The EttЭ cts of Uncertainty and lrreversibility
on lnvestnlent

Koichi Mashiyama
Departinent of Economics

Metti Gakuin University
09‑01

The Efects of Uncertaillty and lrreversibility on lnvestlrlent
Koichi ⅣIashiyama
Department of Econonlics

Metti Gakuin University
Discussion Papers No.09‑01
NIarch,2009: prelillninary version

Abstract
The present paper exaIIunes the robustness of the conclusion derived in the canonical inodel ofinvestment under uncertainty and explore the relationship between uncertainty and investment. We
hether or
have arrived upon the conclusion that, for three dinbrent cases of the demand shock, 1″
not an increase in uncertainty depresses investment depends on the concavity of the operating pront
function with respect to the demand shock However even ifthe operating proflt function is convex invould depress
the demand shock,an increase in uncertainty may not necessarily raise investment and、
investment,depending on the range of lnodel parameters This result implies that the conve対
ty of
the operating pront function need not necessarily to be disnlissed in order to be consistent、vith theemprirical validity

l lntroduction
The efFects of uncertainty on investment have long been extensively exp10red in the literature on theinvestment theory. Taken as whole, di3erent theOries emphasize different channels,some pointing to a
positive relationship and others to a negative relationship.We can classlfy theories of investmelllt under
uncertainty into two strands of theOry.One focuses on the adiustmellt cost and the other emphasizes theirreversibility of investment.
In the models of Hartman(1972),Abel(1983,85),Abel and Eberly(1994),the marginal revenue productrhenヽヽ
of capital is convex and it is the flexibility of labor relative to capital that generates this convexity.
preserving
increases
the Operating proflts are convex in the shock against the rnarket demand,any IInean―in the distribution of the shock raises investment of the nrm.on the other hand,Caballero(1991)
and Pindyck(1993)pointed out that the relationship between uncertaillty and investment should nOt be
expected from the adiustment costs literature alone and when the nrm faces an elastic demand curve inan imperfectly competitie market,the negative relation between investment and uncertaillty is likely tobe reversed l.
In the canonical models of investment、
vith irreversibility based on the real options approach, asshown in Bernanke(1983),McDonald and Siegel(1986),PindyCk(1988),Bert01a and Caballeo(1994),and
Dixit and Pindyck(1994),an inCrease in the volatility of price nucuations leads to a rise of trigger point
to invest and then decelerates investment. In this literature, uncertaillty afFects irreversible investment
in two M″ays: flrst,through the ettects of the risk prellnium component on the marginal profltability ofcapital,and second,through the e∬ ects on the tigger threshold of the value of waiting 2.
Those models mentioned above predict a positive or negative efFect of uncertainty on investment
depending on whether the marginal revenue product of capital is a convex or concave function of thelSee abo Caballero and Pindyck(1996),Abel and Eberly(1996,1999).
2 R e c e t t l y t h i s n e g a t ir e l a t i o n s h i p b e t w e e n u n c e r t a i n t y a nsdt mienn t i s q u e s t i o n e d S a r k a r ( 2 0 0 0 ) , G r y g l e w i c z e t
al(2008)andヽ VOng(2007)have Suggested that an increase in the trigger threshold may not induce a delay in the timmingof investment,i.e. the investment‐uncertainty relationship is not neccessarily monotonic.

exogenous shock, In the case of convexity,an.increase in the variance of the shock、vould raise investment
via Jensent inequality.Leahy and Whited(1996)showed in their empirical studies that uncertainty exerts
a strong negative inluence on investment and that uncertainty a∬ ects investment directly rather thanwOrking through covariances. :rhis result casts doubt on the importnace of theories that emphasize theAbel rnodel.
convexity of the marginal revenue product of capital such as a Hartman―
The present paper exallnines the robustness of the conclusion derived in the canonical rnOdel ofinvest―
ment under uncertainty and explore the relationship between uncertainty and investment. A canOnicalmodel of investment under uncertainty is formulated in Section 2. The section 3 is devoted to analyzing
石ヽ
e explore the invest―
veen uncertaillty and investment in this canonical model. ヽ
the relationship bet、vhen thement behavior、 Ⅳith three different dynamical systems gOverning the shock processes: flrst,、
demand shock is governed by the fallliliar geometric Brownian mOtion;second,、vhen it is governed by avn―Poisson process.
vhen it obeys a geometric Bro、
mean―reverting stochastic process,and third,、

2

A Canonical Model of lnvestrrlent under uncertainty

WVe consider a nrln that produces output using capital K and variable factors of production. It earns theoperating proflt that depends on the random variable X. The operating proflt at tillne t is denoted byitュ
π(K(ι),X(ι))・K(ι)iS the capital stock at time t,and X(t)iS an exogenous shock to the product市
factor prices,or the demand for products.
石ving Cobb―Douglasヽヽe consider the standard case that the production function is given by the follo、
production functiOn:
ν= F ( Z , K ) = L α

K l α, 0 < α

<1.

