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Investment Under Uncertainty: Keeping One's Options OpenAuthor(s): R. Glenn HubbardSource: Journal of Economic Literature, Vol. 32, No. 4 (Dec., 1994), pp. 1816-1831Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/2728795 .

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Journal of Economic Literature Vol. XXXII (December 1994), pp. 1816-1831

Investment Under Uncertainty: Keeping One's Options Open

By R. GLENN HUBBARD

Columbia University and the National Bureau of Economic Research

I am grateful to Avinash Dixit, Mark Gertler, Kevin Hassett, Charles Himmelberg, Anil Kashyap, Gilbert Metcalf, and Robert Pindyck for helpful comments and suggestions, and to the Federal Reserve Bank of New York and the John M. Olin Visiting Profes- sorship at the Center for the Study of the Economy and the State of the University of Chicago for support.

1. Introduction and Overview of Book

CONSIDERABLE INTELLECTUAL attention has been focused on models of de-

rivative financial instruments, including options. The theoretical intuition embodied in early option pricing models has been applied to other financial decisions by individuals and businesses as well. Despite these applications, options as financial instruments are not central to the lives of most of us. However, a wide variety of options is: We all face significant choices about whether we should spend our resources today or wait, thereby "keeping our options open."

In an important new book, Dixit and Pin- dyck (1994) illustrate how modern "options" intuition can be used to analyze a number of individual and business decisions. Focusing their attention on investment decisions, they offer both a cogent methodological discussion of a new view of investment theory (to which they are individually major contributors) and

a cook's tour of practical applications. The book should be required reading both for researchers interested in investment models and for business school professors teaching capital budgeting. Portions of the book are also appropriate for practitioners, particularly given the intuitive presentation of much of the material.1 Readers interested in a non- technical treatment of many of the ideas are offered guidance for so doing, and purely methodological issues (e.g., solution techniques) are treated in separate chapters.

Investment Under Uncertainty provides both economic analysis and the clear message that such analysis be advanced broadly as a new view for studying investment. Accord- ingly, I believe it is appropriate to judge the book's contributions in three respects. First, are the theoretical advances offered in response to practical problems not addressed by earlier approaches? Second, does the theory offer predictions consistent with observed investment decisions? Finally, are there empirical tests that could test the predictions of the new view against those of conventional models? To summarize this review in ad-

Investment Under Uncertainty. By AVINASH K. DIXIT AND ROBERT S. PINDYCK. Princeton: Prince- ton University Press, 1994. Pp. xiv, 486. $39.50. ISBN 0-691-03410-9. Another recent survey of models in this literature can be found in Pindyck (1991).

1 The authors even offer a literary etymology of "one-hoss-shay depreciation" (p. 205).

1816



Hubbard: Investment Under Uncertainty 1817

vance, I will argue that the answers are "yes," "yes," and "quite possibly."

Before examining the book's analytical ar- guments and developing more formal answers to the questions just posed, let me offer a quick guide to reading the book. For all readers, Chapter 1 offers a careful and interesting description of the differences between the familiar neoclassical investment model and the new view embodied in the authors' "real options" approach. The chapter is useful for practitioners as well as academic economists because it offers intuitive applications to such questions as why business investment appears to be relatively unresponsive to interest rates, when a firm should abandon an investment program, why hysteresis (or path depen- dence) may be an important consideration in investment decisions, why effects of public policies (including tax, trade, and antitrust interventions) may have complex effects on investment, and (for the true homo eco- nomicus) why decisions regarding divorce and suicide involve option values.

The second chapter should also be part of the reading by both researchers and practitioners. The chapter uses simple two-and three- period examples to illustrate why, under assumptions of uncertainty and irreversibility, a decision to wait is logically part of a value- maximizing investment decision. It is also in this chapter that Dixit and Pindyck develop an analogy between real options in investment decisions and financial options. Chap- ters 3 and 4, likely of greater interest for researchers (though, again, the writing incor- porates a minimum of technical detail), gen- eralize models of uncertainty hinted at in the earlier numerical examples. Chapter 3 offers an introduction to stochastic processes, while Chapter 4, which addresses optimal sequential decision making under uncertainty, intro- duces the basics of dynamic programming. These chapters present to the interested reader a list of technical references, but they are a relatively self-contained guide to stochastic processes and dynamic optimization under uncertainty.

The crux of the book's analysis lies in Chapters 5-7, which outline the new view's theory of investment. As I explain in more detail in Section 2, the neoclassical invest-

ment model suggests that a firm should increase its capital stock when the market value of the capital assets exceeds their replacement cost. Tobin's q, the ratio of the market valuation of capital to its replacement cost, is a convenient and often-used summary statistic in the neoclassical approach. In the new view, the investment decision is a choice to incur a sunk cost (because investment is assumed to be irreversible), a choice which yields uncertain future returns. Because the firm has the option of delaying the investment, it should increase its capital stock only if q exceeds unity by a margin sufficient to compensate the firm for the loss of the option to delay. Dixit and Pindyck show that, under certain assumptions, threshold values of q are quite large, implying very high "hurdle rates" consistent with those found in interviews with managers responsible for capital budgeting (see Lawrence Summers 1987).

The basic model introduced in Chapter 5 relies on a simplifying assumption of complete irreversibility of capital investments. In fact, firms have a way out; they can temporar- ily or permanently scrap sufficiently unprofit- able projects. Chapter 6 presents an extension to the case of "suspension": A firm may discontinue a project's operation with the option of renewing it in the future. Hence, two options are part of the investment decision, the initial option of delay and a sequence of operating options (related to suspension decisions). Chapter 7 pursues this extension for the case of "abandonment," which entails the extinguishing of the option of restoring the project at some point. In practice, of course, firms typically must choose between suspension (with ongoing, say, maintenance expen- ditures) and abandonment (with costs of severance and the lost option). Dixit and Pin- dyck describe analytically the choice among a firm's operating, suspending, or abandoning a project, then make their points more con- cretely in an illustration of investment decisions in crude oil tankers.

