The uncertainty-investment relationship with endogenous capacity

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The uncertainty-investment relationship with endogenous capacity ✩ Sudipto Sarkar The Michael Lee-Chin Professor in Strategic Business Studies, DeGroote School of Business, McMaster University, 302 DSB, Hamilton, ON L8S 4M4, Canada

a r t i c l e

i n f o

Article history: Received 18 December 2018 Accepted 3 September 2019 Available online xxx JEL classification: D2 G3 Keywords: Uncertainty Volatility Investment Real-option model

a b s t r a c t This paper revisits the uncertainty-investment relationship with a real-option model. Unlike existing models, the firm chooses both timing and size of investment, hence we use a new composite measure of investment that takes into account both timing and size. With this modification, we show that investment is not necessarily a monotonically decreasing function of uncertainty (or volatility) as predicted by the existing literature. For most parameter value configurations, investment is initially an increasing function and subsequently a decreasing function of uncertainty. This result also holds with the newer Knightian measure of uncertainty. Comparative static analysis indicates that uncertainty is more likely to have a positive effect on investment when demand growth rate and demand volatility are low, and interest rate and operating cost are high. The effect also depends on the production technology; with decreasing-returns-to-scale technology it is more likely to be positive. Finally, it is shown that investment is more sensitive to uncertainty for decreasing-returns-to-scale technology, high cost of production capacity, low operating cost and less market power. While there is some empirical evidence in support of a non-monotonic uncertainty-investment relationship and the role of market power, the other implications of the model have not been empirically tested yet. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Capital investment is a critical decision for both the corporate sector and the broader economy. One of the most significant determinants of corporate investment is uncertainty or volatility [2,16,29]. Not surprisingly, there is a substantial literature, both theoretical and empirical, on the effect of uncertainty on investment. The theoretical literature on the uncertainty-investment relationship is not unanimous; to quote Leahy and Whited [[22], p. 64]: “. . . taken as a whole, what theory has to say is ambiguous. Different theories emphasize different channels, some pointing to a positive relationship and some to a negative relationship.” The recent “real options” literature ([13,30], etc.), which views investment as the exercise of a call option, is a well-established approach to investment analysis [14,28]. When viewed this way, the effect of uncertainty on investment timing is unambiguous – higher volatility always results in delayed investment, thus uncertainty has a negative effect on investment ([4,18,21,27], etc.). However, Sarkar [39] and Wong [48], using a different (probability-based) measure of investment, demonstrate that the relationship can be nonmonotonic, and Alvarez and Koskela [1] show that interest-rate

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volatility can have a positive or a negative effect on investment, depending on the interest rate level. The empirical literature on the uncertainty-investment relationship is also mixed. While some papers report a negative relationship between uncertainty and investment, others find a positive, ambiguous or non-monotonic relationship. Kellogg [20] finds that an increase in oil price volatility causes firms to reduce drilling activity, Stein and Stone [45] and Pennings and Altomonte [38] find that higher levels of uncertainty result in reduced capital investment, and Serven [43] finds a negative relationship between exchange-rate volatility and investment in most samples. But Bloom et al. [7] find that the response of investment to demand shocks is convex, and the effect of uncertainty on investment is weaker for firms with high levels of uncertainty; Bloom [6] finds that the real-option effects of uncertainty play no role in the long-run rate of investment; and Misund and Mohn [33] find that uncertainty has a significant effect on investment in the oil and gas industry, but the sign of the effect is not settled. Jeanneret [19] finds a U-shaped relationship between investment and exchange-rate volatility, that is, investment is initially decreasing and subsequently increasing in uncertainty, Driver et al. [15] find that uncertainty might have a positive or negative effect on investment, depending on the industry, and Bo and Lensink [8] and Lensink and Murinde [24] find an inverted-U shaped relationship between investment and uncertainty. Thus, both the theoretical

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predictions and empirical evidence on this issue are not settled in the literature. This paper takes a new look at the uncertainty-investment relationship in a real-option setting, but with one significant departure from the existing literature – the investment size or scale is not fixed but chosen by the firm. Endogeneity of investment size could potentially impact the uncertainty-investment relationship because of two reasons: (i) uncertainty itself might have a direct effect on optimal investment size because of the asymmetric effect of volatility under limited liability [40]; and (ii) greater uncertainty will result in delayed investment, which in turn will lead to larger investment size, as shown by Bar-Ilan and Strange [3] and Huberts et al. [17]. Thus, a higher volatility will cause delayed investment (a negative effect) as well as a larger investment (a positive effect); that is, the “timing” and “size” effects act in opposite directions. The offsetting effects of timing and size make it difficult to determine the overall effect of uncertainty on investment. In order to overcome this difficulty, we have to look for a composite measure that accounts for both timing and size effects, and therefore allows us to make a fair comparison of investment as volatility is varied.1 For this purpose, we choose the measure EPVI (the expected present value of investment), described in Section 2.4. The main result of our paper is that, in contrast to established real-option models ([13,18,21,27,30], etc.), uncertainty does not necessarily have a monotonic negative effect on investment. With endogenous capacity choice, uncertainty can have a nonmonotonic effect on investment. In fact, for most parameter value configurations, EPVI is non-monotonic, initially increasing and subsequently decreasing in uncertainty. Thus, investment is generally (but not always)2 an inverted-U shaped function of uncertainty. The exact point (uncertainty level) at which it switches from increasing to decreasing varies widely with parameter values. It is shown that uncertainty is more likely to have a positive effect on investment when demand growth rate and demand volatility are low, discount rate and operating cost are high, and the technology is decreasing-returns-to-scale. Finally, we show that investment is more sensitive to uncertainty for decreasing-returns-to-scale technology, high cost of production capacity, low operating cost and less market power. Moreover, similar results were obtained using a Knightian measure of uncertainty (instead of the commonly-used measure of volatility), which has been studied in a few recent papers, motivated by ambiguity aversion (that is, uncertainty regarding the underlying probability measure itself, which is assumed to be known with certainty in the traditional literature); some examples being Chen and Epstein [11], Miao and Wang [32], Nishimura and Ozaki [36], and Thijssen [46]. Our model contributes to the real-option investment literature by addressing, for the first time, an important issue, that is, the effect of uncertainty on investment when both timing and size of investment are endogenous, using an appropriate measure of investment that incorporates both degrees of freedom (timing and size). Although the need for such a measure has been suggested in the literature [19], it has not been used to study the effect of uncertainty on investment.

