Auditing in the Presence of Outside Sources of Information

Transcription

1 Journal of Accounting Research Vol. 39 No. 3 December 2001 Printed in U.S.A. Auditing in the Presence of Outside Sources of Information MARK BAGNOLI, MARK PENNO, AND SUSAN G. WATTS Received 29 December 1998; accepted 4 October 2000 ABSTRACT We examine how an auditor s ability to terminate a multi-period client relationship provides the auditor with a real option whose value depends on the nature of informational asymmetry between the incumbent and other potential auditors. In particular, we isolate conditions under which the auditor s private and public sources of information behave as complements rather than substitutes. In such circumstances, increasing the likelihood of publicly provided information induces the auditor to expend more (rather than less)resources in private information gathering activities. 1. Introduction We model an auditor s information gathering choices in a multi-period setting where the auditor s private knowledge of the client firm s type enables the auditor to extract later period quasi-rents while competition forces the auditor to offer to discount (low-ball) the fee charged for services in the first period. We document an interaction between costly private and public sources of information by isolating conditions under which the anticipated level of auditing activity and the increased availability of outside information are positively related (complementary)rather than negatively related. Our model manipulates the level of the auditor s expected future quasirent by introducing a signal which publicly reveals the client firm s type with Purdue University. We thank Bob Eskew, Lynda Thoman, and workshop participants at Carnegie Mellon University, Notre Dame University, and Purdue University for comments. We are especially grateful to the referee for helpful comments. 435 Copyright C, University of Chicago on behalf of the Institute of Professional Accounting, 2001

2 436 M. BAGNOLI, M. PENNO, AND S. G. WATTS a probability strictly greater than zero, but less than one. The public release of this signal effectively eliminates the ex post ability of the auditor to extract a quasi-rent. We show that an option-like feature emerges in such a setting. Because the auditor s eventual cost of providing services is determined by the client firm s underlying type, the auditor has an incentive to learn that type and use that information to decide whether to perform future audits. In particular, the auditor and firm will separate when the next period s equilibrium fee does not permit the auditor to at least break even. Because the fee (and quasi-rent) declines as the probability that the auditor s informational advantage will be lost, the potential loss (not breaking even) of being uninformed increases as the likelihood of public disclosure increases, thereby increasing the expected value of performing an investigation in the first period. 2. The Model In our model, a firm purchases auditing services in each of two periods. The auditing services provided are the mandated attestation by an independent party (the auditor) to required financial disclosures. At the end of the first period, the firm may choose to obtain audit services from another auditor, and the first period auditor may choose not to continue to provide audit services to the firm. That is, separation can occur at the end of the first period at the instigation of either party. We assume that, in the event of separation, the market knows separation has occurred but does not know who instigated it. We also assume that the quality of the audit does not depend on which auditor performs it. The extensive form description of the game is contained in figure 1, and a verbal description is provided below. In each period, t = 1, 2, the firm and the auditor first agree on the price of the audit. Once the agreement is made and the auditor is hired, the auditor chooses whether or not to spend K t and learn the state of the firm, s t [s l, s h ] R for t = 1, 2, where the state of the firm can be thought of as a characteristic of the firm that affects the costs and risks of auditing. We assume that s 1 and s 2 are strictly positively correlated and that the auditor and client firm have identical beliefs at the beginning of the first period. However, unlike the auditor who may learn s t by spending K t, the client firm does not observe information concerning s 1 and s 2 privately. 1 Thus the auditor who obtains private information will have an informational 1 Like Kanodia and Mukherji [1994] we assume that the incumbent auditor s information is superior to that of the client. Morgan and Stocken [1998] examine a two-period setting in which an informed incumbent auditor uses knowledge of the client s type in the second period to bid in a sealed-bid auction against an uninformed (but otherwise identical) prospective auditor. Similar to Englebrecht-Wiggans, Milgrom, and Weber [1983], they show that the informed bidders make an expected profit in such an auction and observe that the first period bid (where all bidders are uninformed) will be discounted to reflect the future expected quasi-rent (a low-ball).

