Voluntary Disclosure of Disaggregated Earnings Information

Voluntary Disclosure of Disaggregated Earnings Information Xiaohong L i u a * and Suil P a e b * "Hong Kong University of Science and Technology bSungkyunkwanUniversity

Received July 2004; Accepted September 2005

Abstract This paper provides an equilibrium analysis of a setting in which a manager is required to disclose GAAP earnings, but has discretion over the provision of information about earnings components that are disaggregated according to their persistence or value relevance. We find that the manager is more likely to disclose disaggregated earnings information when GAAP earnings fall. We derive intuitive comparative static results concerning the impacts of firm characteristics on the manager's equilibrium disclosure choice. We also discuss how the paper's main results are linked to empirical findings on pro forma earnings disclosures.

JEL Classification: M41 Keywords: pro-forma reports; disaggregated earnings information; GAAP earnings; persistence; voluntary disclosure.

1. Introduction Earnings on the basis of generally accepted accounting principles, known as GAAP earnings, provide information about firms' future prospects and thus explain stock prices. Mandatory disclosure of GAAP earnings, however, may not be a sufficiently effective

* The authors appreciate helpful suggestions and comments from Chih Ying Chen, Peter Chen, Jack Hughes, and the seminar participants at the Hong Kong University of Science & Technology (HKUST), the University of British Columbia, and Sungkyunkwan University. Special thanks to Dan Simunic (the editor) and an anonymous reviewer of this paper. The authors also thank the Direct Allocation Grants of HKUST for financial support.

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mechanism for managers to communicate details of information demanded by investors because GAAP earnings aggregate many earnings items that could have different value implications (see, e.g., Lipe, 1986; Ohlson and Penman, 1992). In this regard, managers contend that reporting earnings on a pro forma basis is to overcome such a deficiency of GAAP earnings, because pro forma reports provide information about “core” earnings that is an improved metric for future cash flow (e.g., Alpert, 2000; Bray, 2001; Weil, 2001).‘ In response to recent proliferation of pro forma reports, critics note that investors should be alert to pro forma earnings since that information could be misleading (e.g., Derby, 2001; Dreman, 2001; SEC, 2001). The objective of this paper is to provide an equilibrium analysis of a setting where managers (i) have private information about the components of GAAP earnings that have different persistence and thus different implications for firm value, and (ii) exercise discretion over whether or not to release detailed information about the earnings components according to their persistence on top of GAAP earnings. By conducting a rigorous analysis of this setting, we seek to provide insights into recent debates on pro forma earnings reporting and, more generally, into voluntary disclosure of disaggregated earnings information. To link our study to pro forma reporting, we first note that when managers report pro forma earnings, they in effect disaggregate GAAP earnings into two groups on the basis of earnings persistence and label them “pro forma (core) earnings” and “excluded (non-core or unusual) items”. That is, managers separate individual earnings items comprising GAAP earnings into “pro forma” group and “excluded items” group according to their assessment of each earnings item’s persistence, and in this process, managers may disaggregate an earnings item further if that item contains recumng and non-recurring subcomponents.2 In this sense, pro forma reporting can be viewed as a voluntary disclosure of disaggregated GAAP earnings on the basis of earnings persistence. Also observe that individual earnings items’ persistence is determined by the nature of business transactions that give rise to those items. To the extent that managers in the course of conducting business obtain more knowledge about those transactions (because of their superior knowledge about the operating characteristics of the firms and industries), managers could better assess the likelihood of the recurrence of those transactions and their value implications. Hence, it is reasonable to assume that managers

I Reporting pro forma earnings is not the only way for firms to get around the limitation of GAAP earnings. More detailed financial and non-financial information is often disseminated through Management Discussion & Analysis, press releases, and meetings with analysts (e.g., Brown and Kim, 1993; Clarkson et al., 1994; Bryan, 1997; Hutton et al., 2003).

Observe that whether or not an earnings item is persistent depends on the likelihood of that item’s recurrence in the future. Also note that a restructuring charge disclosed in a firm’s financial statements could be a sum of many types of restructuring charges whose persistence might differ.

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have private information relative to the general public about the persistence of individual earnings component^.^ Another important aspect of reporting pro forma earnings is that it involves significant costs. The SEC issued numerous comments and guidelines on the use of non-GAAP (pro forma) earnings metrics; see, e.g., Accounting Series Release 142 (1973), Carnal1 (1996), and FEI / NIRI guidelines recommended by the SEC (2001).4Recently, pursuant to Section 401 (b) of the Sarbanes-Oxley Act of 2002, the SEC (2003) formalised and imposed additional reporting requirements on firms releasing pro forma earnings information. Specifically, the SEC requires that firms filing earnings releases and documents containing pro forma earnings (i) provide detailed reconciliations between GAAP and pro forma earnings, (ii) justify the estimation methods employed, and (iii) explain in detail the reasons for and usefulness of pro forma reporting, including why pro forma earnings measure is beneficial to investors and additional purposes for which firms use this measure: Given the complexity of disclosure regulations - the rules specifying requirements run over 50 pages - and the burden of proof on the part of firms to justify their pro forma earnings, it is not surprising that many firms created “disclosure committees” after the enactment of the Sarbanes-Oxley Act of 2002 to help them comply with disclosure regulations.6 In addition to direct disclosure costs to comply with regulations as noted above, firms are likely to incur significant indirect disclosure costs because the information that is

In this regard, it must be emphasised that this paper’s main results remain unchanged in so far as investors have residual uncertainty about the firm’s earnings persistence. Indeed, note that the persistence of an earnings item could differ across time and firdindustry characteristics. This means that investors even after reading a firm’s financial statements, footnotes, and other related public disclosures are likely to have difficulty in extracting as precise information as that of the firm’s manager about pro forma (or core) earnings. For example, a restructuring charge that is disclosed in a firm’s income statement and related footnotes does not warrant its exclusion from pro forma earnings unless the firm’s manager has reasons to believe that the impact of the charge on the firm’s future cash flow is immaterial. In this sense, we view our assumption of information asymmetry as realistic. Bhattacharya et al. (2003) provide evidence supporting this view. They find that 65% of their sample firms, which both report pro forma earnings and are covered by IBES, have the same pro forma earnings and “street earnings” (which refer to analysts’ estimates of the firms’ core /operating earnings). This evidence suggests that even analysts, who have more expertise and information than ordinary investors for assessing firms’ future prospects, rely heavily on managers’ pro forma disclosures in forming their estimates. According to a survey by National Investor Relations Institute (January 2002), 90.2% of the surveyed companies complied with the FEI / NIRI guidelines on pro forma reporting. The survey result can be found at FEI’s (Financial Executive International) website, http://www.fei.org.

