Was Japan's "rapid economic recovery" after WWII
miraculous? : A theoretical explanation
Harutaka Takahashi
July 2023
Discussion Paper No. 2312

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANWas Japan's "rapid economic recovery" after WWII miraculous?
: A theoretical explanation
Harutaka Takahashi 1
0F

Abstract This paper is concerned with the existence of optimal recovery paths after an
economy has been destroyed by natural disaster or war. A prominent example is Japanafter World War II. Japan faced massive destruction of its infrastructure and production
facilities and, in fact, starvation and severely high inflation. However, in a relatively shorttime, 10 years, the country achieved economic recovery. This historical fact indicates that
there is an optimal recovery path after economic mass destruction. Under such extreme
conditions, one of the key features is that 1) production technology is fixed, not selective,
and 2) in view of two industry sectors; the capital goods sector and the consumer goods
sector, the capital intensity of the former is much greater than that of the latter. To analyze
such a situation, we return to the two-sector growth model with Leontief technologies
studied by Shinkai and Corden in the 1960s and transform it into a continuous time
optimal growth model with the Leontief technologies. In this study, we focus on
nonlinearity and interior-point solutions, under the assumption that the capital intensityof the capital goods sector is extremely greater than that of the consumption goods sector,
two critical points were derived. These two critical points indicated the unstable focus (orunstable point) and the saddle, respectively. The transversality conditions show that thesaddle-point stable point is the only optimal solution and the optimal recovery path. Then,history-based parameter selection and numerical analysis showed that the initial per
capita capital stock, given historically, must be close to the unstable focus and thatgovernment policy is needed to make the economy jump onto the trajectory which
eventually converges to the optimal steady state path. In other words, Japan's economic
recovery was miraculous. (280)
Key Words: Leontief technologies, two-sector optimal growth model, capital intensity
condition, saddle-path stability, unstable focus, phase diagram
JEL: O22, O41, O53
1 Research Fellow, Graduate School of Economics, Kobe University, Professor Emeritus, Meiji Gakuin University. E-

mail: haru@eco.meijigakuin.ac.jp. I would like to thank Takeo Hori and Masakatsu Nakamura for their detailed
suggestions on an earlier version of this paper and the IMERA-Institute of Advanced Studies at AixMarseille University and the Aix-Marseille School of Economics for providing the best research
environment for the writing of this paper.

11. Introduction
Japan had to take the first steps toward economic recovery from a land devastated in bothname and reality by its defeat in World War II. The loss of national wealth in World War
II amounted to 25.4 percent, or just over a quarter of the country's total wealth. The factory
production base was in a state of ruin, with particularly heavy losses in ships, industrial
machinery and equipment, and buildings. Most of the production facilities that escaped
damage were used for military production, and conversion to peaceful industry was noteasy. The devastation caused by the war and the shortage of materials and labor resulted
in a critical shortage of raw materials and, moreover, hyperinflation. In addition, the rapid
increase in the birth rate, which was accompanied by the repatriation of military personnel
and Japanese living abroad to the interior of the country, led to a rapid increase in the
population and further accelerated the shortage of goods, especially food and clothing. A
rationing system was put in place to distribute the scarce goods equally, but the imbalance
between the supply and demand of goods was so great that black-market goods and blackmarket prices were rampant. This situation is vividly illustrated in Figure 1, where the
100 years of Japan’s per capita real GDP is depicted.
Japan's Real GDP (Ratio scale) :1930~1990
22.5
22.0
21.5
21.0

Old trend

20.5

New trend

20.0
19.5
19.0
30

35

40

45

50

55

60

65

70

75

80

85

90

Figure 1. Postwar recovery period:1946 ~ 1955
From 1945 to 1946, GDP fell sharply due to World War II, returning to prewar levels in
1955. Furthermore, a steeply sloping trend line is observed from 1946 to 1975,
intersecting the old trend line around 1964, when the Tokyo Olympics were held. The
slope of the line indicates the average growth rate, which was about 10%. Such a fact
would give the impression that Japan's rapid postwar economic recovery was inevitable2

