Dynamic incentives and dual-purpose accounting

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Journal of Accounting and Economics 42 (2006) 417–437 www.elsevier.com/locate/jae

Dynamic incentives and dual-purpose accounting$ Gerald Felthama, Raffi Indjejikianb,, Dhananjay Nandac b

a Sauder School of Business, Univesity of British Columbia, Vancouver, Canada BC V6 T 1Z2 Ross School of Business, University of Michigan, 701 Tappan Street, Ann Arbor, MI 48109-1234, USA c Fuqua School of Business, Duke University, Durham, NC 27708-0120, USA

Received 22 January 2004; received in revised form 29 March 2006; accepted 6 April 2006 Available online 1 September 2006

Abstract Ongoing employment relationships often give rise to implicit, dynamic incentives. We describe the implications of implicit incentives when firms use information about both an employee’s past performance and his future productivity in a two-period agency model. We show that when an accounting system serves these dual objectives, an employee’s implicit incentives may be beneficial or detrimental to the firm. As a consequence, firms may prefer an accounting system that reports a single metric that combines information about past performance and future productivity, over one that reports two distinct metrics, one for each purpose. r 2006 Elsevier B.V. All rights reserved. JEL classification: J33; L22; M41 Keywords: Accounting measures; Incentives; Multi-period

1. Introduction Accounting scholars suggest that firms use management accounting systems to serve two broad objectives: facilitate managerial decision making and mitigate organizational

$

This paper was previously titled ‘‘Dual-Purpose Measures.’’ This paper has benefited from workshop comments at Arizona State University, University of California (Berkeley), University of Chicago, Duke University, University of Florida, Harvard University, University of Pittsburg, University of Southern California, 2003 SESARC at Emory University and the 2003 AAA Management Accounting Conference. Corresponding author. Tel.: +734 936 1460; fax: +1 734 647 2871. E-mail address: raffi[email protected] (R. Indjejikian). 0165-4101/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jacceco.2006.04.007

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control problems (e.g., Zimmerman, 2000).1 An important dimension that distinguishes these two objectives is the time frame to which they apply. For instance, while decisionmaking information tends to be forward looking (e.g., information useful for making future production plans), control information is typically retrospective (e.g., information useful for evaluating an employee’s past behavior). As a practical matter, these two roles are intertwined. Reported information serves both a control role with respect to prior behavior and a decision-facilitating role with respect to future actions. The objective of this paper is to show how the interaction between these two roles affects the usefulness of alternative accounting systems. We illustrate the interaction between the control and decision-facilitating roles of accounting information in a two-period linear–exponential–normal (LEN) agency model where a risk-neutral owner (or a board-of-directors acting on behalf of well-diversified owners of a firm) contracts with a risk- and effort-averse manager to perform a single task in each of two periods. We assume the owner commits to hire the manager for both periods, but the terms of the manager’s compensation contract are renegotiable at the end of the first period. We also assume that the payoff to the firm is not contractible information. Instead, the owner evaluates the manager based on performance measures that are informative about the manager’s effort in each period. A distinguishing feature of our model is that the owner is uncertain about the manager’s marginal productivity in the second period. Hence, the information reported at the end of the first period serves two objectives: it helps the owner motivate the manager’s first-period effort (the control role), and it helps the owner determine the optimal level of managerial effort to be induced in the second period (the decision-facilitating role). Renegotiating the second-period contract based on first-period information gives rise to what are termed dynamic or implicit first-period incentives.2 In our model, implicit incentives manifest in two distinct ways. The first effect reflects the manager’s desire to increase his expected second-period compensation. For instance, if random uncontrollable factors affecting reported performance are likely to persist, then a first-period report of high performance raises the owner’s expectation of the manager’s second-period performance. Accordingly, a manager will reduce his first-period effort to dampen his reported performance and, in effect, raise his expected second-period compensation by lowering the principal’s expectation of his future performance.3 The second effect reflects the manager’s desire to reduce the variance of his aggregate compensation. This implicit incentive is a unique feature of our setting and arises only because first-period information serves a dual purpose; it is informative about the manager’s first-period effort and it is also informative about the manager’s second-period 1 Demski and Feltham (1976) and Baiman and Demski (1980) are examples of some early studies that highlight distinct decision-influencing and decision-facilitating roles for accounting information. Arya et al. (1997) and Narayanan and Davila (1998) are more recent studies of the tensions that arise when information is used for both decision making and control. 2 Implicit first-period incentives arise whenever the manager’s expected utility is affected by the difference between his choice of the first-period effort level and the owner’s conjecture with respect to that choice. Of course, in equilibrium, the owner’s conjecture equals the manager’s choice, since in equilibrium the manager is compensated for his effort cost and risk. 3 Recent literature that examines this implicit incentive include Milgrom and Roberts (1992), Gibbons and Murphy (1992), Meyer (1995), Meyer et al. (1996), Meyer and Vickers (1997), Indjejikian and Nanda (1999, 2003), and Christensen et al. (2003a, b, 2004).

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marginal productivity. In particular, since the owner adjusts the strength of the manager’s second-period compensation (the bonus rate in our linear model) based on first-period information, the manager’s first-period compensation (which is also based on first-period information) will be correlated with his anticipated second-period compensation. Accordingly, the manager has an incentive to alter his first-period effort to lower this correlation, thereby reducing the riskiness of his aggregate compensation. The nature and impact of this latter implicit incentive can differ significantly with differences in the reporting systems in place (as well as with the nature of the commitments that can be made by the firm). In the interest of brevity, we provide a short illustration of the implications of this implicit incentive and defer more detailed discussions to Sections 3–5. Consider, for example, a single ‘‘aggregate’’ performance measure that is informative about both the manager’s current effort and his future marginal productivity. If a report of high performance by the aggregate measure raises the owner’s expectation of the manager’s second-period performance, then the manager’s first-period compensation will be negatively correlated with his second-period compensation. Hence, the manager has an incentive to increase his first-period effort because doing so makes his first- and secondperiod compensation more negatively correlated. In contrast, if a report of high performance by the aggregate measure lowers the owner’s expectation of the manager’s second-period performance, then the manager’s first- and second-period compensation will be positively correlated. Hence, the manager has an incentive to decrease his first-period effort to dampen this positive correlation. Ceteris paribus, a decrease in the manager’s first-period effort reduces the firm’s profit. Since the manager’s implicit incentive discussed above arises because first-period information serves a dual purpose, a natural alternative to the aggregate system is a measurement system with two distinct special-purpose metrics, one for each purpose. At a minimum, this offers the prospect of avoiding the correlation between first- and secondperiod compensation if, for example, the firm uses each metric exclusively for its own purpose. Interestingly, we find that if the firm adopts such a ‘‘disaggregate’’ system, then both special-purpose metrics will be optimally used as contracting variables.4 The intuition is that the productivity metric is used by the principal as a contracting variable to induce a negative correlation between the manager’s first- and second-period compensation, thereby giving the manager an incentive to always increase his first-period effort. Our analysis also demonstrates that implicit incentives can influence firms’ preferences for accounting systems to the point where a single ‘‘aggregate’’ performance metric that is informative about both firm productivity and managerial performance are sometimes favored over a disaggregate system that reports two distinct metrics, one for each purpose. This is consistent with a body of literature that links contractual settings characterized by limited commitment with the theme of ‘‘less information is better’’ (e.g., Arya et al., 1997; Indjejikian and Nanda, 1999). These types of effects typically do not arise if managers operate under multi-period contracts that firms commit to uphold. Indeed, in our setting, we show that firm value with a full commitment contract is higher than with a limited commitment contract and that, ceteris paribus, a disaggregate system that reports two 4 This occurs even though the special-purpose productivity metric is not directly influenced by the manager’s effort nor is it a primary means of filtering uncontrollable noise via its correlation with the special-purpose managerial performance metric.

