Disclosure-disciplining mechanisms: capital markets, product markets, and shareholder...
I. INTRODUCTION
This paper examines firms' unaudited voluntary disclosures such as management earnings forecasts. Firms may disclose good news to reduce their cost of capital and bad news to reduce both competition in their product market and exposure to litigation damages, By analyzing these
While the impact of a firm's voluntary disclosures depends critically on their credibility, there is little analytical accounting literature on the determinants of the credibility of unaudited disclosures, with the exception of Newman and Sansing (1993) and Gigler (1994). Disclosure models typically assume that voluntary disclosures must be truthful (Grossman 1981; Milgrom 1981; Verrecchia 1983; Dye 1985; Wagenhofer 1990; Feltham and Xie 1992), and hence, these models do not address the mechanisms that make voluntary disclosures credible.
In contrast to most analytical disclosure literature, empirical research on shareholder litigation concludes that firms sometimes fail to make complete and honest disclosures (Skinner 1994; Francis et al. 1994; Baginski et al. 2002). These studies also suggest that the threat of shareholder litigation affects the nature and extent of firms' voluntary disclosures. Despite this growing empirical interest in the effects of shareholder litigation on the quality of firms' voluntary disclosures, only Trueman (1997) models the relation between shareholder litigation and voluntary disclosures, and his model assumes that any disclosures must be truthful. (1) To address this gap in the literature, we examine how shareholder litigation affects the credibility of disclosures that need not be truthful. We also compare the ability of product market competition and shareholder litigation to complement the capital market in disciplining firms' voluntary disclosures.
Neither the capital markets nor the product markets, acting in isolation, can induce managers to provide truthful disclosures in a single-period environment. A firm would make the most favorable disclosure if only the capital market used the disclosure, and would make the least favorable disclosure if only the product market used the disclosure. However, we show that when both capital and product markets use a firm's disclosures, these offsetting informational demands can induce the firm's manager to issue truthful disclosures under some circumstances.
We begin with a benchmark setting in which shareholders cannot sue the firm. An incumbent firm makes a voluntary disclosure about its future prospects to both the capital and product markets. If the disclosure is credible and favorable, then the capital market provides the firm with equity capital at a lower cost. However, a favorable disclosure also increases the probability that a rival firm will enter the incumbent's product market, imposing a proprietary cost on the incumbent. In contrast, issuing a credible adverse disclosure increases the firm's cost of capital, but decreases the firm's expected proprietary loss by discouraging the rival firm's entry. We demonstrate that if a favorable disclosure would increase the incumbent's proprietary cost sufficiently, the mutual discipline created by the trade-off between reducing the cost of capital versus discouraging the rival from entering can induce the firm to report truthfully. On the other hand, when the capital market consequences of disclosures outweigh the product market effects, favorable disclosures are more common and less credible.
We extend in several ways Gigler's (1994) "mutual discipline" result, whereby opposing informational demands of product and capital markets lead the firm to disclose truthfully. First, we show more precisely how equilibrium disclosures vary as a function of the firm's economic environment. That is, we describe how the firm's disclosures vary with the magnitude of the proprietary costs of disclosure, the amount of capital the incumbent firm must raise, the rival's cost to enter the incumbent's market, and the magnitude of the incumbent's outcomes. Second, our more stylized model permits us to predict when the firm will overreport versus underreport the realized outcome. Finally, we examine the effectiveness of shareholder litigation as an alternative to product market competition in complementing capital markets in disciplining firms' disclosures.
Our analysis of firms' disclosure behavior in the no-litigation regime yields two interesting insights. First, when the firm's favorable prospects are large enough, the firm makes more informative disclosures as its proprietary costs increase, in contrast to earlier studies suggesting that proprietary costs discourage disclosures (e.g., Verrecchia 1983). In contrast, when the firm's favorable prospects are not as large, the firm makes false adverse disclosures to avoid sufficiently large proprietary costs. In this way, we highlight how the interaction of the magnitudes of proprietary costs and the firm's outcomes can produce both truthful and false disclosures.
The second insight is that a greater potential proprietary cost of disclosure can actually benefit an incumbent firm. Increasing the proprietary cost can produce both costs and benefits for the firm. By reducing the firm's net outcome, a greater proprietary cost increases the firm's expected cost of capital, as measured by the expected fraction of the firm that investors demand in return for their investment in the firm. However, a greater proprietary cost can also benefit the firm by reducing its expected proprietary cost. With a greater proprietary cost, the rival may enter the firm's product market only when the firm makes a credible favorable disclosure versus always entering after observing the incumbent's uninformative disclosure that results from a smaller proprietary cost. The shareholders' welfare will increase because the reduced likelihood of the rival's entry decreases the expected proprietary cost by an amount that exceeds the expected increase in capital market costs.
The second part of our paper identifies two ways in which shareholder litigation enhances the credibility of firms' voluntary disclosures by complementing other disclosure-disciplining mechanisms. First, litigation expands the circumstances (i.e., the range of proprietary costs) under which the firm voluntarily makes truthful adverse disclosures. Second, when the firm discloses truthfully with some probability less than 1, the threat of litigation increases that probability.
However, shareholder litigation does not always effectively complement other disclosure-disciplining mechanisms. When proprietary costs are very large, shareholder litigation will be redundant because the firm already discloses truthfully, and when proprietary costs are very small, shareholder litigation will be insufficient to induce truthful disclosures. Likewise, when the rival firm's entry cost is very small the rival will always enter, and when the entry cost is very large the rival will never enter, regardless of the incumbent's disclosure. In these circumstances, shareholder litigation may have no incremental effect on the credibility of the firm's disclosures.
Further, shareholder litigation is less effective in certain ways than product market competition in complementing capital market effects to generate truthful disclosures. Specifically, even without shareholder litigation, the product market effects of disclosures can complement the capital market and induce the manager to issue only truthful disclosures under certain circumstances. This result follows because the incumbent's truthful disclosure never reduces the rival firm's incentive to enter, and hence, never reduces the ability of product market competition to complement capital market effects. In contrast, without product market competition, shareholder litigation cannot complement capital market effects to produce truthful reporting. If the firm always disclosed truthfully, then shareholders would have no incentive to sue. Anticipating this response, the firm would have incentive to overreport to reduce its cost of capital, preventing a truthful disclosure equilibrium.
Although product market competition can sometimes be more effective than shareholder litigation as a complement to capital market influences in disciplining the firm's disclosures, in other circumstances, product market competition can actually inhibit truthful disclosures. For example, when proprietary costs are sufficiently great relative to the capital market benefits of favorable disclosures, the firm may voluntarily understate its actual outcome to discourage the rival from entering. Further, when operating in isolation, either the capital market or the product market would inhibit truthful disclosures. In contrast, the prospect of shareholder litigation never prompts the firm to misreport.
Our analysis relates to several streams of recent empirical research. For example, we establish when mutual discipline makes disclosures credible, consistent with Botosan (1997) and Sengupta (1998) who demonstrate that greater levels or quality of disclosures can reduce the firm's cost of equity and debt, respectively. We also show that mutual discipline requires that the proprietary costs of a favorable disclosure be sufficiently large to offset the capital market costs of an adverse disclosure. Consistent with our results, Clarkson et al. (1994) find that firms are more likely to make adverse disclosures when they face greater proprietary costs from competitive entry. Similarly, Shin (2001) provides empirical evidence that product market competition can discourage firms' favorable disclosures.
Skinner (1994) and Kasznik and Lev (1995) investigate how shareholder litigation influences voluntary disclosures and conclude that the threat of such litigation explains why some firms voluntarily disclose bad news. However, it remains unclear why other firms with bad news choose not to make voluntary adverse disclosures. For instance, Francis et al. (1994) find that only 15 percent of their sample of "at risk" firms voluntarily disclose bad news, suggesting that factors besides litigation influence firms' adverse disclosure decisions. Our analysis offers an explanation for these mixed findings because we show how capital and product market considerations, together with shareholder litigation, can induce some firms to make voluntary adverse disclosures while other firms, faced with higher capital costs and lower proprietary costs, do not.
Our results also have empirical implications for the relation between alternative legal liability regimes and firms' disclosure decisions. An implication of our analysis is that more stringent legal liability in the form of greater litigation damages leads firms to issue more adverse disclosures. Consistent with this result, Baginski et al. (2002) find that more stringent legal liability for disclosures leads U.S. firms to provide a higher incidence of unfavorable earnings forecasts than do Canadian firms, which face a less stringent liability regime.
Further, our analysis suggests that more stringent legal liability can enhance shareholders' welfare by making their firms' disclosures more credible under specific circumstances. Ali and Kallapur (2001) examine the security price consequences of the Private Securities Litigation Reform Act of 1995, which reduced firms' legal liability by restricting private litigants' ability to sue for investment losses from securities fraud. Consistent with our finding that legal liability and shareholder wealth can be positively associated, they conclude that the reduction in legal liability stemming from the Act decreased shareholder wealth. The empirical evidence is mixed, however, because Johnson et al. (2000) examine the same legislation and conclude that the Act (which reduced legal liability) increased shareholder wealth.
This paper also predicts that, under certain conditions, firms raising more capital will issue more false favorable disclosures. This prediction is consistent with Korajczyk et al.'s (1991) empirical findings that disclosures preceding an equity issue are more likely to contain favorable information, but is seemingly inconsistent with Frankel et al.'s (1995) evidence that management forecasts are unbiased when issued in conjunction with the firm raising capital. A potential explanation for the discrepancy between our prediction and Frankel et al.'s (1995) results is that our prediction is contingent on the magnitude of proprietary costs, a factor that their study does not control. Our analysis also predicts that as the firm's good news becomes even more favorable, the firm will issue more false favorable disclosures. While Korajczyk et al. (1991) do not test this prediction directly, they do find that information releases preceding an equity issue contain "unusually good news."
The notion that the simultaneous presence of multiple parties with opposing informational demands can provide a balancing influence on a firm's disclosure decisions has been studied in other contexts. For instance, Gertner et al. (1988) examine a firm's choice of its capital structure to signal its private information to a competing firm. Farrell and Gibbons (1989) also study the relative credibility of disclosures with multiple audiences when one disclosure is potentially private. However, neither of these models examines the effectiveness of shareholder litigation as an alternative disclosure-disciplining mechanism.
As a benchmark setting, Section II analyzes the basic model consisting of capital and product market uses of a firm's disclosures in the absence of shareholder litigation. Section III examines the effect of shareholder litigation on the firm's disclosure behavior. Section IV concludes the paper. Appendix A explains how we derive the pricing equation and litigation damages in the environment with litigation. All proofs appear in Appendix B.
II. DISCLOSURE WITHOUT LITIGATION
Model
Consider a single-period setting in which a firm's first-generation shareholders must raise capital, k, by selling a fraction of their shares to second-generation shareholders. All players are risk neutral. The firm's manager privately observes a perfect signal about the outcome (terminal dividend), x [member of] {[x.sub.1],[x.sub.2]}, where 0 < [x.sub.1] < [x.sub.2]. (2) For expositional convenience, we assume that the two outcomes are equally likely, and that k < [x.sub.1]. We refer to [x.sub.1] as an adverse outcome and to [x.sub.2] as a favorable outcome. Likewise, the firm is a low type when [x.sub.1] occurs and a high type when [x.sub.2] occurs.
