More-on-Bernanke’s-“Bad-News-Principle”.doc

More on Bernanke’s “Bad News Principle”
Yishay D. Maoz*
Department of Economics, University of Haifa
April 2004
Abstract
The role that Bernanke’s Bad News Principle plays in the modern theory of investment under uncertainty is analyzed. The analysis shows that the actual investment dilemma is that by delaying investment firms trade-off a higher present value of earnings for a lower present value of the investment cost, in contrast to previous interpretations of this dilemma. The economic interpretation of the Smooth Pasting Condition is clarified too and found to be representing the trade-off mentioned above. Investment triggers stay intact despite changes in the profit process, if the changes are restricted to the range of sufficiently high profits.

Keywords: Investment, Uncertainty, Option Value, Competition
JEL Classification: D41, D81

*Yishay D. Maoz, Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. Phone: [phone], Fax: [phone], Email: [e-mail]. I am grateful to Moshe Hazan, John Leahy, Joram Mayshar and seminar participants at the Hebrew University of Jerusalem, the University of Haifa and Ben-Gurion University for their help.
1. Introduction
Studying the source of investment cycles, Bernanke (1983) has formulated the “Bad News Principle” which concerns the interaction among the irreversibility of the investment, the uncertainty about its future rewards and the option to delay it. According to the principle: Given the current return, the willingness to invest in the current period depends only on the severity of bad news that may arrive. Just how good is the potential future good news for the investment does not matter at all. (Bernanke, 1983, page 91).
The rationale for the principle is that for any project the policy “enter now” is optimal only if it dominates “delay entry until Good News would arrive”, where “Good News” is the future event in which the value of the project becomes higher than it is now. Since the stream of profits generated once the Good News finally arrives is collected under both policies, the only thing that does make a difference between them is what happens during the Bad News period until the Good News arrives. During that time enter now would generate a stream of profits while the delay policy would lower the present value of the entry cost. Therefore, enter now is optimal only if during this delay time the value of potential stream of profits exceeds the potential reduction in the present value of the entry cost. This is not merely a necessary condition for enter now to be optimal but also a sufficient one because if it holds than it is also asserted that enter now has a positive net value. The reason for that is that if during the delay time the value of the stream of Bad News profits is more than how much the entry cost is discounted, then the value of an infinite stream of such profits exceeds the entire entry cost. In that case the value of the project exceeds the entry cost since this value is based on an infinite stream of profits that is not restricted to the Bad News range.

