Cost of effort in benchmarking: exploration and applications

Abstract The literature on incentives suggests that the cost of effort is a key determinant of production, as an agent’s utility is dependent on both consumption and the cost of effort. However, the benchmarking literature has neglected to consider the cost of effort in performance improvement, as it primarily focuses on selecting best-practice benchmarks for underperforming agents. This paper aims to bridge these two literatures by examining the cost of effort in benchmarking and its applications. Our approach is a direct extension of the rational inefficiency hypothesis. We use the information of slacks regarding technology to make inference about the cost of effort in benchmarking. Importantly, we show that inference about the cost of effort gives new sights into activity planning, incentive provision, and employee layoffs. Our analysis provides a new explanation for benchmarking failure in business practices. It also contributes to the rational inefficiency hypothesis by revealing that inefficiency can be beneficial in benchmarking since it can be regarded as a form of fringe reimbursement provided to stakeholders to offset the cost of effort.


Introduction
It has long been recognized that the productive effort is a determinant of production (Laffont and Martimort 2009).An agent's utility function is dependent on both their consumption and effort, and this will ultimately dictate the specific level of production they choose to pursue.In incentives literature, effort is often non-observable and non-verifiable to the principal who delegates production tasks or third parties like the Court of Justice, thereby agent can loaf on the job if not being fully reimbursed.This hidden action incurs the well-known moral hazard problem, which has gained momentum (Holmstr€ om 1979).The effort variables involved in moral hazard actions have a positive impact on an agent's production level at the cost of exerting effort.
An agent's production level and consumption patterns exhibit valuable information of relative performance for benchmarking purposes.Benchmarking involves selecting efficient agents as potential benchmarks for inefficient agents to learn from.With the help of these benchmarks, managers within organizations are able to obtain production information that allows them to identify the strengths and weaknesses of sub-sectors or sub-units, thereby setting best practice goals to achieve organizational strategies.The goal of benchmarking is to improve performance by identifying and applying best documented practices, as extensively investigated in the fields of production economics and operations research (Djerdjourl 2005;Petersen 2018;F€ are et al. 2019).
To our knowledge, effort receives very limited attention in benchmarking.There is a relationship between the amount of effort and productive characteristics of the agent in the production process.Traditional benchmarking is contingent on the observations (i.e.consumptions and production levels) that aggregate effort and productive characteristics to make inferences about the relative performance of the agent.In other words, effort is often considered negligible or trivial in benchmarking because the observations used to evaluate the performance of an agent typically incorporate partial information about effort.By excluding effort from the analysis, benchmarking can focus on estimating technology through regularity conditions using these observations directly (see for example, Cook et al. 2017).
The objective of this paper is to establish a connection between the literature on incentives and benchmarking by examining the cost of effort in benchmarking and exploring its applications.The ultimate goal of benchmarking is to improve performance by identifying and applying best practices.Identifying best practices has been solved by performance evaluation where Pareto efficient agents represent best practices (Hougaard and Tvede 2002).The effort is negligible as the observations of agents are sufficient to discriminate superior and inferior performance.On the other hand, applying best practices should take agent's utility into account in line with incentive theory (An et al. 2021).The agent's effort should be considered in the issue of applying best practices since it describes the endogenous uncertainty when agent is asked to improve performance.Higher effort brings more revenues and contribute to performance improvement in benchmarking, but also creates a higher cost for the agent.The agent faces a trade-off between revenues (i.e.transfers) and cost when deciding whether to apply best practices.
Incorporating the cost of effort into benchmarking analysis may offer a plausible and suitable rationale for the phenomenon of 'benchmarking failure' observed in practical applications.Global surveys of the most popular management instruments reported by Bain & Company Inc has consistently listed benchmarking among the most important tools.However, a successful implementation of benchmarking is not an easy task for most organizations and enterprises.This failure primarily stems from the agent's reluctance to apply best practices, even after they have been clearly identified.In this paper, we demonstrate that the benchmarking failure can be a rational result due to the trade-off between transfer and the cost of effort when the agent is 'greedy', i.e. maximizes utility first and decides whether applying best practices subsequently.
The rational choice of the agent here reveals agent's response to apply best practices and provides valuable information about agent's cost of effort.The utility function of the agent is the gap between transfer from the principal and the cost of effort.As a consequence, the observed slacks of the agent (i.e.inefficiencies) with regard to the technology identifying best practices can be employed to make inferences about the cost of effort because the transfer is controlled by the principal.The choice of the agent can be modelled as a slack selection process where the effort is a function of the observed slacks.We introduce several slack aggregation functions to operationalize the cost of effort, and the observed slacks are used to estimate the unknown structures of slack aggregation functions.
With the help of the inference about the cost of effort, we discuss more specific applications.The first application is activity planning.The utility function of the agent can be estimated through transfer from the principal and the inference about the cost of effort.Consequently, the principal can make rather precise predictions of agent's rational response to control and plan future production (Bogetoft 2000).The second application is incentive provision.The trade-off between transfer and cost of effort may hinder the agent from applying best practices.Thus, the principal can design contract to minimize the cost of inducing the agent to apply best practices or produce expected output levels (Bogetoft 1994(Bogetoft , 1995)).The third application is employee layoff.The principal gets an explicit or even at least a partial estimate of the cost of effort.Consequently, the principal can decide the employees (i.e.agents) to be dismissed by referring to such estimate above when faced with a labor surplus problem (Folger & Skarlicki 1998;Richter et al. 2018).
The analytic framework in this paper can be regarded as a generalization of Bogetoft and Hougaard (2003) where the cost of effort is reflected by agent's slack selection decision.Specifically, the cost of effort is treated as a variant of the slack aggregation functions proposed by Bogetoft and Hougaard (2003), where the value of slack is represented by the disutility of the effort required to eliminate it.The two applications, namely activity planning and incentive provision, are further refinements of the applications originally proposed by Bogetoft and Hougaard (2003).The advance of this paper lies in demonstrating that both input and output slacks can uniquely reflect the cost of effort.Specifically, when the principal seeks to make accurate estimations about the cost of effort, it is crucial to take into account the individual rationality of the agent.
This paper also enriches the rational inefficiency hypothesis suggested by Bogetoft and Hougaard (2003).They state there are many possible rationales for inefficiency: inefficiency may be a part of the fringe reimbursement paid to stake holders; inefficiency can create loyal employees by incentives; inefficiency can be regarded as a useful resource to buffer uncertainty in organization; inefficiency may be caused by rent seeking behavior.In this paper, we illustrate that inefficiency (i.e. the decision of the agent on rejecting applying best practices) may be a result of the trade-off between the transfer from the principal and the cost of effort.This finding supports the first argument on possible rationales for inefficiency, the agent rationally selects inefficiency as fringe reimbursement to cover the cost of effort.
The structure of the rest of this paper is as follows.The next section reviews the related literature about individual effort and benchmarking.Section 3 first provides the basic definitions of technology, after which we characterize the cost of effort by the observed production mix and the technology.We next model the utility function of the agent as the gap between the transfer from the principal and agent's cost of effort.In line with the rational choice theory, the observed production mix is assumed to be the optimal response to a utility maximization problem of the agent.We introduce four restricted classes of slack aggregation functions, from which we can make at least partial inference about the cost of effort with the observed production mix and the technology.Section 4 presents the applications of the inference about the cost of effort in activity planning, incentive provision and employee layoff.Section 5 concludes.