and independent of government policies. Therefore, we will theoretically examine
whether this impression is true by dating Japan's economic recovery from 1946 to 1955 2.
In order to set up a model to analyze Japan's rapid economic recovery, the following
five characteristics must be met.
1F1) Due to the devastation of the war, the technology of choice was very limited, or the
options were limited or fixed.
2) There was a very large difference in capital intensity when considering the two
sectors: the capital goods sector and the consumption goods sector. The consumption
goods sector had very low capital intensity compared to the capital goods sector, such asiron and steel manufacturing and shipbuilding, because the former imperial military
personnel (7.6 million), military production workers (4 million) and repatriates fromabroad (1.5 million) were mainly engaged in agriculture and self-employment incommerce.
3) Because of the large number of structurally unemployed, total labor input wasdetermined by production technology.
4) The initial stock, which indicates the level of per-capita stock in 1946, is not zero,but very close to the lower limit.
5) In the early stage, the chaotic state immediately after the war, the optimal trajectoryshould be located in the unstable region, but in the later stage, it will converge to the
optimal steady state and show a stable and high economic growth rate.
In this paper, the original Leontief two-sector model studied by Shinkai (1960) and
Corden (1966) is adopted and transformed into a continuous time per capita optimal
growth model with Leontief technologies. Shinkai (1960) and Corden (1966) studied atwo-sector growth model with Leontief technologies and demonstrated that a balanced
growth path is globally stable if the consumer goods sector is capital-intensive, as Uzawa(1961, 1963) showed in a two-sector growth model using neoclassical technology; Uzawa(1964) extended this property to the neoclassical two-sector optimal growth model. Morerecently, Fujio (2008, 2009) applied the geometric method developed in Kahn and Mitra(2007) to a discrete-time two-sector Leontief optimal growth model to study theproperties of the optimal steady state.
To maintain differentiability, we restrict interior solutions. That is, only theequilibrium in which both goods are produced is considered. In doing so, we examine thelocal and global stability of the optimal steady state. In other words, as shown in Uzawa’s
2 Ch. 4 of Flath (2020) and Part 1 of Nakamura (1995) are recommended for a quick look at Japan's postwar

reconstruction. Macroeconomic indicators and major events during the recovery period are shown in Table 1.

3

neoclassical two-sector optimal growth model, labor and capital grow in proportional
to population growth.
The analytical results are as follows: 1) under the capital intensity condition, where
the capital goods sector is extremely capital intensive than the consumer goods sector,
two different optimal steady states are found to exist; 2) as indicated by both analyticaland phase diagram analysis, one steady state is the saddle, and the other is the unstablefocus or an unstable point. Therefore, only the saddle optimal steady state is the optimal
steady state that can be achieved. In other words, there is a unique optimal steady state
under this capital intensity condition. This result is in contrast to the standard result of
optimal growth theory, first shown by Uzawa (1964) in his two-sector optimal growththeory, that the optimal steady state exhibits saddle-point stability when capital intensity
in the consumption goods sector is higher than the capital goods sector.
Based on the analytical results, parameter values were selected to fit the historicalfacts, in particular, the annual growth rate of total labor input was chosen at 10% and thepeople's discount rate at 67%. Numerical analysis was then performed, and as expectedfrom the analytical results, two types of solutions were found to exist: saddle point stable
solutions and unstable central solutions. Applying the backward-shooting method, we
found the optimal path with the initial point near the unstable center solution such that it
eventually converges to the saddle-point stable solution. The saddle-point stable solution
is the optimal steady state in which the level variable grows by 10%. There are two
important points if this is the theoretical explanation for Japan's rapid economic recovery
process shown in Figure 1: 1) the initial per capita capital stock is near the unstable
solution point, and 2) some macroeconomic policies are needed to move the economy
toward a stable solution and eventually a "stable arm" beyond the optimal steady state.
From these two perspectives, we can conclude that the main factors behind Japan's
rapid economic recovery were: (1) the per capita stock at the end of the war happened to
be close to the unstable steady state point, and (2) the government implemented
appropriate policies, such as the "priority production system" and strong inflationary
measures known as the "dodge line," to bring the economy to an optimal steady state. In
other words, Japan's rapid economic recovery was truly a "miracle.

4

YEAR

GNP*

INFLATION**GROWTH

RATE

RATE

(%)

MAJOR EVENTS

(%)

1945

-

51.1 

Surrender of Japan (Aug.)

1946

-

364.5 Financial asset freeze (Feb.),


1947
1948

8.6
13.0

195.9 Land Reform,

Labor Reform

Dodge Line (Feb.),

165.5

1949
2.1

Breakup of the Zaibatsu

63.3 

The exchange rate pegged at 360 yen to thedollar1950

11.0

18.2 

Korean War began (June)

1951

13.0

38.3 

Peace Treaty of San Francisco

1952

11.0

2.0

1953

7.4

5.0 

1954

3.6

6.5

Korean War Armistice Agreement (July)

Table 1. Macroeconomic conditions of Japan, 1945 – 1954
Source: Bank of Japan Major Economic Statistics of Japan since the Meiji Era*) Gross National Expenditure in real terms (Table 011_9_1).
**) Wholesale Price Index (Table 018_2_1).

