Japanese contributions to Dynamic Economic Theories in
the 1940s – the 1970s : A Historical Survey

Second revision February 2009
Toichiro Asada
Faculty of Economics, Chuo University742-1 Higashinakano, Hachioji, Tokyo 192-0393, Japan
E-mail : asada@tamacc.chuo-u.ac.jp

Abstract
Until the end of the World War II, Japan was pure importer of economic theory andJapan had nothing to export except only a few exceptional contributions such as thoseby Kei Shibata on the Marxian Economics and Walrasian/Casselian general equilibrium
analysis and Masazo Sono‟s contribution to mathematical consumer theory. However,the situation dramatically changed soon after the World War II especially in the
research area of mathematical economics. In this paper, we survey the international
contributions of the distinguished Japanese theoretical/mathematical economists of thefirst generation after the World War II, especially focusing on the development of the
dynamic economic theories in the 1940s – the 1970s. We take up the theoreticalcontributions by Takuma Yasui, Michio Morishima, Hirofumi Uzawa, Takashi Negishi,
Nobuo Okishio and Hukukane Nikaido.

11. Introduction
Although this paper is mainly concerned with the international contributions by
Japanese theoretical economists in the period after the World War II (after 1945), in thissection, we shall briefly refer to some important contributions before 1945. Until theend of the World War II, Japan was pure importer of economic theory and Japan had
nothing to export except only a few exceptional contributions.1 Kei Shibata and Masazo
Sono are two such exceptional Japanese scholars who could add something new tomodern economic theory at that period.
Shibata was very creative theoretical economist of Kyoto University. He studied
mathematical Walrasian/Casselian general equilibrium theory as well as Marxian
economic theory. In 1930, He found that Cassel(1918)‟s system of equations of general
economic equilibrium may not have meaningful solution because there is the possibility
that a subsystem of Cassel‟s equations becomes over-determinate. Unfortunately, his
paper(Shibata 1930) was written in Japanese, so that his contribution was unknown
among the western economists. But, his assertion was in fact the same as that of vonStackerberg(1933) which appeared three years after the publication of Shibata(1930).2
Shibata(1933), which was written in English, tried to formulate Marx‟s scheme of
reproduction by means of simplified two sector Walrasian general equilibrium system of
equations, and this paper was referred to by Oscar Lange(1935). He also published acritical review of Keynes‟ General Theory(Keynes 1936) in English only a year after the
publication of Keynes‟ book(cf. Shibata 1937). Masazo Sono was a mathematician rather
than economist, but his mathematical analysis of the „separable goods‟ , which was done
during the World War II, is an important contribution to consumer theory(cf. Sono
1943).3 The contributions by such economists as Shibata and Sono were the pioneering
works which foreshowed the striking international contributions by several Japanesetheoretical economists after the World War II, especially in the period between the
1940s and the 1970s.
The newest imported economics in this period in Japan was Marxian economics, but
the Japanese government oppressed it because the government considered it aspolitically dangerous, and the Japanese mainstream economics in this period was quite
domestic, nationalistic, and illogical economics written in Japanese, which had nointernational influence.
2 Zeuthen(1933) presented correct way to solve this problem by means of the system ofinequalities rather than equations, and Wald(1936) managed to solve the problem
mathematically. See also Yasui, Kumagai and Fukuoka(1977) for the related topics.
3 The goods X and Y are said to be separable from the good Z if the marginal rate of
substitution between X and Y is independent of the quantity of Z. (cf. Ikeo 2006 p. 99.
According to Ikeo 2006, this is the definition by Takuma Yasui..) Sono‟s paper inJapanese(1943) was later translated into English(Sono 1961).
1

2

In the following sections, we shall consider international contributions by Japanese
economists to the development of economic theory after the World War II, especiallyforcusing on the development of dynamic economic theories in the 1940s – the 1970s.
We take up the theoretical contributions by Takuma Yasui, Michio Morishima, Hirofumi
Uzawa, Takashi Negishi, Nobuo Okishio and Hukukane Nikaido.4

2. Takuma Yasui as a pioneer of dynamic analysis of economic stability and nonlinear
analysis of the business cycle
Takuma Yasui, who was a professor of Economics at Tohoku University, Osaka
University and other universities in Japan, is considered to be the father ofmathematical economics in Japan. He began to study Walrasian general equilibrium
theory during the World War II. But, he was frustrated because he could not find thecorrect method to deal with the dynamic analysis of Walras‟ tâtonnement (trial anderror) process of the price adjustment. Soon after the World War II, he found the correct
method in the mathematical theory of differential and difference equations. He
published a series of papers which deal with the mathematical conditions for the
dynamic stability in Japanese(cf. Yasui 1948a, 1948b, 1950). Yasui(1948b) described the
Routh-Hurwitz conditions for the local stability of an n-dimensional system ofdifferential equations5 and provided their economic applications to Walrasian/Hicksian
general equilibrium system and Keynesian system. Yasui(1948b) also clearly described
Other important economists who are not taken up in this paper are Ichiro
Nakayama(study of Walrasian general equilibrium theory and Schumpeter‟s economics),
Eiichi Sugimoto(synthesis of Marshallian and Marxian economics), and Hideo
Aoyama(study of the theory of imperfect competition and the sociological study of Max
Weber), because most of their contributions were written in Japanese and it seems thatthey had little international impact. As for the expositions of their contributions, see
Ikeo(ed.)(1999) and Ikeo(2006). Another influential economist who is not taken up in
this paper is Shigeto Tsuru, a Marxian-oriented „institutional‟ economist who isinternationally well known(cf. Tsuru 1993).
5 Let us consider the following n-th order polynomial characteristic equation.
4n  a1n 1  a 2 n  2    a n 1  a n  0
The Routh-Hurwitz conditions mean that all of the roots of this characteristic equation
have negative real parts if and only if the following set of inequalities are satisfied(thefollowing expressions are the same as those adopted by Yasui 1948b).

a1＞0,

a1

a3

1

a2

＞0,

a1

a3

1

a2

0

a1

a1
1
a 4 ＞0,
0
a3
0

a5

a3
a2
a1
1

a5
a4
a3
a2

a7
a6
＞0,
a5
a4

……

3

the Cohn-Schur conditions for the local stability of an n-dimensional system ofdifference equations without economic application.6 In the postscript to his collected
papers( Yasui 1971), he noted that these papers were written before he read
Samuelson‟s famous book Foundations of Economic Analysis(Samuelson 1947), and
Cohn-Schur conditions are not referred to by Samuelson(1947) although the
Routh-Hurwitz conditions are contained in Samuelson(1947). In Yasui(1950), he
referred to Liapunov‟s theorem on the global stability of a system of differential
equations due to Liapunov(Liapounoff)(1907).7 At that time, this theorem was not wellknown even among western mathematical economists. This means that Yasui is one of

6

The Cohn-Schur conditions mean that all the roots of the characteristic equation

n  a1n 1  a 2 n  2    a n 1  a n  0have the absolute values less than 1 if and only if the following set of inequalities are
satisfied(again the following expressions are the same as those by Yasui 1948b).

