Aggregation in Budgeting_ an Experiment - ProQuest

JOURNAL OF MANAGEMENT ACCOUNTING RESEARCH American Accounting AssociationVol. 24 DOI: 10.2308/jmar-502042012pp. 177199

Aggregation in Budgeting:An Experiment

Steven T. Schwartz

Binghamton University, SUNY

Eric E. SpiresDavid E. WallinRichard A. Young

The Ohio State University

ABSTRACT: We conduct two experiments to study the effect of aggregated budgetproposals on budgetary slack when superiors cannot commit to an acceptance policy.Prior research on similar settings suggests preferences for fairness will lead to retaliatorybehavior by the superior if the subordinates requests are perceived as selfish.Aggregation causes costs to be more closely distributed around the mean, whichdecreases the possibility that subordinates will have to make abnormally large budgetrequests that may be viewed as selfish and hence may reduce the likelihood of rejection.Aggregation also increases the size of each budget decision. This second effect mayalso deter superiors from retaliating, because it becomes more costly to do so. In our firstexperiment we find that aggregation increases the frequency of mutually beneficialbudget approval. In our second experiment we find that, although some of the increase inbudget approval is due to the larger decision unit, the primary driver is the superiorsincomplete appreciation for the statistical effects of aggregation.

Keywords: aggregation; budgeting; management control.

INTRODUCTION

Subordinates often acquire information rents through the budgeting process (Antle and Eppen

1985; Baiman and Evans 1983). Recent theoretical analyses demonstrate that, when the

superior has full and costless powers of commitment, aggregation helps reduce these rents

(Antle et al. 1999; Arya and Glover 2001; Nikias et al. 2009).1 However, in practice superiors are

The authors thank workshop participants at Erasmus University and Indiana University, especially Daniel Beneish, WeiHwa Chua, Joe Fisher, Xanthi Gkougkousi, Frank Hartmann, Max Hewitt, Victor Maas, Marc Picconi, Kenny Reynolds,Marcel van Rinsum, and Geoff Sprinkle. We also gratefully acknowledge research support obtained from The Ohio StateUniversity.

Published Online: May 2012

1 A different stream of research examines the benefits of aggregation in a multi-task setting (Arya et al. 2004;Gigler and Hemmer 2002; Nikias et al. 2005).

177

rarely able to make fully binding, costless commitments. We use an experiment to explore the effect

of aggregation of budget proposals in the absence of commitment. Without the superiors ability to

commit, non-pecuniary preferences come to the fore and should be considered in any experimental

investigation of budgeting (Evans et al. 2001; Hannan et al. 2009; Nikias et al. 2010; Rankin et al.

2003, 2008). Experiments are a particularly useful way to investigate the tensions between the

assumption of perfectly rational selfishness and non-pecuniary preferences such as fairness or

honesty (Brown et al. 2009; Kachelmeier 1996).

The potential advantage of aggregated budgets is quite simple: the sample mean of several

identically distributed, independent costs is more tightly distributed around its population mean

than is a single cost. We refer to this statistical property as the moderating effect of

aggregation. In consequence, the aggregate cost distribution is less likely to force a subordinate to

submit an extremely high budget proposal. As an example, if a department head has private

information about the cost of undertaking a project and submits a budget to his superior, there

may be occasions where his superior rejects the budget out of resentment because it seems

unusually high, even though this may be the result of an unusually high cost and not the

subordinates desire to misuse resources. However, were the budget to cover several projects, the

underlying average cost per project would likely be more moderate, as higher-cost realizations are

on average balanced with lower-cost realizations. Consequently, the subordinate would be in a

position to make a more moderate per-project budget proposal, possibly increasing the chances of

approval by the superior.

On the other hand, a sophisticated superior might internalize the moderating effect of

aggregation, as is assumed in agency models of aggregation. If so, a budget that appears reasonable

to a superior as a single cost would appear less reasonable as the average of several costs. This

would cause an increased chance of rejection for aggregated budgets after controlling for the

average proposal size. Thus, if superiors internalize the change in distribution, the moderating effect

of aggregation may be undone. It is not clear which of these two effects will dominate, as there are

many robust settings where distribution properties are not fully comprehended or, if comprehended,

are not efficiently acted upon, such as the gamblers fallacy (Clotfelter and Cook 1993), the small

sample fallacy (Rabin 2002), and the winners curse (Kagel and Levin 2002). Therefore, empirical

investigations seem warranted.

In addition to its moderating properties, there are several other aspects of aggregation that

make it a particularly interesting approach to mitigating a management control problem. First, it is

not intrusive and therefore unlikely to cause resentment and possible motivational crowding as

might an ex post audit. Second, it has benefits unrelated to management control, such as easingcognitive strain on users. Finally, it is fundamentally an accounting solution. The study ofaccounting is the study of financial information and its structure. Implementation is

straightforward and likely to be part of the budgeting decision, even without control

considerations. For example, each step in or aspect of a factory modernization may be budgeted

separately, or it may be budgeted for on an aggregate basis. Similarly, a travel budget may be for

a trip, a month, a quarter, or a year.

The economic setting we employ for our investigation is a simple one, involving one superior

and one subordinate. It is based on a model of capital investment by Antle and Eppen (1985). There

are three key features of the model that together ensure the presence of a non-trivial management

control problem: (1) the subordinate has private cost information, (2) the subordinate has no

resources, and (3) the subordinate can consume any overfunding as slack. The superior retains the

difference between the funding and the common knowledge revenues, conditional on the project

being approved. We deviate from the Antle and Eppen (1985) model in that we assume the superior

cannot make ex ante commitments as to how she will respond to the subordinates budget proposal,

178 Schwartz, Spires, Wallin, and Young

Journal of Management Accounting ResearchVolume 24, 2012

although she holds the authority to reject proposals at her discretion. Prior literature has found that

superiors will reject proposals they deem unfair, even if rejection is costly (Rankin et al. 2003).

Therefore, if the form of the budget proposal affects the superiors perception of what constitutes a

fair outcome, differences in form can influence the superiors willingness to approve budgets and

hence impact productive efficiency.

