Discussion: Herd Behavior and Investment by Scharfstein and Stein (2)

In this second post we will make a start with the model that Sharfstein and Stein made in their paper Herd Behavior and Investment.

Setup of the model

There are two types of managers: smart and dumb. The smart managers receives informative signals about the value of an investment, although imperfect. The dumb managers receive purely noisy signals, i.e. they don’t have a clue about the value of an investment.

Assumption: initially neither the manager themselves or the labour market can identify the types. Thus, a smart managers initially does not know that he is actually smart and neither does anyone else.

However, after an investment decision the labour market can update his beliefs, based on two pieces of evidence.

      1) Did the manager make a profitable investment?

However this piece of evidence is not conclusive, as the signal a smart manager gets is imperfect. Thus, if there are systematically unpredictable component of the investment value, all smart managers could get unlucky and receive a wrong signal. Hence, making an unprofitable investment.

      2) Was the managers behaviour similar to or different from that of other managers ?

Holding the profitability of the investment fixed, managers will be more favourable evaluated if they follow the decisions of others than if they behave in contrarian fashion. Thus an unprofitable decision is not as bad for reputations when others make the same mistake, they ‘share the blame’. To see this remember that smart managers receive informative signals while dumb managers only receive junk.  Hence smart managers will receive correlated signals, while dumb managers do not.

Thus mimicking each other’s behaviour suggest to the labour market that you have received correlated signals and hence are smart.

Timing and information structure

The economy consists of two firms, firm A and B, run by managers A and B. The managers invest sequentially, with A moving first. At date 1, A decides whether or not to make the investment. At date 2, B decides whether to invest or not. At date 3 there are two possible outcomes. The ‘high’ state occurring with prior probability of $\alpha$, in which case the investment yields a profit of $X_h >0$. Or the low state occurring with prior probability of $(1-\alpha)$, in which case the investment yields a profit of $X_l < 0$

In making his decision A has access to his signal. Which can be either good ($S_G$) of bad ($S_B)$. However, the problem is he does not know if he is dumb or smart. If he is smart, occurring with probability $\theta$, the signal is informative in the sense that a good signal is more likely to occur prior to the high state than to the low state. Formally:

$Prob(S_G | x_h, smart) =p $ (the probability of a good signal, given that a high state occurs and the manager is smart. I.e. getting a right signal)

$Prob(S_G | x_l, smart) = q < p$ ( the probability of a good signal, given that a low state occurs and the manager is smart. I.e. getting a wrong signal)

Hence the probability of receiving a good signal given that you are smart is $Prob(S_G|smart) =\alpha p + (1-\alpha)q$

If the manager is dumb, occurring with probability $(1-\theta)$, he receives completely uninformative signals, i.e. he is as likely to receive a good signal prior to a high or low state, or:

$Prob(S_G | x_h, dumb) = Prob(S_G | x_l, dumb) = z$

Hence $Prob(S_G|dumb) = =\alpha z + (1-\alpha)z = z$

The paper assumes that the ex ante distribution of signals is the same for both smart and dumb managers; both are equally likely to receive $S_G$. Thus the actual signal received thus not communicate managers type. or,

$Prob(S_G|smart) = Prob(S_G|dumb)$

$Z = \alpha p + (1-\alpha)q$

To be clear, both are as likely to receive a good signal. However, smart managers have a higher probability receiving a good signal when the state is high (i.e. getting the right signal)

$Prob(S_G | x_h, smart) > Prob(S_G | x_h, dumb)$

$p > z$

But they have a lower probability receiving a good signal when the state is low (i.e. getting a wrong signal)

$Prob(S_G | x_l, smart) < Prob(S_G | x_l, dumb)$

$q < z$

To easily see this let’s look at a numerical example. Assume that the probability of each state occurring is the same or $\alpha$ is 0.5. Assume that p = 0.8 and q = 0.6, hence filling in z gives z = 0.7 ($\frac{1}{2}*0.8 + (1- \frac{1}{2})*0.6$)

In the next post we will continue the model. We will calculate what probability A attaches to the high state when receiving a good signal.