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distinct metrics for two distinct purposes is preferable. However, we note that full commitment contracts may not be feasible because it may be impossible to preclude mutually agreeable contractual changes.5 The rest of the paper is organized as follows. In Section 2, we describe a two-period agency relationship that benefits from the reporting of two types of information: forward-looking information about firm productivity and ex post information about past managerial actions. In Section 3, we characterize the model assuming a limited commitment contract. In Section 4, we consider the implications and consequences of limited commitment. In Section 5, we characterize the model assuming two-period commitment contracts. Section 6 provides some summary remarks. 2. The model A risk neutral owner-principal hires a risk- and effort-averse agent to operate a firm for two periods.6 The firm’s payoff (gross of the agent’s compensation) in period t is denoted yt at , where yt 2 ð
1; 1Þ is an unobserved normally distributed productivity parameter with prior mean y¯ t X0 and variance s2y , and at 2 ð
1; 1Þ is the agent’s unobserved costly effort in period t. Negative effort is costly to the agent, but it produces a positive payoff to the firm if yt is negative. The agent’s aggregate wages are w and his two-period utility is Uðw; a1 ; a2 Þ ¼
exp½
rðw
12a21
12a22 Þ. We assume that the agent’s reservation wage is zero, and there is no discounting. We assume that the payoffs realized in each period, yt at , are not contractible. Instead, the agent is evaluated and compensated based on a vector y1 of publicly reported firstperiod signals and a single second-period signal y~ 2 ¼ a2 þ u~ 2 .7 We assume y1 is influenced by the agent’s first-period effort as well as by random events that are correlated with both the uncertain second-period productivity and the noise in the second-period report. In particular, we let y~ 1 ¼ ma1 þ qy~ 2 þ u~ 1 ,

(1)

where m and q are vectors (with the same dimension as y1) whose elements are ones and zeros depending on whether a specific signal in y1 is informative about the agent’s effort, firm productivity, or both.8 5 The following excerpt (taken from Johnson and Johnson’s (2001) proxy filing) illustrates the tension between firms’ desires to make long-term commitments and their periodic tendency to restructure compensation arrangement to incorporate new information: ‘‘The annual amount of expected improvement in EVA is fixed. To ensure that the EVA element provides strong incentives for management to annually increase shareholder value and does not reward poor performance by reducing performance standards or penalize superior performance by raising performance standards, the Compensation Committee allowed no recalibration of the expected EVA improvement for the three fiscal years 1997–1999. However, due to the significant changes in the Company’s business since EVA was adopted, such recalibration was performed for fiscal year 2000.’’ 6 The hiring of one agent for two periods is exogenously imposed. We can explicitly consider the hiring choice by introducing nontrivial ‘‘switching’’ costs of (say) hiring and training a new manager. See Christensen and Feltham (2005) and Sabac (2004) for discussions of hiring decisions at the end of a period. 7 Throughout the paper, boldface denotes vectors and matrices. 8 We note that an element of y1 is positively correlated with y2 if the corresponding element of q equals one. We can also accommodate settings where y1 is negatively correlated with y2 , but this additional level of generality offers little incremental insights. It is also important to note that y1 has a linear-additive form even though the firm’s first-period payoff is ‘‘multiplicative’’ as in y1 a1 . Although we acknowledge this as a limitation, we

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Given expression (1), in Section 4 below we illustrate two types of accounting systems distinguished in particular by their dimensionality. One system, which we refer to as an ‘‘aggregate’’ system, generates a single report y~ 1 ¼ a1 þ y~ 2 þ u~ 1 (i.e., m ¼ q ¼ 1). The other system, which we refer to as a ‘‘disaggregate’’ system, generates two special-purpose reports, where y~ 1a ¼ a1 þ u~ 1a (i.e., m1 ¼ 1; q1 ¼ 0) is exclusively informative about the agent’s effort, while x~ ¼ y~ 2 þ u~ 1y (i.e., m2 ¼ 0; q2 ¼ 1) is exclusively informative about second-period productivity.9 We also assume that the additive noise terms (u1,u2) are independent of ðy1 ; y2 Þ, and are joint normally distributed with zero means and variance–covariance matrix ! R11 r12 R¼ . r012 s22 Hence, the variance of the first-period information y1, the covariance between y1 and y2 , and the posterior variance of y2 given y1 are: Ry  var½y1  ¼ qq0 s2y þ R11 , Ryy  cov½y1 ; y2  ¼ qs2y 40;

(2) Ryy ¼ R0yy ,

(3)

and s2y2j1  var½y2 jy1  ¼ s22
r012 ðqq0 s2y þ R11 Þ
1 r12 .

(4)

2.1. First-best setting In a first-best setting where the agent’s efforts are observable, the principal’s choice of a1 is based on his prior expectation of y1 (i.e., y¯ 1 ), and his choice of a2 is based on the expectation of y2 conditional on all publicly available information at the beginning of period two. Thus, given effort costs of 12a21 and 12a22 , the first-best effort levels are a1 ¼ y¯ 1 and a2 ¼ Eðy2 jy1 Þ, where a2 is a linear increasing function of y1. The first-best expected firm profit is 2 Eðpfirst
best Þ ¼ E½y1 y¯ 1 þ y2 Eðy2 jy1 Þ
12y¯ 1
12Eðy2 jy1 Þ2  2

2

¼ 12y¯ 1 þ 12y¯ 2 þ 12Syy S
1 y Syy

ð5Þ

where the last term in (5) reflects the ex ante value of productivity information. 2.2. Second-best setting If a1 and a2 are not publicly observed, the solution to the agency problem depends on the ability of the principal and agent to make contractual commitments over two periods. In a (footnote continued) conjecture that the types of implicit incentives considered here also arise if the information signals are allowed to be multiplicative as in (say) y1 ¼ y1 a1 þ noise, where y1 and y2 are correlated so that y1 is simultaneously informative about the agent’s effort and future productivity. 9 We use the term ‘‘aggregate’’ to refer to systems that generate fewer reports (i.e., lower dimension) than ‘‘disaggregate’’ systems even if the reports in the aggregate system cannot be constructed by a linear combination of reports in the disaggregate system. That said, if u~ 1 ¼ u~ 1a þ u~ 1y , then the aggregate report y~ 1 can be generated as ~ the simple sum of the two disaggregate reports y~ 1a and x.

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Agent chooses action a1

0

Agent chooses action a2

1

Agent offered contract w1 = α+ β'1 (y1- µ1)

y1 observed.

2 y2 observed.

Agent offered contract w2 = α2 +
2 (y2 - 2|1) Fig. 1. Timeline for limited commitment setting.

full-commitment setting, the principal offers the agent a two-period contract (taking into account first-period information but with no renegotiation) and the agent commits to stay for both periods. However, while the principal and agent may agree ex ante to a twoperiod contract, this type of commitment may not be sustainable because there may be gains to reneging ex post. For example, the principal may benefit from replacing the agent or using past information to reset the terms of the contract, and the agent may benefit by leaving the employment relationship after the first period. In what follows, we consider both limited- and full-commitment two-period contracts. 3. Limited commitment A two-period limited commitment relationship can be portrayed in a variety of ways. We consider a setting that calls for active renegotiation at the start of the second period on the basis of first-period information. The initial contract commits both parties to stay in the relationship for two periods, but does not preclude the principal from making a ‘‘takeit-or-leave-it’’ offer to change the contract at the start of the second period.10 Fig. 1 describes the timeline of events. As is standard in dynamic models, we begin by characterizing the last choice and move sequentially until we characterize the first choice. Hence, we begin with the agent’s second-period effort choice, followed by the principal’s second-period contract renegotiation choice, then the agent’s first-period effort choice, and conclude with the principal’s initial contract choice. At the start of the second period, the principal makes a take-it-or-leave-it offer to change the agent’s second-period compensation. The proposed change is assumed to be a linear function of the second-period report, y2, minus the (posterior) expected second-period performance given first-period information y1. That is, w2 ¼ a2 þ b2 ðy2
m2j1 Þ,

(6)

where b2 is the second-period incentive coefficient, a2 is an adjustment to the fixed component of compensation that is sufficient to obtain the agent’s acceptance of the ^ 1 Þ is the principal’s contractual changes and m2j1 ¼ Eðy2 jy1 ; a^ 1 ; a^ 2 Þ ¼ a^ 2 þ r012 R
1 y ½y1
l1 ða 10 The initial contract is sufficient to cover the agent’s reservation wage for two periods assuming he provides some effort in the first period but zero effort in the second period. Equivalently, we can recast this as a renegotiation-proof two-period contract, i.e., there is no incentive to renegotiate at the start of the second period.