Cost of effort
The concept of the 'cost of effort' explores the expenditure, whether physical, mental, or resource-based, that individuals or entities incur when engaging in a particular activity, task, or goal.Understanding the cost of effort is vital in various fields, including economics, psychology, and behavioral sciences, as it sheds light on factors influencing motivation, productivity, and decision-making.For example, Kahneman and Tversky (1979) introduces the concept of cognitive costs related to effort and how they affect decision-making under risk.Fehr and G€ achter (2000) investigates how fairness considerations and concerns about reputation can lead to the incurrence of effort costs in social interactions.Gneezy and Rustichini (2000) studies the relationship between compensation and effort, highlighting how underpayment can lead to a lack of effort.Inzlicht et al. (2018) examines how individuals perceive and balance the costs and benefits of effort, especially in achievement-related contexts.In addition to the aforementioned papers, it is important to acknowledge the extensive body of research dedicated to elucidating the multifaceted aspects of effort-related costs.
In contrast, the individual effort only receives limited attention in performance evaluation and productivity analysis.One early contribution is Bogetoft (1993) who designs incentive schemes for agents producing information in parallel production systems.He examines how the existence of multiple homogenous agents may be exploited to collect underlying production information via relative performance evaluations.The result indicates that more homogenous agents performing parallel investigations can reduce the total incentive costs in the moral hazard problem.To model the characteristics of agents, Bogetoft (1993) introduces the following von Neumann-Morgenstern utility function: where the utility of the agent u t, e ð Þ is additively separable into a transfer u t ð Þ and an effort component v e ð Þ: He assumes that agents possess two key characteristics (i) risk aversion, indicated by u 0 t ð Þ > 0 and u 00 t ð Þ < 0: Here, the first and second derivatives of the utility function uðtÞ reflect the agents' inclination to avoid losses or prioritize the preservation of their wealth, even if this entails accepting a lower expected return; (ii) strictly work aversion, signified by v 0 e ð Þ > 0 and v 0 0 e ð Þ > 0, implying that higher levels of effort in work incur greater costs and marginal costs.Bogetoft (1994) explores how empirical production frontiers (i.e.DEA frontiers) contribute to incentives of similar agents.He points out DEA frontiers overcome the difficulty that the lack of information about the production functions of the agents.The information about production functions incorporates at least two aspects: One is the uncertainty as to the underlying technology; the other is the uncertainty as to the amount of effort put into the production process.Suppose the principal delegates a production task transforming m inputs to generate one output, a formal production function (or technology) can be described as where is the observed production mix, p x ð Þ is denoted as the output per effort unit when input x is consumed, and e is the uncertainty in production.He assumes that e is an independent, nonobservable random variable satisfying a specific density function on R: p x ð Þ can be regarded as the production frontier in production economics.It is unknown to the principal but the set of possible frontiers K (p x ð Þ 2 K) is known and the principal forms belief on K with a probability distribution.
For the multiple-input and multiple-output case, Bogetoft (1995) develops the following correlated technology: with the interpretation that x, y is the production mix of m inputs and h reflects the choice of output mix by the agent where E denotes the unit vector 1, :: and p x, f ð Þ denotes the maximal output along direction f per effort unit when x is consumed.In particular, the actual production frontier p x, f ð Þ is unknown to both the principal and the agents.
The modelling of effort in performance evaluation is mainly employed to elucidate the incentive property of DEA-like efficiency measures, i.e. how to use relative performance evaluations to deal with the moral hazard problem.In the setting of previous studies, the cost of effort is characterized by a generalized function v e ð Þ: v e ð Þ has implicit impacts on the final producing outputs because the observed outputs are determined by the amount of effort put into the production process.However, none of studies above discusses the formulation of v e ð Þ: This paper fills this void theoretically by making inference about v e ð Þ through slack aggregation functions.

