125

Capital investment under output demand and investment costambiguity*
Motoh Tsujimura

Faculty of Commerce, Doshisha University

1

Introduction

Today, the business environment is more complex and uncertain than ever before, and managers
face numerous uncertainties including demand uncertainty and cost uncertainty. In addition tothese uncertainties, expenditure on capital investment is generally a sunk cost. This creates
the problem of when and how much firms should invest in capital in an effort to increase theirvalue. In this study, we investigate a firm’s capital investment problem under output demandand investment cost uncertainty and derive the optimal investment strategy.

Many researchers have investigated the impact of uncertainty and irreversibility on capital

investment problems. Hartman (1972), Able (1983), and Abel and Eberly (1994) investigated
how a firm’s capital investment is influenced by output price uncertainty. They showed that
output price uncertainty promotes capital investment. Meanwhile, real options analysis showsthat uncertainty postpones investment when the investment expenditure is a sunk cost, that is,capital investment is irreversible. See Dixit and Pindyck (1994) for more details on real options
analysis.

In this study, we consider the case in which the firm’s manager does not have sufficientconfidence to predict either future output demand or the price of capital. That is, the firm’s

manager faces output demand and investment cost ambiguity (Knightian uncertainty). See, forexample, Camerer and Weber (1992), Etner et. al. (2012) and Guidolin and Rinaldi (2013) for
further details on decision‐making under ambiguity. Nishimura and Ozaki (2007), Trojanowska
and Kort (2010), Wang (2010), and Thijssen (2011) examine the irreversible capital investment
problem under ambiguity using the framework of \kappa ‐ignorance developed by Chen and Epstein(2002), in which the ambiguity only affects the drift terms of the associated processes. On theother handConversely, we adopt the framework developed by Kast and Lapied (2010) and Kast
et al. (2014), in which the ambiguity affects both the drift terms and diffusion terms of theassociated processes. The ambiguities of output demand and the price of capital are expressedby Choquet‐Brownian motions. Tsujimura (2017) investigated a firm’s capital expansion andreduction problem when output demand is ambiguous. However, he assumed that the price ofcapital was constant. We only consider capital expansion in an ambiguous business environment,
but we extend Tsujimura’s model by incorporating investment cost ambiguity.

In general, if two random variables are incorporated into models, they are usually solved byeither reducing the dimension or employing numerical analysis. McDonald and Siegel (1986)*

This research was supported in part by a Grant‐in‐Aid for Scientific Research (No. 18K01714 ) from the Japan
Society for the Promotion of Science and the Research Institute for Mathematical Sciences, a Joint Usage/ResearchCenter located in Kyoto University.

126
investigated an irreversible project investment problem in which the project value and invest‐
ment cost were both random variables by reducing the dimension by changing one variable andassuming homogeneity of degree one in the boundary condition. Adkins and Paxson (2011)
relaxed the homogeneity and derived an implicit analytical solution, while Stokey (2009, Section
11.5) examined an irreversible investment problem in which the demand and price of capitalwere both random variables and derived an explicit solution without assuming homogeneity.

This study explores a firm’s capital investment problem when future output demand is
ambiguous. To this end, we formulate the firm’s problem as a singular stochastic control problem.
In this study, output demand ambiguity is expressed by a Choquet‐Brownian motion developedby Kast and Lapied (2010) and Kast et al. (2014), whereas the previous studies mentioned above
adopted the framework of \kappa ‐ignorance developed by Chen and Epstein (2002) to incorporate
ambiguity. We solve the firm’s problem by using variational inequalities to derive the optimal

investment strategy, as in Stokey (2009, Section 11.5). This is described by a threshold that
determines the amount of capital expansion. Furthermore, we conduct a comparative static

analysis of some of the parameters. As a result of this analysis, we find that if the firm’s manager
is more ambiguity‐averse, investment is delayed. Furthermore, an increase in the volatility ofthe output price and the price of capital also delays investment.

The rest of the paper is organized as follows. Section 2 describes the setup of the firm’s
problem. Section 3 presents a solution to the firm’s problem. The numerical analysis is presented
in Section 4, and Section 5 concludes.

