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- DR 05005 -
Socially Efficient Managerial Dishonesty Damien BESANCENOT* and Radu VRANCEANU**
May 2005
* **
LEM and University Paris 2, 92 rue d’Assas, 75007 Paris, France. E-mail:
[email protected] ESSEC, Department of Economics, Avenue Bernard Hirsch, BP 50150 Cergy Pontoise, France. E-mail:
[email protected]
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GROUPE ESSEC, ÉTABLISSEMENTS PRIVÉS D'ENSEIGNEMENT SUPÉRIEUR, ASSOCIATION LOI 1901, ACCRÉDITÉ AACSB - THE INTERNATIONAL ASSOCIATION FOR MANAGEMENT EDUCATION, AFFILIÉ A LA CHAMBRE DE COMMERCE ET D'INDUSTRIE DE VERSAILLES VAL D'OISE - YVELINES. WEB : WWW.ESSEC.FR
May 8, 2005
SOCIALLY EFFICIENT MANAGERIAL DISHONESTY Damien Besancenot∗ and Radu Vranceanu†
Abstract As a reaction to the corporate scandals of the early 2000s, the US Administration dramatically tightened sanctions against managers who disclose misleading financial information. This paper argues that such a reform might come with some unpleasant macroeconomic effects. The model is cast as a game between the manager of a publicly listed company and the supplier of an essential input, under asymmetric information about the type of the firm. The analysis focuses on the Hybrid Bayesian Equilibrium where at least some managers choose to communicate a false information about the true type of the firm. We show that by dissuading "virtuous lies", whereby a manager strives to win time for a financially distressed company, a tougher sanction brings about a higher frequency of default. Keywords: Financial distress, Disclosure, Honesty, Corporate regulation, Hybrid Bayesian Equilibrium.
Resumé Au début des années 2000, l’Administration américaine a renforcé de manière significative les sanctions contre les directeurs de grandes entreprises qui, de manière mal intentionnée, communiquent des fausses informations financières. A partir d’un jeu séquentiel entre gestionnaires et fournisseurs en présence d’information asymétrique, l’article démontre qu’un renforcement de la sanction peut provoquer — dans certains cas — un accroissement de la fréquence de faillites. Mots-clef : Politique de communication, Honnêteté des gestionnaires, Réglementation des entreprises, Equilibre Bayesien Hybride JEL Classification Index: D82, G33, G38.
∗ LEM
and University Paris 2, 92 rue d’Assas, 75007, Paris, France. E-mail:
[email protected].
† ESSEC,
Department of Economics, BP 50150, 95021 Cergy Pontoise, France. E-mail:
[email protected]
1
Introduction
After the burst of the US Internet bubble in March 2001, criminal investigations unveiled that in the late nineties several top-level managers had manipulated financial information so as to inflate the share value of their company, then cashed the overvalued stocks just before the company collapse. This proliferation of managerial dishonest behavior brought about significant public distrust in the capitalism system and entailed substantial economic costs connected to the withered reputation of large companies. To address this problem, American authorities took a set of regulatory steps, mainly through the Sarbanes-Oxley Act of 2002. In particular, the law enhanced the responsibility of a CEO with respect to the truthfulness and relevance of compulsory financial statements. Furthermore, changes in the federal sentencing guidelines in 2001 and 2003 significantly raised penalties for financial fraud; economically damaging frauds are now on the same level as armed robberies. Regulation also aimed to dissuade auditors’ complicity with dishonest managers; in particular, an independent organization with national coverage was created to monitor and sanction auditors who do not cope with the standards of the accounting profession.1 While in the bubble years the quest for personal gains appeared to be the main motive for information manipulation, most often managers who engage in such activities pursue less selfinterested goals. As emphasized by Lev (2003, p.36), “the more common reason for earnings manipulation is that managers, forever the optimists, are trying to ‘weather out the storm’ — that is, to continue operations with adequate funding and customer / supplier support until better times come”. A significant literature on corporate financial distress has emphasized that the image clients and suppliers have about a company plays an important role in determining its actual financial stance. More precisely, if creditors start having doubts about the financial position of a company, they may ask for a higher risk premium, which represents an indirect cost for the firm (e.g., Altman, 1984; Wruck, 1990). In difficult times, the manager may well communicate 1 To get a good picture of these corporate scandals of the nineties and the subsequent changes in the US corporate regulation, see: Donaldson (2003), Healy and Palepu (2003), Sims and Brinkman (2003), Economist (2004), Vranceanu (2005).
