The influence of uncertainty on the standard-setting decision between fair value and historical cost accounting under asymmetric information

Abstract

The design of accounting rules by the international standard-setters takes place by considering a trade-off between relevance and reliability. An example for this trade-off is the standard-setting decision between fair value accounting—associated with more relevant information—and historical cost accounting—associated with more reliable information. This paper examines in which way the decision of a standard-setter between fair value and historical cost accounting is influenced by the uncertainty of the underlying assets, if the standard-setter wants to minimize the social costs of his standard-setting decision. As a first step this paper uses a common signaling model: Good firms—i.e. firms with high expected cash flows in the future—signal their firm type to an analyst by using discretionary accruals to manage earnings. As a second step the resulting signaling costs are compared with the analyst’s costs for determining the firm type by using his own valuation technology. The standard-setter chooses the accounting rule that minimizes the social costs.

Appendices

Appendix A: Derivation of the perfect informative limit reporting equilibrium

This appendix contains the proof of Proposition 1.

First some relations from the paper concerning the firm valuation are summarized: The firm value is defined at different levels of over-reporting. If the analyst correctly infers firm type and the good firm over-reports by $\delta _H$ the ($t=1$) expected value of a good firm is:

$$\begin{aligned} V_{H1}(\delta _H)= &\,\theta _1 \mu _H- \theta _2 \delta _H \end{aligned}$$

(25)

$$\begin{aligned} V_{H2}(\delta _H)= & {} [V_{H1}(\delta _H)] (1+r) \end{aligned}$$

(26)

where

$$\begin{aligned} \theta _1= & {} \frac{1}{1+r}\\ \theta _2= & {} \frac{c_{Man}}{\sigma ^2} \end{aligned}$$

Correspondingly, if the analyst correctly infers firm type and the bad firm over-reports by $\delta _L$, the ($t=1$) expected value of a bad firm is:

$$\begin{aligned} V_{L1}(\delta _H)= &\, \theta _1 \mu _L-\theta _2 \delta _L \end{aligned}$$

(27)

$$\begin{aligned} V_{L2}(\delta _H)= &\, [V_{L1}(\delta _H)] (1+r) \end{aligned}$$

(28)

Using theses valuation equations, we can turn to the manager’s compensation functions. With respect to the following analysis the managerial compensation is defined under two scenarios: (1) the analyst correctly infers firm type and (2) the analyst incorrectly infers firm type.

(1) Managerial compensation if the analyst correctly infers firm type:

If the analyst correctly infers firm type, a manager expects to receive:

$$\begin{aligned} W_i(\delta _i)= &\, a_1 V_{i1}(\delta _i) + a_2 V_{i2}(\delta _i) \nonumber \\= &\, a_1 \left( \theta _1 \mu _i-\theta _2 \delta _i\right) + a_2 \left( \theta _1 \mu _i -\theta _2 \delta _i\right) (1+r) \nonumber \\= &\, a \left( \theta _1 \mu _i-\theta _2 \delta _i\right) \end{aligned}$$

(29)

where $W_i(\delta _i)$ is a manager’s compensation level if the report is over-reported by $\delta _i$, where $i=(L,H)$ and where $a=a_1+a_2 (1+r)$.

(2) Managerial compensation if the analyst incorrectly infers firm type:

If a bad firm is mistaken for a good firm at $t=1$, there is an initial over-valuation. That is, if the analyst incorrectly identifies the bad firm as a good firm at $t=1$, the expected value of the manager’s first period compensation is

$$\begin{aligned} a_1(\theta _1 \mu _H-\theta _2 \delta _H) \end{aligned}$$

(30)

where $\delta _H$ is the amount of over-reporting by the good firm. Thus the bad manager’s expected compensation reflects the reporting strategy of the good firm because this is how the analyst prices the good firm.

