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Publicly Available Published by De Gruyter September 17, 2015

The Dynamics of Incentives, Productivity, and Operational Risk

Paul Anglin and Yanmin Gao


This paper develops a dynamic principal-agent model and applies it to understand changes in labor productivity and operational risk. Our analysis demonstrates the importance of matching the terms of the job contract to the technology. Such issues would be especially important in service industries and in the knowledge-based economy where discretionary effort tends to play a greater role. We show that the production technology needs to be characterized by at least two parameters: one parameter which measures output independent of the worker’s effort and a second parameter which measures the effect of the effort. We solve for the Renegotiation-Proof Nash Equilibrium. We show that there can be a tension between increasing expected productivity and controlling costs per worker. Our analysis also adds to the growing interest in “operational risk”, which is associated with human actions. The closed form solutions provided by our model provide a natural way to consider the impact and possibility of this type of risk. Our analysis demonstrates why the effect of a negative event should be considered relative to a concept of normal which is based on an equilibrium, that uncertainty in the external environment enables (but does not cause) operational risk events and that both the equilibrium and the effects vary with the production technology.

JEL Descriptors: D24; D82; M41

1 Introduction

This paper studies the effects of agency contracts on measured productivity and operational risk in dynamic settings. Conventional wisdom suggests that productivity growth and corporate profits are tied to the use of new technologies, and many examples can be used to confirm this wisdom. It is also understood that the effort of workers plays a role, but this role is studied less often and less formally. This vague understanding is unfortunate because measured productivity can vary for many reasons and a precise understanding often requires the use of ideas that are familiar in the principal-agent literature. Despite the common impression that productivity growth is purely a technological issue, we show that the effect of a technological change depends on how well the technology is matched with an appropriate job contract for workers. Thus, the effects of technology cannot be isolated from the effects of other aspects of the contracting environment. Further, this measurement problem implies that shareholders would have trouble allocating blame within the firm if the realized productivity is lower than forecast.

This measurement problem also affects an issue which is attracting increasing attention from international policy forums and in disclosures to shareholders: operational risk. It is defined as “the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events. This definition includes legal risk, but excludes strategic and reputational risk” (Basel Committee on Banking Supervision 2011, fn. 5). As we discuss more fully in Section 5, small events are common and easy to associate with human foibles, such as imperfect memory and trivial carelessness. Large operational risk events are rare and significant in aggregate but their definition and the process of measurement are debated. More careful analysis sometimes measures it statistically, as though it were a mechanical process, despite the definition given above. We demonstrate that any explanation concerning both productivity growth and operational risk requires a principal-agent perspective. We argue that more familiar sources of risk, such as market risk, do not cause operational risk but enable it: if there were no external source of risk then the workers could not hide any mistakes or misbehavior. Our model offers closed form solutions which show how these issues are intimately linked, both in terms of precisely defining any concern and identifying the full implications of any resolution.

Some situations may illustrate the nature of the problem. For example, consider the banking industry and the introduction of computerized spreadsheets programs on a task such as projecting revenue and costs into the future. They were faster but, since they did not do anything that could not be done by hand, otherwise unremarkable. Even so, they had profound implications. The most important may be a simple operational issue: they reduced the number of mechanical mistakes in the arithmetic, such as adding instead of subtracting or shifting a decimal place in a computation or reversing the order of digits in a two-digit number. With increased confidence in the accuracy, more and more complicated What If scenarios could be considered if there was a desire to do so. [1]

Or, consider a sales person. A good sales person tries to learn about their client in order to establish a relationship and, before signing a long term contract with a client, in order to determine whether they are a good client or not. Improvements in information technology make it easier to gather information about somebody more quickly and, in theory, that should help a sales person. In practice, the amount of extra information depends on the effort expended by the sales person: rather than increasing sales directly, this technology is a “force multiplier”. The recent Credit Crisis offers many examples of mortgage loan officers acting as sales people issuing mortgages to clients who were obviously unsuitable after the Crisis, and that fact could have been discovered if they had done due diligence before the Crisis. The incentives facing the sales person may encourage them to build a relationship and to perform a reasonable due diligence, if the principal and agents were to agree to change the terms of the job contract. At the same time, mistakes can occur: e.g., people forget the birthdays of even their spouses or a search might confuse two people with the same name or information which is mostly reliable may not be updated. Even the managers of knowledge workers have limited understanding of the new technology.

These examples show how new technology increases potential output while also altering the effects of a worker’s choices and “mistakes”. In a sense, productivity and operational risk emphasize different dimensions of a change in the environment. Productivity analysis tends to focus on a mean while operational risk tends to focuses on variability (especially negative effects). This sense is too simple since, as we argue, a completely statistical summary of these tendencies would be unstable. An operational risk event is connected to human behavior, where a human may try to hide that connection by invoking uncertainty in the external environment selectively and depending on the technology of production.

We develop a two-period model in which a risk neutral principal hires a risk averse agent and solve for the unique Renegotiation-Proof Nash Equilibrium. This paper focuses on the implications of this equilibrium for changes in measured labor productivity and operational risk in a dynamic setting, when the effort of workers may be “hidden”. In a hidden action setting, one way to make workers work harder or more carefully is to enforce tougher job standards, but monitoring those standards is costly even if workers accept them as part of the job contract. We provide closed form solutions for the equilibrium (linear) incentive contract, if the principal chooses to use incentives, and for the level of effort, if the principal chooses to monitor the workers. A complicating feature is that, in a dynamic setting, workers can also shift their income and consumption over time to mitigate the personal cost of risk in any one period. Our solution anticipates this behavior, which adds to the richness of the comparative statics analysis.

We find that expected productivity can differ between periods, even if the technology does not change. We find that distinguishing between different types of technological change adds a critical, but rarely discussed, insight: some types could increase expected output, but the effect varies with the agent’s effort. If complemented by the equilibrium job contract then the effect of this type of technological change increases at an increasing rate.

Some broad facts from review papers may illustrate that productivity is not merely a matter of technology. Bloom and van Reenan (2011) note that the fraction of the labor force whose compensation is based on some kind of performance related pay system has been rising, despite the decline of the manufacturing sector, to include about 40–50% in the United States and now with comparable figures in the United Kingdom. Bloom and van Reenan link this fact to results repeated in Syverson (2011, 326) concerning an average manufacturing industry: “the plant at the 90th percentile of the productivity distribution makes almost twice as much output with the same measured inputs as the 10th percentile plant” (emphasis in the original; data based on four-digit SIC classification).