2

The Model

We consider a firm’s capital investment problem. The firm produces a single output using capital
K and sells it in a competitive market.

If the output demand increases sufficiently, the firm must

expand its capital to meet the increased demand. In this study, we assume that output demand

and the price of capital are uncertain.

In this context, the firm’s manager is not confident

of predicting either future output demand or the price of capital. In other words, the firm’s

manager faces ambiguity. We adopt the framework developed by Kast and Lapied (2010) and
Kast et al. (2014) to cope with this ambiguity.
Let X_{t} be the output demand at time t . The dynamics of the output demand is governedby the following stochastic differential equation:

dX_{t}=\mu_{X}X_{t}dt+\sigma_{X}X_{t}dW_{t}^{X,c}, X_{0-}=x>0 ,where \mu x>0 and \sigma_{X}>0 are constants.
m=2c-1 and variance(2.1)

W_{t}^{X,c} is a generalized Wiener process with means^{2}=4c(1-c) :dW_{t}^{X,c}=mdt+sdW_{t}^{X} ,where W_{t}^{X} is a standard Brownian motion.

(2.2)

c(0<c<1) is the constant conditional Choquet

capacity, which indicates the firm manager’s attitude toward ambiguity. Then, output demandfollows a Choquet‐Brownian motion as follows:

dX_{t}=(\mu x+m\sigma_{X})X_{t}dt+s\sigma_{X}X_{t}dW_{t}^{X}, X_{0^{-}}=x>0 .

(2.3)

127
If c< \frac{1}{2} (resp. c> \frac{1}{2} ), the firm manager is ambiguity‐averse (resp. ambiguity‐lovingembracing).
If c= \frac{1}{2} , thenm=0 ands=1 .

This means that dW_{t}^{X,c}=dW_{t}^{X} , ambiguity disappears, andthe firm manager has perfect confidence in the dynamics of the output demand.
The firm expands its business by accumulating capital at price P . The dynamics of the priceof capital is governed by:

dP_{t}=\mu pP_{t}dt+\sigma pP_{t}dW_{t}^{P,c}, P_{0-}=p>0 ,where \mu_{P}>0 and \sigma_{P}>0 are constants.
m=2c-1 and variance

W_{t}^{P,c} is a generalized Wiener process with means^{2}=4c(1-c) :dW_{t}^{P,c}=mdt+sdW_{t}^{P} ,
where

(2.4)

(2.5)

E[dW_{t}^{X}dW_{t}^{P}]=\rho dt with \rho\in[-1,1] . Then, the dynamics of the price of capital isrewritten as:dP_{t}=(\mu_{P}+m\sigma_{P})P_{t}dt+s\sigma_{P}X_{t}dW_{t}^{P}, P_{0-}=p>0 .

(2.6)

Let I_{t} be the cumulative expansion of capital until time t . This is right‐continuous withleft‐hand limited adapted processes, nonnegative, and nondecreasing, with I_{0-}=0 . Then, thedynamics of the price of capital is given by:dK_{t}=-\delta K_{t}dt+dI_{t}, K_{0-}=k>0 ,(2.7)

where \delta\in(0,1) is a constant depreciation rate.

The firm’s operating profit \hat{\pi} at time t is given by:\hat{\pi}(K_{t}, X_{t})=K_{t}^{\alpha}X_{t}^{\beta} ,(2.8)

where \alpha\in(0,1) and \beta>0 . The firm’s expected discounted net profit J(k, x,p;I) is given by:\hat{J}(k, x,p;I)=E[\int_{0}^{\infty}e^{-rt}\hat{\pi}(K_{t}, X_{t})dt-\int_{0} ^{\infty}e^{-rt}P_{t}dI_{t}] ,wherer>0 is the discount rate and I(2.9):=\{I_{t}\}_{t\geq 0} is the investment strategy. The investment

strategy is admissible when I\in \mathcal{A} , where \mathcal{A} is the set of all admissible investment strategies. In
this context, it is assumed that:

E[\int_{0}^{\infty}e^{-rt}\hat{\pi}(K_{t}, X_{t})dt]<\infty ,(2.10)

E[\int_{0}^{\infty}e^{-rt}dI_{t}]<\infty .

