degree of optimism. The following lemma shows how these biases will affect the way that imperfect information is interpreted by the manager Lemma 4.1 An overconfident and optimistic manager thinks that P16=1|5=1 A+(1-A)B= T1(A, B), and =1|8=0}=(1-A)B三m0(A,B), (10) where the subscript "b"refers to the fact that the manager is biased Clearly, TI(A, B)is increasing in both A and B, whereas To(A, B)is decreasing in A and increas- ing in B. This is intuitive. The overconfident manager puts too much weight on his information and, as a result, over-adjusts his beliefs towards his information. He therefore thinks that the oject is better(worse) than it really is after observing S=1(8=0). The optimistic manager, on the other hand, always thinks that the project is better than it really is. He does revise his beliefs upwards(downwards) after a positive (negative) signal but, the resulting posterior is still higher than it should b As before, the manager never considers dropping the risky project in favor of the safe investment in the first or second stages: the worst-case scenario from acquiring more information is that the risky project disappears, and the safe investment can then be made anyway. Thus the decision of the manager in each stage is to decide whether the project should be undertaken early, or more information should be gathered before he makes his mind up about the risky project. In the first stage, more information will be gathered if p is large enough; in the second stage, more information will be gathered if q is large enough, with the knowledge of the imperfect signal s E 10, 1. So the manager's strategy can be fully described by the same three thresholds as before 4.2 Overconfidence Let us first concentrate on overconfidence, that is let us assume for now that the manager does not exhibit any optimism (i. e,, B=2), but potentially some overconfidence (i.e, A> a). At the outset this manager correctly assesses the odds of the risky project being successful; his nfidence does not affect his priors, but only the way he processes information. This means that the overconfident manager, like the rational manager, always chooses to gather some information in the first stage undertaking the project at the outset is worth 5(1)-3r to him, which is less than the 3 that can be generated at any stage by making the safe investment. As discussed above, the manager's
degree of optimism. The following lemma shows how these biases will affect the way that imperfect information is interpreted by the manager. Lemma 4.1 An overconfident and optimistic manager thinks that Prb v˜ = 1 | s˜ = 1 = A + (1 − A)B ≡ π1(A, B), and (9) Prb v˜ = 1 | s˜ = 0 = (1 − A)B ≡ π0(A, B), (10) where the subscript “ b” refers to the fact that the manager is biased. Clearly, π1(A, B) is increasing in both A and B, whereas π0(A, B) is decreasing in A and increasing in B. This is intuitive. The overconfident manager puts too much weight on his information and, as a result, over-adjusts his beliefs towards his information. He therefore thinks that the project is better (worse) than it really is after observing ˜s = 1 (˜s = 0). The optimistic manager, on the other hand, always thinks that the project is better than it really is. He does revise his beliefs upwards (downwards) after a positive (negative) signal but, the resulting posterior is still higher than it should be. As before, the manager never considers dropping the risky project in favor of the safe investment in the first or second stages: the worst-case scenario from acquiring more information is that the risky project disappears, and the safe investment can then be made anyway. Thus the decision of the manager in each stage is to decide whether the project should be undertaken early, or more information should be gathered before he makes his mind up about the risky project. In the first stage, more information will be gathered if ˜p is large enough; in the second stage, more information will be gathered if ˜q is large enough, with the knowledge of the imperfect signal ˜s ∈ {0, 1}. So the manager’s strategy can be fully described by the same three thresholds as before. 4.2 Overconfidence Let us first concentrate on overconfidence, that is let us assume for now that the manager does not exhibit any optimism (i.e., B = 1 2 ), but potentially some overconfidence (i.e., A ≥ a). At the outset, this manager correctly assesses the odds of the risky project being successful; his overconfidence does not affect his priors, but only the way he processes information. This means that the overconfident manager, like the rational manager, always chooses to gather some information in the first stage: undertaking the project at the outset is worth 1 2 (1) − 1 2 r to him, which is less than the 1 2 that can be generated at any stage by making the safe investment. As discussed above, the manager’s 14