3 SOURCES OF INFORMATION 437 FIG. 1. Extensive from description of the game. = nature s decision mode; = client firm s decision mode; = auditor s decision mode; y 1, y 2 = price of audits; s 1 = period one state of nature; ĉ b 1 = auditor s period one cost of providing service given no learning; K t = auditor s period t cost of learning the state of the firm. advantage over the client. 2 We assume that the auditor s cost of providing audit services given expenditure of K t to learn the state of the firm in period 2 In conventional relationships involving auditor attestation, circumstances may grant either the auditor or client an informational advantage. However, the potential for the auditor to learn more than the client is more likely if the client is an independent audit committee.

4 438 M. BAGNOLI, M. PENNO, AND S. G. WATTS t is ct a(s) + ε t and that the auditor s cost of providing audit services given that the auditor did not spend K t in period t (and thus does not know the state of the firm) is ct b(s) + ε t. 3 We assume that ε t are iid random variables with mean zero. 4 We also assume that E[ct a(s)] < E[c t b (s)] to represent the idea that the auditor who does not spend K t expends greater effort in the audit to (partially) make up for lack of information. Without loss of generality, we assume that both ct a and ct b are increasing in s. We also assume that: (1) E[c2 b(s 2) c2 a(s 2) s 1 ] is increasing in s 1 ; (2) E[c2 b(s 2) c2 a(s 2) K 2 s l ] < 0; and (3) E[c2 b(s 2) c2 a(s 2) K 2 s h ] > 0. 5 In our game, with probability γ>0, credible and verifiable information revealing the second period state of the firm becomes publicly available after the firm and the first period auditor decide whether to separate. As a technical assumption, let γ<(e[c2 a] E[c 2 a(s 2) s l ])/K 2. Let y 1 and y 2 represent the prices that the firm pays for audit services in the first and second periods respectively. In the first period, the auditor s payoff function is E [ y 1 c1 b(s 1) ] if the auditor does not learn the state of the π 1 = firm, y 1 c1 a(s 1) K 1 if the auditor does learn the state of the firm. The first period auditor knows either the first period state of the firm (by spending K 1 to learn it) or the realized costs of providing the first period audit, ĉ b 1 = c t b(s) + ε t. In the latter case, the auditor can update beliefs about the firm s second period state based on ĉ b 1. In either case, the auditor has more information about the second period state of the firm than a nonincumbent second period auditor would have. If the second period auditor is not the incumbent, then the second period payoff function is E [ y 2 c2 b(s ] 2) separation, K 1 π2 n = y 2 c2 a(s 2) y 2 c2 a(s 2) K 2 if the auditor does not learn the state of the firm, if the auditor publicly learns the state of the firm, if the auditor privately learns the state of the firm. 3 In each period t, we assume that c a t (s) + ε t and c b t (s) + ε t include both direct and indirect costs of providing the audit. Indirect costs can be thought of as potential costs associated with litigation. 4 The extra noise introduced by ε t ensures that the auditor cannot observe the realized costs of providing the period t audit and invert the cost function to infer, without error, the period t state of the firm when K t was not spent. 5 (1)states that the expected benefit of spending K 2 to learn the second period state is larger the worse the first period state of the firm. (2) states that if the auditor learns that the first period state of the firm is as good as it can be, then K 2 is large enough so that the auditor s costs are minimized by not spending it to learn the second period state of the firm. (3) states that if the auditor learns that the first period state of the firm is as bad as it can be, then K 2 is small enough so that the auditor s costs are minimized by spending it to learn the second period state of the firm.