We are not the first researchers to recognize these issues. Jensen and Meckling (1976) note how incentives can affect output, and therefore productivity, by changing the behavior of an agent. Rajiv Banker and several co-authors offer more specific insights in a series of papers. Banker, Datar, and Mazur (1989) use a general model and focus on identifying a theoretically optimal measure of productivity which would be useful as part of a compensation system. They also test their hypothesis for a specific company and find that the company could have used a better alternative. Banker, Datar, and Rajan (1987) and Banker et al. (2002) use field tests to suggest the size of the effect of hidden effort. The first paper estimates a stochastic frontier model and concludes that introducing a “gain-sharing program” has no significant effect on the productivity of direct labor, which is likely to be constrained by machine operating conditions, but that the program increases the productivity of workers whose actions are less constrained, such as supervisors and other forms of indirect labor. This program is noteworthy because, in percentage terms, half of the gains in productivity were passed onto the workers in the form of extra compensation. The second paper argues that the effects of a gain-sharing program are equivalent to 3.7% of costs.

Seshadri and Subrahmanyam (2005, 1) expressed a concern about “a lack of connection between research in operations management and finance.” Incentives may represent a cost on a balance sheet but a well-designed incentive system aligns the interests of a principal and an agent. This broader perspective is important because, as Armitrage and Atkinson (1990) note, many workers oppose “productivity improvements” because they are seen as exhortations to “do more with less”. In other words, the essential question is: if measured productivity were higher in one year relative to the previous year, does management deserve the credit for using better technology or do workers deserve the credit (and appropriate income) or was it the result of luck? We recognize the characteristics of this situation using a richer model which allows for renegotiation of the job contract to show how complementary changes in the technology of production with changes in the agency contract or operating procedures can generate a sustainable long term financial payoff.

Our paper differs from these papers because, where they seek to characterize a given technology in a single period or productivity after a change in agency relationship, we seek to understand the many implications of a change in technology in a more complex environment dynamically. We show that if technology can be summarized by a single parameter then production cost per unit tends to be inversely related to labor productivity. By considering a parameter which measures the marginal productivity of effort in a way that complements the agency model, our model facilitates insights into how a change in technology affects complementary variables that are often associated with operations, such as total cost, unit cost and the variation in productivity over time.

We extend this argument to show that the management of productivity should not be separated from the management of operational risk. The general topic of risk management is becoming more important to managers and, through disclosures in financial statements and other forms of communication, to the relationship between managers and shareholders. One of the least understood forms of risk is called “operational risk”. In contrast to the better-known external sources of risk, such as market risk or credit risk, operational risk focuses on a failure of management to control or workers to implement the company’s business strategy. The fact that the source of a negative event is human behavior implies that this risk factor differs from the risk associated with events external to the firm; a worker is not a machine and does not always do what they are ordered to do. One of the reasons why operational risk is difficult to understand is that “normal” behavior can vary for reasons which statistical or technical measures may not consider. Our model of an equilibrium provides a precise context to discuss “normal” outcomes. In this equilibrium, an external source of risk does not cause operational risk but is a necessary condition which enables behavior-based sources of risk.

The remainder of the paper proceeds as follows. Section 2 develops a two-period agency model with renegotiation between a principal and an agent, for a given technology of production. The two-period model allows us to consider more complex contracts and behaviors, including consumption smoothing by an agent without relying on their employer. Sections 3 and 4 characterize the equilibrium and analyze the effects of contracting on productivity measurement. We note that incentives can vary with other aspects of the contracting environment and, thus, that expected productivity can vary even if the technology of production does not change. Section 5 discusses the issue of operational risk in the context of an overall business strategy. The last section contains our concluding remarks.

2 A Dynamic Agency Model with Renegotiation

We develop a two-period agency model where the principal maximizes expected profit by hiring a group of workers (also referred to as the “agent”) to produce output in each of the two periods. Both the principal and agent commit to stay for two periods. The principal offers the agent a compensation contract which is subject to renegotiation at the start of the second period. For mathematical tractability, we adopt a dynamic LEN model, i.e., Linear contract, Exponential utility, and Normally distributed performance measures. [2] The LEN model specifies the compensation contract as a linear function of the performance measures. Further, based on the concept of renegotiation-proofness of Fudenberg and Tirole (1990), we solve for a linear renegotiation-proof contract.

The number of hours worked by an agent in period t is observable by the principal. Since this contribution is easily measured and has been widely investigated, we normalize the number of hours worked: h1=h2=1. This normalization emphasizes the agent’s second contribution: the hidden intensity or quality of effort at in period t. [3] Effort represents a contribution to the level of output above and beyond the agent’s input of time. Thus, output is


where the technological parameters ϕ and ψ measure different aspects of productivity. ϕ measures the productivity per hour based on a minimal level of effort, i.e., at=0, and ψ represents the added productivity per hour due to increased effort. [4]ϵt, t=1,2, are Normally distributed noise terms with zero mean and covariance matrix


Based on realized demand for the product, and the desire of the firm to produce more or fewer units, an assumption of constant returns to scale in hours implies that the firm can maintain efficiency by asking workers to work more or fewer hours without needing to change any other terms of the contract.

The fact that productivity does not grow smoothly over time suggests that short term random, hidden shocks are important enough for managers to be unsure of the precise cause of a low productivity outcome in a particular year: [5] the situation displays asymmetric information. In such situations, an employer can be expected to offer a job contract which emphasizes incentives plus maybe monitoring and oversight.

If a firm sets a minimum standard of effort, and enforces this standard by monitoring workers, then agents who choose to work for this principal agree to exert at least Et units of effort in period t. If at<Et in a period then the firm has the right to impose a penalty according to the terms of the contract, where this penalty is sufficiently large that all workers exert the expected level of effort in equilibrium. This policy is consistent with the idea that the monitoring scheme is intended to get workers to work “hard enough” without imposing income risk or arbitrary penalties on the workers. The principal’s cost of monitoring is assumed to be proportional to the standard, i.e., that cost is mEt for some m>0. If a firm uses incentives, then an agent’s compensation in period t includes a fixed payment, whose allocation between the periods may vary, and a payment, vtQt which is proportional to output.

The following describes the timeline of the model which highlights the sequence of events and provides important assumptions concerning the agent’s preferences, and the principal’s preferences.

2.1 Timeline of the Model

Stage 1: At date zero, the principal designs an initial compensation contract ci(fi,v1i,v2i), minimum effort standards for the two periods (E1,E2) and offers the job to the agent. fi is the initial fixed payment paid at the end of the first period, and v1i,v2i are the initial incentive rates for the first and second period respectively.

Stage 2: At the start of the first period, the agent chooses her effort a1, which affects the first-period output Q1.

Stage 3: Since the principal cannot observe the agent’s productive effort, the principal uses the output Q1 to determine the agent’s first-period compensation ω1=fi+v1iQ1 if a1E1. If not, the agent is penalized.