(2.11)

and

Therefore, the firm’s problem is to maximize the expected discounted net profit over \mathcal{A} :\hat{V}(k, x,p)=\sup_{I\in A}\hat{J}(k, x,p;I)=\hat{J}(k, x,p;I^{*}) ,where \hat{V} is the value function and I^{*} is the optimal investment strategy.

(2.12)

128
3

Variational Inequalities

The firm’s investment problem (2.12) is formulated as a singular stochastic control problem.
Then, we conjecture that the firm maintains its capital stock level within a given region, so thatwhenever the capital stock is below a certain level, the firm invests in additional capital. Notethat the boundary of the capital stock region depends on the level of demand and the price ofcapital. To verify this conjecture regarding the investment strategy, we solve the firm’s problem(2.12) using variational inequalities.

The variational inequalities of the firm’s problem (2.12) are given as follows.
Definition 3.1 (Variational Inequalities) The following relationships are called the varia‐
tional inequalities in the firm’s problem (2.12):\hat{\mathcal{L}}\hat{V}(k, x,p)+\hat{\pi}(k, x,p)\leq 0 ,(3.1)\hat{V}_{K}(k, x,p)\leq p ,(3.2)[\hat{\mathcal{L}}\hat{V}(k, x,p)+\hat{\pi}(k, x,p)][\hat{V}_{K}(k, x,p)-p]=0 ,(3.3)

where \hat{\mathcal{L} is the operator, defined by:\hat{\mathcal{L} :=-\delta k\frac{\partial}{\partial K}+(\mu x+m\sigma_{X}) x\frac{\partial}{\partial X}+(\mu p+m\sigma p)p\frac{\partial}{\partial P}+ \frac{1}{2}s^{2}\sigma_{X}^{2}x^{2}\frac{\partial^{2} {\partial X^{2} +s^{2} \sigma_{X}\sigma p\rho xp\frac{\partial^{2} {\partial X\partial P}+\frac{1}{2}s^ {2}\sigma_{P}^{2}p^{2}\frac{\partial^{2} {\partial P^{2} -r.

The variational inequalities are summarized as:\max\{\hat{\mathcal{L}}\hat{V}(k, x,p)+\hat{\pi}(k, x,p),\hat{V}_{K}(k, x,p)-p \}=0 .

(3.4)

Let \hat{\mathcal{H} be the continuation region, given by:\hat{\mathcal{H}}:=\{(k, x,p);\hat{V}_{K}(k, x,p)<p\} .

(3.5)

For analytical tractability, we assume that \beta=1-\alpha and the change variables are Y_{t}:=

K_{t}/X_{t} , as in Tsujimura (2017). Then, the profit function and the value function, respectively,
can be rewritten as follows:\hat{\pi}(K_{t}, X_{t})=K_{t}^{\alpha}X_{t}^{1-\alpha}=Y_{t}^{\alpha}X_{t}= \pi(Y_{t})X_{t} .

(3.6)\hat{V}(k, x,p)=x\hat{V}(\frac{k}{x}, 1,p)=xV(y,p) .

(3.7)

It follows from (3.7) that we have \hat{V}_{K}(k, x,p)=V_{Y}(y,p),\hat{V}_{X}(k, x, p)=V(y,p)-yV_{Y}(y,p) ,\hat{V}_{XX}(k, x,p)=(y^{2}/x)V_{YY}(y,p),\hat{V}_{P}(k, x,p)=xVp(y,p),\hat{V}_ {PP}(k, x,p)=xVpp(y,p) , and \hat{V}_{XP}(k, x,p)=Vp(y,p)-yV_{Yp}(y,p) . The variational inequalities (3.1)-(3.3) can also be rewritten as:\mathcal{L}V(y,p)+\pi(y)\leq 0 ,(3.8)

