Choosing among alternative discrete investment projects under uncertainty

Economics Letters 0165.1765/93/$06.00

41 (1993) 265-268 0 1993 Elsevier

265 Science

Publishers

B.V. All rights

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Choosing among alternative discrete investment projects under uncertainty Avinash Department Received Accepted

Dixit* of Economics,

Princeton

University,

Princeton,

NJ 08544, USA

22 September 1992 21 October 1992

Abstract Irreversible choice among are shown to make it optimal wait for a larger project.

mutually exclusive projects under output price uncertainty is considered. Increasing returns to wait for the largest project. Greater uncertainty of the price process makes it optimal to

1. Introduction

Many investment projects like ships, dams, and factories offer an ex ante choice of scale, but prohibitively costly ex post adjustment in scale. When future output price is uncertain, the investor has to consider an extra option value of waiting, because a different scale might turn out to be more profitable. This paper constructs a simple model to quantify this intuition. Thus it adds to the literature on irreversible investment under uncertainty, which has considered the option value of waiting for a single discrete project [McDonald and Siegel (1986)], and the case where capacity can be incrementally augmented [Pindyck (1988)]. The choice among projects depends on how the output changes with scale. If the elasticity of output with respect to capital expenditure is greater than one, it is found not optimal to invest in any but the largest available project. If the elasticity is less than one but increasing, then the optimal choice is an extreme: either the smallest or the largest available project. Greater uncertainty of the price process makes it optimal to wait for a larger project.

2. The model

The firm has available a menu of projects indexed by i ranging from 1 to N, with sunk capital costs K, and output flows Xi ranked in increasing order. ’ A project once installed lasts for ever and incurs no operating cost. Output price P follows the geometric Brownian motion * I thank John Leahy for comments on a preliminary draft, and the National Foundation for financial support. ’ A project that has higher K but lower X is dominated and can be discarded

Science

Foundation

from the menu.

and the Guggenheim

266

A. Dixit

I Economics

Letters

41 (1993) 265-268

Fig. 1.

dP/P=p

(1)

dttudw.

The firm is risk-neutral, and p is the discount rate. ’ Suppose the firm invests in project i when the price is P. The profit flow starts and its expected value grows geometrically at rate CL,so the expected discounted PXi/( p - p). Therefore the net value of this action is K(P) = PXJ(p

- p) - Ki.

A firm contemplating investment time. So the value of investing V(P) = max[V(P)

(2)

at price P will of course choose is

1 i = 1,2,

at the level PX,, present value is

. . . , N].

the project

that looks best at this

(3)

Fig. 1 illustrates this for the case of three projects. Each y(P) 1s . a straight line, and V(P) is their upper envelope. A waiting firm is holding an option to invest. Its value V,(P) is found by noting that this asset pays no dividend, but has an expected capital gain E[dV,(P)]ldt as P fluctuates. This must constitute the normal return pVo(P). Using It‘s Lemma and solving the resulting differential equation, we get V,(P) = BPP, where

p is the positive

’ Extensions bounded difficulties.

(4) root of the quadratic

to allow variable costs, suspension above because of general equilibrium

equation

and abandonment, and a price process that reverts to the mean or is effects, are all conceptually straightforward and present only algebraic

A. Dixif

I Economics

Letters

41 (1993) 265-268

267

(5) In fact, we need p > p to ensure convergence of the expected profit flow, therefore we have fi > 1. The constant B is determined by considering the optimal choice of P at which to invest. For this we must choose the highest option value function V,(P) (equivalently the highest B) that has a point in common with V(P); the value of P at this common point is then the investment threshold. For details in the context of a single project, see Dixit (1992). A fuller survey of the literature is in Pindyck (1991). The solution is simplified by the fact that V(P) is an upper envelope of straight lines and V,(P) is an increasing convex function. Therefore the contact between V,(P) and V(P) at the optimum cannot occur at any of the kinks of V(P). So we need only calculate the tangencies of V,(P) with each of the straight lines K(P), and select the one with the highest Bi. A simple calculation shows that

4 =

((p-*,),)~(y)“-’

Now choosing the largest Bi is equivalent tangency occurs at the price

P

P*_

p-1

(P-P)K, x;

to choosing

(6) that i for which X~lK’~’

.

is largest.

