Irreversibility, Uncertainty, and Investment." Pindyck, Robert S

Published as: “Irreversibility, Uncertainty, and Investment." Pindyck, Robert S. Journal of Economic Literature Vol. 29, No. 3 (1991): 1110-1148.

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Irreversibility,Uncertainty, and Investment

By ROBERT S. PINDYCK MassachusettsInstitute of Technology

My thanks to Prabhat Mehtafor his research assistance, and to Ben Bernanke, Vittorio Corbo, Nalin Kulatilaka,Robert McDonald, Louis Serven, Andreas Solimano, and two anonymous referees for helpful commentsand suggestions. Financial support was provided by MIT's Center for Energy Policy Research, by the World Bank, and by the National Science Foundation under Grant No. SES-8618502. Any errors are mine alone.

I. Introduction ment expenditure can profoundly affect the decision to invest. It also undermines DESPITE ITS IMPORTANCE to economic the theoretical foundation of standard growth and market structure, the neoclassical investment models, and in- investment behavior of firms, industries, validates the net present value rule as and countries remains poorly under- it is usually taught to students in business stood. Econometric models have had school: "Invest in a project when the limited success in explaining and predict- present value of its expected cash flows ing changes in investment spending, and is at least as large as its cost." This rule we lack a clear explanation of why some and models based on it-are incorrect countries or industries invest more than when investments are irreversible and others. decisions to invest can be postponed. One problem with existing models is Irreversibility may have important im- that they ignore two important character- plications for our understanding of aggre- istics of most investment expenditures. gate investment behavior. It makes in- First, the expenditures are largely irre- vestment especially sensitive to various versible; that is, they are mostly sunk forms of risk, such as uncertainty over costs that cannot be recovered. Second, the future product prices and operating the investments can be delayed, giving costs that determine cash flows, uncer- the firm an opportunity to wait for new tainty over future interest rates, and un- information to arrive about prices, costs, certainty over the cost and timing of the and other market conditions before it investment itself. Irreversibility may commits resources. therefore have implications for macro- As an emerging literature has shown, economic policy; if a goal is to stimu- the ability to delay an irreversible invest- late investment, stability and credibility 1110

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could be much more important than tax tential competitors. (Richard Gilbert incentives or interest rates. 1989 surveys the literature on strategic What makes an investment expendi- aspects of investment.) But in most cases, ture a sunk cost and thus irreversible? delay is at least feasible. There may be Usually it is the fact that the capital is a cost to delay-the risk of entry by other firm or industry specific, that is, it cannot firms, or simply foregone cash flows-but be used productively by a different firm this cost must be weighed against the or in a different industry. For example, benefits of waiting for new information. most investments in marketing and ad- An irreversible investment opportu- vertising are firm specific, and hence are nity is much like a financial call option. clearly sunk costs. A steel plant is indus- A call option gives the holder the right, try specific-it can only be used to pro- for some specified amount of time, to pay duce steel. Although in principle the an exercise price and in return receive plant could be sold to another steel com- an asset (e.g., a share of stock) that has pany, its cost should be viewed as mostly some value. Exercising the option is ir- sunk, particularly if the industry is com- reversible; although the asset can be sold petitive. The reason is that the value of to another investor, one cannot retrieve the plant will be about the same for all the option or the money that was paid firms in the industry, so there is likely to exercise it. A firm with an investment to be little gained from selling it. (If the opportunity likewise has the option to price of steel falls so that a plant turns spend money (the "exercise price") now out, ex post, to have been a "bad"invest- or in the future, in return for an asset ment, it will also be viewed as a bad in- (e.g., a project) of some value. Again, vestment by other steel companies, so the asset can be sold to another firm, that the ability to sell the plant will not but the investment is irreversible. As be worth much.) with the financial call option, this option Even investments that are not firm or to invest is valuable in part because the industry specific are often partly irre- future value of the asset obtained by in- versible because of the "lemons" prob- vesting is uncertain. If the asset rises in lem. For example, office equipment, value, the net payoff from investing rises. cars, trucks, and computers are not in- If it falls in value, the firm need not industry specific, but have resale value well vest, and will only lose what it spent to below their purchase cost, even if new. obtain the investment opportunity. Irreversibility can also arise because of How do firms obtain investment op- government regulations or institutional portunities? Sometimes they result from arrangements. For example, capital con- patents, or ownership of land or natural trols may make it impossible for foreign resources. More generally, they arise (or domestic) investors to sell assets and from a firm's managerial resources, tech- reallocate their funds. And investments nological knowledge, reputation, market in new workers may be partly irreversible position, and possible scale, all of which because of high costs of hiring, train- may have been built up over time, and ing, and firing. which enable the firm to productively un- Firms do not always have an opportu- dertake investments that individuals or nity to delay investments. For example, other firms cannot undertake. Most im- there can be occasions in which strategic portant, these options to invest are valu- considerations make it imperative for a able. Indeed, for most firms, a substantial firm to invest quickly and thereby part of their market value is attributable preempt investment by existing or po- to their options to invest and grow in

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions 1112 Journal of Economic Literature, Vol. XXIX (September 1991) the future, as opposed to the capital they This paper has several objectives. already have in place. (For discussions First, I will review some basic models of growth options as sources of firm of irreversible investment to illustrate value, see Stewart Myers 1977; W. Carl the option-like characteristics of invest- Kester 1984; and my 1988 article.) ment opportunities, and to show how op- When a firm makes an irreversible in- timal investment rules can be obtained vestment expenditure, it exercises, or from methods of option pricing, or alter- "kills," its option to invest. It gives up natively from dynamic programming. Be- the possibility of waiting for new informa- sides demonstrating a methodology that tion to arrive that might affect the desir- can be used to solve a class of investment ability or timing of the expenditure; it problems, this will show how the result- cannot disinvest should market condi- ing investment rules depend on various tions change adversely. This lost option parameters that come from the market value is an opportunity cost that must environment. be included as part of the cost of the A second objective is to survey briefly investment. As a result, the NPV rule some recent applications of this method- "Invest when the value of a unit of capital ology to a variety of investment prob- is at least as large as its purchase and lems, and to the analysis of firm and in- installation cost" must be modified. The dustry behavior. Examples will include value of the unit must exceed the pur- the effects of sunk costs of entry, exit, chase and installation cost, by an amount and temporary shutdowns and restartups equal to the value of keeping the invest- on investment and output decisions, the ment option alive. implications of construction time (and the Recent studies have shown that this option to abandon construction) for the opportunity cost of investing can be value of a project, and the determinants large, and investment rules that ignore of a firm's choice of capacity. I will also it can be grossly in error.1 Also, this op- show how models of irreversible invest- portunity cost is highly sensitive to un- ment have helped to explain the preva- certainty over the future value of the lence of "hysteresis," that is, the ten- project, so that changing economic con- dency for an effect (such as foreign sales ditions that affect the perceived riskiness in the U. S.) to persist well after the cause of future cash flows can have a large that brought it about (an appreciation of impact on investment spending, larger the dollar) has disappeared. than, say, a change in interest rates. This Finally, I will briefly discuss some of may help to explain why neoclassical in- the implications that the irreversibility vestment theory has failed to provide of investment may have for policy. For good empirical models of investment be- example, given the importance of risk, havior. policies that stabilize prices or exchange rates may be particulary effective ways 1 See, for example, Robert McDonald and Daniel of stimulating investment. Similarly, a Siegel (1986), Michael Brennan and Eduardo major cost of political and economic insta- Schwartz (1985), Saman Majd and Pindyck (1987), and Pindyck (1988). Ben Bernanke (1983) and Alex bility may be its depressing effect on in- Cukierman (1980) have developed related models in vestment. which firms have an incentive to postpone irrevers- Section II uses a simple two-period ex- ible investments so that they can wait for new information to arrive. However, in their models, this in- ample to illustrate how irreversibility can formation makes the future value of an investment affect an investment decision, and how less uncertain; we will focus on situations in which methods can be used to information arrives over time, but the future is always option pricing uncertain. value a firm's investment opportunity,

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t=0 t= 1 t=2

P1 = $150 - P2= 150 - q

P( = $100

(1--q P1 = $50 -- P2 = 50 Figure 1. Price of Widgets and determine whether or not the firm opportunity can be demonstrated most should invest. Section III then works easily with a simple two-period example. through a basic continuous-time model Consider a firm's decision to invest irre- of irreversible investment that was first versibly in a widget factory. The factory examined by McDonald and Siegel can be built instantly, at a cost I, and (1986). Here a firm must decide when will produce one widget per year forever, to invest in a project whose value follows with zero operating cost. Currently the a random walk. I first solve this problem price of widgets is $100, but next year using option pricing methods and then the price will change. With probability by dynamic programming, and show how q, it will rise to $150, and with probabil- the two approaches are related. Section ity (1 - q) it will fall to $50. The price IV extends this model so that the price will then remain at this new level for- of the firm's output follows a random ever. (See Figure 1.) We will assume that walk, and the firm can (temporarily) stop this risk is fully diversifiable, so that the producing if price falls below variable firm can discount future cash flows using cost. I show how both the value of the the risk-free rate, which we will take to project and the value of the firm's option be 10 percent. to invest in the project can be deter- For the time being we will set I = mined, and derive the optimal invest- $800 and q = .5. (Later we will see how ment rule and examine its properties. the investment decision depends on I Sections III and IV require the use of and q.) Given these values for I and q, stochastic calculus, but I explain the basic is this a good investment? Should we in- techniques and their application in the vest now, or wait a year and see whether Appendix. However, readers who are the price goes up or down? Suppose we less technically inclined can skip directly invest now. Calculating the net present to Section V. That section surveys a num- value of this investment in the standard ber of extensions that have appeared in way, we get the literature, as well as other applica- 00 tions of the methodology, including the NPV -800 + E 100/(1.1)t analysis of hysteresis. Section VI dis- t=O cusses policy implications and suggests - 800 + 1,100 = $300. future research, and Section VII con- The NPV is positive; the current value cludes. of a widget factory is V0 = 1,100 > 800. Hence it would seem that we should go II. A Simple Two-PeriodExample ahead with the investment. The implications of irreversibility and This conclusion is incorrect, however, the option-like nature of an investment because the calculations above ignore a

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cost-the opportunity cost of investing next year, rather than having to invest now, rather than waiting and keeping either now or never? (We know that hav- open the possibility of not investing ing this flexibility is of some value, be- should the price fall. To see this, calcu- cause we would prefer to wait rather than late the NPV of this investment opportu- invest now.) The value of this "flexibility nity, assuming instead that we wait one option" is easy to calculate; it is just the year and then invest only if the price difference between the two NPVs, that goes up: is, $386 - $300 = $86. Finally, suppose there exists a futures NPV = (.5)[ -800/1.1 + > 150/(1.1)t] market for widgets, with the futures price t=1 for delivery one year from now equal to = 425/1.1 = $386. the expected future spot price, $100.2 Would the ability to hedge on the futures (Note that in year 0, there is no expendi- market change our investment decision? ture and no revenue. In year 1, the $800 Specifically, would it lead us to invest is spent only if the price rises to $150, now, rather than waiting a year? The an- which will happen with probability .5.) swer is no. To see this, note that if we The NPV today is higher if we plan to were to invest now, we would hedge by wait a year, so clearly waiting is better selling short futures for five widgets; this than investing now. would exactly offset any fluctuations in Note that if our only choices were to the NPV of our project next year. But invest today or never invest, we would this would also mean that the NPV of invest today. In that case there is no op- our project today is $300, exactly what tion to wait a year, and hence no opportu- it is without hedging. Hence there is no nity cost to killing such an option, so the gain from hedging, and we are still better standard NPV rule would apply. We off waiting until next year to make our would likewise invest today if next year investment decision. we could disinvest and recover the $800 A. Analogy to Financial Options should the price fall. Two things are needed to introduce an opportunity cost Our investment opportunity is analo- into the NPV calculation-irreversibility, gous to a call option on a common stock. and the ability to invest in the future as It gives us the right (which we need not an alternative to investing today. There exercise) to make an investment expendi- are, of course, situations in which a firm ture (the exercise price of the option) and cannot wait, or cannot wait very long, receive a project (a share of stock) the to invest. (One example is the anticipated value of which fluctuates stochastically. entry of a competitor into a market that In the case of our simple example, next is only large enough for one firm. An- year if the price rises to $150, we exercise other example is a patent or mineral resource lease that is about to expire.) The 2 In this example, the futures price would equal the expected future price because we assumed that less time there is to delay, and the the risk is fully diversifiable. (If the price of widgets greater the cost of delaying, the less will were positively correlatedwith the marketportfolio, irreversibility affect the investment deci- the futures price would be less than the expected future spot price.) Note that if widgets were storable sion. We will explore this point again in and aggregatestorage is positive, the marginalconve- Section III in the context of a more gen- nience yield from holding inventory would then be eral model. 10 percent. The reason is that because the futures price equals the current spot price, the net holding How much is it worth to have the flexi- cost (the interest cost of 10 percent less the marginal bility to make the investment decision convenience yield) must be zero.