Stage 4: At the start of the second period, based on the information from the first period, the principal may offer a renegotiated compensation contract cr(Δfr,v2r), where Δfr represents a change in the fixed payment, and v2r is the revised incentive rate for the second period. (The superscript i represents the initial offer and the superscript r represents the renegotiated offer.) The agent can accept or refuse the new contract. If she refuses, then the initial contract stays in effect. Based on Q1, the principal can also revise E2.

Stage 5: The agent chooses her second-period effort a2, which affects the second-period output Q2.

Stage 6: Based on Q2 and the possibly renegotiated contract, the principal pays the agent second-period compensation ω2=Δfr+v2rQ2 if a2E2. If not, the agent is penalized.

2.2 Agent’s Preferences

The agent is assumed to be risk averse and effort averse. Her preferences for the contract period are represented by the following time-additive exponential utility function:


where r>0 is the agent’s risk aversion parameter and ct represents consumption net of effort cost. β=1/R is the agent’s discount factor and R represents the time value of money; since these parameters are necessary to account for inter-temporal tradeoffs but play no essential role in the comparative static analysis, we fix β=0.95 and, approximately, R=1.0526. Let k(at) be the agent’s labor cost where, as is commonly assumed in the agency literature, k(at)=/(1+at2) is a quadratic function. k(at)>0 even if at=0 to recognize the cost of a worker’s time.

The agent can borrow and lend in financial markets. Let t be the amount of borrowing or lending, where an agent’s decision to substitute consumption over time affects the effective risk facing an agent (when measured in terms of consumption), their behavior and the kinds of incentives which are offered in equilibrium. After the agent has received her compensation and paid her labor cost, she has a cumulative-consumption bank balance: [6]


Similarly, at the end of the second period, the cumulative-consumption bank balance is


The agent is assumed to consume her bank balance at the end of the contract, i.e., c2=B2, and, without loss of generality, we assume a zero reservation consumption: c0=0.

2.3 Principal’s Preferences

The principal is assumed to be risk neutral and the discount rate β used by the agent also applies to the principal. Setting the price of output equal to 1 as a numeraire, we let QQ1+βQ2 be the total gross payoff to the principal, Wω1+βω2 be the total compensation payment to the agent, both are stated in period 1 dollars and CMmE1+βmE2 be the cost expended on monitoring. Thus, when negotiating a contract at t = 0, the principal’s preferences are summarized by:


That is, the principal’s expected utility is the expected gross payoff minus expected compensation and monitoring costs.

3 Characterizing the Equilibrium

An equilibrium describes the terms of the contract as well as the actions of the agent and principal during the two periods. Measured costs, output and productivity are consequences of this equilibrium. To incorporate time in an essential way and to explore how productivity can change over time even if the technology does not, we use the concept of renegotiation-proof contracts in the equilibrium. According to Fudenberg and Tirole (1990), a contract is renegotiation-proof if the principal will not choose to alter it at the renegotiation stage. Given the LEN (e.g., Linear contract, Exponential utility, and Normally distributed performance measures) assumption in the model setting, we show why there is no loss of generality in restricting attention to linear renegotiation-proof incentive contracts.

The following two theorems offer closed form solutions for the key variables. An appendix offers a more complete proof, which is based on a backward induction argument and which contains some implications associated with these solutions, such as consumption smoothing behavior by the agent.

Theorem 1

In an equilibrium with monitoring only

Theorem 2


In equilibrium, with incentives only (assuming interior solutions)


where RP0=/{rA1[v1+ρ(σ2/σ1)βv2]2σ12+rβ[v2]2(1ρ2)σ22} represents the overall risk premium and K(a1,a2)k(a1)+βk(a2) represents the total labor cost.

Theorem 2 shows that the solution for v1 in an inter-temporal model varies with the variance of the first-period output σ1and is adjusted for the inter-period correlation ρ. The solution also varies with β because the agent’s behavior in each period (and the effective degree of risk aversion) recognizes the effects of borrowing or lending as she attempts to smooth her consumption.

The equilibrium solution shows how the incentives vary with the technology of production. The realized profit of the principal depends on the agent’s effort and this effort is anticipated as part of the equilibrium. The realized profit is also sensitive to unanticipated increases in productivity, ϵt, by affecting both revenue and the extra compensation paid to the agent. Regardless of the effect on the level of expected profit, an increase in ψ affects this profit sensitivity. More specifically, let QtWt represent the realized profit to the principal in period t.

Corollary 1

Because an increase in ψ increases v1 and v2 in equilibrium, an increase in ψ decreases (QtWt)/ϵt.

Theorems 1 and 2 present solutions conditional on a type of job contract. In principle, a principal would make one of four types of job offers in a two period model:

  • to use incentives in both periods,

  • to monitor effort in both periods,

  • to monitor in the first period and to use incentives in the second period, or

  • to use incentives in the first period and to monitor in the second period

with the levels of incentives or standards determined by marginal conditions. Deriving conditions showing which of these four types of job offers a principal would offer in equilibrium is non-trivial computationally and, besides showing their existence, adds only a little insight. Fortunately, changes in the type of contract are rare and the marginal changes summarized by Theorems 1 and 2 are more relevant empirically. Therefore, an appendix contains a formal proof that an equilibrium exists unconditionally while the following comments note its characteristics informally and briefly.

If monitoring costs are low then a principal is likely to monitor workers. If the productivity of effort is high then the principal may or may not want to use incentives. Two special cases offer some insight: if ψ is low or if ψ and m increase equally.

At a low level of ψ, the cost of monitoring may be so high that the optimal incentives produce a higher level of effort than the optimal standard and only a “small” risk premium to the agent. Therefore, the principal’s preferred contract shares the surplus generated by the agent’s extra effort by offering an incentive contract. Suppose, instead, that ψ is large enough that the effort produced by an optimal effort standard is higher than the effort produced by an optimal incentive contract. In this case, the increase in total surplus as well as the reduction in agent’s risk premium implies that it is more profitable to monitor. In intermediate cases, an optimizing principal must balance the increase in total surplus due to the extra effort against the increased costs due to either the cost of monitoring or the risk premium created by incentives. The participation constraint ensures that the agent is indifferent amongst all equilibria. Therefore, the effect of an increase in ψ on the type of job offer depends on the magnitudes of the different effects on cost of effort, production surplus and the risk premium.

Our second special case considers a situation where two of the parameters vary jointly: at the same time as new technology increases productivity the new technology can also change how workers shirk their effort and, therefore, the costs of monitoring workers. For example, older managers may struggle to keep up with the language of younger specialists or to be fully aware of their skills. Sometimes, the opposite is true: Hubbard (2000) studies an example where the primary effect of a technological change is to make it easier to monitor workers.

Consider a change in technology which increases ψ and m and has no effect on ψm. Theorem 1 shows that this change has no effect on the principal’s preferred level of effort, if he uses a monitoring standard. Even so, the technological change increases output even if effort does not change. Theorem 2 shows that the change increases the optimal incentive, worker’s effort, the worker’s risk premium and the expected compensation. Therefore, the combined increase in (ψ,m) increases the likelihood that a principal offers a contract which includes monitoring of the agent’s effort.