V_{Y}(y,p)\leq p,

(3.9)

129
[\mathcal{L}V(y,p)+\pi(y)][V_{Y}(y,p)-p]=0 ,(3.10)

where \mathcal{L} is the operator defined by:\mathcal{L}:=-(\delta+\mu x+m\sigma_{X})y\frac{\partial}{\partial Y}+(\mu_{P}+ m\sigma_{P}+s^{2}\sigma_{X}\sigma p\rho)p\frac{\partial}{\partial P}+ \frac{1}{2}s^{2}\sigma_{X}^{2}y^{2}\frac{\partial^{2} {\partial Y^{2} -s^{2} \sigma_{X}\sigma p\rho py\frac{\partial}{\partial YP}+\frac{1}{2}s^{2}\sigma_{P} ^{2}p^{2}\frac{\partial^{2} {\partial P^{2} -(r-\mu x-m\sigma_{X})(3.11).

The continuation region (3.5) can be rewritten as:\mathcal{H} :=\{y;V_{Y}(y,p)<p\} .

(3.12)For y\in \mathcal{H} , the variational inequalities (3.8)-(3.10) lead to the following ordinary differential
equation:\mathcal{L}V(y,p)+\pi(y)=0 .

(3.13)Let \phi(y,p)=Ay^{\gamma}p^{\eta} be a candidate function of the homogeneous part of (3.13). If \beta+\gamma=1,
then \phi is homogeneous of degree one. In general, the homogeneity does not necessarily satisfy(Adkins and Paxson, 2011). Then, the general solution of the homogeneous part of (3.13) isgiven by:\phi^{H}(y,p)=A_{1}y^{\gamma_{1}}p^{\eta_{1}}+A_{2}y^{\gamma_{2}}p^{\eta_{2}}+ A_{3}y^{\gamma_{3}}p^{\eta_{3}}+A_{4}y^{\gamma_{4}}p^{\eta_{4}}, y\in \mathcal{H} ,(3.14)

where A_{1}-A_{4} are constants to be determined. \gamma_{1}-\gamma_{4} and \eta_{1}-\eta_{4} are the solutions to the following
characteristic equation:\frac{1}{2}s^{2}\sigma_{X}^{2}\gamma(\gamma-1)-s^{2}\sigma_{X}\sigma_{P} \rho\gamma\eta+\frac{1}{2}s^{2}\sigma_{P}^{2}\eta(\eta-1)-(\mu x+m\sigma_{X}+ \delta)\gamma+(\mu_{P}+m\sigma_{P}+s^{2}\sigma_{X}\sigma p\rho)\eta-(r-\mu x-m\sigma_{X})=0.

(3.15)The discriminant of the characteristic equation (3.15) is:\frac{1}{2}s^{2}\sigma_{X}^{2}\frac{1}{2}s^{2}\sigma_{P}^{2}-\frac{-s^{2} \sigma_{X}\rho}{4}>0.
The characteristic equation (3.15) is the elliptic equation (see Figure 1). \{\gamma_{1}, \eta_{1}\} with \gamma_{1}\geq 0,\eta_{1}\geq 0 are the solutions of the first quadrant, \{\gamma_{2}, \eta_{2}\} with \gamma_{2}\underline{\supset}0, \eta_{2}\leq 0 are the solutions of
the second quadrant, \{\gamma_{3}, \eta_{3}\} with \gamma_{3}\leq 0, \eta_{3}\leq 0 are the solutions of the third quadrant, and\{\gamma_{4}, \eta_{4}\} with \gamma_{4}\leq 0, \eta_{4}\geq 0 are the solutions of the fourth quadrant.
The particular solution of (3.13) is calculated as:\phi^{P}(y,p)=By^{\alpha} ,(3.16)

where B :=(r- \mu x-m\sigma_{X})+(\delta+\mu x+ma_{X})\alpha-\frac{1}{2}s^{2}\sigma_{X} ^{2}\alpha(\alpha-1) . It follows from assumption(2.10) that

B>0 .