1
on better than actual performances, just to get more favorable contracting terms and push down these indirect costs. This paper analyses effects from imposing a tougher sanction for fraudulent disclosure within an equilibrium framework under asymmetric information about the type of firm. If the penalty for false statements gets stronger, two opposite effects are at work; on the one hand, more managers will be honest, thus a manager’s announcement will get more credit; this would raise the survival rate of the high return firms. On the other hand, more honesty increases the indirect costs for distressed firms, thus pushing more of them into bankruptcy. In this elementary but realistic framework, we show that a Hybrid Bayesian Equilibrium, where some of the managers at the head of low-return firms choose to disclose false information, may occur. In this equilibrium, the overall frequency of defaults unambiguously increases with the sanction level.2 A closed form solution is obtained only for a specific income distribution. Despite this limitation, the main result cast some doubts on the optimality of imposing tougher but undifferentiated sanctions for fraudulent disclosure. Wise regulation might made a difference between managerial abuses motivated by selfish reasons — which deserve blame and punishment, and desperate attempts to maintain the company afloat; in this case, a more lenient sanction should apply. The paper is organized as follows. The next section introduces the basic assumptions. Section 3 presents the equilibrium and unveils the relationship between the sanction and the frequency of default. The final section draws the conclusion. (At the end of the paper, a Mathematical Appendix presents all the detailed calculations).
2
The model
The economy is made up of many publicly listed companies which last one-period. The income of the firm i is a random variable y i , following a cumulative distribution F i () supported on [0, τ i ]. In order to keep calculations as simple as possible, we further assume that y i is uniformly distributed 2
See also Besancenot and Vranceanu (2004).
2
on [0, τ i ].3
We also assume that there are only two types of firms, the (H )igh and the (L)ow
expected return firm; formally, i ∈ {H, L} with τ H > τ L . The frequency of H-type firms in total population of firms is denoted by q, with q ∈]0, 1[; this frequency is common knowledge. In order to produce the final good, the manager needs to buy one essential input unit from an external supplier. The manager is a price taker in the input market. Information is asymmetric: the manager, who does not know the future value of y i , knows the type of the firm (i.e., he knows τ i , the rightward bound of the income distribution). The supplier does not know y i or τ i . All he knows is a statement about the type of the firm, denoted by τˆj , with j ∈ {H, L}, made by the manager prior to contracting the input. The manager’s announcement strategy can be represented as a function s(τ i ) : {τ L , τ H } → {ˆ τ L , τˆH } which defines for all types of firms the manager’s statement. The manager of a L − type firm may honestly announce that the firm is of the L − type, or may lye and announce that the firm is of the H−type. In this model, the manager of the H − type firm would never lye.4 Therefore, the irrelevant action τˆL = s(τ H ) will be omitted in the following developments. Let Θ denote the supplier’s beliefs about the manager’s degree of honesty contingent on the type of firm. The probability that the manager at the head of a high-return firm is honest is equal to one. The probability that the manager of a low-return firm is honest (announces that the firm is of type L) is denoted by µ. If µ = 1, the supplier believes that the manager is honest, if µ = 0, the manager is dishonest and if µ ∈]0, 1[, the manager randomizes between the two pure strategies (alternatively, µ can be interpreted as the frequency of honest managers in the total population of managers running L—type firms). ⎧ ⎪ ⎪ ⎨ Pr[ˆ τ L |τ L ] = µ, where µ ∈ [0, 1] Θ= . ⎪ ⎪ H H ⎩ Pr[ˆ τ |τ ] = 1 3
(1)
The structure of the model is inspired by the classical paper by Bhattacharya (1979).
4 The formal proof can be obtained by comparing the H-type firm manager’s payoffs in the two cases. Intuitively, the manager who declares that a good firm is bad would lose twice, since indirect costs go up and he may be fined for false statement.
