Can earnings manipulation create value?
The corporate scandals of late 1990s and early 2000s raised heated debates regarding the earnings manipulations (EM) by firms' insiders. (1) One line of research about EM analyzes factors leading to EM and optimal contracts between shareholders and managers. Examples include Levitt and Snyder (1997), Degeorge et al. (1998), Goldman and Slezak (2006), and Crocker et al. (2007). These papers typically consider models with a static principal-agent relationship, where agent provides costly productive effort and where the agent can manipulate information related to the results of production. EM is socially inefficient. If EM would be impossible, it would improve the social welfare. (2,3) Sarbannes-Oxley act (2002) emerged as a response to negative perception of EM by the public in the beginning of the century. Since then, some progress has been achieved in terms of preventing accounting manipulations and fraud (see, for example, Liu, Liu and Yin, 2014). Debates are still going whether tightening the regulation improves social efficiency (see, for example, Zhang (2007) and Ewert and Waggenhoffer (2013)).
In this paper we analyze a two-period relationship between a board of directors and a CEO. We compare two situations. In the first, the CEO chooses only a costly productive effort--assuming that the CEO cannot be involved in EM. In the second, the CEO is subject to a double-moral hazard problem which includes the choice of productive effort and the EM decision. It is shown that the parties expected payoffs can be higher in the second case. The paper thus contributes to the growing literature that considers EM in the context of long-term principal-agent relationship. Christensen et al. (2013) argue that prohibiting EM can be inefficient because it signals the principal inability to commit in the long-term. Ewert and Waggenhoffer (2013) argue that tightening sanctions for EM can be positive for some types of EM and negative for other types of EM. Drymiotes and Hemmer (2012) study the choice of accrual strategy ("conservative" or "aggressive") and show that in many cases it will be beneficial to delegate the strategy choice to a manager and it should not necessarily be a "conservative" or "no EM" strategy. Also, it's been noted in accounting literature that an incentive for EM and some tolerance towards EM can be a result of better risk-allocation when the agent is risk-averse. (4) In our paper, the agent is risk-neutral. Thus our results can be applied to situations when risk-sharing is possible or when the agent is risk-neutral or close to be risk-neutral which can be the case when the agent is a wealthy top manager of a large company (see Nan (2010) about hedging and EM). (5,6)
Our paper is also related to literature which focuses on real earnings manipulation rather than accounting manipulations or earnings misreporting. We consider earnings manipulation (EM) to be a transfer of funds between periods. This transfer does not create any social value (in contrast to productive effort). In most cases it includes delaying the approval of important decisions, inefficient investments, borrowing in order to manipulate financial results, inefficient discount policy etc. Other examples include allowance for bad debt, cutting expenses on research and development, managing pension plans, and delay in maintenance expenditures and other important decisions (see Degeorge, Patel and Zeckhauser (1999), Ewert et al. (2013), Dutta et al. (2014), Roychowdhury (2005) and Graham et al. (2005)). Graham et al. (2005) and Huang et al. (2008)) suggest that the focus of research should be shifted from accounting manipulation and earnings misreporting towards these kinds of actions.
Our second line of research is related to stock options. Some research argues that stocks options often are the engine of EM. Managers whose remuneration contracts include stock options have stronger incentive for EM compared to managers without stock options. (7) Theoretical literature mostly focuses on accounting manipulation or other fraudulent managerial activities related to stock options. (8) We focus on the link between stock options and real earnings manipulations as described in previous paragraph. (9)
We analyze a two-period model where a contract between a board of directors and a CEO includes cash payments (bonus) and stocks (or stock options). Stock options provide the CEO with a payment contingent on firm value in the second period. However, whether or not the CEO will be granted stock options depends on first-period earnings along with first-period bonus. We compare two situations. In the first, the CEO chooses only a costly productive effort--assuming that the CEO cannot be involved in EM. In the second, the CEO is subject to a double-moral hazard problem which includes the choice of productive effort and the EM decision. It is shown that the parties expected payoffs can be higher in the second case. The following demonstrates the intuitions behind this result. If first-period earnings are below the threshold specified in the contract, the CEO does not get stock options. Since the CEO's effort is costly, the socially optimal level of effort in the first-period involves a trade-off between the firm's expected earnings in the first period and the cost of effort. However, for the CEO this trade-off is biased (compared to the socially optimal level) because of risk of not getting stock options. In the latter case, the CEO loses a large amount of earnings in the second period. As a result, he will usually provide higher than socially optimal level of effort and he will respectively request higher than optimal level of compensation. However, if the CEO is able to transfer earnings between periods and the firm's going concern value is relatively high, he can increase current earnings by reducing the firm's going concern value. This allows him to get stock options and make a positive profit. This in turn increases his ex-ante incentive to provide socially optimal level of effort.