1 Jeanneret [19] also points out the need to look at both timing and size of investment (in a somewhat different context, since he discusses FDI in the context of exporting goods versus exporting capital), stating: “. . . in order to gauge the overall effect of uncertainty on investment, it is more relevant to estimate the expected value of investment within a given time period.” (p. 4). 2 That is, investment is inverted-U-shaped in volatility for most reasonable parameter value configurations. For certain parameter values, however, the relationship might be monotonic. In our comparative static analysis over a wide range of parameter value combinations, we find one such case, where it is monotonically downward-sloping (Section 3.3).

The rest of the paper is organized as follows. Section 2 develops the model, identifies the optimal investment policy, and develops a measure of investment that takes into account both timing and size effects. Section 3 presents the results of the model, and Section 4 summarizes the results and concludes. 2. The model We use a simple real-option model of investment, similar to traditional models such as Dixit and Pindyck [13], to study the uncertainty-investment relationship. The distinguishing feature of our model is that investment size is also endogenous, and chosen optimally by the firm. 2.1. Model specifications A firm holds an option to invest in a project. When it makes the investment, the project is implemented instantaneously, and production starts immediately. The size or scale of the investment is given by the production capacity Q, and the cost of investment is ϕ Qα (as in [3,5], and [40]). The investment cost parameter ϕ is the unit cost of capital, where the amount of capital is given by Qα , and α is a constant that indicates concavity/convexity of the investment cost function (or the production technology’s returns-toscale).3 We call α the returns-to-scale parameter, where a higher (lower) α represents more decreasing (increasing) returns to scale. Production cost is w per unit, and the output price is given by the inverse demand function:

pt = yt −γ qt ,

(1)

where pt is output price, qt is the output level, γ is a non-negative constant, and yt is an exogenous shock to demand. This linear demand function with uncertainty is widely used in the literature, e.g., Lederer and Mehta [23], Sarkar [40]. In Eq. (1), the parameter γ denotes the price-sensitivity of the product, and is a measure of the firm’s market power. If γ = 0, demand is infinitely elastic, and the company has no market power (competitive market); if γ is large, demand is highly inelastic, and the company has substantial market power (uncompetitive market).4 The demand shock y can be viewed as the (random) strength of demand for the product. It is the state variable (source of uncertainty) in our model, as in Dangl [12] and Sarkar [40], and evolves as a lognormal process:

dy/y = μdt + σ dZ,

(2)

where μ and σ are the mean and standard deviation of the process, respectively, and Z is a standard Wiener process. This is a commonly used stochastic process in the literature [12,40]. Thus, the uncertainty in our model is caused by demand volatility, which is common in real-option models [47]. As shown in Section 2.2, the firm will invest when demand strength rises sufficiently, say to yi ; then yi is the investment trigger. All free cash flows are paid to shareholders as dividends, and cash flows are discounted at a constant interest rate of r (we assume r > μ, to ensure non-negative project values and to rule out speculative bubbles). The firm always operates at full capacity, i.e., qt = Q, as in Bar-Ilan and Strange [3], Lederer and Mehta [23], and 3 α < 1 (> 1) means the investment cost function is concave (convex), or the marginal investment cost is decreasing (increasing) in installed capacity, or the technology has increasing (decreasing) returns to scale. 4 Note that the firm need not be a monopolist; all that is required for this analysis is that the firm have some market power or there be some product differentiation; in other words, all that is necessary is that the demand function be downward-sloping, e.g., as in monopolistic competition.

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Seta et al. [44].5 However, if the market turns too negative (that is, if y falls sufficiently, say to yb ), the firm does have the option to abandon operations and exit the business; this ensures that the model is consistent with limited liability. To summarize the firm’s sequence of actions, the first time that y rises to yi the firm invests in the project and starts producing and selling the output; subsequently, when y falls to yb the firm abandons the project, and that is the end of the model. 2.2. Valuation of project and option to invest The instantaneous profit stream is given by revenues (pq) less operating costs (wq); thus, π (q) = (y–w)q–γ q2 = (y–w)Q–γ Q2 (since the firm operates at full capacity). As shown in Appendix A, the project value is given by:

V (y ) =

Qy

(r − μ )

−

( w + γ Q )Q r

+ A y θ2 ,

(3)

where A is a constant to be determined by boundary conditions, and θ 2 is given by:

μ θ2 = 0 . 5 − 2 − σ





0 . 5 − μ/ σ 2

2

+

2r

σ2

The first two terms of the project value in Eq. (3) represent the value of the profit stream from the project if it continued forever. The term Ayθ2 represents the value of the option to abandon the project (therefore, A ≥ 0). The constant A and the optimal abandonment trigger yb are derived in Appendix B, and given by:

A=−

Q ( y b ) 1 − θ2 (w + γ Q )(1 − μ/r ) , and yb = . θ2 ( r − μ ) ( 1 − 1 / θ2 )

Consistent with the real-option literature, investing in the project is equivalent to exercising the option to invest. As is common in this literature, the option to invest is assumed to be a perpetual one, hence the value of the option will be a function of only the state variable y, say f(y). The firm will exercise this option when y rises to the investment trigger yi . Then it can be shown that the value of the option to invest is given by:

f ( y ) = F y θ1 ,

(4)

where F is a constant to be determined by boundary conditions, and θ 1 is given by:

θ1 = 0 . 5 −

μ + σ2





0 . 5 − μ/ σ 2

2

+

2r

σ2

2.3. Optimal investment decision The optimal investment decision with endogenous timing and size is given by Proposition 1 below. Proposition 1. The optimal investment trigger yi is the solution to the equation:

Q y i ( 1 − 1 / θ1 ) ( w + γ Q )Q − + A ( 1 − θ 2 / θ 1 ) ( y i ) θ2 = ϕ Q α (r − μ ) r

(5)

and the optimal capacity Q is the solution to the equation: 5 In real life, the firm might be able to produce at reduced rates (leaving some of the capacity idle) when demand is low. However, as Seta et al. [44] explain, in many industries firms make production plans before the actual realization of market demand, and may find it difficult to produce below capacity due to fixed costs. Also, even when firms can keep some capacity idle, a temporary suspension of production is often costly, because of maintenance costs needed to avoid deterioration of the equipment. It is therefore common to assume that the firm produces at full capacity.

3



yi w+γQ − (w + γ Q ) + Q γ + (1 − μ/r ) 1 − θ2

 y θ2 i

yb

= rϕα Q α −1 (6)

(Proof: See Appendix B and C) If the investment capacity is exogenously specified, optimal timing is determined from Eq. (5); and if the timing (trigger) is specified, optimal capacity is determined from Eq. (6). If both size and timing are endogenous, Eqs. (5) and (6) must be solved simultaneously for the optimal trigger yi and size Q. Both Eqs. (5) and (6) have to be solved numerically since they have no analytical solutions. 2.4. The expected present value of investment (EPVI) The fact that delayed (earlier) investment is larger (smaller) makes it difficult to identify the effect on investment with a onedimensional measure. We therefore use a composite measure that takes into account both dimensions – timing and size of investment. An appropriate measure would be the expected present value of the investment (in dollars), as mentioned in Section 1, since this incorporates both timing and size. It is determined by the investment size, the investment trigger, and the parameters of the stochastic process governing y. The firm will make an investment of size $ϕ Qα when y rises to yi . The expected present value of ϕ Qα invested at the first passage time of y to yi is given by ϕ Q α (y/yi )θ1 (Leland [49], footnote 16). This is a direct measure of the effective magnitude of investment, and can be compared across various volatility scenarios; thus it should be an appropriate measure for our purpose.6 However, this measure varies continuously (since y is a continuous state variable); therefore, for a more meaningful comparison over θ time, we get rid of y and just use the measure ϕ Q α /(yi ) 1 instead. In the rest of the paper, we use the EPVI as the appropriate measure of investment:

EPV I =

ϕQ α (yi )θ1

(7)

2.5. Knightian uncertainty In recent years, the literature has started to recognize two main sources of uncertainty – risk and ambiguity [42]. Risk is measured by the volatility of the underlying state variable (demand, price, profit, etc.) with a known underlying probability measure, and has been well researched, as mentioned in Section 1. Ambiguity (also known as Knightian Uncertainty) refers to a multitude of probability distributions of the state variable. The study of Knightian uncertainty has started more recently. In practice, it is likely that there will be some uncertainty regarding the probability measure, hence the study of Knightian uncertainty seems worthwhile. A few recent papers have examined the effect of this type of uncertainty on investment, e.g., Miao and Wang [32], Nishimura and Ozaki [36], Niu et al. [35] and Thijssen [46]. In this section, we consider Knightian uncertainty and how it might affect the overall investment, in the same spirit as the rest of the paper, that is, we vary the level of Knightian uncertainty and gauge the effect on EPVI. In our approach, Knightian uncertainty is modeled following the method of Chen and Epstein [11] and Nishimura and Ozaki [36].

6 While this seems like an appropriate overall measure of investment, it is not yet in common use in the literature, and has been used only twice that we know of, by Nishihara et al. [34] and Lukas and Thiergart [25], although in a different context.

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Thus, we assume that the firm is ambiguity-averse, that is, its decision is based on the worst-case scenario, or it follows the Maximin criterion. Suppose the actual probability measure is different from the one used by the firm; as in Nishimura and Ozaki [36], we assume the two probability measures are somewhat similar in that they agree with respect to zero-probability events and the differences can be described as a perturbation limited to the range [–κ , κ ], where κ is the measure or magnitude of Knightian uncertainty. Then, (i) from Girsanov’s Theorem, and (ii) given the assumption that the firm follows a Maximin strategy, the lognormal process describing the state variable in our model (Eq. (2)) will be replaced by a lognormal process with the same volatility σ but a drift of (μ–σ κ ) instead of μ.7 Other than this adjustment, the model will remain unchanged. Thus, the only change in the derivations in Sections 2.2, 2.3 and 2.4 will be that the valuation equation (ODE (A1) in Appendix A) will be replaced by:

0.5σ 2 y2V  (y ) + (μ − σ κ )yV  (y ) + Qy − Q (w + γ Q ) = rV (y ) As a result, we will also get different formulas for θ 1 and θ 2 :