However in $t=2$ the true firm type is revealed as the cash flows are realized. Thus the erroneous valuation only affects the first period. This results in a bad manager’s expected compensation for period $t=2$:

$$\begin{aligned} a_2 (\theta _ 1\mu _L-\theta _2 \delta _L)(1+r) \end{aligned}$$

(31)

Combining Eqs. (30) and (31), the bad manager’s expected compensation is:

$$\begin{aligned} W_L^m(\delta _L)= &\, a_1\left( \theta _1 \mu _H-\theta _2 \delta _H\right) + a_2\left( \theta _1 \mu _L-\theta _2 \delta _L\right) (1+r)\nonumber \\= &\, a_1\left( (\mu _H-\mu _L) \theta _1 +\mu _L \theta _1 -\theta _2 \cdot (\delta _L+(\delta _H-\delta _L))\right) \nonumber \\&+\,a_2\left( \mu _L \theta _1-\theta _ 2\cdot \delta _L\right) (1+r)\nonumber \\= &\, a_1\left( \mu _L \theta _1 -\theta _2\cdot \delta _L\right) +a_1\cdot (\mu _H-\mu _L)\theta _1 \nonumber \\&-\,a_1\cdot \theta _2 (\delta _H-(\delta _H+(\mu _H-\mu _L)) + a_2\left( \mu _L \theta _1-\theta _2 \cdot \delta _L\right) (1+r)\nonumber \\= &\, a_1\left( \mu _L \theta _1 -\theta _2\cdot \delta _L\right) +a_1\cdot (\mu _H-\mu _L)\theta _1 \nonumber \\&+\,a_1\cdot \theta _2 (\mu _H-\mu _L) + a_2\left( \mu _L \theta _1-\theta _2 \cdot \delta _L\right) (1+r)\nonumber \\= &\, a \left( \mu _L \theta _1-\theta _2 \cdot \delta _L\right) + a_1(\mu _H-\mu _L)\cdot \left( \theta _1 + \theta _2\right) \end{aligned}$$

(32)

where $W_L^m(\delta _L)$ denotes the bad manager’s compensation if the bad firm is mistaken for a good firm. The first term in Eq. (32) represents the bad manager’s compensation if the firm is correctly identified [see Eq. (29)]. The second term represents the increase in compensation resulting form the over-valuation in $t=1$.

By contrast, if the analyst believes a good firm’s report came from a bad firm, the good manager’s expected compensation in $t=1$ is:

$$\begin{aligned} a_1(\theta _1 \mu _L- \theta _2 \delta _L) \end{aligned}$$

(33)

In $t=2$ the true firm type is revealed and the second period expected compensation is:

$$\begin{aligned} a_2(\theta _1 \mu _H-\theta _2 \delta _H)(1+r)\end{aligned}$$

(34)

Combining both equations results in the following expected compensation:

$$\begin{aligned} W_H^m(\delta _H)= &\, a_1\left( \theta _1 \mu _L-\theta _2 \delta _L\right) + a_2\left( \theta _1 \mu _H-\theta _2 \delta _H\right) (1+r)\nonumber \\= &\, a_1\left( \mu _H \theta _1-(\mu _H-\mu _L) \theta _1 -\theta _2\cdot (\delta _H -(\delta _H-\delta _L)\right) \nonumber \\&+\,a_2\left( \mu _H \theta _1-\theta _2 \cdot \delta _H\right) (1+r)\nonumber \\= &\, a_1\left( \mu _H \theta _1-\theta _2\cdot \delta _H\right) + a_2\left( \mu _H \theta _1-\theta _2\cdot \delta _H\right) (1+r)\nonumber \\&-\,a_1(\mu _H-\mu _L) \theta _1 +\theta _ 2 \cdot (\delta _H-\delta _L)\nonumber \\= &\, a \left( \mu _H \theta _1-\theta _2\cdot \delta _H\right) \nonumber \\&- a_1(\mu _H-\mu _L) \theta _1 +\theta _ 2 \cdot (\delta _H-(\delta _H+\mu _H-\mu _L))\nonumber \\= &\, a \left( \mu _H \theta _1-\theta _2\cdot \delta _H\right) - a_1 (\mu _H-\mu _L) \left( \theta _1 +\theta _2\right) \end{aligned}$$

(35)

where $W_H^m(\delta _H)$ denotes the good manager’s compensation if the good firm is mistaken for a bad firm in $t=1$. The first term in Eq. (35) represents the good manager’s compensation if the firm is correctly identified [see Eq. (29)]. The second term represents the decrease in compensation as the firm is undervalued in $t=1$.

Derivation of the perfectly informative limit reporting equilibrium:

Using the relations above, it is now proved that there exists a perfectly informative equilibrium that satisfies the Intuitive Criterion by Cho and Kreps. The proof consist of three steps which are outlined as follows:

• (1) The level of over-reporting is derived, that makes the bad manager indifferent between his first-best compensation and his compensation under an imitating strategy which blocks information revelation. This level of over-reporting is defined as $\delta _L^{PI}$. If a good manager wants to ensure that the bad manager’s incentive compatibility condition holds, the good manager himself has to over-report earnings. The necessary level of over-reporting by the good manager is denoted by $\delta _H^{PI}$.

