Information acquisition and disclosure by firms in the presence of additional available information

Abstract

In this paper we analyze a model which addresses two stylized facts which have received little attention in disclosure theory. (a) Information that is acquired for internal decision-making can subsequently be disclosed to outside investors who can use it to update their assessment of the firm’s prospects. Thus, the decision to gather information in the first place does not only depend on the decision value of information. (b) Information disclosed by firms is only one element of the information environment upon which investors can draw. This setting creates an interaction between firms’ information gathering and disclosure decisions as well as alternative sources of information. We identify an equilibrium structure which we call a Countersignaling equilibrium in which only average firms acquire information whereas good and bad firms do not. We show that while several equilibria can coexist, a Countersignaling equilibrium is often the economically most efficient one.

Appendix

1.1 Proof of Theorem 2

The proof of this theorem can be done backwards. First we consider the prices (iii) in particular we look only at one, the price in the case of no acquisition of the information system and receiving good additional information. Given market beliefs of $\Pr \left( {\left. j \right| g} \right) =\frac{q_j }{\sum \nolimits _{k\in N} {q_k } }$for $j\in N$ it is a rational decision to assess a firm in this case with $P_N \left( {g,NIS} \right)$ given in (5). The same can be shown for $P_N \left( {b,NIS} \right)$ in (6), $P\left( {ND,g,IS} \right) =P\left( {ND,b,IS} \right) =-K$ and the price in the case of disclosure (2). Next, we analyze how the beliefs are formed. If a manager with a project type of $j\in N$ always decides against an information system and a manager of type $i\in M$ chooses to buy it then the beliefs $\Pr \left( {\left. j \right| g} \right) =\frac{q_j }{\sum \nolimits _{k\in N} {q_k } }$ for $j\in N$ in the case of no acquisition and good additional information fulfill Bayes’ rule. The same has to be true for the beliefs in the other situations. Lastly, the beliefs from (ii) and the resulting prices in (iii) leads to the inequalities in (8) and (9) which lead to the rational decisions of the firms given in (i).

1.2 Proof of Lemma 1

• (a) M = {A, C}

First we consider if $M=\left\{ {A,C} \right\}$. From (5) and (6) we get $P_{\left\{ B \right\} } \left( {g,NIS} \right) =P_{\left\{ B \right\} } \left( {b,NIS} \right) = \overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{B}(x)dx$. Thus, from (8) we get $K\ge \int \nolimits _{\underline{x}}^{0} F_B \left( x \right) dx$ and from (9) we get $K\le \int \nolimits _{\underline{x}}^{0} F_B \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx$. However, these two inequalities cannot hold simultaneously because $F_C \left( x \right) \ge F_B \left( x \right)$ and $F_C \left( x \right) \ne F_B \left( x \right)$. Thus, $M=\left\{ {A,C} \right\}$ is not an equilibrium.

• (b) M = {B, C}

Next we consider if $M=\left\{ {B,C} \right\}$ can constitute an equilibrium. From (5) and (6) we get $P_{\left\{ A \right\} } \left( {g,NIS} \right) =P_{\left\{ A \right\} } \left( {b,NIS} \right) =\overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{A}(x)dx$. From (8), $K\ge \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) dx$ follows and from (9) the following inequality can be obtained: $K\le \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_B \left( x \right) -F_A \left( x \right) dx$. These two inequalities cannot hold simultaneously because $F_B \left( x \right) \ge F_A \left( x \right)$ and $F_A \left( x \right) \ne F_B \left( x \right)$.

1.3 Existence of equilibria

• (a) $M=\left\{ {A,B} \right\}$

If $M=\left\{ {A,B} \right\}$ then from (5) and (6), $P_{\left\{ C \right\} } \left( {g,NIS} \right) =P_{\left\{ C \right\} } \left( {b,NIS} \right) =\overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{C}(x)dx$ follows. From (8) and (9): $\int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx+ \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx\ge K\ge \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$ follows. The assumption of first order stochastic dominance guarantees the existence.