7

1

an

an

1

＞0,

1

a1

a2

an

0

0

1

a1

an

0

0

1

a1

a n 1

an

0

0
an

1
a n1

a n1
1

0

1

a n2

a n 1ana n 1

a n2

1

0

0

0

an

a1

0
an
＞0,
an
0
0
1
0

an

a n 1

a1

1

0

0

an

a2

a1

1

＞0,

……

We can summarize a version of Liapunov‟s theorem as follows.
n
Let x  f (x), x  [ x1 , x 2 ,  , x n ]  R be an n-dimensional system of differential

equations that have the unique equilibrium point x*  [ x1 *, x 2 *,, x n *] such thatf ( x*)  0. Suppose that there exists a scalar function V  V ( x  x*) with continuous
first derivatives and with the following properties (1) – (5).
(1) V ≧ 0,(2) V  0 if and only if xi  xi *  0 for all i  {1,2,, n},(3) V   as(4) V 

n

V

i 1

i

x  x *  , ( x  x *) x ≦ 0,ii(5) V  0 if and only if xi  xi *  0 for all i  {1,2,, n}.
Then, the equilibrium point x * of the above system is globally asymptotically stable.
The above function V  V ( x  x*) is called the „Liapunov function‟. By the way, thisversion of Liapunov‟s theorem is not quoted from Yasui(1950) but it is quoted from
Gandolfo(1996) p. 410. In the postscript to Yasui(1970), he confessed that he found
Liapunov‟s paper in French which was published in 1907 in the library of Tohoku
University and he struggled with this huge paper with 260 pages long to write
Yasui(1950).
4

the economists who realized the analytical power of Liapunov‟s theorem for economic
analysis for the first time, as Weintraub(1987, 1991) correctly pointed out. Although
Yasui(1950) did not provide economic application of this theorem, later this theorem
was effectively used by Arrow and Hurwicz(1958) and Arrow, Block and Hurwicz(1959)
to prove the global stability of Walrasian competitive equilibrium under the assumptionof gross substitutability.
Another important Yasui‟s contribution to dynamic economic theory is areformulation of Kaldor(1940)‟s business cycle theory by means of nonlinear differential
equation. Kaldor(1940) presented an ingenious graphical presentation of his businesscycle theory by means of S-shaped nonlinear investment function(cf. Fig. 1). In Fig. 1,

Y is real national income, the line ff represents the saving function S  S (Y ), andthe S-shaped curves  i (i  1,2,3.) represent the investment function I  I (Y , K ),where the real investment I is supposed to be an S-shaped increasing function of real
national income Y and decreasing function of real capital stock K . Kaldor(1940)
assumes that the speed of the quantity adjustment in the goods market is very high,while the real capital stock moves slowly. In Fig. 1, the downward(upward) shift ofinvestment function is caused by the increase(decrease) of K , and the endogenous
business cycle along the closed curve in Fig. 1 occurs in his model.
Yasui(1953) reformulated Kaldor(1940)‟s theory by means of the following system ofdynamic equations omitting the asymmetry between the movement of income and thatof capital stock.

i  f ( y)  k , f ( y)＞0, ＞0y  i  sy, ＞0, 0＜s＜1(1)k  i  k , 0＜＜1(3)y  Y  Y0 , i  I  I 0 , k  K  K 0

(4)

(2)

where Y  real national income, I  real investment demand, K  real capital stock.
The values Y0 ,I 0 , and K 0 are equilibrium values of these variables. The

parameters  , , s, and  are the sensitivity of investment with respective to thechanges of the capital stock, the reciprocal of the speed of the quantity adjustment in
the goods market, the average propensity to save, and the rate of capital depreciation
respectively. Eq. (1) is the Kaldorian investment function where f ( y ) is S-shaped. Eq.
(2) is the disequilibrium adjustment equation in the goods market. Eq. (3) is the capital
accumulation equation.
Yasui(1953) derived the following single nonlinear second-order differential equationfrom the above system of equations.

56y  [ (   )  s  f ( y)]y  s(   ) y  f ( y)  0(5)Under some additional assumptions, he showed that this equation is mathematicallyequivalent to the following type of the generalized van der Pole equation (or Liénard
equation) with zF ( z )＞0 for z ＞0.

z  F ( z ) z  G( z )  0(6)

Yasui(1953) noted that this type of equation was already treated by Goodwin(1951)
and it produces the dynamic behavior of cyclical fluctuations by means of the limit
cycles as represented by Fig. 2 under some additional assumptions.8
It is worth noting that Kaldor himself referred to Yasui(1953)‟s paper in the preface of
his collected papers(Kaldor 1960). Kaldor wrote as follows.
“ The paper on „A Model of the Trade Cycle‟, written shortly before the war, put
forward a new idea – that the postulate of non-linear investment and/or savings
functions produces a „limit cycle‟ which is neither damped, nor anti-damped, and which
does not presuppose for its operation that the various parameters assume values whichfall within narrow range, as was the case with the earlier models of Frisch, Tinbergen or
Kalecki. This idea attracted relatively little attention until some ten years later when
Goodwin and Hicks published, quite independently, models of the trade cycle based on
non-linear investment functions and Yasui showed that the model could be translated
into mathematical terms with the aid of van der Pol‟s theory of relaxation oscillations.”(Kaldor 1960 p. 9)9
By the way, Yasui(1953)‟s model as well as Kaldor(1940)‟s original model produces thecycles around the constant equilibrium values of income and capital. In other words,
these models cannot produce cyclical growth (cycles around the growth trend).
Yasui(1954) tried to reformulate such a Kaldorian model to produce cyclical growth. We
shall interpret Yasui(1954)‟s model after we interpret Morishima(1955)‟s model ofcyclical growth, because both models are closely related to each other.

