrives, 1 − pb. In equilibrium, the workstation’s mean steady-state through-ut is increasing in: r, the agent’s effort; h, the level of resources allocated to the agent; b, the sizef the input buffer; and �, the mean arrival rate of the intermediate units. Equation �1� illustrateshat the agent’s effort affects throughput via its effect on the probability of blocking, or equiva-ently, the probability of starving.11

Note that the mean arrival rate of incoming intermediate units and therefore the maximumean throughput is �. However, if the principal induced the agent to choose r � �/�1 � h� so that

he workstation was set to process the intermediate units at the same average rate at which theyrrived, then the instantaneous probability of starving the workstation would be ps = 1

1+b � 0,hich in turn would limit the workstation’s mean steady-state throughput to K = �b

1+b � �. That is,nducing the agent to set the workstation to process the intermediate units at the same average ratet which they arrive would significantly limit steady-state throughput and induce a non-trivialrobability of starving. Thus, the principal has a natural incentive to induce the agent to set theorkstation to process units at an average rate that is strictly greater than the average rate at which

he intermediate units arrive, as commonly assumed in the operations management literatureHopp and Spearman 2001�.

Assume that the principal earns R dollars per unit of throughput. The principal chooses the setf decision variables, �w,w0,b�, and a target level of agent setup effort, r*, which maximize herteady-state objective function:

Max�w,w0,b,r*�RK�b,r*� − wK�b,r*� − w0 − cb − h .

iven �w,w0,b�, the agent chooses his setup effort, r, to maximize his steady-state objectiveunction maxr wK�b,r� + w0 − ar. Varying the size of the buffer, b, has three effects on the princi-al’s problem: �1� it incurs a carrying cost of c per unit of buffer, �2� it affects the principal’sxpected revenue through its direct effect on steady-state throughput, K�b,r�, and �3� it indirectlyffects the agent’s effort incentive through its effect on the expected steady-state throughput,�b,r�. Effort and buffers are substitutes in throughput and, therefore, the principal’s choice of bffects the wage, w, required to induce the agent to work. Thus, the latter two effects of therincipal’s choice of b directly influence the effort that the principal wants to induce the agent toake and the cost of inducing that effort.

As noted earlier, we are interested in studying the incentive implications of the principal’shoice of the buffer, b. To isolate that effect, we will analyze the problem in which the principalants to minimize her expected steady-state cost of inducing the agent to take a particular effort,

*. This allows us to hold constant the principal’s desire to choose b so as to directly affectevenue and the carrying cost associated with buffers, and instead concentrate on her desire to

1The two probabilities are directly related in that the probability of starving is ps = 1 − �

r�1+h� �1 − pb�, allowing �1� to be

rewritten as K�b,r� = r�1 + h��1 − ps�. Note that a change in the buffer affects the probability of starving in the samedirection �i.e., increase/decrease� as its effect on the probability of blocking.

he Accounting Review November 2010merican Accounting Association

c�gasp

WlcF

gict�c

io

A

smpcse

H

1

1

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1845

T

hoose b to affect the cost of inducing effort from the agent. Thus, we take a Grossman and Hart1983� approach to the principal’s expected profit maximization problem. This is without loss ofenerality because any incentive effects of inventory buffer size found in this analysis will alsoppear in the principal’s full profit-maximization problem.12 Consequently, our subsequent analy-is is analogous to the first step in a Grossman and Hart �1983� analysis—minimizing the princi-al’s expected steady-state cost of inducing effort r* from the agent.

The problem can be written as:

Min�w,w0,b,r*�wK�b,h,r*� + w0 + h �Program 1�

s.t.

wK̃ + w0 − ar* � 0 ∀ K̃ �LL�

r* � arg maxr�wK�b,r� + w0 − ar� . �IC�

e assume that the labor market is such that the principal must leave the agent with a minimumevel of utility regardless of realized throughput �constraint �LL��.13 The incentive compatibilityonstraint, �IC�, ensures that the agent will select the principal’s desired level of effort r*. In theirst-Best case, in which the agent’s effort is not subject to moral hazard, �IC� can be ignored and,

iven that the minimum possible realized throughput is K̃ = 0, the limited liability constraint, �LL�,mplies that w0 = ar* and w � 0. Hence, the size of the buffer b plays no role in the agent’sompensation in the First-Best solution. In the Second-Best setting, �LL� will again bind, implyinghat w0 = ar*. However, incentive compatibility now requires a strictly positive incentive wage, w

0. Thus, w represents the agency cost per unit of throughput, and it is the behavior of w in b thataptures the incentive effect of b in the Second-Best setting. Note that while the principal could

nfer the agent’s effort from the mean steady-state throughput, only the realized throughput, K̃, isbserved, upon which the principal cannot perfectly infer the agent’s choice of effort.

Lemma 1 characterizes this incentive effect and guarantees the validity of the First-Orderpproach to the agent’s choice of effort:

Lemma 1: The agent’s problem is concave in r. Further, the optimal incentive wage is:

w�b,r*� = �a� �K�b,r��r

�−1�r=r*

which is non-negative and increasing in the principal’s choice of r*.

The optimal incentive wage is given by the agent’s marginal cost of effort divided by theensitivity of mean steady-state throughput to his effort. The more sensitive the performanceetric is to the agent’s effort �i.e., the greater the agent’s marginal productivity�, the lower the

iece-rate required to induce the agent to implement the obedient effort and the lower the agencyost of doing so. The basic idea is that the more sensitive throughput is to variations in the agent’setup effort, the more likely it is that the principal can detect any deviation from the desired setupffort, r*; therefore, the lower the incentive wage necessary to dissuade the agent from shirking.

enceforth, we refer to the sensitivity of throughput to the agent’s effort,�K�b,r�

�r , as the informa-

2 Later we will discuss more fully how our results would manifest in the principal’s full expected profit-maximizationproblem.

3 See Demougin and Garvie �1991� for a similar assumption. Alternatively, we could assume that, because of the agent’slimited liability, the principal cannot use negative payments. Our results remain unchanged with this alternative formu-lation. The only effect is on the optimal fixed wage.

he Accounting Review November 2010American Accounting Association

ttsdcn

toeo

iwpmmdspTaatpt

rHbstiitirac

nit

1

1

1846 Baiman, Netessine, and Saouma

TA

iveness of throughput with respect to the agent’s effort.14 As illustrated by �1�, the agent’s abilityo affect throughput is entirely driven by his ability to affect the probability of blocking andtarving via his choice of effort. Given that the wage required to induce the agent to take theesired action is inversely related to the informativeness of throughput, we turn our attention toharacterizing the relation between the principal’s choice of the buffer size b and the informative-ess of throughput with respect to the agent’s action.

