- Email: [email protected]
Conducting inaccurate audits to commit to the audit policy
International Journal of Industrial Organization 25 (2007) 379 – 389 www.elsevier.com/locate/econbase
Conducting inaccurate audits to commit to the audit policy ☆ Aaron Finkle a , Dongsoo Shin b,⁎ a
Department of Economics, California State University, San Marcos, San Marcos CA 92096, United States b Department of Economics, Santa Clara University, Santa Clara CA 95053, United States Received 23 May 2005; received in revised form 16 May 2006; accepted 17 May 2006 Available online 30 June 2006
Abstract Using a principal–agent model, we study the structure of an optimal audit policy when it is not contractible. In our model, auditing technologies are distinguished according to the accuracy and the frequency of audits. Our analysis reveals that audits take place with the highest frequency, but the accuracy of audits is compromised in equilibrium. We also find that without commitment to the audit policy, the first-best outcome cannot be achieved even when auditing is costless. © 2006 Elsevier B.V. All rights reserved. JEL classification: D82; L23 Keywords: Principal–agent; Accuracy and frequency of audits; Commitment
1. Introduction It is now well understood that auditing is an important device for mitigating incentive problems in organizations. Production outsourcing, corporate governance, lender–borrower relationships, government procurement, and other examples of contracting with auditing mechanisms abound. In selecting an audit policy, the choice in question is between the frequency of audits and the accuracy ☆
We thank Anke Kessler, Fahad Khalil, and Jacques Lawarrée for detailed suggestions and comments. We also thank Sanjiv Das, John Heineke, Michael Kevane, Helen Popper, Bill Sundstrom, and seminar participants at the 2005 Midwest Economic Theory Conference (University of Kansas) for discussions. Comments from the anonymous referees and the editor have substantially improved this paper. The authors acknowledge the financial supports from the Faculty Research Grant at Willamette University and the Presidential Research Grant at Santa Clara University. ⁎ Corresponding author. E-mail addresses: [email protected] (A. Finkle), [email protected] (D. Shin). 0167-7187/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2006.05.005
380
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
of the audits that occur. For example, in outsourcing relationships, buyers can conduct audits by sampling items from different shipments. The buyer can increase the frequency by increasing the chance of auditing a particular shipment, or increase the accuracy by selecting a larger sample size from those shipments that the buyer audits. Similarly, insurance companies can investigate a greater number of claims or increase the time devoted to each individual claim to determine if it is fraudulent. The choice in each of these cases is whether to devote attention to more frequent audits or to a greater accuracy in the audits that take place. Since auditing is a costly process, the allocation of resources between the frequency and the accuracy of audits is an important issue. Organizations choose their audit technologies based on both direct and strategic benefits. While some audits are conducted only once or twice a year,1 there are many situations in which the quantity of audits is emphasized more than the quality. For example, in his report for CFO Magazine, Krass (2002) points out that corporate auditing is often conducted relentlessly, while the effectiveness of such auditing is frequently questioned by practitioners. He reports that there is growing skepticism of the merits of sacrificing quality for a greater quantity of audits. Despite some criticism, many organizations, such as manufacturing companies and financial firms are practicing continuous audits in the controlling process.2 Although numerous studies examine the optimality of audit strategies,3 there are few previous contributions that address the question why organizations sometimes conduct a large number of audits, while compromising the accuracy. Using a principal–agent model, we attempt to shed light on these phenomena by studying the structure of audit strategies with variable accuracy and frequency. We adopt a standard model in which the agent has private information on the cost of production — the agent has an incentive to “cheat” or to exaggerate the production cost in order to command information rent, which the principal reduces through distortions in the output schedule. By conducting an audit, the principal can receive an informative signal about the cost of production. In our model, the principal chooses two distinct probabilities with respect to auditing — first, the probability that the audit takes place, and second, the probability that the signal from the audit is correct. Following Kessler (2004), we interpret the first one as the frequency of audits and the second one as the accuracy of audits. We assume that the principal's ability to commit is limited in that the output levels and the transfers are contractible, but the audit strategy is not. The audit technologies are chosen according to the principal's ex post incentives after production and transfer take place as specified in the contract.4 The main focus of this paper is on the choice of auditing technology in the optimal contract. The distortionary feature of the output schedule is similar to the previous contributions in agency contracting and only briefly discussed here. We derive two main results from allowing both frequency and accuracy to be chosen by the principal ex post: First, the principal chooses the 1 An example is the audits conducted by International Organization of Standardization (ISO) for quality standards. See Harrington and Mathers (1997). 2 See Searcy and Woodroof (2003). 3 For example, see Baron and Besanko (1984), Border and Sobel (1987), Demougin and Garvie (1991), Khalil (1997), and Kessler (2004) among others. 4 In actuality, commitment to audit policies is often limited. For instance, the outsourcing contract for labor and material designed by Consumers Energy, a company that provides utilities in the state of Michigan (http://www. consumersenergy.com/apps/pdf/CSS933-OEP.pdf) states, “[s]aid records and books of account… shall be open to examination during regular business hours by the owner or its agents for the purpose of [auditing].” However, nothing is mentioned of the frequency or accuracy of said audits, which is implicitly left to the discretion of the owner after receipt of the product.
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
381
highest possible level of frequency, while creating some inaccuracy in audits. Second, the principal cannot achieve the first-best outcome even when auditing is costless. When the audit policy is not contractible, the principal will conduct an audit only if the expected net gain from auditing is positive at that point. Since the audit frequency and accuracy are costly, the principal is reluctant to conduct an accurate audit ex post. Thus, by inducing the agent to cheat with some probability,5 the principal provides herself with an ex post incentive to conduct an audit with some level of accuracy. At the same time, to induce the agent to cheat, the contract must induce the principal to decrease one or both of the frequency and the accuracy of audits in equilibrium such that a cheating agent evades punishment with a strictly positive probability. According to our results, the principal conducts an audit with probability one but with some inaccuracy. Since the audit policy is not contractible and accuracy is costly, the principal needs to provide herself with incentive to invest in accuracy ex post. To do so, the principal encourages the agent to cheat with a high enough probability such that the probability of facing a dishonest agent exceeds the probability of facing an agent who has reported costs truthfully, thereby increasing the return from accuracy. Cheating, however, will not occur if the audits are perfectly accurate (i.e. no audit errors). Thus, in equilibrium, the agent will cheat with a strictly positive probability and audits occur with some inaccuracy. Furthermore, the inaccuracy of the audits provide the principal with an additional payoff from penalizing a truthful agent some of the time. The added reward from an inaccurate audit makes the ex post benefit from conducting an audit strictly positive, and as a result audits are conducted with the greatest frequency in equilibrium. We also find that the possibility of collecting a penalty from a truthful agent prevents the principal from implementing the first-best outcome even when there is no cost of conducting an audit. When auditing is costless, the principal does not need to induce cheating by the agent to have ex post incentive to audit. However, the principal will still benefit from auditing the truthful agent by using an inaccurate audit. Thus, even when auditing is costless, the principal must still induce the agent to cheat in order to provide accuracy in the audit. Although there is a long literature in agency contracts with auditing, much of it does not consider the issue of commitment and audit technology. Exceptions are Khalil (1997) and Khalil and Parigi (1998) which also study an optimal audit strategy without commitment. Both papers obtain a result similar to ours showing that mixed strategies are employed in equilibrium. However, they assume that audits are perfectly accurate exogenously. With such a setup, the audit occurs at random. In our model, the principal can choose not only the frequency but the accuracy of audits as well, and the principal conducts the audit with certainty if she cannot commit to auditing strategy. Also, in these papers, the principal implements the first-best outcome when the cost of auditing is zero. In this paper, we show that when the principal chooses the accuracy of audits as well as the frequency, she cannot implement the first-best outcome even when auditing is costless. Other papers that study the issue of auditing without commitment include Dunne and Loewenstein (1995), Graetz et al. (1986), Melumad and Mookherjee (1989), and Strausz (1997). The first paper studies an optimal auditing strategy when agents bid for contracts. The second paper considers income tax compliance when commitment is not possible, but models incomes (output level in our context) exogenously. The last two papers show that the principal uses delegation of the audit policy to a third party as a commitment device when the principal cannot commit to the audit policy. None of the above authors consider the accuracy of audits. 5
As in Khalil (1997), without commitment to the audit policy, the revelation principle no longer applies. See Bester and Strausz (2001) for a general analysis of the revelation principle with limited commitment.
