Cascade Model of Innovative Dynamics with Investment Flows

To understand the phenomenon of economic development, we need a comprehensive study of the innovation process stages, starting with the origin of a basic innovation and ending with the exhaustion of its potential for economic growth and the associated subsequent transition to a new technological structure. Starting with1,2 a significant number of studies have been devoted to the investigation of this phenomenon; their outcome obviously testified that 1) In the innovation process, there are S-type trends, and 2) Between variables, there is a sophisticated system of interrelations, which may be correctly described only by using endogenic evolution models3. A standard in the innovation development theory has long been the metamorphosis theory proposed by Mensch however, after Hirooka showed that the innovation process is described by the cascade model of three consecutive logistic curves, the investigations in this field have been given a new impetus. Actually, the cascade model and its role in the economic growth, namely, its relationship with Kondratieff ’s long-term cycles is widely discussed in the literature6–13. One of the topical issues of the innovation Abstract

development theory is related to the mechanisms of mutual influence of innovations and the economic environment, in particular, the role that investments play in the acceleration of the innovation process.To describe these mechanisms, based on the Hirooka's cascade model, as well as on the analysis of the relationship between the number of registered patents and the scope of investments, we suggest in this paper an extended cascade model that includes investment flows.The dynamics of economic development is characterized in this case by a system of differential equations, the solution of which forecasts the emergence of long-term cycles.For empirical verification of this model, we conduct a spectral and temporal analysis of the U.S.A. GDP dynamics, where we found long-term cycles with duration of about 28 years.We explain the existence of these cycles by the diffusion of trunk innovations according to Hirooka 5 .
The paper is built as follows.In the next section, we enumerate the used data and describe in detail the methods for their analysis.Then, we present the main theoretical results: The extended cascade model, as well as the model of the GDP dynamics on the basis of the Mensch's metamorphosis theory.Further, in the "Empirical results" section, we verify the suggested models for their compliance with the statistical data.To conclude, we sum up the results and discuss the direction of future research.

Data
In this paper, we confirm the constructed models by the example of data on the U.S.A. economy.In particular, we analyze the long-term dynamics of indicators, such as the real GDP, the volume of private investments in R&D and equipment, the amount of personal savings, the index of production of goods for the semiconductor industry and the number of patents used for the first time in the semiconductor industry.The sources of these data are Bureau of Economic Analysis, Board of Governors of the Federal Reserve System and U.S. patent and trademark office.

Methods of Analysis
The investigation of influence of the technological component on economy dynamics is based on the concept of cascade structure of the innovation paradigm, proposed by Hirooka 5 and extended by Akaev 6 .These studies showed that the innovation paradigm has a cascade structure consisting of 3 trajectories: technological, development and diffusion (Figure 1).In this work we make attempt to characterize the mechanism of these trajectories with the aid of time-series analysis of the number of patents first applied, investment in R&D and equipment, as well as production volume.Additionally, based on the G. Mensch's metamorphosis theory, we build a mathematical model linking the cycles in the real GDP with the amounts of investments and savings.
To empirically identify the economic cycles initiated by the innovation development, the paper uses the methods of spectral-temporal analysis 14 , in particular, the Wigner-Weyl transformation.To describe the process of the simultaneous disintegration of the "old" system (some institutes or an entire industry) and breakthroughs in some "new" field, Mensch 4 used the catastrophe theory.However, for the purposes of simulating a simultaneous combination of the processes of destruction of the "old" and breakthrough of the "new", the Wigner-Weyl transformation is more convenient.This idea was suggested to the authors by G. Mensch in 2014, at the conference devoted to cyclical processes and structural transformations in 2014 in the University of Wildau (Germany).According to him, this idea was first proposed as early as 1980 by H. Haken and W. Weidlich, but has not been developed in detail.This method is widely used in the quantum physics for the analysis of incoming signals.In economics, to simulate the dynamics of industrial clusters and a series of diffusions, the Wigner-Weyl's idea of pseudo-particles transformation (or quasi-probabilities) was previously used in Wold, Kaasch 15 .We would like to specify that in our paper, while identifying impulses for the growth and decline of macroeconomic indicators, we use a modification of the Wigner's function -a Wigner-Ville pseudo-transformation, which is widely used in computer application software for spectral-temporal analysis of non-stationary signals 16,17 .