We administer three treatments in our experiments. In the disaggregated treatment, superiors

receive three individual proposals in iterated fashion and must decide whether to accept each project

without knowledge of the subordinates future proposals. In the aggregated treatment, three

projects are bundled together and subordinates submit an aggregate proposal that is intended to

cover the combined cost of all the projects. Therefore, under aggregation, superiors have a single

request to accept or reject.2 Finally, after observing the results from the first two treatments, we

chose to administer a third treatment, large. In large there is a single project with the size of the

combined projects as in aggregated, but with the cost distributed as in disaggregated. The first two

treatments have natural analogs in practice; the third treatment is not a feasible regime given our

basic setup. The purpose of the third treatment is to aid in the interpretation of our results, helping to

determine whether differences between aggregated and disaggregated are the result of the different

cost distributions or the different decision-unit sizes.

The results from our experiments indicate that superiors in aggregated accept significantly

more proposals than in disaggregated, but are less likely to accept high proposals. In response to

or in anticipation of superiors behavior, subordinates in aggregated submit lower proposals for

high-cost projects. However, the difference in superiors responses to high proposals is small

and, pending further experimentation, we tentatively conclude that superiors do not fully

internalize the difference in distributions between the aggregate and disaggregate settings.

Therefore, it appears that the aggregation of budget requests into a single proposal to cover all

costs is efficacious in increasing project acceptance. However, productive efficiency is only

moderately increased by aggregation, because the high-cost projects that mainly benefit from

aggregation do not contribute greatly to overall efficiency. Using large to control for the size of

the superiors decision reveals that although some of the increase in project acceptance in

aggregated is due to the larger decision unit, the primary force is the moderating effect of the

aggregated distribution.

It is important to recognize there may be costs to aggregation unrelated to management control,

such as the ability to plan future activities using past reporting from subordinates; these costs are

absent from the experiment. In this sense, our experiment examines a context favorable for

observing a beneficial effect from aggregation, which seems reasonable for a first investigation.

However, a complete evaluation of aggregation in budgeting should incorporate the potential costs

as well as benefits.

The significance of our results is aggregation allows for the acceptance of more beneficial

budget proposals. Because efficiency gains are modest and would be lower if other unincorporated

costs of aggregation were considered, future research should focus on those circumstances where

aggregation would be most beneficial.

2 A related theoretical literature that also does not allow commitment by the superior finds that coarser informationmay prove useful (Arya et al. 1997; Arya et al. 2000; Pfeiffer 2004). For example, Arya et al. (1997) show that acoarser information system eliminates the potential for the subordinate to report that the project is only slightlyprofitable, and instead forces the subordinate to report the project as either moderately profitable or unprofitable.Hence, even without the ability to commit, the superior only approves projects that would yield more thannegligible profitability, as in Antle and Eppen (1985). However, unlike these analytical studies, aggregationsrole in our experiment is potentially to reduce the temptation for costly retaliation.

Aggregation in Budgeting: An Experiment 179


The outline of the paper is as follows. The next four sections describe our experimental design,

hypotheses, results, and conclusions and limitations.

EXPERIMENTAL TASK AND DESIGN

Our basic experimental task closely resembles that of Rankin et al. (2003). The firm consists of

a superior and a subordinate who interact over three investment projects. The superior is the only

source of project funding and has decision rights over the investments. It is common knowledge

that per-project revenues are deterministic and equal to 200, and that the project costs are

independent and identically distributed U ; f0,. . .,200g.3 The subordinate obtains privateinformation about the cost that can never be verified by the superior. After the subordinate observes

the cost, he submits a budget proposal (request for funds) to the superior. The superior must either

accept the project at the proposed amount or reject it. If the project is accepted, the superior retains

the difference between the revenue and the funding provided the subordinate (200 budgetproposal), and the subordinate retains as slack the excess of the budget proposal over the actual cost(budget proposal cost). If the project is rejected, both superior and subordinate receive nothing.The budget proposal must be weakly greater than the actual cost. These features are retained

throughout all treatments of the experiments.

A standard equilibrium analysis of the settings with and without commitment is as follows. If

the superior can make credible commitments as to how she will respond to the subordinates

proposal, it is optimal ex ante for her to commit to reject proposals above some threshold, eventhough they yield her strictly positive earnings. The subordinate optimally submits proposals equal

to the threshold, as long as the cost is less than or equal to the threshold. In choosing the threshold,

the superior balances the expected loss of earnings from rejected profitable projects with the

expected savings in reduced slack created by the subordinate. With our parameters, the optimal

threshold is 100, yielding superior (subordinate) expected earnings of 50 (25). However, if the

superior does not have the ability to commit, it is optimal for her to accept any proposal leaving her

with positive earnings. The subordinate therefore submits the highest feasible proposal that leaves

the superior with non-negative earnings, and the superior accepts it. We discuss the empirical

validity of this analysis below.

We initially administered two treatments based on this task.4 In the disaggregated treatment,which serves as the control, each subordinate observes an independently drawn cost for each of

three projects. After observing all three costs simultaneously, and without any communication from

the superior, the subordinate submits a proposal for each of the projects, where each projects

proposal is required to be weakly greater than its cost. These proposals are revealed sequentially to

the superior. The superior observes the proposal for Project 1 and decides whether to accept it

before observing the subordinates proposals for Projects 2 and 3. The superior then observes the

proposal for Project 2 and decides whether to accept it, without observing the proposal for Project 3.

The superior cannot go back and alter her decision with respect to Project 1. Finally, the superior

observes the proposal for Project 3 and decides whether to accept it. She cannot go back and alter

3 We assume uniformly distributed and independent costs to keep the task simple for the participants. In practice,proposals may pertain to projects that are not independent. More generally, the variance of the sample mean ofrandom variables with variance r2 and correlation coefficient q is r2 [1/n qn/(n1)]. Thus, positive (negative)correlation decreases (increases) the tightness of the aggregated cost distribution.

4 We also administered another treatment, mainly as a control, in which the superior simultaneously receives threeindividual proposals from the subordinate and then decides whether to accept or reject each project individually.The treatment differed from disaggregated only in the simultaneous receipt of proposals. We do not include theresults here because they were not very insightful and were qualitatively similar to the results for disaggregated.The data and results from this treatment are available upon request.



her decisions with respect to Projects 1 or 2. After the superior has made all three funding decisions,

the decisions are observed by the subordinate and earnings are augmented.