While Chapters 5-7 present the book's core material on real options in firm decisions, they beg the question of how an industry equilibrium (at least outside of a simple monopoly case) might be characterized in the presence of these options. This important



1818 Journal of Economic Literature, Vol. XXXII (December 1994)

subject is treated in Chapters 8 and 9. The essential complication is this: In the context of a competitive industry, two types of uncertainty are relevant and must be distinguished, aggregate (or industry-level) uncertainty and uncertainty specific to the firm (see the discussion in Section 5 that follows).

If, indeed, investment responds sluggishly to news in the presence of uncertainty, one might inquire whether government intervention could increase efficient investment. It depends: Such intervention will improve effi- ciency only if a benevolent government has different opportunity costs of waiting from private agents. Chapter 9 addresses potential market failures that might stimulate intervention. As is common in academic discussions (though, lamentably, not in public policy discussions) of intervention, Dixit and Pindyck show that only very precise policy tools can address market failures associated with incomplete markets for risk.2 Indeed, to the extent that public policy itself becomes uncertain, incentives to postpone investment arise. (I elaborate on this point in the context of tax policy in Section 6.)

The analysis in Chapters 5-9 focuses on investments that involve essentially a single decision. Chapters 10 and 11 extend the exami- nation to "sequential" and "incremental" investment, respectively. In the former case, a firm may complete projects in a sequence as it updates its expectation of future profitability. As the firm commences an investment sequence, most costs are not yet sunk; hence, the firm will proceed only in response to very high expected future profitability. Over time, as more project steps are in place, proceeding is justified by a smaller profitability threshold. An example of this process is the familiar learning curve, in which production costs de- crease with cumulative output. In this setting, an increase in uncertainty diminishes the value of future declines in cost, slowing down the rate of investment. The analysis of incremental investment in Chapter 11 focuses on choices for capacity expansion. Using their

own work as well as recent contributions by others, Dixit and Pindyck illustrate the relationship between optimal capacity expansion in the new view and familiar models based on "adjustment costs."

Chapter 12, the book's concluding chapter, emphasizes applications, including the development of oil reserves and the choice by elec- tric utilities between investing in scrubbers or purchasing tradeable emission permits to meet emission standards. It is only at the very end of the monograph that general empirical implications of uncertainty and irreversibility for analyses of investment are discussed. While econometric testing of the theory against other investment models is in its in- fancy, more direction as to how such tests might be developed would have been useful.

In the interest of brevity, I focus my attention on what I believe to be the book's major contributions. The balance of the review is organized as follows. To fix ideas, Section 2 describes the familiar neoclassical model and summarizes complications introduced by uncertainty and irreversibility. Section 3 reviews the roles played by uncertainty and irreversibility in a simple new view investment model; Section 4 generalizes this discussion of the firm's investment problem. The leap to predictions regarding investment in a competitive industry equilibrium occurs in Sec- tion 5. Section 6 discusses consequences of policy intervention and policy uncertainty in the new view, focusing on tax policy. Section 7 suggests empirical tests that may help distinguish between predictions of irreversibility models and other investment models. Section 8 concludes.

2. From the Neoclassical Model to the New View

Before discussing the analytical advances summarized in Investment Under Uncer- tainty, let me set the stage by reviewing the basic predictions of the traditional neoclassical model.3 Neoclassical models of invest-

2 For example, a competitive equilibrium under uncertainty can be consistent with periods of supernormal profits or losses, calling into question antitrust or trade policies that focus on snapshot measures of industry equilibrium.

3 I, intend the term "neoclassical model" to refer both to the models I discuss herein and to standard net present value rules frequently taught in business schools (see, e.g., Richard Brealey and Stewart Myers 1991).




ment begin with the intuition of marginal valuation basic to economic analysis: In this case, a firm should invest up to the point at which the marginal cost of capital just equals the marginal return to capital. Going from intuition to practice has been the subject of a significant body of applied research on valuation of marginal units of capital and firms' costs of capital, technological descriptions of production, and effects of tax policy on marginal benefits and cost of investing (see, e.g., the reviews in Robert Chirinko 1993, and Jason Cummins, Hassett, and Hubbard 1994).

Applied analyses of neoclassical investment models generally fall into two groups. The first follows the tradition of the "user cost of capital" approach of Dale W. Jorgenson (1963) and Robert Hall and Jorgenson (1967). Treating capital investment as a purchase of a durable good, Jorgenson defines the user of cost of capital to be the rental cost of the capital (determined by the purchase price, opportunity cost of funds, depreciation rates, and taxes). Firms' desired stocks of capital are determined by the equality of the value of the marginal product of capital and the user cost of capital. Transforming the theory's intuition to a model of "investment" requires additional assumptions to generate dynamics, such as "delivery lags" or "costs of adjustment."

The other approach, the origin of which traces to James Tobin (1969), compares the replacement cost of a marginal investment to its capitalized value. Tobin's q, or marginal q, is the ratio of this capitalized value to the replacement cost of the investment.4 Tobin's q approach provides a simple rule to guide investment: If q > 1, the firm should invest, and, if q < 1, the firm should not invest and should shrink its existing capital stock. The firm's equilibrium capital stock is achieved when q = 1. As with the user cost approach, tax parameters can be introduced in the defi- nition of q. Also like the user cost approach,

the q theory yields a testable model of investment when dynamics (e.g., adjustment costs) are imposed.

Both variants of the neoclassical model rely on the net present value rule. A firm should undertake investment projects with positive net present value. They make two subtle assumption as well: First, invested capital can be sold easily to other users (that is, it is reversible). Second, each investment opportunity facing the firm is a once-and-for-all opportunity; if the firm declines the project, it will never have the choice to reconsider.