After realizing the outcome, the manager makes a disclosure [x.sub.i], indicating that [x.sub.i] occurred, for i = 1,2. Consistent with the previous literature on disclosures (e.g., Verecchia 1983; Dye 1985; Gigler 1994), the manager chooses a disclosure to maximize the expected welfare of the first-generation shareholders. We assume that shareholders' responsibility for losses is limited to their investment in the firm (limited liability). Finally, in this section we assume that shareholders cannot sue for false disclosures.
Both the capital market and the product market (a rival firm) observe the firm's disclosure. Based on the firm's disclosure, investors price the firm at [P.sub.i] [equivalent to] P([x.sub.i]), updating their priors about the firm's terminal dividend. The rival firm likewise uses the incumbent's disclosure to update its beliefs before deciding whether to enter the incumbent's product market. If the rival enters the incumbent's product market, then the rival incurs a cost, [k.sup.R], and receives the same outcome as the incumbent. Therefore, the rival will enter if and only if E[x|[x.sub.i]] > [k.sup.R]. The increased competition from the rival's entry produces a proprietary loss of L for the incumbent firm. (3) To avoid uninteresting complications from prospective bankruptcy of the incumbent, we assume that L < [x.sub.1].
After the firm discloses [x.sub.i], investors supply capital of k to the first-generation shareholders in return for a fractional ownership [[alpha].sub.i] [equivalent to] [alpha]([x.sub.i]) in the firm. Capital market competition ensures that k = [[alpha].sub.i][P.sub.i], indicating that investors' expected returns equal their investment. The first-generation shareholders commit to hold their remaining shares until the end of the period. Figure 1 illustrates the timeline of events.
FIGURE 1
Timeline 1: Voluntary Disclosure without Shareholder Litigation
t=0 t=1 t=2
[up arrow] [up arrow] [up arrow]
Original The manager privately The rival firm's
shareholders observes the firm's entry decision is
are endowed terminal dividend, [x.sub.j]. publicly observed.
with shares The manager Shareholders cannot
in a firm. discloses [x.sub.i]. sue the firm for any
The manager must misreporting.
raise k by selling Shareholders
fraction [[alpha].sub.i] of the consume the net
firm at price [P.sub.i]. payoff.
Analysis without Litigation
In a pooling equilibrium, the market prices all firms identically, independent of their disclosures, i.e., the market "pools" all firms together. In contrast, in a separating equilibrium, the market prices firms differentially based on their disclosures, i.e., the disclosures "separate" the different types of firms. Although a pooling equilibrium could involve any disclosure or nondisclosure made by both high- and low-type firms, we assume for ease of exposition that both firm types report [x.sub.2] in a pooling equilibrium. (4)
Let the firm's disclosure strategy be a probability density function, [[rho].sub.ji], indicating that after realizing outcome [x.sub.j] the firm discloses [x.sub.i] with probability [[rho].sub.ji]. Similarly, let [[mu].sub.i] [equivalent to] [mu]([x.sub.i]) be the probability that the rival enters after the incumbent discloses [x.sub.i]. In equilibrium, the firm's disclosure strategy ([[rho].sub.ji]) maximizes the expected payoff to the first-generation shareholders, k + (1 - [[alpha].sub.1])[[x.sub.j] - [[mu].sub.i]L]. In the firm's expected payoff, k is the capital new investors provide, and (1 - [[alpha].sub.1])[[x.sub.j] - [[mu].sub.i]L] indicates that first-generation shareholders retain the fraction 1 - [[alpha].sub.1]) of the expected net payoff, which equals the outcome [x.sub.j] reduced by the expected proprietary cost, [[mu].sub.i]L. All players (the incumbent firm, investors, and the rival) anticipate that other players will act to maximize their own expected net payoffs, given their information sets. That is, each player anticipates sequentially rational strategies of other players, as determined by their consistent, Bayesian-updated beliefs.
The form of the disclosure equilibrium will depend on the firm's trade-off between the capital and product market effects of its disclosure. In a separating equilibrium, a favorable disclosure ([x.sub.2]) reduces the firm's cost of capital (smaller [[alpha].sub.i]), but also increases the expected proprietary cost from the rival's entry. In contrast, an adverse disclosure ([x.sub.i]) discourages entry by the rival but increases the cost of capital (larger [[alpha].sub.i]). In a separating equilibrium, these two countervailing effects create a mutual discipline that makes the disclosures credible even without the threat of an audit, SEC investigation, or shareholder litigation. As shown below, in such a separating equilibrium, if the incumbent issues an unfavorable disclosure, then the rival does not enter and investors price the firm at [P.sub.1] = [x.sub.1]. Conversely, if the incumbent issues a favorable disclosure, then the rival does enter and investors price the firm at [P.sub.2] = [x.sub.2] - L. Investors demand a fraction of the firm that equates their return to the amount of capital, k, that they provide, which yields k = [[alpha].sub.i][P.sub.i], and in a separating equilibrium, k = [[alpha].sub.1][x.sub.1] = [[alpha].sub.2]([x.sub.2] - L). The mutual discipline necessary for a separating equilibrium also requires that the rival's entry decision depend on the firm's disclosure. That is, the rival's cost to enter, [k.sup.R], must be neither so small that the rival always enters nor so large that the rival never enters.
The following analysis considers mixed-strategy equilibria only when no pure-strategy equilibrium exists. Proposition 1 predicts how the firm's disclosure behavior will vary with the economic environment as characterized by the parameters L, [k.sup.R], and [x.sub.2] for circumstances most conducive to credible disclosures. We focus initially on these combinations of parameter values because they reflect the richest interaction of capital market and product market influences on the firm's disclosures. Corollary 1 then completes the characterization for the remaining parameter values.
Proposition 1 (No Shareholder Litigation): When the rival's cost to enter falls in an intermediate range ([k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)) and the incumbent's favorable outcome is sufficiently large ([x.sub.2] [greater than or equal to] [x.sup.*.sub.2]): (5)
(a) There exists a threshold, [L.sup.*], such that for all proprietary costs, L [member of] [[L.sup.*], [x.sub.1]), the firm will always report truthfully (i.e., disclose [x.sub.1] after realizing [x.sub.1] and [x.sub.2] after realizing [x.sub.2]); (6)
(b) For proprietary costs less than or equal to a second threshold, [L.sup.**], where [L.sup.**] < [L.sup.*], only a pooling equilibrium will occur in which the firm will always disclose [x.sub.2]; and
(c) For all intermediate proprietary costs, L [member of] ([L.sup.**], [L.sup.*]), no pure-strategy disclosure equilibrium exists. However, a partial separating equilibrium will exist in which the firm will report truthfully after realizing [x.sub.2], but will mix reports of [x.sub.1] and [x.sub.2] after realizing [x.sub.1].
In all three cases, the rival will enter only when the incumbent firm discloses [x.sub.2].
Although previous research suggested that potential proprietary costs discourage complete and truthful disclosures (e.g., Verrecchia 1983), Proposition 1 demonstrates that disclosures can become more complete and truthful as the proprietary cost (L) increases. (7) With [x.sub.2] [greater than or equal to] [x.sup.*.sub.2], when the proprietary cost is sufficiently small (i.e., for L [less than or equal to] [L.sup.**]), the incumbent's disclosures have no credibility in the resulting pooling equilibrium (see Region A of Figure 2). As the proprietary cost increases (i.e., for L [member of] ([L.sup.**], [L.sup.*])), firms make partially credible disclosures (Region B of Figure 2). Over this interval, adverse disclosures ([x.sub.1]) are fully credible, but favorable disclosures ([x.sub.2]) are only partially credible. Finally, for large proprietary costs (i.e., for L [member of] [[L.sup.*], [x.sub.1])), all disclosures are fully credible (Region C of Figure 2). As the proprietary cost increases, the incumbent's incentives to report adverse information to discourage the rival's entry (i.e., the product market effects of disclosure) increase relative to the incumbent's incentives to report favorable information to reduce the cost of raising new capital (i.e., the capital market effects). When proprietary costs become sufficiently large, the mutual discipline created by the combination of product market and capital market effects gives the firm sufficient incentive to disclose completely and truthfully. In contrast, when proprietary costs are smaller, mutual discipline is limited or absent because the capital market incentives to report favorable information dominate the product market incentives to report adverse information, and as a result, the firm's disclosures are not fully credible.
[FIGURE 2 OMITTED]
The assumption in Proposition 1 that [x.sub.2] is sufficiently large ([x.sub.2] [greater than or equal to] [x.sup.*.sub.2]) ensures that for a high-type firm, the capital market benefit from disclosing a favorable outcome exceeds the product market proprietary cost from the rival's entry. Corollary 1, below, shows that when [x.sub.2] is not sufficiently large, product market disclosure consequences could dominate so that a firm with a favorable outcome ([x.sub.2]) would underreport, i.e., disclose an unfavorable outcome, [x.sub.1], to avoid the proprietary cost.
Finally, the equilibria in Proposition 1 require bounds on the rival's cost of entry, [k.sup.R] (i.e., [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)) to ensure that the rival's entry decision is sensitive to the firm's report. Specifically, for costs of entry within the specified interval, the rival always enters except following a credible adverse disclosure ([x.sub.1]). A separating equilibrium requires that the incumbent's adverse disclosure be credible, thereby discouraging the rival from entering the market.
Proposition 1 integrates the strategic interests of first-generation shareholders, potential investors (second-generation shareholders), and the rival firm in equilibrium. In doing so, it demonstrates how the informational conflict between the capital and product markets can both (1) lend credibility to a firm's voluntary disclosures via the mutual discipline of opposing capital and product market incentives, and (2) provide incentives for the firm to make voluntary adverse disclosures.
The results in Proposition 1 help us provide insights into how a firm's proprietary costs interact with its cost of capital to determine the welfare of its first-generation shareholders. First, Remark 1 characterizes the effect of product market competition on the firm's cost of capital.
Remark 1: Assume that the rival's cost to enter, [k.sup.R], falls in the intermediate range (i.e., [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)). Then the incumbent firm's expected cost of capital, as measured by the expected fraction of the firm that must be sold to raise capital of k, is strictly greater with product market competition than without.
Remark 1 follows because the potential loss to the firm from product market competition reduces the firm's expected terminal value, thereby increasing the fraction of the firm that new investors require in return for their investment. Next, Remark 2 establishes that notwithstanding the result in Remark 1, greater proprietary cost can actually increase the welfare of the first-generation shareholders.
Remark 2: Let the rival's cost to enter, [k.sup.R], fall in the intermediate range (i.e., [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)), the incumbent's favorable outcome be sufficiently large ([x.sub.2] [greater than or equal to] [x.sup.*.sub.2]), and the initial proprietary cost be L [less than or equal to] [L.sup.**]. Then increasing the proprietary cost from L to L', where L' [greater than or equal to] [L.sup.*] > [L.sup.**] [greater than or equal to] L, strictly increases the first-generation shareholders' welfare if and only if L > 0.5L'.