In the aggregated treatment the subordinate is presented with the individual cost realizationsfor each of the three projects and then submits a single aggregate budget proposal. The budget

proposal may not be less than the sum of the three projects costs. The superior determines whether

to accept or reject the aggregate project proposal, receiving revenue of 600 minus the budget

proposal if she accepts the budget.5

After observing the results of the first two treatments, we administered a third treatment, large. Inthis treatment there is a single project. The size of the large project is the same as three aggregatedprojects: revenue of 600 and costs ranging from 0 to 600. However, the cost is distributed uniformly

from 0 to 600, as opposed to the bell shape that obtains from aggregating three uniformlydistributed costs. Thus, the size of the superiors decision in aggregated and large is held constant, sothat the payoff consequences of the superiors decision are identicalonly the underlying cost

distribution is different. The purpose of this treatment is to determine the extent to which differences

between aggregated and disaggregated are due to the difference in the probability distribution ordecision size. Importantly, unlike the first two treatments, large is not a feasible budgeting regimegiven our maintained assumptions. That is, aggregation of three projects with independent uniform

cost distributions will not produce a total cost that is uniformly distributed.6

The experiment was administered at a large Midwestern university. All participants were

undergraduate students enrolled in business courses. There were two sessions administered for each

of the treatments. The total number of participants (superiors and subordinates) was 34, 30, and 36

for aggregated, disaggregated, and large, respectively. Participants were given written instructions,which were subsequently read aloud by one of the experimenters. They were then shown examples

of earnings calculations and passed written quizzes on the experimental task. Finally, they received

training on the computer interface that did not affect their experimental earnings. In each session, 20

rounds of play-for-pay were administered over a computer network, with each round consisting of

three projects (except for large, which consisted of one project) for the superior-subordinate pair.Participants were randomly re-paired after each round. Experimental points were accumulated

throughout the experiment and converted to cash at a rate of US$0.01 for each experimental point.

Sessions lasted one and one-half hours, and participant remuneration averaged $27. Participants

were paid in private, outside of the room where the experiment was conducted.

HYPOTHESES

Assuming rational and selfish behavior by all participants, our setting yields a straightforward

economic prediction that is invariant to the treatment. The superior earns virtually nothing and the

subordinate obtains nearly all the surplus. However, prior experimental research provides strong

evidence that participants will behave differently. A vast literature on ultimatum games suggests (1)

budget proposals that are indicative of a highly unequal sharing of the profits are likely to be rejected by

superiors and (2) subordinates tend to avoid making such proposals (Roth 1995). The conventional

5 We do not include a treatment in which three distinct project proposals are made and the superior is required to acceptprojects on an all-or-none basis. There are two reasons for the exclusion of such a treatment, one concerning externalvalidity and one concerning internal validity. With respect to external validity, there is the question of why there wouldbe project-by-project bankruptcy constraints if the superior cannot cherry pick projects. However, without project-by-project bankruptcy constraints, the subordinate can simply assign each project a proposal equal to the total cost dividedby three, essentially replicating the aggregation setting; in this sense such a treatment is merely a slightly veiled versionof aggregation. With respect to internal validity, even if project-by-project bankruptcy costs were enforced, the onlypayoff-relevant number for the superior is the sum of the proposals, so that the results of the treatment may be verysensitive to how prominently the sum is displayed relative to the individual project proposals.

6 A uniform distribution on total costs would result only if the individual costs were perfectly correlated.



explanation for the rejection of profitable offers is that individuals have a preference for fairness and are

willing to punish unfair acts.7 Especially relevant is the literature on bargaining with incomplete

information, where proposers (subordinates) know the total available surplus, but responders

(superiors) are uncertain (Guth et al. 1996; Mitzkewitz and Nagel 1993; Rapoport and Sundali 1996).

Mitzkewitz and Nagel (1993) and Rapoport and Sundali (1996) provide evidence that responders tend

to accept proposals offering at least one-third of the mean surplus, and increasingly reject proposals as

the surplus offered decreases from that point. The evidence from prior research also suggests proposals

are often a best response to responders rejection behavior (Winter and Zamir 2005).

The question of how the probability distribution of the surplus affects fairness perceptions in

bargaining games with incomplete information is only lightly touched upon in the literature. The

question fundamental to our study is whether only the mean of the distribution is relevant or

whether higher-order moments such as the variance are also relevant. At one extreme, superiors

may simply ignore the lower variance accompanying aggregation and consider only the percent of

expected surplus offered. Croson (1996) provides evidence of the importance of the proportion of

the expected surplus offered on perceptions of fairness as evidenced by responder behavior.

Further, consideration of only the mean requires a lower order of reasoning than consideration of

higher-order moments and would therefore ease cognitive strain. With respect to our experiment, if

superiors consider only the mean of the surplus, they would view a single-project proposal of 170

and a three-project proposal of 510 (an average of 170 per project) identically, because both

proposals represent 30 percent of the mean surplus. In aggregated, on average there will be fewerextreme high and more mid-range cost realizations, allowing the subordinate to submit a greater

proportion of moderate proposals. If superiors care only about the proportion of the mean surplus

offered, we would expect higher overall acceptance rates in aggregated. It is also worth mentioningthat aggregation will make extremely low costs less likely; however, we do not expect this to affectacceptance rates as it is only high costs that preclude sending an acceptable proposal.

A more sophisticated view would take into consideration higher-order moments of the

distribution of the surplus. Specifically, superiors may recognize that, as indicated in Figure 1, the

probability that the cost of a single project is at least 170 is 15 percent (Panel A), whereas the

probability that the average cost of three projects is at least 170 is about 1.6 percent (Panel B).8 Table

1, Panel A, presents the probability of observing average costs above selected thresholds for both the

aggregated (three-project) and single-project distributions. As an extreme example, the probability of a

single cost exceeding 195 is 357 times that of a three-cost mean exceeding 195. Table 1, Panel B, also

provides the more general cumulative distribution function for the sum of the costs of three

independent uniformly and identically distributed projects. Returning to our example, it is therefore

conceivable that an aggregate proposal of 510 is considered by sophisticated superiors to be more

extreme than a single-project proposal of 170, and for this reason they may be more likely to accept

the latter than the former.9

7 By fairness, we refer to the extent to which an outcome is deemed to be equitable given the available choices,property rights, entitlements, etc., of the players.

8 This is a basic result of the central limit theorem. The conclusion follows unambiguously because costs areassumed independent.