For instance, if a gain or loss is excluded from pro forma earnings calculation but there was a similar item that occurred within two prior years, then firms should provide at least five reasons to justify and demonstrate the usefulness of their pro forma earnings. For details of the requirements, see Item 10(e) of Regulation S-K (SEC, 2003) and the FAQ section on the SEC’s website http://www.sec.gov/divisions/corpfin/ faqs/nongaapfaq.htm#item 1Oe. According to a survey by National Investor Relations Institute (January 2003), 85% of the 386 polled companies have established disclosure committees in response to Sarbanes-Oxley Act. The disclosure committee is responsible for reviewing the company’s SEC fillings, earnings releases, conference call scripts, and other material disclosure issues. The survey result is available at http://www.niri.org/irresource-pubs/alerts/ ea20030 129.pdf.

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required to supplement a pro forma earnings release could be highly proprietary. For example, firms reporting pro forma earnings are required to disclose not only the details of business transactions related to each of the earnings items excluded from pro forma earnings calculation, but also the types of internal decisions based on this pro forma measure such as planning activities and performance evaluation. In sum, firms reporting pro forma earnings in essence choose to supplement aggregate GAAP earnings with information about earnings persistence, thereby providing more value-relevant information to investors. In doing so, firms incur disclosure costs. To conduct an analysis of this setting in a tractable manner, we present a stylised model, in which a firm’s manager seeking to maximise investors’ current valuation of the firm is privately informed of two earnings items, y , and y,, that comprise the firm’s aggregate (GAAP) earnings y = y , + yz. The earnings item y , is more persistent than the other item y,, so that y,’s impact on the firm’s future cash flow is greater than that of y2. While the manager is required to disclose the aggregate earnings y , she has discretion over the disclosure of individual earnings components, i.e., (y,, y J . Releasing detailed earnings information to the public, however, gives rise to a disclosure cost, and thus, the manager weighs that cost against the benefit of disclosure in her voluntary disclosure decision. As shown in the analysis, the benefit comes from avoidance of under-valuation of the firm. Our main results are as follows. First, we establish that there exists a unique equilibrium, in which the manager’s decision on whether to disclose detailed earnings information (yI, y 2 ) is contingent on the aggregate earnings y. More specifically, the equilibrium is characterised by a threshold function such that for a given aggregate earnings y , the manager reveals the highly persistent earnings item y1 if and only if it exceeds the value of the threshold function for the given aggregate earnings y . Observe that disclosing y, is the same as disclosing both y1 and y2 because the aggregate earnings y = y , + y 2 is public information. Second, it is shown that the equilibrium threshold function increases with the aggregate earnings. This implies that, ceteris paribus, the manager is more likely to reveal the highly persistent earnings item y, when the aggregate earnings y is low than high. Intuitively, given the positive association between y and yl, a decrease of the aggregate earnings y when the earnings item y, is undisclosed, translates into investors’ inference that y , is low. Such a downward inference of the highly persistent earnings item y , in turn motivates the manager to reveal y1 to avoid under-valuation of the firm that would arise otherwise. Third, we provide intuitively appealing comparative static results on the impacts of firm characteristics on the manager’s equilibrium disclosure decision. We show that the manager tends to provide more detailed earnings information (i.e., disclose y , ) when the volatility of the firm’s core earnings increases and its aggregate earnings become low. That is, low aggregate earnings and a large variance of core earnings have a reinforcing effect on the manager’s incentive to release detailed earnings information. We also show that for any given level of the aggregate earnings, the manager is more likely to disclose detailed earnings information when the difference in the persistence of earnings components increases. Finally, it is shown that an increase of disclosure costs motivates the manager to withhold private information about earnings components. Our model is closely related to Verrecchia (1983, 1990) in the sense that the presence of disclosure costs (associated with pro forma reporting) sustains partial disclosure equilibrium for a given level of the aggregate GAAP earnings. By introducing further structure

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into Verrecchia’s model, the principal novelty of this paper is to examine pro forma reporting from the perspective of voluntary disclosure of disaggregate earnings information, and especially, to investigate how the equilibrium varies with the GAAP earnings and with other parameters of the model, such as the divergence of the persistence of the earnings components and the volatility of the earnings items.’ Our results shed light on recent debates and empirical findings on pro forma reporting. The former SEC chairman (Pitt, 2001) has commented that managers could have legitimate incentives to communicate value-relevant information that may not be available from GAAP earnings disclosures, such as the persistence of individual earnings components. Without such information, mispricing could arise and that could motivate managers to release their private information through pro forma earnings reporting although doing so might be costly. If such asymmetry of information does exist, then pro forma earnings could provide incremental information to the market in so far as investors rationally take into account managers’ disclosure incentives. Empirical evidence is consistent with this argument. That is, pro forma earnings are found to have incremental explanatory power for stock returns over GAAP earnings (e.g., Bradshaw and Sloan, 2002; Brown and Sivakumar, 2003; Bhattacharya et al., 2003). While market reactions to pro forma disclosures could be attributed to investors’ cognitive bias or limited rationality (see Frederickson and Miller, 2004; Hirshleifer and Teoh, 2003), our analysis offers an alternative explanation by fully rationalising those reactions in the presence of information asymmetry between managers and investors concerning earnings persistence. It is also documented that pro forma announcers tend to be firms suffering GAAP losses, and pro forma earnings are on average greater than their corresponding GAAP earnings (Bhattacharya et al., 2003; Bradshaw and Sloan, 2002; Lougee and Marquardt, 2004). While these findings could be attributed to managers’ opportunistic behaviour to mislead investors, our study offers an alternative explanation that the findings could be a result of managers’ legitimate disclosure incentives because the benefit of disclosing disaggregated earnings information increases when aggregate (GAAP) earnings fall. Furthermore, our analysis of the impact of earnings volatility, in conjunction with the levels of aggregate earnings, offers an intuitive explanation for Bhattacharya et al.’s (2003) finding that firms reporting pro forma earnings tend to be high-tech firms that frequently report GAAP losses. We provide more detailed discussions of our results and their links to empirical findings in sections 3 and 4. The remainder of this paper is organised as follows. Section 2 presents the model. Sections 3 and 4 characterise the equilibrium and provide comparative static analyses. Section 5 concludes the paper.