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our option by paying $800 and receive moment, however, we need not be con- an asset that will be worth V1 = $1,650 cerned with the actual implementation (= 115O0/i.it). If the price falls to $50, of this portfolio.) The value of this portfo- this asset will be worth only $550, so we lio today is (Do = Fo - nP0 = Fo - 100n. will not exercise the option. We found The value next year, (I4 = F1 - nPI, that the value of our investment opportu- depends on P1. If P1 = 150 so that nity (assuming that the actual decision F1 = 850, (DI = 850 -150n. If P1 = 50 to invest can indeed be made next year) so that F1 = 0, DI = -50n. Now, let is $386. It will be helpful to recalculate us choose n so that the portfolio is risk- this value using standard option pricing free, that is, so that (DI is independent methods, because later we will use such of what happens to price. To do this, just methods to analyze other investment set problems. 850 - 150n =-50n, To do this, let Fo denote the value today of the investment opportunity, that or, n = 8.5. With n chosen this way, is, what we should be willing to pay today (DI = -425, whether the price goes up to have the option to invest in the widget or down. factory, and let F1 denote its value next We now calculate the return from year. Note that F1 is a random variable; holding this portfolio. That return is the it depends on what happens to the price capital gain, (DI - (DO, minus any pay- of widgets. If the price rises to $150, then ments that must be made to hold the F1 will equal S150/(1. 1)t - $800 = $850. short position. Because the expected rate If the price falls to $50, the option to of capital gain on a widget is zero (the invest will go unexercised, so that F1 will expected price next year is $100, the equal 0. Thus we know all possible values same as this year's price), no rational infor F1. The problem is to find Fo, the vestor would hold a long position unless value of the option today. he or she could expect to earn at least To solve this problem, we will create 10 percent. Hence selling widgets short a portfolio that has two components: the will require a payment of . IPo = $10 investment opportunity itself, and a cer- per widget per year.3 Our portfolio has tain number of widgets. We will pick this a short position of 8.5 widgets, so it will number of widgets so that the portfolio have to pay out a total of $85. The return is risk-free, that is, so that its value next from holding this portfolio over the year year is independent of whether the price is thus (DI (Do - 85 = (- (Fo - of widgets goes up or down. Because the nP0) -85 = -425 - Fo + 850 -85 = portfolio will be risk-free, we know that 340 -Fo. the rate of return one can earn from hold- Because this return is risk-free, we ing it must be the risk-free rate. By set- know that it must equal the risk-free rate, ting the portfolio's return equal to that which we have assumed is 10 percent, rate, we will be able to calculate the cur- times the initial value of the portfolio, rent value of the investment opportunity. (Do= Fo - nP0: Specifically, consider a portfolio in 340 - - 850). which one holds the investment opportu- Fo= .1(Fo nity, and sells short n widgets. (If widgets were a traded commodity, such as oil, 'This is analogousto selling short a dividend-pay- one could obtain a short position by bor- ing stock. The short position requires payment of rowing from another producer, or by go- the dividend, because no rationalinvestor will hold the offsettinglong positionwithout receiving that div- ing short in the futures market. For the idend.

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We can thus determine that Fo = $386. and dynamic programming in more de- Note that this is the same value that we tail. before the NPV obtained by calculating B. Changing the Parameters of the investment opportunity under the assumption that we follow the optimal So far we have fixed the direct cost strategy of waiting a year before deciding of the investment, 1, at $800. We can whether to invest. obtain further insight by changing this We have found that the value of our number, as well as other parameters, and investment opportunity, that is, the calculating the effects on the value of the value of the option to invest in this proj- investment opportunity and on the in- ect, is $386. The payoff from investing vestment decision. For example, by go- (exercising the option) today is $1,100 - ing through the same steps as above, it $800 = $300. But once we invest, our is easy to see that the short position option is gone, so the $386 is an opportu- needed to obtain a risk-free portfolio de- nity cost of investing. Hence the full cost pends on I as follows: of the investment is $800 + $386 = n = 16.5 - .011. $1,186 > $1,100. As a result, we should wait and keep our option alive, rather The current value of the option to invest than invest today. We have thus come is then given by to the same conclusion as we did by com- = 750 - .4551. paring NPVs. This time, however, we Fo calculated the value of the option to in- The reader can check that as long as vest, and explicitly took it into account I > $642, Fo exceeds the net benefit from as one of the costs of investing. investing today (rather than waiting), Our calculation of the value of the op- which is V0 - I = $1,100 -1. Hence if tion to invest was based on the construc- I > $642, one should wait rather than tion of a risk-free portfolio, which re- invest today. However, if I = $642, quires that one can trade (hold a long Fo = $458 = V0 - 1, so that one would or short position in) widgets. Of course, be indifferent between investing today we could just as well have constructed and waiting until next year. (This can also our portfolio using some other asset, or be seen by comparing the NPV of invest- combination of assets, the price of which ing today with the NPV of waiting until is perfectly correlated with the price of next year.) And if I < $642, one should widgets. But what if one cannot trade invest today rather than wait. The reason widgets, and there are no other assets is that in this case the lost revenue from that "span" the risk in a widget's price? waiting exceeds the opportunity cost of In this case one could still calculate the closing off the option of waiting and not value of the option to invest the way we investing should the price fall. This is did at the outset-by computing the NPV illustrated in Figure 2, which shows the for each investment strategy (invest to- value of the option, Fo, and the net pay- day versus wait a year and invest if the off, V0 - 1, both as functions of 1. For I price goes up), and picking the strategy > $642, Fo = 750 - .4551 > V0 - 1, that yields the highest NPV. That is es- so the option should be kept alive. How- sentially the dynamic programming ap- ever, if I < $642, 750 - .4551 < V0 - proach. In this case it gives exactly the 1, so the option should be exercised, and same answer, because all price risk is di- hence its value is just the net payoff, Vo versifiable. In Section III we will explore - I. this connection between option pricing We can also determine how the value

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L,100 , a . l l l l l L,000_ \

900 - Vo- I -

800 2 4

700 __\-

F00oI For) F ~~~750-.4551 I

3000- 200: -

100 _

0 200 400 600 562 800 1000

Figure 2. Option to Invest in Widget Factory of the investment option depends on q, troduced in this problem. There will be the probability that the price of widgets a value to waiting (i.e., an opportunity will rise next year. To do this, let us once cost to investing today rather than wait- again set I = $800. The reader can verify ing for information to arrive) whenever that the short position needed to obtain the investment is irreversible and the net a risk-free portfolio is independent of q, payoff from the investment evolves sto- that is, n= 8.5. The payment required chastically over time. Thus we could have for the short position, however, does de- constructed our example so that the un- pend on q, because the expected capital certainty arises over future exchange gain on a widget depends on q. The ex- rates, factor input costs, or government pected rate of capital gain is [E(P1) - policy. For example, the payoff from in- PO]IPO= q - .5, so the requiredpayment vesting, V, might rise or fall in the future per widget in the short position is .1 - depending on (unpredictable) changes in (q - .5) = .6 - q. By followingthe same policy. Alternatively, the cost of the in- steps as above, it is easy to see that the vestment, I, might rise or fall, in re- value today of the option to invest is sponse to changes in materials costs, or Fo = 773q. This can also be written as a to a policy change, such as the granting function of the current value of the or taking away of an investment subsidy project, V0. We have V0 = 100 + or tax benefit. 2(1OOq + 50)/(1. 1)t = 600 + lOOOq,so In our example, we made the unrealis- Fo = .773Vo - 464. Finally, note that it tic assumption that there is no longer any is better to wait rather than invest today uncertainty after the second period. In- as long as Fo > V0 - I, or q < .88. stead, we could have allowed the price There is nothing special about the par- to change unpredictably each period. For ticular source of uncertainty that we in- example, we could posit that at t = 2,

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if the price is $150, it could increase to and Siegel (1986). They considered the $225 with probability q or fall to $75 with following problem: At what point is it probability (1 - q), and if it is $50 it optimal to pay a sunk cost I in return could rise to $75 or fall to $25. Price could for a project whose value is V, given that rise or fall in a similar way at t = 3, 4, V evolves according to a geometric and so on. One could then work out the Brownian motion: value of the option to invest, and the dV = otVdt + aVdz optimal rule for exercising that option. (1) Although the algebra is messier, the where dz is the increment of a Wiener method is essentially the same as for the process, that is, dz = E(t)(dt)"2, with E(t) simple two-period exercise we carried a serially uncorrelated and normally dis- out above. (This is the basis for the bino- tributed random variable. Equation (1) mial option pricing model. See John Cox, implies that the current value of the proj- Stephen Ross, and Mark Rubinstein 1979 ect is known, but future values are log- and Cox and Rubinstein 1985 for detailed normally distributed with a variance that discussions.) Rather than take this ap- grows linearly with the time horizon. proach, in the next section we extend (See the Appendix for an explanation of our example by allowing the payoff from the Wiener process.) Thus although in- the investment to fluctuate continuously formation arrives over time (the firm ob- over time. serves V changing), the future value of The next two sections make use of con- the project is always uncertain. tinuous-time stochastic processes, as well McDonald and Siegel pointed out that as Ito's Lemma (which is essentially a the investment opportunity is equivalent rule for differentiating and integrating to a perpetual call option, and deciding functions of such processes). These tools, when to invest is equivalent to deciding which are becoming more and more when to exercise such an option. Thus, widely used in economics and finance, the investment decision can be viewed provide a convenient way of analyzing as a problem of option valuation (as we investment timing and option valuation saw in the simple example presented in problems. I provide an introduction to the previous section). I will rederive the the use of these tools in the Appendix solution to their problem in two ways, for readers who are unfamiliarwith them. first using option pricing (contingent Those readers might want to review the claims) methods, and then via dynamic Appendix before proceeding. (Introduc- programming. This will allow us to com- tory treatments can also be found in Rob- pare these two approaches and the as- ert Merton 1971; Gregory Chow 1979; sumptions that each requires. We will John Hull 1989; and A. G. Malliaris and then examine the characteristics of the William Brock 1982.) Readers who would solution. prefer to avoid this technical material al- A. The Use of Option Pricing together can skip directly to Section V, although they will miss some insights by As we have seen, the firm's option doing so. to invest, that is, to pay a sunk cost I and receive a project worth V, is analo- III. A More General Problem of gous to a call option on a stock. Unlike Investment Timing most financial call options, it is perpetual-it has no expiration date. We can One of the more basic models of irre- value this option and determine the opti- versible investment is that of McDonald mal exercise (investment) rule using the

