Journal of Natural Disaster Science, Volume 26, Number 2, 2004, pp81-85

Economic Restoration after a Catastrophic Event:
Heterogeneous Damage to Infrastructure and Capitaland Its Effects on Economic Growth
Hirokazu TATANO1, Toshitsura HOMMA2, Norio OKADA1 and Satoshi TSUCHIYA3
Dr. Eng., Professor, Integrated Management for Disaster Risk, Disaster Prevention Research Institute,
Kyoto University (Gokasho, Uji, 611-0011, JAPAN)
2
M. Eng., Mitsui-Sumitomo Bank (1-5-1 Otemachi, Chiyoda-ku, Tokyo, 100-0004, JAPAN)
3
M. Eng., Dept. of Urban Management, Graduate School of Engineering, Kyoto University(Yoshida-Hommachi, Sakyo-ku, Kyoto, 606-8501, JAPAN)

1(Received for 10 Jun., 2004 and in revised from 7 Jan., 2005)

ABSTRACT
Economic restoration after a large catastrophic event in which the economy is described by an endogenous economic growth model consisting of two regions was investigated. Since a natural catastrophe is local and a largescale event, it destroys the accumulated infrastructure and production capital in the economy instantaneously, butdamages to capital is not homogeneously distributed in space. Two types of capital are assumed; infrastructure
and production capital. Infrastructure is formulated as a common capital good in all the area’s economic regions.
Production capital consists of the private goods used for production in each region. This paper investigates howheterogeneous damage to different types and locations of capital affects economic restoration after a catastrophic
event. Implications for disaster risk management policy are discussed based on analysis of the model.

1. INTRODUCTION
A natural catastrophe causes a discrete downward drop in the
production capacity of an economy. This is because some portionof the capital accumulated in the economy is destroyed. Tatano etal. [2000] reported that economic losses caused by a natural disaster consist of “stock losses” and “flow losses”. Stock losses are thelost values of economic stocks.
Flow losses are secondary ones caused by the shift in economic growth triggered by the occurrence of a disaster. These two
types of losses are interconnected with restoration effort investments after the event. If no investment to restore destroyed stocks
can be made, i.e., no investment for the accumulation of capital,
the economy will remain at the same level or lower. To avoid suchlosses, investment to restore destroyed capital must be made in thereal world. Economic losses observed as flow in the real world
therefore reflect the lost value of products of destroyed capitalminus the benefit of the restoration investment.
Reducing both stock and flow losses is important when
designing effective disaster risk management strategies.
Mitigation investment contributes to decrease damage to stock, andrisk financing arrangements and the restoration policy utilized after
the disaster affect the levels of production and economic growth ofthe economy. This research focused on restoration policies implemented after a catastrophic event in order to identify desirable
policies.
The economy is described by an endogenous economic

growth model consisting of two regions. The natural catastrophe islocal but a large-scale event. It destroys both the accumulated
infrastructure and production capital of the economy instantaneously, but the damage to capital is not homogeneously distributed in space. It is assumed that there are two types of capital in
the economy; infrastructure and production capital. Infrastructureis defined as a common capital good used for production in bothregions of the economy, and production capitals as private goods
which can be used for production exclusively in a region. This
study investigated how the heterogeneity of damage to the different types and allocations of capital affects the economic restorationand growth of the economy as a whole. It also provides clues fordisaster risk management policies based on an analysis of themodel.
2. RESTORATION POLICIES AND ECONOMICGROWTH
Disaster risk management options consist of “risk control”
and “risk financing” countermeasures. Fig. 1 shows the disasterrisk management options for seismic risk management. These
options consist of two categories; risk avoidance and mitigation
countermeasures. Risk avoidance countermeasures center on thedecrease in population and assets exposed to a disaster risk (“exposure”); e.g., land use regulation. Mitigation countermeasures can
aid in decreasing “vulnerability” to that exposure. Risk control
countermeasures can decrease damage to capital stocks in the