The general solution of (3.13) is:\phi(y,p)=\phi^{H}(p, y)+\phi^{P}(y,p) , y\in \mathcal{H} .

(3.17)

It follows from the definition of the firm’s problem that the function \phi satisfies the following
inequality:\phi(y)>By' .

(3.18)

130

Figure 1: Graph of the characteristic equation (3.15).

The general solution of the homogeneous part of (3.13) \phi^{H} represents the option value toinvest in capital. This implies that the constant to determine A_{i} must be nonnegative.

Suppose that the output demand X decreases to 0 , or the level of capital K goes to \infty , i.e., ygoes to \infty . Then, there is no economic justification for expanding capital at any price of capitalp:yarrow\infty 1\dot{ \imath} m\phi^{H}(y,p)ar ow 0, \foral p.
This implies that \gamma<0 . Next, suppose that the price of capital P goes to \infty . Then, there is no
economic justification for expanding capital for any y :\lim_{parrow\infty}\phi^{H}(y,p)arrow 0, \forall y.
This means that \eta<0 . Then, the homogeneous solution (3.14) is:\phi^{H}(y,p)=A_{3}y^{\gamma_{3}}p^{\eta_{3}} .

(3.19)

Thus, we obtain the following form for \phi :\phi(y,p)=\{ begin{ar y}{l \psi(\underline{y}(p), -p(\underline{y}(p)-y , y\leq\underline{y}(p), \psi(y,p):=A_{3}y^{\gam a_{3}p^{\eta_{3}+By^{\alpha}, y>\underline{y}(p). \end{ar y}(3.20)

The four unknowns A_{3}, \underline{y}(p), \gamma_{3} , and \eta_{3} are determined by the following simultaneous equa‐tions:\psi_{Y}(\underline{y}(p),p)=p, \forall p ,(3.21)\psi_{YY}(\underline{y}(p),p)=0, \forall p .

(3.22)

Equation (3.21) is the smooth‐pasting condition and equation (3.22) is the super‐contact condi‐tion (see Dumas (1991) for details).
It follows from (3.21) and (3.22) that we have:\underline{y}(p)=[\frac{\gam a_{3}-1}{(\gam a_{3}-\alpha)\alpha}\frac{1}{B}]^{ \frac{1}{\alpha-1}p^{\frac{1}{\alpha-1},

A_{3}= \frac{1}{\gamma_{3} p^{-\eta_{3} [p\underline{y}(p)^{1-\gamma_{3} - \alpha B\underline{y}(p)^{\alpha-\gamma_{3} ] .

(3.23)(3.24)

131
131
Substituting (3.23) into (3.21) yields:\gam a_{3}A_{3}[\frac{\gam a_{3}-1{(\gam a_{3}-\alpha)\alpha}\frac{1}B}] ^{\frac{\gam a_{3}-1{\alpha-1}p^{\eta_{3}+\frac{\gam a_{3}-1{\alpha-1}+\frac {\alpha-1}{\gam a_{3}-\alpha}p=0.

(3.25)To Satisfying the equality in hold (3.25) gives the following equations:\eta_{3}+\frac{\gamma_{3}-1}{\alpha-1}=1 ,(3.26)\gam a_{3}A_{3}[\frac{\gam a_{3}-1{(\gam a_{3}-\alpha)\alpha}\frac{1}B] ^{\frac{\gam a_{3}-1{\alpha-1}=\frac{\alpha-1}{\gam a_{3}-\alpha}.

(3.27)

It follows from (3.26) that we obtain:\eta_{3}=\frac{\alpha-\gamma_{3} {\alpha-1} .
4(3.28)

Numerical Analysis

In this section, we calculate the four unknowns, A_{3}, \underline{y}(p), \gamma_{3} , and \eta_{3} , and investigate the effects
of changes in the parameters on the threshold, \underline{y}(p) . The basic parameter values are set out
as follows:

r=0.05, \delta=0.1, \mu x=0.01, \sigma_{X}=0.2, \mu_{P}=0.01, \sigma_{P}=0.2, \rho=0.3, \alpha=0.4,

p=10 , and c=0.4 . Then, we obtain A_{3}=14.0086,\underline{y}(p)=0.058602, \gamma_{3}=-0.232449 , and\eta_{3}=-1.05408.