Benchmarking
Benchmarking has been widely explored in performance evaluation and productivity analysis.Bogetoft (2013) shows benchmarking originates from the concept of key performance indicators, and DEA is the most prevalent and practical management instrument for benchmarking recently.The superiority of DEA is its ability to identify best practices under unknown prior preferences and production possibilities.Recent contributions are Le et al. (2021) who use benchmarking to select benchmarks for Vietnamese Education System for performance improvement, and Piran et al. (2021) who conduct an internal benchmarking analysis for a broiler production system through cost efficiency, to mention just a few.
A crucial task of benchmarking is to identify best practices which relates to the notion of technical efficiency in production theory.Early contributions of measuring technical efficiency also identify best practices for benchmarking simultaneously.For example, the seminal work of Farrell (1957) defines the radial input-oriented technical efficiency as where T denotes the technology set, i.e. the set of combinations of inputs and outputs such that the outputs can be produced through consuming the inputs, h is a scalar that scales all elements of the inputs vector, and TE x, y; T ð Þ is the well-known Farrell measure.The corresponding best practice x BP , y BP À Á is with the interpretation that best practice is identified by radial projection from Farrell measure.Similar projections (i.e.best practices) can be derived from Russell measure (Russell 1985), additive measure (Charnes et al. 1985), and hyperbolic measure (F€ are et al. 2019).
In the sense of Pareto-Koopmans, best practices are revisited through Pareto efficiency to ensure all inefficiencies are eliminated.Denote Eff T as the set of Pareto efficient production mixes with technology T, i.e.
best practices under Pareto optimality x BP , y BP À Á satisfy the following condition: with the interpretation that best practices belong to the set of Pareto efficient production mixes with technology T: This aspect has been widely examined in benchmarking, see for example, Aparicio et al. (2007) and Korhonen et al. (2018).
In summary, the ultimate goal of benchmarking is to improve performance, thereby inefficiencies are undesirable and should be eliminated.However, as noted by Bogetoft and Hougaard (2003), inefficiency may be 'rational' because there are gains from inefficiency and costs of improve efficiency.This argument still holds in benchmarking, but to the best of our knowledge, no major methodological innovation has developed in this regard.This paper fills this gap by introducing the cost of effort in benchmarking where inefficiency can be desirable in performance improvement is conceptualized.

Technology: Basic definitions
We consider a principal who delegates the production of h outputs consuming m inputs to n homogenous and comparable agents.Let y j 2 R h þ and x j 2 R m þ be the produced outputs and consumed inputs of agent j (j ¼ 1, 2, :::, n), respectively.The technology T can be defined as Accordingly, the efficient subset of T can be denoted as Eff T identifies best practices in benchmarking because any increase in outputs and any reduction in inputs are impossible with technology T: The production mixes in Eff T are Pareto efficient because inefficiencies in inputs and outputs are fully eliminated.Without loss of generality, we employ the notion of Pareto efficiency to make inferences about the cost of effort in benchmarking hereinafter.
A successful implementation of benchmarking indicates the agents improve performance to Pareto efficiency, i.e. x j , y j ð Þ 2 Eff T, j ¼ 1, 2, :::, n: However, numerous empirical evidences confirm that benchmarking failure are widely common.For example, Bloom et al. (2010) find developing countries have lower productivity than developed ones.Note that while firms in developing countries may have knowledge of best practices from those in developed countries, the implementation of best practices may not be realized.The reasons for such benchmarking failures are multifaceted.This paper seeks to introduce the concept of the cost of effort in the production process as a potential explanation for benchmarking failure.

Characterization of the cost of effort
The characterization of benchmarking failure is reflected by the fact that the agent has produced an inefficient production mix x, y ð Þ 2 T Eff T: We think of this as a result of trade-off between the cost of effort put into the production process and transfer from the principal.First, there is a positive correlation between the amount of effort and the observed production mix.Intuitively, an agent who exerts effort to accomplish a production task is likely to generate higher outputs or consume fewer inputs compared to an agent who is not fully committed to the task.This suggests the effort e can be described as a function of input-output vector x, y ð Þ with a specific technology T, i.e.
where e x, y ð Þ jT À Á is (weakly) increasing with y and (weakly) decreasing with x, i.e.
@e @y r !0, 8r ¼ 1, 2, :::, h, and @e @x i 0, i ¼ 1, 2, :::, m: The production mix x, y ð Þ also can be elucidated by the slacks with technology T: Suppose production mix x, ỹ ð Þ belongs to T and weakly dominates production mix Þ can be regarded as the underlying production mix of the agent if effort is not considered in benchmarking.The set of underlying production mix can be defined as The difference Àx, ỹ ð ÞÀ Àx, y ð Þ represents the overinvestment of inputs and the underproduction of outputs, or we refer it as the organizational slacks 1 , i.e.
With no further information, we cannot know which production mix x, ỹ ð Þ is selected as the underlying production mix of the agent, thereby we cannot exactly obtain which slack vector is used.However, we can induce that the slack vector belongs to the following slack possibility set S x, y ð Þ T À Á given observed production mix x, y ð Þ and technology T, which is defined as The graphical representations about Z x, y ð Þ T À Á and S x, y ð Þ T À Á are presented in Figure 1.For ease of exposition, Figure 1 considers three agents (i.e.A, B, and C) who consume one input to produce one output, where agents B and C are efficient.To illustrate, we take agent A as an example.The technology T is enveloped by the red curve BC and the x-axis.For agent A, the set of underlying production mix Z x, y ð Þ T À Á and the slack possibility set S x, y ð Þ T À Á are described by the marked region AA 0 BCA 00 : The slight difference is the latter is depicted by the blue axes AA 0 and AA 00 in the s x , s y ð Þ space rather than in the x, y ð Þ space.
The slack possibility set S x, y ð Þ T À Á is contingent on the observed production mix x, y ð Þ and technology T, it also provides an indirect measure to the effort e x, y ð Þ T À Á : Intuitively, it is suggested that the amount of effort put into the production process will not decrease at least if the agent reduces organizational slacks.Therefore, we can reformulate the effort e as a function of organizational slacks s as with the interpretation that e s ð Þ is (weakly) decreasing with s, i.e. @e @s l 0, 8l ¼ 1, 2, :::, m þ h: In particular, we assume s is strictly positive in describing effort, i.e. s > 0: This assumption reflects each slack plays a unique role in describing effort that cannot be negligible 2 .This assumption also ensures the existence of analytic solution to the programs used to make inferences about the cost of effort hereinafter.
There is no free lunch when the agent exerts effort to finish the production task assigned by the principal.The cost of agent's effort v is a function of effort e, i.e.
the agent is strictly effort averse.By analogy, an agent is said to be effort neutral if v 0 e ð Þ ¼ L where L > 0 is a constant.The function of the cost of effort is depicted in Figure 2.
The formulations of e s ð Þ and v e ð Þ above are rather conceptual, we will introduce additional assumptions and elucidate how to use these assumptions to make inferences about e s ð Þ and v e ð Þ hereinafter.This is also the main task of this paper, from which benchmarking failure can be rationalized.