’

While the settings are different, this paper also shares the spirit of Kirschenheiter (1997), Hughes and Pae (2004), and Pae (2005) in the sense that the analysis focuses on investors’ inference about undisclosed information based on disclosed information. Such an inference, of course, is a rational reaction to information asymmetry and thus must be fully reflected in the analysis.

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2. Model Consider a firm trading in a competitive capital market, where investors are risk neutral and have a zero discount rate for their future payoffs. The firm’s manager seeks to maximise investors’ current valuation of the firm, whose future liquidating value is represented by a random variable x.The manager is required to make a public disclosure of the firm’s aggregate earnings, which are denoted by y . As explained below, we assume that individual earnings items comprising the aggregate earnings y have different value implications. For simplicity, we let Y =YI + Y ,

and assume that each earnings item yi, i E ( 1,2}, is independent and follows a normal distribution, i.e., yi

- N ( p , T ~for) all i and Cov[y,,y,] = 0.8

In addition, assume that the firm’s future liquidating value x is related to the earnings items b,,y,) in the following manner:

-

where p, > p, 2 0 are known constants and E is pure noise with E N(0, 0:).Note that the magnitude of the coefficient pito each earnings item yi indicates how much impact each earnings item has on the firm’s liquidating value x, and p, is assumed to be strictly greater than p,; i.e., y , has a bigger impact than y,. This means that y , is a more persistent earnings item than y2in the sense that y , has a greater likelihood of recurrence in the future than yz. Also note that we allow 0, to be 0, in which case y , is a completely transitory earnings item (i.e., has no impact on the firm’s future value). Given the properties of the earnings components (y,, y,) and their relations to the firm’s future liquidating value, we consider the following sequence of events. The manager privately observes ( y , , y,). While it is mandatory for the manager to release the aggregate earnings y to the market, she has discretion over the provision of more detailed earnings information in the sense that she may or may not release the earnings items (y,, y,) on top of y . As is commonplace in the voluntary disclosure literature (e.g., see Verrecchia, 1983, 1990; Kirschenheiter, 1997; Hughes and Pae, 2004), we assume that if the manager chooses to disclose detailed information about the earnings items, she does so truthfully; in other words, while she can choose not to disclose her private information about the earnings

*Assuming independence does not affect our results materially; in particular, it can be shown that all of our results remain qualitatively unchanged under the assumption that y , and y 2 are positively correlated, i.e., Covb,. Y,l 2 0.

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items, she cannot disclose it ~ntruthfully.~ Also observe that since y = y , + y2 and y must be disclosed, the market, given a disclosure of either one of y1 and y2, can infer perfectly the remaining undisclosed earnings item. Thus, we hereafter focus on the manager’s decision on her voluntary disclosure of y l , and assume that the manager withholds y1 if she is indifferent between disclosing and withholding it. Finally, we assume that making detailed earnings information public is costly (in the sense of Verrecchia, 1983);i.e., the firm incurs a disclosure cost, denoted by c > 0, if it releases its earnings item yl. The model structure is common knowledge. Before proceeding to the analysis of the model, a remark on how to interpret the model in the real world context is in order. Note that the manager’s disclosure of yl could be viewed as “pro forma reporting”. As explained above, releasing y1 is equivalent to reporting y , and yz separately, rather than aggregating them into y that could be viewed as the firm’s GAAP earnings. In this sense, y , and y , represent “pro forma (or core) earnings” and “excluded (or non-core) items”, respectively, given their differing persistence or value relevance.’O Notice that whether or not investors observe the entire earnings items in the firm’s financial statements does not affect this interpretation. For instance, we could let (zl, z,, ..., z,) be the earnings items disclosed in the income statement and say that the manager decomposes y = Zizi into y I = ZjEcEzjand yz = Ck e N C E z k , where C E and NCE represent the set of core and non-core earnings items, respectively, which is the manager’s private information.

3. Equilibrium This section examines the manager’s voluntary disclosure decision concerning the earnings item yl, conditional on the aggregate earnings y that she has to release. Suppose that the manager discloses y , in addition to y. Since disclosing Cy, y , ) is equivalent to disclosing Cyl, y,), investors’ expectation of the firm value in this case equals

While we do not explicitly consider in the model, regulatory disclosure guidelines and potential litigation threats may justify this assumption because they may effectively deter managers from providing misleading information. For example, in the context of pro forma earnings reporting, firms are required to justify and reconcile differences between pro forma and GAAP earnings in accordance with disclosure guidelines (SEC, 2003). Furthermore, antifraud provisions of the federal securities laws also apply to firms issuing pro forma earnings (SEC, 2001). For example, the SEC has brought enforcement actions against Trump Hotels & Casino Resorts Inc. regarding its inconsistent pro forma disclosures. Details about this legal case can be found at Fenwick & West LLP’s website at http://www,fenwick.com. See Section 5 for more discussions related to this assumption. l o This interpretation is consistent with managers’ contention that pro forma reporting provides a clearer picture of firms’ core operations by excluding earnings items that are expected to be non-recurring or less persistent in the future. Doyle et al. (2003) provide concrete evidence for differing persistence of “pro forma earnings” versus “excluded items”: they report that for their sample firms $1 of “pro forma earnings” ($1 of “excluded items”, respectively) in a quarter predicts $7.895 ($4.567) of future cash flows from operations over the next three years.

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where the last equality follows from the relation y2= y - y , . On the other hand, suppose that the manager withholds y , . Investors in this case form an expectation of the firm value by using only the aggregate earnings y. In doing so, they take into account the manager's incentive to withhold detailed earnings information (i.e., the incentive to suppress y,). Denoting N D b ) to be the nondisclosure set of the earnings item y , conditional on the aggregate earnings y , we note that investors' expectation of the firm value equals

where

and f(y, I y ) is the density function of y , conditional on y . Comparing the firm values stated in (2) and (3), we define

to be the benefit of disclosing the earnings item y , along with the aggregate earnings y. When deciding whether to disclose y , , the manager weighs that benefit against the cost of disclosure, and chooses to suppress y , if and only if the benefit is less than the cost, i.e., only when B(y, y , ) I c holds. The next proposition establishes the existence of a unique equilibrium.

