Information Models for Forecasting Nonlinear Economic Dynamics in the Digital Era

The aim of this study was to develop an adequate mathematical model for long-term forecasting of technological progress and economic growth in the digital age (2020-2050). In addition, the task was to develop a model for forecast calculations of labor productivity in the symbiosis of “man + intelligent machine”, where an intelligent machine (IM) is understood as a computer or robot equipped with elements of artificial intelligence (AI), as well as in the digital economy as a whole. In the course of the study, it was shown that in order to implement its goals the Schumpeter-Kondratiev innovation and cycle theory on forming long waves (LW) of economic development influenced by a powerful cluster of economic technologies engendered by industrial revolu-tions is most appropriate for a long-term forecasting of technological progress and economic growth. The Solow neoclassical model of economic growth, synchronized with LW, gives the opportunity to forecast economic dynamics of technologically advanced countries with a greater precision up to 30 years, the time which correlates with the continuation of LW. In the information and digital age, the key role among the main factors of growth (capital, labour and technological progress) is played by the latter. The authors have developed an information model which allows for forecasting technological progress basing on growth rates of endogenous technological information in economics. The main regimes of producing technological information, corresponding to the eras of information and digital economies, are given in the article, as well as the Lagrangians that engender them. The model is verified on the example of the 5 th information LW for the US economy (1982-2018) and it has had highly accurate approximation for both technological progress and economic growth. A number of new results were obtained using the onstrated that the symbiosis of human and intelligent machine (IM) is the driving force in the digital economy, where man plays the leading role organizing effective and efficient mutual work. Authors suggest a mathematical model for calculating labour productivity in the digital economy, where the symbiosis of “human + IM” is widely used. The calculations carried out with the help of the model show: 1) the symbiosis of “human + IM” from the very beginning lets to realize the possibilities of increasing work performance in the economy with the help of digital technologies; 2) the largest labour productivity is achieved in the symbiosis of “human + IM”, where man labour prevails, and the lowest labour productivity is seen where the largest part of the work is performed by IM; 3) developed countries may achieve labour productivity of 3% per year by the mid-2020s, which has all the chances to stay up to the 2040s.


Introduction
The widespread use of digital technologies has led to increasing attention being paid to issues such as the relationship between the digital economy and the development of new business models [1], the role of new breakthrough technologies in changing economic relations [2], and reducing the main costs associated with digital economic activity [3]. As a first step in understanding the nature of digital transformation, it is proposed to distinguish between the "digital sector" and the increasingly expanding trend of digitalization of the modern economy, which is often called the" digital economy" [4]. New approaches to classifying sectors are being formed depending on the degree of their transition to digital technologies [5]. A true "digital economy" is now defined as "a part of the economic output generated exclusively or predominantly by digital technologies and business models based on digital goods or services" [6]. Since digital technologies cause accelerated changes in the organization of labor, proposals for new social approaches to the development and use of innovative models in the digital economy are of great interest [7]. One of the most recent publications [8] addresses the issues of creating a unified system for measuring the digital economy, including a set of existing indicators for measuring jobs, employee skills, as well as the infrastructure for implementing digital technologies. However, we did not find any work on modeling and forecasting the dynamics of the digital economy for the future, which is necessary for developing a long-term development strategy.
It should also be noted that certain aspects of the digital economy are evaluated differently by the expert community. Some believe that the future belongs to robots and people working together, as automation gives people the opportunity to focus on more skilled and highly paid tasks [9]. However, measured productivity growth in the US decreased by half over the past decade, whereas the real income of most Americans has not changed since the 1990s. In this regard, the authors of the study [10] believe that the impressive capabilities of intelligent machines will not be fully realized until special additional innovations are developed and implemented. That is why it is proposed to solve the discrepancy between the skill requirements for working with new technologies and the possibility of implementing automation with elements of artificial intelligence at the expense of other technologies that increase productivity [11]. Unfortunately, there is still no model for calculating labor productivity in the symbiosis of "man + intelligent machine", in order to evaluate the effectiveness of the methods of organizing labor in the digital economy proposed in the aforementioned studies. As for the impact of IT on economic growth and the growth of aggregate factor productivity, it should be noted that such estimates already exist [12]. There is also a study on quantitative impact of IT on economic growth based on data from 39 African countries for 2012-2016 [13]. However, models for calculating and predicting growth in the digital age do not yet exist.
As can be seen from the brief review of the above-stated studies, many problematic issues of digital economy formation have already been investigated and quite interesting results, which answer these questions, have been obtained. In our work, we aimed to supplement and expand the existing methods of studying the digital economy, in particular on the basis of the information model developed by us [14], which allows us to predict technological progress on a longterm basis, using the growth rate of endogenous technological information in the economy. The article presents the main modes of production of technological information corresponding to the epochs of informatization and digitalization of the economy, as well as the Lagrangians that generate them. The model was tested (verified) on the example of the 5th information LW for the US economy  and showed a very high accuracy of approximation both for technological progress and for calculating economic growth. Using the developed model, an example of calculating the forecast trajectory of economic growth for the United States at the stage of the digital economy formation (2018-2042), as well as a forecast calculation of labor productivity in the symbiosis of "man + intelligent machine", are given.

