Do Joint Audits Improve or Impair Audit Quality?

Transcription

1 Do Joint Audits Improve or Impair Audit Quality? Mingcherng Deng Baruch College Tong Lu University of Houston Dan A. Simunic University of British Columbia Minlei Ye University of Toronto September 14,

2 ABSTRACT The conventional wisdom holds that joint audits would improve audit quality by enhancing audit evidence precision, because Two heads are better than one, and by preserving auditor independence, because it is more expensive for a company to bribe two firms than one. Our paper challenges this wisdom. We show that joint audits by one big firm and one small firm may impair audit quality, because joint audits (1) induce a free-riding problem between audit firms, lowering audit evidence precision, and (2) creates an opportunity for internal opinion shopping, compromising auditor independence. We further derive a set of empirically testable predictions comparing audit evidence precision, auditor independence, and audit fees under joint and single audits. This paper, the first theoretical study of joint audits, contributes to a better understanding of the economic consequences of joint audits on audit quality. Keywords: joint audits, audit quality, precision, auditor independence, audit fees. 2

3 1 Introduction Are two heads better than one? The answer to this question seems straightforward and it is the most commonly-cited reason in support of joint audits Two audit firms simultaneously and yet separately audit a company to sign a common audit report. Proponents of joint audits argue that two pieces of audit evidence produce higher total information precision than just a single piece. For example, in auditing a company s fair value estimate of a certain asset, auditors are weakly better off with another piece of audit evidence about the fair value estimates, because they always have an option of ignoring the second piece of evidence if it is completely uninformative. Proponents of joint audits also argue that joint audits enhance auditor independence, because it is more expensive for a company to bribe two audit firms in joint audits than one single firm in single audits. Under joint audits, the audit report must be co-signed by both firms and if one of the two refuses to sign, the audit report cannot be released. Thus, as the company has to buy off two audit firms rather than one, it is much more costly for the company to compromise auditor independence. Joint audits are not uncommon. For example, France has mandated by law joint audits of public companies since The same thing is true for the financial services sector in South Africa. Various countries, such as India, Germany, Switzerland, and the U.K., have proposed voluntary joint audits. In 2010, the European Commission seriously considered mandating joint audits, and continues to debate the issue at this time (EC 2010). If two heads were indeed better than one, and if two firms were more expensive to buy off than one, then joint audits probably would have been more prevalent in the world. However, single audits are still the norm, with the U.S. being a notable example. An interesting about-face change occurred in Denmark, which mandated joint audits in 1930 but abolished this requirement in

4 We identify two economic forces that may work against joint audits that are absent in the extant literature. The first one is free-riding. In joint audits, one of the audit firms may save its audit resource cost by investing less in its own audit work and taking advantage of the other audit firm s hard work. The second one is opinion shopping. Joint audits provide the company with an opportunity of opinion shopping internally between its two audit firms, because having more audit firms on board is like having more draws in a lottery. Furthermore, in contrast to external opinion shopping, internal shopping is unobservable to the capital market. Thus, joint audits can endanger auditor independence. In theory, joint audits entail significant trade-offs in both audit evidence precision and auditor independence, and in practice, the audit institutions are diverse. Thus it is unclear ex ante under what conditions joint audits dominate single audits and under what conditions the converse is true. We investigate audit evidence precision, auditor independence, and audit fees in three regimes: single audits by one big firm (Regime B); joint audits by two big firms (Regime BB); joint audits by one big firm and one small firm (Regime BS). While the first two regimes serve as benchmarks, the third regime is particularly interesting because it is the European Commission s target regime. Specifically, the European Commission states that, Regime BS could act as a catalyst for dynamising the audit market by allowing small and medium-sized firms to participate more substantially in the segment of large audits (EC (2010), p.17). We make two assumptions to capture the differences between a big audit firm and a small one. First, a big audit firm has an advantage in its auditing technology in the sense that it has a lower marginal cost of audit evidence precision than a small firm. Second, a big audit firm bears a larger proportion of misstatement costs (such as litigation risk and reputation loss) than a small firm. Comparing Regime BB with Regime B, we find the joint audits generate the same audit 4

5 evidence precision as the single audits. We also find that it costs more to compromise auditor independence under the joint audits. However, when companies do not have budgetary concerns, auditor non-independence is more likely to occur under the joint audits. Audit fees are lower under the joint audits. Comparing Regime BS with Regime B, we find that the total precision of audit evidence under the joint audits is lower than that under single audits. Additionally, similar to Regime BB, auditor independence is more likely to be compromised under joint audits. Furthermore, the audit fees under joint audits are less than that under single audits if the technological difference between the two audit firms is small and the big firm bears a large proportion of misstatement cost. Our research makes the following contributions. First, to our knowledge, there is no prior theoretical study of joint audits. Joint audits provide a unique setting for analyzing both audit evidence precision and auditor independence, two major components of audit quality. Second, we extend the theoretical literature on audit quality by introducing two new strategic interactions into an auditing game. One is the company s strategic shopping of audit opinions between its two joint auditors. The other is the joint auditors strategic freeriding incentives between each other. Previous research analyzed different factors that may impair auditor independence (e.g., DeAngelo 1981a, Antle 1984, Simunic 1984, Magee and Tseng 1990, Kanodia and Mukherji 1994, Lu 2006) or may influence audit evidence precision (Dye 1993, Schwartz 1997, Pae and Yoo 2001, Zhang 2007). These papers focus on single audits. Taking advantage of the institutional setting of joint audits, our paper enriches the literature by identifying additional strategic factors that are absent in single audits. Third, the existing empirical research provides mixed evidence on the impact of joint audits on audit quality and audit fees (e.g., Gonthier-Besacier and Schatt 2007, Piot 2007, 5