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same methods used to value financial op- r + 4)pvmO,c where r is the risk-free rate tions. (For an overview of option pricing and 4) is the market price of risk. We methods' and their application, see Cox will assume that a-, the expected percent- and Rubinstein 1985; Hull 1989; and age rate of change of V, is less than its Scott Mason and Merton 1985.) risk-adjusted return Vt. (As will become This approach requires one important clear, the firm would never invest if this assumption, namely that stochastic were not the case. No matter what the changes in V are spanned by existing as- current level of V, the firm would always sets. Specifically, it must be possible to be better off waiting and simply holding find an asset or construct a dynamic port- on to the option to invest.) We denote folio of assets (i.e., a portfolio whose by 8 the difference between It and cx, holdings are adjusted continuously as as- that is, 8 = - at. set prices change), the price of which is A few words about the meaning of 8 perfectly correlated with V. This is equiv- are in order, given the important role it alent to saying that markets are suffi- plays in this model. The analogy with a ciently complete that the firm's decisions financial call option is helpful here. If V do not affect the opportunity set available were the price of a share of common to investors. The assumption of spanning stock, 8 would be the dividend rate on should hold for most commodities, which the stock. The total expected return on are typically traded on both spot and fu- the stock would be pt = 8 + a-, that is, tures markets, and for manufactured the dividend rate plus the expected rate goods to the extent that prices are corre- of capital gain. lated with the values of shares or portfo- If the dividend rate 8 were zero, a call lios. However, there may be cases in option on the stock would always be held which this assumption will not hold; an to maturity, and never exercised prema- example might be a new product unre- turely. The reason is that the entire related to any existing ones. turn on the stock is captured in its price With the spanning assumption, we can movements, and hence by the call op- determine the investment rule that maxi- tion, so there is no cost to keeping the mizes the firm's market value without option alive. But if the dividend rate is making any assumptions about risk pref- positive, there is an opportunity cost to erences or discount rates, and the invest- keeping the option alive rather than exer- ment problem reduces to one of contin- cising it. That opportunity cost is the div- gent claim valuation. (We will see shortly idend stream that one forgoes by holding that if spanning does not hold, dynamic the option rather than the stock. Because programming can still be used to maxi- 8 is a proportional dividend rate, the mize the present value of the firm's ex- higher the price of the stock, the greater pected flow of profits, subject to an arbi- the flow of dividends. At some high trary discount rate.) enough price, the opportunity cost of Let x be the price of an asset or dy- forgone dividends becomes high enough namic portfolio of assets perfectly corre- to make it worthwhile to exercise the op- lated with V, and denote by Pvm the cor- tion. relation of V with the market portfolio. For our investment problem, pt is the Then x evolves according to expected rate of return from owning the completed project. It is the equilibrium dx = [Lxdt + uxdz, rate established by the capital market, and by the capital asset pricing model and includes an appropriate risk pre- (CAPM), its expected return is pt = mium. If 8 > 0, the expected rate of

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions 1120 Journal of Economic Literature, Vol. XXIX (September 1991) capital gain on the project is less than will help to clarify the main effects of Vt.Hence 8 is an opportunity cost of de- irreversibility and uncertainty. We will laying construction of the project, and discuss more complicated (and hopefully instead keeping the option to invest alive. more realistic) models later. If 8 were zero, there would be no opportunity cost to keeping the option alive, B. Solving the Investment Problem and one would never invest, no matter Let us now turn to the valuation of how high the NPV of the project. That our investment opportunity, and the op- is why we assume 8 > 0. On the other timal investment rule. Let F = F(V) be hand, if 8 is very large, the value of the the value of the firm's option to invest. option will be very small, because the To find F(V) and the optimal investment opportunity cost of waiting is large. As rule, consider the return on the following -) 8 oc, the value of the option goes to portfolio: Hold the option, which is worth zero; in effect, the only choices are to F(V), and go short dF/dV units of the invest now or never, and the standard project (or equivalently, of the asset or NPV rule will again apply. portfolio x). Using subscripts to denote The parameter 8 can be interpreted derivatives, the value of this portfolio is in different ways. For example, it could P = F - FvV. Note that this portfolio reflect the process of entry and capacity is dynamic; as V changes, Fv may change, expansion by competitors. Or it can sim- in which case the composition of the port- ply reflect the cash flows from the proj- folio will be changed. ect. If the project is infinitely lived, then The short position in this portfolio will equation (1) can represent the evolution require a payment of 6VFv dollars per of V during the operation of the project, time period; otherwise no rational inves- and 5V is the rate of cash flow that the tor will enter into the long side of the project yields. Because we assume 8 is transaction. (To see this, note that an in- constant, this is consistent with future vestor holding a long position in the proj- cash flows being a constant proportion ect will demand the risk-adjusted return of the project's market value. 4 ptV, which includes the capital gain plus Equation (1) is, of course, is an abstrac- the dividend stream 8V. Because the tion from most real projects. For exam- short position includes Fv units of the ple, if variable cost is positive and the project, it will require paying out 8VFv.) project can be shut down temporarily Taking this into account, the total return when price falls below variable cost, V from holding the portfolio over a short will not follow a log-normal process, even time interval dt is if the output price does. Nonetheless, equation (1) is a useful simplification that dF - FVdV - 8VFvdt.

4A constant payout rate, 8, and required return, We will see shortly that this return is ji, imply an infinite project life. Letting CF denote risk-free. Hence to avoid arbitrage possi- the cash flow from the project, bilities it must equal r(F - FvV)dt: T T VO= CFte-1tdt fAVOe(R-8)te-1tdt, 0 0 dF - FVdV - 6VFvdt which implies T = oo. If the project has a finite life, = r(F - FvV)dt. (2) equation (1) cannot represent the evolution of V during the operating period. However, it can represent To obtain an expression for' dF, use its evolution prior to construction of the project, which is all that matters for the investment decision. Ito's Lemma: See Majd and Pindyck (1987, pp. 11-13), for a detailed discussion of this point. dF = FVdV + (1/2)Fv(dV)2. (3)

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(Ito's Lemma is explained in the Appen- where a is a constant, and ,3 is given dix. Note that higher-order terms van- by6 ish.) NoW substitute (1) for dV, with ox = 1/2 - (r - )/cr2 replaced by > - 8, and (dV)2 = cr2V2dt + {[r - 6)/U2 - 1/2]2 + 2r/cu2}1/2. (8) into equation (3): The remaining boundary conditions, dF = ( - + )VFvdt crVFvdz (6b) and (6c), can be used to solve for + (1/2)cu2V2Fvdt. (4) the two remaining unknowns: the con- Finally, substitute (4) into (2), rearrange stant a, and the critical value V* at which terms, and note that all terms in dz can- it is optimal to invest. By substituting cel out, so the portfolio is indeed risk- (7) into (6b) and (6c), it is easy to see free: that V*= (1/2)U2V2Fw + (r - 5)VFV PI/(P- 1) (9) -rFO= . (5) and a = (V* - I)/(V*)3. (10) Equation (5) is a differential equation Equations (7-10) give the value of the that F(V) must satisfy. In addition, F(V) investment opportunity, and the optimal must satisfy the following boundary con- investment rule, that is, the critical value ditions: V* at which it is optimal (in the sense F(O) = 0. (6a) of maximizing the firm's market value) F(V*) = V* - I. (6b) to invest. We will examine the character- istic of this solution below. Here we sim- FV(V*) = 1. (6c) ply point out that we obtained this solu- Condition (6a) says that if V goes to zero, tion by showing that a hedged (risk-free) it will stay at zero-an implication of the portfolio could be constructed consisting process (1)-so the option to invest will of the option to invest and a short posi- be of no value. V* is the price at which tion in the project. However, F(V) must it is optimal to invest, and (6b) just says be the solution to equation (5) even if that upon investing, the firm receives a the option to invest (or the project) does net payoff V* - I. Condition (6c) is called not exist and could not be included in the "smooth pasting" condition. If F(V) the hedge portfolio. All that is required were not continuous and smooth at the is spanning, that is, that one could find critical exercise point V*, one could do or construct an asset or dynamic portfolio better by exercising at a different point.5 of assets (x) that replicates the stochastic To find F(V), we must solve equation dynamics of V. As Merton (1977) has (5) subject to the boundary conditions shown, one can replicate the value func- (6a-6c). In this case we can guess a function with a portfolio consisting only of tional form, and determine by substitu- the asset x and risk-free bonds, and be- tion if it works. It is easy to see the solution to equation (5) that also satisfies condition (6a) is 'The general solution to equation (5) is F(V) = a1VP' + a2V02, (7) F(V) = aV3 where I31= 1/2 - (r- 2 + {[(r - 8) - 1/2]2 + 2rl(/2}"12 > 1, and P2= 1/2- (r- 8)/u2 - {[(r - 8)(J2- 1/2]2 + 2rl/o2}'12 < 0. 5Avinash Dixit (1988) provides a heuristic deriva- Boundary condition (6a) implies that a2 = 0, so the tion of this condition. solution can be written as in equation (7).

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions 1122 Journal of Economic Literature, Vol. XXIX (September 1991) cause the value of this portfolio will have (1/2)cu2V2Fw + acVFv - = 0 the same dynamics as F(V), the solution = - to (5), F(V) must be the value function or, substituting ac 8, to avoid dominance. (1/2)cu2V2Fv As discussed earlier, spanning will not + (Vt- 8)VFV - VF= 0. (13) always hold. If that is the case, one can Observe still solve the investment problem using that this equation is almost identical to equation (5); the only differ- dynamic programming. This is shown be- ence is that the discount rate ,u replaces low. the risk-free rate r. The boundary condi- C. Dynamic Programming tions (6a-6c) also apply here, and for the same reasons as before. (Note that (6c) To solve the problem by dynamic follows from the fact that V* is chosen programming, note that we want a rule to maximize the net payoff V* - I.) that maximizes the value of our invest- Hence the contingent claims solution to ment opportunity, F(V): our investment problem is equivalent to a dynamic programming solution, under F(V) = max EJ[(VT- I)e 1T] (11) the assumption of risk neutrality.7 where Et denotes the expectation at time Thus if spanning does not hold, we can t, T is the (unknown) future time that still obtain a solution to the investment the investment is made, ,u is the discount problem, subject to some discount rate. rate, and the maximization is subject to The solution will clearly be of the same equation (1) for V. We will assume that form, and the effects of changes in u- or ,u > a-, and as before denote 8 = ,u- 8 will likewise be the same. One point (x. is worth noting, however. Without span- Because the investment opportunity, ning, there is no theory for determining F(V), yields no cash flows up to the time the "correct" value for the discount rate T that the investment is undertaken, the ,u (unless we make restrictive assump- only return from holding it is its capital tions about investors' or managers' utility appreciation. As shown in the Appendix, functions). The CAPM, for example, the Bellman equation for this problem would not hold, so it could not be used is therefore to calculate a risk-adjusted discount rate. uF = (1/dt)E,dF. (12) D. Characteristics of the Solution Equation (12) just says that the total in- Assuming that spanning holds, let us stantaneous return on the investment op- examine the optimal investment rule portunity, ,uF, is equal to its expected given by equations (7-10). A few numeri- rate of capital appreciation. 'This result was first demonstrated by Cox and We used Ito's Lemma to obtain equa- Ross (1976). Also, note that equation (5) is the Bell- tion (3) for dF. Now substitute (1) for man equation for the maximization of the net payoff dV and (dV)2 into equation (3) to obtain to the hedge portfolio that we constructed. Because the portfolio is risk-free, the Bellman equation for the following expression for dF: that problem is dF = cxVFvdt+ zrVFvdz rP =-- VFv + (jIdt)EtdP. (i) + (1/2)o2V2Fwvdt. That is, the return on the portfolio, rP, equals the per period cash flow that it pays out (which is nega- Because = 0, = tive, because iVFVmust be paid in to maintain the Et(dz) (lldt)EtdF short position), plus the expected rate of capital gain. oaVFv+ (1/2)u2V2FW, and equation (12) By substituting P = F - FvV and expanding dF as can be rewritten as: before, one can see that (5) follows from (i).