The Cascade Model
The long wave theory of N. Kondratieff significantly contributed to development of the economy innovation development theory.He discovered cyclic nature of interchanging industrial stages, established an idea of multiplicity of cycles and developed cycle models.N. Kondratieff revealed empirical regularities characterizing long-term economic fluctuations represented (in one combination or another) in profound changes of production engineering and trade (in turn, preceded by significant innovations and discoveries), conditions of money circulation, as well as in strengthening of the role of emerging nations in the world economic system 1 .Thus, the formulated wave theory of N. Kondratieff and its improvement by S. Kuznets 18 allowed seeing the possibility to accelerate overcoming of another cyclic economy crisis with the aid of implementation of radical technical and economic innovations.On the basis of N. Kondratieff 's works J. Schumpeter proved existence of innovation cycles 2 .
Mensch 4 inspired a renewed interest in the innovation development theory after certain inactivity in this area and proposed a metamorphosis model of cycles of structure change.The further development of innovation theory was provided by Rogers 3 , Forrester 19 , Freeman 20 , Kingston 21 , Kleinknecht 22 , Perez 23 , Grubler 24 , Van Duijn 25 .At the same time Korotayev and Tsirel 11 , Korotayev and Grinin 12 , Denison 26 , Modelski 27 , Glaz' ev 28 and other economists researched different business cycles.In course of development of innovation theory Hirooka 5 established the three-cascade model of the innovation paradigm.
In this paper, we develop the Hirooka's three-cascade model 5 by adding two investment trajectories thereto: the amount of investments in research and development and the amount of investments in equipment (Figure 2).Following Hirooka 5 before an innovation enters the market, it must be adapted with the help of improving innovations.This process corresponds to the technological trajectory (Figure 2).As a result, new products (a development trajectory) will be created.However, both trajectories need an investment; that is why we introduce into consideration a trajectory of investments in research and development as a required motivating factor for the existence of a technological trajectory and a trajectory of development.For the same reason, we introduce into consideration a trajectory of investments in equipment.It facilitates the transition of the innovation process from the trajectory of development to the trajectory of diffusion.This mechanism clearly shows that the emergence of a trunk innovation itself is not the key to economic prosperity.The required economic institutes ensuring an investment inflow to separate parts of the system are necessary to make this prosperity the most complete.Thus, a cascade model of innovation development is created, completed by investment flows, as shown in Figure 2.An empirical confirmation of this model, based on the example of the semiconductor industry, is provided in section 4.1.

GDP Dynamics Model on the Basis of the Metamorphosis Theory of G. Mensch
To model gross product dynamics, let us consider differential equation system: Here Y (t) -Production volume (GDP level), U (t) -Investment volume, V (t) -Savings volume (in regard to available cash assets), μ, λ 1 , λ 2 -Parameters of response of investment and savings' delay to change in economics, α, β, γ -Coefficients of proportionality.
A compact mathematical notation in the form of a system (3.1) has the following economic interpretation.The dynamics of the gross products depends on the imbalance of investments and savings: The growth of investments ensures the acceleration of the GDP growth rate, the growth of savings leads to a slowdown in the GDP growth rate, the equality in the growth rates of investments and savings creates a state of equilibrium (see the first equation).At the same time, the dynamics of investments and savings is proportional to the rate of change in the gross product, but with a certain shift caused by inertia and delayed reaction to the changes in the macroeconomic situation (the second and third equations).
The GDP growth against the background of a large amount of free cash ensures an increase in investments.If the production output volume increases and available funds reduce, it results in the economy's "overheating", leading to investment weakening.The decline in production and the growth of savings indicate the economy's recession, resulting in investment cutback.A simultaneous decline in the GDP and the amount of free cash forces to take actions for overcoming the crisis, which begins to stimulate the investment growth.
The GDP and investment growth determines the inexpediency to keep the money dormant, which reduces the amount of savings.A simultaneous decline in the production and investments leads to an economic crisis, resulting in the reduction of available funds.The production output growth and decline in investment predict an imminent reversal of the economic trend, which forces to increase savings.The decline in production with a high level of investments indicates economy's inefficiency, which forces to increase the savings.
The solution of the system (3.1) for Y (t) is the sum of the linear function Ȳ (t), that characterizes the main trend and the composition of tangent and arctangent functions Y (t), describing cyclic S-shaped fluctuations: A detailed solution of the system (3.1) is provided in Appendix 1.
The main problem of model (3.1) is a linear trend in the solution of the differential equation system for Y(t), which does not correspond to dynamics of GDP index.This problem is solved, if we apply the initial system to the GDP, investment and saving logarithms.In sections 4.2 and 4.3, we present the empirical results of applying this model.

