Strategic Interaction in Auditing: An Analysis of Auditors' Legal Liability, Internal Control...

I. INTRODUCTION

This paper studies the economic consequences of auditors' legal liability in a setting where the quality of a firm's internal control system and an auditor's effort jointly determine the informativeness of the auditor's report on the firm's financial statements.(1)

Recent studies in the auditor legal liability literature--such as Dye (1993, 1995), Narayanan (1994), Schwartz (1997), Chan and Pae (1998), Hillegeist (1999), Zhang and Thoman (1999)--have focused on audit effort, but have not considered the link between audit effort and the firm's internal control system. Our study considers this link. This feature of the paper provides new insight into the efficiency implications of auditors' legal liability.

The costly nature of the audit effort and internal control system raises a question of how to balance the owner's investment in the internal control system with the auditor's effort. We show that when the auditor is legally liable to financial statement users, there is a nontrivial strategic interaction between the auditor's provision of effort and the firm owner's investment in the internal control system. If the owner and auditor fully cooperate, then they could achieve an efficient resource allocation. However, in a noncooperative setting in which the owner and auditor maximize their own personal payoffs, an efficiency loss arises from a misallocation of resources. We first examine how a change in the auditor's legal liability to outside investors (who trade based on audit reports) affects the firm owner's choice of internal control system quality and the auditor's effort choice. We then investigate how changes in these equilibrium choices (induced by a change in the auditor's legal liability) influence social welfare, which we define as the sum of the owner's, auditor's, and outside investors' ex ante expected payoffs.

Our model is similar to Schwartz (1997), but we extend her model as follows. At the beginning of the period, a firm's owner makes an investment that determines the quality of her firm's internal control system. The owner then hires an auditor, who (after assessing the quality of the internal control system) provides unobservable effort and issues an audit report. The audit report is the auditor's public attestation to the firm's future prospects, which we refer to as the firm's type.(2) In contrast to prior research, we allow not only the auditor's effort, but also the quality of the firm's internal control system to affect the informativeness of the audit report about the firm's type. In particular, we assume that: (1) the higher the quality of the internal control system, the higher the probability that the auditor correctly attests to the firm's type; and (2) the higher the quality of the internal control system, the lower the marginal impact of audit effort on the probability of the correct audit report. The first assumption states that, for a given level of audit effort, an increase in the firm's internal control system quality improves the audit report's informativeness about the firm's type; whereas the second assumption states that an increase in audit effort has a greater incremental effect on the informativeness of the audit report when the quality of the internal control system is low (relative to when it is high). The second assumption implies that audit effort and internal control system quality are strategic substitutes.(3) Investors in a competitive capital market price the firm based on an audit report, and if they purchase this firm, they make an additional capital investment. At the end of the period, the firm's cash flow is realized, and the firm's true type is publicly revealed if the firm goes bankrupt. The investors sue the auditor if the firm's true type is different from the type attested to by the auditor (i.e., if an audit failure has occurred). In that case, the auditor must pay damages to investors.

The paper's main results are as follows. First, we show that an increase in the auditor's legal liability reduces the owner's investment in the internal control system and motivates the auditor to provide more effort. The intuition is straightforward. The auditor, faced with increased legal liability, has a stronger ex ante incentive to avoid an audit failure by exerting more effort. On the other hand, as the auditor's legal liability increases, outside investors pay more for the firm because of the audit's increased insurance value. This reduces the owner's ex ante incentive to choose a high-quality internal control system.

Second, for all levels of the auditor's legal liability, the equilibrium allocation (i.e., the owner's investment in the internal control system and the auditor's effort provision) differs from the first-best allocation that maximizes social welfare. As a result, there always exists an efficiency loss. The way in which resources are misallocated depends on the size of the auditor's legal liability. For example, when the auditor's legal liability is sufficiently large, the owner has little incentive to choose a high-quality internal control system, but the auditor has a strong incentive to avoid an audit failure by providing more effort. In equilibrium, the owner underinvests while the auditor overinvests, relative to their first-best investments. On the other hand, when the auditor's liability is sufficiently small, incentives are reversed so that the efficiency loss arises from the owner's overinvestment and the auditor's underinvestment. Indeed, we show that there does not exist a level of auditor legal liability that induces both the owner and auditor to choose the first-best investments. In particular, a liability level that induces one party's first-best investment always results in the other party's underinvestment.

Our result that an efficiency loss is inevitable contrasts with Schwartz's (1997) main result. Specifically, she concludes that a regulator can achieve a socially efficient resource allocation (i.e., the first-best outcome in her model) by choosing an appropriate damage award to investors. Our extension of Schwartz's (1997) model to include a firm owner's strategic choice of the internal control system quality that affects the informativeness of the audit report, leads to a very different conclusion: there is no damage award that induces both the socially optimal internal control system quality and the socially optimal audit effort. In other words, regulators cannot achieve the first-best outcome by choosing the damage award alone.(4)

Our third main result illustrates how regulation of the auditor's legal liability affects the equilibrium social welfare. We show that if the auditor's legal liability is sufficiently large, then reducing the liability improves social welfare by alleviating the owner's extreme underinvestment and the auditor's overinvestment.(5) On the contrary, if the auditor's liability is sufficiently small, increasing liability improves social welfare because it alleviates the auditor's extreme underprovision of effort and the owner's overinvestment. Indeed, we show that there exists a level of auditor liability that minimizes (but does not eliminate) the efficiency loss.

The rest of this paper is organized as follows. Section II presents the model. In Section III, we characterize the socially optimal allocation of resources in our model. In Section IV, we derive the equilibrium allocation. We present our main results in Section V and examine the economic consequences of a change in auditors' legal liability and discuss some regulatory implications of our analysis. Section VI summarizes the paper.

II. THE MODEL

All parties in the model have risk-neutral preferences and a zero discount rate for their future payoffs. Consider a setting in which a firm's owner wants to sell the firm to outside investors in a competitive capital market. The firm has a risky project, which requires a fixed amount of capital, I [is greater than] 0. If outside investors purchase this firm, they supply I and undertake the project. The gross return from the project, which is the firm's only cash flow, is R [is greater than] I if the project succeeds, but zero if it fails. The probability that the project will succeed depends on its type, which can be either high (H) or low (L). A priori, all parties in the model have common prior beliefs that the project type is high with probability [Lambda] [element of] (0, 1).

Let [p.sub.t] [element of] (0, 1), t = H, L, be the probability that a type-t project will succeed, and assume that a high-type project is more likely to succeed (i.e., [p.sub.L] [is less than] [p.sub.H]). Further assume that:

(1) [p.sub.L]R [is less than] I [is less than] [p.sub.H]R and [NPV.sub.[Lambda]] [equivalent] [[Lambda][p.sub.H] + (1 -[Lambda])[p.sub.L]]R - I [is greater than] 0.

That is, a high- (low-) type project has a positive (negative) net present value while, based on prior beliefs, the project has a positive net present value on average. Given that the capital market is competitive, the owner sells the firm at the price of [NPV.sub.[Lambda]], which is the owner's payoff. However, the lack of information about the project type gives rise to an ex ante efficiency loss. Specifically, if the project type were common knowledge, investors would undertake only a high-type project, in which case the firm value would equal [p.sub.H]R - I. This implies that the owner's ex ante welfare loss amounts to [Lambda]([p.sub.H]R - I) - [NPV.sub.[Lambda]] = (1 - [Lambda])(I - [p.sub.L]R) [is greater than] 0, and thus, she has an incentive to obtain more precise information about the project type than prior beliefs.

