سرمایه گذاری غیر قابل برگشت و عدم قطعیت Knightian

When firms make a decision about irreversible investment, they may not have complete confidence about their perceived probability measure describing future uncertainty. They may think other probability measures perturbed from the original one are also possible. Such uncertainty, characterized by not a single probability measure but a set of probability measures, is called “Knightian uncertainty.” The effect of Knightian uncertainty on the value of irreversible investment opportunity is shown to be drastically different from that of traditional uncertainty in the form of risk. Specifically, an increase in Knightian uncertainty decreases the value of investment opportunity while an increase in risk increases it.

The investment decision of any firm typically involves three features. First, future market conditions are uncertain. Second, the cost of investment is sunk and thus investment is irreversible.Third, investment opportunity does not vanish at once and when to invest becomes a critical decision. This irreversibility of investment under uncertainty and resulting optimal investment timing problem have attracted considerable attention in recent years, especially after McDonald and Siegel [15] successfully applied financial option pricing techniques to this problem and Dixit and Pindyck [4] related option-theoretic results to neoclassical investment theory. Most irreversible investment studies, however, assume more than that future market conditions are uncertain. In these studies, future uncertainty is characterized by a certain probability measure over states of nature. This amounts to assuming that the firm is perfectly certain that future market conditions are governed by this particular probability measure. However, this assumption may be farfetched: the firm may not be so sure about future uncertainty. It may think other probability measures are also likely and have no idea of the relative “plausibility” of these measures. Uncertainty that is not reducible to a single probability measure and thus characterized by a set of probability measures is often called Knightian uncertainty (see [14,12,13]), or ambiguity in some cases. In contrast, uncertainty that is reducible to a single probability measure with known parameters is referred to as risk . That is, a firm may face Knightian uncertainty in contemplating investment, facing not a single probability measure but a set of probability measures. The purpose of this paper is to show that the effect of uncertainty on the value of irreversible investment opportunity differs drastically between risk and Knightian uncertainty. Specifically, the standard result that increase in uncertainty increases the value of irreversible investment opportunities is reversed if uncertainty is not risk but Knightian uncertainty. That is, an increase in Knightian uncertainty (properly defined) reduces the value of an irreversible investment opportunity, while the opposite is true for an increase in risk in the form of an increase in variance. In contrast, both of them have the same effect on the value of waiting: they increase the value of waiting and make it more likely. In this paper,we take a patent as an example of irreversible investment. To highlight the effect of Knightian uncertainty, the firm is assumed to be risk-neutral but uncertainty-averse in the sense that it computes the expected profit by using the “worst” element in the set of the probability measures characterizing Knightian uncertainty and chooses its strategy to maximize it (maximin criterion). 1 Following the standard procedure of irreversible investment studies, we assume that (1) to utilize a patent, the firm has to build a factory and construction costs are sunk after its completion, and (2) the profit flow after the construction is characterized by a geometric Brownian motion with a drift. Then, the firm first calculates the value of the utilized patent, and then contemplates when to build a factory by taking into account the value of the utilized patent and the cost of investment. The firm’s problem is thus formulated as an optimal stopping problem in continuous time. 2 Unlike the standard case, however, we assume that the firm is not perfectly certain that the profit flow is generated by a particular geometric Brownian motion with say, variance 2 and drift , or equivalently, by a probability measure underlying this geometric Brownian motion,say P. The firm may think that the profit flow is generated by other probability measures slightly different from P. The firm has no idea about which of these probability measures is “true.” Thus, the firm faces Knightian uncertainty with respect to probability measures characterizing the profit flow. We assume that the firm thinks these probability measures are not far from P. Firstly,we assume that these probability measures agree with P with respect to zero probability events. (That is, if a particular event’s probability is zero with P, then it is also zero with these probability measures.) Then, these probability measures can be shown as a perturbation of P by a particular “density generator.” Second, the deviation of these probabilities from P is not large in the sense that the corresponding density generator’s move is confined in a range, [−, ], where  can be described as a degree of this Knightian uncertainty. This specification of Knightian uncertainty in continuous time is called -ignorance by Chen and Epstein [3] in a different context. These