Optimal Disclosure Decisions When There are Penalties for Nondisclosure

Transcription

1 Optimal Dislosure Deisions When There are Penalties for Nondislosure Ronald A. Dye August 16, 2015 Abstrat We study a model of the seller of an asset who is liable for damages to buyers of the asset if, after the sale, the seller is disovered to have failed to dislose an estimate of the asset s value that the seller knew prior to sale. The model yields some surprising preditions onerning how the seller s dislosure deision hanges with hanges in the severity of this liability, and with other parameters of the model, inluding the preision of the estimate and whether the seller behaves myopially or nonmyopially. 1 Introdution This paper ontains an eonomi model of voluntary dislosures where the seller of an asset is presumed to have a duty to dislose whatever private information he has about the asset s value to potential buyers prior to the asset s sale, and where - if the seller violates that duty - the seller may be subjet to damage payments. Thus, the model studies situations in whih the seller s dislosure is mandatory, but the seller s ompliane with the mandatory requirements is voluntary. There are a variety of settings in whih the theory applies, from selling seurities in IPO/investment settings to selling produts in manufaturing/onsumer settings. The model is founded on previous models of a seller s optimal voluntary dislosure behavior, where the premise is that the seller sometimes privately reeives information about an asset s value prior to the asset s sale whih the seller The paper is urrently undergoing revision. Comments welome! 1

2 may dislose or withhold (as in Dye [1985], Farrell [1986], Jung and Kwon [1988], Shavell [1994], Hughes and Pae [2004], Hughes and Pae [forthoming], among others). In these prior works, when the seller makes no dislosure, potential buyers of the asset are unsure whether the seller did not reeive information about the asset prior to sale or whether he eleted to withhold information that he reeived. This unertainty regarding the reason for the seller s nondislosure allows a seller who reeived unfavorable information about the asset s value to "pool" with sellers who reeived no pre-sale information about the asset s value by not making a dislosure. After the sale, if buyers disover that the seller withheld information, then buyers will know that they overpaid for the asset based on the information in the seller s possession. In the prior literature, buyers who made suh a disovery were posited to have no reourse. The innovation in the present paper is that we presume that buyers do have reourse: sellers aught having previously withheld information are obliged to pay the buyers damages. We study a base model along with two variants. In the base model, the asset being sold is indivisible and the seller annot take any ations that influene the subsequent value of the asset to buyer. In one variant, the asset ontinues to be indivisible but the seller an enhane the asset s value to buyers by taking some ation prior to sale; in the seond variant, the seller an also enhane the asset s value prior to sale, but the asset is divisible and the seller an deide what fration of the asset to retain for his own use. In both the base model and its variants, we assume that the seller s penalties for withholding take the natural form of being proportional to the buyer s overpayment for the asset, where "overpayment" is alulated based on the differene between what buyers paid for the asset and what they would have paid for the asset had the seller dislosed his information. The proportionality fator that determines what (possibly frational) multiple of this overpayment onstitutes the damages penalty is a parameter of the model that we refer to as the "damages multiplier." 2

3 In the base model and its variants, we obtain a variety of results onerning the seller s equilibrium behavior. Among these results, at least four stand out. To desribe the first of these results, all the seller s dislosure deision myopi if he ignores the potential damage payments he is subjet to if he is aught withholding information and instead bases his dislosure deision just on whether the selling prie of the asset is highest with or without dislosure. Call the seller s dislosure deision nonmyopi if in deiding whether to dislose information he reeives, he takes into aount these damage payments. Our first result is that for a wide range of the model s parameter values, the seller s optimal myopi dislosure deision oinides with his optimal nonmyopi dislosure deision. As part of this first result, besides demonstrating the robustness of the oinidene of myopi and nonmyopi dislosure deisions for a wide range of parameter values, we also show that myopi and nonmyopi dislosure poliies oinide both when the seller is subjet to lawbak provisions and when he is not subjet to them. (A "lawbak provision" is a requirement that the seller disgorge any overpayment made to him by buyers of the asset as a onsequene of his withholding, and is a soure of redution in the seller s net reeipts from the asset s sale distint from the damage payments.) In our seond result, we show, also for a wide range of parameter values, that the seller optimally disloses the information he reeives less often as the size of the damages multiplier inreases. One might have expeted that the seller would dislose his information more often as this penalty inreases, but this turns out not to be the ase. In our third result, we show that as the damages multiplier inreases, the seller hooses to sell a larger fration of the asset to buyers. One might have expeted the seller to sell a smaller fration of the asset as the damages multiplier gets larger, sine the seller is not liable for failure to dislose information about portions of the asset that he retains for himself, and so the benefits to the seller of retaining a large stake in the asset would seem to inrease with the seller s liability for nondislosure, but this also 3

4 turns out not to be orret. We defer presenting the intuition for the preeding results to the body of the paper. In the last of the results we highlight here, we show that in the model variant where the seller an both sell just a fration of the asset and undertake ations that enhane the value of the asset prior to sale, the seller optimally retains a smaller stake in the asset as the quality or preision of the estimate he reeives, and sometimes disloses, inreases. Among the four highlighted results, this last is perhaps the most intuitive, sine as the preision of the estimate improves, the estimate provides a more aurate indiator of the seller s pre-sale ativities to enhane the asset s value, and so it beomes less important to rely on the seller s retention of the asset to motivate the seller to undertake the pre-sale value-enhaning ativities. When "the asset" is a firm that an entrepreneur starts up and an inrease in the preision of the dislosed estimate is interpreted as an improvement in the quality of the entrepreneur s firm s finanial aounting reports, this last result suggests the empirially testable impliation that improvements in the quality of aounting information leads entrepreneurs to retain smaller ownership stakes in the firms they found. We dedue many other results besides those highlighted here from the model in the paper, but we leave the presentation and disussion of those results to the main text below. This paper is part of the growing literature that studies firms voluntary dislosure deisions that was initiated by the fundamental ontributions of Grossman [1981] and Milgrom [1981] in their "unravelling" result. This literature is too large to summarize here, but see for example, the surveys of Gertner [1988], Milgrom [2008], and Dranove and Jin [2010]. Pae [2005] proposed an analysis similar to the one ontained here, but Pae did not arry out the analysis. There is also a large empirial literature on voluntary dislosures, surveyed by Healy and Palepu [2001]. The empirial work most losely tied to the present paper is that of Heitzman, Wasley, and Zimmerman [2010] who study empirially firms 4

