A Note on Competitive Investment under Uncertainty by

This paper clarifies how uncertainty affects irreversible investment in a competitive market equilibrium. With free entry, irreversibility affects the distribution of future prices, and thereby creates an opportunity cost of investing now rather than waiting. As with an imperfectly competitive firm, uncertainty can also increase the value of a marginal unit of capital. I show that with an infinite horizon, the opportunity cost is larger than this increase in value, so that uncertainty reduces investment. A Note on Competitive Investment under Uncertainty by Robert S. Pindyck* (Revised: August 1991) Uncertainty over future output prices or input costs can affect investment by a risk-neutral firm in two opposing ways. First, it can increase the value of the marginal unit of capital, which leads to more investment. This only requires that the stream of future profits generated by the marginal unit be a convex function of the stochastic variable; by Jensen's inequality, the expected present value of that stream is increased. This result was demonstrated by Hartman (1972), and later extended by Abel (1983) and others. In their models, convexity in output price and input costs is due to capital's substitutability with other factors. But even with fixed proportions, convexity in ensured by the ability of the firm to vary output, so that the marginal unit of capital need not be utilized.1 If investment is irreversible and can be postponed, a second effect of uncertainty is to create an opportunity cost of investing now, rather than waiting for new information to arrive before committing resources. This increases the full cost of the marginal unit of capital, which reduces investment.2 Hence the net effect of uncertainty on irreversible investment depends on the size of this opportunity cost relative to the increase in the value of the marginal unit of capital. Caballero (1991) has recently argued that for a perfectly competitive firm with constant returns to scale, this opportunity cost is zero, so that uncertainty over future demand unambiguously increases current investment, even if that investment is irreversible. This is in contradiction to the negative relationship between uncertainty and irreversible investment found by Pindyck (1988) and Bertola (1989), and is significant because of possible policy implications. Caballero suggests that such a negative relationship requires either decreasing returns to scale or imperfect competition. Caballero's result is based on a model with convex adjustment costs. An interesting and innovative aspect of the model is that these costs can be asymmetric, thereby allowing for partial or complete irreversibility; complete irreversibility corresponds to a cost of downward adjustment that is infinite. The model is particularly useful in that it allows one to study the sensitivity of the investment-uncertainty relationship to the extent of asymmetry in adjustment costs. An important aspect of Caballero's model, shared with Abel's (1983) and other models of this kind, is that the size of the firm would be unbounded were it not for adjustment costs. In fact it is only adjustment costs that determine firm size. As I show below, this role of adjustment costs is crucial to the results of Caballero and earlier authors regarding the effect of uncertainty on investment. While it is helpful for studying the behavior of a firm in isolation, this adjustment cost framework is inconsistent with a competitive market equilibrium, and hence with the behavior of a competitive firm. To study a competitive market equilibrium, one must make price and industry output endogenous, and doing so restores the positive opportunity cost associated with irreversible investment. 1. Adjustment Costs and Irreversibility. A firm that has constant returns to scale everywhere and faces an infinitely elastic demand curve will have a profit function that is linear in the capital stock. Hence convex costs of some kind are needed to bound the size of the firm; otherwise the firm would expand indefinitely if its marginal profit exceeded the cost of a unit of capital. In Caballero's model, convex adjustment costs serve this role by making the cost of investment an increasing function of the level of investment. But because these adjustment costs are a function of only the level of investment, investment in each period is independent of investment or the stock of capital in any other period -there are no "intertemporal links." This, however, necessarily eliminates irreversibility from the problem. Irreversibility matters when it causes decisions made now to constrain decisions in the future under some states of nature but not under others. For example, a firm that invests a large amount this period would not want to disinvest next period if demand expands, and so would not be constrained by irreversibility. This large investment would lead it to be constrained, however, if demand were to contract, because then it would want to disinvest. This is why irreversibility creates an opportunity cost, which leads the firm to invest somewhat less this period. This can never arise when the size of the firm is constrained only by adjustment costs. Then investment next period depends only on the realization of demand that period and on the adjustment cost function; it is completely independent of investment this period. Hence the firm need only compare the marginal cost of investing to current and expected future marginal profits. Since uncertainty increases expected future marginal profits, it necessarily increases investment. Convex adjustment costs may indeed affect the rate at which firms invest (although simple "time to build" and the lumpiness of investment are likely to be more important constraints). It seems unrealistic, however, to treat adjustment costs as the sole or main determinant of firm and industry size in equilibrium. In fact, a pure adjustment cost model is inconsistent with a competitive market equilibrium. In principle, free entry will ensure that a very large number of very small firms come into the industry. (Very