The counterintuitive result in Remark 2 follows because increasing the proprietary cost produces both costs and benefits for the firm. On the cost side, the increased proprietary cost means that the firm must give investors a larger fraction of the firm in order to raise a given amount of capital. The corresponding benefit is that with a larger proprietary cost, L', the rival will enter only when the firm reports [x.sub.2] (which occurs with probability 0.5) in the resulting separating equilibrium as opposed to always entering (probability 1) in a pooling equilibrium produced by the smaller proprietary cost, L. That is, the larger proprietary cost, L', reduces the likelihood the rival will enter, thereby reducing the probability that the incumbent firm will incur the proprietary cost L'. The impact of the reduction in the probability of incurring L' outweighs that of the increase from L to L', such that the expected value of the proprietary cost declines. Remark 2 identifies a necessary and sufficient condition under which the reduction in the incumbent firm's expected proprietary cost exceeds the increase in expected capital market costs, so that the greater proprietary cost actually improves the shareholders' overall expected welfare.
Next, Corollary 1 analyzes equilibria for the remaining parameter values not covered by Proposition 1.
Corollary 1 (No Shareholder Litigation; other parameter values)
(a) For intermediate values of the rival's cost to enter (i.e., [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)), if the favorable outcome is not large (i.e., if [x.sub.2] < [x.sup.*.sub.2]), then in addition to the thresholds [L.sup.*] and [L.sup.**] identified in Proposition 1, there also exists a third threshold [bar]L, where [L.sup.*] < [bar]L < [x.sub.1]. Furthermore:
(i) For all L [member of] [[L.sup.*], [bar]L], the firm will always report truthfully;
(ii) For all L [member of] ([bar]L, [x.sub.1]), a partial separating equilibrium will occur in which the low-type firm always reports truthfully while the high-type firm mixes truthful (favorable) and false (adverse) disclosures;
(iii) for all L [member of] ([L.sup.**], [L.sup.*]), a partial separating equilibrium will exist in which the high-type firm reports truthfully, but the low-type firm mixes truthful (adverse) and false (favorable) disclosures, and for all L [less than or equal to] [L.sup.**] only a pooling equilibrium will exist.
(b) For all other values of [k.sup.R] (i.e., for [k.sup.R] < [x.sub.1] and [k.sup.R] [greater than or equal to]([x.sub.1] + [x.sub.2])/2), a pooling equilibrium will exist. (8)
Given conditions identical to those in Proposition 1 except that the favorable outcome is not large ([x.sub.2] < [x.sup.*.sub.2]), part (a) of Corollary 1 shows that, as in Proposition 1, disclosures initially become more credible as the proprietary cost of disclosure increases. That is, Figure 2 shows that as L increases, disclosures become more truthful over Regions 1-3 under Corollary 1 just as they do over Regions A-C under Proposition 1. However, Figure 2 also shows that a new threshold, [bar]L, now divides Region C from Proposition 1 into Regions 3 and 4 under Corollary 1. Significantly, in Region 4 the firm underreports; i.e., after realizing a favorable outcome the firm makes an adverse disclosure with a strictly positive probability. In Region 4 the combination of large proprietary costs and a smaller favorable outcome ([x.sub.2]) means that product market influences dominate capital market influences, leading the incumbent firm to make a false adverse disclosure to deter the rival's entry. Hence, from Region 3 to Region 4 disclosures become less credible.
Part (b) of Corollary 1 addresses environments with no tension between the informational demands of capital and product markets. In these environments, the rival's cost of entry is either so low that the rival always enters or so high that the rival never enters. Because the rival firm does not react to the firm's disclosure, the disclosures are not credible, producing a pooling equilibrium.
Next, Corollary 2 provides empirical predictions concerning how changes in the economic environment alter the firm's disclosure behavior.
Corollary 2: When the rival's cost to enter falls in an intermediate range ([k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)):
(a) If the firm's favorable outcome is sufficiently large ([x.sub.2] [greater than or equal to] [x.sup.*.sub.2]), then thresholds [L.sup.*] and [L.sup.**] and [[rho].sub.12] (the probability of overreporting for L [member of] ([L.sup.**], [L.sup.*])) all increase in k and [x.sub.2], and decrease in [x.sub.1].
(b) If the firm's favorable outcome is not large ([x.sub.2] < [x.sup.*.sub.2]), then the threshold [bar]L increases in k and [x.sub.2], and decreases in [x.sub.1].
Corollary 2(a) demonstrates how the economic environment determines the extent of mutual discipline created by product and capital markets influences, yielding alternative disclosure equilibria. Increasing k means the firm raises more capital and hence has more to gain by overreporting and reducing the cost of capital. Increasing [x.sub.2] increases the spread between the favorable and unfavorable outcomes, making the firm's cost of capital more sensitive to its disclosures. Therefore, increases in k and [x.sub.2] enhance the importance of the capital market and increase the two thresholds, [L.sup.*] and [L.sup.**], producing less credible disclosures. An increase in [L.sup.*] means less credible disclosures because proprietary costs that originally produced fully informative disclosures now yield only partially informative disclosures. Similarly, as [L.sup.**] increases, pooling equilibria with uninformative disclosures occur for a wider interval of proprietary costs, and hence become more likely. Conversely, increasing [x.sub.1] reduces the capital cost of an adverse disclosure, and therefore the firm overreports less frequently (i.e., voluntary adverse disclosures become more frequent). The intuition is similar for the remaining cases.
III. LITIGATION
Model
Extending the model from Section II to incorporate shareholder litigation is important for several reasons. First, by imposing significant costs on firms (including the cost of frequent out-of-court settlements; Alexander 1991), shareholder litigation affects firms' disclosure decisions (AICPA 1994). Second, only disclosure models with shareholder litigation relate directly to the growing empirical literature on the relations among firms' financial disclosures, shareholder litigation and various antecedent and consequent conditions (Skinner 1994; Francis et al. 1994; Bamber and Cheon 1998; Johnson et al. 2000, 2001; Ali and Kallapur 2001; Baginski et al. 2002). However, most prior disclosure models do not address the determinants of truthful disclosures, and those that do (Newman and Sansing 1993; Gigler 1994) have not examined shareholder litigation. We address this gap by developing theoretical predictions concerning how shareholder litigation affects a firm's disclosure behavior. Third, shareholder litigation is itself an interesting phenomenon because the shareholders' decision to sue is noncontractual, discretionary, and sequentially rational. These features distinguish shareholder litigation from an alternative disciplining mechanism, contractual monitoring, in which a firm ex ante commits to a monitoring policy that is suboptimal once the agent chooses his effort (e.g., Baiman and Demski 1980; Kanodia 1985).
As shown in the timeline in Figure 3, we generalize the model from Section II by assuming that following the firm's disclosure and subsequent trading, first-generation or second-generation shareholders can sue the firm for distorting the stock price through false disclosures (x [not equal to] x). Shareholders sue with probability [[sigma].sub.i] = [sigma]([x.sub.i],[[alpha].sub.i],e/ne), reflecting the fact that when they sue, shareholders know the firm's disclosure ([x.sub.i]), the fraction of the firm sold ([[alpha].sub.i]), and whether the rival entered (e) or did not (ne). We assume that shareholders incur a cost, [k.sup.s] > 0, to sue and that the litigation-prompted investigation perfectly reveals the manager's private information ([x.sub.j]). (9) When litigation reveals a false disclosure, the firm must pay the litigants damages of [D.sub.ji] [equivalent to] D([x.sub.j],[x.sub.i],[[alpha].sub.i]), depending on the actual outcome ([x.sub.j]), the disclosed outcome ([x.sub.i]), and the fraction of the firm sold ([[alpha].sub.i]). We use the widely recognized "market model" (Hurd and Wagner 1990; Posner 1992) to determine damages. According to this model, damages equal the difference between the realized market price of the shares sold based on the firm's actual disclosure and what would have been the "true" price of the shares based on a full and honest disclosure. In a rational expectations equilibrium, the market price will incorporate expected future litigation damages. Appendix A explains the joint determination of market prices and damages. (10)
FIGURE 3
Timeline 2: Voluntary Disclosure with Shareholder Litigation
t = 0 t = 1 t = 2
[up arrow] [up arrow] [up arrow]
Original The manager privately The rival firm's
shareholders are observes the firm's entry decision is
endowed with shares terminal divided, publicly observed.
in a firm. [x.sub.j]. Shareholders may sue.
The manager The shareholders
discloses [x.sub.i]. consume
The manager must the net payoff
raise k by selling after the firm
fraction pays any litgation
[[alpha].sub.i] of damages.
the firm at price
[P.sub.i]
With litigation, a firm with an unfavorable outcome chooses a disclosure by trading off the potentially higher stock price (lower cost of capital) associated with a favorable disclosure against the corresponding proprietary and litigation costs. The manager's disclosure strategy, [[rho].sub.ji], maximizes the expected payoff to first-generation shareholders, k + (1 - [[alpha].sub.i])[[x.sub.j] - [[mu].sub.i]L - [[sigma].sub.i][D.sub.ji]], based on the sequentially rational strategies of all other players. This objective function parallels that in Section II but now includes expected litigation damages.
Analysis
Shareholder litigation produces four related changes in the analysis of equilibrium disclosures. First, shareholders choose suing strategies based on the firm's disclosure. Second, the firm incorporates expected litigation damages in its disclosure decision. Third, the rival considers how potential litigation damages could affect the firm's disclosure. Fourth, investors incorporate expected litigation damages in pricing the firm's shares. Lemma 1 characterizes disclosure equilibria with litigation.
Lemma 1 (Shareholder Litigation):
1. When the rival's cost to enter falls in an intermediate range ([k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)) and the incumbent's favorable outcome is sufficiently large ([x.sub.2] [greater than or equal to] [x.sup.*.sub.2]), then:
(a) For proprietary cost L [member of] [[L.sup.*], [x.sub.1]), the firm will always report truthfully;
(b) For proprietary costs less than or equal to a second threshold, L, where L < [L.sup.*], only a pooling equilibrium will occur; and
(c) For all intermediate proprietary costs, L [member of] (L,[L.sup.*]), only a mixed-strategy equilibrium will occur, in which the firm will report truthfully after realizing [x.sub.2] but will mix reports of [x.sub.1] and [x.sub.2] after realizing [x.sub.1].
2. When the rival's cost to enter falls in an intermediate range ([k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)) and the incumbent's favorable outcome is not large ([x.sub.2] < [x.sup.*.sub.2]), then:
(a) For all proprietary costs in the interval L [member of] [[L.sup.*], [bar]L], the firm will always report truthfully;
(b) For all proprietary costs, L [member of] ([bar]L, [x.sub.1]), a partial separating equilibrium will occur in which the firm will report truthfully after realizing [x.sub.1] but will mix reports of [x.sub.1] and [x.sub.2] after realizing [x.sub.2];
(c) For all L [member of] (L, [L.sup.*]), a partial separating equilibrium will exist in which the firm will report truthfully after realizing [x.sub.2] but will mix reports of [x.sub.1] and [x.sub.2] after realizing [x.sub.1], and for all L [less than or equal to] L only a pooling equilibrium will exist.
3. For all other values of [k.sup.R], a pooling equilibrium will exist.
In parts 1 and 3 of Lemma 1 the firm trades off the lower capital costs from overreporting against both increased proprietary costs and expected shareholder litigation costs. When capital market incentives dominate product market concerns, the incumbent firm overreports in equilibrium. In contrast, in part 2 of Lemma 1, when L [member of] ([bar]L, [x.sub.1]), the firm trades off avoiding proprietary costs against both higher capital costs and expected shareholder litigation costs. In this case, product market concerns dominate capital market influences, and the firm underreports. (11) In this way, Lemma 1 distinguishes circumstances under which the dominant factor in disclosure decisions is the cost of capital (parts 1 and 3) versus product market effects (part 2).