Financial statements provide valuable information about a firm's future prospects because they summarize the firm's current economic status, which is a leading indicator of its future profitability (such as the project type in our model). Following the literature (e.g., Dye 1993, 1995; Narayanan 1994; Schwartz 1997; Chan and Pae 1998; Hillegeist 1999; Lee et al. 1999; Zhang and Thoman 1999), we assume that an auditor issues an attestation report on the project type. However, in contrast to this prior research, we postulate that the audit report's informativeness about the project type hinges on both the auditor's effort and the quality of the firm's internal control system (hereafter ICS) because the auditor's assessment depends not only on his own expertise, but also on the quality of the firm's financial statements. Ceteris paribus, financial statements' quality (i.e., how well they represent a firm's economic reality) depends on the quality of the firm's ICS.(6)

Formally, the sequence of events is as follows. At the beginning of the period, the owner makes an investment, [q.sub.o] [is greater than or equal to] 0, which represents the quality of the firm's ICS. The owner then hires an auditor from a competitive audit market at a noncontingent audit fee, K [is greater than or equal to] 0, and the audit process begins.(7) The auditor's costly effort throughout the audit process is denoted by [q.sub.a] [is greater than or equal to] 0. We assume that [q.sub.a] is unobservable to the other parties in the model. There are two stages in the audit process. At the first stage, the auditor evaluates the firm's ICS, after which he gains knowledge of [q.sub.o]. At the second stage, the auditor collects and verifies evidence of the firm's economic reality. The auditor's decision at this second stage depends on his knowledge of the firm's ICS quality obtained at the first stage. At the end of the audit process, the auditor issues a report on [q.sub.o] and the firm's future prospects, i.e., the project type. We assume that the auditor's report on [q.sub.o] is without error. On the other hand, his report on the project type may be erroneous. Specifically, given the project's true type, we assume that the pair of ICS quality and audit effort, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], translates into the conditional probabilities of the auditor's correct attestation to the project type as follows:

(2) Pr[h | H, q] = 1 and Pr[l | L, q] = [Phi] (q),

where h (l) is the auditor's attestation report that the project type is high (low). The audit technology given above captures the unique feature of our model in that both the audit effort, [q.sub.a], and ICS quality, [q.sub.o], affect the probability distribution of audit reports.(8)

We impose the following regularity conditions on audit technology [Phi]. For all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

(i) [Phi](q) [element of] [0, 1];

(ii) [[Phi].sub.i](q) [is greater than] 0 and [[Phi].sub.ii](q) [is less than] 0 for all i [element of] {o, a} and [[Phi].sub.oo](q)[[Phi].sub.aa](q) - [[[Phi].sub.oa](q)].sup.2] [is greater than] 0,

where:

[[Phi].sub.i](q) [equivalent] [differential][Phi](q)/[differential][q.sub.i] and [[Phi].sub.ij](q) [equivalent] [[differential].sup.2][Phi](q)/[differential][q.sub.i][differential][q.sub.j] for i, j [element of] {o, a};

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i [element of] {o, a}

(iv) [[Phi].sub.oa](q) = [[Phi].sub.ao](q) [is less than] 0.

The first condition states that [Phi] is a probability function. The second condition states that [Phi] is an increasing and concave function of q. Hence, ceteris paribus, an increase in ICS quality enhances [Phi] (at a decreasing rate), and so does the audit effort. In this sense, [q.sub.o] and [q.sub.a] are complements in improving the audit report's informativeness about the project type. The third condition ensures a unique interior optimum for a social surplus maximization problem, which we will use as a benchmark in our analysis. The fourth condition states that an increase in [q.sub.o] reduces [q.sub.a]'s marginal effect on [Phi], and vice versa. This implies that [q.sub.o] and [q.sub.a] are strategic substitutes (in the sense of Bulow et al. 1985).(9)

After observing the auditor's report on [q.sub.o] and the project type (i.e., l or h), outside investors price the firm accordingly. Investors' knowledge of [q.sub.o] implies that they can correctly infer the auditor's unobservable effort [q.sub.a] since, as will be shown, there is a one-to-one correspondence between [q.sub.o] and [q.sub.a]. When the auditor issues report l, the owner cannot sell the firm at a positive price because Pr[L | l, q] = 1 for any q and the net present value of a low-type project is negative. Thus, investors do not purchase the firm and the game is over. On the other hand, if the auditor issues report h, equations in (2) imply that investors revise their beliefs about a high-type project to:

(3) [Mu](q) [equivalent] Pr[H | h, q] = [Lambda]/[Lamda] + (1 - [Lambda])(1 - [Phi](q)).

Since [Mu](q) [is greater than or equal to] [Lambda] for any q, investors purchase the firm and supply capital I to undertake the project. The return from the project is realized at the end of the period and it is publicly observed. If the project succeeds (i.e., the return is R), the game is over. However, if the firm goes bankrupt (i.e., the return is zero), the project's true type becomes public information. As a result, investors know whether the auditor correctly attested to the project type (through the audit report). Given that the auditor issued report h, we say that an audit failure has occurred if the project's true type is L. We assume that the auditor is held liable for the audit failure (i.e., a strict liability rule prevails) and D [is greater than] 0 is the auditor's expected liability payment to investors. The game structure is common knowledge. Appendix A lists the notation and Figure 1 is an extensive-form game tree of the model.

[ILLUSTRATION OMITTED]

Before proceeding, we note that the present model directly applies to settings where the owner raises capital I from outsiders rather than selling the firm in its entirety. For example, one might consider a case in which the owner raises capital from investors in return for the firm's equity, or another case in which the owner funds the project through a loan contract with a bank. None of this paper's results change in these cases. What matters in our analysis is that third parties make their capital supply decision based on the auditor's report, and they sue the auditor in the case of an audit failure.

III. SOCIAL SURPLUS AND BENCHMARK

Social Surplus

We define social surplus as the sum of the owner's, investors', and auditor's ex ante expected payoffs. Let q = ([q.sub.o], [q.sub.a]) be given. Since investors will purchase the firm and undertake the project if and only if audit report h is issued, the competitive market value of the firm given audit report h is:

(4) V(q) = {[[Mu](q)[p.sub.H] + (1 - [Mu](q))[p.sub.L]]R - I} + [1 - [Mu](q)](1 - [p.sub.L])D,

where [Mu](q) is given by equation (3). The expression in the braces is the investors' net expected return from the project, and the last term is their expected payoff from litigation against the auditor. It follows that, for any given K and q, the owner's ex ante expected payoff is:

(5) [[Pi].sup.O](q) = Pr[h | q]V(q) - K - [q.sub.o] = [NPV.sub.[Lambda]] + (1 - [Lambda])[Phi](q)(I - [p.sub.L]R) + (1 - [Lambda])(1 - [Phi](q))(1 - [p.sub.L])D - K - [q.sub.o],

where Pr[h | q] [equivalent] [Lambda] + (1 - [Lambda])(1 - [Phi](q)) is the probability that audit report h is issued, and we use equations (1), (3), and (4) in the second equality. On the other hand, given K and q, the auditor's ex ante expected payoff is:

(6) [[Pi].sup.A](q) = K - [q.sub.a] - (1 - [Lambda])(1 - [Phi](q))(1 - [p.sub.L])D,

where the last term is his ex ante expected liability loss.

Since investors' expected payoff is zero in the competitive capital market (which we have already taken into account in equation [4]), social surplus in our model is equal to:

(7) [[Pi].sup.S](q) [equivalent] [[Pi].sup.O](q) + [[Pi].sup.A](q) = [NPV.sub.[Lambda]] + (1 - [Lambda])[Phi](q)(I - [p.sub.L]R) - ([q.sub.o] + [q.sub.a]),

where we use equations (5) and (6). It is clear that the audit fee, the auditor's liability loss, and the equilibrium value of the firm do not appear in equation (7) because they are mere wealth transfers; instead, social surplus is solely characterized by q. The second term in equation (7) represents the ex ante social expected benefit from detecting a low-type project (i.e., savings on wasteful investment), and the last term is the social cost associated with that benefit.

Benchmark: The First-Best Case

Before proceeding to a formal analysis of the model, we consider a hypothetical case, referred to as the first-best case, in which a regulator (or a social planner) can directly enforce the owner and auditor to choose particular levels of [q.sub.o] and [q.sub.a] to maximize the social surplus given in equation (7). That is, consider a regulator's problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given our assumptions on [Phi], a socially optimal pair of ICS quality and audit effort is uniquely characterized by the following first-order conditions:

(8) (1 - [Lambda])(I- [p.sub.L]R)[[Phi].sub.o](q) = 1;

(9) (1 - [Lambda])(I- [p.sub.L]R)[[Phi].sub.a](q) = 1.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the solution to equations (8) and (9), and we call it the first-best allocation. That is, [q.sup.[dagger]] is an allocation at which the social marginal cost and benefit of q are equal.