Proposition 1. For any given aggregate earnings y E (-m, m), there exists a unique equilibrium characterised by a threshold z(y) such that the manager withholds y , 3 and only 3 y , E N D ( y )= (-m, z ( y ) ] , where z(y) is the value of z that solves

To explain the equilibrium disclosure threshold conditional on the aggregate earnings y," we first note that for any nondisclosure set ND(y), the benefit function B b , y , ) defined in (4) is strictly increasing in y , . This implies that if the manager's nondisclosure set exists, then it must be characterised by a threshold z such that the manager chooses not to disclose y , if and only if y , I z. Since the manager with y , = z is actually indifferent between disclosure and nondisclosure, it must be true that B(y, z ) = c for this indifferent

I ' While for simplicity we denote the equilibrium threshold by ~ ( y )it , is clear from equation (5) that the equilibrium threshold depends not only on y hut also on (PI, p,, c) and the parameters characterising the distribution of earnings. We will examine how the parameters other than y affect z(y) in section 4.

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manager; i.e., ( 5 ) must hold. The next question is whether or not such a threshold z exists. We show in the appendix that for any given y, (i) B(y, z ) > 0 for all z, and (ii) B(y, z ) is a strictly increasing function of z with B(y, z ) + 0 as z + -m. These two properties of the benefit function B(y, z ) imply that there exists a unique value z satisfying the equilibrium condition (3,which is denoted by z(y). The intuition for the equilibrium is as follows. Consider a manager who disclosed y and has private information about y,. Given that N D b ) = (-m, z ( y ) ] ,we can restate the benefit of disclosing y, stated in (4) as

Recall that this expression represents the difference in investors’ expectation of the firm value x with and without the manager’s disclosure of the earnings item y,. Therefore, B(y, y,) being positive (negative) means that the firm is under-valued (over-valued) when y, is undisclosed relative to when it is disclosed. Given this relation, it is useful to partition the possible values of y, into three intervals as depicted in Figure 1 and explain the manager’s disclosure decision. Figure 1 The cost and benefit of disclosing the earnings item y , and the manager’s equilibrium disclosure decision for a given aggregate earningsy.

disclosure cost c.

First, consider the case where y, is high enough to exceed z(y). In this case the manager seeking to maximise investors’ valuation of the firm reveals y, because y, > z ( y ) implies that the benefit from disclosing y, is greater than the disclosure cost c, i.e., B(y, y,) > c.I2 In other words, the manager chooses to reveal y, and thereby bear the disclosure cost, because if y, were undisclosed, then under-valuation would be too severe. Therefore, in equilibrium, the firm is fairly valued at E[x I y, y,] less the disclosure cost c; i.e., its value equals V(y,y,) stated in (2). Second, consider the case where y, is intermediate in that it

Recall that B(y, y , ) = c > 0 at y , = z(y) and B(y, y , ) is strictly increasing in y,.

Iz

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Journal of Contemporary Accounting & Economics Vol I , No 2 (Dec 2005) I47-I70 belongs to the interval ( E [ y , I y , y , 5 z ( y ) ] , z ( y ) ] . Clearly, we know from (4’) that B(y, y , ) > 0 here, which means that with no disclosure of y , the firm is under-valued. However, the magnitude of under-valuation in this case is not so large as to justify the disclosure cost, i.e., B(y, y , ) 5 c. As such, the manager suppresses yI and the firm is valued at V(y, N D ( y ) ) stated in (3). Finally, consider the manager with y , < E [ y , I y , y , I z(y)]. Since B(y, y , ) < 0 in this case, the firm is over-valued if yI is undisclosed. This means that, even without any disclosure cost, the manager has no incentive to release detailed earnings information, which results in the firm value equal to V(y,ND(y)) stated in (3). Note that in equilibrium investors, given no disclosure of y , , cannot distinguish between the case where the manager suppresses y, because it is too low (i.e., below E[y, I y , y , I z(y)]), and the case where the manager withholds relatively high y , (but not exceeding zb)) because of her incentive to avoid the cost of disclosure. Taking expectation of y , across those two cases, investors’ posterior expectation of the undisclosed earnings item y, conditional on y equals E [ y , I y , y , I z(y)l. It is evident from the above discussion that for a given aggregate earnings y , the partial disclosure equilibrium characterised by Proposition 1 is qualitatively similar to that in Verrecchia (1983, 1990). That is, only when the highly persistent earnings item y , exceeds a threshold value, the manager releases that information. In our subsequent analysis, we will examine how the equilibrium nondisclosure set of y , conditional on the aggregate earnings y vanes with y as well as the other parameters, which is the main focus of this paper.

Proposition 2. The equilibrium threshold z(y) is increasing in the aggregate earnings y. Specifically, z ( y ) is a linear function of y with its slope coefficient being equal to o,’/(o,’+ o;),where o,’ is the variance of the earnings item y,, i E { 1,2}. Figure 2 The equilibrium disclosure threshold function zo).

The shaded region is the nondisclosure set of the earnings item Y , The manager’s disclosure decision depends on the level ot the aggregate earnings y For example, y, = 9 , is disclosed when y = y”, but is undisclosed when y = )

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Since the upper bound of the nondisclosure set (4, z(y>]changes with y , managers with the same y , may exhibit different preferences toward disclosure depending on the level of the aggregate earnings y , Proposition 2 offers insights into this issue by characterising how the equilibrium threshold z ( y ) changes with y . In particular, as shown in Figure 2, z(y) is a linear increasing function of y and the shaded region represents the nondisclosure set of the earnings item y , . The monotonicity of the threshold function z(y) implies that when the aggregate earnings y becomes higher (lower), the manager is motivated to suppress (release) the earnings item y I that would be disclosed (undisclosed) otherwise. For instance, consider the earnings item y , = 9, in Figure 2. That item is disclosed when the aggregate earnings y equals y". However, the manager chooses to withhold it when the aggregate earnings become higher, say, y = yb. To explain the intuition for the monotonicity of z b ) , recall that z(y) is the level of y , at which the manager is indifferent between disclosing and withholding y , , as characterised by the equilibrium condition (5). Differentiating (5) with respect to y yields

The first term is negative ( i . We .Y , z> L < 0). That is, with z being fixed, the benefit of aY disclosure decreases (i.e., under-valuation of the firm in the case of nondisclosure becomes less severe) as the aggregate earnings y increases. This occurs because investors' posterior expectation of the undisclosed earnings item y , conditional on y , i.e., E[yl I y , y , I z],becomes higher when y increases, which is intuitive given that y and y , are positively related (see the appendix for a formal proof). Such a reduced benefit of disclosure translates into an increase of the threshold z because the indifferent manager now strictly prefers nondisclosure, thereby saving her disclosure cost. Notice that an increase of the threshold, however, has a positive impact on the benefit of disclosure; that is,

az

> 0 in the second term of (6). As a result, the new equilibrium disclosure thresh-

old, say, z' corresponding to a higher level of the aggregate earnings must exceed the old equilibrium threshold z, so that the decrease in the benefit of disclosure induced by an increase of y (the first term) is offset by the increase in the benefit of disclosure due to the increase of the threshold (the second term), i.e., (6) holds. We now let pUp be the mean of the disclosed earnings item y , , i.e.,

and let p; be the mean of the aggregate earningsy whose component y , is disclosed, i.e.,

where z-' is the inverse function of the equilibrium threshold function z.