3
Given his beliefs and the observed signal, the supplier determines the price of the input, denoted by c(ˆ τ j ). This price is due at the end of the period. If the actual income of the firm is too small, the manager might not be able to fully comply with his obligations, i.e. might not pay the full price; therefore, c(ˆ τ j ) must include a premium related to the risk of payment default. The supplier is the residual claimant, hence, in the worst of cases, he gets the firm’s income. If the firm obtains a positive profit (y i − c(ˆ τ j ) > 0), the manager obtains a reward from work proportional to this profit (in order to keep the model as simple as possible, the manager gain is set equal to the profit). If the firm defaults on its obligations (y i < c(ˆ τ j )), the firm’s profit and the manager’s reward become zero.5
In order to study a non trivial problem, it is
assumed that τ L > max{c(ˆ τ H ), c(ˆ τ L )} if else the strategy of the L-type firm’s manager degenerates to a single action (if τ L ∈ [min{c(ˆ τ H ), c(ˆ τ L ), max{c(ˆ τ H ), c(ˆ τ L )}],, or is inexistent (indeed, if τ L < min{c(ˆ τ H ), c(ˆ τ L )} the manager has no reason to enter this trade). Finally, an external inspection body will check files of all the bankrupt firms and impose a fine θ (with θ > 0) on managers who have delivered false statements. Figure 1 presents the basic sequence of decisions. At time t = 0, Nature chooses the type τ i of the firm; at time t = 1, depending on the type of the firm, the manager makes his optimal statement, τˆj . At time t = 2, given the signal issued by the manager, the supplier upgrades his prior beliefs and posts a price c(ˆ τ j ) for the input. At time t = 3, Nature decides on the income of the firm, y i . Depending on whether the firm’s resources suffice (or not) to pay the contracted ¯ or not solvent (D); in this latter case, the firm is pulled out price, the firm is either solvent (D) of the market and the residual revenue is transferred to the supplier. Remark that both H and L-type firms may be subject to default. The manager’s payoff at the end of the game depends on the type of the firm, his announcement and Nature’s choice of output. Finally, the objective probability of payment default of the i−type firm whose manager has 5 During the nineties, the share of performance-based pay increased tremendously, to account for 60 to 70 % of total manager compensation (Murphy, 1999; Perel, 2003).
4
c(τ^H )
τ^H
(chooses y) _ D
Frequency: q τH
τL Frequency: 1-q Legend Nature Manager
D
Pr=1-µ τ^H
^ τL Pr=µ
c(τ^H ) (chooses y)
Manager's payoff 0
y - c(τ^H )
D
-θ
_ D
y - c(τ^H )
D
0
c(τ^L ) (chooses y) _ D
Supplier
y - c(τ^L )
Figure 1: Decision Tree announced τˆj is defined as Pr[y i < c(ˆ τ j )].
3
Equilibrium
A Bayesian equilibrium is a pair (s, Θ) such that a manager’s announcement strategy s maximizes his expected payoff given the supplier’s beliefs Θ, and the supplier’s beliefs are correct given s. A separating configuration implies that the manager’s announcement unambiguously reveals the firm’s type (s(τ H ) = τˆH and s(τ L ) = τˆL ). A pooling configuration appears when all managers deliver the same signal whatever the type of the firm; in this model, s(τ i ) = τˆH , ∀τ i . The game presents a Hybrid Bayesian Equilibrium (HBE), where some managers running low-return firms will communicate truthfully, and some will lye; in this case, the equilibrium frequency of honest L-type firms’ managers (µ) belongs to the interval ]0, 1[. Pure strategy equilibria can then be interpreted as special cases of this hybrid equilibrium, which obtain for µ → 1 (the pooling case) and µ → 0 (the separating case). To determine the equilibrium of the game, we first have to define the input price depending on 5
the announcement. This price is needed in order to determine the objective probability of default which, in turn, has a bearing on the manager’s expected payoff.