• (2) After deriving the level of over-reporting which is necessary to prevent the imitating strategy of the bad manager, it is shown that $\delta _H^{PI}$ is incentive compatible for the good manager.

• (3) Finally it is demonstrated that the perfectly informative equilibrium satisfies the Intuitive Criterion by Cho and Kreps.

(1) Bad manager incentive compatibility:

In a perfectly informative equilibrium, the bad manager must find the outcome under perfect information incentive compatible:

$$\begin{aligned} W_L(\delta _{L}^{*} =0) \ge W_L^m(\delta _L) \end{aligned}$$

(36)

Using Eq. (36), you can find $\delta _L^{PI}$ such that (36) holds as an equality:

$$\begin{aligned} W_L(\delta _{L}^{*} =0) = W_L^m(\delta _L^{PI}) \end{aligned}$$

(37)

Substituting Eqs. (29) and (32) in (37) and solving for $\delta _L^{PI}$ yields:

$$\begin{aligned} a \left( \mu _L \theta _1 \right)= &\, a \left( \mu _L \theta _1-\theta _2\cdot \delta _L\right) + a_1(\mu _H-\mu _L)\cdot \left( \theta _1+\theta _2\right) \nonumber \\ a\theta _2 \delta _L= &\, a_1 (\theta _1 + \theta _2)(\mu _H-\mu _L) \nonumber \\ \delta _L^{PI}= &\, \frac{a_1}{a} \cdot \left( \frac{\theta _1}{\theta _2}+1\right) \cdot (\mu _H-\mu _L) \end{aligned}$$

(38)

$\delta _L^{PI}$ is strictly positive. Since $\delta _L^{PI}= \mu _H - \mu _L + \delta _H^{PI}$, the corresponding level of over-reporting by the good firm is

$$\begin{aligned} \delta _H^{PI}= & {} \left( \frac{a_1}{a} \cdot \left( \frac{\theta _1}{\theta _2} +1 \right) -1\right) \cdot (\mu _H-\mu _L) \end{aligned}$$

(39)

The sign of $\delta _H^{PI}$ can be positive or negative. The sign is negative if the corresponding amount of the bad manager’s manipulation ($\delta _L^{PI}$) is smaller than the difference between the expected values ($\mu _H-\mu _L$). That is the amount is not large enough to overcome the difference between the expected values. In order to release identical reports the good manager would have to manipulate his report downwards. But a downward manipulation can not beneficial for the good manager because it simplifies an imitating strategy by the bad manager. Therefore in the perfectly informative limit reporting equilibrium the amount of manipulation by the good manager is the maximum of $\delta _H^{PI}$ and zero:

$$\begin{aligned} \delta _H^{LRE}= &\,max(0,\delta _H^{PI}) \end{aligned}$$

(40)

Since the good manager always selects an amount of over-reporting that makes the bad manager’s outcome under perfect information incentive compatible, the bad manager does not manipulate his report in the limit reporting equilibrium, i.e. $\delta _L^{LRE}=0$.

(2) Good manager incentive compatibility:

It will be shown that the determined level of over-reporting is incentive compatible for the good manager. Over-reporting by $\delta _H^{LRE}$ is incentive compatible if the good manager’s compensation, $W_H(\delta _H^{LRE})$, exceeds the compensation if the analyst believes the firm is a bad firm with probability one, $W_H^m(\delta _H^m)$.

Lemma 1

If a good manager over-reports income by$\delta _H^{LRE}$,

$$\begin{aligned} W_H(\delta _H^{LRE}) > W_H^m(\delta _H^m) \end{aligned}$$

(41)

where$\delta _H^{LRE}$is the level of over-reporting defined by Eq. (40) and$\delta _H^m$is the level of over-reporting if the analyst believes the firm is a bad firm with probability one.

