福岡大学先端経済研究センター ワーキング・ペーパーシリーズ

Okishio's Theorem, Innovation and Economic Growth
Yoshihiro Yamazaki

福岡大学経済学部

WP-2024-001

福岡大学先端経済研究センター〒814-0180

福岡県福岡市城南区七隈八丁目 19 番 1 号

Okishio’s Theorem, Innovation and Economic Growth
Yoshihiro Yamazaki

Abstract
Marx proposed the proposition that the advancement of capital’s organic composition
would reduce the rate of profit. He thought that this tendency brought the end of capitalist
economy, Just after World War II, Nobuo Okishio, a Japanese Marxian economist, proposed
Okishio’s theorem. The theorem told that an adoption of a new technology raises the rate ofprofit under the condition of fixed real wage rate. The difference between Marx’s proposition
and Okishio’s theorem can be made clear from the view point of the trade-off relation between
wage and profit, which Sraffa investigated closely. While Marx and Okishio adopted value
approach, contemporary economics adopts value added approach concerning innovations. A
Schumpeterian model by Aghion=Howitt is an example of contemporary treatment ofinnovations. Their model is a macroeconomic model while Okishio’s was a two sectors model.
Schumpeter also thought that innovations would exhaust finally, which is the same conclusion
as Marx’s proposition.

JEL classifications: B140, B310, B410.

Keywords: Okishio’s theorem, Sraffa, Schumpeter, value added approach, innovation.

i

1. Introduction
Karl Marx wrote that in the process of the capitalist economy the rate of profit would decline andthat this made the competition among capitalist severer. Thus he predicted the end of the capitalist economy.
Marxian economists have continually insisted this proposition of Marx.
Against such a tendency, Schumpeter wrote that Marx’s proposition was not true. He wrote about
Marx’s proposition like this;The fact is that he had no simple theory of business cycles. And none can be made to follow logicallyfrom the “laws” of the capitalist process. Even if we accept his explanation of the emergence of surplus
value and agree to allow that accumulation, mechanization (relative increase of constant capital) andsurplus population, the latter inexorably depending mass misery, do link up into a logical chain that
ends in catastrophe of the capitalist system – even then we are left without a factor that would
necessarily impart of cyclical fluctuation to the process and account for an immanent alteration ofprosperities and depressions (Schumpeter, 2009, 49)
Schumpeter wrote that because Marx could not build a complete theory of business cycle, heconfused a panic phase of cyclical business fluctuation with the final scene of the capitalist economy itself.
Marx’s proposition bases on the thesis that the speed of constant capital’s accumulation surpasses theprogress of productivity. Whether you agree to Marx’s proposition or not depends on the acceptance of thisthesis.
Nobuo Okishio is a modern Marxian economist. He tried to study the contemporary problem of theeconomy. Because of this, he could not help treating the progress of productivity. He watched a rapidly
rising of productivity in Japan after World War II. Thus he proposed Okishio’s theorem, which told that therate of profit would not decrease when capitalist raised the productivity.
Okishio’s theorem was proposed within the Marxian economics. Marxian economics inherited value
approach from classical economics. There the progress of productivity is represented as the diminishmentof value. This image of economic growth does not fit with contemporary economics’ one.
There are several models which include Schumpeterian innovation process. In the models,innovations happen stochastically. Of course contemporary economics presents the economic growth asincrease of value added. Differing from Marxian economics, contemporary economics adopts value addedapproach.
The validity of Okishio’s theorem can be proved when interpreting from the viewpoint of
Schumpeterian innovation model. And finally it is interesting that Schumpeter told the exhaustion ofinnovation in the future. Marx’s proposition revives in Schumpeterian model.

12. Okishio’s Theorem
Supposing an economy consisted of production goods sector and consumption goods sector.
Variables pi, li, w, r are prices of the two sectors’ goods, labor input to the two sectors, wage rate and profit
rate respectively. Here the following definition is introduced;

ρ=11+𝑟(1)The equations (2) are obtained.