Utility maximization problem
In addition to the cost of effort v e ð Þ, the agent is reimbursed with a monetary transfer t from the principal.Consequently, the utility function of the agent u t, e ð Þ can be formulated as We assume the utility function u t, e ð Þ is separable between money t and effort e: A similar separation of money and effort is widely introduced in incentive theory (Lambert 1983;Laffont 1995).We also assume that u t, e ð Þ is strictly increasing in t but decreasing in e, i.e.
For analytical purpose, it is useful to assume that the transfer t consists of two components, a fixed compensation component b and a pay-for-performance compo- with the interpretation that b can be elucidated as the wage paid to the agent in the labor market (Leontief 1946;Fehr and Falk 1999), and I x, y ð Þ as a complement to b is a bonus-pay relative to performance (Anderson et al. 2010;Lemieux et al. 2012).This structure of the transfer is commonly used in universities, hospitals and for-profit organizations.It also facilitates the analysis below since it allows us to investigate the trade-off between the cost of effort v e ð Þ and the incentives from the principal I x, y ð Þ : For ease of exposition, we further assume the bonus-pay is proportional to the profit of the production mix (Lewellen and Huntsman 1970), i.e.
with the interpretation that a 2 ð0, 1 denotes the scale factor and (linear P m l¼1 p l x l that aggregates m inputs and h outputs through price coefficients p l > 0 ð Þ: The profit function reveals the underlying preferences of the principal regarding the production mix for pay-for-performance incentives, thereby influencing the selection of input-output combinations in benchmarking 3 .
The agent controls the bonus-pay aRðx, yÞ and the cost of individual effort v e ð Þ by selecting production mix x, y ð Þ and effort e: In line with the rational choice theory, it is natural to further assume that the agent exploits possibilities in a rational manner.In other words, the agent selects production mix x, y ð Þ and effort e to maximize utility.This can be expressed as the following optimization program: With regard to ðx, yÞ and e, we have illustrated that they can be represented by the underlying production x, ỹ ð Þ and the slack vector s ¼ s x , s y ð Þ through Formulas ( 12) and ( 14).We therefore can rewrite the above program into This program depicts that the agent picks an underlying production x, ỹ ð Þ and a slack vector s x , s y ð Þ compatible with the actual production mix As a consequence, the program can be reorganized as To solve the program above, it is particularly useful to introduce the notion of local allocative efficient input-output combination.The set of local allocative efficient input-output combinations G is defined as With regard to the local allocative efficient input-output combination x AE , y AE À Á , the following proposition records the agent's choice on the underlying production mix x, ỹ ð Þ to Program (21).
Proposition 1.In an optimal solution to the utility maximization problem, the agent will select a local allocative efficient input-output combination as the underlying production mix, i.e. x, ỹ The proof for this proposition and all subsequent results are provided in the Appendix.Proposition 1 shows whatever slack possibility is selected, the agent intends to use input to produce outputs in the most profitable way using x, ỹ ð Þ ¼ x AE , y AE À Á 2 G to maximize utility.To illustrate this, let us consider the same example as shown in Figure 1.
In Figure 3, the line R x, y ð Þ ¼ p 2 y À p 1 x passing through agent A represents the actual profit given price vector p 1 , p 2 ð Þ : The blue line parallel to R x, y ð Þ and tangent to T reflects the maximal profit of agent A when agent A improves performance to the best practice frontier.The optimal choice of underlying production mix of agent A is then captured by the intersection of the blue line and the set of underlying production mix characterized by the marked region AA 0 BCA 00 , i.e. the red point in Figure 3 where x, ỹ ð Þ ¼ x AE , y AE À Á holds.The optimal choice of underlying production mix also relates to best practices in benchmarking.We record this relation a proposition.
Proposition 2. The optimal choice of underlying production mix x, ỹ ð Þ belongs to the efficient subset of T, i.e. x, ỹ ð Þ 2 Eff T: Proposition 2 reveals that the optimal choice of underlying production also represents best practices in benchmarking.It shows the agent will select a Pareto-efficient input-output combination as the benchmark when the agent maximizes utility in benchmarking.This result can be elucidated by the red point in Figure 3. Proposition 2 links the agent's behavior of utility maximization to the benchmark selection problem.The intuition and interpretation of this property is the price information from the profit function forces the agent to select the most profitable inputoutput combination as the benchmark because the agent expects to maximize the bonus-pay relative to performance aRðx þ s x , ỹ À s y Þ: The best practices, however, may not be realized by the agent because the utility function is also determined by the slack vector selected by the agent.Although the underlying preference of slack vector s does not affect the underlying production mix x, ỹ ð Þ, it affects the actual production mix x, y The following proposition records the choice of actual production.
Proposition 3. In an optimal solution to the utility maximization problem, the agent will select the actual production mix as x, y Proposition 3 implies that the agent will locate himself in a position dominated by at least one optimal underlying production mix in G: The final location is contingent on the value of slack aggregation function ÀaR Às 21).If a rational agent selects a non-zero slack vector s x , s y ð Þ 2 R mþh þ , then the so-called benchmarking failure occurs.
Wrapping up, a rational agent make decision on input-output combination in the following steps.First, given the profit function R x, y ð Þ and technology T, the agent identifies the set of profit maximization points G according to the prospective production mix x, y ð Þ : In particular, the prospective production mix x, y ð Þ belongs to Next, the agent faces the trade-off between the potential slackbased bonus-pay aR Às x , s y ð Þ and the cost of effort v e s x , s y ð Þ ð Þ which allows for slack possibilities.Finally, the rational agent selects the input-output combination x, y ð Þ that maximizes utility.
We now return to the utility maximization problem.With the help of Proposition 1, Program (21) can be rewritten as the following slack selection problem.
The agent's utility maximization problem is now contingent on the selected slack vector merely.For ease of exposition, define The agent's observed production mix x, y ð Þ , together with technology T, enable us to induce agent's response to the slack selection problem.Recall that we assume that the agent is rational, that is, whether applying best practices in benchmarking is a result of the agent's behavior of utility maximization, which is reflected by the optimal solution of the slack selection problem above.Therefore, the agent's observed production mix can be used to make at least partial inference about the trade-off between the transfer from the principal W À aR Às x , s y ð Þ and the cost of effort of the agent v e s x , s y ð Þ ð Þ : This trade-off is illustrated by two slack aggregation functions aR Às x , s y ð Þ and v e s x , s y ð Þ ð Þ in Program (24).