Proof

In order to proof this lemma, an upper bound for $W_H^m(\delta _H^m)$ is defined, which is denoted as ${\bar{W}}$. If the analyst believes a firm is a bad firm with probability one, there is no incentive for the good manager to over-report earnings. Hence an upper bound for $W_H^m(\delta _H^m)$ is obtained by setting $\delta _H^m=0$:

$$\begin{aligned} {\bar{W}}=W_H^m(\delta _H^m=0)=a \mu _H\theta _1 - a_1 (\mu _H-\mu _L) (\theta _1 + \theta _2) \end{aligned}$$

(42)

In order to show that $W_H(\delta _H^{LRE}) > {\bar{W}}$ consider the two cases (1) $\delta _H^{LRE} > 0$ and (2) $\delta _H^{LRE} = 0$ separately.

• (1) Suppose $\delta _H^{LRE} > 0$: Taking the difference between $W_H(\delta _H^{LRE})$ and ${\bar{W}}$, we obtain:

  $$\begin{aligned}&W_H(\delta _H^{LRE})- {\bar{W}}\nonumber \\&\quad =a (\mu _H \theta _1-\theta _2 \cdot \delta _H^{LRE}) -\left( a \mu _H\theta _1 - a_1 (\mu _H-\mu _L) (\theta _1 +\theta _2)\right) \nonumber \\&\quad = -\,a\theta _2\left( \frac{a_1}{a}\left( \frac{\theta _1}{\theta _2}+1\right) +1\right) (\mu _H-\mu _L)+a_1(\mu _H-\mu _L)(\theta _1+\theta _2)\nonumber \\&\quad = (-\,a_1 \theta _1 - a_1 \theta _2 +a \theta _2) (\mu _H-\mu _L)+a_1(\mu _H-\mu _L)(\theta _1+\theta _2)\nonumber \\&\quad = -\,a_1(\theta _1+\theta _2)(\mu _H-\mu _L) +a\theta _2 (\mu _H-\mu _L)+a_1(\mu _H-\mu _L)(\theta _1+\theta _2)\nonumber \\&\quad =a \theta _2 (\mu _H-\mu _L) > 0 \end{aligned}$$

  (43)

  The inequality follows because all terms are strictly positive.

• (2) Suppose $\delta _H^{LRE} = 0$:

  $$\begin{aligned}&W_H(\delta _H^{LRE})- {\bar{W}}\nonumber \\&\quad =a \mu _H \theta _1 - \left( a \mu _H \theta _1 - a_1(\mu _H-\mu _L)(\theta _1+\theta _2)\right) \nonumber \\&\quad =a_1(\mu _H-\mu _L)(\theta _1+\theta _2) > 0 \end{aligned}$$

  (44)

  The inequality follows because all terms are strictly positive. This completes the proof.

(3) The perfectly informative equilibrium satisfies the Intuitive Criterion by Cho and Kreps:

In the described perfect informative equilibrium bad firms do not over-report earnings, they report truthfully. That is:

$$\begin{aligned} R_{L}^{LRE}(\delta _L^{LRE})=\mu _L \end{aligned}$$

In order to prevent the bad manager from imitating the good manager has to over-report by $\delta _H^{LRE}$ in the perfect informative equilibrium. That is:

$$\begin{aligned} R_{H}^{LRE}(\delta _L^{LRE})=\mu _H + \delta _H^{LRE} \end{aligned}$$

To show that this equilibrium satisfies the Intuitive Criterion by Cho and Kreps, consider how the analyst would respond to feasible defections. The defections may occur in three different cases:

• 1. the defection may occur in the interval ${\tilde{R}} \in [R_{L}^{LRE}(\delta _L^{LRE}) , \mu _H)$,

• 2. the defection may occur in the interval ${\tilde{R}} \in [\mu _H , R_{H}^{LRE}(\delta _H^{LRE}))$,

• 3. or it may occur the case ${\tilde{R}} > R_{H}^{LRE}(\delta _H^{LRE})$

(1) First, consider a defection in the interval ${\tilde{R}} \in [R_{L}^{LRE}(\delta _L^{LRE}) , \mu _H)$. Since the good manager would not manipulate his report downwards, the analyst would assign the belief that

$$\begin{aligned} Pr(\it{{{defector \, is\, a\, good\, firm}}}\, |\,{\tilde{R}} \in [R_{L}^{LRE}(\delta _L^{LRE} , \mu _H))=0 \end{aligned}$$

and the defector is valued as a bad firm . Therefore, defections over this interval satisfy the Intuitive Criterion by Cho and Kreps.