(μ − σ κ ) θ1 = 0 . 5 − + σ2 (μ − σ κ ) θ2 = 0 . 5 − − σ2



0.5 −

(μ − σ κ ) σ2

0.5 −

(μ − σ κ ) σ2



2 +

2 +

2r

σ2 2r

σ2

Therefore, the main results of the model, that is, Proposition 1 and Eqs. (5), (6) and (7), will be unchanged under Knightian uncertainty, with the only change being the above formulas for θ 1 and θ 2 . The question of interest now is: with the above changes, how is EPVI affected by Knightian uncertainty κ ? Brief numerical results are presented in Section 3.4, where it is shown that the economic channels by which Knightian uncertainty impacts investment are quite different from the case of volatility. However, the overall effect on investment is quite similar, that is, Knightian uncertainty also generally has a non-monotonic (inverted-U-shaped) effect on EPVI. Nishimura and Ozaki [36] show that a higher level of Knightian uncertainty results in lower project value and lower option value. Using a similar measure of Knightian uncertainty, Niu et al. [35] study its effect on the expansion decision, and show that greater Knightian uncertainty makes the firm more likely to choose smaller capacity. Knightian uncertainty has also been examined using different measures. Cartea and Jaimungal [10] derive optimal investment rules under ambiguity aversion and market incompleteness, using “robust indifference prices,” and show that greater ambiguity can result in earlier or delayed investment. Schroder [42] incorporates varying degrees of managerial risk-aversion by using a measure known as α -MEU preferences, which views manager’s ambiguity preferences as a combination of the two extreme attitudes towards ambiguity (i.e., best and worst cases). He shows that Knightian uncertainty makes the manager more eager to invest. 3. Results Since the results are derived numerically, the values of the input parameters need to be specified. We start with a reasonable “base case” set of parameter values, and repeat the numerical procedure with a wide range of values, to ensure robustness of the results and also to identify the comparative static relationships. 7 This implies that the effect of Knightian uncertainty on investment will depend on volatility σ , because the adjustment term contains σ .

3.1. Base-case parameter values The base-case parameter values are taken from Sarkar [41]: discount rate r = 5%, and demand growth rate μ = 0%. In addition, we use a price sensitivity of γ = 1, unit cost of capital ϕ = 1, variable cost w = 1, and returns-to-scale parameter α = 1. The results are derived for various values of volatility σ , to study the effect of volatility on investment. 3.2. Base-case results when both investment trigger and capacity are chosen optimally When investment size is exogenously specified in a real-option model, investment trigger yi is increasing in the level of uncertainty [13,30], and when investment timing is exogenously specified, investment size is an increasing function of uncertainty [40]. Both are established results in the literature. However, we are interested in the case when neither investment timing nor investment size is exogenously specified. This paper focuses on the overall effect of uncertainty on investment (measured by EPVI) when both investment timing and size are chosen optimally by the firm. The optimal investment trigger and optimal size are determined by solving Eqs. (5) and (6) simultaneously, and the resulting EPVI computed from Eq. (7). With the base-case parameter values, we get the following output with three different levels of volatility σ : For σ = 10%, we get yi = 2.0620, Q = 0.6256, and EPVI = 0.0430; For σ = 15%, yi = 3.5551, Q = 1.6030, and EPVI = 0.0544; For σ = 20%, yi = 11.3441, Q = 6.7771, and EPVI = 0.0359. We note that both investment trigger yi and capacity Q are monotonically increasing functions of volatility, as expected from the above discussion. Thus, the effect of greater uncertainty is larger but delayed investment; but since these two effects work in opposite directions, the overall effect of uncertainty on investment is not clear. For that, we turn to the EPVI. As volatility is increased from 10% to 15% and then to 20%, EPVI first rises from 0.0430 to 0.0544 and then falls to 0.0359. Thus, the effect of uncertainty on investment is non-monotonic, in sharp contrast to the real-option literature which predicts a monotonic negative effect. For instance, Mauer and Ott [[27], p. 582] state: “This leads to the important implication that an increase in uncertainty raises the option value and thereby discourages new investments,” Caballero [[9], p. 279] states “. . . the recent literature on irreversible investment has shown that increase in uncertainty lowers investments,” Metcalf and Hassett [[31], p. 1472] state “. . . based on previous findings by researchers of an inverse relationship between uncertainty and investments,” and Dixit and Pindyck [[13], p. 192] state: “Hence, greater uncertainty reduces investment.” What we have shown is that, when investment size is endogenous, the effect of uncertainty is not necessarily monotonically negative. With the baseline parameter values, investment is initially an increasing function and subsequently a decreasing function of uncertainty, that is, investment is an inverted-U-shaped function of uncertainty. Fig. 1 shows the results over a range of volatility. We first note that both investment trigger and investment size rise at an increasing rate with volatility, reaching very high levels for large volatility. This happens because the two reinforce one another; a higher Q results in a higher yi , which in turn results in a higher Q, and so on, until very high levels are reached. Thus, for high volatility, investment can be postponed indefinitely [12], resulting in very small EPVI. It is clear that high levels of uncertainty can be very unfavorable for corporate investment; not surprisingly, politicians are very reluctant to make dramatic policy changes once they

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Fig. 1. Shows the optimal investment decision (yi and Q) and the resulting EPVI as functions of volatility σ . The gray line shows Q, the broken line shows yi and the solid black line shows (100∗ EPVI). The base-case parameter values are used: r = 5%, μ = 0%, γ = 1, ϕ = 1, w = 1 and α = 1.

achieve power, since abrupt policy changes represent high volatility and tend to freeze corporate investment. The measure of overall investment (EPVI) is an inverted-Ushaped function of uncertainty. The non-monotonicity arises from the two conflicting effects of a higher volatility: (i) delayed investment (timing effect or discounting effect), which has a negative effect on investment because of discounting or time value, and (ii) larger capacity (size effect), which has a positive effect on investment. The discounting effect will be smaller and thus less important if investment is made early (i.e., yi is low). As we can see from Fig. 1, yi is small when volatility is low. For small volatility, therefore, the discounting effect is small, and as a result the size effect dominates; therefore, EPVI is increasing in volatility when volatility is small. For large volatility, on the other hand, the investment trigger is higher, hence investment is delayed; this increases the importance of the timing effect, hence the timing effect dominates, and EPVI is decreasing in volatility. This explains why EPVI is an increasing function when volatility is low and a decreasing function when volatility is high. The above inverted-U relationship was illustrated numerically, because the model is too complicated to allow analytical solutions. With numerical solutions, there is no guarantee that the result will hold for all possible scenarios, hence it is not clear how robust the result is. We therefore repeat the computations with a wide range of parameter values in Section 3.3. As we see in that section, the inverted-U relationship is obtained for almost all the parameter values examined (with one exception), hence it appears to be quite a robust relationship.8 We now state our first major result. Observation 1. When both the timing and size of investment are chosen optimally by the firm, the investment trigger and investment size are increasing functions of volatility; however, the effective magnitude of investment (EPVI) is generally (that is, with most parameter value configurations) non-monotonic, initially increasing and subsequently decreasing in volatility.

