This paper develops a model for optimal capital investment in continuous time when both existing and new capital stocks are subject to uncertainty. The model is generalized to allow for large and infrequent changes in the dynamics of the capital stock, which may arise as a result of natural and man-made disasters.

Hyeon-seung Huh

Inwon Jang

Richard Wong

English

Research Papers in Economics

Optimal capital investment under uncertainty: An extension 
Inwon Jang
Merrimack College
Richard Wong Hyeon-seung Huh
Hartford Investment Management Company Yonsei University
Abstract
This paper develops a model for optimal capital investment in continuous time when both
existing and new capital stocks are subject to uncertainty. The model is generalized to allow
for large and infrequent changes in the dynamics of the capital stock, which may arise as a
result of natural and man-made disasters.
Citation: Jang, Inwon, Richard Wong, and Hyeon-seung Huh, (2008) "Optimal capital investment under uncertainty: An
extension." Economics Bulletin, Vol. 5, No. 4 pp. 1-7
Submitted: February 3, 2008.  Accepted: February 12, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume5/EB-08E20002A.pdf1. Introduction 
 
Studies on the issue of optimal capital investment typically assume that only a new capital 
investment is subject to uncertainty. The uncertainty arises as a result of variations in the cost 
of capital, financial intermediation, or capital-embodied technological innovation (see the 
survey by Dixit and Pindyck, 1994). Recently, several studies in the business-cycle literature 
suggest that the existing capital stock is also subject to uncertainty and may change for 
reasons other than natural depreciation (Ambler and Paquet, 1994; Cooley et al., 1997; 
Dueker et al., 2002). Possible sources of uncertainty are economic obsolescence, physical 
break-down, compositional shifts, among others. Presumably, some may also affect the 
portfolio decision on new capital investment concurrently. To take account of this 
development, the present paper constructs a version of the Ramsey model under the condition 
that the existing and new capital stocks are subject to different but correlated random 
fluctuations. We do so in continuous time and are able to derive an analytic solution for the 
optimal level of new capital investment, which is also in contrast to the studies mentioned 
above.  
Another feature of this paper is a generalization to allow for large but infrequent 
changes in the existing and new capital stock. These types of changes may be a result of, say, 
natural and man-made disasters such as earthquakes, tsunamis, terrorism, or wars, which are 
likely to cause substantial and prolonged damages the physical capital. The effects can be 
particularly devastating in developing countries, washing out a major portion of their capital 
stock, as in the 1998 floods in Bangladesh and the 2004 tsunami in Indian Ocean countries. 
Tol and Leek (1999), Skidmore and Toya (2002), and Okuyama (2003) have examined the 
economic effects of disasters. In particular, Okuyama shows in a Solow model that 
investment decisions under disaster conditions are rather complex and different from those 
made in the absence of disaster. We take one step further by analyzing the effects of such 
catastrophic events in a stochastic optimizing setting. This extension is important as an 
increase in the saving rate necessary to return the capital stock to its previous steady state can 
only be accomplished through a decrease in consumption. In conclusion, some policy 
implications are drawn for those countries prone to natural and man-made disasters.   
 
2. Optimal capital investment under log-normally distributed uncertainty 
 
We first consider a baseline case: that the random fluctuations of existing and new capital 
stocks are log-normally distributed. The analytic model is closely related to those of Merton 
(1975) and Williams (1979), which develop a stochastic version of the Ramsey model in 
continuous time. Uncertainty in Merton is due to the random dynamics of the labor force, 
which follows a log-normal distribution. Williams introduces several sources of uncertainty 
to examine portfolio selection when investing in education and marketable assets. In our 
model, physical capital investment, either existing or new, is posited to be the source of 
uncertainty for the reasons discussed in the Introduction.   
Consider a one-sector neoclassical model with a constant-returns-to-scale, strictly 
concave production function,  , where Y is total output, K the stock of physical 
capital, and L the stock of labor. The production function can be expressed in intensive form, 
(,) YF K L =
() y fk = , where  / y YL =   is the output per worker and  / kK L =   is the capital per worker. 
It is assumed throughout the paper that  () f k  is concave (i.e.   and 
) and satisfies the Inada conditions (i.e. li  and 
). We further assume that a linear production function for new capital of the 
f o r m   , where  , is the percentage of output that is invested in capital formation 
'( ) 0 k fk f =>
0 mkk f → =∞ ''( ) = fk
limk f →∞
() st
0 < kk f
0 k =
() yt 0 s ≥at time t. The existing capital stock is depreciated at a constant rate  δ . In the absence of 
uncertainty the capital stock thus grows according to the following equation: 
 