• (b) $\left\{ B \right\}$

In order to calculate the minimum of the left side of (10), we set two constrained maximization problems. The first one is: maximize: $-\Psi _A$ with respect to $q_A$ and $q_C$ under the constraints: $\Psi _A \ge \Psi _C , q_A \ge q_C , q_A \le 1$ and $q_C \ge 0$. The second one is: maximize $-\Psi _C$ with respect to $q_A$ and $q_C$ under the condition $\Psi _A \le \Psi _C$, the rest stays the same. This can be solved with the Kuhn-Tucker conditions. The minimum of $max\left[ {\Psi _A ,\Psi _C } \right]$ is attained if:

$$\begin{aligned} q_C&\le \chi \equiv \frac{ \int _{\underline{x}}^{\overline{x}} F_C -F_A dx- \int _{0}^{\overline{x}} F_C -F_A dx}{ \int _{\underline{x}}^{\overline{x}} F_C -F_A dx+ \int _{0}^{\overline{x}} F_C -F_A dx}\quad \hbox { and} \end{aligned}$$

(13)

$$\begin{aligned} q_A&= \phi \nonumber \\&\equiv \frac{q_C \int _{\underline{x}}^{\overline{x}} F_C \!-\!F_A dx\!+\!\left( {1-q_C } \right) \int _{0}^{\overline{x}} F_C \!-\!F_A dx\!+\!\sqrt{\left( { \int _{0}^{\overline{x}} F_C \!-\!F_A dx} \right) ^{2}\!+\!4\left( {1-q_C } \right) q_C \left( { \int _{0}^{\overline{x}} F_C -F_A dx} \right) \left( { \int _{\underline{x}}^{\overline{x}} F_C -F_A dx} \right) }}{ \int _{\underline{x}}^{\overline{x}} F_C -F_A dx+ \int _{0}^{\overline{x}} F_C -F_A dx}\nonumber \\ \end{aligned}$$

(14)

Inserting (14) in $\Psi _A$ or $\Psi _C$ lead to the minimal cost of: $\frac{1}{2} \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) +F_A \left( x \right) dx$. $\Psi _A <\Psi _C$ if $0\le q_C <\chi$ and $\phi <q_A \le 1$. $\Psi _A >\Psi _C$ if $0\le q_C <\chi$ and $q_C <q_A <\phi$ or $\chi <q_C <q_A$ and $q_A \le 1$.

• (c) $M=\left\{ A \right\}$

The boundaries for $K$ can be calculated using Eq. (8) and (9). The minimum value of the lower boundary equals: $\frac{1}{2} \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) +F_B \left( x \right) dx$. The upper limit follows from (9): $\tau \int \nolimits _{\underline{x}}^{\overline{x}} F_B \left( x \right) dx+\left( {1-\tau } \right) \int \nolimits _{\underline{x}}^{\overline{x}} F_C \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_A \left( x \right) dx\ge K$ with: $\tau =\frac{q_A q_B \left( {2-q_B -q_C } \right) +\left( {q_B +q_C } \right) \left( {1-q_A } \right) \left( {1-q_B } \right) }{\left( {q_B +q_C } \right) \left( {2-q_B -q_C } \right) }$.

• (d) $M=\left\{ {A,B,C} \right\}$

For $M=\left\{ {A,B,C} \right\}$ we must first define the out-of-equilibrium beliefs for $P_{\left\{ \right\} } \left( {g,NIS} \right)$ and $P_{\left\{ \right\} } \left( {b,NIS} \right)$. In this equilibrium, the market believes that all firms buy the information system. But what will be the market price if the market observes no acquisition of the information system? If a deviation from the equilibrium strategies is observed the market must hold beliefs about which firm type $i$, with $i\in \left\{ {A,B,C} \right\}$, deviates. Denote the probability that it will be a firm of type $i$ with $\lambda _i$, where $\sum \nolimits _{i\in \left\{ {A,B,C} \right\} } \lambda _i =1$. Thus, in case of good and bad information the market price will be:

$$\begin{aligned} P_{\{\}} \left( g,NIS \right)&= \sum \limits _{i\in \left\{ A,B,C \right\} } \frac{\lambda _i q_i \left( \overline{x}-\int _{\underline{x}}^{\overline{x}}F_{i}(x)dx \right) }{\sum _{j\in \left\{ A,B,C \right\} }\lambda _i q_i}\quad \hbox {and } \nonumber \\ P_{\{\}} \left( b,NIS \right)&= \sum \limits _{i\in \left\{ A,B,C \right\} } \frac{\lambda _i \left( 1-q_i \right) \left( \overline{x}-\int _{\underline{x}}^{\overline{x}}F_{i}(x)dx \right) }{\sum _{j\in \left\{ A,B,C \right\} } \lambda _i \left( 1-q_i \right) }. \end{aligned}$$

