Clients determine an optimal level of external audit services

24Modeling auditors’ investment strategies consists of describing the means by which auditors cater to their clients’ demands for services. That is why we first characterize the demand for audit services.

25Clients maximize the value of their firm by deciding on, among others, the optimal level of external audit services and the optimal level of other governance mechanisms, complementary or alternatives control activities, subject to certain budget constraints.

26In our opinion, it is relevant to consider that the external audit generates firm value primarily from three components: (1) assurance (i.e., control function of the audit which limits management’s bias, improves the quality of financial disclosures and brings credibility to the financial statements), (2) implicit financial claim stakeholders have on the auditor’s wealth in the event of an audit failure and (3) the value of the audit service itself (for example, the formal and informal advice provided by the auditor over the course of the audit). Only ‘assurance’ value, or component, is considered here to the extent that we focus on audit quality, as previously defined. Because, as stated above, auditor independence does not depend on the quantity of inputs, independence issues are excluded from our analysis so that the ‘assurance’ value of an external audit can simply be modeled as a function of an external auditor’s capability to detect material misstatements through the use of audit effort and audit technology.

27To simplify the analysis, we consider a very simple setting in which all client firms are identical and exclude other determinants of their market value from the model. Therefore, for all client firms we have:

28

29where π corresponds to the market value of the client firm as a function of monitoring activities (external auditing purchased from an auditor i and other governance mechanisms put in place, noted respectively as AE_i and y). In accordance with the audit and governance literature, we assume that the value of the client firm is increasing in monitoring activities, although at a decreasing rate.

30As previously explained, we consider that the value of an external audit to the client firm can be modeled as a function of the auditor’s capability to detect material misstatements given their level of audit effort and use of audit technology. To illustrate, we model the value of the client firm generated from monitoring activities using a simple Cobb-Douglas function:

31

32The first term of equation (2) captures the value of external auditing purchased from a given auditor i and is a function of total audit effort x_i (i.e., number of audit hours) and the audit technology δ_i used by that auditor, taking into account the proportion α ∈ (0,1) of the firm’s optimal budget on monitoring activities B (introduced below) that the client have chosen to allocate to the external audit. The technology δ_i is normalized to be greater than or equal to 1, with δ_i = 1 corresponding to the minimum of audit technology required to provide an audit of ‘standard’ quality (i.e., an audit that just complies with generally accepted auditing standards, or a GAAS audit). Consistent with the definition of audit quality presented above, the product δ_ix_i corresponds to audit quality and is strictly increasing in effort and technology.

33It should be emphasized that the inputs – effort and technology – are neither perfect substitutes, nor perfect complements and that a minimum level of both effort (i.e., labor) and technology is required to deliver an audit of standard quality. When a client hires an external auditor at a given time, the audit technology δ_i is considered to be fixed. In other words, in the short term the technology is a fixed input in the audit production process, whereas the effort is a variable input.

Clients choose their auditor

34As we have explained above, clients determine an optimal level of external auditing, taking into account their budget constraint. Clients buy external audit services from an auditor i [5] for a total cost of p_ix_i. That is, total audit fees are equal to auditor’s i weighted average hourly billing rate, p_i, multiplied by the total number of audit hours, x_i. Other monitoring activities can be purchased at an average hourly price p_y. The budget constraint is reflected in the following equation:

35

36where B > 0 is an optimal total monitoring budget.

37B is determined like any other investment decision the clients face and is set at a stage prior to our analysis. This decision process is not explicitly modeled here and, for simplicity, B is simply taken as given. The client firms spend a portion α ∈ (0,1) et (1 – α) of their optimal monitoring budget B on external auditing and other monitoring activities respectively. The client’s characteristics, such as size or ownership structure affect B and α. The largest clients, for example, have a larger overall budget B. Likewise, those facing more marked agency conflicts tend a priori to allocate a greater share α of their budget to the external audit (Simunic and Stein 1996; Piot 2001). Certain external factors, such as the regulatory and legal environment, can also affect B and α. Our focus here is on audit quality so B and α are considered exogenous to the model and assumed sufficient to satisfy mandatory (minimum) audit requirements. We relax this assumption in Section 3.2 and examine what impact variations of B and α can have on the implications of the model.