382
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
Our paper is also related to Kessler (2004) and Nalebuff and Scharfstein (1987). Like ours, both the accuracy and the frequency of audits are considered in their models. Using a model without a cost of auditing, Kessler shows that the frequency of audits is not a concern for the principal. The second paper, adopting a competitive market framework, shows that the first-best outcome can be achieved only when the auditing signal is accurate enough and the penalty is unbounded. Unlike ours, the authors do not cope with the issue of commitment. Finally, Krasa and Villamil (2000) study lender–borrower relationships with limited commitment to enforcement to show that a simple debt contract with deterministic enforcement is optimal. In their model, as in ours, the optimal contract induces the agent to withhold information which makes deterministic enforcement attractive ex post. Stochastic enforcement induces information revelation but also eliminates incentive for enforcement ex post. As in our model, the principal must induce the agent to withhold information if enforcement is to be used ex post. While we also find deterministic audits are optimal, it is because of the accuracy choice. The rest of the paper is organized as follows. The next section presents the model. Section 3 discusses our results. Section 4 concludes. All proofs are in the Appendix. 2. The model A risk-neutral principal offers a contract to a risk-neutral agent to carry out a project that yields an output q. The cost parameter of the project, θ, can be either θL or θH, with Δθ ≡ θH − θL > 0. We denote by ϕi the principal's prior belief that θ = θi, where i ∈ {L, H}. The principal values the output of the project according to a strictly concave function, V(q), satisfying the Inada condition. The agent bears the cost of the project given by θq and receives a transfer t from the principal in return for producing the output level q. If the agent rejects the contract, he ends up with his reservation payoff, which is normalized to zero. If the agent accepts the contract, he produces qi according to the state i ∈ {L, H }. Under full information, the output schedule is said to be the first best and is characterized by V Vðq⁎i Þ ¼ hi ; iafL; Hg; and the agent obtains no information rent in either state. Under asymmetric information, the realized state i ∈ {L, H} is private information to the agent, and it is not known to the principal. Thus, the agent may have incentive to “cheat” or misrepresent the state i by producing qj ( j ≠ i) to command information rent. We denote by τi the probability that the agent is “truthful” or produces qi in state i ∈ {L, H}. Thus, the probability that the agent produces qi (regardless of whether he is truthful or not) is given by pi u/i si þ /j ð1−sj Þ; i pj and i; jafL; Hg: As usual in the models of this type, the agent has an incentive to exaggerate the production cost for a higher compensation when true state i = L, but has no incentive to cheat when i = H.6 This implies that τH = 1. To simplify the notation, define τL ≡ τ. This implies πL ≡ ϕLτ and πH ≡ ϕH + ϕL (1 − τ).
6
This will be clear from the result that the agent receives no rent in the optimal contract with auditing.
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
383
Using Bayes' rule, we derive the following conditional probabilities from the principal's point of view. After receiving qH, her posterior belief that the agent was truthful is: pHjH u
/H ; /H þ /L ð1−sÞ
ð1Þ
and the probability that the agent cheated is 1−pHjH ¼ pLjH u
/L ð1−sÞ : /H þ /L ð1−sÞ
ð2Þ
By conducting an audit, the principal can verify the true state i ex post. The principal conducts an audit with probability f ∈ (0, 1] after receiving qH.7 We interpret f as the frequency of audits. By conducting an audit, the principal receives a signal s ∈ {L, H } on state i. At the time of auditing, the principal also chooses the accuracy of the audit, denoted by α ∈ (1/2, 1]. An audit contract is one in which 1 ≥ f > 0 and 1 ≥ α > 1/2 (therefore a no audit contract is one in which f = 0 and/or α = 1/2). To ease the exposition, we follow Kofman and Lawarrée (1993) and assume that α = Pr (sL|θL) = Pr(sH |θH) and 1 − α = Pr(sH|θL) = Pr(sL|θH). The cost of auditing depends on the frequency and accuracy, and is given by C( f, α) = cf (2α − 1), with c > 0. We only focus on the parameters for which the audit contract is optimal. Specifically, we assume ϕL > ϕH and c is sufficiently small such that the principal prefers the audit contract to the no audit contract.8 We assume that the principal's audit strategy is not contractible. That is, the contract only specifies the transfers and the output schedule, and the principal chooses f and α after receiving the output from the agent. We briefly present optimal contract when no audit is conducted. As the revelation principle applies, we can restrict our attention to the contract that induces the agent to report truthfully. This means that τ = 1 in equilibrium, and thus πL = ϕL, πH = ϕH, and πH | H = 1. Without auditing, the contract specifies {qL, qH, tL, tH} to maximize the following problem faced by the principal. Max
/L ½V ðqL Þ−tL þ /H ½V ðqH Þ−tH
subject to tL −hL qL z0;
ðPCL Þ
tH −hH qH z0;
ðPCH Þ
tL −hL qL ztH −hL qH :
ðICL Þ
tH −hH qH ztL −hH qL :
ðICH Þ
7 An audit after receiving qL has no effect except by increasing tL in response to the ex post penalty that occurs following the audit. We rule out such audits since they do not affect the results in meaningful ways. Specifically, the agent is compensated ex ante for the precise penalty that results ex post without any effect on the overall incentives. This trivial consequence is easily avoided by assuming that the penalty for under-reporting costs is zero since it entails a lower compensation for the agent. pffiffiffi 8 For example, if V ðqÞ ¼ 100 q, θL = 1, θH = 2, ϕL = 3/4 and ϕH = 1/4, then the audit contract is optimal for c not too large. In general, for ϕL < ϕH, the audit contract is not optimal in our model.
384
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
The first two constraints assure that the agent's payoff in any state is higher than his reservation payoff (zero). The last two constraints assure that the agent's payoff from being truthful is higher than his payoff from misrepresenting the state. As in the standard principal–agent problem, (PCH) and (ICL) are binding in equilibrium, and from these two binding constraints the agent's rent when i = L is expressed as ΔθqH. To reduce the rent, the principal distorts qH downward and the optimal b output schedule is characterized by V′(qLb) = θL and V′(qH ) = θH + Δθ(πL / πH), and the principal's b b b b b expected payoff is: ϕL[V(qL) − θLqL − ΔθqH] + ϕH[V(qH) − θHqH ]. This is the standard second9 best outcome with no audits. The timing of the game is summarized as follows. Nature chooses the state i ∈ {L, H } and the agent learns it privately. The principal offers the contract followed by the agent's decision whether or not to accept it. The contract {qi, ti} is executed. The principal conducts an audit with probability f and accuracy α if she received qH. When auditing takes place, the signal s ∈ {L, H } is realized. If s = L, then the principal collects a lump sum penalty P from the agent. As many studies have shown, when the principal can impose unbounded penalties, she can always achieve the first-best outcome.10 We follow Laffont and Tirole (1993, Ch12) and assume that the agent is wealth constrained and the principal cannot impose a penalty larger than the transfer to the agent: the penalty P cannot exceed the transfer to the agent. As typical in the literature, the maximal punishment principle11 applies, and thus in the optimal contract, P = tH. 3. The results Recall that since the principal cannot commit to the auditing policy, f and α are not included in the contract. In the first stage, the principal chooses {qL, qH, tL, tH} to maximize her payoff. The contract must guarantee the agent his reservation payoff in each state of i ∈ {L, H}. The optimal contract also induces a second stage equilibrium in the choice of the agent, τ (the probability of being truthful), and the choice of the principal, f and α. These choice variables in the second stage will determine the probabilities πL, πH, πH | H, and πL|H. When writing the principal's problem, the second stage equilibrium conditions act as constraints. This means that the principal can choose {qL, qH, tL, tH, f, α, τ} as long as she respects the optimality conditions for the second stage. Therefore, the principal's problem can be written as follows. Max
pL ½V ðqL Þ−tL þ pH ½V ðqH Þ−tH þ f fapLjH tH þ ð1−aÞpHjH tH −cð2a−1Þg
subject to s½tL −hL qL þ ð1−sÞ½ f ð1−aÞtH þ ð1−f ÞtH −hL qH z0;
ðPCL Þ
f atH þ ð1−f ÞtH −hH qH z0;
ðPCH Þ
̂ LjH tH þ ð1−aÞpHjH tH −cð2a−1Þg; f aarg maxf ̂ f fap
ðFIÞ
9 When α = 1/2, auditing provides no information and the expected payoffs for both parties will be identical to the second-best (without audits). The difference will be an additional transfer to the agent ex ante equal to the expected ex post penalty. Specifically, the contract that results in α = 1/2 will be {qbL, qbH, tbL, tBH} where tBH = 2tbH, which induces truthful revelation (τ = 1) and auditing with f = 1. 10 See, for example, Riordan and Sappington (1988) and Demougin and Garvie (1991) among others. 11 See Baron and Besanko (1984) and Bolton (1987) for a formal analysis. If P < tH, the principal will have a less incentive to audit ex post, and the agent will have a greater incentive to misrepresent the state.