The Schumpeter-Kondratiev Innovation and Cycle Theory of Economic Development
had an uneven, unstable and cyclic character of development. Crises and depression are logical, and the governments must forecast them and try to mitigate them. In such periods of cyclic crisis recessions and depressions in the world economy the scholars tend to turn to the Kondratiev long wave theory (LW) of economic development [15], lasting for 30 -40 years in the modern era. It was so in the period of the Great Depression of the 1930s, then it happened in the 1970s and 1980s during the structural crisis of the world economy, when profound works on the LW theory were created. The compound analysis of these works can be found in the research [16]. We have developed a complete closed mathematical model in the work [17] to describe and calculate the Kondratiev long wave of economic development, which allows us to forecast all major economic variables with a long-term perspective up to 30 years, which corresponds to the length of LW. We can say that nowadays there exists a satisfactory and comprehensive explanation of the LW theory or long economic cycles. Kondratiev stated that long economic cycles have an endogenous character, i.e. they are internally characteristic of capital economy. It is important to state that he was the first to understand that wavy cyclic movements of the economy deviate from a balance which the capitalist economy tries to preserve. Consequently, most of the time a healthy dynamic economy develops in imbalance, while the classical economic theory stated the contrary. Kondratiev himself identified first (1780/90-1845/51), second (1845/51-1890/96) and third (1890/96-1940/46) long economic cycles. Besides, at the beginning of the 1920s he made an assumption that a cyclic crisis should take place at the end of the 1920s followed by the depression, which indeed happened in 1929 and the 1930s. Another great economist of the 20 th century, Joseph Schumpeter, continued Kondratiev's study of LW and developed an innovation theory of long waves integrating it in his general innovation theory of economic development [18]. Schumpeter saw cycles as the direct consequence of innovative processes determined by technological progress, and, as well as Kondratiev, he believed that the cyclical movement the output was a deviation from the balance. It should be mentioned that Schumpeter highlighted that the main driving force of the capitalist economy was innovations and entrepreneurship, not capital per se, as many economists of that time thought. Schumpeter stated that capital is useless and powerless to cause economic growth without innovations and initiative, will and perseverance of the entrepreneur. Schumpeter believed that spontaneous "blobs" of innovation by forming large clusters, caused radical changes in economy, taking it away from the original balanced trajectory only seen in the periods of stagnation. Moreover, the system never returns to the original balance. A new cycle begins at the end of another depression at a new level of balance.
According to Schumpeter, the change in balance levels determine the longterm trajectory of economic development, during which economic system is in a dynamic, not a stationary, balance. Both Kondratiev and Schumpeter believed that there existed three types of balance, and, consequently, three oscillating movements, which consist of short Kitchin cycles (3 -5 years), provoked by oscillations of inventories; medium industrial Juglar cycles (7 -11 years) and long Kondratiev cycles (30 -40 years). These three waves overlap the trend trajectory of economic growth and, according to Schumpeter, their superposition shows the general state of economic development at every moment [18].
Schumpeter was the first to assume that innovations appear unevenly in time, and then spontaneously unite in clusters of innovations. He distinguished basic and ameliorative innovations. He highlighted the key role of basic innovations in the cyclic dynamics of long waves of economic development, seeing them as the main driving force of the capital economy. In fact, he predicted the 4th afterwar LW  generated by a large cluster of epoch-making basic innovations, including computers, electronics, televisions, jet and rocket engines as well as nuclear energy. Since the long economic cycle concept plays a highly important role in the Schumpeter innovative theory of economic development, and taking into account the fact that Schumpeter himself saw it as a cornerstone of his theory, we decided to call the latter "the Schumpeter-Kondratiev innovation and cycle theory of economic development." The Schumpeter-Kondratiev innovation and cycle theory of economic development is valuable because it suggests an effective mechanism to overcome the global cyclic crisis followed by the depression through "the launch and comprehensive stimulating of storm of the new generation of highly effective basic technological innovations" [19], aiming at substitution of outdated industrial technologies and forms of organizing production. It is also important that this theory shows in a certain way the period of crises and depressions, and it also has an innovative paradigm for forecasting the beginning of a new cycle [20].
The success of the Schumpeter-Kondratiev theory was already evident in the 1980s. Firstly, it should be noted that Mensch then predicted the appearance of a cyclic structural crisis of the world economy at the end of the 1970s in its beginning. Secondly, he accurately pointed out the feature characteristic of the forthcoming crisis-"stagflation", which meant that economic stagnation would be coupled with rise in prices, i.e. inflation, and not their decrease as it was before.
Thirdly, he clarified that under such circumstances monetary and credit policy could not help in resolving the problem of overcoming the crisis [19]. Mensch and other adherents to the Schumpeter-Kondratiev theory suggested launching the process of mastering the basic innovations of a new technological paradigm based on the achievements of microelectronics and information technology.
The Schumpeter-Kondratiev innovation and cycle theory was validated during the 5 th LW . The core for the 5 th technological paradigm was microelectronics, personal computers, information and communication technologies (ICT) and biotechnology [20]. Microprocessors and computers became widely spread and their usage turned out to be a breakthrough in goods production in each field of the economy and control of dynamic objects. It was demonstrated in the work [21] that the beginning of the 5 th long economic cycle dated back to 1982. Indeed, it was 1982 when there was a rise in the world economy, which then transformed into a lengthy (1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994) period of a stable and quite rapid economic growth with average annual rates of 3.4 percent, which ended with a slight fall in 1995. Then, economy flourished from 1996 to 2006 when labour productivity rates reached 2.8% per year and they almost twice exceeded the same figure for the previous decade (1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995).
Mentioned achievements could be explained by the usage of ICT and an upsurge in investments in the sphere. It explains the phenomenon of an intermittent labour productivity growth in the second half of the 1990s in developed countries. At the beginning of the 21 st century the global ICT market exceeded 1 billion US dollars. At that time the talks on the emergence of new economy-"knowledge economy"-began in developed countries. However, by the mid-2000s the growth of production caused by ICT stopped. According to Mensch, it meant that the 5 th long economic cycle reached its peak and it was necessary to search for innovative technologies and next generation products. In 2006-2007 there was a recession in economic growth rates in OECD countries, which meant the transfer from an upward stage of the 5 th LW to a downward. Thus, 2004-2005 were the upper turning point for the 5 th LW. The continuity of the upward stage of the 5 th LW predictably constituted 22 -24 years (1982-2006).
Less than three years had passed before a sudden world financial and economic crisis of 2008-2009 broke out, which resembled the crisis of 1929, followed by the Great Depression of the 1930s, and thus called "The Great Recession". In the work [22] we have shown that a bursting rise in prices on such highly liquid commodities as oil and gold is a forerunner of global cyclic financial and economic crises and we have also developed a nonlinear dynamic model for forecasting the onset of the crisis. With the help of this model we successfully predicted the date of the second wave of the global financial crisis of August 2011 nine months before it actually took place with an error of only two weeks and it showed that the crisis of 2008-2009 could have also been predicted beforehand.
Then, with the help of the Hirooka innovation paradigm [20] we also described the trajectory of developing basic technologies of the 6 th technological paradigm and predicted the beginning of the 6 th LW-2017-2018. Next, we calculated economic potential of NBIC-technologies [23]. Since NBIC-technologies are mutually convergent, a significant synergetic effect is achieved due to their cooperative action, and this effect is to accelerate the rates of technological progress in developed countries up to 3% and higher by 2030, which is much better than the same rate during the rise of the 5 th long economic cycle (1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994), which was 2.3%. Consequently, basic technologies of the 6 th technological paradigm will be able to ensure record rates of economic growth close to those back in the 1950-1960s.
Indeed, as was predicted in the Schumpeter-Kondratiev theory, in 2017 there was a simultaneous growth of leading world economies. According to the IMF, by the end of 2017 all eight world economies (the USA, China, Japan, Germany, India, Russia, France and the UK) increased by over 1.5% [IMF, 2018]. All the world economy also increased by 3.8% in 2017 against 3.2% in 2016. In 2018 a simultaneous growth of GDP was seen in almost each of those 45 countries observed by the OECD. The IMF points out that such an event has occurred only twice over the past 40 years and adds that previously such periods of a simultaneous growth lasted, as a rule, for several years. For example, the world economy grew at a rate of 4% annually in 1984-1989 and 2004-2007, i.e. at the beginning of the development and on the peak of the upward stage of the 5 th LW. It is evident that since 2018 we have observed the development of the 6 th LW, which will last for about thirty years.
The question arises: will the current simultaneous growth of developed world economies be stable in the medium term? Kondratiev and Schumpeter noted in their classical works that the economic growth at the beginning of the development at the upward stage of the long economic cycle is subjected to various risks, which make it unstable, and recommended that the governments assist entrepreneurs with overcoming such risks. The main risks that the current stable economic growth faces are: a large scope of a total debt of governments, households, corporate and financial sectors; aggravating gap between the real economy and the financial sphere; accelerating growth of the excessive income inequality; an acute shortage of consumer demand; instability of the financial system and sharp increase in protectionism from developed countries, transformed in trade wars, and increased environmental threats. It is possible to solve all these problems only globally, for instance, at the G20 summits. However, contrary to the crisis years of 2008-2009, there is no spirit of global cooperation nowadays. At that time the G20 countries acted as a single whole, which played a unique role in preventing the worst consequences of the financial crisis. Now we can see trade and ideological conflicts between leading countries, which hinder constructive cooperation in the context of the G20. That is why there are reasons to believe that the development of the 6 th LW will be quite unstable and may be interrupted by crisis recessions, not so lengthy and deep as "The Great Recession" of 2009.

