Natural Disaster, Migration, and Regional Development+
Daiuske IKAZAKI
Japan Women’s University,e-mail: ikazaki@fc.jwu.ac.jp

1．Introduction
In this paper, we extend a simple matching theory to consider how natural disasters affect regional
economic activities and interregional migration.
In section 2, we introduce a simple model of matching theory based on the previous studies.1 This
theory explains how unemployment rate, a measure of market tightness, wage rate, and otherimportant variables are determined.
In section 3, we integrate the elements of natural disasters into the model of section 2. First, we
assume agglomeration increases productivity. Second, natural disasters pull down production factors
and thus deteriorate productivity. Then, population drain occurs.
In section 4, we extend the model of section 3. In 4.1, we consider regional loyalty. Damages caused
by natural disasters decreases the utility of each household. However, suppose that the utility
difference between domicile (hometown) and other regions are comparatively low. And so, it is taken
for granted that people tend to stay in their hometown even if monetary gains becomes better off when
they migrate to other areas. Under this premise, there are multiple steady states. In this case, natural
disasters do not need to raise population drain.
In 4.2, we assume that productivity depends on public capital, which will be devastated by natural
disasters. Just after the natural disaster, public capital decreases and people in this region may
migrate to other regions. We also discuss the effects of fiscal policies to recover public capital. We showthat once migration and a decline in population occur, such fiscal policies may deteriorate the regional
economy. That is, excess supply of public capital increases the onus of the region and declines theutility of household. If so, fiscal policies may pose further population outflow.

2. Basic model of Matching
In this section, we show a simple matching model. The matching technology determines the total
number of matches in the economy. Following the basic framework of matching theory (for example,
Diamond, 1982; Mortensen, 1982; Pissarides, 1985), we specify the matching function as
M=mUαV1-α,(1)where M is the total number of matches, U denotes the number of unemployed, V represents thenumber of vacancies, m and α are the parameters (m＞0 and 0＜α＜1). Let us define θ ≡v/u as ameasure of market tightness.
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This work was supported by JSPS KAKENHI Grant Numbers 25245042 and 25516007.

1 See Diamond (1982), Mortensen and Pissarides (1994), Pissarides, C. (1985), (2000).

1The jobs are assumed to be broken at a rate λ per period. Then the dynamic behavior ofunemployment rate is given as
du/dt =λ (1‐u)‐muαv1-α,(2)where u≡U/L and v≡V/L (L denotes the number of population). In the steady state, the unemployment
rate becomesu =λ / (λ+mθ1-α).

(3)

It is assumed that this economy has only one factor of production, labor. If a firm hires a worker, it
can produce y units of output and pay wage which is denoted as w. Each firm can earn net profit (y‐w)
in every period until the match is dissolved. Let us represent the present discounted value of each firmthat produces goods as Πe, the present discounted value of a vacant job as Πv, and the search cost forfirm as δ. Free entry condition means Πv =0. Then we obtainΠe = (y‐w)/(r＋λ)＝δ/ mθ-α.

(6)Equation (6) is regarded as a labor demand curve in the matching theory.2
Let Ve denote the represent discounted value of each employee, Vu be the present discounted value ofeach unemployed person who searches a job. Bellman equations are given as(7)

rVe = w + λ(Vu‐Ve),
1-αrVu = b + mθ (Ve‐Vu).

(8)

We assume that w is determined endogenously through a process of bargaining between the worker
and the firm (see Nash, 1953). In this paper, bargaining solution is to determine w to maximize (Ve‐Vu)γ (Πe‐Πv)1- γ, where γ is the bargaining power of the worker. Conditions for the maximum are givenas γ(Πe‐Πv)} = (1‐γ) (Ve‐Vu). Equations (6), (7), and Πv = 0 implyw = (1‐γ)b +γy +γδθ(13)Equation (13) is regarded as a labor supply curve in the matching theory. In our model, equations (6)and (13) determine the wage rate and the measure of market tightness. Once θ is determined, we canderive the steady state values of u and v. We can easily show that ∂θ*/∂y＞0, ∂v*/∂y＞0, and ∂u*/∂y＜0.
Furthermore, Vu and Ve are given as
Vu = (1/r)・{b + γδθ*/(1-γ)},

(14)
1-α

Ve = (b /r) +γδθ*/(1-γ){ 1/r + 1/mθ* }.