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Corollary 1. (a) The mean of the earnings item y , that is disclosed is greater than its prior mean: i.e.,

Pf

’P,.

(b) The mean of the aggregate earnings y whose component y , is disclosed is less than its prior mean: i.e., pD < py= p,+ p2, (c) Ifthe prior mean i f the earnings item y2 is zero, i.e., if& = 0, then pf > p;.

Parts (a) and (b) are immediate consequences of the fact that the equilibrium threshold

z is an increasing function of y. To explain part (a), we return to Figure 2. Fix any y and observe that the mean of the earnings item y , disclosed exceeds its mean conditional on the given aggregate earnings y, i.e.,

where the inequality follows because the left-hand side is the mean of the earnings item y , that is truncated below by the threshold &). Taking expectation with respect to the aggregate earnings y in both sides of the above inequality preserves the inequality, i.e.,

For part (b), first note that monotonicity of the threshold function z implies that its inverse function z-’ is increasing in the earnings item y,. Thus, in Figure 2, fix any y , and observe that the mean of the aggregate earnings y for which the manager reveals the given earning item y l , is less than its mean conditional on the given earnings item y , , i.e.,

where the inequality holds because the left-hand side is the mean of y that is truncated above by the threshold z-’(y,). Taking expectation with respect to the earnings item y, in both sides of the above inequality establishes part (b), i.e.,

Finally, part (c) follows immediately from parts (a) and (b): if p 2 = 0, then

We now discuss how our results in Proposition 2 and Corollary 1 are related to recent empirical findings on pro forma reporting. Empirical regularities documented by numerous studies include: (i) firms are more likely to report pro forma earnings when their GAAP earnings fall; (ii) pro forma earnings announcers are less profitable than average firms in terms of GAAP earnings; and (iii) pro forma earnings are on average greater than their corresponding GAAP earnings (see, e.g., Bhattacharya et al., 2003; Bhattacharya et al., 2004; Bradshaw and Sloan, 2002; Brown and Sivakumar, 2003; Lougee and Marquardt, 2004). Based on a conventional wisdom that firms with disappointing GAAP earnings

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have an incentive to manipulate financial reports opportunistically, these findings are often regarded as evidence that firms use pro forma earnings reports to manipulate the market perception of the performance of firms. It is also documented that (iv) the market reacts to pro forma earnings incrementally to GAAP earnings. One explanation for the market reaction to pro forma earnings is that investors are not fully rational or have cognitive bias, and thus firms attempting to mislead the market are successful even though pro forma earnings report per se may have no intrinsic informational value (e.g., Hirshleifer and Teoh, 2003; Frederickson and Miller, 2004). Turning to our model, as mentioned in section 2, disclosing y , could be interpreted as releasing a pro forma earnings report where earnings components that have different persistence are disclosed separately in accordance with regulatory guidelines on disclosure. With this interpretation, our results in Proposition 2 and Corollary 1 are consistent with the aforementioned four empirical findings, but offer alternative explanations without appealing to investors' limited rationality or cognitive bias. That is, in our model, all market participants are fully rational. In the presence of information asymmetry concerning the earnings components whose value implications differ and consequential mispricing of the firm in the case of nondisclosure, the manager seeking to maximise the market valuation of the firm has a legitimate incentive to voluntarily disclose detailed earnings information in spite of disclosure costs, and investors are fully aware of that incentive and react accordingly. More specifically, consistent with the empirical finding (i), Proposition 2 implies that the manager is more likely to release pro forma earnings (i.e., disclose y , ) when the aggregate (GAAP) earnings y becomes lower. l 3 The reason is that when y falls, the benefit from disclosing y , becomes larger. As explained earlier, such an increased benefit comes from the fact that by revealing the highly persistent earnings item yI separately in the face of falling aggregate earnings y , the manager avoids severe under-valuation that would be borne by the firm otherwise. Also note that the empirical finding (ii) is consistent with Corollary 1 (b). Furthermore, if the prior mean of the less persistent earnings item y 2 is zero (i.e., if y2= 0), then the empirical finding (iii) is consistent with Corollary 1 (c). That is, the average pro forma earnings y , disclosed is greater than the average GAAP earnings y of the firms that release pro forma earnings, although these two earnings metrics apriori have the same average values for the population firms (i.e., if p2= 0, then pv= y,).Finally, our analysis suggests that the empirical finding (iv) - i.e., the market reactions to pro forma earnings are incremental to GAAP earnings - could be consistent with investors' rational behaviour, in so far as they take account of managers' incentives to communicate private information about differing earnings components that have heterogeneous value implications.

l 3 That is, as illustrated in Figure 2, a manager with the earnings item y, = 9, chooses to withhold it if the aggregate earnings y equals yb, but discloses it if y equals yo.

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4. Comparative Static Analysis We have shown in the previous section that a threshold function z(y) uniquely characterises the manager’s equilibrium disclosure choice. In this section, we will examine how the equilibrium threshold function z(y) is affected by the exogenous parameters of the model. In particular, since the tradeoff relation between the cost and benefit of disclosure specified in (5) determines ~ ( y )we , focus on the parameters affecting that tradeoff.

Proposition 3. The equilibrium threshold z(y) has the following properties: (a) zb)is increasing in the disclosure cost c; (b) z(y) is decreasing in p, and increasing in p,; be two levels of the variance of yI where o:H> of,,and let zj(Y) be (c) Let ofHand the equilibrium threshold associated with a:? jE { H, L } . Then, there exists a cutoff value of the aggregate earnings y denoted by y* such that z,(y) < z,(y) $ and only $

Y p,; i.e., the earnings item y , is more persistent so that its impact on the firm value is bigger than that of the earnings item y,. As evident from the equilibrium condition (3,this implies that the benefit of disclosure B(y, z ) increases when p, (p,) increases (decreases); i.e., under-valuation in the case of nondisclosure becomes more severe when PI and p, become farther apart. Consequently, the manager is motivated to release y, despite the accompanying disclosure cost, which leads to a lower equilibrium threshold z(y). To sharpen the intuition further, consider the case where the earnings items y, and y2 are highly homogeneous in terms of their persistence; i.e., the magnitudes of p, and p, are almost the same. To illustrate, we fix p, and let p, + p, in (5). Since B(y, z ) + 0 for any given p,, there is no disclosure benefit in this case. The reason for why the disclosure benefit vanishes as p, + p, is that the aggregate earnings y alone provides sufficient information about the firm value; in other words, note in (2) and (3) that

when 0, + &.I4In the presence of the disclosure cost, and given that there is no differential effect of the earnings items on the firm value (which means no disclosure benefit), it follows that z(y) + M, i.e., yI is never disclosed.