It is useful to view equations (8) and (9) as indicating how the owner and auditor should coordinate [q.sub.o] and [q.sub.a] to maximize social surplus. In particular, solve equation (8) for [q.sub.o] as a function of [q.sub.a] and denote it by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, solve equation (9) for [q.sub.a] as a function of [q.sub.o] and denote it by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We call [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] coordination functions in the first-best case. Given the concavity of [Phi] and [[Phi].sub.oa] = [[Phi].sub.ao] [is less than] 0, totally differentiating the first-order conditions reveals that both [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are decreasing functions. This means that when ICS quality is higher, the auditor invests less effort, and vice versa. In the space of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the first-best allocation [q.sup.[dagger]] is the point at which the two coordination functions, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], intersect (see Figure 2).

[ILLUSTRATION OMITTED]

IV. EQUILIBRIUM ALLOCATION OF ICS QUALITY AND AUDIT EFFORT

In the previous section, we characterized the socially optimal allocation of [q.sub.o] and [q.sub.a] in the first-best case, i.e., [q.sup.[dagger]]. However, the first-best case is hypothetical because we have assumed that a regulator can directly enforce the owner and auditor to choose [q.sup.[dagger]] to maximize social surplus. We now return to our original model, referred to as the second-best case, in which the owner and auditor choose [q.sub.o] and [q.sub.a] based on their private incentives--i.e., to maximize their own surpluses. In this case, the owner's incentive for ICS quality and the auditor's incentive for audit effort are not aligned to maximize social surplus. Consequently, the equilibrium allocation of q is different from [q.sup.[dagger]], which implies that an efficiency loss exists in the second-best case. We formalize this argument in what follows.

We derive the equilibrium of the game by using backward induction. First, for any given q, the firm value in the competitive capital market when the auditor issues report h equals V(q) in equation (4). Next, consider the game between owner and auditor. The owner moves first by choosing an ICS quality, [q.sub.o]. Then, after observing [q.sub.o], the auditor determines his audit effort, [q.sub.a]. This implies that the owner and auditor play the roles of a Stackelberg leader and follower, respectively. To derive a Stackelberg equilibrium, consider the auditor's problem first.

Given an audit fee and his ex ante expected liability loss, as stated in equation (6), the auditor solves:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the auditor's best response to [q.sub.o] given D. Since D [is greater than] 0, it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the value of [q.sub.a] that solves the first-order condition:

(10) (1 - [Lambda])(1 - [p.sub.L])D[[Phi].sub.a](q) = 1.

Given equation (10) and the concavity of [Phi], the auditor increases effort when ICS quality is lower (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) because [q.sub.o] and [q.sub.a] are strategic substitutes (i.e., [[Phi].sub.ao] [is less than] 0). Also, given equation (10) and the monotonicity of [Phi] (i.e., [[Phi].sub.a] [is greater than] 0), we see that the auditor increases effort when his expected liability is larger (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). In addition, a comparison of equations (9) and (10) reveals that:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In other words, when D = [bar]D, the auditor's reaction function is the same as the coordination function in the first-best case. This follows because [bar]D is the value of the auditor's expected liability that induces the auditor to fully internalize the economic consequence of investment in a low-type project.

Now consider the owner's problem. Given D [is greater than] 0, the owner anticipates that the auditor will provide effort [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if she chooses [q.sub.o]. This implies that the auditor will issue report h with probability Pr[h | q] = [Lambda] + (1 - [Lambda])(1 - [Phi](q)), in which case outside investors price the firm at V(q) given in equation (4). Thus, using equation (5) along with the fact that the auditor's effort choice hinges upon ICS quality, we can state the owner's problem as:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by equation (10). Note that if D = [bar]D, the owner's objective function reduces to:

(13) [[Pi].sub.O] = [NPV.sub.[Lambda]] + (1 - [Lambda])(I - [p.sub.L]R) - K - [q.sub.o] = [Lambda]([p.sub.H]R - I) - K - [q.sub.o],

which indicates that the owner is fully insured against the negative net present value investment in a low-type project.(10) If so, as evident in equation (13), the owner's optimal choice of ICS quality is zero. To focus on the interior solution to the owner's problem, we restrict our subsequent analysis to the case of D [is less than] [bar]D.

Given D [element of] (0, [bar]D), the owner's optimal choice of ICS quality is characterized by the first-order condition for the owner's maximization problem:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the solution to equation (14), which is the owner's equilibrium choice of ICS quality. Substituting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into the auditor's best response function yields the equilibrium audit effort; i.e., it is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is characterized by equation (10). Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the equilibrium allocation given D.

Figure 2 depicts the equilibrium allocation q*(D) and the first-best allocation [q.sup.[dagger]]. First, we explain the first-best allocation. As explained in Section III, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the coordination functions in the first-best case so that the first-best allocation [q.sup.[dagger]] is the intersection of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denoted by F. Tilted ellipses are iso-social-surplus curves, and the bigger ones (i.e., the ones farther from F) correspond to lower levels of social surplus.

Next, we explain the second-best case. In Figure 2, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the auditor's best response function defined by equation (10). For an expositional purpose, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the value of [q.sub.o] defined by equation (14) with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the owner's best response function if the auditor does not observe [q.sub.o]. In contrast to the coordination functions in the first-best case, both the owner's and the auditor's best response functions hinge upon D. In particular, for any given D [is less than] [bar]D, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.o]. This is because, for any given [q.sub.o], the auditor's private marginal benefit of audit effort (given by the left-hand side of equation [10]) is strictly less than the social marginal benefit of audit effort (given by the left-hand side of equation [9]). Similarly, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.a] because the owner's private marginal benefit of [q.sub.o] (given by the left-hand side of equation [14] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is strictly less than the social marginal benefit of [q.sub.o] (given by the left-hand side of equation [8]). Finding the owner's optimal ICS quality given the auditor's optimal effort selection is equivalent to finding a point at which the owner's iso-surplus curve is tangential to the auditor's best response function. U-shaped dashed parabolas are the owner's iso-surplus curves; the higher the parabola, the higher the owner's expected payoff. It thus follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denoted by S in Figure 2, is the Stackelberg equilibrium allocation given D.(11)

As we discussed at the beginning of this section, when the owner and auditor maximize their individual surplus, their private incentives are not aligned to maximize social surplus. As a result, we have q*(D) [is not equal to] [q.sup.[dagger]], which shows that there is an efficiency loss. The equilibrium depicted in Figure 2 is a case in which the efficiency loss arises from the owner's underinvestment in ICS quality and auditor's overprovision of effort relative to the first-best levels; i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Generally speaking, however, the equilibrium levels of ICS quality and audit effort depend on D, and so does the equilibrium social surplus.

V. IMPLICATIONS OF AUDITORS' LEGAL LIABILITY FOR THE EQUILIBRIUM ALLOCATION AND SOCIAL SURPLUS

Analysis of the Economic Consequences of Auditors' Legal Liability

Having derived the equilibrium allocation given D, we now turn to the paper's main objective. Specifically, we examine how a change in auditors' legal environment (e.g., legislation affecting auditors' legal liability) alters the equilibrium social surplus through a change in the equilibrium allocation of ICS quality and audit effort. We model a change in auditors' legal environment as a change in D. In the previous section, we showed that the auditor reduces effort when ICS quality increases (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). However, since the auditor's optimal effort also depends on D (recall that the auditor's best response function is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), the audit effort's sensitivity to ICS quality depends on the legal environment, D. In this regard, we assume a regularity condition throughout the paper that the auditor's optimal reaction to [q.sub.o] is more sensitive when D is larger, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].(12) The next proposition summarizes the effect of a change in D on the equilibrium allocation under this condition.

Proposition 1: Suppose that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds. Then, an increase in the auditor's expected liability loss results in a lower-quality internal control system and a higher audit effort, i.e.:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all D [element of] (0, [bar]D).

Proof: See Appendix B for the proof of all propositions in the paper.