9 Nikias et al. (2010) also examine the effects of budget proposal aggregation, but in a non-strategic game. That is,superiors make no choicesall projects are implemented; therefore, Nikias et al. (2010) study only subordinatebehavior. An important deviation from our setting is the interactions in their experiment were face to face. Infact, they attribute their finding that aggregation increases the self-interested behavior of subordinates to thedecreased interaction of superior and subordinate under aggregation. Related to this, Hirst et al. (2007) also use anon-strategic setting to look at how the level of aggregation affects the credibility of a forecast. Their theorywould predict fewer acceptances from aggregation, and the rationale would not rely on statistical properties ofthe setting.



Rapoport and Sundali (1996) provide a somewhat more refined model of the distribution of the

surplus and fairness perceptions. They suggest that responders may frame their acceptance decision

in terms of the likelihood that an offer is fair, given the information they possess. It is not

unreasonable to assume superiors define a fair offer as one that is equal to at least one-half of the

surplus; this is the assumption made by Rapoport and Sundali (1996).10 Under this definition, the

FIGURE 1Single-Cost and Aggregate-Cost Probability Density Functions

Panel A: Control (Single-Project Cost)

Panel B: Treatment (3-Project Mean Cost)

10 The importance to responders of an equal split has been highlighted in the ultimatum game literature (Guth et al.2001) and may be focal to the superiors in our setting. More generally, we view our setting as consistent withFehr and Schmidts (1999) notion that a responder deciding whether to reject an unequal offer must weigh theutility gain from rejecting the unequal offer against the utility loss from less wealth.



likelihood that a single-project proposal of 170 is fair and the likelihood that an aggregate proposal

of 510 is fair are quite different. A single-project proposal of 170 is fair if and only if the

single-project cost is greater than 140 (200 170 170 single-project cost). However, due to thestructure of our experimental task, a single-project proposal of 170 is possible only if the cost is less

than or equal to 170. Therefore, the conditional probability that a single-project proposal of 170 isfair is equal to:

Pr140 cost 170Prcost 170 17:6%:

On the other hand, an aggregate proposal of 510 is fair if and only if the aggregate cost is greater

than 420 (600 510 510 aggregate cost). The likelihood that an aggregate proposal of 510 isfair is equal to:

Pr420 aggregate cost 510Praggregate cost 510 10:8%;

substantially lower than for the single-project proposal.11

TABLE 1

Probabilities of Observing Single and Average Costs within a Range

Panel A: Uniform Distribution Parameters in Experimental Task

Range

Probabilityfor a Single Cost

(ci )

Probabilityfor the

Average Cost[(c1 c2 c3)/3]

160200 0.200 0.036

170200 0.150 0.016

180200 0.100 0.005

190200 0.050 0.0006

195200 0.025 0.00007

Each single cost ci is independent and distributed uniformly on [0, 200].

Panel B: General Uniform Distribution

Range Probability Density Cumulative Distribution

0 Z 1 Z2 Z3

6

1 Z 2 (2Z2 6Z 3) 16 1

22

3Z3 3Z2 3Z 2

3

2 Z 3 (Z2 6Z 9) 56 1

2

1

3Z3 3Z2 9Z 82

3

The calculations in Panel A are derived from the density function for the sum, Z, of three independent uniform randomvariables distributed from 0 to 1 (Grinstead and Snell 1997, 299) as shown in Panel B.

11 Computations similar to those in the text show that the indifference proposal is 160.7 (aggregate of 482.1).Proposals above (below) these values have a higher probability of being fair for the single-project (aggregate)case.



Rapoport and Sundali (1996) utilized a design similar to ours in order to test their theory. In

particular, in each treatment of their ultimatum experiment, the surplus was uniformly distributed

with identical means but different supports. They found that when a low offer was made with wider

supports, it had a greater chance of being accepted than when the same offer was made with

narrower supports. The idea is that with wider supports the low offer might have been fair, but with

narrower supports the low offer could not possibly have been fair because the surplus could not

have been low enough to justify fairness. Given the higher-order reasoning necessary to adjust for

aggregation compared to the relatively transparent manipulation of Rapoport and Sundali (1996),

there is considerable uncertainty regarding whether the effect of aggregation on superiors

acceptance rates in our setting will be analogous to that found in Rapoport and Sundali (1996).

Despite the uncertainty as to whether superiors will adjust for the moderating effects of

aggregation, we present the following hypothesis:

H1: As proposal size increases, the probability of acceptance will be lower with aggregatedproject proposals.

We now consider the behavior of subordinates. As we have previously noted, proposers in

ultimatum games generally behave as if they maximize their wealth conditional on responders

strategies. Therefore, there are two possibilities regarding the effects of aggregation on

subordinates behavior. If superiors treat an aggregate proposal the same way superiors treat a

disaggregated proposal of one-third the size, we would expect no differences in subordinate

behavior. However, if superiors take into consideration that higher-cost outcomes are less likely

with aggregation, and hence reject high proposals more often, we would expect lower proposals

with aggregation. H2 is found below:

H2: After controlling for (average) cost, proposal size will be smaller with aggregated projectproposals.

Finally, we consider the effect of aggregation on productive efficiency, measured by the

percentage of potential profits realized. Here we are faced with interpretive issues, in that efficiency

will surely be a function of the experimental parameters. Therefore it is important to understand the

factors affecting efficiency, so that some attempt to generalize is possible. It is for high-cost

outcomes that aggregation may play a role in increasing the likelihood of acceptance, because for

low-cost projects the subordinate always has the opportunity to submit an acceptable proposal.

However, high-cost projects yield the least benefit. That being the case, if aggregation increases

project acceptance for some costs while not decreasing project acceptance for any others, we can

unequivocally say aggregation increases efficiency. On the other hand, superiors may be more

demanding of the subordinates for aggregate proposals, which would reduce efficiency, ceterisparibus. If superiors do consider the change in cost distribution caused by aggregation, andsubordinates respond by reducing their proposals, this would weaken our predictive ability

regarding the effects of aggregation on efficiency. To summarize, if superiors do not take intoconsideration the distributional effects of aggregation, we would expect aggregation to increaseproductive efficiency, mainly through increased acceptance of high-cost projects. If superiors dotake into consideration the distributional effects of aggregation, we are uncertain about thedirectional effect on productive efficiency. Our final hypothesis is found below:

H3: Productive efficiency will be unaffected by aggregation.