(15)

In this equilibrium equation (9) must hold for all $i\in \left\{ {A,B,C} \right\}$ with $P_{\left\{ \right\} } \left( {g,NIS} \right)$ and $P_{\left\{ \right\} } \left( {b,NIS} \right)$ given in (15). This equilibrium does not exists if $K> \int \nolimits _{\underline{x}}^{0} F_C dx$. In this case it is independent of the beliefs, $\lambda _i$, at least for the worst type, $C$, not profitable to acquire the information system.

• (e) $M=\left\{ \right\}$

In order that this equilibrium exists the parameter must fulfill the inequalities in (8). The minimum value for this lower boundary for $K$ equals: $\frac{1}{3} \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) +F_B \left( x \right) +F_A \left( x \right) dx$, this is reached for some combination of $q_A , q_B$ and $q_C$. This pooling equilibrium needs no out of equilibrium beliefs because equation (2) denotes the price in case of deviating.

• (f) $\left\{ C \right\}$

In order that $M=\left\{ C \right\}$ could be an equilibrium, the costs must lie in the following interval:

$$\begin{aligned} max\left[ {\Psi _A ,\Psi _B } \right] \!\le \! K \!\le \! \overline{x}-\int \limits _{0}^{\overline{x}}F_{C}(x)dx - \left( q_{C}P_{AB}(g, NIS)\!+\!(1-q_{C})P_{AB}(b, NIS) \right) \end{aligned}$$

(16)

The lowest possible value of the left side equals $\frac{1}{2} \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) +F_B \left( x \right) dx$. The upper limit follows from (9): $\varphi \int \nolimits _{\underline{x}}^{\overline{x}} F_A \left( x \right) dx+\left( {1-\varphi } \right) \int \nolimits _{\underline{x}}^{\overline{x}} F_B \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) dx\ge K$. The upper boundary has to be higher than $\Psi _A$ and $\Psi _B$. From rearranging these two conditions we get:

$$\begin{aligned} \frac{\left( {q_A -q_B } \right) \int _{\underline{x}}^{\overline{x}} F_B -F_A dx}{\left( {q_A +q_B } \right) \left( {2-q_A -q_B } \right) }\ge \frac{ \int _{0}^{\overline{x}} F_C -F_j dx}{\left( {q_j -q_C } \right) },\hbox { for }j\in \left\{ {A,B} \right\} \end{aligned}$$

(17)

Both inequalities can hold simultaneously, f. ex. suppose $F_C \left( x \right) =F_B \left( x \right) =F_A \left( x \right)$ for $x\in \left[ {0,\overline{x}} \right]$.

1.4 Proof of Theorem 3

1.4.1 Coexistence of equilibria

This subsection shows that some equilibria can coexist with others. To prove the coexistence, it is sufficient to show that there exists parameter values for which equilibria can simultaneously exist. It is not essential to define the whole set of parameter values for which coexistence occur.

• (a) Coexistence with $M=\left\{ B \right\}$

First, we compare $M=\left\{ B \right\}$ and $M=\left\{ {A,B} \right\}$. Suppose the additional information is good in distinguishing between the best and the worst project types and bad in separating a medium project from the worst project, $q_A \rightarrow 1$ and $q_C \rightarrow 0$ and $q_B \rightarrow q_C$. The boundaries for the existence of the equilibrium $M=\left\{ B \right\}$ simplify as well to $\int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx\le K\le \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx+ \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx$. Thus, coexistence of the two equilibria is possible. Comparing the efficiency of these equilibria, we get: $Z\left( {\left\{ B \right\} } \right) -Z\left( {\left\{ {A,B} \right\} } \right) =\frac{1}{3}\left( {K- \int \nolimits _{\underline{x}}^{0} F_A dx} \right) >0$. While the inequality follows from cost domain which is needed for the considered equilibria.