3. Micho Morishima as a pioneer of nonlinear analysis of business cycle and cyclical
In Fig 2, we have x  y.
At the footnotes on this sentence, Kaldor referred to Yasui(1953) and Morishima(1958)
as well as Hicks(1950) and Goodwin(1951). Although he did not refer to the paper byKalecki in the footnote, apparently he talked about Kalecki(1935) in the text which wequoted.
8
97

growthMichio Morishima is a distinguished Japanese mathematical economist, although he
was a professor of London School of Economics during the period 1970 – 1988. Heentered into Kyoto University(Kyoto Imperial University at that time) in 1942 as a
student, and studied Hicks‟ Value and Capital (Hicks 1939) under the guidance of his
teacher Hideo Aoyama. After the World War II, he taught economic theory in Kyoto
University, Osaka University, University of Essex and London School of Economics. In
the 1950s and the 1960s, he was a colleague of Takuma Yasui at Osaka University. In
1964, he became the first Japanese economist who played the role of the president of the
Econometric Society. At that time, he was 41 years old.10 Since the 1950s, he publishednumerous quite original and creative papers and books on dynamic economic theory
both in English and in Japanese (cf. Morishima 1950, 1953, 1955, 1958, 1964, 1969,1996). In particular, he studied nonlinear macrodynamic models of economic
fluctuations and multisectoral dynamic models of economic growth à la von Neumann.
He also studied mathematical approaches to Marxian, Ricardian, and Walrasian
economic theories (cf. Morishima and Seton 1957, Morishima and Catephores 1978, andMorishima 1973, 1977, 1989).11
In this section, we take up Morishima‟s contribution to nonlinear business cycle
theories in the 1950s which are closely related to Yasui‟s works which we considered in
the previous section. First, we shall take up the model of Morishima(1953, 1958), which
was called „Ichimura-Morishima model‟ by Velupillai(2008).12
Morishima(1953, 1958)‟s model consists of the following system of equations(cf.
Velupillai 2008).

Ct  Yt    (t ), 0＜＜1
I t  v(Qt  Lt  K t ), v＞0(7)Q t   (Yt ),  ＞0(9)

Yt  C t  I t  l (t ), l (t )  Lt

(10)

(8)

where Y  national income, C  consumption expenditure, I  induced investment
The second and the third Japanese economists who became the president of the
Econometric Society are Hirofumi Uzawa and Takashi Negishi.
11 With respect to mathematical study of Marxian economics, it seems that Morishima
was much influenced by another very creative Japanese economist Nobuo Okishio, who
is true pioneer of mathematical Marxian economics.
12 This name stems from the fact that Shinichi Ichimura, who was a colleague ofMorishima at Osaka University, published a paper with the similar idea(cf. Ichimura
1955).
108

expenditure, Q  desired capital stock, L  capital stock required for innovational
investment,

K  actual capital stock, and l (t )  autonomous or innovational

investment expenditure. The subscript denotes time period.
Eq. (7) is the consumption function with expenditure time lag. Eq. (8) together with
Eq. (9) is the investment function of the nonlinear accelerator type. Eq. (10) is theequilibrium condition of the goods market.
The above system is a nonlinear system of mixed difference and differential equations.
But, Morishima approximates Eq. (7) by the following equation expanding Eq. (7) in a
Taylor series and neglecting the terms of higher order than the first.13

C t  Yt  Yt   (t ),   ＞0(11)

Furthermore, he adopted the simplifying assumption that  (t ) and l (t ) areconstant Then, he derived the following single nonlinear second order differential
equations from the system of equations (8) – (11).

y  [  (1   )   ( y ) / y ] y  (1   ) y  0 ;   1 / v, y  Y  Y *(12)

where Y * is the equilibrium value of Y . 14 As Morishima(1958) noted, this is aRayleigh type nonlinear differential equation rather than Liénard equation. He proved
the existence of the limit cycle in this model under some restrictive assumptions. As
Velupillai(2008) pointed out, however, it is worth to note that we can prove the existenceof the limit cycle in Morishima(1953, 1958)‟s model as well as Yasui(1953)‟s model underless restrictive assumptions than those imposed by them by applying Hopf-bifurcation
theorem.15
If it is assumed that the autonomous expenditures  (t ) and l (t ) are increasing
functions of time, the equilibrium level of national income itself becomes the increasing
function of time. In this case, the system of equations (8) – (11) can produce the cyclical
growth around the moving equilibrium. Morishima(1955) Chap. 4 applied the same idea
to Kaldor(1940)‟s model of business cycle. He wrote as follows.
“ Basic consumption increases as the population and the standard of living increase,
while the population increases as the time passes, and the standard of living also tends
to increase as the time passes, so that we can suppose that the basic consumption
increases as the time passes. Therefore, b(t ) becomes the increasing function of t ,This procedure was also adopted by Goodwin(1951). Needless to say, this procedurecan be rationalized only when the time lag  is sufficiently small.
14 In this expression, we omit the subscript which describes time period for simplicity of
the notation.
15 See Gandolfo(1996) p. 477 for the exposition of the Hopf-bifurcation theorem.
13

9

10

where b(t ) is the basic consumption at the time period t." (Morishima 1955 p. 102,
original text is in Japanese)
Then, Morishima(1955) presented Fig. 3 which describes the process of cyclical growthin Kaldorian setting. In this figure, the increase of basic consumption induces therightward shift of the saving function S i *, so that the cyclical growth along the
trajectory abcdef  is caused.
Yasui(1954) provided more complex version of Kaldorian cyclical growth, which isrepresented by Fig. 4. It must be noted that Yasui(1954)‟s model is an adaptation ofMorishima(1955)‟s model and not the other way round although Yasui‟s paper was
published a year before the publication of Morishima‟s book. In fact, Yasui(1954)
referred to Morishima‟s forthcoming book at that time. In Morishima(1955)‟s model, it issimply assumed that the saving function continues to shift rightward because of theincrease of the autonomous consumption expenditure, while Yasui(1954) adopted moresophisticated Duesenberry(1949)‟s „ratchet effect‟ approach, which means that theconsumption does not return to the previous low level even if the national income
decreases because of the inertia of consumers‟ custom. Another difference between these
two models is that in Yasui(1954)‟s model the cyclical growth can persist forever because
the trajectory moves toward the north-east direction, while in Morishima(1955)‟s model
the process of the cyclical growth must stop before long because the trajectory moves
toward the south-east direction.

