3.3 Case II: No Fraud is Detected in Period One
3.3.5 Client Opinion Shopping vs. Incumbent Resignation in Case II
In Case I, where fraud is detected in period one, auditor switches are expected to occur whenever Yi > Y 1- When no fraud has been detected in period one we have seen that the expected switching behavior may be independent o f these second period beliefs under certain conditions related to the previous period, in particular, beliefs about 0 and
Yx-
From Corollaries 3.2 and 3.3 we have seen that for low values of 6 e (0,0], auditor switches will never occur because a low type incumbent can profitably risk adjust her second period fee, while for high values Y\ 6 \y\ d) switches will occur for given parameter combinations whenever the incumbent auditor is a low type, which is a positive probability event.
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The above implies that comparisons between the likelihood of opinion shopping versus low type incumbent resignation after a clean report in period one, will differ across the above scenarios as follows:
(A) For y 2 < 0, the probability o f observing an auditor switch is given by:
/ . \ y , ) ( l - t y ) ( l- 4 > , O y , ) , ,
PrN^V,)- + i X - e - r , V n e (0 J ) (3'19)
(B) For y 2 > 1, we have that: Pr^S'wj.D^, ) = 0, V y 2 e (0 ,l).
(C) For y 2 e (0 ,l), Pt(Sm\D„f i ) = 0 V y 2 < y 2
i | \ <27-( l — O „ y , ) (1 - c t ) ) { \ - < ^ . d y . )
P f o - K . , ) - r V y 2>r 2, (3.20)
The first term in both (3.19) and (3.20) is the ex-ante probability that an auditor switch will occur after a no fraud report due to client opinion shopping, denoted
P r^G S ]/)^,), and the second term the ex-ante probability that an auditor switch will occur after a no fraud report due to a low type incumbent resignation, denoted
P r ( ô |0 * F.,)-
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We have that Pr^OSjD^-,) = Pr(/i?|Dw/ri j whenever y t = y “ , where y 2 is given by:
( l- a > ) ( l- < E , fl ^ )
2 co d (\-<& H y ^ (3-21)
Note that y 2 > 0 , but that it can also be greater than one. For, y 2 > 1, we will have
This leads to the following proposition regarding the likelihood of auditor switches after a no fraud report.
Proposition 3.7: For any combination o f auditor fra u d detection technologies
<t>H ><& L, there exists a a)11 e (0,1) such that fo r any CO < 0)", we have that:
The interpretation of Proposition 3.7 is that when auditor-client pre-alignment is lower than a given threshold, then auditor switches after a clean period one report are more likely to be caused by a low type incumbent resignation, regardless of the second period beliefs concerning the client’s opportunities to commit fraud. Note that for the case where y 2 e (0,l), that the relevant interval for y 2 will be ( y 2,l), since for
y 2 < y2 no auditor switches occur.
V y 2 e (0,1). Note also that y 2 is decreasing in CD.
Pr(0 SjD(VF,) < Pr(/i?|D^F1) f o r any y 2 e (0,1)
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The most probable cause for an auditor switch when w > a 11 will depend on whether y 2 < 0, or y 2e (0,1) • In the case where y 2 < 0 there exists no upper bound on co because any y 2e (0,1) is feasible, while when y 2e (0,l), an upper bound condition will exist given that y 2is restricted to the (y 2,l) interval.
Corollary 3.4: I f y 2 < 0, then fo r any co > to'1 there exists a y 2 (co) e (0,1) such that:
Pr((9S|£>WF,) < Pr(fl2[£>flp i ) V y2 € ( o ,y r ( ® ) )
P r(o s|z v ,)> P r(/.R |z > .VF.1) V y 2 e [ f ;
Note that J'" fthw) = i ằ and that (ty) is decreasing in to, with ( i ) =0.
This implies that as co increases, the interval y 2e (0, y 2 (cofj for which low type incumbent resignation is more likely than client opinion shopping decreases. However, over the entire range co e (co" , l ) , the inequality can go either way depending on the value of y 2. That is, there is no upper bound for © as of which opinion shopping is more likely than low type incumbent resignation.
An upper bound on © will exist, however, in the scenario where y 2e ( 0 ,l) , as stated in Proposition 3.8.
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Proposition 3.8: When y 2 e ( 0 ,l) , then fo r any combination o f auditor fra u d detection technologies there exists a Wa e (0,1) such that fo r any
0) > Q}11, we have that:
Pr(OS|DFFI) > Pr(/j?|f)FF1) fo r any y 2 e ( y 2 ,l)
Proposition 3.8 states that if auditor-client pre-alignment is greater than a given threshold, then auditor switches after no fraud has been detected in period one are more likely to be caused by client opinion shopping than by a low type incumbent resignation, again regardless o f second period beliefs concerning the client’s opportunities to commit fraud.
Corollary 3.5: I f y 2 e ( 0 ,l) , then fo r any Q)u <(D< a " , there exists a y 2 (co) e ( y 2,1) such that:
, ) < Pr(/i?|Z)WF,) v
P r(O S |Ê V ,)> P r(Z R |Z V ,) V n *& "(< ằ),l)
Again, the interval ( y2, y " (<ằ)) for which incumbent resignation is more likely than client opinion shopping decreases as to increases, which is consistent with previous results.
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Propositions 3.7, 3.8, and Corollary 3.5, are represented schematically in Figure 3.3.
Figure 3.3: Client Opinion Shopping versus Incumbent Resignation in Case II
Where: Area A is where ,, y z > y2) > Pr^OSjD^-,, y 2 > y2) ,
A reaB is where P r(O S |£V .i>Yi > Y i)> Pr( ^ | A v F . i > ^2)- Area C is where no auditor switches are observed because y 2 ^ y 2, L I is the line that represents y 2(co), and L2 represents y 2 (a>).66
66 In Figure 3.3, LI is represented by a straight line for ease o f presentation. It is, however, strictly convex.
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