(15)

3. Simple model of Natural Disaster and Interregional Migration
From now on, we consider how natural disasters affect important variables such as per capita income,2 The numbers of equations are based on the original paper. So some numbers are not included in this resume.

2

population, unemployment rate and so on.
3.1 Marshallian Externalities
In this subsection, we focus on production function and utility. First, suppose that production functionof firm j is given as Yj=ALLξNj, where A and ξ are the parameters (A＞0 and 0 ＜ ξ ＜ 1), Nj represents thenumber of workers employed in firm j, LL denotes positive externalities from the regional population and LL= L in equilibrium (L is regional population). Note that we use the idea of Marshallian externality.3
Each firm takes the value of LL as given. Output per worker (which is denoted as y) is given as y=ALLξ.
So, labor productivity increases with regional population.
Let us describe households. If he is employed, his utility, We is given as We =Ve‐h(L), where h(L)captures the negative externalities of congestion. We assume that h’(L)>0, h’’(L)>0. If he is aunemployed person, his utility, Wu is represented as Wu =Vu‐h(L).4
From equations (6) and (13), and (14), it is shown that ∂Vu/∂L > 0. Furthermore, we assume that2

∂ Vu/∂L2 < 0. Then, the utility of unemployed (Wu) will be an inverted-U shape with respect to regional
population, L.

3.2 Population distribution before natural disasterWe assume that the common utility level of households is established for other regions. Let W’represent that common utility level. This view is similar to the open city model. This means thathouseholds that go to live in other regions can enjoy the welfare level, W’. Households in a region
consider this utility level W′ as given. We draw this case in figure 4. Then we can obtain the following
results:(1) If the population given in the initial stages is smaller than L1 or larger than L2, the households here
have an incentive to migrate to other regions because the utility level established in this region is lessthan the W′ provided in other regions.
(2) If the population in this region is given as between L1 and L2, the households in other regions havean incentive to migrate to this region.
(3) So, population in this region becomes 0 or L2 in the long run.
We assume that population level is L2 at time 0. If so, regional population is L2, per capita output isgiven as y=AL2ξ for all t if natural disasters do not occur.
3.2 Population dynamics after natural disaster
Suppose that natural disaster occurs at time τ and it devastates economic activities. It is assumed thatper capita output becomes sALξ rather than ALξ, where 0＜ s ＜1. Labor productivity for a given value ofregional population declines in consequence of natural disaster because productive factor is affected by
3 Generally speaking, these positive externalities come from the number of employed worker rather than regional

population. However, such setting does not affect our main results.
4 More precisely, the present discounted value of negative externalities is defined as h(L)≡∫exp(-rt)h (L )dt. Suppose thataha’(Lt)>0, ha’’(Lt)>0, and Lt =L for all t. Then, ∫exp(-rt)ha(Lt)dt =(1/r) ha(L). Here, we define (1/r) ha(L) = h(L).

3tnatural disaster.5 In that case, per capita output as well as regional population decline. Per capita
output declines through two channels. First, natural disaster alters production function. Second, a fallin population counteracts the positive effect of agglomeration. Output per worker decrease from AL2ξ tosAL3ξ , where L2 is as before and L3 is given in figure 4. Note that this fall in population will increase an
unemployment rate. The utility of representative unemployed remains unchanged because population
drain alleviates congestion.

4. Extensions of the model
4.1 Regional Loyalties
In this subsection, we extend the model introduced in section 3. First, suppose that migrating to otherregions involves some costs. This reflects regional loyalties, social capital that one has constructed intheir life, moving costs including psychological burden, and so on. We write this cost as F. So, If
households in this region migrate to other regions , they can enjoy the utility level denoted as W’‐F
(we define W’ in section 3). Households in other regions have an incentive to migrate to this region if
the utility level of unemployed in this region is higher than W’＋F. We maintain other assumptions
that we made in the previous section. Per capita output is given as ALξ before the natural disaster. If anatural disaster strikes, then per capita output becomes sALξ instead of ALξ.
First of all, we will focus on the case of before the natural disaster (see figure 6).
(1) If the population given in the initial stages is smaller than L1, the households here have an
incentive to migrate to other regions and this region disappears in the long run.
(2) If the initial value of the regional population is larger than L6, regional population converges to L6.
(3) If the population in this region is given as between L1 and L3, or between L4 and L6, the population
remains unchanged.
(4) If the population in this region is given as between L3 and L4, regional population converges to L4.
5 We implicitly assume the existence of production factors other than labor. Presuming that the parameter A depends onthese factors, it is natural to consider A becomes sA by natural disasters. In section 4, we will consider this point in detail.