8 In all the cases examined in Section 3.3 (comparative statics), only in one instance (low w) is the relationship monotonic downward-sloping. In all other cases the relationship is non-monotonic and inverted-U-shaped. This issue is discussed further in Section 3.3.

5

This is a new finding, because our measure of investment takes into account both timing and size of investment. An important implication of the inverted-U relationship is that both low and high levels of uncertainty will hurt corporate investment, because EPVI is lowest for the two extremes (low and high volatility). While it has not been formally tested, there is some supportive evidence in the literature; for instance, Bo and Lensink [8] and Lensink and Murinde [24] find the inverted-U shaped relationship in a sample of Dutch firms and British firms, respectively. Although the above result provides the general shape of the uncertainty-investment relationship, it says nothing about the relative importance of the rising and falling segments. This might be important because it would affect the frequency with which we observe positive or negative relationships. For instance, if the critical volatility (the point at which it switches from increasing to decreasing) is high, we should more frequently observe an increasing relationship. Similarly, if the critical volatility is low, we would more likely observe a negative relationship. In extreme cases (very high or very low critical volatility) we might, for practical purposes, get a monotonic relationship: monotonically increasing (decreasing) if critical volatility is very high (low). In Section 3.3, we repeat the numerical computations with a wide range of parameter values, to see (i) if the general shape is unchanged, and (ii) how the point at which the relationship turns from positive to negative varies with the different parameter values. We find that (i) the general shape is different in only one case, and (ii) the critical volatility varies widely with parameter values. 3.3. Comparative static results Fig. 2 displays the results with different parameter values. In the general case (that is, for the vast majority of parameter value configurations), EPVI is an initially increasing and subsequently decreasing function of uncertainty, switching from increasing to deceasing at the critical volatility (say σ c ). If σ c is small enough, the EPVI curve will, for practical purposes, be monotonically decreasing in uncertainty (as in the case of w = 0.5, shown in Fig. 2(f) below). Similarly, if σ c is large enough, EPVI will be monotonically increasing in uncertainty. Thus, in the general case, investment is non-monotonic and inverted-U-shaped in uncertainty, but in special cases (for certain parameter values) it is possible that investment is monotonically increasing or decreasing in uncertainty. Based on our comparative static results, the inverted-U relationship is quite robust. However, there is significant variation in the critical volatility (or σ c ), i.e., the exact point at which the relationship switches from increasing to decreasing. This is important because it affect the likelihood of observing a positive or negative relationship between uncertainty and investment. The practical implication of this is discussed in detail below. (a) Returns-to-scale parameter α : Fig. 2(a) shows that for higher α (more decreasing returns to scale) EPVI starts falling later (at higher σ ). Thus, when α is higher, the uncertainty-investment relationship is positive over a wider range of volatility, that is, we are more likely to observe an increasing relationship. The explanation for this result lies in how α impacts yi : a larger α increases the cost of capacity and therefore results in a smaller Q, which results in a lower investment trigger yi ; earlier investment reduces the impact of the discounting effect described in Section 3.2 and thereby causes EPVI to start falling later. Therefore, a higher (lower) α causes EPVI to be increasing in σ over a wider (narrower) range of σ . We also note from Fig. 2(a) that the effect of α on EPVI is ambiguous; for small (large) volatility, EPVI is decreasing (increasing) in α . This is because a higher α results in a smaller Q (i.e., the size effect has a negative effect on investment) along with a lower

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Fig. 2. Comparative static results. Shows the effect of volatility σ on investment (EPVI) for a range of parameter values. Base-case parameter values: r = 5%, μ = 0%, γ = 1, ϕ = 1, w = 1 and α = 1. In (a), the black, gray and broken lines show the relationship for α = 0.5, 1 and 1.5 respectively. In (b), the black, gray and broken lines show the relationship for γ = 0.5, 1 and 1.5 respectively. In (c), the black, gray and broken lines show the relationship for ϕ = 1.5, 1 and 0.5 respectively. In (d), the black, gray and broken lines show the relationship for r = 3%, 5% and 7% respectively. In (e), the black, gray and broken lines show the relationship for μ = 1%, 0 and –1% respectively. In (f), the black, gray and broken lines show the relationship for w = 0.5, 1 and 1.5 respectively.