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posterior expected value of y2 given y1 and his conjectures a^ 1 and a^ 2 with respect to the agent’s first and second-period effort levels, respectively. Given the first-period information, y1, the agent’s first-period effort a1, and the proposed changes in the contract, the agent chooses a2 to maximize the expected utility of his net second-period wage, w2
12a22 . Expressing the agent’s objective in terms of his certainty equivalent, his second-period decision is Max a2

CE2 ðy1 ; a1 ; a2 Þ ¼ a2 þ b2 ½Eðy2 jy1 ; a1 ; a2 Þ
m2j1 
12 a22
12rb22 s2y2j1 ,

(7)

where Eðy2 jy1 ; a1 ; a2 Þ ¼ a2 þ r012 R
1 y ½y1
l1 ða1 Þ.

(8)

Hence, the solution to (7) is characterized by a2 ¼ b 2 .

(9)

That is, the agent’s second-period effort choice depends solely on the second-period incentive rate chosen by the principal. Given the first-period information y1 and the principal’s conjecture of the agent’s effort levels a^ 1 and a^ 2 ¼ b2 (from 9 above), the principal chooses the terms of the renegotiated second-period contract to maximize his expected incremental second-period payoff, p2 ¼ y2 b2
w2 , subject to the requirement that the renegotiated contract be conjectured to be acceptable to the agent, i.e., subject to CE 2 ðy1 ; a^ 1 ; a^ 2 ¼ b2 Þ ¼ a2
12b22
12rb22 s2y2j1 X0:

(10)

Setting (10) as an equality, we can solve for a2 . Hence, the principal’s problem can be expressed as an unconstrained choice of b2 given y1 and a^ 1 : Max b2

Eðp2 jy1 ; a^ 1 Þ ¼ b2 Eðy2 jy1 ; a^ 1 Þ
12b22 ½1 þ rs2y2j1 ,

(11)

for which the solution is11 b2 ðy1 ; a^ 1 Þ ¼ Eðy2 jy1 ; a^ 1 ÞB;

and a2 ðy1 ; a^ 1 Þ ¼ 12Eðy2 jy1 ; a^ 1 Þ2 B,

(12)

where ^ 1 Þ Eðy2 jy1 ; a^ 1 Þ ¼ y¯ 2 þ Ryy R
1 y ½y1
l1 ða

(13)

B  ½1 þ rs2y2j1 
1 ,

(14)

and

where B is the second-period risk premium adjustment that appears frequently throughout our analysis. We note that the optimal second-period incentive rate, b2 in (12), parallels the solution to a typical one-period LEN model in that it equals the risk-adjusted expected marginal payoff to the principal. The only difference is that both the marginal payoff Eðy2 jy1 ; a^ 1 Þ and the risk adjustment B take into account reported first-period information. Since 11

Additional details for all the derivations are in Appendix A.

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Eðy2 jy1 ; a^ 1 Þ is a linear combination of the performance measures in y1, and a1 ðb1 Þ is linear in y1, a2 ðy1 ; a^ 1 Þ in (12) implies that the proposed adjustment to the second-period fixed compensation (i.e., the reimbursement for the second-period effort cost and risk premium) is a quadratic function of y1.12 We complete the characterization of the second-period by deriving the principal’s expected second-period profit (evaluated ex ante in the first period). 2

Eðpsecond
best Þ ¼ 12B½y¯ 2 þ Ryy R
1 2 y Ryy .

(15)

This expression is the same as the second-period profit in the first-best case (expression 5), except that it is multiplied by the risk adjustment B which is less than one. Next we consider the agent’s first-period effort choice assuming that the initial contract offered by the principal is a linear function of the first-period information set y1. That is, we assume w1 ¼ a þ b01 ðy1
l1 Þ,

(16)

where a represents the fixed compensation, b1 represents the vector of first-period incentive coefficients, and l1 ¼ Eðy1 ; a^ 1 Þ is the prior expectation of the first-period information given the principal’s conjecture of the agent’s first-period effort a^ 1 . In the first period, the agent chooses a1 to maximize the expected utility of his net aggregate wages, Ef
exp½
rðw1 þ w2
12a21
12a22 Þg, anticipating that the principal will renegotiate at the end of the first period based on the reported information y1 (and conjecture a^ 1 ). That is, the agent anticipates that w2 is given by (6), and a2 and b2 in (6) are selected by the principal according to (12). Renegotiation based on y1 implies that, conditional on first-period information, the net incremental second-period compensation, w2
12a22 , is independent of first-period compensation, w1
12a21 . Hence, we can write the agent’s objective as Max Ef
exp½
rða þ b01 ðy1
m1 Þ
12a21 þ CE2 ðy1 ; a1 ; a2 ÞÞg, a1

(17)

where CE2 , the agent’s second-period certainty equivalent at the time of renegotiation, is CE2 ðy1 ; a1 ; a2 Þ ¼ a2 ðy1 Þ þ b2 ðy1 Þ½Eðy2 jy1 ; a1 ; b2 ðy1 Þ
m2j1 ðy1 ; a^ 1 Þ
12b2 ðy1 Þ2 ½1 þ rs2y2j1 , ^ 1 Þ. ¼
b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða

ð18Þ

In equilibrium, the agent’s second-period certainty equivalent is equal to his reservation wage of zero (i.e., a1 ¼ a^ 1 in (18) since the agent is fully compensated for his effort cost and risk). However, when the agent chooses his first-period effort he considers the possibility of earning more than his reservation wage by choosing an effort different from the principal’s conjecture. We refer to the impact of CE2 on the agent’s first-period effort choice via (18) as implicit or dynamic incentives. Differentiating (17) with respect to a1 yields the following characterization of the agent’s first-period effort: 0 0
1 a1 ðb1 Þ ¼ b01 m
y¯ 2 Br012 R
1 y m þ rb1 cov½y1 ; b2 ðy1 Þr12 Ry m, 12

(19)

This suggests that in a setting where managerial effort and firm productivity are multiplicative, a long-term full commitment contract cannot achieve the same result as a limited commitment linear contract unless the former contract is not constrained to be linear. This observation will prove useful when we characterize a twoperiod commitment contract in Section 5 and Appendix A.

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where m ¼ dl1 ða1 Þ=da1 ;

and cov½y1 ; b2 ðy1 Þ ¼ Ryy B.

(20)

Before discussing the agent’s first-period effort incentives in (19), we characterize the principal’s first-period contract choice. Expression (15) implies that the principal’s expected second-period net profit is independent of his initial contract choice. Hence, the principal’s first-period problem is to select the first-period incentive coefficients, b1, so as to maximize his expected first-period payoff minus a, which is set equal to the cost of the agent’s induced first-period effort and the cost of the agent’s first-period incentive risk. Thus, the principal’s problem is Max b1

E½y1 a1 ðb1 Þ
12½a1 ðb1 Þ2
2r b01 Ry b1 þ Eðpsecond
best Þ, 2

(21)

where a1 ðb1 Þ is given by (19) and Eðpsecond
best Þ is given by (15). Substituting for a1 ðb1 Þ, the 2 optimal b1 is b1 ¼ ðWW0 þ rRy Þ
1 W½y¯ 1 þ y¯ 2 m0 R
1 y r12 B,

(22)

where W  ðI þ rRyy Br012 R
1 y Þm. Finally, the principal’s expected first-period profit can easily be obtained by substituting (22) into (21), which provides the following result. Proposition 1. The equilibrium agency relationship with limited commitment and linear contracts is characterized by first-period effort level a1 ¼ a1 ðb1 Þ given by (19) and secondperiod effort level a2 ¼ b2 , where (b1 ; b2 ) are given by (22) and (12), respectively. The equilibrium profit is therefore given by 2
1 0 0 ¯ EðpÞ ¼ 12y¯ 1 W0 ½WW0 þ rRy 
1 W
r½y¯ 1 m0 þ 12y¯ 2 Br012 R
1 y mm ½WW þ rRy  r12 y2 B 2

þ 12B½y¯ 2 þ Ryy R
1 y Ryy .