III. RESULTSIn what follows, we will sometimes refer to the mean rate at which the workstation processes

he intermediate units relative to their mean arrival rate, r�1 � h�/�, as , the agent’s relative effortr the production schedule. We parameterize our discussion with respect to the agent’s relativeffort since the amount of help resources, h, and the mean arrival rate of the intermediate units, �,nly affect the principal’s incentive problem as they relate to the agent’s effort via .

As noted earlier, the principal may choose to avoid excessive blocking and starving bynducing the agent to set the workstation to process intermediate units at a rate exceeding that athich they arrive. Therefore, consider first the case in which � 1. In the agency model, therincipal compensates the agent as if she were inferring the agent’s effort from the performanceetric �Holmstrom 1979�. But recall that throughput is subject to two sources of uncertainty thatake inferring the agent’s setup effort from it difficult: the stochastic arrival rate of the interme-

iate units and the workstation’s stochastic processing rate. If the buffer size, b, is sufficientlymall, then, in expectation, the probability of starving is large, resulting in relatively few units pereriod available for the workstation to process into throughput �i.e., a small potential sample size�.herefore, when � 1 and the buffer is small, throughput is relatively uninformative about thegent’s effort. By increasing the buffer, b, the principal can increase the expected number of unitsvailable to the workstation, thereby making throughput less sensitive to the stochastic arrivals ofhe intermediate units and more sensitive to the agent’s setup effort. Thus, when b is small, therincipal can improve the informativeness of throughput, and reduce the compensation weight onhroughput, by increasing b.

The above intuition is based on a small increase in b starting at a buffer that, given � 1 �i.e.,�1�h� � ��, is sufficiently small to severely restrict throughput due to blocking and starving.owever, the effect of increasing the buffer on the informativeness of throughput differs once theuffer is already large enough so that blocking and starving are both unlikely. Increasing bufferize beyond some point must result in a decrease in the informativeness of throughput with respecto the agent’s effort choice. To see this, note that in the extreme, when the buffer is infinite, theres zero probability of blocking and, given � 1, the mean steady-state throughput is K � �, whichs independent of the agent’s effort in excess of �. Thus, when there is no blocking, throughput isotally insensitive to and uninformative about any agent effort beyond the cut-off level �. Thismplies that some probability of blocking is necessary for the throughput to be informative withespect to the agent’s effort. Therefore, increasing b beyond some point decreases the relativemount of information about the agent’s setup effort. So when b is relatively large, the principalan increase the informativeness of throughput by decreasing b.

The intuition above suggests that increasing the size of the buffer may cause the informative-ess of throughput to initially increase and later decrease. The following proposition confirms ourntuition and guarantees that once the informativeness of throughput is decreasing in the size ofhe buffer, a larger buffer will never make throughput more informative.15

4 Our notion of informativeness is closely related to that in Laux �2001� and Laffont and Martimort �2002, 156�.5 All proofs are in Appendix B.

he Accounting Review November 2010merican Accounting Association

se

tTtpbpts1

aawrT

�iwi

tWtmmtleC

�mBb

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1847

T

Proposition 1: For any fixed � 1, the informativeness of throughput with respect to theagent’s choice of effort, r, is either decreasing in the size of the buffer, b, orinitially increasing in b and then decreasing in b.

Proposition 1 supports the preceding intuition by establishing that when buffers are relativelymall �large�, the informativeness of throughput is increasing �decreasing� in buffer size. Forxample, consider the case in which h � 0, � � 1, and r* � 1.1. The informativeness of

hroughput �� �K�b,r�

�r�r=r*� is 0.226757 for b � 1, 0.292075 for b � 2, and 0.0286627 for b � 50.

hus, when h � 0, � � 1, and r* � 1.1, informativeness is initially increasing then decreasing inhe buffer size. The proposition also establishes the possibility that the informativeness of through-ut is monotonically decreasing in buffer size. This case arises because no matter how small theuffer size �b ≥ 1�, there always exists a sufficiently large level of setup effort that would cause therobability of blocking and starving to be very small, in which case the informativeness ofhroughput would always decrease in buffer size. For example, consider a case where the work-tation is set up to process intermediate units twice as fast as they arrive: r* � 2 and h � 0, � �. The informativeness of throughput is 0.111111 for b � 1, 0.102041 for b � 2, and 1.088

10−14 for b � 50; i.e., the informativeness of throughput is always decreasing in buffer size.The intuition for Proposition 1 suggests that, when a relatively large buffer causes the prob-

bility of blocking to be very small, the principal would want to decrease the buffer, while whenrelatively small buffer causes the probability of blocking to be very large, the principal wouldant to increase the buffer. Since the probability of blocking is always decreasing in the agent’s

elative effort, , this intuition suggests that the optimal buffer size should also be decreasing in .his logic is formalized in Proposition 2:

Proposition 2: For � 1, the buffer size, b, which maximizes the informativeness ofthroughput with respect to the agent’s choice of effort, is decreasing in theagent’s relative effort, .

Lemma 1 and Proposition 1 lead to the empirically testable result that the incentive wage ratei.e., the marginal cost of effort divided by the informativeness of throughput� should be eitherncreasing or U-shaped in buffer size. Combining Lemma 1 with Proposition 2 implies that oneould expect to see smaller buffers in settings with relatively aggressive production schedules;

.e., � 1.To this point, we have only addressed the case in which the principal wants the agent to set

he workstation to operate, on average, at a faster rate than intermediate goods arrive � � 1�.hen � 1, on average, the workstation processes the intermediate units at a slower rate than

hey arrive. Unlike the case with � 1, when � 1, even as the buffer size tends to infinity, thearginal effect of the agent’s effort on the probability of blocking does not go to zero, because noatter how large the buffer, the principal can never eliminate the threat of blocking. In this setting,

he principal continues to increase the informativeness of throughput by increasing b, as thisowers the probability of blocking, which in turn allows the agent a greater opportunity to influ-nce the probability of blocking and, consequently, throughput. This logic is summarized inorollary 1:

Corollary 1: When � 1, the informativeness of throughput, K, with respect to the agent’seffort, r, is everywhere increasing in the buffer size, b.