The starting point for the "new view" stressed by Dixit and Pindyck is that many real-world investment decisions violate these subtle assumptions, and irreversibility and a chance for delay are important consider- ations. This importance reflects the observa- tion that the possibility of delay gives rise to a call option: The firm has the right, though not the obligation, to buy an asset (the investment project) at some future time at its dis- cretion. To the extent that investment is irreversible-a feature of the new view models-making an investment extinguishes the value of the call option, or "real option," in the terms of Dixit and Pindyck. (If investment were reversible, the neoclassical model's guidance would be more applicable.) The value of the lost option is a component of the opportunity cost of investment. In the terminology of the Tobin's q approach, the threshold criterion for investment requires that q exceed unity by the value of maintaining the call option to invest. Indeed, this additional component may account for the high "hurdle rates" required by corporate managers actually making investment decisions.

An easy, though not satisfying, response to this argument is that the neoclassical model could be modified to incorporate the real option component. Even with this semantic change, it is still necessary, to the extent that the real option is valuable, to analyze how it might be priced in firms' decisions.

3. Irreversibility, Delay, and the New View

The "real option" approach to studying investment under uncertainty relies on the con-

4 Empirical research generally focuses on average q, the ratio of the market value of a firms' capital stock to the replacement cost of the capital stock. Under certain assumptions, average q and marginal q are equivalent (see, e.g., Fumio Hayashi 1982).




cepts of irreversibility and delay. The significance of each is, of course, an empirical question. The illustrations of irreversibility (or, equivalently, sunk costs) are not hard to pro- vide in many cases, including firm-specific marketing costs. The costs of even plant and equipment investments may be sunk. This is easy to see in the case of industry-specific capital when aggregate uncertainty is important. Even less specific capital such as com- puters or general use machinery are (at least partially) irreversible if secondary markets are inefficient. Such markets may be inefficient because of classic "lemons" problems of adverse selection or because regulatory or in- stitutional impediments. As with irreversibility, a value to "delay" is also plausible on a priori grounds. While delay entails possible costs (say, lost returns from a project or entry by competitors), it confers benefits in the form of new information about the project's value during the period of delay.

3A. A Simple Example

To make the significance of irreversibility and delay more concrete, let me use a simple two-period example, followed by a more formal basic model; similar motivation can be found in Chapters 2 and 5. The managers of the Ilova Watch Company are deciding whether to invest in a new watch factory. To fix ideas, let us assume that the investment is totally irreversible; that is, Ilova will not be able to recover any of the investment. A new watch factory costs $800 and produces one watch each year in perpetuity with no operating cost. (Oh, those efficient Swiss!) The current price of Ilova watches is $100, but the price will change next year to $150 with probability one half and to $50 with probability one half, the new price then remains forever. The firm's discount rate is 10 per- cent.

Using the neoclassical model's intuition, note that the expected future price of watches is $100, so that the present value of the factory's revenue is $100 + 100/0.10, or $1100. Because the factory costs $800, the proposed project's net present value is $300, so that the factory should be built. In the context of the formal neoclassical models to

which I referred earlier, note that: (1) the marginal return from the investment each year ($100) exceeds the user cost (0.10 x $800 = $80), and (2) the project's q ($1100/$800) exceeds unity. Those criteria support the decision to invest.

These simple calculations ignore the opportunity cost of investing now in the form of the lost option to delay. As an alternative, suppose that Ilova's managers wait a year and build the factory only if the price of watches rises from $100 to $150. Now, the present value of the cost of the investment is ($800/1.1), and the present value of returns

00

from investing is I 150/(l.l)t yielding a net t=1

present value of $386, which is greater than the net present value of $300 in the "invest now" case. Likewise, Tobin's q in the "delay" case (1650/727, or 2.27) exceeds the q for the "invest now" case. Its value comes jointly from uncertainty about returns and the assumptions of irreversibility and preservation of the investment project even with delay. The "real option"-the value of the call option of delay-is $386. The less completely irreversible is the investment, the lower is the value of waiting. An increase in uncertainty over price (a mean-preserving spread in the distribution for the next-period-and-forever- more watch price) increases the value of the option to delay. This description of changes in the value of the option to delay follows closely related effects on the value of financial call options (see Dixit and Pindyck 1994, Ch. 2; Brealey and Myers 1991; and Hubbard 1993, Ch. 9).

One could add other sources of uncertainty to the two-period example just described, including uncertainty over costs (as in most research and development projects) or over interest (discount) rates. The latter is of additional interest because it provides a way to consider effects of uncertain tax policy on the level and timing of investment. In all cases, though, the intuition is that the value- maximizing investment decision must consider the value of the option of delay when comparing the marginal benefit and cost of investing (or, equivalently, when calculating the project's net present value).




3B. A Simple Formalization

A simple and instructive formalization of the roles played by irreversibility and delay is the model of Robert McDonald and Daniel Siegel (1986).5 McDonald and Siegel analyzed when a monopoly firm should pay a sunk cost I to obtain an investment project value V, where V is uncertain in future periods. This evolution is governed by a Wiener process, or Brownian motion, which is a continuous-time stochastic process commonly used for three reasons of analytical tractabil- ity. First, the process is a Markov process; that is, the probability distribution of future values of V is a function only of the current value of V. Second, the probability distribution for a change in the process over a time interval is independent of other time inter- vals (as long as they do not overlap). Third, over any given interval of time, the changes in the process are normally distributed, and the variance increases linearly with the time interval.6

McDonald and Siegel use a process of geometric Brownian motion with drift:

dV=aVdt+aVdz, (1)

where dt is a time increment, dz is the increment of a Wiener process, and ot and a are constants. The process for V describes the arrival of new information over time and is consistent with an uncertain future value of V.

The analytical advantage of this setup is clear from the perspective of the new view. The investment opportunity considered by the firm is a call option with no expiration date; that is, the firm has the right, though not the obligation to undertake the project at a prespecified price 1. At the same time, this simple setup has some drawbacks in describing typical investment projects. For example, if there are variable costs and the firm can close the factory in periods when price is below variable cost, the process for V is not geo-

metric Brownian motion even if that for the output price is. Second, if the firm had competitors, the price could not depart too much from the industry's long-run marginal cost, suggesting that the process for V may be more complicated. Let me defer these con- cerns for now. (In Investment Under Uncer- tainty, they are addressed in Chs. 6-9.)