The litigation equilibria in Lemma 1 and the no-litigation equilibria in Proposition 1 and Corollary 1 share similar general structures. Proposition 2 identifies the differences created by incorporating potential shareholder litigation. Let [P.sup.SL.sub.i] and [P.sub.i] denote prices after the firm discloses [x.sub.i] with and without shareholder litigation (SL), respectively.
Proposition 2:
When the rival's cost to enter falls in an intermediate range ([k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)), shareholder litigation:
1. (a) Reduces the likelihood of uninformative disclosures (a pooling equilibrium) by reducing the threshold proprietary cost below which pooling occurs (L < [L.sup.**]), and
(b) Reduces the frequency of overreporting for all intermediate proprietary costs, L [member of] (L, [L.sup.*]);
2. Increases the price for a favorable disclosure, [P.sup.SL.sub.2] > [P.sub.2] for all L [member of] (L, [L.sup.*]), when the incumbent's favorable outcome is large ([x.sub.2] [greater than or equal to] [x.sup.*.sub.2]); and
3. (a) Reduces the frequency of underreporting for proprietary costs L [member of] ([bar]L, [x.sub.1]) when the incumbent's favorable outcome is not large ([x.sub.2] < [x.sup.*.sub.2]), and
(b) Reduces prices for unfavorable disclosures ([P.sup.SL.sub.1] < [P.sub.1]) for all L [member of] ([bar]L, [x.sub.1]) and increases prices for favorable disclosures ([P.sup.SL.sub.2] > [P.sub.2]) for all L [member of] (L, [L.sup.*]) when [x.sub.2] < [X.sup.*.sub.2].
Proposition 2 establishes that litigation can boost credibility by making false disclosures more costly. Parts 1(a), 1(b), and 3(a) of Proposition 2 describe how greater credibility takes the form of either a reduced range of proprietary costs for which disclosures have no credibility (i.e., pooling occurs) or an increased frequency of truthful disclosures in mixed-strategy equilibria. The greater credibility imparted by litigation results in higher prices for favorable disclosures (parts 2 and 3b) and lower prices for unfavorable disclosures (part 3b). Figure 4 illustrates how shareholder litigation increases the informativeness of disclosures by reducing the region over which a pooling equilibrium occurs (Region A with shareholder litigation is less than Region 1 without shareholder litigation.) Consequently, for L [member of] (L, [L.sup.**]) the firm sometimes makes voluntary adverse disclosures because of the incremental effect of shareholder litigation.
[FIGURE 4 OMITTED]
At the same time, the effect of litigation on the firm's disclosure behavior is limited in several ways. First, with [x.sub.2] [greater than or equal to] [x.sup.*.sub.2] and intermediate values of the rival's entry cost ([k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2)), one might expect litigation to yield more truthful disclosures by reducing the threshold proprietary cost ([L.sup.*]) above which a truthful-reporting equilibrium obtains. However, we find that the threshold [L.sup.*] remains unchanged with litigation because when L [greater than or equal to] [L.sup.*], sufficient product market incentives to induce truthful disclosure already exist without litigation. Further, if shareholder litigation were to reduce the threshold for a truthful-reporting equilibrium to, say, [L.sup.*]' < [L.sup.*], then for L [member of] [[L.sup.*]', [L.sup.*]) the firm would report truthfully. But in any truthful-reporting equilibrium, shareholders would not sue because suing is costly and yields no benefit given that the firm discloses honestly. Therefore, the cost of overreporting for L [member of] [L.sup.*]', [L.sup.*]) would consist only of the proprietary cost. In turn, this would contradict the derivation of the original threshold [L.sup.*] in a no-litigation regime. Likewise, when [x.sub.2] < [x.sup.*.sub.2], litigation does not alter the threshold [bar]L and the associated range of proprietary losses that sustains a truthful-reporting equilibrium, [[L.sup.*], [bar]L].
Second, if there were no rival firm or if proprietary cost were zero, then shareholder litigation alone could never produce a truthful-reporting equilibrium. If the firm always disclosed honestly in equilibrium, then rational shareholders would never sue in equilibrium. In turn, facing neither proprietary costs nor litigation damages, the incumbent would deviate and always disclose [x.sub.2] to reduce its cost of capital. (12) In contrast, Proposition 1(a) shows that, even without shareholder litigation, the threat of the rival's entry alone is sometimes sufficient to produce truthful reporting in equilibrium. In this way, the product market alone (without shareholder litigation) can support a truthful-reporting equilibrium, but shareholder litigation alone (with no product market) cannot.
The greater disclosure discipline imposed by product market forces vis-a-vis shareholder litigation stems from fundamental differences between the two processes. Product market competition can impose proprietary costs on the firm even when the rival believes the incumbent has truthfully disclosed its outcome. In contrast, shareholder litigation can be effective only when second-generation shareholders (correctly) believe that the firm's report differs from the actual outcome with some positive probability. When second-generation shareholders believe that the firm has disclosed truthfully, they will never sue in equilibrium and, as a consequence, shareholder litigation will be ineffective as a disclosure-disciplining mechanism because it imposes no costs on the firm.
Finally, when the rival's entry cost is sufficiently high ([x.sub.2] [less than or equal to] [k.sup.R]) or sufficiently low ([k.sup.R] < [x.sub.1]), the rival's entry decision is insensitive to the incumbent's disclosure and shareholder litigation alone is again insufficient to support a truthful-reporting equilibrium for similar reasons.
While the product market is sometimes more effective in complementing capital markets to discipline a firm's disclosure behavior than shareholder litigation, product market effects can also induce the firm to misreport in some circumstances. For instance, under the conditions in Corollary 1(a)(ii), where the high outcome is not sufficiently large and [k.sup.R] falls in the intermediate range, the firm will sometimes underreport to discourage the rival from entering. If there were no rival firm, then the incumbent would report truthfully after realizing [x.sub.2] under those circumstances. However, the absence of the rival firm would also prompt the low-type incumbent to overreport, and thus a pooling equilibrium would obtain, rendering all the disclosures uninformative. Although potential proprietary costs lead the firm to report truthfully after realizing the low outcome under the conditions specified in Corollary 1(a)(ii), the same product market considerations can also lead the firm to underreport after realizing the high outcome (Corollary 1(a)(ii)). In contrast, the prospect of shareholder litigation never prompts the firm to misreport.
Our model assumes that the first-generation shareholders retain a residual interest in the firm. However, in practice some first-generation shareholders may sell all of their shares, while other first-generation shareholders retain an interest in the firm. The existence of first-generation shareholders who sell all of their shares would further reduce the effectiveness of shareholder litigation as a disclosure-disciplining mechanism if these first-generation shareholders can influence the design of the manager's compensation contract. Such shareholders would design the manager's contract to produce a disclosure that would maximize the firm's price at time t = 1 because they would enjoy the benefits of a potentially higher price while bearing no subsequent litigation damages. This would further reduce the effectiveness of shareholder litigation relative to product market competition as a disclosure-disciplining mechanism. On the other hand, if the manager were held personally liable for misreporting or if he were to suffer reputation loss following a shareholder lawsuit, then one would expect shareholder litigation to become a more effective disclosure-disciplining mechanism. The effectiveness of shareholder litigation as a disclosure-disciplining mechanism therefore depends on which shareholders retain the rights to design the manager's contract, and whether the manager is personally liable for misreporting or bears costs such as the loss of reputation from misreporting.
IV. CONCLUSION
To decrease its cost of capital, a firm would prefer to overreport its outcome, whereas the proprietary costs associated with product market competition (i.e., a rival's potential entry into the incumbent firm's product market) gives the incumbent firm incentive to underreport. When shareholder litigation is not feasible, disclosures become more informative as the product market's influence becomes stronger relative to the capital market's influence. The resulting disclosure equilibrium moves from being completely uninformative (Proposition 1(b), Corollary 1(a)(iii), and Regions A and 1 in Figure 2) when the product market's effect is weakest to being fully informative (Proposition 1(a), Corollary 1(a)(i), and Regions C and 3 in Figure 2) when the product market's effect is strongest. (13) For intermediate levels of product market influence, disclosures are partially informative.
Further, we establish the counterintuitive result that under certain circumstances, a greater proprietary cost increases both the credibility of voluntary disclosures and the welfare of first-generation shareholders. When a greater proprietary cost increases the credibility of the incumbent firm's disclosure, a more credible adverse disclosure can reduce the probability that a rival firm will enter the firm's market, so that the expected value of the proprietary cost declines. This benefit of a smaller expected proprietary cost can exceed the increased capital market costs, and thus benefit first-generation shareholders.
Introducing shareholder litigation generally reduces the likelihood of false disclosures, and thereby leads to higher prices for favorable disclosures and lower prices for adverse disclosures. However, product market competition and shareholder litigation in some cases interact to render shareholder litigation less effective in promoting truthful disclosure. In particular, shareholder litigation has no incremental effect on voluntary disclosures when: (1) the rival's cost to enter is either sufficiently small or sufficiently large, or (2) the proprietary cost from the rival's entry is sufficiently small or sufficiently large. These results suggest circumstances under which shareholder litigation has a limited effect in disciplining a firm's disclosure behavior. However, we demonstrate how the product market can sometimes prompt the firm to underreport, an effect that shareholder litigation never produces.
Our analysis is subject to several limitations associated with specific features of our model. First, relaxing our assumption that litigation perfectly reveals the firm's actual outcome would produce an environment with imperfect litigation. Determining litigation damages will be more complex, and it is not clear how the interaction between shareholder litigation and product market competition will be affected. Second, a more general analysis would allow our fixed proprietary cost, L, to depend on the firm's actual outcome. We would also expect this change to affect the interaction of shareholder litigation and product market competition as disclosure-disciplining mechanisms, although the precise effect is likely to depend on the assumed form of the proprietary cost function.
The additional credibility provided by shareholder litigation is robust to expanding the outcome space to include any finite number of outcomes. In this more general environment, there will exist parameter values L and [k.sup.R] such that the market price is strictly increasing in the firm's disclosure and the rival firm will enter only after observing a report greater than a given threshold level. Unfortunately, the resulting analytical complexity in an environment with three strategic players without commitment, endogenous determination of litigation damages and prices, etc., prevents us from fully characterizing disclosure equilibria for a range of parameter values.
A multiperiod extension would allow managers and firms to build a reputation for prompt and complete disclosures, potentially increasing the likelihood of adverse disclosures. Likewise, examining the influence of vague disclosures, ambiguous legal provisions (Trueman 1997), and uncertain costs of suing are other potential avenues for future research. Finally, the analysis could be generalized to examine how contracting to mitigate managerial moral hazard would affect the firm's disclosure behavior.