(1)

The demand curve faced by the nrln is assumed to be given by)/φP(ιx(ι)9,φ≧1,9>0,)=0(ι )(1 φ(2)where,P(t)iS the market price at time t,o(ι)iS the aggregate quantity of product supplied in thee,we set φ=1・ When 9=1,the demandmarket.When the product markt is perfectly competit市
shock is proportional to the market price, so that the demand shock is interpreted as a shock totmarket price.Assume that o=Ⅳ ν(Ⅳ =l iS the number of Arms),and then the operating pro■π(κ,X)=P(t)五 αKl α ―υL(ι)is giVen byK(ι)β,
π(K(ι),X(ι))=ん χ (ι)ν
where,υ

0)is the wage rate andにん= に一か囁瀞り つ,ν=ψφ >0,

β = :モ =≦ 1・
Here,it is clear that πκ >0,7rKK≦ 0,民x>0. The operating proflt is a concave function of capital stockアχ. ヽ
hen νく 1,the proflt
ヽ必
ν>1,the pront function is a convex function of the exogenOus shock̀. Iffunction
is a concave function of the exogenous shock.ν
>l if and only if 9>1‑α /φ.ThiS fact implies
that the marginal revenue product of capital is a decreasing function Of capital but convex or concaveァ、、
hen the market isin the value of the exogenous shock depending on the value of the parameter,3.
perfetly competitive,the proflt function depends linearly on the capital stock K. On the other hand,when the market is ilnperfetly cOInpetitive,the pront function is strictly concave in the capital stock κ .
3ン =l iS assumed in deriving theL conclusion in Abel(1983),Abel and Eberly(1994),and Caballero and Pindyck(1996)
π in K and X is presupposed in the model of Abel and
The speciflcation that ν<l coupled with the linear homogeneity of
ν>1
Eberly(1996,199)・ CaballerO(1991)used the presupposition that石ヽヽe utilize three diferent stochastic di∬ erential equations for the exogenous shock processes: thefallniliar geometric Bro、
vnian motion,a rnean― reverting stochastic prOcess,and a geometric Bro、 vn̲POissonprocess.
The geometric Brownian case. The random variable X is governed by the following stochatic di3erential
equation:Z,αχ (ιι+σ (X(ι))α)=μ (χ(ι))α

解)nian motion
nian IInotion. The randorn variable X obeys the geometric BroM″、
vhere z is the standard BroM″
if the drift and vairance coefncients are expressed byμ(X)=μX,σ (X)=σX,

er̲ving mean―reverting di∬
The llnean―
reverting case. The randoln、rariable X is governed by the follo、
ential equation:αχ (ι)=ι (μ― X(ι))読 +σ X(ι)αZ(ι),(5)

term demand level,and
σ is the volatility of the prOcess.
where,ι is the speed Of reversion,μ is the long―
The geometric Brown― Poisson case.The random variable X is governed by the folloving geometric
Brown― Poisson difFerential equation:(0

こχ =(μ ― λん)X(ι)αιtt σχ (ι)αZ(ι)十 たX(t)αⅣ (t),where Ar is the Poisson process, λ is the arrival rate or intensityrate of Poisson process, andんis the
expected amplitude of size on jump.
The capital stock gro、vs according to
(7)

ακ (ι
ι
,
)=(I(ι )一 δK(ι))α

where,I(ιis t,δ
the rate of depreciation.In order to build a new capi
)iS the investment at time
stOck,the adiustment cost is inquired.The direct cost of investmellt is composed of theprice of capital goods and the installing/detaChing costs.The adiustment costs of invest
training M/orkers and expanding the operating capacity to manage and operate the plant and machines atthe larger scale. The nxed cost is independent of the amount ofinvestment but this cOst may be linearlyhOmOgeneous of investment and capital if it renects stopping the operation and/or Starting up the plantcOnditions in the efncient、vay and on the digerent scale. The grOss cost Of investment can be therefore
The gross investmelllt cost fllnction c(I,K)iS aSSumed strictly
expressed in the function form c(I,K)・
convex for r and continuous except for at the origin f=0. The right― hand derivative on the origin ishand derivative on that pOillt. That is,
assumed to be larger than the left――ん,K), ん>0)≧1鳴CIω
1嶋C10+ん,κThis assulnption illnplies that the sale of capital g00ds cannOt be accolnphshed at the same price as thetheir purchase and there are also installation costs,which are added to the purchase price but cannot berecovered on sale. There rnay be additional costs of detaching and lnoving to other places,and sumcieFltly
specialized machinery and plants may have little value to others.
WVe assume that the flrln is risk neutral or risk averse and maxllnizes the expected presellt value ofbyπ( K , X ) m i n u S C ( r , κ ) ・T h e p r e s e n t m l u e y o f t h e n r m i s gni ℃∞y(K(→,X← ))=野X Es嘱{ π( κO + →,XO+→) 一 C ば ← + ⇒ , κ← + → ) } θρt d 」

where, Es is the conditional expectation operator at time s, andρ > O is the discount rate that theve haveinvestors or stockhholders require. Using dynamic prograllnllning technique,、

yに
=Ψ{<氏
+れα
,均
ρ
η Cば
η
'η

0

鰻)

The left hand side is the required rate of retllrn and the right hand side is the maxirnized rate of returnon investment.The ma対zimed rate of return is composed of the grOss prolt π(X,X)一 C(I,K)and thecapital gain Eαy/at・ Applying the ltoヽ lemma,we have the Hamilto卜Jacobi equatiOn
4+干

PXIン

(S)y+π(κ ,X)一C(I,K)]=ρ

7(κ

,X),

w h e r e / i s t h e i n A n i t e s i m a l g e n e r a t o r ( s e C O n d o r d e r d i f F e r e n t i a lW hoepne rtahteo rs)t。
Ochastic difFer̲
ential equation(4)is used,the ininitesimal generator is given by 4/ ←) y = ( I 一δK ) y k + μ( X ) y x + : σ(χ)2yxx.
Since L=0,
maXレ (K,X)C(I,K)+ば

δK ) レ+ μ ( χ) y x + : σ(X)2yxx]=ρ 7(κ ,X),

(1のHere, ん denOtes the partial derivative ofノ by χ, ん。iS the secOnd Order partial derivative by χ. lrhe
*is the amOunt Of investment
marginal value 9 of the nrm is deined by 9=ン
缶 .Optimal investmelllt」
which maxilnizes the function:

缶 ―C ( I , K )ψ( I ; K , X ) ≡」レ
Optimal investment depends On g(=レ2)andκ.Therefore,we can express this relationship by f*=f*(9,K).
If the grOss investment cost function is ditterentiable,the flrst Order cOndition is given by
*,κ
CI(」
)=9.