5 SOURCES OF INFORMATION 439 where the conditional expression indicates that the new auditor is replacing an incumbent with cost K 1 who has decided to separate. If the second period auditor is an incumbent, then the second period payoff function is E [ y 2 c2 b(s ] 2) ĉ b 1 E [ y 2 c2 b(s ] 2) s 1 π2 i = y 2 c2 a(s 2) y 2 c2 a(s 2) K 2 if the auditor does not learn the state of the firm in either period, if the auditor does not learn the state of the firm in the second period but did in the first period, if the auditor publicly learns the state of the firm in period two, if the auditor privately learns the state of the firm in period two. If an auditor does not provide audit services in a period, the auditor s payoff for that period is zero. The payoff to the game is the sum of the payoffs in each of the two periods. The setting is competitive in the sense that, in equilibrium, the expected sum of the two payoffs equals zero. The firm s payoff in each period is the difference between the value of the audit services to the firm and the price it pays to acquire them. We assume that this difference is always positive that is, the firm always finds it worthwhile to hire an auditor and its payoff to the game is the sum of its payoffs in each of the two periods. If the first period auditor learned the first period state of the firm, we have a two-period game of incomplete information and so we seek a perfect Bayes- Nash equilibrium. To solve for such an equilibrium, we backward induct. 2.1 ANALYSIS OF THE SECOND PERIOD Lemma 1 describes, in part, the auditors decisions of whether to spend K 2. 6 LEMMA 1. (1) If the outside source of information reveals the firm s second period state, the second period auditor does not spend K 2. (2) If the outside source of information does not reveal the firm s second period state, then: (a) given that the second period auditor is the incumbent, the auditor spends K 2 and learns the second period state of the firm when doing so lowers the expected costs of providing the second period audit, given knowledge of the first period state of the firm (K 2 E[c2 b(s 2) c2 a(s 2) s 1 ] if the auditor learned the first period state of the firm or K 2 E [c2 b(s 2) c2 a(s 2) ĉ1 b] if the auditor only learned the realized cost of providing the first period audit). 6 Lemma 1 is offered without proof as it summarizes the conditions for profit maximization. Lemmas 2 and 3 are proved in the appendix.

6 440 M. BAGNOLI, M. PENNO, AND S. G. WATTS (b) given that the second period auditor is not the incumbent, the auditor spends K 2 and learns the second period state when doing so lowers the expected costs of performing the second period audit (K 2 E[c2 b(s 2) c2 a(s 2) separation, K 1 ]). The next step is to determine the equilibrium price of audit services in the second period. Competition forces the non-incumbent second period auditor to charge c2 a(s 2) in the event that the outside source of information reveals the second period state of the firm. If the outside source does not reveal that information, the price the firm pays depends on what the market infers from a separation of the first period auditor and the firm. If K 1 is small (K 1 K, a threshold defined below), the market infers that the first period auditor spent K 1 and learned the first period state of the firm whereas, if K 1 is large (K 1 > K ), the market infers that the first period auditor did not spend K 1. Thus, in the former case, separation provides direct information about s 1 whereas in the latter case, separation provides direct information about the first period auditor s realized costs and, consequently, only indirect information about s 1. As a result, the expectation of a new auditor s cost of performing the second period audit depends on K 1, and competition among auditors ensures that the price the firm pays equals the expected cost of performing the second period audit given that separation occurred. Formally, y 2 (K 2 ) = γ E [ c a 2 (s 2) ] + (1 γ )E[C i (s 2 ) separation, K 1 ] (1) where C i = { c a 2 + ε 2 + K 2 if the auditor spends K 2 to learn s 2 c b 2 + ε 2 if the auditor does not spend K 2 Equation (1) reflects both (a) that if the firm and the first period auditor separate, with probability γ, the outside source will reveal the state of the firm prior to contracting with the second period auditor, and (b)that the expected costs of performing the second period audit depend both on the cost of learning the second period state of the firm (K 2 ) and what the market infers the first period auditor learned, s 1 or ĉ1 b, which, in turn, depends on whether K 1 is less than or greater than K. To determine the final term in (1), we must understand the circumstances in which the first period auditor chooses not to provide the second period audit. Recall that competition among auditors forces the incumbent auditor to charge y 2 to win the right to perform the second period audit. Thus, the first period auditor will perform the second period audit if expected profits are nonnegative when the auditor charges y 2. Since expected costs depend on whether or not the auditor intends to spend K 2 and learn the second period state of the firm, we have the following lemma.