5 dislosure deisions in the presene of mandatory dislosure requirements. The paper proeeds as follows. The next setion, Setion 2, introdues the model. Setion 3 desribes the seller s preferenes along with a formal speifiation of damages payments. Setion 4 examines the seller s optimal dislosure poliy, and it ontains the general findings regarding the equilibrium dislosure probability mentioned above. Setion 5 extends the model to a setting where the seller s investment hoie is endogenous. Setion 6 ontains a summary of some of our main findings, and the appendix ontains proofs of many of the results not proven in the text or aompanying footnotes. 2 Base model setup In the base model, S (for "seller") has an asset he wants to sell. multiple homogenous potential buyers of the asset. There are The value of the asset is unertain to all of these potential buyers at the time S sells the asset and is given by the realization z of the random variable z, whih is taken to be normally distributed with mean m and variane 1 τ, heneforth written as z N(m, 1 τ ). z s realization ours after the sale. Before the sale takes plae, with probability p (0, 1) S privately reeives an estimate ṽ of z. This estimate ṽ is taken to be unbiased and given by ṽ = z + ε, where ε N(0, 1 ) is independent of z. (1) r Here, r denotes the preision of the estimate ṽ. It is apparent that the prior distribution of ṽ is normal, with mean m and variane σ 2 1 τ + 1 r. In the following, the prior density and df of ṽ are denoted by g(v) and G(v) respetively. With probability 1 p, S reeives no estimate before the sale. Consistent with the now onventional assumptions of the voluntary dislosure literature (see, e.g., Dye [1985] and Jung and Kwon [1988]), we assume that: if S reeives no information, he makes no dislosure; in partiular, S is presumed inapable of redibly dislosing that he did not reeive information; 5

6 and if S disloses information, the dislosure is onfined to be truthful. The new feature of the model is that, after the sale takes plae, if S dislosed nothing prior to the asset s sale, then a fat finder (investor, reporter, auditor, et.) undertakes an investigation and, if the reason S dislosed nothing turns out to be that S withheld information, then the fat finder with probability q (0, 1) both detets and reports that S withheld information along with what the withheld information was. In the latter event, S is fored to pay damages to the buyer of the asset, as desribed below. Also, the fat finder fails to disover that S withheld information with probability 1 q, but the fat finder never wrongly asserts that S withheld information when that was not the ase. 3 Preferenes and damage payments S s dislosure, or nondislosure, is the only soure of information about the asset s value to the asset s potential buyers. and they all behave ompetitively. All the of buyers are risk neutral, Consequently the equilibrium selling prie of the asset is its expeted value based on what potential buyers learn or an infer about the asset s value prior to sale. If S learns v and disloses it, the asset s selling prie is denoted P (v). Sine buyers are risk neutral, this prie P (v) is the asset s expeted value onditional on ṽ = v. By Bayes rule applied to normal distributions (see, e.g., DeGroot [1970]), we know that this onditional expeted value is given by: P (v) E[ z v] = τm + rv. (2) This prie is a weighted average of buyers initial beliefs about the asset s value m and the dislosed estimate v, with the weights the relative preisions of the prior and the estimate. 1 τ τ+r and r τ+r determined by 1 Even though at this point in the expostion we have yet to desribe what onstitutes an equilibrium of the model, it is worth pointing out for future referene that the speifiation of the prie P (v) in (2) impliitly entails making off-equilibrium speifiations, beause this prie is being speified for all v, and not just for those v that S is expeted to dislose in 6

7 We let P nd denote the selling prie of the asset when the seller makes no dislosure. If S withheld v and the fat finder subsequently finds that out, then the differene P nd P (v) onstitutes the amount the buyer overpaid for the asset based on the seller s withheld information. 2 that S has to pay the damages payment In this event, we suppose β (P nd P (v)) (3) to the buyer. "damages multiplier." β is formally what we referred to in the Introdution as the The speifiation of the damages payment does not indiate the full onsequenes to S of getting aught withholding information from buyers, beause it does not address the "lawbak" question. If there is a lawbak, S has to return the overpayment P nd P (v) to buyers, so when S is aught having withheld information, S will reeive net (equivalently, the buyer will pay net 3 ) P (v) β (P nd P (v)). If β > 0, the buyer winds up owning an asset worth (in expetation) P (v) but pays net only P (v) β (P nd P (v)) for it. If β = 0, the buyer ends up paying P (v) for the asset whih is exatly what the asset is worth. If β ( 1, 0), the buyer ends up losing on the purhase, beause he reeives an asset worth P (v) but pays P (v) β (P nd P (v)) > P (v) for it. Nevertheless, in this last event, onditional on having purhased the asset, the buyer is better off having the fat finder detet the seller s withholding (than not having the withholding deteted), beause the buyer reeives a positive amount from the fat finder s efforts. If β = 1, the buyer is stuk with paying P nd for the asset (sine P nd = P (v) β (P nd P (v)) when β = 1) and hene equilibrium. Clearly, the off-equilibrium speifiation of P (v) in (2) is the uniquely natural way to speify P (v) when v is dislosed, regardless of how other faets of the equilibrium are defined. 2 We shall show below that when the seller withholds information, it is the ase that P nd > P (v), so P nd P (v) is indeed an overpayment. 3 The equivalene between amounts paid for by the seller and reeived by the buyer holds only when there is no "slippage" due to attorneys fees or other legal osts. The analysis an be extended to over ases of slippage. 7