6 Thinggaard and Kiertzner 2008, Francis et al. 2009, Lesage et al. 2011). Our model reconciles these mixed findings and provides new empirical predictions, which we elaborate on in Section 6. Fourth, this study provides timely policy implications for regulators. To encourage the growth of small-sized audit practices, the European Commission is considering mandating large companies to hire at least one audit firm outside the Big-Four firms to conduct joint audits (EC 2010). Our analysis suggests such a mandate could damage audit quality. The paper proceeds as follows. Section 2 provides institutional background on joint audits. Section 3 presents the structure and ingredients of the model under the three regimes, B, BB, andbs. Section 4 establishes the equilibrium audit quality and audit fees in these regimes. Section 5 compares those regimes. Section 6 develops the empirical predictions. Section 7 concludes and discusses alternative model assumptions. We relegate all proofs to the Appendix. 2 Institutional Background This section describes the current joint audit practice in France to provide a foundation for our model assumptions. 1 Any listed company, any bank or other financial institution, and any company that prepares consolidated financial statements is required by law in France to appoint two different audit firms, who share the audit work and jointly sign the audit report. The law has evolved into a professional standard of practice, requiring a balanced division of the work of both auditors in order to ensure an efficient dual control mechanism (Gonthier-Besacier and Schatt 2007). However, in practice, auditors may not balance their work allocation. For example, if one Big 4 firm and one small firm conduct joint audits, it is 1 This description is based on an interview of an audit firm senior partner conducted by one of the authors in Paris on 12/13/

7 harder to share the workload equally, because the small audit firm cannot completely cover the client s businesses. For instance, the small firm may not have an audit network abroad. In contrast, the work allocation is likely to be balanced if joint audits are conducted by two Big 4 firms. After accepting an audit engagement, the two audit firms first agree on their work allocation. Their work is usually allocated either by regions (e.g., one audits America and the other audits Europe) or by divisions. Then they start to audit the financial statements simultaneously; they typically work this way because auditors need to meet a deadline of releasing audit reports. After finishing their part of the audit, they review each other s audit work and prepare relevant documentation. At the end of the audit, each joint auditor signs the audit report on the whole financial statements, not just on the work he has done. However, the courts evaluate each audit firm s responsibility based on auditing standards. Fault, if found, may not be at the same level for each audit firm. For example, if the audited inventory is materially misstated, then the auditor who is responsible for inventory could be held more responsible than the other auditor, who simply reviewed the work. These basic features of the French joint audit regime independent collection of audit evidence by the two audit firms with a review of each other s work, a joint agreement on the report to be issued, and separate and proportionate liability for undetected material misstatements are incorporated into our model in the next section. 3 Model This section sets up the structure and ingredients of the model under three regimes: single audits by one big firm (Regime B); joint audits by two big firms (Regime BB); joint 7

8 audits by one big firm and one small firm (Regime BS). In what follows, we first set up the model for Regime B and then articulate how Regime BB and Regime BS deviate from Regime B. Regime B Let x denote the fundamental value of a company, 2 which is distributed normally with mean x 0 and precision h: x N(x 0, 1 ). (1) h Since our focus is on both dimensions of audit quality, the precision of audit evidence and auditor independence (DeAngelo (1981a)), we model both the audit evidence accumulation process and the subsequent company-auditor negotiation. The audit evidence accumulation process produces audit evidence y B about x: ỹ B x N(x, 1 ), (2) eb that is, conditional on x, ỹ B is distributed normally with mean x and precision e B. Thus, the audit evidence is an unbiased but noisy estimate of x. An increase in the quantity of resources utilized by the auditor can reduce the noisiness of, or enhance the precision of, audit evidence. To capture this effect, we assume that the audit resource cost is k B C(e), where k B > 0 is a parameter and the precision e is a choice variable. We make the following standard assumptions about the cost function: C(0) = 0, C > 0 (but C (0) = 0), C > 0, and C = 0. A quadratic function C(e) =e 2 /2, commonly used in the literature, satisfies all of the above assumptions (Chan and Pae 1998, Laux and Newman 2010, etc.). 2 As a general rule in this paper, a symbol with a indicates a random variable and the same symbol without a indicates the realized value of that random variable. For example, x is a realized value of the random variable x. 8

9 To model the company-auditor negotiation, we introduce a pair of (Q, r) representing the give-and-take between the company and its auditor. Specifically, a company may offer its auditors an amount of Q in return for a certain report r the company prefers. Following DeAngelo (1981a), we also label Q as quasi-rent. An independent report, r I E[ x y B ], is the auditor s best estimate of x conditional on the audit evidence y B. However, if auditor independence is compromised, the released report r will exceed r I. We assume that when auditor independence is compromised (when r>r I ), r cannot exceed the audit evidence (r y B ). This assumption is justified by the observation that the audit evidence documented in the audit firms working papers is the only admissible evidence in court (Dye and Sridhar 2004), implying that the misstatement cost is extremely high if the audit report is higher than the observed audit evidence. Following the standard assumption in the literature (e.g., Antle and Nalebuff 1991, Dye and Sridhar 2004), we assume that the audit firm will bear a cost of (r x) 2 if a misstatement occurs; that is, if the certified report r is different from the company s fundamental value x. This misstatement cost includes legal liability and reputation losses resulting from an audit failure. 3 The sequence of events is as follows: The company offers an audit fee of F and hires a big audit firm in a competitive audit market. The company proposes an unaudited report r 0 to its auditor. The auditor chooses her desired precision e B of audit evidence in her audit plan. The audit evidence accumulation process produces audit evidence y B. 3 To focus on auditing issues, we abstract away from companies misstatement cost. If instead the companies legal liability and/or reputation loss due to misstatement is sufficiently high, the company may not propose an inflated report in the first place, thereby making auditing a moot issue. 9