Likewise, equilibria $M=\left\{ B \right\}$ and $M=\left\{ A \right\}$ can coexist. Suppose $q_A \rightarrow 1$ and $q_C \rightarrow 0$ and $q_B \rightarrow q_C$ then the boundaries for the existence of the equilibrium $M=\left\{ A \right\}$ are given by: $\frac{1}{2} \int \nolimits _{\underline{x}}^{\overline{x}} F_B \left( x \right) +F_C \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_B \left( x \right) dx\le K\le \frac{1}{2} \int \nolimits _{\underline{x}}^{\overline{x}} F_B \left( x \right) +F_C \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_A \left( x \right) dx$. The boundaries for the existence of equilibrium $M=\left\{ B \right\}$ simplify to $\int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx\le K\le \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx+ \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx$. Thus, in order that both equilibria exist, the following inequality must hold: $\frac{1}{2} \int \nolimits _{\underline{x}}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx\le \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_A \left( x \right) dx$, otherwise both boundaries for $K$ cannot hold simultaneously. The efficiency of $M=\left\{ B \right\}$ is higher because: $Z\left( {\left\{ B \right\} } \right) -Z\left( {\left\{ A \right\} } \right) =\frac{1}{3}\left( { \int \nolimits _{\underline{x}}^{0} F_B -F_A dx} \right) >0$. The inequality follows from the assumption of first order stochastic dominance.

The existence of parameter constellation for which both $M=\left\{ \right\}$ and $M=\left\{ B \right\}$ hold can be shown by calculating the boundary value for $K$ if $F_B \left( x \right) \rightarrow F_A \left( x \right)$ and $q_B \rightarrow q_C$. In this case the upper boundary converges to: $\overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{A}(x)dx-\left( q_{C} P_{\{A,C\}} (g)+(1-q_{C}) P_{\{A,C\}} (b)\right)$ and the lower boundary converges to the maximal value of $\overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{A}(x)dx-\left( q_{C} P_{\{A,B,C\}} (g)+(1-q_{C}) P_{\{A,B,C\}} (b)\right) , \overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{A}(x)dx-\left( q_{C} P_{\{A,C\}} (g)+(1-q_{C}) P_{\{A,C\}} (b)\right)$ and $\overline{x}-\int \nolimits _{\underline{x}}^{\overline{x}}F_{C}(x)dx-(q_{C} P_{\{A,C\}} (g)+ (1-q_{C}) P_{\{A,C\}} (b))$. All of this three values are smaller than the upper boundary for $K$. This proves the coexistence of these two equilibria. $M=\left\{ B \right\}$ is preferred if the following inequality holds: $Z\left( {\left\{ B \right\} } \right) -Z\left( {\left\{ \right\} } \right) =\frac{1}{3}\left( { \int \nolimits _{\underline{x}}^{0} F_B dx-K} \right) >0$. This inequality shows that compared to $M=\left\{ \right\} , M=\left\{ B \right\}$ is only preferred if the benefits of having the information are higher than the cost of acquiring it for a manager of type $B$.

$M=\left\{ B \right\}$ and $M=\left\{ {A, B,C} \right\}$ can coexist. From the analysis above, we know that there exists out of equilibrium beliefs which lead to the existence of this equilibrium as long as $K\le \int \nolimits _{\underline{x}}^{0} F_C dx$. Likewise, there exist parameter values for which the equilibrium $M=\left\{ B \right\}$ is possible and also $K\le \int \nolimits _{\underline{x}}^{0} F_C dx$ is fulfilled. Comparing the efficiency of the equilibrium $M=\left\{ {A,B,C} \right\}$ with $M=\left\{ B \right\}$ we get: $Z\left( {\left\{ B \right\} } \right) -Z\left( {\left\{ {A,B,C} \right\} } \right) =\frac{1}{3}\left( {2K- \int \nolimits _{\underline{x}}^{0} F_A +F_C dx} \right) >0$. The difference is positive because we assumed that the equilibrium $M=\left\{ B \right\}$ exists, which is only the case if $K\ge \frac{1}{2} \int \nolimits _{\underline{x}}^{0} F_A +F_C dx$. Note that the right side of this inequality is the minimum of $max\left[ {\Psi _A ,\Psi _C } \right]$.