KEY WORDS: natural catastrophe, endogenous economic growth, economic restoration path, heterogeneous damage

82

H. TATANO, T. HOMMA, N. OKADA AND S. TSUCHIYA

Production capital

Production capital
Infrastructure

Fig. 3 Illustration of the model’s assumed economy

Log GDP

Fig. 1 Possible options for disaster risk management

Secondary Effect(Level Effect)
Flow losses
Restoration Policy
Risk FinanceRisk Control
Mitigation

Restoration
PeriodTime

Fig. 2 Roles of risk control and risk financing in economic growtheconomy. Risk financing countermeasures are ex ante financial
preparations for a catastrophic event in order to decrease losses
that ensue after the event such as a decrease in production and consumption in flow terms. Representative measures include insurance and alternative risk transfer (ART); e.g., CAT security. The
purpose of risk financing is to increase the economy’s capacity for
rapid recovery after the event.
Fig. 2 shows the role of risk control and risk finance in relation to economic growth. A risk control countermeasure, if properly designed and implemented, contributes to a decrease in damage to stock in the economy. A risk financing countermeasurecontributes to a decrease in flow losses within the economy.
Restoration policy also contributes to decreased flow losses, subject to available resources after the event. This study focused on
optimal restoration policies, especially the relationship between theheterogeneity of damage and restoration period length.
The literature dealing with catastrophic losses and economic
growth basically has had a Neo-classical framework [Tatano et al.,2000, Kobayashi et al., 2002]. Catastrophic events in the economy
have been modeled as unexpected downward drops in production

due to the sudden decrease in capital as discussed above. Suchmodels show that a catastrophic event brings about a permanent
shift in the growth of an economy, called the “level effect”.
These models assume only one type of capital in an economy,
therefore heterogynous damage cannot occur. This assumption isjustifiable to obtain basic figures such as the “level effect” [Barroand Sala-i-Martin (1995)], which explains the relationship between
economic growth and catastrophic risk. On the basis of the models, we cannot analyze the restoration path after an event. In a single capital economy, the shock produced by a catastrophe simply
would return the economy to its past stage of capital accumulation,and growth after the shock would be exactly the same as that in thepast. The restoration period is defined as the transition period during which economic growth of an economy catches up with theoriginal growth trajectory, turned downward by the catastrophe. In
this sense, the literature does not discuss the restoration path afteran event, and it can be said that the published studies implicitly
assume that restoration can be finished immediately after the
event.
Heterogeneous damage to capital is divided to two groups inthis paper; heterogynous damage to different types of capital andlocalization differences in damage. Fig. 3 shows the economy
modeled in this study. There are two regions. One may suffer
direct catastrophic losses. Without loss of generality, it is assumed
that such a catastrophic event could occur only in area 1, and thatarea 2 is always safe in such an event. Each area has accumulated
capital for production. Production efficiency also depends on theinfrastructure level and labor for in an area. To avoid excessive
complexity, people and firms are assumed not to change their locations, and that the population of each area is fixed. Production
technology in each area is assumed to be constant return-to-scale
and a function of infrastructure and production capital. Based onthese assumptions, the economy grows steadily at a constant
growth rate if the initial values of the types of capital are in the“optimal capital component ratio” (defined later). The capitalcomponent ratio is assumed to be optimal in the normal state(before the catastrophe), but a reduction in capital may occur heterogeneously in stock types and places when the event happens,
and the capital component ratio after the event therefore will not beat the optimal ratio, at least for some time. The restoration period
is the interval in which the capital component ratio varies from theoriginal ratio.
The restoration policy is the resource allocation policy utilized
during the transient period. Here, investigation of the optimal
restoration policy is based on a two-region, endogenous economic

ECONOMIC RESTORATION AFTER A CATASTROPHIC EVENT

subject togrowth model with heterogeneous capital damage.