Mathematical Models for Describing and Forecasting the Dynamics of Information and Digital Economy
To describe a long-term economic dynamics in a technologically developed country we can successfully use a classical Cobb-Douglas production function (PF) with a labour-saving technological (technical) progress [14]: where ( ) Y t is a current national income (GDP); ( ) K t is a productive capital; ( ) L t is a number of employees in the economic sphere; h is the average level of human capital; ( ) A t is technological progress; α is the capital share in GDP; δ is the parameter characterizing the growing impact of production scale ( 0 δ > ); γ is a constant normalizing index. It is explained by the fact that informationand the future digital economy-is the same high-technology economy where the key role for increasing the production of main factors, capital and labour, is played by ICT, digital technologies and platforms. To calculate in the retrospective area with the help of PF (1), we used the data from the following sources: In PF (1) and in the time of information and digital economy capital and labour may be described with the help of traditional methods and models ( [24], §2.1). However, technological progress ( ) A t cannot be described using traditional models ( [24], ch.5) since they do not have the main factor of informational and digital economy-the speed and scope of the production of technological information. In what follows we give an account of mathematical models for calculating and forecasting the dynamics of technological progress in information and digital economy, based on (speed) rates of the production of technological information. These information models first appeared in our work [14] and were developed in the work [25]. For practical application it is better to write and use PF (1) in a tempo-based form by means of its logarithmic differentiation: where (2а) If the functions describing the main economic variables in PF (1) are known, it is easy to find their rates of growth by using Formulas (2a). Vice versa, if the growth rates of variables are given (2a), there is no difficulty in determining variables by formulas We should note that here and above it has concerned an average technological level ( ) A t over the whole economy, which is clearly defined by high technological level of newly formed innovative fields of the economy. In our case these are the fields of information and digital economies.
We took up in (1) 1 h = supposing that its real value will be based on the assessed value of normalizing multiplier/factor γ. In Figure 1 you can see the trajectory of the US GDP calculated on the basis of PF (1) with certain values of constant parameters (4) together with a factual trajectory. As shown in the figure, there is a perfect coincidence in each of the stages. Besides, mean square error of approximation did not exceed 0.05%.
Thus, PF (1) may be successfully employed in long-term forecasting of the trajectory of economic growth in case when there are reliable long-term forecasts of the dynamics of key growth factors-K, L and A. It is necessary to take into account the changes in long-term tendencies in the development of modern economy, which were profoundly explored by a French economist Thomas Piketty [26]. In the 20 th century there was a range of empirical regularities corresponding to the process of long economic growth, which are justified in a longterm period, when the results of various economic and financial crises are smoothed. A number of these regularities were first formulated by an American economist Nicholas Kaldor [27], and some of them still remain in force even nowadays. However, some of these empirical regularities do not work anymore, which signals the changes in tendencies in the development of the leading economy of the 21 st -century.
For our future analysis the following Kaldor's regularities are of special importance: 1) The ratio of physical capital to output is nearly constant, i.e. а) where K σ is index of capital intensity; K k is the index of capital productivity; in this connection, 2) The shares of capital and labour in national income are nearly constant, i.e. const α = and 1 const 3) The wages for workers if labour share in GDP is constant grow in proportion to labour productivity i.e.
where ( ) w t is a current average wage; 0 a is a normalizing index. Such wage growth was seen in the golden age of the capitalist economy flourishing .