4 The Effects of Technology on Labor Productivity with Incentive Contracts

This section uses the equilibrium derived above to analyze the effects of incentive contracting on productivity measurement. In any period t, labor productivity is measured using a ratio of output to measurable input, such as hours of worker time:


Since we normalize ht to be 1, eq. [1] implies that


Since labor productivity is a well-known, widely-studied and commonly-used measure, this normalization helps to focus attention on the implications of variables which are harder to measure. The possibility or impossibility of measurement is an important aspect of our analysis because, if sufficient data were available, it is well-known that other measures (such as “total factor productivity” or “multi-factor productivity”) are better able to account for issues not represented in our model of production: e.g., economies of scale and economies of scope (i.e., the production of a mix of types of output at a single production facility). It would also be possible to study whether multiple measurable inputs are being combined efficiently to produce any given type of output. [8] Other authors, e.g., Banker, Datar, and Kaplan (1989), provide such an excellent insight into the differences between different measures that we choose to focus on the special implications of the principal not being able to measure one of the inputs into production.

In our model, measured productivity represents the combined effects of technological parameters (e.g., engineering specifications in a manufacturing context), behavioral choices and random shocks. In other words, if measured productivity were higher in one year relative to the previous year, one should determine whether the principal deserves the credit for using better technology, or whether the agent deserves the credit (and appropriate wages), or whether it increased because of luck. Using Theorems 1 and 2, we can express expected productivity based on the parameters of the model:

Proposition 1

: i) E[P1]=ϕ+ψ2v1 and E[P2]=ϕ+ψ2v2 in equilibrium, if using incentives.

ii) E[P1]=ϕ+ψE1 and E[P2]=ϕ+ψE2 in equilibrium, if monitoring.

The traditional perception that productivity varies only if the technology varies is a special case of our model. Corollary 2 shows that this important special case has implications for the link between productivity and the average cost of producing a unit.

Corollary 2

i) If and only if ψ=0, then E[Pt]=ϕ.

ii) Define UnitCost as the ratio of total compensation paid to an agent over the two periods to the total production over the two periods, Q1+Q2. If ψ=0, then E(UnitCost)>x/E(P1) for some constant x that does not depend on ϕ.

This Corollary offers an significant benchmark because the discussion below, and especially the simulations, show that the qualified logic that “reduced cost is the corollary of increased productivity” is reasonable only in some circumstances: i.e., when ψ=0. The key difference between the effects of an increase in ψ or in ϕ is that the principal optimizes profit from the increase in ψ by changing the job offer made to an agent. Our model of hidden effort shows when basing decisions on costs alone would not fully exploit the improved technology. Trivially, the inequality in E(UnitCost)>x/E(P1) is due to Jensen’s Inequality and would not be true if there was no randomness in output. Still, the randomness enables the principal-agent relationship and the case of ψ=0 represents a lower bound on E(UnitCost)x/E(P1). An increase in ψ would increase output (since the agent would exert some effort in equilibrium) but the solution in Theorem 2 indicates that the multiplicative effect of changing effort declines as ψ rises. On the other hand, the cost per worker would increase in equilibrium since both the cost of effort and the agent’s risk premium would increase at an increasing rate. Without a clear accounting of the link between revenue and cost, uninformed senior managers in a firm may see the increase in total labor cost as an example of wasteful spending on inefficient workers.

The model also shows how the negotiated incentive varies with non-technological variables such as the environmental parameters (σ1,σ2,ρ). The following Proposition explores the implications of these added variables.

Proposition 2

Suppose that ψ>0 and that the principal uses incentives. In equilibrium, E[P2]>E[P1] if and only if A1(σ12+βρσ1σ2)(1ρ2)σ22>0.

Proposition 2 shows that expected productivity can differ between the periods even if the technology does not change. This result is true only with the use of incentives: Theorem 1 shows that effort and expected productivity would not change if the principal monitors effort in both periods. The extra condition on the exogenous parameters identifies conditions under which the agent’s hidden effort is higher in the second period than in the first because the incentive is higher in the second period than in the first. What may appear to be the effects of learning-from-experience, i.e., an increase in productivity over time, tends to be true in the equilibrium to our model if ρ2 is close to 1 or if σ12 is large relative to σ22.

Propositions 3 and 4 help to understand the trade off facing a firm which must choose between two different types of technological change: one which increases output regardless of the effort of the agent and one whose effect depends on the hidden effort of the agent. If the cost of acquiring both types is the same, the relevant question is which has the bigger effect on expected productivity.

Proposition 3

: Suppose that the principal uses incentives in both periods. If (2ψ1)[ψ2+2r(1ρ2)σ22]ψ2+[r(1ρ2)σ22]2>0, then E[P2]/ψ>E[P2]/ϕ in equilibrium.

Proposition 4

: If an increase in ψ does not alter the choice between incentives and monitoring within a period then an increase in ψ increases E[P2] at an increasing rate.

Proposition 3 shows that the effect would vary across industries and market conditions. A unit increase in ψ would always have a bigger effect than a unit increase in ϕ if ψ>1/2. This result implies that, when studying the kinds of technological advances used to improve productivity, industries where hidden effort is already a significant feature would tend to adopt technologies which exaggerate that feature. For example, methods of accounting for the cost and benefits of productivity change in a manufacturing industry, where engineering issues would tend to dominate and monitoring of effort is relatively easy, may be unsuitable in a service industry or in the knowledge-based economy, where the contribution of the discretionary behavior of an agent would be more significant. [9]

Proposition 5

: Suppose that the principal uses incentives in period 2. An increase in σ22 decreases E[P2]/ψ. An increase in ρ increases E[P2]/ψ if and only if ρ>0.

Proposition 5 reinforces the idea that the contract is negotiated based on the characteristics of the technology offered by the principal and the dynamic contracting environment in which both the principal and agent live. An increase in environmental risk discourages the risk averse agent. As the proof of Proposition 5 shows, the effect of an increase in ρ on the incentive depends on whether it reduces the conditional risk in the second period, i.e., (1ρ2)σ22.

5 Operational Risk

This model emphasizes the idea that, in an environment of asymmetric information, the principal offers a contract and anticipates that agents will behave in a certain way in equilibrium. Reality shows that people do not always act as anticipated. Such deviations are identified as examples of “operational risk”: “Operational risk is defined as the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events. This definition includes legal risk, but excludes strategic and reputational risk.” (Basel Committee on Banking Supervision 2011, fn. 5, emphasis added). Operational risk is distinguished from other sources of risk, such as market risk (evidence of which may be seen in flow measures, such as profit, revenue or costs) or credit risk (evidence of which may be seen in stock measures, such as the value of an asset).