RESULTS

Table 2 provides summary statistics by treatment for all rounds (Panel A) and by halves

(Panels B and C) for the aggregated and disaggregated treatments. Because there were no



noteworthy within-treatment differences between sessions, we combined both sessions of each

treatment. The total number of projects was 1,020 (3 projects per round 3 20 rounds 3 17superior-subordinate pairs) for aggregated and 900 (3 projects per round 3 20 rounds 3 15superior-subordinate pairs) for disaggregated.

In all panels the overall (unconditional) mean acceptance rate is greater in aggregated than in

disaggregated. Considering all rounds, aggregateds acceptance rate of 87.1 percent is significantly

greater than disaggregateds 75.9 percent, p, 0.05.12 Panels B and C indicate that both treatmentsacceptance rates increased in the second half, but the pattern of acceptance rates is the same as when

considering all rounds.

Acceptance Rates Conditioned on Proposal Size (H1)

Figure 2, Panel A graphs acceptance rates conditional on proposal size for each treatment. (The

large treatment is discussed below.) For aggregated, the proposal is divided by three to put it on the

same scale as disaggregated. As the proposal increases, acceptance rates decrease for all treatments.

For proposals , 140, acceptance rates approach 100 percent for all treatments, whereas forproposals . 180, acceptance rates are within the 2060 percent range.

H1 addresses whether the relation between proposal size and acceptance rate differs across

treatments. Figure 2, Panel A indicates the presence of differences among treatments. To formally

test the hypothesis, we estimated a repeated-measures logistic regression model as shown below in

(1):

Accept b0 b1PROPOSAL b2AGG b3PROPOSAL AGG e; 1where Accept 1 if the project is accepted, 0 otherwise; PROPOSAL is the proposal size; and AGG 1 for the aggregated treatment, and 0 for disaggregated.13 The logit model enables us tocondition acceptance rates on proposal size, as required by H1, and to test for differences across

treatments. The interaction coefficient measures whether the relationship between proposal size and

acceptance differs between the two treatments, and is used for hypothesis testing. Model

coefficients are shown below:

Accept 17:83 0:099 PROPOSAL 13:11 AGG 0:074 PROPOSAL AGG:All of the coefficients are significant (p , 0.01 for b0 and b1, and p , 0.10 for b2 and b3).

14 The

logit functions for each treatment are presented in Figure 2, Panel B.

The significant negative coefficient on the interaction term b3 indicates that the superiors inaggregated became more stringent as proposal size increased than did superiors in disaggregated.

For lower proposals, acceptance was greater in aggregated, but for large proposals, acceptance was

greater in disaggregated. The crossover point at which the superiors in aggregated became more

stringent is an (average) proposal of approximately 177 (see Figure 2, Panel B). This increased

stringency is consistent with the argument developed for H1 that if superiors have at least an

12 All reported p-values are two-tailed, unless otherwise indicated. Also, because the disaggregated treatmentinvolved three proposals and the aggregated treatment only one, we ran supplemental analyses using only thefirst proposal in disaggregated. The results, which are similar to the three-proposal analyses, are not reportedherein.

13 We used the GENMOD procedure in SAS, which provides analyses using Generalized Estimating Equations(Liang and Zeger 1986) to account for any non-independence of repeated measures.

14 To test the robustness of our results, we also estimated models that tested for differences across halves of theexperiment. Compared to the first half, the second-half intercepts are higher and the slope coefficients lower forboth treatments, but not significantly so. Tests of H1 are consistent across halves.



TABLE 2

Summary Statistics

Panel A: All Rounds

n(proj) n(pair) CostBudgetProposal

AcceptanceRate

SuperiorEarningsper Project

SubordinateEarnings per

ProjectProductiveEfficiency

aggregated 1,020 17 101.0 163.6 87.1 33.2 56.7 0.908(58.4) (3.7) (13.0) (4.5) (10.5)

disaggregated 900 15 100.5 160.2 75.9 35.6 52.6 0.887(59.2) (8.9) (18.0) (6.2) (7.1)

Panel B: First Half


AcceptanceRate




aggregated 510 17 102.4 163.7 85.3 32.6 53.6 0.884(58.8) (4.3) (17.0) (6.0) (14.7)

disaggregated 450 15 98.9 158.4 72.4 36.3 49.9 0.853(59.3) (10.7) (22.8) (8.7) (12.4)

Panel C: Second Half


AcceptanceRate




aggregated 510 17 99.6 163.6 88.8 33.8 59.7 0.931(57.9) (4.8) (13.6) (5.2) (10.4)

disaggregated 450 15 102.1 162.1 79.3 34.9 55.3 0.922(59.1) (8.5) (17.8) (6.4) (10.0)

n(proj) total number of projects by treatment.n(pair) number of superiors and subordinates by treatment.Cells contain means and (standard deviations). For cost, standard deviations use the project as the unit of observation.For budget proposal and subordinate earnings, standard deviations use the subordinate as the unit of observation. Foracceptance rate and superior earnings, standard deviations use the superior as the unit of observation.

superior earnings per projecti 200 budget proposaliIi:

subordinate earnings per projecti budget proposali ciIi:

productive efficiency RN

i200 ciIi

RN

i200 ci

:

where ci cost of project i; Ii1 if project i is accepted, and 0 if rejected; N 1,020 (510) for aggregated and 900 (450)for disaggregated for all rounds (first and second half ).



intuitive feel for the effects of the central limit theorem, the acceptance rate for high proposals

would be lower in aggregated than in disaggregated.