ð23Þ

Proof. The proof follows by substituting for a1 ðb1 Þ in (21) and using the solution for (b1 ; b2 ) from (22) and (12), respectively. Additional details are in Appendix A. 4. Implications and consequences of limited commitment To understand the implications of the implicit incentives that arise in a dynamic environment, we restate expression (19) as follows: 0 0
1 a1 ¼ b01 m
y¯ 2 Br012 R
1 y m þ rBb1 Ryy r12 Ry m.

(24)

The first term on the right-hand side of (24) reflects the direct (or explicit) incentive from first-period compensation, while the second and third terms reflect the dynamic or implicit incentives due to impact of the second period on the agent’s first-period effort choice. We note that these implicit incentives reinforce or counteract the direct incentive if, and only if, the performance measures are intertemporally related via r12 (where r012 R
1 y in the implicit incentive terms represents the coefficient on y1 in Eðy2 jy1 Þ). Corollary 1. In a two-period agency relationship, the agent faces implicit incentives if, and only if, the information is intertemporally dependent (i.e., r12 a0).

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If r12 a0, then (24) suggests that the implicit incentives manifest in two distinct ways.13 The first effect, due to the term
y¯ 2 Br012 R
1 y m, is well known in the literature. It reflects the agent’s implicit incentive to alter his first-period effort to increase the ‘‘mean’’ of his aggregate compensation; in particular, to increase the mean of his anticipated secondperiod certainty equivalent, i.e., the mean of CE2 in expression (18). To illustrate, consider a setting where r12 is positive. Since decreasing a1 decreases y1, it also dampens the principal’s expectation of the agent’s second-period performance (i.e., if r12 is positive, then decreasing y1 decreases m2j1 ). From (18), this translates to a higher mean CE2 for the agent since, in effect, this implies that he is being held to a lower standard of future performance.14 Of course, if y1 and y2 are negatively correlated (i.e., r12 is negative), then the agent’s implicit incentive would be to increase a1 in order to ensure a lower future performance standard. The second effect, represented by the term rBb01 Ryy s012 R
1 y m, is a unique feature of our model. It arises only because first-period information y1 serves a dual purpose; it is informative about second-period productivity y2 (i.e., Ryy 40) and it is informative about first-period effort a1. In contrast to the implicit incentive to increase his mean compensation discussed above, the agent’s implicit incentive here is to alter a1 to decrease the riskiness of his aggregate compensation; in particular, by reducing the ‘‘covariance’’ between his first-period compensation and his second-period certainty equivalent, i.e., the covariance between b01 y1 and CE2 in expression (17).15 The impact of this second type of implicit incentive also depends on r12. To illustrate, consider a setting where w1 is increasing in y1 (i.e., b0 1 is positive) and CE2 is decreasing in y1 (because r12 is positive) so that w1 and CE2 in (17) are negatively correlated. It follows that the agent has an implicit incentive to increase a1 because increasing a1 increases y1, which makes w1 and CE2 even more negatively correlated. Hence, this implicit incentive is favorable in the sense that it reinforces the agent’s direct incentive to increase a1. On the other hand, if w1 and CE2 in (17) are positively correlated (e.g., if b01 is positive but r12 is negative), then the agent’s implicit incentive is to decrease a1 in order to reduce the positive correlation, thereby reducing the variance of his aggregate compensation. Hence, in this case, the agent’s implicit incentive is unfavorable in the sense that it offsets the agent’s direct incentive to increase a1. The preceding discussion suggests that firms may prefer or explicitly design accounting measurement systems to account for the favorable or unfavorable consequences of managers’ implicit incentives. To illustrate how the properties of accounting systems affect the implicit incentive that arises when first-period information y1 serves a dual role, we consider two alternative measurement systems that we label as aggregate and disaggregate systems. Since r12 a0 plays a central role in our analysis, it is useful to note that there is empirical evidence supporting both positive (i.e., r12 40) and negative (i.e., r12 o0) time-series correlation of accounting-based performance measures (see Christensen et al. 2005 for a discussion of this evidence). 14 In this spirit, Indjejikian and Nanda (1999) suggest that when r12 is positive, this mean adjustment is reminiscent of a ‘‘ratchet effect’’ phenomenon where an agent provides less effort in earlier periods to avoid being held to higher standards in the future. In contrast, Gibbons and Murphy (1992) interpret this mean adjustment as a ‘‘career concerns’’ phenomenon that motivates an agent to provide greater effort to enhance his future market wage. 15 Note that even though the agent’s effort in any period does not affect the variances of his performance measures, in a dynamic setting the agent’s effort can affect the variance of his aggregate compensation. 13

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Table 1 Aggregate accounting system Basic elements: y1 ¼ a1 þ y~ 2 þ u~ 1 (i.e., m ¼ q ¼ 1), y~ 2 ¼ a2 þ u~ 2 , R11 ¼ varðu1 Þ ¼ s21 , r12 ¼ s12 . First-best actions and payoff: a1 ¼ y¯ 1 and a2 ¼ Eðy2 jy1 Þ ¼ y¯ 2 þ

s2y

s2y ðy
a1 Þ, þ s21 1

1 2 1 2 1 1 ¯ 2 1 ¯ 2 1 s4y . Eðpfirst
best Þ ¼ y¯ 1 þ y¯ 2 þ Syy S
1 y Syy ¼ y1 þ y2 þ 2 2 2 2 2 2 s2y þ s21 Second-best contract: Second-period risk premium adjustment: 

1 s2 , B  1 þ r s22
2 12 2 sy þ s1 Bonus rates    y¯ 1 þ y¯ 2 Bðs12 =ðs2y þ s21 ÞÞ 1 þ rBs2y s12 =ðs2y þ s21 Þ b1 ¼ ,   2 1 þ rBs2y s12 =ðs2y þ s21 Þ þ rðs2y þ s21 Þ Eðb2 Þ ¼ y¯ 2 B.

(T1.1)

(T1.2)

(T1.3)

Induced effort: a1 ðb1 Þ ¼ b1


y¯ 2 B rBs2 s12 þ 2 y 2 b1 s12 , sy þ s1 s2y þ s21

E½a2 ðb2 Þ ¼ Eðb2 Þ.

(T1.4) (T1.5)

4.1. Aggregate system Consider the setting and results described in Table 1. We refer to this as an aggregate accounting system since there is a single first-period report y1 that contains information about both the agent’s first-period effort and his second-period marginal productivity. Our focus is on T1.4. As before, the first term, b1 , represents the direct incentive that motivates a1 while the second and third terms represent the two types of implicit incentives discussed above. We note that, if s12 40, then the direct incentive b1 is positive (from T1.2) but the two types of implicit incentives have countervailing effects. The agent has an incentive to decrease a1 to increase the mean of his compensation which offsets the direct incentive, but at the same time has an incentive to increase a1 to reduce the riskiness of his compensation which reinforces the direct incentive. The two types of implicit incentives are also countervailing if s12 o0 (and b1 is positive); now the agent has an incentive to increase a1 to increase the mean of his compensation which reinforces the direct incentive, but at the same time has an incentive to decrease a1 to reduce the riskiness of his compensation which offsets the direct incentive.16 This latter Although we typically expect b1 in (T1.2) to be positive, in a two-period model with s12 o0, b1 may be negative if the second period is sufficiently more important than the first period (e.g., y¯ 2 by¯ 1 ). In this latter case, 16