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
385
aaarg maxâ ap ̂ LjH tH þ ð1−aÞp ̂ HjH tH −cð2a−1Þ; ̂
ðAIÞ
saarg maxŝ s½t̂ L −hL qL þ ð1− sÞ½ ̂ f ð1−aÞtH þ ð1−f ÞtH −hL qH :
ðICÞ
From the objective function, when auditing takes place, the principal collects penalty P = tH from the agent in the following two cases: the principal received the correct signal when i = L, or she received the incorrect signal when i = H. As before, (PCL) and (PCH) induce participation in any state of i. Conditions (FI ), (AI ), and (IC ) represent the second stage equilibrium and determine the values of f, α, and τ given the contract in the first stage. The requirement of sequential optimality restricts the principal's choice of these variables: f and α are chosen to maximize her gain, and τ is chosen to maximize the agent's payoff when i = L. In an audit contract ( f > 0 and α > 1/2), as in Khalil (1997), the revelation principle no longer applies without commitment to the audit policy. If the principal were to choose fα = 1, the agent would be truthful with certainty (τ = 1). Then, the principal has an ex post incentive to save the entirety of the audit cost by reducing frequency and/or accuracy. This can be anticipated by the agent, and hence cheating by the agent occurs with a strictly positive probability. As a result, the second stage equilibrium is comprised of mixed strategies in the principal's choice of audit policy and agent's choice of compliance. We present our first result in the following proposition. Proposition 1. In the audit contract without commitment, ˙ The principal always audits, i.e., f = 1, but with imperfect accuracy: α = θH / (θH + θL) < 1. ˙ The agent's truth-telling is random, i.e., 1 > τ > 0. ˙ The agent obtains no rent in either state of nature. Proof. See Appendix. The proposition says that when the audit policy is not contractible, the principal chooses to audit with the highest frequency, but allows for audit errors to occur. Clearly, the principal will not conduct a costly accurate audit unless it is beneficial to her ex post. Therefore, the principal will increase accuracy of the audit only if the probability of collecting the penalty P = tH from cheating agent is higher than the probability of collecting the penalty from truth-telling agent by receiving a wrong signal (πL|H > πH | H).12 This means that the principal must let the agent cheat with a strictly positive probability, which will occur only if the accuracy and/or the frequency of audits are reduced from their highest levels ( fα < 1). Reducing accuracy has another effect which reducing frequency does not — the gain from penalizing the truthful agent. The additional effect from reducing accuracy is the key difference from Khalil (1997), where α = 1 exogenously. In his paper, the principal must be indifferent toward the frequency of audits in equilibrium, and hence an audit yields a zero expected return (ex post) even though she collects the penalty with positive probability. This can be seen here by considering the (FI ) constraint with α = 1, which would imply that 1 > f > 0 as in Khalil (1997), since the principal employs a mixed strategy in equilibrium in order to induce the agent to cheat. With α = 1, the first order condition of (FI ) gives: πL|HtH = c. As this equation implies, in Khalil (1997), where α = 1, the expected gain and cost of conducting an audit are equal. When the accuracy is a choice variable, however, the principal can collect πH | HtH from the honest agent by decreasing α from 1, while at
12
(AI) implies that (πL|H − πH|H)tH ≥ 2c. Therefore, πL|H > πH|H must hold for c > 0.
386
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
the same time, reduce the cost of auditing. Thus, with α < 1, the principal strictly benefits from conducting the audit ex post, and consequently, auditing takes place with certainty ( f = 1). Some previous studies have shown that the principal prefers more accurate to more frequent audits when commitment to audit policy is possible. In particular, Kessler (2004) demonstrates that only the accuracy of audits affects the agent's rent, which implies that when auditing is costly the frequency of audits is minimized.13 Our result shows that, without commitment, the result is reversed. When the audit policy is not contractible, generating some inaccuracy in auditing provides an incentive to the principal to conduct a costly audit ex post since the principal's ex post gain from conducting an audit becomes strictly positive. Consequently, audits are conducted with the highest frequency. The specific level of α comes from the principal's effort to not provide a positive rent to the agent. With f = 1, the indifferent condition of the agent from (IC ) gives the following expression for his rent: 1−a tL −hL qL ¼ hH −hL qH : 14 ð3Þ a As the agent's rent directly influences the principal's payoff, she will not leave any rent to the agent. From Eq. (3), the agent's rent increases as α decreases. In the optimal contract, the principal decreases α just to the level at which the RHS of Eq. (3) does not become strictly positive, and hence is α = θH / (θH + θL). We now briefly discuss the distortions in the output schedule. As in Khalil (1997), the output level is efficient when i = L(qL = qL ⁎), but there is overproduction when i = H(qH > qH ⁎). The first order condition for qH after replacing the choice variables with their values is: V VðqH Þ ¼ ðhH þ hL Þ=2 < hH : Since 1 > α > 1/2 in equilibrium, (AI ) implies that (πL|H − πH | H)tH = 2c. This equation implies that the principal must increase tH in order to reduce πL|H thereby reducing the probability of cheating. Since the agent's rent is zero in either state, the principal increases tH while increasing qH as well. Therefore, from the first-best levels she reduces cheating by increasing the transfer and output in the high-cost state. The endogenous choice of accuracy of audits brings about another important difference from Khalil (1997). With α set at 1 exogenously, the frequency of audits f equals 1 only when auditing is completely costless (c = 0). In our model, the result of f = 1 is independent of the cost of auditing (unless c is too large that the principal prefers the no-audit contract). Again, in our model, the result of auditing with certainty ( f = 1) stems from the strategic effect of conducting an inaccurate audit — collecting penalty from the honest agent. By letting c = 0 in our model, we have the next result. Proposition 2. Without commitment to the audit policy, the principal cannot implement the firstbest outcome even when c = 0.