8 is no better off than he would have been had the fat finder not deteted S s withholding. In the following, when studying the lawbak ase, we onfine attention to damage multipliers bounded below by β 1 and bounded above so that the inequality q(1 + β) < 1 (4) holds. Additional disussion onerning this bound may be found in the aompanying pair of footnotes. 4 5 Before S knows whether or not he will reeive the estimate ṽ - S s expeted proeeds from the buyer net of his ost of damage payments is given by: E[(1 p)p nd + p max{p (ṽ), (1 q)p nd + q(p (ṽ) β(p nd P (ṽ)))}]. (5) We onlude this setion with a short disussion of the "no-lawbak" ase. In this ase, S keeps the original payment P nd from buyers if he is aught withholding information, and pays the damages payment β(p nd P (v)), and so gets net P nd β(p nd P (v)). For there to be ontent for the damages payment (3) to be a penalty in this ase, the damages multiplier β must be positive. As was true of the lawbak ase, our analysis of the no-lawbak ase also requires imposing an upper bound on β. In this ase, we require β to be positive and that the following (weaker) upper bound hold: qβ < 1. (6) 4 In the ontext of seurities litigation, buyers/investors are typially not made even lose to being made whole by the damages payment when firms are found to have improperly withheld information, but investors usually reeive some payment following a fat finder s disovery that a firm withheld information, so in the seurities litigation ontext, β ( 1, 0). See, e.g., Ryan and Simmons [2009]. 5 The inequality (4) has an intuitive eonomi motivation. If one ombines the lawbak with the penality speified in (3), the total penalty (i.e., drop in prie from P nd ) for S derived from getting aught withholding information is P nd P (v) + β (P nd P (v)) = (1 + β) (P nd P (v)), and sine the fat finder detets S s withholding with probability q, the expeted total penalty is q(1+β) (P nd P (v)). The inequality (4) restrits S s expeted total penalty for withholding to be no larger than the absolute amount by whih the firm s market value is overstated as a onsequene of S s withholding. 8

9 4 The equivalene of nonmyopi and myopi dislosure strategies Suppose the seller reeives information v. As we desribed in the Introdution, we all the seller s dislosure deision myopi if the seller deides whether to dislose v solely based on whether P (v) is bigger or smaller than P nd. That is, the seller s dislosure deision is myopi if the seller disregards the potential damages he is subjet to were he to withhold v and the fat finder subsequently detets his withholding. In ontrast, we all the seller s dislosure deision nonmyopi if he takes into aount these potential damage payments when making his dislosure deision. Our first result below asserts that: Lemma 1 As long as the parameter restrition (4) holds in the lawbak ase, or as long as the parameter restrition (6) holds in the no-lawbak ase, the seller s optimal nonmyopi dislosure deision oinides with the seller s optimal myopi dislosure deision. The proof is simple for both ases. In the lawbak ase, the seller s optimal nonmyopi dislosure entails nondislosure iff (1 q)p nd + q(p (v) β(p nd P (v))) > P (v). (7) Colleting terms in LHS(7) that involve P nd together, and also olleting terms in LHS(7) involving P (v) together, nondislosure is S s best nonmyopi hoie iff:: (1 q qβ)p nd + (q + qβ)p (v) > P (v),or equivalently, iff (1 q qβ)p nd > (1 q qβ)p (v). When the bound (4) holds, this last inequality is obviously equivalent to: P nd > P (v). (8) This proves the lemma in the lawbak ase. 9

10 In the ase of no-lawbak, the seller s optimal nonmyopi dislosure entails nondislosure iff P nd qβ(p nd P (v)) > P (v), i.e., iff P nd P (v) qβ(p nd P (v)) > 0, i.e., iff (P nd P (v))(1 qβ) > 0. (9) Clearly, when the bound 1 qβ > 0 holds, this last inequality is equivalent to: P nd > P (v). This proves the lemma in the no-lawbak ase. The result in both ases obtains beause when damage payments are proportional to buyers overpayments, then the optimal nonmyopi dislosure deision is based on the differene between the "no dislosure" prie of the asset and the prie of the asset had S dislosed his information, and so results in the optimal nonmyopi dislosure deision oiniding with the optimal myopi dislosure deision. 6 To avoid studying a proliferation of ases in the following, for the most part, we onfine attention to the "non-lawbak" ase and, unless the ontrary is expliitly stated, we assume the bound (6) holds. Given those maintained assumptions, it follows, in view of the preeding lemma, that we do not have to provide additional separate analyses depending on whether S is believed to behave myopially or nonmyopially in making his dislosure deision. 6 It is perhaps useful to mention what happens if the multiplier β is so large that exeeds the upper bounds ((4) or (6)) referened above. Consider the no-lawbak ase when qβ > 1. Then aording to (9), the optimal myopi and the optimal nonmyopi dislosure deisions are the opposite of eah other: when S behaves in the optimal nonmyopi fashion, it is optimal for S to dislose v when P (v) is smaller than P nd, and it is optimal for S to withold v when P (v) is larger than P nd. 10