10 The company-auditor negotiation determines a pair of (Q, r), where the company offers its auditor an amount of Q in return for an audit report r. The company s fundamental value x is realized and the auditor bears a misstatement cost of (r x) 2. The company s payoff is a benefit function M(r) from the audited report r net of its audit fee F and its quasi-rent Q paid to the auditor. Following the disclosure literature (for example, Dye 1985 ), we assume that the benefit M(r) is a linear function of the audited report (i.e., M(r) =α + β r). The auditor s payoff is the receipt of F and Q net of the audit resource cost of k B C(e B ) and the misstatement cost of (r x) 2. Regime BB and Regime BS Having laid out the model setup for Regime B, we now turn to Regimes BB and BS. The three regimes differ in the following ways: Regime B: The big audit firm has an audit resource cost function of k B C(e) andmust bear 100% of the misstatement cost. Regime BB: Each of the two big audit firms has an identical cost function of k B C(e) and each must bear 50% of the misstatement cost. 4 Additionally, y B and y B2 denote the audit evidence accumulated by the two big firms. Regime BS: The big audit firm has a cost function of k B C(e) and the small audit firm has a cost function of k S C(e), where k S k B m>1 so that the big firm is more cost efficient than the small firm. Furthermore, the big audit firm will bear a proportion of α B and the small firm will bear a proportion of α S =1 α B of the total misstatement 4 Alternatively, we assume that one auditor shares α 1 (0, 1) proportion of the misstatement cost and the other auditor shares α 2 (0, 1), where α 1 + α 2 = 1. We derive qualitatively similar results, which are available upon request. 10

11 cost (r x) 2. We assume that the big audit firm bears a larger misstatement cost than the small audit firm, i.e., α B >α S, consistent with DeAngelo (1981b). The assumptions imply that α B ( 1, 1) and α 2 S (0, 1). Additionally, y 2 B denotes the audit evidence accumulated by the big firm and y S denotes the audit evidence accumulated by the small firm. The audit evidence accumulation process produces audit evidence about x: ỹ B x N(x, 1 e B ); ỹ S x N(x, 1 e S ). (3) In practice, because of tight deadlines for the release of audited financial statements, joint audit firms must work simultaneously rather than sequentially. Given that, we model the joint audit firms evidence accumulation processes as simultaneous moves the big firm and the small firm choose e B and e S simultaneously and therefore both e B and e S are hidden actions. 4 Equilibrium We use backward induction to conduct the equilibrium analysis. We first analyze the quasi-rent Q and the certified report r, then the precision e of the audit evidence, and finally the audit fee F. We first derive the results for regime B and then for regimes BB and BS. Before the analysis begins, a discussion of the company s initial unaudited report r 0 proposed to the auditor is in order. In our model, the company has an incentive to engage in earnings management to boost its report. Thus initially it will propose a high unaudited report r 0 to its auditors. Because it is in the best interest of the company to propose a r 0 as high as possible regardless of its private information (if any), r 0 is uninformative and 11

12 so is ignored by the auditors. Knowing that, the auditors use their own audit evidence to estimate the company value. Such an assumption aims at focusing our attention exclusively on auditing issues and is prevalently adopted in the existing literature (e.g., Antle and Nalebuff 1991, Lu and Sapra 2009, Laux and Newman 2010). In the conclusion, we fully discuss this assumption. 4.1 Auditor Independence In Regime B, at the stage of company-auditor negotiation, the company may offer its auditor an amount of Q in return for a certain audit report r. The pair of (Q, r) represents the give-and-take between the company and its auditor. The auditor weighs the amount of the quasi-rent Q she expects from the company against her expectation of the misstatement cost (r x) 2. Her expectation is conditional on audit evidence accumulated, that is, E[(r x) 2 y B ]. If Q = 0, the auditor will concern only about the misstatement cost and so she will certify the independent report, r I E[ x y B ]. As a result, her expectation of the misstatement cost will be E[(r I x) 2 y B ]. If the auditor chooses to certify an inflated report r>r I, her expectation of the misstatement cost will increase to E[(r x) 2 y B ]. Naturally, to overcome the auditor s objection to an inflated report r>r I, the company must offer a Q large enough to cover the increase in the misstatement cost, E[(r x) 2 y B ] E[(r I x) 2 y B ]. In equilibrium, the company has no incentive to overpay the auditor, so it will set Q = E[(r x) 2 y B ] E[(r I x) 2 y B ]. (4) On the other hand, the company must determine the level of audit report r it prefers. A higher r will bring about a higher benefit M(r) but will require a higher amount of the 12

13 quasi-rent Q offered to the auditor. To balance this trade-off, the company solves max r Q(r)+M(r), (5) subject to r max{y B,r I }. Because the audit evidence documented in the audit firm s working papers is the only admissible evidence in court, the audit report r, if inflated, cannot exceed the audit evidence y B. The auditor independence is preserved when the auditor issues the independent report r = r I. Given the feasibility constraint, this suggests that auditor non-independence occurs when audit evidence exceeds the independent report, that is, y B >r I. Therefore, the probability of auditor independence Pr(AI) is the probability that the independent report exceeds the audit evidence, i.e., Pr(r I >y B ). In Regime BB, besides y B, the joint audits generate one piece of additional evidence by the second big audit firm, denoted by y B2. Analogously, the company will set Q equal to E[(r x) 2 y B, y B2 ] E[(r I x) 2 y B, y B2 ] to induce a report r. The audit report r is limited by the feasibility constraint r max{y B,y B2, r I }. Similarly, in Regime BS, thecompany will set Q equal to E[(r x) 2 y B,y S ] E[(r I x) 2 y B, y S ]wherey S is the audit evidence accumulated by the small firm. Proposition 1. [Auditor Independence] The negotiated pair of report r and quasi-rent Q is as follows: (i) Under Regime B, define r I E[ x y B ]= hx 0+e B y B and r such that r r h+e B I = β/2 ( ) 2 β If y B >r, r = r and Q = ; 2 If y B [r I,r ], r = y B and Q =(y B r I ) 2 ; If y B <r I, r = r I and Q =0. The probability of auditor independence is Pr(AI) =Pr(y B <x 0 ). 13