New Empirical Regularities of Economic Development in the First Half of the 21 st Century
The new tendencies of capital accumulation, economic growth, and income inequality emerging at the beginning of the 21 st century were comprehensively studied by Thomas Piketty .
As we can see, changes in capital intensity in the 20 th -century United States were more limited than in Western European countries, which gave the impression of its stationarity, which was recorded by Kaldor in the first empirical regularity (5).
The return of the value of capital intensity K σ in developed countries in the 21 st century to its maximum value means that it stabilizes again, at least until the middle of the century ( [26], ch.5). Hence, the first empirical regularity of Kaldor (5) remains valid in the first half of the 21 st century and will be the determining factor in transformations of PF (1). Our calculations of the value of capital intensity in the US economy for the entire afterwar period (1946-2017), presented in Figure 2 with two graphs (current and comparable prices) show that the US capital intensity remained practically constant throughout the whole period, stabilizing at the beginning of the 21 st century around the stationary value As for the second empirical regularity of Kaldor (6), in the 21 st century it ceases to operate in practice: the share of capital income in GDP (α) will no longer be a constant value, but will grow as Piketty claims [26]. For example, in Western European countries it has already risen from 20% -25%, characteristic of the mid-20 th century, to 25% -30% by the beginning of the 21 st century.
Piketty believes that the share of capital income at the global level will reach 30% -40% by the middle of the century, i.e. the level close to the indicators of the 18 th -19 th centuries ( [26], ch.6) with average profitability of capital of 4% -5%.
Such precedents have already taken place in economic history: an increase of 10 percentage points from 35% -40% at the turn of the 18 th -19 th centuries up to 45% -50% by the middle of the 19 th century. As for the labour share, it fell during such periods accordingly. For example, in the United States, the share of labour in GDP fell from 65% to 55% from 1970 to 2015, i.e. in just 45 years. This is exactly what caused the stagnation of worker wages in the United States, which has been observed since the 1970s, instead of their growth in accordance with Equation (7).