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effect arises because, with a single metric of the form y1 ¼ a1 þ y~ 2 þ u~ 1 that combines information about a1 and y2 , the firm is compelled to use a single coefficient b1 to provide the agent with a direct incentive, but in doing so bears the cost of an offsetting implicit incentive via the third term in T1.4. 4.2. Disaggregate system A natural alternative to the aggregate system discussed above is a measurement system that decouples performance information from productivity information. In particular, consider the disaggregate system described in Table 2 where the first-period report y1 consists of two special-purpose signals fy1a ; xg. Signal y1a is informative about the agent’s first-period effort, while signal x is informative about second-period marginal productivity; in this way, explicit incentives to motivate a1 are, in principle, separate from the implicit incentive that arises because first-period information is informative about second-period productivity y2 . Our focus is on T2.5. As before, the first term, b1y , represents the direct or explicit incentive that motivates a1 while the second and third terms represent the two types of implicit incentives discussed above. Note that b1x a0 which means that the principal optimally uses the special-purpose productivity signal x ¼ y2 þ u1y as a contracting variable to motivate a1 even though the signal is not directly influenced by the agent’s effort nor is it correlated with y1a ¼ a1 þ u1a . The important insight here is that, with two special-purpose signals fy1a ; xg, b1y is optimally set to directly motivate a1 while b1x is optimally set so that increasing a1 always induces a greater negative correlation between the agent’s first-period compensation and second-period certainty equivalent at the time of renegotiation. That is, the agent’s implicit incentive, the rs2y b1x Bðcovðu1a ; u2 Þ=varðu1a ÞÞ term in (T2.5), equals b1y

½rs2y B covðu1a ; u2 Þ2 , varðu1a Þ½s2y þ varðu1y Þ

which always reinforces the direct incentive b1y . 4.3. A comparison of aggregate and disaggregate systems As our final illustration of the consequences of using dual-purpose accounting systems, we compare the aggregate and disaggregate accounting systems illustrated in the previous two subsections. In general, when there are no implicit incentives, we expect firms to prefer disaggregate accounting systems that report two special-purpose signals as described in Table 2. For instance, relative to aggregate accounting systems, disaggregate systems have the potential to generate more reliable signals of future productivity as well as generate less ‘‘noisy’’ indicators of current managerial performance uncontaminated by other information. For example, if var(u~ 1 ) in Table 1 is comparable to var(u~ 1a ) in Table 2, then ðy1 ; y2 Þ in Table 1 are clearly less ‘‘informative’’ than ðy1a ; y2 Þ in Table 2 in the sense implied by Holmstrom (1979)—i.e., var(y1)4var(y1a) and varðy2 j y1 ÞXvarðy2 j y1a Þ. Similarly, if we assume that (footnote continued) the direct incentive via b1 motivates the agent to decrease a1 but both implicit incentives motivate the agent to increase a1 .

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Table 2 Disaggregate accounting system Basic elements: y1 ¼ ðy1a ; xÞ, y~ 1a ¼ a1 þ u~ 1a , x~ ¼ y~ 2 þ u~ 1y , y~ 2 ¼ a2 þ u~ 2 m ¼ (1,0)0 , q ¼ (0,1)0 , u1 ¼ ðu1a ; u1y Þ0 ), which is independent of y2 , u1y is independent of u1a and u2 so that R11 is a diagonal matrix, and r12 ¼ ½covðu1a ; u2 Þ; 0. First-best actions and payoff: s2y x, a1 ¼ y¯ 1 and a2 ¼ Eðy2 jy1 Þ ¼ y¯ 2 þ 2 sy þ varðu1y Þ 1 2 1 2 1 1 ¯2 1 ¯2 1 s4y . Eðpfirst
best Þ ¼ y¯ 1 þ y¯ 2 þ Ryy R
1 y Ryy ¼ y1 þ y2 þ 2 2 2 2 2 2 s2y þ varðu1y Þ Second-best contract: Second-period risk premium adjustment: 

1 covðu1a ; u2 Þ2 . B  1 þ r s22
varðu1a Þ Bonus rates: b1y ¼



 y¯ 1 þ y¯ 2 Bðcovðu1a ; u2 Þ=varðu1a ÞÞ

1 þ rvarðu1a Þ þ ½rBs2y covðu1a ; u2 Þ2 =varðu1a Þ½s2y þ varðu1y Þ

b1x ¼ b1y

(T2.1)

,

rs2y B covðu1a ; u2 Þ , s2y þ varðu1y Þ

Eðb2 Þ ¼ y¯ 2 B.

(T2.2)

(T2.3) (T2.4)

Induced effort: covðu1a ; u2 Þ covðu1a ; u2 Þ þ rs2y b1x B , a1 ðb1y ; b1x Þ ¼ b1y
y¯ 2 B varðu1a Þ varðu1a Þ

(T2.5)

E½a2 ðb2 Þ ¼ Eðb2 Þ.

(T2.6)

u~ 1 in Table 1 equals u~ 1a þ u~ 1y in Table 2 so that the aggregate report is a simple sum of the two special-purpose signals (i.e., y1 ¼ a1 þ y~ 2 þ u~ 1a þ u~ 1y ), then the special purpose signal x in Section 4.2 is clearly a more reliable signal of y2 than y1 .17 That said, we seek a comparison between the two systems that revolves around the consequences of implicit incentives (since that is the focus of our paper) without exogenously conferring an informational advantage to either the aggregate or disaggregate system. In this spirit, we assume that var(y1) in Table 1 equals var(y1a) in Table 2 and similarly varðy2 j y1 ) in Table 1 equals varðy2 j y1a ) in Table 2 so that the informational properties of the metrics used to evaluate managerial performance are a priori identical. At the same time, we assume that var(y1) in Table 1 equals var(x) in Table 2 so that the two systems are equally reliable signals about y2 . Given the notation in Tables 1 and 2, this calibration requires that varðu1y Þ ¼ s21 , covðu1a ; u2 Þ ¼ s12 , and varðu1a Þ ¼ s2y þ s21 . These assumptions jointly ensure that (i) the first-best payoff is the same for both systems and (ii) 17

This is obvious from the first-best payoffs in Tables 1 and 2 since s2y þ varðu1 Þ4s2y þ varðu1y Þ.

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Table 3 Comparison of induced first-period effort for aggregate versus disaggregate accounting systems, with limited commitment
 s12 ¯ ¯ D, (T3.1) aI1
aD 1 ¼ y1 þ y2 B 2 sy þ s21 

where D  

2  2 1 þ rBs2y s12 =ðs2y þ s21 Þ 1 þ rBs2y s12 =ðs2y þ s21 Þ
2  2 1 þ rBs2y s12 =ðs2y þ s21 Þ þ rðs2y þ s21 Þ 1 þ rBs2y s12 =ðs2y þ s21 Þ þ rðs2y þ s21 Þ


and B  1 þ r s22


s2y

s212 þ s21

(T3.2)


1 .

in a second-best setting, the agent’s second-period effort, and consequently second-period firm profit, are the same under both systems. Thus, a firm’s preference for one system over another is solely attributed to the agent’s induced first-period effort. Table 3 above illustrates the difference between the agent’s first-period effort under an aggregate system (illustrated earlier in Table 1, expression T1.4) and his first-period effort under a disaggregate system (illustrated earlier in Table 2, expression T2.5). We note that D in Table 3 is positive when s12 is positive and vice versa. Hence for y¯ 2 not too large relative to y¯ 1 , we obtain the following result. Observation 1. With limited commitment, the principal prefers an aggregate accounting system if s12 40, but prefers a disaggregate accounting system otherwise. The intuition for this result is straightforward. When s12 40, the agent’s implicit incentive to reduce the variance of his aggregate compensation reinforces his direct incentive under both systems. For an aggregate system, this reinforcing implicit incentive is a (‘‘free’’) byproduct of providing the agent a direct incentive via b1 . The corresponding incentive in a disaggregate system is optimally induced via the choice of b1x , but the use of x as a contracting variable requires the payment of an additional risk premium. Hence, when s12 40, a firm prefers an aggregate system because it is a less costly way to provide comparable first-period incentives. When s12 o0, the agent’s implicit incentive to reduce the variance of his aggregate compensation offsets his direct incentive under an aggregate system. This is because with a single performance metric (that combines information about a1 and y2 ), the firm is compelled to use a single incentive coefficient to provide a direct incentive, but in doing so bears the cost of an offsetting implicit incentive as discussed earlier in Section 4.1. In contrast, with a disaggregate system the agent’s induced incentive to reduce the variance of his aggregate compensation always reinforces his direct incentive as evident from Table 2. Hence, when s12 o0, a firm prefers a disaggregate system. 5. Two-period contracts 5.1. Renegotiation-proof contracts In Sections 3 and 4 we assumed that the second-period component of the initial contract specified only a fixed wage, and the second-period incentive wage was determined by

