Discussion: Herd Behavior and Investment by Scharfstein and Stein (3)

The next step is looking when A should invest. He therefore need to determine given a certain signal what is the  probability of a high state (or low state)?  We have the use bayes rule to calculate this. 

Let us first look at the probability of a high state give a good signal (your signal is right!). Remember you can be either smart or dumb.

$Prob (X_h| S_G) = \mu_G$ =

={ probability of being smart *probability of receiving a good signal, given high state when smart + probability of being dumb *probability of receiving a good signal given high state when dumb} *probability of high state / probability of receiving a good signal

$Prob (X_h| S_G) = \mu_G =\frac{[ \theta p + (1-\theta)z]\alpha}{z}$

Second what is the probability of a high state when your signal is bad (your signal is wrong!)

$Prob (X_h| S_B) = \mu_B =$

= {probability of being smart *probability of receiving a bad signal, given high state when smart + probability of being dumb *probability of receiving a bad signal, given high state when dumb}* probability of high state / probability of receiving a bad signal

$Prob (X_h| S_B) = \mu_B =\frac{[ \theta (1-p) + (1-\theta)(1-z)]\alpha}{(1-z)}$

To make the investment problem interesting it is assumed that if you receive a good signal the investment is attractive. While you receive a bad signal it is not. This implicates two inequalities:

1) when receiving a good signal it is attractive to invest: $\mu_Gx_G + (1-\mu_G)X_L > 0$  or $\mu_Gx_G > -(1-\mu_G)X_L$ (remember $X_L$ is negative, thus this term becomes positive. It may be interpreted as the absolute value of making an investment in a low state (the wrong decision, its a loss, you do not want to invest in a low state!)

Note that $(1-\mu_G)$ is equal to the probability of a low state given a good signal $prob (X_L | S_G)$. Check it yourself !

2) Subsequently, when receiving a bad signal it is not attractive to invest:

$\mu_BX_H +(1-\mu_b)X_L < 0$ or $\mu_BX_H < -(1-\mu_b)X_L$

To clarify lets continue our numerical example from our previous post. We had

$p = 0.8; q = 0.7; z =0.6, \alpha = 0.5$

Further assume:

$\theta = 0.5, X_H =30, X_L = -30$

1)

Calculating $Prob (X_H| S_G) = \mu_G =$

$\frac {[ 0.5(1-0.8)+(1-0.5)(0.6) 0.5]}{(0.6)} \approx 0.536$

$(1-\mu_G) \approx 0.464$

Filling in assumption:

$=0.536*30 > -0,464*-30$

$\approx 16.07 >13.93$

Thus presuming that a good signal indicates a high state, your investment is expected to gain you 16.07 (indeed high state) and losses you 13.93 (your signal was wrong, although you got a good signal there was actually a low state). When getting a good signal you thus want to invest, the expected gain is larger than the expected loss.

2)

Calculating $Prob (X_H| S_B = \mu_B = 0.417$

$(1-\mu_b) = 0.583$

$= 0.417*30 <-0.583*-30$

$\approx12.5 < 17.5$

Thus presuming that a bad signal indicates a low state, not investing 'saves' you an expected 17.5 (there was indeed a bad state), however if you had invested you could have gained an expected 12.5 (your signal was wrong, although you got a bad signal there was a high state). When getting a bad signal you do not want to invest. The expected loss is greater that the expected gain.

Thus following your signal, when good signal presuming a high state, when a bad signal presuming a low state, gives larger absolute payoffs than not doing so. So manager A when getting a good signal should invest, and when getting a bad signal he should not!

A nice little exercise may be to calculate what the payoffs are when managers do the opposite, i.e. when there is a good signal presume a low state, when there is a bad signal presume a high state. Tip: the payoff should be negative !

I hope you still follow..

Now, manager B can besides looking at his signal also observe what manager A did. Now before solving the model we need one more assumption.

Like manager A, B can be either dumb or smart. If one manager is smart and the other is dumb the signals are independently drawn. Thus the probability that both the dumb as smart manager receive a good signal in the high state is $pz$. If both are dumb this is also the case, so for example the probability that two dumb managers both observe a good signal is $z^2$.

However, if both managers are smart, they are assumed to observe exactly the same signal. Thus the probability that two smart managers both get a good signal and the state is high is $p$, and not $p^2$ if they were independently drawn.  Without this assumption there would not be herd behaviour in the model. The assumption implicates that there are unpredictable factor influencing the future state that nobody can know about.

In the next post we will define the managers objectives, which will finally allow us to look at the equilibriums of the model. If you have any question do not be afraid to comment.