The intuition is straightforward. When D becomes larger, the auditor invests more effort because of a larger expected loss in the case of audit failure. On the other hand, an increase in D motivates the owner to reduce her investment in ICS quality because it lowers the marginal benefit of ICS quality. In addition, due to strategic substitutability, an increase in audit effort (induced by an increase in D) further reduces the owner's incentive to invest in ICS quality, which again reinforces the auditor's incentive to increase effort, and so on.

To be more precise, we can trace the effects of a change in D on the owner's and the auditor's private incentives by examining the first-order conditions. A change in D alters the owner's marginal benefit of [q.sub.o] in three ways. Consider the left-hand side of equation (14). First, holding [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constant, an increase in D reduces the marginal benefit of ICS quality, thereby motivating the owner to cut down her investment in [q.sub.o]; this resembles the "insurance" effect in Schwartz (1997). Second, we know from equation (10) that an increase in D induces the auditor to exert more effort (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Such an increase in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] introduces a change in the left-hand side of equation (14) through [[Phi].sub.o] and [[Phi].sub.a]. However, this indirect effect through [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] equals zero because the effect of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [[Phi].sub.o] exactly offsets its effect on [[Phi].sub.a].(13) The third effect of the auditor's expected liability loss on the left-hand side of equation (14) is through its impact on the slope of the auditor's best response function (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Under the regularity condition ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), this indirect effect is nonpositive. In sum, an increase in D translates into a decrease in the marginal benefit of ICS quality so that the owner reduces her investment in [q.sub.o]. A lower ICS quality induced by an increase in D strengthens the auditor's incentive to increase effort because [q.sub.o] and [q.sub.a] are strategic substitutes. That is, this indirect effect reinforces D's positive direct effect on the auditor's marginal benefit of effort (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), which we discussed earlier.(14)

Proposition 2: There exist two constants [D.sub.o] [element of] (0, [bar]D) and [D.sub.a] [element of] (0, [bar]D) such that the equilibrium ICS quality is equal to the first-best level if, and only if, D = [D.sub.o]; and the equilibrium audit effort is equal to the first-best level if, and only if, D = [D.sub.a]. In addition, [D.sub.o] [is less than] [D.sub.a].

Proposition 2 identifies two levels of the auditor's expected liability at which either the equilibrium ICS quality or audit effort is equal to the first-best level. Specifically, the owner's equilibrium investment in ICS quality is the same as the socially optimal level when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), whereas the auditor provides the first-best effort when D = [D.sub.a] (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Proposition 2 also shows that the level of legal liability that induces the owner's first-best investment is strictly less than the level that induces the auditor's first-best effort (i.e., [D.sub.o] [is less than] [D.sub.a]). Two implications immediately follow. First, for any given D [element of] (0, [bar]D), it is impossible that both the equilibrium ICS quality and audit effort are equal to the first-best levels, i.e., q*(D) [is not equal to] [q.sup.[dagger]]. Thus, an efficiency loss is inevitable. Second, since [D.sub.o] [is less than] [D.sub.a] (and given the results in Proposition 1), it also follows that whenever one party's equilibrium choice equals the first-best level, the other party underinvests.

Propositions 1 and 2 allow us to compare the equilibrium allocation q*(D) with the first-best allocation [q.sup.[dagger]] for various levels of the auditor's expected liability loss D. Corollary 1 summarizes this comparison.

Corollary 1: For any given D [element of] (0, [bar]D), the equilibrium allocation relative to the first-best allocation is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 1 shows that, for any given D [element of] (0, [bar]D), the case in which both the owner and auditor overinvest in ICS quality and audit effort, respectively, does not occur in our model. Instead, both the owner and auditor underinvest if D [element of] ([D.sub.o], [D.sub.a]), whereas one overinvests and the other underinvests otherwise. Figure 3 graphically illustrates the results in Propositions 1 and 2 and Corollary 1.

[GRAPH OMITTED]

The dashed curve in Figure 3 is a locus of the equilibrium allocation q*(D) for various levels of D. As D moves from zero to [bar]D, q*(D) travels in a northwest direction starting from [E.sub.1], passes through [E.sub.o] when D equals [D.sub.o] and then through [E.sub.a] when D equals [D.sub.a], and finally reaches [E.sub.2]. To understand this locus, suppose that the auditor's expected liability loss is arbitrarily close to zero. Then, from the auditor's problem in Section IV, we can see that the auditor's best response function becomes arbitrarily close to the horizontal axis (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.o] as D [right arrow] 0). On the other hand, when D [right arrow] 0, it is easy to verify that the owner's reaction function, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], becomes identical to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.a]. Thus, as the auditor's expected liability loss becomes arbitrarily small, the equilibrium allocation converges to the point at which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] intersects the horizontal axis, denoted by [E.sub.1] in Figure 3. Obviously, the auditor underinvests effort while the owner overinvests in ICS quality in this case.

From Proposition 1, we know that an increase in D motivates the auditor to provide more effort and induces the owner to reduce her investment in ICS quality. This is equivalent to saying that, as D increases, the equilibrium allocation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] moves in a northwest direction in ([q.sub.o], [q.sub.a]) space. Therefore, as D increases from zero to [D.sub.o], the equilibrium allocation travels from [E.sub.1] to [E.sub.o], where [E.sub.o] is the equilibrium allocation corresponding to [D.sub.o]. That is, the owner's equilibrium investment in ICS quality is the same as the first-best level at [E.sub.o]. Nonetheless, since [D.sub.o] [is less than] [D.sub.a], the auditor's underinvestment problem persists. A further increase of D starting from [D.sub.o] induces the owner to underinvest. However, unless D reaches [D.sub.a], the auditor continues underinvesting. As a result, when D [element of] ([D.sub.o], [D.sub.a]), the efficiency loss arises from both parties' underinvestments. When D becomes equal to [D.sub.a], the equilibrium allocation reaches [E.sub.a] at which the auditor provides the first-best audit effort. However, since [D.sub.a] [is greater than] [D.sub.o], the owner's underinvestment problem becomes more severe. When the expected liability loss exceeds [D.sub.a], the equilibrium allocation continues its move in a northwest direction. Thus, the auditor begins to overinvest effort, while the owner's underinvestment problem becomes further exacerbated.

Now suppose that D becomes close to [bar]D. In this case, the auditor's best response function becomes identical to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because the auditor fully internalizes the consequence of the negative NPV investment to a low-type project (see equation [11]). We also know that if D = [bar]D, the owner has no incentive to make a positive investment in ICS quality. Hence, as D [right arrow] [bar]D, the equilibrium allocation converges to the intersection of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the vertical axis, denoted by [E.sub.2] in Figure 3. From the above discussion, one can see that the Stackelberg equilibrium in Figure 2, denoted by S, corresponds to an equilibrium allocation when D [element of] ([D.sub.a], [bar]D), in which the auditor overinvests and the owner underinvests.

We now investigate how a change in D affects the equilibrium social surplus through its impact on the equilibrium allocation. Formally, the equilibrium social surplus is given by:

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is identical to equation (7) except that we use the equilibrium allocation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead of q. Differentiating [[Pi].sup.S](q*(D)) with respect to D yields:

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Define:

(17) [Alpha](D) [equivalent] [differential][[Pi].sup.S](D))/[differential][q.sub.o]/ [differential][[Pi].sup.S](q*(D))/[differential][q.sub.a] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

That is, [Alpha](D) is the slope of the iso-social-surplus curve at a given equilibrium allocation q*(D). On the other hand, [Beta](D) is the ratio between the sensitivity of the equilibrium audit effort in response to an increase in D and that of the equilibrium ICS quality.

Proposition 3:

(i) Social surplus increases with D if, and only if, [Alpha](D) is greater than [Beta](D).

(ii) There exist two constants [D.sub.1] [element of] (0, [bar]D) and [D.sub.2] [element of] (0, [bar]D) where [D.sub.1] [is less than or equal to] [D.sub.2] such that social surplus increases with D for all D [is less than] [D.sub.1], while it decreases with D for all D [is greater than] [D.sub.2].

(iii) If [Beta]'(D) [is greater than] 0, then there exists a unique constant [D.sub.s] [element of] (0, [bar]D) such that social surplus attains its second-best maximum at D = [D.sub.s].