Many readers may not be familiar with the term “operational risk”. Increasingly, the financial statements offered by many companies report on risk management issues in general and operational risk issues in particular to explain why the performance differed from last year’s expectation. Comments in these statements include qualifiers such as “breakdown”, “exceeding”, “inadequate”, “inappropriate”, “misconduct”, “errors”, “interruptions”, and “planned”. These qualifications to an event indicate that whether an event is worthy of consideration depends on a comparison relative to some kind of persistent or normal solution.

Some evidence may give a sense of the problem. [10] The Basel Committee on Banking Supervision within the Bank for International Settlements (BCBS 2009) used reports from 119 banks in 17 countries during 8 years (“pre-2002” to 2008); they report on 172,000 events which were worth more than €20,000 individually and were worth €54,000 million in total. The Operational Riskdata eXchange Association (ORX 2014) report on the experience of about 60 large financial institutions during 2008–2013; they report on 241,000 events which were worth more than €20,000 individually and were worth €134,000 million in total. Both the BCBS and the ORX data show that the distribution of losses is highly skewed, with many medium value events (e.g., €20,000–€50,000) and very few high value operational risk events (e.g., more than 10 million euros). These events are in addition to many millions of small mistakes worth tens or hundreds of euros.

Looking into the data more deeply suggests our model is relevant to this situation. The examples in the Introduction note that mistakes happen, that workers do not always do what they are told. Obviously, there has been technological changes in banking sector that has led to changes in client expectations and, if only because it is so hard to convict misbehavior in a court of law, there is evidence that effort can be hidden. Fraud due to an internal and external activities (e.g., unauthorized transactions and bank robberies) is common in the retail banking sector, as expected, but events in the categories most closely associated with the activities of an “agent” (i.e., “Clients, products, and business practices”, “Employee practices and workplace safety” and “Execution, delivery and process management”) are more numerous and are much more expensive individually. If we assume that a “large” bank or financial institution has 50,000 employees, which is less than many large banks, then the average employee would make a mistake leading to a medium or large sized operational risk event less than once every 10 years (using the more recent ORX data). For these reasons, we think that our model is at least descriptively accurate, that the medium and high value events are rare relative to the day to day routine activities and that these types of events are significant enough in aggregate for the principal to worry about.

We suggest that the “normal solution” which is projected in a company’s financial statement is an equilibrium. With this perspective, it makes sense that market risk and credit risk can create the asymmetry of information that leads to a principal-agent relationship, even if neither is a direct cause of an operational risk event. The closed form solutions derived from our model allow us to explore the significance of an operational risk event and its consequences for different players. We consider the effects of an operational risk event where effort is 10% less than the equilibrium.

This presentation is “ad hoc” in the most informative sense of the term: a good model should account for everything relevant and any departure from the model should be for a reason that is independent of the rest of the model. We observe that small, medium and large unanticipated deviations occur (e.g., BCBS 2009; ORX 2014) and, while we label the deviation as “ad hoc” in order to focus on the practical effects and to show how the contract varies with the technology, the deviations can be justified by appealing to any one of several theories. In addition to the ancient debates about whether it is possible to resolve conflicts of interest in a principal-agent relationship and the growing literature on time-inconsistent decisions (e.g., O’Donoghue and Rabin 2015), large literatures can be interpreted as refining the idea of the “cost of effort”: e.g., the literatures on “self-control”, “temptation”, “opportunistic behavior” and simple carelessness (e.g., Heidhues and Koszegi 2010; Belot and Schroder; 2013; Myrseth and Wollbrant 2013). The use of any specific theory does not affect our numerical analysis, since it emphasizes differences in production technology. To the extent that the principal and the agent anticipate these types of behavior on average by way of the incentive compatibility and participation constraints, they are already represented in the agency contract. Being able to anticipate the average does not eliminate the kinds of deviations that we consider below. Focusing on the effects of a 10% decrease is arbitrary. Fortunately, some of the important effects are linear in the size of the deviation: a 10% deviation is exactly 10 times larger than a 1% deviation. The costs of those effects tend to non-linear in the sense that an increase in the deviation increases the cost at an increasing rate since the deviation is a difference from an equilibrium. In this sense, 10% should be seen as big but not too big. [11]

To estimate the effects and discuss them, we need to compute the equilibrium solution in a not-unrealistic scenario. Our model uses two parameters to describe the production technology, (ϕ,ψ), and three variables describing the contracting environment, (σ1,σ2,ρ). Other parameters (β,r) describe the agent’s patience and degree of risk aversion. Our model offers closed form equilibrium solutions for the terms of the principal-agent contract and the choice of hidden effort by the agent as well as other derived measures such as measured productivity.

To provide a common benchmark, we focus mostly on the outcome for the second period. Most importantly, we normalize the level of measured productivity in the second period, E(P2)=100. We also use


The condition on ρ offers a neutral starting point and the propositions above can be used to explore the significance of an alternative value. The condition on (σ1,σ2) is a conservative estimate for the variability facing a firm because the variability should be much greater at a more disaggregated level of industrial aggregation. This parameter reveals how much measured productivity can vary even if effort and technology do not change. The measure of risk aversion is chosen to represent a normal level of risk aversion. The value of β is reasonable given long term real interest rates but, as noted during the development of the model, changes in its value play no role in the following discussion: β=0.95 implies that R=1.0526 and that A1=(1+β)1=0.5128 approximately.

These conditions are not sufficient to uniquely identify the fundamental technology parameters. To avoid over-emphasizing the role of hidden effort, we deliberately choose parameter values where hidden effort appears to play a minor role on average: we set ϕ=99.0. Then, to reproduce the benchmark level of measured productivity in equilibrium, it must be true that


As part of this solution, it is also true that v1=0.3012, and a1=0.7079. [12]

Table 1:

Effects of an operational risk event, when using an equilibrium incentive contract.

Change in Expected Output =(Δa2)ψ0.
Change in Worker’s Expected Compensation =(Δa2)ψv2r0.0180.6250.0180.625
Change in Expected Profit =(Δa2)ψ(1v2r)0.0820.6250.0820.625

Table 1 shows the effects of a 10% decrease in effort in period 2, given an equilibrium incentive contract based on different values of the fixed technology parameters. The first row indicates the equilibrium level of a2, which changes with the technology, shown in each column and the second row shows the expected value of labor productivity in equilibrium based on the technology and the equilibrium level of worker’s effort. The third row shows the size of a 10% decrease in the level of effort while the remaining rows show its consequences of that decrease for the expected output, expected compensation, and expected profit.