The author's result, however, relies on the audit-dependent transfers — the principal can discriminate the transfer to the agent depending on whether an audit is conducted. We employed the audit-independent transfers in this paper for expositional purpose. Our results do not change in any significant way with audit-dependent transfers. 14 From binding (PCH), tH = (θH / α)qH. Substituting for tH by its value gives the expression in Eq. (3). 13
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
387
Proof. See Appendix. The result above differs from the conventional wisdom that the first-best outcome is achieved when auditing is costless. When the principal can choose both the frequency and the accuracy of audits without commitment, the first-best outcome can never be achieved even if c = 0. Because of the temptation to collect penalty from an honest agent, the principal still conducts an inaccurate audit even when auditing is costless. This implies that there is a strictly positive probability that the agent does not comply. As in Khalil (1997), the principal distorts qH upward when she cannot commit to her auditing strategy.15 In his paper, however, the accuracy of audits is exogenously set at the perfect level, and thus the first-best outcome is implemented when c = 0. When the principal chooses the accuracy of audits, qH is still distorted even with c = 0. 4. Conclusion In this paper, we have analyzed the optimal audit strategy in a principal–agent relationship when audit technologies are distinguished according to the frequency and the accuracy of audits and the audit policy is not contractible. Because the principal cannot commit to auditing, the optimal contract induces the agent to cheat with some probability, which in turn induces a costly audit ex post. Our result suggests that audits are conducted with a highest level of frequency, but there are some inaccuracies in audits in equilibrium. This result has another interpretation, comparing two different types of probability. One type determines the chance that an audit results in a signal, and the other determines the accuracy when a signal occurs. As our result suggests, without commitment, the principal prefers the former type of probability. That is, the possibility of an incorrect signal is preferred to the possibility of no signal. Our analysis also revealed that when the audit policy is not contractible, the first-best outcome cannot be implemented even if the principal does not incur any cost to conduct an audit. Appendix A Proof of Proposition 1. Suppose the principal always audits with perfect accuracy, i.e., fα = 1. Then, by (IC ) and (PCL), the agent would always be truthful (τ = 1), which in turn gives πL|H = 0 and πH|H = 1. Then, (AI ) becomes: ̂ H −cð2a−1Þ: ̂ aaarg maxâ̂ ð1− aÞt Clearly, with τ = 1, the principal would set α as small as possible (α = 1/2), which implies that the no audit contract is preferred. Thus, when the audit contract ( f > 0 and α > 1/2) is preferred by the principal, 1 > fα and 1 > τ must hold. Next, we show that if α > 1/2 then f = 1. With 1 ≥ α > 1/2, (AI ) implies that: pLjH tH −pHjH tH −2cz0:
ðA1Þ
Also, with f > 0, the following condition must hold from (FI ): aðpLjH tH −pHjH tH −2cÞ þ pHjH tH þ cz0;
15
ðA2Þ
By increasing qH, the principal must pay a higher tH. This gives her more incentive to increase the accuracy of the audit ex post.
388
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
If α = 1, then the expression in Eq. (A2) becomes πL|HtH − c = 0, where the equality comes from the fact that f < 1 with α = 1. However, with πL|HtH − c = 0, we have tH ≤ 0 from Eq. (A1), which is contradiction. If 1 > α > 1/2, then πL|HtH − πH|HtH − 2c = 0 in Eq. (A1), which makes the expression in Eq. (A2) strictly positive, i.e., f = 1. We now show that τ > 0. Suppose the agent always cheats (τ = 0). The first order condition of (AI ) gives ϕLtH − ϕHtH − 2c = 0 or c = (ϕL − ϕH)tH / 2. With these expressions and f = 1, we can write the principal's problem as: Max
V ðqH Þ−tH =2
subject to ð1−aÞtH −hL qH z0; atH −hH qH z0: From the objective function and the constraints, it is clear that when α = 1/2 the principal's payoff is at the highest level, V(qH⁎) − θH qH⁎, and with α > 1/2, her payoff is strictly lower than V(qH ⁎) − θH qH ⁎. However, the principal can improve this payoff by offering the second best no audit contract. Therefore, if τ = 0, then the no-audit contract gives a strictly higher payoff to the principal than the audit contract, and τ > 0 must hold in the audit contract. Finally, we show that the agent receives no rent and α = θH / (θH + θL). Since 1 > α, Eq. (A1) becomes: pLjH tH −pHjH tH −2c ¼ 0:
ðA3Þ
From Eq. (A3), we have the expression for τ by using the relation between τ and πH|H (and πL| from Eqs. (1) and (2) in Section 2: τ = 1 − (ϕH / ϕL)[(tH + 2c) / (tH − 2c)]. From this equation, we can express πL ≡ ϕLτ and πH ≡ ϕH + ϕL(1 − τ) as πL = ϕL − ϕH [(tH + 2c) / (tH − 2c)] and πH = ϕH [2tH / (tH − 2c)]. Also, together with f = 1, (IC ) with 1 > τ > 0 implies that H)
tL −hL qL ¼ ð1−aÞtH −hL qH :
ðA4Þ
The expression in Eq. (A4) means that the agent with i = L is indifferent to cheating. Therefore, the participation constraints (PCL) and (PCH) with f = 1 become: tL −hL qL z0:
ðA5Þ
atH −hH qH z0:
ðA6Þ
Using the expressions πL = ϕL − ϕH[(tH + 2c) / (tH − 2c)] and πH = ϕH[2tH / (tH − 2c)], we can write the principal's problem as follows: Max
tH þ 2c 2tH 1 /L − /H ½V ðqL Þ−tL þ /H V ðqH Þ− tH tH −2c 2 tH −2c
subject to Eqs. (A5) and (A6), where V(qH) − 1/2tH in the second term of the objective function is the simplified expression by using πL|HtH − πH | HtH − 2c = 0. From the objective function and the
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
389
constraints, Eqs. (A5) and (A6), the principal can extract the agent's rent by decreasing tL and increasing qH until the agent's rent is zero. From binding (A6), tH = θHqH / α, and Eq. (A4) and binding (A5) imply that (1 − α)tH − θLqH = 0. Solving these two equations, α = θH / (θH + θL). □ Proof of Proposition 2. From Proposition 1, we have tL = θLqL, tH = (θH + θL)qH, f = 1, and α = θH / (θH + θL). After substituting these by their values, the objective function of the principal is expressed as: ðhH þ hL ÞqH þ 2c /L −/H ½V ðqL Þ−hL qL ðhH þ hL ÞqH −2c 2ðhH þ hL ÞqH hH þ hL qH þ /H V ðqH Þ− ðhH þ hL ÞqH −2c 2 Even with c = 0, the first order condition for qH is V′(qH) = (θH + θL) / 2, which means that qH is still distorted upward (qH > qH⁎). □ References Baron, D., Besanko, D., 1984. Regulation, asymmetric information, and auditing. Rand Journal of Economics 15, 447–470. Bester, H., Strausz, R., 2001. Contracting with imperfect commitment and the revelation principle: the single agent case. Econometrica 69, 1077–1098. Bolton, P., 1987. The Principle of Maximum Deterrence Revisited. Working Paper #8749, U.C. Berkeley. Border, K., Sobel, J., 1987. Samurai accountant: a theory of auditing and plunder. Review of Economic Studies 54, 525–540. Demougin, D., Garvie, D., 1991. Contractual design with correlated information under limited liability. Rand Journal of Economics 22, 477–489. Dunne, S., Loewenstein, M., 1995. Costly verification of cost performance and the competition for incentive contracts. Rand Journal of Economics 26, 690–703. Graetz, M., Reinganum, J., Wilde, L., 1986. The tax compliance game: toward an incentive theory of law enforcement. Journal of Law, Economics, & Organization 2, 1–32. Harrington, H., Mathers, D., 1997. ISO 9000 and Beyond: From Compliance to Performance Improvement. McGraw-Hill. Krasa, S., Villamil, A., 2000. Optimal contracts when enforcement is a decision variable. Econometrica 68, 119–134. Krass, P., 2002. The never-ending audit. CFO 18, 25–26. Kessler, A., 2004. Optimal auditing in hierarchical relationship. Journal of Institutional and Theoretical Economics 160, 210–231. Khalil, F., 1997. Auditing without commitment. Rand Journal of Economics 28, 629–640. Khalil, F., Parigi, B., 1998. Loan size as a commitment device. International Economic Review 39, 135–150. Kofman, A., Lawarrée, J., 1993. Collusion in hierarchical agency. Econometrica 61, 629–656. Laffont, J.-J., Tirole, J., 1993. A Theory of Incentives in Procurement and Regulation. MIT press, Cambridge, Massachusetts. Melumad, N., Mookherjee, D., 1989. Delegation as commitment: the case of income tax audits. Rand Journal of Economics 20, 139–163. Nalebuff, B., Scharfstein, D., 1987. Testing in models of adverse selection. Review of Economic Studies 5, 265–277. Riordan, M., Sappington, D., 1988. Optimal contract with public ex post information. Journal of Economic Theory 45, 189–199. Searcy, D., Woodroof, J., 2003. Continuous auditing: leveraging technology. CPA Journal 73, 46–48. Strausz, R., 1997. Delegation of monitoring in a principal–agent relationship. Review of Economic Studies 64, 337–357.