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2.0 I ' '

1.8

1.6 -

1.4-

1.2-

5~1.0 I 4.1 0 =.3 0.8-

0.6 0.2

0.4- O I I _ 1/ 1 l I I I ~ ~ =0 ~~~~~~~~~~II 0.2-

0 0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 v

Figure 3. F(V) for a = 0.2, 0.3

Note: 8 = 0.04, r = 0.04,1 = 1

cal solutions will help to illustrate the include the opportunity cost of investing results and show how they depend on now rather than waiting. That opportu- the values of the various parameters. As nity cost is exactly F(V). When V < V*, we will see, these results are qualitatively V < I + F(V), that is, the value of the the same as those that come out of stan- project is less than its full cost, the direct dard option pricing models. Unless oth- cost I plus the opportunity cost of "kill- erwise noted, in what follows we set r ing" the investment option. = .04, 8 = .04, and the cost of the invest- Note that F(V) increases when of in- ment, I, equal to 1. creases, and so too does the critical value Figure 3 shows the value of the invest- V*. Thus uncertainty increases the value ment opportunity, F(V), for r = .2 and of a firm's investment opportunities, but .3. (These values are conservative for decreases the amount of actual investing many projects; in volatile markets, the that the firm will do. As a result, when standard deviation of annual changes in a firm's market or economic environment a project's value can easily exceed 20 or becomes more uncertain, the stock mar- 30 percent.) The tangency point of F(V) ket value of the firm can go up, even with the line V - I gives the critical value though the firm does less investing and of V, V*; the firm should invest only if perhaps produces less! This should make V ? V*. For any positive or, V* > L. it easier to understand the behavior of Thus the standard NPV rule, "Invest oil companies during the mid-1980s. when the value of a project is at least as During this period oil prices fell, but the great as its cost," must be modified to perceived uncertainty over future oil

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2.0 ,I , l

1.8 -

1.6-

1.4 -

1.2 -

> 1.0 _

0.8 -

0.6 - 6=.04

0.4 -6= 08

0.2 -

0 0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 V

Figure 4. F(V) for 8 = 0.04, 0.08

Note: a = 0.2, r= 0.04, 1 = 1 prices rose. In response, oil companies as 8 becomes larger, the expected rate paid more than ever for offshore leases of growth of V falls, and hence the ex- and other oil-bearing lands, even though pected appreciation in the value of the their development expenditures fell and option to invest and acquire V falls. In they produced less. effect, it becomes costlier to wait rather Finally, note that our results regarding than invest now. To see this, consider the effects of uncertainty involve no as- an investment in an apartment building, sumptions about risk preferences, or where 5V is the net flow of rental income. about the extent to which the riskiness The total return on the building, which of V is correlated with the market. Firms must equal the risk-adjusted market rate, can be risk-neutral, and stochastic has two components-this income flow changes in V can be completely diversifi- plus the expected rate of capital gain. able; an increase in cr will still increase Hence the greater the income flow rela- V* and hence tend to depress invest- tive to the total return on the building, ment. the more one forgoes by holding an op- Figures 4 and 5 show how F(V) and tion to invest in the building, rather than V* depend on 8. Observe that an in- owning the building itself. crease in 8 from .04 to .08 results in a If the risk-free rate, r, is increased, decrease in F(V), and hence a decrease F(V) increases, and so does V*. The rea- in the critical value V*. (In the limit as son is that the present value of an invest- 8 -* co, F(V) ->O for V < I, and V* -, ment expenditure I made at a future time as Figure 5 shows.) The reason is that T is Ie- rT, but the present value of

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7 I ,I I

1~~~~~~~~~~~

o L I IX I 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 6

Figure 5. V* as a Function of 8

Note: a = 0.2, r = 0.04, I = 1

the project that one receives in return walk. It would also allow for the project for that expenditure is Ve 8T. Hence with to be shut down (permanently or tempo- a fixed, an increase in r reduces the pres- rarily) if price falls below variable cost. ent value of the cost of the investment The model developed in the previous but does not reduce its payoff. But note section can easily be extended in this that while an increase in r raises the way. In so doing, we will see that option value of a firm's investment options, it pricing methods can be used to find the also results in fewer of those options be- value of the project, as well as the opti- ing exercised. Hence higher (real) inter- mal investment rule. est rates reduce investment, but for a Suppose the output price, P. follows different reason than in the standard the stochastic process: model. dP = c.Pdt + rPdz. (14) IV. The Value of a Project and the We will assume that (x < ,u, where ,u is Decision to Invest the market risk-adjusted expected rate As mentioned earlier, equation (1) ab- of return on P or an asset perfectly corre- stracts from most real projects. A more lated with P. and let 8 = ,u- ( as before. realistic model would treat the price of If the output is a storable commodity the project's output as a geometric ran- (e.g., oil or copper), 8 will represent the dom walk (and possibly one or more fac- net marginal convenience yield from stor- tor input costs as well), rather than mak- age, that is, the flow of benefits (less stor- ing the value of the project a random age costs) that the marginal stored unit

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provides. We assume for simplicity that value the project (as well as the option 8 is constant. (For most commodities, to invest) using contingent claim meth- marginal convenience yield in fact fluctu- ods. Otherwise, we can specify a discount ates as the total amount of storage fluctu- rate and use dynamic programming. We ates.) will assume spanning and use the first We will also assume the following: (i) approach. Marginal and average production cost is As before, we construct a risk-free equal to a constant, c. (ii) The project portfolio: long the project and short Vp can be costlessly shut down if P falls be- units of the output. This portfolio has low c, and later restarted if P rises above value V(P) - VpP, and yields the instanta- c. (iii) The project produces one unit of neous cash flow j(P - c)dt - 5VpPdt, output per period, is infinitely lived, and where j = 1 if P c so that the firm is the (sunk) cost of investing in the project producing, and j = 0 otherwise. (Recall is L. that 6VpPdt is the payment that must be We now have two problems to solve. made to maintain the short position.) The First, we must find the value of this proj- total return on the portfolio is thus ect, V(P). To do this, we can make use dV - VpdP + j(P - c)dt - 6VpPdt. Be- of the fact that the project itself is a set cause this return is risk-free, set it equal of options.8 Specifically, once the project to r(V - VpP)dt. Expanding dV using has been built, the firm has, for each Ito's Lemma, substituting (14) for dP, and future time t, an option to produce a unit rearranging yields the following differen- of output, that is an option to pay c and tial equation for V: receive P. Hence the project is equivalent to a large number (in this case, infi- (1/2)&P2VPP + (r - 5)PVp - rV - = nite, because the project is assumed to + j(P c) 0. (15) last indefinitely) of operating options, This equation must be solved subject and can be valued accordingly. to the following boundary conditions: Second, given the value of the project, we must value the firm's option to invest V(0) = 0. (16a) in it, and determine the optimal exercise V(c-) = V(c+). (16b) (investment) rule. This will boil down to finding a critical P*, where the firm in- VP(c-)= Vp(c+). (16c) vests only if P - P. As shown below, lim V = PI - clr. (16d) the two steps to this problem can be P -c o solved sequentially by the same methods Condition (16a) is an implication of equa- used in the previous section.9 tion (14); that is, if P is ever zero it will remain zero, so the project then has no A. Valuing the Project value. Condition (16d) says that as P be- If we assume that uncertainty over comes very large, the probability that P is spanned by existing assets, we can over any finite time period it will fall be- 8This point and its implications are discussed in low cost and production will cease be- detail in McDonald and Siegel (1985). comes very small. Hence the value of 9Note that the option to invest is an option to the project approaches the difference be- purchase a package of call options (because the project is just a set of options to pay c and receive P at tween two perpetuities: a flow of revenue each future time t). Hence we are valuing a com- (P) that is discounted at the risk-adjusted pound option. For examples of the valuation of com- rate but is expected to grow at rate pound financial options, see Robert Geske (1979) and li Peter Carr (1988). Our problem can be treated in a (x, and a flow of cost (c), which is constant simpler manner. and hence is discounted at the risk-free

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400 'I ' I

350 -

300 -

250-

200 -=.4

150-

100

50- 0

0 0 2 4 6 8 10 12 14 16 18 20

p Figure 6. V(P) for a = 0, 0.2, 0.4

Note: 8 = .04, r= .04, c= 10

rate r. Finally, conditions (16b) and (16c) 31 = 1/2 - (r- )/u2 say that the project's value is a continu- + - + 2r/ ous and smooth function of P. {[(r-)/U2 1/2]2 2}"/2 The solution to equation (15) will have and two parts, one for P < c, and one for P = 1/2 - - -c. The reader can check by substitu- 12 (r )u2 - + 2r/u tion that the following satisfies (15) as -{[(r- b)/U2 1/2]2 2}/2. well as boundary conditions (16a) and The constants A1 and A2 can be found (16d): by applying boundary conditions (16b) and (16c):

r - 132(r- w A2Pr 2 + plb-Clr , P ' C Al- A1 =r P(138- c(l-2).) where:1? 2 r-(r - P2) 10 By substituting(17) for V(P)into (15), the reader The solution (17) for V(P) can be inter- can check that PI and 32 are the solutions to the following quadraticequation: preted as follows: When P < c, the project is not producing. Then, A1PP' is the -1) + (r - - r = 0. (1/2)g2pl(p3 8)PI value of the firm's options to produce in Because V(O)= 0, the positive solution (1I > 1) must the future, if and when P increases. apply when P < c, and the negative solution (12 < P - 0) must apply when P > c. Note that P is the same When c, the project is producing. as 1 in equation (8). If, irrespective of changes in P. the firm

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400

350 -

300 -

250-

8= .02 200

150- / .04

100-

50 =~~~~~~~~~~~~~.08

0 0 2 4 6 8 10 12 14 16 is 20 p Figure 7. V(P) for 8 = 0.02, 0.04, 0.08

Note: r = .04, cr = .2, c = 10 had no choice but to continue producing side is limited to zero, the greater is a, throughout the future, the present value the greater is the expected future flow of the future flow of profits would be of profit, and the higher is V. given by Plb - clr. However, should P Figure 7 shows V(P) for a = .2 and fall, the firm can stop producing and 8 = .02, .04, and .08. For any fixed risk- avoid losses. The value of its options to adjusted discount rate, a higher value of stop producing is A2P2. 8 means a lower expected rate of price A numerical example will help to illus- appreciation, and hence a lower value trate this solution. Unless otherwise for the firm. noted, we set r = .04, 8 = .04, and c = 10. Figure 6 shows V(P) for cr = 0, B. The Investment Decision .2, and .4. When r = 0, there is no possibility that P will rise in the future, so in Now that we know the value of this this case the firm will never produce (and project, we must find the optimal invest- has no value) unless P > 0. If P > 10, ment rule. Specifically, what is the value V(P) = (P - 10)/.04 = 25P = 250. How- of the firm's option to invest as a function ever, if a > 0, the firm always has some of the price P, and at what critical price value as long as P > 0; although the firm P* should the firm exercise that option may not be producing today, it is likely by spending an amount I to purchase the to produce at some point in the future. project? Also, because the upside potential for fu- By going through the same steps as ture profit is unlimited while the down- above, the reader can check that the

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500 . I I I I

400

300

> 22t}}100 ~~~~~F(P)

0 0 t - 8 < 1lI 14V4 1 I 20 1 1 2 0

-100 0 4 8 12 16 20 24 28 P*= 23.8 p

Figure 8. V(P) -I and F(P) for a = 0.2, 8 = 0.04

Note: r = .04, c= 10, I = 100

value of the firm's option to invest, F(P), where 1P is given above under equation must satisfy the following differential (17). To find the constant a and the criti- equation: cal price P*, we use boundary conditions (19b) and (19c). By substituting equation + (r - - rF = 0. (18) (1/2)u2P2Fpp 8)PFp (20) for F(P) and equation (17) for V(P) F(P) must also satisfy the following (for P - c) into (19b) and (19c), the reader boundary conditions: can check that the constant a is given by F(0) = 0. (19a) F(P*) = V(P*) - I. (19b) a = 2A2 (p*)(P2-PI) + j (P*)(l-P) (21) Fp(P*) = Vp(P*). (19c) These conditions can be interpreted in and the critical price P* is the solution the same way as conditions (6a-6c) for to the model presented in Section III. The - difference is that the payoff from the in- A2(PI 12) (p*)P2 vestment, V, is now a function of the price P. The solution to equation (18) and + P* -I = 0. (22) boundary condition (19a) is Equation (22), which is easily solved nu- F(P) = [aP 'I, P ' P* (20) V(P - I>,P>P* merically, gives the optimal investment

