Stepwise investment and capacity sizing under uncertainty

Abstract

The relationship between uncertainty and managerial flexibility is particularly crucial in addressing capital projects. We consider a firm that can invest in a project in either a single (lumpy investment) or multiple stages (stepwise investment) under price uncertainty and has discretion over not only the time of investment but also the size of the project. We confirm that if the capacity of a project is fixed and the investment premium associated with stepwise investment is positive, then lumpy investment becomes more valuable than a stepwise investment strategy under high price uncertainty. By contrast, if a firm has discretion over capacity, then we show that the stepwise investment strategy always dominates that of lumpy investment. In addition, we show that the total amount of installed capacity under a stepwise investment strategy is always greater than that under lumpy investment.

Appendix

Proposition 1

The optimal investment threshold and the corresponding optimal capacity under lumpy and stepwise investment are:

$$\begin{aligned} P^*_{j}\left( K^*_{j}\right)= & {} \frac{I\left( K^*_{j}\right) }{K^*_{j}}\frac{\beta (\rho - \mu )}{\beta - 1} \ \ and \ \ K^*_{j} = \left[ \frac{a_{j}}{b}\frac{1}{\gamma (\beta - 1) - \beta }\right] ^{\frac{1}{\gamma -1}}, \nonumber \\&\quad \gamma (\beta - 1) - \beta >0 \end{aligned}$$

(A.1)

Proof

By maximising the value of the now-or-never investment opportunity, we obtain the expression for the optimal capacity, $\overline{K}^*_{j}$, corresponding to the current output price P, as indicated in (A.2) for $j = \ell , s_{i}$.

$$\begin{aligned} \max _{\overline{K}_{j}}\overline{F}_{j}\left( P, \overline{K}_{j}\right)\Rightarrow & {} \overline{K}^*_{j}\left( P\right) = \left[ \frac{1}{b\gamma }\left( \frac{P}{\rho - \mu } - a_{j}\right) \right] ^{\frac{1}{\gamma - 1}} \end{aligned}$$

(A.2)

Next, the value of the option to invest is described in (A.3).

$$\begin{aligned} F_{j}\left( P\right) = {\left\{ \begin{array}{ll} (1 - \rho \mathrm{d}t)\mathbb {E}_P\left[ F_{j}(P + \mathrm{d}P)\right] , &{}\quad P< P^*_{j}\\ \frac{PK^*_{j}}{\rho - \mu } - I\left( K^*_{j}\right) , &{}\quad P\ge P^*_{j} \end{array}\right. } \end{aligned}$$

(A.3)

By expanding the first branch on the right-hand side of (A.3) using Itô’s lemma, we obtain the differential equation (A.4)

$$\begin{aligned} \frac{1}{2}\sigma ^{^{2}}P^{^{2}}F^{''}_{j}(P) + \mu PF^{'}_{j}(P) - \rho F_{j}(P) = 0 \end{aligned}$$

(A.4)

which, for $P<P^*_{j}$, has the general solution that is indicated in (A.5).

$$\begin{aligned} F_{j}(P) = A_{j}P^{\beta } + B_{j}P^{\delta } \end{aligned}$$

(A.5)

Notice that $\delta$ is the negative root of the quadratic $\frac{1}{2}\sigma ^2x(x - 1) + \mu x - \rho = 0$, and therefore, $P \rightarrow 0\Rightarrow B_{j}P^{\delta } \rightarrow \infty$. Consequently, we must have $B_{j} = 0$, and thus, we finally obtain (A.6).

$$\begin{aligned} F_{j}\left( P\right) = {\left\{ \begin{array}{ll} A_{j}P^{\beta }, &{}\quad P< P^*_{j}\\ \frac{PK^*_{j}}{\rho - \mu } - I\left( K^*_{j}\right) , &{}\quad P\ge P^*_{j} \end{array}\right. } \end{aligned}$$

(A.6)

By applying value-matching and smooth-pasting conditions between the two branches of (A.6), we obtain the expression for the endogenous constant and the optimal investment threshold that are indicated in (A.7) and (A.8), respectively.

$$\begin{aligned} A_{j}= & {} \frac{1}{P^{^{*^{\beta }}}_{j}}\left[ \frac{P^*_{j}K^*_{j}}{\rho - \mu } - I\left( K^*_{j}\right) \right] \end{aligned}$$

(A.7)

$$\begin{aligned} P^*_{j}\left( K^*_{j}\right)= & {} \frac{I\left( K^*_{j}\right) }{K^*_{j}}\frac{\beta (\rho - \mu )}{\beta - 1} \end{aligned}$$