Since the late 1980s, an extensive literature has returned to the study of the interaction between uncertainty, irreversibility and the option to delay investment, this time with a focus on the impact that uncertainty exerts on investment. For surveys of this literature see Pindyck (1991), Dixit (1992) and Dixit and Pindyck (1994).
In addition to this departure from Bernanke’s focus on the creation of business cycles, most articles in that literature differ from Bernanke’s article in three main technical features. First, they assume that the firm is facing a single investment project, unlike Bernanke who assumes that the firm chooses between several projects. Second, they assign an important role to the irreversibility of adjustment costs, while in Bernanke’s article the irreversibility springs from the property that once a certain project is taken, the firm cannot enter any of the other projects. Third, they focus on cases where the profit flow is characterized by gradual changes, unlike Bernanke’s modeling of the profit flow as a general Markov process. To grasp the distance between Bernanke (1983) and the recent investment literature note that the most influential articles of the first years of this literature do not refer to Bernanke (1983) at all. See for example McDonald and Siegel (1986), Pindyck (1988) and Dixit (1989).
Despite all these differences, the key elements underlying the Bad News Principle – and therefore the principle itself - still exist in the typical articles of this literature. Yet, so far this literature has demonstrated the role of this Principle only in cases of simplified stochastic processes. Dixit, Abel, Eberly and Pindyck (1995) and Dixit and Pindyck (1994) show the Bad News Principle in a two-period model. Drazen and Sakellaris (1999) use a two-state process for that purpose.
My purpose here is to analyze in detail the role that the Principle plays in the typical cases of this literature using a discrete-time version of Dixit’s 1989 influential investment model. See Cox, Ross and Rubinstein (1976), Dixit (1991a) and Dixit (1993) for the presentation of the continuous-time Itô processes as limiting cases of discrete-time processes of the type used in this paper.
, See Leahy (1993), Bar-Ilan and Strange (1996), Kongsted (1996) and Sabarwal (2004) as examples for the numerous articles that contain models based on the main features of Dixit’s 1989 model.
This focus on the Bad News Principle yields several results:
First, it deepens our understanding of the problem of investment under uncertainty. So far, the economic rationale for the mathematically derived main results of the relevant literature has not been fully understood. In the most recent effort to attend to this lack, Dixit, Pindyck and Sodal (1999) have presented a new approach that highlights the similarity of the investment decision to the pricing decision of a firm facing a downward slopping demand curve. They find that the economic meaning of the optimal decision rule is that it represents “a trade-off between a larger versus a later net benefit”. However, this result is restricted to the case where the investment yields a one-time reward. In the more common case, where the investment yields a flow of profits, the application of the Bad News Principle reveals that this higher but later net benefit is Good News that is irrelevant to the investment dilemma and that the firm’s actual trade-off is between a larger benefit and later cost. By delaying investment, the firm gives up the flow of operative profit that could be gained during that time, but lowers the present value of the irreversible investment cost. I also show that this dilemma is in fact what the Smooth Pasting condition represents in the typical continuous-time models of this literature. Although rigorously derived, so far there have been no attempts to assign an economic meaning to this condition.For rigorous derivations of the smooth pasting condition see, among others, McKean’s 1965 appendix to Samuelson (1965), Merton (1973) and Dixit and Pindyck (1994, pp. 130-132). More recently, Sodal (1998) has simplified the derivation of this condition. Yet, he too has abstracted from its economic meaning: the non-technical explanations he provides are restricted to the necessary condition for optimum of differentiating the value function with respect to the investment thresholds.

Second, using the Bad News Principle I show that investment thresholds may remain intact even when the original profit process is changed, as long as the changes are restricted to the good news range. In particular, this result is relevant to the study of the effect of changes in the government regulations on firms that make sufficiently
large returns or, in the individual level, the effect of changes in the upper tax brackets. Dixit (1991b) and Dixit and Pindyck (1994, pages 296-303) have identified a particular case of this result, namely the case of an exogenously enforced price ceiling. Another possibility for changes that are restricted to the Good News range is the endogenous truncation of the price process caused by free entry under competition. Studying such a case, Leahy (1993) has shown that the optimal entry triggers under competition are identical to those of the case where the price process is not truncated by competitive entry. As the analysis preformed here shows, these results by Dixit (1991b), Dixit and Pindyck (1994) and Leahy (1993) are closer than
previously noticed, to the Bad News Principle.
In section 2 I develop the model. In section 3 I use this model to show the applications described above. Some technical proofs are left to the appendix.
2. The Model
Consider a risk-neutral firm with an infinite planning horizon that has an option to enter a certain production project. The entry cost to the project is I. Once the firm enters the project it cannot exit it. Production yields an operating profit, denoted by Pi, in each period. The index i stands for the size of the profit and it is not a time index. The time index is omitted since I assume that the profit process is time-stationary. If at a certain period the profit is Pi, then at the next period the profit is Pi+1 with probability qi, or Pi-1 with probability 1-qi, where Pi ³ Pi-1 "i. Figure 1 is a scheme of this generalized random walk.
The value of the project is the expected sum of the discounted profits. The value of the project when the profit is Pi will be denoted as V(Pi). The following Proposition 1 establishes that V(Pi) is an increasing function of i.
Proposition 1: V(Pi+1) > V(Pi) "i.
Proof : In the appendix. ð

Pi+3

(qi+2)

Pi+2

(qi+1) (1-qi+2)

Pi+1 Pi+1

(qi) (1-qi+1) (qi)

Pi Pi

(1-qi) (qi-1) (1-qi)

Pi-1 Pi-1

(1-qi-1) (qi-2)

Pi-2

(1-qi-2)

Pi-3

t t+1 t+2 t+3

Figure 1: The profit process, starting at period t with the profit Pi.