ρ𝑝1 = 𝑎1 𝑝1 + 𝑤𝑙1 𝑝2 , ρ𝑝2 = 𝑎2 𝑝1 + 𝑤𝑙2 𝑝2(2)When a new production technology is introduced, the inequality (2) must be held𝑎1 𝑝1 + 𝑤𝑙1 𝑝2 > 𝑎2 ′𝑝1 + 𝑤𝑙2 ′𝑝2(3)

Of course, the equations (4) will be held after the introduction of a new technology supposing thatthe consumption goods are numéraire.

ρ′𝑝1 ′ = 𝑎1 ′𝑝1 ′ + 𝑤𝑙1 ′𝑝2 , ρ′𝑝2 = 𝑎2 𝑝1 ′ + 𝑤𝑙2 𝑝2(4)

From the second equations of (2) and (4), the equation (5) can be obtained.
(𝜌′ − 𝜌)𝑝2 > 𝑎2 (𝑝1′ − 𝑝1 )(5)

From the inequality, the equivalence (6) follows.
𝜌′ < 𝜌 ⟺ 𝑝1′ < 𝑝1(6)On the other hand, the equation (7) is obtained from the first equation of (4).
ρ′𝑝1 ′ = (𝑎1′ 𝑝1 + 𝑤𝑙1′ 𝑝2 ) + 𝑎1 ′(𝑝1′ − 𝑝1 )Using the inequality (2), the inequality (8) is obtained from the equation (7).

2(7)

ρ′ 𝑝1 ′ < (𝑎1 𝑝1 + 𝑤𝑙1 𝑝2 ) + 𝑎1 ′(𝑝1′ − 𝑝1 )(8)

Substituting the first equation of (2) into the inequality (8), the inequality (9) follows.
ρ′ 𝑝1 ′ < 𝜌𝑝1 + 𝑎1 ′(𝑝1′ − 𝑝1 )(9)

The inequality (9) is equivalent to the inequality (10).
𝑝1 (ρ′ − 𝜌) < (𝑎1′ − 𝜌′)(𝑝1′ − 𝑝1 )(10)

From the first equation of (4), the inequality (11) follows.
𝑎1′ < 𝜌′(11)From the inequality (11), the condition (12) must be held.
𝜌′ ≥ 𝜌 ⟹ 𝑝1′ < 𝑝1(12)The condition (12) contradicts the equivalence (6). Because of this, the left side of the condition (12)is denied. Thus the conclusion that when a new technology is introduced, the rate of profit must rise isproved.
In Marx’s proposition, the accumulation of capital must deduce the rate of profit. When capitalists
accumulate capital, they normally adopt a new technology. Okishio’s theorem insists that adoptedtechnology will raise the rate of profit. The conclusion opposes Marx’s proposition.

3. Sraffian Interpretation of Okishio’s Theorem
The conclusion of Okishio’s theorem depends on the stickiness of wage rate. Okishio thought thatwhen capitalist introduced a new technology, this introduction must reduce the production goods price. This
deduction of the production goods price will mitigate the rise of organic composition. Instead of that, therise of organic composition reduce the real rate of wage. This effect brings the reduction of profit rate.
The difference between Marx’s proposition and Okishio’s theorem, therefore, originates from thesupposition about wage rate. To make clear this point, the trade-off relation between wage rate and profit
rate. Sraffa investigated it closely.

3

Sraffian system of production was represented as the equations (13).
(13)(1 + r) ∑ 𝑎𝑖𝑗 𝑝𝑗 + 𝑤𝑙𝑖 = 𝑝𝑖

Because Sraffa thought the wage paid after production, only physical input is multiplied by profit
rate. Here when workers obtain all the products, profit rate must be zero. On the contrary, when capitalists
obtain all the products, wage rate must be zero. Sraffa constructed a system where composition of net
products was equal to the composition of total products and called it the standard system. In the system, thetrade-off between profit rate and wage rate follows the linear relation (14).
(14)

r = R(1 − w)

Here R means maximum rate of profit. The equation (14) can be represented like figure 1.