Mathematical Models for Forecasting the Trajectory of Capital Accumulation in the 21 st Century
Taking into account the new tendencies in the development of leading economies in the 21 st century mentioned above, we shall consider models for long-term where ( ) We estimated K µ value as the regression index in Equation (9) within the 5 th LW (1982-2018), and the K s value was taken from the above World Bank database (1a).
Taking into account that the first empirical rule of Kaldor (5) remains valid in the first half of the 21 st century, Equation (9) can be rewritten as: The solution for this equation is exponential capital growth: If we take 0 2018 T = as the beginning of the 6 th LW in the US economy, then we find in the World Bank database that 0 59.3 K = trillion US dollars. Taking into account the specific values of the parameters K k (8), K µ , K s (9а), we find out that 0.021 . Thus, under the conditions of the Kaldor empirical regularity (5), in the first half of the 21 st century there will be an exponential growth of accumulated capital (11). However, according to the LW theory [17], the effect of capital saturation should happen at its downward stage in the 2040s. Therefore capital accumulation will take place according to a logistic trajectory:

Forecast Models for Calculating Potential Workplaces in Digital Economy
Let us now pass on to the models for predictive calculation of potential workplaces in the economy. There are two possible forecasting options: 1) based on a theoretical model; 2) based on empirical regularities. We shall begin with the first option. Let us take PF (1) as a theoretical model. At the beginning of §2 it was verified for the 5 th Kondratiev LW (1982-2018) on the example of the US economy and it was shown that the PF can be employed in the long-term forecasting within the 6 th Kondratiev LW (2018-2050). We also took up 1 h ≡ there.
If we now assume that the first Kaldor regularity (5) will be fulfilled, which means balanced economic growth, when Y K q q = , then, substituting relation (5в) into PF (1), we obtain the equation for determining the effective amount of labour: Solving this equation with respect to L, we get the formula for the predictive calculation of workplaces in the economy in terms of its balanced growth: Thus, according to Formulas (13) and (14), we can carry out a predictive calcu-  (14) with the already known function of technological progress ( ) A t is shown in Figure 3. As is seen, the number of workplaces is growing until the early 2030s, and then will begin to sharply decline. Sometimes it is possible to obtain empirical regularities in connection with the creation of new workplaces and the reduction of existing ones, by means of their technological replacement. For example, in the work ( [28], ch.3) the authors provide the results of empirical extrapolation forecasting for the next 10 -15 years: a new stage of automation will, on average, reduce 12% of existing workplaces and create 13% of new vacancies, which will eventually lead to the creation of 2 million additional workplaces by 2030. This empirical regularity can be most easily approximated by a linear predictive function: where bd L is the number of employed (workplaces) in the economy in the initial year ( calculated using Formula (14) and built on the same Figure 3, also supports the conclusion about the significant technological replacement of workplaces in the digital economy [29].

Forecasting the Trajectory of Economic Growth
Since from the very beginning we allowed for long-term balanced growth for the economies of developed countries, it is obvious that the forecast trajectory of the GDP movement ( ) Y t can be calculated using the empirical Kaldor formula K Y k K = ⋅ (5c) through the known forecast trajectories of capital accumulation (11) or more exactly (12). If we take a look only at balanced trajectories of economic growth with effective labour (13), then the forecast trajectory of the GDP movement will naturally be the only one. Such a balanced trajectory of economic growth Y 2 , calculated according to the exponential trajectory of capital accumulation (11) and Formula (5c), is shown in Figure 4. As a comparison, there you can see the forecast trajectory of the GDP movement    , which is directly seen in Figure 5. This was to be expected, since we took into account only technological progress, but did not consider its impact on the productivity of workers. We shall return to this issue later. Above, we forecasted long-term balanced economic growth using the wellknown Solow neoclassical model in the form of the Cobb-Douglas PF (1). Since the two main factors of growth, capital and labour, can be described using the classical models discussed above, it is now necessary to find a model to be able to reliably forecast technological progress ( ) d A t in the digital age, which can no longer be described with the help of traditional models. This is explained by the peculiarities of the digital economy and technologies. In the second half of the 20 th century technological progress ( ) A t in the economy was determined mainly by the efficiency of the R & D system. In this regard, various R & D models have been proposed by a number of authors ( [24], ch.5). We have developed an improved R & D equation, invariant to the scale of the economy, which turned out to be very effective and provided a sufficiently accurate solution that allows us to reliably predict both technological progress and long-term economic dynamics [30]. However, in the future digital economy, technological progress will already be determined by the dynamics of the production of technological information. That is why an information formula will be required to determine technological progress, which we obtained in the work [14] and verified for the information age . The mathematical models of economic dynamics and technological progress that are most suitable for the age of the digital economy are given below [25].
The digital economy is a new paradigm for accelerating economic development, improving the quality and usefulness of goods and services. The digital economy is a real developed economy in which a key role is played by digital platforms, platform business models and digital technologies, designed to increase the productivity of economic factors, minimize the costs of materials and resources, and most importantly, improve the accuracy of forecasting demand on the part of consumers and ensure their full compliance with their preferences and requirements regarding the characteristics of goods and services. Therefore, in describing the economic dynamics in the digital age, technological progress, which determines the total productivity of factors, will play a key role, and it is important that it is directly determined by the dynamics of the production of technological information, since the main factors of the digital economy are knowledge and know-how embodied in digital technologies.