Y1 + Y2 = L1C1 + L2C2 + I K1 + I K 2 + IG
K˙ = I - d K (i = 1,2)

3. THE MODELi

A social planner’s model is formulated to investigate the optimal restoration policy. In this model, the social planner decides
investment policies for the allocation of production capital andinfrastructure resources from now on to maximize household welfare in both areas subject to production technology and the initialendowments of production factors. People and firms are assumed
not to change their locations and that the population of each area isfixed.
Production technology is represented by a constant return-toscale production function of capital Ki (i = 1,2) and infrastructureG;
Yi = F ( Ki ,G ) = ( A i L i G )1-a Kia (i = 1,2)

(1)where L i is the labor available in area i and A i and a (0 £ a £ 1) areparameters. Infrastructure is regarded as a common production
factor in all areas of the economy and contributes to the production
efficiency of labor.
The instantaneous utility function of a household in area i,u (C i ), is a non-decreasing function of consumption C i . Eachhousehold in each area is assumed to supply unit labor for production. A social welfare function is set as a Bentham type one:L 1 u(C1 ) + L 2 u (C2 ). The social planner is assumed to maximize thediscounted present value of the social welfare function:•{L1 u (C1 ) + L 2 u (C2 )}e- rt dt
0

U =Ú

(2)

The social planner can decide what parts of the economy’s
production should be invested in capital IKi (i = 1,2) and infrasructure IG and consumed in each area. The balance for production,
consumption, and investment is
Y1 + Y2 = L1 C1 + L 2 C2 + IK1 + IK2 + IG

(3)

Capital accumulation conditions are
K˙ i = I Ki - d K Ki (i = 1,2)

(4)

Ġ = IG - d G G(5)where δK and δG are the respective depreciation rates of production capital and infrastructure, respectively.
The social planner also faces the problem of expenditures andis allowed to spend no more for capital than the rate of that capital’s return in the long run. These conditions are given byÈÏÔ t Ê ∂Yˆ ¸Ô˘lim ÍK1 ◊ exp Ì - Ú0 Á 1 - d K ˜ dt ˝˙ ≥ 0t Æ•¯ Ô˛˙˚ÔÓ Ë ∂K1ÍÎÈÏÔ t Ê ∂Yˆ ¸Ô˘lim ÍK 2 ◊ exp Ì - Ú0 Á 2 - d K ˜ dt ˝˙ ≥ 0t Æ•¯ Ô˛˙˚ÔÓ Ë ∂K 2ÍÎÈ
Ï t Ê ∂Y ∂Yˆ ¸˘lim ÍG ◊ exp Ì - Ú0 Á 1 + 2 - d G ˜ dt ˝˙ ≥ 0
t Æ•
Ë
¯ ˛˙˚
∂
G
∂
G
ÍÎ
Ó
Based on all these conditions, the social planner’s behavior is•maxU = Ú0 {L1u(C1 ) + L2u(C2 )}e - rt dt

83

Ki

K

i

G˙ = IG - d G G
Yi = F ( Ki , G ) = ( Ai Li G )1-a Ki

a(i = 1,2)Èˆ Ô¸˘ÔÏ t Ê ∂Ylim ÍK1 ◊ exp Ì - Ú0 Á 1 - d K ˜ dt ˝˙ ≥ 0t Æ•¯ Ô˛˙˚ÔÓ Ë ∂K1ÍÎÈÏÔ t Ê ∂Yˆ ¸Ô˘lim ÍK 2 ◊ exp Ì - Ú0 Á 2 - d K ˜ dt ˝˙ ≥ 0t Æ•¯ Ô˛˙˚ÔÓ Ë ∂K 2ÍÎÈ
Ï t Ê ∂Y ∂Yˆ ¸˘lim ÍG ◊ exp Ì - Ú0 Á 1 + 2 - d G ˜ dt ˝˙ ≥ 0
t Æ•
¯ ˛˙˚
ÍÎ
Ó Ë ∂G ∂G
I K1 , I K 2 , IG , C1 , C2 ≥ 0(I)