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renegotiation at the end of the first period. Observe that the same result can be achieved with a two-period contract that has the following form: w ¼ a þ b01 ðy1
l1 Þ þ b2 ðy1 Þ½y2
m2j1 ðy1 Þ þ a2 ðy1 Þ,

(25)

where b2 ðy1 Þ and a2 ðy1 Þ are as specified by (12), and b1 is as specified by (22). It is straightforward to prove that if (25) is the initial contract, then renegotiation at the end of the first period will not result in any change in the contract—it is renegotiationproof. Importantly, we note that the two-period renegotiation-proof contract in (25) is not linear. The first three components are linear functions of y1 and y2, but the fourth component, a2(y1), is not. It is a constant if there is no productivity information (i.e., the vector q is zero), but otherwise a2 ðy1 ; a^ 1 Þ ¼ 12Eðy2 jy1 ; a^ 1 Þ2 B is a quadratic function of y1. This implies that a two-period renegotiation-proof linear contract cannot achieve the same result as the renegotiable short-term linear contracts considered in Section 3.18 To achieve the same results with a renegotiation-proof two-period contract as we did in Section 3, the feasible two-period contract must be nonlinear. But what nonlinearities do we permit? Consistent with (25), our answer is to restrict the two-period contracts to those that take the following form:19 w ¼ a þ b01 ðy1
l1 Þ þ b2 ðy1 Þ½y2
m2j1 ðy1 Þ þ 12a^ 2 ðy1 Þ2 þ 2r b2 ðy1 Þ2 s2y2j1

(26)

with b2 ðy1 Þ ¼ b0 þ b01 ðy1
l1 Þ restricted to be a linear in y1. This approach allows the twoperiod contract to provide the agent with full insurance against two types of risk: ‘‘effortcost risk,’’ which occurs because the agent’s action varies with y1, and a ‘‘risk-premium risk,’’ which occurs because the second-period incentive risk premium depends on the second-period incentive rate b2(y1), which also varies with y1. The insurance corresponding to the conjectured levels of effort-cost and risk-premium risk are therefore 12a^ 2 ðy1 Þ2 and 2 2 r 2b2 ðy1 Þ sy2j1 , respectively. 5.2. Full-commitment contracts We now consider a two-period full-commitment contract of the form in (26) in which the principal and agent commit to the employment relationship for two periods, thereby precluding the possibility of renegotiation at the end of the first period. In this setting, the principal selects all the compensation coefficients a, b1, b0 and b1 in (26) to maximize the firm’s ex-ante value taking into account the agent’s opportunity wage, and his optimal first- and second-period effort choices. In the interest of brevity, we relegate the derivation of the two-period full-commitment contract to Appendix A. 18

Previous multi-period LEN models have usually retained the linearity assumption on any two-period contracts that are considered. This is because, with no productivity information, linear renegotiation-proof twoperiod contracts are equivalent to linear short-term contracts, and the optimal linear two-period full-commitment contract is superior to short-term contracts (see for example, Christensen et al., 2003b). 19 Expanding the set of feasible long-term contracts to permit only some non-linearities, but not all, is admittedly ad hoc. However, it is consistent with the spirit of restricting compensation contracts to be linear. Effectively, the long-term contract in (26) is sequentially linear, except that the ‘‘fixed’’ component of the secondperiod is a quadratic function of the first-period information.

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It suffices to say however that the principal’s objective in this setting corresponds to his objective in the limited commitment setting considered in Section 3 if we constrain the choice of b0 and b1 coefficients so that b2 ðy1 Þ ¼ b0 þ b01 ðy1
l1 Þ in (26) equals Eðy2 jy1 ÞB given by (12). Hence, consistent with the prior literature, it follows that firm value with a two-period full-commitment contract is (weakly) greater than firm value with a limited commitment contract. Importantly, we note that with a full commitment contract, a comparison of the sort illustrated in Section 4.3 typically favors a disaggregate system. This occurs because the principal’s ex ante choice of the second-period compensation coefficients, (b0,b1), affords him an extra degree of freedom when the first-period information set is comprised of two signals rather than one (i.e., both elements of b1 are nonzero). This degree of freedom is unavailable with a limited commitment contract since sequential rationality in setting the second-period compensation coefficient restricts the choice of b1. Of course, as in Section 4.3, this comparison is reasonable as long as neither system is inherently advantaged when the firm’s informational demands are considered in isolation. Thus, for the calibrated comparison described in Section 4.3, we have the following observation. Observation 2. Ceteris paribus, with a two-period commitment contract, a firm prefers a disaggregate accounting system. Proof. A detailed proof is provided in Appendix A. 6. Conclusion Recent literature has recognized the role of implicit incentives in shaping organizational design and responsibility accounting practices. Insights from this literature suggest that ‘‘more’’ information is not always preferred in dynamic agency relationships since ‘‘less’’ information may be a commitment device that ameliorates the negative implicit incentives that may arise. In the same vein, we present a two-period linear principal agent model to study how the multi-period nature of an employer-employee relationship affects the demand for multi-purpose accounting information. Although our model is stylized, it captures three salient features of practical relevance. First, we explicitly consider the multipurpose nature of the demand for accounting information. Second, we highlight the presence of an intertemporal information link by allowing reported managerial performance to be correlated through time. And third, we illustrate the role of contractual commitments in multi-period relationships. We find that a firm’s choice among different measurement systems (e.g., between a dualpurpose aggregate measure and a disaggregate system with two single-purpose measures) is sensitive to the level of commitment a firm enjoys with its employees or managers. For instance, if the parties commit to a multi-period employment relationship, a firm unambiguously prefers a disaggregate system, i.e., ‘‘more’’ information is better. However if employment commitments are limited, a dual-purpose aggregate measure may indeed be preferable, i.e., ‘‘less’’ information is better. More generally, our results suggest that a firm’s choice of a measurement system depends on indirect, and in particular dynamic, benefits (or costs) that are likely to be as consequential as more visible out-of-pocket expenditures.

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Appendix A This appendix provides details of some of the analysis in the text. Limited commitment (Section 3) The principal’s second-period problem: The principal’s second-period problem is Max Eðp2 jy1 ; a^ 1 ; a^ 2 Þ ¼ Eðy2 a2
a2
b2 ðy2
m2j1 Þjy1 ; a^ 1 ; a^ 2 Þ, a2 ; b2 subject to

a 2 ¼ b2 ,

(A.1)

(A.2)

CE 2 ðy1 ; a^ 1 ; a^ 2 ¼ b2 Þ ¼ a2
12b22
12rb22 s2y2j1 X0.

(A.3)

0 2 where s2y2j1  var½y2 jy1  ¼ s22
r012 S
1 y r12 and Sy  qq sy þ S11 . Solving (A.3) as an equality, we have a2 ¼ 12b22 þ 12rb22 s2y2j1 . Hence, (A.1) can be restated as

Max b2

Eðp2 jy1 ; a^ 1 ; a^ 2 Þ ¼ b2 Eðy2 jy1 ; a^ 1 Þ
12b22 ð1 þ rs2y2j1 Þ.