(2) Next, consider a defection in the interval ${\tilde{R}} \in [\mu _H , R_{H}^{LRE}(\delta _H^{LRE}))$. If the analyst assigns the belief that

$$\begin{aligned} Pr(\it{{{defector\, is\, a\, good\, firm}}}\, |\,{\tilde{R}} \in [\mu _H , R_{H}^{LRE}(\delta _H^{LRE})))=1 \end{aligned}$$

the defector is valued as a good firm . In this case we cannot rule out either firm type as a potential defector and therefore defections over this interval satisfy the Intuitive Criterion by Cho and Kreps.

(3) Finally, consider the case ${\tilde{R}} > R_{H}^{LRE}(\delta _H^{LRE})$. Obviously no firm would wish to defect at levels in excess of $R_{H}^{LRE}(\delta _H^{LRE})$. For the bad manager such a report is not incentive compatible and the good firm manager strictly prefers to report $R_{H}^{LRE}(\delta _H^{LRE})$. Therefore, defections over this interval also satisfy the Intuitive Criterion by Cho and Kreps.

The considerations above show that the described perfect informative equilibrium survives the Intuitive Criterion by Cho and Kreps.

The uninformative equilibrium

It will be shown that an uninformative equilibrium does not satisfy the Intuitive Criterion by Cho and Kreps under a certain condition. An uninformative equilibrium results if both firm types render identical earnings reports. By rendering the same report as the good firm, the bad firm blocks the information revelation and turns the earnings reports completely uninformative. This means if a good firm manipulates by $\delta _H^{UI}$ the bad firm must manipulate by $\delta _L^{UI}$ where $\delta _L^{UI}=\mu _H - \mu _L + \delta _H^{UI}$.

If the bad firm follows this reporting strategy earnings reports are rendered uninformative, so that the analyst makes use of his analyzing technology which reveals the firm type with probability q. As a consequence the ($t=0$) expected market values of the good and the bad firm are not identical in the uninformative equilibrium after the valuation technology has been used. The ($t=0$) expected market value of a good firm is:

$$\begin{aligned}&q V_{H1}^{*} + (1-q) \cdot (p V_{H1}^{*} + (1-p) V_{L1}^{*}) - \nonumber \\&\qquad q \cdot \theta _2 \delta _H^{UI} - (1-q) \cdot (p \theta _2 \delta _H^{UI} + (1-p)\theta _2 \delta _L^{UI})\nonumber \\&\quad =(q +(1-q)\cdot p) \cdot V_{H1}^{*} + (1-q)(1-p) \cdot V_{L1}^{*} \nonumber \\&\qquad - (q+(1-q)p \cdot \theta _2 \delta _H^{UI} - (1-q)(1-p) \cdot \theta _2 \delta _L^{UI}\nonumber \\&\quad =(q +(1-q)\cdot p) \cdot V_{H1}^{*} + (1-q)(1-p) \cdot V_{L1}^{*} \nonumber \\&\qquad - \theta _2 \delta _H^{UI} - (1-p)(1-q) \cdot \theta _2 (\mu _H-\mu _L) \end{aligned}$$

(45)

Correspondingly, the ($t=0$) expected market value of a bad firm is inferred as follows:

$$\begin{aligned}&q V_{L1}^{*} + (1-q) \cdot (p V_{H1}^{*} + (1-p) V_{L1}^{*}) \nonumber \\&\qquad - q \cdot \theta _2 \delta _L^{UI} - (1-q) \cdot (p \theta _2 \delta _H^{UI} + (1-p) \theta _2 \delta _L^{UI})\nonumber \\&\quad =((1-q)p \cdot V_{H1}^{*} + (q+(1-q)(1-p)) \cdot V_{L1}^{*} \nonumber \\&\qquad - (1-q)p \cdot \theta _2 \delta _H^{UI} - (q+(1-q)(1-p)) \cdot \theta _2 \delta _L^{UI}\nonumber \\&\quad =((1-q)p \cdot V_{H1}^{*} + (q+(1-q)(1-p)) \cdot V_{L1}^{*} \nonumber \\&\qquad - \theta _2 \delta _H^{UI} - (q+(1-q)(1-p)) \cdot \theta _2 (\mu _H-\mu _L) \end{aligned}$$

(46)