These results provide some evidence that superiors did respond to the change in distribution

from disaggregated to aggregated; however, further analysis shows that the superiors did not fully

adjust. First, as mentioned earlier, the proposal size giving equal probability of fairness is

approximately 160.7. The observed crossover point of 177 is greater than this, indicating the

superiors in aggregated did not fully adjust. Second, the acceptance rates for proposals greater than

the observed cutoff point of 177 (41.9 percent for disaggregated and 33.3 percent for aggregated)

do not statistically significantly differ (p . 0.30). Thus, although the superiors in aggregated weremore stringent for higher proposals, the increased stringency did not lead to significant differences

in acceptance rates for the high proposals. Third, the superiors adjustment is much smaller than

reported in Rapoport and Sundali (1996). To see this, consider the idea discussed earlier that

superiors may focus on the probability that a particular offer is fair, which is the probability that the

offer represents at least one-half of the expected surplus. To investigate whether superiors fully

internalize the moderating aspect of aggregation, we considered offers that have between a 5

percent and 15 percent probability of being fair. This equates to offers in the range of 174 to 190 for

disaggregated and in the range of 167 to 177 for aggregated.15 For those ranges, the acceptance

rates by superior are 83.6 percent and 53.4 percent for aggregated and disaggregated, respectively

FIGURE 2Acceptance Rates Conditioned on Proposal Size

Panel A: Acceptance Rates by Treatment Conditional on Proposal (H1)

(continued on next page)

15 The range for aggregated may be calculated using the cumulative distribution functions found in Table 1, Panel B.



(p , 0.05). The difference indicates superiors are not fully adjusting for the statistical property of

aggregated costs, at least in the manner theorized by Rapoport and Sundali (1996).

Subordinate Proposals Conditioned on Project Cost (H2)

Figure 3, Panel A graphs subordinates proposals conditional on cost, where cost is classified

into deciles. For aggregated, the proposals are classified according to where the average of the three

costs lies. So, for example, the 020 cost range would include aggregate costs of 60 or less. (As it

turned out, there are no observations in this range for aggregated.) As cost increases, proposals

increase (more noticeably for high costs), for all treatments. H2 deals with whether the relation

between proposal and cost differs for the treatments. The graph in Figure 3, Panel A indicates only

slight differences, with subordinates proposals in aggregated being a little less sensitive to changes

in cost than in disaggregated. However, care should be exercised in across-treatment comparisons

of these graphs, because even with the stratification, the samples of realized costs within ranges

differ across the treatments. The regression model we use to test H2 does not suffer from this

drawback.

To test H2, we regressed proposal size on cost and a treatment dummy variable using

generalized estimating equations as described previously. Because subordinates proposals are less

affected by low costs than by high costs, the model includes a quadratic component related to cost:

FIGURE 2 (continued)

Panel B: Logistic Regressions of Acceptance Rates on Proposal

AGG aggregated; DIS disaggregated; LAR large.For aggregated and large, proposal is proposal/3.



Proposal b0 b1COST b2AGG b3COST AGG b4COSTSQ b5COSTSQ AGG e; 2

where COSTSQ COST COST, and AGG is a dummy variable equal to 1 if the treatment isaggregated, and 0 if disaggregated. The fitted model follows:

Proposal 141:8 0:0324 COST 13:45 AGG 0:0074 COST AGG 0:0016 COSTSQ 0:0005 COSTSQ AGG:

Because COST and its interaction with AGG are insignificant (p . 0.60 for both coefficients)16

and, in the presence of COSTSQ add virtually nothing to explaining the variance in proposal size,we use the following reduced model (20) for hypothesis testing:

Proposal b0 b1COSTSQ b2AGG b3COSTSQ AGG e: 2 0The reduced model is reasonable because subordinates know that superiors are unlikely to reject

moderate-sized proposals. Thus, for a wide range of lower costs, subordinates are likely to make

moderate-sized proposals; that is, the cost will have little effect on the proposals, indicating that a

FIGURE 3Subordinate Proposals Conditioned on Project Cost

Panel A: Proposal Size by Treatment Conditional on Project Cost (H2)


16 The negative sign for the insignificant COST coefficient is likely due to multicollinearity. A model includingonly COST, AGG, and COST AGG as independent variables yields a significant positive coefficient on COST.



linear term is not useful for lower costs. To illustrate, for the disaggregated treatment, costs in the

020 range and costs in the 4060 range both result in average proposals of 142; the cost has

virtually no effect on proposals. However, when costs are higher, subordinates must make high

proposals because their proposals cannot be less than the cost, and cost has a larger effect on

proposals. For the disaggregated treatment, costs in the 140160 range and costs in the 180200

range result in average proposals of 173 and 195, respectively; the cost has a large effect on

proposals. Model coefficients are shown below:17

Proposal 140:5 0:0015 COSTSQ 12:80 AGG 0:0006 COSTSQ AGG:All coefficients are significant (p , 0.001).18 The model is graphed in Figure 3, Panel B for alltreatments.

The significantly negative interaction coefficient b3 indicates that as cost increases,subordinates in aggregated make relatively smaller proposals than subordinates in disaggregated,

consistent with H2. This is also consistent with the H1 result that superiors in aggregated reject

high proposals more often than superiors in disaggregated, and subordinates anticipation of or

reaction to this behavior.


Panel B: Regressions of Proposal Size on Project Cost

AGG aggregated; DIS disaggregated; LAR large.For aggregated, proposal is proposal/3 and cost is average cost. For large, proposal is proposal/3 and cost iscost/3.

17 Care should be exercised in interpreting the aggregated model at both extremes of the cost range due to theexistence of only a few data points in those ranges.

18 The results are qualitatively consistent for the first and second half of the experiment.



Interpreting these results, aggregation induces more moderate proposals. It does not affect the

average proposal (the average proposals shown in Table 2 do not differ across treatments, p .0.15). Instead, aggregation reduces the variance of proposals; Table 2 shows the standard deviation

of average proposals across subordinates for aggregated is 3.7, versus 8.9 for disaggregated. Thisis further illustrated in Figure 4. In the 161170 range are 49 percent of the aggregated proposals,but only 14 percent of the disaggregated proposals. In the upper tail (. 180) lie only 6 percent ofaggregated proposals, but 18 percent of disaggregated proposals. To formally test thesedifferences, we compared the average standard deviation of proposals by subordinate across

treatments; the average standard deviation for aggregated is significantly lower than fordisaggregated (p , 0.001). The higher frequency of mid-range proposals is a large factor in theoverall acceptance rate in aggregated being higher than in disaggregated.

Productive Efficiency (H3)

Productive efficiency is measured by total profits realized as a percentage of total profits

possible. Table 2 shows that efficiency is higher for aggregated (0.908) than for disaggregated(0.887), with a similar pattern in each half (but dampened in the second). There are two

determinants of efficiency: (1) whether a project is accepted, and (2) the cost mix of projects

accepted; high-cost projects do not contribute as much to overall efficiency as low-cost projects.