Part (i) states the necessary and sufficient condition under which an increase in the auditor's expected liability loss, D, results in an increase in the equilibrium social surplus. It shows that D's impact on social surplus solely depends on the relative size of [Alpha](D) and [Beta](D). To see why, consider the equilibrium allocation S = q*(D) depicted in Figure 2. From Proposition 1, we know that [Beta](D) is negative, and thus, q*(D) moves in a northwest direction when D increases. Such a move increases social surplus if, and only if, q*(D) moves inside the iso-social-surplus curve passing though S. This occurs if, and only if, the iso-social-surplus curve at S is flatter than the locus of q*(D). Since [Alpha](D) is also negative (see the proof in Appendix B), an increase in D improves social surplus if, and only if, [Alpha](D) is greater than [Beta](D).

Even though Part (i) is a general statement about the social welfare consequence of an increase in D in our model, note that the relative size of [Alpha](D) and [Beta](D) varies across different values of D. In contrast, Part (ii) shows that if D is sufficiently small (in the sense of D [is less than] [D.sub.1]), an increase in D always improves social surplus (i.e., [Alpha](D) [is greater than] [Beta](D) for all D [is less than] [D.sub.1], and the converse is true if D is sufficiently large (i.e., [Alpha](D) [is less than] [Beta](D) for all D [is greater than] [D.sub.2]). To explain the intuition, suppose that D is sufficiently close to zero. In this case, the auditor severely underinvests effort while the owner severely overinvests in ICS quality. Thus, an increase in D results in an increase in audit effort and a decrease in ICS quality. This alleviates the misallocation problem, thereby improving social surplus. On the other hand, when D is sufficiently large, say, close to [bar]D, the efficiency loss arises from the owner's underinvestment and the auditor's overinvestment. An increase in D in this situation strengthens the owner's underinvestment incentive and at the same time induces the auditor to provide more effort, thereby exacerbating the misallocation problem and further lowering social surplus.(15)

Part (iii) states a sufficient condition under which the equilibrium social surplus given in equation (15) attains its maximum at a unique value [D.sub.s] [element of] (0, [bar]D). That is, [[Pi].sup.S](q*([D.sub.s])) [is greater than or equal to] [[Pi].sup.S](q*(D)) for all D [element of] (0, [bar]D). This is equivalent to saying that, when [Beta]'(D) [is greater than] 0, [D.sub.1] and [D.sub.2] in Part (ii) shrink to a single point ([D.sub.1] = [D.sub.2] [equivalent] [D.sub.s]) so that the equilibrium social surplus is increasing in D if, and only if, D [is less than] [D.sub.s]. To explain the condition [Beta]'(D) [is greater than] 0, we return to Figure 3. Given that there is a one-to-one correspondence between D and any point on the locus of the equilibrium allocation (i.e., the dashed curve in Figure 3), one can find a social-welfare-maximizing D by locating an allocation on the locus at which social surplus is maximized. However, as evident in Figure 3, such an allocation may not be unique in general. Nonetheless, if [Beta]'(D) [is greater than] 0, then one can verify that the locus of the equilibrium allocation must be concave in ([q.sub.o], [q.sub.a]) space (unlike the locus depicted in Figure 3). If so, the locus is tangential to an iso-social-surplus curve at a unique point, say, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], because [[Pi].sup.S] is a strictly concave function of q. The value of D that corresponds to [q.sup.s] is [D.sub.s].(16) Here, we must emphasize that [D.sub.s] is the second-best optimum in the sense that the equilibrium social surplus given [D.sub.s] is strictly less than the first-best social surplus; that is, [[Pi].sup.S](q*([D.sub.s])) [is less than] [[Pi].sup.S]([q.sup.[dagger]]). This is because, as shown in Corollary 1, an efficiency loss from the misallocation of ICS quality and audit effort is inevitable in the second-best case for any given D.(17)

Some Regulatory Implications of the Analysis

We conclude this section by interpreting recent changes in auditors' legal environment in the context of our model. Corollary 1 shows that if legal liability is excessive (as often claimed by practicing auditors, e.g., O'Malley [1993] and Weinback [1993]), an efficiency loss arises from the owner's underinvestment in ICS quality and the auditor's overprovision of effort. The SEC's recent emphasis on strengthening firms' internal controls(18) would ameliorate any effects of excessive legal liability on firms' disincentive to invest in internal controls and auditors' overprovision of effort, thereby achieving a better allocation of resources.

Second, Proposition 3 suggests that if auditors' legal liability was excessive, then the recent Private Securities Litigation Reform Act of 1995 (the Reform Act) might enhance social efficiency. The Reform Act curbs financial statement users' litigation against auditors, thereby reducing auditors' expected liability loss. In light of Part (ii) of Proposition 3, if auditors' legal liability prior to the Reform Act was sufficiently high, then lowering auditors' expected liability loss (i.e., a decrease in D) improves social surplus by simultaneously alleviating the misallocation of firms' investment in ICS quality and auditors' effort provision.

Nevertheless, insofar as an efficiency loss arises from two different sources (i.e., firms' and auditors' private incentives), our analysis also illustrates that regulators cannot completely eliminate the efficiency loss by simply changing auditors' legal liability. To elaborate, recall that [D.sub.s] given in Part (iii) of Proposition 3 is the second-best optimal liability level; i.e., it never achieves the first-best social surplus. The dual source of incentive problems is crucial to this result. To be more precise, suppose that a regulator can directly enforce a certain level of [q.sub.o], and that only the audit effort is uncontrollable. If so, the regulator can achieve the first-best outcome by enforcing [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and setting D = [bar]D. This is because, when D = [bar]D, the auditor's best response function becomes identical to the coordination function in the first-best case (see equation [11]). Given that ICS quality is equal to the first-best level [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the auditor's optimal effort investment is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence, the first-best allocation [q.sup.[dagger]] is achieved.(19) However, absent regulatory intervention in the owner's ICS quality choice, setting D = [bar]D induces the owner to choose zero ICS quality, which in turn leads to an extreme overprovision of audit effort. Therefore, as Proposition 3 shows, the optimal audit liability in the second-best case must be strictly lower than the liability level that induces the auditor to fully internalize a wasteful investment in a low-type project (i.e., [D.sub.s] [is less than] [bar]D).

Our result that a regulatory agency cannot achieve the first-best outcome by using a single regulatory instrument D contrasts with Schwartz's (1997) result that, under the strict liability rule, the first-best outcome (in her model) is attainable if a regulator chooses an appropriate damage award. This difference arises because we focus on different incentive problems. Schwartz (1997) considers an auditor's incentive problem associated with effort provision and outside investors' incentive problem associated with their capital investment. On the other hand, our paper addresses the owner's and the auditor's incentive problems associated with their joint production of information. In Schwartz's (1997) model, the regulator can eliminate the outside investors' incentive problem by making their potential ex post damage recovery independent of their ex ante capital investment. Thus, the regulator's remaining concern is to choose an appropriate damage award (independent of the investors' actual investment) that induces the auditor to provide the first-best effort. Schwartz (1997) shows that there exists such a damage award in her model. However, in our model, no damage award can eliminate both the owner's and the auditor's incentive problems in their joint production of information, because any damage award has an insurance effect on the owner's payoff. This insurance effect distorts the owner's investment in ICS quality, which in turn affects the auditor's effort decision.(20)

VI. SUMMARY

This paper presents a model of imperfect auditing in which a firm's owner, an auditor, and outside investors strategically interact. An audit benefits the owner because it increases the ex ante market value of the firm by detecting an unprofitable project. The distinguishing feature of our study is that the informativeness of audit reports depends not only on the auditor's effort, but also on the owner's costly investment in the internal control system. We show that if the auditor's expected liability loss is sufficiently large, the owner underinvests in the internal control system and the auditor overinvests effort, leading to an efficiency loss. While reducing the auditor's legal liability in this situation improves social efficiency by alleviating the misallocation of the owner's and auditor's investments, insofar as the owner's and auditor's incentive problems arise from their joint production of information, regulators cannot completely eliminate the efficiency loss by changing legal liability alone.