Mathematical Models for Describing and Forecasting Technological Progress in the Information and Digital Age
Technological progress A(t), which determines the total productivity of the main economic factors-physical capital K(t) and labour L(t), in the information and digital age should naturally depend on the dynamics of production of technological information S(t). Since the rate of economic growth by 80% or more is determined exactly by technological progress (the total productivity of factors) ( [31], ch.1), it is extremely important to establish the desired functional relationship. We shall use the information model first proposed by A. I. Yablonsky ([32], p.163) as a basic model for calculating the rate of technological progress in the information and digital age ( ) Although the ideas underlying the construction of Formula (16) are absolutely correct, however, Formula (16) itself is incorrect, since it does not take into account the correspondence between the dimensions of the right and left sides of the equation. Based on the π-theorem of the theory of dimensions [33], Formula (16) can be correctly written as: Since information and digital technologies are exponentially growing, it can be assumed that ( ) Ad S t will also be an exponential function. Indeed, we have defined (for more details see [25]) that the S Ad function is an exponential function written as The fact that the accumulation of industrial technological knowledge in the information and digital sectors of the economy occurs according to the exponential law was previously noted by Raymond Kurzweil ([34], pp. 491-496).
Most works usually consider the simplest version of the function, when g(t) = g 0 t and g 0 = const, i.e. the case of a linear function. In this paper, we consider the general case when g(t) is a nonlinear but differentiable function. Hence, If we substitute representations (19) and (18) into Formula (17), we get a compact formula for calculating the required growth rates of technological progress: Thus, rates q Ad of technological progress, which determine the dynamics of the total productivity growth of the main economic factors, in the information and digital age will mainly depend on the rates of production of technological information If we know q Ad (t), it is quite easy to calculate the trajectory of the technological progress A d (t). itself. Indeed, it follows from Equation (17) If we put here the expression for q Ad (20), we will finally have: In the work [14] we considered a number of real dynamical regimes of pro-

The Invariable Dynamical Regime
Information is produced at a constant growth rate . The Lagrange function has the form of ( ) 2 , , . L g g t g = The corresponding Lagrange equation has the solution In this case, The process of accumulation of industrial technological knowledge occurs according to the simplest exponential law which characterizes the steady state within the framework of one Kondratiev long economic cycle-a long wave of economic development lasting for 30 -40 years, formed under the influence of another technological revolution. The simplest dynamical regime of production of technological information described above was characteristic of the symbiosis "human + computer" at the initial stage of the development of computer technology in the period of the 4 th long economic cycle (1946-1982).

The Dynamical Regime with Aggravation
Growth rates of information production grow exponentially with its accumulation ( [34], p. 492), i.e. ~e g g . The following Lagrangian leads to this regime ( ) 2 2 , , e . g L g g t g − = The corresponding Lagrange equation has the following form: leads to a hyperbolic increase in growth rates of information production: where T S is a point of singularity.
Equation (25a) resembles the hyperbolic equation of demographic dynamics, first obtained by Heinz von Foerster, Patricia Mora and Lawrence Amiot [36], with a point of singularity at T S = 2026. In reality, the explosive demographic growth was replaced with the stabilization dynamical regime-demographic transition [37]. The same happened with the dynamical regime of information production in the 5 th LW (1982-2018), when instead of explosive growth (25a), smooth logistic growth was realized. We assume that during the 6 th LW (2018-2050), the dynamical regime with aggravation is implemented with subsequent stabilization.

The Dynamical Regime with Stabilization
This regime is a combination of the dynamical regime with aggravation ge g , which is implemented at the initial stage of development, and the invariable dynamical regime const g = at the final stage: and Lagrange equation ( ) Usually the solution needs to be scaled to make it adequate for the problem at hand. This is done by introducing new variables  Here, as we can see from Equation (27a), rates of production of technological information monotonically increase according to the logistic law with a variable rate, because, as follows from Equation (27b), g(t) is not a strictly linear function of argument t.
This is the dynamical regime of production of technological information which was observed during the 5 th LW. Indeed, as can be seen in Figure 6, which shows the graph of the q Ad (t) function calculated by Formula (20) with respect to (27), and the curve with points characterizing the actual values of the contribution of ICT to the rates of technical progress [38], they coincide with a high index of determination (R 2 = 0.998).

The Aggravation Dynamical Regime with Return to the Stable Level
In this scenario, at the initial stage, the process will be sharply escalated ( ge g ) and, due to inertia, it skips the stationary mode ( const g = ), and then, having reached a certain maximum value m g , it returns asymptotically to the stable level. This regime can be described by the relation ( ) Scaled solution of such an equation has the following form: a) ( ) are the initial values for the process of production of technological information in the 6 th LW; s g is the scaled factor. It is this scenario of growth with a return according to which the number of population in separate countries is stabilized in the 21 st century [39]. avoid the stationary level, due to powerful acceleration, and then, due to the activation of the deceleration mechanism, return asymptotically to the stationary level above. The method of calculating constant parameters and indexes in Equations (27) and (28) will be explained in the following paragraphs. In the next section, by verifying technological progress and economic dynamics at the 5 th informational LW (1982-2018), we will show that curve