4. Hirofumi Uzawa‟s investment function and Tobin‟s q theory
Hirofumi Uzawa is one of the most distinguished Japanese mathematical economists
together with Michio Morishima and Hukukane Nikaido. He studied mathematics inUniversity of Tokyo, and moved to Stanford University to complete joint research ofmathematical programming with Kenneth Arrow and Leonid Hurwicz in 1956. Outputsof this joint research were published in Arrow, Hurwicz and Uzawa(1958). He became aprofessor of Economics at University of Chicago in 1964. At that time, he was 36 years
old. In 1969 he went back to Japan and became a professor of Economics at Universityof Tokyo. In 1977 he was selected as the president of the Econometric Society. Afterretiring teaching Economics at University of Tokyo, he taught Economics at Niigata
University and Chuo University. Most of his important theoretical works are collectedin three books (Uzawa1988a, 1988b, 1990). In the 1960s he published a series ofimportant papers on Neoclassical growth theory (Uzawa1961a, 1961b, 1963, 1965,111969) and Walrasian general equilibrium theory (Uzawa1962). In particular we shallrefer to three important papers (Uzawa 1962, 1965, 1969) in this section.
It is well known that the existence of Walrasian competitive equilibrium can beproved by means of Brouwer‟s fixed point theorem.16 This means that Brouwer‟s fixed
point theorem implies Walras‟s existence theorem. On the other hand, Uzawa(1962)
proved that the existence of Walrasian competitive equilibrium implies Brouwer‟s fixed
point theorem. This implies that Brouwer‟s fixed point theorem and Walras‟s existence
theorem are logically equivalent. This is the famous Uzawa‟s theorem.
In Uzawa(1961a), he studied the mathematical condition for the „neutral‟ technicalprogress. In this paper, however, the rate of technical progress was treated as an
exogenous variable. On the other hand, in Uzawa(1965), the rate of technical progress
becomes an endogenous variable which depends on the amount of the investment on
human capital(education of the workers). In this sense, Uzawa(1965) is a pioneering
work which became a predecessor of the so called „endogenous growth theory‟ byLucas(1988), Romer(1990) etc. which flourished in the 1980s and the 1990s. In fact,
Lucas(1988) quoted Uzawa(1965) as the direct source of his idea.17
The above Uzawa‟s works which we referred to are very famous among economic
theorists. Now, we shall take up relatively less known Uzawa‟s contribution, that is,
Uzawa(1969). In this paper, he presented a dynamic optimization model of Keynesiantype investment function by adopting the hypothesis of increasing adjustment cost ofinvestment.18 This contribution is quite important because later Yoshikawa(1980) andHayashi(1982), both of them were Uzawa‟s students at University of Tokyo, showd that
Uzawa(1969)‟s investment function provides a microeconomic foundation of Tobin‟s qtheory due to Tobin(1969, 1980). In fact, they proved that Uzawa(1969)‟s investment
theory implies Tobin‟s q theory.
Uzawa(1969) formulated the following typical firm‟s dynamic optimization problem ofinvestment.

Maximize {r   ( g (t ))} p K (t )e
0

K

 t

dt

subject to K (t )  g (t ) K (t ), K (0)  K 0  constant .

(13)
(14)

See Nikaido(1968) for the exposition of Brouwer‟s fixed point theorem.
Other two important predecessors of endogenous growth theory which entails
endogenous technical progress are Kaldor(1957) and Arrow(1962).
18 Uzawa(1968)‟ approach is called the „adjustment cost approach‟ along with thesimilar approaches by Lucas(1967) and Treadway(1969).
16
17

12

where K  real capital stock, g  rate of investment(rate of capital accumulation),r  rate of profit, p K  price of the capital goods,   rate of discount, and t denotes
time period. It is assumed that r , p K , and are positive parameters. The function (g ) is the adjustment cost function of investment with the following properties.19(15)
 (0)  0,  ( g )＞0,  ( g )＞0,  (0)  1
The control variable is the rate of investment (g ).
Uzawa(1969) derived the following optimal solution of this problem.20
For all t ≧ 0, keep g (t )  g * at the level such thatr   ( g*)  ( g* ) .
 g*

(16)

Uzawa(1969) illustrated this solution by using a graph such as Fig. 5.
It follows from Fig. 5 thatg*  g * (r ,  ) ;g * / r＞0, g * / ＜0,(17)which is nothing but Uzawa(1969)‟s investment function.
By the way, Yoshikawa(1980) showed that this investment function is equivalent toTobin‟s q theory. We can summarize his argument as follows. Suppose thatg (t )  g *＜ for all t ≧ 0. In this case, the market value of the firm becomesV (t )   {r   ( g*)} p K K ( s )e   ( s t ) dst  {r   ( g*)} p K K (t )e (   g *)(s t ) ds tr   ( g*)p K K (t ).
 g*

(18)

This means that the marginal adjustment cost as well as the total adjustment cost ofinvestment increases as the rate of investment increases. Uzawa(1969) called this effect„Penrose effect‟ after Penrose(1959).
20 We can solve this problem by means of Pontryagin‟s maximum principle as follows(cf.
Gandolfo 1996 for Pontryagin‟s maximum principle). First, we can define the current
value Hamiltonian as19

H (t )  {r   ( g (t ))} p K K (t )   (t ) g (t ) K (t ),where  (t ) is the co-state variable. Then, a set of necessary conditions for optimal
solution becomes as follows.

K (t )  H (t ) /  (t ).
 (t )   (t )  H (t ) / K (t ).
Select g (t ) which maximizes H (t ) for all t ≧ 0.

l i m (t )e  t  0.
t Solving this system of equations, we have the result which is described in the text.
13

14

Therefore, Tobin‟s q, which is defined as the ratio of the market value of the firm to thevalue of the real capital stock, becomes as follows.

q

V (t )r   ( g *)

p K K (t )
 g*

(19)

From equations (16) and (19) we haveq   (g* ) .