38Note that clients will take auditor characteristic vectors, and , as given at the time of purchase. Consequently, the optimal auditor choice for clients is such that the technology/price (average hourly fee) ratio, δ_i/p_i, is maximized [6]. That is, out of the set of available auditors, clients only choose among the subset of auditors where δ_i/p_i is maximized since, for a given average billing rate p, clients strictly prefer more technologically capable auditors as this yields the highest level of audit quality [7] that can be purchased at total cost αB. Therefore, at equilibrium, auditors enjoying positive sales must have set a price proportional to their technological abilities: δ_i/p_i = δ_j/p_j, ∀ i, j. Since the level of technology δ_i ultimately determines the quality of the audit, we refer to the technology/price ratio as “audit value”.

Modeling of the competition in the audit market

39Following our discussion above, we now model audit firm competition as a three-stage game:

40– In the first stage, auditors who decide to enter a given audit market face an entry cost σ > 0, determined exogenously and constituted, for example, of the cost of basic professional insurance.

41– In the second stage, each audit firm chooses the optimal level of technology it will employ for its audits. At this stage, the auditor incurs an additional fixed cost:

42

43Where δ ≥ 1 is the technology index with δ = 1 (A(δ=1) = 0) corresponding to minimal technology [8], a > 0 is the unit cost of technology improving the quality of the audit and γ > 1 determines at what speed fixed costs increase with technology δ. Higher values of γ correspond to more rapidly diminishing returns in technology.

44– The third stage corresponds to the production and delivery of audit services. Total costs incurred up to this stage are fixed and equal to F(δ_i) = σ + A(δ_i). We assume that unit effort cost is constant and independent of technology, which implies that the marginal audit production cost is constant [9]: c_i(δ_i, x_i) = c > 0, ∀i (we relax this assumption somewhat hereafter, see below, as well as Section 3.2.1).

45For simplicity, competition is modeled as Cournot. With technology level fixed and assuming all audit firms possess equivalent technology, , a deviant firm employing technology will earn a variable profit in the final-stage of the subgame equal to [10]:

46

47where S is equal to market size (i.e., total market audit fees, determined by the sum of clients’ budget allocated to external audit services, determined by both B and α), and N is the number of audit firms entering at the first stage. Because parameters B and α are taken as given, market size S is itself exogenous. Note as well that final-stage variable profit is increasing in S and δ_i (holding constant), and decreasing in N. In the simple case where all auditors set , then Π = S/N²; that is, variable profits are independent of technology as there is no vertical differentiation between the services offered by the auditors.

48In equilibrium, it must be that all entrants earn a non-negative final (net) payoff. With respect to technological investments determined at the second stage, entrants will face one of the following two situations:

49– The first possibility occurs when the marginal gain for a deviant auditor who sets δ_i slightly above is insufficient to recover the additional investment . In this case, all entrants will offer simple audits with a minimum level of technology and incur total fixed costs .

50– The second possibility occurs when it becomes profitable to invest above the minimum level of audit technology. In this case, all entrants undertake to invest in audit technology (therefore ), up to the point where the marginal profit obtained from the investment is offset by its cost A(δ).

51A movement from the first situation to the next identifies the key points at which capital investments in audit technology above the required minimum becomes an essential element of competition between audit firms. These points depend on market size S and fixed costs parameters σ, a and γ. Once it becomes profitable to invest in audit technology, audit firms will compete by investing increasingly in audit technology in order to offer a higher level of audit quality. Solving the game with backward induction, as Sutton (1991), yields equilibrium price:

52

53and technology:

54

Implications for audit quality

55This leads to the first observation:

56Observation 1. Ceteris paribus, audit quality is increasing in market size: .

57The proof of Observation 1 comes directly from equation (7) and the definition of audit quality retained here, namely δ_ix_i. Essentially, investments in audit technology δ are increasing in market size S as these fixed costs can be more easily supported in larger markets (i.e., economies of scale). Holding the cost of audit effort constant, as well as the number of entrants, further implies that the amount of audit hours x_i purchased by individual clients remains unchanged. As a result, the level of audit quality increases through greater use of audit technology.