Conducting inaccurate audits to commit to the audit policy ☆ Aaron Finkle a , Dongsoo Shin b,⁎ a
Department of Economics, California State University, San Marcos, San Marcos CA 92096, United States b Department of Economics, Santa Clara University, Santa Clara CA 95053, United States Received 23 May 2005; received in revised form 16 May 2006; accepted 17 May 2006 Available online 30 June 2006
Abstract Using a principal–agent model, we study the structure of an optimal audit policy when it is not contractible. In our model, auditing technologies are distinguished according to the accuracy and the frequency of audits. Our analysis reveals that audits take place with the highest frequency, but the accuracy of audits is compromised in equilibrium. We also find that without commitment to the audit policy, the first-best outcome cannot be achieved even when auditing is costless. © 2006 Elsevier B.V. All rights reserved. JEL classification: D82; L23 Keywords: Principal–agent; Accuracy and frequency of audits; Commitment
1. Introduction It is now well understood that auditing is an important device for mitigating incentive problems in organizations. Production outsourcing, corporate governance, lender–borrower relationships, government procurement, and other examples of contracting with auditing mechanisms abound. In selecting an audit policy, the choice in question is between the frequency of audits and the accuracy ☆
We thank Anke Kessler, Fahad Khalil, and Jacques Lawarrée for detailed suggestions and comments. We also thank Sanjiv Das, John Heineke, Michael Kevane, Helen Popper, Bill Sundstrom, and seminar participants at the 2005 Midwest Economic Theory Conference (University of Kansas) for discussions. Comments from the anonymous referees and the editor have substantially improved this paper. The authors acknowledge the financial supports from the Faculty Research Grant at Willamette University and the Presidential Research Grant at Santa Clara University. ⁎ Corresponding author. E-mail addresses: [email protected] (A. Finkle), [email protected] (D. Shin). 0167-7187/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ijindorg.2006.05.005
380
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
of the audits that occur. For example, in outsourcing relationships, buyers can conduct audits by sampling items from different shipments. The buyer can increase the frequency by increasing the chance of auditing a particular shipment, or increase the accuracy by selecting a larger sample size from those shipments that the buyer audits. Similarly, insurance companies can investigate a greater number of claims or increase the time devoted to each individual claim to determine if it is fraudulent. The choice in each of these cases is whether to devote attention to more frequent audits or to a greater accuracy in the audits that take place. Since auditing is a costly process, the allocation of resources between the frequency and the accuracy of audits is an important issue. Organizations choose their audit technologies based on both direct and strategic benefits. While some audits are conducted only once or twice a year,1 there are many situations in which the quantity of audits is emphasized more than the quality. For example, in his report for CFO Magazine, Krass (2002) points out that corporate auditing is often conducted relentlessly, while the effectiveness of such auditing is frequently questioned by practitioners. He reports that there is growing skepticism of the merits of sacrificing quality for a greater quantity of audits. Despite some criticism, many organizations, such as manufacturing companies and financial firms are practicing continuous audits in the controlling process.2 Although numerous studies examine the optimality of audit strategies,3 there are few previous contributions that address the question why organizations sometimes conduct a large number of audits, while compromising the accuracy. Using a principal–agent model, we attempt to shed light on these phenomena by studying the structure of audit strategies with variable accuracy and frequency. We adopt a standard model in which the agent has private information on the cost of production — the agent has an incentive to “cheat” or to exaggerate the production cost in order to command information rent, which the principal reduces through distortions in the output schedule. By conducting an audit, the principal can receive an informative signal about the cost of production. In our model, the principal chooses two distinct probabilities with respect to auditing — first, the probability that the audit takes place, and second, the probability that the signal from the audit is correct. Following Kessler (2004), we interpret the first one as the frequency of audits and the second one as the accuracy of audits. We assume that the principal's ability to commit is limited in that the output levels and the transfers are contractible, but the audit strategy is not. The audit technologies are chosen according to the principal's ex post incentives after production and transfer take place as specified in the contract.4 The main focus of this paper is on the choice of auditing technology in the optimal contract. The distortionary feature of the output schedule is similar to the previous contributions in agency contracting and only briefly discussed here. We derive two main results from allowing both frequency and accuracy to be chosen by the principal ex post: First, the principal chooses the 1 An example is the audits conducted by International Organization of Standardization (ISO) for quality standards. See Harrington and Mathers (1997). 2 See Searcy and Woodroof (2003). 3 For example, see Baron and Besanko (1984), Border and Sobel (1987), Demougin and Garvie (1991), Khalil (1997), and Kessler (2004) among others. 4 In actuality, commitment to audit policies is often limited. For instance, the outsourcing contract for labor and material designed by Consumers Energy, a company that provides utilities in the state of Michigan (http://www. consumersenergy.com/apps/pdf/CSS933-OEP.pdf) states, “[s]aid records and books of account… shall be open to examination during regular business hours by the owner or its agents for the purpose of [auditing].” However, nothing is mentioned of the frequency or accuracy of said audits, which is implicitly left to the discretion of the owner after receipt of the product.
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
381
highest possible level of frequency, while creating some inaccuracy in audits. Second, the principal cannot achieve the first-best outcome even when auditing is costless. When the audit policy is not contractible, the principal will conduct an audit only if the expected net gain from auditing is positive at that point. Since the audit frequency and accuracy are costly, the principal is reluctant to conduct an accurate audit ex post. Thus, by inducing the agent to cheat with some probability,5 the principal provides herself with an ex post incentive to conduct an audit with some level of accuracy. At the same time, to induce the agent to cheat, the contract must induce the principal to decrease one or both of the frequency and the accuracy of audits in equilibrium such that a cheating agent evades punishment with a strictly positive probability. According to our results, the principal conducts an audit with probability one but with some inaccuracy. Since the audit policy is not contractible and accuracy is costly, the principal needs to provide herself with incentive to invest in accuracy ex post. To do so, the principal encourages the agent to cheat with a high enough probability such that the probability of facing a dishonest agent exceeds the probability of facing an agent who has reported costs truthfully, thereby increasing the return from accuracy. Cheating, however, will not occur if the audits are perfectly accurate (i.e. no audit errors). Thus, in equilibrium, the agent will cheat with a strictly positive probability and audits occur with some inaccuracy. Furthermore, the inaccuracy of the audits provide the principal with an additional payoff from penalizing a truthful agent some of the time. The added reward from an inaccurate audit makes the ex post benefit from conducting an audit strictly positive, and as a result audits are conducted with the greatest frequency in equilibrium. We also find that the possibility of collecting a penalty from a truthful agent prevents the principal from implementing the first-best outcome even when there is no cost of conducting an audit. When auditing is costless, the principal does not need to induce cheating by the agent to have ex post incentive to audit. However, the principal will still benefit from auditing the truthful agent by using an inaccurate audit. Thus, even when auditing is costless, the principal must still induce the agent to cheat in order to provide accuracy in the audit. Although there is a long literature in agency contracts with auditing, much of it does not consider the issue of commitment and audit technology. Exceptions are Khalil (1997) and Khalil and Parigi (1998) which also study an optimal audit strategy without commitment. Both papers obtain a result similar to ours showing that mixed strategies are employed in equilibrium. However, they assume that audits are perfectly accurate exogenously. With such a setup, the audit occurs at random. In our model, the principal can choose not only the frequency but the accuracy of audits as well, and the principal conducts the audit with certainty if she cannot commit to auditing strategy. Also, in these papers, the principal implements the first-best outcome when the cost of auditing is zero. In this paper, we show that when the principal chooses the accuracy of audits as well as the frequency, she cannot implement the first-best outcome even when auditing is costless. Other papers that study the issue of auditing without commitment include Dunne and Loewenstein (1995), Graetz et al. (1986), Melumad and Mookherjee (1989), and Strausz (1997). The first paper studies an optimal auditing strategy when agents bid for contracts. The second paper considers income tax compliance when commitment is not possible, but models incomes (output level in our context) exogenously. The last two papers show that the principal uses delegation of the audit policy to a third party as a commitment device when the principal cannot commit to the audit policy. None of the above authors consider the accuracy of audits. 5
As in Khalil (1997), without commitment to the audit policy, the revelation principle no longer applies. See Bester and Strausz (2001) for a general analysis of the revelation principle with limited commitment.