APPENDIX A
Determination of Prices and Damages with Multiple Markets and Shareholder Litigation
Second-generation shareholders (investors) in a competitive market pay k in return for a fraction [[alpha].sub.i] of the firm at time t = 1 following a disclosure of [x.sub.i]. These investors pay the expected value of their future net cash flow, yielding:
(1) k = [[[alpha].sub.i] {[E.sub.1][x - [[mu].sub.i]L - [D(x,[x.sub.i].[[alpha].sub.i]|[x.sub.i], [[alpha].sub.i]]}] + [[E.sub.1][D(x,[x.sub.i],[[alpha].sub.i]) - [k.sup.s]|[x.sub.i],[[alpha].sub.i]],
where [E.sub.t][*] refers to expectations taken at time t. On the right-hand side (RHS), the first expectation represents investors' share of the firm's expected net residual value. The second expectation indicates that second-generation shareholders receive all expected litigation damages less the cost of suing in the event of overreporting in equilibrium. (14) Dividing Equation (1) by [[alpha].sub.i], using the relation k = [[alpha].sub.i][P.sub.i], and collecting common terms yields the following pricing equation for the firm:
(2) [P.sub.i] = [[E.sub.i][x - [[mu].sub.i]L|[x.sub.i],[[alpha].sub.i]]] + [(1 - [[alpha].sub.i]) [E.sub.1][D(x,[x.sub.i],[[alpha].sub.i])|[x.sub.i],[[alpha].sub.i] - [E.sub.1][k.sup.s]|[x.sub.i],[[alpha.sub.i]]/[[alpha].sub.i].
Litigation damages, D([x.sub.j],[x.sub.i],[[alpha].sub.i]), are determined by the "market model" as the difference between the realized price, [P.sub.i], and the "true" or "intrinsic" price of the firm, [P.sub.T]([x.sub.j]). (15) Thus, litigation damages are:
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where the true price [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
APPENDIX B
Proof for Proposition 1
(a) Separating Equilibrium: Given the assumptions that (1) the rival's payoffs are identical to that of the firm, and (2) the condition in Proposition 1 that [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2), the rival will enter only following a report of [x.sub.2] in a separating equilibrium. A separating equilibrium requires first, that the firm (i.e., the first-generation shareholders) prefers to disclose an unfavorable outcome ([x.sub.1]) honestly to overreporting, which means that:
(4) k + (1 - [[alpha].sub.1]) x [x.sub.1] [greater than or equal to] k + (1 - [[alpha].sub.2]) x ([x.sub.1] - L),
where k is the capital the first generation of shareholders receives from investors in return for the fraction [[alpha].sub.i] of the firm, i = 1,2, based on the disclosure [x.sub.i] by the firm. The terms (1 - [[alpha].sub.1])[x.sub.1] on the left-hand side (LHS) of Equation (4) and (1 - [[alpha].sub.2]) x ([x.sub.1] - L) on the RHS of (4) represent first-generation shareholders' net expected residual interests in the firm from disclosures of [x.sub.1] and [x.sub.2], respectively. The term ([x.sub.1] - L) reflects the rival's sequentially rational strategy of entering the incumbent's product market after observing a report of [x.sub.2] in a separating equilibrium. Then, with equilibrium prices [P.sub.1] = [x.sub.1] and [P.sub.2] = [x.sub.2] - L reflecting investors' consistent beliefs in a separating equilibrium, [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] and [[alpha].sub.2] = k/[P.sub.2] = k/([x.sub.2] - L). Substituting these values for [[alpha].sub.1] and [[alpha].sub.2] in (4) yields:
(5) k + (1 - [k/[x.sub.1]] x [x.sub.1]) [greater than or equal to] k + (1 - k/[[x.sub.2] - L]) x ([x.sub.1] - L).
Subtracting k from both sides of (5) and some simplifying yields:
(6) ([x.sub.1] - k) [greater than or equal to] ([x.sub.2] - L - k/[x.sub.2] - L) x ([x.sub.1] - L).
Define [L.sup.*] as the proprietary cost that leaves the firm indifferent between disclosing honestly ([x.sub.1]) vs. disclosing [x.sub.2], given [x.sub.1]:
(7) ([x.sub.1] - k) = ([x.sub.2] - [L.sup.*] - k/[x.sub.2] - [L.sup.*]) x ([x.sub.1] - [L.sup.*]).
Claim 1 below demonstrates that there is always a solution to (7) such that 0 < [L.sup.*] < k. (Claims 1-6 appear at the end of the proof for Proposition 1.) The LHS of (6) is independent of L, and Claim 2 demonstrates that the RHS of (6) is decreasing in L. Therefore, for any L [greater than or equal to] [L.sup.*], (6) is satisfied.
A separating equilibrium also requires that following a favorable outcome ([x.sub.2]), the firm prefers to disclose honestly rather than disclosing [x.sub.1]. Given the assumed parameter values, the rival would, in a separating equilibrium, enter following a disclosure of [x.sub.2] but not following a disclosure of [x.sub.1]. Thus, truthful disclosure of [x.sub.2] requires:
(8) k + (1 - [[alpha].sub.2]) x ([x.sub.2] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x [x.sub.2],
where, as before, [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] and [[alpha].sub.2] = k/[P.sub.2] = k/([x.sub.2] - L). Substituting these values for [[alpha].sub.1] and [[alpha].sub.2] in (8), subtracting k from both sides, and rearranging yields:
(9) ([x.sub.2] - L - k) [greater than or equal to] ([x.sub.1] - k/[x.sub.1]) x [x.sub.2].
Define [bar]L as the proprietary cost that leaves the firm indifferent between disclosing [x.sub.2] honestly versus disclosing [x.sub.1], given [x.sub.2]:
(10) ([x.sub.2] - [bar]L - k) = ([x.sub.1] - k/[x.sub.1]) x [x.sub.2] [??] [bar]L = k ([x.sub.2] - [x.sub.1]/[x.sub.1]).
The RHS of (9) is independent of L, whereas the LHS decreases in L, implying that (9) is satisfied for:
(11) L [less than or equal to] [bar]L = k ([x.sub.2] - [x.sub.1]/[x.sub.1]).
The preceding analysis indicates that (6) and (9) constitute necessary and sufficient conditions for a separating equilibrium. Claim 3 establishes that [bar]L = k([[x.sub.2] - [x.sub.1]]/[x.sub.1]) [greater than or equal to] [x.sub.1] when [x.sub.2] [greater than or equal to] [x.sup.*.sub.2], as assumed in Proposition 1. In combination with the assumption that k < [x.sub.1], these relations together with the Claim 1 result that [L.sup.*] [member of] (0,k) imply that 0 < [L.sup.*] < k < [x.sub.1] [less than or equal to] [bar]L. Then, because (6) is satisfied for any L [greater than or equal to] [L.sup.*] and condition (9) is satisfied for any L [less than or equal to] [bar]L, it follows that both (6) and (9) will be met for any L [member of] [[L.sup.*], [x.sub.1]), thereby ensuring that a separating equilibrium will exist.
(b) Pooling Equilibrium: We assume that in a pooling equilibrium, firms with both favorable and unfavorable outcomes disclose [x.sub.2], and that investors' off-equilibrium beliefs are such that an off-equilibrium disclosure of [x.sub.1] implies that an unfavorable outcome ([x.sub.1]) occurred. Therefore, given the assumption that [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2), the rival will enter following the equilibrium disclosure of [x.sub.2], but not following the off-equilibrium disclosure of [x.sub.i]. A pooling equilibrium then requires that following the realization of [x.sub.1], the firm prefers to report [x.sub.2]:
(12) k + (1 - [[alpha].sub.2]) x ([x.sub.1] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x ([x.sub.1]),
where [[alpha].sub.2] = k/[P.sub.2] = k/(0.5([x.sub.1] + [x.sub.2]) - L) and [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] represent investors' and the rival's consistent equilibrium and off-equilibrium beliefs, respectively, in a pooling equilibrium. The price [P.sub.1] = [x.sub.1] reflects investors' off-equilibrium belief that the low report comes from a low-type firm. Substituting for [[alpha].sub.1] and [[alpha.sub.2] in (12) yields:
(13) ([0.5([x.sub.1] + [x.sub.2]) - L - k]/[0.5([x.sub.1] + [x.sub.2]) - L]) x ([x.sub.1] - L) [greater than or equal to] ([x.sub.1] - k).
Define [L.sup.**] as the proprietary cost that leaves the firm with an unfavorable outcome ([x.sub.1]) just indifferent between disclosing [x.sub.2] versus [x.sub.1]:
(14) ([0.5([x.sub.1] + [x.sub.2]) - [L.sup.**] - k]/[0.5([x.sub.1] + [x.sub.2]) - [L.sup.**]]) x ([x.sub.1] - [L.sup.**] = ([x.sub.1] - k).
Claims 4 and 5 establish that there exists a solution to (14) such that 0 < [L.sup.**] < [L.sup.*]. Claim 6 establishes that the LHS of (13) is decreasing in L, whereas the RHS of (13) is independent of L, which implies that (13) will be satisfied for all L < [L.sup.**]. Hence, given L < [L.sup.**], a firm with an unfavorable outcome will disclose [x.sub.2].
Similarly, after realizing [x.sub.2], the firm will prefer the pooling disclosure [x.sub.2] if:
(15) k + (1 - [[alpha].sub.2]) x ([x.sub.2] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x [x.sub.2],
where, as before, [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] and [[alpha].sub.2] = k/[P.sub.2] = k/(0.5([x.sub.1] + [x.sub.2]) - L). Substituting for [[alpha].sub.1] and [[alpha].sub.2] in (15), multiplying by ([x.sub.1]/[x.sub.2]), and rearranging yields:
(16) ([0.5([x.sub.1] + [x.sub.2]) - L - k]/[0.5([x.sub.1] + [x.sub.2]) - L]) x ([x.sub.1] - L ([x.sub.1]/[x.sub.2])) [greater than or equal to] ([x.sub.1 - k).
The RHS's of (13) and (16) are identical and the LHS of (16) > LHS of (13) because ([x.sub.1]/[x.sub.2]) < 1 and L > 0. Thus, (13) implies (16), and hence a pooling equilibrium will exist for all L < [L.sup.**].
(c) Mixed-Strategy Equilibrium: Parts (a) and (b) have established that no pure-strategy separating equilibrium can exist for L < [L.sup.*] and that no pure-strategy pooling equilibrium can exist for L > [L.sup.**]. Therefore, for L [member of] ([L.sup.**], [L.sup.*]), we next establish that at least one mixed-strategy equilibrium will exist. Conjecture a mixed-strategy equilibrium in which the firm discloses honestly after realizing [x.sub.2], but follows a mixed disclosure strategy after realizing [x.sub.1]. After realizing [x.sub.j], the firm's expected payoff from reporting [x.sub.1] and [x.sub.2] with probabilities [[rho].sub.j1] and [[rho].sub.j2], respectively, will be [[pi].sub.j] [equivalent to] [[rho].sub.j1][[pi].sub.j1] + [[rho].sub.j2][[pi].sub.j2], where[[pi].sub.ji] represents the firm's expected payoff (excluding k) from reporting [x.sub.i] after realizing outcome [x.sub.j]. Define [r.sub.ji] = Pr([x.sub.j]|[x.sub.i]) to be the posterior probability that [x.sub.j] occurred given that the firm disclosed [x.sub.i]. Consistent beliefs of the market and the rival firm imply that:
(17) [[pi].sub.j1] = (1 - k/[P.sub.1]) ([x.sub.j] - [[mu].sub.1]L); and
(18) [[pi].sub.j2] = (1 - k/[P.sub.2]) ([x.sub.j] - [[mu].sub.2]L),
where [[alpha].sub.i] = k/[P.sub.i] and the market price [P.sub.i] = [r.sub.1i][x.sub.1] + [r.sub.2i][x.sub.2] - [[mu].sub.i]L for i = 1,2, reflects the capital and product markets' consistent beliefs in the conjectured mixed-strategy equilibrium. After realizing [x.sub.1], the firm will mix reports [x.sub.i] and [x.sub.2] only if:
(19) [[pi].sub.11] = [[pi].sub.12].