overconfidence makes him reach biased beliefs upon learning s. In particular, the manager thinks that the project is worse(better)than it really is after he observes s=0(s= 1. when s 0 therefore, the overconfident manager views the project as less likely to be successful than an otherwise rational manager, and so he has nothing to lose by acquiring more information On the other hand, when s= l, the overconfident manager values the risky project more than an otherwise rational manager. As the following result shows, this affects his choice of a threshold for q Lemma 4.2 Suppose that the firm is managed by an overconfident individual with risk aversion r>0. The information acquisition strategy of this manager is given by thresholds of pov(r, A)=0, Qo(r, A)=0, and 2[A-(1-A) Notice that Qp(r, A)is increasing in A. This is because an overconfident manager believes that his imperfect information is better than it actually is, and so tends to rely on imperfect information more that an otherwise rational manager. More precisely, for q E(Q1(r),Q0(r, A)] the overconfident manager chooses to rely on imperfect information, whereas an otherwise identical but rational manager would choose to gather perfect information before making a decision. Recall from section 3.3 that risk aversion has the opposite effect: when s= l, the manager relies on perfect information more than is optimal for firm value. Therefore, it is possible for overconfidence to have a positive effect on firm value. This will be the case for example when Q9(, A)E(Q1(r),QF] that is when the manager's overconfidence offsets his risk aversion in such a way that his decisions become similar to that of a rational profit-maximizing manager or owner. This positive role of erconfidence is stated more precisely in the following proposition Proposition 4.1 For any risk-averse manager, there is a level of overconfidence, a+(1+a)r A=1+(1+a)∈,1 (12) such that the value of the firm is equal to the first-best value f. The value of the firm is strictly increasing(decreasing) in A for A< A'(A>A%) The last part of this proposition has important implications. In particular, for a given level of risk aversion, it is always the case that some overconfidence helps restore some of the value that is lost to decisions made to reduce risk. This process is not monotonic. Too much overconfidence distorts the decision-making process in that the manager may over-rely on his imperfect information
overconfidence makes him reach biased beliefs upon learning ˜s. In particular, the manager thinks that the project is worse (better) than it really is after he observes ˜s = 0 (˜s = 1). When ˜s = 0 therefore, the overconfident manager views the project as less likely to be successful than an otherwise rational manager, and so he has nothing to lose by acquiring more information. On the other hand, when ˜s = 1, the overconfident manager values the risky project more than an otherwise rational manager. As the following result shows, this affects his choice of a threshold for ˜q. Lemma 4.2 Suppose that the firm is managed by an overconfident individual with risk aversion r ≥ 0. The information acquisition strategy of this manager is given by thresholds of P¯OV(r, A)=0, Q¯OV 0 (r, A)=0, and Q¯OV 1 (r, A) = 2 A − (1 − A)r 1 + A . (11) Notice that Q¯OV 1 (r, A) is increasing in A. This is because an overconfident manager believes that his imperfect information is better than it actually is, and so tends to rely on imperfect information more that an otherwise rational manager. More precisely, for ˜q ∈ Q¯1(r), Q¯OV 1 (r, A) , the overconfident manager chooses to rely on imperfect information, whereas an otherwise identical but rational manager would choose to gather perfect information before making a decision. Recall from section 3.3 that risk aversion has the opposite effect: when ˜s = 1, the manager relies on perfect information more than is optimal for firm value. Therefore, it is possible for overconfidence to have a positive effect on firm value. This will be the case for example when Q¯OV 1 (r, A) ∈ Q¯1(r), Q¯FB 1 , that is when the manager’s overconfidence offsets his risk aversion in such a way that his decisions become similar to that of a rational profit-maximizing manager or owner. This positive role of overconfidence is stated more precisely in the following proposition. Proposition 4.1 For any risk-averse manager, there is a level of overconfidence, A∗ ≡ a + (1 + a)r 1 + (1 + a)r ∈ [a, 1], (12) such that the value of the firm is equal to the first-best value F¯FB. The value of the firm is strictly increasing (decreasing) in A for A<A∗ (A>A∗). The last part of this proposition has important implications. In particular, for a given level of risk aversion, it is always the case that some overconfidence helps restore some of the value that is lost to decisions made to reduce risk. This process is not monotonic. Too much overconfidence distorts the decision-making process in that the manager may over-rely on his imperfect information. 15