382
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
Our paper is also related to Kessler (2004) and Nalebuff and Scharfstein (1987). Like ours, both the accuracy and the frequency of audits are considered in their models. Using a model without a cost of auditing, Kessler shows that the frequency of audits is not a concern for the principal. The second paper, adopting a competitive market framework, shows that the first-best outcome can be achieved only when the auditing signal is accurate enough and the penalty is unbounded. Unlike ours, the authors do not cope with the issue of commitment. Finally, Krasa and Villamil (2000) study lender–borrower relationships with limited commitment to enforcement to show that a simple debt contract with deterministic enforcement is optimal. In their model, as in ours, the optimal contract induces the agent to withhold information which makes deterministic enforcement attractive ex post. Stochastic enforcement induces information revelation but also eliminates incentive for enforcement ex post. As in our model, the principal must induce the agent to withhold information if enforcement is to be used ex post. While we also find deterministic audits are optimal, it is because of the accuracy choice. The rest of the paper is organized as follows. The next section presents the model. Section 3 discusses our results. Section 4 concludes. All proofs are in the Appendix. 2. The model A risk-neutral principal offers a contract to a risk-neutral agent to carry out a project that yields an output q. The cost parameter of the project, θ, can be either θL or θH, with Δθ ≡ θH − θL > 0. We denote by ϕi the principal's prior belief that θ = θi, where i ∈ {L, H}. The principal values the output of the project according to a strictly concave function, V(q), satisfying the Inada condition. The agent bears the cost of the project given by θq and receives a transfer t from the principal in return for producing the output level q. If the agent rejects the contract, he ends up with his reservation payoff, which is normalized to zero. If the agent accepts the contract, he produces qi according to the state i ∈ {L, H }. Under full information, the output schedule is said to be the first best and is characterized by V Vðq⁎i Þ ¼ hi ; iafL; Hg; and the agent obtains no information rent in either state. Under asymmetric information, the realized state i ∈ {L, H} is private information to the agent, and it is not known to the principal. Thus, the agent may have incentive to “cheat” or misrepresent the state i by producing qj ( j ≠ i) to command information rent. We denote by τi the probability that the agent is “truthful” or produces qi in state i ∈ {L, H}. Thus, the probability that the agent produces qi (regardless of whether he is truthful or not) is given by pi u/i si þ /j ð1−sj Þ; i pj and i; jafL; Hg: As usual in the models of this type, the agent has an incentive to exaggerate the production cost for a higher compensation when true state i = L, but has no incentive to cheat when i = H.6 This implies that τH = 1. To simplify the notation, define τL ≡ τ. This implies πL ≡ ϕLτ and πH ≡ ϕH + ϕL (1 − τ).
6
This will be clear from the result that the agent receives no rent in the optimal contract with auditing.
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
383
Using Bayes' rule, we derive the following conditional probabilities from the principal's point of view. After receiving qH, her posterior belief that the agent was truthful is: pHjH u
/H ; /H þ /L ð1−sÞ
ð1Þ
and the probability that the agent cheated is 1−pHjH ¼ pLjH u
/L ð1−sÞ : /H þ /L ð1−sÞ
ð2Þ
By conducting an audit, the principal can verify the true state i ex post. The principal conducts an audit with probability f ∈ (0, 1] after receiving qH.7 We interpret f as the frequency of audits. By conducting an audit, the principal receives a signal s ∈ {L, H } on state i. At the time of auditing, the principal also chooses the accuracy of the audit, denoted by α ∈ (1/2, 1]. An audit contract is one in which 1 ≥ f > 0 and 1 ≥ α > 1/2 (therefore a no audit contract is one in which f = 0 and/or α = 1/2). To ease the exposition, we follow Kofman and Lawarrée (1993) and assume that α = Pr (sL|θL) = Pr(sH |θH) and 1 − α = Pr(sH|θL) = Pr(sL|θH). The cost of auditing depends on the frequency and accuracy, and is given by C( f, α) = cf (2α − 1), with c > 0. We only focus on the parameters for which the audit contract is optimal. Specifically, we assume ϕL > ϕH and c is sufficiently small such that the principal prefers the audit contract to the no audit contract.8 We assume that the principal's audit strategy is not contractible. That is, the contract only specifies the transfers and the output schedule, and the principal chooses f and α after receiving the output from the agent. We briefly present optimal contract when no audit is conducted. As the revelation principle applies, we can restrict our attention to the contract that induces the agent to report truthfully. This means that τ = 1 in equilibrium, and thus πL = ϕL, πH = ϕH, and πH | H = 1. Without auditing, the contract specifies {qL, qH, tL, tH} to maximize the following problem faced by the principal. Max
/L ½V ðqL Þ−tL þ /H ½V ðqH Þ−tH
subject to tL −hL qL z0;
ðPCL Þ
tH −hH qH z0;
ðPCH Þ
tL −hL qL ztH −hL qH :
ðICL Þ
tH −hH qH ztL −hH qL :
ðICH Þ
7 An audit after receiving qL has no effect except by increasing tL in response to the ex post penalty that occurs following the audit. We rule out such audits since they do not affect the results in meaningful ways. Specifically, the agent is compensated ex ante for the precise penalty that results ex post without any effect on the overall incentives. This trivial consequence is easily avoided by assuming that the penalty for under-reporting costs is zero since it entails a lower compensation for the agent. pffiffiffi 8 For example, if V ðqÞ ¼ 100 q, θL = 1, θH = 2, ϕL = 3/4 and ϕH = 1/4, then the audit contract is optimal for c not too large. In general, for ϕL < ϕH, the audit contract is not optimal in our model.
384
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
The first two constraints assure that the agent's payoff in any state is higher than his reservation payoff (zero). The last two constraints assure that the agent's payoff from being truthful is higher than his payoff from misrepresenting the state. As in the standard principal–agent problem, (PCH) and (ICL) are binding in equilibrium, and from these two binding constraints the agent's rent when i = L is expressed as ΔθqH. To reduce the rent, the principal distorts qH downward and the optimal b output schedule is characterized by V′(qLb) = θL and V′(qH ) = θH + Δθ(πL / πH), and the principal's b b b b b expected payoff is: ϕL[V(qL) − θLqL − ΔθqH] + ϕH[V(qH) − θHqH ]. This is the standard second9 best outcome with no audits. The timing of the game is summarized as follows. Nature chooses the state i ∈ {L, H } and the agent learns it privately. The principal offers the contract followed by the agent's decision whether or not to accept it. The contract {qi, ti} is executed. The principal conducts an audit with probability f and accuracy α if she received qH. When auditing takes place, the signal s ∈ {L, H } is realized. If s = L, then the principal collects a lump sum penalty P from the agent. As many studies have shown, when the principal can impose unbounded penalties, she can always achieve the first-best outcome.10 We follow Laffont and Tirole (1993, Ch12) and assume that the agent is wealth constrained and the principal cannot impose a penalty larger than the transfer to the agent: the penalty P cannot exceed the transfer to the agent. As typical in the literature, the maximal punishment principle11 applies, and thus in the optimal contract, P = tH. 3. The results Recall that since the principal cannot commit to the auditing policy, f and α are not included in the contract. In the first stage, the principal chooses {qL, qH, tL, tH} to maximize her payoff. The contract must guarantee the agent his reservation payoff in each state of i ∈ {L, H}. The optimal contract also induces a second stage equilibrium in the choice of the agent, τ (the probability of being truthful), and the choice of the principal, f and α. These choice variables in the second stage will determine the probabilities πL, πH, πH | H, and πL|H. When writing the principal's problem, the second stage equilibrium conditions act as constraints. This means that the principal can choose {qL, qH, tL, tH, f, α, τ} as long as she respects the optimality conditions for the second stage. Therefore, the principal's problem can be written as follows. Max
pL ½V ðqL Þ−tL þ pH ½V ðqH Þ−tH þ f fapLjH tH þ ð1−aÞpHjH tH −cð2a−1Þg
subject to s½tL −hL qL þ ð1−sÞ½ f ð1−aÞtH þ ð1−f ÞtH −hL qH z0;
ðPCL Þ
f atH þ ð1−f ÞtH −hH qH z0;
ðPCH Þ
̂ LjH tH þ ð1−aÞpHjH tH −cð2a−1Þg; f aarg maxf ̂ f fap
ðFIÞ
9 When α = 1/2, auditing provides no information and the expected payoffs for both parties will be identical to the second-best (without audits). The difference will be an additional transfer to the agent ex ante equal to the expected ex post penalty. Specifically, the contract that results in α = 1/2 will be {qbL, qbH, tbL, tBH} where tBH = 2tbH, which induces truthful revelation (τ = 1) and auditing with f = 1. 10 See, for example, Riordan and Sappington (1988) and Demougin and Garvie (1991) among others. 11 See Baron and Besanko (1984) and Bolton (1987) for a formal analysis. If P < tH, the principal will have a less incentive to audit ex post, and the agent will have a greater incentive to misrepresent the state.