From the assumption that [k.sup.R] [member of] ([x.sub.1], ([x.sub.1] + [x.sub.2])/2), we have [[mu].sub.1] = 0 (no entry following [x.sub.1]) and [[mu].sub.2] = 1 (entry following [x.sub.2]) for all values of [[rho].sub.11], given that [[rho].sub.22] = 1. Applying these values of [mu] in (17) and (18) yields:
(20) [[pi].sub.11] = (1 - k/[P.sub.1]) ([x.sub.1] - [[mu].sub.1]L) = (1 - k/[x.sub.1]) [x.sub.1] = [x.sub.1] - k; and
(21) [[pi].sub.12] = ([[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L - k]/[[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L]) ([x.sub.1] - L).
Because [[pi].sub.11] = [[pi].sub.12] in equilibrium, equating the RHS's of (20) and (21) yields:
(22) [x.sub.1] - k = ([[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L - k]/[[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L]) ([x.sub.1] - L).
Bayesian updating yields [r.sub.12] = 0.5[[rho].sub.12]/(0.5[[rho].sub.12] + 0.5[[rho].sub.22]) = [[rho].sub.12]/([[rho].sub.12] + 1) because in the conjectured equilibrium [[rho].sub.22] = 1, and (22) becomes:
(23) [x.sub.1] - k = ([[x.sub.2] - [[[rho].sub.12]/[[[rho].sub.12] + 1]] ([x.sub.2] - [x.sub.1]) - L - k]/ [[x.sub.2] - [[[rho].sub.12]/[[[rho].sub.12] + 1]] ([x.sub.2] - [x.sub.1]) - L]) ([x.sub.2] - L).
It can be shown that there always exists a value of [[rho].sub.12] [member of] (0, 1) that solves (23). Such a value of [[rho].sub.12] defines the mixed disclosure strategy for the firm after realizing [x.sub.1]. (16)
After realizing [x.sub.2], the firm will report [x.sub.2] with probability 1 ([[rho].sub.22] = 1) only if:
(24) k + (1 - [[alpha].sub.2]) x ([x.sub.2] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x [x.sub.2].
As before, based on the markets' consistent beliefs and the rival's entry strategy in the conjectured mixed-strategy equilibrium, we have [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] and [[alpha].sub.2] = k/[P.sub.2] = k/([r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L). Substitute these values in (24) and multiply by ([x.sub.1]/[x.sub.2]) to yield:
(25) ([[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L - k]/[[r.sub.12][x.sub.1] + (1 - [r.sub.12]) [x.sub.2] - L]) ([x.sub.1] - L [[x.sub.1]/[x.sub.2]]) [greater than or equal to] ([x.sub.1] - k).
Given that ([x.sub.1] - L ([x.sub.1]/[x.sub.2])) > ([x.sub.1] - L), LHS (25) > RHS (23) because [x.sub.1] < [x.sub.2] and L > 0. This implies that (25) is met, and the firm will always report [x.sub.2] after realizing [x.sub.2], as conjectured. This establishes that at least one mixed-strategy equilibrium will exist for L [member of] ([L.sup.**], [L.sup.*]). Finally, we provide the following claims without proofs. (17)
Claim 1
A solution to (7) exists such that 0 < [L.sup.*] < k.
Claim 2
d ([[x.sub.2] - L - k]/[[x.sub.2] - L]) x ([x.sub.1] - L)/dL [less than or equal to] 0.
Claim 3
[x.sub.2] [greater than or equal to] [x.sup.*.sub.2] [equivalent to] [x.sup.2.sub.1]/k + [x.sub.1] [??] [bar]L [greater than or equal to] [x.sub.1].
Claim 4
A solution to (14) exists such that 0 < [L.sup.**].
Claim 5
[L.sup.**] < [L.sup.*].
Claim 6
d ([0.5([x.sub.1] + [x.sub.2]) - L - k]/[[0.5([x.sub.1] + [x.sub.2]) - L]) x ([x.sub.1] - L)/dL < 0.
This concludes the proof for Proposition 1.
Proof for Remark 2
Compare the expected welfare of first-generation shareholders given a lower proprietary cost of L (which yields a pooling equilibrium) to their welfare given a higher proprietary cost of L' (which yields a separating equilibrium). With a proprietary cost of L, expected welfare is:
(26) (1 - [[alpha].sub.P])(0.5([x.sub.1] + [x.sub.2]) - L) = (0.5([x.sub.1] + [x.sub.2]) - L - k).
because in a pooling equilibrium, [[alpha].sub.P] = k/(0.5([x.sub.1] + [x.sub.2]) - L). With a higher proprietary cost of L', the expected welfare in the resulting separating equilibrium is:
(27) 0.5 x [(1 - [[alpha].sub.1])[x.sub.1] + (1 - [[alpha].sub.2])([x.sub.2]-L')] = 0.5 x [([x.sub.1] - k) + ([x.sub.2] - L' - k)].
(27) is greater than (26) if and only if:
(28) (0.5([x.sub.1] + [x.sub.2]) - L - k) < 0.5 x [([x.sub.1] - k) + ([x.sub.2] - L' - k)] [??] L > 0.5L'.
So, if and only if L > 0.5L', increasing L to L' benefits the first-generation shareholders.
Proof for Corollary 1
Consider the mixed-strategy equilibrium conjectured in Corollary 1(a)(ii). The firm's expected payoff [[pi].sub.ji] from observing [x.sub.j] and disclosing [x.sub.i] will be [[pi].sub.ji] = (1 - [[alpha].sub.i])([x.sub.j] - [[mu].sub.i]L) = (1 - (k/[P.sub.i])([x.sub.j] - [[mu].sub.i]L) for i, j = 1,2, where the market price [P.sub.i] = [r.sub.1i][x.sub.1] + [r.sub.2i][x.sub.2] - [[mu].sub.i]L for i = 1,2, reflects the markets' consistent beliefs in the conjectured mixed-strategy equilibrium. After realizing [x.sub.2], the firm will follow a mixed-disclosure strategy only if:
(29) [[pi].sub.21] = [[pi].sub.22].
Given the assumption that [k.sup.R] [member of] ([x.sub.1], ([x.sub.1] + [x.sub.2])/2), the rival's equilibrium strategy will be to enter with probability [[mu].sub.1] [greater than or equal to] 0 and [x.sub.2] = 1 for all values of [[rho].sub.21], given that [[rho].sub.11] = 1. Applying these strategies for the rival in the conjectured equilibrium yields:
(30) [[pi].sub.21] = (1 - [[alpha].sub.1])([x.sub.2] - [[mu].sub.1]L) = (1 - k/[P.sub.1]) ([x.sub.2] - [[mu].sub.2]L) = (1 - k/[r.sub.11][x.sub.1] + [r.sub.21][x.sub.2] - [[mu].sub.1]L) ([x.sub.2] - [[mu].sub.2]L); and
(31) [[pi].sub.22] = (1 - [[alpha].sub.2])([x.sub.2] - [[mu].sub.2]L) = (1 - k/[P.sub.2]) ([x.sub.2] - L) = (1 - k/[x.sub.2] - L) ([x.sub.2] - L) = [x.sub.2] - L - k.
Given that [[pi].sub.21] = [[pi].sub.22] in equilibrium, equating the RHS's of (31) and (30) yields:
(32) [x.sub.2] - L - k = (1 - k/[r.sub.11][x.sub.1] + [r.sub.21][x.sub.2] - [[mu].sub.1]L) ([x.sub.2] - [[mu].sub.1]L).
Using Bayesian updating, [r.sub.21] = 0.5[[rho].sub.21]/(0-5[[rho].sub.11] + 0.5[[rho].sub.21]) = [[rho].sub.21]/([[rho].sub.21] + 1) (because in the conjectured equilibrium [[rho].sub.11 ]= 1); and using the result that [[mu].sub.1] < 1 in (32) yields: (18)
(33) [x.sub.2] - L - k = (1 - k/[[x.sub.1] + [[[rho].sub.21]/[[rho].sub.21] + 1] ([x.sub.2] - [x.sub.1]) [[mu].sub.1]L).
Allowing for the rival also to adopt mixed strategies (when no pure strategies exist in equilibrium), the rival's equilibrium indifference between entering and not entering implies that [r.sub.21][x.sub.2] + [r.sub.11][x.sub.1] = [k.sup.R], which yields:
(34) [[rho].sub.21]/[[rho].sub.21] + 1 = [k.sup.R] - [x.sub.1]/[x.sub.2] - [x.sub.1],
where [[rho].sub.21], and [[mu].sub.1] are determined as the solutions to (34) and (33), respectively. We next confirm the conjecture that [[rho].sub.11] = 1. This requires that:
(35) k + (1 - [[alpha].sub.1]) x ([x.sub.1] - [[mu].sub.1]L) > k + (1 - [[alpha].sub.2]) x ([x.sub.1] - L).
Subtracting k and dividing both sides of (35) by (1 - [[alpha].sub.1]) ([x.sub.1] - L) gives:
(36) ([x.sub.1] - [[mu].sub.1]L)/([x.sub.1] - L) > (1 - [[alpha].sub.2])/(1 - [[alpha].sub.1]).
From (29) we know that:
(37) (1 - [[alpha].sub.2])/(1 - [[alpha].sub.1]) = ([x.sub.2] - [[mu].sub.1]L/[x.sub.2] - L).
Using the result [[mu].sub.1] < 1 we get d([x.sub.2] - [[mu].sub.1] L)/([x.sub.2] - L)/d[x.sub.2] < 0, and therefore, ([x.sub.1] - [[mu].sub.1]L)/([x.sub.1] - L) > ([x.sub.2] - [[mu].sub.1]L)/([x.sub.2] - L) = (1 -[[alpha].sub.2])/(1 - [[alpha].sub.1]). Thus, (37) implies that (36) always holds, implying that the firm will always report [x.sub.1] after realizing [x.sub.1], as conjectured. Equilibria in other cases can be characterized on the lines of the proof for Proposition 1.
Proof for Corollary 2
Part (a): First, we establish the results relating to the sensitivity of [L.sup.*]. Based on (7), define the function:
[phi]([L.sup.*],k,[x.sub.1],[x.sub.2]) [equivalent to] ([x.sub.1] - k) - ([x.sub.2] - [L.sup.*] - k/[x.sub.2] - [L.sup.*]) ([x.sub.1] - [L.sup.*]).
Then by the implicit function theorem, d[L.sup.*]/dk = - dk/d[phi](L.sup.*],k,*)/d[L.sup.*] Differentiating [phi]([L.sup.*],k,*) with respect to k and [L.sup.*] yields, respectively:
(38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(39) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Using (38) and (39) yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Along similar lines, the implicit function theorem yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Next using (14), define the function:
(40) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Using the implicit function theorem in a similar manner yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Similarly, using (23), define the function:
(41) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Using the implicit function theorem yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Part (b): Using (10), define the function of [phi]([bar]L,k,[x.sub.1],[x.sub.2]) [equivalent to] [bar]L - k ([x.sub.2] - [x.sub.1]/[x.sub.1]) and apply
the implicit function theorem to obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Proof for Lemma 1 and Proposition 2 (sketch)
Now assume that shareholders can sue the firm for misreporting. First, we provide Lemma 2, which establishes pricing equations for the equilibria examined in subsequent proofs.