14 (ii) Under Regime BB, define r I E[ x y B,y B2 ]= hx 0+e B y B +e B2 y B2 h+e B +e B2 r I = β/2. and r such that r If max{y B,y B2 } >r, r = r ( ) 2 β and Q = ; 2 If max{y B,y B2 } [r I,r ], r =max{y B,y B2 } and Q =(max{y B,y B2 } r I ) 2 ; If max{y B,y B2 } <r I, r = r I and Q =0. The probability of auditor independence is ( (h + e B )y B2 hx 0 Pr(AI) =Pr e B <y B < hx ) 0 + e B2 y B2. h + e B2 (iii) Under Regime BS, define r I E[ x y B,y S ]= hx 0+e B y B +e S y S and r such that r r h+e B +e S I = β/2. If max{y B,y S } >r, r = r ( ) 2 β and Q = ; 2 If max{y B,y S } [r I,r ], r =max{y B,y S } and Q =(max{y B,y S } r I ) 2 ; If max{y B,y S } <r I, r = r I and Q =0. The probability of auditor independence is ( (h + e B )y S hx 0 Pr(AI) =Pr e B <y B < hx ) 0 + e S y S. h + e S 4.2 Audit Evidence Precision Under Regime B, when the auditor plans the audit, she chooses her desired level of audit evidence precision e B. The benefit of higher precision is to obtain more precise audit evidence about the fundamental value of the company and thus to reduce the possibility of costly misstatement, E[(r x) 2 y B ]. Higher precision requires a larger audit resource cost, kc(e). This benefit-cost trade-off determines the auditor s optimal precision decision. Note from the preceding subsection that the company offers Q to induce the auditor to deviate from 14

15 certifying the independent report r I. In other words, Q = E[(r x) 2 y B ] E[(r I x) 2 y B ]. Therefore, the auditor s remaining expected cost of misstatement is only E[(r I x) 2 y B ]. Formally, the big audit firm s optimization program is min e B k B C(e B ) }{{} + E[E[(r I x) 2 y B ] e B ]. }{{} (6) audit resource cost expected misstatement cost Under Regime BB, the two auditors total expected cost of misstatement is E[(r I x) 2 y B,y B2 ]. Each auditor shares this cost equally, because they are economically and technologically identical. The two auditors optimization programs are, respectively, and min e B k B C(e B )+ 1 2 E[E[(r I x) 2 y B,y B2 ] e B,e B2 ], (7) min e B2 k B C(e B2 )+ 1 2 E[E[(r I x) 2 y B,y B2 ] e B,e B2 ]. (8) Under Regime BS, since the small audit firm receives y S, the two audit firms total expected cost of misstatement is E[(r I x) 2 y B,y S ]. The big audit firm bears a proportion α B of the misstatement cost whereas the small audit firm bears the remaining proportion of α S where α B >α S. The big audit firm s optimization program is min e B k B C(e B )+α B E[E[(r I x) 2 y B,y S ] e B,e S ]. (9) Analogously, the small audit firm s optimization program is min e S k S C(e S )+α S E[E[(r I x) 2 y B,y S ] e B,e S ]. (10) Proposition 2. [Audit Evidence Precision] (i) Under Regime B, the big audit firm s optimal choice of evidence precision e B B is determined by the following equation: k B C (e B 1 B) =0. (11) (h + e B B )2 15

16 (ii) Under Regime BB, each of the big audit firm s optimal choice of evidence precision e B BB is determined by the following equation: k B C (e B BB ) 1/2 =0. (h +2e B BB )2 (iii) Under Regime BS, the big audit firm s optimal choice of evidence precision e B BS and the small audit firm s optimal choice of evidence precision e S BS are jointly determined by the following pair of equations: k B C (e B BS ) k S C (e S BS ) α B (h+e B BS +es BS )2 =0; α S (h+e B BS +es BS )2 =0. (12) 4.3 Audit Fees In a competitive audit market, the audit fee F covers both the auditors resource cost and their expected cost of misstatement. Therefore, the total audit fees in each regime are characterized by: F = k B C(e B )+E[E[(r I x) 2 y B ] e B ]. (13) F = k B C(e B )+k B C(e B2 )+E[E[(r I x) 2 y B,y B2 ] e B,e B2 ]. (14) F = k B C(e B )+k S C(e S )+E[E[(r I x) 2 y B,y S ] e B,e S ]. (15) Proposition 3. [Audit Fee] (i) Under Regime B, the equilibrium total audit fee F B is as follows: F B = k B C(e B B )+ 1. (16) h + e B B (ii) Under Regime BB, the equilibrium total audit fee F BB is as follows: F BB =2k B C(e B 1 BB)+. (17) h +2e B BB 16