Recent changes in the regulatory environment can be interpreted in the context of our model. Our analysis suggests that if auditors' legal liability was excessive, then the Private Securities Litigation Reform Act of 1995 (which reduces auditors' legal liability) may enhance social efficiency. In addition, the SEC's recent effort to enhance corporate internal controls may be beneficial if excessive auditors' legal liability has led firms to underinvest in their internal control systems.

We appreciate the comments and suggestions from Richard Sansing (associate editor) and two anonymous referees, which have significantly improved the paper's quality. We also gratefully acknowledge helpful comments from the seminar participants at Ajou University and HKUST.

(1) We broadly define an internal control system to include hiring appropriate personnel, installing a reliable treasury/ accounting system, having competent audit committee members, etc. Although a high-quality internal control system has many benefits, our main focus in this paper is on its spillover effect on external auditing.

(2) Auditors, in reality, do not make a direct statement about their clients' future prospects; instead, they express an opinion about whether a firm's financial statements are presented in accordance with Generally Accepted Accounting Principles (GAAP). However, outsiders (e.g., investors, creditors, financial analysts, investment banks, etc.) then use the audited financial statements to predict firms' future prospects. For example, when investors assess a firm's future earnings and/or its viability as a going concern, they rely on the information in financial statements. When a firm applies for a loan, the bank requires the firm to submit financial statements and uses that information to make a loan decision. In this paper, for the sake of expositional simplicity, we refer to the auditor as directly reporting on the firm's future prospects.

(3) See Bulow et al. (1985) for detailed explanations of strategic substitutability in a gaming situation. The key implication of this assumption in our model is that, ceteris paribus, when the client firm has a higher-quality internal control system, it is rational for the auditor to reduce his effort (measured by the auditor's costs of conducting an audit). Such an inverse relation between the quality of the internal control system and audit effort is consistent with practice (e.g., see Wallace 1984).

(4) To be more precise, the regulator in Schwartz's (1997) model has two problems. One is to control investors' incentive to overinvest, which arises when the amount of their potential ex post damage recovery from the auditor depends on their ex ante investment. This incentive problem disappears if the regulator makes the damage award independent of the investors' actual investment. The regulator's remaining problem is then how to motivate the auditor to provide the socially efficient level of audit effort. Schwartz (1997) shows that the regulator can do so by choosing an appropriate damage award (independent of the actual investment), and this achieves the first-best outcome in her model. In contrast to Schwartz's (1997) model, the informativeness of the audit report in our model is a joint product of the owner's investment in the internal control system and the auditor's effort investment. Consequently, the regulator cannot simultaneously control the owner and auditor's incentives by using a single regulatory instrument (i.e., the damage award).

(5) Shibano (1996) and Chan and Pae (1998) establish a similar result that a decrease in the auditor's legal liability may increase social efficiency. However, they solely focus on the efficiency implication of a change in audit effort (implied by a change in the legal liability) without considering how a firm's endogenous choice of the internal control system affects social efficiency. In contrast, our analysis shows that the efficiency gain comes not only from attenuating the auditor's overprovision of effort, but also from an improvement in the internal control system.

(6) A presumption here is that, ceteris paribus, when a firm has a high-quality ICS, its financial statements are less likely to contain (intentional and/or unintentional) misreporting and, thus, they are more useful in predicting the firm's future prospects. Although our model exclusively focuses on how the owner's choice of an ICS quality affects the auditor's effort choice in an external reporting context, a high-quality ICS can lead to other benefits. For example, a high-quality ICS can reduce agency costs within the firm. An earlier version of this paper considered a setting in which the firm's owner hires a manager and the manager's unobservable effort improves the project's future cash flow in the sense of first-order stochastic dominance. In that setting, we showed that when the owner has a higher-quality ICS, she induces the manager's costly effort more efficiently (i.e., with a lower expected wage).

(7) For analytic simplicity, we treat K as an exogenous variable. However, K could be endogenous without affecting the paper's results. This follows because, as long as the audit fee is not contingent on any subsequent events and observables (including audit reports, the firm's ICS quality and its selling price, the return from the project, etc.), it does not affect the owner and auditor's decisions.

(8) Although the audit technology assumed here implies no type-I error, a weaker assumption would not change the paper's results. As Dye (1995, 80) noted, one could introduce a type-I error (i.e., Pr[l | H, q] [is greater than] 0), and if this type-I error rate is sufficiently small that the project is implemented only when audit report h is issued, then the paper's analysis follows through.

(9) As we show in Section IV, the key implication of [[Phi].sub.ao] [is less than] 0 is that when ICS quality is high, it is optimal for the auditor to reduce his costly effort. This assumption is plausible (e.g., Wallace 1984, 20; O'Keefe et al. 1994, 251-252). For example, consider two levels of ICS quality, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In an audit sampling to verify the account balance of a financial statement item, [q.sub.a] is positively related to the sample size. The condition, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then implies that increasing the sample size has a smaller incremental informational value (i.e., a smaller positive impact on [Phi]) when ICS quality is high than when ICS quality is low. This is reasonable if a higher-quality ICS screens out intentional/unintentional errors so that the reported account balance is already closer to the true balance, and there is less scope for audit effort to improve the accuracy of the financial statements.

(10) Audit fee K in this case is actually an insurance fee. More precisely, it is the outside investors who are insured against investment in a low-type project. However, the firm's market price impounds such an insurance value of the audit, and it is passed to the owner. This is clear in equation (4), i.e., V(q) = [Mu](q)([p.sub.H]R - I) if D = [bar]D. In fact, any value of D greater than [bar]D overinsures investors and imposes punitive damages on the auditor. We rule out such a case in our model.

(11) Our model assumes that prior to choosing audit effort [q.sub.a], the auditor observes the firm's ICS quality [q.sub.o]. This assumption is consistent with the Generally Accepted Auditing Standards (GAAS) requirement that auditors evaluate the quality of firms' internal control systems (SAS No. 55). If, in contrast to GAAS, we allow the auditor's observation decision to be endogenous (i.e., the auditor chooses whether to observe [q.sub.o] prior to determining [q.sub.a]), then it can be shown that the auditor would be better off if he were able to pre-commit not to observe [q.sub.o]. This is because if the auditor does not observe [q.sub.o], then a Nash equilibrium prevails (determined by the intersection of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Figure 2) instead of the Stackelberg equilibrium. However, it can be also shown that the auditor cannot credibly make such a pre-commitment because, once engaged in auditing and as long as he has the ability to assess the firm's ICS quality at no cost, the auditor has an unavoidable incentive to observe [q.sub.o]. In this sense, we can view the auditor's observation of [q.sub.o] as equilibrium behavior, rather than as an assumption.

(12) Observe that since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] becomes more sensitive to [q.sub.o] if, and only if, the slope of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the space of ([q.sub.o], [q.sub.a]) becomes more negative when D increases. In other words, this condition implies that as the liability becomes smaller, the auditor's effort choice becomes less elastic to the changes in the client firm's ICS quality. One can use equation (10) to verify that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] holds if [[Phi].sub.aao] [is less than or equal to] 0 and [[Phi].sub.aaa] [is greater than or equal to] 0. In the remainder of the paper, we also maintain a technical assumption that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are all continuously differentiable.

(13) Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(14) The above discussion shows that the regularity condition (i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is sufficient but not necessary for the monotonicity of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is, insofar as D's indirect effect on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not positively large enough to offset the direct negative effect, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] decreases with D.

(15) In general, [D.sub.1] and [D.sub.2] in Proposition 3 are not directly comparable to [D.sub.o] and [D.sub.a] in Proposition 2. As shown in the proof, [D.sub.1] and [D.sub.2] are the values of D at which [II.sup.S]'(q*(D)) = 0, i.e., [Alpha](D) = [Beta](D). Such values of D have nothing to do with values of D that induce either the owner's or the auditor's equilibrium choice to be the same as his or her first-best choice, i.e., [D.sub.o] and [D.sub.a]. For example, at D = [D.sub.o], [Alpha]([D.sub.o]) might be greater or less than [Beta]([D.sub.o]).