(, ) () () () () ktt t kt kt styt δ +Δ = − +                            ( 1 )  
 
We now allow for the impact of random fluctuations on both the existing and the 
newly installed capital by formulating the following dynamics for the capital stock: 
 
(, ) (, ) () () (, )() ( ktt t xtt tkt kt ztt tsty ) t δ +Δ = +Δ − + +Δ                ( 2 )  
 
where x and z are log-normally distributed random variables on the interval [0,1], with means 
x μ  and  z μ , variance  2 '
x xx σ ππ =  and  2'
zz z σ ππ =   respectively, and covariance 
'
xzx z σ ππ = . Note that  x π  and  z π  are two-dimensional vectors. Consequently, the exiting 
and new capital stocks are subject to different but correlated uncertainties. As in Merton and 
Williams, the stochastic processes, x and z, may be interpreted as representing the 
productivity or efficiency of existing and new capital stocks, respectively. The parameters 
x μ  ( z μ ) and  x π  ( z π ) measure the expected productivity and uncertainty in the existing 
(new) capital stock. In a continuous-time setting the dynamics of the capital stock may be 
represented as 
 
'
z
xx
sy sy dk
dt dB
kk k
μ
μδ π ⎡⎤ ⎡ =+ − + + ⎢⎥ ⎢ ⎣⎦ ⎣
z π ⎤
⎥ ⎦
                   ( 3 )  
 
subject to  . Here B is a two-dimensional Brownian process. See Williams (1979, 
p. 526) for a similar expression. Eq. (3) gives the dynamics of the state variable k. The 
control variables are the amount of investment (s) and the amount of consumption (c). The 
two control variables are connected through the equation  =
0 (0) 0 kk =>
() ct (1 ( ) ) st) ( yt − , where s∈(0,1).  
     There  exists  a  benevolent  utility-maximizing agent with utility function,   or 
. By assumption, the instantaneous utility function U is continuously 
differentiable, strictly increasing, and strictly concave in consumption (i.e.   and 
). We also assume that 
(() ) Uct
0 c U >
((1 ( )) ( ) Us t y −
0 cc U <
t
0 cc lim U → = ∞ and that lim 0 c c U →∞ = . These conditions 
guarantee a positive rate of consumption and a positive savings function. Thus the 
optimization problem can be summarized as follows: 
 
{ } {} 0
(() )
T
s max  U c t dt ∫ 0 Ε                                 ( 4 )  
 
subject to Eq. (3), where  ,  , and E is the conditional expectation operator. Define 
the indirect utility function: 
0 c ≥ 0 s ≥
 
{ } [( ) ,] [() ]
T
t t
Jkt t m a x  E Uc d τ τ = ∫   .                           ( 5 )  
 
Note that   and  . The former can be obtained by taking  / 0 k JkJ ∂∂ = > 22 / kk Jk J ∂∂ =< 0
  2the derivative of Eq. (5) with respect to k and acknowledging that  ,   and 
. The latter follows from the assumption that 
() yf k = 0 k f >
0 c U > 0 kk f < .  
Using theory of stochastic optimal control we define the Hamilton-Jacobi-Bellman 
(HJB) equation as follows 
 
2
2
{}
1
0( ) 2
2
⎛⎞ ⎛⎞ ⎛⎞ ⎜⎟ =+ + + − + ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠
z 22 + +
⎧⎫ ⎪⎪
⎨⎬
⎪⎪ ⎩⎭
s tx kz x z
sy sy
U c J kJ k J
k
μ
μδ σ x
sy
kk
σσ
22
k k sy J
k k max      ( 6 )  
 
where    is the partial derivative of the indirect utility function with respect to t. Taking the 
derivative of Eq. (6) with respect to s yields the first-order condition for a regular interior 
maximum as: 
t J
 
() 0 c z kx z k kz y U yu J ykJ σσ −+ + + = .                   ( 7 )  
 
Simplifying and solving for s gives an expression for the optimal capital investment. 
 