Slack aggregation functions
To further examine such trade-off, we shall impose several assumptions on functions v e ð Þ and e s x , s y ð Þ , from which a set of parametric slack aggregation functions is characterized.Note that the slack aggregation function aR Às x , s y ð Þ is known to both the principal and the agent, we only need to focus on the specifications of v e ð Þ and e s x , s y ð Þ in the analysis hereinafter.For ease of exposition, we particularly assume functions v e ð Þ and e s x , s y ð Þ are differentiable.We then consider four restricted classes of slack aggregation functions (effort neutral vs strictly effort averse, strong monotonic in slack vs weak monotonic in slack) according to the characteristics of v e ð Þ and e s x , s y ð Þ : Effort neutral implies that the cost of effort is a linear, increasing function of the individual effort level, meaning that the cost per unit of effort remains constant regardless of the amount of effort invested in the production process.For example, if the cost of effort when the agent exerts unity effort is V, the cost of effort doubles (i.e.2V) when the agent exerts two units of effort.On the other hand, strong monotonic in slack shows that any reduction in the slack vector results in an increase in the level of effort.This characteristic suggests that an increase in the actual output combination or a decrease in the actual input combination leads to an increase in the level of effort, which is consistent with common sense.
We now are in a position to make further assumptions on the specifications of v e ð Þ and e s x , s y ð Þ to make inferences about the two functions with the observed outputs.For ease of exposition, we assume It is easy to verify the two functions satisfy the characteristics in Class 1. v e ð Þ ¼ e implies the cost of effort is equal to the amount of effort.P mþh l¼1 b l ¼ 1 normalizes the attached weights on slack vector s x , s y ð Þ : In game theory, Q mþh l¼1 s b l l denotes the asymmetric (Nash) bargaining value where b l > 0, l ¼ 1, 2, :::, m þ h represents the relative bargaining power of different slacks (Lozano et al. 2019) l suggests that the effort is expressed as the gap between an unknown maximal effort level g and the result of negotiation among all slacks.g is denoted as the maximal effort since e s x , s y ð Þ ¼ g when the slack vector equals zero, i.e. the observed production mix is Pareto efficient with technology T: In addition, an underlying result of e s x , s y l because the effort e s x , s y ð Þ is non-negative, i.e. e s x , s y ð Þ !0: With the help of specifications of v e ð Þ and e s x , s y ð Þ , the slack aggregation function v e s x , s y ð Þ ð Þ can be derived as Weak monotonic in slack implies that the level of effort may not decrease even if there is a reduction in the slack vector.This allows for some slacks to not contribute to the level of effort, thus relaxing the requirement of strong monotonic in slack.An example of this could be an agent engaging in an 'inside job' if the principal lacks effective monitoring and control mechanisms over the production process.In such a scenario, the agent could reduce slack without exerting additional effort, simply by changing their behavior.These slacks would then reflect the risk faced by the agent, rather than the level of effort required to complete the task.
Combining effort neutral and weak monotonic in slack, we thus assume It is easy to verify the two functions satisfy the characteristics in Class 2. min b 1 s 1 , b 2 s 2 , :::, b mþh s mþh È É represents the smallest weighted slack coordinate, which is in line with the weighted fairness model (Rawls 2017).e s x , s y ð Þ ¼ g À min b 1 s 1 , f b 2 s 2 , :::, b mþh s mþh g states that the effort is expressed as the gap between the unknown maximal effort level g and the minimum of weighted slack b l s l , l ¼ 1, 2, :::, m þ h: Note that min b 1 s 1 , b 2 s 2 , :::, b mþh s mþh È É indicates a conservative estimation on the individual effort since the minimum of weighted slack b l s l reflects the maximum effort in all output dimensions.To illustrate, suppose e s l ð Þ ¼ g À b l s l represents the partial effort of maintaining slack s l , and the total effort e s x , s y ð Þ is an aggregation of all partial effort e s l ð Þ : with the interpretation that the maximal partial effort is selected as the total effort.By analogy, an underlying result of e s x , Strictly effort averse shows the cost of effort v e ð Þ is convex.It suggests that the cost of one additional unit of effort is higher as the agent puts more effort into the production process.For example, if the cost of effort when the agent exerts unity effort is V, the cost of effort strictly exceeds 2V when the agent exerts two units of effort.Strictly effort averse is widely employed in incentive literature because it reflects the agent's tendency to resist expending effort for the benefit of the principal (Bogetoft 1994).
For ease of exposition and to facilitate comparison, we thus assume It is clear that the above functions satisfy the characteristics in Class 3. The major difference to the two functions in Class 1 is the cost of effort is a quadratic function of effort, i.e. v e ð Þ ¼ e 2 : This quadratic function implies that the cost of effort is the square of effort.The choice of representing effort aversion using vðeÞ ¼ e 2 in this paper is grounded in two primary considerations.Firstly, it simplifies the estimation of key parameters such as b l , g lb and g ub as shown in Proposition 4. Secondly, vðeÞ ¼ e 2 aligns with the classical setting for the disutility of effort in well-established incentive literature.This classical setting has been previously utilized in works like Bogetoft (1995), lending further credence to its suitability in the context of this research.As a consequence, the slack aggregation function v e s x , s y ð Þ ð Þ can be derived as By combining strictly effort averse and weak monotonic in slack, we assume The functions v e ð Þ and e s x , s y ð Þ above are parts of functions used in Classes 2 and 3.With the help of such function combination, the slack aggregation function (29)