The above corollary suggests that minimizing buffers, as advocated by the JIT literatureSchonberger 1986, 67� exacerbates the incentive problem when � 1. However, when � 1,inimal buffers �b � 1� may be optimal from an incentive perspective under certain conditions.ecause buffer sizes are integer-valued, when � 1, it is difficult to characterize the optimaluffer in closed form, though Proposition 3 establishes an interval in which the optimal b must lie.

he Accounting Review November 2010American Accounting Association

mrcojc

rdtbisdbsotfi

eoiieasteapmi

1

1848 Baiman, Netessine, and Saouma

TA

Proposition 3: When � 1:

a. the optimal b lies in the interval: max� 1−1

,2� − 1 � b � max� 1−1 ,1�.

b. ≥ 2 is a sufficient condition for minimal buffers �b � 1� to be optimal.

The first part of Proposition 3 indicates that the common objective in the JIT literature ofinimizing buffers would incur unnecessary agency costs if the agent is not being asked to work

elatively hard; i.e., 1 � � 2. More generally, our results point to the complementarities that arereated between the choice of buffer size and the choice of the agent’s relative effort, , as a resultf the effect of both on the informativeness of throughput as the performance metric.16 As we haveust shown, changes in buffer size without associated changes in the agent’s desired relative effortan lead to unintended agency costs.

Our analysis of the effect of buffer size on the informativeness of throughput is based on theole of buffers in blocking and starving the production process, with the other organizationalesign variables �r*,�,h� held fixed. However, we can also view these design variables in terms ofheir potential effects on the informativeness of throughput via their effects on the probabilities oflocking and starving. Situations can exist in which it may be more efficient to manipulate thenformativeness of throughput by varying these other design choices than by varying the bufferize. For example, increasing buffer size has space utilization implications that the other variableso not; i.e., one cannot increase buffer size if there is no additional space available, whereas it maye possible to upgrade the workstation machinery �increase h� or speed up/slow down the up-tream workstations �vary ��. Similarly, buffer size increments come in discrete steps while thether design variables may allow for finer adjustments. This motivates us to consider the effect ofhese other design choices on the informativeness of throughput while holding the buffer sizexed, as considered next.

Proposition 4: Holding the buffer fixed, the informativeness of throughput with respect tothe agent’s setup effort is:

a. increasing in the mean arrival rate of intermediate units, �,b. decreasing in the principal’s desired level of agent effort, r*, andc. single-peaked in the level of help resources allocated to the agent, h.

Increasing the arrival rate, �, decreases the likelihood of starving, which in turn reduces theffect of the variation in arrivals on throughput, making throughput more informative about thether source of variation, the agent’s effort, r. The same logic holds for decreases in r. The morenteresting result concerns productivity-enhancing investments, h. Proposition 4 implies that evengnoring out-of-pocket costs, there are two counter-acting effects to increasing productivity-nhancing investments for the agent. The direct positive effect is that such investments make thegent more productive, thereby increasing his ability to influence throughput via blocking andtarving. The indirect, negative effect is that such investments may render the agent so productive,hat holding effort fixed, the threat of blocking tends to zero as h increases, even if the agent’sffort is only negligible. As �1� indicates, throughput only reflects effort to the extent that thegent’s effort affects the probability of blocking. Beyond some point, increases in h make through-ut less sensitive to the agent’s effort and therefore less informative about the agent’s effort,aking it more costly to motivate him. In a sense, resources, h, play the same role as buffer size

n our earlier analysis. Both h and b can be looked at as productivity-enhancing investments and

6 Thus, our results are consistent with the complementarities literature, for example, Milgrom and Roberts �1990, 1992�.

he Accounting Review November 2010merican Accounting Association

bo

iaeaotsig

btwtsiitammg

mlmaaopHo

ltsrraic

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1849

T

oth can be the source of additional agency costs as a result of their effects on the informativenessf throughput as a performance metric.

One can interpret our results as highlighting a value to creating bottlenecks, either by choos-ng a finite buffer size or a finite amount of productivity-enhancing investments. Because we havebstracted from both the out-of-pocket costs associated with inventory buffers, �c � 0� and theffect of the principal’s choice of buffer size on the expected throughput and revenue, these resultsrise solely for incentive reasons. Allowing for these two additional effects will mitigate the forcef our results; however, because all of our findings are obtained using arbitrary parameter values,he incentive issues identified would persist. Appendix A provides a numerical example where weolve the principal’s more general, expected steady-state, profit-maximizing problem. The solutionllustrates that the incentive effect of buffer size identified will continue to hold in this moreeneral setting.

IV. CONCLUSIONThe present study analyzes the relation between the choice of the firm’s production line

uffers and optimal incentives. The operations management literature has focused on the impor-ance of minimizing buffers and reducing inventory. A major insight from the present work is thate identify an additional incentive value associated with the firm’s choice of buffers. We show

hat under reasonable conditions, the informativeness of throughput as a performance metric isingle-peaked in both buffer size and production-enhancing resources, such that it is optimal tonduce non-zero, albeit bounded, probabilities of blocking and starving. We show further that thenformation-maximizing buffer is decreasing in the effort that the principal wants to induce fromhe agent. Therefore, we would expect to observe smaller buffer sizes in settings with moreggressive production schedules �i.e., � 1�. Our findings thus emphasize the incentive comple-entarity between buffer size and production schedules, and provide a possible explanation for theixed empirical evidence regarding the profitability of work-in-process inventory reductions �Na-

ar et al. 2009�.This research spans two different disciplines, managerial accounting and operations manage-

ent, each with its own nuances and paradigms. It represents an attempt to bridge these twoiteratures by weaving together some of the basic models of each. To do so, we made a number of

odeling assumptions. We ignored some of the factors that are identified in the operations man-gement literature as influencing the inventory buffer decision in order to focus on incentive issuesnd the informational effect of the buffer size decision. For example, some of the claimed benefitsf small buffers under JIT that we have ignored include the increased ability to identify productionroblems early, shorter lead-times, and faster responsiveness to changes in market conditions.owever, we are confident that the incentive consequences of inventory buffers brought to light byur analysis will be present notwithstanding these other factors.