The path of economic growth can be specified if the initial value ofeach type of capital is set.
4. HETEROGENEOUS DAMAGE TO CAPITALAND OPTIMAL RESTORATION POLICIESi) Economic growth before a catastrophic event
The following conditions are derived from the first order condition of optimization of problem (I):ÊGˆÁ ˜Ë K1 ¯a ( A1L1 )

1-a= a ( A2 L2 )

1-a

1-a

ÊGˆ
Á ˜
Ë K2 ¯= (1 - a )( A1L1 )

1-a

1-a

(6)

Ê K1 ˆ
Á ˜
ËG¯

1-a

+ (1 - a )( A2 L2 )1-a

Ê K2 ˆ
Á ˜
ËG¯

1-a

This means that the productivity of each type of capital should
equal the optimal path. Equation (6) can be written*

Ê K2 ˆ
A2 L2*
Á ˜ =l =A1L1
Ë K1 ¯

(7)

*

ÊGˆ
1-a Ê
A2 L2 ˆ*Á ˜ =m =Á1 +˜aKA1L1 ¯Ë 1¯ËThis means that capital component ratios are constant on the
optimal growth path. In contrast, the economy’s growth is always
the optimal if the initial value of each type of capital is within the
optimal component ratio. On the path, production and consumption in both regions grow at a constant rate:(g * = (1 / s ) a ( A1L1 )

1-a

(m )
*

1-a

-d - r)(8)

That is, the economy grows at a constant rate before the event.
ii) Economic growth after a catastrophe
A catastrophic event brings about a discontinuous, heterogeneous decrease in capital. In such a situation, it is useless toassume that all capital component ratios remain optimal.
Moreover, the social planner does not necessarily determine
investment such that the optimal component ratios of capital are

84

H. TATANO, T. HOMMA, N. OKADA AND S. TSUCHIYA

kept, but should prioritize investment for the capital that has thehighest marginal productivity.
Table 1 gives the combinations of possible damage patterns in
the economy and changes in marginal productivity for each type of
capital. It provides the following findings.
There is a decrease in the marginal productivity of capitalwhich has not been damaged, whereas the marginal productivity ofdamaged capital is increased by the event. This means that themost severely damaged capital should be given the highest priority
in making the restoration policy. In some cases, restoration of theinfrastructure should be promoted even if damage is not as severeas that to the production capital.
The first result is common sense, but the last is not explainedin a straightforward way. The last result is related to the presenceof spillovers of infrastructure to the economy in area 2. Fig. 4
shows changes in investment priorities during the restoration period. The indifferent marginal productivity curve signifies the critical capital component ratio in area 1 where the marginal productivities of the production and infrastructure capital are same. Priority
for investment in the infrastructure is focused on where the capitalcomponent ratio is smaller than the critical value, g, a decreasefunction of available infrastructure ex post. The optimal capitalcomponent ratio m* equals g when the infrastructure is maintainedat the value before the event; when no loss of infrastructure has

occurred. Hence, for the case when some damage occurs to theinfrastructure, if the capital component ratio G/K1 is smaller than g,priority of investment is placed on the infrastructure, even for G/K1
> m*.
Fig. 5 shows changes in the marginal productivities of thetypes of capital and production levels in each area in damage patterns, in which either the infrastructure or production capital suffers losses when the optimal restoration policy is adopted. Pattern
1 is the case when production capital suffers losses. Pattern 2 isthe case for infrastructure capital. In both patterns, damaged capital has the highest marginal productivity right after the event, andinvestment during the restoration period takes place only for damaged capital. Restoration paths in the affected area (area 1) aresimilar, whereas those in the non-affected area (area 2) differ.
Damage done only to production capital in the affected area has noeffect on the productivity of the non-affected area. The effect ofMarginal
capitalproductivity