The coexistence of $M=\left\{ B \right\}$ and $M=\left\{ C \right\}$ is not possible. If equilibrium $M=\left\{ B \right\}$ exists, the following inequality must hold:

$$\begin{aligned} \frac{ \int _{0}^{\overline{x}} F_C -F_B dx}{\left( {q_B -q_C } \right) }\ge \frac{\left( {q_A -q_C } \right) \left( { \int _{\underline{x}}^{\overline{x}} F_C -F_A dx} \right) }{\left( {q_A +q_C } \right) \left( {2-q_A -q_C } \right) }. \end{aligned}$$

(18)

From (17) and (18) we get:

$$\begin{aligned} \frac{\left( {q_A -q_B } \right) \int _{\underline{x}}^{\overline{x}} F_B -F_A dx}{\left( {q_A +q_B } \right) \left( {2-q_A -q_B } \right) }\ge \frac{\left( {q_A -q_C } \right) \left( { \int _{\underline{x}}^{\overline{x}} F_C -F_A dx} \right) }{\left( {q_A +q_C } \right) \left( {2-q_A -q_C } \right) }. \end{aligned}$$

(19)

However, this inequality cannot hold for $0\le q_C <q_B <q_C \le 1$ and $F_C \ge F_B \ge F_A$.

• (b) Coexistence with $M=\left\{ C \right\}$

The equilibrium $M=\left\{ {A,B} \right\}$ requires $K\ge \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$. If costs are that high, $M=\left\{ C \right\}$ is not possible. Likewise, the cost limits for equilibrium $M=\left\{ A \right\}$ and $M=\left\{ C \right\}$ are not compatible.

According to the cost interval for equilibrium $M=\left\{ {A,B,C} \right\}$ and $M=\left\{ C \right\}$, a coexistence is possible. Comparing the efficiency between these equilibria lead to: $Z\left( {\left\{ C \right\} } \right) -Z\left( {\left\{ {A,B,C} \right\} } \right) =\frac{1}{3}\left( {- \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) +F_B \left( x \right) dx+2K} \right) >0$ while the inequality follows from the lowest possible value of $K$ for the existence of the equilibrium $M=\left\{ C \right\}$.

The coexistence from $M=\left\{ C \right\}$ and $M=\left\{ \right\}$ is not possible. From $M=\left\{ C \right\}$ we get: $\varphi \int \nolimits _{\underline{x}}^{\overline{x}} F_A \left( x \right) dx+\left( {1-\varphi } \right) \int \nolimits _{\underline{x}}^{\overline{x}} F_B \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) dx\ge K$ while from $M=\left\{ \right\}$ we receive: $\lambda \int \nolimits _{\underline{x}}^{\overline{x}} F_A \left( x \right) dx+\gamma \int \nolimits _{\underline{x}}^{\overline{x}} F_B \left( x \right) dx+\left( {1-\lambda -\gamma } \right) \int \nolimits _{\underline{x}}^{\overline{x}} F_C \left( x \right) dx- \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) dx\le K$. Because $\varphi >\lambda$ and $\left( {1-\varphi } \right) > \gamma$ both inequalities cannot hold at the same parameter values.

• (c) Coexistence with $M=\left\{ \right\}$

$M=\left\{ \right\}$ and $M=\left\{ {A,B,C} \right\}$ can coexist because there exists parameter values for which the lower boundary for $K$ equals: $\frac{1}{3} \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) +F_B \left( x \right) +F_A \left( x \right) dx$ and there exists out of equilibrium beliefs which support the equilibrium $M=\left\{ {A,B,C} \right\}$ as long as $K\le \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$. $Z\left( {\left\{ \right\} } \right) -Z\left( {\left\{ {A,B,C} \right\} } \right) =\frac{1}{3}\left( {- \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) +F_B \left( x \right) +F_C \left( x \right) dx+3K} \right) >0$.