It should be reminded that the function
not diferentiable at the origin )十and> c.(0,κ
の iS c(I,ス=0,opumd
iwestmellt
must
be
zero
as far as
C・
針
nCe
c(I,κ
nOt
dittrenthbb
at」①,K) 。
)お(0,κ)+≧ 9≧ CI(0,κC・)iS Satisied.Hence optimal investment is characterized as follows:*,K)=9, When g>c.(0,κC ・( 」)+,

w h e n c I (の0十, ≧9ス ≧c I ( 0 , K ) ,I*=0,

C ・( I * , K ) = 9 , W h e n c I ( 0 , K ) > g

When 9>cf(0,ス0+,the gross investment is positive but when g<cI(0,K) ,the gross investment isnegative,that is,the old machinery and equipments are on sale, If the investment cost function is kinked
at the Origin,the investment policy is the trigger strategy of Ss type.The ttlue gu=cI(0,κ)+iS theupper trigger point and 9ι =cI(0,K)iS the lower trigger point,Since the investment cost functiOn isassumed strictly cOnvex,in℃ stment is an increasing functiOn of 9 as 10ng as 9>cI(0,K)+・
In otherword,investment is positively related to the value of 9 but is not lnonotonically related.
Next, Nヽ derive the fOrmula describing the dynamics of the marginal value of the nrm. Taking thederitttive of eq,(10)with respectκ ,weto obtainπκ ( κ , X ) 一

CK(I*,κ

) 一 δα + 9 K ( I * ―δK ) + μ(χ ttk+:σ

(X)2牧

x=ρ

α.

(11)

It should be noted that g is a function 【of」
and X and so the functiOnal forlln canbe expressed as
g=9(t,X,X)・ Eq.(11)Can be expressed in the f0110wing fOrm:
必 気 ι) g t t

π K(K,X)一CK(I*,K)一

(ρ +δ

)9=0・

∞ +… cKl(→
抑Ⅷ
+朝
に 詢=馴
,聯
パ
倒
レ
ズケ

Using the Feynman―(12)

Kac theOrem5,vre have

(13)

4The ininitesimal generator for the stochastic ditterential equation(5)or(6)is similarly gi
n The analysis for thesecases、 ′ill be conducted in the next section
5 see the textbook on stochstic difFerential equations,fOr instance,Karatzas and Shreve(1991),P364‐365

(ρ+δ)T=0.Thus,9 is the present
where we utilized the reasonable condition that hmT→
∞ 9(T,κ ,X)θvhicll consists of the marginal revenue productvalue of the stream of expected marginal proflt Of capital、
πI Ofinvestment and the marginal investment cost cI.
We assume that the grOss adiustment cost ofinvestment can be formulated in the following form:
ｒ
ｉ
く

Ｊ

κ

ｌ
ｔ

lκ γ2, when f>0
αlK+bl」 +η lJγ0,(the grOSS Cost funabn ofiⅣestment)0,when I〓α2κ +ら 2f+η 21」γlκ γ2, when f<0

κ,α2スr are interpreted aswhere,αl,α2>0,bl>b2>0,γ2≧ 0・The flrst terlns a71」l,γ2≧ 1,and ηl,ηthe flxed cost inquired on investment which depends on caital stock but is independent ofthe amount ofv capital is installedvhile ne、
investment. This flxed cost may renect the cost of stopping and restated、「
rhe assumption that bl > b2 >O Captures a partial irreversibility ofor existing capital is removed 6。
the past investment. η2 =∞ COrresponds to the complete irreversibility. The third term captures the
attuStment cost which is commonly assumed linearly homOgeneous in the amount ofinvestment and theexisting amount of capital stock in the traditional literature. The functional fbrm described above is anextended version used by Abel and Eberly(1994).Whenγl+鯵 =1,the gross investment cost function iscapital ratiO
vhich implies that optillnal investment―
linearly homogeneous ofinvestment r and capital]で ,、depends only on the marginal value of capita1
Ъr. In 9=レthiS case,it can be shown that the marginal
=0,
if
the
operttingt
pro■
isa linear function of capital.When第equalstheⅣerage
9,7/κ
9=ン 安
石
e cannot analytically deriveヽCκ =COSntant,in which case the analytical form of 9 can be easily derived. ヽ
the characteristics of optimal investment rule in general and so we need to analyze the probem for two
後0 .
s e p a r a t e c a s e s : t h e c a e l i n αwl h= iαc h2 = 後 = 0 , a n d t h e c a s e H i n w h i c h≠

3

The relationship between uncertainty and investrrlent

石ヽe consider the case l flrst. The cost function 6f investment is given by
ヽ
一
一 ＜

０
０

＞

ｆ ｆ
ｎ ｎ

０

ｎ

ｆ

ｅ
ｈ
Ｗ

ｅ
ｈ
Ｗ

範

η

岡

ｋ

＋

ｌ

Ｊ
︐Ｌ

く

０

ｅ
ｈ
Ｗ

︐
ｌ

Ｆ

ｌ

η

＋

ｒ

Ｊ
﹄

Ｃ

Ｊ

Ｊ

κ

Then,cK(I,X)=0.Optimal investment is characterized by( I * )l γ1 = ( 9 ‑ b l ) /l(γlη) > 0 , W h e n g > b l ,I* =0,
when bl≧9 ≧b 2 ,(一I * )lγ1 = ̲ ( g ̲ ら2 ) / (2 ηγl)>0, When b2>9
Eq。(10)iS Simplined int。