7 SOURCES OF INFORMATION 441 LEMMA 2. If the outside source of information does not reveal the second period state of the firm and if the first period auditor learned the first period state of the firm, the auditor is willing to perform the second period audit at the price y 2 if (i) E[y 2 c2 a (s 2) K 2 s 1 ] 0 when the auditor spends K 2 to learn s 2 or (ii) E[y 2 c2 b (s 2) s 1 ] 0 when the auditor does not. If the first period auditor did not learn the first period state of the firm, the auditor is willing to perform the second period audit at the price y 2 if (i) E[y 2 c2 a (s 2) K 2 ĉ1 b ] 0 when the auditor spends K 2 to learn s 2 or (ii) E[y 2 c2 b (s 2) ĉ1 b ] 0 when the auditor does not. To determine the information that can be inferred from separation when the first period auditor learned the first period state of the firm (K 1 K ), it is convenient to more succinctly describe the first period auditor s expected costs of doing the second period audit. Let 7 { [ ] E c a 2 (s 2 ) + K 2 s1 if the auditor spends K 2 to learn s 2 TC(s 1 ; K 2 ) E [ c2 b(s 2) ] s1 if the auditor does not spend K 2. Recall that c a and c b are both increasing in s 2 and we have assumed that s 1 and s 2 are strictly positively correlated. Because the first period auditor separates if and only if TC(s 1 ; K 2 ) > y 2, the first period auditor separates only if s 1 is large enough that is, if the first period state of the firm is sufficiently bad. Note, the equilibrium second period price is a function of K 1 (because it depends on what the first period auditor learns) and K 2 (because it depends on the cost of performing the second period audit). LEMMA 3. (i) If K 1 K (the period one auditor spends K 1 ), the equilibrium price of the second period audit is 8 y2 (K 1, K 2 ) = γ E [ c2 a (s 2) ] + (1 γ )Pr { K 2 E [ c2 b (s 2) c2 a (s 2) ] s 1 s 1 > s1 (K 2) } E [ c2 a (s 2) + K 2 s 1 > s1 (K 2) ] + (1 γ )Pr { K 2 > E [ c2 b (s 2) c2 a (s 2) ] s 1 s 1 > s1 (K 2) } E [ c b 2 (s 2) s 1 > s 1 (K 2) ], where, s 1 (K 2) is implicitly defined by TC (s 1 (K 2); K 2 ) = y 2 (K 1, K 2 ). 7 Note, TC(s 1 ; K 2 ) is continuous in s 1 because at K 2 = E [c2 b (s 2) c2 a(s 2) s 1 ], E[c2 a(s 2) + K 2 s 1 ] = E [c2 b (s 2) s 1 ]. 8 Note that in the expression for y2, we have substituted for TC(s 1; K 2 ).