The cost or severity of an operational risk event is minor if ψ is small because a2 would be small. Thus, if the agent exerts 10% less effort than the manager anticipates, and even though they would be penalized by it under the contract, then 10% of a small number is small. If ψ is larger, then the cost of a deviation is larger but, when using the equilibrium contract, the cost in terms of lost wages is higher for two reasons: because the change in output is larger and because the equilibrium incentive rate is higher. [13] Although the principal always loses if the agent exerts less effort, the fact that an equilibrium incentive contract is designed to be incentive-compatible means that the size of the loss varies with the technology.

The propositions above show how these conclusions can be applied to a wider range of scenarios. If using an incentive contract and if ϕ were higher, then columns 4 and 5 shows that (f,v1,v2) would not change. When using an incentive contract, an increase in (σ1,σ2) increases the worker’s risk premium, and would cause the level of effort exerted in equilibrium to fall; in such cases, a 10% deviation from that equilibrium would have a smaller effect.

Some agency relationships are resolved by monitoring and oversight rather than incentives. The consequences of a 10% decrease in effort are easy to predict under this type of contract since the agent bears no risk due to external uncertainty: a worker who fails to meet the standard is easily identified and penalized. This rule may deter a worker enough that the possibility of accidental underperformance is minimal. The most relevant difference between the two types of job contracts is that, if a worker anticipates being fired for underperforming even a little then, rather than shirking a little or accidentally, they may not show up for work at all. Thus, the cost or severity of the event to a principal who monitors may depend on a worker’s average productivity: i.e., both ϕ and ψ. Setting standards and monitoring workers may decrease the possibility of operational risk but increase its severity if a negative event happens. [14]

An alternative perspective on monitoring would start by asking: if it could vary, what would the cost of monitoring need to be to reproduce an equivalent output as an incentive contract? [15] The algebra in Theorem 1 shows that this question has a simple answer: a1=a2=E1=E2=ψm. Thus, in equilibrium, knowing ψ and the value of a2 implied by an incentive contract makes it easy to solve for the value of m which produces an equivalent effort. Specifically, for the case of ψ=2.35 the equivalent value of m is 1.92 and the total cost of monitoring is 1.92* 0.43 = 0.83 or less than 1% of the benchmark output. If ψ increases to 5.00 then the equivalent value of m is 2.5 and the total cost of monitoring is 2.5* 2.5 = 6.25 or more than 5% of expected output. Thus, an increase in ψ increases the cost of monitoring which is equivalent to an incentive contract, at an increasing rate. (Note that this cost is incurred by the principal but has no effect on the agent’s payoff, since it is fixed by the participation constraint.)

The specification of the model suggests an additional scale which could be used to judging the magnitude or relevance of an operational risk event. The model characterizes the existence of a risky environment. Even though our discussion focuses on the expected outcome, the measured productivity would differ between periods without any deviation from the equilibrium behavior and would be less than expected with some probability: with no change in technology and using our assumption that random shocks are distributed Normally, there is a 16% probability that the measured productivity would be at least one standard deviation below its expectation. Our calculations assume that one standard deviation is approximately 5% of expected output (i.e., 5 units of output in Table 1). Therefore, this computation suggests that the effects of the operational risk events discussed in Table 1 are possible but that they are less serious than the negative “probable” events.

6 Concluding Remarks

This paper uses a dynamic agency model with renegotiation to study the relationships amongst incentives, monitoring, worker effort and productivity. We develop a two period model which shows how job offers vary with technology and how measured productivity can vary even if the technology does not. Since we are able to provide closed form solutions for critical variables, we are also able to discuss the impact of the issue of an operational risk event, where workers do not act as anticipated.

Our model uses two parameters to model the different types of technology. We show that increases in the marginal productivity of effort, ψ, have a nonlinear effect. Estimating these parameters separately may help to distinguish the productivity characteristics of a manufacturing industry (where the engineering processes and the presence of supervisors allow agents relatively little opportunity for discretionary effort) from the productivity characteristics of service industries. For a firm in the service sector or the knowledge-based economy, such as consulting or restaurants, hidden effort by its workers may be the source of its competitive advantage.

We show that an increase in ψ would increase productivity and increase total cost per worker in equilibrium. This finding differs from the traditional view of productivity, expressed in Corollary 2, where the percentage effect on unit cost of an increase in measured productivity is equal but in the opposite direction. While the technology-based model of increasing productivity focuses on the idea that the “boss” deserves the credit because he buys the better technology used by the agent, our model focuses on the idea that the agent deserves the credit for an increase in productivity because of hard work. These ideas are not mutually exclusive and our model of negotiations (followed by potential re-negotiations) should serve to remind readers that any productivity gain produces an expected surplus which can be bargained over: good management ensures that the terms of the job contract are appropriate to the technology. This idea is consistent with the conclusion of Bloom and van Reenan (2011) that the surprisingly large productivity dispersion amongst producers in a given industry may be due to the slow adoption of appropriate human resource management practices by some of them.

Other papers, e.g., Banker, Datar, and Mazur (1989), note that measurement is necessary before one can recognize whether productivity has changed and before one can institute policies to manage it. Our model shows that, even if the technology is fixed, productivity can appear to vary between periods, due to changes in the conditions, changes in behavior over time or luck (since the standard deviation of Pt equals σ). [16] Therefore, while we do not offer any direct predictions concerning how productivity would evolve over time, our closed-form solutions would be useful to anybody trying to determine whether a change in the measured value of Pt is statistically significant. We find that distinguishing between the different types of technological change adds insight since, if matched with the appropriate equilibrium job contract, the effect of a certain type of technological change increases expected productivity at an increasing rate.

Beyond the issue of measuring productivity, we note that the significance of operational risk. The issue is not well understood. While many people agree that it exists although the events occur, researchers disagree about how best to measure it and about how common it is. Some people argue that good risk management aims to control the “possibility” and “impact” of an event. According to this definition, our paper argues that the analysis should recognize how the terms of the agency contract can alter the possibility of risks associated with human behavior. By proposing to study this phenomenon in a principal-agent setting, we note that an external source of risk is a necessary feature of the situation: market risk is a natural candidate. Based on the environment, an incentive contract provides appropriate incentives and shares the risk. If an agent shirks and an operational risk event occurs then the impact on the principal varies with the negotiated terms of the job contract. Or, with an appropriate investment, the principal can choose to take on that risk by monitoring the work of the agent. [17] In other words, the technology and the conditions determine an equilibrium which represents “normal” behavior. Therefore, the impact of an operational risk event on a company’s profit would vary with the type of production technology as well as with the human dimensions of the operating environment.