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions 700 ! ' I

600

500

400

300

200

100

-100 l 10 20 t30

P:,0= 14.0 P __ 2 = 23.8 P*:,4 = 34.9

Figure 9. V(P) - I and F(P) for r = 0.0, 0.2, 0.4

Note: r = .04, 8 = .04, c = 10, I = 100

500 , I I , . I I

400 =.04

300

200

100

-10 0 4 8 12 16 20 24 28 f 32 p P 04 = 23.8 PL 08 = 29.2 Figure 10. V(P) - I and F(P) for 8 = 0.04, 0.08

Note: r = .04, r = .2, c = 10, I = 100

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions Pindyck: Irreversibility, Uncertainty, and Investment 1131 rule. (The reader can check first, that nates, so that a higher 8 results in a (22) has a unique positive solution for P* higher Pt. This is illustrated in Figure that is larger than c, and second, that 10, which shows F(P) and V(P) - I for V(P*) > I, so that the project must have 8 = .04 and .08. Note that when 8 is an NPV that exceeds zero before it is increased, V(P) and hence F(P) fall optimal to invest.) sharply, and the tangency at P* moves This solution is shown graphically in to the right. Figure 8, for a = .2, 8 = .04, and I = This result might at first seem to con- 100. The figure plots F(P) and V(P) - . tradict what the simpler model of Section Note from boundary condition (19b) that III tells us. Recall that in that model, P* satisfies F(P*) = V(P*) - I, and note an increase in 8 reduces the critical value from boundary condition (19c) that P* is of the project, V*, at which the firm at a point of tangency of the two curves. should invest. But while in this model The comparative statics for changes in P* is higher when 8 is larger, the corre- a or 8 are of interest. As we saw before, sponding value of the project, V(P*), is an increase in a results in an increase lower. This can be seen from Figure 11, in V(P) for any P. (The project is a set which shows P* as a function of a for of call options on future production, and 8 = .04 and .08, and Figure 12, which the greater the volatility of price, the shows V(P*). If, say, cu is .2 and 8 is in- greater the value of these options.) But creased from 0.4 to .08, P* will rise from although an increase in ar raises the value 23.8 to 29.2, but even at the higher P*, of the project, it also increases the critical V is lower. Thus V* = V(P*) is declining price at which it is optimal to invest, that with 8, just as in the simpler model. is, aP*/&o. > 0. The reason is that for This model shows how uncertainty any P, the opportunity cost of investing, over future prices affects both the value F(P), increases even more than V(P). of a project and the decision to invest. Hence as with the simpler model pre- As discussed in the next section, the sented in the previous section, increased model can easily be expanded to allow uncertainty reduces investment. This is for fixed costs of temporarily stopping illustrated in Figure 9, which shows F(P) and restarting production, if such costs and V(P) - I for u = 0, .2, and .4. When are important. Expanded in this way, u - 0, the critical price is 14, which just models like this can have practical appli- makes the value of the project equal to cation, especially if the project is one that its cost of 100. As u is increased, both produces a traded commodity, like cop- V(P) and F(P) increase; P* is 23.8 for per or oil. In that case, ou and 6 can be u = .2 and 34.9 for u = .4. determined directly from futures and An increase in 8 also increases the criti- spot market data. P* which the firm should in- cal price at C. Alternative Stochastic Processes vest. There are two opposing effects. If 8 is larger, so that the expected rate of The geometric random walk of equa- increase of P is smaller, options on future tion (14) is convenient in that it permits production are worth less, so V(P) is an analytical solution, but one might be- smaller. At the same time, the opportu- lieve that the price, Pa is better repre- nity cost of waiting to invest rises-the sented by a different stochastic process. expected rate of growth of F(P) is For example, one could argue that over smaller-so there is more incentive to the long run, the price of a commodity exercise the investment option, rather will follow a mean-reverting process (for than keep it alive. The first effect domi- which the mean reflects long-run mar-

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= .08

30-

25 - 8 =.04 20 -

10 _

0 I I 0 0.2 0.4

Figure 11. P* vs. a for 8 = 0.04, 0.08

1,000 ,

900

800

700

600

> 500

400 -

300 -

200 -

100,

0 0 0.2 0.4

Figure 12. V(P*) vs. or for 8 = 0.04, 0.08

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ginal cost, and might be time-varying). ture is a sunk cost. Because the future Our model can be adapted to allow for value of the project is uncertain, this cre- this or foValternative stochastic processes ates an opportunity cost to investing, for P. However, in most cases numerical which drives a wedge between the cur- methods will then be necessary to obtain rent value of the project and the direct a solution. cost of the investment. As an example, suppose P follows the In general, there may be a variety of mean-reverting process: sunk costs. For example, there may be a sunk cost of exiting an industry or aban- dP/P = X(P - P)dt + adz. (23) doning a project. This could include Here, P tends to revert back to a "nor- severance pay for workers, land reclama- mal" level P (which might be long-run tion in the case of a mine, and so on.11 marginal cost in the case of commodity This creates an opportunity cost of shut- like copper or coffee). By going through ting down, because the value of the proj- the same arguments as we did before, ect might rise in the future. There may it is easy to show that V(P) must then also be sunk costs associated with the op- satisfy the following differential equation: eration of the project. In Section IV, we assumed that the firm could stop and re- (1/2)o2P2Vpp + [(r - ,- X)P + XP]PVp start production costlessly. For most -rV +j(P-c) = O (24) projects, however, there are likely to be substantial sunk costs involved in even together with boundary conditions (16a- temporarily shutting down and restart- 16c). The value of the investment option, ing. F(P), must satisfy The valuation of projects and the deci- (1/2)o2P2Fpp + [(r - - X)P + XP]PFp sion to invest when there are sunk costs of this sort have been studied by Brennan -rF= O (25) and Schwartz (1985) and Dixit (1989a). with boundary conditions (19a-19c). Brennan and Schwartz (1985) find the ef- Equations (24) and (25) are ordinary dif- fects of sunk costs on the decision to open ferential equations, so solution by nu- and close (temporarily or permanently) merical methods is relatively straightfor- a mine, when the price of the resource ward. follows equation (14). Their model accounts for the fact that a mine is subject V. Extensions to cave-ins and flooding when not in use, and a temporary shutdown requires ex- The models presented in the previous penditures to avoid these possibilities. two sections are fairly simple, but illus- Likewise, reopening a temporarily closed trate how a project and an investment mine requires a substantial expenditure. opportunity can be viewed as a set of Finally, a mine can be permanently options, and valued accordingly. These closed. This will involve costs of land rec- insights have been extended to a variety lamation (but avoids the cost of a tempo- of problems involving investment and rary shutdown). production decisions under uncertainty. This section reviews some of them. " Of course the scrap value of the project might A. Sunk Costs and Hysteresis exceed these costs. In this case, the owner of the project holds a put option (an option to "sell" the In Sections III and IV, we examined project for the net scrap value), and this raises the project'svalue. This has been analyzedby Myers models in which the investment expendi- and Majd (1985).

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Brennan and Schwartz obtained an exit if P ' w - pl, where p is the firm's analytical solution for the case of an infi- discount rate. 13 However, if a > 0, there nite resource stock. (Solutions can also are opportunity costs to entering or exit- be obtained when the resource stock is ing. These opportunity costs raise the finite, but then numerical methods are critical price above which it is optimal required.) Their solution gives the value to enter, and lower the critical price be- of the mine as a function of the resource low which it is optimal to exit. (Further- price and the curent state of the mine more, numerical simulations show that (i.e., open or closed). It also gives the r need not be large to induce a significant decision rule for changing the state of effect.) the mine (i.e., opening a closed mine These models help to explain the prev- or temporarily or permanently closing an alence of hysteresis-effects that persist open mine). Finally, given the value of after the causes that brought them about the mine, Brennan and Schwartz show have disappeared. In Dixit's model, firms how (in principle) an option to invest in that entered an industry when price was the mine can be valued and the optimal high may remain there for an extended investment rule determined, using a con- period of time even though price has tingent claim approach like that of Sec- fallen below variable cost, so they are tion IV. 12 losing money. (Price may rise in the fu- By working through a realistic example ture, and to exit and later reenter in- of a copper mine, Brennan and Schwartz volves sunk costs.) And firms that leave showed how the methods discussed in an industry after a protracted period of this paper can be applied in practice. But low prices may hesitate to reenter, even their work also shows how sunk costs of after prices have risen enough to make opening and closing a mine can explain entry seem profitable. Similarly, the the "hysteresis" often observed in extrac- Brennan and Schwartz model shows why tive resource industries: During periods many copper mines built during the of low prices, managers often continue 1970s when copper prices were high to operate unprofitable mines that had were kept open during the mid-1980s been opened when prices were high; at when copper prices had fallen to their other times managers fail to reopen lowest levels (in real terms) since the seemingly profitable ones that had been Great Depression. closed when prices were low. This insight The fact that exchange rate movements is further developed in Dixit (1989a, during the 1980s left the U. S. with a per- 1991), and is discussed below. sistent trade deficit at the end of that Dixit (1989a) studies a model with sunk decade can also be seen as a result of costs k and 1, respectively, of entry and hysteresis. For example, Dixit (1989b) exit. The project produces one unit of models entry by Japanese firms into the output per period, with variable cost w. U. S. market when the exchange rate fol- The output price, P, follows equation lows a geometric Brownian motion. (14). If ar = 0, the standard result holds: Enter (i.e., spend k) if P - w + pk, and 13 As Dixit points out, one would find hysteresis if, for example, the price began at a level between 12 Jerey MacKie-Mason (1990) developed a re- w and w + pk, rose above w + pk so that entry lated model of a mine that shows how nonlinear tax occurred, but then fell to its original level, which is rules (such as a percentage depletion allowance) affect too high to induce exit. However, the firm's price the value of the operating options as well as the in- expectations would then be irrational (because the vestment decision. price is in fact varying stochastically).

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Again, there are sunk costs of entry and goods, on income and wealth. Most pur- exit. The Japanese firms are ordered ac- chases of consumer durables are at least cording to their variable costs, and all partly irreversible. Poksang Lam (1989) firms are price takers. As with the models developed a model that accounts for this, discussed above, the sunk costs com- and shows how irreversibility results in bined with exchange rate uncertainty a sluggish adjustment of the stock of du- create opportunity costs of entering or rables to income changes. Sanford Gross- exiting the U.S. market. As a result, man and Guy Laroque (1990) study con- there will be an exchange rate band sumption and portfolio choice when within which Japanese firms neither en- consumption services are generated by ter nor exit, and the U.S. market price a durable good and a transaction cost will not vary as long as exchange rate must be paid when the good is sold. Un- fluctuations are within this band. Richard like in standard models (e.g., Merton Baldwin (1988) and Baldwin and Paul 1971), optimal consumption is not a Krugman (1989) developed related mod- smooth funciton of wealth; a large change els that yield similar results. These mod- in wealth must occur before a consumer els help to explain the low rate of changes his holdings of durables and exchange rate passthrough observed dur- hence his consumption. As a result, the ing the 1980s, and the persistence of consumption-based CAPM fails to hold the U. S. trade deficit even after the (although the market portfolio-based dollar depreciated. (Baldwin 1988 also pro- CAPM does hold). vides evidence that the over- empirical B. Sequential Investment valuation of the dollar during the early 1980s was indeed a hysteresis-inducing Many investments occur in stages shock.) that must be carried out in sequence, Sunk costs of entry and exit can also and sometimes the payoffs from or costs have hysteretic effects on the exchange of completing each stage are uncertain. rate itself, and on prices. Baldwin and For example, investing in a new line of Krugman (1989), for example, show how aircraft begins with engineering, and the entry and exit decisions described continues with prototype production, above feed back to the exchange rate. testing, and final tooling stages. And an In their model, a policy change (e.g., a investment in a new drug by a pharma- reduction in the money supply) that ceutical company begins with research causes the currency to appreciate sharply that (with some probability) leads to a can lead to entry by foreign firms, which new compound, continues with extensive in turn leads to an equilibrium exchange testing until FDA approval is obtained, rate that is below the original one. (These and concludes with the construction of ideas are also discussed in Krugman a production facility and marketing of the 1989.) Similar effects occur with prices. product. In the case of copper, the reluctance of Sequential investment programs like firms to close down mines during the these can take substantial time to com- mid-1980s, when demand was weak, al- plete five to ten years for the two exam- lowed the price to fall even more than ples mentioned above. In addition, they it would have otherwise. can be temporarily or permanently aban- Finally, sunk costs may be important doned midstream if the value of the end in explaining the dependence of con- product falls, or the expected cost of com- sumer spending, particularly for durable pleting the investment rises. Hence