5

2. Notation and Model
We follow the two-sector model, consumption goods and capital goods sectors with
Leontief technologies, which was studied originally by Shinkai (1960) and Corden (1966).
Let us define some notation which are used in Coden (1966). The indices “k” and “c”
indicate capital goods sector and consumption goods sector respectively.
(1) Capital coefficient: K k / Yk , K c / Yc ,=ak Y=
Yc / Lc ,(2) Outputs per person:k / Lk , ac
K c / Lc ,(3) Capital-labor=
ratio: bk K=
k / Lk , bc
Lk + Lc and K =Kk + Kc ,(4) Full employment conditions: L =nt(5) Total labor input growth:=
L L=
e nt ,
0eRemark 1. As explained in Introduction, there exists a huge amount of structurally
unemployed labor outside labor market, the technologies of both sectors determine thelabor inputs L. The initial labor input is normalized as one.
(6) Capital Accumulation:
The following important assumptions are made:Assumption 1. All the technological coefficients: ac , ak , bc , bk are constant.
Assumption 2. i) The capital goods sector is more capital intensive than the consumption
goods sector () and ii).
In Assumptions 1and 2 ii), let's draw a familiar diagram consisting of the constraint lines
of labor and capital. The slope of this line is indicative of the capital intensity of eachsector. Based on the calculations below, a diagram like Figure 2 of Corden (1966) can be
drawn.

6


1
 1 
a 
−  c  Yk + ac L
L =
  Yc +   Yk ( L − const.) ⇒ Yc =
 ac 
 ak 
 ak  bc 
b 
 b  a 
a 
K =
−  k   c  Yk +  c  K
Yc +  k  Yk ( K − const.) ⇒ Yc = ac 
 ak 
 ak   bc  bc 


Assume that the total labor input and total capital stock are increasing at the rate of totallabor input growth (n). Then, both constraint lines shift upward simultaneously, and theinternal equilibrium point Q also shifts to point Q'. Let's trace only such a continuous shift.
From (1) through (4), we obtain the output ratio. Further rewriting in per-capita terms
yields (7) derived as Eq. (3) in Corden (1966).

(7)

Remark 2. Note that (7) is an important function. It indicates that when given andchosen through the resource allocation between two sectors,
is automaticallydetermined. Assumption 2 ii) guarantees that
.

7

Q’
Q

Figure 2.
3. Optimal growth problem
Now let us consider the problem that keeping (7), chooseso that the utility function
is maximized. From (7), to choose
implies simultaneously to choose . Due toaccumulation equation (6) it increases per capita capital stock. In the next period, since
the capital goods sector is more capital intensive than the consumption goods sector fromi) of Assumption 2,
increases andreduces by Rybczynski theorem 3 . In otherwords, to choose greater
implies that it directly increases today’s utility, but it
increases
more and reducesin the next period. Thus, it has a negative effect to
2F

the utility in the next period. We have a trade-off between today’s utility and tomorrow’s
utility to choose today’s . Thus, under the trade-off, the problem is to find out theproper path ofin the long run to maximize the objective function:.
We can set up the following intertemporal optimization problem to solve the problem:

3

Rybczynski theorem implies that an increase in the factor endowment of a factor of production under a constantrelative price of a good increases the output of the good produced using that factor of production in an intensive
manner and decreases the output of other goods.

8

We make an extra assumption for making the objective function definable:
Assumption 3. ρ > n .
Then the Hamiltonian for the problem is defined as follows:

Then the first order conditions concerned with the Hamiltonian are obtained as follows:
(8)

(9)

(10)

(11)

.

Note that Eq. (11) indicates the transversality condition.
3.1 Solving the optimal problem
Due to the last relation in (10), it is convenient to define the following new variable:
Definition.

.
9Remark 3. It is worth noting that, by Rybczynski theorem, to increase

has a negative

impact on the objective function, as discussed in Remark 2, so the Lagrangianisnegative, as shown in (10). Also note that P is considered the relative price of theconsumption good measured in utility units.
Substituting the result of (10) into the equations (8) and (9) yields

And

Note that due to ii) of Assumption 1, ( k − bk )( bc − k ) ≠ 0.
We finally obtain:.

Differentiating P w.r.t. time yields:(11)

Furthermore, substituting

into the above relation (11) yields,

10Then we have finally obtained the following dynamical system (*) constituted of two
differential equations:(*)

3.2 Multiple steady states
To obtain the optimal steady state, we need to solve the following simultaneous equations(**):

The origin (0, 0) is one of the critical points of (*). The other two critical points are
obtained as roots of the following quadratic equation:

11

Note that

. The discriminant of the equation:

turns to be as

follows.