*
22
cz k x z
zk k z
UJ k
s
yJy
μ σ
σσ
−
=−   () 2
1
/ k
kk
J
kJ
⎛⎞
⎜⎟
⎝⎠
zc k x z
z
k
UJ
y
μ
σ
⎛⎞
− − ⎜⎟ ⎜⎟
⎝⎠
σ =−      ( 8 )  
or,  
 
*
2
/ kz c k
kk
JU J
s
kJ
μσ
σσ
⎛⎞ ⎛⎞ −
⎜⎟ =− − ⎜ ⎜⎟ ⎜ ⎜⎟ ⎝⎠ ⎝⎠
k
2
x z
zz
k
y
⎛⎞
⎟ ⎟
⎝⎠
                      ( 9 )  
 
Several comments are in order. First, the term  / kk Jk J −   represents the Arrow-Pratt 
index of relative risk aversion for gambles with existing capital. Note that   
because   and   (see discussion below Eq. (5)). As it rises and hence the 
investor becomes more risk averse to the existing capital, the optimal level of new investment 
rises. Second, the term   measures the marginal rate of substitution of capital 
accumulation for consumption. The difference between the expected productivity of new 
capital (
( 0 k −> / ) kk Jk J
0 k J >
z
0 kk J <
/ c U k J
μ ) and the marginal rate of substitution of capital for consumption ( ) 
divided by the variance of the uncertainty in new capital (
/ U ck J
2
z σ ) can be interpreted as the risk-
adjusted excess return on new capital investment. As the risk-adjusted excess return on new 
capital investment increases, there is a rise in the optimal level of new capital investment.   
Third, the greater the uncertainty associated with new capital stock ( z σ ), the lower 
the amount of optimal investment in new capital. This can be more easily seen from the last 
expression Eq. (8) by noting that the figure in the largest parentheses should be positive to 
ensure that  . Fourth, the greater the expected productivity of the new capital stock 
(
* 0 s >
z μ ), the greater the amount of optimal investment in new capital. Finally, the greater the 
covariance between the uncertainty of existing and new capital ( xz σ ), the lower the amount 
of optimal investment in new capital if  xz σ >0 and the greater the amount of optimal 
investment in new capital if  xz σ <0. This result may be a reflection of the investor’s desire to 
  3spread the risks across the existing and new capital investments. 
 
3. Optimal capital investment with catastrophic losses 
 
Now, the model is generalized to allow for large and infrequent changes as an additional 
source of uncertainty in the existing and newly installed capitals. We do so by assuming that 
the arrival of such catastrophic events follows a Poisson process. Then, the capital 
accumulation in discrete time can be defined as follows: 
 
(, ) +Δ = ktt t  (, ) () () (, )() () +Δ−+ + Δ x tt tkt kt ztt tstyt δ              
             
                                    ( 1 0 )   () [ ( ) ( ) ( ) ] ( , −+ Δ + + Δ l t tk t s t y tN t t t )
 
where    is an increment of the Poisson process. The random variable   is 
the loss rate associated with the catastrophic event. The dynamics of capital stock in 
continuous time may be represented as: 
(, ) +Δ Ntt t (0,1) ∉ l
 
1 ⎡⎤ ⎡⎤ ⎛ ⎞ =+ −+ + − + ⎜⎟ ⎢⎥ ⎢⎥ ⎣⎦ ⎝ ⎠ ⎣⎦
z
xx z
sy sy sy
dk kdt kdB l kdN
kk
μ
μδ ππ
k
}
        ( 1 1 )  
 
where dN is an increment of the Poisson process. Thus dN=1 if there is a catastrophic event 
and dN=0 otherwise. The process    can also be represented as  , where  t N { 01
∞
≤ = =∑
j tt j N τ
j τ  represent the arrival of the  th j  catastrophe. When the Poisson process has such a 
representation, the probability of having j events in the time interval    is given by  (0, ] t
()
rob( ) exp( )
!
Ρ= = −
j
t
t
Nj t
j
λ
λ                                   
where  λ   is the intensity of arrival of a catastrophe, which is constant. 
The expected utility is now maximized subject to Eq. (11),  , and  . Thus 
the HJB equation is given as 
0 ≥ c 0 ≥ s
 