~

l4

The notation “=” denotes “approximately the same”.

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Parts (a) and (b) could also be seen from the fact that the slope of the equilibrium threshold function z(y) is solely characterised by the variances of the earnings items (see Proposition 2). This fact implies that any changes of the disclosure cost c and/or the earnings persistence parameters p, and p, affect only the intercept of z b ) , so that any changes of (c,PI,p,) lead to apara2ZeZ shift of the threshold function z(y). Combining this property with the equilibrium condition (3,it is easy to see that an increase of c or p, (or a decrease of p,) results in an upward parallel shift of z(y). In this regard, observe that for any y and no matter what (c, PI,p,) is, z(y) always exceeds investors’ posterior expectation of the undisclosed earnings item y1 conditional on the aggregate earnings y , i.e., E [ y , I y , yl I z(’y)];see Figure 1. However, the threshold z(y) may be greater or less than investors’ prior expectation of the undisclosed earnings item yi conditional on the aggregate earnings y , i.e.,

In particular, viewing piI, as a function of y , we note that ,ulyhas the same slope as that of z(y) and thus is parallel to z(y). Hence, the comparison of z(y) and p,lydepends on the magnitudes of (c, p,, p,) that affect the intercept of z(y) but not the intercept of P , , ~It. can be shown that z0,) lies above pIiy for ally if the disclosure cost c is sufficiently large, or if the difference in the earnings persistence (pi - p,) is sufficiently small. We now turn to part (c), a comparative static result concerning the impact of the variance of the highly persistent earnings item y , on the equilibrium threshold. Basically, it is stated that a large variance of y , and a low aggregate earnings y reinforce each other in motivating the manager to release detailed earnings information; i.e., she is more likely to disclose yl when its variance is large and y is low, relative to when its variance is small and y is high. While mathematically this result follows directly from the fact that an increase of y,’s variance 6;changes the threshold function z(y) to be steeper (which is obvious from Proposition 2), the reasons for why that occurs and how that change is linked to the manager’s disclosure incentives are subtle and require careful explanations. To illustrate, we refer to Figure 3 depicting two threshold functions corresponding to two different values of of,i.e., o:H> o:~.

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Figure 3 The effect of a change in the variance of the highly persistent earnings item y , on the equilibrium.

The equilibrium threshold function zjy) is associated with the variance of earnings item y, denoted by a;,,where j E { H, L] and oyH> o : ~Note . that A belongs to the nondisclosure region , does not when a: when a: = o ; ~but = The converse is true for B.

Y

We first explain why a larger value of 0:leads to a steeper threshold function. Recall that the equilibrium depends on how much information investors can infer about the undisclosed earnings item y , from the mandatory disclosure of the aggregate earnings y , because that information determines the benefit of disclosure B(y, z ) as specified in the equilibrium condition (5). Furthermore, the amount of that information depends on the aB aiUII, level of y and the magnitude of 0:.To be precise, first observe that - = __ - , aY aPlly aY where plly = E [ y , Iy] is given by (9). This relation indicates that, with z being held constant, a change of y affects the benefit of disclosure B (y, z ) for the threshold firm (i.e., the firm with y , = z ) only through the market expectation of y , conditional on y , namely plly. Second, as shown in the proof of Proposition 2 (see the appendix), we have

-

aB

< o . This means that the decrease of B b , z ) due to an increase of pllrexactly

a h " az offsets the increase of B(y, z ) due to an increase of 2. Using the two properties of B(y, z ) explained above, we can rewrite (6) as aB

< 0 , the above equation implies that in equilib-

3 4 I>, rium y's effect on the threshold z must be the same as its effect on pl,, (i.e., -= 1. aY aY

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Now recall that, as stated in (9), pl,, has a slope coefficient

0:

163

, which increases

~

0;+0;

with

0:. It thus follows that the equilibrium threshold z

the aggregate earnings y when 0:is large than when as

is more responsive to a change of

0:is small;

az

i.e., - becomes larger aY

0:increases.

We next explain how the change in the slope of the threshold function induced by a change of 0:affects the manager’s disclosure incentives across different levels of y . Consider the two points A and B depicted in Figure 3. When 0: = o:~,A belongs to the nondisclosure region, but it does not when 0:= c ~ : ~ >~7:~.Conversely, B is in the disclosure region when 0:= C T : ~but , it belongs to the nondisclosure region if 0:increases to ~7:~. It is clear that we can always find points like A and B depending on whether or not the aggregate earnings y exceeds the cutoff value y*.To explain the link between the change of 0:and the manager’s disclosure incentives, suppose that yl’s variance is sufficiently large relative to the given variance of the earnings item y,. Then, the variation of the aggregate earnings y = y1 + y z (from its prior mean p, E p, + p 2 )is more likely to be caused by changes of y , relative to changes of y2. This implies that investors, after observing a realisation of the aggregate earnings y, infer the undisclosed earnings item y, more confidently when 0:is sufficiently large rather than small (relative to the variance of y2).15 Now consider the point A where the aggregate earnings y is low. In this case, if yl’s variance is sufficiently large, then it is more likely that the low aggregate earnings y is due to a low realisation of the earnings item y,. Since y1 has a bigger impact on the firm’s future cash flow, investors’ inference of a low y , results in a more negative impact on the valuation of the firm. This in turn motivates the manager with (y, y,) = A , who suppresses y , when 0:is small, to release yl and thereby avoid the severely negative impact on the firm value. Using the same argument, it is easy to see that the incentive effect of a change of 0:when the aggregate earnings y is high (e.g., the manager with (y, y , ) = B ) is reversed. In sum, when yl’s variance 0:becomes larger, the aggregate earnings y becomes more informative about the undisclosed earnings item yl. And given that y and y , are positively correlated, the manager’s incentive to disclose (withhold) the highly persistent earnings item y , is intensified when the aggregate earnings y falls (rises).16 Our comparative static analysis provides insights into empirical findings on managers’ voluntary disclosure of detailed earnings information. According to a survey by National Investor Relations Institute, 16.5% of the polled companies, 99 out of 600, said that they have changed from reporting both GAAP and non-GAAP (pro forma) earnings to

Is Note that the slope coefficient a: / (a:+ a :) in (9), which increases with a:, is the square of the correlation coefficient of y and y,, a well-known measure of the association between two variables. l6 Since the slope coefficient of z(y), i.e., a : / (a:+ a:), decreases with y2’s variance 0:.it is easy to see that a change of a’,has an effect on the manager’s disclosure incentive that is the opposite of the effect of a change of 0:.