(A.4)

The solution to (A4) is b2 ðy1 ; a^ 1 Þ ¼ Eðy2 jy1 ; a^ 1 ÞB;

and

a2 ðy1 ; a^ 1 Þ ¼ 12Eðy2 jy1 ; a^ 1 Þ2 B,

(A.5)

where ^ 1 Þ; and B  ½1 þ rs2y2j1 
1 . Eðy2 jy1 ; a^ 1 Þ ¼ y¯ 2 þ Ryy R
1 y ½y1
l1 ða

(A.6)

Substituting (A.5) into (A.4), we have Eðp2 jy1 ; a^ 1 ; a^ 2 Þ ¼ 12Eðy2 jy1 ; a^ 1 Þ2 B

(A.7)

and thus expression (15) in Section 3 can be derived as 2 Eðp2sec ond
best Þ ¼ E½Eðp2 jy1 ; a^ 1 ; a^ 2 Þ ¼ 12B½y¯ 2 þ Ryy R
1 y Ryy ,

(A.8)

where Ryy  cov½y1 ; y2  ¼ qs2y and Ry is defined above. The agent’s first-period effort choice: The general form of the agent’s expected utility at t ¼ 0 is given in (17) using the agent’s first-period compensation and effort cost, plus the second-period certainty equivalent (conditional on the first-period report), which is derived in (18). The agent chooses a1 (given the principal’s conjecture a^ 1 ) to solve (17) for which the certainty equivalent is CEða^ 1 ; a1 Þ ¼ a þ b01 ½l1 ða1 Þ
l1 ða^ 1 Þ
12a21
2r b01 Ry b1 ^ 1 Þ
E½b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða ^ 1 Þ
2r var½b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða ^ 1 Þ. þ rb01 cov½y1 ; b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða

ðA:9Þ

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Differentiating with respect to a1 yields the following first-period effort20: a ðb1 Þ ¼ b0 m
y¯ 2 Br0 R
1 m þ rb0 cov½y1 ; b2 ðy1 Þr0 R
1 m. 1

1

12

y

1

12

y

The principal’s first-period problem: The principal’s first-period problem is Max Eðp1 Þ ¼ E½y1 a1
a
b01 ðy1
l1 Þ þ Eðpsecond
best Þ 2 a; b1

(A.10)

(A.11)

subject to a1(b) in (A.10) and a
12½a1 ðb1 Þ2
2r b01 Ry b1 X0.

(A.12)

Solving (A.12) as an equality, we have a ¼ 12½a1 ðb1 Þ2 þ 2r b01 Ry b1 . Hence, (A.11) becomes Max E½y1 a1 ðb1 Þ
12½a1 ðb1 Þ2
2r b01 Ry b1 þ Eðpsecond
best Þ. (A.13) 2 b1 The solution to (A.13) is b1 ¼ ðWW0 þ rRy Þ
1 W½y¯ 1 þ y¯ 2 m0 R
1 y r12 B,

(A.14)

where W  ðI þ rRyy Br012 R
1 y Þm and m is a vector of ones and zeros with the same dimension as y1. Substituting (A.14) into (A.13) and using (A.8) we have 2 EðpÞ ¼ 1y¯ W0 ½WW0 þ rRy 
1 W
r½y¯ 1 m0 þ 1y¯ 2 Br0 R
1 mm0  2 1

2

 ½WW þ rRy  r12 y¯ 2 B þ 0


1

1 ¯2 2B½y2

þ

12

y

Ryy R
1 y Ryy .

ðA:15Þ

Full commitment (Section 5) Agent’s two-period problem: As discussed in Section 5, we assume the two-period contract has the following form w ¼ a þ b01 ðy1
l1 Þ þ b2 ðy1 Þ½y2
m2j1 ðy1 Þ þ 12a^ 2 ðy1 Þ2 þ 2r b2 ðy1 Þ2 s2y2j1 .

(A.16)

Given (A.15), the agent selects his second-period effort to maximize the following: Max Ef
exp½
rðw
12a21
12a22 Þjy1 g. a2

(A.17)

Integrating with respect to y2, we have Max a2


expf
r½a þ b01 ðy1
l1 Þ þ b2 ðy1 Þ½m2 ða2 Þ
m2 ða^ 2 Þ
12a21

^ 1 Þg.
12a22 þ 12ða^ 2 Þ2
b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða

ðA:18Þ

Hence, a2 ¼ b2 ðy1 Þ as in Section 3. Setting a2 ¼ a^ 2 , the agent’s optimal certainty equivalent at the end of the first (or beginning of the second) period can be expressed as ^ 1 Þ. CE1 ¼ a þ b01 ðy1
l1 Þ
12a21
b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða 20

(A.19)

Note that the agent’s first-order condition that solves (A.9) depends on a^ 1 . Expression (A.10) reflects the fact that the principal has rational expectations, and hence in equilibrium a1 ¼ a^ 1 .

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In the first period, the agent solves Max E½
expð
rCE1 Þ. a1

(A.20)

Since b2 ðy1 Þ is linear in y1, the certainty equivalent in (A.19) is also linear in y1. We integrate (A.20) with respect to y1 and obtain an expression that mirrors (A.9) in the limited commitment setting above: CEða^ 1 ; a1 Þ ¼ a þ b01 ½l1 ða1 Þ
l1 ða^ 1 Þ
12a21
2r b01 Ry b1 ^ 1 Þ
E½b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða ^ 1 Þ
2r var½b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða ^ 1 Þ, þ rb01 cov½y1 ; b2 ðy1 Þr012 R
1 y ½l1 ða1 Þ
l1 ða

ðA:21Þ

where E½b2 ðy1 Þ ¼ b0 and cov½y1 ; b2 ðy1 Þ ¼ Ry b1 . Differentiating (A.21) with respect to a1 yields the following: 0 0
1 a1 ðb1 ; b0 ; b1 Þ ¼ b01 m
b0 r012 R
1 y m þ rb1 Ry b1 r12 Ry m,

(A.22)

which parallels (A.10) or expression (19) in Section 3, except that the compensation coefficients b1, b0 and b1 will (optimally) differ. The principal’s problem: The principal selects a, b1, b0 and b1 to maximize the firm’s exante value y¯ 1 a1 ðb1 ; b0 ; b1 Þ þ E½y2 a2 ðb2 ðy1 ÞÞ
EðwÞ taking into account the agent’s opportunity wage, a ¼ 12ða1 Þ2 þ 2r b01 Ry b1 , and his optimal effort choices a1 ðb1 ; b0 ; b1 Þ in (A.22) and a2 ¼ b2 ðy1 Þ ¼ b0 þ b01 ðy1
l1 Þ. Substituting for a, we have Max b1 ; b0 ; b1

y¯ 1 a1 ðb1 ; b0 ; b1 Þ
12½a1 ðb1 ; b0 ; b1 Þ2
2r b01 Ry b1

þ E½y2 ðb0 þ b01 ðy1
l1 ÞÞ
12E½ðb0 þ b01 ðy1
l1 ÞÞ2 ½1 þ rs2y2j1 .

ðA:23Þ

In the interest of brevity, we do not present the optimal coefficients nor do we characterize the firm’s profit in the full commitment setting. It suffices to say however that the objective function above corresponds to (A.13) in the limited commitment setting if we 21 constrain ðb0 ; b1 Þ in b2 ðy1 Þ ¼ b0 þ b01 ðy1
l1 Þ such that b0 ¼ y¯ 2 B and b01 ¼ Ryy R
1 y B. Hence, it follows that firm value with a two-period full-commitment contract is (weakly) greater than firm value with a limited commitment contract. Of course, if the optimal b0 does not equal y¯ 2 B and/or the optimal b01 does not equal Ryy R
1 y B, then firm value with a two-period full-commitment contract is strictly greater than firm value with a limited commitment contract. 21 Christensen and Feltham (2005, Chapter 27) analyze both optimal full-commitment contracts (i.e., with no restrictions on the contract form) and quadratic–exponential–normal (QEN) full-commitment contracts (which are similar to the contracts considered in this paper) for a setting similar to the aggregate system examined in this paper. Unlike a limited commitment setting, they establish that it is optimal to use contracts in which b2 varies with y1 (i.e., b1 is non-zero) even if y1 is uninformative about y2.