(A.8)

By inserting (A.8) into (A.2), we obtain the expression for the optimal capacity

$$\begin{aligned} K^*_{j}= & {} \left[ \frac{a_{j}}{b}\frac{1}{\gamma (\beta - 1) - \beta }\right] ^{\frac{1}{\gamma -1}}, \ \ \gamma (\beta - 1) - \beta >0 \end{aligned}$$

(A.9)

while the final expression for the optimal investment threshold is obtained by inserting (A.9) into (A.8) and is indicated in (A.10).

$$\begin{aligned} P^*_{j}\left( K^*_{j}\right)= & {} \frac{a_{j}\beta (\rho - \mu )\left( \gamma - 1\right) }{\gamma (\beta - 1) - \beta } \end{aligned}$$

(A.10)

$\square$

Proposition 2

$P^*_{s_{1}}$ is independent of $P^*_{s_{2}}$.

Proof

If we assume that $\tau _{s_{2}}\ge \tau _{s_{1}}$, then $P^*_{s_{1}}\le P^*_{s_{2}}$ and the maximised option value in the case of staged investment is indicated in (A.11).

$$\begin{aligned} F_{s}(P)= & {} \left( \frac{P}{P^*_{s_{1}}}\right) ^{\beta } \left[ \frac{P^*_{s_{1}}K^*_{s_{1}}}{\rho - \mu } - I\left( K^*_{s_{1}}\right) + \left( \frac{P^*_{s_{1}}}{P^*_{s_{2}}}\right) ^{\beta }\left[ \frac{P^*_{s_{2}} K^*_{s_{2}}}{\rho - \mu } - I\left( K^*_{s_{2}}\right) \right] \right] \quad \quad \quad \end{aligned}$$

(A.11)

Hence, $p_{s_{1}}$ satisfies the first-order necessary condition (A.12)

$$\begin{aligned} \beta \left( -\frac{1}{P^*_{s_{1}}}\right) \left[ \frac{P^*_{s_{1}}K^*_{s_{1}}}{\rho - \mu } - I\left( K^*_{s_{1}}\right) \right] + \frac{ K^*_{s_{1}}}{\rho - \mu } = 0 \end{aligned}$$

(A.12)

from which we have:

$$\begin{aligned} P^*_{s_{1}} = \frac{I\left( K^*_{s_{1}}\right) }{K^*_{s_{1}}}\frac{\beta (\rho - \mu )}{\beta - 1} \end{aligned}$$

(A.13)

Consequently, $P^*_{s_{1}}$ is independent of $P^*_{s_{2}}$, i.e. the presence of the second stage does not affect the decision to invest in the first one. Note that the assumption $P^*_{s_{1}}< P^*_{s_{2}}$ can be expressed as in (A.14).

$$\begin{aligned} \frac{I\left( K^*_{s_{1}}\right) }{I\left( K^*_{s_{2}}\right) }<\frac{K^*_{s_{1}}}{K^*_{s_{2}}} \end{aligned}$$

(A.14)

$\square$

Proposition 3

$\frac{\partial K^*_{j}}{\partial \sigma }>0$ and $\frac{\partial P^*_{j}}{\partial \sigma }>0$.

Proof

By differentiating the expression of the optimal capacity in (A.9), we have:

$$\begin{aligned} \frac{\partial K^*_{j}}{\partial \sigma }= & {} K^*_{j}\left[ \frac{-\frac{\partial }{\partial \sigma }\beta }{\gamma (\beta - 1) - \beta }\right] , \ \ \gamma (\beta - 1) - \beta >0 \end{aligned}$$

(A.15)

Since $\frac{\partial }{\partial \sigma }\beta <0$, we have $\frac{\partial K^*_{j}}{\partial \sigma }>0$. Additionally, the expression of the optimal investment threshold is:

$$\begin{aligned} P^*_{j}\left( K^*_{j}\right)= & {} \frac{I\left( K^*_{j}\right) }{K^*_{j}}\frac{\beta (\rho - \mu )}{\beta - 1}\nonumber \\= & {} \frac{a_{j}K^*_{j} + bK^{^{*^{\gamma }}}_{j}}{K^*_{j}}\frac{\beta (\rho - \mu )}{\beta - 1}\nonumber \\= & {} \left( a_{j} + bK^{^{*^{\gamma -1}}}_{j}\right) \frac{\beta (\rho - \mu )}{\beta - 1} \end{aligned}$$

(A.16)

Since $\frac{\partial }{\partial \sigma } \frac{\beta }{\beta - 1}>0$ and $\frac{\partial }{\partial \sigma }K^*_{j}>0$ we have $\frac{\partial }{\partial \sigma }P^*_{j}>0$. $\square$

Proposition 4

If a firm has discretion over capacity, then $F_{s}(P)> F_{\ell }(P)$.