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reporting only GAAP earnings after the SEC’s (2003) new regulatory rules came into effect (see Plitch, 2003). Part (a) of Proposition 3 suggests that such a decreased use of pro forma reports is a rational equilibrium behaviour in response to a change in the regulatory environment, because that change translates into a significant increase in direct and indirect disclosure costs (see our discussions in Section 1 for details about the impacts of new regulations on disclosure costs). Regarding industry characteristics of pro forma announcers, while Bhattacharya et al. (2003) provide descriptive evidence that firms reporting pro forma earnings tend to be high-tech firms that frequently report GAAP losses, they provide little e~p1anation.l~ Our result in part (c) of Proposition 3 could be used to explain this finding because it predicts that, ceteris paribus, high volatility of core earnings combined with disappointing aggregate (GAAP) earnings provides a stronger incentive for managers to release more detailed information about core earnings in the form of pro forma reporting. Finally, note that part (b) offers an empirical hypothesis concerning the impact of the heterogeneity of earnings persistence on corporate pro forma earnings disclosure decisions.

5. Summary and Concluding Remarks This study offers an equilibrium analysis of a setting where a manager is better informed about her firm’s individual earnings components whose value implications differ. While disclosing detailed earnings information is costly, the manager has incentives to release that information on top of the firm’s aggregate earnings because doing so has a benefit of avoiding under-valuation. We show that there exists a unique equilibrium, in which the manager releases or suppresses detailed earnings information depending on the level of the aggregate earnings. Also shown is that the manager is more likely to disclose detailed earnings components when the aggregate earnings become lower. The paper then provides a comparative static analysis. Intuitive explanations are offered for the impacts of the variances of earnings items, the divergence of the persistence of earnings items, and disclosure costs on the manager’s equilibrium disclosure incentives. The results established in this paper provide new insights into recent debates on pro forma earnings reporting. In particular, recall that the argument based on managers’ opportunism could explain empirical findings on pro forma disclosures. That is, as noted earlier, managers might be purely opportunistic in reporting pro forma earnings (in the sense that those earnings have no information value incremental to GAAP earnings), and investors might be misled by such behaviour if they have limited rationality or cognitive bias. In contrast with that argument, our analysis offers an alternative explanation on the basis of an informativeness argument, which fully rationalises the empirical findings. That is, when there is information asymmetry between managers and investors concerning the

Similarly, Bhattacharya et al. (2004) and Lougee and Marquardt (2004)document that firms reporting pro forma earnings tend to be concentrated in high-tech industries and are less profitable than average firms.

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components of earnings that have different value implications, managers could have legitimate incentives to communicate value-relevant information in order to avoid mispricing of firms, and investors fully understanding those incentives react accordingly. We believe that the equilibrium analysis we offer in this paper helps us gain more insights into voluntary disclosure of disaggregated earnings information. Our study, of course, is not without limitations. In particular, our analysis is based on the assumption of truthful disclosure; the manager with private information about individual earnings items’ persistence is assumed to release that information truthfully if she chooses to disclose it. While we make this assumption to focus on the issue of disclosure vs. nondisclosure of the earnings components that have different value implications, managers in reality might have opportunistic incentives to misreport pro forma earnings; i..e., the reported core earnings might not be the same as the true core earnings privately known to the manager. Anther limitation of this study is that we only consider a single-period setting and thus have no room to explore strategic intertemporal considerations that firms might have in their pro forma reporting choice. We leave potential manipulation of pro forma earnings in a multi-period setting as a future research topic.

References Alpert, B., 2000, “The numbers game: Reporting of pro forma earnings is rising, and so is the debate about it”, Baryon’s (September 1l), 22-24. Bagnoli, M. and T. Bergstrom, 1989, “Log-concave probability and its applications”, Working paper, University of Michigan. Bhattacharya, N., E. Black, T. Christensen, and C.R. Larson, 2003, “Assessing the relative informativeness and permanence of pro forma earnings and GAAP operating earnings”, Journal of Accounting and Economics 36 (1-3), 285-319. Bhattacharya, N., E. Black, T. Christensen, and R. Mergenthaler, 2004, “Empirical evidence on recent trends in pro forma reporting”, Accounting Horizons (forthcoming). Bradshaw, M. T. and R. G. Sloan, 2002, “GAAP versus the street: An empirical assessment of two alternative definitions of earnings”, Journal of Accounting Research 40 (l), 41-66. Bray, C., 2001, “SEC looking at pro forma earnings with ‘purpose in mind”’, Dow Jones Newswires, November 8. Brown, L. D. and K. J. Kim, 1993, “The association between non-earnings disclosures by small firms and positive abnormal returns” The Accounting Review 68 (3), 668-680. Brown, L. D. and K. Sivakumar, 2003, “Comparing the value relevance of two operating income measures”, Review of Accounting Studies 8 (4), 561-572. Bryan, S. H., 1997, “Incremental information content of required disclosures contained in management discussion and analysis”, Accounting Review 72 (2), 285-301. Carnall, W. E., 1996, “Developments in division of corporation finance”, American Institute of Certified Public Accountants twenty-third annual conference on current SEC developments, Washington, DC, February 16. Clarkson, P. M., J. L. Kao, and G. D. Richardson, 1994, “The voluntary inclusion of forecasts in the MD&A section of annual reports”, Contemporary Accounting Research 11 (l), 423-450.