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436

Proof of observation 2: To compare a disaggregate system with an aggregate system under full commitment, we rewrite (A.23) for the two respective systems using the parameterizations in Tables 1 and 2 in Section 4. For an aggregate system, we have y1 ¼ a1 þ y~ 2 þ u~ 1 (i.e., m ¼ q ¼ 1), y~ 2 ¼ a2 þ u~ 2 , R11 ¼ varðu1 Þ ¼ s21 , and r12 ¼ s12 . Hence, (A.23) is restated as Max bI1 ; bI0 ; bI1

y¯ 1 a1 ðbI1 ; bI0 ; bI1 Þ
12a1 ðbI1 ; bI0 ; bI1 Þ2
2r ðbI1 Þ2 ðs2y þ s21 Þ

þ y¯ 2 bI0 þ bI1 s2y
12½ðbI0 Þ2 þ ðbI1 Þ2 ðs2y þ s21 Þ½1 þ r varðy2 jyI1 Þ,

ðA:24Þ

where a1 ðbI1 ; bI0 ; bI1 Þ ¼ bI1 ð1 þ rs12 bI1 Þ
bI0 s12 =ðs2y þ s21 Þ

(A.25)

and the superscript ‘‘I’’ indicates an aggregate system. For a disaggregate system, we have (from Table 2) y1a ¼ a1 þ u~ 1a , x ¼ y~ 2 þ u~ 1y (i.e., m ¼ ð1; 0Þ0 and q ¼ ð0; 1Þ0 ), y~ 2 ¼ a2 þ u~ 2 , u1 ¼ ðu1a ; u1y Þ0 is independent of y2 , R11 is a diagonal matrix with elements varðu1a Þ and varðu1y Þ, and r12 ¼ ½covðu1a ; u2 Þ; 0. Hence, (A.23) is restated as Max b1 ; bD 0 ; b1

2 y¯ 1 a1 ðb1 ; b0 ; b1 Þ
12a1 ðb1 ; b0 ; b1 Þ2
2r ðbD 1 Þ varðu1a Þ

2
2r ðb1x Þ2 ½s2y þ varðu1y Þ þ y¯ 2 bD 0 þ b1x sy 2 D 2 2 2 D
12½ðbD 0 Þ þ ðb1y Þ varðu1a Þ þ ðb1x Þ ðsy þ varðu1y ÞÞ½1 þ rvarðy2 jy1 Þ,

ðA:26Þ

where D D 2 a1 ðb1 ; b0 ; b1 Þ ¼ bD 1y ð1 þ rs12 b1y Þ
b0 s12 =varðu1a Þ þ b1x b1x rs12 ½sy þ varðu1y Þ=varðu1a Þ,

(A.27) 0 ðbD 1y ; b1x Þ ,

0 ðbD 1y ; b1x Þ ,

b1 ¼ b1 ¼ and the superscript ‘‘D’’ indicates a disaggregate system. Now, if we calibrate the disaggregate and aggregate systems as in Section 4.3 with varðu1y Þ ¼ s21 , covðu1a ; u2 Þ ¼ s12 , and varðu1a Þ ¼ s2y þ s21 , then varðy2 jy1a Þ ¼ varðy2 jy1 Þ ¼ s2y2j1 and (A.26) can be restated as Max b1 ; bD 0 ; b1

2 2 2 y¯ 1 a1 ðb1 ; b0 ; b1 Þ
12a1 ðb1 ; b0 ; b1 Þ2
2r ðbD 1y Þ ðsy þ s1 Þ

2
2r ðb1x Þ2 ðs2y þ s21 Þ þ y¯ 2 bD 0 þ b1x sy 2 D 2 2 2 2 2 12½ðbD 0 Þ þ ðsy þ s1 Þððb1y Þ þ ðb1x Þ Þð1 þ rsy2j1 Þ.

ðA:28Þ

To show that the disaggregate system weakly dominates the aggregate system, it suffices to show that the objective function for an aggregate system in (A.24) can be expressed as a constrained version of the objective function for a disaggregate system in (A.28). If we constrain the maximization problem in (A.28) by setting b1x ¼ 0 and b1x
12b21x

ð1 þ s21 =s2y Þ ¼ bD 1y , ð1 þ rs2y2j1 Þ

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437

then (A.27) can be expressed as D D D D D 2 2 a1 ðb1 ; b0 ; b1 Þ ¼ a1 ðbD 1y ; b0 ; b1y Þ ¼ b1y ð1 þ rs12 b1y Þ
b0 s12 =ðsy þ s1 Þ

(A.29)

and (A28) can be rewritten as Max D D bD 1y ; b0 ; b1y

D D D D D 2 2 1 r D 2 2 y¯ 1 a1 ðbD 1y ; b0 ; b1y Þ
2a1 ðb1y ; b0 ; b1y Þ
2ðb1y Þ ðsy þ s1 Þ

D 2 D 2 D 2 2 2 2 1 þ y¯ 2 bD 0 þ b1y sy
2½ðb0 Þ þ ðb1y Þ ðsy þ s1 Þð1 þ rsy2j1 Þ,

ðA:30Þ

which is equivalent to (A.24). Hence, under full commitment, a disaggregate system is weakly preferable over an aggregate system. Finally, we can easily show that the optimal solution to (A.28) implies that b1x a0 which means that the constraint suggested in the previous paragraph is binding. This is sufficient to ensure that the disaggregate system strictly dominates the aggregate system. References Arya, A., Glover, J.C., Sivaramakrishnan, K., 1997. The interaction between decision and control problems and the value of information. The Accounting Review 72, 561–574. Baiman, S., Demski, J.S., 1980. Economically optimal performance evaluation and control systems. Journal of Accounting Research 18 (Suppl.), 184–220. Christensen, P.O., Feltham, G.A., 2005. Economics of Accounting: Volume II—Performance Evaluation. Kluwer Academic Publishers, Boston. Christensen, P.O., Feltham, G.A., Sabac, F., 2003a. Dynamic incentives and responsibility accounting: a comment. Journal of Accounting and Economics 35, 423–436. Christensen, P.O., Feltham, G.A., Hofmann, C., Sabac, F., 2003b, Timeliness, accuracy, and relevance in dynamic incentive contracts. Working Paper, University of British Columbia. Christensen, P.O., Feltham, G.A., Sabac, F., 2004. A contracting perspective on earnings quality. Working Paper, University of British Columbia. Demski, J.S., Feltham, G.A., 1976. Cost Determination: A Conceptual Approach. Iowa State University Press, Ames. Gibbons, R., Murphy, K.J., 1992. Optimal incentive contracts in the presence of career concerns: theory and evidence. Journal of Political Economy 100, 468–505. Holmstrom, B., 1979. Moral hazard and observability. Bell Journal of Economics 10, 74–91. Indjejikian, R.J., Nanda, D., 1999. Dynamic incentives and responsibility accounting. Journal of Accounting and Economics 27, 177–202. Indjejikian, R.J., Nanda, D., 2003. Reply to: dynamic incentives and responsibility accounting: a comment. Journal of Accounting and Economics 35, 437–441. Johnson and Johnson, 2001. Securities and Exchange Commission Proxy filings for the year 2000. Meyer, M.A., 1995. Cooperation and competition in organizations: a dynamic perspective. European Economic Review 39, 709–722. Meyer, M.A., Vickers, J., 1997. Performance comparisons and dynamic incentives. Journal of Political Economy 105, 547–581. Meyer, M.A., Olsen, T.E., Torsvik, G., 1996. Limited intertemporal commitment and job design. Journal of Economic Behavior and Organization 31, 401–417. Milgrom, P., Roberts, J., 1992. Economics, Organization and Management. Prentice Hall, Englewood Cliffs, NJ. Narayanan, V.G., Davila, A., 1998. Using delegation and control systems to mitigate the trade-off between the performance-evaluation and belief-revision uses of accounting information. Journal of Accounting and Economics 25, 255–282. Sabac, F., 2004. Dynamic moral hazard with multiple correlated periods and renegotiation: managerial tenure and incentives. Working Paper, University of Alberta. Zimmerman, J.L., 2000. Accounting for Decision Making and Control, third ed. Irwin McGraw Hill, Chicago.