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these investments can be viewed as com- uncertainty. The lower the maximum pound options; each stage completed (or rate of investment (i.e., the longer it dollar invested) gives the firm an option takes to complete the project), the higher to complete the next stage (or invest the is the critical V*(K) required for construc- next dollar). The problem is to find a con- tion to proceed. This is because the tingent plan for making these sequential project's value upon completion is more and irreversible expenditures uncertain, and the expected rate of Majd and Pindyck (1987) solve this growth of V over the construction period problem for a model in which a firm in- is less than p, the risk-adjusted rate of vests continuously (each dollar spent return (8 is positive). Also, unlike the buys an option to spend the next dollar) model of Section III where the critical until the project is completed, invest- value V* declines monotonically with 8, ment can be stopped and later restarted with time to build, V* will increase with costlessly, and there is a maximum rate 8 when 8 is large. The reason is that while at which outlays and construction can a higher 8 increases the opportunity cost proceed (i.e., it takes "time to build"). of waiting to begin construction, it also The payoff to the firm upon completion reduces the expected rate of growth of is V, the value of the operating project, V during the construction period, so that which follows the geometric Brownian the (risk-adjusted) expected payoff from motion of equation (1). Letting K be the completing construction is reduced. Fi- total remaining expenditure required to nally, by computing F(V, K) for different complete the project, the optimal rule maximum rates of investment, one can is to keep investing at the maximum rate value construction time flexibility, that as long as V exceeds a critical value is, what one would pay to be able to build V*(K),with dV*/dK < 0. Using the meth- the project faster.'5 ods of Sections III and IV, it is straight- In the Majd-Pindyck model, invest- forward to derive a partial differential ment occurs as a continuous flow; that equation for F(V, K), the value of the is, each dollar spent gives the firm an investment opportunity. Solutions to this option to spend another dollar, up to the equation and its associated boundry con- last dollar, which gives the firm a com- ditions, which are obtained by numerical pleted project. Often, however, sequen- methods, yield the optimal investment tial investments occur in discrete stages, rule V*(K).4 as with the aircraft and pharmaceutical These solutions show how time to build examples mentioned above. In these magnifies the effects of irreversibility and cases, the optimal investment rules can be found by working backward from the 14 Letting k be the maximumrate of investment, this equation is 15 The production decisions of a firm facing a learn- '/2u2V2Fvv + (r - 8)VFv - rF - x(kFK + k) = 0 ing curve and stochastically shifting demand is another example of this kind of sequential investment. where x = 1 when the firm is investing and 0 other- Here, part of the firm's cost of production is actually wise. F(V, K) must also satisfy the following boundary an (irreversible) investment, which yields a reduction conditions: in future costs. Because demand fluctuates, the fu- F(V, 0) = V, ture payoffs from this investment are uncertain. Majd and Pindyck (1989) introduce stochastic demand into limv ),OFV(V, K) = e bKlk a learning curve model, and derive the optimal pro- F(O, K) = 0 duction rule. They show how uncertainty Over future and F(V, K) and FV(V, K) continuous at the boundary demand reduces the shadow value of cumulative pro- V*(K). For an overview of numerical methods for duction generated by learning, and thus raises the solving partial differential equations of this kind, see critical price at which it is optimal for the firm to Geske and Kuldeep Shastri (1985). produce.

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions Pindyck: Irreversibility, Uncertainty, and Investment 1137 completed project, as we did with the of a stage. Then the problem must be model of Section IV. solved numerically, using a method like To see how this can be done, consider the one in Majd and Pindyck (1987).17 a two-stage investment in new oil pro- In all of the models discussed so far, duction capacity. First, reserves of oil no learning takes place, in the sense that must be obtained, through exploration future prices (or project values, V) are or outright purchase, at a cost 11. Second, always uncertain, and the degree of un- development wells (and possibly pipe- certainty depends only on the time hori- lines) must be built, at a cost 12. Let P zon. For some sequential investments, be the price of oil and assume it follows however, early stages provide informa- the geometric Brownian motion of equation about costs or net payoffs in later tion (14). The firm thus begins with an stages. Synthetic fuels was a much de- option, worth F1(P), to invest in reserves. bated example of this; oil companies ar- Doing so buys an option, worth F2(P), gued that demonstration plants were to invest in development wells. Making needed (and deserved funding by the this investment yields production capac- government) to determine production ity, worth V(P). costs. The aircraft and pharmaceutical in- Working backward to find the optimal vestments mentioned above also have investment rules, first note that as in the these characteristics. The engineering, model of Section IV, V(P) is the value prototype production, and testing stages of the firm's operating options, and can in the development of a new aircraft all be calculated accordingly. Next, F2(P) provide information about the ultimate can be found; it is easy to show that it cost of production (as well as the aircraft's must satisfy equation (18) and boundary flight characteristics, which will help de- conditions (19a-19c), with 12 replacing I, termine its market value). Likewise, the and P* becoming the critical price at R & D and testing stages of the develop- which the firm should invest in development of a new drug determine the effi- ment wells. Finally, F1(P) can be found. cacy and side effects of the drug, and It also satisfies (18) and (19a-19c), but hence its value. with F2(P) replacing V(P) in (19b) and Kevin Roberts and Martin Weitzman (19c), 1 replacing I, and P** replacing (1981) developed a model of sequential P*. (P** is the critical price of oil at which investment that stresses this role of infor- the firm should invest in reserves.) If mation gathering. In their model, each marginal production cost is constant and stage of investment yields information there is no cost to stopping or restarting that reduces the uncertainty over the production, an analytical solution can be value of the completed project. Because obtained. 16 the project can be stopped in midstream, In this example there is no time to it may pay to go ahead with the early build; each stage (obtaining reserves, and stages of the investment even though ex building development wells) can be com- ante the net present value of the entire pleted instantly. For many projects each project is negative. Hence the use of a stage of the investment takes time, and simple net present value rule can reject the firm can stop investing in the middle

17 In a related paper, Carliss Baldwin (1982) ana- 16 James Paddock, Siegel, and James Smith (1988) lyzes sequential investment decisions when invest- value oil reserves as options to produceoil, but ignore ment opportunities arrive randomly, and the firm the development stage. OctavioTourinho (1979) first has limited resources to invest. She values the se- suggested that natural resource reserves can be quence of opportunities, and shows that a simple valued as options. NPV rule will lead to overinvestment.

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions 1138 Journal of Economic Literature, Vol. XXIX (September 1991) projects that should be undertaken. This the economics literature on investment, is just the opposite of our earlier finding however, focuses on incremental invest- that a simple NPV rule can accept proj- ment; firms invest to the point that the ects that should be rejected. The crucial cost of the marginal unit of capital just assumption in the Roberts-Weitzman equals the present value of the revenues model is that prices and costs do not it is expected to generate. The cost of evolve stochastically. The value of the the unit can include adjustment costs (re- completed project may not be known (at flecting the time and expense of installing least until the early stages are com- and learning to use new capital) in addi- pleted), but that value does not change tion to the purchase cost. In most mod- over time, so there is no gain from wait- els, adjustment costs are a convex func- ing, and no opportunity cost to investing tion of the rate of investment, and are now. Instead, information gathering adds thus a crucial determinant of that rate. a shadow value to the early stages of the (For an overview, see Stephen Nickell investment. 18 1978 or the more recent survey by An- This result applies whenever informa- drew Abel 1990.) tion gathering, rather than waiting, Except for work by Kenneth Arrow yields information. The basic principle (1968) and Nickell (1974), which is in a is easily seen by modifying our simple deterministic context, this literature gen- two-period example from Section II. erally ignores the effects of irreversibil- Suppose that the widget factory can only ity. As with discrete projects, irrevers- be built this year, and at a cost of $1,200. ibility and the ability to delay investment However, by first spending $50 to re- decisions change the fundamental rule search the widget market, one could de- for investing. The firm must include as termine whether widget prices will rise part of the total cost of an incremental or fall next year. Clearly one should unit of capital the opportunity cost of in- spend this $50, even though the NPV vesting in that unit now rather than wait- of the entire project (the research plus ing. the construction of the factory) is nega- Giuseppe Bertola (1989) and Pindyck tive. One would then build the factory (1988) developed models of incremental only if the research showed that widget investment and capacity choice that ac- prices will rise. count for irreversibility. In Pindyck's model, the firm faces a linear inverse de- C. Incremental Investment and mand function, P = 0(t) - yQ, where 0 Choice Capacity follows a geometric Brownian motion, So far we have examined decisions and has a Leontief production technol- to invest in single, discrete projects, for ogy. The firm can invest at any time at example, the decision to build a new fac- a cost k per unit of capital, and each unit tory or develop a new aircraft. Much of of capital gives it the capacity to produce up to one unit of output per period. The investment problem is solved by first de- '8 Weitzman (1981) used this model to evaluate the case for building demonstrationplants for syn- termining the value of an incremental thetic fuel production, and found that learning about unit of capital, given 0 and an existing costs could justify these early investments. Much of K, and then finding the the debate over synthetic fuels has had to do with capital stock, the role of government, and in particularwhether value of the option to invest in this unit subsidies (for demonstrationplants or for actual pro- and the optimal exercise rule. This rule duction)could be justified. These issues are discussed is a invest whenever in Paul Joskow and Pindyck (1979) and Richard function K*(0)- Schmalensee (1980). K < K*(o) which determines the firm's

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optimal capital stock. Pindyck shows that Hartman (1972) pointed out, it is also the an increase in the variance of 0 increases case for a competitive firm that combines the value of an incremental unit of capital capital and labor with a linear homoge- (that unit represents a set of call options neous production function. Hartman on future production), but increases the shows that as a result, price uncertainty value of the option to invest in the unit increases the firm's investment and capi- even more, so that investment requires tal stock. a higher value of 0. Hence a more volatile Andrew Abel (1983) extends Hartman's demand implies that a firm should hold result to a dynamic model in which price less capital, but have a higher market follows a geometric Brownian motion and value. 19 there are convex costs of adjusting the In Bertola's model, the firms' net reve- capital stock, and again shows that uncer- nue function is of the form AK'-Z, with tainty increases the firm's rate of invest- O < 3 < 1. (This would follow from a ment. Finally, Ricardo Caballero (1991) Cobb-Douglas production function and introduces asymmetric costs of adjust- an isoelastic demand curve.) The de- ment to allow for irreversibility (it can mand-shift variable Z and the purchase be costlier to reduce K than to increase price of capital follow correlated geomet- it), and shows that again price uncer- ric Brownian motions. Bertola solves for tainty increases the rate of investment. the optimal investment rule, and shows However, the Abel and Caballero results that the marginal profitability of capital hinge on the assumptions of constant re- that triggers investment is higher than turns and perfect competition, which the user cost of capital as conventionally make the marginal revenue product of measured. The capital stock, K, is nonsta- capital independent of the capital stock. tionary, but Bertola finds the steady-state Then the firm can ignore its future capital distribution for the ratio of the marginal stock (and hence irreversibility) when de- profitability of capital to its price. Irre- ciding how much to invest today. As Ca- versibility and uncertainty reduce the ballero shows, decreasing returns or im- mean of this ratio; that is, on average perfect competition will link the marginal capital intensity is higher. Although the revenue products of capital across time, firm has a higher threshold for invest- so that the basic result in Pindyck (1988) ment, this is outweighed on average by and Bertola (1989) holds.20 low outcomes for Z. The assumption that the firm can in- The finding that uncertainty over fu- vest incrementally is extreme. In most ture demand can increase the value of a industries, capacity additions are lumpy, marginal unit of capital is not new. All and there are scale economies (a 400 that is required is that the marginal reve- room hotel usually costs less to build and nue product of capital be convex in price. operate than two 200 room hotels). This is the case when the unit of capital can go unutilized (so that it represents 20 Even if firms are perfectly competitive and have a set of operating options). But as Richard constant returns, stochastic fluctuations in demand will depress irreversible investment if there can be entry by new firms in response to price increases. See Pindyck (1990b) for a discussion of this point. 19This means that the ratioof a firm'smarket value Also, Abel, Bertola, Caballero,and Pindyckexamine to the value of its capital in place should always ex- the effects of increased demand or price uncertainty ceed one (because part of its market value is the holding the discount rate fixed. As Roger Craine value of its growth options), and this ratio should (1989) points out, an increase in demand uncertainty be higher for firms selling in more volatile markets. is likely to be accompanied by an increase in the Kester's (1984) study suggests that this is indeed the systematic riskiness of the firm's capital, and hence case. an increase in its risk-adjusteddiscount rate.