Verification of the Information Model of Technological Progress and Economic Dynamics at the 5 th LW (1982-2018)
In the previous paragraph ( §2.5), mathematical models were given for describing and forecasting technological progress in the age of the information and digital economy. The models were based on various production regimes of technological information. As a result, they are called information models. The main verified formula is the formula for calculating the growth rates of technological progress (20): where ( ) g t is the production rates of technological information in the economy; is relative investments of ( ) d I t in the fixed productive capital ( ) d K t of the information and digital economy.
Here and in what follows, the normalizing index ξ is assumed to be equal to unity ( 1 ξ = ), since normalization can be carried out in terms of constant indexes of the function ( ) g t . We will verify Formula (29) for the 5 th Kondratiev LW in the world economy , which is characterized as the age of formation of the information economy or the "knowledge economy" in developed countries and for which there exist verified and sufficiently accurate data on both technological progress and economic growth for all OECD countries.
We obtained the actual data on the growth rates of technological progress ( ) Ad q t by aggregating the geometric mean series according to the initial data from the two most reliable sources [Bureau of Labor Statistics USA; University of Groningen]. They are presented graphically in Figure 8. In our work [14] an approximate analytical expression was given for approximating for the US economy: Substituting here numerical values r T , 0 T , 0 g (32) and r g (33d), we get Substituting the expression for the production rate of technological information ( ) g t (31а) and (31b) in Formula (29) we get: where Since the left side of Equation (37) is set by its actual values, which have already been presented in Figure 8, and only the value of the scaled index g s remained the unknown and undefined on the right side, it can be estimated using the method of least squares. It turned out to be where ( ) (40а) The corresponding part of the trend trajectory of technological progress in the phases of recession and depression is also shown in Figure 8 (2004-2014). So, it is necessary to find an approximate analytical description of technological progress in the recovery phase (2014-2018) of the 5 th LW, which is best approximated by an exponential curve: where ( ) The initial value 0 d A is calculated by the production function (1) using the method of least squares, so that at the initial timepoint Y 0 , K 0 and L 0 would coincide in their actual values with the best approximation of the formula The trajectory of the actual technological progress is obtained by numerical integration (42), using the actual data The calculation results are presented graphically in Figure 9, in the form of a growth trajectory of the normalized level of technological progress . The mean square error turned out to be 1.95% As we can see, the proposed information model of technological progress (29) provides a sufficiently high accuracy of approximation. Now let us consider how this model error can affect the further calculations of the trajectory of economic growth by the production function (1): Here we have just calculated and verified  After having calculated these both variants, we obtained a mean square error at current prices and at comparable prices The trajectories of economic growth are shown in Figure 10 and Figure 11 respectively.
As can be seen from the graphs in Figure 10 and Figure 11, the accuracy of model (49) in calculating the trajectory of economic growth is much higher than for calculating technological progress, and the accuracy is two times higher for calculating economic growth in comparable prices than in current prices. Therefore, forecast calculations are also recommended to be carried out at comparable prices.
In conclusion, let us calculate the growth trajectory of the production of technological information ( ) g t from Formula (40):  Here, in the integrand, we successively use approximating functions (37), (40) and (

Forecasting the Technological and Economic Dynamics at the Upward Stage of the 6 th Kondratiev LW (2018-2042)
The dynamics of technological progress at the upward stage of the 6 th Konrdatiev LW in world economic development is determined by the accelerated production regime of technological information with a return to the stationary Equation (28): Since the production regime of technological information in question is a regime with a return, the upper turning point of the 6 th LW will be higher than the stationary level, therefore, Equation (47a) Here all values except for rt g are known. Solving the equation, we find out that 12.72 Finally, we turn to forecasting the dynamics of economic growth at the upward stage of the 6 th Kondratiev LW (2018-2042) using PF (1): The predicted growth trajectory of technological progress in the digital age Employment dynamics is forecasted using Formulas (14) or (15). Let us choose the predictive Formula (15) based on an empirical pattern:  In Figure 15 you can see the trajectory of the US GDP, calculated using PF (52) based on the predicted trajectories of the main growth factors A d (51), K (53) and L pe (54) with constant parameters (45b) at the upward stage of the 6 th LW (2018-2042). As is shown here, there is an inertial economic growth, continuing the growth trend that developed during the depression years (2010-2016) after the Great Recession of 2009, although digital technologies are expected to provide a significant acceleration. However, we have not yet considered the possible increase in labour productivity through the effective use of the symbiosis "human + intelligent machine".

The Main Driving Force of the Digital Economy-The
Symbiosis "Human + IM" There are enough reasons to believe that most of the work in the future cannot be done without humans. Indeed, any cognitive work can be fragmented into a certain set of tasks, some of which are programmable, and therefore can be automated and transferred for the execution by the IM or bot programmes, while others, in fact, cannot be automated and people will fulfill them. Moreover, the latter, as noted above, are likely to be supplemented and expanded. The book cited above [28] gives the results of special studies indicating that in many cases about 20% of routine tasks will go to the IM. In general, the dispersion covers from 25% to 50%. Thus, it is concluded that the upcoming digital automation would take away from people a maximum of 25% -50% of routine and boring work ( [28], ch.3). On the other hand, by freeing themselves from such work, humans can redouble efforts to do the rest of the work and significantly increase labour productivity or the quality of performance. These considerations mentioned above will form the basis of our mathematical model for calculating the impact of highly qualified human capital on labour productivity in the digital economy.
The abovementioned resembles the processes of fragmentation of industrial production that began in the 1980s and the transfer of some of them, usually the most labour-intensive, to developing countries with relatively cheap labour. Of course, a progressive step on the part of developed countries was that the process Figure 15. The trajectory of the US GDP movement in the information and digital age. was accompanied by the transfer of innovative technologies and know-how, albeit with the aim of ensuring the high-quality standards adopted in developed countries. The correlation of high technologies and low wages drastically reduced the cost of manufacturing products and sharply increased the volume of profits. Good workplaces with high wages, requiring highly skilled workers, remained in the developed countries. Such a situation combined the sources of competitiveness of developed countries-their technological, managerial and marketing know-how-and the comparative advantage of developing countriestheir cheap labour. In the end, everyone won. Moreover, the practice of transferring a labour-intensive routine work from developed to developing countries has convincingly shown that in the end this part of production could be fully automated. In the age of digital intelligent machines, work will be fragmented into tasks, and routine tasks will be transferred to the IM, and an interesting creative part of it will remain for humans. In addition, the overall supervision of any work performance will certainly remain with people. There highly qualified managers will be required.