The management and prevention of EM is an important part of an efficient corporate governance structure. A possible strategy is to move in the direction of a zero-tolerance policy via internal control and strict regulation. This paper points out that such a policy will not always lead to an improvement in production efficiency of corporations due to a double-moral hazard problem. The key is to understand the nature of managerial moral hazard which depends on firms' specific conditions including industrial conditions. If the firm operates in an environment with a high degree of managerial moral hazard, where managerial level of effort and earnings are difficult to verify by third parties, some degree of earnings manipulation can be tolerable. Examples may include high-tech, internet, research, biotechnology, venture firms, and financially distressed firms. On the other hand other industries can definitely benefit from zero-tolerance policy. Examples may include public utilities, energy producers, real estate. Our model also predicts that EM is not optimal if the outside options of managers are weak. Industries with low degree of competiveness are examples of industries where it is hard for managers to find a new position in case of losing their job. We also find that stronger corporate governance can lead to less EM but it does not necessarily improves firm's value and management compensation.
Recent empirical papers provide some evidence consistent with the spirit of our findings. Linck, Netter and Shu (2009) argue that EM may improve efficiency of managerial actions in financially constrained firms. Julio and Yook (2009) find that intertemporal transfer of earnings can help to improve efficiency of corporate investment decisions. Jiraporn et al. (2008) find that EM is more likely to occur if the cost of EM is low.
The rest of this paper is organized as follows: Section 2 describes the model and explains optimal contracting without EM. Section 3 discusses optimal contracting when the agent is subject to a double moral hazard problem which includes EM. A comparison of the outcomes is presented in Section 4. Section 5 discusses the model's implications with regard to empirical evidence and Section 6 presents the conclusions.
2. Basic Model without Earnings Manipulation
Consider a firm that is owned by a Principal (P). The firm's performance depends on the effort provided by a Manager (M). M 's effort is denoted by e, e [member of] [0,1]. M and P are risk neutral. The cost of effort is [e.sup.2]. The interim first-period cash flow [r.sub.0] equals 1 with probability e and 0 otherwise. The company's assets which remain at the end of first period may yield the revenue [v.sub.0] in the second period. It is assumed that E[v.sub.0] = 2. (10) P cannot observe e. The first-best level of effort [e.sup.*] maximizes the firm's expected value. This can be written as E[[r.sub.0] + [v.sub.0]--[e.sup.2]] = e + 2-[e.sup.2]. Obviously, [e.sup.*] = 1/2. The reservation payoff of M is w. We assume that the project's net present value is positive or that surplus can be created from parties' cooperation, i.e.
[E.sub.e=1/2] + [v.sub.0]--[e.sup.2]] = 9/4 > w (1)
To induce productive effort by M, P offers M a bonus b in the case the firm is successful in the first period and stock options which will give M a fraction d of earnings in the second period. (11) If r = 1,M's overall payoff is b + dv and P's payoff is 1--b + (1--d)v. If r = 0, M's overall payoff is 0 and P 's payoff is v.
The game is as follows:
1. P offers contract to M. M accepts or rejects the contract. If the contract is rejected, P gets 0 and M gets w.
2. M chooses e.
3. [r.sub.0] and [v.sub.0] become known.
4. The parties get their payoffs according to the contract signed.
When choosing which contract to offer, P maximizes the expected value of his net earnings (payoff from the project minus the payment to M). On the one hand, the contracts should provide M with the optimal incentive to choose e. On the other hand, the expected value of M's payoff must cover the reservation payoff, w, in order for M to accept the contract.
P 's problem can be written as follows (problem P1).
[max.sub.b,d] [EV.sub.P] subject to
e = arg[max.sub.e] [EV.sub.M]
0 [less than or equal to] e [less than or equal to] 1
[EV.sub.M] [greater than or equal to] w
where [V.sub.M] and [V.sub.P] denote the payoffs of M and P respectively.