Figure 1. Wage and Profit in Sraffa’s Standard Systemw1Rr

Different from Sraffa, Marx’s model follow classical one. There wage is paid before production.
Because of this, both physical input and labor input are multiplied by profit rate. The trade-off between
wage and profit is represented by the equation (15) in Marx’s system.

r=R(1 − w)
1 + 𝑅𝑤

The equation (15) can be rearranged into the equation (16).

4(15)

(16)

1𝑅
r+1=
1𝑤+
𝑅
1+The equation (16) can be represented as figure 2.

Figure 2. Wage and Profit in Marx’s Modelw

1R-1r-1/R

Even in Marx’s model, the trade-off relation between wage and profits can be confirmed. Because ofthis, the fact that real rate of wage rises as the accumulation of capital proceeds is important in Marx’sproposition. On the contrary, the condition that the real rate of wage doesn’t rise at least is crucial in
Okishio’s theorem.

4. Aghion=Howitt Model and Okishio’s Theorem
In Okishio’s theorem, the introduction of a new technology will cause the reduction of production
goods price. People may think an innovation will raise the prices of products. This difference comes fromthe fact that classical economists including Marx adopted a different approach to innovations.
Marx and classical economists adopted value approach. Value is determined by labor cost ofproduction. Because of this, an innovation means a cost cut for them. When the cost is cut, the price will
be reduced. This was the image of innovation for classical economists including Marx.
On the contrary, modern economists adopts value added approach. For them, an innovation produces
a new addition of value added. Because of this, an innovation could possibly raise the product price incontemporary economics.

5Such a modern image of innovation was disseminated by Schumpeter. Schumpeterian way of thetreatment has been integrated into several macroeconomic models of economic growth. HereAghion=Howitt model is considered.
In Aghion=Howitt model, an innovation happens following Poison process of probability. Production
function is represented as the equation (17).
Y = Af(𝐿𝑌 )(17)

Here A is total factor productivity and follows Hicks neutral technological advances. Thoughinnovation happens stochastically, it is necessary for the firm to continue research activities employing
research workers. Total workers are divided into production workers and research workers.
L = 𝐿𝑌 + 𝐿𝑅

(18)

The level-up degree of productivity per an innovation is constant.

𝐴𝑡+1 = 𝐴𝑡 𝛾, 𝛾 > 1(19)

The probability of successful innovation at each point of time is proportional to the number ofresearch workers.
λ𝐿𝑅

(20)

A successful innovator produces goods following the production function (17). The profit can becalculated by subtracting wage from revenue.
Π = Af(𝐿𝑌 ) − 𝑤𝐿𝑌(21)

The condition of profit maximizing is represented as the equation (22).
w = Af′(𝐿𝑌 )(22)

Substituting the equation (22) into the equation (21), the equation (23) is obtained.
Π = A[f(𝐿𝑌 ) − 𝑓 ′ (𝐿𝑌 )𝐿𝑌 ]6(23)Considering the future profit’s discounted present value, the arbitrage condition (24) is obtained.
i𝑉𝑡 = Π𝑡 − 𝜆𝐿𝑅𝑡 𝑉𝑡(24)

The second term of the right side means the mathematical expectation of capital loss for the present
successful innovator.
(1 − λ𝐿𝑅 ) × 0 + λ𝐿𝑅 × (−𝑉𝑡 ) = −λ𝐿𝑅 𝑉𝑡(25)

On the contrary, the arbitrage condition for the future successful innovator is presented as theequation (26).
𝑤𝐿𝑅 = λ𝐿𝑅 × 𝑉𝑡+1 + (1 − λ𝐿𝑅 ) × 0(26)From the equation (26), the equation (27) follows.

𝑤𝑡 = λ𝑉𝑡+1(27)From the equations (24) and (27), the wage rate can be represented as the equation (28).

𝑤𝑡 =λΠ𝑡+1𝑖 + 𝜆𝐿𝑅𝑡+1(28)The equation (28) can be rearranged into the equation (29).