11 5 Determination of the equilbrium in the base model We now turn to a disussion of how the equilibrium no dislosure prie P nd is set. Clearly, if S prefers to dislose rather than withhold v, then S will also prefer to dislose rather than withhold any v > v, and so the nondislosure set is given (for both myopi andor nonmyopi sellers) by a left-tailed interval of the form: or equivalently by: ND {v P nd > τm + rv }, (10) ND = {v v > v}. (11) where v is defined impliitly by the solution to the equation τm + rv = P nd. (12) To speify P nd more preisely, we next alulate buyers pereptions of the expeted value of the asset given that S makes no dislosure and also given that the buyer thinks S uses the utoff v in deiding whether to dislose the private information he reeives. There are three events related to the seller s nondislosure that must be handled separately. (1). The seller did not reeive any information. (2). The seller reeived and withheld information and the fat finder subsequently fails to detet that S withheld information. (3). The seller reeived and withheld information and the fat finder subsequently detets that S withheld information. Sine buyers ex ante pereptions of the probability S will make no dislosure is given by 1 p + pg(v ) (this is the probability S reeives no information plus the probability that S reeives information that is below the utoff v ), buyers an apply Bayes Rule to onlude that the probability S reeived no information, onditional on S making no dislosure, i.e., buyers pereptions of the probability of event (1) as desribed above given 11

12 no dislosure is: 1 p 1 p + pg(v ). (13) Likewise, the buyer s pereptions of the probability of events (2) and (3) above, given S makes no dislosure, are respetively: p(1 q)g(v ) 1 p + pg(v ) (14) and pqg(v ) 1 p + pg(v ). (15) Next notie that, onditional on event (1), buyers expet to reeive an asset whose value is given by E[ z] = m. (16) (Were buyers to know that the reason S made no dislosure was that S reeived no information, buyers would have no reason to make a "lemons" inferene about the asset s value from S s nondislosure, so the asset s expeted value in that event is just the asset s unonditional expeted value.) Buyers pereptions of the asset s expeted value onditional on no dislosure and event (2) is given by: E[ z ṽ < v ] = E[E[ z ṽ] ṽ < v τm + rṽ ] = E[ ṽ < v ]. (17) (If buyers knew S withheld information and that withholding would not be subsequently deteted by the fat finder, buyers would alulate the onditional expeted value of the asset to be E[ z ṽ < v ].) Finally, to alulate buyers pereptions of the asset s expeted value onditional on no dislosure and event (3), first fix a partiular v < v and suppose that buyers knew the fat finder was going to disover that S withheld this partiular v. Then, the value the buyers would attah to purhasing the asset inlusive of the damage payments they expet to reeive in this ase is given by: E[ z v] + β(p nd E[ z v]) = τm + rv + β(p nd τm + rv ). 12

13 So, onditional only on no dislosure and event (3) (but not also onditioning on a partiular v < v ), the value buyers expet to reeive, net of expeted damage payments, is given by: τm + rṽ E[ + β(p nd τ ˆm + rṽ ) ṽ < v ]. (18) Thus, when buyers believes the seller uses the utoff v in deiding whether to dislose the estimate he reeives, the total value buyers expet to reeive, onditional only on no dislosure by the seller, onsists of the sum of the produts of (13) and (16), (14) and (17), (15) and (18), i.e., onsists of: (1 p)m + p(1 q)g(v )E[ τm+rṽ τ+r ṽ < v ] + pqg(v )E[ τm+rṽ τ+r + β(p nd τm+rṽ τ+r ) ṽ < v ] 1 p + pg(v, ) whih may be rewritten as: (1 p)m + pg(v )E[ τ ˆm+rṽ τ+r ṽ < v ] + pqg(v )βe[p nd τ ˆm+rṽ τ+r ṽ < v ] 1 p + pg(v. (19) ) When S makes no dislosure, ompetition among the buyers will drive the prie P nd that S reeives to this onditional expeted value (19). Thus, the no dislosure prie P nd will satisfy: P nd = (19). (20) Now, realling the link between v and P nd desribed in (12) above, it follows from (19) and (20) that in equilibrium, the utoff v must satisfy the equation: τm + rv = (1 p)m + pg(v )E[ τm+rṽ τ+r ṽ < v ] + pqg(v )βe[ τm+rv τ+r τm+rṽ τ+r ṽ < v ] 1 p + pg(v ) (21) Rearranging this last equation (see the appendix for details) leads to the following theorem. In the statement of the theorem, φ( ) and Φ( ) refer to the density and df of a standard normal random variable x respetively, and α is defined by α 1 qβ.. 13

14 Theorem 2 The v that solves (21) is given by where x = x is the unique solution to the equation v = σx + m, (22) x(1 p + αpφ(x)) + αpφ(x) = 0. (23) This theorem, ombined with the next three orollaries below, onstitutes the main result of this setion. It establishes the link between the equilibrium utoff v and a "standardized" equilibrium utoff x defined in terms of the density and df of a standard normal random variable assoiated with the solution to (23). The theorem has several onsequenes. First, sine the equilibrium probability that S will make no dislosure, given that it reeived information, is Pr(ṽ v ), and the transformed random variable ṽ m σ distribution, it follows from (22) that Pr(ṽ v ) = Pr(ṽ m σ has a standard normal v m ) = Φ(x ). (24) σ Sine both the df Φ(x ) and the density φ(x ) are defined independently of eah of: the mean m and variane σ 2 of ṽ, and also the preision r of the estimate, we onlude: Corollary 3 The equilibrium probability of dislosure (and hene also the equilibrium probability of no dislosure) is independent of eah of: the mean m and variane σ 2 (= 1 τ + 1 r ) of the estimate ṽ, and also independent of ṽ s preision r. In partiular, this last orollary implies that were, say, the prior mean m of ṽ (equivalently, the prior mean of the asset s value z) to be made endogenous, then the equilibrium probability of dislosure is unaffeted by whatever turns out to be the equilibrium value of that investment hoie. This observation will be important in extensions of the model to be presented in later setions of the paper. 14