$M=\left\{ \right\}$ and $M=\left\{ {A,B} \right\}$ can coexist because $M=\left\{ {A,B } \right\}$ requires $\int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx+ \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx\ge K\ge \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$ whereas there exists parameter values for which $M=\left\{ \right\}$ is an equilibrium as long as $K\ge \frac{1}{3} \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) +F_B \left( x \right) +F_A \left( x \right) dx$. Comparing the efficiency, we get the following inequality: $Z\left( {\left\{ \right\} } \right) -Z\left( {\left\{ {A,B} \right\} } \right) =\frac{1}{3}\left( {- \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) +F_B \left( x \right) dx+2K} \right) >0$. The inequality follows from the equilibrium conditions for $K$.

For the coexistence of $M=\left\{ \right\}$ and $M=\left\{ A \right\}$ we can apply a similar argument. Comparing the efficiency of $M=\left\{ \right\}$ and $M=\left\{ A \right\}$ we get: $Z\left( {\left\{ \right\} } \right) -Z\left( {\left\{ A \right\} } \right) =\frac{1}{3}\left( {- \int \nolimits _{\underline{x}}^{0} F_A \left( x \right) dx+K} \right) >0$ while the inequality follows from lower boundary for $K$ in equilibrium.

• (d) Coexistence with $M=\left\{ A \right\}$

As we know from above the equilibrium $M=\left\{ {A,B} \right\}$ is independent from the additional information, $r\in \left\{ {g,b} \right\}$. Further, in order that this equilibrium occur the inequality, $\int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx+ \int \nolimits _{0}^{\overline{x}} F_C \left( x \right) -F_B \left( x \right) dx\ge K\ge \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$ must hold. There exist parameter values such that these inequalities together with the inequalities for $M=\left\{ A \right\}$ lead to no contradiction. Comparing the efficiency of the two equilibria leads to the conclusion that $M=\left\{ A \right\}$ is preferred, because $Z\left( {\left\{ A \right\} } \right) -Z\left( {\left\{ {A,B} \right\} } \right) =\frac{1}{3}\left( {- \int \nolimits _{\underline{x}}^{0} F_B \left( x \right) dx+K} \right) >0$, while the inequality follows from $K\ge \int \nolimits _{\underline{x}}^{0} F_B \left( x \right) dx$ which is required for both equilibria.

The upper bound for the cost for the equilibrium $M=\left\{ {A,B,C} \right\}$ is given by $\int \nolimits _{\underline{x}}^{0} F_C dx$ which is independent of the additional information, $r\in \left\{ {g,b} \right\}$ and base on the out of equilibrium beliefs $\lambda _C =1$. Dependent on the parameter values, $K\le \int \nolimits _{\underline{x}}^{0} F_C dx$ is also possible for the equilibrium $M=\left\{ A \right\}$. $Z\left( {\left\{ A \right\} } \right) -Z\left( {\left\{ {A,B,C} \right\} } \right) =\frac{1}{3}\left( {- \int \nolimits _{\underline{x}}^{0} F_B \left( x \right) +F_C \left( x \right) dx+2K} \right) >0$, the inequality follows from the lowest possible $K$ value for equilibrium $M=\left\{ A \right\}$.

• (e) Coexistence with $M=\left\{ {A,B} \right\}$

For $M=\left\{ {A,B,C} \right\} K\le \int \nolimits _{\underline{x}}^{0} F_C dx$ is required while for $M=\left\{ {A,B} \right\} , K\ge \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$ is required. Except for $K= \int \nolimits _{\underline{x}}^{0} F_C \left( x \right) dx$ and out of equilibrium beliefs $\lambda _C =1$, the two equilibria cannot coexist.