two assumptions, though they seem quite general, have strong implications. Under the first assumption, for each of the probability measures constituting the firm’s Knightian uncertainty, the profit flow is characterized by a “geometric Brownian motion” of the same variance 2 with respect to this probability measure. Thus, “geometric Brownian motions” corresponding to these probability measures differ only in the drift term. (In fact, this is a direct consequence of wellknown Girzanov’s theorem in the literature of mathematical finance. See for example, [11].) Under the second assumption, the minimum drift term among them becomes  − . Note that the uncertainty-averse firm evaluates the present value of the patent according to the “worst” scenario. Loosely speaking, this amounts to calculating the patent’s value using the probability measure corresponding to a geometric Brownian motion with variance 2 and minimum drift  − . Thus, an increase in , the degree of Knightian uncertainty, leads to a lower value of the utilized patent at the time of investment, since it is evaluated by a less favorable Brownian motion process governing the profit flow from the utilized patent. Consequently, the value of the unutilized patent is also reduced. This is in sharp contrast with the positive effect of an increase in risk (that is, an increase in ) on the value of the unutilized patent, when there is no Knightian uncertainty. An increase in  under no Knightian uncertainty implies, when the firm waits, it can undertake investment only when market conditions are more favorable than before (since it does not have to undertake investment when market conditions are less favorable). Consequently, an increase in  increases the value of the unutilized patent. Despite such differences, both an increase in risk and in Knightian uncertainty similarly raise the value of waiting and thus make the firm more likely to postpone investment. However, the reason for waiting is critically different. An increase in risk () under no Knightian uncertainty leaves the value of a utilized patent unchanged but increases the value of an unutilized patent, and thus makes waiting more profitable. An increase in Knightian uncertainty () reduces both the value of the utilized patent and that of the unutilized patent, but it lowers the former more than the latter. This is because the value of the unutilized patent depends not only on the proceeds from undertaking investment (the utilized patent), but also on the proceeds from not undertaking investment, which is independent of the value of the utilized patent. Since the value of the utilized patent is reduced more than that of the unutilized patent, the firm finds waiting more profitable. While in the current paper an increase in Knightian uncertainty raises the value of waiting, the opposite holds true in Nishimura and Ozaki’s [19] search model, which is set up in a discretetime infinite-horizon framework. They show that an increase in Knightian uncertainty lowers the reservation wage and hence shortens waiting. The value of waiting is thus reduced.Although both the job search model in Nishimura and Ozaki [19] and the irreversible investment models in the current paper are formulated as optimal stopping problems, there is a fundamental difference between the two in the nature of uncertainty. In Nishimura and Ozaki [19], the decision-maker determines when she stops the search and resolves uncertainty. Thus, an increase in Knightian uncertainty makes the uncertainty-averse decision-maker more likely to stop the search and to resolve uncertainty. In contrast, in the current paper, the decision-maker contemplates when to begin investment and face uncertainty. Thus, an increase in Knightian uncertainty makes the uncertainty-averse decision-maker more likely to postpone investment to avoid facing uncertainty. This paper is organized as follows. In Section 2, we present a simple two-period, two-state example and explain intuitions behind the result of this paper. In Section 3, we formulate the firm’s irreversible investment problem in continuous time. In the same section, we formally define Knightian uncertainty in continuous time, derive an explicit formula for a utilized patent, and investigate the optimal investment timing problem. In Section 4, we conduct a sensitivity analysis and present the main result of this paper: differing effects of uncertainty between an increase in risk and Knightian uncertainty. Appendix A contains derivations of important formulae in Section 3. The concept of rectangularity of a set of density generators, 3 of which the -ignorance is a special case, plays an important role in our analysis. Appendix B provides some results on rectangularity for the sake of readers’ convenience and to make exposition self-contained.

Although both the job search and irreversible investment models are formulated as optimal stopping problems, there is a fundamental difference between the two as to the nature of uncertainty. In the job search model, the decision-maker determines when to stop the search and thus resolve uncertainty. Thus, an increase in Knightian uncertainty makes the uncertainty-averse decision-maker more likely to stop the search and resolve uncertainty. In contrast, in the irreversible investment model, the decision-maker contemplates when to begin investment and face uncertainty. Thus, an increase in Knightian uncertainty makes the uncertainty-averse decision-maker more likely to postpone investment to avoid facing uncertainty.