The

(7)

In fig. 1 there are three projects, and the tangency occurs for i = 2. If the initial price is P < P*, the firm should wait, and invest in Project 2 when the price rises to this level. Note that if Project 1 were the only one available, it would have been chosen at a lower threshold price given by the tangency of V,,(P) with a lower B (shown by dashes) and V,(P). The larger Project 2 is sufficiently better in its (K, X) tradeoff that the firm finds it preferable to bypass Project 1 and wait for the possibility that P rises sufficiently high to make it worthwhile to invest in Project 2. But with the magnitudes embodied in the figure, it is not similarly preferable to wait for Project 3. Of course if the initial price is above P*, the firm should invest immediately and choose that project which is best at the current price. For sufficiently high P this could be Project 3.

3. A continuum

of projects

The simple characterization of the optimum as maximizing Xf3/Kf3-’ can be generalized to a continuum of projects. That in turn offers more precise results concerning the tradeoff between capital cost and output flow. Suppose the continuum is indexed by the capital cost K, and the corresponding output flows are given by a ‘production function’ X = F(K). Then the optimum project maximizes X6/K@-’ subject this function. This is simpler in terms of the logarithms of X and K, say x and k respectively. The objective function is px - (p - l)k, or equivalently, x - [/3l(p - l)]k. Write the production function as x =f(k). This simple formulation yields several implications. (1) For an interior

optimum

f’(k) = (P - 1)/P.

we have the first order-condition

(8)

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A. Dixit

I Economics

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41 (1993) 265-268

The right-hand side is less than 1. In terms of the original production function, f’(k) = KF’(K)IF(K). This elasticity is less than one when F(K)IK is a decreasing function of K. Thus a necessary condition for an interior solution is that larger projects have less than proportionate outputs, or there are decreasing returns to scale. If larger projects have larger average product, as is often the case with dams and ships because of ‘engineering’ economies of scale, then the optimum is a corner solution at the right-hand extreme. This corresponds to an optimal policy of waiting for the largest in the spectrum of projects. (2) The second-order condition for an interior optimum is f”(k) < 0, for which the elasticity of output with respect to capital should be decreasing. If that is not the case, the optimum is found returns are a by a binary comparison of the smallest and the largest projects. 3 Thus diminishing necessary but not sufficient condition for a project in the interior of the range to be chosen. (3) Now suppose an interior optimum and examine some comparative statics. If p increases or lo falls, p increases and so does (p - 1)/p. Using the first- and second-order conditions, this means a smaller k. Thus a larger discount rate, or a smaller expected rate of growth of price, induces the firm to settle for a smaller project. This is quite intuitive. More interestingly, a larger v lowers p and therefore (p - 1)/p, which means a larger optimal k. Greater uncertainty makes it optimal to wait until a larger threshold price is reached, and then invest in a larger project.

4. Concluding This simple

comments example

can be generalized

in several

ways; I briefly

mention

two.

(1) If projects have finite lives, then the investment opportunity will reopen when an existing project expires. This will lead to a ‘renewal’ model. (2) I have assumed the whole menu of projects to be known in advance. But in reality the firm may have an R&D program with uncertain outcomes, or even a subjective probability that the future will reveal other projects. This uncertainty will contribute to the option value, augmenting the effect of the price uncertainty. Finally, note that very general models of choice among risky alternatives have been considered under the heading of ‘bandit problems’; see Berry and Fristedt (1985). The special feature that permits a very simple solution in the above model is that the risks of all projects are perfectly special case in the context of correlated through the output price. This is an interesting investment, but introducing additional project-specific risks will complicate the analysis considerably.

References Berry, D.A. and B. Fristedt, 1985, Bandit problems (Chapman and Hall, London). Dixit, A., 1992, Investment and hysteresis, Journal of Economic Perspectives 6, 107-132. McDonald, R. and D.R. Siegel, 1986, The value of waiting to invest, Quarterly Journal of Economics 101, 707-728. Pindyck, R.S., 1988, Irreversible investment, capacity choice, and the value of the firm, American Economic Review 969-985. Pindyck, R.S., 1991, Irreversibility, uncertainty, and investment, Journal of Economic Literature, 29, 1110-1152. ’ If we take the production function requiring an elasticity of substitution

terminology at face value, less than 1.

then

the second-order

condition

can be interpreted

79,

as