We expect that M's effort is not equal the first-best level of effort. This is because he bears the full cost of the effort while the results of the effort must be shared with P (in the spirit of Jensen and Meckling, 1976).
1) If w [less than or equal to] 1/16, any contract that satisfies the following condition is optimal
b/2 + d = 1/4 (2)
2) If w > 1/16, any contract that satisfies the following condition is optimal
b/2 + d = [square root of w] (3)
Proof. The M's expected payoff is
[EV.sub.M] = e(b + 2d)--[e.sup.2] = be + 2de--[e.sup.2] (4)
This is because the first-period earnings are 1 with probability e, in which case M gets bonus b and stocks options that give him a fraction d of firm's value in period 2 . Otherwise, M gets nothing. Hence the optimal level of effort is
e' = (b + 2d)/2 (5)
P's expected payoff is
[EV.sub.P] = e(1--b + 2(1--d)) + 2(1--e) (6)
Substituting (5) into (6) we get:
[EV.sub.p] = b/2--[b.sup.2]/2--2bd + d--2[d.sup.2] + 2 (7)
The optimal b maximizes P's expected payoff, [EV.sub.P], under the condition that EVM is not less than w. From (4) and (5) we get:
[EV.sub.M] = [(b + 2d).sup.2]/4 (8)
d > 1/4-b/2 (9)
In this case [EV.sup.P] is decreasing in both d and b and so the condition [EV.sub.M] [greater than or equal to] w should be binding. Otherwise one can marginally decrease b and/or d since at least one of them should be greater than 0 . (9) will hold and [EV.sub.p] will improve. Since [EV.sub.M] = w, [EV.sub.P] = b /2--2w + d + 2 and also
d = 2[square root of w]--b/2 (10)
Then [EV.sub.P] = b/2-2w + 2[square root of w-b]/2 + 2 = [square root of w] + 2-2w. From (5) and (10), e' = [square root of 1+w]/2. It follows from (9) that this case holds only if d + b/2 > 1/4 or [square root of w] > 1/4.
Case 2: d [less than or equal to] 1/4--b/2. In this case [EV.sub.P] is increasing in both d and b and so the condition (9) should be binding (otherwise one can simply increase b and d). Thus, P's expected payoff is b/2--[b.sup.2]/2--2b(1/4--b/2) +1/4--b/2--2[(1/4--b/2).sup.2] + 2 = 17/8.
Since d = 1/4--b/2 and respectively 2d + b = 1/2, (5) implies e' = 1/4. Since [EV.sub.M] [greater than or equal to] w, it follows from (8) that this case holds only if w [less than or equal to] 1/16. End proof.
Note that according to (2), (3) and (5), M 's effort can be greater or less than 1/ 2 (the first-best effort) depending on w .
Figure 1 illustrates intuitions behind Proposition 1. In Figure 1a e' < [e.sup.*] and in Figure 1b e' > [e.sup.*]. This follows from comparing the slopes of lines describing M 's payoff with firm's value.
[FIGURE 1 OMITTED]
If w is low, b and d should also be low in most cases (otherwise P will pay "too much"). It results in small slope for [EV.sub.M] and therefore M's effort is below the socially optimal level of effort. In contrast, if w is large, P has to increase b and d. It increases the slope of [EV.sub.M] that in turn can make M's effort greater than the socially optimal level of effort. Finally, note that Innes (1990) analyzes a similar environment (where a manager's effort is costly and EM is not allowed) with only one period (in terms of our model this means v = 0) and demonstrates that "live-or-die" contract is the best one.
3. Optimal Contracting with Earnings Manipulation
In this section we assume that M may engage in EM. If the firm is unsuccessful in the first period or if [r.sub.0] = 0, M will shift earnings upward (otherwise he gets nothing) that will subsequently reduce the second-period earnings. In this case, the firm's first-period earnings are r = [r.sub.0] + 1 = 1. The firm's earnings in the second period are then v = [v.sub.0]--1--c, where c is the cost of EM, 0 < c < 1. (12) Since c > 0, EM is socially inefficient. To insure that earnings are non-negative in the second period we assume
c [less than or equal to] [v.sub.0]--1 [for all][v.sub.0] (11)
P's problem (P2) is similar to P1. However when [r.sub.0] = 0, r = 1and v = [v.sub.0]--1--c . We also expect that M 's effort can be more efficient than in the case without EM. The opportunity to manipulate earnings protects M against the risk of a low payoff when the results of production are low. This provides an incentive for M to improve effort.