The results of the comparative static analysis of the threshold, \underline{y}(p) , are shown in Figures
3‐7. Note that the blue circle represents the result of comparative static analysis when the firm’s
manager is ambiguity‐averse (c=0.4) , while the red cross represents the results of comparative
static analysis when the firm’s manager is ambiguity‐neutral (c=0.5) in Figures 3‐7.
Figure 2 shows how investment decision‐making is influenced by the manager’s attitude
toward ambiguity,c. Figure 2 shows that the capital investment threshold, \underline{y}(p) , is increasing

with . This means that if the firm’s manager is more ambiguity‐averse, capital investment iscdelayed. An ambiguity‐averse manager tends to hesitate to invest in capital in an ambiguous
business environment.
\underline{\ve }(p)
0
0
0
0
0
0

Figure 2: The effect of changing the ambiguity attitude parameter, c , on the threshold \underline{y}(p) .
Figure 3 shows the impact of volatility of both output demand, \sigma_{X} , and the price of capital,\sigma_{P} .

If the firm’s manager is ambiguity‐averse (c=0.4) , the threshold \underline{y}(p) is decreasing in132both volatilities. This means that investment is delayed when either volatility increases. Thisis consistent with the standard result of real options analysis. Conversely, if the firm’s manager(c=0.5) , the threshold \underline{y}(p) is initially increasing in both volatilities.
However, beyond a certain level in each volatility (about \sigma_{X}=0.091 and \sigma_{P}=0.037 ), theis ambiguity‐neutral

thresholds start to decrease.

\underline{y}1p) \underline{y}(p)
0
0

0
0
0
0

0

0

0

ec=04 xc=0.5\bullet cM.4(a) \sigma_{X}

x

ct.5

(b) \sigma_{P}Figure 3: The effect of changing the volatility of output demand and capital price, \sigma_{X} and \sigma_{P},on the threshold\underline{y}(p) .

Figure 4 shows the impact of the correlation between the two Brownian motions, W_{t}^{X} and

W_{t}^{P} , on investment decision‐making. The investment threshold \underline{y}(p) is increasing with thecorrelation coefficient \rho.

\underline{v}lp)\bullet c=0.4 xc=05

Figure 4: The effect of changing the correlation coefficient between W_{t}^{X} and W_{t}^{P}, \rho , on thethreshold \underline{y}(p) .

Figure 5 shows how investment decision‐making is influenced by the expected rate of changeof both the output demand and the price of capital, \mu_{X} and \mu_{P} . The investment threshold \underline{y}(p)is increasing in both the expected rates of change. This implies that investment is hastened

133
when the expected rate of change of the output demand and the price of capital is increasing.

\underline{v}(p)\bullet c=0.4 xc=0.55(a) \mu_{X}\bullet cM.4 xc=0.5(b) \mu_{P}Figure 5: The effect of changing the drift coefficient of output demand and the price of capital,\mu_{X} and \mu_{P} , on the threshold\underline{y}(p) .

Figure 6 shows the impact of the output elasticity of capital, \alpha , on investment decision‐
making. Recall that output per unit of capital increases in \alpha . Figure 6 shows that the threshold\underline{y}(p) is initially increasing with the output elasticity of capital. However, beyond a certain level(about \alpha=0.3586 if c=0.4 , and \alpha=0.3759 if c=0.5 ), the threshold starts to decrease.
\underline{v}lp)
0

0

0

0

\bullet c=04 Kc=05

Figure 6: The effect of changing the output elasticity of capital, \alpha , on the threshold \underline{y}(p) .
Figure 7 shows how the price of capital affects investment decision‐making. Figure 7 showsthat the threshold \underline{y}(p) decreases as the price of capital decreases. This implies that a higher
price of capital delays investment.
These results provide useful insights into investment decision‐making when output demand andthe price of capital are ambiguous.

