2.2 Infinite Horizon Model without Retention/Rotation Regulation
2.2.2 Model with a Credible Threat to Replace Auditors that Report Fraud
I now analyze the case where the auditor faces a credible threat of being replaced by the client if fraud is detected and reported. This replacement threat does not violate the ban on contingent fees. The auditor’s incumbency, not the fee, is contingent on the auditor’s report. The client will announce such a replacement threat if it is ex-ante efficient for him to do so. This threat will be credible only if it is also ex-post efficient for the client to follow-through with it.18 In determining their equilibrium bid prices, auditors must account for the possibility of being replaced. The incumbent auditor will charge a price that is as high as possible, but low enough to prevent entry due to price. The client, however, does not only consider the magnitude of the comparative equilibrium bids, but also the effect that replacing an auditor has on effort choice, which affects his expected benefits and the probability o f a government audit. I will refer to this case as the replacement case.
Binding multi-period engagements are permitted. However, contrary to the no
replacement case (where contracts of any length are optimal for the client), the optimal
18 The conditions under which it is both ex-ante and ex-post optimal for the client to adopt the replacement strategy will be determined below.
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contract length, for the client, in the replacement case is one period. This is formalized in Lemma 2.3.
Lem m a 2.3: In the replacement case, the client w ill only commit to single-period engagements.
The intuition behind Lemma 2.3 is the following: If the client engages with an auditor for more than one period, the conditions for replacement would occur with positive probability before the end of the engagement. The client cannot, however, replace the auditor until the end of the multi-period engagement given the binding nature o f the contract. The auditor will have no further threat of replacement in the remaining periods and will revert to the no-replacement effort level, leaving the client worse off.19 If the contract length is set to one period, the auditor will exert the replacement effort level in that one period and thus, in every period, leaving the client better off.
Lemma 2.4 presents the equilibrium effort that will be adopted and the equilibrium prices that will be charged, by the incumbent auditor and by the candidate auditor in the replacement case, when the auditor and the client agree to single-period engagements. The subscript r, denotes equilibrium values for the replacement case.
19 Recall that the client will adopt the replacement strategy only if the increase in benefits from inducing a lower effort level is greater than the increase in audit fees and expected switching costs caused by the replacement threat.
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Lemma 2.4: In the replacement case:
(i) Both candidate and incumbent auditors will exert the same effort in every period i f engaged The effort choice is given by:
et e argmin C(e) + L-qr-\h - /7(e)] + p(e)-6-(5+c) (2.6) ee[g,e]
(ii) The price bid by all candidate auditors in any given period, denoted <$r , is given by:
<£r = C(er) + L-q-\h - /?(er)] + (l-6)-£ - 5-c + p(er)-6-({+c) (2.7) (iii) The price bid by the incumbent auditor in any given period where
incumbency is retained, denoted is given by:
& c = C(er) + L-q-[h -p (e f[ + (l-6)*(C+c) +/7(er)-6-(ô+c) (2.8)
The value of incumbency is independent of the discount rate and is given by (J+c), as in the no-replacement case. The incumbent auditor, however, will lose the incumbency when fraud is detected, which occurs with probability p(er). In light of this positive probability event, the auditor’s effort choice is adjusted accordingly as given in (2.6). Thus, the expected value of incumbency is given by [1 -p(er)]'(H+c).
If we interpret future quasi-rents in expected value terms, low-balling is still obtained in equilibrium. The auditor adjusts the fee to account for the possibility of being replaced, by raising the price by/?(er)-8-(fi+c) in each period. As in the no-replacement case, the auditor also raises the audit price by (1 -8)*C in each period to recover the initial learning cost C, and provides a discount of 8-c in the first period which is recovered in subsequent
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periods by raising the price by (l-S)-c.20
The client will pay the auditor <&r in the first period. With probability [1 - p(er)], no fraud will be detected and the client will retain the auditor and pay <E>[r in the next period.
W ith probability p(er) the client will switch auditors incurring the switching cost c and paying the new auditor $ r The incumbency fee, <£>',, is equal to <Er + c, which implies that the client will pay <£r + c, in every subsequent period. Let Y r denote the present value of fees and expected switching costs in an infinite horizon model with a credible replacement threat. Y r is given by:21
Yr = fi + 1 / (1-8) (C(er) + L -q ^h -pCeJ] +p(eT)-6-(t+c)} (2.9) where (C + 1 / (1-8) (C(er) + L-qr-[h -p(er)] + p(e,)'8*6}) corresponds to the present value o f fees, and [6 / (1-8) p(er)'c] is the present value o f expected switching costs.