(20)

Solving this equation with respect to g*, we obtain the following famous „q investment
function‟.

g*  f (q) ; f (q)＞0, f (1)  0(21)This means that Uzawa(1969)‟s approach can serve as a microeconomic foundation of
Tobin‟s q theory.

5. Takashi Negishi as a pioneer of dynamic stability analysis of the Walrasiancompetitive equilibrium
Takashi Negishi, who became a professor of Economics at University of Tokyo after
studying Economics at University of Tokyo, published a series of pioneering
mathematical papers on general equilibrium theory in the 1950s and the 1960s (cf.
Negishi 1958, 1960, 1961, 1962). In the 1970s, he utilized the method of generalequilibrium analysis to investigate the microeconomic foundation of Keynesian
Macroeconomics(cf. Negishi 1979). In the 1980s, he shifted his attention to the study ofthe history of economic theory(cf. Negishi 1989). After retiring teaching Economics at
University of Tokyo, he taught Economics at Aoyama Gakuin University. He became thepresident of the econometric society in 1993. In this section, we shall focus on Negishi‟s
remarkable contributions on the dynamic stability analysis of the Walrasiancompetitive equilrium in the 1950s and the 1960s.
Walras‟s tâtonnement process is the price adjustment process in which the prices of
the goods with positive (negative) excess demand increase (decrease). But, there is no apriori reason to guarantee that this process is dynamically stable unless we impose
some restrictions on the properties of the excess demand functions. Hicks(1939)
proposed a set of particular mathematical „stability conditions‟, but Samuelson(1944)
pointed out that Hicks‟s stability conditions are not identical to the „true‟ Routh-Hurwitz
conditions for local stability of a system of differential equations. Metzler(1945) foundthat Hicks‟s stability conditions become identical to the „true‟ local stability conditions if

15

it is assumed that all goods are „gross substitutes‟ each other.21
In 1958, remarkable progress was made in the related research area by two papers
which were published in Econometrica, that is, Hahn(1958) and Negishi(1958). They
arrived at the same conclusion independently of each other. Namely, the „true‟ local
stability conditions are satisfied automatically in the system with „gross substitutes‟.
Next, we shall summarize Negishi(1958)‟s approach. He effectively used thePerron-Frobenius theorem concerning the indecomposable nonnegative matrix to prove
the local stability of the Walrasian price adjustment in the system with grosssubstitutes.22
Negishi(1958) considers the following Walrasian price adjustment process by meansof a system of differential equations.

p j  E j ( p1 , p 2 ,  , p m 1 , p m ), j  1,2,, m  1(22)

pm  1(23)m

 p E ( p , p ,, p
j 1

j

j

1

2

m 1

, pm )  0(24)

where p j is the price of the good j , and E j is the excess demand (the difference
between demand and supply) of the good j. It is assumed that the excess demand
function is differentiable and the following conditions for the gross substitutability aresatisfied.

E j
p j

＜0,

E j
pk

＞0 if j  k ( j, k  1,2,, m  1, m)(25)

Eq. (23) means that the m‟th good is selected as the numéraire and its price is always
fixed at the level 1. In addition, it is assumed that there exists the unique strictly
positive equilibrium price vector p  [ p1 , p 2 , , p m 1 ] which satisfies23

E j ( p )  0 for all j  {1,2,, m  1, m}.

(26)

Eq. (24) is the so called „Walras law‟, which means that the total excess demand is

21

The good i is said to be the gross substitute of the good j if the excess demand of

the good i ( E i ) is an increasing function of the price of the good j ( p j ).
See also Negishi(1965) for the restatement of the results.
In this expression, the dash is the notation of transposition of the vector. In otherwords, we define the price vector p as the column vector.
22
23

16

always zero even at the disequilibrium situation.24
Expanding the right hand side of Eq. (22) in a Taylor series at the equilibrium point
and neglecting the terms of higher order than the first, Negishi(1958) derived thefollowing linear system of differential equations.
m 1

E j ( p)

k 1

pk

p j  ( pk  pk ), j  1,2,, m  1(27)

By using matrix notation, we can rewrite this system as follows.

p  A( p  p),(28)

p  [ p1, p 2 ,  , p m 1 ]

(29)

 a11
a
A   21 a m11a1m1  a 2 m1   a m1m1 (30)( j, k  1,2,, m  1)

(31)

a jk 

a12
a 22

a m12

E j ( p)p k

The characteristic equation of this linear system of differential equation becomes( )  I  A  0.

(32)

It is well known that the equilibrium point of the original (nonlinear) dynamic system(22) is locally stable if all the roots of Eq. (32) have negative real parts. As Yasui(1948b)
pointed out, such stability condition is equivalent to the quite complicated
Routh-Hurwitz conditions. The direct application of the Routh-Hurwitz conditions tosuch a high-dimensional system is, however, highly impractical. Negishi(1958) solved
theproblemingeniously

byapplying

thePerron-Frobenis

theoremontheindecomposable nonnegative matrix rather than the Routh-Hurwitz theorem to thissystem of equations. In his derivation, the Walras law (Eq. 24) as well as the grosssubstitutability postulate (inequalities 25) plays the crucial role.
By differentiating Eq. (24) with respect to p k , he obtains the following expression.

E j ( p)

m

p
j 1

j

pk

 Ek ( p)  0 (k  1,2,, m  1)

(33)

At the equilibrium point E k ( p )  0 is satisfied, so that he obtains

Usually this equality is derived from the budget constraints of the economic agents(cf. Negishi 1965).
24

17

m 1

E j ( p)

j 1

pk

 pj



Em ( p)(k  1,2,, m  1)
pk

(34)

or in matrix notation,

 xA  z,x  p   [ p1 , p 2 ,, p m1 ]＞0,
z [

(35)
(36)

E m ( p ) E m ( p )E ( p )
,
,, m
]＞0.
p1
p 2
p m 1(37)

Since all of the non-diagonal elements of the matrix A are positive, he can define thefollowing positive matrix M by selecting the sufficiently large positive scalar t.

M  tI  A＞0(38)

Substituting this equation into Eq. (35), he obtains the following equation.

x[tI  M ]  z, M＞0, x＞0, z＞0, t＞0(39)Now, let us consider the following characteristic equation.