Proof

The relative value of the two strategies is indicated in (A.17).

$$\begin{aligned} \frac{F_{s}(P)}{F_{\ell }(P)}= & {} \sum _{i}\frac{F_{s_{i}}(P)}{F_{\ell }(P)} \end{aligned}$$

(A.17)

to show that $F_{s}(P)> F_{\ell }(P)$, we will show that each term on the right-hand side of (A.17) is greater than one, i.e.

$$\begin{aligned} \frac{F_{s_{i}}(P)}{F_{\ell }(P)} = \frac{\left( \frac{P}{P^*_{s_{i}}}\right) ^{\beta }\left[ \frac{P^*_{s_{i}} K^*_{s_{i}}}{\rho - \mu } -I\left( K^*_{s_{i}}\right) \right] }{\left( \frac{P}{P^*_{\ell }}\right) ^{\beta }\left[ \frac{P^*_{\ell } K^*_{\ell }}{\rho - \mu } - I\left( K^*_{\ell }\right) \right] }=\left( \frac{K^*_{s_{i}}}{K^*_{\ell }}\right) ^{^{\beta }}\frac{I\left( K^*_{\ell }\right) ^{^{\beta - 1}}}{I\left( K^*_{s_{i}}\right) ^{\beta - 1}}>1,&i = 1,2 \nonumber \\ \end{aligned}$$

(A.18)

By manipulating the expression of the relative value in (A.18) we obtain (A.19)

$$\begin{aligned} \left( \frac{K^*_{s_{i}}}{K^*_{\ell }}\right) ^{^{\beta }}\frac{I\left( K^*_{\ell }\right) ^{^{\beta - 1}}}{I\left( K^*_{s_{i}}\right) ^{\beta - 1}}>1\Leftrightarrow \frac{\frac{I\left( K^*_{\ell }\right) ^{^{\beta - 1}}}{K^{^{*^{\beta } }}_{\ell }}}{\frac{I\left( K^*_{s_{i}}\right) ^{\beta - 1}}{K^{*^{\beta }}_{s_{i}}}}>1 \Leftrightarrow \frac{K^*_{s_{i}}}{K^*_{\ell }}\frac{\frac{I\left( K^*_{\ell }\right) ^{^{\beta - 1}}}{K^{^{*^{\beta -1}}}_{\ell }}}{\frac{I\left( K^*_{s_{i}}\right) ^{\beta - 1}}{K^{^{*^{\beta -1}}}_{s_{i}}}}>1 \end{aligned}$$

(A.19)

where the expression for $\frac{K^*_{s_{i}}}{K^*_{\ell }}$ is indicated in (A.20).

$$\begin{aligned} \frac{K^*_{s_{i}}}{K^*_{\ell }} = \frac{\left[ \frac{a_{s_{i}}}{b}\frac{1}{\gamma (\beta - 1) - \beta }\right] ^{\frac{1}{\gamma -1}}}{\left[ \frac{a_{\ell }}{b}\frac{1}{\gamma (\beta - 1) - \beta }\right] ^{\frac{1}{\gamma -1}}} = \left( \frac{a_{s_{i}}}{a_{\ell }}\right) ^{^{\frac{1}{\gamma - 1}}} \end{aligned}$$

(A.20)

By substituting the expression for $\frac{K^*_{s_{i}}}{K^*_{\ell }}$ into (A.19) and by inserting the expression for the optimal capacity from (A.9) into (A.19), we finally obtain (A.21).

$$\begin{aligned} \frac{K^*_{s_{i}}}{K^*_{\ell }}\frac{\frac{I\left( K^*_{\ell }\right) ^{^{\beta - 1}}}{K^{^{*^{\beta -1}}}_{\ell }}}{\frac{I\left( K^*_{s_{i}}\right) ^{\beta - 1}}{K^{^{*^{\beta -1}}}_{s_{i}}}}= \left( \frac{a_{s_{i}}}{a_{\ell }}\right) ^{^{\frac{1}{\gamma - 1}}}\left( \frac{a_{\ell }}{a_{s_{i}}}\right) ^{^{\beta - 1}} = \left( \frac{a_{\ell }}{a_{s_{i}}}\right) ^{^{\beta - \frac{\gamma }{\gamma - 1}}} \end{aligned}$$