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Derby, M. S., 2001, “Investors getting more data, but is it the right data?’, Dow Jones Newswires, September 3. Doyle, J. T., R. J. Lundholm, and M. T. Soliman, 2003, “The predictive value of expenses excluded from pro forma earnings”, Review ofAccounting Studies 8 (2-3), 145-174. Dreman, D., 2001, “Fantasy earnings”, Forbes (October l), 134. Frederickson, J. R. and J. S . Miller, 2004, “The effects of pro forma earnings disclosures on analysts’ and nonprofessional investors’ equity valuation judgments”, The Accounting Review 79, 667-686. Greene, W. H., 2003, Econometric Analysis (Pearson Education, Inc.: Upper Saddle River, NJ). Hirshleifer, D. and S. H. Teoh, 2003, “Limited attention, information disclosure, and financial reporting”, Journal of Accounting and Economics 36 (1-3), 337-386. Hughes, J. S. and S. Pae, 2004, “Voluntary disclosure of precision information”, Journal of Accounting and Economics 37 (2), 261-289. Hutton, A. P., G. S. Miller, and D. J. Skinner, 2003, “The role of supplementary statements with management earnings forecasts”, Journal of Accounting Research 41 ( 5 ) , 867-890. Kirschenheiter, M., 1997, “Information quality and correlated signals”, Journal of Accounting Research 35 ( I ) , 43-59. Lipe, R., 1986, “The information contained in the components of earnings”, Journal of Accounting Research 24 (Supplement), 37-64. Lougee, B. and C. Marquardt, 2004, “Earnings quality and strategic disclosure: An empirical examination of ‘pro forma’ earnings”, The Accounting Review (forthcoming). Ohlson, J. and S . Penman, 1992, “Disaggregated accounting data as explanatory variables for returns”, Journal of Accounting, Auditing and Finance 7,553-573. Pae, S., 2005, “Selective disclosures in the presence of uncertainty about information endowment”, Journal ofAccounting and Economics (forthcoming). Pitt, H. L., 2001, “Remarks before the AICPA governing council”, Miami Beach, FL, October 22. Plitch, P., 2003, “More companies say they avoid releasing pro forma statements”, The Wall Street Journal (June 12), C9. Securities and Exchange Commission (SEC), 2001, “Cautionary advice regarding the use of “pro forma” financial information in earnings releases”, Release Nos. 33-8039; 34-45 124; FR-59 (December 4). (http://www.sec.gov/rules/other/33-8039.htm). Securities and Exchange Commission (SEC), 2003, “Conditions for use of non-GAAP financial measures”, Release No. 33-8 176; 34-47226; FR-65 (January 22). (http://www.sec.gov/rules/final/33-8 176.htm). Verrecchia, R. E., 1983, “Discretionary disclosure”, Journal of Accounting and Economics 5 (3), 179-194. Verrecchia, R. E., 1990, “Information quality and discretionary disclosure”, Journal of Accounting and Economics 12,365-380. Weil, J., 2001, “Companies pollute earnings reports leaving P/E ratios hard to calculate”, The Wall Street Journal (August 21), Al.

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Appendix Proof of Proposition 1: It is clear from the discussion below Proposition 1 that for any given y , if a nondisclosure set exists, it must be characterised by a threshold z such that the manager withholds y , if and only if yI Iz. Here we establish that for any given y such a threshold z exists uniquely. In the equilibrium condition ( 5 ) , observe that

where F(. I y) is the cumulative distribution function of y , conditional on y , and the last equality follows from integrating by parts. First, note that B(y, z ) > 0 for all (y, z ) because F(. I y ) > 0 and Dl - P, > 0. Second, fix y and let z vary. Using L‘Hospital’s rule, we have

= lim z+-c

(PI

-

P M z I Y>= lim (P, - PI) V d z I Yl = 0.

f ‘(2 I Y >

z+--

-E[zIYl

:!

Third, for any given y , observe that F(y, I y ) dy, is a log-concave function of z. Then, we know from Bagnoli and Bergstrom (1989) that B(y, z ) is strictly increasing in z, i.e.,

The three properties of B(y, z ) established above imply that for any y and positive c, there exists a unique value z solving B(y, z ) = c. Denoting that value by z ( y ) establishes the equilibrium. Q.E.D.

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Proof of Proposition 2: Recognizing the equilibrium threshold z as a function of y, we differentiate the equilibrium condition (5) with respect to y: aB dB az -+--=o (-4.1) ay az ay Define 0 2

and consider the first term in (A.1). Since B depends on y only through pllv, we have

In addition.

where the second equality and the last inequality are shown in the proof of Proposition 1. Hence, (A.1) can be rewritten as

aB az aPll, - 0: Since -< 0 , we must have -= -- ___ > 0, where the last equality

ay

aPIl,

ay

o;+0;

follows from (A.2). We thus conclude that z is a linear function of y with its slope coefficient being equal to

IT2 - 1

,

~

0:+0;

Q.E.D.

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Proof of Corollary 1: Given the joint normal distribution of (yl, y), we know that the distribution of y, 0;0;

-

conditional on y is yl I y N(plly,o : ~ , )where , pllyis given by (A.2) and o : ~=, ___ . 0;+0;

n(a)

, where q5 and Q, are the standard I -@(a) normal density and cumulative distribution functions, respectively.Then,

Next, define a = z ( Y ) - P i b and

_____

C1l.v

where the inequality follows from the fact that A(a)is positive (see Greene 2003,759, for details). where Similarly, the distribution of y conditional on y1 is y I y , N (pYu, pyli= y1 + p, and o2Yl,= 0,”. Denoting z-I to be the inverse function of the equilibrium

-

E,, [E b I Y , ,y < z-’(Y~)l] = Ey,

Then, P:

- Ovl~17(@>1

= PI + P2 - 0”llEY, [77(@)1
where the inequality follows from the fact that 7 (0) is positive. Finally, if p, = 0, then p I D >p y Dbecause pID>p1and pyD
Proof of Proposition 3: (a) From the equilibrium condition B (y, z ) = c, we have

aZ az ac aB

--= 1.

aB az Since -> 0, we must have -> 0.

aZ

ac

aZ

aB aB (b) From the equilibrium condition B (y, z) = c, we have -+ --= 0

ap, az ap,

as JB az Since -> 0 and - > 0, we must have -< 0 . Similarly, since

az

ap1

aB

aZ

az

ap,

aB

aB

aZ

aZ

ap, >

+ --= 0 , where -< 0 and -> 0, we must have -

-

ap,

aB

ap1

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(c) From Proposition 2, we know that z(y) = S(CT ;,c;) y + k ( a :,o;, /?,,P2,c), where =

s(O

0: ~

0;+0;

as -

where --

ao;

az a0;

-< 0

aZ

as

- -y + and k (.) is a term independent of y . Since --

aq aq

ak

ao;

0;

[ q + q>*0 , there exists a cutoff value y*

if, and only if, y < y*. Q.E.D.

such that