A limitation of our model is our restricted contracting space. While we limited our analysis toinear contracts for analytical reasons, the crucial assumption underlying our results is the mono-onicity of compensation in the agent’s marginal expected productivity, which is either assumed orhown to be optimal in the majority of agency models. Our analysis also assumes that the agent isisk-neutral. However, recall that the principal chooses b based on the information content ofealized throughput with respect to the agent’s effort. Therefore, to the extent that a risk-aversegent must be compensated for risk, intuition would suggest that our results would be strengthenedn the presence of risk-aversion. However, introducing risk-averse agents imposes several analyti-al complications, and we leave the resolution of such issues to future research.

he Accounting Review November 2010American Accounting Association

mlbviattitpi

ismtiwclm

wtpcssiptm

1

1

1850 Baiman, Netessine, and Saouma

TA

Another simplifying assumption is that the agent’s compensation is based on a single perfor-ance metric. Clearly, other performance metrics are possible, including realized inventory

evels.17 However, the mean steady-state inventory level does not offer any information above andeyond the mean steady-state throughput, K, since both measures are functions of exactly the sameariables. On the other hand, if we allowed for continuous monitoring, as with radio frequencydentification �RFID�, then we could isolate the agent from the stochasticity of the arrival processnd mitigate the agent’s moral hazard problem.18 In this case, our problem reduces to the tradi-ional moral-hazard setting where the agent always has work to process, and the principal infershe agent’s effort via stochastic output. It would therefore be interesting to empirically analyze thencentive effect of buffers both before and after the installation of an RFID monitoring applicationo see the extent to which they are used as substitutes to buffers for filtering noise in availableerformance measures. We leave to future research such inquiries, as compensation data surround-ng RFID installations are not yet readily available.

APPENDIX AOur results are based on analyzing a model in which the principal minimizes her cost of

nducing the agent to exert a particular setup effort. In particular, our analysis of the optimal bufferize ignored both the effect of buffers on the out-of-pocket cost of maintaining inventory and,ore importantly, the direct revenue effect of the choice of buffer size. In this appendix we show

hat the insights and qualitative results derived from our analysis continue to hold when we takento consideration both of these additional effects. To do so, we provide a numerical example inhich the principal’s objective is to maximize her steady-state expected profit. As before, her

hoice variables are buffer size, b, and the compensation contract, �w0,w�, determining the desiredevel of agent effort, r*, which is subject to moral hazard. Including the explicit revenue and

aintenance cost effects associated with the choice of buffer size, the principal’s problem is now:

Max�r*,w,w0,b�RK�b,r*� − wK�b,r*� − w0 − cb

s.t.

wK̃ + w0 − ar* � 0 ∀ K̃ �LL�

r* � arg maxr�wK�b,r� + w0 − ar� �IC�

here we again assume a linear demand schedule for throughput. We first solve the above problemo find the compensation-buffer size-effort quadruple �w,w0,bMH,r*�. We then solve the sameroblem where we again induce the agent to choose r*, but we assume that the agent’s effort isontractible and therefore not subject to moral hazard. Thus, in this second problem we are onlyolving for the optimal compensation and buffer size. We label the optimal buffer size in thisecond, no moral hazard problem, bNMH. Comparing the two optimal buffer sizes allows us tosolate the incentive effects of the principal’s choice of buffer size within this more generalrofit-maximizing setting. The example demonstrates that our findings continue to hold. In par-icular, the incentive effect of buffers continues to exist and can bias the principal’s profit-

aximizing buffer away from the no moral hazard solution.

7 We thank a reviewer for raising this issue.8 RFID involves “an integrated circuit with an antenna, known as a ‘tag,’ attached to a …product. Product information as

well as other relevant information can be stored in the tag … Using wireless technologies, readers can be set up to readthe information on the tags” �Lee and Ozer 2007, 40�.

he Accounting Review November 2010merican Accounting Association

epb

mCdwbetttnn

L

Fem2

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1851

T

The example assumes that the principal’s revenue per unit of throughput is R dollars. In ourxample we set R � 350, c � 10, a � 1, h � 0 and � � 1. In the moral hazard problem, therincipal finds it optimal to induce the agent to work relatively hard � � r* � 2.11� and to set theuffer size at bMH = 3.

Figure 2 plots both the expected steady-state profits for the moral hazard case and the infor-ativeness of throughput, while fixing the induced effort at r* � 2.11 and varying the buffer size.onsistent with Proposition 3 and the fact that � r* � 2.11 � 2, informativeness is everywhereecreasing in the size of the buffer, implying that the optimal buffer from an incentive perspective,hich we label bI, is equal to 1. However, unlike in Proposition 3, the optimal buffer size is

MH = 3. The reason for this difference is that we are now trading off the revenue and holding costffects of buffer size �the smaller the buffer size, the smaller the expected revenue and the lowerhe holding costs� against the agency cost �the smaller the buffer size, the more informative ishroughput and, hence, the smaller the agency cost�. The logic of Proposition 3, however, indicateshat when r* � 2.11, the optimal buffer size in the moral hazard case should be smaller than in theo moral hazard case. In fact, that is exactly what we find: the profit-maximizing buffer size in theo moral hazard case is bNMH = 4, as indicated in Figure 2.

APPENDIX Bemma 0

The function f � Cka,b� has a root of multiplicity k at r if and only if:

FIGURE 2Second-Best Profit and Informativeness

r = 2.11

190

210

230

250

270

290

310Profit

Second-Best Profit and Informativeness, r = 2.11ProfitInformativeness

150

170

0 1 2 3 4 5 6 7 8

Buffer Size

bI bMH bNMH

or R = 350, c = 10, a = 1, � = 1, the solution to the profit-maximizing problem with moral hazard sets the agent’sffort at r* = = 2.11 and the optimal buffer size at bMH = 3, though a buffer of size bI = 1 maximizes the infor-ativeness of throughput when = r* = 2.11. For the no moral hazard case in which the principal chooses = r* =

.11, her profit (not drawn) is maximized at the buffer size bNMH = 4.

he Accounting Review November 2010American Accounting Association

P

P

S

a��a

li

wtsf

i

1

1852 Baiman, Netessine, and Saouma

TA

0 = f�r� = f��r� = f��r� = . . . = f �k−1��r� but f �k��r� � 0.

roof of Lemma 0See Theorem 3.20 in Vinberg �2001� and Corollary 3.21 in Artin �1991, 508�. �

roof of Lemma 1The agent’s utility is given by u = wK�b,r� + w0 − ar. Hence:

�u

�r= w

�K

�r− a

�2u

�r2 = w�2K

�r2 .

ubstituting the functional form of K from �1� yields:

�2u

�r2 = �w��2K

�r2

= �1 + h�2�w��b + 1���/r�1 + h��b+1

�b�r�1 + h� − ������/r�1 + h��b + r�1 + h�� + 2r�1 + h�����/r�1 + h��b − 1��

����/r�1 + h��b − r�1 + h��3 ,

nd the denominator is non-positive if and only if r�1 � h� ≥ �. Further,w��b + 1��� / r�1 + h��b+1 � 0. Hence, if we can show thatb�r�1 + h� − ������ / r�1 + h��b + r�1 + h�� + 2r�1 + h����� / r�1 + h��b − 1�� is non-negative ifnd only if r�1 � h� ≥ �, we will then have established that �2u

�r2 � 0.To this end, we can re-write the expression as:

�r�1 + h��−bb�r�1 + h��2+b + �r�1 + h��b+1�− b� − 2�� + r�1 + h��b+1�2 + b� − b�b+2�

abeling the bracketed expression, N. According to Descartes’ rule, N can have at most three rootsn r. Using Lemma 0, note that:

�N�r= �1+h

= � �N

�r�

r= �1+h

= � �2N

�r2 �r= �

1+h

= 0 � b�1 + h�3�b−1�2 + 3b + b2� = � �3N

�r3 �r= �

1+h

,

hich shows that all three roots are at r � �/�1 � h�. Finally, since the third derivative is positive,he Extremum Test assures that N has a saddle point at r � �/�1 � h�;19 therefore N alternates inign as r crosses �/�1 � h�. However, since N is negative at r�1 � h� � 0, N must be non-negativeor r�1 � h� ≥ � � 0.

The constraint �IC� can be expressed as:

�

�r�w

1 − �r�1 + h�/��−b

1 − �r�1 + h�/��−�b+1�� − ar� = 0,

mplying that:

9 For an explanation of the Extremum Test, see http://mathworld.wolfram.com/ExtremumTest.html.

he Accounting Review November 2010merican Accounting Association

w

P

w

b

tm

w

Wbsf

Witiaib

T

Tt

t

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1853

T

w�b,r� = a� �K�b,r��r

�−1

=a��r�1 + h�/��b+1 − 1�2

�1 + h��1 − �r�1 + h�/��b�1 − b�1 − r�1 + h�/����,

hich is both positive and increasing in r. �

roof of Proposition 1Because we have defined informativeness as a constant times the inverse of the incentive

age, it suffices to show that the incentive wage is either U-shaped in buffer size, or increasing in

. In particular, we will show that if the wage is increasing in b, at say b̂, then the wage continues

o increase for all b � b̂. Then we will show that there exists a finite b beyond which the wageust be increasing in b.

In general, we can write:

w�b,� =a�b+1 − 1�2

�1 + h��bb+1 − �b + 1�b − 1�, �2�

here, henceforth to simplify, we omit the constant a�1+h� . Hence:

�

�bw�b,� = b�b+1 − 1�

− � − 1��b+1 − 1� + �b� − 1��1 + b+1� − �b − 2� − 1�log���bb+1 − �b + 1�b + 1�2 . �3�

e first claim that the denominator of �3� is strictly positive. To see this, note thatb+1 − �b + 1�b + 1 has a root at � 1; its first derivative evaluated at � 1 is zero and itsecond derivative is strictly positive for all ≥ 1, implying that the denominator is strictly positiveor all � 1. Thus, for � 1, the sign of �

�bw�b,� is determined by the function t�b,�, where:

t�b,� = − � − 1��b+1 − 1� + �b� − 1��1 + b+1� − �b − 2� − 1�log�� .

e will show that if t�b,� is positive, it is increasing in b, implying that once the wage isncreasing in buffer size b, it will continue to increase in b for larger buffers. To this end, note thathe first term in t�b,� is strictly negative for � 1. We claim that the second term in t�b,�s strictly positive for � 1. To see this, note that we can write the coefficient of log��s −1 − b + �2 + b� − �b + 1�1+b + b2+b, which at � 1, is both equal to 0 and increasingn . Moreover, the second derivative in is strictly positive for � 1, since�1 + b��b−1�2 + b� − 1� − 1�� � 0.

Thus, if t�b,� is positive, it must be the case that:

log�� �� − 1��b+1 − 1�

b� − 1��1 + b+1� − �b − 2� − 1. �4�

o show that if t�b,� � 0, t�b,� is increasing in b, we difference t�b,� in b:

t�b + 1,� − t�b,� = � − 1��− � − 1�1+b + �1 − �b + 1�1+b + �b + 1�2+b�log��� . �5�

he coefficient of log�� in �5� is always positive, hence we can use the bound for log�� from �4�o form a lower bound for t�b � 1,� � t�b,�:

�b + 1,� − t�b,� = � − 1��− � − 1�1+b + �1 − �b + 1�p1+b + �b + 1�2+b�log���

� � − 1��− � − 1�1+b + �1 − �b + 1�p1+b + �b + 1�2+b�� − 1��b+1 − 1�

b� − 1��1 + b+1� − �b − 2� − 1�

= � − 1�2− 1 + �3 + 2b�1+b − �3 + 2b�2+b + 3+2b

− 1 − b + �2 + b� − �b + 1�1+b + b2+b .

he Accounting Review November 2010American Accounting Association

Tpt�

c

P

i

t

b

T

w

Wh0ii

hit�m

t

a

P

P

ip

1854 Baiman, Netessine, and Saouma

TA

he denominator above is identical to that in �4�, which we have already shown to be strictlyositive for � 1. Thus, we need only sign the numerator, which has at most three roots accordingo Descartes’ rule. However, in accordance with Lemma 0, the numerator has all three roots at

1, and since it has a strictly positive third derivative in , the expression is strictly positive for

� 1, implying that if at some b = b̂ the wage is increasing in buffer size �t� b̂,� � 0�, then it willontinue to be increasing in buffer size for larger buffers.

To prove that the wage is eventually increasing in buffer size, note that for an arbitrary, finite

� 1, limb→�

t�b,� = �, implying that there exists a finite b̃ for which t� b̃,� � 0. �

roof of Proposition 2In order to characterize the most informative buffer size, it suffices to characterize the lowest

ncentive compatible wage, w. We will show that if the wage is minimized at b̂ when = ̂, then

he wage will be increasing in b at b̂ when � ̂; thus by Proposition 1, the wage-minimizing

uffer for � ̂ is less than or equal to b̂.