Marginal
capitalproductivity

Pattern 1

G

K1
r*

K2

r*

K1 K2

G

t

t

*

t

t+

1

1

Restoration path in area 1

In Y1

Pattern 2

Restoration path in area 1

In Y1

Table 1. Possible damage patterns and marginal capital productivities
t

t+

t

t+

1

1

Y2

Restoration path in area 2

In Y2

Restoration path in area 2

In Y2

t

t+

Fig. 5

Marginal
capitalproductivity

1

Changes in marginal capital productivities and production in each area in damage patterns in which
either production capital or infrastructure suffers
losses when the optimal restoration policy is adopted

Marginal
capitalproductivity

Pattern 3

1st phase ofrestoration

2nd phase ofrestoration

2nd phase ofrestoration

G
r*

G

K1

K1

G/K1

Pattern 4

1st phase ofrestoration

K1
r*

t

t+

1

K1

t

*

1

t

G/K1

t*

t

t*

t

1

G(t1+)
K1(t1+)

g
m*

g
m*
t*
1

Fig. 4 Changes in investment priority during the restoration periodt1Fig. 6 Changes in Marginal Capital Productivities and theCapital Component Ratio in the Restoration Period
when Both Types of Capital Suffer Losses

ECONOMIC RESTORATION AFTER A CATASTROPHIC EVENT

the event on economic growth of the non-affected area therefore islimited by the effect produced by the reduction of capital investment in area 2 during the restoration period. In contrast, damage
to the infrastructure in the affected area has a spillover effect on
the productivity of the non-affected area because it is capital common to both areas. Therefore, production in the non-affected area
also decreases with the extent of damage to the infrastructure, andits restoration path is similar to that in the affected area.
Fig. 6 shows that changes in the marginal productivities of thetypes of capital and in the capital component ratio when both typesof capital suffer damage during the restoration period. In both patterns, the restoration period has two phases. In the first, investment is implemented only for the most severely damaged capital.
Once the capital component ratio reaches the critical value g, therestoration policy is changed, and investment is made in both typesof damaged capital to maintain the same marginal productivities
for the infrastructure and production capital. This is the second
phase of the restoration policy, and it continues until the capitalcomponent ratio reaches the optimal ratio, m*.
5. CONCLUSION
Economic restoration after a large catastrophic event wasinvestigated in an endogenous economic growth model. How theheterogeneity of damage done to different types of capital and the
localization affect on the economic restoration path and growth ofthe economy as a whole were studied.
The heterogeneity of the damage to capital was shown to be85

crucial to describe what restoration processes should be implemented after a catastrophic event. The analysis conducted clarifies
the basic principle of restoring capital after a catastrophe and thestructures of the necessary restoration processes are shown for thedifferent damage patterns.
Infrastructure capital should be prioritized over a wider rangeof the economic environment rather than production capital. Thereis greater resistance to catastrophic risks for the infrastructure
because damage to it has spillover effect on non-affected areas. To
investigate mitigation policies for the use of different types of capital in different locations, it is necessary to conduct a cost benefit
analysis.
The uncertainty of the occurrence of a catastrophic event was
not considered in order to avoid excessive complexity. In the
future, the model must be extended to include uncertainty in orderto estimate the benefit of alternative mitigation measures.
REFERENCES
Barro, R. J. and X. Sala-i-Martin (1995). Economic Growth. New York:
McGraw-Hill.
Tatano, H., Isobe, W. and Okada, N., 2000. Economic Evaluation ofSeismic Risks, Proc. of the 2000 Joint Seminar on Urban Disaster
Management, 36-39, Beijing.
Yokomatsu, M. and Kobayashi, K., 2002. Economic Growth and Dynamic
Mitigation Policy, Proc. of the 3rd IIASA-DPRI annual meeting on“Integrated Disaster Risk Management: Mega-city Vulnerability and
Resiliency”, Kyoto.