8 442 M. BAGNOLI, M. PENNO, AND S. G. WATTS (ii) If K 1 > K, the equilibrium price of the second period audit is y2 (K 1, K 2 ) = γ E [ c2 a (s 2) ] + (1 γ )Pr { K 2 E [ c2 b (s 2) c2 a (s 2) ĉ1 b ] ĉ1 b >κ (K 2 ) } E [ c2 a (s 2) + K 2 ĉ 1 b >κ (K 2 ) ] + (1 γ )Pr { K 2 > E [ c2 b (s 2) c2 a (s 2) ĉ1 b ] ĉ1 b >κ (K 2 ) } E [ c2 b (s 2) ĉ 1 b >κ (K 2 ) ], where, κ (K 2 ) is implicitly defined by TC (κ (K 2 ); K 2 ) = y 2 (K 1, K 2 ). 9 Lemmas 1 3 are summarized as: Second period equilibrium (SE): (i) If the incumbent auditor did not learn the first period state of the firm, the firm expects to purchase the second period audit for y2 (K 1, K 2 ). The audit is performed by the incumbent auditor if, at that price, the incumbent expects to make a nonnegative profit, given the first period s realized costs. Otherwise, the audit is performed by another auditor, or (ii) If the incumbent auditor learned the first period state of the firm, the firm expects to purchase the second period audit for y2 (K 1, K 2 ). The audit is performed by the incumbent auditor if, at that price, the incumbent expects to make a nonnegative profit, given the first period state of the firm. Otherwise, the audit is performed by another auditor. 2.2 ANALYSIS OF THE FIRST PERIOD Having solved for equilibrium in the second period for all possible first period outcomes, we can now solve for equilibrium actions in the first period. To determine the payoff to each alternative (spend K 1 or not), we first compute the expected payoff for that alternative in the second period and add it to the payoff in the first period. If the first period auditor does not spend K 1, by (SE), the auditor performs the second period audit for y 2 (K 1, K 2 ) only if ĉ b 1 κ (K 2 ). Expected second period profits are Pr{ĉ b 1 κ (K 2 )}(y 2 (K 1, K 2 ) E[TC(ĉ b 1 ; K 2) ĉ b 1 κ (K 2 )]). Expected first period profits are y 1 E[c b 1 (s 1)]. Thus, the auditor s expected profits from choosing not to spend K 1 in the first period are π n 1 (K 1, K 2 ) which equals y 1 E [ c b 1 (s 1) ] + Pr { ĉ b 1 κ (K 2 ) }( y 2 (K 1, K 2 ) E[TC ( c b 1, K 2) ĉ b 1 κ (K 2 )] ). (2) On the other hand, if the first period auditor spends K 1, then by (SE), the auditor performs the second period audit for y 2 (K 1, K 2 ) only if s 1 s 1 (K 2). Expected second period profits are Pr{s s 1 (K 2)}(y 2 (K 1, K 2 ) E[TC(s 1 ; K 2 ) s 1 s 1 (K 2)]). Expected first period profits are y 1 E[c a 1 (s 1)] K 1, and so expected profits from choosing to spend K 1 in the 9 If there is sufficient noise in the cost realization, y 2 (K 1, K 2 ) > TC(ĉ b 1 ; K 2) for all ĉ b 1 in the support of the cost realizations. In this case, κ (K 2 ) is the supremum of the support.

9 SOURCES OF INFORMATION 443 first period are π i 1 (K 1, K 2 ) = y 1 E [ c a 1 (s 1) ] K 1 + Pr{s 1 s 1 (K 2)}(y 2 (K 1, K 2 ) E[TC(s 1 ; K 2 ) s 1 s1 (K 2)]). (3) Since the auditor maximizes profits, the auditor spends K 1 if π1 i(k 1, K 2 ) π1 n(k 1, K 2 ). Further, the critical value of the cost of learning the first period state that determines whether the first period auditor spends K 1, K, solves π1 i(k, K 2 ) = π1 n(k, K 2 ). To better understand the first period auditor s decision, we examine the difference between the expected profits from spending K 1 and not spending it: π1 i (K 1, K 2 ) π1 n (K 1, K 2 ) = ( E [ c1 b (s 1) ] E [ c1 a (s 1) ] ) K 1 + Pr { s 1 s1 and ĉ 1 b κ } E [ TC ( c1 b ; K ) 2 TC(s1 ; K 2 ) s1 s1 and ĉ 1 b κ ] + Pr { s 1 s1 and ĉ 1 b >κ } E [ y2 (K 1, K 2 ) TC(s 1 ; K 2 ) s1 s1 and ĉ 1 b >κ ] Pr { s 1 > s1 and ĉ 1 b κ } E [ y2 (K 1, K 2 ) TC ( c1 b ; K 2) s1 > s1 and ĉ 1 b κ ]. (4) This equation shows that the value of spending K 1 to learn the first period state of the firm contains an option value associated with knowing the first period state of the firm. 10 This option value arises when the first period auditor knows the first period state of the firm since the auditor can avoid performing a second period audit in those states where the auditor expects to earn negative profits. Said differently, knowing the first period state allows the first period auditor to use the possibility of separation as if it were a call option that is, to undertake the second period audit only if, given the auditor s information about the first period state of the firm, the auditor expects to earn nonnegative profits. This idea is reflected in the last two terms of equation (4): They represent the value of knowing s 1 times the probability that a state occurs in which that knowledge is valuable. 11 Finally, observe that competition among auditors in the first period forces the first period auditor s expected total profits to be zero. Thus, the 10 Numerical simulations, available from the authors on request, indicate that the value of the option varies from zero to more than 100% of π n The next-to-the-last term represents the auditor s expected profits from performing profitable second period audits that would have been avoided based on realized first period costs. The last term represents the value of avoiding unprofitable second period audits that would have been performed based on the auditor s realized first period costs.