The Cascade Model by the Example of Semiconductor Industry
Throughout the 1990s, the leadership in the innovation development belonged to the semiconductor industry.It ensured a phenomenal jump in the U.S.A. economic growth and gave a powerful impetus to the development of other high-tech sectors.However, the development of this industry began long before 1990, the process of the product diffusion to the market had been preceded by a long period of scientific and technological development.
First of all, let us consider dynamics of the number of patents first applied in production of conductors and other electronic components (NAICS -3344) since 1964, provided by the USPTO (Figure 3).This distribution can be split into the number of new technologies (technological trajectory) and new products (development trajectory).The results of approximation of the number of patents by the sum of 2 normal distributions are represented in Figure 3. Additionally, Figure 4 shows the cumulative function of the number of technologies and the number of new products (the technological trajectory and the development trajectory), corresponding to the Hirooka's cascade model.Now, let us compare these two trajectories with the data on the amount of investments in research and development and the data on the amount of investment into equipment (Figure 5).Let us also compare the development trajectory and the amount of investments in equipment with the index of industrial production in the semiconductor industry (Figure 6).After determination of correlation between the basic innovation and investment flows we shall study influence of investment flows on economy growth.To that effect, we begin with checking of the proposed mathematical model described in cl.3.2.

Spectral-Temporal Analysis
Let us consider US GDP in comparable prices from 1929.Let us move on to the GDP logarithm, eliminate the linear trend and analyze the series of first remainder using the spectral-temporal analysis.For the analysis, we use the software provided by PSEL 29 .With the aid of pseudo Wigner-Ville distribution we construct a time and frequency series enabling to extract non-stationary signals.
The obtained distribution of impulses in the first remainder of the series of US GDP is represented in Figure 7.Here (Figure 7) time in natural units is laid off along the X-axis and frequency is laid off along the Y-axis.Horizontal zero level variations are considered to be the signals.
Let us compare results of spectral-temporal analysis with dynamics of the first remainder of the series.Let us plot on the graph of the first remainder of the series border lines of the signal beginning and end in accordance with the Wigner-Ville distribution.Similar to technical analysis used to study results of stock market trading, we determine the signal type ("growth" or "decline") observed within the borders.The results are represented in Figure 8.

Modeling of Innovative Waves
According to the solution of system (3.1) for GDP, combination of arctangent of tangent mathematically represents innovative waves.Here arctg is responsible for the wave trajectory and its S-shape and tg is responsible for cyclicality and jumps characterizing the end of one innovation wave and beginning of the other. Where: A -Innovative wave force (the more its value, the more is economy growth after depression, provided by innovations); В -Economy recovery speed (the more its value, the shorter is depression interval); С -Wave angle (the less is its value, the smoother is transition from wave generation to saturation); D -Time adjustment, shifting wave graph along the X-axis; E -Wave adjustment, shifting wave graph along the Y-axis.
Let us model innovative waves for the first remainder of the series.Moreover, we construct the classic sinusoid for the series, representing cyclic motion of economy.The results (remainder, innovative waves and cyclic motion) are shown in Figure 9.As a result of modeling, we obtain 3 innovative waves at intervals of 28 years.The first wave characterizes US recovery from the Great Depression and is related to military production increase during the World War II.Transition from defense economy to peacetime economy is related to the next innovative wave.Technological studies are applied in other industries.The oil crisis of the 1970s broke off economy expansion.Development of computer technologies was a trigger to recovery (the last innovative wave).As these waves closely correlate to the trunk innovations, according to Hirooka there is no doubt that their origin is based on the innovation process.
In regard to the sinusoid, the modeled S-shaped waves exhibit the following characteristics: The beginning of waves corresponds to sinusoid valleys (this characteristic confirms the basic hypothesis related to activation of innovation processes during a crisis or depression), the end corresponds to the point after a sinusoid peak.
Increase of the S-shaped wave corresponds to a growth signal; jumps and transition correspond to a decline signal.Thus, we confirm our assumption related to correlation between signals in the first remainder of the series with innovative waves.
It is necessary to point out that the innovative wave with the beginning in 1990s pretty accurately characterizes the world crisis of 2008.GDP decline corresponds to transition from one trajectory to another.
Comparison of remainder variances allows concluding that equation 4.2 characterizes cycles in GDP dynamics more accurately (Table 1).