(A.21)

Notice that $(\beta -1)(\gamma - 1)>1\Leftrightarrow \beta >\frac{\gamma }{\gamma - 1}$, which is the required condition so that $K^*_{j}\in \mathbb {R}^{^{+}}$. Additionally, by differentiating (A.18) with respect to $\sigma$ as in (A.22), we can determine the relationship between uncertainty and the relative value of the two strategies.

$$\begin{aligned} \frac{\partial }{\partial \sigma } \frac{F_{s_{i}}(P)}{F_{\ell }(P)}= \frac{\partial }{\partial \sigma }\left( \frac{a_{\ell }}{a_{s_{i}}}\right) ^{^{\beta - \frac{\gamma }{\gamma - 1}}} = \left( \frac{a_{\ell }}{a_{s_{i}}}\right) ^{^{\beta - \frac{\gamma }{\gamma - 1}}}\ln \left( \frac{a_{\ell }}{a_{s_{i}}}\right) \frac{\partial \beta }{\partial \sigma } \end{aligned}$$

(A.22)

Notice that if $a_{s_{i}}<a_{\ell }$, then greater uncertainty decreases the relative value of the stepwise investment strategy. Consequently, if $a_{s_{i}}<a_{\ell }$ $\forall i\in \mathbb {N}$, then $\sigma \nearrow \ \Rightarrow \ \frac{F_{s}(P)}{F_{\ell }(P)}\searrow$. In addition, from (A.21) we conclude that the stepwise investment strategy is always more valuable than lumpy investment.

$$\begin{aligned} \frac{F_{s}(P)}{F_{\ell }(P)} \ = \ \sum _{i}\frac{F_{s_{i}}(P)}{F_{\ell }(P)} \ = \ \sum _{i}\left( \frac{a_{\ell }}{a_{s_{i}}}\right) ^{^{\beta - \frac{\gamma }{\gamma - 1}}}>1,&\hbox {if}&a_{s_{i}}<a_{\ell }, \ \forall i\in \mathbb {N}\quad \quad \quad \end{aligned}$$

(A.23)

$\square$

Proposition 5

$K^*_{\ell }<\sum ^{n}_{i = 1} K^*_{s_{i}} \ \Leftrightarrow \ a^{^{\frac{1}{\gamma - 1}}}_{\ell }<\sum ^{n}_{i = 1} a^{^{\frac{1}{\gamma - 1}}}_{s_{i}}$.

Proof

The optimal capacity of the project under lumpy and stepwise investment is described in (A.24).

$$\begin{aligned} K^*_{j}= & {} \left[ \frac{a_{j}}{b}\frac{1}{\gamma (\beta - 1) - \beta }\right] ^{\frac{1}{\gamma -1}} \end{aligned}$$

(A.24)

From (A.24) we have $K^*_{\ell }< K^*_{s_{1}} + K^*_{s_{2}} \Leftrightarrow a^{^{\frac{1}{\gamma - 1}}}_{\ell } < a^{^{\frac{1}{\gamma - 1}}}_{s_{1}} + a^{^{\frac{1}{\gamma - 1}}}_{s_{2}}$. Notice that this inequality cannot be solved analytically for all values of $\gamma$. However, since $\gamma>\frac{\beta }{\beta - 1}>1$, it is easy to see that $\gamma = 2\Rightarrow a_{\ell } < a_{s_{1}} + a_{s_{2}}$, which holds by assumption. Similarly, by substituting the expression for $K^*_{j}$ into (3), we find that the assumption $I(K^*_{\ell })<\sum _{i}I (K^*_{s_{i}})$ is equivalent to (A.25).

$$\begin{aligned} a^{^{\frac{\gamma }{\gamma - 1}}}_{\ell } < a^{^{\frac{\gamma }{\gamma - 1}}}_{s_{1}} + a^{^{\frac{\gamma }{\gamma - 1}}}_{s_{2}} \end{aligned}$$

(A.25)

Consequently, it is not sufficient to assume that $a_{\ell } < a_{s_{1}} + a_{s_{2}}$ to ensure that at the optimal solution, the modular project is more expensive than the lumpy one. Instead, this assumption depends on a more complex relationship between the cost parameters $a_{j}$ and the convexity of the cost function. $\square$

Corollary 1

The MB is steeper than the MC.