Suppose the wage is minimized at b̂ when = ̂, Then, by definition:

w�b̂, ̂� − w�b̂ − 1, ̂� � 0 � w�b̂ + 1, ̂� − w�b̂, ̂� .

o complete the proof, we will prove that w� b̂ + 1,̂� − w� b̂,̂� � 0 implies

� b̂ + 1,� − w� b̂,� � 0 for � ̂.To this end, consider the difference w�b � 1,� � w�b,�:

� − 1�b1 + b − 3�b + 1� + 32+b + 3+b − �2 + b�3+2b + b4+2b

�1 − �b + 1�b + b1+b��1 − �2 + b�1+b + �b + 1�2+b�. �6�

e first show that the denominator of �6� is always positive. Using Descartes’ rule, the first termas at most two roots. The term and its first derivative in evaluated at � 1 are both equal to. Therefore, Lemma 0 implies that its only positive root is � 1 and since its leading coefficients positive, the term is strictly positive for � 1. The same argument shows that the second termn the denominator is strictly positive for � 1.

Thus, the sign of �6� is determined by the numerator. Applying Descartes’ rule, the numeratoras at most four positive roots in . Because the numerator and both its first and second derivativesn are equal to 0 when � 1, Lemma 0 implies that � 1 is a root with multiplicityhree. Because the third derivative of the numerator in evaluated at � 1 is negative−3b2 − 9b − 6� but the leading coefficient is positive, the final root in , which we label �b�,ust be greater than 1.

Now, suppose w� b̂ + 1,̂� − w� b̂,̂� � 0. Then it must be the case that ̂ is greater than or equal

o the final root, � b̂�; therefore, if � ̂, then � � b̂�, in which case w� b̂ + 1,� − w� b̂,� � 0,

nd thus the wage-minimizing buffer for � ̂ must be no larger than b̂. �

roof of Corollary 1The proof is similar to that of Proposition 2 and is omitted. �

roof of Proposition 3By definition, the information-maximizing b will equivalently minimize the wage, w. We

dentify a sufficient upper bound on b, as parameterized by , for w�b � 1,� � w�b,� to beositive, and a sufficient lower bound on b for w�b � 1,� � w�b,� to be negative. Combining the

he Accounting Review November 2010merican Accounting Association

tw

L

tx

P

ffifc

L

f

P

Wf

W�a

Hwt

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1855

T

wo bounds, we identify a region for b where w�b � 1,� � w�b,� crosses zero in b, defining theage-minimizing buffer size.

We begin with two additional Lemmas.

emma 2Let f�x� = anxn + . . . +a1x + a0 be a real polynomial with f�1� � 0. Furthermore, assume that

here exists a k � n � 1 such that for i � k, ai � 0 and for i ≤ k, ai � 0. Then f�x� � 0 for� 1 and f�x� � 0 for x � 1.

roof of Lemma 2For all x � 1 we have f�x� � �an + an−1 + . . . +ak+1�xk+1 − �−ak − ak−1 − . . . −a0�xk. Since

�1� � 0, the sum of the terms in the parentheses are equal, and we denote the sum �either in therst or the second set of parentheses� by s. Clearly, s � 0 and sxk�x − 1� � 0 for x � 1; hence,

�x� � 0 when x � 1. Similarly, for all 0 ≤ x � 1, we have f�x� � sxk+1 − sxk = sxk�x − 1� � 0. Thisoncludes the proof of the lemma. �

emma 3Let b be a natural number and � 1, then:

2

b + 1 �

�i=0

b

�i + 1�i

�i=0

b−1

�i + 1�i

�b + 1

b +

1

b

or � 1, and the opposite weak inequalities hold when � 1.

roof of Lemma 3First consider the inequality on the right-hand side. Set:

f�� = �b + 1

b +

1

b��

i=0

b−1

�i + 1�i − �i=0

b

�i + 1�i.

e want to show that f�� � 0 for � 1 and f�� � 0 for � 1. Thus, it is sufficient to show that�� satisfies the hypothesis of the previous lemma. First, to show that f�1� � 0, we have:

f�1� = �b + 1

b+

1

b��

i=0

b−1

�i + 1� − �i=0

b

�i + 1� = �b + 2

b��b

2��b + 1� −

�b + 1��b + 2�2

= 0.

e must now find a k as in the prior lemma. If f�� = ann + . . . a1 + a0, then n � b � 1although there is a b term in the construction of f��, note that it has a coefficient of zero� and forrbitrary m ≥ 1, the coefficient am is given by:

b + 1

bm +

1

b�m + 1� − �m + 1� =

2m − �b − 1�b

.

ence, am � 0 if and only if m �b−1

2 . In particular, a threshold k exists when n = b − 1 �b−1

2 ,hich is valid for b ≥ 2. If b � 1, then f�� = 1

2 � − 1�, which is trivially greater than zero for� 1 and less than zero for � 1. Hence, Lemma 2 holds and we have established Lemma 3 for

he right-hand side inequality.Next, consider the left-hand side inequality in the statement of the lemma. Let:

he Accounting Review November 2010American Accounting Association

N

Mats

c

T

T

I

1856 Baiman, Netessine, and Saouma

TA

g�� = �i=0

b

�i + 1�i − �2

b + 1��

i=0

b−1

�i + 1�i.

ote that:

g�1� = �i=0

b

�i + 1� − �2

b+ 1��

i=0

b−1

�i + 1� = 0.

oreover, the leading coefficient of g�� is positive for b � 1 and all the remaining coefficientsre always negative; hence, the conditions of Lemma 2 are always satisfied for b � 1. If b � 1,hen g�� � 2� � 1�, which is again positive for � 1 and negative for � 1, as was to behown. This concludes the proof of the lemma. �

Continuing with the proof of Proposition 3, when � 1, we can rewrite w�b,� without theonstant multiplier a

�1+h� as:

w�b,� =�b+1 − 1�2

�bb+1 − �b + 1�b + 1�=

2b + 22b−1 + . . . + �b + 1�b + bb−1 + . . . + 2 + 1

bb−1 + �b − 1�b−2 + �b − 2�b−3 + . . . + 2 + 1.

o see this, we begin by rewriting the numerator:

�b+1 − 1�2 = ��b+1 + b + b−1 + . . . + � − �b + b−1 + . . . + 1��2

= �� − 1��b + b−1 + . . . + 1��2

= � − 1�2�b + b−1 + . . . + 1�2

= �p − 1�2��i=0

b

i�2

= 2b + 22b−1 + . . . + �b + 1�b + bb−1 + . . . + 2 + 1.