We make the following assumption.
Assumption 4.

Under Assumption 1,
is established. Furthermore, it implies that the parabola
expressed by (12) and the line (13) intersect each other as depicted in Figure 1 below.
Therefore, we have two different roots α and β (α > β ) that are possible optimal steadystates, and they satisfy

because. This property

contrasts with the one derived by Bruno (1966), where there exists a globally uniquesaddle optimal steady state. We also make an additional assumption regarding theparameters.

4. Local stability
Let us study local properties at each steady state point derived in Section 3. The Jacobianand the trace of the differential equation system (*) at two critical points k * is obtained asfollow:

12

Due to ii) of Assumption 2 and Assumption 4, it follows that

We have.

Proposition. At the critical value
denoted by Point A, the saddle-point stability is
exhibited, while the critical value
denoted by Point B exhibits the unstable focus or
unstable point.
Proof. Substituting
intoyields

The fact that
implies a saddle point at k * = α denoted by Point A.
To show the proprietary of Point B, we need to explicitly derive the solution , whichis expressed as follows.

Substitute this into the Jacobian to be evaluated,13

implies that the following two cases occur.

Point A has a "stable arm" indicated by arrows c-A and d-A in Figure 1, and any path on
the stable arm converges to the steady state indicated by point A, which satisfies thetransversality condition and is the optimal steady state. Conversely, the paths in thevicinity of point B are divergent and violate the transversality condition and are not
optimal.
To reconfirm our results, let's draw a phase diagram and analyze the global dynamic
behavior of the system (*). To do this, we need to identify the locilocus of. The

is a straight line: P= (n + δ )k . The following differential equations should

be analyzed to derive the locus of.

The last relation shows that the locus ofis a quadratic equation with the vertex
14

( b + b 2, (δ + ρ ) ( b − b ) 4 ) . Now we can draw the following phase diagram below:
k

c

k

c

Phase portraits in Figure 1 below are drawn based on the following properties.

Points A and B represent two different possible optimal steady-state solutions α and β
respectively. As shown in Figure 3, the possible steady-state points A and B represent thesaddle and the unstable focus (or instability) respectively. This phase diagram contrasts
with the figure shown as Figure 4 in Bruno (1966), where the optimal steady state is
unique and saddled-point stable in the same region as ours. Our model, on the other hand,
has complex dynamics, even though there is only a single fixed production technology
for each sector.

5. Numerical analysis 4
3F

In Section 4, the properties of the optimal steady state of the dynamical system (*) werequalitatively examined. To demonstrate that the analytical results obtained in the previous
section can explain the miracle of Japan's postwar recovery, a numerical analysis was
performed using values of parameters selected to be in line with historical facts.
5.1 Historical parameter values
For this purpose, each parameter should be carefully selected as follows.

4

Mathematica 10.3 was used here. The Mathematica source codes will be provided upon request.

15

e
d
A

c

B

Figure 3: Phase Diagram: Takahashi et al. (2012) measured the capital intensity of two sectors inJapan from 1955 to 1995 based on the I-O tables 5 and reported that the capitalintensity of the consumption goods sector in 1955 was 1.14 (million/person) and thatof the investment goods sector was 1.18 (million/person). The value of capitalintensity varies with monetary units. Therefore, the value of capital intensity isarbitrarily chosen in monetary units. In other words, only the capital intensity ratio
and the differential are meaningful. In fact, in 1955 it was 1.14/1.18
0.97.
Although the ratio was close to one in 1955, but it can be inferred that during earlystage of the postwar reconstruction period from 1946 to 1955, there was a significant
difference in the level of capital intensity between the two sectors. The reasons forthis are as follows. After the war, 7.61 million troops went to war and 1.5 million
civilians were repatriated from abroad. This meant that more than a quarter of thelabor force was structurally unemployed. Many of them worked in agriculture or selfemployment in the service sector. On the other hand, there were also workers who
repaired machinery used for munitions and produced peacetime supplies. For
example, producing merchant ships rather than combat vessels. Assuming that thecapital intensity of the consumption sector is normalized as 1, the capital intensity of
4F

5 The 1955 I-O table is the earliest postwar table.

16the capital goods sector is measured as a ratio. For example, assigning 6 to the capital
goods sector implies that the capital goods sector is six times more capital intensive
than the consumption sector. We assign here 35 to the capital intensity ratio.
The initial per capita stock must satisfy Assumption 2:

.