Although the overall acceptance rate for aggregated (87.1 percent) is much higher than fordisaggregated (75.9 percent), the increase in efficiency for aggregated is not as great because theincreased acceptance occurs mostly for high-cost projects.19 However, it is mainly for these high-

cost, yet profitable, projects that productive efficiency can be enhanced, because low-cost projects

are routinely accepted, no matter the budgeting procedure.

Because efficiency for a given project is either 1 or 0, depending on whether a project is

funded, tests of differences in efficiency using the project as the unit of observation will yield the

same result as tests of project acceptance. To address this problem, we calculated efficiency by

individual superior, as the total profits realized as a percentage of the total profits possible for the

projects involving the superior, that is:

R60

i200 ciIi

" #R60

i200 ci

" #;

where ci cost of project i, and Ii 1 if project i is accepted, and 0 if rejected. We used t-tests tocompare the average efficiency across superiors in the aggregated treatment with the disaggregatedtreatment.

Considering all projects, productive efficiency of 0.908 in aggregated is not significantlyhigher than disaggregateds 0.887 (p . 0.50). For projects with cost 100, efficiency measuresfor aggregated and disaggregated are 0.925 and 0.940, respectively; the difference is not significant(p. 0.70). However, for projects with cost. 100, aggregateds efficiency of 0.853 is significantlygreater than disaggregateds 0.722 (p , 0.10). Thus, as indicated above, aggregation increasesefficiency for high-cost, but not low-cost projects, providing only limited support for H3. We do not

view this result as an artifact of the parameters used. In general, aggregation will allow for greater

acceptance of lower-quality projects (Nikias et al. 2009).

19 Another potential cause of only modest gains in efficiency in aggregated is that relatively more low-cost projectsmay be rejected if superiors become more demanding, especially if a low-cost project is paired with two high-cost projects. Although this may have occurred, the magnitude is not significant. Considering all projects withcost of 100 or lower, the rejection rates for aggregated and disaggregated are 8.0 percent and 6.5 percent,respectively (p . 0.50).



Figure 5 presents graphs of superior and subordinate earnings. For aggregated, cost refers tothe average cost, so that we can unequivocally identify a cost decile. Superior earnings (Figure 5,

Panel A) decline as cost increases for both treatments (p , 0.01). Regression models (resultsneither graphed nor tabulated) indicate that the disaggregated intercept is significantly greater thanaggregateds (p , 0.01). In addition, the relationship between cost and superior earnings differs (p, 0.02). In particular, as project cost increases, superiors in aggregated receive greater incrementsthan superiors in disaggregated. This finding shows that the benefits of aggregation for superiorsoccur for the high-cost projects, similar to productive efficiency.

Subordinate earnings (Figure 5, Panel B) also decline as cost increases for both treatments (p

, 0.01). Regression models (results not graphed nor tabulated) indicate that the intercept foraggregated is larger than disaggregateds (p , 0.02), but the slopes do not differ significantly.Thus, after controlling for differences in cost realizations, subordinates in aggregated obtainedhigher mean earnings.

Additional Treatment (Large)

To aid in interpreting our results, we administered a third treatment, large, after viewing theresults of the first two treatments. As noted earlier, large is not a feasible reporting regime, as itcannot result from aggregation; it is simply a tool to aid in the interpretation of our results. Our

hypothesis development led us in two directions. The first is that superiors would understand the

moderating effects of aggregation and adjust their proposal tolerance accordingly. Under this

assumption, aggregation would not yield increases in productive efficiency. The second is that

superiors would not fully appreciate the change in the distribution of the surplus, and hence the

moderating effects of aggregation would lead to a conditional (on proposal size) increase in project

FIGURE 4Distribution of Proposal Size by Treatment



acceptance. So far we have found that superiors in aggregated accept a higher percentage ofprojects, primarily because the moderating effects of aggregation allow a high percentage of

subordinate proposals (even for high-cost projects) to be moderately sized. Superiors in aggregateddo show greater willingness to reject high-proposal projects, but this effect is small and does not

completely undo the effects of aggregation.

However, there is another possibilitysuperiors understood and were willing to react fully to

the distributional change brought on by aggregation, but did not want to reject projects of a larger

size. In other words, it is possible that the higher overall acceptance in aggregated and mildness ofthe superiors reaction to high proposals are both due to the superiors in aggregated deciding onproposals with 600-point revenues versus 200-point revenues in disaggregated. To investigate thispossibility we administered a third treatment, one that had the distributional properties of

disaggregated, but the project size of aggregated. In large there were 20 decisions made, each on aproject with revenues of 600 and a uniform cost distribution from 0 to 600. The total amount at

stake in all three treatments was the same (either 60 decisions on 200-point projects or 20 decisions

on 600-point projects); only the size per decision differed. If size, as opposed to the moderating

effect of aggregation, were driving the results of the first two treatments, we would expect large andaggregated to have (1) approximately the same overall acceptance rate, (2) approximately the sameacceptance rate for high-proposal projects, and (3) similar distributions of proposals, so that the

acceptance rates for high-cost projects are approximately the same. On the other hand, if size is not

FIGURE 5Superior and Subordinate Earnings Conditioned on Project Cost

Panel A: Superior Earnings by Treatment Conditional on Project Cost




relevant we would expect (1) the overall acceptance rate to be higher in aggregated than in large;(2) aggregated to have lower acceptance rates for high-proposal projects, but not to the extent ofcompletely undoing the moderating effects of aggregation; and (3) the distribution of proposals for

aggregated to have lower variance, so that acceptance of high-cost projects is greater inaggregated.

With respect to (1), the mean acceptance rate in large is 82.8 percent; the rate for aggregated(87.1 percent) is higher, but not significantly so (p 0.32).20 For (2), Figure 2 shows, whenconditioning on proposal size, aggregated has a lower acceptance rate than large for all levels ofproposal size, and for high proposals is much lower.21 For proposals in the 171180 and 181190

ranges, the difference is significant (p , 0.10 and 0.02, respectively).22 Also, comparingacceptance rates using the range of proposals that have a 5 percent to 15 percent chance of being


Panel B: Subordinate Earnings by Treatment Conditional on Project Cost

AGG aggregated; DIS disaggregated; LAR large.For aggregated, cost is average cost. For large, cost is cost/3.

20 The acceptance rate for large is not significantly greater than the 75.9 percent rate for disaggregated (p 0.19).Recall that aggregated has a significantly higher rate than disaggregated.