3.1
The input price defined
Let us consider a supplier who observes the signal issued by the manager. Given his beliefs (Eq. 1), the contingent probability he assigns to the type of the contracting firm is: ⎧ ⎪ ⎪ ⎨ Pr[τ L |ˆ τ L] = 1 ⎪ ⎪ ⎩ Pr[τ H |ˆ τH] =
Pr[ˆ τ H |τ H ] Pr[τ H ] Pr[ˆ τ H |τ H ] Pr[τ H ]+Pr[ˆ τ H |τ L ] Pr[τ L ]
=
(2)
q 1−µ(1−q)
Let c(ˆ τ j ) denote the price posted by the supplier depending on the signal (ˆ τ j ) issued by the manager. If the firm’s income is large enough (y i > c(ˆ τ j )), the supplier will receive the full price c(ˆ τ j ). If the firm income is lower than the contracted price (y i < c(ˆ τ j )) the supplier, who is the residual claimant, will get y i . We denote by c be the price that the the supplier might get in a risk-less trade. a) If the supplier receives the signal τˆL , he can unambiguously infer that he deals with a lowreturn firm (type L). The price c(ˆ τ L ) posted by a risk neutral supplier is implicitly defined by the zero trade-off condition: c=
Z
c(ˆ τ L)
ydF L (y) +
Z
τi
c(ˆ τ L )dF L (y)
(3)
c(ˆ τ L)
0
Considering an uniform distribution for the income of the L-type firm, c(ˆ τ L ) is the solution of: c = c(ˆ τ L) −
c(ˆ τ L )2 , 2τ L
(4)
where c(ˆ τ L ) > c. It can be checked that a price c(ˆ τ L ) exists if τ L > 2c. Remark also that the solution c(ˆ τ L ) is independent of θ. b) If the supplier gets the signal τˆH , he must take into account the possibility that the good signal might have been issued by a low return firm. Hence, the posted price c(ˆ τ H ) is implicitly
6
defined by: H
H
τ ] c = P [τ |ˆ
ÃZ
c(ˆ τH) H
ydF (y) +
0
Z
τH H
c(ˆ τH)
H
!
H
L
c(ˆ τ )dF (y) +P [τ |ˆ τ ]
ÃZ
c(ˆ τH) L
ydF (y) +
0
Z
τL H
or, given the uniform distribution of the firm’s income, by: Ã
c(ˆ τ H )2 c = P [τ |ˆ τ ] − + c(ˆ τH) 2τ H H
!
à ! ³ ´ c(ˆ τ H )2 H H H + 1 − P [τ |ˆ τ ] − + c(ˆ τ ) 2τ L
(6)
τ H ] appears to be a function of θ; the solution c(ˆ τ H ) depends on this In the next Section, P [τ H |ˆ variable. Finally, it can be checked that c(ˆ τ L ) > c(ˆ τ H ). Writing the difference between Eq. (4) and Eq. (6), it turns out that: h i L τ H ] 1 − ττH c(ˆ τ H )2 P [τ H |ˆ h i >0 c(ˆ τ L ) − c(ˆ τH) = 2τ L − c(ˆ τ L ) + c(ˆ τH)
(7)
The supplier would ask a higher price if he observes the signal "low return" than if he observes the signal "high return". The manager of the low return firm has an incentive to lye.
3.2
Conditions of existence of a HBE
Let us denote by U [ˆ τ j |τ L ] the expected payoff of the manager at the head of the L-type firm who issues a signal τˆj . This payoff is related to the manager’s reward for profits (identical to profits if these are positive, zero if else), less the fine for fraudulent statements, to be charged only in the case of default. Formally, the expected payoff of the manager of a L − type firm who announces τˆL is: τ L] = U [ˆ τ L |ˆ
Z
τL
c(ˆ τ L)
[y − c(ˆ τ L )]dF L (y) =
∙
¸ 1 L 1 c(ˆ τ L )2 τ L) + . τ − c(ˆ 2 2 τL
(8)
and the expected payoff of the manager who announces τˆH is: U [ˆ τ H |τ L ] =
Z
τL
c(ˆ τH)
[y − c(ˆ τ H )]dF L (y) − θ
Z
c(ˆ τH)
dF L (y)
0
c(ˆ τH) 1 H 1 τ H )] + [ c(ˆ τ ) − θ]. = [ τ L − c(ˆ 2 τL 2 7
!
c(ˆ τ )dF (y)
c(ˆ τH)
(5)
H
L
(9)
A hybrid equilibrium exists if the L-type firm’s manager is indifferent between announcing τˆL or τˆH : U [ˆ τ H |τ L ] = U [ˆ τ L |τ L ]
h in h io ⇔ c(ˆ τ L ) − c(ˆ τ H ) 2τ L − c(ˆ τ H ) + c(ˆ τ L) = 2θc(ˆ τ H ).