Furthermore, because of the catastrophic destruction of the capital stock due to thewar, the initial stock is assumed to have been near the lower bound .
: Since along the optimal steady state, all the level variables grow at , we may becautioned to assign the value. According to Okawa and Shinohara's data reported inTable 4.1 of Flath (2013), GDP per capita grew at a rate of almost 10% from 1946 to
1955. Therefore, the growth rate of total labor input, n, is set to 0.1.

: After World War II, people were busy with their day-to-day lives, and they
probably had fairly myopic preferences. The values here range from 0.2 to 0.9. Sinceis substituted for 0.1, the future discount rate of the objective function varies from
90% to 45%.

: No data are available. The standard peacetime value is about 0.15, but it isconsidered relatively high during the reconstruction period. The main reason for thisis that it was very difficult to find undamaged machinery and the resources to replace
it during the economic reconstruction period, when the war had caused extensive
damage. Here we assume 0.20. This means that the depreciation would be almost
complete in five years.

Determining the discount rate during the restructuring period is difficult, but according toTable 2, 90% is very optimistic, and 55% is too pessimistic. Assume a discount rate of
67% and a capital intensity ratio of 35%.
Futurediscount

90

81

74

67

60

55

50

45

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

factor(%)

Table 2. Future discount factors
17

Now, we implement the chosen scenario as indicated in Table 3.
Parameters
Values

bk

bc

35

1

ρ

n

δ

0.5

0.1

0.20

Table 3. Parameter values of the chosen scenario
In this scenario, the discount rate of the objective function is about 0.67 from Table 1. It
means that people discount future heavily. This is reasonable under the post war situation
because people were just trying to survive day by day during this period and they couldn’t
think about future. The difference in capital intensity by sector indicates that the capital
goods sector was 35 times more capital intensive than the consumption sector. In fact,
during this period, the so-called "priority production scheme" was implemented by the
Reconstruction Finance Bank, which provided subsidized loans to heavy industries suchas coal mining, electricity, steel, and chemical fertilizers, in accordance with the Japanese
government's economic policy. This policy clearly further widened the gap in capitalintensity.
Numerical analysis yielded the results shown in the three figures in Figure 3 below.
It can be seen that there are two solutions, A:(19.6471, 5.8941) and B:(1.7814, 0.5344),
indicated by the red dots in Figure 3 a). To
clarify the local dynamics near points A and B, we examined the dynamics in theneighborhood of the two solutions. The results are shown in Figures 3 b) and 3 c).

a)18b) Around the Point Ac) Around the Point B*) Stable arms are shown in red and unstable arms in green 6.
5F

Figure 3. Local dynamics
As the theory presented in Section 4 indicates, Point A exhibits saddle-point stability andPoint B exhibits an unstable focus. Figure 3 a) shows that every trajectory violates thetransversality condition given by equation (10), except that the trajectories converge atpoint A.
It is important to note the simultaneous existence of saddle point stability andunstable focus points. Global numerical analysis applying the backward shooting method
reveals the existence of interesting trajectories, indicated in red in Figure 4. This trajectory
starts from the neighborhood of point B, shown in red in Figure 3 b), and eventuallyconverges to the optimal stationary point A. This trajectory is optimal because it satisfies
the transversality condition, and exhibits the recovery process of the Japanese economy
after World War II. As mentioned before, the initial per capita stock just after the war wasnear Point B. Due to chaotic economic situations after the war, Japan experienced hyperinflation as indicated Table 1, especially from 1945 to 1948. However, after Doge Linewas implemented in 1949, it ended by 1952. After that the Japanese economy grewsteadily under the stable price changes.

6

The stable arm is obtained by starting from a neighborhood of the OSS and solving the system of differential
equations backward. The unstable arm is obtained by solving the system of differential equations forward.

19

Figure 4. An optimal trajectory

5. Conclusion
Our two main results are follows.
1)
According to standard optimal growth theory, there is no stable steady state whencapital intensity in the capital goods sector is higher than in the consumer goods sector.
However, the Leontief two-sector optimal growth model shows that if capital intensity inthe capital goods sector is extremely higher than in the consumer goods sector, and thediscount rate and depreciation rate are appropriately given, two different steady states
exist: a saddle-point steady state and an unstable steady state.
2)
According to the numerical results and their discussion, two miracles occurred
during Japan's postwar recovery period. One was that the initial per capita stock
remaining at the end of the war happened to be close to the unstable focus point, and theother was that the trajectory eventually converged to a steady-state path as a result of
appropriate government policies. In other words, the rapid postwar recovery of theJapanese economy was truly a miracle.

20References
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21