2.2 Infinite Horizon Model without Retention/Rotation Regulation
2.2.1 Model without Credible Threat to Replace Auditors that Report Fraud
I first analyze the case where the incumbent auditor faces no report-related threat o f replacement. That is, the client cannot credibly threaten to terminate an auditor who reports fraud. The conditions that lead to this scenario will be formally stated in what follows, but the intuition behind this case is straightforward. If the increase in audit fees
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and expected switching costs caused by a replacement threat is greater than the increase in benefits to the client obtained by inducing a lower audit effort level, then the client will not announce such a threat, nor will it be credible. Thus, the chance that an auditor be engaged in any given period depends exclusively on the nature of the competitive market, and no past behavior is relevant. The incumbent auditor will charge as high a price as possible consistent with remaining the incumbent. I assume that if the client is indifferent between retaining the incumbent auditor and hiring a candidate auditor, the client will retain the incumbent. I will denote this benchmark case as the no-replacement case.
Lemma 2.1 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 no
replacement case, assuming that the auditor and the client agree to binding engagements with a duration of A^-periods, where M=[l ,ô>), is determined endogenously.15 The subscript nr denotes equilibrium values for the no-replacement case.
Lemma 2.1: In the no-replacement case:
(i) Both candidate and incumbent auditors will exert the same effort in every period, i f engaged, regardless o f the contract length N. The effort choice
is given by:
enr e argmin C(e) +L-qm\ h -p(e)] (2.2)
ee[g,e]
,s Without loss o f generality, I assume stationarity in the engagement length N. That is, if an initial engagement o f N periods is agreed upon, all subsequent engagements will be for N periods also.
Equivalent expressions can be obtained if we let the contract length vary over time.
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(ii) The price bid, in present value terms, by all candidate auditors fo r an N- period audit, denoted Q j f t ) , is given by:
^(A O = (1-6N)I(1-6) { C ( 0 + L - q ^ h - p ( e j ]} + (1-6NH - 6N-c (2.3) (Hi) The price bid, in present value terms, by the incumbent auditor fo r an N-
period audit, denoted & m(N), is given by:
fcUAO = (1-6N) / (1-6) { C (e J + L -q^ih - p ( e j ]} + (l-6 NH<?+c) (2.4)
The above audit-pricing equations are generalizing those of Magee and Tseng [1990] by allowing for variable auditor effort and multi-period engagements. The "value of incumbency" as defined in Magee and Tseng [1990], is independent of the discount rate and the number of periods N the engagement lasts, and is given by (H+c).16 As expected, low-balling and price cutting in the first AAperiod engagement do occur. The auditor charges (1-6N)-C in each Af-period engagement to recover the initial learning cost H. The auditor also provides a discount of 6N-c in the first A/’-period, which is recovered in subsequent A-period engagements by raising the price by (l-6 N)-c.17 At the start of the world, all auditors are candidates, and consequendy, the price bid will be ^(A O -
The above pricing relations are set up to leave the client indifferent, in terms of
16 Magee and Tseng [1990] define the value o f incumbency as the present value o f being the incumbent, assuming optimal pricing (and reporting) decisions over the remaining horizon. In the current
formulation, the engaged auditor becomes the "incumbent" at the end o f the iV-period relationship. The value o f incumbency, (t+ c), is for the entire N periods.
17 Low-balling in the first iV-period is given by [-6N-(J+c)], followed by an infinite series o f
[(l-6 N)-((+c)] every iV periods. The //-period discount rate is given by (1-6N)/5 N. Thus, the auditor earns zero profits in present value.
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audit price, between the incumbent and candidate auditor. In addition, both incumbent and candidate auditors will exert the same effort in every period if engaged. Consequently, there will never be any switches between auditors in the no-replacement case.
Note that if the auditor and client engage in binding multi-period contracts of N z 2, there may be considerable mismatches between current period fees and current period costs, because there is no unique fee for any given period within the iV-period engagement. The only equilibrium condition regarding the distribution of fees is that the present value o f fees over the A'-period engagement must add up to $>a£N) or If fees are back-loaded, quasi-rents within the jV-period engagement will exist, but these "quasi-rents" do not affect auditor behavior: the auditor will exert em in each period as given in (2.2).
The audit-pricing model of Lemma 2.1 suggests that the engagement length N has some effect on the client’s audit cost, and thus, N will be chosen optimally by the client.
Lemma 2.2 shows, however, that the client is indifferent to the engagement length N. Let denote the discounted present value of fees in an infinite horizon no-replacement model.
Lemma 2.2: The client is indifferent to the engagement length N, in cm. infinite horizon no-replacement model. The present value o f fees, T np is given by:
¥ „ = (! + l / ( l - 6 ) [ a e J + L - q ^ lh - p ie J ] } (2.5)
Note that is independent of the engagement length N, and the switching cost c.
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However, each potential auditor switch would increase the client’s cost by c. The intuition behind this result is straightforward. Once an auditor is engaged, she will remain with the client in perpetuity. Thus, how this infinite horizon is partitioned into finite contracts is irrelevant.