(  )  I  M  0(40)

Since the positive matrix M is a special case of the indecomposable nonnegative
matrix, we can use the following results of the Perron-Frobenius theorem concerning
the indecomposable nonnegative matrix.25(1) The characteristic root of Eq. (40) which has the largest absolute value (  M ),which is called the „Frobenius root‟, is positive real root.
(2) Suppose that the positive vectors x, z , and positive real number t satisfy the
relationship (39). Then, we have the inequality t＞ Ｍ .
Therefore, we have the following inequality for any characteristic roott＞ Ｍ ≧ 

 of Eq. (40).
(41)

By the way, it follows from equations (32) and (38) that     t , so that inequality(41) implies thatt＞  t ≧ R e (  t )  ( R e )  t(42)

where Re  means the real part of the complex numberthat we have Re ＜0 for any characteristic root . 26 Inequality (42) implies

 of Eq.(32). This completes

See Nikaido(1968) chap. 2 for the exposition of the Perron-Frobenius theorem.
Note that the absolute value of a complex number is not smaller than the value of its
real part.
25
26

18

Negishi(1958)‟s proof of local stability of the equilibrium point of the Walrasian dynamic
adjustment process with gross substitutes (22) – (24).
Arrow and Hurwicz(1958) proved the global rather than local stability of thecompetitive

equilibrium

under

Walrasianadjustment

process

with

gross

substitutability in a paper which was also published in Econometrica in 1958 by using
the Liapunov‟s theorem, so that it seems that the proofs of local stability of the same
system by Hahn(1958) and Negishi(1958) immediately became obsolete in 1958. 27
Nevertheless, Negishi(1958)‟s proof has its own value as an example of the ingenious
application

of

the

Perron-Frobenius

theoremconcerningtheindecomposable

nonnegative matrix to economic analysis in the area other than Leontief-type
input-output analysis. A few years later, Negishi published an exhaustive survey article
on the stability of the Walrasian competitive equilibrium including the evaluations of
the contributions by himself, Hahn, Arrow and Hurwicz and others, and his survey
article(Negishi 1962) had much influence on the researchers of the related topics(cf.
Weintraub 1991 chap. 7).
By the way, in the disequilibrium situation in the Walrasian tâtonnoment process,transaction is not allowed for. On the other hand in the so-called non-tâtonnement
process the transaction is allowed for even in the disequilibrium situation. Hahn and
Negishi(1962) proved the dynamic stability of equilibrium in such a non-tâtonnement
process.

6. Nobuo Okishio on the mathematical formulation of the Harrodian Instability
Nobuo Okishio, who was a professor of Economic theory at Kobe University, is aworld-famous pioneer of the mathematical Marxian economics. Mathematical study ofthe classical and Marxian economic systems began in the early period of the 20th
century by a few economists such as Dmitriev and Bortkiewicz(cf. Dmitriev 1904 for the
Ricardian system and Bortkiewicz 1907 for the Marxian system). But, up to the 1950s,
much progress in this research area did not occur. In 1960, Sraffa published a small but
important book on the mathematical study of the Ricardian system (Sraffa 1960) andthe revival of the mathematical study of the classical economics occurred. But, in the
1950s, Okishio already began to study the Marxian theory of value and price by using
the mathematical theorems on nonnegative matrices which became famous among the
economists through the study of the Leontief-type input-output system. Earlier
mathematical works by Okishio on Marxian value and price theory were published in27

See also Arrow, Block and Hurwicz(1959).
19

Japanese(cf. Okishio 1957, 1965), but later his works which include the essence of hisanalytical results were published in English(cf. Okishio 1963 and Okishio 1993). In fact,
Morishima‟s

works

on

the

mathematical

MarxianEconomics,

forexample,Morishima(1973), are much indebted on Okishio‟s works and not the other way round.
Another notable contribution by Okishio is the mathematical formulation of theHarrodian instability principle in the models of economic growth and economic
fluctuations(cf. Okishio 1964, 1976, 1977). Okishio reconsidered Harrod(1948)‟s
dynamic theory of economic growth from a Marxian point of view. He thought that
Harrod‟s idea on the instability of the capital accumulation process can be utilized toconstruct a mathematical Marxian theory of the reproduction process of the capitalist
economy. Next, we shall reproduce the simplest version of Okishio‟s analysis ofHarrodian instability by means of the difference equation which is contained in
Okishio(1964).
Okishio(1964) summarizes the steady-growth solution of the so-called „Harrod-Domarmodel‟ as follows.

sY  IY  K

(43)
(44)

(45)
I  K
where Y  real national income, I  real investment expenditure, K  real capitalstock, s  average propensity to save (0＜s＜1), and   output-capital ratio(average
productivity of capital) in case of the full utilization of capital stock＞0. It is assumed
that s and  are constant parameters.
The system of equations (43) – (45) can determine the movement of the three
endogenous variables Y , I , and K . In fact, he obtains the following solution of thegrowth rates of these variables from the above system of equations.

s K I Y


K
I
Y

(46)

This equation determines the equilibrium growth rates of capital, investment andincome at the steady state with full capacity utilization. Okishio(1964) correctly pointed
out, however, that there is no investment function in this model, and he wrote as
follows.
“ Though this model determines the value of new investment I , this value is therequired value of new investment to guarantee the steady growth and is not the actual
value which capitalists invest. Therefore it remains as a problem whether the actual
growth path converges to the steady growth.” (Okishio 1993 p. 191)

20

He pointed out that in the actual situations, there is no guarantee that the full
utilization of the capital stock is realized and the degree of capital utilization must
become an endogenous variable. Therefore, according to Okishio(1964), the actual
growth process must be reformulated as follows.28

sYt  I t

(47)

Yt   t K t

(48)

I t  K t

(49)

where  is the degree of the utilization of the capital stock. This is a system of threeequations with four endogenous variables (Y , I , K ,  ). Okishio(1964) pointed out that
one more equation is required to close the system and the missing additional equation is
nothing but the firms‟ investment function. He proposed to close this system by adding
the following type of the investment function.