In what follows, I describe the solution to the McDonald-Siegel problem7 by a dynamic programming, and draw an analogy to the predictions of the standard neoclassical model. Recall that irreversibility and delay lead to the creation of an option to invest;8 call its value F(V). The firm would like to maximize the expected present value of this option:

F(V) = max E[(VT - I) e-pT],

where VT is the value of the project at the unknown future time T of which the investment is made (so VT - I is the payoff from investing), p is the appropriate discount rate, and E is the expectation operator. To solve the problem assume that ot < p; otherwise, delay is always sensible and V increases inde- finitely with T.

Because the option value of the investment opportunity yields no dividends until it is exercised, the return from holding it takes the form of capital appreciation. Hence, for values of V for which the firm should not yet invest, Bellman's equation is given by:9

pF dt = E(dF). (2)

Using Ito's Lemma,10 we can expand dF:

dF = F'(V)d V + '/2 F" (V)(dV)2 (3)

s Predecessors of this formalization in the macroeconomics literature include the models of Alex Cukierman (1980) and Ben Bernanke (1983).

6 For a more detailed description of stochastic processes and their properties, see Dixit and Pin- dyck (1994, Ch. 3), Dixit (1993), Darrell Duffie (1988), and Samuel Karlin and Howard Taylor (1975, 1981).

7 For a more thorough discussion of the model, see Dixit and Pindyck (1994, Ch. 5) or McDonald and Siegel (1986).

8While uncertainty over future returns affects the value of the option, the option may still be valuable even under certainty.

9 For an intuitive description of Bellman's equation (also known as the fundamental equation of optimality), see Dixit and Pindyck (1994, Ch. 3).

10 Consider a function F(x,t) that is (at least) twice differentiable in x (a state variable) and once in t (time). Ito's Lemma gives the total differential dF as:

dF = aF/Jt + DF/Ax dx + %/2 (dX)2 ~DX2 pJ




In the McDonald-Siegel problem, dV = axV dt + oVdz, so that we can rewrite (3) (because

E(dz) = 0) as:

E (dF) = aV F'(V) dt + 1/2a2 V2 F" (V) dt.

Dividing through by dt produces Bellman's equation:

1/2 a2 V2 F"(V) + aVF'(V) - pF = 0. (4)

While the Bellman equation is a second-or- der differential equation, F(V) must also sat- isfy three boundary conditions. First, given the stochastic process used, V will remain at zero if it goes to zero (hence, F(M) = 0). Sec- ond, if V4 is the value at which it is optimal to invest, upon investing the firm receives a net payoff V - I (hence, F(V0) = V' - I); see Dixit and Pindyck (1994) or Dixit (1993) for a discussion. The third condition, known as the "smooth-pasting" condition requires F' (V4) = 1, so that F(V) is continuous at the threshold of V4. The need for a third condition arises because of the existence of a "free boundary." While the first boundary is given by V = 0, the second boundary position V4 must be determined as part of the solution. As Dixit and Pindyck show, such "free boundary" problems occur in many applications.

The solution to the McDonald-Siegel problem is relatively straightforward. One can guess a functional form and verify its success. Given the first boundary condition mentioned above, the solution must assume the form:

F(V) = AV1,

where A is a constant to be determined and 1 > 1 is a known constant, the value of which

depends on the parameters ot, p, and (r of the differential equation (4).

The other two boundary conditions help us solve for the other unknowns: V4 (the threshold value at which the firm should invest) and A (the constant mentioned above). Specifi- cally:

V*= [PAN - 1)] I, (5)

and

A = (V*- I)/V*pI = (l - 1)P1/[03 i l-P1].

The basic intuitive point is this: Irre- versibility and the possibility of delay generate a range of inaction (not present in the neoclassical model) in which V> I, yet the firm does not invest. (This is because,

pi>1,PiA/pi - ) > l, and V* > L) How large is this wedge between the tradi-

tional net present value investment criterion and that in the new view? Without going into algebraic detail, let me indicate some chan- nels. First, as uncertainty about future returns (measured by a) rises, the wedge 1/(P1 - 1) also rises. Second, ceteris paribus,

an increase in the discount rate p increases the wedge. Third, given p, an increase in trend value growth a increases the wedge.

To compare these results with the predictions of the neoclassical model, it is useful to revisit predictions of the q approach. If we define q as the ratio of the expected value of profits from an investment-conditional on its completion-to its construction cost, then q = V/I. A firm should invest when V > I. If we acknowledge the loss of the option to invest when the project is completed, the in- stalled project should raise firm value by V - F(V), not by V. While in this alternative setup the threshold criterion for investment is V> I + F(V), the threshold q expressed in the more conventional q terms defined earlier is 01/(Pl - 1) > 1. That is, fluctuations in V can still yield a region of inaction in which q exceeds unity with no investment response. 1

11 In some respects, the wedge between the investment criteria in the neoclassical model and the new view is likely to be at its upper bound in geometric Brownian motion examples in which de- fay can be arbitrarily long. An alternative model (also considered by Dixit and Pindyck) would re- flect the likelihood that project value will decline discontinuously at some point in the future (because of, e.g., patent expiration or entry generally). Analytically, the example just described would be augmented by the possibility of a down- ward Poisson jump:

dV = a Vdt + a Vdz - Vdq, where dq, an increment of a Poisson process with a mean arrival rate X, is independent of dz. When the "event" (of patent expiration, entry, etc.) occurs, q declines with probability one by some fraction 4 (between zero and one). Intuitively, while V fluctuates (with geometric Brownian motion), over each interval (dt), there is a probability ( Adt) that it will drop a fraction of the original value (1 - 4) and continue fluctuating (with geometric




4. Generalizing a (Monopoly) Firm's Problem in the New View

Dixit and Pindyck offer many generaliza- tions of the basic problem of investment under uncertainty for a monopoly firm. Here, I focus on four arising in many applications, encompassing: (a) more explicit descriptions of price uncertainty, (b) consequences of depreciation, (c) combined effects of price and cost uncertainty, and (d) costs of exit and scrapping.