(10)
τ H )] as defined by Eq. (7) into Eq.(10), the necessary condition After replacing the term [c(ˆ τ L )−c(ˆ for a HBE becomes: P [τ H |ˆ τH] =
(τ H
2θτ H . − τ L )c(ˆ τH)
(11)
We can now be more specific about the nature of the equilibrium. Recall the definition of Pr[τ H |ˆ τ H ] building on supplier’s beliefs (Eq. 2): Pr[τ H |ˆ τH] =
q ∈ [q, 1]. 1 − µ(1 − q)
(12)
Writing the equality between Eq. (11) and Eq. (12), the equilibrium frequency µ of honest L−type firms’ managers can be written: ¡ ¢ ¡ ¢ 2θτ H − q τ H − τ L c(ˆ τH) 1 − q σθ µ= = 2θτ H (1 − q) 1−q
¢ τH) τ H − τ L c(ˆ > 0. with σ ≡ 2τ H ¡
(13)
A HBE would exist for µ ∈]0, 1[: this can happen if θ ∈]qσ, σ[. Given that we study the case where there are at least some low-return firms (q < 1), the HBE can exist. If θ ∈ [0, qσ], the strategy of honesty cannot be optimal for the manager of the L-type firm. A pooling equilibrium where all managers announce that their firm is a high return one occurs. Hence, the antifraud policy would become effective only if the sanction exceeds a critical threshold, θ > σq. If θ ∈ [σ, ∞, [, the separating equilibrium emerges: all managers announce the true type of their firm, honesty is generalized.
3.3
The consequence of a tougher sanction
Let us denote by ν the frequency of defaulting firms in this economy. It depends on the distribution of firms between high and low return firms (q and 1 − q), on the frequency of dishonest managers 8
(1 − µ) in the population of L − type firms, and the default rate recorded in each population of firms: ν
n o = qF H [c(ˆ τ H )] + (1 − q) µF L [c(ˆ τ L )] + (1 − µ)F L [c(ˆ τ H )] =
o q (1 − q) n H L H H c(ˆ τ ) + ) − c(ˆ τ )] + c(ˆ τ ) µ[c(ˆ τ τH τL
(14)
Proposition 1 The economy wide frequency of default is increasing with the sanction level; dν/dθ > 0. Proof. Given that
dc(ˆ τ L) dθ
= 0,
¸ ∙ i (1 − q)(1 − µ) dµ (1 − q) h L dν dc(ˆ τH) q H + ) − c(ˆ τ ) . + c(ˆ τ = dθ dθ τH τL dθ τ L
(15)
τ H ] by its expression in Eq. (11) in the implicit definition of Let us replace the probability P [τ H |ˆ the price c(ˆ τ H ) (Eq.6); Differentiating this equation, it turns out that: dc(ˆ τH) c(ˆ τH) . =− dθ θ + τ L − c(ˆ τH)
(16)
# " dµ (τ H − τ L ) c(ˆ τ H ) 2θ + τ L − c(ˆ τH) q . = dθ (1 − q) 2τ H θ2 θ + τ L − c(ˆ τH)
(17)
From Eq. (13), we obtain:
Replacing these derivatives and (1 − µ) by its equilibrium value (cf. Eq. 13) in Eq. (15), we obtain: ih i o nh dν qc(ˆ τ H )(τ H − τ L ) i 2θ + τ L − c(ˆ h τ H ) c(ˆ τ L ) − c(ˆ τ H ) − θ[c(ˆ τ H ) − 2θ] = dθ 2θ2 τ L τ H θ + τ L − c(ˆ τH)
(18)
This derivative is positive if:
ih i h τ H ) c(ˆ τ L ) − c(ˆ τ H ) > θ[c(ˆ τ H ) − 2θ] ≡ RHS(θ) LHS(θ) ≡ 2θ + τ L − c(ˆ
(19)
The inequality is fulfilled for θ = 0. It is fulfilled for any θ > 0 given that: dLHS(θ)/dθ > dRHS(θ)/dθ. Indeed, dLHS(θ) dRHS(θ) − = dθ dθ
h i τ H )] 2θ + c(ˆ τ L ) − c(ˆ τH) [2θ + 2τ L − c(ˆ θ + τ L − c(ˆ τH)
9
> 0.
(20)
4
Conclusion
The corporate scandals of the early 2000s brought about substantial public distrust about managers of large, publicly listed companies. In reaction to this phenomenon, the US Administration changed the existing corporate regulation or adopted new rules, aiming to dissuade managerial dishonest behavior. In particular, the new activism strengthened sanctions against managers who communicate false or incomplete information about the financial stance of their companies. We argue in this paper that this reform may bring about some unpleasant and probably unintended consequences. By further dissuading “virtuous lies”, whereby managers disclose false information only to “win time” for a financially distressed company, the frequency of defaulting firms might well increase. While sanctioning greedy managers who manipulate information in order to obtain a personal benefit is quite a sensible policy, extending the sanction to all the situations of information manipulation may be socially harmful.