58Observation 1 highlights that audit quality is also driven by audit technology, rather than just audit effort (and/or auditor independence), as implicitly assumed in the audit quality literature. Furthermore, it is clear that offering higher audit quality only through greater audit effort x is not the optimal strategy for auditors. To see this, it is important to understand that when a client firm purchases an external audit, it essentially purchases a “bundle” of audit effort from a given auditor utilising a specific level of audit technology δ_i. The total number of hours purchased depends not only on client characteristics, but also on the technology of the chosen auditor.

59To illustrate, consider a market with two auditors: auditor i with technology δ_i = 1 (minimum level of technology); and auditor j with technology δ_j > 1. In this setting, auditor i may ‘compensate’ for its inferior technology by providing more effort on its audits such that total audit quality supplied by auditors i and j is equivalent (i.e., δ_ix_i = δ_jx_j, avec x_i > x_j). Yet, because average marginal costs are equal for both auditors, c_i = c_j = c, their average billing rates will also be equal, p_i = p_j = p. The clients of auditor i therefore bear the full cost of the additional audit effort (i.e., px_i > px_j). Even if both auditors supply equal (total) audit quality, clients will opt for auditor j’s services as total audit fees will be lower. In other words, although greater effort increases audit quality, it does not enhance audit value. Finally, auditor j is strictly preferred and will be able to recover its additional investment in audit technology from the additional market shares gained at the expense of auditor i.

Implications for audit value: extending the interpretation of ‘δ’

60Thus far, we have limited our interpretation of audit technology to investments that enhance the real quality of audits performed. However, the key results of the EFC model hold under a more general setting where ‘technology’ can also refer to investments that increase the perceived quality of products or services, such as advertising, or diminish the marginal cost of these, such as process innovations (Sutton 1991, 1998). Whatever the nature of these investments, what is important is that: (1) the level of such investments is a strategic auditor-specific decision (i.e., endogenous); (2) these investments constitute costly production inputs that are fixed at the time of audit production and delivery; and (3) these investments are valued by clients and/or allow auditors to lower marginal effort costs (i.e., is reflected by an increase in the δ_i/p_i ratio).

61When audit firms compete, they likely engage in a combination of innovations set to improve the quality – real and/or perceived – of their audits and/or the efficiency of the audit production process. Even if the general results of the EFC model remain unchanged, explicitly allowing for the possibility of process innovations requires some clarification. In the basic model, investments that enhance audit quality, real and/or perceived, are explicitly included in the model through the parameter δ. That is, auditors enhance the value of their audits (i.e., ratio δ_i/p_i) by investing to increase δ_i while marginal effort cost c, and therefore price p_i, remain virtually constant. However, as Sutton (1998) shows, it is also possible for firms to successfully compete and improve the relative value of their product or service by lowering their average marginal cost of production (in the case of an audit, the cost of effort) through process innovations. Consequently, the ratio δ_i/p_i is improved by lowering p_i.

62Accordingly, to interpret the results of the model, it is more accurate to focus on the relative value, δ_i/p_i, of an audit firms’ audits rather than just on the level of audit quality resulting from δ_i. Because of this, it is difficult to predict which of audit quality or production efficiency is improving in market size. This is further complicated by the fact that some investments can improve both audit effectiveness, or quality, and efficiency. However, we can reformulate Observation 1 to account for the possibility of efficiency gains resulting from process innovations. This is formalised with Observation 1’:

63Observation 1’. Ceteris paribus, audit value, or relative audit quality, is increasing in market size: .

64This means that investments in audit technology (represented here as fixed costs A) are increasing in market size, with some of these investments improving audit quality, real and/or perceived, or δ, while others can lower the marginal cost of audit production, or the cost of audit effort c, and therefore the price p of the audit.