4A. Price Uncertainty

The first step toward generalizing the basic model for the firm is to specify sources of value fluctuations. Suppose that the proposed project produces a physical flow of one unit of output each period in perpetuity. The price depends on the inverse demand function P = YD(1), where D is a measure of nonstochastic flow demand and Y is a stochastic shift variable. In the simplest case, with no variable costs of operating the project, we can write a process for P reminiscent of the geometric Brownian motion used earlier to study V:

dP = a Pdt + a Pdz (6)

The relationship to V is a simple one. Dis- counting future revenues at rate p gives V = P/(p - a), where P is the current price. The solution for the threshold price at which investment should occur is P* = [01/(01- 1)] (p - a) I, so that the threshold value is again V*= [01/(01 -1)] 1. Or, in the context of the q model, the firm should invest if q > q* = 1/(3iI - 1).

Placing the problem in terms of a threshold price for investment serves as a useful segue to the more realistic consideration of the case with operating costs. In this case, the firm will want to suspend operations if operating costs cannot be covered. For simplicity, assume that a project's operation can be costlessly ceased when P is less than operating cost C, and can be costlessly restarted if P

rises above C. McDonald and Siegel (1985) note that such a project gives the firm an infi- nite set of options: to receive P by paying C at any time t the option is exercised. Dixit and Pindyck (1994, ch. 6) show the reader a simple way of valuing the project as a single claim that is a function of P. There are now two regions for V(P), depending on whether P - C or P < C. Once V(P) is determined, it is straightforward to solve for the option value F(P) for the case of a geometric Brownian motion process for P. As in the earlier examples, an increase in uncertainty increases F(P), increasing the threshold criterion for investment.

4B. Depreciation

A second generalization of the basic model addressed in Chapter 6 considers depreciation. A priori, this is an important extension because the option to invest in a depreciating project may seem less valuable than the option to invest in a nondepreciating project. The simplest case is the one most generally used in applied research, exponential decay. In the language of stochastic processes, the depreciating project has a random lifetime following a Poisson process with arrival rate X. The probability that the project depreci- ates to zero before any period T is 1' - eT.

Returning to the simple process for the output price P with no variable costs, let the initial price be P, which evolves according to a geometric Brownian motion with trend growth rate a; the discount rate is p. The expected value of the project's profits if it lasts T periods is:

T

EJ e-PtPtdt = P[ 1 - e-P - a)T/(p - a). 0

To calculate the expected value of the project V(P) we apply the probability density function of the lifetime in a Poisson process:

C00

V(P) = f Xe - TP[l - e(p-a)T]/(p - a)dT 0

=P/(X+p-a). (7)

Intuitively, though the project can be analyzed as infinitely lived, future profits are dis- counted at a higher rate reflecting the probability of "death."

Brownian motion) until the next event occurs. In this case, holding other parameters constant, the option value F(V) falls relative to pure geometric Brownian motion example (see, e.g., Dixit and Pindyck 1994, ch. 5).




To obtain the option value in this case F(P), we have to decide whether the firm has the right to invest in the project once or in perpetuity. In the former case, the investment threshold P4 can be calculated analo- gously to the price process example in Sec- tion 4A:

P = [*1/(l - 1)] (p - a + X) 1, (8)

where "p - a + X" replaces "p - a." Deprecia- tion per se does not affect the option value in this case. That is, the threshold investment

criterion is V* = I/(PI - 1) T > I, where PI is as before. This is because the firm's opportunity to invest is assumed to be available only once; hence, exercising the option is irreversible even though the project has a finite life.

The option value can be reduced, however, if the option to invest is available in perpetuity (that is, when the firm has the right to start a new project at some point after the first one dies). This is because exercising an option to invest is less irreversible when the option to invest again is always available. While, simulated threshold q values in this case remain above unity, they fall significantly in the presence of modest rates of depreciation.

4C. Uncertainty Over Both Price and Cost

Chapter 6 also considers the possibility that both price and the cost of investing are uncertain. While the intuition for arriving at a threshold investment criterion does not differ substantively in this case, the solution process is mathematically more difficult. Both the value of the project and the value of the investment option are now functions of both P and 1. The solution must then find regions of (P,I) values in which investment will occur. Dixit and Pindyck present a sketch of a solution when both P and I follow geometric Brownian motion. It is still the case that the threshold q criterion for investment exceeds unity on account of the option value.

4D. Costs of Exit and Scrapping

The final aspects of investment decisions by a monopoly firm I want to address are

costs of exit and scrapping (see Ch. 7). The examples in Section 4A, while nicely con- nected to the simple solutions with which we began in Section 3, rely on the unrealistic assumptions of costless suspension and restarting. One way to gauge the impact of such costs on investment options is to consider an example in which the investment cost I must be reincurred if a project's operation has been previously suspended, as if the project had rusted or decayed in the interim. Intui- tively, a new option value arises-the value of maintaining the project's operation given the cost of restarting it. Now it is the abandonment decision which has a higher threshold on account of the option value (in the sense that the rate of loss must exceed a certain critical positive level rather than simply zero).

This case considered by Dixit and Pindyck is one in which operating costs are constant and demand uncertainty affects the output price, which follows a geometric Brownian motion as before. Investment, as before, requires a cost I; the firm must pay an amount E if it wishes to abandon the project.12 We can build upon the discussion in Section 4A of the option value to invest under demand uncertainty. Having invested, the firm's live project still has an option component related to abandonment. If the firm exercises that option, it reacquires one of the original type, the option to invest.