4So, regional population becomes between L1 and L3 or between L4 and L6 in the steady state. In themodel we consider in section 3, the steady state values of regional population are 0 or L2 in figure 3.
However, if we consider the term ‘regional loyalties’, there are lots of steady states.
Suppose that natural disaster occurs at some date and production function moves to sALξ. Thenutility curve shifts downward (see figure 6) and we obtain the following results：(5) If the population before the natural disaster is between L1 and L2, regional population converges to 0.
That is, if population before the natural disaster is relatively small, the natural disaster makes it
impossible for the afflicted region to maintain the economic activities.
(6) If the population in this region before the natural disaster is given as between L2 and L3, or L4 and L5,
Regional population remains unchanged.
(7) If regional population before the natural disaster is larger than L5, post-disaster regional
population converges to L5.
Note that the utility level decreases after the natural disaster unless the initial level of population is
L6. In section 3, natural disaster decreases regional population, although the utility level of
unemployed remains unchanged. However, in the extended model here, not only population
distribution but also utility level will become altered. The unemployment rate will increase because
natural disaster lowers labor productivity.

4.2 Production Function with Infrastructure
In section 3, we have assumed that per capita output, y, is given as y= ALξ. Here, we introduce anotherproduction function. Suppose that output per worker is defined as y= AGβ, where G is public capital or,5infrastructure. We assume that public capital is provided by the central government and maintenanceis undertaken by the local government. So, the local government takes the values of G as given.
To maintain G units of public capital which is provided by a central government, the local
government must collect εG units of final good in every period. We assume that the local government
imposes a tax on firms and each firm must incur τ units of output. Budget constraint of the local
government is given as τ (1‐u*) L = εG. Let us call (y‐τ ) as net output per worker.
Some assumptions insure the inverted U shaped relationship between regional population and theutility of an unemployment person. So, we can use figure 6 to analyze the interregional migration.
The Natural disaster occurs at some date and public capital is devastated. Suppose that public
capital becomes s’G rather than G in the aftermath of the natural disaster. We assume that before thenatural disaster, the regional population, L0, is between L1 and L3 or between L4 and L6.
The effects of the natural disaster are similar to those of section 4.1. Hereafter, we mainly focus on
the case where L5 ＜L0＜ L6. As discussed above, natural disaster decreases regional population from
L0 to L5.
Next, we analyze how fiscal policies to recover public capital destroyed by the natural disaster affectpeople’s economic activities. Suppose that the government promotes public investment to recover
public capital to G*(L0)* from s’G*(L0). However once regional population becomes L5, G*(L0) is nolonger optimal. If regional population is L5, and ∂Ynet /∂L ＜0 at L=L5, the reconstruction plans for public
investment accelerate the population drain. If ∂Ynet /∂L ＞0, public investment to recover public capital
may increase the net output. In this case, the utility of residents in the region will increase. However,once people migrate to the other regions, it may be hard to increase the number of households to theoriginal level.

Main References
Diamond, P. (1982)‘‘Wage Determination and Efficiency in Search Equilibrium,’’ Review of

Economic Studies, Vol. 49, pp. 217-227.
Mortensen, D. and C. Pissarides (1994)‘‘Job Creation and Job Destruction in the Theory of
Unemployment,’’Review of Economic Studies, Vol. 61, pp. 397-415.
Nash, J. F. (1953) ‘‘Two person cooperative games,’’Econometrica, Vol. 21, pp.128-140.
Pissarides, C. (1985) ‘‘Short-run Equilibrium Dynamics of Unemployment, Vacancies and Real
Wages,’’American Economic Review, Vol. 75, No. 4, pp. 676-690.

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