134
when the expected rate of change of the output demand and the price of capital is increasing.

\underline{v}(p)\bullet c=0.4 xc=0.55(a) \mu_{X}\bullet cM.4 xc=0.5(b) \mu_{P}Figure 5: The effect of changing the drift coefficient of output demand and the price of capital,\mu_{X} and \mu_{P} , on the threshold\underline{y}(p) .

Figure 6 shows the impact of the output elasticity of capital, \alpha , on investment decision‐
making. Recall that output per unit of capital increases in \alpha . Figure 6 shows that the threshold\underline{y}(p) is initially increasing with the output elasticity of capital. However, beyond a certain level(about \alpha=0.3586 if c=0.4 , and \alpha=0.3759 if c=0.5 ), the threshold starts to decrease.
\underline{v}lp)
0

0

0

0

\bullet c=04 Kc=05

Figure 6: The effect of changing the output elasticity of capital, \alpha , on the threshold \underline{y}(p) .
Figure 7 shows how the price of capital affects investment decision‐making. Figure 7 showsthat the threshold \underline{y}(p) decreases as the price of capital decreases. This implies that a higher
price of capital delays investment.
These results provide useful insights into investment decision‐making when output demand andthe price of capital are ambiguous.

135
Camerer, C. and M. Weber, (1992). Recent developments in modeling preferences: uncertainty
and ambiguity, Journal of Risk and Uncertainty, 5(4), 325‐370.
Chen, Z. and L. Epstein, (2002). Ambiguity, risk, and asset returns in continuous time, Econo‐

metrica, 70(4), 1403‐1443.
Dixit, A. and R. S. Pindyck, (1994). Investment under uncertainty, Princeton University Press,
New Jersey, USA.

Dumas, B., (1991). Super contact and related optimality conditions, Journal of Economic Dy‐
namics and Control 15(4), 675‐685.
Etner, J., M. Jeleva, and J.‐M. Tallon, (2012). Decision theory under ambiguity, Journal of
Economic Surveys, 26(2), 234‐270.
Guidolin, M. and F. Rinaldi, (2013). Ambiguity in asset pricing and portfolio choice: A reviewof the literature, Theory Decision, 74(2), 183‐217.
Hartman, R., (1972). The effects of price and cost uncertainty on investment, Journal of Eco‐nomic Theory, 5(2), 258‐266.
Kast, R. and A. Lapied, (2010). Dynamically consistent Choquet random walk and real invest‐
ments, Working Paper LAMETA, DR 2010‐21.

Kast, R., A. Lapied, and D. Roubaud, (2014). Modeling under ambiguity with dynamicallyconsistent Choquet random walks and Choquet‐Brownian motions, Economic Modelling, 38,
495‐503.

McDonald, R., and D. Siegel, (1986). The value of waiting to invest. The Quarterly Journal of
Economics, 101(4), 707‐727.
Nishimura, K. G. and H. Ozaki., (2007). Irreversible investment and Knightian uncertainty,
Journal of Economic Theory, 136(1), 668‐694.

Stokey, N. L. (2009). The Economics of Inaction: Stochastic Control models with fixed costs.
Princeton University Press.

Thijssen, J. J., (2011) Incomplete markets, ambiguity, and irreversible investment, Journal of
Economic Dynamics and Control 35(6), 909‐921.
Trojanowska, M. and P. M. Kort, (2010). The worst case for real options, Journal of optimization
Theory and Applications, 146(3), 709‐734.
Tsujimura, M., (2017) Partially Reversible Capital Investment under Demand Ambiguity. forth‐coming in RIMS Kôkyûroku.

Wang, Z., (2010). Irreversible investment of the risk‐ and uncertainty‐averse DM under k‐ignorance: The role of BSDE, Annals of Economics and Finance, 11(2), 313‐335.

136

Faculty of Commerce, Doshisha University
Kamigyo‐ku, Kyoto, 602‐8580 JapanE‐mail address: mtsujimu@mail.doshisha.ac.jp