The solution to this problem conforms with one's intuition. Imagine two threshold prices PH and PL* On the one hand, if a firm does not have a project in operation, it will remain idle when the price P < PH, and will invest when P 2 PH On the other hand, if a firm has an active project, it will keep the project going when the price exceeds PL. These ranges of inaction stand in contrast to the more knife-edge investment and abandonment decisions featured in most intermediate microeconomics texts.

Dixit and Pindyck illustrate these ideas in an analysis of capacity investment in mines

12 The example I consider here focuses on abandonment. Chapter 7 also considers temporary abandonment, in which maintenance costs are required during the period in which a project is mothballed.




and smelters by U.S. copper producers in the 1970s and 1980s. In the boom times of the 1970s, firms did not appear to restart previously profitable mines (that had been shut- tered in earlier low-price periods) or invest in new ones. In the period of very low copper prices in the mid-1980s, the firms continued to operate mines and smelters that are no longer profitable. This behavior, with very broad ranges of inaction, is consistent with the new view's predictions.

5. Extending the New View: Competitive Equilibrium

As I noted at the outset, Chapters 5-7 pro- vide the core of the intuition of the new view analyzed in Investment Under Uncertainty. For researchers and practitioners interested in empirical applications and potential policy implications, however, the lessons of the new view would be more compelling if couched in an industry equilibrium. This is because most firms constantly struggle with the prospect of competition in investment from other incum- bents and potential entrants. The concept of irreversibility is also perhaps more appropri- ately analyzed at the industry level because the liquidity of most assets in place is surely greater within than outside the industry.

To see the need for this distinction, suppose that investment is completely irreversible and firms expect an industry-wide expansion of demand. A given firm recognizes the qualitative implications of the favorable demand shift for the industry price, and would like to increase its capital stock. Of course, the firm understands that other firms in the industry are making a similar calculation. As a result, the competitive firm will not increase irreversible investment by as much as it would in the case of a monopoly. By contrast, an unfavorable shift in industry demand has a larger effect relative to the monopoly case. Given the assumption of complete irreversibility, all firms suffer because exit is not possible. There is, then, an asymmetry in the competitive response to uncertainty, an asymmetry that makes firms, all else equal, more cautious in undertaking incremental investment. The asymmetry does not arise in the case of firm-specific uncer-

tainty. This is because a firm understands that an idiosyncratic demand shift does not imply similar fortunes for other firms; hence, the firm behaves like a monopolist.

More formally, what is the effect of competition within an industry on the option value of investment and, by extension, on the new view's criticism of predictions of conventional investment models? To answer these questions in a competitive setting, we must go a bit further in characterizing demand or cost uncertainty, and differentiate between firm- specific uncertainty and industry-wide uncertainty.'3

Focusing on demand uncertainty, consider both firm industry shocks to a firm's price. Any given firm's price for a unit of output is:

P = XY D(Q), (9)

where Q is the current output flow; D(Q), a decreasing function, is the nonstochastic por- tion of the inverse demand curve faced by the industry; and X and Y are firm-specific and industry-wide shocks, respectively. If we nor- malize firm output to unity, Q is a summary statistic for the number of active firms in the industry.

Let's suppose that X-shocks are symmetric firm-specific price uncertainty; that is, firms face, with equal probability, equal up or down shifts of X. Note that if the firm delays investing, it can reduce its exposure to an adverse shock; it preserves the option to invest if the price rises while choosing not to invest if the price falls (as in the Ilova watch example in Section 2). By waiting, the firm receives a payoff that is a convex function of X; hence, the expected value of waiting rises with an uncertainty in X. This is the now familiar option value of delay.

Let's alternatively consider industry-wide uncertainty in which symmetric Y-shocks are permitted. Unlike the X-shock case, when Y rises, each firm understands that entry is now attractive for all firms. When other firms indeed enter, the supply shift leads the price to rise disproportionately less than Y. Price becomes a concave function of Y; an increase in

13 In general, of course, both firm-specific and industry-wide uncertainty exist; see, e.g., Ricardo Caballero and Pindyck (1992).




uncertainty in the aggregate shock Y reduces the expected value of investing (relative to that of not investing). Hence, while the threshold investment criterion can still be described in terms that "q must exceed unity by some given amount," an option value is not responsible for the wedge.

Let me elaborate on the effects of industry- wide uncertainty on investment decisions in the competitive equilibrium. In this case, X = 1, and the industry's inverse demand curve is:

P = Y D(Q). (10)

Let the aggregate shock Y evolve according to a geometric Brownian motion process:

dY=aYdt+ sYdz.

Before new entry takes place, Q is fixed, ren- dering P proportional to Y, and:

dP = aP dt + aPdz. (11)

Intuitively, there is some price P above which new entry occurs. Hence, any one firm views the price process as described by (11) as long as P < P; the price cannot go higher without triggering entry and a subsequent price decline.

Dixit and Pindyck solve first for P, which is identical to that faced by a monopolist pro- ducing one unit of output subject to the same demand process. There are key differences, however. First while the monopolist's view of the price process is not affected by potential entry (at least in the example), the competi- tor's is. An opposing second effect arises because the monopolist has an option value of waiting, while the competitive firms do not. These differences offset one another pre- cisely. Dixit and Pindyck solve formally, then, for the competitive equilibrium for the industry by setting the entry threshold P? equal to P.

Because of this effective upper barrier on the price process under competition, the threshold price for entry is above the usual Marshallian level. The authors show that under assumptions of modest average industry growth and demand uncertainty, the required rate of return for a risk-neutral firm can be significantly higher than the real interest

rate. This qualification, also expanded to include the possibility of exit in the authors' application to copper prices, implies that the industry price can exceed long-run average cost for a long period of time in a competitive industry without stimulating entry.