Implications for audit fees

65Interestingly, equation (6) indicates that audit firms will price their services as oligopolists according to their average marginal cost c (e.g., weighted average cost of labor) and the number of entrants N. This has important implications for the specification of audit fee models used in empirical research. The additional fixed costs, A(δ > 1) > 0, incurred by auditors investing in audit technology are, for the most part, not passed on to their clients. Rather, auditors who choose, at stage two, to invest above the minimum in audit technology can hope to recover these investments in stage three because of greater market shares. Competition for these market shares drives auditors entering a market to offer greater audit quality by investing in audit technology, while pricing their audits at the most competitive rate (i.e., billing rate p close to average marginal cost c). The equilibrium condition guaranties that auditors recover their total investments F(δ) given the number of entrants in the first stage of the game.

Implications for audit market structure

66The key result from the EFC model is that audit firms will compete on audit technology as a means to offer highly valued high-quality audits (i.e., best quality, or technology, to price ratio). This, in turn, has a profound impact on the structure of the audit industry. Generally, the concentrated structure of the audit industry and its domination by the Big 4 audit firms is argued to be the result of significant barriers to entry (for example, GAO 2003, 2008; Rose and Hinings 1999; The Economist 2004). These barriers arise, for example, from regulatory requirements, the high cost of legal liability, or the complexity of engagements and accounting rules, and are commonly assumed to evolve exogenously. As such, the current literature generally fails to account for the dynamics of the industry and ignores the role competition between audit firms plays in raising fixed costs and naturally limiting entry in the industry. The EFC framework, on the other hand, explicitly takes into consideration the interaction between audit firms and allows for fixed costs to evolve endogenously as a result of competition.

67As market size continues to increase, audit firms will compete on the respective value of their services by investing ever more aggressively in audit technology in order to enhance the quality or improve the efficiency of their audits. As firms try to gain market shares at the expense of others by investing in audit technology, there comes a point where the number of audit firms in the market does not increase with market size since further entry would drive variable profits below the total fixed costs that audit firms have engaged. Eventually, this ‘race for quality’ leads to a natural oligopoly where only a limited number of large, technology intensive, high-quality audit firms service the market, no matter how large it gets. At this stage, market structure is independent of setup costs while technological investments create an endogenous and strategic barrier to entry. Formally:

68Observation 2. The audit industry is characterized as a natural oligopoly and the minimum level of auditor concentration, measured by the market share of the dominant audit firm (C₁), does not converge to 0 as the market size approaches infinity: 1/N = C₁ > 0. [11]

69It should be noted here that Observation 2 helps explain the competitive environment that leads to the emergence of a group of large audit firms, the Big 4, without justifying the presence of non-Big 4. In the next section, we adapt the model to allow for the presence of the non-Big 4 with the aim of making it more representative of the true audit market in which both type of audits firms coexists.

Big 4 audits are too expensive for some clients

82The simplest argument to explain the presence of non-Big 4 auditors relates to budget constraints clients may face that can limit their ability to purchase Big 4 audits. While a Big 4 auditor can reduce audit effort, and thus audit fees, by employing superior audit technology, audits always require a minimum amount of labor (i.e., effort) to comply with auditing standards. Hence, even if possibly all clients strictly prefer a Big 4 audit (i.e., ), the (minimum) ‘quantity’ x_B4 of audit effort purchased can be such that the total cost of a Big 4 audit may exceed some clients’ optimal monitoring budget (i.e., the client firm would purchase more audit quality than it optimally requires) [13]. In settings where audits are mandatory, these clients prefer alternative, more affordable, albeit lower quality auditors. Of course, this assumes that the Big 4 billing rate is sufficiently higher than that of non-Big 4 auditors: p_B4 > p_NB4. There are two explanations consistent with the EFC model which can account for this.

83First, because they enhance the value of their audits through a series of capital investments, Big 4 audit firms successfully vertically differentiate their audits from those of non-Big 4 auditors. Consequently, Big 4 auditors can extract monopoly rents proportional to the number of Big 4 auditors entering a market (recall equation (6)), while still supplying more highly valued audits: , with p_B4 > p_NB4. In fact, the Big 4 will charge more than the non-Big 4 for their audits (p_B4 > p_NB4), even when average marginal costs are equal for both types of auditors: c_B4 = c_NB4 (see equations (6) and (8)). Essentially, Big 4 audit firms optimally exclude some clients from their client pool by increasing their mark-up on their remaining clients. This ‘quality premium’ is consistent with the concept of the Big 4 premium documented in the audit fee literature (Hay et al. 2006).