Inference about the cost of effort
By operationalizing the four restricted classes of slack aggregation functions above, we shall make inferences about v e s x , s y ð Þ ð Þ through determining g and b l , l ¼ 1, 2, :::, m þ h based on the observed outputs and individual rationality of the agent hereinafter.For ease of exposition, we shall impose additional assumptions on the utility function of the agent.An intuitive and stylized assumption is that the optimal utility of the agent in the utility maximization problem ( 23) should be at least no less than the reservation utility Qð! 0Þ: This assumption is consistent with the individual rationality constraint in incentive theory, which can be expressed as The next proposition records (i) the optimal estimates of relative importance or bargaining power of the factors b l ; (ii) the lower and upper bounds of the underlying maximal effort level g lb and g ub ; and (iii) the inference about the cost of effort in the four restricted classes of slack aggregation functions.
Proposition 4. Suppose the agent is rational, i.e. the observed slack vector s ¼ s x , s y ð Þ is the best response to the utility maximization problem (23).Let Qð! 0Þ be the reservation utility of the agent.
l , the optimal estimates of bargaining power of the factors are The lower and upper bounds of the underlying maximal effort level g lb and g ub are The inference about the cost of effort is , g 2 g lb , g ub Â Ã : , the optimal estimates of relative importance of the factors are b l ¼ 1 The lower and upper bounds of the underlying maximal effort level g lb and g ub are The inference about the individual effort is , the optimal estimates of bargaining power of the factors are The lower and upper bounds of the underlying maximal effort level g lb and g ub are The inference about the individual effort is Á 2 , the optimal estimates of relative importance of the factors are b l ¼ 1 The lower and upper bounds of the underlying maximal effort level g lb and g ub are The inference about the individual effort is , g 2 g lb , g ub Â Ã : An explicit finding from Proposition 4 is the characteristic of the cost of effort v e ð Þ (i.e.effort neutral vs strictly effort averse) has no impact on the optimal estimates of relative importance or bargaining power of the factors b l , l ¼ 1, 2, :::, m þ h in the setting above.This finding is supported by comparing parts (i) and (iii) or parts (ii) and (iv) of the proposition.In particular, the optimal estimates of relative importance or bargaining power of the factors b l are uniquely determined by the observed slack vector s: Proposition 4 also suggests that the characteristic of the cost of effort v e ð Þ (i.e.effort neutral vs strictly effort averse) affects the upper bound of the underlying maximal effort level g, although it has no impact on the optimal estimates of relative importance or bargaining power of the factors b l , l ¼ 1, 2, :::, m þ h: This can be elucidated by parts (i) and (iii) or parts (ii) and (iv) of this proposition.Specifically, the set of g under strictly effort averse is a subset of that under effort neutral due to W À a P mþh l¼1 p l s l À Q > 0: However, it is worth highlighting that g cannot be uniquely determined by the individual rationality constraint because the optimal utility of the agent (denoted by Q Ã ) is unknown to the principal.The principal only knows the optimal utility of the agent must be no less than the reservation utility, i.e.Q Ã !Q: This forces the principal to make inference about g with uncertainty, as shown in Proposition 4.
If the optimal utility of the agent (denoted by Q Ã ) is known to the principal, we then can make unique inference about g: In this case, the optimal estimates of the underlying maximal effort level g are where g ub is the upper bound of g derived from Proposition 4. To avoid repetition, we do not record this finding as a new proposition here.Proposition 4 demonstrates how slack aggregation functions can be used to make inference about the cost of effort.The observed slack vector s contributes to the estimates of the cost of effort using specific prior functions.This elucidates why the agent may refuse to apply best practices in benchmarking, i.e. the benchmarking failure takes place.Specifically, the agent maximizes utility first and then decides whether applying best practices in benchmarking subsequently.The observed slack vector s (i.e.not applying best practices) is the best response to the trade-off between transfer from the principal and the cost of effort.
Proposition 4 also enriches the rational inefficiency hypothesis suggested by Bogetoft and Hougaard (2003).The observed slack vector is regarded as the source of inefficiency in production economics.Proposition 4 suggests that such inefficiency may be an underlying compensation for the cost of effort in benchmarking.The agent can select an inefficient input-output combination to maximize utility when the agent faces the trade-off between the transfer from the principal and the cost of effort.This result supports one possible rationale for inefficiency stated in rational inefficiency hypothesis that inefficiency may be a part of fringe reimbursement to stake holders, i.e. the agent participating in benchmarking.
Last but not least, Proposition 4 reflects the heterogeneity of individual effort.For agents with different observed slacks, the inferences about the cost of effort are distinct because the optimal estimates of relative importance or bargaining power of the factors b l , l ¼ 1, 2, :::, m þ h are determined by the observed slack vector.With the help of any restricted classes of slack aggregation functions, the inferences about the cost of effort exhibit similar structures (for example, v e s x , ) but derive heterogenous results (i.e.different estimates of b l ).This heterogeneity of individual effort is consistent with the fact that 'no two leaves are alike'.It advances traditional benchmarking studies that only focus on the homogeneity of production combinations of the agents to ensure performance evaluation.

Cost of effort in benchmarking: applications
In the previous section, we have explored individual effort in benchmarking and made inference about the cost of effort.It has been suggested that the agent may refuse to apply best practices because there is an underlying cost of effort put into the production process.Rather, the observed slacks from not applying best practices can be the optimal choice for the agent who maximizes utility in benchmarking.We model the utility function of the agent as the gap between the transfer from the principal and the cost of effort.We also have demonstrated how to use observed production together with the technology to determine the trade-off between the transfer and the cost of effort.This allows us to make inference about the cost of effort in benchmarking through some slack aggregation functions, which is the primary purpose of this paper.
The next natural step is to consider some applications that use the inference about the cost of effort above.The inference is valuable for the principal as it provides partial estimate of the non-observable and non-verifiable individual effort at least.In this section, we sketch its applications in activity planning, incentive provision and employee layoff.