6. Policy Intervention and Policy Uncertainty

The ninth chapter of Investment Under Un- certainty addresses whether policy interventions are suggested by the new view investment models. The existence of delay and inertia per se does not, of course support the desirability of intervention in the absence of a clearly articulated market failure. It is indeed easy to imagine real-world interventions that do more harm than good. For example, Dixit and Pindyck consider potential market failures associated with incomplete markets for sharing risk. Studying price controls as a way to reduce risk, they find that such controls may be welfare-decreasing in many cases. Another rich area for application lies in industrial organization and antitrust policy. Returning to the description in the previous section of competitive equilibrium, policy makers might see incumbent firms in an industry earning supernormal profits, but with no entry occurring. With Marshallian intuition, they might take inappropriate action against assumed barriers to entry, when they might be observing a competitive industry. In the same regard, observing a price below the minimum level of average cost need not indicate predation; the "dumping" firms might be maintaining the option to keep their (sunk) investments.

In this section, I focus on the chance that policy uncertainty may have unintended consequences; to be concrete, I frame the discussion in terms of business investment incentives. In describing price uncertainty in Section 4, I noted that different charac- terizations of uncertainty can have different effects on the responsiveness of investment to changes in the net return to investing. The same holds for tax policy uncertainty. An instructive case is the U.S. investment tax credit (ITC), which, following its introduc-




tion in 1962, has been changed many times.14 The example that follows, draws largely on the work of Hassett and Metcalf (1994) and the analysis of their work in Dixit and Pin- dyck (1994, Ch. 9).

It is useful to begin with the basic geometric Brownian motion model. The firm chooses its capital stock to produce a stream of output in perpetuity to be sold at an after- tax price P, where:

dP= apPdt+ op Pdz,

where P subscripts refer to the output price process. To capture the idea of an ITC (or accelerated depreciation), we can add tax incentives which reduce the cost of capital. Suppose the price of capital also evolves according to a geometric Brownian motion process:

dPK = aK PK dt + AK PK dz,

where the K subscript refers to the price of capital. This setup yields a result similar to that we saw earlier for price uncertainty; the threshold Tobin's q value for investing, q, is:

q* = PI/(I - 1) >1,

and PI1 is defined as in (5). That is, the firm will not invest until the returns are sufficient to compensate it for the lost option of delay.

To focus on tax policy uncertainty, consider a mean-preserving spread in the price of capital: da > 0. One can show that

1- )]/aa > 0; that is, an increase in uncertainty over the price of capital increases the threshold q criterion for investment. In this sense, uncertain tax policy can lead to an increase in the hurdle rate for investment projects. 15

Using the description of investment tax policy changes over the postwar period from Cummins, Hassett, and Hubbard (1994), Hassett and Metcalf argue that it is more realistic to model investment incentives as a Poisson process. (Recall the description of these processes in Section 3.) This is because investment incentives appear to change ran- domly and in discrete amounts. Consider an ITC at rate k that reduces the price of capital from PK to (1 - k) PK. The tax process follows a Poisson process, switching between "high" (kH) and "low" (kL) values of the ITC. In particular, the ITC in the Hassett-Metcalf example switches between kH and kL with transi- tion probabilities XH and XL. If no credit is currently in effect, the expected time until one is introduced is known, though the actual time is not. In this structure:

IkH-kL ,XHdt

dkt= 0 ,(1- XH)dt kL-kH ,XLdt

0 ,(1 - XL)dt.

The solution to such a problem describes, among other things, threshold prices PH and PL for investment in the absence and presence of the ITC, respectively.

Hassett and Metcalf examine a mean-preserving spread to consider the effect of an increase in uncertainty on investment (given the Poisson description of investment incentives). In particular, they fix XH and XL, but change kH and kL, keeping E(k) constant. In this case, Tobin's q threshold for investment actually decreases with an increase in uncertainty, and the median time to investment falls.16 Put simply, investment is bunched in periods in which the high ITC is in effect. If

14 Since its introduction, the average duration of periods with no ITC is three years, and the average duration of a period in which a specific ITC is in effect is 3.67 years (see Cummins, Hassett, and Hubbard 1994).

15 In another type of exercise, Dani Rodrik (1991) analyzed consequences for investment of uncertainty over the enactment of policy reforms designed to stimulate investment. In his model, if the reform policy can be reversed with some probability each year, the consequent uncertainty reduces the salutary effect of 'he reform on investment.

16 Another case in which the interaction of uncertainty and irreversibility may increase investment is that of learning about some types of uncertain costs-as, for example, in large-scale R&D projects. Pindyck (1993) calls this technical uncertainty: If prices of inputs were known, how many inputs and what level of effort are required to complete the project? This type of uncertainty (as opposed to uncertainty about the cost of individual inputs) can be resolved only by actually proceeding with the project, and hence can stimulate investment.




policy makers increase the frequency with which changes in the ITC occur, then it is more likely that the high-ITC state (in which most investment occurs) is reached quickly. (One note of caution: While this experiment holds E(k) constant, the average cost of capital for investment changes because investment is bunched in high-ITC periods.)

Hassett and Metcalf (1994) and the related discussion in Dixit and Pindyck (1994, ch. 9) usefully illustrate the significance of choosing carefully the underlying stochastic process if normative analysis is the object. Two additional steps seem promising here. The first is to extend the consideration to a competitive industry equilibrium. The second is to inves- tigate whether the analytical and simulation results can help empirical researchers to explain findings of insignificant effects of tax parameters on investment (as in many of the studies surveyed by Chirinko 1993) or significant effects of tax parameters on investment (as in Cummins, Hassett, and Hubbard 1994).

7. Empirical Implications

At the beginning of this review, I raised three questions. I think it is clear that the literature summarized and analyzed in Invest- ment Under Uncertainty addresses practical problems in modeling investment and offers predictions consistent with observations of many case studies of investment. Let me now turn to the question of general empirical tests of predictions of the new view. Because many of the theoretical advances in this literature are recent, empirical work is not as well developed, though many studies are emerging. Below, I comment briefly on studies in the new view literature17 and an alternative generalization of neoclassical models to incorporate capital-market imperfections.