―δKい μ
X " + : σ 2 x 2 政x = ρy に , 約.
:乙W ψ ば 均 + πに' 駒
f*≠Note that when f*=0,maxlψ(I:κ,X)=一C(0),and when 0,Ψ ψ「: 氏均=げ0=玲←⇒卜1 ( 7 μ(10

O r ⇒‑ 6 0 , に L a=2 if f<0. Since the marginal value of capitalン2 is in general a function Of
where t=l if r>O andを
c a p i t a l s t o c k K , m a xψl( I ) = 」* ンk― C(I*)depends on the leve1 0f capital stock.It is difncult to deri
the analytical solution Of the partial diSerential equation(14)unleSS the marginal value of capital yK is
independent of capital stock. In other wclrds,the operating proflt must be a linear functiOn of capital,、
vhich requires the perfect competition in the product llnarket. To obtain the analytical solution,we needto assume thatβ =1,6This presumption implies that the nrst terms should be expressed by

αlπ(X,K),α2π(X,K)Where O<αl,α2<13.l The Casel with
β=1
4e assume thatヽヽ,X)= that y(κβ〓 1,1.e.the product market is perfectly competitive.Suppose
ス(χ)K+B(X)・

Substituting this ftlnction into the ab∝e equation(14),we hNe

降X ν δ
バη + μχ〃的 + ン X η 賓均 ―ρバ到 K

XF的+ンX2Fて
m額
●
Ц
到=0
η̲ρ
」
lfl+μ

(1つ

This relationship must hold for arbitrary values of J(and so the value l″ithin each bracket llnust be zero.

Hence the folloM″
ing equations hold:んχ ν一δバ 駒 + μ χ〃 的

(10

X 物 貿豹 ―ρバ 均 = Q

+ン

maXα
χ
F因+ンX2FК
+μ
η̲ρ
Ц
均=Q
⊃

(1つ

The Feynman― Kac formula leads to the solutions

Ц 颯 羽 =島J∞

図JIStt aЦ
「 lp・
射 =島

げ

J∞

+ 朝 「 ″ふ

mⅨ α

(10

・

The marginal value of capital is given by
9=レ 缶 =ス (χ).
It is obvious that if the market is perfectly competitive,the marginal value of capital does not dependア
hen the market is imperfectlyヽon the existing level of capital stock.(Dur pressumption is supported.ヽ
btain an analytical expressiOn for the expected present ttlue y(K,X)
competitiК,it is very dimcult 。
t。
Of the flrlln.
To examine the efFects of uncertailllty on investment,we must specify the nature of the dynamic pathgoverning the state variable X. First lve assume that the state variable X obeys the geolnetric Brownian
motion. We can set the expected growth rate Of X tO be zero without the loss of generality Then wehave
島 [nXO+S)]=hχ2s,Lrt[nX←
C)一

:σ

+→

]=:σ

2s,

so that vre have

E t l l l l l XηC=+イ⇒h X O ―:れ

2 v a r tn口x(t+S)ンχ( ιV a r tn口+S)]・
1=νNoting th乱Z ‖X ← + →ν
]=eXpIEt hX←+→ν+:Vart{hX←ν
十→ H

we have―=XO%覇EtlXO+→
η
デ・
ンν

(1の

Therefore the marginal value of capita1 9 is given by
9(ι
)=yx=

んX(ι)ν
2・
ρ+δ ― ;ν(ν‑1)σThis formula is exactly the same result derived by Abel and Eberly(1994),whiCh induces the posi―tive relationship between uncertainty and investment identined in Hartman(1972),Abel(1982),and Ca̲ν =l if 9=1,that is,the shock is aν >l is presupposed.It is alwas true thatballerO(1991)Where(20)

price shock.When ν >1(i・e.α >φ (ヮー 1)),an increase in
σ increases the marginal ttlue of capital.
I n o t h e rv、
o r d s , i n v e s t m e n t i s n o t a d e c r e a s i n g f u nσc tfioorn gOifv eχn (ι).ThiS pOsitive relationshiporiginates in the conve対ty of marginal revenue products of capital with respect to χ (t)(interpreted as
the market prices in the present setting)・ On the other hand,ifν <1,an increase inσ decreses themarginal value of capital.In other words,investment is not an increasing function of
σ for given χ (ι)・This negative relationship originates in the concavity of rnarginal revenue prOducts of capital
ith respectM″ν<1).
tO χ (t)(under the condition
raluesIn conclusion,an incease in uncertainty raises
vell
as、
as decreases investment depdending on the、アOf 9・ψ =(φ ― α)/φ iS the critical value.When ψ <の ,an increase in uncertainty decelerates in、
estment
and othewise,it accelerates investment. There exists no monotonic relationship between uncertainty andinvestment in the canonical investment model with the convex attuStment cost.
For the model to be consistent with canonical real options models of investment such as developed
by Brennan and Schwartz(1985),McDonald and Siegel(1986),and Dixit and Pindyck(1994),the rate ofdiscount ρ used in the preceeding analysis above shOuld be questioned. In any real options models ofinvestment,the discount rate is utilized sO as to be consistent、vith the no―arbitrage condition in asset
markets.For instance,CAPⅣ
I says that
ρ= r + λ ρt れσ,
where,λ is the market rice of risk andveen the rate of return on theρtm is the correlation coemcient bet、ア
investment project and the rate of return on the market portfolio in the anancial
henasset market,ヽヽ
we use this relatiohship,
んX ( ι) ν