Proof

The result follows from differentiating the MB and MC of delaying investment with respect to the output price. Notice that the MC is positive and independent of the output price, while $\frac{\partial }{\partial P}\hbox {MB}<0$. $\square$

Proposition 6

If $K_{j}$ is fixed, then $\frac{\partial }{\partial \sigma }\hbox {MB}<0$, $\frac{\partial }{\partial \sigma }\hbox {MC}<0$, and $|\frac{\partial }{\partial \sigma }\hbox {MB}|<|\frac{\partial }{\partial \sigma }\hbox {MC}|$, whereas if $K_{j}$ is scalable, then $\frac{\partial }{\partial \sigma }\hbox {MB}>0$, $\frac{\partial }{\partial \sigma }\hbox {MC}>0$, and $\frac{\partial }{\partial \sigma }\hbox {MB}>\frac{\partial }{\partial \sigma }\hbox {MC}$.

Proof

The MB and MC of delaying investment is indicated in (A.26).

$$\begin{aligned} \hbox {MB} = \hbox {MC}\Leftrightarrow & {} \frac{\beta I\left( K^*_{j}\right) }{P^*_{j}} + \frac{K^*_{j}}{\rho - \mu } = \frac{\beta K^*_{j}}{\rho - \mu } \end{aligned}$$

(A.26)

Notice that if the capacity of the project is fixed, i.e. $K^*_{j}\equiv K_j$, then an increase in $\sigma$ lowers both the MB and the MC as indicated in (A.27).

$$\begin{aligned} \frac{\partial \hbox {MB}}{\partial \sigma } = \frac{I\left( K_{j}\right) \frac{\partial }{\partial \sigma }\beta }{ P^*_{j}}<0 \quad \hbox {and} \quad \frac{\partial \hbox {MC}}{\partial \sigma } = \frac{K_{j}\frac{\partial }{\partial \sigma }\beta }{ \rho - \mu }<0 \end{aligned}$$

(A.27)

However, from (A.13) we know that $\frac{I(K_{j})}{P^*_{j}}< \frac{K_{j}}{\rho - \mu }$, and therefore, the MC of delaying investment decreases by more than the MB.

$$\begin{aligned} \left| \frac{\partial \hbox {MC}}{\partial \sigma }\right| > \left| \frac{\partial \hbox {MB}}{\partial \sigma }\right| \end{aligned}$$

(A.28)

By contrast, if the capacity of the project is scalable, then the MB and MC of delaying investment increase with greater price uncertainty. Indeed, for $P < P^*_{j}$, we have:

$$\begin{aligned} \frac{\partial \hbox {MB}}{\partial \sigma } = \frac{I\left( K^*_{j}\right) \frac{\partial }{\partial \sigma }\beta + \beta \frac{\partial }{\partial \sigma }I\left( K^*_{j}\right) }{ p_{j}} + \frac{\frac{\partial }{\partial \sigma }K^*_{j}}{\rho - \mu }>0 \end{aligned}$$

(A.29)

The second term on the right-hand side of (A.29) is positive since $\frac{\partial }{\partial \sigma }K^*_{j}>0$. Notice also that even though $\frac{\partial }{\partial \sigma }\beta <0$, $\beta$ is bounded from below since $\beta >1$. By contrast, since the capacity of the project is not bounded and $\frac{\partial }{\partial \sigma }I(K^*_{j})>0$, the decrease in $\beta$ is mitigated by the increase in the investment cost. The impact of $\sigma$ on the MC is indicated in (A.30).

$$\begin{aligned} \frac{\partial \hbox {MC}}{\partial \sigma } = \frac{K^*_{j}\frac{\partial }{\partial \sigma }\beta + \beta \frac{\partial }{\partial \sigma }K^*_{j}}{\rho - \mu } \end{aligned}$$

(A.30)

Notice that the reduction in $\beta$ makes the impact of $\sigma$ on $K^*_{j}$ less pronounced, and therefore, the second term on the right-hand side of (A.29) is greater than right-hand side of (A.30). Since the first term on the right-hand side of (A.29) is positive, we have:

$$\begin{aligned} \frac{\partial \hbox {MB}}{\partial \sigma } > \frac{\partial \hbox {MC}}{\partial \sigma } \end{aligned}$$

(A.31)

Finally, notice that the second term on the left-hand side of (A.26) is constant when the capacity is fixed and increasing when the capacity is scalable. Similarly, the reduction of the first term due to the decrease in $\beta$ with greater uncertainty is offset by the increase in $K^*_{j}$. Consequently, the impact of $\sigma$ on the MB of delaying investment is not only reversed when the firm has discretion over capacity but it is also more pronounced. $\square$