10 444 M. BAGNOLI, M. PENNO, AND S. G. WATTS equilibrium price of a first period audit is y1 (K 1, K 2 ) E [ c1 a(s 1) ] + K 1 Pr { } s 1 s1 ( y2 = E[ TC(s 1 ; K 2 ) ]) s 1 s1 if π1 i(k 1, K 2 ) π1 n(k 1, K 2 ) E [ c1 b (s 1) ] Pr { c1 b κ } ( y2 E[ TC ( ) c1) b (s1 ); K 2 ĉ 1 b κ ]) otherwise. Collecting our results, Equilibrium: In equilibrium, the firm purchases the audit in the first period for y1. The first period auditor chooses to spend K 1 and learn the first period state of the firm if π1 i(k 1, K 2 ) π1 n(k 1, K 2 ). The firm expects to pay y2 for the second period audit. The first period auditor performs the second period audit if the auditor did not spend K 1 to learn the first period state of the firm (π1 i(k 1, K 2 ) <π1 n(k 1, K 2 )) or if the auditor spent K 1 to learn the first period state of the firm (π1 i(k 1, K 2 ) π1 n(k 1, K 2 )) and expected costs from performing the audit given the realized first period costs of performing the audit or given the first period state of the firm respectively are not larger than y2. Otherwise, the firm purchases the second period audit for y2 from another auditor. 3. Results We have solved our model for all exogenous K 2 meeting our technical assumptions. If this set is labeled K, we can define a probability distribution g on this set. The distribution can be interpreted in at least two ways. First, it can be thought of as a description of the set of auditors and their associated costs of learning the firm s state. Second, it can be thought of as the probability that a given auditor will have each of the feasible costs of learning the firm s state. Under this interpretation, g(k 2 ) is the probability that the auditor s costs are K 2 for all K 2 K. Under either interpretation, we say that if the set of K 2 s for which the auditor makes a particular decision increases after an exogenous change, it is more likely that the auditor will make that particular decision. The proposition to follow analyzes the effect of a change in γ on equilibrium prices and actions. PROPOSITION. If the probability that the outside information source reveals the second period state of the firm increases, (i) the equilibrium price of the second period audit declines, (ii) it is more likely that the first period auditor does not perform the second period audit, (iii) it is more likely that the first period auditor spends K 1 to learn the first period state of the firm, and (iv) the equilibrium price of the first period audit rises if the first period auditor chooses to expend resources to learn the first period state of the firm.