𝑤𝑡λ𝛾
Π𝑡+1=𝑅𝐴𝑡 𝑖 + 𝜆𝐿𝑡+1 𝐴𝑡+1(29)From the equation (22), when the number of production workers increases, wage rate divided by totalfactor productivity decreases. Also considering the equation (18), the function (30) can be considered.

𝑤𝑡= 𝜑(𝐿𝑅𝑡 ), 𝜑 ′ > 0𝐴𝑡(30)On the other hand, from the equation (23), when the number of the production workers increases, theprofit divided by total factor productivity will also increases. Considering this, the right hand of the equation

7(29) can be represented like the function (31).
(31)λ𝛾
Π𝑡+1= 𝜙(𝐿𝑅𝑡+1 ), ϕ < 0𝑖 + 𝜆𝐿𝑅𝑡+1 𝐴𝑡+1Considering the functions (30) and (31), the relation about the number of research workers can beobtained from the equation (29).
𝐿𝑅𝑡+1 = 𝑔(𝐿𝑅𝑡 ), 𝑔′ < 0(32)The dynamics of Aghion=Howitt model can be expressed as figure 3.

Figure 3. Dynamics of Aghion=Howitt Model
LRt+145ºLRt

In the stationary state, the number of production workers is fixed because the number of research
workers is fixed like figure 3. In Aghion=Howitt model, the present innovator can obtain profit until thenext innovation happens. Once the next innovation happens, his profit will disappear. Concerning the newinnovator, the same thing can be told.
The profit rate is kept on a level by continual innovation in Okishio’s theorem, too. In this meaning,
the images of economic growth are almost the same both in Okishio’s theorem and Aghion=Howitt model.
The difference is that an innovation reduces the price of production goods in Okishio’s theorem while therelative price is not considered in Aghion=Howitt model because it is a macroeconomic model.
There is another difference. The wage rate is fixed in Okishio’s theorem while the wage rate changes
according to the frequency of innovation like the equation (27) in Aghion=Howitt model. The reason is thatthe same wage rate is applied to both production workers and research workers in Aghion=Howitt model.

85. Conclusion
Though Ashion=Howitt did not consider the tendency, Schumpeter wrote that the frequency ofinnovations would decrease. In this meaning, Schumpeter also reached the same conclusion as Marx finally.
Finally, there is what may be described as the “material” the capitalist engine feeds on, i.e., theopportunities open to new enterprise and investment. The theory under discussion puts so muchemphasis on the element as to justify the label we have affixed to it. The main reason for holding that
opportunities for private enterprise and investment are vanishing are these: saturation, population, newlands, technological possibilities, and the circumstance that many existing investment opportunities
belong to the sphere of public rather than private investment (Schumpeter, 2009, 123).
Here Schumpeter discussed the vanishing opportunities of innovations. The circumstances whichcauses innovations become poorer than before. This is the objective reason of innovations’ decrease.
The great bursts of economic activity that used to spread the symptoms of prosperity all over theeconomic organism have of course always been associated with expansions of producers’ expenditurethat were in turn associated with the construction of additional plant and equipment. Now some
economists have discovered, or think they have discovered, that at the present time new technologicalprocesses tend to require less fixed capital in this sense than they used to in the past, particularly inthe epoch of railroad building. The inference is that spending for capital construction will henceforthdecrease in relative importance (Schumpeter, 2009, 129).
Here a subjective reason of vanishing innovation was written. He wrote that the tendency ofdecreasing capital construction could be seen. The tendency, however, can be told to hold back the processof diminishing rate of profit that Marx’s proposition. Even so, when the speed of decreasing innovation
excesses the advancement of capital’s organic composition, the rate of profit will be reduced

References:Aghion, P. & P. Howitt, ‘A Model of Growth through Creative Destruction,’ Econometrica, Vol.60, No.2,323-351, 1992.
Okishio, N., ‘Technical Changes and the Rate of Profit,’ Kobe University Economic Review, 7, 85-99.
Schumpeter, J. A., Capitalism, Socialism and Democracy, Corporate Governance Vol.3, Pickering &Chatto, London, 2009.

9

Sraffa, P., Production of Commodities by Means of Commodities, Cambridge University Press, 1960.

10