1) If w < 1 + [(1--c).sup.2]/4, b = 0 and
d = 2c--2 + 2 [square root of [(1--c).sup.2] + w [(1 + c).sup.2]/[(1 + c).sup.2]
w [greater than or equal to] 1 + [(1--c).sup.2]/4 (12)
d = 1 and
b = w + c/2--[c.sup.2]/4--5/4
Proof. See Appendix.
To illustrate the proof of Proposition 2, consider c = 1/2 and w = 17/16. M 's payoff is:
b + d[v.sub.0], if [r.sub.0] = 1 (13)
b + d ([v.sub.0]--1--c) = b + d ([v.sub.0]--3/2), if [r.sub.0] = 0 (14)
This means that if the interim earnings are low ([r.sub.0] = 0), M will be involved in EM. Otherwise, M gets nothing. P's expected earnings are:
[EV.sub.P] = e(2(1--d) + (1--b)) + (1--e)(1--b +1/2(1--d)) (15)
The choice of e maximizes
[EV.sub.M] = e(2d + b) + (1--e)(b + 1/2d)--[e.sup.2] (16)
The maximand of this expression is
e" = 3/4d (17)
If [EV.sub.M] = w, then from (16),
b = 17/16--2de -1/2(1--e)d + [e.sup.2] (18)
Using (15), (16) and (17), we get:
[EV.sub.P] = 7/16 + 9d/8--9[d.sup.2]/16 (19)
We have to maximize this expression under the condition:
0 [less than or equal to] b [less than or equal to] 1 (20)
Using (17) and (18), we have:
b = 17/16--1/2 d--9/16 [d.sup.2] (21)
The right side of (20) holds for d [greater than or equal to] [d.sub.1] = 1/9. The left side holds if d [less than or equal to] [d.sub.2] = 1. Note that [d.sub.1] [less than or equal to] [d.sub.2] and 0 [less than or equal to] [d.sub.2]. Optimal d = 1 and from (21)
b = 0
Proposition 2 is intuitive. First, d is crucial since M always has b in the first period regardless his level of effort. Higher d implies higher effort. When c is relatively large and respectively the condition (12) is more likely to hold, P gives maximal incentive to M in order to improve effort and avoid losses related to EM in case when first-period earnings are low.
Note that the level of effort (e" = d(1 + c) /2) can be closer to the first-best level (e = 1/2) than in the case without EM (e'). This insight is illustrated in Figure 2. This is because the slope of the line describing M's expected payoff (given by equation (16)) is less than that without EM (see (6)).
[FIGURE 2 OMITTED]
Corollary 1. Earnings manipulation is more probable as c decreases and w decreases.
Proof. EM is more probable when M's effort is smaller because it increases the probability of low earnings in period 1. Consider changes in c for a given value of w. An increase in c has two effects: first it increases chances that w [greater than or equal to] 1 + [(1-c).sup.2]/4. In this case M's effort is stronger than in case w < 1 + [(1-c).sup.2]/4. Secondly, it increases the value of M's effort in both cases.
So ultimately, higher c implies lower probability of EM. An increase in w has exactly the same effect as an increase in c . End proof.
It follows from Corollary 1 that EM is more likely if cost of EM is low. Also, managers with low outside opportunities are more likely to be involved in EM.
4. Can Earnings Manipulation Enhance a Firm's Value?
Now we compare firms that are involved in EM (Section 3) with those that are not (Section 2). If EM is possible, the manager's effort is usually closer to the socially efficient level of effort. At the same time the manager is often involved in EM in order to achieve earnings threshold and to get stock options. It follows then that there is a trade-off in social efficiency between the benefits from EM improving the manager's effort and the costs of EM.
Proposition 3. If w < 1/16, firms that manipulate earnings have a lower value than firms that do not. 2) If 1/16 [less than or equal to] w < 1 + [(1-c).sup.2]/4, for any c, there is [w.sup.*] (c) such that for any w < [w.sup.*] (c), firms that manipulate earnings have a lower value than firms that do not and otherwise firms that manipulate earnings have on average a higher value than firms that do not. 3) If w [greater than or equal to] 1 + [(1-c).sup.2]/4, firms that manipulate earnings have on average a higher value than firms that do not.