1.5 Non-observability of the acquisition of the information system

In this extension we assume that the market only observes with probability $1-p$ whether the manager bought an information system or not. In case of observing, we attain the same approach as above. Thus, if the market learns that the manager refrains from buying the information system and the additional information was good (bad) then market price is given by (5) ((6)). If the market detects that the manager produced the information but decides not to disclose, it values the firm with $-K$. In case of a voluntary disclosure it values the firm according to (2). However, if the market is unable to verify if the information system was installed or not a manager who receives the information discloses it if:

$$\begin{aligned} x-K\ge t_r \end{aligned}$$

(20)

Where $t_r$ is manager’s beliefs about the market price if no acquisition of the information system is observed and the public information turns out to be $r\in \left\{ {b,g} \right\}$. In case of a disclosure the market price is given by (2). In equilibrium the beliefs coincide with actual market price. To simplify notation, we do not distinguish between conjectured and actual market price. The rational market evaluates the firm with its expected value, which is given by:

$$\begin{aligned} t_g&= \frac{ \sum _{i\in N} q_i E_i \left[ {{\tilde{x}}} \right] }{ \sum _{i\in N} q_i + \sum _{j\in M} q_j F_j \left( {t_g +K} \right) } \nonumber \\&\quad +\frac{\sum _{j\in M} q_j \left( {F_j \left( {t_g +K} \right) -F_j \left( u \right) } \right) E_j \left[ {\left. {\tilde{x}} \right| u\le {\tilde{x}} \le t_g +K} \right] }{ \sum _{i\in N} q_i + \sum _{j\in M} q_j F_j \left( {t_g +K} \right) } \nonumber \\&\quad -\frac{ \sum _{j\in M} q_j F_j \left( {t_g +K} \right) K}{ \sum _{i\in N} q_i + \sum _{j\in M} q_j F_j \left( {t_g +K} \right) } \end{aligned}$$

(21)

$$\begin{aligned} t_b&= \frac{ \sum _{i\in N} \left( {1-q_i } \right) E_i \left[ {\tilde{x}} \right] }{ \sum _{i\in N} \left( {1-q_i } \right) + \sum _{j\in M} \left( {1-q_j } \right) F_j \left( {t_b +K} \right) } \nonumber \\&\quad +\frac{ \sum _{j\in M} \left( {1-q_j } \right) \left( {F_j \left( {t_b +K} \right) -F_j \left( u \right) } \right) E_j \left[ {\left. {\tilde{x}} \right| u\le {\tilde{x}} \le t_b +K} \right] }{ \sum _{i\in N} \left( {1-q_i } \right) + \sum _{j\in M} \left( {1-q_j } \right) F_j \left( {t_b +K} \right) } \nonumber \\&\quad - \frac{ \sum _{j\in M} \left( {1-q_j } \right) F_j \left( {t_b +K} \right) K}{ \sum _{i\in N} \left( {1-q_i } \right) + \sum _{j\in M} \left( {1-q_j } \right) F_j \left( {t_b +K} \right) } \end{aligned}$$

(22)

$t_g$ and $t_b$ represent the market’s beliefs about manager’s disclosure strategy, for the situation with a good public signal and for the bad situation, respectively. $u\in \left\{ { {\underline{x}} ,0} \right\}$ reflects two scenarios: $u= \underline{x}$ displays the situation where the abandonment decision is observable by the market and has to be undertaken before the date at which the manager cares about the market price. In this case the manager would not abandon an unprofitable project in order to convince the market that he has no information because a manager with no information would never abandon a project. In the other cases we would have: $u=0$, because the informed manager would abandon the project if expected payoff is negative. This threshold value is unique and $\frac{\sum \nolimits _{i\in N} q_i E_i \left[ {\tilde{x}} \right] }{\sum \nolimits _{i\in N} q_i }>t_g >t_b >u$. Existence and uniqueness follows from the intermediate value theorem. Inequality $t_g >t_b$ follows from the assumption that a firm with higher quality receives the good public signal more likely.¹²

In order that this is an equilibrium, a manager in a firm of type $i\in N$ must have a higher utility from not buying the information system:

$$\begin{aligned}&p\left( {q_i t_g +\left( {1-q_i } \right) t_b } \right) +\left( {1-p} \right) \left( {q_i P_N \left( {g,NIS} \right) +\left( {1-q_i } \right) P_N \left( {b,NIS} \right) } \right) \nonumber \\&\quad \ge p\left\{ q_i F_i \left( {t_g +K} \right) t_g +\left( {1-q_i } \right) F_i \left( {t_b +K} \right) t_b \right. \nonumber \\&\qquad +q_i \left( {1-F_i \left( {t_g +K} \right) } \right) \left( {E_i \left[ {\left. {\tilde{x}} \right| t_g +K\le x} \right] -K} \right) \nonumber \\&\qquad \left. +\left( {1-q_i } \right) \left( {1-F_i \left( {t_b +K} \right) } \right) \left( {E_i \left[ {\left. {\tilde{x}} \right| t_b +K\le x} \right] -K} \right) \right\} \nonumber \\&\qquad +\left( {1-p} \right) \left( {\left( {1-F_i \left( 0 \right) } \right) E_i \left[ {\left. {\tilde{x}} \right| 0\le x} \right] -K} \right) \hbox { for }i\in N \end{aligned}$$

(23)

The left side shows the expected market price in case of not producing the information. With probability $p$ the market is unable to verify if the manager installed the information system or not. Market prices in these cases are given in Eqs. (21) and (22). If the market detects that no information was produced then market price is given by either (5) or (6) depending on whether there was a good or bad signal. The right side reflects the ex ante expected value if the manager bought the information system. If the market is unable to verify the existence of an information system the manager can pool with the firms who have no information system. The manager will do so if inequality (20) does not hold. Thus, with probability $pq_i F_i \left( {t_g +K} \right)$ he will achieve an evaluation of $t_g$ and with probability $p\left( {1-q_i } \right) F_i \left( {t_b +K} \right)$ market price will be $t_b$. However, if market price in case of a disclosure, $x-K$, exceeds the market price in case of no disclosure $t_r$, he prefers disclosing. The expected market prices of theses cases weighted with the probability of occurrence is given by the third and fourth term in curly brackets. With probability $\left( {1-p} \right)$ we have the same situation as already discussed in the previous sections. Simplifying Eq. (23) we derive that an equilibrium requires:

$$\begin{aligned}&\left( {1-p} \right) \left( {K-\Psi _i } \right) \nonumber \\&\quad \ge p\left\{ q_i \left( {1-F_i \left( {t_g +K} \right) } \right) \left( {E_i \left[ {\left. {\tilde{x}} \right| t_g +K\le x} \right] -\left( {t_g +K} \right) } \right) \right. \nonumber \\&\qquad \left. +\left( {1-q_i } \right) \left( {1-F_i \left( {t_b +K} \right) } \right) \left( {E_i \left[ {\left. {\tilde{x}} \right| t_b +K\le x} \right] -\left( {t_b +K} \right) } \right) \right\} \hbox { for }i\in N\nonumber \\ \end{aligned}$$

(24)

while a manager in a firm of type $j\in M$ has to prefer buying the information system:

$$\begin{aligned}&\left( {1-p} \right) \left( {K-\Psi _j } \right) \nonumber \\&\quad \le p\left\{ q_j \left( {1-F_j \left( {t_g +K} \right) } \right) \left( {E_j \left[ {\left. {\tilde{x}} \right| t_g +K\le x} \right] -\left( {t_g +K} \right) } \right) \right. \nonumber \\&\qquad \left. +\left( {1-q_j } \right) \left( {1-F_j \left( {t_b +K} \right) } \right) \left( {E_j \left[ {\left. {\tilde{x}} \right| t_b +K\le x} \right] -\left( {t_b +K} \right) } \right) \right\} \hbox { for }j\in M\nonumber \\ \end{aligned}$$

(25)

Note that the term in curly brackets of the inequalities is positive because the expected value that $x$ is larger than the lower boundary minus this lower boundary is always positive. Thus, if for example the market never observes the installation of the information system (and the manager is not able to communicate the not acquisition) $p=1$, then a manager acquires the information system because this is the dominant strategy independent of the type. For $p\rightarrow 0$ we have the same situation analyzed above. Thus, for $p$ sufficiently small the same equilibria exists as above. Note further, that the left side of the inequality is increasing in $K$ while the right side is decreasing. Additionally, we know that a manager of type $i$ is indifferent if: $K=\Psi _i$ if $\hbox {p}=0$. Therefore, the lower and upper threshold values for the costs $K$ obtained through the inequalities (24) and (25) will be higher than the ones for $\hbox {p}=0$.