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Hence firms must decide when to add vary stochastically. It can produce these capacity, and how large an addition to products by (irreversibly) installing and make. This problem was first studied in utilizing product-specific capital, or by a stochastic setting by Alan Manne (irreversibly) installing a more costly flex- (1961). He considered a firm that must ible type of capital that can be used to always have enough capacity to satisfy produce either or both products. The demand, which grows according to a sim- problem is to decide which type and how ple Brownian motion with drift. The cost much capital to install. Hua He and Pin- of adding an amount of capacity x is kXa, dyck (1989) solve this for a model with with 0 < a < 1; the firm must choose x linear demands by first valuing incre- to minimize the present value of ex- mental units of capital (output-specific pected capital costs. Manne shows that and flexible), and then finding the opti- with scale economies, uncertainty over mal investment rule, and hence optimal demand growth leads the firm to add ca- amounts of capacity. By integrating the pacity in larger increments, and in- value of incremental units of specific and creases the present value of expected flexible capital, one can determine the costs. preferred type of capital, as well as the In Manne's model (which might apply value (if any) of flexibility. to an electric utility that must always sat- In all of the studies cited so far, the isfy demand) the firm does not choose stochastic state variable (the value of the when to invest, only how much. Most project, the price of the firm's output, firms must choose both. Pindyck (1988) or a demand- or cost-shift variable) is determined the effects of uncertainty on specified exogenously. In a competitive these decisions when there are no scale equilibrium, firms' investment and out- economies in construction by extending put decisions are dependent on the price his model to a firm that must decide process, but also collectively generate when to build a single plant and how that process. Hence we would like to large it should be. 21 As with Manne's know whether firms' decisions are consis- model, uncertainty increases the optimal tent with the price processes we specify. plant size. However, it also raises the At least two studies have addressed critical demand threshold at which the this issue. Steven Lippman and Richard plant is built. Thus demand uncertainty P. Rumelt (1985) model a competitive should lead firms to delay capacity addi- industry where firms face sunk costs of tions, but makes those additions larger entry and exit, and the market demand when they occur. curve fluctuates stochastically. They find Sometimes capacity choice is accompa- an equilibrium consisting of optimal in- nied by a technology choice. Consider a vestment and production rules for firms firm that produces two products, A and (with uncertainty, they hold less capac- B, with interdependent demands that ity), and a corresponding process for market price. John Leahy (1989) extends 21 The firm has an option, worth G(K, 0), to build Dixit's (1989a) model of entry and exit a plant of arbitrary size K. Once built, the plant has to an industry setting in which price is a value V(K, 0) (the value of the firm's operating endogenous. He shows that price will be options), which can be found using the methods of Section IV. G(K, 0) will satisfy equation (18), but driven by demand shocks until an entry with boundary conditions G(K*, 0*) = V(K*, 0*) - or exit barrier is reached, and then en- kK* and GO(K*,0*) = VO(K*, 0*), where 0* is the try or exit prevent it from moving fur- critical 0 at which the plant should be built, and K* is its optimal size. See the Appendix to Pindyck ther. Hence price follows a regulated (1988). Brownian motion. Surprisingly, it makes

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions Pindyck: Irreversibility, Uncertainty, and Investment 1141 no difference whether firms take entry be of only secondary importance as a de- and exit into account, or simply assume terminant of aggregate investment that price will follow a geometric spending; interest rate volatility may be Brownian motion; the same entry and more important. exit barriers result. This suggests that In fact, investment spending on an ag- models in which price is exogenous may gregate level may be highly sensitive to provide a reasonable description of in- risk in various forms: uncertainties over dustry investment and capacity. future product prices and input costs that directly determine cash flows, uncer- VI. Investment Behavior and Economic tainty over exchange rates, and uncer- Policy tainty over future tax and regulatory policy. This means that if a goal of macro- Nondiversifiable risk plays a role in economic policy is to stimulate invest- even the simplest models of investment, ment, stability and credibility may be by affecting the cost of capital. But the more important than the particular levels findings summarized in this paper sug- of tax rates or interest rates. Put another gest that risk may be a more crucial det- way, if uncertainty over the economic en- erminant of investment. This is likely to vironment is high, tax and related incen- have implications for the explanation and tives may have to be very large to have prediction of investment behavior at the any significant impact on investment. industry- or economy-wide level, and for Similarly, a major cost of political and the design of policy. economic instability may be its depress- The role of interest rates and interest ing effect on investment. This is likely rate stability in determining investment to be particularlyimportant for the devel- is a good example of this. Jonathan Inger- oping economies. For many LDCs, in- soll and Ross (1988) have examined irre- vestment as a fraction of GDP has fallen versible investment decisions when the dramatically during the 1980s, despite interest rate evolves stochastically, but moderate economic growth. Yet the suc- future cash flows are known with cer- cess of macroeconomic policy in these tainty. As with uncertainty over future countries requires increases in private in- cash flows, this creates an opportunity vestment. This has created a catch-22 cost of investing, so that the traditional that makes the social value of investment NPV rule will accept too many projects. higher than its private value. The reason Instead, an investment should be made is that if firms do not have confidence only when the interest rate is below a that macro policies will succeed and critical rate, r*, which is lower than the growth trajectories will be maintained, internal rate of return, r?, which makes they are afraid to invest, but if they do the NPV zero. Furthermore, the differ- not invest, macro policies are indeed ence between r* and ro grows as the vola- doomed to fail. It is therefore important tility of interest rates grows. to understand how investment might de- Ingersoll and Ross also show that for pend on risk factors that are at least partly long-lived projects, a decrease in ex- under government control, for example, pected interest rates for all future periods price, wage, and exchange rate stability, need not accelerate investment. The rea- the threat of price controls or expropria- son is that such a change also lowers the tion, and changes in trade regimes.22 cost of waiting, and thus can have an am- 22 Caballero and Vittorio Corbo (1988), for exam- biguous effect on investment. This sug- ple, have shown how uncertainty over future real gests that the level of interest rates may exchange rates can depress exports.

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The irreversibility of investment also since. Other things equal, we would ex- helps to explain why trade reforms can pect this to increase the value of land turn out to be counterproductive, with and other resources needed to produce a liberalization leading to a decrease in the commodity, but have a depressing aggregate investment. As Rudiger Dorn- effect on construction expenditures and busch (1987) and Sweder Van Wijnber- production capacity. Most studies of the gen (1985) have noted, uncertainty over gains from price stabilization focus on ad- future tariff structures, and hence over justment costs and the curvature of de- future factor returns, creates an opportu- mand and (static) supply curves. (See Da- nity cost to committing capital to new vid Newbery and Joseph Stiglitz 1981 for physical plants. Foreign exchange and an overview.) The irreversibility of in- liquid assets held abroad involve no such vestment creates an additional gain commitment, and so may be preferrable which must be accounted for. even though the expected rate of return The existing literature on these effects is lower.23 Likewise, it may be difficult of uncertainty and instability is a largely to stem or reverse capital flight if there theoretical one. This may reflect the fact is a perception that it may become more that models of irreversible investment difficult to take capital out of the country under uncertainty are relatively compli- than to bring it in. cated, and so are difficult to translate into Irreversibility is also likely to have pol- well-specified empirical models. In any icy implications for specific industries. case, the gap here between theory and The energy industry is an example. empiricism is disturbing. While it is clear There, the issue of stability and credibil- from the theory that increases in the vola- ity arises with the possibility of price con- tility of, say, interest rates or exchange trols, "windfall" profit taxes, or related rates should depress investment, it is not policies that might be imposed should at all clear how large these effects should prices rise substantially. Investment de- be. Nor is it clear how important these cisions must be made taking into account factors have been as explanators of invest- that price is evolving stochastically, but ment across countries and over time. also the probability that price may be Most econometric models of aggregate capped at some level, or otherwise regu- economic activity ignore the role of risk, lated. or deal with it only implicitly. A more A more fundamental problem is the explicit treatment of risk may help to bet- volatility of market prices themselves. ter explain economic fluctuations, and es- For many raw commodities (oil is an ex- pecially investment spending.24 But sub- ample), price volatility rose substantially in the early 1970s, and has been high 24 sharp jumps in energy prices in 1974 and 1979-80 clearly contributed to the 1975 and 1980- 82 recessions. They reduced the real incomes of oil- 23 But Van Wijnbergen is incorrect in claiming importing countries, and caused adjustment prob- (1985, p. 369) that "there is only a gain to be obtained lems-inflation and further drops in income due to by deferring commitment if uncertainty decreases rigidities that prevented wages and nonenergy prices over time so that information can be acquired about from quickly equilibrating. But energy shocks also future factor returns as time goes by." He bases his raised uncertainty over future economic conditions; analysis on the models of Bernanke (1983) and it was unclear whether energy prices would fall or Cukierman (1980), in which there is indeed a reduc- keep rising, what impact higher energy prices would tion in uncertainty over time. But as we have seen have on the marginal products of various types of from the models in Sections III and IV of this paper, capital, how long-lived the inflationary impact of the this is not necessary. In those models, the future shocks would be, and so on. Much more volatile value of the project or price of output is always uncer- exchange rates and interest rates also made the eco- tain, but there is nonetheless an opportunity cost nomic environment more uncertain, especially in to committing resources. 1979-82. This may have contributed to the decline

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions Pindyck: Irreversibility, Uncertainty, and Investment 1143 stantial empirical work is needed to de- ing, training, and sometimes firing work- termine whether the theoretical models ers. (Samuel Bentolila and Bertola 1990 discussed in this paper have predictive have recently developed a formal model power. 25 that explains how hiring and firing costs One step in this direction is the recent affect employment decisions.) paper by Bertola and Caballero (1990). Another important set of applications They solve the optimal irreversible capi- arises in the context of natural resources tal accumulation problem for an individ- and the environment. If future values of ual firm with a Cobb-Douglas production wilderness areas and parking lots are un- function, and then characterize the be- certain, it may be better to wait before havior of aggregate investment when irreversibly paving over a wilderness there are both firm-specific and aggre- area. Here, the option value of waiting gate sources of uncertainty. Their model creates an opportunity cost, and this does well in replicating the behavior of must be added to the current direct cost postwar U.S. investment. of destroying the wilderness area when Simulation models may provide an- doing a cost-benefit analysis of the park- other vehicle for testing the implications ing lot. This point was first made by Ar- of irreversibility and uncertainty. The row and Anthony Fisher (1974) and structure of such a model might be simi- Claude Henry (1974), and has since been lar to the model presented in Section IV, elaborated upon in the environmental and parameterized so that it "fits" a par- economics literature.26 It has become es- ticular industry. One could then calcu- pecially germane in recent years because late predicted effects of observed changes of concern over possible irrevers- in, say, price volatility, and compare ible long-term environmental changes them to the predicted effects of changes such as ozone depletion and global warm- in interest rates or tax rates. Models of ing. this sort could likewise be used to predict While this insight is important, actu- the effect of a perceived possible shift ally measuring these opportunity costs in the tax regime, the imposition of price can be difficult. In the case of a well- controls, and so on. Such models may defined project (a widget factory), one also be a good way to study uncertainty can construct a model like the one in of the "peso problem" sort. Section IV. But it is not always clear what the correct stochastic process is for, say, VII. Conclusions. the output price. Even if one accepts equation (14), the opportunity cost of in- I have focused largely on investment vesting now (and the investment rule) in capital goods, but the principles illus- will depend on parameters, such as a trated here apply to a broad variety of and r, that may not be easy to measure. problems involving irreversibility. For The problem is much greater when ap- example, as Dornbusch (1987) points out, plying these methods to investment deci- the same issues arise in labor markets, where firms face high (sunk) costs of hir- 26Recent examples are Fisher and W. Michael Hanemann (1987) and Hanemann (1989). This con- in investment spending that occurred, a point made cept of option value should be distinguished from by Bernanke (1983) with respect to changes in oil that of Schmalensee (1972), which is more like a risk prices. Also, see Paul Evans (1984) and John Tatom premium that is needed to compensate risk-averse (1984)for discussionsof the effects of increased inter- consumers because of uncertainty over future valua- est rate volatility. tions of an environmental amenity. For a recent dis- 25 See Pindyck (1990a)for a more detailed discus- cussion of this latter concept, see Mark Plummer sion of this issue. and Hartman (1986).