Lemma 2: Assume that shareholders can sue the firm for misreporting. Then: (19)
(a) The firm's market price in a separating equilibrium or in a mixed-strategy equilibrium where the firm overreports sometimes and where [[sigma].sub.i] assumes an interior value is:
(42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(b) The firm's market price in a pooling equilibrium or in a mixed-strategy equilibrium with overreporting and where [[sigma].sub.i] = 1 is:
(43) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Proof for Lemma 2 (a) In any given equilibrium, using the result that k = [[alpha].sub.i][P.sub.i] in the pricing Equation (1) yields:
(44) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
For a given report, [x.sub.i], the above equation implies:
(45) [[alpha].sub.i][P.sub.i] = [[alpha].sub.i] ([summation over X([x.sub.i])] [r.sub.ji][x.sub.j] - [[mu].sub.i]L) + [[sigma].sub.i] ((1 - [[alpha].sub.i]) [summation over X([x.sub.i])] [r.sub.ji][D.sub.ji] - [k.sup.s]),
where X([x.sub.i]) = {x[absolute value of [rho]([x.sub.i]]x) > 0}. Next, the second-generation shareholders' expected payoff from suing with probability [[sigma].sub.i] following a report of [x.sub.i] is:
(46) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In the first line above, the probability of suing, [[sigma].sub.i], is multiplied by the sum of three terms. The first term is the second-generation's share of the firm's outcome net of damages and proprietary costs. The second term is the damages to be paid to second-generation shareholders, and the third term is the cost of suing. If the shareholders' suing strategy is in the interior, then set the derivative of (46) with respect to [[sigma].sub.i] equal to zero to determine the firm's optimal disclosure strategy in a mixed-strategy disclosure equilibrium:
(47) (1 - [[alpha].sub.i]) [summation over X([x.sub.i])] [D.sub.ji][r.sub.ji] = [k.sup.s].
Letting [r.sub.ji] = 0.5[[rho].sub.ji]/(0.5[[rho].sub.ji] + 0.5[[rho].sub.j'i]) for j,j',i = 1,2 and j [not equal to] j', the firm's equilibrium disclosure strategy, [[rho].sub.ji], is derived as the solution to (47). (20) Substituting from (47) in (45) results in the damages dropping out of the pricing equation, yielding:
(48) k = [[alpha].sub.i] ([summation over X([x.sub.i])] [x.sub.j][r.sub.ji] - [[mu].sub.i]L) = k/[P.sub.i] ([summation over X([x.sub.i])] [x.sub.j][r.sub.ji] - [[mu].sub.i]L).
Multiplying both sides of (48) by [P.sub.i]/k establishes Lemma 2(a). In a separating equilibrium, the firm is never sued because it always reports truthfully, and therefore, the term containing damages again drops out of the pricing equation leaving [P.sub.i] = [[summation of].sub.j][r.sub.ji][x.sub.j] - [[mu].sub.i]L.
(b) Next, consider a pooling equilibrium. In the pricing equation (45), the upper bound on [k.sup.s], together with the sequential rationality of the shareholders in deciding whether to sue, yields ((1 - [[alpha].sub.2]) [summation over X([x.sub.2])] [r.sub.j2][D.sub.j2] - [k.sup.S]) [greater than or equal to] 0, so that second-generation shareholders will always sue ([[sigma].sub.2] = 1). Then, using the result that k = [[alpha].sub.2][P.sub.2] in (45) and dividing both sides of (45) by [[alpha].sub.2] yields:
(49) [P.sub.2] = ([summation over X([x.sub.2])] [r.sub.j2][x.sub.j] - [[mu].sub.2]L) + [[sigma].sub.2][(1 - [[alpha].sub.2])[r.sub.12][D.sub.12] - [k.sup.s]]/[[alpha].sub.2].
Next, substituting [D.sub.12] = [[alpha].sub.2] ([P.sub.2] - ([x.sub.1] - [[mu].sub.1]L)) and [[alpha].sub.2] = k/[P.sub.2] in (49) gives:
(50) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The pricing equation in the case of a mixed-strategy equilibrium with [[sigma].sub.i] = 1 follows the lines of the pooling equilibrium above. (21) This completes the proof for Lemma 2.
Part 1 of Lemma 1
Assume that [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2) and [x.sub.2] [greater than or equal to] [x.sup.*.sub.2] = ([x.sup.2.sub.1]/k) + [x.sub.1].
Lemma 1.1(a) Separating Equilibrium: We establish the existence of a separating equilibrium for all L [greater than or equal to] [L.sup.*]. Given that the rival's payoffs are identical to the firm's payoff and the assumption that [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2), the rival will enter only following a report of [x.sub.2] in a separating equilibrium. A separating equilibrium requires that after realizing [x.sub.1] and [x.sub.2], respectively, the following relations must hold:
(51) k + (1 - [[alpha].sub.1])([x.sub.1] - [[sigma].sub.1][D.sub.11]) [greater than or equal to] k + (1 - [[alpha].sub.2])([x.sub.1] - [[sigma].sub.2][D.sub.12] - L); and
(52) k + (1 - [[alpha].sub.2])([x.sub.2] - [[sigma].sub.2][D.sub.22] - L) [greater than or equal to] k + (1 - [[alpha].sub.1])([x.sub.2] - [[sigma].sub.1][D.sub.21]).
In a separating equilibrium consistent beliefs require that shareholders do not sue, i.e., [[sigma].sub.1] = [[sigma].sub.2] = 0. Further, [D.sub.11] = [D.sub.22] = 0, so (51) and (52) reduce, respectively, to:
(53) k + (1 - [[alpha].sub.1])[x.sub.1] [greater than or equal to] k + (1 - [[alpha].sub.2])([x.sub.1] - L); and
(54) k + (1 - [[alpha].sub.2])([x.sub.2] - L) [greater than or equal to] k + (1 - [[alpha].sub.1])[x.sub.2].
These expressions are identical to (4) and (8), the corresponding expressions in the proof of Proposition 1 [no litigation], respectively. Therefore, the threshold levels, [L.sup.*] and [bar]L, with litigation are identical to those without litigation, as determined by equations (7) and (10), respectively. As established in the proof of Proposition 1, [x.sub.2] [greater than or equal to] ([x.sup.2.sub.1]/k) + [x.sub.1] [??] [bar]L [greater than or equal to] [x.sub.1], and thus for all L [member of] [[L.sup.*], [x.sub.1]), a separating equilibrium exists.
Lemma 1.1(b) Pooling Equilibrium: As before, we assume that, in a pooling equilibrium, both firm types report [x.sub.2] and that investors interpret any off-equilibrium report ([x.sub.1]) as coming from a firm with an unfavorable outcome ([x.sub.1]). Thus, a pooling equilibrium requires that when [x.sub.1] is realized:
(55) k + (1 - [[alpha].sub.2]) x ([x.sub.1] - [[sigma].sub.2][D.sub.12] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x ([x.sub.1]),
where [[alpha].sub.2] = k/[P.sub.2], with [P.sub.2] as determined in Lemma 2(b). This price reflects investors' and the rivals' consistent beliefs in a pooling equilibrium given the assumption that [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2). Similarly, [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] represents the investors' off-equilibrium belief that the adverse report comes from the low-type firm. Because [D.sub.11] = 0, damages drop out of the RHS of (55). Substituting in (55) for [[alpha].sub.1] and [[alpha].sub.2] and subtracting k from both sides yields:
(56) ([P.sub.2] - k/[P.sub.2]) x ([x.sub.1] - [[sigma].sub.2][D.sub.12] - L) [greater than or equal to] ([x.sub.1] - k).
Let L be the solution to:
(57) ([[P.sub.2](L) - k]/[P.sub.2](L)) x ([x.sub.1] - [[sigma].sub.2][D.sub.12] - L) = ([x.sub.1] - k)
where [P.sub.2](L) = (0.5([x.sub.1] + [x.sub.2]) - L) + (0.5 x (1 - (k/[P.sub.2](L)))([P.sub.2](L) - [x.sub.1]) - ([k.sup.s][P.sub.2](L)/k)), using the results from Lemma 2(b) and the conditions in Lemma 1.1 (b). Because the LHS of (56) can be shown to decrease in L, the inequality (56) will be met for all L < L. Finally, comparing (14) and (57), Claim 9 (Claims 7-9 appear at the end of this proof) establishes that L < L** < L*. This proves Proposition 2.1(a).
Similarly, after realizing [x.sub.2], a pooling equilibrium requires that:
(58) k + (1 - [[alpha].sub.2]) x ([x.sub.2] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x ([x.sub.2] - [[sigma].sub.1][D.sub.21]),
where (as before) [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] and [[alpha].sub.2] = k/[P.sub.2], with [P.sub.2] as determined in Lemma 2(b). Given the off-equilibrium belief that the report [x.sub.1] came from a firm with an unfavorable outcome, not suing ([[sigma].sub.1] = 0) is the optimal off-equilibrium response; and inequality (58), after some algebraic manipulation, reduces to:
(59) ([[P.sub.2] - k]/[P.sub.2]) x ([x.sub.1] - L ([x.sub.1]/[x.sub.2])) [greater than or equal to] ([x.sub.1] - k).
If (56) is satisfied, then (59) will be also be met because LHS (59) > LHS (56), while the RHS's are the same. Therefore, a pooling equilibrium will exist for all L < L < L**. Finally, from the proof of part (a), no pure-strategy separating equilibrium can exist for L < L.
Lemma 1.1(c) Mixed-Strategy Equilibrium: Because we know from parts (a) and (b) that for L [member of] (L, L*) no pure-strategy equilibrium exists, we now conjecture a mixed-strategy equilibrium in which the firm discloses honestly after realizing [x.sub.2], but follows a mixed-disclosure strategy after realizing [x.sub.1]. From the proof of Lemma 2 (see (47)), assuming that the shareholders' suing strategy is in the interior, we can establish that:
(60) ([[alpha].sub.1] - [[alpha].sub.2])[D.sub.12][r.sub.12] = [k.sup.s].
With [[rho].sub.22] = 1 in the conjectured equilibrium, Bayesian updating gives [r.sub.12] = 0.5[[rho].sub.12]/ (0.5[[rho].sub.12] + 0.5[[rho].sub.22]) = [[rho].sub.12]/([[rho].sub.22] + 1). Substituting this value for [r.sub.12] in (60) yields [[rho].sub.12], the firm's equilibrium reporting strategy as the solution to (60).
From the assumption that [k.sup.R] [member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2), we have [[mu].sub.1] = 0 and [[mu].sub.2] = 1 for all values of [[rho].sub.12], given that [[rho].sub.22] = 1. Based on the investors' consistent beliefs and the rival's entry strategy in the conjectured mixed-strategy equilibrium and using the prices as determined by Lemma 2, we have [[alpha].sub.1] = k/[P.sub.1] = k/[x.sub.1] and [[alpha].sub.2] = k/[P.sub.2] = (k/[[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L]). Thus, the first-generation shareholders' expected payoff from reporting [x.sub.1] given [x.sub.1] (excluding k) is:
(61) [[pi].sub.11] = (1 - [[alpha].sub.1])[x.sub.1] = (1 - k/[x.sub.1]) [x.sub.1] = [x.sub.1] - k.