I t 1I t  F ( t ), F (1)  0, F ＞0
K t 1 K t
Thisequation

means

that

the

(50)
entrepreneurs

determine

their investment

expenditures according to the degree of the capital utilization. In fact, it is possible tointerpret this type of investment function as a mathematical formalization ofHarrod(1948)‟s idea.29
From equations (47) and (48) Okishio(1964) obtains the following expression.

s t 

It
Kt

(51)

Substituting this equation into Eq. (50), he derived the following difference equation
with the single endogenous variable  .

 t 1   t 

1
F ( t )s

(52)

Differentiating this equation with respect to  t , he derived the following
28

The subscript t denotes time period in this formulation.

29

It follows from Eq. (48) that Ct  K t / Yt  1 /( t  ), C*  ( K t / Yt )  1  1/  , whereC t is the „actual‟ capital-output ratio and C * is the „required‟ capital-output ratio.
Therefore, we can rewrite Eq. (50) as follows.

I t 1I t  H (C t ), H (C*)  0, H ＜0
K t 1 K t
This means that the entrepreneurs accelerate (decelerate) investment expenditure incase of the shortage (surplus) of the capital stock, which is nothing but the investment
attitude of the entrepreneurs which Harrod(1948) postulated in his verbal explanationof the model.
21

22inequality.

d t 11
 1
F ( t )＞1
d t
s

(53)

This inequality means that the equilibrium point of this system which satisfies   1
is dynamically unstable. Okishio(1964) illustrated this instability proposition by using
such a graph as Fig. 6.
One of the causes of instability in this model is the dual characteristics of theinvestment. In other words, the increase of the investment means the increase of theeffective demand as well as the increase of the productive capacity. The increase(decrease) of the rate of capacity utilization of capital stock which is caused by theincrease (decrease) of the effective demand induces the increase (decrease) of theinvestment expenditure, which causes the increase (decrease) of the rate of capacity
utilization through the increase (decrease) of the effective demand. This is thedestabilizing positive feedback mechanism in this model. Another cause of instability is
the postulated myopic bounded-rational investment behavior of the entrepreneurs,
which exclusively depends on the current rate of capacity utilization of the capital stock.
This process of the „cumulative disequilibrium‟ will continue until the floor and theceiling which are created by financial, labor or other markets prevent the investment
expenditure from the upward or downward infinite divergence. Okishio(1976) developed
his theory of business cycle by introducing such ceiling and floor into the model, whichsomewhat resembles Hicks(1950)‟s theory of business cycle.

7. Hukukane Nikaido on the Harrodian instability and the irrelevance of smooth factor
substitution
Hukukane Nikaido, who was a professor of mathematical economics at Osaka
University, Hitotsubashi University and other universities in Japan, is one of the mostbrilliant Japanese mathematical economists. After studying mathematics at Universityof Tokyo, he began to write a series of brilliant papers on mathematical economics in the
1950s. His earlier important contribution is Nikaido(1956), which provided a rigorous
but elegant proof of the existence of the Walrasian competitive equilibrium by means of
Kakutani‟s fixed point theorem independent of the papers by Arrow and Debreu(1954)
and Gale(1955). He also wrote a joint paper with Uzawa on the stability of thecompetitive equilibrium (Nikaido and Uzawa 1960) and a joint paper with Gale on
sufficient conditions for the uniqueness of the equilibrium solution (Gale and Nikaido1965).

He

also

succeededtopresentacompletemathematicallyrigorous23

characterization of Radner(1961)‟s turnpike theorem (cf. Nikaido 1964). Nikaido(1968)
is the exhaustive advanced textbook of mathematical economics, in which his originalcontributions up to this period are contained.
After the 1970s, he shifted his attention from the mathematical study of theWalrasian/Neoclassical microeconomic theories to the mathematical study of theimperfect competition and Keynesian/Harrodian/Marxian macrodynamic theories (cf.
Nikaido 1975a, 1975b, 1979, 1980, 1983). In the introduction of his collected papers on
economic dynamics which was published in 1996, he wrote as follows.
“These early results30 dealt with statics and dynamics within the Hicksian week of ageneric Walrasian Arrow-Debreu economy, one that is solely composed of production,exchange and consumption but devoid of capital formation, credit, and money. In myview, such models describe ideal resource allocation in an institutional framework ofcomplete private ownership, rather than the working of the market economy in the realworld. They are best thought of as normative, rather than positive, construction.”(Nikaido 1996 p. xiii)“The principal view, which drives me most, is that the working of the real world
capitalist economy can better be elucidated along the lines suggested by Keynes(1936)
in his criticism of the Classical school rather than along the lines currently pursued by
the rational expectations and new Classicals as descendants of the old Classicals. It isespecially his grand vision that the economy is unstable and can be ill-behaved that
seems most important.” (Nikaido 1996 p. xiv)
In this section, we consider Nikaido(1975b) which proved that the assumption ofsmooth factor substitution is not sufficient to deny the Harrodian instability principle
contrary to the common belief among the economists.31 Nikaido(1975b) formulates thefollowing system of equations.

L / L  ny  min[ I / s, F ( K , L)]K  sy(54)

I / s  F ( K , L)  F (K ,L)

(57)

(55)
(56)

„These early results‟ mean the analytical results which Nikaido derived in his earlyworks.
31 Nikaido(1980) is the more complicated sequel to Nikaido(1975).
30

24

d
(58)
( I / K )   (  1),  (0)  0,  ＞0dt
where L  labor supply, n  growth rate of labor supply＞0, y  real national income,I  real investment demand, K  real capital stock, s  average propensity tosave (0＜s＜1),   measure of the degree of the capital shortage, and F ( K , L) is the
Neoclassical linear homogeneous production function with smooth factor substitution. It
is assumed that n and s are constant parameters.
Eq. (54) means that the labor supply grows at the constant rate. Eq. (55) means thatthe real national income ( y) is determined by the „short side‟ between the Keynesianeffective demand ( I / s ) and the potential full employment level ( F ( K , L)). Eq. (56)means that the capital accumulation is determined by the following rule.

sF ( K , L) if
K  
Iif

I / s＞F ( K , L)(59)

I / s ≦ F ( K , L)

Eq. (57) defines the measure of the degree of the capital shortage ( ). The inequality

＞1 means capital shortage, while ＜1 means excess capacity of capital. Eq. (58) is
the Harrod-type investment function, which implies that the entrepreneurs accelerate(decelerate) investment in case of capital shortage (excess capacity of capital). The
dynamic system of five equations (54) – (58) determines the movement of fiveendogenous variables ( L, y, K , , I ).
Nikaido(1975b) proved that the rate of investment ( I / K ) continues to decreaseindefinitely if the following set of inequalities are satisfied at the initial period (period
0).
(60)