Conclusions
In the course of our investigation, we have analyzed the technological and investment components of the economic growth.We managed to identify interrelations between them and improve the cascade model of the innovation paradigm developed by M. Hirooka and completed by A. A. Akaev, due to adding investment flows to it.
Based on the Mensch's metamorphosis theory, we suggested a mathematical model of innovation dynamics, which describes cyclic asynchronous fluctuations in the GDP index against the trend and allows for predicting with a high accuracy the dynamics of the macro-indicator in the short term, as well as for estimating the nearest crisis occurrence timelines.We suppose this result to be of the utmost importance from a practical point of view as it allows developing economy policy in accordance with upcoming changes.
In the course of the empirical part of the investigation, we came to the following conclusions: • There is a correlation between technological development, investment activity and economy growth, which can be characterized by a cascade model.The basic innovation requires investment flows for its study and development, thus forming the technological trajectory.These 2 interrelated processes lead to development of new products requiring investments as well.Further we observe mass production affecting other industries and economy growth.• At the certain stages of the life cycle the innovation waves develop growth and decline signals in regard to macro indicators.Thus, uprising section of the innovation trajectory characterizes the stage of economy recovery and its further growth.On the last stage of the innovation wave we observe market saturation ended with a crisis and depression.• The performed studies can be considered basic.
In our view, there are several options of further development.Firstly, to confirm and develop the proposed ideas it is necessary to study economies of other countries.Secondly, the cascade model requires further improvement: additional factors must be added, and the bifurcation point needs to be considered more closely.

Acknowledgement
The study was conducted with the support of Russian Science Foundation (Grant No.14-28-00065).

Appendix
The original differential equation system is represented as follows: Let us study solvability of (1) system of equations.Let us consider λ 1 = λ 2 = λ case.Multiplying the second equation of (1) by γU and adding the result to the third equation multiplied by βV , we obtain: After integration we obtain Due to substitution of Y(t) in (3) the solution of system (1) takes its final form.Here A, B, C -random real numbers, where α 2 B 2 (β + γ) < (μ -λ) 2 βγ .

Figure 1 .
Figure 1.Three-cascade model of the innovation paradigm.

Figure 2 .
Figure 2. Cascade model of the innovation paradigm with investment flows.Basic innovations lead to occurrence of new structural industries, development of which initiates economy recovery and growth.Consequently, breakthrough innovations help to overcome depression and therefore, lead to growth in demand.A long-term growth cycle is being formed.But, in the process of exhaustion of the innovation potential and the associated increase in the level of savings within a system, a financial crisis occurs and to overcome it, a new wave of trunk innovations is needed.Thus, investments and savings play an endogenic role in the innovation development.This logic has led us to the creation of the mathematical model presented in

Figure 3 .
Figure 3. Number of patents first applied in the semiconductor industry and approximation.

Figure 4 .
Figure 4. Technological and development trajectories in the semiconductor industry

Figure 6 .
Figure 6.Number of commodities, investment volume and production level.

Figures 4 -
Figures 4-6 confirm our assumptions in regard to correlation between investment flow and technological changes.Appearance of the basic innovation leads to increase of the number of technologies via researches and further increase of the number of new products.As a result, the innovation leads to development of an industry or group of industries, as well as structure changes and growth of macroeconomic indicators.After determination of correlation between the basic innovation and investment flows we shall study influence of investment flows on economy growth.To that effect, we begin with checking of the proposed mathematical model described in cl.3.2.

Figure 7 .
Figure 7. Signals in the first remainder series.

Figure 8 .
Figure 8.First remainder of the series with signals.