5
References
Altman, E. I., 1984, A further empirical investigation of the bankruptcy cost question, Journal of Finance, 39, 4, pp. 1067-1089. Besancenot, D. and R. Vranceanu, The information limit to honest managerial behavior, ESSEC Working Paper 04008, Online at www.essec.fr. Bhattacharya, S., 1979, Imperfect information, dividend policy and "the bird in the hand" fallacy, Bell Journal of Economics, 10, 1, pp. 259-270. Donaldson, W. H., 2003, Corporate governance: What has happened and where we need to go?, Business Economics, July, pp. 16-20. Economist, 2004, Bosses behind the bars, The Economist, June 12th, 2004. Healy, P. M. and K. G. Palepu, 2003, The fall of Enron, Journal of Economic Perspectives, 17, 2, pp. 3-26.
10
Lev, B., 2003, Corporate earnings: Facts and fiction, Journal of Economic Perspectives, 17, 2, pp. 27-50. Murphy, K. J., 1999, Executive compensation, in O. Ashenfelter and D. Card, (Eds.), Handbook of Labour Economics, vol. 3B, Elsevier Science, North Holland: Amsterdam, New York and Oxford, pp. 2485-2563. Perel, M., 2003, An ethical perspective on CEO compensation, Journal of Business Ethics, 48, pp. 381-391. Sims, R. R. and J. Brinkmann, 2003, Enron ethics (or culture matters more than codes), Journal of Business Ethics, 45, 3, pp. 243-256. Vranceanu, R., 2005, Deregulating dishonesty. Lessons from the US corporate scandals, in D. Daianu and R. Vranceanu, (Eds.), Ethical Boundaries of Capitalism, Ashgate, Aldershot, pp. 219-238. Wruck, K. H., 1990, Financial distress, reorganization, and organizational efficiency, Journal of Financial Economics, 27, pp. 419-444.
11
A
Mathematical appendix: detailed calculations
A.1
Determining the expression of [c(ˆ τ L ) − c(ˆ τ H )]
Claculus of Eq. (4). To determine c(ˆ τ L ), we write the zero trade-off condition: c =
Z
c(ˆ τ L) L
ydF (y) +
Z
τL
c(ˆ τ L )dF L (y)
c(ˆ τ L)
0
= c(ˆ τ L) +
Z
c(ˆ τ L)
(y − c(ˆ τ L ))dF L (y)
0
¸c(ˆτ L ) ∙ 1 1 L = c(ˆ τ ) + L y( y − c(ˆ τ )) τ 2 0 ¸ ∙ 1 1 L L L L = c(ˆ τ ) + L c(ˆ τ )) τ )( c(ˆ τ ) − c(ˆ τ 2 L
= c(ˆ τ L) −
c(ˆ τ L )2 2τ L
(A.21)
We get: c(ˆ τ L) = c +
c(ˆ τ L )2 2τ L
Calculus of Eq.(6). In the same way, c(ˆ τ H ) results form the zero trade-off condition: c = P [τ H |ˆ τH] H
H
= P [τ |ˆ τ ]
ÃZ
c(ˆ τH)
ydF H (y) +
0
ÃZ Ã∙
c(ˆ τH)
0
Z
τH
c(ˆ τ H )dF H (y) c(ˆ τH)
y dy + c(ˆ τH) τH
τH
1 dy H c(ˆ τH) τ !
!
+ P [τ L |ˆ τH] L
H
+ P [τ |ˆ τ ] Ã∙
ÃZ
ÃZ
c(ˆ τH)
ydF L (y) +
0
τL
c(ˆ τ H )dF L (y)
c(ˆ τH)
0
c(ˆ τH)
Z
y dy + c(ˆ τH) τL
Z
τL c(ˆ τH)
!
1 dy τL
¸c(ˆτ ) y2 c(ˆ τ H) τL + [y]c(ˆτ H ) 2τ L 0 τL 0 # # Ã" ! Ã" ! i i c(ˆ τH) h H c(ˆ τH) h L c(ˆ τ H )2 c(ˆ τ H )2 H H H H L H = P [τ |ˆ + H τ − c(ˆ + τ ] τ ) + P [τ |ˆ τ ] τ ) τ − c(ˆ 2τ H τ 2τ L τL Ã ! Ã ! c(ˆ τ H )2 c(ˆ τ H )2 H H H H L H = P [τ |ˆ τ ] − + c(ˆ τ ) + P [τ |ˆ τ ] − + c(ˆ τ ) 2τ H 2τ L
τH] = P [τ H |ˆ
We get:
y2 2τ H
¸c(ˆτ
Z
!