15 The independene of the equilibrium utoff highlighted in this last orollary from the prior mean and variane of ṽ also implies that if the model were altered so that additional publi information, say ỹ, were dislosed before S had an opportunity to reeive or dislose the private estimate ṽ, and this publi information ỹ was suh that buyers posterior beliefs about ṽ were still normally distributed (as would be true, for example, if ỹ = z + ω for some normal random variable ω that is independent of all other variables in the model), then the release of this publi information has no impat on S s subsequent voluntary dislosure deisions. That is to say, in this model: Corollary 4 The dislosure of publi information before S has an opportunity to reeive or dislose his private information has no effet on S s dislosure deision, and hene publi dislosures are neither omplementary to nor a substitute for voluntary dislosures. This orollary speaks to a ommon onern in the dislosure literature as to whether mandatory dislosures may "drive out" voluntary dislosures or more generally how the release of publi information may influene private information prodution and/or dislosure deisions. 7 Third, the theorem is the basis for the following additional omparative statis. Corollary 5 As long as the bound qβ < 1 holds, the unique x that solves (23): p/2; (i) is negative; (ii) implies that the ex ante probability S will make a dislosure is at least (iii) is stritly dereasing in p; (iv) is stritly inreasing in β; 7 In models related to, but distint from, the present model, where the emphasis is not on voluntary dislosure but rather on the related phenomenon of traders ostly private information aquisition ativities, sometimes very different onlusions emerge. E.g., Diamond [1985] shows that publi dislosures an redue investors inentives to aquire information on private aount if the publi dislosures are suffi iently informative. 15

16 (v) is stritly inreasing in q. Before disussing these omparative statis individually, we make two observations that apply to all of them. First, in view of (24) and sine Φ( ) is an inreasing funtion, eah of these omparative statis is also a omparative stati about the probability that S does not make a dislosure in equilibrium. Thus, for example, sine part (iv) of the orollary asserts that x inreases in β, it follows that the probability that S makes no dislosure in equilibrium also inreases in β. 8 Seond, eah of these omparatives statis is also a omparative stati about how the equilibrium utoff v that solves (21) hanges in any parameter, in view of the monotone relationship between x and v desribed in Theorem 2, namely that v = σx + m. Thus, for example, sine the orollary asserts that in equilibrium x inreases in β, it follows that the equilibrium utoff v also inreases in β. Now, as to the individual onlusions of the orollary: part (i) implies that the equilibrium utoff x is below the prior mean of x (and hene the equilibrium utoff v is below the prior mean of ṽ). Part (ii) then follows immediately: sine x < 0, the symmetry of the density of a standard normal random variable around x = 0 implies that p(1 Φ(x )) > p/2. In words, we obtain the robust onlusion that for any probability p that S reeives information, the ex ante probability S disloses his information always exeeds p/2 regardless of the preision of the information S reeives, regardless of the damages multiplier β appliable if S is aught withholding information, et. Part (iii) shows that the utoff is dereasing in the prior probability S reeives 8 As applied to parameter p, the ex ante probability S reeives information, this last statement in the text is also true (that is, sine (iii) asserts that x is stritly dereasing in p, it follows that the ex ante probability that S will not dislose information is also dereasing in p), but the laim requires the following extra observation beyond what is stated in the orollary to onfirm it: the ex ante probability that S does not make a dislosure is 1 p + pg(x (p)). We ontend that this probability delines in p. The derivative of this probability with respet to p is given by: p (1 p+pg(x (p))) = 1+G(x (p))+pg(x (p)) x (p) = p (1 G(x (p)))+pg(x (p)) x (p). Sine the orollary asserts that x (p) < 0, we an further p p onlude that p (1 p + pg(x (p))) < 0 too, whih is the substane of the laim made in the text. 16

17 information. This result extends Jung and Kwon [1988] to situations where a value-maximizing firm (or S) is subjet to potential damage payments when the (potentially dislosed) estimate of the firm s ash flows is normally distributed. Part (iv) of the orollary asserts that x inreases in β. At first blush, this result seems so ounterintuitive as to be wrong, as it asserts that an inrease in the damages multiplier β redues the probability S will dislose the information he reeives. But, the result has an easy explanation. Reall from Lemma 1 that, as long as the inequality in (6) is maintained, S s optimal nonmyopi dislosure poliy is determined by the (myopi) omparison of P nd and P (v) in (8). While the parameter β does not appear expliitly in (8), β does appear impliitly in (8) through P nd. P nd inreases in β (as long as the bound qβ < 1 is preserved) beause as β inreases, buyers will reeive a larger damages payment in the event the fat finder athes S withholding information from them. Sine P nd inreases in β, it follows that inequality (10) will hold for more values of v as β inreases, and hene S will withhold the information he reeives more often. The explanation for why x q is positive in part (v) is similar to the explanation for part (iv). When S makes no dislosure, the value buyers attahes to purhasing the asset is the sum of the buyers pereptions of the value of the asset itself ombined with the expeted value of the damages laim. Sine the expeted value of the damages laim inreases as the probability q the fat finder detets the withholding inreases, it follows that P nd inreases with q, so (as in part (iv)) of the orollary) inequality (10) will hold for more values of v as q inreases 5.1 Extension 1: When S an make a pre-sale investment that affets the expeted value of the asset In this setion, we extend the base model to a setting where the prior mean m of the asset is endogenous and is affeted by an investment I S selets prior to 17