7A. Empirical Tests of Option-Based Models

A key prediction of the new view investment models is that the threshold expected return required to trigger investment is influ-

enced by uncertainty and irreversibility. Test- ing this prediction is a bit less straightforward than tests of conventional neoclassical models, however. This is because the new view models, while offering a rigorous description of threshold q values for investment (to draw an analogy to the neoclassical model), do not offer specific predictions about the level of investment. To go this extra step requires the specification of structural links between the marginal profitability of capital and the desired capital stock (the usual research focus in the traditional neoclassical literature). Nonetheless, option-based models offer some testable hypotheses. For example, a fall in the average rate of growth of profitability or rise in the volatility of profitability should depress investment over some period. Dixit and Pin- dyck offer some applications in Chapter 12.

A potentially informative literature testing new view models using disaggregated data is in its early stages. Caballero and Pindyck (1992) use data on U.S. manufacturing indus- tries to study the determinants of the return required to trigger investment. They employ a proxy for the required return using extreme values of the marginal profitability of capital.18 Supportive of the intuition of option- based models, they find the proxies for the required return depend positively on the volatility of the marginal profitability of capital.19 In an analogous cross-sectional test, Pindyck and Andres Solimano study the relationship between uncertainty and investment for a set of thirty countries. For each country, they calculate a time-series for the marginal profitability of capital over a 28-year period. Decomposing the period into three nine-year subperiods, Pindyck and Solimano calculate the mean and standard deviation of each country's investment-GDP ratio and the annual log change in the marginal profitability of capital. Using the panel of data, Pindyck and Solimano relate the investment-GDP ra-

17A related line of inquiry (not reviewed here) generalizes conventional models of adjustment costs; see Andrew Abel and Janice Eberly (1993) and Dixit and Pindyck (1994, Ch. 11).

18 This calculation requires making assumptions about the production technology, of course; they assume a Cobb-Douglas technology with constant returns to scale.

19 Such a test is only suggestive, because the maximum value of the constructed marginal profitability and its variance are correlated even if there is no causal link between investment and uncertainty.




tio to the mean and standard deviation of the (log change in the) marginal profitability of capital in each period. Consistent with the prediction of new view models, they find that the investment-GDP ratio varies positively with the mean of the log change in the marginal profitability of capital, and negatively with the standard deviation. While such approaches offer results supportive of the new view predictions, they are only reduced-form tests. Nonetheless, they are suggestive of potentially promising results from more structural analysis (using, say, techniques to study dynamic firm decisions and industry equilibrium described in Ariel Pakes 1993).

While empirical research has not quite caught up with the rapidly changing theoretical developments in this literature, applied researchers have an opportunity to apply new view models rigorously in industry-level studies of investment dynamics and in panel data studies of investment using firm-level or industry-level data. The most promising such research would focus on testable differences between conventional neoclassical models and option-based models.20

7B. Empirical Tests of the Role of Capital-Market Imperfections

At least part of the motivation for the option-based investment models in the new view is the problem raised in many empirical studies that investment's responses to changes in the user cost of capital or q are implausibly small (or, equivalently, that "ad-

justment costs" are implausibly large). This claim is subject to debate. Some recent studies find that tests of neoclassical investment models using panel data on large firms produce quite sensible estimates of adjustment costs (see, e.g., Hubbard, Anil K. Kashyap, and Whited, forthcoming; and Cummins, Hassett, and Hubbard 1994).

As with the option-based models, some models of "financing constraints" on investment predict ranges of inaction; that is, the user cost of capital or Tobin's q can fluctuate in a given range with no (or an attenuated) response of investment. In these models, problems of adverse selection or moral haz- ard impart a wedge between the cost of external finance and internal finance. The "range of inaction" can be explained as follows. For firms with high levels of internal net worth relative to investment opportunities, the neoclassical model holds; shifts in q or the user cost of capital change desired investment. For firms with lower levels of net worth, costs of external finance vary inversely with the level of internal net worth: When borrowers' net worth improves, lenders become more willing to lend, and additional investment can be financed. Hence, while shifts in internal net worth affect investment in such firms, observed movements in q or the cost of capital may not. Empirical studies in this literature have focused on theoretically consistent ways to group high-net-worth and low-net-worth firms in firm-level data or high- and low-net- worth periods in industry data (see, e.g., Steven M. Fazzari, Hubbard, and Bruce C. Petersen 1988; Takeo Hoshi, Kashyap, and David Scharfstein 1991; and Hubbard and Kashyap 1992).

Additional research on firm-level investment decisions could attempt to distinguish between the predictions of neoclassical models augmented by informational imperfections on the one hand and option-based models on the other hand. Such an integration might proceed in two steps: (1) analyzing effects of "finance constraints" in the continuous-time-stochastic-process models described earlier (as for example, the analysis of "bor- rowing constraints" and consumption in Hub- bard and Kenneth Judd 1986), and (2) deriv- ing empirical tests to discriminate between

20 John Leahy and Toni Whited (1994) study the relationship between uncertainty and investment using firm-level panel data from Comiustat. After discussing alternative models of links between uncertainty and investment, they perform some reduced-form tests, regressing firms' investment- capital ratio on uncertainty (measured by the variance of the firms' daily stock price for each year) and for Tobin's q. They find that, holding q constant, the variance term has no statistically significant effect on firms' investment. However, in another test, they note that the variance term has a negative and statistically significant effect on the value of q, suggesting an indirect impact of uncertainty on investment. This stylized fact indicates potential gains from testing structural repre- sentations of the alternative "uncertainty" approaches.




the "range of inaction" predictions of the two classes of models.

8. Conclusions

Let me conclude where I began. Invest- ment Under Uncertainty, by Dixit and Pin- dyck, should be required reading for researchers interested in investment, professors teaching capital budgeting, and many practitioners. The book not only summarizes a growing literature on investment under irreversibility and uncertainty, but bridges methodological and intuitive gaps among alternative approaches. This integration, arguably the book's greatest strength, will likely stimulate a substantial body of future research. The authors' careful and enjoyable writing style also ensures that, conditional on exercising your option to read the book, you will not abandon it.

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