Our formal model emphasizes ideas which are more familiar to financial analyses and that fact may seem to limit the insights into models which are more descriptively accurate of the operational processes. For example, Hatzakis, Nair, and Pinedo (2010) offer a number of insights based on specific features of the financial industry. They, as well as the descriptive data from the BCBS (2009) and ORX (2014), note certain issues with using the Normal distribution to describe uncertainty about operational risk. While there are many small events which could be considered as a cost of doing business, most of the attention focuses on the distribution of medium and large value events which tends to have fatter tails and to be skewed toward a few catastrophic events. Incorporating such issues into the analysis coherently would need to consider the extra effects of the preferences of a principal and an agent with respect to higher moments of a distribution, such as “skewness-loving” or “kurtosis-aversion” or even uncertainty aversion and ambiguity aversion. We conjecture that the basic insights of our comparative statics analysis with respect to the technology would remain valid. Having fat tails would have little direct effect on the mean productivity but the cost of a risk, and the associated consumption smoothing behavior, would become a more significant consideration when the worker and manager negotiate the terms of the job contract. If an agent has limited liability, it may not be possible to impose the penalty implied by a linear incentive contract in the event of a catastrophe. If true then a principal may prefer to use some kind of costly monitoring system. In all cases, the choice of contract needs to be matched with the technology. The match affects the expected productivity, the agent’s anticipated level of effort as well as affecting how the losses are distributed between principal and agent when people do not behave as anticipated.


This paper has benefited from discussions with Tom Ross, Xiaopeng Yin, Kam Yu, seminar audiences at the University of Alberta, University of British Columbia, University of California− Los Angeles, University of Illinois− Urbana Champaign, Peking University, University of Toronto, the CAAA Annual Conference, the Canadian Economics Association Annual Meeting and the AAA FARS Midyear Conference and the comments of two referees. We gratefully acknowledge the financial support from City University of Hong Kong.

Appendix: Proofs

Proof of Theorem 1: Behavior with Monitoring Only

Workers who exert the required effort know that the monitoring technology is accurate enough to confirm that they are doing the job as agreed in the contract. Therefore, they have no reason to work any less hard than the standard. Without incentives, i.e., v1i=v1r=0, workers have no reason to exert more than the minimum set by the standard. Thus, a1=E1 and a2=E2. Knowing this, the principal pays f=k(E1)+βk(E2) to workers and incurs the cost of monitoring separately.

The principal can increase (E1,E2) and the effect on output varies with the technology. In our model, ψ represents the incremental benefit of effort and a profit maximizing principal would set the effort standard at a point where the incremental benefits of an increase balance the incremental costs. Given k(at)=/(1+at2), the first order condition showing the optimum to eq. [6] implies that the optimal (E1,E2) satisfies


(if there is an interior solution, else Ei=0). This solution depends on the fixed parameters; it would not be renegotiated at the beginning of the second period based on output in the first period or on any change in the worker’s bargaining position between the first and second period. Q.E.D.

Proof of Theorem 2: A Renegotiation-proof Incentive Contract

Solving for the appropriate incentives is more complex because the possibilities for a worker’s behavior are more varied and complex. As is commonly done in dynamic models, we construct an equilibrium solution using backward induction, as follows. Since the agent is assumed to consume all the remaining bank balance at the end of the second period, the second-period consumption c2 is not a decision variable (i.e., c2=B2) and we can start by characterizing the agent’s second-period effort choice a2 and the principal’s renegotiated incentive rate v2r at t=2. Given these solutions, we can characterize the agent’s first-period consumption decision c1, and first-period effort a1. Finally, we solve for the principal’s incentive contract at t=0, described by (f,v1).

The initial linear compensation contract is ci(fi,v1i,v2i) and, at the start of t=2, (i.e., stage 4 in the time line), fi and v1i are irrevocable. At this stage, the principal can offer a renegotiated linear compensation contract cr(Δfr,v2r) with an (additional) fixed payment Δfr and a revised incentive rate for the second period v2r. If the agent accepts the new contract, then her second-period compensation will be ω2(cr)=Δfr+v2rQ2. The LEN assumption implies that, conditional on the information at the end of t=1, her expected compensation net of labor cost is


and that the risk premium for the net compensation based on the posterior variance of Q2 is


Therefore, the agent’s certainty equivalent with respect to the net compensation, conditional on the information at the end of the first period, is


The agent is assumed to consume her bank balance at the end of t=2, i.e., her second-period consumption c2=B2, and B2=R(B1c1)+ω2k(a2). Thus, the agent’s utility maximization decision with respect to her second-period effort a2 is


The first-order condition for the problem yields a2r=ψv2r. This condition indicates that the agent chooses her second-period effort optimally based on the revised incentive rate v2r.

Given the initial contract, the output Q1, and the principal’s conjecture of the agent’s first-period effort aˆ1, the principal chooses Δfr and v2r so as to maximize his second-period expected utility, subject to acceptance by the agent. Formally, the principal solves


If the participation constraint, eq. [15], is binding then


Substituting eqs [17] and [16] into the objective function [14], the principal’s decision problem becomes an unconstrained optimization problem:


The first-order condition with respect to v2r is


Therefore, the revised incentive rate is


The solution indicates that Δfr is a linear function of (Q1,aˆ1). Furthermore, if the initial contract specifies v2i=v2r, then Δfr=0. For any optimal initial linear contract ci(fi,v1i,v2i) that is subject to renegotiation, there exists an equivalent initial linear contract that is renegotiation-proof crp(f,v1,v2r).

At the end of t=1 and before the contract might be renegotiated, the agent decides on her consumption c1 to maximize her expected utility over the two-period consumption, conditional on the information available at this time. Borrowing and lending opportunities are available to the agent. Given c2=B2=R1+ω2k(a2) and 1B1c1, the agent’s decision problem with respect to c1 is


The first-order condition for this problem is satisfied by c1=R(B1c1+CE1). Rearranging it and using the annuity amortization factor A1(1+β)1 gives c1=A1(B1+CE1). Because the agent smoothes consumption over time, it is optimal for the agent to consume an amount equal to an annuity based on the sum of first-period bank balance and second-period certainty equivalent. Relative to a one-period model of the principal-agent relationship, consumption smoothing alters the effectiveness of incentives within a period by reducing the agent’s utility of a good outcome in that period and increasing the utility of a bad outcome in that period. Consumption smoothing also implies that the conditions anticipated in the future and the terms of the contract at that time can affect behavior in the current period.

At the beginning of t=1, and based on a given compensation contract, the agent chooses a1 to maximize her expected utility over the two periods. This is equivalent to maximizing her total certainty equivalent CE0 under the LEN assumption,

Wω1+βω2 is the total compensation payment to the agent, stated in period 1 dollars, and K(a1,a2)k(a1)+βk(a2) represents the total labor cost.

represents the agent’s total risk premium, which takes account of the effect of inter-temporal correlation, captured by ρ, and the effective risk aversion parameter, rA1. The first-order condition for the problem implies that a1=ψv1.

Finally, we consider the compensation terms of the incentive contract, (f,v1), offered by the principal at t=0. Anticipating the optimal solutions in the next two periods, the principal chooses (f,v1) to maximize his expected utility at date zero subject to the agent’s participation and incentive constraints. That is, the principal solves the following problem


where QQ1+βQ2 represents the expected gross payoff stated in period 1 dollars and Wω1+βω2, represents the expected compensation payment stated in period 1 dollars.