H
)
+
c(ˆ τ H) τH [y]c(ˆτ H ) τH
" # # c(ˆ τ H )2 c(ˆ τ H )2 L H c(ˆ τ ) = c + P [τ |ˆ τ ] τ ] + P [τ |ˆ 2τ H 2τ L H
H
H
"
τH] + P [τ L |ˆ
H
12
!
!
(A.22)
Determining Eq. (7). The difference is: L
H
c(ˆ τ ) − c(ˆ τ ) = =
=
=
" " # # c(ˆ τ L )2 c(ˆ τ H )2 c(ˆ τ H )2 H H L H c+ − c − P [τ |ˆ τ ] τ ] − P [τ |ˆ 2τ L 2τ H 2τ L " " # # ´ c(ˆ ³ c(ˆ τ L )2 c(ˆ τ H )2 τ H )2 H H H H − P [τ |ˆ τ ] τ ] − 1 − P [τ |ˆ 2τ L 2τ H 2τ L i h " # τ H )2 c(ˆ τ L )2 − c(ˆ c(ˆ τ H )2 c(ˆ τ H )2 H H − P [τ |ˆ τ ] − 2τ L 2τ H 2τ L i h ∙ ¸ i τH) h c(ˆ τ L ) + c(ˆ 1 1 L H H 2 H H ) − c(ˆ τ ) − P [τ |ˆ τ ]c(ˆ τ ) − c(ˆ τ 2τ L 2τ H 2τ L
or, in an equivalent way: " # ∙ H ¸ h i c(ˆ τ L ) + c(ˆ τH) − τL L H H 2 τ H H c(ˆ τ ) − c(ˆ τ ) 1− τ ]c(ˆ τ ) = P [τ |ˆ 2τ L 2τ L τ H ∙ ¸ i h i h H − τL L H L H H 2 τ L H H τ ) [2τ − c(ˆ τ ) + c(ˆ τ ) ] = P [τ |ˆ τ ]c(ˆ τ ) c(ˆ τ ) − c(ˆ τH We obtain
A.2
h i L τ H ] 1 − ττH c(ˆ τ H )2 P [τ H |ˆ h i . c(ˆ τ L ) − c(ˆ τH) = 2τ L − c(ˆ τ L ) + c(ˆ τH)
Defining the HBE
Calculus of Eq. (8). The expected payoff of the L-type firm manager who fairly announces τ L , U [ˆ τ L |ˆ τ L] = = = = = =
Z
τL
c(ˆ τ L)
[y − c(ˆ τ L )]dF L (y)
L 1 1 [y( y − c(ˆ τ L ))]τc(ˆτ L ) τL 2 ¾ ½ 1 L L 1 L L L 1 L ( − c(ˆ τ )) − c(ˆ τ )( ) − c(ˆ τ )) τ τ c(ˆ τ τL 2 2 ½ ¾ 1 1 1 L L L L L τ ( − c(ˆ τ )) − c(ˆ τ )(− )) τ c(ˆ τ τL 2 2 ½ ¾ 1 1 1 τ L )) + c(ˆ τ L ( τ L − c(ˆ τ L )2 τL 2 2 ∙ ¸ 1 L 1 c(ˆ τ L )2 τ L) + τ − c(ˆ 2 2 τL
13
(A.23)
Calculus of Eq. (9). The expected payoff of the manager who announces τˆH (the dishonest one) is: H
L
U [ˆ τ |τ ] =
Z
τL H
c(ˆ τH)
L
[y − c(ˆ τ )]dF (y) − θ
Z
c(ˆ τH)
dF L (y) 0
¸τ L ∙ c(ˆ τH) 1 1 H )] − θ y[ y − c(ˆ τ τL 2 τL c(ˆ τH) ¸ ¸ ∙ ∙ 1 1 c(ˆ τH) H H 1 H H L 1 L [ − c(ˆ τ )] − )[ ) − c(ˆ τ )] − θ τ c(ˆ τ τ c(ˆ τ τL 2 τL 2 τL
= =
1 c(ˆ τH) 1 H = [ τ L − c(ˆ τ H )] + [ c(ˆ τ ) − θ] 2 τL 2
(A.24)