(16) Using the definition of [Beta](D), one can check that [Beta]'(D) [is greater than] 0 if both [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are concave in D. In fact, this concavity condition implies that [II.sup.S](q*(D)) is concave in D so that, given the result in Part (ii), [II.sup.S](q*(D)) attains a unique interior maximum.

(17) Thus far, our analysis has been based on the assumption of [[Phi].sub.ao] [is less than] 0, which implies that ICS quality and audit effort are strategic substitutes. If one alternatively assumes [[Phi].sub.ao] [is greater than] 0 (in which case [q.sub.o] and [q.sub.a] exhibit strategic complementarity in the sense of Bulow et al. [1985]), then some results in Propositions 1 and 2 change. For example, the audit effort may increase or decrease as the legal liability increases. Also, even if the first-best outcome is still unattainable, the efficiency loss in this case arises from both parties' underinvestments; i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any D [element of] (0, [bar]D). However, one can verify that Proposition 3 remains unchanged under the alternative assumption [[Phi].sub.ao] [is greater than] 0; that is, the results in Proposition 3 are robust against the sign of the cross partial derivative of [Phi]. A formal proof of the above arguments is available from the authors upon request.

(18) For example, the SEC called for improving the effectiveness of corporate audit committees, which constitute a key element of internal controls (see the report of the Blue Ribbon Committee on Improving the Effectiveness of Corporate Audit Committees [BRC 1999]).

(19) This may not be the only way to achieve the first-best allocation. That is, other regulatory policies may lead to the first-best outcome. For example, decoupling audit liability (a la Polinsky and Che 1991) may be an alternative. While it may be difficult--if not impossible--to enforce a particular level of internal control system quality, our main point is that to completely eliminate the efficiency loss, regulators must control both the owner's and the auditor's incentive problems simultaneously. This requires at least two regulatory instruments.

(20) Although we simplify the model by assuming that outside investors' capital investment is a fixed amount I, our result generalizes to a setting where the investors' capital investment is endogenous. To be specific, to make our model directly comparable to Schwartz (1997), suppose [p.sub.L] = 0 and let outside investors choose any level of I [is greater than or equal to] 0. Also, let R(I) be the return from investing I in a high-type project where R is an increasing and concave function of I. Since we assume an exogenous damage award D (i.e., independent of I) under the strict liability regime, it follows from Schwartz (1997) that investors fully internalize the consequences of investment I; i.e., their optimal choice of I is the same as the socially optimal one. Then, the question is whether it is possible for a regulator to choose an arbitrary level of damage award that induces both the owner and the auditor to choose the socially optimal levels of internal control system quality and audit effort, respectively. This paper shows that there does not exist such a damage award. More precisely, note that the market value of the firm in this case consists of the investors' optimized expected return from the project and their expected liability recovery. Then, using the fact that both components translate into the owner's payoff function, one can verify that (1 - [Lambda])(1 - [Phi](q))D is the owner's ex ante benefit from the audit's insurance value. Since [q.sub.o] and D are multiplicatively separable, D affects the owner's optimal choice of [q.sub.o] in the manner we have analyzed in Section IV.

REFERENCES

Blue Ribbon Committee (BRC) on Improving the Effectiveness of Corporate Audit Committees. 1999. Report and Recommendations of the Blue Ribbon Committee on Improving the Effectiveness of Corporate Audit Committees. New York, NY:NYSE and NASD.

Bulow, J. I., J. D. Geanakoplos, and P. D. Klemperer. 1985. Multimarket oligopoly: Strategic substitutes and complements. Journal of Political Economy 93 (June): 488-511.

Chan, D. K., and S. Pae. 1998. An analysis of the economic consequences of the proportionate liability rule. Contemporary Accounting Research 15 (Winter): 457-480.

Dye, R. A. 1993. Auditing standards, legal liability, and auditor wealth. Journal of Political Economy 101 (October): 887-914.

--. 1995. Incorporation and the audit market. Journal of Accounting and Economics 19 (February): 75-114.

Hillegeist, S. A. 1999. Financial reporting and auditing under alternative damage apportionment rules. The Accounting Review 74 (July): 347-369.

Lee, C.-W. J., C. Liu, and T. Wang. 1999. The 150-hour rule. Journal of Accounting and Economics 27 (April): 203-228.

Narayanan, V. G. 1994. An analysis of auditor liability rules. Journal of Accounting Research 32 (Supplement): 39-59.

O'Keefe, T. B., D. A. Simunic, and T. S. Stein. 1994. The production of audit services: Evidence from a major public accounting firm. Journal of Accounting Research 32 (Autumn): 241-261.

O'Malley, S. F. 1993. Legal liability is having a chilling effect on the auditor's role. Accounting Horizons 7 (June): 82-87.

Polinsky, A. M., and Y.-K. Che. 1991. Decoupling liability: Optimal incentives for care and litigation. RAND Journal of Economics 22 (Winter): 562-570.

Schwartz, R. 1997. Legal regimes, audit quality, and investment. The Accounting Review 72 (July): 385-406.

Shibano, T. 1996. Over-guarding the guardians: Increasing auditor liability decreases new investment. Working paper, University of Chicago.

Wallace, W. A. 1984. Internal auditors can cut outside CPA costs. Harvard Business Review 62 (March/April): 16, 20.

Weinback, L. A. 1993. The liability crisis: Its impact on clients. Journal of Economics, Management, and Strategy 2 (Fall): 361-366.

Zhang, P., and L. Thoman. 1999. Pre-trial settlement and the value of audits. The Accounting Review 74 (October): 473-491.

APPENDIX A
NOTATION

                         I   Required capital investment to undertake
                             a project.

                         R   Return in the case of project success.

     t [element of] {L, H}   Type of project, where L represents a low
                             type and H represents a high type.

                  [Lambda]   Common prior probability that the project
                             type is high (t = H).

                 [p.sub.t]   The probability that type-t project
                             succeeds.

        [NPV.sub.[Lambda]]   Expected return (net present value) of a
                             project given [Lambda].

                    {l, h}   Set of audit reports on the project type,
                             where l (h) represents the auditor's
                             attestation that the project type is low
                             (high).

q [equivalent] ([q.sub.o],   Pair of the firm's internal control system
                [q.sub.a])   (ICS) quality, [q.sub.o], and the
                             auditor's effort, [q.sub.a].

                  [Phi](q)   The probability that audit report l is
                             correctly issued to a type-L project
                             given q.

D [element of] (0, [bar]D)   Damage award measured by the investors'
                             (auditor's) expected gain (loss) in the
                             case of an audit failure where [bar]D
                             [equivalent] (I - [p.sub.L]R)/(1 -
                             [p.sub.L]).

                   [Mu](q)   Investors' posterior beliefs that the
                             project type is H given audit report h.

                      V(q)   Competitive market value of the firm given
                             audit report h.

           [[Pi].sup.j](q)   The owner's (j = O) and auditor's (j = A)
                             ex ante expected payoff given q.

           [[Pi].sup.S](q)   Social surplus given q.

  [MATHEMATICAL EXPRESSION   Coordination function, i.e., the
NOT REPRODUCIBLE IN ASCII]   first-best [q.sub.i] given [q.sub.j] for
                             i [is not equal to] j [element of] {o, a}.

  [MATHEMATICAL EXPRESSION   The first-best allocation of q.
NOT REPRODUCIBLE IN ASCII]

  [MATHEMATICAL EXPRESSION   Auditor's best response to [q.sub.o]
NOT REPRODUCIBLE IN ASCII]   given D.

  [MATHEMATICAL EXPRESSION   Owner's best response to [q.sub.a] given D
NOT REPRODUCIBLE IN ASCII]   if the auditor does not observe [q.sub.o].

  [MATHEMATICAL EXPRESSION   The equilibrium internal control system
NOT REPRODUCIBLE IN ASCII]   quality (i = o) and audit effort (i = a).

                     q*(D)   The equilibrium allocation of q given D,
                             i.e., [MATHEMATICAL EXPRESSION NOT
                             REPRODUCIBLE IN ASCII.

                 [D.sub.i]   The value of D that induces the first-best
                             ICS quality (i = o) and the first-best
                             audit effort (i = a).