he denominator can be rewritten as:

bb+1 − �b + 1�b + 1 = b�b+1 − b� − �b − 1�

= bb� − 1� − � − 1��i=1

b

b−i

= � − 1��bb − �i=1

b

b−i�= � − 1��

i=1

b

�b − b−i�

= � − 1�2�i=0

b−1

�i + 1�i

= bb−1 + �b − 1�b−2 + �b − 2�b−3 + . . . + 2 + 1.

n light of this representation, we can continue simplifying to obtain:

he Accounting Review November 2010merican Accounting Association

U

Wer

Slf�b

ad

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1857

T

w�b,� =2b + 22b−1 + . . . + �b + 1�b + bb−1 + . . . + 2 + 1

bb−1 + �b − 1�b−2 + �b − 2�b−3 + . . . + 2 + 1

=2b + 22b−1 + . . . + �b + 1�b

bb−1 + �b − 1�b−2 + �b − 2�b−3 + . . . + 2 + 1+ 1

= 2b 1 + 2−1 + . . . + �b + 1�−b

bb−1 + �b − 1�b−2 + �b − 2�b−3 + . . . + 2 + 1+ 1. �7�

sing �7�, w�b � 1,� � w�b,� is equivalent to:

2b+21 + 2−1 + . . . + �b + 2�−b−1

�b + 1�b + . . . + 2 + 1+ 1 � 2b1 + 2−1 + . . . + �b + 1�−b

bb−1 + . . . + 2 + 1+ 1

⇔ 21 + 2−1 + . . . + �b + 2�−b−1

�b + 1�b + . . . + 2 + 1�

1 + 2−1 + . . . + �b + 1�−b

bb−1 + . . . + 2 + 1

⇔ 2� �b + 1�b + . . . + 2 + 1

1 + 2−1 + . . . + �b + 1�−b��1 + 2−1 + . . . + �b + 2�−b−1

�b + 1�b + . . . + 2 + 1�

� � �b + 1�b + . . . + 2 + 1

1 + 2−1 + . . . + �b + 1�−b��1 + 2−1 + . . . + �b + 1�−b

bb−1 + . . . + 2 + 1�

⇔ 21 + 2−1 + . . . + �b + 2�−b−1

1 + 2−1 + . . . + �b + 1�−b = 2

�i=0

b+1

�i + 1�−i

�i=0

b

�i + 1�−i

��b + 1�b + . . . + 2 + 1

bb−1 + . . . + 2 + 1=

�i=0

b

�i + 1�i

�i=0

b−1

�i + 1�i

.

�8�

e can use Lemma 3 with � 1 on the right-hand side of �8� to conclude that it is less than orqual to b+1

b + 1b . Since we have assumed that � 1, −1 � 1, and we can again apply Lemma 3,

eplacing with −1 on the left-hand side of �8�, to conclude that it is greater than or equal to2� b+2

b+1−1 + 1b+1

�. Thus, a sufficient condition on b and for w�b � 1,� � w�b,� is given by:

1

b + 12 +

b + 2

b + 1 �

b + 1

b +

1

b⇔

1

b + 12 +

b + 2

b + 1 −

b + 1

b −

1

b� 0.

olving the second quadratic above for equality gives two roots in : �1 and b+1b . Since the

eading coefficient is positive, we know that �8� holds whenever �1+b

b , or a sufficient conditionor the wage to be increasing in b is given by b �

1−1 . Because our analysis applies to all � 1,

8� implies that for ≥ 2, the wage-minimizing buffer is given by b � 1, the smallest feasibleuffer size. This establishes part �b�.

Next we establish a sufficient upper bound for the wage to be decreasing in b. When a � 0nd � 1, −a � 1; thus, Lemma 3 implies that the left-hand side of �8� is less than2� 2

b+1−1 + 1�, whereas the right-hand side is bounded below by 2b + 1. Hence, a sufficient con-

ition for �8� to be violated �for the wage to be decreasing in b� is given by:

he Accounting Review November 2010American Accounting Association

T

c

1

a+

1

P

si

ThtofTnn

T�−t1np

A

AB

1858 Baiman, Netessine, and Saouma

TA

2

b + 1 + 2 −

2

b − 1 � 0. �9�

he two roots to the polynomial on the left-hand side of �9� are given by =

2b�b+1� − 4

b2�b+1�2 +4

2 and

=

2b�b+1� + 4

b2�b+1�2 +4

2 , since the latter is negative and the polynomial has a positive leadingoefficient; a sufficient condition for the wage to be decreasing in b is given by

� �

2b�b+1� + 4

b2�b+1�2 +4

2 . A lower bound for

2b�b+1� + 4

b2�b+1�2 +4

2 is given by

2b�b+1� +2

2 = 1 + 1b�b+1� ,

nd since b�b + 1� � �b + 1�2, a sufficient condition for w�b,� to be decreasing is thus � 11

�b+1�2 or 1−1

− 1 � b. The lower bound on b is redundant once �54 , as b is bounded below by

. �

roof of Proposition 4In order to prove that the sensitivity of K to r is increasing in �, decreasing in r, and

ingle-peaked in h, we will show that the wage w is decreasing in �, increasing in r and U-shapedn h. Differentiating w with respect to h yields:

�w

�h= a

�− 1 + b�1 + 2b + b2� − b+1�4b + b2� + 2b+2�1 + 4b� − 3b+2�1 + b�2 + 3b+3b2��2�1 + b�b − b − 1���2 . �10�

he denominator is unambiguously non-negative, whereas Descartes’ rule implies the numeratoras at most five positive roots. Because the numerator and the first three derivatives with respecto have a root at � 1, only one root remains unidentified. If b � 1, then the fourth derivativef the numerator in �10� is zero at � 1, and thus the fifth root is in fact � 1. If b � 1, then theourth derivative of the numerator in �10� is strictly positive at � 1, hence, by the Extremumest, � 1 is a relative minimum and there exists a sufficiently small � � 0 such that theumerator is positive at � 1 � � when b � 1. However, because the numerator in �10� is strictlyegative at � 0, the final root must belong to the interval �0,1�.