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
385
aaarg maxâ ap ̂ LjH tH þ ð1−aÞp ̂ HjH tH −cð2a−1Þ; ̂
ðAIÞ
saarg maxŝ s½t̂ L −hL qL þ ð1− sÞ½ ̂ f ð1−aÞtH þ ð1−f ÞtH −hL qH :
ðICÞ
From the objective function, when auditing takes place, the principal collects penalty P = tH from the agent in the following two cases: the principal received the correct signal when i = L, or she received the incorrect signal when i = H. As before, (PCL) and (PCH) induce participation in any state of i. Conditions (FI ), (AI ), and (IC ) represent the second stage equilibrium and determine the values of f, α, and τ given the contract in the first stage. The requirement of sequential optimality restricts the principal's choice of these variables: f and α are chosen to maximize her gain, and τ is chosen to maximize the agent's payoff when i = L. In an audit contract ( f > 0 and α > 1/2), as in Khalil (1997), the revelation principle no longer applies without commitment to the audit policy. If the principal were to choose fα = 1, the agent would be truthful with certainty (τ = 1). Then, the principal has an ex post incentive to save the entirety of the audit cost by reducing frequency and/or accuracy. This can be anticipated by the agent, and hence cheating by the agent occurs with a strictly positive probability. As a result, the second stage equilibrium is comprised of mixed strategies in the principal's choice of audit policy and agent's choice of compliance. We present our first result in the following proposition. Proposition 1. In the audit contract without commitment, ˙ The principal always audits, i.e., f = 1, but with imperfect accuracy: α = θH / (θH + θL) < 1. ˙ The agent's truth-telling is random, i.e., 1 > τ > 0. ˙ The agent obtains no rent in either state of nature. Proof. See Appendix. The proposition says that when the audit policy is not contractible, the principal chooses to audit with the highest frequency, but allows for audit errors to occur. Clearly, the principal will not conduct a costly accurate audit unless it is beneficial to her ex post. Therefore, the principal will increase accuracy of the audit only if the probability of collecting the penalty P = tH from cheating agent is higher than the probability of collecting the penalty from truth-telling agent by receiving a wrong signal (πL|H > πH | H).12 This means that the principal must let the agent cheat with a strictly positive probability, which will occur only if the accuracy and/or the frequency of audits are reduced from their highest levels ( fα < 1). Reducing accuracy has another effect which reducing frequency does not — the gain from penalizing the truthful agent. The additional effect from reducing accuracy is the key difference from Khalil (1997), where α = 1 exogenously. In his paper, the principal must be indifferent toward the frequency of audits in equilibrium, and hence an audit yields a zero expected return (ex post) even though she collects the penalty with positive probability. This can be seen here by considering the (FI ) constraint with α = 1, which would imply that 1 > f > 0 as in Khalil (1997), since the principal employs a mixed strategy in equilibrium in order to induce the agent to cheat. With α = 1, the first order condition of (FI ) gives: πL|HtH = c. As this equation implies, in Khalil (1997), where α = 1, the expected gain and cost of conducting an audit are equal. When the accuracy is a choice variable, however, the principal can collect πH | HtH from the honest agent by decreasing α from 1, while at
12
(AI) implies that (πL|H − πH|H)tH ≥ 2c. Therefore, πL|H > πH|H must hold for c > 0.
386
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
the same time, reduce the cost of auditing. Thus, with α < 1, the principal strictly benefits from conducting the audit ex post, and consequently, auditing takes place with certainty ( f = 1). Some previous studies have shown that the principal prefers more accurate to more frequent audits when commitment to audit policy is possible. In particular, Kessler (2004) demonstrates that only the accuracy of audits affects the agent's rent, which implies that when auditing is costly the frequency of audits is minimized.13 Our result shows that, without commitment, the result is reversed. When the audit policy is not contractible, generating some inaccuracy in auditing provides an incentive to the principal to conduct a costly audit ex post since the principal's ex post gain from conducting an audit becomes strictly positive. Consequently, audits are conducted with the highest frequency. The specific level of α comes from the principal's effort to not provide a positive rent to the agent. With f = 1, the indifferent condition of the agent from (IC ) gives the following expression for his rent: 1−a tL −hL qL ¼ hH −hL qH : 14 ð3Þ a As the agent's rent directly influences the principal's payoff, she will not leave any rent to the agent. From Eq. (3), the agent's rent increases as α decreases. In the optimal contract, the principal decreases α just to the level at which the RHS of Eq. (3) does not become strictly positive, and hence is α = θH / (θH + θL). We now briefly discuss the distortions in the output schedule. As in Khalil (1997), the output level is efficient when i = L(qL = qL ⁎), but there is overproduction when i = H(qH > qH ⁎). The first order condition for qH after replacing the choice variables with their values is: V VðqH Þ ¼ ðhH þ hL Þ=2 < hH : Since 1 > α > 1/2 in equilibrium, (AI ) implies that (πL|H − πH | H)tH = 2c. This equation implies that the principal must increase tH in order to reduce πL|H thereby reducing the probability of cheating. Since the agent's rent is zero in either state, the principal increases tH while increasing qH as well. Therefore, from the first-best levels she reduces cheating by increasing the transfer and output in the high-cost state. The endogenous choice of accuracy of audits brings about another important difference from Khalil (1997). With α set at 1 exogenously, the frequency of audits f equals 1 only when auditing is completely costless (c = 0). In our model, the result of f = 1 is independent of the cost of auditing (unless c is too large that the principal prefers the no-audit contract). Again, in our model, the result of auditing with certainty ( f = 1) stems from the strategic effect of conducting an inaccurate audit — collecting penalty from the honest agent. By letting c = 0 in our model, we have the next result. Proposition 2. Without commitment to the audit policy, the principal cannot implement the firstbest outcome even when c = 0.