I (0) / s＜F ( K (0), L(0))
K (0) / K (0) ≦ L (0) / L(0)  n(61)

His reasoning is as follows. Since Eq. (57) implies thatI / s  F ( K , L)  (  1) F ( K , L),(62)

we have ＜1 at the initial period if the inequality (60) is satisfied. In this case, wehaved( I / K )＜0 at t  0. This result assures that there exists the time period ＞0
dtsuch that we have the set of inequalities

I (t ) / s＜F ( K (t ), L(t )), K (t ) / K (t )＜n
for all t  (0, ), which also implies that we have(63)d
( I / K )＜0 for all t  (0, ). On
dt

the other hand, we have the expressiond I
F ( K , L)
1d I
L
K

[

]
( )  ( )( n  )F (1, L / K ).
dt sK
Ks dt KKK ( L / K )

(64)

25

It follows from the inequalities (63) and Eq, (64) that we haved I
F ( K , L)
(65)
[

]＜0
dt sK
K
for all t  (0, ). This inequality means that we can extend the time period  which
satisfy a set of inequalities (63) for all t  (0, ) indefinitely. Therefore, the rate ofinvestment I / K continues to decrease indefinitely if a set of initial conditions (60)and (61) are satisfied. This completes Nikaido(1975b)‟s proof of the Harrodian
instability in a system with smooth factor substitution. This is the rationale of Nikaido‟sfollowing assertion.
“The stability of the Solovian neoclassical growth equilibrium heavily relies upon its
particular investment habits of capitalists in which investment automatically equals
savings, rather than factor substitution.32
On the other hand, one of the most crucial points in Harrod‟s analysis is that theinvestment behaviors of capitalists are primarily responsible for the growth equilibriumbalanced on the knife-edge. Thus instability is likely to occur even under flexible smoothfactor substitution, depending on the investment habit.” (Nikaido 1996 p. 159)

8. Theoretical contributions of other Japanese economists in the 1950s – the 1960s
In this final section, we shall refer to some other important theoretical contributionsof Japanese economists in the 1950s – the 1960s.
Kenjiro Ara, who was a professor of Economics at Hitotsubashi University, is the firstJapanese economist who published a paper in Economic Journal (cf. Ara 1958).
Ara(1958) is one of the earliest papers on the Neoclassical aggregative theory of
economic growth which were formulated independent of Solow(1956) and Swan(1956).
Ara(1959), which was published in Econometrica, is a theoretical contribution to theaggregation problem of the Leontief-type multisectoral input-output system, which issomewhat related to the paper by Morishima and Seton(1957).
Kenichi Inada and Yoichi Shinkai, both of them were professors of Economics at
Osaka University, also contributed to the theory of economic growth in the 1960s (cf.
Inada 1964, 1966 and Shinkai 1960). In particular, the name of Inada is well known
because of the so called „Inada conditions‟ concerning the shape of the NeoclassicalNeedless to say, the word „Solovian‟ that was used by Nikaido stems from
Solow(1956)‟s famous paper that formulated the prototype of the neoclassical growthmodel. See also Swan(1956) as an example of the similar formulation.
32

26

production function.33
In this paper, we concentrated on some notable contributions of the Japanese
economists to dynamic economic theories in the 1940s – the 1970s. Although thecontributions which were taken up in this paper are by no means exhaustive, it isunquestionable that all of them are important contributions to economic theory.

References[1] Ara, K.(1958) : “Capital Theory and Economic Growth”. Economic Journal 118, pp.
511 – 527.
[2] Ara, K.(1959) : “The Aggregation Problem in Input-Output Analysis.” Econometrica
27, pp. 257 – 262.
[3] Arrow, K. J.(1962) : “The Economic Implications of Learning by Doing”. Review of

Economic Studies 29, pp. 155 – 173.
[4] Arrow, K. J. and G. Debreu(1954) : “Existence of an Equilibrium for a Competitive
Economy”. Econometrica 22, pp. 265 – 290.
[5] Arrow, K. J. and L. Hurwicz(1958) : “On the Stability of the Competitive
Equilibrium”. Econometrica 26, pp. 522 – 552.
[6] Arrow, K. J., L. Hurwicz and H. Uzawa(1958) : Studies in Linear and Nonlinear

Programming. Stanford University Press, Stanford.
[7] Arrow, K. J., H. D. Block and L. Hurwicz(1959) : “On the Stability of the Competitive
Equilibrium, II”. Econometrica 27, pp. 88 – 109.
[8] Bortkiewicz, L. V.(1949) : “On the Correction of Marx‟s Fundamental Theoretical
Construction in Third Volume of Capital”. P. M. Sweezy (ed.) Karl Marx and the

Close of His System, Kelly, New York, pp. 197 – 221. (original German version 1907)[9] Cassel, G.(1918) : Theoretische Sozialökonomie. C. F. Winter, Leipzig.
[10] Dmitriev, V. K.(1974) : Economic Essays on Value, Competition and Utility. D. M.
Nuti (ed.) Cambridge University Press, Cambridge. (original Russian version 1904)[11] Duesenberry, J. S.(1949) : Income, Saving and the Theory of Consumer Behavior.
Harvard University Press, Cambridge, Massachusetts[12] Gale, D.(1955) : “The Law of Supply and Demand”. Mathematica Scandinavia 3, pp.

f (k )＜0,
where y  Y / L is the output-labor ratio (average labor productivity) and k  K / L is33

Neoclassical production function is formulated as y  f (k ), f (k )＞0,

the capital-labor ratio (capital intensity). The „Inada conditions‟ are the additional
conditions such that lim f (k )   and lim f (k )  0. For example, the Cobb
k 0

k 



Douglas production function y  Ak , A＞0, 0＜＜1 satisfies these conditions.
27

155 – 169.
[13] Gale, D. and H. Nikaido(1965) : “The Jacobian Matrix and Global Univalence ofMappings”. Mathematischen Annalen 159, pp. 81 – 93.
[14] Gandolfo, G.(1996) : Economic Dynamics (Third Edition). Springer, Berlin.
[15] Goodwin, R.(1951) : “The Nonlinear Accelerator and the Persistence of Business
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