g(ι
)=7x=

(21)

r+λ ρtれσ+δ ― ;ν(ν‑1)σ 2・

It is easy to see that g is a decreasing function
σ ofifν<l but、 vhen ν>l and the follo、
ving inequalityholds
σ <δ =ν(ν̲1),.An increase in uncertainty of prOducts demand(priCe)χdecelerates
9 is a decreasing function σOf
4alue a but an increase in uncertaillty accelerates investment
investment if the v01atilityσ is less than the、
ifν>l and σ is greater than the valueδ. Since an increase in uncertainty raises the discount rate via therisk premiuln forllnula and an increase in the dicount rate leads to a decrease in the present value of future
expected proflts,otherthings being equal,it comes with little surprise that an increase in uncertalllty may
decrease the present investment.「 rhis efFect llnight be named the discounting effect. Thus,uncertaillty
may accelerates as、 vell as decelerates investment depending on the value of the v01atility of price fluctu―
ations faced the nrm. This cOnclusiOn is supprted by assunling that the market is perfectly competitive,
the production functiOn is linearly homogeneous in labOr and capital,and demand fluctuations obey thegeometric Brownian motion 7.
What happens when the diferent stOchastic processes are assumed for the dynamical systenl gOverning
the future pathes of the state variable X. Suppose that the state variable X obeys the folloM/ing mean―
reverting process:αX(ι )=ι (μ― X(ι ))dt+σ X(ι )αz(t),(22)

where,ι is the speed of reversion,μ is the long―term demand level,and
σ is the v01atility Of the prOcess.
z is the standard Brownian motion.The HamiltOn―
Jacobi equttion(14)can be rewritten illlto

干Fψ

ば :κ

'X)+π

(κ ,X)一

δK驚

+ι O一

Xltl)政+:σ

2x27xx=ρ

y(κ

,X)・

(2o

7Leahy and Whited(1996)deteCted the empirical evidence that the discount erect is not a major channel through which
uncertainty a ects investment. In the subsequent analysis,we will not explicitely discuss about this discount efFect

(χ)K+B(χAs before,supposing that y(K,X)=五)and utilizing the solving technique used above,wehave the same formula of the marginal value function:∞ 「AIXltl)=島
吼榊+ギい州・vn to bereverting stochastic equation for the state variable X is℃、
kⅥ
Note that the solution Of the rnean―
given by 8
諷 ι+ → = μ I 和

一め ど

+∝

れが
IS潤

囲

ズけ

The expected value and the variance of」 F are colnputed as follows:
島 区 01=μ

+lXO―

De

ち

耽 L区 ③ ]=∫;に

一 θ―% μり・

In the present setting of the mOdel the operating pro■ t functiOn is convex in the demand shock(1・ e.,ン>1)Or concⅣ ein χ (i.e.ν<1)・ First,to caputure the nature of convexittt we can assume that ν=2
to simplify our analysis.Then the formula for the marginal value of capital is simplined int。

alsと
lp・
銭区
い萌「
0=J∞

Since
+ S ) 2 1 = yartlχ(t+S)]+[島 X(ι+S)12,
ユ IX(ι
we obtain∂島 IX(ι+S)2]
∂σ

μ
「χ
り> Q
号に一

Therefore,

物〆°
→鼈
ん
S>0
平 =I∞
号。「⊃
1t is clearly seen that an increase σinraises the marginal value of capital and so accelerates investment
as 10ng as the discoullting efFect through the risk prerniunl component is ignored. This result comes fromthe convexity of the proflt fllnctionithv″respect to the value of the demand shock. The operating proflt
is cowex if and only if ψ >1‑α /φ ・When the product market is perお ctly competiti℃ ,this condition. A sumcient condition fOr the prolt function to be convex is that
reduces to the inequality: 9>1‑αve can clailln that thethe nuctuations of the market price are proportional to the demand shock. Hence、
convexity of the proflt function vrith respect to the demand shock plays the crucial role to assure thepositive relationship between the uncertaillty and investment.
PoissonNow we suppose that the state variable X is governed by the following geometric Brown―
process:αχ =(μ ― λん)X(t)洸 +σ X(ι)αZ(ι),)+ん X(ι)αⅣ (ιwhere Ar is the Poisson process,(25)

λ is the arrival rate or intensity rate of Poisson process, and た is the

Ve assume thatた>‑1.The time path of this stochoastic process
expected amplitude of size on jump.ヽ
is given by―スλ―σ2/2)ι
X ( t ) =X(0)eXp{(μ+ σz(t)}(た+1)N(t).
8see Musiela and Rutkowski(2007),Lemma 10 12

As well known,the inflnitesirnal generator for this stochastic difFerential equation is expressed by9/7(κ ,X)=稔

2x2+スκ)+yx櫻 一人OX+:yxxσ「 ―δ

where,7(X(ι ))一 y(x(t̲))=y((たbyly(Xltl)y(Xlt)月

,

y(X(ι
+1)χ (ι))一))・Eq。(14)for the Brown̲Poisson case is written

2x2
κい 玲ゆ―わ X + : 政x σΨ ψ「: 氏η + π ' η一δ+λ ly((ん+1)X(t))一
Supposing that y(κ
have

,X)=ス

y(x(ι

))]=ρ

y(K,X)・(X)κ +B(X),and substituting this function into the above equation,weνレχ 一 δバ η + し ― わ X 個」 mⅨ Ollfl+し ―わ X3〃 的