17 (iii) Under Regime BS, the equilibrium total audit fee F BS is as follows: F BS = k B C(e B BS)+k S C(e S 1 BS)+ h + e B BS +. (18) es BS 5 Comparison Having derived the results on auditor independence, audit evidence precision, and audit fees, now we are ready to compare Regimes B, BB, andbs along those three dimensions. 5.1 Audit Evidence Precision To facilitate comparisons among Regimes B, BB, and BS, we juxtapose the optimal choices of audit evidence precision described in Proposition 2 in the following table. Table 1 Regime B k B C (e B B ) 1 (h+e B B )2 =0. Regime BB Regime BS k B C (e B BB ) 1/2 =0, (h+2e B BB )2 k B C (e B BB ) 1/2 =0. (h+2e B BB )2 k B C (e B BS ) k S C (e S BS ) α B (h+e B BS +es BS )2 =0, α S (h+e B BS +es BS )2 =0. On appearance, it seems that the argument of two heads are better than one makes sense. If the big audit firm s evidence precision were fixed across the three regimes (that is, if e B B = eb BB = eb BS ), then indeed the total evidence precision under joint audits would exceed that under single audits, that is, e B BS + es BS >eb B and 2eB BB >eb B. However, it turns out that the big audit firm s optimal choice of audit evidence precision is not fixed across the three regimes. Corollary 1. The big audit firm s optimal choice of audit evidence precision is lower under joint audits than under single audits: e B BS <eb B and eb BB <eb B. 17

18 Contrary to the conventional wisdom, the big audit firm s audit evidence precision is lowered under joint audits. This result is due to two factors: (i) the misstatement cost sharing with another auditor (the numerator 1 2 or {α B,α S } of the second term in the jointaudit equations in Table 1), and (ii) free-riding (the additional term e B BB or es BS in the denominator of the second term in the joint audit equations in Table 1). An audit firm may enjoy the benefits of reduction in audit risk brought by her joint-audit counterpart without exerting her fair share of effort, resulting in a free-riding problem. Formally, Proposition 4. [Free Riding] Under Regime BS, the small firm free rides the big firm: k S C(e S BS ) k B C(e B BS ) < αs h+e B BS +es BS α B h+e B BS +es BS. The ratio k SC(e S BS ) k B of the two firms audit resource cost captures the cost of audit ev- C(e B BS ) idence precision borne directly by the firms, whereas the ratio α S h+e B BS +es BS α B h+e B BS +es BS of the two firms misstatement cost captures the benefit of precision received by the firms. The inequality indicates that the small firm free rides the big firm s effort. Two reasons lead to this result. The small firm is technologically less efficient than the big firm and so the small firm has an incentive to withhold some resources and free ride the big firm s effort. The big firm of course anticipates the small firm s free riding incentive, but the big firm is concerned about its own large share of the misstatement cost and thus, is willing to incur large audit resource cost to reduce the total misstatement risk and is (grudgingly) free-ridden by the small firm. While the free-riding problem reduces an individual auditor s audit evidence precision, the main focus should be on whether joint audits give rise to higher total audit evidence precision than single audits. The next proposition answers this question. Proposition 5. [Comparison: Audit Evidence Precision] (i) The total evidence precision in Regimes BB and B are the same: 2e B BB = eb B. (ii) The total evidence precision in Regime BS is less than that in Regimes B and BB: 18

19 e B BS + es BS <eb B =2eB BB. Proposition 5(i) is straightforward because in Regime BB no technological difference between two big firms exists and both big firms bear an equal proportion of the misstatement costs. Therefore, each audit firm exerts one half of the effort that would have been provided by a single big firm audit, thereby leading to the same total evidence precision as in Regime B. Proposition 5(ii) shows a dimmer picture in Regime BS, because a technological difference between big and small firms exists. On one hand, since the marginal resource cost is high for the small audit firm, it saves costs by choosing low precision, which decreases the total evidence precision. On the other hand, the big firm needs to provide more effort than its share to reduce the misstatement risk. Thus, the small firm free-rides on the big firm, as demonstrated by Proposition 4. However, the increase in precision by the big firm is not large enough to make up for the decrease in precision by the small firm. This causes a lower total audit evidence precision in Regime BS than in Regime B. In summary, while the joint audits under Regime BB provide the same audit evidence precision, the joint audits under Regime BS impair precision due to free riding. 5.2 Auditor Independence To facilitate comparisons among Regimes B, BB, andbs, we juxtapose the equilibrium quasi-rent and probability of auditor independence described in Proposition 1 in the following table. For the sake of comparison, we present only the quasi-rent given r = y B because neither r = y S nor r = y B2 is present in all the three regimes. 19

20 Table 2 Regime B Q B = Regime BB Q BB = Regime BS Q BS = ( h(yb x 0 ) ) 2 Pr(AI B )=Pr(y B <x 0 ) h+e B B ( h(yb x 0 )+e B BB (y B y B2 ) h+2e B BB ( h(yb x 0 )+e S BS (y B y S ) h+e B BS +es BS ) 2 ( (h+e B Pr(AI BB )=Pr BB )y B2 hx 0 ) 2 Pr(AI BS )=Pr e B BB ( (h+e B BS )y S hx 0 e B BS ) <y B < hx 0+e B BB y B2 h+e B BB) <y B < hx 0+e S BS y S h+e S BS Table 2 shows that if the total audit evidence precision were the same across regimes (e B BS +es BS =2eB BB = eb B ), the quasi-rent required to bribe the auditors would be higher under joint audits, because joint audits by definition involve more auditors than single audits. The bribe is indicated by the additional term in the numerator of Q in the table. Additionally, as we show in Proposition 5, the total audit evidence precision is lower in Regime BS than in Regimes B and BB (e B BS + es BS < 2eB BB = eb B ). Taken together, we confirm that it is more expensive to buy off joint auditors than one single auditor. However, a higher cost of bribe does not necessarily imply a higher likelihood of auditor independence. Contrary to conventional wisdom, joint audits impair auditor independence. Proposition 6. [Opinion Shopping] The likelihood of auditor independence under joint audits is lower than that under single audits: Pr(AI BS ) < Pr(AI B ) and Pr(AI BB ) < Pr(AI B ). The result of Proposition 6 may sound surprising. If it is more expensive to buy off two auditors, why would the probability of auditor independence be lower under joint audits? This is because joint audits provide companies an opportunity of internal shopping for a favorable audit opinion from its two auditors. But such an opportunity does not exist under single audits. Thus far we have demonstrated that joint audits give rise to two conflicting forces: buying off auditors becomes more expensive, but opinion shopping between auditors is easier. A question follows naturally: ultimately will auditor independence be maintained or impaired 20