0 =  {} s max  
2
22 1
() 2
2
⎧ ⎛⎞ ⎪ ⎛⎞ ⎛ ⎞ ⎜⎟ ++ + − + + + ⎨ ⎜⎟ ⎜ ⎟ ⎜⎟ ⎝⎠ ⎝ ⎠ ⎪ ⎝⎠ ⎩
tz x k x x z z
sy sy sy
Uc J k J kJ
kk
μμδ σ σ σ 2
k k k
  
          1, ( ,
⎫ ⎛⎞ ⎛⎞ +− + −⎬ ⎜⎟ ⎜⎟
⎝⎠ ⎝⎠ ⎭
sy
Jkl k t J k t
k
λ ) .                            ( 1 2 )  
 
See Merton (1990, Section 5.8) for a similar derivation of this HJB equation. Taking the 
derivative of the HJB equation with respect to s gives 
 
22 () ( ) ()0 −+ + + +− = cz kx z z k kk yU y uJ k y s y J J l y σσ λ .             ( 1 3 )  
 
Simplifying and solving for s yields the optimal capital investment as 
 
  4*
22
−
=+ − cz k k x z
zk k zk k z
UJ l J
s
2
k
yJy J y
μ λσ
σσ σ
                             
          
or, in a more intuitive form 
 
*
2
/ ⎛⎞ ⎛⎞ ⎛⎞ −−
⎜⎟ =− − ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠
kz c k x z
kk zz
JU J l k
s
kJ y
μλ σ
σσ 2 .                   ( 1 4 )  
 
Eq. (14) is similar to the form derived earlier in Eq. (9). In fact, it encompasses all the 
properties implied by Eq. (9). It is also worth noting that the risk-adjusted excess return on 
new capital investment is now reduced by  2 / z l λ σ . The terms  l λ  can be regarded as the 
expected fractional loss rate of capital when the model allows for large and infrequent 
changes. A rise in  l λ  will make the excess return on new capital investment smaller and 
hence the optimal level of new capital investment lower. In addition, Eq. (14) offers two 
interesting results in relation to the parameters λ  and l. They are: (1) the greater the 
likelihood of a catastrophe (measured by  λ ), the lower the amount of optimal investment in 
new capital, and (2) the greater the loss rate (l), the lower the amount of optimal investment 
in new capital. Hence, those countries that are prone to greater likelihood of natural or man-
made disasters will have a lower capital investment than countries that are not, ceteris 
paribus.  
     This  model  prediction  may  be  used  to  shed some light on the current issue of why 
capital accumulation in the African countries has been chronically low, often blamed as a 
major cause of their dismal economic performance. Some studies find that geography, climate, 
ethnic fractionalization, political turbulence, and civil war are the key factors (see Bloom and 
Sachs, 1998). Devarajan et al. (2003) conclude that such a low level of capital accumulation 
is an inevitable and rational outcome after all. Arising from the same factors, some countries 
in this region would be exposed to a greater possibility of natural and man-made disasters. 
They may also face a greater expected loss rate due to lack of infrastructure and hazard 
management. In our model, these will bring the optimal capital investment to a lower level. 
Low capital investment accumulation may hence be an optimal response to such catastrophic 
risks, along the lines of Bloom and Sachs and Devarajan et al. An implication is that policy 
makers should be more careful about calling for an investment boom to obtain a resumption 
of growth. Unless some or all of the underlying factors that made investment unattractive in 
the past are addressed, the results may be disappointing.   
 
4. Conclusion 
 
This paper derives an analytic solution for the optimal level of capital investment in 
continuous time when both existing and new capital stocks are subject to uncertainty. Two 
forms of uncertainty are considered. One is the log-normal specification, as typically assumed 
in previous studies. The other one is a Poisson distribution to allow for large and infrequent 
changes that may arise as a result of natural and man-made disasters. Several implications are 
drawn, including that countries prone to catastrophic risks will have lower capital investment 
and hence lower economic growth.   
  5  6
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Optimal capital investment under uncertainty: An extension

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