11 SOURCES OF INFORMATION 445 Our main result is (iii): One might expect that as it becomes more likely that the outside information source will reveal the second period state of the firm, the first period auditor s incentives to acquire information about the first period state of the firm are reduced. Said differently, one might expect that the outside information source is a substitute for private information gathering by the auditor. In our model, however, an increase in the probability that the outside information source reveals the second period state of the firm combines with the competition among auditors to lower the equilibrium second period price of the audit. This makes the first period auditor s separation option more valuable because the losses that are avoided by separating grow. Since the option value has increased, part (iii) states that purchasing the option by spending K 1 to learn the first period state of the firm becomes more attractive the auditor is more likely to gather private information. Also note the effect on pricing: As γ increases, the first period auditor s expected quasi-rents from performing the second period audit decline, making the auditor less willing to absorb losses on the first period audit. There is no offsetting benefit from the increase in the option value because all of that value arises from the first period auditor s ability to separate and thereby avoid expected losses from performing the second period audit. Thus, an increase in the probability that the outside information source reveals the second period state increases the equilibrium first period price of an audit (decreases the low-ball). 4. Conclusion Like previous multi-period audit pricing models, (e. g., DeAngelo [1981] and Magee and Tseng [1990]), this paper assumes an ex ante competitive market for audits with the ex post creation of quasi-rents. We extend this literature by modeling an auditor s multi-period information gathering choices and examining how these choices will be non-trivially influenced by market forces in the presence of outside information. The results suggest that there are interesting interactions between these sources if auditing and the outside source behave as complements rather than substitutes. In our model, the increased presence of outside information reduces the second period quasi-rent, with competition increasing the period one audit fee. Consequently, we illustrate how increasing levels of low-balling may be evidence of decreasing incentives to gather information. These results should add some new perspective to the role of quasi-rents (and their controversial low-balling manifestations)in audit markets. APPENDIX PROOF OF LEMMA 2. When s 2 is not revealed, by Lemma 1, if the first period auditor learned s 1, the auditor spends K 2 if K 2 E[c2 b(s 2) c2 a(s 2) s 1 ] and performs the second period audit only if E[y 2 c2 a(s 2) K 2 s 1 ] 0. If K 2 > E[c2 b(s 2) c2 a(s 2) s 1 ], the auditor does not spend K 2 but performs

12 446 M. BAGNOLI, M. PENNO, AND S. G. WATTS the audit if E[y 2 c2 b(s 2) s 1 ] 0. If the first period auditor did not learn s 1, the auditor updates using ĉ b 1, spends K 2 if K 2 E[c2 b(s 2) c2 a(s 2) ĉ b 1 ] and performs the second period audit only if E[y 2 c2 a(s 2) K 2 ĉ b 1 ] 0 or if K 2 > E[c2 b(s 2) c2 a(s 2) ĉ b 1 ] and E[y 2 c b (s 2 ) ĉ b 1 ] 0. PROOF OF LEMMA 3. Part (i): When K 1 K, the first period auditor separates if TC(s 1 ; K 2 ) > y 2. Since no other auditor knows s 1, their inference from separation depends on whether TC(s 1 ; K 2 ) equals E[c2 a(s 2) + K 2 s 1 ] or E[c2 b(s 2) s 1 ]. The probabilities that TC equals each (respectively) are Pr{K 2 E[c2 b(s 2) c2 a(s 2) s 1 ] s 1 > s1 (K 2)} and Pr{K 2 >E[c2 b(s 2) c2 a(s 2) s 1 ] s 1 > s1 (K )}. Putting the pieces together yields the expression for y2. Next, we must show that s 1 (K 2) exists. To do so, recall that TC(s 1 ; K 2 ) is continuous in s 1 and observe that y 2 is too. Letting φ(s1 ) Pr{K 2 E[c2 b(s 2) c2 a(s 2) s 1 ] s 1 > s1 (K 2)} and substituting for y2 in TC(s 1; K 2 ) = y2 and evaluating at s yields γ E[c2 a(s 2)] + (1 γ ) [φ(s1 )E[c 2 a(s 2) + K 2 s 1 > s1 (K 2)] + (1 φ(s1 ))E[c 2 b(s 2) s 1 > s1 (K 2)]]. Next, since K 2 < E[c2 b(s 2) c2 a(s 2) s h ], φ(s h ) = 1. Substituting and rearranging, E[c2 a(s 2) + K 2 s h ] (1 γ )E[c2 a(s 2) + K 2 s h ] >γe[c2 a(s 2)] because E[c2 a(s 2) s 1 ] is increasing in s 1. Finally, K 2 > E[c2 b(s 2) c2 a(s 2) s l ] and so E[c2 a(s 2) s l ] + γ K 2 < (γ + (1 γ )φ(s l ))E[c2 a(s 2)] + (1 γ )(1 φ(s l ))E[c2 b(s 2)] if E[c2 a(s 2) s l ] + γ K 2 < E[c2 a ] which we have assumed. Thus, since s [s l, s h ] R, by Brouwer s fixed point theorem, there exists a fixed point which defines s1 (K ). Part (ii): The proof is essentially identical to the proof of lemma 3 when all expectations are conditioned on ĉ b 1 rather than s 1 and so is omitted. PROOF OF THE PROPOSITION. Parts (i)and (ii): If K 1 K, then lemma 3 describes equilibrium in period 2. Given this, E[c2 a(s 2)] is smaller than both E[c2 a(s 2)+ K 2 s 1 > s1 (K 2)] and E[c2 b(s 2) s 1 > s1 (K 2)] because both c2 a and c 2 b are increasing in s 2 and s 2 is strictly positively correlated with s 1. Therefore, for a fixed s1, y 2 (K 1, K 2 ) declines in γ. Next, consider the equation defining s1 (K 2), TC(s1 (K 2); K 2 ) = y2 (K 2). Since an increase in γ shifts the rhs of this equation down and since TC is increasing is s 1, s1 (K 2) declines in γ. If K 1 > K, then lemma 3 again describes equilibrium in period 2. Since the only difference is that the auditor conditions on ĉ b 1 rather than s 1 the above arguments are readily adapted to show that y2 and κ (K 2 ) decline in γ. Part (iii): Differentiating (4) with respect to γ, (( [ ( E TC c b 1 (s1 ) + ε ) 1; K 2 c1 b (s 1 ) + ε 1 κ ] TC(s1 ; K 2) ) + (y2 TC(s 1 )) + ( y 2 E [ TC ( c b 1 (s 1 ) + ε 1; K 2 ) c b 1 (s 1 ) + ε 1 κ ])) ( s 1 / γ ) + ( Pr { s 1 s1 and ĉ b 1 >κ } Pr { s 1 > s1 and ĉ b 1 κ }) ( y 2 / γ) ( sh + E [ TC ( c κ 1 b (s ) 1) + ε 1 ; K 2 c1 b (s 1) + ε 1 κ ] f (s 1 )ds 1 s l + s l (y κ 2 TC(s 1; K 2 ))Pr { c1 b (s 1) + ε 1 >κ } ) f (s 1 ) ds 1 ( κ / γ ). s l

13 SOURCES OF INFORMATION 447 Of the terms multiplied by s1 / γ, the first is negative, the second is zero, the third is negative and since s1 / γ < 0, the first term is positive. The second term is also positive because the term multiplying y2 / γ is negative so long as there is sufficient noise in the cost realization. Of the terms multiplied by κ / γ (which is negative by (ii)), the second term is negative and the first is positive but small if there is sufficient noise in the cost realization. Therefore, an increase in γ increases the value of spending K 1. Part (iv): Differentiating y1 (K 1, K 2 ) with respect to γ when π1 i π 1 n < 0 yields ( y 2 / γ ) which is negative. When π1 i π 1 n 0, the terms that depend on γ increase (see the proof of (iii)). REFERENCES DEANGELO, L. Auditor Independence, Low-balling, and Disclosure Regulation. Journal of Accounting & Economics (August 1981): ENGLEBRECHT-WIGGANS, R.; P. MILGROM; AND R. WEBER. Competitive Bidding and Proprietary Information. Journal of Mathematical Economics (April 1983): KANODIA, C., AND A. MUKHERJI. Audit Pricing, Lowballing, and Auditor Turnover: A Dynamic Analysis. The Accounting Review (October 1994): MAGEE, R., AND M. TSENG. Audit Pricing and Independence. The Accounting Review (April 1990): MORGAN, J., AND P. STOCKEN. The Effects of Business Risk on Audit Pricing. Review of Accounting Studies (December 1998):