21 under joint audits? To answer this question, we utilize the magnitude of earnings management, reflected by the absolute value of the difference between the published report r and the independent report r I, to measure auditor independence in three regimes. Proposition 7. [Comparison: Auditor Independence] On average, the magnitude of ex-post earnings management is larger under joint audits than under single audits: E[ r r I Regime BS] > E[ r r I Regime B] and E[ r r I Regime BB] > E[ r r I Regime B]. This surprising finding results from two factors. First, the opportunity of internal opinion shopping provides another draw in a lottery, thereby increasing the chance of a higher audit report to be released. Second, lower audit evidence precision in Regime BS implies larger volatility of audit evidence. This volatility in turn exacerbates earnings management, because of the well-known auditor s curse in Antle and Nalebuff (1991) the auditors will discard lower values of audit evidence (those below r I ), but exploit higher values of audit evidence (those above r I ). 5.3 Audit Fees The opponents of joint audits often claim that joint audits will result in higher audit costs and thus higher audit fees. To check out this claim, we juxtapose the equilibrium audit fees in Regimes B, BB, andbs given in Proposition 3 in the following table: Table 3 Regime B F B = k B C(e B B )+ 1 h+e B B Regime BB F BB =2k B C(e B BB )+ 1 h+2e B BB Regime BS F BS = k B C(e B BS )+k SC(e S BS )+ 1 h+e B BS +es BS The audit fees cover both the audit resource cost and the expected cost of misstatement. Higher evidence precision necessarily requires a higher audit resource cost but decreases the likelihood of misstatement. Thus, an increase in precision increases one component of the 21

22 audit fees but decreases the other component. Therefore, the audit fees are not monotonic in audit precision and it is not ex ante straightforward to rank the audit fees across the three regimes. Proposition 8. [Comparison: Audit Fees] (i) The audit fees in Regime BB are lower than those in Regime B: F BB <F B. (ii) The audit fees in Regime BS are lower than those in Regime B if and only if the big firm and small firm have similar technology efficiency and/or the big firm bears a sufficiently large proportion of misstatement cost: there exists a m > 1 such that F BS <F B for m<m and there exists an 1 2 <α < 1 such that F BS <F B for α B >α. Proposition 8(i) states that the total audit fees under joint audits by two big firms are lower than those under single audits by one big firm. This result is due to the convexity of the resource cost function, a standard assumption in economics. For example, one audit firm doing all the work under a completion time constraint (Regime B) may experience a higher cost of staff supervision and coordination than if the work was split between two firms (Regime BB). Proposition 8(ii) compares the audit fees across Regimes B and BS in the dimension of the audit firms technological advantage and share of misstatement costs. Recall that the small audit firm suffers technological inefficiency in the sense that its marginal resource cost is larger than the big firm s, that is, k S k B m>1, where m represents the extent of the small firm s technological inefficiency. When the small audit firm s marginal resource cost is smaller (a smaller m), its audit resource cost will be lower. Moreover, when the difference in the technological efficiency is smaller between the large firm and the small firm, the total audit evidence precision is higher, which leads to a lower likelihood of misstatement. As a result of these two factors, the total audit fees in Regime BS will be lower than their counterparts in Regime B when m is sufficiently small. 22

23 Second, when α B is sufficiently large, the big firm has a stronger incentive to boost her precision because she bears a larger proportion of misstatement cost. This implies a large total audit evidence precision and thus a small misstatement cost, which is reflected in the total audit fees. In brief, the total audit fees in Regime BS are lower than those in Regime B when the big firm bears a sufficiently large proportion of misstatement cost. Additionally, as α B continues to increase, the increase in resource cost from higher precision will dominate the reduction in misstatement cost, causing an increase in audit fees. As α B is approaching 1, the audit fees in Regime BS are approaching the audit fees in Regime B. 6 Empirical Prediction We present our model s empirical predictions and relate them to the extant empirical evidence on joint audits and highlight the new hypotheses yet to be tested. Audit Quality. Audit quality is determined by both audit evidence precision and auditor independence (DeAngelo 1981a). Our theoretical analysis generates predictions on both dimensions. With respect to audit evidence precision, our model predicts that: (1) Joint audits where the two audit firms have comparable technology efficiency provide the same audit quality as single audits (Proposition 5(i)); (2) adding a firm with lower technology efficiency to form a joint audit will reduce the overall audit quality, which implies adding a firm with higher technology efficiency will improve audit quality (Proposition 5(ii)). With respect to auditor independence, our analysis shows that the ex ante probability of non-independence is higher under joint audits than under single audits (Proposition 6) and the ex post earnings management is larger under joint audits than under single audits (Proposition 7). The existing empirical studies on joint audits have not separated the effect of audit 23