+ン
+ン

Xη

X2Bノ

ttη ‑0+乃

AIXl+乃̲0+乃Ц 均 十わЦ け的Ц 鮨 +llXllκ
llXll=0This relationship must hold for arbitrary values(andof』so the value v/ithin each bracket lnust be zero.
Hence the fo1lo、
ving equations hold:んXν δバ詢 十ゆ―渤 χ姻
mⅨ 州+ し ―わ X 3 〃 因( 鮨+ l l X l = Qス+ : ′X 2 ノ的 ̲ ぃ ゎA I X l + 乃+ン

X2Bノ 的̲0+わЦ 均 +秋o+llXl=aυ0(2つ=α Xν into the equation(26)giveS rise to
S u b s t i t u t i n g t h e f o( rX m) ス

バη =

+口

軒 明

・

Since the Poisson p■ocess has the variance λιduring tirne period t, the uncertaillty arising frolln the
demand shock driven by the Poisson process can be represented by the quantity λ. It is obvious that if
ν>1,the inequality:
1 + た ν< ( 1 + た ) νν>(1+た )νfOr any posit市e number λ.Therefore,
is satisned for any positive numberた .If ν<1,1+た
an increase λi nr a i s e s t h e t t l u e fo(f=‰t h e m a r g i n a l t t l u e o f c a p i t a 1 ν>9 ) 1i ,f a n d t h e n a c c e l e r e r a t e sinvestment. Other、 vise,an increase in tt decelerates investment. The uncertainty arising fronl the Poissonprocess also raises as M″
ell as depresses investment depending on the parameter value of 9,. It is clear that、vhen 9 is close to one,uncertainty arising fron■the Poisson process raises investment.
ve have arrived on the conclusion that,for three di∬erent cases ofthe demand shock,、 vhether
In sun■,、or not an increase in uncertainty raises investment depends on the convexity of the operating proflt
function、vith respect to the demand shock. For the Operating pront function to be convex in the demand
shOck,the folloMring inequality must be satisied:
9>1‑テ

・

tF‖
lⅧ背t鵬
ふ
:Ⅲ
lttT響
蹴
∫
胤
蹴宙
篇露施Ⅸ
:LF層
賄 L脚
誌Υ 冨
Otherwise,the proflt function is concave in the demand shock and an increase in uncertainty depresses

functiOn reduces to 9>1 α

10・

9See Shre(2004),TheOrem l1 5 1
10α is the share oflabor and is arround O.7‑08,which implies thatψ >02‑03

3 . 2 T h e C a s e l w i tβ
h <1
石
e.ヽ ヽ
e also assume that the
We assume that
β <1,1.e.the product market is imperfectly competit市
nian motion(4).The Hamilton―
JacobiB̈ellman equation
emed by the geometric Bro″demand shock is g∝
for the marginal ttlue of capital is given by Eq.(11):πK ( κ , X ) 一

CKば

,K)一

δ9 + 9 κ

ぽ

― δκ

)+μ

(χ )9x+:σ(X)29xx=ρ

g・

「
ヽ
ヽ
hen the marginal value of capital,9,is between the sale price of capital,b2and the purchase price ofvhenonly、
capital, bl, it is optillnal not to purchse nor to sell capital. It is optirnal to purchase capitalアhen｀
the marginal value of capital is greater than bl and to sell capital only when it is less than ｀b2・b2<g<bl,I*=O so that the equation above is sillnplifled into

ν 1‑δκ+μX9x+:σ 2x29χx=ρ g・βんχ κβ
9‑9Kδ 」

(2o

The general solution Of eq.(28)can be expressed in the form
9(K,X)〓スXν κ βl+B(X),石ヽヽe suppose、vhere the flrst term is the special solutiOn and the second term is the homogenous solutiOn.

must satistt the f0110wing quadraticthat the homogenous solutiOn is in theBXθ
form
B(X)〓
・Then,θ
equation:2θ(θ̲1)￨卜卜ρ)==0・μθ― (δ―O②:σ

There exist two distinct roots of this equation,θ
l>1,θ 2く 0・ The value of B(X)must remain inite
2 shOuld vanishes. Therefore,
when the value of X apprOaches zero,、vhich means that the term Of Xθ
B(X)=BXθ

l・T h e c O e f n c i e n t B i s a c o n s t a n t t h a t i s y e t t o b e d e t e r m i n e d . S u b s t i t u t i n g t h e s p e c i a l( 2 s8 o) l, u t i O n i n t o e q
we haveス=「
rhe values
zerO gross investment only if 9 reaches one of boundaries bl or b2・
The flrnl、ァ
11l undertake non―pasting and the high―cOntact conditions. The
Of X at these bOundaries,Xl,X2,are given by the smooth―smooth―pasting conditions for the solutiOn of eq.(28)are9(κl,Xl)=bl,9(κ

2,X2)=b2,

and the high―contact conditions are

T=QΨ

=0

Using these boundary conditions gives rise to

Xl ltt

r」

B〓
静満

(抑 資
l17 κ
allれ

and

叱

The trigger threshold to invest,Xl,is a function of bl,θl,ν and the capital stock K.

10

Suppose that the pront function is concave in X,1.e.

blν<1, Then,‑1)σ2/2
ρ+βδ―ν
μ一ν(ν
>0

1‑ν/θ
l

βん

2/2>O for θs i n c eν/ θ
l<l andl > ν > θ 2 ・T h e t r i g g e r t h r e s h 0 1 d X l i s
ρ+ β δ― νμ ― ν( ν‑ 1 ) σan increasing function of the direct cost of investment, i.e., the cost of purcllasing capital goods. An
increase in the purchase cOst of capital g00ds raises the trigger level so that it、″ould depress investment.
If l≧ β tt ν,an increase of the existing capital stock decreases investment. The inequality l≧
β +ン
holds、vhen the operating proflt function is linearly homogenous inχ and K. This is an intuitively
reasonable implication.What happens when uncertainty increases? An increase in
σ increases the value
O f ρ+ β δ― νμ― ν( ν‑ 1 ) σ2 / 2 . S i n c eαθl
ασ2σθl(θl‑1)

<0,
σ2(θ
l̲1/2)+μア
an increaseinσdecreases
theθalue ofThus, an increase inl valueand soofincreases
the、
万.T=リ
ァ
uncertainty exerts a positive efect on the trigger threshold to invest.
The critical leve1 0f the operating prOflt cOrrespOnding to the trigger threshold to invest is given byπ(κ,Xl)=んχrκβ=κ.