Mathematical Models for Describing and Calculating the Dynamics of Labour Productivity Growth in the Symbiosis "Human + IM"
As can be seen in Figure 14, technological progress based on digital technologies will grow very slowly in the 2020s, reaching 1% by the end of the decade, and only in the 2030s it will significantly accelerate and its rates by the beginning of the 2040s will double and exceed 2% per year. However, in the 2020s it is possible to significantly increase labour productivity in the economy effectively using the symbiosis "human + IM". The main PF (1)   ( ) h q t is a function determining the transition process in establishing effective joint work of the human and the IM. Usually this process takes one medium business cycle of 6 -10 years. It can be roughly described with the help of a logistic function: After graduation a person has some knowledge described by the formula ( [24], ch.3): where u is an average number of years of study; ψ is the index of return on education.
Here, the growth rate of an employee's labour productivity depends solely on digital technological progress, since a person does not participate in work performance, but observes or, at the best, controls the work of IM.
In the first case fragments of work performed by a person prevail, in the thirdby a machine, while in the second they are equally distributed. It is important for us to know in which of these options we can speak of the maximum labour productivity. The solution to Equation (55) has the form: Since the employee's labour productivity at the initial moment 0 h A is determined by the human capital accumulated in the learning process (59) and further practice, assuming that a highly qualified person has completed a master's program (who has studied for 18 years or more, i.e. In practice, for a comparative analysis, it is not the trajectories of labour productivity growth (63) that are more suitable, but their growth rates ( ) Ah q t , which can be most easily obtained from Equation (55):  The corresponding curves of labour productivity growth in the digital economy are indicated by the symbols q 1 , q 2 and q 3 (see Figure 16).
As is seen in the growth curves of labour productivity in Figure 16, the highest labour productivity is achieved at 1 3 β = , when human labour dominates, and the lowest labour productivity is observed at 2 3 β = , when the share of work performed by machines prevails. In any case, we can see that the symbiosis of "human + IM" allows for effective use of the potential of digital technologies at the initial stage of the formation of the digital economy, ensuring the growth of labour productivity to potential values already in the mid-2020s (see Figure 16).
At the same time, digital technologies are maximally manifested only since the mid-2030s, i.e. a decade later.
If we now use the predictive Formulas (52)-(54) to calculate the trajectory of GDP in the digital era (2018-2042), replacing ( ) d A t in the production function (52) by ( ) h A t (63), we get graphs Y 1 , Y 2 and Y 3 , shown in Figure 17. As we can see directly from Figure 17, economic growth receives an additional acceleration compared to the basic trajectory of economic growth ( basic Y ). Using these graphs, it is already easy to calculate the rates of economic growth q 1 , q 2 and q 3 , which are shown in Figure 18. The graphs of the movement of economic growth rates (see Figure 18) demonstrate that with the effective use of the symbiosis "human + IM" (q 1 ), the growth rates of the digital economy in the 2020s may confidently exceed 3% and maintain this level in the 2030s. The US economy has a great chance to grow at a high rate, as in the 2000s, already in the long-term period.   , , q q q in the digital age under different dynamical regimes of the use of the symbiosis "human + IM". 3) Most of the cognitive work in the digital age will be fulfilled by people, since all of it, as a rule, is fragmented into non-programmable tasks that require creative, highly skilled human labour to solve them, and routine programmable tasks that can be automated and transferred to IM. In this regard, digital competencies and a person's ability to work in a symbiosis with IM are in great demand. All this leads to the inconsistency of numerous predictions and hypotheses that in the digital age, most workplaces will be taken by IM.

Conclusions
4) The main driving force of the digital economy will be the symbiosis "human + IM", which effectively works under the leadership of a person. On the basis of a mathematical model it is shown that, from the very beginning of the formation of the digital economy, precisely due to the high level of human capital and its effective interaction with IM the potential of digital technologies to increase labour productivity is realized. Moreover, it turned out that the highest labour productivity is achieved in the symbiosis "human + IM", where highly skilled human labour dominates, and the lowest labour productivity is observed where the programmed work performed by IM prevails. It is also calculated that for developed economies, which are the leaders in the formation of the digital economy, growth rates of labour productivity equal to 3% per year can be achieved by the mid-2020s, and it has great chances to remain like this until the 2040s.

5)
Since the main driving force of the digital economy is the symbiosis of "human + IM", scholars and developers need to ensure that IM is extremely friendly to people and serve to improve and complement human labour, and enhance its cognitive ability. On the other hand, the education system in the digital age should, along with the formation of deep professional knowledge and solid work skills in people, provide them with good mathematical knowledge, engineering thinking, teamwork skills and sufficient competencies in the field of digital technologies in order to let future specialists successfully and effectively work together with IM.