                [Alpha](D)   The slope of the iso-social-surplus curve
                             at a given equilibrium q*(D).

                 [Beta](D)   The rate of change in the equilibrium
                             audit effort relative to that of the
                             equilibrium internal control system
                             quality when D increases.

                 [D.sub.1]   The value of D, below which social surplus
                             increases with D.

                 [D.sub.2]   The value of D, above which social surplus
                             decreases with D.

                 [D.sub.s]   The value of D, at which the equilibrium
                             (i.e., second-best) social surplus
                             attains its maximum.

APPENDIX B PROOFS

Proof of Proposition 1

To simplify notation, define:

(B.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In equilibrium, equation (14) is equivalent to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Totally differentiating [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with respect to D yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The second-order condition implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, to show [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we only need to show that [differential]f/[differential]D [is less than] 0.

From equation (B.1), we have:

(B.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since D [is less than] [bar]D, equation (14) implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence, the first term in equation (B.2) is strictly negative. Next, by differentiating equation (10) with respect to [q.sub.o] and D, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. As a result, the second term in equation (B.2) is equal to 0. It thus follows that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in equation (B.2), then [differential]f/[differential]D [is less than] 0, which in turn implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, consider the equilibrium audit effort. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the inequality follows from the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Proposition 2

Let D [right arrow] 0. Then, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.o], it must be true that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [q.sub.o]. From equation (14), this implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [q.sub.o] is defined by:

(B.3) (1 - [Lambda])(I - [p.sub.L]R)[[Phi].sub.o]([bar][q.sub.o], 0) = 1.

From equation (8), we also have:

(B.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, equation (B.3) and equation (B.4) imply that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the inequality follows from [[Phi].sub.oa] [is less than] 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Given that [[Phi].sub.oo] [is less than] 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have:

(B.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly, let D [right arrow] [bar]D. Then, we know [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which in conjunction with equation (10) implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[bar]q.sub.a] is defined by:

(B.6) (1 - [Lambda])(I - [p.sub.L]R)[[Phi].sub.a](0, [[bar]q.sub.a]) = 1.

From equation (9), we have:

(B.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, using equation (B.6) and equation (B.7), we must have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the inequality follows from [[Phi].sub.oa] [is less than] 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Given that [[Phi].sub.aa] [is less than] 0 and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have:

(B.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 1 shows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all D [element of] (0, [bar]D). Given equation (B.5) and the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as D [right arrow] [bar]D, it follows from the intermediate value theorem that there exists a unique constant [D.sub.o] [element of] (0, [bar]D) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if, and only if, D [is less than] [D.sub.o]. Similarly, using the intermediate value theorem together with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Proposition 1, equation (B.8), and the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as D [right arrow] 0, we conclude that there exists a unique constant [D.sub.a] [element of] (0, [bar]D) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if, and only if, D [is greater than] [D.sub.a].

It remains to show that [D.sub.o] [is less than] [D.sub.a]. Consider the auditor's first-order condition (10) in equilibrium. When D = [D.sub.o], we must have:

(B.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where we use [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since [D.sub.o] [is less than] [bar]D, equation (B.7) and equation (B.9) imply that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [[Phi].sub.aa] [is less than] 0, the above inequality implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now suppose that [D.sub.o] [is greater than or equal to] [D.sub.a]. Then, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is strictly increasing in D, we must have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the last equality follows from the property of [D.sub.a]. However, this is a contradiction to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This completes the proof of Proposition 2.

Proof of Corollary 1

Consider any D [element of] (0, [D.sub.o]]. From Proposition 2, we know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also, from Proposition 1, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all D. Hence, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all D [element of] (0, [D.sub.o]]. On the other hand, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [D.sub.o] [is less than] [D.sub.a] (Proposition 2), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all D (Proposition 1), it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all D [element of] (0, [D.sub.o]]. This completes the comparison of q*(D) with [q.sup.[dagger]] when D [element of] (0, [D.sub.o]]. We omit the proof of the remaining cases because it is similar to the case of D [element of] (0, [D.sub.o]].

Proof of Proposition 3

(i) From equation (16), note that:

(B.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Alpha](D) and [Beta](D) are defined in equation (17). We know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In addition:

(B.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we use equation (14) in the second equality and the last inequality is due to the fact that [[Phi].sub.o] [is greater than] 0, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and D [element of] (0, [bar]D). Thus, from equation (B.10), we have:

(B.12) d[[Pi].sup.S]/dD [is greater than] 0 ?? 1 [is less than] [Beta](D)/[Alpha](D).

From Proposition 1, we know that [Beta](D) is negative for all D [element of] (0, [bar]D). On the other hand, note that:

(B.13) [differential][[Pi].sup.S](q*(D))/[differential][q.sub.a] = (1 - [Lambda])(I - [p.sub.L]R)[[Phi].sub.a] - 1 = (1 - [Lambda])(I - [p.sub.L]R)[[Phi].sub.a] - (1 - [Lambda])(1 - [p.sub.L])D[[Phi].sub.a] [is greater than] 0,

where we use Equation (10) in the second equality and the last inequality is due to D [is less than] [bar]D. Equations (B.11) and (B.13) thus imply that [Alpha](D) is also negative for all D [element of] (0, [bar]D). Rearranging the second inequality in (B.12) establishes the result.

(ii) First, let D [right arrow] 0. Using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[Phi].sub.a] [right arrow] [infinity] (since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as D [right arrow] 0), it is easy to verify that [differential][[Pi].sup.S](q*(D))/[differential][q.sub.o] [right arrow] 0 in equation (B.11) and [differential][[Pi].sup.S](q*(D))/[differential][q.sub.a] [right arrow] [infinity] in equation (B.13). On the other hand, observe that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, using equation (16), we conclude that d[[Pi].sup.S](q*(D))/dD [right arrow] [infinity] as D [right arrow] 0.

Second, let D [right arrow] [bar]D. Then, it is easy to verify that [differential][[Pi].sup.S](q*(D))/[differential][q.sub.o] [right arrow] [infinity] in equation (B.11) (since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies [[Phi].sub.o] [right arrow] [infinity]) and [differential][[Pi].sup.S](q*(D))/[differential][q.sub.a] [right arrow] 0 in equation (B.13), respectively. In addition, from the proof of Proposition 1, we know [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using equation (B.2), one can also check that [differential]f/[differential]D [right arrow] -[infinity] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from below (recall that [differential]f/[differential][q.sub.o] [is less than] 0 due to the second-order condition). As a result, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and thus d[[Pi].sup.S](q*(D))/dD [right arrow] -[infinity] as D [right arrow] [bar]D.

Finally, given that d[[Pi].sup.S](q*(D))/dD [right arrow] [infinity] as D [right arrow] 0 whereas d[[Pi].sup.S](q*(D))/dD [right arrow] -[infinity] as D [right arrow] [bar]D, the intermediate value theorem dictates that there exist [D.sub.1] [element of] (0, [bar]D) and [D.sub.2] [element of] (0, [bar]D) where [D.sub.1] [is less than or equal to] [D.sub.2] such that d[[Pi].sup.S](q*(D))/dD [is greater than] 0 for all D [is less than] [D.sub.1], while d[[Pi].sup.S](q*(D))/dD [is less than] 0 for all D [is greater than] [D.sub.2].

(iii) For any given D, define a compact set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consider the following optimization problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that if [Beta]'(D) [is greater than] 0 for all D, then [Gamma] is a convex subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This, in conjunction with the fact that [[Pi].sup.S] is a strictly concave function of q, implies that there exists a unique optimum, say, [q.sup.s], at the frontier of [Gamma]. Now observe that the frontier of [Gamma] and D has a one-to-one correspondence. The proof of Proposition 3 (ii) shows that the optimum cannot be an allocation that corresponds to either D = 0 or [bar]D. Hence, [q.sup.s] must be an allocation corresponding to a value of D [element of] (0, [bar]D). Denoting such a value of D as [D.sub.s] completes the proof.

Suil Pae Seung-Weon Yoo Hong Kong University of Science and Technology

Submitted January 2000 Accepted November 2000