Proof. See Appendix.
To illustrate the proof of Proposition 3 consider
1 + [(1--c).sup.2]/4 < w (22)
From Proposition 1, [V.sub.N] = [square root of w] + 2--2w. According to Proposition 2, [V.sub.EM] = 2 + [(1+c).sup.2]/2--w--[(1+c).sup.2]/4--c. We have:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This is positive because from (22) w > 1. Thus [V.sub.EM] > [V.sub.N].
5. Empirical Evidence and Policy Implications
1. We have shown that EM can be optimal in the relationship between a firm's board of directors and the CEO. This holds even if the cost of EM is relatively high (as follows from Proposition 3) and it would be possible to implement a system of control that would eliminate EM completely. Principal (shareholders) accepts some degree of EM because it increases the managers' incentive to provide an optimal level of productive effort.
2. From Corollary 1, EM is more likely if the cost of EM is relatively low. If the cost of EM is relatively high, the opportunity to engage in EM either does not affect firms' values (when they do not use EM in equilibrium) or is detrimental to firms' values (when firms engage in EM in equilibrium). Some recent evidence in the study by Jiraporn et al. (2008) is consistent with this prediction although they are not specifically focused on real earnings manipulations. So additional research is required here.
3. EM should more frequently be observed in industries with high level of managerial moral hazard. These are more likely to be industries characterized by a high degree of technological or market uncertainty (such as software, internet, biomedical etc.). In similar spirit Francis et al. (2004) find that EM is inconsistent with efficient contracting. Also firms in financial distress can be considered as ones with higher degree of managerial moral hazard.
4. As implied by Corollary 1, EM should more frequently be observed among managers with low outside opportunities (low w). This is a new prediction which has never been tested to the best of our knowledge. Intuitively firms in more concentrated industries have lower outside opportunities. Also, in firms where managers have low outside opportunities, EM is more likely to be socially inefficient.
5. Firms which manipulate earnings issue stock options for managers which are contingent on firms' intermediate results (Lemmas 3 and 4).
Since EM can be socially efficient, the question of its regulation depends on the industry and any parameters related to the firm's projects. If the cost of EM is relatively low, putting in place an expensive public system of EM prevention cannot be efficient: entrepreneurs will invest less funds in socially efficient projects and will not provide high levels of productive effort. According to our analysis (proof of Proposition 3), such a system should target firms with high degree of managerial moral hazard and where managers have low outside opportunities.
Many debates regarding EM by firms' insiders have still not been resolved even after Sarbannes-Oxley act (2002). Graham et al. (2005) found that EM is used more frequently than accounting fraud and misreporting. Existing literature usually considers EM to be a negative social phenomenon and suggests measures for its elimination. In the present paper, we argue that zero tolerance policy towards EM may be socially inefficient. We analyze a model where a manager's productive effort is not observable by the firm's owner. The optimal contract should provide the manager with the optimal incentive to provide productive effort. The equilibrium level of effort is not socially optimal. Following this, we analyze the case where in addition to productive effort the manager can be engaged in EM that reduces the firm's total value. EM consists of transferring cash flow between periods. Our main finding is that the existence of EM can lead to increased output and therefore, improved social efficiency. It is shown that EM should be observed more often among firms with low profitability, low costs of EM, and where managers have low outside opportunities.