Obviously, in an uninformative equilibrium, the good manager does not over-report earnings because the firm value is strictly decreasing in $\delta _H^{UI}$. In the uninformative equilibrium a good manager sets $\delta _H^{UI}$ equal to zero ($\delta _H^{UI}=0$). In contrast the bad manager blocks the information revelation by choosing the amount of manipulation $\delta _L^{UI}=\mu _H-\mu _L$. If the good manager does not over-report an over-reporting on the part of the bad manager in the amount of $\mu _H-\mu _L$ results in identical reports from both types of firms and renders the reports completely uninformative. This results in a ($t=0$) expected value for the good firm of

$$\begin{aligned} (q +(1-q)\cdot p) \cdot V_{H1}^{*} + (1-q)(1-p) \cdot V_{L1}^{*} - (1-q)(1-p) \cdot \theta _2 (\mu _H-\mu _L) \end{aligned}$$

(47)

The ($t=0$) expected value for the bad firm is

$$\begin{aligned} (1-q) p \cdot V_{H1}^{*} + (q+(1-q)(1-p)) \cdot V_{L1}^{*} - (q+(1-q)(1-p)) \cdot \theta _2 (\mu _H-\mu _L) \end{aligned}$$

(48)

Having characterized the uninformative equilibrium, it is now shown, that it fails to satisfy the Intuitive Criterion by Cho and Kreps under a certain condition concerning parameter q:

Proposition

The described uninformative equilibrium characterized by a manipulation of$\delta _L^{UI}=\mu _H-\mu _L$by the bad manager and$\delta _H^{UI}=0$by the good manager does not satisfy the Intuitive Criterion by Cho and Kreps if parameterqfulfills the following condition:

$$\begin{aligned} q< {\bar{q}} \end{aligned}$$

where${\bar{q}}= \frac{(1-p) V_{H1}^{*} - (1-p)\cdot V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)- \theta _2 \delta _H^{'}}{(1-p)V_{H1}^{*}-(1-p) V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)}$.

Proof

Let $\delta _L^{'}=\mu _H-\mu _L+\delta _H^{'}$ denote the level of over-reporting such that a bad manager will not wish to defect from the characterized uninformative equilibrium even if the analyst believes with probability one that the defector is a good firm . That is, find $\delta _H^{'}$ such that the bad manager is indifferent between the market value in the uninformative equilibrium and the market value if the report is manipulated by $\delta _H^{'}+\mu _H-\mu _L$ and the analyst believes the firm is a good firm with probability one. That is:

$$\begin{aligned}&(1-q) p \cdot V_{H1}^{*} + (q+(1-q)(1-p)) \cdot V_{L1}^{*} \nonumber \\&\quad - (q+(1-q)(1-p)) \cdot \theta _2 (\mu _h-\mu _L)=V_{H1}^{*}-\theta _2 (\delta _H^{'}+\mu _H-\mu _L) \end{aligned}$$

(49)

The effective cost to a good firm from over-reporting by $\delta _H^{'}$ is $\theta _2 \delta _H^{'}$, so the good firm is worth

$$\begin{aligned} V_{H1}^{*} - \theta _2 \delta _H^{'} \end{aligned}$$

(50)

We have to analyze under which condition the good manager strictly prefers to defect from the uninformative equilibrium. This is the case if the following inequality is fulfilled:

$$\begin{aligned}&(q +(1-q)\cdot p) \cdot V_{H1}^{*} + (1-q)(1-p) \cdot V_{L1}^{*} \nonumber \\&\quad - (1-q)(1-p) \cdot \theta _2 (\mu _H-\mu _L) < V_{H1}^{*} - \theta _2 \delta _H^{'} \end{aligned}$$

(51)

The fulfillment of this inequality is particularly dependent on the parameter q which indicates the quality of the analyst’s valuation technology. It is clear that if $q=0$, that is the analyzing technology is worthless, the inequality is always fulfilled. That means in this case the good manager always prefers to defect from the described uninformative equilibrium. Otherwise if $q=1$ the analyzing technology is perfect and the good firm manager would not defect from the uninformative equilibrium because he can trust on the perfect valuation technology. An information transport by signaling the firm type is not necessary in this case. So if the good firm manager wishes to defect from the uninformative equilibrium is obviously dependent on q. We investigate the condition under which the good manager strictly prefers to defect from the uninformative equilibrium, i.e.