Activity planning
Activity planning is a crucial task for firms and organizations that produce goods or services for the market (Bogetoft 2000).The primary objective of activity planning is to forecast or determine the optimal quantity of products and services to be produced in a rational way.In situations where there is an information asymmetry and the agent possesses superior knowledge about the production costs, the principal also employs activity planning to adjust the production mix, minimizing the information rents earned by the agent (Antle and Bogetoft 2019).
In this context, if an explicit estimate of the cost of effort is available, the principal can rather make precise inference about the utility of the agent because the principal takes charge of the transfer paid to the agent.As a consequence, the principal can make precise predictions about the behavior of the agent when the factors controlled by the principal are altered.In particular, the principal controls the fixed compensation b, the scale factor of the profit from the production a, and the prices of inputs and outputs p l , l ¼ 1, 2, :::, m þ h in the utility function of the agent 4 .Using the inference about the cost of effort v e s ð Þ ð Þ, we now can predict the rational response of the agent if the fixed compensation is altered.The rational response of the agent is the optimal solution to the following utility maximization problem.
The first constraint follows Proposition 1.The second constraint implies that the actual input-output combination is dominated by a local allocative efficient input-output combination x AE , y AE À Á and belongs to technology T: The last constraint is the individual rationality constraint.Denote the optimal solution of the program above as s Ã x , s Ã y À Á , a rational agent will select input-output combination x AE þ s Ã x , y AE À s Ã y as the production mix.This result creates explicit connections between the rational response of the agent x AE þ s Ã x , y AE À s Ã y and the altered fixed compensation b, which provides valuable information for activity planning in the future.
Second, let us consider the case that the principal alters the scale factor of the revenue from the produced outputs a to a: In particular, a > a can be a result of the fact that the principal increases the bonus-pay to induce the agent to work hard by strong incentives.On the other hand, a typical example of a < a is the principal lowers the bonuspay due to difficulties in operation (i.e.shrinking demand and financial problems).With the help of the inference about the cost of effort v e s ð Þ ð Þ, we now can predict the rational response of the agent by solving the following utility maximization problem.
Let the optimal solution of the above program be s Ã x , s Ã y À Á , the rational response of the agent will be The rational response contributes to activity planning in the case that the principal intends to lower or increase the bonus-pay to the agent.Third, suppose the price are altered from p l , l ¼ 1, 2, :::, m þ h to p l , l ¼ 1, 2, :::, m þ h: The new prices can be consequences of the invisible hand of the free market (Smith 1937;Bishop 1995).In this case, the new profit function is R x, y ð Þ ¼ P mþh l¼mþ1 p l y lÀm À P m l¼1 p l x l : Combining the inference about the cost of effort v e s ð Þ ð Þ, the rational response of the agent can be predicted by the following utility maximization problem.
The local allocative efficient output combination x AE , y AE À Á is now determined by the new revenue function R x, y ð Þ : By analogy, denote the optimal solution of the above program as s Ã x , s Ã y À Á , a rational agent will select input-output combination as the production mix.The prediction of rational response of the agent facilitates activity planning in a competitive market.Finally, we stress that the rational response of the agent can still be predicted using the inference about the cost of effort v e s ð Þ ð Þ when the above factors or any combinations of them are altered simultaneously.The corresponding utility maximization problem is contingent on the context of activity planning.Wrapping up, the inference about the cost of effort offers an opportunity to estimate the maximal utility of the agent, thereby predicting the number of products produced in the future.

Incentive provision
Incentive provision is a critical task for firms and organizations that work with external agents or outsource various production tasks to downstream agents in a supply chain (Wang et al. 2021).The primary goal of providing incentives is to motivate or incentivize the agent to perform the delegated tasks as expected by the principal.Incentive provision has been a significant focus for some time, and designing optimal incentive provision with information asymmetry has been widely explored in operations management (see, for example, € Ozer and Raz 2011;Ryan et al. 2022).
In our setting above, when the principal intends to induce the agent to act as specific production mix, the principal should take the incentive effects into account in terms of both the transfer paid to the agent and the cost of effort.The specific production mix can represent the best practices in benchmarking (i.e.x, y ð Þ 2 Eff T), or be forced by regulatory documents (i.e.x, y ð Þ 2 T).Except for the individual rationality constraint, the incentive compatibility constraint should be introduced to design the optimal incentive provision.
To illustrate the optimal incentive provision, let us consider the problem that the principal contracts upon the fixed compensation b to minimize the cost (i.e.transfer) of inducing the agent to produce output combination y by consuming input combination x: Without loss of generality, we assume that the agent can use any input-output vector from a set of acceptable input-output combination A: The contract design problem of the principal can be modelled as The objective of the program above is to minimize the transfer paid to the agent t ¼ b þ aRðx, yÞ: The first two constraints are consistent with Proposition 1, i.e. the optimal choice of the underlying production is a local allocative efficient one.The next two constraints are the individual rationality constraints.Constraint (35-3) ensures that the utility of the agent from selecting input-output combination in A is no less than the reservation utility.Constraint (35-4) indicates the fixed compensation b is no less than the historical fixed compensation b: It suggests that the observation that agents in the labor market tend to prefer tasks that bring higher income than their previous tasks aligning with the notion of rational behavior.This individual rationality constraint places limitations on the possible selection of input-output vectors from the set A: The last constraint is the incentive compatibility constraint.It illustrates that the rational response of the agent is to produce y with x that minimizes the transfer.
On the other hand, if the principal contracts upon the fixed compensation b to maximize the revenue of inducing the agent to produce output combination y by consuming input combination x, the contract design problem reads max The objective of the above program is to maximize the revenue of the principal R x, y ð Þ À t: It says that the principal will select an input-output combination y from set A to maximize the proportional revenue 1 À a ð ÞR x, y ð Þ and minimize the fixed compensation b paid to the agent simultaneously.This differs from the former program that the principal will select an input-output combination x, y ð Þ from set A to minimize the proportional revenue aRðx, yÞ:

Employee layoff
The issue of layoffs is an economic and social reality for both managers and employees (Bennett et al. 1995;Gu et al. 2020).In recent years, the numbers of layoffs are staggering because of the COVID-19 recession in the global supply chain or other factors like regulatory crackdown (Chodorow-Reich and Coglianese 2021).For example, a survey by TechNode reports nearly 73,000 workers were let go between mid-April and July 2022 alone in Beijing, China 5 .A report published in March by Reuters also shows that Alibaba and Tencent are preparing to cut tens of thousands of jobs in 2022 to cope with China's sweeping regulatory crackdown 6 .In these contexts, how to make the layoff decision is of great importance for firms and organizations (Flanagan and O'Shaughnessy 2005).
The inference about the cost of effort contributes to the layoff decision when the principal has to dismiss some employees.If the principal can predict or even partial estimate the minimal costs (i.e.transfers) of a group of agents, the principal can select the cheapest agents as layoff survivors while the rest are dismissed.To illustrate the employee layoff problem, let us first consider two agents, namely agents 1 and 2 where one agent will be the layoff victim.Suppose the inferences about their costs of effort are denoted as v l e s ð Þ ð Þ, l ¼ 1, 2: We further assume that the principal appoints production mix as the future production of the layoff survivor.The layoff survivor l Ã is thus determined by where x AE l , y AE l À Á , b l and Q l denote the optimal choice of the underlying production, the fixed compensation and the reservation utility of agent l, respectively.The layoff survivor is the agent who requires the minimal transfer to finish the production task x, y ð Þ : That is, the cheapest agent will be selected as the layoff survivor in the employee layoff problem.

Conclusions
The aim of this paper is to examine the cost of effort in benchmarking and explore its applications in different contexts.Despite the extensive research on individual effort in incentives, its role in benchmarking has received little attention.Benchmarking aims to improve performance by identifying and applying best practices.The individual effort is a key factor to applying best practices because it determines the utility of the agent participating in benchmarking directly.However, to our knowledge, the cost of effort in benchmarking has seemingly never been explored explicitly.This paper attempts to dig deeper into this issue in the context of benchmarking.
We first formally define the technology to identify best practices in benchmarking.Thereafter, we characterize the effort of the agent with the help of the observed production mix and the technology.To describe the strategic behavior of the agent, the utility function is described as the gap between the transfer from the principal and the cost of effort.In particular, the transfer consists of a fixed compensation component and a pay-for-performance component.In line with the rational choice theory, the observed production mix can be regarded as the optimal response to the utility maximization problem of the agent.By introducing four restricted classes of slack aggregation functions to operationalize the cost of effort, we can make at least partial inference about the cost of effort with the observed production mix and the technology.We show that the inference about the cost of effort is consistent with the heterogeneity of individual effort.More importantly, the inference about the cost of effort also provides a rational and appropriate explanation to the 'benchmarking failure', i.e. why the agents may refuse to apply best practices in benchmarking.
With the help of the inference about the cost of effort, we examine its applications in three contexts.The first application is activity planning.The principal can make precise predictions about the behavior of the agent when some factors (i.e. the prices or the fixed compensation) are altered because the principal can use the inference about the cost of effort to estimate the utility function of the agent.The second application is incentive provision.The inference about the cost of effort enables the principal to design a contract to minimize the transfer inducing the agent to finish a specific production task.The third application is employee layoff.The inference about the cost of effort provides valuable information for layoff decision making.The principal can select the cheapest agents as the layoff survivors if the principal knows or even partial estimates the minimal costs of a group of agents.
The analysis in this paper can be extends in several ways.First, the inference about the cost of effort needs further deliberation.The inference in this paper is only available for inefficient agents as we use slack aggregation functions to operationalize the cost of effort.Consequently, it would be important to explore how to make inference about the cost of effort for the efficient agents in the future.The individual rationality constraints of the efficient agents can be used to make partial inferences.However, additional assumptions should be imposed on the cost function of individual effort to make precise inferences.
Second, it is worthwhile to extend and refine the slack aggregation functions adopted in this paper.In the multiple outputs case, we normalize the relative importance or bargaining power of the factors to estimate the cost of effort.The normalization facilitates the inference about the cost of effort as shown in Propositions 4. Nevertheless, it has non-negligible impacts on applications in activity planning, incentive provision and employee layoff.In practice, the relative importance or bargaining power of the factors may be complicate that cannot be normalized.In this case, the normalization constraint results in sub-optimal decision on activity planning, incentive provision and employee layoff.Therefore, it will be fruitful to relax the normalization constraints in slack aggregation functions depending on the contexts.
Third, it would be interesting to introduce more applications with the inference about the cost of effort.For example, the principal can use this information to reallocate resources among all agents from the point of view of centralized management.Analogous to the employee layoff problem, the inference about the cost of effort may contribute to the employee recruitment problem if the relevant information of candidates is available.Another productive application is to examine whether the inference about the cost of effort can used to copy with the well-known moral hazard problem.

Notes
1. Note that the organization slacks here differ from the slacks representing inefficiencies in production economics.The latter stresses the distance to the efficient production frontier, whereas the former represents agent's choice on the production mix.

Figure 1 .
Figure 1.The set of underlying production mix and the slack possibility set.

Figure 2 .
Figure 2. The function of the cost of effort.

Figure 3 .
Figure 3. Optimal choice of the underlying production.
l À Q: First, suppose the principal alters the fixed compensation b to b where either b > b or b < b holds.Rather, an illustrative example of b > b is the government raises the minimum wage to eliminate property or protect human rights by legislation.On the other hand, b < b can be a consequence of pay cuts due to economic crisis (i.e.2008-2009 global financial crisis), global epidemic (i.e.COVID-19), and war (i.e.Russia-Ukraine War).