Proof of Proposition 2. M's payoff is:
b + d[v.sub.0], if [r.sub.0] = 1 (23)
b + d([v.sub.0]--1--c), if [r.sub.0] = 0 (24)
This means that if the interim earnings are low ([r.sub.0] = 0), M will be involved in EM. Otherwise, M gets nothing. P's expected earnings are:
[EV.sub.P] = e(2(1--d) + (1--b)) + (1--e)(1--b + (1--d)(1--c)) (25)
The choice of e maximizes
[EV.sub.M] = e(2d + b) + (1--e)(b + d(1--c))--[e.sup.2] (26)
The maximand of this expression is
e" = d (1 + c)/2 (27)
If [EV.sub.M] = w, then from (26),
b = w--2de--(1--e)d(1--c) + [e.sup.2] (28)
Using (25), (26) and (27), we get:
[EV.sup.P] = 2 + d[(1 + c).sup.2]/2--w--[d.sup.2][(1 + c).sup.2]/4--c (29)
We have to maximize this expression under the condition:
0 [less than or equal to] b [less than or equal to] 1 (30)
Using (27) and (28), we have:
b = w--d (1--c)--[d.sup.2] (1/4 + c/2 + [c.sup.2]/4) (31)
The right side of (30) holds for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The left side holds if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that [d.sub.1] [less than or equal to] [d.sub.2] and 0 [less than or equal to] [d.sub.2]. Two cases are possible:
w < 1 + [(1--c).sup.2]/4
In this case [d.sub.2] < 1. The maximand of (29) is d = 1. Therefore, when [d.sub.2] < 1, [EV.sub.P] is increasing in d for any feasible d. Hence, the optimal d = [d.sub.2] and b = 0. Also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)
w [greater than or equal to] 1 + [(1--c).sup.2]/4
Then optimal d = 1 and from (31)
b = w + c/2--[c.sup.2]/4--5/4
If [EV.sub.M] > w, then from (25),
b = 0 (33)
Using (25) and (27), we get:
[EV.sub.P] = 2--[[d.sup.2](1 + c)]/2 + d (3c -1)/2 + d (1 + c)c/2--[d.sup.2] (1 + c)/2--c (34)
We have to maximize this expression under the condition that [EV.sub.M] > w. The maximand of (34) is:
d = 4c--1 + [c.sup.2]/2[(1+c).sup.2] (35)
If c < [square root of 5]--4, (35) is negative. Thus, this can work only for c [greater than or equal to] 2[square root of 5]--4. In this case:
EVP = 2--c--[(4c--1 + [c.sup.2]).sup.2]/8[(1+c).sup.2] (36)
This works only if w < (4c-1+[c.sup.2])(c+3)([c.sup.2]+3)/8[(1+c).sup.3]. Otherwise [EV.sub.M] [less than or equal to] w.
Now we have to compare two situations analyzed above: [EV.sub.M] = w and [EV.sub.M] > w . First consider w < 1 + [(1-c).sup.2]/4. The difference between (32) and (36) can written as
[DELTA] = [[DELTA].sub.1] + [[DELTA].sub.2]
where [[DELTA].sub.1] = [(1 + c).sup.2] [square root of [(1 -c).sup.2] + w[(1 + c).sup.2]] -8[(1 + c).sup.2] (1 -c + w) and [[DELTA].sub.2] = [(4c -1 + [c.sup.2]).sup.2]--8((c--1 + [[square root of [(1--c).sup.2] + w[(1 + c).sup.2])]).sup.2]. The sign of [[DELTA].sub.1] depends on the sign of w(4c + [c.sup.2] -1--w). Since w < 1 + [(1-c).sup.2]/4, 4c + [c.sup.2]--1--w > [c.sup.2] + 6c--3 > 0. The latter follows from c [greater than or equal to] 2[square root of 5]--4 > 2[square root of 3]--3. The sign of [[DELTA].sub.2] depends on the sign of 4c -1 + [c.sup.2]--2[square root of 2](c -1 + [square root of [(1--c).sup.2] + w[(1 + c).sup.2]])). For the case w < 1, this is positive because 2[square root of 2] > 1 + c. The case 1 < w < 1 + [(1-c).sup.2]/4 does not work because w < (4c-1+c,)(c+3)(c +3) < 1. The latter follows from: 1) 16[(1 + c).sup.3] [greater than or equal to] (c + 3)([c.sup.2] + 3) and 2) [(1 + c).sup.3] [greater than or equal to] 2(4c -1 + [c.sup.2]). End proof.
Proof of Proposition 3. Let [V.sub.EM] denote the value of firms that can manipulate earnings and let [V.sub.N] denote the value of firms that cannot manipulate earnings. As follows from Proposition 1, if w [less than or equal to] 1/16, [e.sub.N] = 1/4 and [V.sub.N] = 17/8. According to Proposition 2, if w < 1 + [(1--c).sup.2]/4, [V.sub.EM] = 1+ [square root of [(1--c).sup.2] + w[(1 + c).sup.2]]--w--(c-1+ [[square root of [(1- c).sup.2]+w[(1+c).sup.2]].sup.2]). [V.sub.EM] is decreasing in w so maximum is obtained when w = 0. In this case [V.sub.EM] = 2--c. This is less than 17/8.