24 evidence precision from that of auditor independence. They use abnormal accruals or disclosures to measure the overall audit quality. Some studies support our prediction regarding audit evidence prediction (Proposition 5(ii)). For example, examining auditor choices for listed companies in France, Francis et al. (2009) find that companies using one Big 4 auditor paired with a non-big 4 auditor have smaller income-increasing abnormal accruals than those using no Big 4 auditors and that such an effect is even stronger for companies that use two Big 4 auditors. Using a disclosure score for companies composing the French SBF 120 index from 2006 to 2009, Paugam and Casta (2012) provide evidence that the combination of Big 4/non-Big 4 auditors generates higher impairment-related disclosure levels than other combinations, i.e., two Big 4 or two non-big 4. These results are partly consistent with Proposition 5(ii): adding a firm with higher technology efficiency will improve audit quality (i.e., Big 4/non-Big 4 is better than two non-big 4). Their reason for the finding on the comparison between Big4/non-Big 4 and two Big 4 is that Big 4 firms fully bear reputation cost under joint audits of Big 4/non-Big 4. Echoing Paugam and Casta (2012), Marmousez (2008) provides evidence that the presence of two Big 4 audit firms is associated with lower reporting quality (measured by the timeliness of earnings) than the combination of Big 4/non-Big 4 firms, using a sample of 177 French listed companies on December 31, These results are inconsistent with Proposition 5(ii), but the inconsistency could be due to the auditor independence effect. Further empirical study is needed to separate the effect of audit evidence precision from that of auditor independence. Comparing audit quality under joint audits and single audits in general, two studies did not find significant or stable audit quality differences. Lesage et al. (2011) find no stable relationship between joint audits and abnormal accruals. Holm and Thinggaard (2010) find that there is no difference in the auditors ability to constrain earnings management between joint and single audits in general. These findings highlight the importance of considering the pairing of auditors when comparing audit quality under these two regimes. 24

25 In summary, our theoretical predictions can explain most of the current empirical studies and generates additional predictions on auditor independence (Proposition 6 and 7), which calls for future studies that separate the effect of audit evidence precision from that of auditor independence. Audit Fees. Proposition 8(i) proposes that the total audit fees under joint audits by two big firms are lower than that under single audits by one big firm. The empirical evidence is consistent with the direction of our prediction. Francis et al. (2009) find French audit fees are not higher under joint audits compared to other European countries that do not require joint audits. However, Proposition 8 (ii) predicts that when a small firm is involved in joint audits, the comparison is not clear-cut: The audit fee in Regime BS is lower than that in Regime B if and only if the big firm and small firm have similar technology efficiency and/or the big firm bears a sufficiently large proportion of misstatement cost. Indeed, Lesage et al. (2011) confirm the absence of any stable relationship between joint audit and audit fees. Thinggaard and Kiertzner (2008) examines audit fees paid by all 126 non-financial companies listed on the Copenhagen Stock Exchange in They find that joint audits reduce audit fees compared with audits where one auditor is dominant, albeit only for larger companies. Gonthier-Besacier and Schatt (2007) find that when two Big Four firms audit company accounts, the fees charged (adjusted for company size) are significantly lower in comparison with those paid in joint audits participated by one or two small firms. However, the opposite evidence is documented in a different setting by Holm and Thinggaard (2010). They use the data for the whole population of non-financial Danish companies listed on the Copenhagen Stock Exchange in the five-year period surrounding the abolishment of joint audits in They find discounts (of around 25%) in audit fees for companies under single audits. 25

26 Moreover, we discover that when joint auditors bear the same proportion of misstatement costs, the total audit fee under Regime BS exceeds that under Regime BB because of small auditors technology inefficiency (see the proof of Proposition 8). Consistent with this prediction, Audousset-Coulier (2012) shows joint audits by two big auditors do not require a fee premium compared to joint audits by one big firm and one small firm. 7 Conclusion To restore trust in financial reporting, in the wake of the recent financial crisis, the European Commission, among others, is re-examining the role of auditing. One of its proposed actions is to mandate joint audits. We develop a theory of joint audits and identify the following two trade-offs. Though two heads may be better than one, free riding can reduce the information precision produced by joint audits. Regarding auditor independence, even though it is more expensive to buy off two audit firms than one, adding another firm provides the companies an opportunity of shopping for a more favorable opinion. We identify the conditions under which joint audits impair audit quality. Our research extends the theoretical literature on audit quality and provides timely policy implications to regulators. Our model also provides a theoretical framework that can explain the seemingly inconsistent and diverse empirical findings to date, and further predictions that can be tested empirically. In the following we discuss two issues on our model assumptions. First, we assume that the company s initial unaudited report r 0 is uninformative and thus the auditors use their own audit evidence to estimate the company value. This is true in our model whether the company has hidden information or not. Another way of 26

27 information transmission is signaling: If the company has private information, it can signal it by real actions or by audit fees. If the signaling were effective, the auditing would be moot. In a nutshell, we assume away the signaling issue in order to focus on the auditing issue, consistent with the prior literature. If there exists information asymmetry between the company and investors, the auditor s report could be a mechanism by which to alleviate the lemons problem as in Myers and Majluf (1984). In this case, the company s initially unaudited report may convey noisy information about its fundamental value. The auditor then would determine the audit evidence precision depending on the unaudited report. The economic forces identified in our model, including free riding and internal opinion shopping, nonetheless, will continue to be present in this more complicated model. Thus we conjecture that the main results would still hold in the presence of information asymmetry. Second, one of the European Commission s objective of mandating joint audits is to reduce the extent of audit market concentration. By mandating joint audits by one big firm and one small firm, the EC hopes that more and more small firms can have access to big companies and thus reduce the dominance of the Big Four. Such a mandate, in our view, would effectively in the long run increase the audit market competition. This market structure effect of joint audits warrants a full-scale treatment in a sequel paper. 27