βπ(κ ,X)>π(κ ,Xl)if and only if X>Xl independently of the values ofν and β.The nrm will
undertake non― zero investlnent only when the operating pront
π(K,x)reaches the criticaHe℃lπ(κ ,Xl).
The critical trigger level of the proflt is linearly homOgenous in the existing capital stock, An increasein the capital stock induces in one― to―one a rise in the critical proflt level. It shOuld be renlinded that2 / 2 i s a n i n c r e a s i n g f l l n c tσi oinfν<ofl a n d t h a t θl i s a d e c r e a s i n g f t l n c t i o n o f
ρ+ β δ一 νμ 一 ν( ν‑ 1 ) σ
σ. Therefore,an increase in uncertaillty depresses investment and so there exists a negative relationshipbetween uncertainty and investment as far asν<1,i.e.,the proflt functiOn is concave functiOn in
χ.

Nextヽ
、suppose that the operating pront function is convex in,νthe
>1. demand
In thisshOck,i.e。
2/2 is a decreasing functiOnof σdecreases‑1)σ(ν
case,ρ
σ・An increasein+βδ一ν
the vttue ofμ―ν
2/2,so that a rise of uncertaillty ectexertsa negative‑1)σon thetrigger e■thresholdρ+βδ一ν
μ―ν(νto invest.(Dn the other hand,an increase in σ decreases the value ofθl and so it increases the value ofヽ opposing e■ects On the trigger threshold to invest. A
l=ちラ,「・ThuS'an increase in uncertainty exerts tミ
rise of uncertainty may decrease as s、
1l as raise the critical trigger level of the pront to invest,depending

On the values of such parametersνandasβ. TherefOre,when
ν>1,there exists no sillnple lnonOtonic
relationship bet、
veen uncertainty and investment.
The relationship betvreen uncertainty and investment discussed above is based on the ettect
Ⅵ″
hich
石
unceratainty exerts on investment through changes in the trigger threshold to invest.
ヽヽe have nOt
analyzed the traditional famous efFect that changes in the marginal value of capital cause the amount ofアζinvestment M/hen positive gross investment is being undertaken.
ヽヽe、vill explore this issue.ヽ
hen theヽ
marginal value of capital is greater than the trigger thresh01d Xl,optillnal investmellt is given

つ・F=(需μいThe Hanlilton― Jacobi equation is1‑δνβんχ Kβ9+9K{(9 bl)1/(γl 1)一 δκ}■ μX9x+:σThis partial difFerential equation is dimcult to s。
l ve analytically.

2x29xχ

=ρ 9・0の

3.3 The Case II
W e c o n s i d e r t hⅡe , ic na s we h i c h≠
範0 . T h e c O s t f u n c t i O n o f i n v e s t m e n t i s g i v e n b y={III軍
<ム
均
i:￨・
III胤
,熊

Optimal investment is characterized by( I * ) lγ 1 = 話
r==

詩 方 >0,when g>bl=cI(0,κo,←」
ツ1 1 ‑ 論)+,
when bl≧ 9≧ b2=CI(0,κ ) ,>Q Whenb2>g

When the marginal value Of capital,9,is between the sale price of capital(b2)and the purchase price ofcapital(bl),it iS Optimal not to purchse nor tO sell capital.It is optimal to buy capital only if g reaches
石the boundary bl and tO sell capital only if 9 reaches the bOundary
eb2・
assume that the demand｀｀
shock is governed by the geometric Brownian motion(4).The Hamilton―」acobi―Bellman equation for themarginal value of capital,9,is given by Eq。 (11):π嵐乙 均

軟 ばヽ約 一¨+ 吹 ぱ ―δ
η + バ均牧 + : ∝功 χx = ″ .

アヽヽhen I*=0,the equation above is sillnphfled into1‑δf+μX9x+:σ 2x29xx=ρ 9・βんXνκβ9‑δ9KЙ

01)

The general solution of eq。(31)can be expressed in the fOrmg(κ ,X)=4Xν

κ βl+B(χ),

re 1ハsupposewhere the flrst term is the special solutiOn and the secOnd term is the homOgeneous solution.
石that the homogeneOus solution is in the form B(X)=BXθ .ヽヽe suppose that the hOmOgeneous solution
θ.Then,θ must satistt the f0110wing quadratic equation:is in the form B(χ)=Bχ
2θ(θ̲1)■:σ卜ρ)==0.
μθ― (δ―(32)

There exist two distinct roots of this equation,θ l>1,θ 2<0.The
when the value of X apprOaches zero,whidl means that the term Of Xθ
B(χ )=BXθ

Ⅶ lue Of B(X)must remain anite
2 shOuld vanishes. Therefore,l・The cOefncient B is a constant that is yet tO be determined.Substituting the special solutiOn i、ve have4=

βん

ρ+β δ―νμ一 ;ν(ν‑1)σ2・

The■ rm v″ill undertake non―zero gross investment only if a reaches one of boundaries bl or b2・lrhe values
Ofじてat these boundaries,Xl,X2,are given by the sm00th―
pasting and the high―contact conditions. The
same analysis as undertaken in the previous section can be used here. Therefore, the basically same
result concerning with the relationship between uncertainty and the trigger thresh01d tO invest will apply.
An increase in uncertainty depresses investment and so there exists a negative relationship bet、
/een
uncertainty and investment as far asν < 1, i.e., the prolt functiOn is concave functiOn in X. On
the other hands, lvhen ν > 1, the negative relationship between uncertaillty and investment does notnecessarily hOld.