18 the asset s sale. Examples of settings where m is naturally endogenous inlude: the asset is a used ar S drove prior to sale whose value is affeted by how muh are S took in maintaining the ar; the asset is a business started by S whih he wants to sell beause he is retiring, and the value of that business is affeted by marketing, prodution, and other ations that S took while the business was under his management, et. Formally, we now suppose that if S selets private investment I 0 at personal ost.5i 2, this investment results in the distribution of the asset s value at the time of sale z being distributed z N(m(I), 1 τ ), for m(i) = w I. Here, w is the marginal produtivity of investment. S s atual investment I is taken to be a private hoie of S, so buyers must make a onjeture about I when trying to assess the asset s value. We let buyers onjeture (assumed to be ommon to all buyers) be denoted by Î. In equilibrium we require that buyers onjetures be orret: I = Î. The rest of the model is the same as the base model desribed above: subsequent to seleting I but before the sale of the asset, with probability p S reeives estimate v, the realization of the random variable ṽ distributed as in (1) above. The prior density and df of ṽ are now written as g(v I) and G(v I) to aknowledge their dependene on I. dislose or withhold v. If S reeieves v, S an elet whether to If S withholds v, a fat finder detets the withholding with probability q, and if the fat finder does so, then S is liable for damages as desribed above. Taking as given buyers onjeture Î about S s initial investment hoie, then equation (21) above still uniquely defines the utoff v now denoted by v (Î) that S will use in deiding whether to dislose the information he reeives, provided the prior mean m is replaed by m(î) ˆm and G(v ) is replaed by 18

19 G(v (Î) Î). = That is, the utoff v (Î) is now defined by the equation: τ ˆm + rv (Î) (25) τ ˆm+rṽ (1 p) ˆm + pg(v (Î) Î)E[ τ+r ṽ < v (Î), ˆm] + pqg(v (Î) Î)βE[ τ ˆm+rv τ+r τ ˆm+rṽ τ+r ṽ < v (Î), ˆm] 1 p + pg(v (Î) Î). nd The "no dislosure" prie of the asset, now written as P (Î), is onneted to the utoff v (Î) just as the no dislosure prie P nd was onneted to the utoff v in the base model above via (12), i.e., P nd τ ˆm + rv (Î) (Î) =. (26) At the time S initially hooses I, S takes buyers onjeture Î, along with both nd the utoff v (Î) and the no dislosure prie P (Î) implied by that onjeture, as given when deiding what investment I atually to adopt. Speifially, S will hoose I so as to maximize: OBJ(I Î) (1 p + pg(v (Î) I))P nd (Î) + p pqβ v (Î) τ ˆm + rv v (Î) nd τ ˆm + rv (P (Î) )g(v I)dv.5I2. g(v I)dv (27) nd The first term in (27), (1 p + pg(v (Î) I)) P (Î), is the produt of the ex ante probability S will make no dislosure (taking into aount both buyers onjetures and S s atual investment hoie) and the prie S will get if he makes no dislosure; the seond term in (27), p τ ˆm+rv v (Î) τ+r g(v I)dv, is the ex ante probability S will make a dislosure, p(1 G(v (Î) I)), times the onditional expeted value of the asset if he makes a dislosure, E[ṽ ṽ > v (Î), I].9 The term in (27), pqβ v (Î) third (P nd (Î) τ ˆm+rv )g(v I)dv, is the ex ante probability τ+r S will be subjet to damages payments, pqg(v (Î) I), times the onditional nd expeted value of those damages payments E[β (P (Î) ṽ) ṽ v (Î), I].10 p 9 Sine, when multiplied out: p(1 G(v (Î) I)) E[ṽ ṽ > v (Î), I] = τ ˆm+rv v (Î) τ+r g(v I)dv) 10 Sine, when multiplied out: pqg(v (Î) I) E[β(P nd (Î) ṽ) ṽ v (Î), I] = pqβ v (Î) (P nd (Î) v)g(v I)dv. 19

20 Given the preeding definition of OBJ(I Î), an equilibrium is given as follows. Definition 6 An equilibrium investment level onsists of an I that satisfies: I = arg max I OBJ(I I ). Identifying the equilbrium investment level I leads to a omplete speifiation of the equilibrium sine it determines both the equilibrium utoff v (I ) and the equilibrium no dislosure prie P nd (I ). Realling that α 1 qβ and that x was speified in Theorem 2 above, and setting X pr (1 αφ(x )), (28) we prove in the appendix the following theorem, whih fully haraterizes the equilibrium value I. Theorem 7 The equilibrium value I is given by: I = wx. (29) The theorem is best understood in terms of its omparative statis impliations, summarized in the following orollary. Corollary 8 I is: (a) inreasing in r; (b) inreasing in p; () dereasing in τ; (d) inreasing in β; (e) inreasing in q. All the results in the orollary intuitive: onsider part (a). It asserts that S s investment optimally inreases in the preision r of the estimate ṽ. As the preision of the estimate inreases, ṽ reflets more aurately the ation I S took 20