Differentiating w with respect to r yields:

�w

�r= − a

�1 + b�b−1�1 − b+1��− b + �b + 2� − b+1�b + 2� + b+2b���1 − b�1 + b� + bb+1�2 . �11�

he denominator of �11� is non-negative. Descartes’ rule implies that in addition to � 1,11� may have up to three additional positive roots. Using Lemma 0, the expressionb + �b + 2� − b+1�b + 2� + b+2b and its first two derivatives have a root at � 1, whereas the

hird derivative is strictly positive at � 1, implying that the expression has a saddle point at �

. Since the pre-multiplier, 1 − b+1, also alternates in sign at � 1, the entire expression, �w�r , is

on-positive. Finally, differentiating w with respect to � yields �w�� = − r

��w�r , and since �w

�r is non-ositive, �w

�� must be non-negative. �

REFERENCES

lles, M., S. Datar, and R. A. Lambert. 1995. Moral hazard and management control in just-in-time settings.Journal of Accounting Research 33 �Supplement�: 177–204.

rtin, M. 1991. Algebra. Upper Saddle River, NJ: Prentice Hall.alachandran, K., and S. Radhakrishnan. 1996. Cost of congestion, operational efficiency and management

accounting. European Journal of Operational Research 89 �2�: 237–245.

he Accounting Review November 2010merican Accounting Association

B

B

B

B

C

D

D

F

G

G

GH

H

—

H—

HH

L

L

L

L

M

M

——

N

R

Informativeness, Incentive Compensation, and the Choice of Inventory Buffer 1859

T

alakrishnan, R., N. Nagarajan, and K. Sivaramakrishnan. 1998. The effect of property rights and auditinformation quality on team incentives for inventory reduction. Management Science 44 �9�: 1193–1204.

alsamo, S., V. Persone, and R. Onvural. 2001. Analysis of Queuing Networks with Blocking. Boston, MA:Kluwer Academic Publishers.

erg, N., and N. Fast. 1975. The Lincoln Electric Company. Harvard Business School Case 9-376-028.Cambridge, MA: Harvard Business School.

erkley, B. 1992. A review of the kanban production control research literature. Production and OperationsManagement �4�: 393–411.

remer, J. 1995. Towards an economic theory of incentives in just-in-time manufacturing. European Eco-nomic Review 39 �3–4�: 432–439.

atar, S., and M. Rajan. 1995. Optimal incentive schemes in bottleneck-constrained production environ-ments. Journal of Accounting Research 33 �Spring�: 33–58.

emougin, D., and D. Garvie. 1991. Contractual design with correlated information under limited liability.RAND Journal of Economics 22 �4�: 477–489.

eltham, G., and J. Xie. 1994. Performance measure congruity and diversity in multi-task principal/agentrelations. The Accounting Review 69 �3�: 429–454.

ietzmann, M., and T. Hemmer. 2002. On the relation between optimal incentive structures and the cost andbenefits of bottlenecks. Journal of Labor Economics 20 �2, part 2�: 34–56.

roenevelt, H. 1993. The just-in-time system. In Handbooks in OR & MS, edited by Graves, S., A. RinnooyKan, and P. Zipkin, 629–670. Amsterdam, The Netherlands: Elsevier.

rossman, S., and O. Hart. 1983. An analysis of the principal agent problem. Econometrica 51 �1�: 7–45.all, B., C. Madigan, and E. Lazear. 2000. Performance pay at Safelite Auto Glass. Harvard Business School

Case 800291. Cambridge, MA: Harvard Business School.emmer, T. 1995. On the interrelation between production technology, job design, and incentives. Journal of

Accounting and Economics 19 �2–3�: 209–245.—–. 1998. Performance measurement systems, incentives, and the optimal allocation of responsibilities.

Journal of Accounting and Economics 25 �3�: 321–347.olmstrom, B. 1979. Moral hazard and observability. The Bell Journal of Economics 10 �1�: 74–91.—–, and P. Milgrom. 1991. Multi-task principal-agent analyses: Incentive contracts, asset ownership and

job design. Journal of Law Economics and Organization 7 �Spring�: 24–52.opp, W., and M. Spearman. 2001. Factory Physics. 1st edition. Chicago, IL: Irwin.orngren, C., S. Datar, M. Rajan, and C. Ittner. 2008. Cost Accounting: A Managerial Emphasis. 13th

edition. Upper Saddle River, NJ: Pearson Prentice Hall.affont, J.-J., and D. Martimort. 2002. The Theory of Incentives: The Principal-Agent Model. Princeton, NJ:

The Princeton University Press.aux, C. 2001. Limited-liability and incentive contracting with multiple projects. RAND Journal of Econom-

ics 32 �3�: 514–526.ee, H., and O. Ozer. 2007. Unlocking the value of RFID. Production and Operations Management 16 �1�:

40–64.u, L., J. Van Mieghem, and R. Savaskan. 2009. Incentives for quality through endogenous routing. Manu-

facturing & Service Operations Management 11 �2�: 254–273.elumad, N., D. Mookherjee, and S. Reichelstein. 1995. Hierarchical decentralization of incentive contracts.

RAND Journal of Economics 26 �4�: 654–672.ilgrom, P., and J. Roberts. 1990. The economics of modern manufacturing: Technology, strategy, and

organization. The American Economic Review 80 �3�: 511–528.—–, and ——–. 1992. Economics, Organization & Management. Englewood Cliffs, NJ: Prentice Hall.—–, and ——–. 1995. Complementarities and fit: Strategy, structure, and organizational change in manu-

facturing. Journal of Accounting and Economics 19 �2–3�: 179–208.agar, V., M. Rajan, and R. Saouma. 2009. The incentive value of inventory and cross-training in modern

manufacturing. Journal of Accounting Research 47 �4�: 991–1026.adhakrishnan, S., and K. Balachandran. 1995. Delay cost and incentive schemes for multiple users. Man-

agement Science 41 �4�: 646–652.

he Accounting Review November 2010American Accounting Association

—

R

S

V

1860 Baiman, Netessine, and Saouma

TA

—–, and ——–. 2004. Service capacity and incentive compatible cost allocation for reporting usage fore-casts. European Journal of Operational Research 157 �1�: 180–195.

iordan, M., and D. Sappington. 1987. Information, incentives, and organizational mode. The QuarterlyJournal of Economics 2 �May�: 243–264.

chonberger, R. 1986. World Class Manufacturing: The Lessons of Simplicity Applied. New York, NY: TheFree Press.

inberg, E. B. 2001. A Course in Algebra. Vol. 56. Providence, RI: American Mathematical Society.

he Accounting Review November 2010merican Accounting Association

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.