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yi (i.e., the timing effect has a positive effect on investment). For small volatility, as shown above, the timing effect is smaller, hence the former effect above dominates, and the net effect of higher α on investment is negative. For large volatility, the net effect of higher α on investment is positive. Thus, the overall effect of α on investment is ambiguous. This is very different from traditional real-option models, where a higher α has an unambiguous negative effect on investment. Finally, we also note that investment (EPVI) is more sensitive to volatility when α is high. (b) Price-sensitivity or Market Power γ : The point at which EPVI starts falling does not change with γ , hence γ has no effect on the exact shape of the uncertainty-investment relationship. We also note the following from Fig. 2(b): (i) EPVI is a decreasing function of γ , and (ii) EPVI is more (less) sensitive to volatility when γ is low (high). The explanation is as follows. For (i), recall that a higher γ means greater market power, and it is well known in Microeconomics that a firm with greater market power will keep output low. For (ii), it is because a higher γ (greater market power) better shields the company from the effects of uncertainty, thus investment is less sensitive to uncertainty. Therefore, the market structure or degree of competitiveness affects how uncertainty affects investment, consistent with the empirical findings of Maioli [26] and Patnaik [37]. (c) Cost of capacity ϕ : The point at which EPVI starts falling does not change with ϕ , hence unit investment cost has no impact on the shape of the uncertainty-investment relationship. As expected, EPVI is an increasing function of ϕ , since the dollar investment amount is directly and positively related to ϕ . Also, EPVI is more sensitive to volatility when ϕ is large. (d) Interest rate r: For higher r, EPVI starts falling later (at higher σ ), thus the uncertainty-investment relationship is more likely to be positive when r is higher. This is because, with a higher discount rate r, the benefits from the investment (which come in the future) will be smaller but the investment cost (incurred at the beginning) will remain unchanged; thus, a higher r will result in a smaller optimal capacity. The smaller capacity results in earlier investment, which reduces the impact of the discounting effect, hence the EPVI starts falling at a later stage (for higher σ ). Therefore, we are more likely to observe an increasing relationship when interest rate is high. We also note from Fig. 2(d) that the effect of r on EPVI is ambiguous; for small σ EPVI is decreasing in r, and for large σ EPVI is increasing in r. The explanation is that a higher r results in a smaller Q (i.e., the size effect has a negative effect on investment) and therefore also a lower yi (i.e., the timing effect has a positive effect on investment); for small volatility, the timing effect is smaller, hence the former effect above dominates, and the net effect of higher r on investment is negative. For large volatility, the net effect of higher r is positive. Thus, the overall effect of r on investment is ambiguous, and depends on the volatility; a higher interest rate will not necessarily have a negative effect on investment. (e) Demand growth rate μ: For higher μ, EPVI starts falling earlier (at lower σ ). This is because a higher growth rate leads to larger capacity Q, and the larger capacity results in delayed investment, which increases the impact of the discounting effect. Hence, when μ is higher, EPVI starts falling earlier (for lower σ ); that is, the relationship is more likely to be a negative one. Therefore, a higher (lower) μ raises the likelihood of observing a negative (positive) uncertainty-investment relationship. The effect of μ on EPVI depends on the volatility; for small (large) σ , EPVI is increasing (decreasing) in μ. A higher μ results in a larger Q (i.e., the size effect has a positive effect on investment) and therefore also a higher yi (i.e., the timing effect has a negative effect on investment). For small volatility, the timing effect is smaller, hence the net effect of higher μ on investment is

7

positive. For large volatility, the timing effect is stronger, hence the net effect of higher r is negative. Thus, the overall effect of r on investment is ambiguous, and a higher expected growth rate will not necessarily have a positive effect on investment. (f) Operating cost w: A higher w results in a higher critical volatility (σ c ), that is, EPVI starts falling later; thus the observed uncertainty-investment relationship is more likely to be positive (negative) when w is higher (lower). A higher operating cost leads to a smaller capacity Q, and the smaller capacity results in earlier investment, which reduces the impact of the discounting effect. As a result, the EPVI starts falling later (σ c is higher). Therefore, a higher (lower) w causes the EPVI curve to be increasing in σ over a wider (narrower) volatility range. For low enough w, the critical volatility σ c might be small enough to make the entire EPVI curve monotonically downward-sloping (for practical purposes), in contrast to the other cases examined in this paper. This is indeed the case for w = 0.5, as Fig. 2(f) shows. As expected, EPVI is a decreasing function of w. Finally, EPVI is more sensitive to volatility when w is small. This is because, w being the variable cost, a lower w means higher operating leverage, which increases the sensitivity to volatility. The comparative static results of this section are summarized below: (a) The effect of uncertainty on investment is more likely to be positive for decreasing-returns-to-scale technology, for low demand growth rate and volatility, and for high interest rate and operating cost. (b) Investment is more sensitive to volatility for decreasing-returnsto-scale and low-operating-cost technology, for costly capacity, and for competitive industries. The comparative static results specify the conditions under which the relationship is more likely to be positive or negative, as well as the conditions under which investment will be more sensitive to uncertainty. The latter is of considerable interest because uncertainty is substantially impacted by government policies. If government policies are unstable, economic uncertainty increases. Our results indicate that for costly capacity, decreasing returns to scale, low operating cost (or high margin) and more competitive industries, government policies are more relevant because investment is more sensitive to uncertainty. In these cases, therefore, government decisions and policies have larger impact on corporate investment, and therefore have to be set more carefully. The effect of uncertainty on investment has important policy implications for governments, because governments around the world encourage corporate investment, and government policies have a significant impact on the level of uncertainty. Thus the government would be very interested in how EPVI might change if volatility σ were to be higher or lower. There are two issues of interest – the direction and sensitivity of EPVI. Suppose σ were to increase from 10% to 15% (this might occur if, for instance, if the stringency of regulation is reduced). Then, from Fig. 2(f), EPVI would go up if w is high (i.e., for a lowmargin product), but EPVI would go down for a high-margin product; thus the direction can vary in different circumstances. Also, for the same change in σ , Fig. 2(b) shows that EPVI will rise for all values of γ , but will rise more slowly if γ is lower (a more competitive industry); thus the sensitivity can also vary considerably in different circumstances. 3.4. The uncertainty-investment relationship for Knightian uncertainty As discussed in Section 2.5, uncertainty can also be represented by Knightian uncertainty κ (instead of the traditional measure, volatility σ ). We repeated the numerical computations with the

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Fig. 3. (a). Shows investment trigger (yi ), investment size (Q) and EPVI as functions of Knightian Uncertainty κ . The base-case parameter values are used: r = 5%, μ = 0%, γ = 1, ϕ = 1, w = 1 and α = 1, along with a volatility of σ = 20%. The thick blue line shows Q, the broken red line shows yi , and the thin black line shows (100∗ EPVI). (b). Shows EPVI as a function of κ for three volatility levels; the thick blue line shows σ = 15%, the broken red line shows σ = 20%, and the thin black line shows σ = 25%. The base-case parameter values are used: r = 5%, μ = 0%, γ = 1, ϕ = 1, w = 1 and α = 1.