21 when S makes a dislosure. Antiipating this, S is enouraged to hoose a higher level of investment. Virtually the same explanation applies to explain part (b): as the probability S reeives the estimate ṽ inreases, S is more likely to dislose the estimate, and antiipating this, S is more inlined to pik a high value for I. Next, onsider the result in part (): as buyers priors beliefs about the asset s value beome tighter (τ goes up), then buyers will plae relatively less weight on the estimate S sometimes disloses to them and more weight on their prior beliefs about the asset s value when valuing the asset after S makes a dislosure. Sine S an do nothing to affet buyers prior beliefs about the asset s value, inreases in τ disourage S from hoosing a high level of investment. Consider part (d). Inreases in β inrease the liability S has to pay in the event his withholding gets aught. S an redue the probability of having to pay any damages if he works harder, i.e., hooses a higher level of investment, beause the realized value of ṽ is more likely to be above the utoff, and hene S is less likely to have an inentive to withhold the information and be subjet to the liability payment. Likewise, as part (e) reports, inreases in q inrease the likelihood that S will have to pay damages, holding I fixed. As was the ase for the explanation of part (d), S an redue the likelihood of having to pay suh damages by inreasing I. 5.2 Extension 2: When S sells a divisible asset In this setion, we onsider another variant of the first extension above, where now the asset S sells is divisible, and so S an hoose, if he so desires, to sell only a fration of the asset. We now also suppose that the realized value of the asset is determined far into the future after the sale takes plae, so disounting beomes important to assess the expeted present value of the asset. This extension is appropriate for onsidering, among other things, dislosures in an IPO setting, where S is viewed as an entrepreneur who sells some fration of his asset/firm to outsider investors, and these outside investors share with him 21

22 in the eventual ash proeeds generated by the firm. In this extension, unlike the first extension, no sale need take plae: S an retain all of the asset for himself. Sine we ontinue to assume that both S and the buyers of the asset are risk neutral, we need to introdue some additional feature to the model to motivate S to sell a nonzero fration of the asset to the buyers. The feature we now add is that we suppose that S disounts the ash flows generated by the asset at a higher rate than do the buyers. 11 Speifially, we assume that the eventually realized ash flows z produed by the asset have present value δz to S, for some positive onstant δ < 1. Without loss of generality, we normalize the present value of those same ash flows to buyers of the asset to be z, and we further assume that the ash flows reeived or paid by S related to the (possibly frational) sale of the asset and ensuring liability assessed on S for withholding information, if any, arrive or are paid suffi iently lose in time to S s dislosure deision that S evaluates those reeipts and payments at their undisounted values. These latter onventions are irrelevant to the onlusions we reah below and are adopted solely to eliminate the notational lutter assoiated with the introdution of additional disount fators were these onventions not adopted. With these assumptions and onventions in plae, we see that if S deides to retain fration f of the asset, hooses investment I, learns ṽ = v and withholds it, then his expeted utility before knowing whether his withholding will be deteted is given by: δ(1 f) E[ z I, v] + f P nd qfβ(p nd P (v)).5i 2. Similar expressions hold for other possible senarios, e.g., if S deides to retain fration f of the asset, hooses investment I, learns ṽ = v and disloses it, then his expeted utility at the time of dislosure is given by δ(1 f) E[ z I, v] + 11 The differene between S s and buyers disount rates is natural, and an be motivated by S s life yle or liquidity demands or S s relative lak of diversifiation. Demarzo and Duffi e [1999] use this same exogenous assumption in their liquidity based model of seurity design. 22

23 f P (v).5i 2, et. In ongoing work, we allow for S s deision about the fration of the asset to sell to buyers to be delayed until S learns the realization of the estimate ṽ or learns that he is not going to learn the realization of the estimate. In the present analysis, we adopt the simplifiation that S hooses the fration f of the asset to sell at the same time he makes his investment hoie I. The tradeoff S faes in deiding what fration of the asset to retain is the following: sine buyers attah a higher present value to the ash flows eventually produed by the asset than S does (beause of their lower disount rate), that enourages S to sell a large fration of the asset so as to, in effet, arbitrage the differene between his and buyers disount rates. But, offsetting that pressure to sell most or all of the asset is the pressure to retain the asset due to the positive inentive effets on S s pre-sale investment hoie arising from retention (due to the fat that the ash flows eventually produed by the asset are more informative about S s initial investment hoie than is the estimate S oasionally disloses, and the extra informativeness of those ash flows influenes S s pre-sale investment only when S retains ownership of those ash flows). S s optimal frational sales of the asset balanes these two ompeting effets. We start the analysis of this extension by observing that the ex ante value to S of hoosing investment I when buyers onjeture he has hosen investment Î, and selling fration f of the asset to the buyers is given by: where OBJ δ(1 f)m(i) + f Ψ(I, Î).5I2, (30) τm(î) + rv (Î) Ψ(I, Î) (1 p + pg(v (Î) I)) τm(î) + rv p g(v I)dv pqβ v (Î) v (Î) + (31) τm(î) + rv (Î) ( Here, f Ψ(I, Î) is the prie S antiipates reeiving from buyers when they τm(î) + rv )g(v I)dv. 23

24 purhase fration f of the asset from him net of the expeted damage payments he subsequently expets to pay buyers. 12 The equilibrium value of OBJ in (30) an be obtained in two steps: first determine the equilibrium value of the investment I (f) for any fixed f, and seond, determine the optimal f. Using derivations similar to those whih led to Theorem 7 of the previous setion, one an show that the solution to this first step yields the following natural parallel to Theorem 7. Theorem 9 If S hooses to retain fration f of the asset, then S s equilibrium investment level is unique and given by: I (f) = δw + fw (X δ). (32) Note that when δ = 1 and f = 1, then I (f) = wx, whih is the same as the equilibrium level of investment in the first extension above of the base model above. It is easy to show that all of the omparative statis assoiated with Theorem 7 hold here too, and in addition, in the present setting we get two additional omparative statis: I (f) inreases in δ, and I (f) inreases or dereases in f depending on whether X is bigger or smaller than δ. The former result is straightforward: as S s disount rate falls (or equivalently, as S s disount fator rises), the present value of the portion of the asset S retains inreases in present value, whih auses S to work harder. The latter result indiates that when S disounts the future more than buyers (i.e., when δ < 1), then it is not always true that S has a greater inentive to work hard as his retained ownership stake in the asset inreases. Whether that result obtains depends, as (32) shows, on how big S s disount fator δ is relative to X. The next theorem ompletely haraterizes S s optimal hoie of what fration f of the asset to sell to buyers. The statement of the theorem makes use 12 Eah of the omponents of Ψ(I, Î) has a natural ounterpoint to eah of the omponents τm(î)+rv (Î) in (27) above, when we substitute for P nd (Î), and so we forego explaining eah τ+r of the omponents of Ψ(I, Î) here. 24