$$\begin{aligned}&(q +(1-q)\cdot p) \cdot V_{H1}^{*} + (1-q)(1-p) \cdot V_{L1}^{*} - (1-q)(1-p) \cdot \theta _2 (\mu _H-\mu _L)\\&\quad< V_{H1}^{*} - \theta _2 \delta _H^{'} \\&(q+p-qp)\cdot V_{H1}^{*}+(1-q-p+qp)\cdot V_{L1}^{*}-(1-q-p+qp)\cdot \theta _2 (\mu _H-\mu _L)\\&\quad < V_{H1}^{*} - \theta _2 \delta _H^{'} \\& q(1-p)\cdot V_{H1}^{*}- q(1-p)\cdot V_{L1}^{*}+q(1-p) \cdot \theta _2 (\mu _H-\mu _L)\\&\quad< V_{H1}^{*} - \theta _2 \delta _H^{'} -p \cdot V_{H1}^{*}- (1-p)\cdot V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)\\ & q \cdot \left( (1-p)V_{H1}^{*}-(1-p) V_{L1}^{*}+(1-p)\theta _2 (\mu _H-\mu _L)\right) \\ &\quad< (1-p) V_{H1}^{*} - \theta _2 \delta _H^{'} - (1-p)\cdot V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L) \\& q <\frac{(1-p) V_{H1}^{*} - (1-p)\cdot V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)- \theta _2 \delta _H^{'}}{(1-p)V_{H1}^{*}-(1-p) V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)} \\& q<{\bar{q}} \end{aligned}$$

(52)

where ${\bar{q}}=\frac{(1-p) V_{H1}^{*} - (1-p)\cdot V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)- \theta _2 \delta _H^{'}}{(1-p)V_{H1}^{*}-(1-p) V_{L1}^{*}+(1-p)\cdot \theta _2 (\mu _H-\mu _L)}.$

Obviously ${\bar{q}}<1$. The interpretation of this condition is as follows: If the analyst’s valuation technology is not too accurate, i.e. $q<{\bar{q}}$, the good manager always prefers to defect from the described uninformative equilibrium.²⁸ In this case the probability that the defector is a bad firm is zero according to the Intuitive Criterion by Cho and Kreps. Therefore the only reasonable posterior belief is that the defector is a good firm so that the market’s best response is to price the defector as a good firm. Hence, the good firm will defect, and the described uninformative equilibrium fails the Intuitive Criterion by Cho and Kreps.

Appendix B: Conditions for the existence of an intersection point

This appendix contains the proof of Proposition 2:

The analyst’s cost function provides positive values right from the origin. Costs related to a manipulated report do not arise until the minimum level of uncertainty ($\sigma _{min}^{2}$) is exceeded. Therefore with low uncertainty the analyst’s costs exceed the costs for a manipulated report. Consequently both cost functions intersect, if the following applies: The limit of the analyst’s cost function when applying high values of $\sigma ^2$ has to be lower than the costs of manipulation:

$$\begin{aligned} \lim _{\sigma ^2 \rightarrow \infty } \frac{C^{An}}{q}< & {} \lim _{\sigma ^2 \rightarrow \infty } p \cdot C^{Man}\nonumber \\ \lim _{\sigma ^2 \rightarrow \infty } \left( \frac{-\frac{c_{An}}{\sigma ^2 + 1}+c_{An}}{q}\right)< & {} \lim _{\sigma ^2 \rightarrow \infty } \left( p \frac{c_{Man}}{\sigma ^2}{\delta _H}\right) \end{aligned}$$

(53)

The signaling costs on the right-hand side only affect the good firm , the costs are therefore weighted with probability p. The analyzing technology is not perfect and reveals the firm type with probability q. Therefore the analyzing costs on the left-hand side are divided by q.

Substituting $\delta _H$ with the manipulation in equilibrium $\delta _H^{LRE}$ subject to Eq. (17) yields:

$$\begin{aligned} \lim _{\sigma ^2 \rightarrow \infty } \left( \frac{-\frac{c_{An}}{\sigma ^2 + 1}+c_{An}}{q}\right) < \lim _{\sigma ^2 \rightarrow \infty } \left( p \theta _2 \left( \frac{a_1}{a}\left( \frac{\theta _1}{\theta _2} +1\right) -1\right) (\mu _H-\mu _L)\right) \end{aligned}$$

(54)

Rearranging yields the following condition:

$$\begin{aligned} c_{An} < q p \frac{a_1}{a}\theta _1 (\mu _H-\mu _L) \end{aligned}$$

(55)