## Of a Project and the Decision to Invest

The basic model of irreversible investment in Chapter 5 demonstrated a close analogy between a firm's option to invest and a financial call option. In the case of a call option, the price of the stock underlying the option is assumed to follow an exogenously specified stochastic process, usually a geometric Brownian motion. In our model of real investment, the corresponding state variable was the value of the project, K, for which we stipulated an exogenous stochastic process.

However, as we explained at the beginning of Chapter 5, letting V follow an exogenous stochastic process, and particularly a geometric Brownian motion, is an abstraction from reality. First, if the project is a factory and there are variable costs of operation, V will not follow a geometric Brownian motion. Second and more important, the value of a project depends on future prices of outputs and inputs, interest rates, etc. These in turn can be explained in terms of the underlying demand and technology conditions in various markets. Hence fluctuations in V can be traced back to uncertainty in these more basic variables. How deep one goes depends on the purpose of the analysis. To understand a firm's behavior, we might be satisfied to work with an exogenous process for the output and input prices. At the industry level, we must make the output price endogenous. At an even more general equilibrium level, the input prices must also be determined simultaneously by considering all industries' factor demands. In this chapter we take some first steps along this route.

For most of the time in this chapter, we consider a firm that has the privileged opportunity or monopoly right to invest in a single discrete project that will produce a given output flow. The basic uncertainty is over the demand for this output, but given a fixed scale, there is an immediate correspondence between demand and price. Therefore we allow the output price P to be exogenous, and determine the value V of the project, and the value F of the option to invest, in terms of the stipulated stochastic process for P. The methods are the same as those employed in Chapter 5, namely, contingent claims analysis or dynamic programming. We will see that once again, the value of the option to invest includes a holding premium, which implies a stiffer test for investment than the traditional Marshallian criterion.

In Section 1, we begin with the simplest case where production has no operating costs. Then the value of a completed project is just the discounted present value of the revenue flow, and the formulas of Chapter 5 that were expressed in terms of V translate immediately into corresponding formulas in terms of P.

In Section 2 we introduce an operating cost C. Hence the project will generate a flow of operating profit equal to (P — C) per period. This raises a new issue—the output price can go below C from time to time, which would make the operating profit negative. We must specify what happens then. We will consider two, somewhat extreme, possibilities. The first, which is the subject of this chapter, is that the project can be costlessly shutdown if P falls below C, and later costlessly restarted if P rises above C. In effect, this makes the project an infinite sequence of instantaneous operating options, each of which is exercised if P > C, and can be valued accordingly. The other extreme possibility, which we will consider in Chapter 7, prohibits such temporary suspension by supposing that the full investment cost I must be incurred over again if operations are ever resumed. Then some losses will be sustained to keep the option of future operations alive, but if the losses grow sufficiently large, the project will be abandoned. Of course reality lies somewhere between these two extremes. Ongoing projects generally build up specific assets—workers' skills, customers' loyalty, etc.—that will gradually disappear, or "rust," if operation is suspended. Thus resumption involves a cost, but less than the cost of starting anew, and the difference depends on the nature of the product and the duration of the suspension. Our analysis of the extreme cases yields results that can be suitably combined to fit particular applications that lie in between.

In Section 3 we allow some instantaneous variation of inputs like labor and raw materials, to vary the output flow from the project in response to transient price fluctuations. Now the profit flow becomes a nonlinear function of the price, which alters the effect of uncertainty on investment.

All of the analysis up to this point assumes that the project, once installed, goes on producing the output flow forever. This unrealistic assumption is made only to convey the basic ideas of option values in a simpler manner. In Section 4 we relax this assumption by introducing depreciation. We show that the effect on option values depends not on the mortality of one project, but on how we specify the opportunities available to the firm after its initial project has reached the end of its life. We also show that option values remain of considerable significance even with fairly rapid depreciation.

In the concluding section of this chapter we consider a situation where two variables that affect the firm's investment decision—the output price and the investment cost—are both random. Here the value of the option to invest is a function of both of these independent variables, and therefore it satisfies a partial differential equation. In general such equations are difficult and must be solved numerically. A special feature of homogeneity helps us reduce the problem to an ordinary differential equation and solve it analytically. Now investment is optimal only when the ratio of output price to investment cost exceeds a threshold influenced by the option value of waiting.

Throughout this chapter, the insights about option values which we gained from the analysis of Chapter 5 will remain valid and valuable as we introduce new features into the model. The techniques developed there will also continue to be useful. In future chapters we will continue the program of generalizing the models and posing new issues. In Chapter 7 we will consider the possibility of temporarily mothballing or permanently abandoning a project if its cash flow turns negative. Then, in Chapters 8 and 9 we will move to the level of industry equilibrium, where each firm has the opportunity to invest in a single project. In Chapter 10 we will return to the perspective of a single firm, but generalize the nature of the project, letting it consist of a number of investment steps, all of which must be completed before the profit flows begin. Finally, in Chapter 11 we consider incremental investment, where each unit of addition to capacity begins to yield its marginal revenue product as soon as it is installed.

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