The HBE condition can be written: τ L |τ L ] U [ˆ τ H |τ L ] = U [ˆ ½ ¾ 1 1 1 H 1 1 L 2 L L 1 L [ τ L − c(ˆ τ H )] − L c(ˆ τ H )[− c(ˆ ( − c(ˆ τ )) + ) τ τ ) + θ] = τ c(ˆ τ 2 τ 2 τL 2 2 h i 1 c(ˆ τ L ) − c(ˆ τ H ) τ L + [c(ˆ τ L )][c(ˆ τ H ) + c(ˆ τ L )] = θc(ˆ τH) τ H ) − c(ˆ 2 h i c(ˆ τ L ) − c(ˆ τ H ) 2τ L + [c(ˆ τ H ) − c(ˆ τ L )][c(ˆ τ H ) + c(ˆ τ L )] = 2θc(ˆ τH) h in h io c(ˆ τ L ) − c(ˆ τ H ) 2τ L − c(ˆ τ H ) + c(ˆ τ L) = 2θc(ˆ τH)
But we know that (Eq. 7): h i L τ H ] 1 − ττH c(ˆ τ H )2 P [τ H |ˆ h i τH) = c(ˆ τ L ) − c(ˆ 2τ L − c(ˆ τ L ) + c(ˆ τH)
thus the HBE condition becomes:
∙ ¸ τL τ H )2 P [τ H |ˆ τH] 1 − H 2θc(ˆ τ H ) = c(ˆ τ H 2θτ τH] = ⇔ P [τ H |ˆ H (τ − τ L )c(ˆ τH) which is Condition (11) in the text.
A.3
Consequences of a tougher sanction
We start from Eq. (14): ν=q
o c(ˆ τ H ) (1 − q) n L H H + ) − c(ˆ τ )] + c(ˆ τ ) µ[c(ˆ τ τH τL 14
Given Eq. (4), it turns out that
dc(ˆ τ L) dθ
= 0. The derivative of the frequence of defaults with
respect to θ can be written: dν dθ
= =
( ) dc(ˆ τ H ) dc(ˆ q dc(ˆ τH) τ H ) (1 − q) dµ L H τ )] − µ + [c(ˆ τ ) − c(ˆ + τ H dθ τL dθ dθ dθ ¸ ∙ i (1 − µ)(1 − q) dc(ˆ τH) q dµ (1 − q) h L H + ) − c(ˆ τ ) + c(ˆ τ dθ τH τL dθ τ L
and was referred to in the text as the Eq. (15). dc(ˆ τH) τ H ] such as defined by Eq. (11) in Eq. (6): .We replace P [τ H |ˆ dθ ¸" H 2# ∙ ¸" H 2# ∙ c(ˆ τ ) c(ˆ τ ) 2θτ H 2θτ H H c(ˆ τ ) = c+ + 1− 2τ H 2τ L (τ H − τ L )c(ˆ τH) (τ H − τ L )c(ˆ τH)
a. Calculus of
= c+ = c+
τH) τ H ) c(ˆ 2θτ H c(ˆ 2θτ H c(ˆ τ H )2 − H + L H L L − τ ) 2τ (τ − τ ) 2τ 2τ L
(τ H
θc(ˆ τH) c(ˆ τ H )2 θc(ˆ τ H )τ H + − (τ H − τ L ) (τ H − τ L )τ L 2τ L
Differentiating the former expression: dc(ˆ τ H ) = dθ
c(ˆ τH) θ θτ H τH) c(ˆ τ H )τ H H H H c(ˆ ) ) ) +dc(ˆ τ −dc(ˆ τ −dθ +dc(ˆ τ (τ H − τ L ) (τ H − τ L ) τ L (τ H − τ L ) τ L (τ H − τ L ) τL
dc(ˆ τ H )[1 −
(τ H
θ c(ˆ τH) θτ H c(ˆ τH) c(ˆ τ H )τ H ] = dθ[ H + L H − − L H ] L L L L − τ ) τ (τ − τ ) τ (τ − τ ) τ (τ − τ L )
τ H )(τ L − τ H )] = dθc(ˆ τ H )[τ L − τ H ] dc(ˆ τ H )[−τ L (τ L − τ H ) − θ(τ L − τ H ) + c(ˆ we get Eq.(16) in the text: dc(ˆ τH) c(ˆ τH)