The author's result, however, relies on the audit-dependent transfers — the principal can discriminate the transfer to the agent depending on whether an audit is conducted. We employed the audit-independent transfers in this paper for expositional purpose. Our results do not change in any significant way with audit-dependent transfers. 14 From binding (PCH), tH = (θH / α)qH. Substituting for tH by its value gives the expression in Eq. (3). 13
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
387
Proof. See Appendix. The result above differs from the conventional wisdom that the first-best outcome is achieved when auditing is costless. When the principal can choose both the frequency and the accuracy of audits without commitment, the first-best outcome can never be achieved even if c = 0. Because of the temptation to collect penalty from an honest agent, the principal still conducts an inaccurate audit even when auditing is costless. This implies that there is a strictly positive probability that the agent does not comply. As in Khalil (1997), the principal distorts qH upward when she cannot commit to her auditing strategy.15 In his paper, however, the accuracy of audits is exogenously set at the perfect level, and thus the first-best outcome is implemented when c = 0. When the principal chooses the accuracy of audits, qH is still distorted even with c = 0. 4. Conclusion In this paper, we have analyzed the optimal audit strategy in a principal–agent relationship when audit technologies are distinguished according to the frequency and the accuracy of audits and the audit policy is not contractible. Because the principal cannot commit to auditing, the optimal contract induces the agent to cheat with some probability, which in turn induces a costly audit ex post. Our result suggests that audits are conducted with a highest level of frequency, but there are some inaccuracies in audits in equilibrium. This result has another interpretation, comparing two different types of probability. One type determines the chance that an audit results in a signal, and the other determines the accuracy when a signal occurs. As our result suggests, without commitment, the principal prefers the former type of probability. That is, the possibility of an incorrect signal is preferred to the possibility of no signal. Our analysis also revealed that when the audit policy is not contractible, the first-best outcome cannot be implemented even if the principal does not incur any cost to conduct an audit. Appendix A Proof of Proposition 1. Suppose the principal always audits with perfect accuracy, i.e., fα = 1. Then, by (IC ) and (PCL), the agent would always be truthful (τ = 1), which in turn gives πL|H = 0 and πH|H = 1. Then, (AI ) becomes: ̂ H −cð2a−1Þ: ̂ aaarg maxâ̂ ð1− aÞt Clearly, with τ = 1, the principal would set α as small as possible (α = 1/2), which implies that the no audit contract is preferred. Thus, when the audit contract ( f > 0 and α > 1/2) is preferred by the principal, 1 > fα and 1 > τ must hold. Next, we show that if α > 1/2 then f = 1. With 1 ≥ α > 1/2, (AI ) implies that: pLjH tH −pHjH tH −2cz0:
ðA1Þ
Also, with f > 0, the following condition must hold from (FI ): aðpLjH tH −pHjH tH −2cÞ þ pHjH tH þ cz0;
15
ðA2Þ
By increasing qH, the principal must pay a higher tH. This gives her more incentive to increase the accuracy of the audit ex post.
388
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
If α = 1, then the expression in Eq. (A2) becomes πL|HtH − c = 0, where the equality comes from the fact that f < 1 with α = 1. However, with πL|HtH − c = 0, we have tH ≤ 0 from Eq. (A1), which is contradiction. If 1 > α > 1/2, then πL|HtH − πH|HtH − 2c = 0 in Eq. (A1), which makes the expression in Eq. (A2) strictly positive, i.e., f = 1. We now show that τ > 0. Suppose the agent always cheats (τ = 0). The first order condition of (AI ) gives ϕLtH − ϕHtH − 2c = 0 or c = (ϕL − ϕH)tH / 2. With these expressions and f = 1, we can write the principal's problem as: Max
V ðqH Þ−tH =2
subject to ð1−aÞtH −hL qH z0; atH −hH qH z0: From the objective function and the constraints, it is clear that when α = 1/2 the principal's payoff is at the highest level, V(qH⁎) − θH qH⁎, and with α > 1/2, her payoff is strictly lower than V(qH ⁎) − θH qH ⁎. However, the principal can improve this payoff by offering the second best no audit contract. Therefore, if τ = 0, then the no-audit contract gives a strictly higher payoff to the principal than the audit contract, and τ > 0 must hold in the audit contract. Finally, we show that the agent receives no rent and α = θH / (θH + θL). Since 1 > α, Eq. (A1) becomes: pLjH tH −pHjH tH −2c ¼ 0:
ðA3Þ
From Eq. (A3), we have the expression for τ by using the relation between τ and πH|H (and πL| from Eqs. (1) and (2) in Section 2: τ = 1 − (ϕH / ϕL)[(tH + 2c) / (tH − 2c)]. From this equation, we can express πL ≡ ϕLτ and πH ≡ ϕH + ϕL(1 − τ) as πL = ϕL − ϕH [(tH + 2c) / (tH − 2c)] and πH = ϕH [2tH / (tH − 2c)]. Also, together with f = 1, (IC ) with 1 > τ > 0 implies that H)
tL −hL qL ¼ ð1−aÞtH −hL qH :
ðA4Þ
The expression in Eq. (A4) means that the agent with i = L is indifferent to cheating. Therefore, the participation constraints (PCL) and (PCH) with f = 1 become: tL −hL qL z0:
ðA5Þ
atH −hH qH z0:
ðA6Þ
Using the expressions πL = ϕL − ϕH[(tH + 2c) / (tH − 2c)] and πH = ϕH[2tH / (tH − 2c)], we can write the principal's problem as follows: Max
tH þ 2c 2tH 1 /L − /H ½V ðqL Þ−tL þ /H V ðqH Þ− tH tH −2c 2 tH −2c
subject to Eqs. (A5) and (A6), where V(qH) − 1/2tH in the second term of the objective function is the simplified expression by using πL|HtH − πH | HtH − 2c = 0. From the objective function and the
A. Finkle, D. Shin / Int. J. Ind. Organ. 25 (2007) 379–389
389
constraints, Eqs. (A5) and (A6), the principal can extract the agent's rent by decreasing tL and increasing qH until the agent's rent is zero. From binding (A6), tH = θHqH / α, and Eq. (A4) and binding (A5) imply that (1 − α)tH − θLqH = 0. Solving these two equations, α = θH / (θH + θL). □ Proof of Proposition 2. From Proposition 1, we have tL = θLqL, tH = (θH + θL)qH, f = 1, and α = θH / (θH + θL). After substituting these by their values, the objective function of the principal is expressed as: ðhH þ hL ÞqH þ 2c /L −/H ½V ðqL Þ−hL qL ðhH þ hL ÞqH −2c 2ðhH þ hL ÞqH hH þ hL qH þ /H V ðqH Þ− ðhH þ hL ÞqH −2c 2 Even with c = 0, the first order condition for qH is V′(qH) = (θH + θL) / 2, which means that qH is still distorted upward (qH > qH⁎). □ References Baron, D., Besanko, D., 1984. Regulation, asymmetric information, and auditing. Rand Journal of Economics 15, 447–470. Bester, H., Strausz, R., 2001. Contracting with imperfect commitment and the revelation principle: the single agent case. Econometrica 69, 1077–1098. Bolton, P., 1987. The Principle of Maximum Deterrence Revisited. Working Paper #8749, U.C. Berkeley. Border, K., Sobel, J., 1987. Samurai accountant: a theory of auditing and plunder. Review of Economic Studies 54, 525–540. Demougin, D., Garvie, D., 1991. Contractual design with correlated information under limited liability. Rand Journal of Economics 22, 477–489. Dunne, S., Loewenstein, M., 1995. Costly verification of cost performance and the competition for incentive contracts. Rand Journal of Economics 26, 690–703. Graetz, M., Reinganum, J., Wilde, L., 1986. The tax compliance game: toward an incentive theory of law enforcement. Journal of Law, Economics, & Organization 2, 1–32. Harrington, H., Mathers, D., 1997. ISO 9000 and Beyond: From Compliance to Performance Improvement. McGraw-Hill. Krasa, S., Villamil, A., 2000. Optimal contracts when enforcement is a decision variable. Econometrica 68, 119–134. Krass, P., 2002. The never-ending audit. CFO 18, 25–26. Kessler, A., 2004. Optimal auditing in hierarchical relationship. Journal of Institutional and Theoretical Economics 160, 210–231. Khalil, F., 1997. Auditing without commitment. Rand Journal of Economics 28, 629–640. Khalil, F., Parigi, B., 1998. Loan size as a commitment device. International Economic Review 39, 135–150. Kofman, A., Lawarrée, J., 1993. Collusion in hierarchical agency. Econometrica 61, 629–656. Laffont, J.-J., Tirole, J., 1993. A Theory of Incentives in Procurement and Regulation. MIT press, Cambridge, Massachusetts. Melumad, N., Mookherjee, D., 1989. Delegation as commitment: the case of income tax audits. Rand Journal of Economics 20, 139–163. Nalebuff, B., Scharfstein, D., 1987. Testing in models of adverse selection. Review of Economic Studies 5, 265–277. Riordan, M., Sappington, D., 1988. Optimal contract with public ex post information. Journal of Economic Theory 45, 189–199. Searcy, D., Woodroof, J., 2003. Continuous auditing: leveraging technology. CPA Journal 73, 46–48. Strausz, R., 1997. Delegation of monitoring in a principal–agent relationship. Review of Economic Studies 64, 337–357.