2.1 The optimal policy
I now turn to characterizing the optimal policy of the firm. Let the current profit be Pm. Assume that the firm has not entered the project yet and it considers delaying entry until the profit is Pn. At the entry time the expected return from this policy is V(Pn)-I, but in the current period the firm does not know the date in which it will enter, and therefore - what is the value of the discount factor that multiplies this expected return. The firm also must consider what is the return that can be collected until the profit reaches Pn. These variables are defined and denoted below:

Em,n - The present value of the stream of profits that the project yields while the profit evolves from its current level, Pm, until it reaches Pn for the first time (excluding the profit Pn received when the process finally hits Pn).There is a positive probability that the price process will never reach Pn.

Bm,n - The value of the discount factor, (1+r)-T, where T is the number of periods until the profit is Pn for the first time, starting at Pm (excluding the period in which the process finally hits Pn).

These definitions enable the following dynamic programming presentation of the project’s value function for each m and n:

(1) V(Pm) º Em,n + Bm,nV(Pn).

Suppose that the current profit is Pi. The standard N.P.V. rule tells us that the policy “enter now” is better than the policy “never enter” if V(Pi)-I>0. However, other policies are possible too, including the policy “enter when the profit is Pi+1 or above it”. The value of this policy when the current profit is Pj, denoted Fi(Pj), satisfies:

(2) Fi(Pj) = Bj,i+1[V(Pi+1) - I].

When the current profit is Pi, “Enter now” is better than this last policy only if:

(3) V(Pi) - I > Fi(Pi).

Fi(Pi), therefore is an alternative cost that can be added to the direct cost of entry, I. It follows from (2) and Proposition 1 that if indeed V(Pi)-I>0, then this alternative cost is positive too. Applying (2) in (3), immediate entry is preferred if:

(4) V(Pi) - I > Bi,i+1[V(Pi+1) - I].

Looking at an equation similar to (4), Dixit, Pindyck and Sodal (1999) have concluded that economic meaning of the optimal decision rule is that it represents “a trade-off between a larger versus a later net benefit”. However, their result is restricted to the case where the investment yields a one-time reward. Here, modifying this investment rule through the Bad News Principle, using (1), we can rewrite it as:

(4’) Ei,i+1 + Bi,i+1V(Pi+1) - I > Bi,i+1[V(Pi+1) - I].

V(Pi+1) appears on both sides of (4’), multiplied by the same term and therefore cancels out. Rearranging this expression we get the following condition for “enter now” (when the profit is Pi) to be better than “delay entry until the profit is Pi+1”:

(5) Ei,i+1 > (1-Bi,i+1)I.

Thus, the exact magnitudes of the elements of Ci º {(Pj, qj); " j > i} are irrelevant to the question whether “enter now” is better than “delay entry until the profit is Pi+1” , when the profit is Pi. The only relevant factors are the cost of the delay – namely, the forgone profits during the delay time, on the left side of (5) - and the benefit form the delay – the expected decrease in the present value of the entry cost. Both are related only to the parameters of the profit process below Pi+1. Note that the case where Pi+1 is not reached at all and the delay saves the entire entry cost is a just one component of the reduction, due to the delay, in the present value of the entry cost.
So far the discussion was limited to the choice between “enter now” and “delay entry until the profit is Pi+1”, given that the value of the current profit is Pi. Next, it is shown that it is unnecessary to discuss policies that suggest a longer delay of entry. To do so, rewrite condition (5) as:

(5’) Mi > I,

where the function Mi is defined by:

(6) Mi º .

To gather an economic meaning for Mi note from (6) that:

(7) .

(7) leads to interpreting Mi as the value of the project in the case where there are no better news than the current profit, Pi. More specifically, assume that the original profit process is replaced by a new one in which: (i) for all j£i all the values of the pairs {Pj, qj} are the same as in the original process; (ii) when the profit is Pi in a certain period then in the next period with probability qi the value of the profit remains Pi, instead of rising to Pi+1.
Under this process, truncated at Pi, it is impossible to continue focusing the analysis on the event in which the profit process hits Pi+1 for the first time, starting at Pi. For this truncated process, the equivalent event is that the value of the profit is Pi for a second consecu