Next, we have:
(62) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and because the firm will randomize over reports [x.sub.1] and [x.sub.2] only if the reports yield the same expected payoffs, it must be that [[pi].sub.11] = [[pi].sub.12]. Substituting the value of [r.sub.12] = [[rho].sub.12]/([[rho].sub.12] + 1) in (62) and equating (62) and (61) yields:
(63) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Given [[rho].sub.12] as defined by (60), the value of [[sigma].sub.2] that solves (63) defines the shareholders' suing strategy after observing [x.sub.2]. (22)
After realizing [x.sub.2], for the firm to report [x.sub.2] with probability one ([[rho].sub.22] = 1) it must be true that:
(64) k + (1 - [[alpha].sub.2]) x ([x.sub.2] - L) [greater than or equal to] k + (1 - [[alpha].sub.1]) x [x.sub.2],
where [[sigma].sub.1][D.sub.12] has been dropped on the RHS of (64) because [[sigma].sub.1] = 0, given the off-equilibrium belief that a report of [x.sub.1] came from the low-type firm. Similarly, [[sigma].sub.2][D.sub.22] is dropped on the LHS of (64) because [D.sub.22] = 0. Rearranging (64) yields:
(65) ([[r.sub.12][x.sub.1] + (1 - [r.sub.12])[x.sub.2] - L - k]/ [[r.sub.12][x.sub.1] + (1 - [r.sub.12][x.sub.2] - L]) ([x.sub.1] - L ([x.sub.2]/[x.sub.2])) [greater than or equal to] ([x.sub.1 - k).
Because LHS (65) > LHS (63), while the RHSs are the same, the inequality in (65) is also met, implying that the firm will always report [x.sub.2] after realizing [x.sub.2], as conjectured.
Finally, comparing the LHS of (63) to the RHS of (23), in order for both expressions to equal ([x.sub.1] - k) with [[sigma].sub.2][D.sub.12] > 0 (see Claim 7), it must be that [[rho].sub.12] in (63) is strictly less than [[rho].sub.12] in (23). That is, the probability of false disclosure must be smaller when shareholder litigation is feasible. This establishes Proposition 2.1(b).
The existence of the equilibria in all other cases can be established in a similar fashion. We next provide Claims 7-9 without proofs. (However, proofs are available from the authors.)
Claim 7
[k.sup.s] < ([([x.sub.2] + [x.sub.1] - 2L - 2k)]/[[x.sub.2] + [x.sub.1] - 2L]) ([k([x.sub.2] - [x.sub.1])]/[([x.sub.1] + [x.sub.2]) - 2L]) [??] [[sigma].sub.2] > 0 in a pooling equilibrium.
Claim 8
[[rho].sub.12] in (63) is strictly less than [[rho].sub.12] in (23); i.e., the firm overreports less frequently with litigation.
Claim 9
L < [L.sup.**].
This concludes the proof for Lemma 1. The firm's equilibrium disclosure behavior, as characterized by Lemma 1, yields the results in Proposition 2.2 and Proposition 2.3. This concludes the proof for Proposition 2.
We thank Sunil Dutta, Ronald Dye, Jerry Feltham, Esther Gal-Or, Vasu Krishnamurthy, Ken Lehn, Michael Mikhail, Arijit Mukherji, Katherine Schipper, Martin G. H. Wu, the seminar participants at the University of British Columbia, Duke University, the University of Illinois at Chicago, the University of Pittsburgh, and the Fifth Annual Conference on Financial Economics and Accounting at the University of Michigan, and particularly, two anonymous reviewers and Associate Editor Richard Sansing. Professor Sridhar is grateful to the Accounting Research Center at Northwestern University for financial assistance.
Submitted September 1999 Accepted February 2002
(1) In further contrast to Trueman (1997), market prices and litigation damages are endogenous in our model.
(2) Although we describe the manager as directly observing a perfect signal about the outcome, the manager's private information could also reflect any event that affects her estimate of the value of the firm. Examples of such events include an ongoing merger negotiation, development of a new product or process, obtaining a new order, geological determination of underground resources, and a private contract of strategic significance such as a joint venture.
(3) For simplicity we describe L as a fixed parameter, but none of our analysis would change if the proprietary loss, L, were stochastic and drawn from a support of (0,[x.sub.1]), provided the realized value of the proprietary loss, L, were publicly observable before the firm made its disclosure decision.
(4) Because all parties know that the manager always observes the firm's outcome (x), nondisclosure is another possible report that the firm could make in a pooling equilibrium without litigation. In such an environment, nondisclosure and a pooling equilibrium produce the same posterior beliefs about the firm's outcome because investors have no basis for updating their beliefs in either case.
(5) The threshold is [x.sup.*.sub.2] = ([x.sup.2.sub.1]/k) + [x.sub.1]; e.g., for k = 0.5[x.sub.1], [x.sup.*.sub.2] = 3[x.sub.1].
(6) For certain off-equilibrium beliefs, a pooling equilibrium could also exist in the setting specified in Proposition 1(a). However, common reasonableness checks on off-equilibrium beliefs such as the Cho and Kreps' (1987) intuitive criterion eliminate such a pooling equilibrium. Likewise, using a criterion that focuses on the more informative equilibrium would also eliminate the pooling equilibrium as being less informative than the truthful reporting equilibrium.
(7) Although the underlying intuition for voluntary disclosures in our analysis is similar to the rationale in the signaling literature, where a firm uses a costly signal to communicate its type, important distinctions also arise. First, in the signaling literature, the informed party makes a decision imposing the cost of the signal on itself, whereas in our model the uninformed party (the rival firm) makes a decision that imposes a cost on the informed party (the incumbent firm). Second, multiple users responding to the firm's disclosure in our setting can render the costs and benefits imposed on the firm stochastic, whereas the self-imposed cost of signaling is deterministic in most signaling models.
(8) Furthermore, this pooling equilibrium is the unique pure-strategy disclosure equilibrium for all [k.sup.R] [not member of] [[x.sub.1], ([x.sub.1] + [x.sub.2])/2), except when ([x.sub.1] + [x.sub.2])/2 [less than or equal to] [k.sup.R] < [x.sub.2]. When ([x.sub.1] + [x.sub.2])/2 [less than or equal to] [k.sup.R] < [x.sub.2] and L [member of] [[L.sup.*], Max{L,[x.sub.1]}), a separating equilibrium also exists, where [x.sub.2] [greater than or equal to] [x.sup.*.sub.2] [??] [bar]L [greater than or equal to] [x.sub.1].
(9) To ensure a positive probability of suing in the event of misreporting, we assume that the cost of suing is bounded from above,, i.e., that [k.sup.s] < (([x.sub.2] + [x.sub.1] - 2L - 2k)/([x.sub.2] + [x .sub.1] - 2L)) (k([x.sub.2] - [x.sub.1]))/(([x.sub.1] + [x.sub.2]) - 2L).
(10) Lemma 2 in Appendix B establishes equilibrium prices with litigation.
(11) While most shareholder lawsuits involve alleged overreporting, some lawsuits have also alleged that the firm has understated future economic prospects. For example, the widely cited Basic Incorporated v. Levinson, 108 S.Tr. 978 (1988) case involved Basic's denial that certain value-enhancing transactions had taken place (Davidson et al. 1993).
(12) Without a rival firm, a mixed-strategy equilibrium could still exist in which the firm would randomize over reports of [x.sub.1] and [x.sub.2] after realizing [x.sub.1] and the shareholders would sue the firm with a strictly positive probability after observing [x.sub.2]. Such a mixed-strategy equilibrium would produce more frequent misreporting than the truthful-reporting equilibrium produced by product market competition alone under conditions described in Proposition 1(a).
(13) Corollary 1(a)(iii) identifies an exception to this general rule. As shown in Figure 2, when the favorable outcome is not large (i.e., if [x.sub.2] < [x.sup.*.sub.2]) and the proprietary cost (L) increases from Region 3 to Region 4, the firm's disclosure shifts from truthful reporting in Region 3 to underreporting with a strictly positive probability in Region 4.
(14) If the firm underreports in equilibrium, then the sellers (from the first-generation shareholders) potentially receive the damages and the investors (or buyers of these shares) receive no damages in equilibrium, because it is only the sellers and not the investors who suffer from underreporting in equilibrium. In this case, the expressions in the pricing equation in (1) can be modified to be consistent with equilibrium underreporting.
(15) The true price is based on the legal premise under Rule 10b-5 that all disclosures must be complete and accurate. Hence, the true price reflects the result that in a first-best setting the rival firm would act only when the firm's terminal dividend exceeded its cost to enter, [k.sup.R]. Hurd and Wagner (1990, 563) define true value as the "value that would be observed but for the fraud."
(16) This mixed-strategy equilibrium exists for all values of [k.sup.R] such that [x.sub.1] [less than or equal to] [k.sup.R] < [r.sub.12][x.sub.1] + [r.sub.22][x.sub.2]. Because [[rho].sub.12] < 1, ([x.sub.1] + [x.sub.2])/2 < [r.sub.12][x.sub.1] + [r.sub.22][x.sub.2], and hence this mixed-strategy equilibrium exists for all [x.sub.1] [less than or equal to] [k.sup.R] < ([x.sub.1] + [x.sub.2])/2, as in Proposition 1.
(17) The proofs are straightforward and are available from the authors.
(18) To demonstrate that [[mu].sub.1] < 1 in the conjectured equilibrium, we establish that assuming [[mu].sub.1] = 1 produces a contradiction. That is, substituting [[mu].sub.1] = 1 in (32) and subtracting ([x.sub.2] - L) from both sides yields - k = - (([x.sub.2] - L)/[[x.sub.1] + [r.sub.21]([x.sub.2] - [x.sub.1]) - L])k. This equation cannot hold because the fraction in parentheses is greater than 1 as our conjectured equilibrium reporting strategies imply that [r.sub.21] < 1. The contradiction implies that [[mu].sub.1] < 1.
(19) The following analysis is based on the firm's overreporting in equilibrium, in which case the aggrieved party is the second-generation shareholders who buy the shares at an exaggerated value. When the firm underreports in equilibrium, the sellers would be the aggrieved party and the following analysis can be easily adapted to address such underreporting in equilibrium. To save space, we do not provide the details of determining the pricing equation when the firm underreports in equilibrium.
(20) The shareholders' optimal suing strategy in a given equilibrium is established below in a subsequent part of the proof for Lemma 1.
(21) The existence of well-specified pricing equations and damages can also be established from the previous analysis.
(22) If [k.sup.s] is such that the shareholders' suing strategy is not in the interior (i.e., if [[sigma].sub.2] = 1), the firm's equilibrium disclosure strategy is determined as the solution to (63) with [[sigma].sub.2] = 1 and [P.sub.2] determined as in Lemma 2(b). When [[sigma].sub.2] = 1, the argument for the firm's disclosing [x.sub.2] after realizing [x.sub.2] follows the lines provided below, with the modification that [P.sub.2] is as determined in Lemma 2(b).
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