25 of the funtion: ψ(r) (1 X(r)) 2 for all r > 0. Here, X(r) is the same X as defined in (28) above, where the dependene of X on r is now emphasized. 13 Theorem 10 (A) If (1 p(1 αφ(x ))) 2 < 1 δ, then there exists a unique r > 0, all it r δ, suh that ψ(r δ ) = 1 δ. (Ai) For all r < r δ, the equilibrium f is given by f = (Aii) for all r > r δ, the optimal f is f = 1. (1 δ)δ (1 X) 2 (1 δ) 2 ; (33) (B) If (1 p(1 αφ(x ))) 2 1 δ, then for all r > 0, the equilibrium f is given by (33). Using the theorem, we an make speifi preditions about how S s optimal share retention 1 f varies with various parameters of the model, as summarized in the following orollary. Corollary 11 S s equilibrium retained ownership stake in the asset, 1 f, always (at least weakly): (i) delines as the preision r of the estimate ṽ inreases; (ii) delines as the probability p S reeives information inreases; (iii) delines as the damages multiplier β inreases. The first two omparative statis (i) and (ii) are intuitive. Holding his investment hoie fixed, as we already noted, S has an inentive to sell 100% of the 13 The ritial observation that underlies this next theorem is the alulation of the derivative of OBJ with respet to f, when evaluated at the equilibrium value Î(f) = I (f). It an be shown to be given by: f OBJ = (1 δ)δw2 + (2 δ X) w 2 f(x δ). 25

26 asset to buyers, sine they value the asset more than he does (beause of their lower disount rates). But, sine S s investment hoie is endogenous, and depends in part on what fration of the asset he retains, his ex ante investment hoie may be ineffi iently low unless he retains a substantial ownership stake in the asset. However, as the quality of the estimate ṽ (as measured by the preision of the estimate) S reeives inreases, or as the probability S reeives the estimate inreases, the estimate ṽ an be relied on more to ensure that S has good inentives to work hard to invest in the asset even if he sells some fration of his original ownership stake. the asset as either r or p inreases. Hene, S an profitably sell a larger stake in As inreases in the preision r of the estimate an be interpreted in pratie as an improvement in the quality of the aounting information S provides to investors, the first result has the empirially testable impliation that improvements in finanial reporting result in lower retained equilibrium ownership stakes by entrepreneurs who found a ompany. The explanation for the third omparative stati (that S s optimal retained ownership stake delines as the damages multiplier β inreases) is somewhat more ompliated. One might think that an inrease in β would inrease S s optimal retained ownership stake in the asset, sine an inreased ownership stake, i.e., a redued sale to buyers, eonomizes on S s liability risk. For example, S is obviously exposed to no liability for nondislosure if he sells none of the asset to buyers. Further reinforing this effet is that, as was noted above in Corollary 5 (iv), an inrease in β leads to an inrease in the threshold x, 14 and hene a redution in S s propensity to dislose information, and in turn, more exposure to liability for having withheld information. ombined "the liability effet." Call these effets But, there is an offsetting benefit to selling a higher fration of the asset to 14 It is easy to hek that that orollary remain valid if we replae the assumption there that S sells 100% of the asset by the assumption that S sells any fixed fration of the asset. 26

27 buyers when β inreases, beause of the tight onnetions between x and v (reall (22)), and between v and P nd (reall (12)). These onnetions imply that an inrease in the damages multiplier β inreases the no dislosure prie P nd. This latter effet, whih we shall all "the prie effet," by itself enourages S to inrease the fration of the asset he sells to buyers. 15 Thus whether inreasing β will lead to an inrease in the optimal fration f of the asset S sells to outsiders depends on whih of the "liability effet" or the "prie effet" is larger. The proof of part () shows that the prie effet is the larger of these two effets, and so S optimally inreases the fration of the asset he sells as β inreases. 6 Summary We have studied a dislosure problem where the seller of an asset sometimes reeives private information regarding an estimate of the asset s value prior to sale, whih he may deide to dislose to, or withhold from, potential buyers of the asset. If the seller elets to withhold the estimate from buyers, he is liable for damage payments in the event his withholding is subsequently deteted after the sale. These damage payments are taken to be SEC 10b-5 like in that they are proportional to the amount the buyer(s) of the asset overpaid for the asset (alulated based on what the asset would have sold for were the seller s estimate made publi). The analysis shows that for a broad range of parameter values, the seller s optimal dislosure poliy is the same whether the seller is myopi and hooses to dislose his estimate just based on whether the dislosure inreases or dereases the asset s market value at the time of the dislosure, or whether the seller is nonmyopi and also takes into aount the possible damage payments he may be liable for if his withholding is subsequently deteted. A olletion of omparative statis were obtained, some of whih are unexpeted. For example, if the damages multiplier determining what fration of the buyer s overpayment 15 I wish to thank Xu Jiang of Duke University for providing me with intuition for this last result. 27