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sions involving resources and the envi- mean 0, and variance nAt = T. This last point which ronment. Then one must model, for ex- follows from the fact that Az depends on Ai and not on At, is particulary important; the variance of ample, the stochastic evolution of soci- the change in a Wiener process grows linearly with ety's valuation of wilderness areas. the time interval. On the other hand, models like the Letting the At's become infinitesimally small, we write the increment of the Wiener process as dz = ones discussed in this paper can be E(t)(dt)"2. Because E(t) has zero mean and unit stan- solved (by numerical methods) with al- dard deviation, E(dz) = 0, and E[(dz)2] = dt. Finally, ternative stochastic processes for the rel- consider two Wiener processes, zl(t) and z2(t). Then we can write E(dz,dz2) = p12dt, where P12 is the evant state variables, and it is easy to coefficient of correlation between the two processes. determine the sensitivity of the solution We often work with the following generalization to parameter values, as we did in Sec- of the Wiener process: tions III and IV. These models at least dx = a(x, t)dt + b(x, t)dz. (A. 1) provide some insight into the importance The continuous-time stochastic process x(t) repre- of irreversibility, and the ranges of op- sented by equation (A. 1) is called an Ito process. Consider the mean and variance of the increments portunity costs that might be implied. of this process. Because E(dz) = 0, E(dx) = a(x, t)dt. Obtaining such insight is clearly better The variance of dx is equal to E{[dx - E(dx)]2} = than ignoring irreversibility. b2(x, t)dt. Hence we refer to a(x, t) as the expected drift rate of the Ito process, and we refer to b2(x, t) APPENDIX as the variance rate. This appendix provides a brief introduction to the An important special case of (A. 1) is the geometric tools of stochastic calculus and dynamic programming Brownian motion with drift. Here a(x, t) = ox, and that are used in Sections III and IV. For more de- b(x, t) = ax, where ot and a are constants. In this tailed introductory discussions, see Stuart Dreyfus case (A.1) becomes (1965), Merton (1971), Chow (1979), Malliaris and dx = otxdt + uxdz. (A.2) Brock (1982), or Hull (1989). For more rigorous treatments, see Harold Kushner (1967) or Wendell Flem- (This is identical to equation (1) in Section III, but ing and Raymond Rishel (1975). I first discuss the with V replaced by x.) From our discussion of the Wiener process, then Ito's Lemma, and finally sto- Wiener process, we know that over any finite interval chastic dynamic programming. of time, percentage changes in x, Ax/x, are normally distributed. Hence absolute changes in x, Ax, are Wiener Processes log-normally distributed. We will derive the ex- A Wiener process (also called a Brownian motion) pected value of Ax shortly. is a continuous-time Markov stochastic process whose An important property of the Ito process (A. 1) is increments are independent, no matter how small that while it is continuous in time, it is not differentia- the time interval. Specifically, if z(t) is a Wiener pro- ble. To see this, note that dxldt includes a term with cess, then any change in z, Az, corresponding to a dzldt = E(t)(dt)-2, which becomes infinitely large time interval At, satisfies the following conditions: as dt becomes infinitesimally small. However, we will often want to work with functions of x (or z), (i) The relationship between Az and At is given and we will need to find the differentials of such by functions. To do this, we make use of Ito's Lemma. AZ = Et t Ito's Lemma where Et is a normally distributed random variable with a mean of zero and a standard devia- Ito's Lemma is mostly easily understood as a Taylor tion of 1. series expansion. Suppose x follows the Ito process (A. 1), and consider a function F(x, t) that is at least (ii) Et iS serially uncorrelated, that is, = 0 E(EtE,) twice differentiable. We want to find the total differ- for t $4 s. Thus the values of Az for any two ential of this dF. The different intervals of time are independent, so function, usual rules of calculus define this differential in z(t) follows a Markov process. terms of first-order changes in x and t: dF = F,dx + Ftdt, where subscripts denote Let us examine what these two conditions imply partial derivatives, that is, F, = aF/ax, and so on. for the change in z over some finite interval of time But suppose that we also include higher-order terms T. We can break this interval into n units of length for changes in x: At each, with n = T/At. Then the change in z over this interval is dF = + Ftdt + given by F,dx (1I2)F,,(dx)2 (A.3) n + (116)Fxxl(dX)3 + ., . z(s + T) -_z(s) = Ei(At)1/2 ,=1 In ordinary calculus, these higher-order terms all Because the Ei's are independent of eacn other, the vanish in the limit. To see whether that is the case change z(s + T) - 'z(s) is normally distributed with here, expand the third and fourth terms on the right-

This content downloaded from 18.7.29.240 on Wed, 5 Nov 2014 09:36:58 AM All use subject to JSTOR Terms and Conditions Pindyck: Irreversibility, Uncertainty, and Investment 1145 hand side of (A.3). First, substitute (A.1) for dx to dx = cLxdt + ouxdz, determine (dx)2: dy = oL,ydt + uyydzy (dx)2= a2(:X,t)(dt)2 + 2a(x, t)b(x, t)(dt)312+ b2(x,t)dt. with E(dz,dzy) = p. We will find the process followed Terms in (dt)312and (dt)2 vanish as dt becomes infini- by F(x, y), and the process followed by G = log F. tesimal, so we can ignore these terms and write Because F = Fyy = 0 and F"y = 1, we have (dx)2= b2(x, t)dt. As for the fourth term on the right- from (A.7): hand side of (A.3), every term in the expansion of dF = xdy + ydx + (dx)(dy). (A. 10) (dx)3 will include dt raised to a power greater than 1, and so will vanish in the limit. This is likewise Now substitute for dx and dy and rearrange: the case for any higher-orderterms in (A.3). Hence dF = (ox + oty + puruy)Fdt (A. 1) Ito's Lemma gives the differentialdF as + (u,dz, + uydzy)F. dF = F,dx + Ftdt + (1I2)Fxx(dx)2, (A.4) Hence F also follows a geometric Brownian motion. What about G = log F? Going through the same or, substitutingfrom (A.1) for dx, steps as in the previous example, we find that dF = + a(x, + ?/2b2(x, [Ft t)F, t)Fx]dt (A 5) dG = (ox + ofy) - 1/2u 2 - ?/2U2)dt +b (x, t)F,dz. +u1z od .(A. 12) We can easily extend this to functions of several From (A. 12) we see that over any time interval T, Ito processes. Suppose F = F(xl, . . ., xm, t) is a the change in lo~ F is normally distributed with mean function of time and the m Ito processes, x,. (Ot + Ot - 1/2 - 1/2 2)T and variance (uw + + Xm,where 2puxuy)f'1. Stochastic dxi= ai(x., xm,t)dt (A.6) Dynamic Programming + bi(x1. Xmt)dzi, i = 1, m Ito's Lemma also allows us to apply dynamic programming to optimization problems in which one and E(dzidzj)= pg.dt.Then, letting Fi denote aF/ax, or more of the state variables follow Ito processes. and FY-denote a2lIax,jaxpIto's Lemma gives the dif- Consider the following problem of choosing u(t) over ferential dF as time to maximize the value of an asset that yields a flow of income fl = fl [x(t), u(t)]: dF = Ftdt + E Fidxi + 1/2E E Fiidx,dx, (A.7) 1 i J max Eo H[x(t),u(t)]e-tdt, (A. 13) or, substitutingfor dxi: dF = [Ft + ai(xl,. t)F, where x(t) follows the Ito process given by dx = a(x, u)dt + b(x, u)dz. (A.14) + 1/2 b2(x1, t)F, (A. 8) Let J be the value of the asset assuming u(t) is + ppjb(xi, t)bj(xl,. t)FI,]dt chosen optimally, i?J + E b(xl,. t)Fidzi. J(x) = max Etf H [x(T),u(T)]e1`dT. (A.15)

Example:Geometric Brownian Motion. Let us re- Because time appears in the maximand only through turn to the process given by equation (A.2). We will the discount factor, the Bellman equation (the funda- use Ito'si Lemma to find the process followed by mental equation of optimality) for this problem can F(x) = log x. Because Ft = 0, Fx = lIx, and F = be written as -l/x2, we have from (A. 4): ,J= max (A.16) dF = (llx)dx - (112x2)(dx)2 " [H(x, ta) + (Ildt)EtdJ]. = oLdt+ adz - ?/2a2dt (A.9) Equation (A. 16) says that the total return on this has two the income flow = (Ot - ?/2u2)dt + adz. asset, ,uJ, components, fl(x, u), and the expected rate of capital gain, (1/ Hence, over any finite time interval T, the change dt)EtdJ. (Note that in writing the expected capital in log x is normallydistributed with mean (ot- ?/2a2)T gain, we apply the expectation operator Et, which and varianceau2T. eliminates terms in dz, before taking the time deriva- The geometric Brownian motion is often used to tive.) The optimal u(t) balances current income model the prices of stocks and other assets. It says against expected capital gains to maximize the sum returnsare normallydistributed, with a standardde- of the two components. viation that grows with the squareroot of the holding To solve this problem, we need to take the differ- period. ential dJ. Because J is a function of the Ito process Example:Correlated Brownian Motions. As a sec- x(t), we apply Ito's Lemma. Using equation (A.4), ond example of the use of Ito's Lemma, consider a function F(x, y) = xy, where x and y each follow dJ = J,dx + ?2J,x(dx)2. (A. 17) geometric Brownianmotions: Now substitute (A. 14) for dx into (A. 17):

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dJ = [a(x, u)J, + 1/2b2(x, u)J,J]dt + b(x, u)J,dz. (A. 18) REFERENCES Using this expression for dJ, and noting that ABEL, ANDREW B. "Optimal Investment Under Un- E(dz) = 0, we can rewrite the Bellman equation certainty," Amer. Econ. Rev., Mar. 1983, 73(1), (A. 16) as pp. 228-33. . "Consumption and Investment," in Hand- ,J = max [H(x, u) + a(x, u)Jh book of monetary economics. Eds.: BENJAMIN + ?/b2(x, u)JXJ. (A. 19) FRIEDMAN AND FRANK HAHN. NY: North-Holland, 1990, pp. 726-78. In principle, a solution can be obtained by going ARROW, KENNETH J. "Optimal Capital Policy with through the following steps. First, maximize the ex- Irreversible Investment," in Value, capital and pression in curly brackets with respect to u to obtain growth, papers in honour of SirJohn Hicks. Ed.: = an optimal u* u*(x, J, J,,). Second, substitute JAMES N. WOLFE. Edinburgh: Edinburgh U. this u* back into (A. 19) to eliminate u. 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