12

4Concluding Remarks

We have shown that an increase in uncertaillty decelerates as well as accelerates investment dependingon the、 ralue of the model paratemets. In particular, as far as the operating proflt functiOn is concavein the demand shock, an increase in uncertaillty depresses investment, but even if the operating proflt
functiOn is convex in the demand shock,an increase in uncertainty may not necessarily raise investment
and would depress investment,depending on the range ofrnOdel parameters. This result implies that theconvexity of the operating proflt function need not necesarily tO be disIInissed in order to be consistentuハth the elnprirical validity.

Reference(1).Andrew B.Abel(1983),Optimal ln、
228‑233.

にstment under Uncertaint脇

ス鶴θ
rdcα
η Ecοηο鶴,c Rθυづθυ,73(1),(2)。Andrew B.Abel(1985),A StOchastic Model oflnvettment,Marginal q and the Market Value oftheFirm,Lι θrπαι
づOηα:Ecοηθ鶴,c Rθυり
θttl,26(2),305‑322.
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ス鶴θrづ
cα2 Ecθηθmづc Rcυづθυ,84(1).
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stment Under Uncertainty,

anice C.Eberly(1996),Optimal lnvestment with Costly Reversibilittt Rθ

υづ
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ηο
m o c st ιtづαθs,63,581‑593.
げE C ο

(5).Andrew B.Abel and Janice C.Eberly(1999),The Effects Of Reversibility and Uncertainty on Cap

A c c u m u l a t i o nz,t力
t Jαげ″ο
t tι
θ
α
ο
鶴づ
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η E cη
ο
(6).Ben S.Bernanke(1983),Irre℃
Ecοηο鶴づ
cs,98(1),85‑106.

rsibilittt Uncertainttt and Cyclical lnvestment,Ottαrterι
ν 力 包r2αJげ(7).Guiseppe Bertola and Ricardo J.Caballero(1994),IrreVersibility and Aggregate lnvestmellt,Rθ

υづ
θυ

ηο
鶴づ
C S t tづetsα,61,223‑246.
げE C ο
(8).Michael J.Brennan and EduardO S.Schwartz(1985),Ettluating Natural Resources lnvestment,

J o/Bzsれ力包鶴αθSS,58(2),135‑157.
(9).Ricardo」.Caballero(1991),On the sign of the lwestment―Ecοηο鶴づc Rθυづettl,81(1),279‑288.

Uncertainty RelatiOnship,ス

鶴erづ
εαη

(10).Ricardo」.Caballero and Robert S.Pindyよ(1996),Uncertainty lnvestment,and lndustry Evolution,
17tterηαれοηαιEcοηο鶴づ
6 Rθυづθυ,37(3),641‑662.

(11).Avinash K.D破Press.

υθStttθηιιrndcr ιttη6ertαれ ιν,Princeton UniКit and Robert S.Pindyck(1994),九(12).Sebatian Gryglewicz,Kuno Jo M.Huisman,and Peter M.KOrt(2008),Finite Project Liおcertailllty Ettcts on ln(stment,力

rsity

and Un―arηαJげ ECοπο鶴づ
C Dνηα鶴づcs αηα θOttι
知 ら32,2191‑2213.

(13).Riぬard Hartman(1972),The EfFects of Price and Cost Uncertainty on lwestment,」協ο鶴づc Tんθοrν,5,258‑266.

οttrnαJげ ECο―( 1 4 ) . I o a n n i s K a r a t z a s a n d S t e v e n E . S h r e v e ( 1 9 9 1 ) , B "協υづ
αn ″ οιづοη αηα S t ο
c んαs ιづ
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力
rη
α
J げ″θ
ηθ
αλ
づ
ν
, θt t α ηd B ηη, 2 8 ( 1 ) , 6 4 ‑ 8 3 .
̀t,α

13

(16)。Robert McDonald and Daniel Siegel(1985),InveStment and the Valuation of Firms When There isc Rθυづθυ,26(2),331‑349.
ttαれOηα:Ecοnοπづan Option to Shut Down,L̀θ

ttθ
ttνJo鶴鶴αι(17).Robert McDonald and Daniel Siegel(1986),The Value of Watting to lnvestment,Ottα
ECο
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鶴
づ
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rtづηθα:θyctんοαsれ 几 ηαncづαJ yoαθJあ(18).Marek Musiela and Marek Rutkowski(2007),Mα(19)・Bernt Oksendal(2007),StθSんαι
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ιEgしαι
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(20).Robert S.Pindyck(1988),IrreVersible lnvestment,Capacity Choice,and the Value of the Firm,■鶴θttcαtt Ecοttο
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θ‰―
Uncertaintぁ and lnvestment,」 o鶴鶴 αιげ EヒθηθttjC Ljι

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a i n t y R e l a t i o n s h i p i n a R e a 1 0 p t i o n s M鶴o αdιel,Joし

πづ
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鶴づ
6s α
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げE"πο"ら( 2 4 ) . S t e V e n E . S h r e v e ( 2 0 0 4 ) , 乱οεんαS t づ
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C Dνπα鶴づ
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力 鶴rηαJげ Ecοηο鶴づ

14