The fixed wage f is sufficient to induce the agent to accept the job offer, i.e., the participation constraint is binding. Without loss of generality, we assume a zero reservation consumption: i.e., c0=0. This condition implies that


where ω1v=v1Q1 is the variable component of the first-period compensation. Using the binding participation constraint enables us to express the principal’s decision problem as the following unconstrained optimization problem:

Ex ante, the principal chooses v1 so as to maximize his expected gross payoff net of the agent’s total labor cost and total consumption risk premium. The first derivative of Π0p with respect to v1 is:

Setting this derivative equal to zero in order to characterize the optimal solution yields v1=[ψ2]/Δλ[βv2], where λrA1ρσ1σ2/Δ and Δψ2+rA1σ12.

In a Nash equilibrium, aˆ1=a1 by definition. To characterize an equilibrium, we focused on the case where the initial contract does not need to be renegotiated (i.e., v2r=v2i=v2). Substituting these equalities in the equations above produces the Theorem stated in the main text. Q.E.D.

Proof of the Existence of an Equilibrium

Theorem 2 identifies algebraic solutions for the appropriate renegotiation-proof contract crp(f,v1,v2) if using an incentive contract and Theorem 1 identified solutions for the appropriate monitoring standard if using a monitor. This proof shows that the principal-agent equilibrium exists unconditionally.

A key step in the argument is that the monitoring standard imposes a lower bound on effort and, if the contract creates an incentive to work harder then the standard has no other effect. If the standard is binding in either period in equilibrium, then using incentives also adds income risk for the agent and the agent would demand compensation. Thus, the principal would use either a binding standard of effort in a period or an incentive contract in that period (with Et=0) but not both. The essential question is to identify conditions under which each is used in equilibrium.

We note that there is no benefit and some cost to a principal who both monitors workers and introduces income risk by using incentives. Due to the LEN assumptions and our consideration of renegotiation, the choice of incentives vs. monitoring in the first period has no effect on the choice of incentives vs. monitoring in the second period. If the principal chooses to monitor in the second period then this choice has no effect if the principal chooses to monitor in the first period. Any effect of this choice on the level of incentives in the first period, if the principal chooses to monitor in the second period, depends on the total risk faced by a worker and the effective degree of risk aversion (which varies with the worker’s decision to smooth income over time). Therefore, for any given set of parameters, it is possible for the principal to choose their best contract in period 2. Given that choice and the parameters, it is possible for the principal to choose their best contract in period 1. Thus, an equilibrium exists. Q.E.D.

Proof of Corollary 2ii)

Corollary 2 ii) Define UnitCost as the ratio of total compensation paid to an agent over the two periods to the total production over the two periods, Q1+Q2. If ψ=0, then E(UnitCost)>x/E(P1) for some constant x that does not depend on ϕ.


Jensen’s Inequality implies that E(1/z)>1/E(z) for any random variable z. Therefore, E(UnitCost)=E(f+E0(ω1v+βω2))/(Q1+Q2)>E(f+E0(ω1v+βω2))/EQ1+Q2. If ψ=0 is the agent’s cost of effort minimal and, with vt=0, there is no risk premium: E(f+E0(ω1v+βω2))=K(0,0) in equilibrium. If ψ=0 then EQ1+Q2=2ϕ=2E(P1) in equilibrium. Therefore, E(UnitCost)>K(0,0)/2E(P1) in equilibrium. Q.E.D.

Proof of Proposition 2

Proposition 2: E[P2]>E[P1] if and only if A1(σ12+βρσ1σ2)(1ρ2)σ22>0.


In equilibrium, E[P1]=ϕ+ψ2v1 and E[P2]=ϕ+ψ2v2, where

  1. v1=1Δ[ψ2]λ[βv2r], where Δψ2+rA1σ12, λ1ΔrA1ρσ1σ2, and A1(1+β)1;

  2. v2r=ψ2ψ2+r(1ρ2)σ22.

Therefore, E[P2]>E[P1] if and only if v2r>v1. The equilibrium incentive rates give


Substituting Δψ2+rA1σ12 and λ1ΔrA1ρσ1σ2, it follows that


Thus, we have


Proof of Proposition 3

Proposition 3

If(2ψ1)[ψ2+2r(1ρ2)σ22]ψ2+[r(1ρ2)σ22]2>0, then E[P2]/ψ>E[P2]/ϕ.


Using the equilibrium incentive rates in the expected productivity expressions, and differentiating with respect to ϕ and ψ give


Since v2rψ=2ψr(1ρ2)σ22[ψ2+r(1ρ2)σ22]2, it follows that


Hence, E[P2]ψ>E[P2]ϕ if the following condition is satisfied


Proof of Proposition 4

Proposition 4

If an increase in ψ does not alter the choice between incentives and monitoring within a period then an increase in ψ increases E[P2] at an increasing rate.


In equilibrium, the first derivative of the expected productivity with respect to ψ is


Since v2rψ=2ψr(1ρ2)σ22[ψ2+r(1ρ2)σ22]2>0, so E[P2]ψ>0.

Further, the second derivative with respect to ψ is


Substituting 2v2r[ψ]2=2r(1ρ2)σ22[r(1ρ2)σ223ψ2][ψ2+r(1ρ2)σ22]3 into [35] gives


where Mψ2+r(1ρ2)σ22.

If the principal monitors effort, then an increase in ψ increases E1 and E2. Since E[P1]=ϕ+ψE1 and E[P2]=ϕ+ψE2 when the principal monitors, an increase in ψ increases E[Pt] at an increasing rate.

Therefore an increase in ψ increases E[Pt] at an increasing rate independent of whether the principal monitors or uses incentives within a period. Q.E.D.

Proof of Proposition 5

Proposition 5

An increase in σ22 decreases E[P2]/ψ. An increase in ρ increases E[P2]/ψ if and only if ρ>0.


From the last proof, we have


Differentiating the above with respect to σ22 and ρ respectively, yields


Since v2rσ22=ψ2r(1ρ2)M2 and 2v2rσ22ψ=2ψr(1ρ2)[ψ2r(1ρ2)σ22]M3, it follows that 2E[P2]σ22ψ=4ψ3[r(1ρ2)]2σ22M3<0, where Mψ2+r(1ρ2)σ22.

Substituting 2v2rρψ=4ψrσ22ρ[ψ2r(1ρ2)σ22]M3 and v2rρ=2ψ2rσ22ρM2 into [38], we have


Therefore, the sign of 2E[P2]ρψ depends on the sign of ρ. Q.E.D.


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Published Online: 2015-9-17
Published in Print: 2016-1-1

©2016 by De Gruyter

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