84Second, it may be that the average marginal cost is greater for the Big 4 than for non-Big 4: c_B4 > c_NB4 ⇒ p_B4 > p_NB4. For example, surveys suggest that audit staff are more highly paid in Big 4 firms than in non-Big 4 firms (e.g., Robert Half International Inc. 2006; Public Accounting Report 2007; Hays Specialist Recruitment 2008). Yet, it must be that Big 4 average marginal costs c_B4 are not so high, relative to c_NB4, that it eliminates the Big 4 auditors’ competitive advantage. In other words, Big 4 auditors’ superior audit technology compensates for higher prices in such a way that their audits are still more highly valued.

85Differences in Big 4 and non-Big 4 labor costs indicate differences in their respective audit production process and the technical abilities (i.e., ‘quality’) of staff employed. In fact, Big 4 and non-Big 4 audit production processes are expected to differ if these firms use different audit technologies. One implication is that Big 4 and non-Big 4 audit firms likely employ staff auditors, managers and partners in different proportions on their engagements. Evidence of this come from Blokdijk et al. (2006) who find that Big 4 and non-Big 4 auditors essentially exert the same amount of total audit effort (i.e., total hours spent) for similar clients, but with a different allocation of audit hours among the engagement team members. Consistent with the EFC model, Blokdijk et al. (2006) conclude that Big 4 auditors actually deliver higher audit quality than the non-Big 4 as a result of the differences in their audit production processes. The Big 4 devote more effort to the tasks of planning and risk assessment, usually performed by more qualified and experienced auditors, and less effort on substantive testing and audit completion. Accordingly, the weighted average marginal cost of labor should be higher for the Big 4 than that of the non-Big 4.

86Another reason why average marginal cost may be higher for Big 4 than for non-Big 4 auditors is because the majority of investments to improve real audit quality are made a priori in human capital (e.g., employee training). As a result, the Big 4 pay higher wages to recruit high quality candidates in whom it will be profitable to ‘invest’. Also, the human capital developed by the Big 4 has value not only for the firms but also for their employees. In effect, ownership of human capital cannot be solely restricted to Big 4 employers. The skills and experience acquired by working for a Big 4 improve individuals’ career prospects (inside and outside the firms) and, consecutively, the minimum wage that they are willing to accept (reservation wage). As Big 4 audit firms invest more in human capital, they must also raise the wage they pay their auditors, which in turn increases C_B4.

87Importantly, however, this does not alter the results of the EFC model. In fact, Sutton (1991) shows that in a more general setting, the results of the EFC model remain unchanged even if average marginal cost c(δ) is increasing in technology, just so long as the increase is small. The argument here is that the additional variable costs are more than offset by the superior audit quality (i.e., greater audit value) which comes from investing in human capital. Moreover, human capital investments likely allow for efficiency gains that dampen, compensate for, or even, in the largest markets, outweigh the effect of the (small) increase in the average marginal cost.

Determinants of the optimal budget allocated to an external audit

88The determinants of a client’s optimal budget for external auditing, αB, are almost entirely specific to that client. In the basic model developed above, all clients are assumed identical. However, in a more realistic setting, clients possess different characteristics. For example, smaller clients will have a smaller budget for monitoring activities which may limit the ability of some firms to purchase a (possibly) more costly Big 4 audit. Likewise, private companies may find it optimal to allocate a larger proportion (1 – α) of their budget for monitoring activities to improving internal financial systems and internal auditing. In effect, these companies tend to use financial information mainly for internal decision-making, rather than for external financing. Overall, it is expected that θ will be smaller in the small to medium-sized client market segment (as well as the ‘private company’ segment), which is consistent with the fact that the presence of the non-Big 4 is limited almost exclusively to this market segment (see, for example, GAO 2003, 2008 ; Oxera Consulting 2006).