Consider 1/16 [less than or equal to] w < 1 + [(1-c).sup.2]/4. From Proposition 1, [V.sub.N] = [square root of w] + 2-- 2]w. According to Proposition 2,
[V.sub.EM] = 1 + [square root of [(1--c).sup.2] + w[(1 + c).sup.2]]--w--(c--1 + [square root of [(1--c).sup.2] + [w[(1 + c).sup.2]).sup.2]/[(1+c).sup.2].
Consider [V.sub.N]--[V.sub.EM]. The derivative of this with respect to w equals:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
One can see that [square root of [(1--c).sup.2] + w[(1 + c).sup.2]] < (3 + [c.sup.2]) [square root of w]. This is equivalent to [(1--c).sup.2] + w[(1 + c).sup.2] < [(3 + [c.sup.2]).sup.2] w. One can rewrite this as (6w--1) + (2w--2cw) + (c--[c.sup.2]) + c + 5[c.sup.2]w + [c.sup.4]w. This is positive because c < 1 and w [greater than or equal to] 1/16.
So [V.sub.N]--[V.sub.EM] is decreasing in w. Also when w = 0, we have [V.sub.N] = 2 > [V.sub.EM] = 2--c
Consider 1 + [(1-c).sup.2]/4 [less than or equal to] w. From Proposition 1, [V.sub.N] = [square root of w] + 2--2w. According to Proposition 2, [V.sub.EM] = 2 + [(1+c).sup.2]/2--w--[(1+c).sup.2]/4--c. We have:
[DELTA] = [V.sub.EM]--[V.sub.N] = ([(1 + c).sup.2]/4 -c) + (w--[square root of w]) = [(1--c).sup.2]/2 + [square root of w] ([square root of w)--1)
This is positive because w > 1. Thus [V.sub.EM] > [V.sub.N]. End proof.
School of Business, Faculty of Applied and Professional Studies, Nipissing University
Received 17 December 2014 * Received in revised form 17 April 2015
Accepted 18 April 2015 * Available online 25 January 2016
I would like to thank Hikmet Gunet, Pierre Lasserre, Pierre Liang, Nicolas Marceau, Michel Robe, and the seminar and conference participants at 2009 American Finance Association Annual Meeting, 2010 European Economic Association Annual Meeting, UQAM, and University of Canterbury for their comments. Many thanks to Bennett Minchella for editing assistance.
(1.) For a review of empirical literature about EM, see, for instance, Deshow et al. (2010) and Healey et al. (1999).
(2.) There also exists literature analyzing EM in the context of costly communication problem (see Arya (1988), Dye (1988), Demski (1998)). Similarly to previously mentioned literature, a complete absence of opportunities for EM would typically improve social welfare.
(3.) Empirical literature focusing on negative effect of EM include Asciouglu et al. (2012), Theo et al. (1998), Chen and Yuan (2004), Jaggi (2006), Palmrose (2004).
(4.) See, for example, Liang (2004).
(5.) One should also note other papers that analyze long-term principal-agent relationship with EM and that focus on different from our paper issues. An example is Laux (2014). The focus in that paper is the choice of convexity level for a CEO contract. Miglo (2010) considers the link between EM and capital structure. Miglo (2013) analyzes a model with incomplete contracts in the spirit of Hart (1995) and focuses on promotion/firing decision regarding a manager involving in EM.
(6.) Empirical literature demonstrating positive effect of EM includes Bartov (1993), Trueman and Titman (1988), Bartov et al. (2002), Duh et al. (2015), Healey and Palepu (1993), Holthausen (1990) and Subramanyam (1996)
(7.) See Cheng and Warfield (2005), Erikson, Halon and Mayden (2003), Gao and Shrieves (2002), Hall and Murphy (2002), Ke (2001), Laux (2014), Magnan et al. (2008).
(8.) See Magnan et al. (2008).
(9.) Dutta et al. (2014) analyze the difference between convex and non-convex contracts.
(10.) For simplicity it is assumed that the firm's second-period earnings do not depend on e. The model can easily be generalized by allowing this. As far as we can see, no intuitions will be affected by this change. The specific value for E[v.sub.0] is chosen arbitrarily although it assures that the going-concern value of the firm is large enough compared to current earnings.
(11.) In Miglo (2013) we consider a more narrow set of contracts where the second-period sharing rule follows "0-1" rule.
(12.) The model can be generalized by allowing different cost functions for EM.
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|Date:||Mar 1, 2016|
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