Knightian uncertainty model of Section 2.5, for various levels of κ , so as to gauge its effect on investment (EPVI). With the base-case parameter values and volatility σ = 20%, the output is summarized below:

κ = 0.01, yi = 8.4533, Q = 4.7983, EPVI = 0.0416; κ = 0.04, yi = 5.1243, Q = 2.5223, EPVI = 0.0476; κ = 0.10, yi = 3.2809, Q = 1.2628, EPVI = 0.0417. In these results, the first point worth noting is that both yi and Q are decreasing functions of κ ; this is exactly the opposite of the traditional uncertainty (volatility) effect (Section 3.2). The intuitive explanation is as follows. As discussed in Section 2.5, Nishimura and Ozaki [36] showed that a higher κ results in lower option value and lower project value. The lower option value leads to earlier option exercise (earlier investment) or lower trigger yi , and the lower project value leads to smaller project size Q. Thus, a higher κ results in earlier but smaller investment. EPVI, however, is initially increasing and subsequently decreasing in κ . Therefore, for the base case, the effect of Knightian uncertainty on EPVI is very similar to that of volatility. Although this inverted-U-shaped relationship is the same as in the case of volatility, the explanation is different. As mentioned above, a higher κ will result in earlier but smaller investment; the earlier investment has a positive effect on investment (the “discounting effect”) but the smaller investment has a negative effect (the “size effect”). Because of the two opposing effects of a higher κ , the overall effect on EPVI is ambiguous. For small κ , the trigger yi is high, hence the discounting effect dominates and thus the net effect is positive; therefore, for small κ the EPVI will be increasing in κ . For large κ , yi is lower, hence the discounting effect is smaller, and the size effect will dominate, resulting in a net negative effect; therefore, for large κ the EPVI will be decreasing in κ . Overall, then, EPVI will be an increasing (a decreasing) function of κ for small (large) κ , giving the familiar inverted-U-shaped relationship again. These results are illustrated graphically in Fig. 3(a) for a more complete range of Knightian uncertainty κ . It shows Q, yi and EPVI as functions of κ for the base-case parameter values and σ = 20%. Consistent with the above results, yi and Q are found to be mono-

tonically decreasing functions of κ , while EPVI is initially increasing and subsequently decreasing in κ . To summarize: With endogenous investment timing and size, both investment trigger and size are decreasing functions of Knightian uncertainty κ ; however, the effective magnitude of investment (EPVI) is generally non-monotonic, initially increasing and subsequently decreasing, in κ. Thus the non-monotonic nature of the uncertainty-investment relationship holds, in general, for both measures of uncertainty – the traditional measure (volatility) and the more recent Knightian uncertainty described in this section. The inverted-U relationship between uncertainty and investment seems to be quite a general one. Finally, we look at how volatility affects the Knightian uncertainty-investment relationship. This is of interest because, as mentioned in footnote 7, the effect of κ on EPVI will depend on σ . Fig. 3(b) shows the EPVI as a function of Knightian uncertainty κ for three volatility levels, σ = 15%, 20% and 25%. While we get the familiar inverted-U-shaped relationship for σ = 20% and 25%, the EPVI with low volatility (σ = 15%) is found to be a decreasing function of κ . Thus, Knightian uncertainty also generally has a non-monotonic inverted-U-shaped effect on EPVI, except for low volatility levels (when it has a negative effect). 4. Summary and conclusions This paper uses a real-option model to study how uncertainty affects corporate investment, when both timing and size of investment are determined by the firm (as is usually the case in practice). Such a study does not currently exist in the literature. With endogenous timing and size, the effect on investment needs to capture both the dimensions, thus a composite measure of investment is necessary. We propose such a measure, the expected present value of investment or EPVI (in dollars). For most parameter value configurations (with one exception), EPVI is found to be an initially increasing and subsequently decreasing function of uncertainty. Thus, in contrast to the traditional real-option investment literature, we show that uncertainty can have a non-monotonic effect on investment. An important prac-

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tical policy implication is that extremes in uncertainty (in either direction) hurt corporate investment. This is of relevance to government policymakers because government policies and decisions generally play an important role in impacting uncertainty levels in the economy. The point at which EPVI switches from being an increasing to a decreasing function of uncertainty (hence the likelihood of observing a positive or negative relationship) can vary widely with input parameter values. We are more likely to observe a positive relationship between uncertainty and investment for decreasingreturns-to-scale technology, low demand growth rate and volatility, or high interest rate and operating cost. In these situations, therefore, greater uncertainty can be good for investment. However, there are other situations where greater uncertainty will have a negative impact on investment. We also show that the effect of certain parameters on investment can differ greatly from the traditional fixed-capacity case; for instance, investment is a decreasing function of interest rate when uncertainty is low but an increasing function when uncertainty is high. Finally, investment is more sensitive to volatility for decreasing-returns-to-scale technology, competitive markets, costly capital and high operating margins. As in most models, we make simplifying assumptions for reasons of tractability. For instance, we consider only one investment opportunity that requires a lumpy investment, with no option of future expansion. Thus our model will be useful for lumpy, oneshot investments, but not necessarily for investments with expansion opportunities or for a company considering multiple investment opportunities at the same time. Our model could be extended to analyze these scenarios. We also take a brief look at Knightian uncertainty or ambiguity as a measure of uncertainty, where the firm is uncertain about the underlying probability measure driving uncertainty in the economy. With Knightian uncertainty, we find that the overall results are very similar. Thus, the finding that uncertainty might have a non-monotonic or ambiguous effect on investment seems to be quite a general result.