28 Appendix ProofofProposition1 (i) Regime B. The auditor s expectation of the misstatement cost given a report r and the audit evidence y B is E[(r x) 2 y B ] = E[r 2 2r x + x 2 y B ] = r 2 2rE[ x y B ]+E[ x 2 y B ] = r 2 2rE[ x y B ]+(E[ x y B ]) 2 + Var[ x y B ] = (r E[ x y B ]) 2 + Var[ x y B ] = (r r I ) 2 + Var[ x y B ] = (r r I ) h + e, B where r I E[ x y B ]. Because both x and ỹ follow the normal distribution specified in (1) and (2) respectively, applying the standard formula of the conditional mean for the multivariate normal distribution yields r I = hx 0+e B y B h+e B. cost is If the independent report r I is issued, then the auditor s expectation of her misstatement E[(r I x) 2 y B ]=(r I r I ) h + e B = 1 h + e B. (19) Therefore, to induce the auditor to certify r instead of r I, the company must offer the auditor 28

29 an amount of Q to compensate her for the increase in the misstatement cost: Q = E[(r x) 2 y B ] E[(r I x) 2 y B ] (20) = (r r I ) 2. The company s optimization program is given by (5) in the text. Using (20), we rewrite the program as max r (r r I ) 2 + M(r), subject to the feasibility constraint r max{y B,r I }. First, suppose the feasibility constraint does not bind. The company s most preferred report r is characterized by the first-order condition: r r I = M (r ) 2 = β 2 > 0. The second-order condition is also satisfied 2+M (r) 0. Second, when the feasibility constraint binds, the company s most preferred report r is not feasible. In this case, the company must settle for less by choosing r = y B.Ifinstead y B <r I, r is again no longer feasible and therefore the company must settle for less by choosing r = r I. Using (20), we can derive a specific expression of Q foraparticularvalueofr: If r = r, Q = ( ) β 2; 2 If r = y B, Q =(y B r I ) 2 ; If r = r I, Q =0. 29

30 Because the independent report r I will be released if and only if y B <r I, the probability of auditor independence Pr(AI) is the probability that r I >y B.Thus,wehave Pr(AI) = Pr(r I >y B ) = Pr(y B <x 0 ). (ii) Regime BB. The auditors expectation of their misstatement costs, given a report r and their audit evidence y B and y B2, is E[(r x) 2 y B,y B2 ] = (r r I ) 2 + Var[ x y B,y B2 ] = (r r I ) h + e B + e B2, where r I E[ x y B,y B2 ]= hx 0+e B y B +e B2 y B2,sinceboth x and ỹ follow the normal distribution. h+e B +e B2 If the independent report r I is issued, then the auditors expectation of their misstatement cost is E[(r I x) 2 y B,y B2 ]=(r I r I ) h + e B + e = 1. (21) B2 h + e B + eb2 Therefore, to induce the auditors to certify r instead of r I, the company must offer the auditors an amount of Q to compensate them for the increase in the misstatement cost: Q = E[(r x) 2 y B,y B2 ] E[(r I x) 2 y B,y B2 ] (22) = (r r I ) 2. Similar to the analysis for Regime B, the company s most preferred report is r. However, an inflated report (r >r I ) cannot exceed the most favorable audit evidence max{y B,y B2 }. 30

31 Thus, if y B [r I,r ], r is no longer feasible and therefore the company must settle for less by choosing r =max{y B,y B2 }.Ifinsteady B <r I, r is again no longer feasible and therefore the company must settle for less by choosing r = r I. Using (22), we derive a specific expression of Q for a particular value of r: If r = r, ( ) 2 β Q = ; 2 If r =max{y B,y B2 }, Q =(max{y B,y B2 } r I ) 2 ; If r = r I, Q =0. Now we solve for the probability of auditor independence Pr(AI) under Regime BB. Auditor independence occurs when the independent report exceeds the audit evidence. Hence, it is the probability of r I > max{y B,y B2 }. Recall r I = hx 0+e B y B +e B2 y B2.Thus,wehave h+e B +e B2 Pr(AI) = Pr(r I > max{y B,y B2 }) ( (h + e B )y B2 hx 0 = Pr e B <y B < hx ) 0 + e B2 y B2. h + e B2 (iii) Regime BS. The expressions for (Q, r) andpr(ai) in Regime BB are given above. Replacing y B2 by y S and replacing e B2 by e S yields the counterparts in Regime BS. In particular, note that E[(r I x) 2 y B,y S ]=(r I r I ) 2 + where r I E[ x y B,y S ]= hx 0+e B y B +e S y S h+e B +e S. 1 h + e B + e = 1 S h + e B + e, (23) S ProofofProposition2 In Regime B, using (19), we can rewrite (6) as min e B k B C(e B )+ 1 h + e B. (24) 31

32 Differentiating it with respect to e B and setting it equal to 0 yields and 0 yields k B C (e B ) 1 (h + e B ) 2 =0. In Regime BB, using (21), we can rewrite (7) and (8) as min k B C(e B 1/2 )+ (25) e B h + e B + e B2 min k B C(e B2 1/2 )+. (26) e B2 h + e B + eb2 Differentiating them with respect to e B and e B2 respectively and setting them equal to k B C (e B ) k B C (e B2 ) 1/2 (h+e B +e B2 ) 2 =0 1/2 (h+e B +e B2 ) 2 =0. Obviously, e B = e B2 e B BB. So we can rewrite the above pair of equations as two identical equations: k B C (e B BB ) 1/2 (h+2e B BB )2 =0. In Regime BS, using (23), we can rewrite (9) and (10) as and min k B C(e B α B )+ (27) e B h + e B + e S min k S C(e S α S )+ e S h + e B + e. (28) S Differentiating them with respect to e B and e S respectively yields α B k B C (e B ) (h + e B + e S ) 2 = 0 k S C (e S α S ) (h + e B + e S ) 2 = 0. 32