21 Because superiors in large were so acquiescent in their acceptance, subordinates submitted higher proposals(average of 168.5; see Table 2 for other treatments averages and Figure 3 for graphs) than in any other treatment,all with p , 0.05. As a result, the average superiors earnings in large are 29.2 (refer to Table 2 for othertreatments and Figure 5 for plots), significantly lower than in any other treatment (all with p , 0.05).Subordinate earnings are highest in large (61.5), but significantly so only when compared to disaggregated (p ,0.05).

22 The shapes of the logit functions in Figure 3, Panel B for aggregated and large are similar (p . 0.5 for both theintercept and slope coefficient). Nonetheless, as indicated in the text, aggregated has significantly loweracceptance rates for high ranges of proposal size.



fair, as we did earlier, shows higher acceptance in aggregated, 83.6 percent versus 65.2 percent forlarge (p , 0.11). Thus, superiors in aggregated are reacting to, but are not fully undoing, theeffects of aggregation.

With respect to (3), Figure 4 shows that the distribution of proposals for aggregated has lowervariance than for large. For aggregated (large), 49 percent (33 percent) of the proposals are in the161170 range (after dividing proposal by 3), whereas 6 percent (23 percent) are in the . 180 tail.Using the test described earlier, the standard deviation is lower for aggregated than large (p ,0.005). The less frequent use of high-request proposals leads to higher acceptance of high-cost

projects in aggregated (p , 0.01 for project costs of 140 and higher).23

These results suggest that although size may have impacted the results for the original

treatments, it was not the primary factor. Compared to superiors in large, superiors in aggregatedreacted to the moderating effects of aggregation by becoming stricter in their acceptance decisions,

but their reaction was not sufficient to negate completely the beneficial effects of aggregation, such

that the overall rate of acceptance is higher in aggregated, especially for high-cost projects. This issimilar to the conclusion based on the first two treatments. In addition, when comparing aggregatedto large, we see much of the effect found in Rapoport and Sundali (1996)decreased acceptanceconditional on proposal size for the tighter distribution. However, unlike Rapoport and Sundali

(1996), the effect was not as concentrated in the very low surplus offers. Perhaps the more subtle

nature of aggregation, as opposed to the more transparent contraction of the supports in Rapoport

and Sundali (1996), caused a more diffused response in our experiment.

CONCLUSIONS AND LIMITATIONS

We conduct two experiments designed to investigate the effects of aggregating budgeting

proposals. The setting has important elements of a management control problem: information

asymmetry, a subordinate with limited resources and a taste for slack, and a superior with expost authority over project approval. Within this setting we administer three treatments: (1)disaggregated proposals, wherein the superior receives and decides upon each of the proposals

in sequence; (2) aggregated proposals, wherein the superior observes only an aggregate

proposal and is constrained to either accept or reject all projects; and (3) large single proposals,

involving projects with the size of aggregated projects, but the cost distribution characteristics

of the disaggregated treatment. We administered the third treatment after viewing the results of

the first two treatments. The total revenue available in the experiments was constant across all

treatments.

We find that aggregated budget proposals lead to a greater acceptance rate than disaggregated

budget proposals, especially for the difficult-to-fund high-cost projects. Although superiors

appeared to sense the difference in aggregated and disaggregated distributions and hence were

stricter with similar-sized aggregate proposals, this effect was small and not enough to counteract

the statistically induced moderating effects of aggregation. Controlling for the size of the decision

unit does not change this basic result, although the magnitudes of some of the differences are

smaller.

One implication of the experiment is that aggregation may be efficacious in increasing

productive efficiency, but for very different reasons than those found in theoretical papers such as

Antle et al. (1999). To the extent that superiors do not fully undo the effects of aggregation, our

results are consistent with superiors suffering from false consensus bias, wherein they attribute the

same motives to others they would to themselves (Ross et al. 1977). That is, people think others are

23 The acceptance rate for projects with costs of 140 and higher in large does not differ from disaggregated (p 0.25).



being fair because they would be fair. In Rapoport and Sundali (1996), it was impossible for

responders to (rationally) believe others were acting fairly for some low offers. In contrast, our

manipulation (aggregation) implies a constant support, and so fair-minded people might tend to

think others are acting fairly despite how improbable (but not impossible) it may be. Failure to

completely adjust for the statistical properties of aggregation is also consistent with a large body of

literature in cognitive psychology that shows that individuals tend to underestimate the effect of

sample sizes in many contexts (Kahneman and Tversky 1972; Rabin 1998).

Our results also suggest that the benefit of increased project acceptance and enhanced

productive efficiency with aggregation mainly accrues to the subordinate. It is possible, as

discussed earlier, that these findings are a function of the particular parameters we used to

characterize the revenue and cost distributions. More importantly, a subordinate may receive some

of his reservation utility in the form of perquisite consumption as opposed to a wage, as for example

suggested by Fama (1980, 296). Therefore, increased efficiency accruing mainly to subordinates

may benefit the owners of the firm in addition to the subordinate or even to the exclusion of the

subordinate, depending on the relative amount of slack produced and the subordinates reservation

utility. Along a different dimension, aggregation has been touted as a control mechanism that, in the

presence of full and costless commitment, increases efficiency and benefits the superior. It is ofinterest that if full powers of commitment are not present, aggregation may enhance subordinates

earnings more than superiors.

As with virtually all experiments, there are important limitations to consider when interpreting

results and conducting future research. One limitation of our experiment is that all projects were

weakly profitable. As the potential for negative earnings increases, the value of aggregation

decreases (Nikias et al. 2009). Also the cost draws were independent within subordinates.

Independent cost draws increase the potential for aggregation to be useful; correlated cost draws

would not tighten the distribution around the center as much as independent cost draws. Somewhat

related, we only consider uniformly distributed costs; other cost distribution may produce different

results. We also note that in our experiment the budgeting structure was exogenous. A potential

avenue of investigation is to conduct an experiment wherein the budgeting structure is endogenous.

That is, superiors would be given a choice as to the form in which proposals would be submitted.

Conceivably, superiors might have an innate desire for greater detail or reduced uncertainty, and

would therefore elect to receive disaggregated proposals, even though we seem to have

demonstrated a benefit to aggregation. This may have economic consequences, which cannot be

addressed with our experiment.

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