9

μθ − r = 0, and are given by:9

μ θ1 = 0 . 5 − 2 + σ θ2 = 0 . 5 −

μ − σ2









0 . 5 − μ/ σ 2

0 . 5 − μ/ σ 2

2 2

+ +

2r

σ2

,

2r

σ2

Thus the complete solution to the ODE is: V (y ) = A0 yθ1 + Q ( w+γ Q ) Qy θ Ay 2 + r− , where the last two terms represent the μ − r value if the project is operated forever (without abandonment). As y reaches very high levels (y → ∞), it becomes very unlikely that the project will be abandoned, i.e., it will be in operaQ ( w+γ Q ) Qy tion forever. That is, Limy→∞ V (y ) = r− , which implies μ − r

Limy→∞ (A0 yθ1 + Ayθ2 ) = 0. Since θ 1 > 0, this means A0 = 0. Thus the project value is given by Eq. (3) of the paper. B. Boundary conditions for optimal abandonment and investment triggers The project value V(y) is given by Eq. (3), where the constant A and the optimal abandonment trigger yb are determined from the boundary conditions. There are two boundary conditions associated with the abandonment trigger y = yb :10 Value-matching (continuity) condition: V (yb ) = 0, and Smooth-pasting (optimality) condition: V’(yb ) = 0. The two boundary conditions can be solved for the constant A and the optimal abandonment trigger yb :

A=−

Q ( y b ) 1 − θ2 (w + γ Q )(1 − μ/r ) , and yb = θ2 ( r − μ ) ( 1 − 1 / θ2 )

Similarly, for the option to invest, the optimal investment trigger yi can be determined from two the boundary conditions at the trigger y = yi : Value-matching (continuity) condition: f (yi ) = V (yi ) − ϕ Q α Smooth-pasting (optimality) condition: f  (yi ) = V  (yi ) These boundary conditions give Eq. (5) for the optimal investment trigger yi (in Proposition 1):

Appendix

Q y i ( 1 − 1 / θ1 ) ( w + γ Q )Q − + A(1 − θ2 /θ1 )(yi )θ2 = ϕ Q α (r − μ ) r

A. Project value

C. Optimal capacity

The project value will be a function of just the state variable y (it is not a function of time because of the infinite-horizon setting); let the project value be V(y). If dV is the change in project value over the next instant dt, then dV = V’(y)dy + 0.5V”(y)(dy)2 , from Ito’s lemma. Substituting for dy from Eq. (2), we get:

The optimal capacity will be such that it maximizes the payoff at investment, which is given by: {V (yi ) − ϕ Q α } or {Q yi /(r − μ ) − (w + γ Q )Q /r + A(yi )θ2 − ϕ Q α }, where yi is the state variable when the investment is made. Differentiating this expression with respect to Q and equating the derivative to zero gives us the equation for optimal capacity (in Proposition 1):

dV = V  (y )[μydt + σ ydZ ]+0.5V  (y )σ 2 y2 dt . Taking expectations (and keeping in mind that E(dZ) = 0), we get the expected value of dV:

E (dV ) =



μyV  (y )+0.5σ 2 y2V  (y ) dt .

 y θ2 i

yb

= rϕα Q α −1 .

D. Second-order conditions

This is the expected capital gain from holding the project over the next instant dt. There is also a cash inflow of (Qy–Qw–γ Q2 ) dt. Thus the total instantaneous return is 0.5σ 2 y2V  (y )+μyV  (y )+Q y−Q (w+γ Q ) . V (y )

In order to ensure the existence of internal optima for investment trigger and size, the following second-order conditions 2 2 2 2 need to be satisfied: (i) ∂∂y f2 < 0, (ii) ∂∂Q 2f < 0 and (iii) ∂∂y f2 ∂∂Q 2f − i i

( ∂∂y ∂fQ )2 > 0, where f(y) is the payoff to the firm prior to invest2

From the Local Expectations Hypothesis, this is equal to the risk-free rate r. Setting the above expression equal to r, we get the ordinary differential equation (ODE):

0.5σ 2 y2V  (y ) + μyV  (y ) + Qy − Q (w + γ Q ) = rV (y )



yi w+γQ − (w + γ Q ) + Q γ + (1 − μ/r ) 1 − θ2

(A1)

The particular solution to the above ODE is: Qy/(r – μ) – Q(w + γ Q)/r, and the general solution is: A0 yθ1 + Ayθ2 , where θ 1 and θ 2 are solutions to the quadratic equation: 0.5σ 2 θ (θ − 1 ) +

i

ment (which is the function to be maximized):

f (y ) =

  y  θ1  Q y (w + γ Q )Q −Q yb 1−θ2 θ2 i − + yi − ϕ Q α . yi r−μ r θ2 ( r − μ )

Note that θ 1 > 1 and θ 2 < 0. See Dixit and Pindyck [13] for a detailed discussion of the value-matching and smooth-pasting conditions. 9

10

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After some simplification, these second-order conditions can be stated as follows: (i)

2γ r

+ ϕα (α − 1 )Q α −2 > 0,

θ −1 (ii) θ 1−θ > 1 2

( yyi )θ2 −1 , and b

y

1−θ2

(iii) {θ1 − 1 − (θ1 − θ2 )( yb ) (1−θ1 )2 yi . r−μ

i

}{ 2γr Q + ϕα (α − 1 )Q α −1 } >

Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.omega.2019.102115.

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